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University of Southern California Dissertations and Theses
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New electronic structure methods for electronically excited and open-shell species within the equation-of-motion coupled-cluster framework
(USC Thesis Other)
New electronic structure methods for electronically excited and open-shell species within the equation-of-motion coupled-cluster framework
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Content
NEW ELECTRONIC STRUCTURE METHODS FOR ELECTRONICALLY EXCITED AND
OPEN-SHELL SPECIES WITHIN THE EQUATION-OF-MOTION COUPLED-CLUSTER
FRAMEWORK
by
Sahil Gulania
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMISTRY)
May 2021
Copyright 2021 Sahil Gulania
Acknowledgements
First of all, I would like to thank my research adviser, Prof. Anna Krylov, who has con-
tributed greatly to my research progress. Her passion and enthusiasm for science has been
a source of inspiration throughout my Ph.D. Valuable and based on experience, her advices
helped me a lot. I appreciate that she did not just teach me science but also taught me how to
become a good scientist. It was a great honor for me to be part of her lab and contribute to
iOpenShell science.
I would like to thank all members of Anna’s group: Samer, Kaushik, Wojciech, Alexan-
dre, Atanu, Anastasia, Tirthendu, Pavel, Sven, Maristella, Goran, Madhubani, and Ronit, for
creating a friendly and supportive environment.
I would also like to thank Prof. Richard Mabbs from the Washington University in St.
Louis and Prof. Andrei Sanov from the University of Arizona for teaching and sharing their
deep knowledge of experimental spectroscopy. I would also like to thank Prof. Thomas Jagau
from the Katholieke Universiteit Leuven, Prof. John Stanton from the University of Florida
and Prof. Henrik Koch from Scuola Normale Superiore, Italy, for sharing their insight into
many-body theory.
ii
Prof. Aiichiro Nakano from the Department of Physics and Prof. James Whitfield from
Dartmouth College also contributed greatly to my scientific development. They taught me how
to approach problems in science. I would also like to thank my undergraduate advisor Prof.
Lourderaj, who introduced me to quantum chemistry, optimization algorithms, and program-
ming languages.
The list would be incomplete without mentioning my life partner Arpita Patra, who has been
supportive throughout my Ph.D. Finally, I would like to thank my parents and my sister, who
always encouraged me throughout my life.
iii
Table of contents
Acknowledgements ii
List of tables vii
List of figures x
Abstract xiv
Chapter 1: Introduction 1
1.1 Quantum chemistry of open-shell species . . . . . . . . . . . . . . . . . . . . 1
1.2 Bound and temporary anions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Equation-of-motion coupled-cluster formalism . . . . . . . . . . . . . . . . . . 4
1.4 Electronic resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Chapter 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2: Equation-of-motion coupled-cluster method with double electron-attaching
operators: Theory, implementation, and benchmarks 14
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 CH
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.3 Benzynes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.4 Cyclobutadiene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.5 Ozone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.6 Ethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.7 Butadiene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.8 Potential energy surfaces and conical intersections in retinal chromophore 43
2.3.9 Excited states in water and ammonia . . . . . . . . . . . . . . . . . . . 46
2.3.10 Conical intersection in HeH
2
. . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
iv
2.6 Chapter 2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter 3: The quest to uncover the nature of benzonitrile anion 70
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2 Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 Computation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.1 The benzonitrile anion . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.2 The C
6
H
5
CN
H
2
O complex . . . . . . . . . . . . . . . . . . . . . . . 81
3.4.3 Photodetachment spectra . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6 Appendix A: Computational details . . . . . . . . . . . . . . . . . . . . . . . . 89
3.7 Appendix B: Normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.8 Appendix C: Absolute cross sections . . . . . . . . . . . . . . . . . . . . . . . 90
3.9 Appendix D: Basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.10 Appendix E: Relevant Cartesian geometries . . . . . . . . . . . . . . . . . . . 99
3.11 Chapter 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Chapter 4: EOM-CC guide to Fock-space travel: The C
2
edition 107
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Molecular orbital framework and essential features of electronic structure of
C
2
species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.1 C
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.2 C
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4.3 C
2
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.6 Chapter 4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Chapter 5: Dissociative Electron Attachment in C
2
H 134
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.2 Theoretical methods and computational details . . . . . . . . . . . . . . . . . 136
5.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4.1 Electronic structure of C
2
. . . . . . . . . . . . . . . . . . . . . . . . 141
5.4.2 Electronic structure of C
2
H
. . . . . . . . . . . . . . . . . . . . . . . 142
5.5 Chapter 5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Chapter 6: Future directions 150
6.1 Spin-orbit coupling for EOM-DEA and EOM-DIP . . . . . . . . . . . . . . . . 150
6.2 Analytical gradients for EOM-DEA and EOM-DIP . . . . . . . . . . . . . . . 151
v
6.3 CVS-EOM-DIP-CCSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.4 Chapter 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Bibliography 154
vi
List of tables
1.1 EOM operators in second quantization.a;b;c;::: denote virtual orbitals,i;j;k;:::
denote occupied orbitals andr denotes EOM amplitudes. Occupied and virtual
spaces are defined by the choice of the reference determinant
0
. . . . . . . . 6
2.1 Total energies (hartree) for the ground
~
X
3
B
1
state of CH
2
and adiabatic excita-
tion energies (eV) for the three lowest singlet states
a
. ZPE not included; TZ2P
basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 One-electron properties
a
for the lowest electronic states of CH
2
computed using
EOM-SF-CCSD and EOM-DEA-CCSD wave functions; aug-cc-pVTZ basis
set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Adiabatic singlet-triplet gaps (eV) in benzynes (no ZPE); cc-pVTZ basis set. . 31
2.4 Head-Gordon index (n
u;nl
) for the lowest singlet and triplet states in benzynes
computed using the EOM-DEA-CCSD and EOM-SF-CCSD (numbers in paren-
thesis) wave functions; cc-pVTZ basis set. . . . . . . . . . . . . . . . . . . . . 31
2.5 Total energies (hartree) of the ground stateX
1
A
g
of cyclobutadiene and vertical
excitation energies (eV) at the X
1
A
g
equilibrium geometry (D
2h
symmetry, 4
frozen core orbitals); cc-pVTZ basis set. . . . . . . . . . . . . . . . . . . . . . 33
2.6 Total energies (hartree) of the ground state X
1
B
1g
of cyclobutadiene and vertical
excitation energies (eV) at the X
1
B
1g
equilibrium geometry (D
4h
symmetry);
cc-pVTZ basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Orbital occupations in dominant electronic configurations of the ground and
low-lying excited states in ozone
a
. . . . . . . . . . . . . . . . . . . . . . . . . 35
2.8 Ozone. Vertical
a
excitation energies (eV) relative to the X
1
A
1
state computed
by EOM-EE-CCSD and EOM-DEA-CCSD; aug-cc-pVTZ. . . . . . . . . . . . 35
2.9 Ethylene torsion barrier (eV) computed with various methods and a DZP basis
set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.10 Ethylene. Vertical excitation energies (eV) from the lowest singlet state (
1
A
2
)
at the twisted geometry using EOM-DEA-CCSD/aug-cc-pVTZ. . . . . . . . . 38
2.11 Ethylene. Vertical excitation energies (eV) at the equilibrium geometry. . . . . 39
2.12 One-particle state and transition properties
a
for ethylene computed with EOM-
EE-CCSD and EOM-DEA-CCSD; aug-cc-pVTZ basis set. . . . . . . . . . . . 39
2.13 Vertical excitation energies (eV) for the 1
1
B
+
u
and 2
1
A
g
in butadiene . . . . . . 40
vii
2.14 One-particle state and transition properties
a
of butadiene computed with EOM-
EE-CCSD, EOM-EE-CC3, and EOM-DEA-CCSD. . . . . . . . . . . . . . . . 43
2.15 The S
0
and S
1
energy gaps (kcal/mol) at TS
CT
, TS
DIR
, and cis-PSB3 geometry
of retinal; 6-31G
basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.16 Ammonia. Vertical excitation energies (eV) for the four lowest singlets and
the lowest triplet states with different methods; aug-cc-pVQZ basis. Geometry
from Ref. ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.17 Ammonia. One-particle state and transition properties
a
computed with EOM-
EE-CCSD and EOM-DEA-CCSD; aug-cc-pVQZ basis set. . . . . . . . . . . . 48
2.18 Programmable expressions for the right () and left (~ ) vectors in EOM-DEA-
CCSD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.19 Intermediates used in EOM-DEA-CCSD-vectors. . . . . . . . . . . . . . . . 54
2.20 Programmable expressions for EOM-DEA-CCSD density matrices. . . . . . . 55
2.21 Ethylene torsion, DZP basis. Total energies (hartree) for the SF-TDDFT(5050),
MR-CI (TCSCF-CISD), CCSD, EOM-SF-CCSD, and EOM-DEA-CCSD mod-
els. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.22 Comparing
^
R
2
2
(EOM,
^
R
2
amplitude contribution) for 2
1
A
g
excited state in
butadiene from EOM-EE-CCSD and EOM-EE-CCSDT . . . . . . . . . . . . 56
2.23 Water. Vertical excitation energies (eV) for the 3 lowest singlet and 3 lowest
triplet states computed with different methods; aug-cc-pVQZ basis. Geometry
from Ref. ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.24 Water. One-particle state and transition properties
a
computed with EOM-EE-
CCSD and EOM-DEA-CCSD; aug-cc-pVQZ basis set. . . . . . . . . . . . . . 57
3.1 Attachment and detachment energies (in eV) for valence (
2
B
1
) and dipole-bound
(
2
A
1
) anion of benzonitrile. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Attachment and detachment energies (in eV) for the valence (
1
A
00
) and dipole-
bound (
2
A
0
) states of the benzonitrile-water complex. . . . . . . . . . . . . . . 83
3.3 Peak positions and assignments for the photoelectron spectrum of C
6
H
5
CN
(
1
A
1
2
B
1
).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.4 Frequencies (cm
1
) for the neutral and valence anion computed with RI-CCSD
and RI-EOM-EA-CCSD using the aug-cc-pVDZ basis set. . . . . . . . . . . . 89
4.1 Equilibrium bond lengths (r
e
,
˚
A) and term energies (T
ee
, cm
1
) of the low-lying
states of C
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 Equilibrium bond lengths (r
e
,
˚
A) and term energies (T
ee
, cm
1
) of bound elec-
tronic states of C
2
. EOM-IP-CCSD vertical excitation energies (E
ex
, cm
1
) and
oscillator strengths (f
l
) are also shown. . . . . . . . . . . . . . . . . . . . . . 118
4.3 First-order corrected energies of C
2
2
at optimal values of the parameter and
the corresponding trajectory velocities (in a.u.) computed with CAP-CCSD/aug-
cc-pVTZ+3s3p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
viii
4.4 First-order corrected energies of C
2
2
at optimal values of the parameter com-
puted with CAP-CCSD/aug-cc-pCVTZ+6s6p6d and CAP-HF/aug-cc-pCVTZ+6s6p6d.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5 Calculation of partial widths using Coulomb wave and Dyson orbitals from real-
valued and complex-valued EOM-IP-CCSD calculations. . . . . . . . . . . . . 124
5.1 Excitation energies (eV) and oscillator strength for C
2
at R
CC
= 1.219
˚
A using
EOM-IP-CCSD/aug-cc-pVTZ with the dianionic reference (C
2
). . . . . . . . . 141
5.2 Resonance positions E
R
(eV) and widths (eV) at equilibrium geometry of
neutral C
2
H
using CAP-EOM-EE-CCSD/aug-cc-pVTZ+3s3p1d(3s3p). . . . . 143
5.3 NTOs (real part) at equilibrium geometry of neutral (R
CC
= 1.204
˚
A, and R
CH
= 1.062
˚
A.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4 Resonance positions E
R
(eV) and widths (eV) at a stretched geometry (R
CC
=
1.204
˚
A, and R
CH
= 1.662
˚
A.); CAP-EOM-EE-CCSD/aug-cc-pVTZ+3s3p1d(3s3p).144
5.5 NTOs (real part) at stretched geometry (R
CC
= 1.204
˚
A, and R
CH
= 1.662
˚
A.) . 144
5.6 Resonance positions E
R
(eV) and widths (eV) at a stretched and bend geom-
etry (R
CC
= 1.204
˚
A, R
CH
= 1.662
˚
A, and \CCH = 170
o
); CAP-EOM-EE-
CCSD/aug-cc-pVTZ+3s3p1d(3s3p). . . . . . . . . . . . . . . . . . . . . . . . 145
5.7 NTOs (real part) at stretched and bend geometry (R
CC
= 1.204
˚
A, R
CH
= 1.662
˚
A, and\CCH = 170
o
). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
ix
List of figures
1.1 Examples of closed-shell and open-shell species. Dinitrogen in ground state is
closed-shell. Vinyl has one unpaired electron and is radical. Methylene has two
unpaired electrons and is di-radical. ReproducedfromRef. ?. . . . . . . . . . . 2
1.2 Examples of processes leading to formation of open-shell species. The left panel
shows the formation of an open-shell excited state. The middle panel shows
the formation of meta-stable excited state (resonance). The right panel shows
ionization. The ionization continuum is shown by blue. . . . . . . . . . . . . . 2
1.3 Formation of bound and temporary anion. Formation of bound and temporary
anions. The left panel shows the formation of bound anion by attaching an elec-
tron to a closed-shell precursor. The extra electron is attached to an orbital,
which is below the detachment energy of the anion and lies below the contin-
uum. Right panel shows the formation of a temporary anion. The extra electron
occupies a continuum orbital and has a finite lifetime. Blue shows the electronic
continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Different types of target states can be accessed by different combinations of the
reference state and EOM operators. ReproducedfromRef. ?. . . . . . . . . . . 5
1.5 The transformation of the spectrum of the Hamiltonian upon complex scaling
of all coordinates as described by the Balslev–Combes theorem. Reproduced
from Ref. ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Different types of target states can be accessed by different combination of the
reference state and EOM operators. ReproducedwithpermissionfromRef. ?. . 17
2.2 Benzynes. Frontier NOs and their occupation numbers in the lowest singlet and
triplet states (triplet-state occupations are given in parenthesis) computed for the
EOM-SF-CCSD (black) and EOM-DEA-CCSD (red) wave functions using the
cc-pVTZ basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Geometries of the 1
3
A
2g
(left) and X
1
A
g
(right) states optimized at the CCSD(T)/cc-
pVTZ level of theory. Bond lengths are in angstroms and angles are in degrees;
the structures are from Ref. ?. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Frontier natural orbitals of the X
1
A
1
state of ozone. computed with EOM-DEA-
CCSD/aug-cc-pVTZ for the ground state. In the ground state, 2b
1
orbital is
vacant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
x
2.5 Ethylene torsion barrier computed with various methods and a DZP basis set.
All curves are shifted such that the energy at 0
0
is zero. . . . . . . . . . . . . . 37
2.6 Potential energy curves along torsion coordinate for several electronic states of
ethylene computed with EOM-DEA-CCSD/aug-cc-pVTZ. . . . . . . . . . . . 38
2.7 Butadiene. Natural frontier orbitals and their occupations computed using the
EOM-DEA-CCSD/aug-cc-pVTZ wave functions (using dication reference orbitals). 41
2.8 Potential energy surface in retinal showing the location of the conical intersec-
tion (CoIn or CI) between the charge transfer and diradical states. The two
coordinates are bond-alternation (BLA) and twisting reaction coordinate (RC).
Relevant mechanistic paths are shown by white dashed lines. MEP
CT
: minimum
energy path on the ground state that connects the cis and trans retinal equilib-
rium structures through a transition state (TS
CT
). MEP
DIR
: connects cis, TS
DIR
and trans structures of the
DIR
state. The BLA path connects the TS
CT
and
TS
DIR
transition states and also intercepts a CoIn (CI) point. Atomic charges
of the two transition states are shown by bubble diagrams. Reproduced with
permission from Ref. ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.9 The S
0
and S
1
energy profiles (retinal) along the BLA coordinate computed
with EOM-DEA-CCSD (with orbitals from the +3 and +1 charge Hartree-Fock
reference) and MRCISD+Q using the 6-31G
basis set. The energy values are
relative to cis-PSB3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.10 HeH
2
molecule. One He-H bond length is varied (R). . . . . . . . . . . . . . . 49
2.11 Natural orbitals and their occupation numbers for ground and first excited state.
Black color corresponds to geometry at He-H = 0.6
˚
A. Red color corresponds
to geometry at He-H = 0.8
˚
A. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.12 Conical intersection betweenX
1
A
0
and 1
1
A
0
computed using EOM-DEA-CCSD,
EOM-EE-CCSD, and full configuration interaction (FCI) methods with the aug-
cc-pVDZ basis set. The left pane shows all three limits for the full range of
bond lengths (0.6 to 0.8
˚
A). The right pane zooms in on the near-degeneracy for
EOM-DEA-CCSD and EOM-EE-CCSD. . . . . . . . . . . . . . . . . . . . . 50
2.13 Relevant MOs of ethylene at 90
twisted geometry. . . . . . . . . . . . . . . . 56
3.1 Photoelectron spectra of C
6
H
5
CN
(top) and C
6
H
5
CN
H
2
O (bottom) obtained
with two different energy photons. Red and blue lines correspond to the spectra
obtained with 1.165 eV and 4.661 eV photons, respectively. Reproduced with
permission from Ref. ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2 EOM-EA target configurations generated from a closed-shell reference state.
The first two configurations are 1p ones and the third one is 2p1h. . . . . . . . 75
3.3 Dyson orbitals for the two lowest electronic states of the benzonitrile anion,
2
A
1
and
2
B
1
, computed at the respective optimized geometries. Isovalue 0.007. . . 79
3.4 Schematic representation of the energy levels of the neutral, valence and dipole-
bound anionic states of benzonitrile (see text). Note that the V A adiabatically
drops below the neutral and the DBS due to zero-point energy. . . . . . . . . . 79
xi
3.5 Potential energy curves along the butterfly mode for V A, DBS, and the neu-
tral, showing the relaxation of V A. Energies along the scan are computed with
EOM-EA-CCSD/aug-cc-pVDZ+6s3p(3s). The scan was generated by taking
the displacement along the butterfly normal mode of the anion (mode #2 of
209.18 cm
1
). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.6 Dyson orbitals for the two lowest electronic states of the C
6
H
5
CN
H
2
O com-
plex,
2
A
0
and
2
A
00
, computed at the respective optimized geometries. Isovalue
0.007. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.7 Schematic representation of the neutral, valence anion, and dipole-bound states
of the benzonitrile-water complex. . . . . . . . . . . . . . . . . . . . . . . . . 82
3.8 Ratio of the Boltzmann population of V A (N
VA
) relative to the total population
(N
total
) as a function of temperature. . . . . . . . . . . . . . . . . . . . . . . . 83
3.9 Computed photoelectron spectrum for the V A and the experimental spectrum
obtained by Sanov and co-workers
?
. In the computed spectrum, the Franck-
Condon factors were convoluted with gaussians of width 0.05 eV . . . . . . . . 84
3.10 Franck-Condon active modes: #0 (out-of-plane), #2 (butterfly), and #26 (ring
breathing). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.11 Top: Ratio of the cross sections (DBS versus V A) for electron detachment from
the benzonitrile anion. Bottom: Convolution of the Franck–Condon factors
using Eq. (3.15) with gaussian (width=0.05 eV) and assuming equal populations
of the DBS and V A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.12 Absolute cross sections for detachment from the dipole bound (left) and valence
(right) states of C
6
H
5
CN
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1 Molecular orbital diagram. The three lowest orbitals that remain doubly occu-
pied in the low-energy manifold of electronic states of C
2
and C
2
are denoted
as ’core’. The electronic states of C
2
are derived by distributing six additional
electrons over four upper orbitals,
2s
,
2px
=
2py
, and
2pz
. Shown is the lead-
ing electronic configuration of the ground state, X
1
g
. Low-lying states of C
2
are derived by distributing five electrons over the four upper orbitals. In C
2
2
,
all four upper orbitals are doubly occupied. Shown are Dyson orbitals (iso-
value = 0.05) computed with EOM-IP-CCSD/aug-cc-pVTZ from the dianionic
reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 EOM-IP (left) and EOM-DIP (right) manifolds generated from the dianionic
reference (center). Only configurations generated by
^
R
1
from the top four
orbitals from Fig. 4.1 are shown. EOM-IP enables access to the ground and
electronically excited states of C
2
, whereas EOM-DIP describes the ground
and excited states of C
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3 Potential energy curves of low-lying singlet and triplet states of C
2
. . . . . . . 115
4.4 Potential energy curves of the three lowest states of C
2
. . . . . . . . . . . . . 119
xii
4.5 Potential energy curves of C
2
2
and C
2
. Total electronic energies are shown.
Solid lines show CCSD/aug-cc-pVTZ and EOM-IP-CCSD energies. Orange
squares show the results from CAP-CCSD/aug-cc-pvTZ+3s3p (first-order cor-
rected energy). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6 Uncorrected and first-order corrected CAP-CCSD using aug-cc-pCVTZ+3s3p
(top) and aug-cc-pCVTZ+6s6p6d (bottom)-trajectories for C
2
2
at equilibrium
bondlengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.7 Squared overlap between Dyson orbitals and a Coulomb wave with charge=-
1. Solid lines correspond to Dyson orbitals from EOM-IP-CCSD/aug-cc-pVTZ
(scale on the left). Dashed lines correspond to Dyson orbitals (real part) from
CAP-EOM-IP-CCSD/aug-cc-pVTZ+6s6p6d (scale on the right). . . . . . . . . 125
4.8 First-order corrected resonance width of C
2
2
as a function of bond length com-
puted with CAP-CCSD and CAP-HF and the aug-cc-pCVTZ+6s6p6d basis set.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.1 Left panel: electronic states at C
2
H
at the ground state equilibrium geome-
try of the of anion. Middle: electronic states of C
2
H
at elongated C-H bond
geometry. Right panel: electronic states of C
2
at equilibrium geometry of C
2
(X
2
+
g
). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 Manifolds of target states accessed by EOM-IP and EOM-EE. In both cases, the
same closed-shell reference is used. . . . . . . . . . . . . . . . . . . . . . . . 137
5.3 Natural transition orbitals for X
2
+
u
!A
2
u
and X
2
+
u
!B
2
+
u
in C
2
at equi-
librium geometry of C
2
using EOM-IP-CCSD/aug-cc-pVTZ with dianionic ref-
erence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4 Ionization energy and the respective Dyson orbitals for the C
2
H anion at differ-
ent geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
xiii
Abstract
This thesis focuses on the electronic structure and spectroscopy of open-shell and electron-
ically excited species, which are crucial for understanding chemistry and difficult to simulate.
This thesis provides several examples of challenging open-shell systems and discusses their
roles in chemistry. To study open-shell species, one needs reliable methods to describe their
physics accurately. The equation-of-motion coupled-cluster (EOM-CC) approach is a pow-
erful tool for accurate calculations of electronically excited and open-shell molecules. This
work shows how different variants of EOM-CC treat electronic structure of various open-shell
species.
Chapter 1 introduces open-shell species and explains their importance in chemistry. It
defines different types of anions and explains the complexities in the simulation of their elec-
tronic structure. Chapter 1 also contains a brief introduction to the EOM-CC theory and its
extensions to electronic resonances. Chapter 2 provides derivations and benchmark results for
the double electron attachment (DEA) variant of EOM-CC. This chapter describes the imple-
mentation and benchmarking of the method in the Q-Chem electronic structure package. The
EOM-DEA-CC method can treat diradicals, doubly excited states, some types of conical inter-
section and bond breaking.
Chapter 3 reports our study of the electronic structure of the benzonitrile anion. It compares
experimental electron attachment energies with calculations using the electron attachment (EA)
xiv
variant of EOM-CC. In this study, we were able to resolve the long-standing controversy about
the nature of the lowest bound anion state of benzonitrile. The dipole-bound and valence-
bound anion states are close in energy, making it difficult to resolve the character of the ground
state. The experiment suggested the formation of the valence-bound anion. EOM-EA-CCSD
calculations with the inclusion of zero-point energy corrections supported the conclusion drawn
from the experimental measurements. Chapter 4 shows the application of two variants, double
ionization potential (DIP) and ionization potential (IP). The EOM-DIP method was used to
study the electronic structure of the neutral carbon dimer and the EOM-IP method was used
to study its anion. This chapter shows that EOM-DIP-CCSD offers a simple computational
approach based on a single-reference formalism to study the electronic states of the carbon
dimer. In the same study, complex-variable extension of EOM was employed to study the
metastable dianion of the carbon dimer.
Chapter 5 describes the electronic resonances in the C
2
H
anion. This study was inspired
by an experimental finding of the formation of excited carbon dimer anion in plasma. The
EOM-CC method augmented with the complex absorbing potential was used to study elec-
tronic resonances. The calculations explain the formation of excited C
2
via dissociative electron
attachment to C
2
H molecule. Chapter 6 discusses the future directions and possible extensions
of the work presented in this thesis.
xv
Chapter 1: Introduction
1.1 Quantum chemistry of open-shell species
In most stable molecules all electrons are paired, giving rise to closed-shell type of electronic
structure. Due to Pauli’s exclusion principle
1
, only two electrons can occupy one molecular
orbital and their spins should be opposite. Molecules or atoms with unpaired electrons are called
open-shell species
2
. Fig. 1.1 shows examples of closed and open-shell structure. Depending on
the number of unpaired electrons, one can classify open-shell species into different categories.
Radicals have one unpaired electron, di-radicals have two unpaired electrons, tri-radicals have
three unpaired electrons, and so on. Open-shell species are ubiquitous in chemistry
3–10
. They
often result from the light-matter interaction and bond breaking. Open-shell species are also
formed upon ionization or electron attachment. Hence, cations and anions often have open-shell
character. Fig. 1.2 illustrates the formation of open-shell species from closed-shell precursors
as a result of the interaction with a photon. Reliable theoretical methods should treat open-shell
and closed-shell species on an equal footing. The theoretical modeling of open-shell species is
a major challenge because of the electronic degeneracies prevalent in open-shell species.
Equation-of-motion (EOM) coupled-cluster (CC) methods
2, 11–20
represent an effective strat-
egy for describing open-shell species. EOM-CC can describe multi-configurational target states
within a single-reference formalism. It is also capable of computing open-shell and closed-shell
1
Figure 1.1: Examples of closed-shell and open-shell species. Dinitrogen in ground state is
closed-shell. Vinyl has one unpaired electron and is radical. Methylene has two unpaired
electrons and is di-radical. ReproducedfromRef. 2.
h! h! h! e
-
Figure 1.2: Examples of processes leading to formation of open-shell species. The left
panel shows the formation of an open-shell excited state. The middle panel shows the
formation of meta-stable excited state (resonance). The right panel shows ionization. The
ionization continuum is shown by blue.
species on an equal footing. The EOM-CC ansatz has sufficient flexibility to describe different
types of open-shell structures by utilizing different EOM operators.
2
1.2 Bound and temporary anions
A neutral N-electron molecule can attach an electron, forming N+1 electron system, i.e., an
anion. If the anion’s energy lies above the neutral ground state, such systems are metastable
to electron detachment. Such species are called temporary anions. They have finite lifetime
and auto-ionize to form either the ground state or an excited state of the neutral. Anions that
lie below the neutral’s ground state are electronically bound. Fig. 1.3 shows two types of
anions produced by electron attachment to a closed-shell precursor, forming open-shell dou-
blets. Bound anions can be of two types, valence-bond and non-valence-bound.
In a valence-bound anion, the extra electron occupies a valence-like orbital. Generally, the
attachment leads to some structural distortions, making the anion’s geometry different from the
neutral’s. In non-valence-bound anions, the extra electron is bound by long-range electrostatics
or dispersion-type correlation effects
21–24
and often resides on a diffuse orbital, far from the
molecular core. For example, dipole-bound anions are non-valence-bound anions formed due
to electrostatic interactions of the extra electron with the dipole moment of the neutral molecular
core. For the dipole anion to exist, the neutral molecule should have large dipole moment
21, 22, 24
( 2.5 Debye). Examples of molecules that can form dipole-bound anions are polar molecules
such as HCN and HF. The dipole-bound states also exist in metastable anions
25
derived from
strongly polar AgF and CuF which have dipole moment of 5.5 and 6.2 Debye respectively. In
some cases the long-range correlation effects can bind an extra electron, forming correlation
bound anions. Examples for correlation bound anions, in which correlation effects dominate
the binding energy, are large Xe
n
clusters
26
, s-type anion states of C
60
27
, and large acenes
28
.
Theoretical modeling of bound anions is feasible but requires careful selection of a method.
Valence-bound anions are relatively easy to stud. They can be described with a method that
3
e
-
e
-
Figure 1.3: Formation of bound and temporary anion. Formation of bound and tempo-
rary anions. The left panel shows the formation of bound anion by attaching an electron
to a closed-shell precursor. The extra electron is attached to an orbital, which is below
the detachment energy of the anion and lies below the continuum. Right panel shows the
formation of a temporary anion. The extra electron occupies a continuum orbital and has
a finite lifetime. Blue shows the electronic continuum.
can treat closed-shell and open-shell patterns on an equal footing. In dipole-bound anions and
correlation-bound anions, one should include a high level of correlation treatment.
1.3 Equation-of-motion coupled-cluster formalism
EOM-CC
2, 11–20, 29
is a robust theoretical framework capable of treating diverse types of elec-
tronic structure. EOM-CC can treat open-shell and closed-shell systems on an equal footing.
The method yields size-intensive excitation, ionization and attachment energies. One can sys-
tematically converge to the exact solution by including higher-order excitations. EOM-CC is
closely related to the linear response theory. The EOM states are found by diagonalizing the
similarity transformed Hamiltonian. The EOM-CC wave function is expressed as
j i =Re
T
j
0
i; (1.1)
4
where the linear operatorR acts on the reference CC wave functione
T
j
0
i. The operatorT is
an excitation operator satisfying the CC equations for the reference state,
h
j
Hj
0
i = 0; (1.2)
where
H =e
T
He
T
and
are the-tuply excited determinants with respect to the reference
determinant
0
. In EOM-CCSD, the CC operator is truncated as
30
IP
DIP
EA
DEA
EE
SF
Figure 1.4: Different types of target states can be accessed by different combinations of
the reference state and EOM operators. ReproducedfromRef. 29.
TT
1
+T
2
; (1.3)
5
where T
1
and T
2
are spin- and particle-conserving single and double excitation operators of
1-hole-1-particle (1h1p) and 2-holes-2-particles (2h2p) types:
T
1
=
X
ia
t
a
i
a
y
i; T
2
=
1
4
X
ijab
t
ab
ij
a
y
b
y
ji (1.4)
and the truncation of R, i.e, R
0
+R
1
+R
2
is done in a consistent manner. Different vari-
ants
20, 31, 32
of EOM-CC are defined by different choices of the reference state and the type of
EOM operators, R as illustrated in Fig. 1.4. The respective EOM operators for the CCSD level
of theory are shown in Table 1.1.
EOM
^
R
0
^
R
1
^
R
2
EE r
0
P
ia
r
a
i
a
y
i
1
4
P
ijab
r
ab
ij
a
y
b
y
ji
IP 0
P
i
r
i
i
1
2
P
ija
r
a
ij
a
y
ji
EA 0
P
a
r
a
a
y 1
2
P
iab
r
ab
i
a
y
b
y
i
DIP 0
1
2
P
ij
r
ij
ji
1
6
P
ijka
r
a
ijk
a
y
kji
DEA 0
1
2
P
ab
r
ab
a
y
b
y 1
6
P
iabc
r
abc
i
a
y
b
y
c
y
i
Table 1.1: EOM operators in second quantization. a;b;c;::: denote virtual orbitals,
i;j;k;::: denote occupied orbitals andr denotes EOM amplitudes. Occupied and virtual
spaces are defined by the choice of the reference determinant
0
.
If the target state is an open-shell doublet, the electron attachment (EA) variant of EOM-CC
can be used with an even-electron closed-shell singlet reference. Alternatively, a doublet target
state can be described by the ionization potential (IP) variant with an N+1 electron closed-shell
singlet reference. Singlet and triplet excited states can be described by using the excitation
energy (EE) variant of EOM with a closed-shell reference. Double ionization potential (DIP)
and double electron attachment (DEA) can be used with an N-electron closed shell reference to
6
describe N-2 and N+2 electron target states of open-shell character. The quality of the target-
state description depends on the quality of the reference, therefore, it is preferable to use spin-
pure solutions and closed-shell references when possible.
1.4 Electronic resonances
Electronic resonances
33–39
are electronic states that are metastable with respect to electron
detachment and decay by auto-ionization. They are located above electron-detachment or ion-
ization thresholds and belong to the continuum part of the spectrum. Resonances play impor-
tant roles in plasma physics, atmospheric chemistry, fusion reactors, interstellar mediums, and
many other high-energy environments. These states can be produced by excitation or ionization
of core electrons and electron attachment.
Depending on how the resonance decay, electronic resonances can be classified as either
shape or Feshbach resonances. Shape resonances decay by a one-electron process. Transient
anions, such as electron-attached states of N
2
, CO, CO
2
, are shape resonances. Feshbach reso-
nances decay by two-electron processes.
In Hermitian quantum mechanics, resonances belong to the continuum and cannot be asso-
ciated with a single state but rather with the density of states. Their wavefunctions are not
L
2
-integrable and cannot be represented by a finite basis, such as gaussian basis sets. In non-
Hermitian quantum mechanics, resonances can be described by single state and their treatment
does not require explicit description of continuum
40
. Two most common non-Hermitian tech-
niques are complex scaling
41–46
(CS) and complex absorbing potential
47, 48
(CAP) method. In
complex scaling, all Hamiltonian coordinates are scaled by a complex number e
i
(dilation
7
Figure 1.5: The transformation of the spectrum of the Hamiltonian upon complex scaling
of all coordinates as described by the Balslev–Combes theorem. Reproduced from Ref.
38.
transformation). This makes the Hamiltonian non-Hermitian and the resulting expectation val-
ues complex. Bound states areL
2
-integrable at all the values and do not change their position
on dilation, whereas the resonances becomeL
2
-integrable after some critical value of
c
=
1
2
atan
2(E
R
E
t
)
; (1.5)
whereE
r
is resonance energy andE
t
is threshold energy. For >
c
, the resonance energies
are independent of. This condition is an outcome of Balslev–Combes theorem
43–45
.
CAP is another method of representing resonances byL
2
-integrable functions. The regular
Hamiltonian
^
H
0
is augmented with an imaginary potentiali
^
W :
^
H() =
^
H
0
i
^
W (r); (1.6)
where describes the strength of the CAP. Adding this imaginary potential absorbs the diverg-
ing tail of the metastable state, making itL
2
-integrable. Similar to the CS approach, the
8
eigen-energies of the CAP-augmented Hamiltonian are complex. The real part tells about the
metastable state’s position in the spectrum, and the imaginary part provides information about
the lifetime of the resonance.
The research presented in this thesis employed real-valued and complex-valued EOM-CC
methods to describe different types of electronic structures. I also extended the EOM-CC suite
of methods by implementing the EOM-DEA-CCSD equations for energy and density matrices.
The results of the research presented in this thesis were published in the following papers:
1. S. Gulania, T-C. Jagau, A. I. Krylov, “EOM-CC guide to Fock-space travel: the C
2
edi-
tion”,FaradayDiscuss., 217, 514-532 (2019).
2. S. Gulania, T-C. Jagau, A. Sanov and A. I. Krylov, “The quest to uncover the nature of
benzonitrile anion”,Phys. Chem. Chem. Phys., 22, 5002-5010 (2020).
3. S. Gulania, E. F. Kjønstad, J. F. Stanton, H. Koch, and A. I. Krylov, “Equation-of-motion
coupled-cluster method with double electron-attaching operators: Theory, implementa-
tion, and benchmarks.” J.Chem. Phys., 154, 114115 (2021).
I would also like to mention my work, which was published during my Ph.D. but not included
in this thesis:
1. J. Lyle, O. Wedig, S. Gulania, A. I. Krylov and R. Mabbs, “ Channel branching ratios in
CH
2
CN
photodetachment: Rotational structure and vibrational energy redistribution in
auto detachment”,J.Chem. Phys., 147, 234309 (2017).
2. W. Skomorowski, S. Gulania and A. I. Krylov, “Bound and continuum-embedded states
of cyanopolyyne anions”,Phys. Chem. Chem. Phys., 20, 4805-4817 (2018).
9
3. M. Ivanov, S. Gulania and A. I. Krylov, “Two cycling centers in one molecule: Com-
munication by through-bond interactions and entanglement of the unpaired electrons”,J.
Phys. Chem. Lett., 11, 1297-1304 (2020).
10
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1
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J. Schiedt and R. Weinkauf, J. Chem. Phys. 110, 304 (1999).
38
K. B. Bravaya, D. Zuev, E. Epifanovsky, and A. I. Krylov, J. Chem. Phys. 138, 124106
(2013).
39
T.-C. Jagau, K. B. Bravaya, and A. I. Krylov, Annu. Rev. Phys. Chem. 68, 525 (2017).
40
N. Moiseyev, Non-Hermitianquantummechanics. Cambridge University Press, 2011.
41
W. P. Reinhardt, Annu. Rev. Phys. Chem. 33, 223 (1982).
42
N. Moiseyev, Phys. Rep. 302, 212 (1998).
43
J. Aguilar and J. M. Combes, Commun. Math. Phys. 22, 269 (1971).
12
44
E. Balslev and J. M. Combes, Commun. Math. Phys. 22, 280 (1971).
45
B. Simon, Commun. Math. Phys. 27, 1 (1972).
46
P.-O. L¨ owdin, Adv. Quantum Chem. 19, 87 (1988).
47
G. Jolicard and E. J. Austin, Chem. Phys. Lett. 121, 106 (1985).
48
U. V . Riss and H.-D. Meyer, J. Phys. B 26, 4503 (1993).
13
Chapter 2: Equation-of-motion
coupled-cluster method with double
electron-attaching operators: Theory,
implementation, and benchmarks
2.1 Introduction
The robust black-box treatment of open-shell and electronically excited species remains
elusive, despite the progress in our understanding of electron correlation and advances in algo-
rithms and computer hardware. The essential challenge in designing a general strategy is a
great diversity of open-shell patterns, the same trait that is responsible for chemical versatility
of transition metals, multiple roles of reaction intermediates, and vast variety of photoinduced
reactions. The great variety of open-shell patterns can be adequately captured, in general, by
only the exact treatment, full configuration interaction (FCI). More compact formulations rely
on specific approximations, which are grounded in concrete physics and, therefore, have a lim-
ited scope of applicability. Nevertheless, quantum chemistry has developed several uniform and
well-defined approaches for treating broad classes of electronic structure.
14
The coupled-cluster (CC) hierarchy of approximations
1, 2
provides the best set of tools
for ground states of molecules with moderate non-dynamical correlation, such as closed-shell
molecules with large spectral gaps or simple open-shell species (i.e., such as high-spin states
with the maximal spin projection). If the wave function is well described by a single Slater
determinant, then CC methods account for dynamical correlation with an impressive accuracy
achieved already at low rungs of many-body treatments.
3, 4
Equation-of-motion coupled-cluster (EOM-CC) theory
5–15
provides an extension of CC the-
ory to open-shell species and electronically excited states. EOM-CC theory is not a single
method; rather, it is a general framework. Using the Fock-space formalism, it provides a strat-
egy for describing different classes of electronic structure in an efficient and robust manner.
Different variants of EOM-CC provide access to different types of target electronic states, such
as electronically excited, electron attached, or ionized states. It is important to note that the
description of the target states relates to electronic structure patterns and not to a particular phe-
nomenon or experiment. For example, in the EOM terminology, electron-attached states refer
to the states that can be described as the result of adding an electron to a closed-shell reference
and not to the states produced by actual attachment of an electron to a molecule.
In this contribution, we focus on a variant of EOM-CC targeting electronic structure that can
be described as two-electrons-in-many orbitals pattern, as observed in diradicals and molecules
with small spectral gaps (often colloquially referred to as HOMO-LUMO gaps). This approach,
which was introduced
16
by Nooijen and Bartlett for similarity transformed EOM in 1997, was
later further developed
17
and used in several illustrative calculations.
18–21
However, its full
potential has not been appreciated by the computational chemistry community. Here we report
an efficient implementation, including energies and properties, and illustrate the capabilities of
this method by a broad set of examples. The examples highlight the ability of EOM-CC to treat
15
a variety of situations, often described as ’multi-reference’, in an efficient and robust single-
reference framework
12, 14, 22
. In particular, we show that EOM-DEA is a useful tool for treating
diradicals, doubly excited states, bond-breaking, and some types of conical intersections.
2.2 Theory
The EOM-CC wave function is expressed as
j i =Re
T
j
0
i; (2.1)
where the linear operatorR acts on the reference CC wave function,e
T
j
0
i. The operatorT is
an excitation operator satisfying the CC equations for the reference state,
h
j
Hj
0
i = 0; (2.2)
where
H =e
T
He
T
and
are the-tuply excited determinants with respect to the reference
determinant
0
. In EOM-CCSD, the CC operator is truncated as
3
TT
1
+T
2
; (2.3)
where T
1
and T
2
are spin- and particle-conserving single and double excitation operators of
1-hole-1-particle (1h1p) and 2-holes-2-particles (2h2p) types:
T
1
=
X
ia
t
a
i
a
y
i; T
2
=
1
4
X
ijab
t
ab
ij
a
y
b
y
ji (2.4)
and the truncation ofR is done in a consistent manner. Here and belowp
y
andq are electron
creation and annihilation operators corresponding to orbitals
p
and
q
(following the standard
16
convention, indicesi;j;k::: denote orbitals occupied in
0
,a;b;c;::: denote virtual orbitals,
andp;q;r;::: denote orbitals from both subspaces).
IP
DIP
EA
DEA
EE
SF
Figure 2.1: Different types of target states can be accessed by different combination of the
reference state and EOM operators. ReproducedwithpermissionfromRef. 81.
Different variants
12–14
of EOM-CC are defined by different choices of the reference state
and the type of EOM operators R, as illustrated in Fig. 2.1. The focus of this work is on
EOM-DEA.
17, 21, 23
In this method, the operatorsR
1
andR
2
are of 2p and 1h3p types:
R
1
=
1
2
X
ab
r
ab
a
y
b
y
R
2
=
1
6
X
iabc
r
abc
i
a
y
b
y
c
y
i:
17
The EOM amplitudes and the corresponding energies are found by diagonalizing the matrix of
the similarity transformed Hamiltonian,
H, in the basis of determinants generated by the EOM
operatorsR acting on the reference
0
. Since
H is a non-Hermitian operator, its left and right
eigenstates are not Hermitian conjugates but form a biorthonormal set:
HRj
0
i =ERj
0
i (2.5)
h
0
jL
H =h
0
jLE (2.6)
h
0
jL
M
R
N
j
0
i =
MN
(2.7)
whereM andN denote theMth andNth EOM states and
L =L
1
+L
2
=
1
2
X
ab
l
ab
ba +
1
6
X
iabc
l
abc
i
i
y
cba: (2.8)
For energy calculations right eigenstates are sufficient but for property calculations both left and
right eigenstates need to be computed.
The left and right amplitudes are found by diagonalizing the corresponding matrix repre-
sentation of
H. For example, an EOM-EE-CCSD calculation entails the diagonalization of
the effective Hamiltonian
H in the basis of the reference and singly and doubly excited deter-
minants, in an EOM-DEA-CCSD calculation,
H is diagonalized in the basis of 2p and 1h3p
determinants, and so on. Schematically, this can be written in the matrix form as
H
SS
E
cc
H
SD
H
DS
H
DD
E
cc
!
R
1
R
2
!
=!
R
1
R
2
!
(2.9)
and
L
1
L
2
H
SS
E
cc
H
SD
H
DS
H
DD
E
cc
!
=!
L
1
L
2
(2.10)
18
where! is the energy difference with respect to the reference coupled-cluster state. Note that
the structure of the effective Hamiltonian ensures size-intensivity, but not size-extensivity.
24
In
practice, Eqns. (2.9) and (2.10) are solved iteratively, using a generalization of Davidson’s
diagonalization procedure. This procedure requires calculation of the Hamiltonian acting on
trial vectors; the relevant expressions are given in Appendix A.
When a spin-pure closed-shell reference is used, the coupled-cluster amplitudesT are natu-
rally spin-adapted and, consequently, diagonalization of
H yields spin-pure states without any
constraints. Hence, EOM-EE/IP/EA/SF/DEA-CCSD yield naturally spin-adapted states when
using closed-shell references. In contrast, CCSD wave-functions for open-shell references are
not spin-pure, unless explicit spin-adaptation
25–28
is enforced (in most cases, spin-contamination
is rather minor)
29–32
. Hence, in such calculations, e.g., EOM-SF-CCSD with triplet or quartet
references, the target states are not spin-pure and spin-contamination should be monitored
15
.
The scaling of the EOM part in the standard EOM-DEA-CCSD implementation is
N
6
(O
1
V
5
), to be compared withO
2
V
4
in EOM-EE/SF-CCSD,O
3
V
2
in EOM-IP-CCSD,O
4
V
2
in EOM-DIP-CCSD, and O
1
V
4
in EOM-EA-CCSD. The EOM-DEA-CCSD scaling is deter-
mined by the contraction between the transformed two-electron integral andR
2
orL
2
vectors
(
P
de
I
5
abde
r
cde
i
or
P
de
l
ade
i
I
5
debc
) entering the calculation of the doubles-doubles block (see Table
2.18). The storage requirements of the standard implementation reported here are roughly sim-
ilar to those of EOM-EE-CCSD. The size of the EOM vectors is larger (OV
3
in EOM-DEA-
CCSD versusO
2
V
2
in EOM-EE-CCSD), but fewer intermediates are used: e.g.,I
2
,I
4
, andI
6
intermediates
11, 33
are dropped, but the largest ones,I
7
andI
5
, are still needed (see Table 2.19).
As in the case of other EOM-CCSD models, the storage and memory footprint of EOM-DEA-
CCSD can be significantly reduced by using resolution-of-identity or Cholesky decomposition
of the electron-repulsion integrals and avoiding explicit calculation and storage of the interme-
diates
33, 34
.
19
To analyze EOM-CCSD wave functions, we employ reduced quantities such as state and
transition one-particle density matrices. These objects are related to concrete physical observ-
ables
35
and, therefore, provide insight into underlying electronic structure and facilitate com-
parison between different methods
36
. They also provide a way to derive a molecular orbital
picture
35
of many-body wave functions by using concepts such as natural orbitals (NOs) and
natural transition orbitals (NTOs).
The NOs
37
provide a compact one-particle representation of many-electron wave functions.
They are defined as eigenstates of one-particle (state) density matrices (1PDMs). In second
quantization, the 1PDM (
I
) for the
I
state is defined as
I
pq
=h
I
jp
y
qj
I
i: (2.11)
The eigenvalues of 1PDM are called occupation numbers (n
i
); they range from 0 to 1 and add
up to the total number of the electrons. 1PDMs contain all information necessary to compute
expectation values of any one-particle operator
^
O
1
, e.g., dipole moment (^ ), centroids (
^
X,
^
Y ,
^
Z) and second moments (
^
X
^
X,
^
X
^
Y ,:::) of the electron density, etc:
h
I
j
^
O
1
j
I
i =
X
pq
I
pq
h
p
j
^
O
1
j
q
i: (2.12)
1PDMs can be also used to extract quantities that are not related to physical observables,
but provide insight into electronic structure, such as the number of effectively unpaired elec-
trons.
38–40
. Here we usen
u;nl
index, proposed by Head-Gordon,
39
, which can be used to quantify
a polyradical character associated with a given electronic state:
n
u;nl
=
X
i
n
i
2
(2 n
i
)
2
; (2.13)
20
where n
i
are spin-averaged occupation numbers. As one can see, only the orbitals with large
fractional occupations contribute ton
u;nl
whereas the contributions of orbitals withn
i
0 or
n
i
2 are suppressed.
Similarly, one can define the one-particle transition density matrix
36, 41–45
(1PTDM) between
two many-electron wave functions
I
and
J
:
IJ
pq
=h
I
jp
y
qj
J
i: (2.14)
1PTDM can be used to compute one-electron transition properties such as oscillator strengths,
transition dipole moments, non-adiabatic
46
and mean-field spin-orbit couplings
47, 48
. The norm
of
IJ
(defined as
q
P
pq
(
IJ
pq
)
2
) provides a measure of one-electron character of the transition
(e.g.,jj
jj=1 for transitions between
0
and a CIS excited state). The 1PTDM provides a com-
pact representation of the difference between the two states in terms of hole-particle excitations.
Singular-value decomposition of
IJ
yields a set of left and right eigenvectors
IJ
=UV
T
; (2.15)
which define hole (
h
K
) and particle (
e
K
) orbitals corresponding to singular value
K
(elements
of the diagonal matrix )
h
K
=
X
q
U
qK
q
; (2.16)
e
K
=
X
q
V
qK
q
: (2.17)
21
Such pairs of hole and particle orbitals are called NTOs. Usually, only a small number of
singular values are significant. To quantify the collectivity of excitation (i.e., how many NTO
pairs are needed to describe the transition), we use the participation ratio defined as
PR
NTO
=
(
P
i
2
i
)
2
P
i
4
i
=
jj
jj
4
P
i
4
i
: (2.18)
where the sum runs over all singular values. For example, for a CIS wave function withN non-
zero equal-magnitude amplitudes (
1
p
N
), PR
NTO
=N. The participation ratio is closely related to
the number of entangled states, an alternative metric
49
also defined on the basis of the 1PTDMs:
Z
HE
= 2
S
HE
= 1=
Y
i
i
i
; (2.19)
where
i
=
2
i
=jj
jj are squares renormalized singular values of 1PTDM (i.e., weights of
respective NTO pairs) andS
HE
is the hole-electron entanglement entropy:
S
HE
=
X
i
i
log
2
i
: (2.20)
To properly account for spin entanglement in spin-adapted basis, we use the extensivity of
entropy, i.e., the total S
HE
is the sum of the and parts of the transition. Therefore,
one needs to addS
HE
from the and spin-sectors to obtain totalS
HE
. If spin-averaged
1PTDMs are used (as is in Q-Chem/libwfa implementation
50
), thenS
HE
should be computed
as
S
HE
=2
i
i
2
log
2
i
2
; (2.21)
where
i
are renormalized singular values from the spin-averaged 1PTDM.PR
NTO
’s computed
from spin-averaged 1PTDM can be interpreted as the number of contributing configurational
22
state functions; to convert them to the number of contributing Slater determinants, one should
multiply the spin-averaged values by 2.
Somewhat less intuitive than PR
0
NTO
s, Z
HE
’s also report on both spin and spatial entan-
glement. For the case when all weights are equal,Z
HE
andPR
NTO
values are identical. For
example, an excited state produced by an excitation of a single unpaired electron (as in the
hydrogen atom) hasZ
HE
=1 (no entanglement), whereas a singlet excited state corresponding
to an excitation between a single pair of orbitals (such as a HOMO-LUMO excitation in H
2
)
hasZ
HE
=2 (two entangled states due to spin coupling), for a wave-function with 4 equal ampli-
tudes,Z
HE
=4.0, and so on. Z
HE
’s also reflect the entanglement (i.e., correlation) in the states
involved in the transition
51
: an excited state in a bichromophoric system (two H
2
molecules far
apart) can have maximumZ
HE
=4. As was pointed in Ref. 51, an excited state in a stretched H
2
also hasZ
HE
=4, reflecting the entanglement between local excitations of the two separated but
entangled moieties.
Note that the NTOs and the respective exciton descriptors provide convenient tools to ana-
lyze the transitions between states
35, 36, 52
, as long as they have predominantly one-electron char-
acter (i.e., whenjj
jj1).
2.3 Results and discussion
For benchmark purposes, we compare EOM-DEA-CCSD results with higher-level meth-
ods (up to FCI) and, in selected cases, with experiments. When appropriate, we also com-
pare EOM-DEA-CCSD with other EOM models. For singly excited states, EOM-DEA-CCSD
is expected to be comparable to EOM-EE-CCSD. EOM-EE-CCSDT and EOM-CC3 provide
highly accurate results for singly and doubly excited states
53–56
. For diradicals, EOM-DEA-
CCSD is expected to be comparable to EOM-SF-CCSD or EOM-SF-OOCCD (optimized-
orbitals coupled-cluster doubles
57
); in this case, EOM-SF-CCSD with triple corrections, (dT)
23
or (fT), provides sub-kcal/mol accuracy for energy differences
58
. For selected cases, we also
compare against multi-reference configuration interaction (MR-CI).
To illustrate the scope of applicability of EOM-DEA-CCSD, we consider the following
examples:
Low-lying states in molecules with diradical character
11, 58–60
: CH
2
, benzynes, cyclobu-
tadiene. For these systems, we compare EOM-DEA with EOM-SF. We consider both
energy differences and properties. For the diradical manifold (such as low-lying singlets
and triplets), we anticipate similar performance; however, EOM-DEA can also access
higher excited states, as was illustrated in recent studies of Rydberg diradicals
51, 61
.
Ground and excited states of ozone. This example illustrates both the advantages and the
limitations of the EOM-DEA ansatz. Depending on the chosen reference state, different
target manifolds are accessible by EOM-DEA.
Ground and excited states of ethylene at the equilibrium geometry and along the torsional
coordinate. This example illustrates () bond-breaking and the ability of EOM-DEA to
treat doubly excited states.
Excited states of butadiene. This examples illustrates the extent of doubly excited char-
acter in the lowest dark state in polyenes.
To assess the ability of EOM-DEA to treat conical intersections, we consider a well stud-
ied retinal example
62–64
and HeH
2
.
Excited states in small molecules (water and ammonia), where we compare EOM-
DEA with EOM-EE and other methods (including FCI). Similarly to the ozone case,
these examples illustrate that states that are dominated by excitations from HOMO are
24
described well by EOM-DEA, but states derived by excitations from lower orbitals are
not.
Section 2.3.1 provides computational details for each example.
2.3.1 Computational details
All EOM-CCSD calculations were performed using the Q-Chem electronic structure pro-
gram
65, 66
. The reported symmetry labels of electronic states and MOs correspond to Mulliken’s
convention
67
, which differs from the standard molecular orientation used in Q-Chem (hence,
some state labels had to be changed). The EOM-CCSDT calculations were performed with
CFOUR
68
and the EOM-CC3 calculations with eT.
69, 70
For HeH
2
, EOM-EE-CCSD calcula-
tions were performed with eT and FCI calculations with DALTON.
71
In EOM-DEA-CCSD cal-
culations, we use defaultN-2 electron references. To investigate effects of orbital relaxation,
we also carried out calculations with orbitals computed for N-electron states. In cases when
different orbitals were used, the charge of the reference state is indicated as follows: ‘EOM-
DEA-CCSD/+X’, where ‘+X’ denotes the charge of the Hartree-Fock determinant defining the
orbitals used in the EOM-CC calculations.
Methylene calculations were performed using the equilibrium geometries and TZ2P basis
set from Ref. 72. Pure angular momentum polarization functions were employed. In CCSD
and EOM calculations, all orbitals were active. Note that the reference FCI energies
72
were
computed with one frozen core and one frozen virtual orbital.
Cyclobutadiene calculations were performed using the same equilibrium geometries as in
Ref. 11. All calculations were performed with the cc-pVTZ basis set. Pure angular momentum
polarization functions were employed; all orbitals were active.
25
Calculations of benzynes were performed using the same equilibrium geometries as in
Ref. 84 and with the cc-pVTZ basis set. Pure angular momentum polarization functions were
employed; all orbitals were active.
Butadiene calculations were performed using the geometries from Ref. 86. Pure angular
momentum polarization functions were employed; core orbitals were frozen.
Ethylene calculations were performed using the geometries from Ref. ? (r
CC
= 1:330
˚
A, r
CH
= 1:076
˚
A, and
HCH
= 116
o
; the torsion profile was computed by varying the
torsional angle and keeping the rest of the structural parameters frozen at their equilibrium
values). Cartesian polarization functions were employed; core electrons were frozen.
Ozone calculations were performed using the geometries from Ref. 83. Pure angular
momentum functions were employed. Core electrons were frozen.
Water and ammonia calculations were performed using the geometries from Ref. 87. Pure
angular momentum functions were employed; core electrons were frozen.
Retinal chromophore calculations were performed using the geometries from Ref. 62. Carte-
sian polarization functions were employed; all electrons were active.
HeH
2
calculations are performed using pure angular momentum functions; all electrons
were active.
Computation ofZ
HE
requires additional clarification. When using restricted Hartree-Fock
references, the Q-Chem/libwfa printout of renormalized entanglement entropy (S
RHF
HE
) for the
transition should be used to compute fullS
HE
as follows:
S
HE
=S
RHF
HE
+ 1 (2.22)
26
which follows from Eq. (2.21) above. When using unrestricted Hartree-Fock references,S
HE
should be computed using Eq. (2.19) and taking care of proper normalization of the two spin-
blocks. This leads to the following expression forS
HE
:
S
HE
=S
HE
+S
HE
=
f
S
UHF
f
log
2
f
+f
S
UHF
f
log
2
f
=
f
S
UHF
+f
S
UHF
(f
log
2
f
+f
log
2
f
) (2.23)
where
f
=
jj
=
jj
2
jj
jj
2
+jj
jj
2
: (2.24)
ThenZ
HE
is computed as 2
S
HE
. Note thatlibwfa prints squares of singular values (
2
K
).
2.3.2 CH
2
Methylene is an example of a simple diradical with a triplet ground state. Its low-lying
electronic states are derived by distributing two electrons over two frontier orbitals (p
z
andsp
2
-
hybridized orbitals of carbon). Table 2.1 shows the total energy for the
~
X
3
B
1
state (high-spin)
and adiabatic excitation energies for the ~ a
1
A
1
,
~
b
1
B
1
, and
~
b
1
B
1
states computed with EOM-DEA-
CCSD and several other methods.
As expected, Hartree-Fock calculations overestimate the singlet-triplet gap because of an
unbalanced description of the singlet state (which needs more correlation because of the diradi-
cal character) and the triplet state (which is well described by the single determinant). However,
the diradical character in the lowest singlet state is modest (n
u;nl
< 0:1), so that CCSD cal-
culation yields a reasonable gap, which is further improved by including triples corrections.
However, CCSD and CCSD(T) cannot be used to describe the open-shell singlet state,
~
b
1
B
1
;
they also yield large errors for the energy gap between the two closed-shell singlet states.
27
Table 2.1: Total energies (hartree) for the ground
~
X
3
B
1
state of CH
2
and adiabatic exci-
tation energies (eV) for the three lowest singlet states
a
. ZPE not included; TZ2P basis set.
Method
~
X
3
B
1
~ a
1
A
1
~
b
1
B
1
~ c
1
A
1
HF
b
-38.937956 1.236 2.772
CCSD
b
-39.080919 0.545 2.054
CCSD(T)
b
-39.083856 0.505 1.907
EOM-EE-CCSD
b
-39.08066 0.538 1.566 3.843
SF-CIS
b
-38.93254 0.883 1.875 3.599
SF-CIS(D)
b
-39.05586 0.613 1.646 2.953
EOM-SF-OOCCD
b
-39.08045 0.514 1.564 2.715
EOM-SF-CCSD/UHF
c
-39.080453 0.517 1.565 2.718
EOM-SF-CCSD(fT)/UHF
c
-39.08184 0.500 1.552 2.688
EOM-SF-CCSD(dT)/UHF
c
-39.08217 0.496 1.548 2.678
EOM-DEA-CCSD
d
-39.069681 0.481 1.461 2.518
FCI
e
0.483 1.542 2.674
a
MR-CISD/TZ2P optimized geometries. All electrons were active in CC/EOM-CC
calculations.
b
From Ref. 59 (EOM-EE-CCSD values computed using closed-shell singlet
reference).
c
From Ref. 58.
d
This work.
e
From Ref. 72; one frozen core and one frozen
virtual orbital.
Table 2.2: One-electron properties
a
for the lowest electronic states of CH
2
computed using
EOM-SF-CCSD and EOM-DEA-CCSD wave functions; aug-cc-pVTZ basis set.
State (a.u) R
2
(a.u) n
u;nl
EOM-SF-CCSD
~
X
3
B
1
0.239 24.24 2.00
~ a
1
A
1
0.692 25.07 0.07
~
b
1
B
1
0.275 24.84 2.00
~ c
1
A
1
0.099 25.41 1.96
EOM-DEA-CCSD
~
X
3
B
1
0.327 23.96 2.00
~ a
1
A
1
0.773 24.84 0.06
~
b
1
B
1
0.327 24.67 2.00
~ c
1
A
1
0.105 24.74 1.96
a
=dipole moment, R
2
=h
^
X
2
i +h
^
Y
2
i +h
^
Z
2
i,
n
u;nl
= Head-Gordon index.
We note that the EOM-EE-CCSD calculation (performed using a closed-shell reference)
yields a better value of the singlet-triplet gap than CCSD, because of a more balanced descrip-
tion of the two states. Another advantage of the EOM-CC ansatz is that it yields not just the
28
lowest triplet, but also two other singlet states. The energy of the open-shell singlet,
~
b
1
B
1
is
reproduced well by EOM-EE-CCSD, but the error for the ~ c
1
A
1
state is off by1 eV because
of its doubly excited character. As one can see, the EOM-SF-CCSD and EOM-SF-OOCCD
ans¨ atze yield excellent values for all three gaps (within 0.03 eV from FCI) and the inclusion of
triples correction reduces the errors relative to FCI to 0.01 eV range. The EOM-DEA-CCSD
performs similarly to EOM-SF-CCSD, but the errors are slightly larger (0.002, 0.081, and 0.156
eV). This slightly deteriorated performance can be attributed to orbital relaxation effects—the
EOM-SF calculation uses the triplet-state orbitals, which are nearly optimal for all four states
from the diradical manifold whereas EOM-DEA-CCSD uses dication orbitals. Consequently,
theR
2
operator in the EOM-SF-CCSD ansatz can be fully employed to describe the differen-
tial correlation of the EOM states whereas in EOM-DEA-CCSD R
2
needs to deal with both
correlation and orbital relaxation effects.
Table 2.2 compares EOM-SF-CCSD and EOM-DEA-CCSD wave functions. We observe
small but noticeable differences in permanent dipole moments and the size of electron density
distribution. The EOM-DEA-CCSDhR
2
i values, which quantify the size of the electron den-
sity, appear to be somewhat smaller than the EOM-SF-CCSD ones, which is consistent with
using more compact dicationic orbitals. The diradical character, as quantified by n
u;nl
, is nearly
the same: for the lowest singlet state it is 0.06 for the EOM-EE-CCSD and 0.07 for EOM-DEA-
CCSD; for the triplet state both methods yield 2.00; and for the ~ c
1
A
1
state both methods yield
1.96.
Overall, despite slightly larger errors in energy gaps than in EOM-SF-CCSD, EOM-DEA-
CCSD performs rather well and is free from spin-contamination, as it relies on the closed-shell
reference. We anticipate that performance of EOM-DEA-CCSD can be brought up to sub-
kcal/mol range by perturbative account of higher excitations (i.e., 2h4p).
29
In addition, EOM-DEA-CCSD can describe higher excited states, which are not accessi-
ble by EOM-SF-CCSD, as was illustrated in recent studies of Rydberg diradicals
51, 61
, a class
of molecules in which the two unpaired electrons reside in two diffuse orbitals. These exotic
species came into a spotlight due to their potential utility in quantum information science
51, 61, 73
.
In the context of laser cooling, which is an essential step in utilizing these molecules in quan-
tum information applications, one needs to describe not only the low-lying diradical states (as
those discussed above), but also higher excited states (excitations from HOMO to higher-lying
orbitals) through which optical cycling is carried out. Hence, EOM-DEA-CCSD offers an
advantage over EOM-SF-CCSD.
2.3.3 Benzynes
Benzynes, aromatic diradicals, are popular benchmark systems for theory
58, 59, 74–78
,
owing to the availability of the high-quality experimental data
79, 80
. Table 2.3 shows adiabatic
singlet-triplet gaps in ortho-, meta-, and para-benzynes; the frontier NOs and their occupa-
tions are shown in Fig. 2.2. The results follow an anticipated trend: as the distance between the
diradical centers increases, the gap shrinks and the diradical character increases, as evidenced
by the occupations of the two frontier NOs in the ground singlet state. Table 2.4 compares the
diradical character (as characterized by the Head-Gordon index) of the singlet and triplet states
computed using EOM-DEA-CCSD and EOM-SF-CCSD wave functions. As in the methylene
example, both EOM-SF-CCSD and EOM-DEA-CCSD yield accurate gaps for all three iso-
mers. The differences between the two methods do not exceed 0.05 eV . The character of the
wave functions, e.g., as illustrated by the number of effectively unpaired electrons, is also very
similar.
30
Table 2.3: Adiabatic singlet-triplet gaps (eV) in benzynes (no ZPE); cc-pVTZ basis set.
Method o-benzyne m-benzyne p-benzyne
EOM-SF-CCSD
a
1.578 0.782 0.147
EOM-SF-CCSD(fT)
a
1.615 0.875 0.169
EOM-SF-CCSD(dT)
a
1.619 0.892 0.172
EOM-DEA-CCSD
b
1.625 0.799 0.145
ZPE
d
-0.028 0.043 0.021
Expt. - ZPE 1.656 0.868 0.144
a
From Ref. 58.
b
This work.
c
From Ref. 79, 80.
d
From Ref. 59.
o m p
1.85 (1.00)
0.14 (0.99)
1.80 (1.00)
0.19 (0.99) 0.60 (0.98)
1.38 (1.00)
0.11 (0.98) 0.18 (0.98) 0.59 (0.98)
1.86 (0.98) 1.78 (0.98) 1.38 (0.98)
Figure 2.2: Benzynes. Frontier NOs and their occupation numbers in the lowest sin-
glet and triplet states (triplet-state occupations are given in parenthesis) computed for the
EOM-SF-CCSD (black) and EOM-DEA-CCSD (red) wave functions using the cc-pVTZ
basis set.
Table 2.4: Head-Gordon index (n
u;nl
) for the lowest singlet and triplet states in benzynes
computed using the EOM-DEA-CCSD and EOM-SF-CCSD (numbers in parenthesis)
wave functions; cc-pVTZ basis set.
Type Singlet Triplet
o-benzyne 0.11 (0.16) 2.00 (2.00)
m-benzyne 0.26 (0.26) 2.00 (2.00)
p-benzyne 1.43 (1.45) 2.00 (2.00)
2.3.4 Cyclobutadiene
Cyclobutadiene is a popular benchmark system
81
. At square (D
4h
) structures, symmetry
requires that the two frontier orbitals be exactly degenerate, giving rise to a perfect diradical
31
pattern, whereas symmetry lowering to D
2h
lifts the degeneracy and results in a closed-shell
pattern. The lowest electronic state is a singlet state. Due to the second-order Jahn-Teller
effect, the equilibrium ground state structure is rectangular (D
2h
), with alternating double and
single bonds. The lowest triplet state is not affected by the Jahn-Teller effect; its equilibrium
geometry is D
4h
. Fig. 2.3 shows the geometries of the 1
3
A
2g
and X
1
A
g
states optimized at the
CCSD(T)/cc-pVTZ level of theory.
The ground singlet state of cyclobutadiene shows variable extent of the diradical character,
which also affects low-lying electronic states. Table 2.5 shows the total energy for
~
X
1
A
g
ground
state and vertical excitation energies for the 1
3
B
1g
, 1
1
B
1g
, and 2
1
A
g
computed with EOM-
DEA-CCSD and other methods. Table 2.6 shows the total energy for the
~
X
1
B
1g
ground state
and vertical excitation energies for 1
3
A
2g
, 2
1
A
1g
, and 1
1
B
2g
for EOM-DEA-CCSD and other
methods.
C C
C C
H
H H
H
C C
C C
H
H H
H
1.439
1.073
135.0
1.566
1.074
1.343
134.91
Figure 2.3: Geometries of the 1
3
A
2g
(left) and X
1
A
g
(right) states optimized at the
CCSD(T)/cc-pVTZ level of theory. Bond lengths are in angstroms and angles are in
degrees; the structures are from Ref. 11.
Total EOM-DEA-CCSD energies for the ground state are significantly above EOM-SF-
CCSD ones, by about0.02 hartree, which we attribute to using the dication orbitals in the for-
mer. However, the differences in the respective excitation energies are relatively small. At both
geometries (D
4h
and D
2h
structures) the vertical excitation energies obtained with EOM-DEA-
CCSD are underestimated relative to EOM-SF-CCSD (by 0.2-0.4 eV); this can also be attributed
32
Table 2.5: Total energies (hartree) of the ground stateX
1
A
g
of cyclobutadiene and vertical
excitation energies (eV) at theX
1
A
g
equilibrium geometry (D
2h
symmetry, 4 frozen core
orbitals); cc-pVTZ basis set.
Method E
tot
(X
1
A
g
) 1
3
B
1g
1
1
B
1g
2
1
A
g
EOM-EE-CCSD -154.354 95 1.349 3.314 7.874
EOM-EE-CCSDT -154.390 67 - 3.264 4.512
UHF-EOM-SF-CCSD -154.362 85 1.652 3.411 4.354
UHF-EOM-SF-CCSD(fT) -154.367 45 1.516 3.257 4.203
UHF-EOM-SF-CCSD(dT) -154.367 44 1.474 3.210 4.174
ROHF-EOM-SF-CCSD -154.363 39 1.656 3.408 4.348
ROHF-EOM-SF-CCSD(fT) -154.367 56 1.515 3.253 4.197
ROHF-EOM-SF-CCSD(dT) -154.367 37 1.467 3.200 4.168
EOM-DEA-CCSD -154.339 22 1.403 3.120 4.127
Table 2.6: Total energies (hartree) of the ground state X
1
B
1g
of cyclobutadiene and vertical
excitation energies (eV) at the X
1
B
1g
equilibrium geometry (D
4h
symmetry); cc-pVTZ
basis set
Method E
tot
(X
1
B
1g
) 1
3
A
2g
2
1
A
1g
1
1
B
2g
EOM-EE-CCSD
a
-154.380 58 -0.590 - 1.534
UHF-EOM-SF-CCSD
a
-154.413 01 0.369 1.824 2.143
UHF-EOM-SF-CCSD(fT)
a
-154.414 78 0.163 1.530 1.921
UHF-EOM-SF-CCSD(dT)
b
-154.413 90 0.098 1.456 1.853
ROHF-EOM-SF-CCSD
a
-154.413 42 0.369 1.814 2.137
ROHF-EOM-SF-CCSD(fT)
b
-154.414 77 0.159 1.521 1.915
ROHF-EOM-SF-CCSD(dT)
b
-154.413 58 0.088 1.438 1.837
EOM-DEA-CCSD
c
-154.386 00 0.023 1.406 1.751
a
From Ref. 11.
b
From Ref. 58.
c
This work.
to the use of compact Hartree-Fock orbitals of the +2 reference state. Interestingly, EOM-
DEA-CCSD is closer to UHF/ROHF-EOM-SF-CCSD(dT) than EOM-EE-CCSD or EOM-SF-
CCSD: the EOM-DEA-CCSD gaps are within 0.1 eV from the reference UHF/ROHF-EOM-
SF-CCSD(dT) values, likely due to a fortuitous error cancellation.
2.3.5 Ozone
Ozone has been extensively studied because of its role in atmospheric chemistry
82, 83
. Owing
to its non-classical bonding pattern, which cannot be described by a single Lewis structure,
ozone features the ground state of diradical character and low-lying excited states.
33
Ozone has 24 electrons and belongs to C
2v
symmetry. Electronic configurations of its low-
lying states are summarized in Table 2.7; Fig. 2.4 shows relevant frontier NOs. Table 2.7
lists the occupation numbers of the leading configurations in the ground and several excited
states. The [core] denotes 9 molecular orbitals that are doubly occupied in the ground state
and in the excited state discussed here. Electronic configuration of ozone’s ground state
is [core]
18
(4b
2
)
2
(6a
1
)
2
(1a
2
)
2
, with relatively weak diradical character: n
u;nl
=0.013 (EOM-
DEA-CCSD value). To compute the ground and excited states of ozone by EOM-DEA-
CCSD, we need to use a +2 charge reference. One can consider different choices, e.g.,
REF1=[core]
18
(4b
2
)
2
(1a
2
)
2
and REF2=[core]
18
(4b
2
)
2
(6a
1
)
2
. Both references are suitable for
describing the ground state, but the accessibility of excited states differs. When starting from
REF1, X
1
A
1
,
3
B
1
,
1
B
1
, and 2
1
A
1
excited states can be described accurately, because their lead-
ing electronic configuration can be generated by the 2p part of the EOM-DEA operator. In
contrast, when using REF2, only X
1
A
1
and
3
B
2
can be described accurately. Neither REF1 nor
REF2 is suitable for the
1
A
2
excited state, because from either one the 3h1p DEA operators are
required to reach this state.
4b
2
1b
1
6a
1
1a
2
2b
1
Figure 2.4: Frontier natural orbitals of the X
1
A
1
state of ozone. computed with EOM-
DEA-CCSD/aug-cc-pVTZ for the ground state. In the ground state, 2b
1
orbital is vacant.
Table 2.8 compares vertical excitation energies in ozone computed with EOM-EE-CCSD
and EOM-DEA-CCSD using the aug-cc-pVTZ basis set; for the singlet states, we also report
EOM-EE-CCSDT values. As expected, for EOM-EE the effect of triple excitations is small for
the
1
A
2
and
1
B
1
states, but is about 4 eV for the
3
A
2
state because of its doubly excited character.
Relative to EOM-EE-CCSD, EOM-DEA-CCSD underestimates the excitation energy of the
1
B
1
34
Table 2.7: Orbital occupations in dominant electronic configurations of the ground and
low-lying excited states in ozone
a
.
Configurations
State [core] 4b
2
6a
1
1a
2
2b
1
X
1
A
1
18 2 2 2 0
3
B
2
18 2 2 1 1
3
B
1
18 2 1 2 1
3
A
2
18 1 2 2 1
1
A
2
18 1 2 2 1
1
B
1
18 2 1 2 1
2
1
A
1
18 2 0 2 2
a
Geometry used:r
OO
= 1.2724
˚
A,
OOO
= 116:82
o
[core] = (1a
1
)
2
(2a
1
)
2
(3a
1
)
2
(4a
1
)
2
(5a
1
)
2
(1b
2
)
2
(2b
2
)
2
(3b
2
)
2
(1b
1
)
2
state, whereas the excitation energies of the
3
B
2
and
3
B
1
states are overestimated (the differences
are 0.1-0.3 eV). The advantage of EOM-DEA-CCSD is that it captures the doubly excited state,
2
1
A
1
—the error against EOM-EE-CCSDT is less than 0.2 eV . However, EOM-DEA cannot
access the
3
A
2
state.
Table 2.8: Ozone. Vertical
a
excitation energies (eV) relative to the X
1
A
1
state computed
by EOM-EE-CCSD and EOM-DEA-CCSD; aug-cc-pVTZ.
State EE-CCSD
b
EE-CCSDT
b
DEA-CCSD
c
DEA-CCSD
d
3
B
2
1.313 - - 1.605
3
B
1
1.707 - 1.965 -
3
A
2
1.873 - - -
1
A
2
2.281 2.138 - -
1
B
1
2.311 2.200 2.265 -
2
1
A
1
9.112 5.174 4.992 -
a
Geometry used:r
OO
= 1.2724
˚
A,
OOO
= 116:82
o
b
HF reference = [core]
18
(4b
2
)
2
(6a
1
)
2
(1a
2
)
2
c
HF reference = [core]
18
(4b
2
)
2
(1a
2
)
2
d
HF reference = [core]
18
(4b
2
)
2
(6a
1
)
2
35
2.3.6 Ethylene
Ethylene features a dense manifold of low-lying excited states of valence and Rydberg char-
acter. The lowest states are derived by the excitations from the HOMO. The key valence
states are traditionally called: N (
1
()
2
), T(
3
()
1
(
)
1
), V (
1
()
1
(
)
1
), and Z (
1
(
)
2
). At the
twisted geometries, the overlap between the two p-orbitals is reduced and the gap between
and
shrinks. At 90
, the double bond is broken and and
should be exactly degenerate
by symmetry.
Because of its perfect diradical character, twisted ethylene (at 90
) presents a challenge to
standard single-reference methods. Fig. 2.5 shows ground-state potential energy profiles along
the twisting coordinate computed with different methods; the respective torsion barriers are
summarized in Table 2.9 (total energies for the entire scan are given in Table S1).
MRCI (here, 2x2 CASSCF with with single and double excitations) captures both static and
dynamic correlation, yielding a smooth potential curve; we regard these values as the reference.
In the closed-shell Hartree-Fock reference, the two orbitals are not treated in a balanced way,
which results in a high barrier and a cusp on the torsional potential. Inclusion of the correlation
in the CCSD ansatz reduces the barrier height, but cannot fully eliminate the cusp, because of
an unbalanced treatment of ()
2
and (
)
2
configurations. In contrast, EOM-SF-CCSD (with
a high-spin triplet reference) and EOM-DEA-CCSD (with a dication reference) are treating the
two frontier orbitals and the respective configurations in a balanced way, yielding smooth poten-
tial energy curves and accurate barrier heights (as compared to the reference MRCI values). For
comparison, we also show the results of the SF-TDDFT calculations
84
(using recommended
B5050LYP functional); this uncorrelated approach yields a smooth curve, but the barrier is
overestimated by0.2 eV .
We note that the cusp reappears in the EOM-DEA-CCSD calculations that use the neutral
reference orbitals. These calculations also overestimate the barrier height (i.e., the difference
36
between +2 and 0 calculation is 0.89 eV). The analysis of the relevant orbitals (shown in the
SI) attributes this to the scrambling of the HOMO with the low-lying orbitals (in the neutral
reference), leading to symmetry breaking.
0.1
0.11
0.12
0.13
0.14
75 80 85 90 95 100 105
Energy (hartree)
Angle (Θ)
SF-5050
MR-CI
CCSD
SF-CCSD
DEA-CCSD
Figure 2.5: Ethylene torsion barrier computed with various methods and a DZP basis set.
All curves are shifted such that the energy at 0
0
is zero.
Table 2.9: Ethylene torsion barrier (eV) computed with various methods and a DZP basis
set.
Method Barrier
SF-TDDFT/5050 3.48
MR-CISD 3.27
CCSD 3.91
EOM-SF-CCSD/UHF 3.23
EOM-DEA-CCSD/+2 3.20
EOM-DEA-CCSD/+0 4.09
EOM-DEA-CCSD
/+0 3.41
Dihedral angle = 89.9999
o
. This allows
the symmetry of molecule remain inD
2
point group.
Table 2.11 compares vertical excitation energies for relevant excited states of ethylene at
its equilibrium geometry (0
torsion angle) computed with EOM-EE-CCSD and EOM-DEA-
CCSD using the aug-cc-pVTZ basis set. Relevant properties are given in Table 2.12. As in the
37
0
2
4
6
8
45 90 135
VEE (eV)
Angle (Θ)
N-state
Z-state
V-state
T-state
Figure 2.6: Potential energy curves along torsion coordinate for several electronic states
of ethylene computed with EOM-DEA-CCSD/aug-cc-pVTZ.
Table 2.10: Ethylene. Vertical excitation energies (eV) from the lowest singlet state (
1
A
2
)
at the twisted geometry using EOM-DEA-CCSD/aug-cc-pVTZ.
State EOM-DEA-CCSD
1
3
A
2
(T-state) -0.011
1
1
A
1
(V-state) 2.485
2
1
A
1
(Z-state) 2.507
previous examples, EOM-DEA-CCSD excitation are comparable to the EOM-EE-CCSD ones
for all excited states that are dominated by one-electron excitation. The differences between the
two methods are 0.02-0.4 eV . In this case, EOM-DEA-CCSD seems to be consistently closer to
the experimental values. The characters of all computed singly excited states are very similar
for EOM-EE and EOM-DEA. In particular, both methods agree in the extend of the Rydberg
character, which can be conveniently quantified by average electron-hole separation. Z
HE
val-
ues are close to two for the Rydberg states, indicating their pure character (one NTO pair is
sufficient to describe these transitions), whereas for the states with dominant!
charac-
ter Z
HE
are larger due to Rydberg-valence interactions
85
(i.e., contributions of the ! Ry
configurations). EOM-DEA-CCSD can also describe doubly excited state,
1
A
g
, which is not
accessible by EOM-EE-CCSD.
38
Ethylene torsion has significant effect on the valence excited states, e.g., T, V , and Z states
come down in energy, as expected from the orbital energetics. Fig 2.6 shows EOM-DEA-
CCSD/aug-cc-pVTZ potential energy curves along the torsional coordinate for the four valence
states (N, T, V , and Z); Table 2.10 shows the vertical excitation energy from lowest singlet state
at 90
. As one can see, at 90
, the lowest singlet and triplet states become nearly degenerate,
and the two excited singlet states are also nearly degenerate.
Table 2.11: Ethylene. Vertical excitation energies (eV) at the equilibrium geometry.
State Orbital assign Exp.
a
DEA-CCSD
b
EE-CCSD
b
3
B
1u
3
(;
) 4.36 4.48 4.50
3
B
3u
3
(; 3s) 6.98 6.82 7.31
1
B
3u
1
(; 3s) 7.11 6.92 7.44
1
B
1u
1
(;
) 7.68(8.0) 7.79 8.04
2
B
3u
2
() 10.5 10.43 10.43
1
A
g
1
(
;
) - 12.86 -
a
Exp. from Ref. ?.
b
Basis set: aug-cc-pVTZ basis set.
Table 2.12: One-particle state and transition properties
a
for ethylene computed with
EOM-EE-CCSD and EOM-DEA-CCSD; aug-cc-pVTZ basis set.
State n
u;nl
f
l
jj
jj Z
HE
e-h sep
EE-CCSD
3
B
1u
2.00 0.00 0.94 2.36 1.82
3
B
3u
2.00 0.00 0.91 2.03 3.60
1
B
3u
2.00 0.08 0.90 2.01 3.71
1
B
1u
2.00 0.37 0.92 2.68 2.59
1
A
g
- - - - -
DEA-CCSD
3
B
1u
2.00 0.00 0.95 2.34 1.87
3
B
3u
2.00 0.00 0.91 2.02 3.64
1
B
3u
2.00 0.09 0.91 2.01 3.74
1
B
1u
2.00 0.38 0.93 2.38 2.81
1
A
g
0.55 0.00 0.19 3.41 2.41
a
n
u;nl
= Head-Gordon’s index,f
l
= oscillator strength,jj
jj= norm of 1PTDM,Z
HE
=
hole-particle entanglement, e-h sep = electron hole separation (in
˚
A).
39
2.3.7 Butadiene
Table 2.13: Vertical excitation energies (eV) for the 1
1
B
+
u
and 2
1
A
g
in butadiene
Basis
EE-CCSD
a
DEA-CCSD/+2
b
DEA-CCSD/0
b
EE-CC3
b
EE-CCSDT
a
1
1
B
+
u
2
1
A
g
1
1
B
+
u
2
1
A
g
1
1
B
+
u
2
1
A
g
1
1
B
+
u
2
1
A
g
1
1
B
+
u
2
1
A
g
cc-pVDZ 6.918 7.648 6.893 7.412 6.857 7.449 6.776 6.968 6.794 6.830
cc-pVTZ 6.660 7.555 6.643 7.206 6.632 7.296 6.514 6.870 6.535 6.763
cc-pVQZ 6.562 7.458 6.520 7.074 - - - - - 6.722
aug-cc-pVDZ 6.389 7.057 6.265 6.684 6.353 6.867 6.269 6.661 6.285 6.577
aug-cc-pVTZ 6.365 7.093 6.241 6.679 6.343 6.883 6.238 6.654 6.241
c
-
a
Ref. 86.
b
This work.
c
Estimated with nine frozen orbitals and extrapolation
86
.
Butadiene is the smallest polyene, representing an important motif commonly occurring in
photoactive molecules and dyes. The theoretical description of the two lowest states in polyenes
is challenging due to their different character. In butadiene, the lowest excited state, 1
1
B
+
u
, is
a bright, dipole-allowed, singly excited state. The second excited state, 2
1
A
g
is dark, dipole-
forbidden state, which is believed to have substantial doubly excited character. UV-VIS absorp-
tion spectrum of butadiene places the 1
1
B
+
u
state at 5.92 eV above the ground state (vertically).
Because butadiene is able to fluoresce, the consensus is that the dark state is located above
the bright state. In longer polyenes, the dark state drops below the bright state, which leads
to fluorescence quenching. The exact positions, and even the ordering, of the two states in
short polyenes, butadiene and hexatriene, has been debated, as summarized, for example, in
Ref. 86. The difficulties in resolving this issue theoretically stem from the strong dynamical
correlation effects in the bright state and some doubly excited character of the dark state. Only
high-level methods, such as EOM-EE-CCSDT, can provide an accurate description of these two
effects. Moreover, because of the contributions of Rydberg excitations into the dark state, the
results (including wave function composition) are sensitive to the basis set. Consequently, the
results of approximate treatments vary widely. Multi-reference methods tend to overestimate
the doubly excited character of the dark state and underestimate the contributions from Rydberg
40
excitations. In contrast, single-reference methods, such as EOM-CCSD or ADC, underestimate
doubly excited contributions in the dark state.
X
1
A
-
g
1.99
1.88
0.08 0.02
2
1
A
-
g
1.97 0.25 0.80 0.93
1
1
B
+
u
1.99 0.98 0.96 0.02
Figure 2.7: Butadiene. Natural frontier orbitals and their occupations computed using
the EOM-DEA-CCSD/aug-cc-pVTZ wave functions (using dication reference orbitals).
Our results show that the EOM-DEA-CCSD ansatz is capable of describing both states
on the same footing, yielding excitation energies of the two states in good agreement with the
reference EOM-EE-CCSDT values. This is illustrated by Table 2.13, which compares excitation
energies for the bright (1
1
B
+
u
) and dark (2
1
A
g
) excited states of trans-butadiene computed with
the EOM-EE-CCSD, EOM-DEA-CCSD, EOM-EE-CC3, and EOM-EE-CCSDT methods with
different basis sets. For EOM-DEA-CCSD, we report the results obtained with two references:
one constructed using the default dication orbitals and one constructed using the orbitals from
the neutral system.
Excitation energies for both states using EOM-DEA-CCSD are comparable to EOM-EE-
CCSDT and EOM-EE-CC3. For the bright state, the effect of triple excitations is small and the
difference between EOM-EE-CCSD and EOM-EE-CCSDT is0.1 eV; as usual, inclusion of
triples brings the excitation energies down. The EOM-DEA-CCSD values are slightly below
EOM-EE-CCSD and, therefore, are slightly closer to EOM-EE-CCSDT. Using neutral orbitals
increases the excitation energy, which can be attributed to better description of the ground state.
The effect of the basis set is noticeable — including diffuse functions lowers the excitation
energy of the bright state by 0.2 eV .
41
The results for 2
1
A
g
show larger differences between EOM-EE-CCSD and EOM-EE-
CCSDT. In the small basis set (cc-pVDZ), the difference is 0.8 eV , but it shrinks by 0.3 eV
when diffuse functions are included. This illustrates the importance of Rydberg contributions
and that the effect of double excitations is exaggerated when using compact basis sets (this is
why most multi-reference calculations significantly overestimate doubly excited character of
the dark state). EOM-DEA-CCSD energies for the dark state are closer to EOM-EE-CCSDT,
yielding smaller errors than those of EOM-EE-CCSD. In the aug-cc-pVDZ basis, the energy
of the dark state is overestimated by only 0.1 eV by EOM-DEA-CCSD (relative to EOM-EE-
CCSDT). The CC3 results show similar trends as EOM-EE-CCSDT and EOM-DEA-CCSD.
These observations are supported by the wave function analysis of the EOM-EE-CCSD,
EOM-EE-CC3, and EOM-DEA-CCSD wave functions, summarized in Table 2.14 (see also
Table S2 in the SI). Fig. 2.7 shows the occupancy of natural orbitals for the ground and two
excited states (described by EOM-DEA-CCSD with dicationic reference). Occupations of the
frontier NOs clearly show singly excited character of the 1
1
B
+
u
state and contributions from
doubly excited configurations in the 2
1
A
g
state. The comparison ofjj
jj computed with EOM-
EE-CCSD, CC3, and EOM-DEA-CCSD shows that doubly excited character of the 2
1
A
g
state
increases in CC3 and EOM-DEA-CCSD relative to EOM-EE-CCSD, but overall is not domi-
nant. Similar conclusions can be drawn from the values of the norm of the 2h2p EOM ampli-
tudes (R
2
2
, collected in Table S2 in the SI). The comparison between the regular and augmented
basis sets also show that the doubly excited character is overestimated when the basis does
not capture substantial Rydberg character in the 2
1
A
g
state. The participation ratios indicate
relatively pure!
character of the bright state for both EOM-EE-CCSD and EOM-DEA-
CCSD. In contrast, for the dark state bothPR
NTO
andZ
HE
differ between EOM-EE-CCSD,
EOM-CC3, and EOM-DEA-CCSD, indicating that the character of this state is sensitive to the
42
method employed. For example, large values of Z
HE
for EOM-CC3 reflect a larger diradi-
cal character of the EOM-CC3 ground state, consistent with an overestimated doubly excited
character of the dark state (reflected by smallerjj
jj).
Table 2.14: One-particle state and transition properties
a
of butadiene computed with
EOM-EE-CCSD, EOM-EE-CC3, and EOM-DEA-CCSD.
State n
u;nl
f
l
jj
jj Z
HE
PR
nto
EE-CC3/cc-pVDZ
1
1
B
+
u
- 0.71 0.80 2.77 2.23
2
1
A
g
- 0.00 0.62 4.09 3.99
EE-CC3/aug-cc-pVDZ
1
1
B
+
u
- 0.67 0.79 2.63 2.18
2
1
A
g
- 0.00 0.65 3.64 3.26
EE-CC3/aug-cc-pVTZ
1
1
B
+
u
- 0.66 0.79 2.65 2.19
2
1
A
g
- 0.00 0.64 3.68 3.32
EE-CCSD/aug-cc-pVTZ
1
1
B
+
u
2.00 0.71 0.88 2.61 2.17
2
1
A
g
2.10 0.00 0.82 2.93 2.50
DEA-CCSD/+2/aug-cc-pVTZ
1
1
B
+
u
2.00 0.79 0.91 2.39 2.11
2
1
A
g
2.11 0.00 0.78 2.28 2.09
a
n
u;nl
= Head-Gordon’s index,f
l
= oscillator strength,jj
jj= norm of 1PTDM,Z
HE
=
hole-particle entanglement, PR
NTO
= participation ratio.
2.3.8 Potential energy surfaces and conical intersections in retinal chro-
mophore
The penta-2,4-dieniminium cation (PSB3) is a protonated-imine with three conjugated dou-
ble bonds. It has been used extensively as a computational model of the retinal protonated
Schiff base (rPSB) chromophore. Retinal is a well studied model system
62–64
, featuring conical
intersections between the two lowest electronic states relevant to the cis-trans photoinduced
isomerization of rhodopsin. The two lowest states are the ground state and a bright
excited
state. Photoexcitation changes the conjugation pattern and initiates twisting, which imparts
diradical character into the ground state, somewhat similar to the ethylene torsion. We use this
43
PSB3 also reproduces several features of the S
0
potential
energy surface of opsin-embedded rPSB. In both chromo-
phores, a loop constructed with the branching plane vectors
and encompassing the CI passes through regions of different
electronic character (see the bottom of Figure 1 and the
legend).
19,20
In one region, the molecule has its positive charge
fully localized on the Schiff-base-containing moiety, similar to
the S
0
reactant (cis-PSB3) and the product (trans-PSB3); thus,
the underlying wave function has predominantly a covalent/
diradical character (Ψ
DIR
). In the other region, the positive
charge is almost completely translocated to the other end of the
molecule (the allyl group in the case of PSB3, or the β-ionone-
containing moiety in the case of rPSB). In this region the wave
function is predominantly of charge-transfer character (Ψ
CT
). A
schematic representation of the S
0
energy surface around the CI
point is given in Figure 1a. Moreover, in both PSB3
10
and
rPSB,
19
each region also hosts a transition state (TS) that could
mediate thermal (i.e., proceeding on the ground state)
isomerization of the chromophore. One TS (TS
DIR
) lies in
the Ψ
DIR
region and, therefore, corresponds to the homolytic
cleavage of the isomerizing double bond. The other TS (TS
CT
)
is in theΨ
CT
region and is reached by heterolytic cleavage of
the double bond. Both TSs are ca. 90° twisted, similar to the
CI, and the main structural difference between TS
DIR
, TS
CT
,
and the CI is along the bond length alternation (BLA)
coordinate (see Scheme 1B), with the CI situated between the
two TSs at the CASSCF level of theory.
The S
0
CASSCF energy surface near the CI of PSB3 was
characterized by mapping the surface along three potential
energy paths (see Figure 1).
10
The first path (the BLA path)
connects the two TSs and intercepts the CI point shown in
Scheme 1B. The other two paths are minimum energy paths
(MEPs) connecting cis-PSB3 to trans-PSB3 through TS
DIR
and
TS
CT
(MEP
DIR
and MEP
CT
paths, respectively). The MEP
CT
path, therefore, starts and ends in the Ψ
DIR
regions while
intersecting the Ψ
CT
region, whereas the MEP
DIR
path is
confined to the Ψ
DIR
region of the S
0
surface (therefore, the
molecule maintains a covalent/diradical character in the S
0
state
along this path). The two-root SA-CASSCF/6-31G* energy
profiles and the corresponding wave functions along the three
paths are shown in Figure 2. Details regarding the generation of
these paths are provided in the Methods section.
Owing to the complexity of its potential energy surfaces,
sensitivity to the methodology, as well as small molecular size
and chemical relevance as a model for rPSB, PSB3 is a useful
benchmark system for testing different computational methods.
In the present contribution, we extend previous benchmark
studies to include single-reference EOM-CC methods.
21−24
The EOM-CC (or linear response CC) methods allow one to
Scheme 1. (A) The Structures of the 11-cis-Retinal
Protonated Schiff Base (rPSB) Connected to the Lys296
Residue in Bovine Rhodopsin and Its Reduced Model, the
cis-Penta-2,4-dieniminium Cation (PSB3) and (B) Selected
CASSCF/6-31G* Geometrical Parameters (Bond Lengths in
Ångstroms and C1−C2−C3−C4 Dihedrals in Degrees) for
the cis-PSB3, trans-PSB3, TS
CT
, TS
DIR
, and the CI
a,b
a
The resonance formula also provides a qualitative representation of
the singlet electron pairing and charge distribution.
b
The CI structure
shown is the one intercepted by the Bond Length Alternation (BLA)
coordinate.
Figure 1. Top. Schematic two-dimensional cut of the S
0
potential
energy surface of PSB3. The two coordinates can be described as bond
length alternation (BLA) and the C2−C3 twisting reaction coordinate
(RC), respectively. The region in which the wave function has
predominantly a charge-transfer character (ψ
CT
) is displayed in brown,
whereas the part corresponding to a covalent/diradical wave function
(ψ
DIR
) is displayed in green. The electronic structure of the two
transition states is illustrated by a bubble diagram showing the values
of the CASSCF Mulliken charges along the backbone (charges
summed onto heavy atoms). The three paths used in the present study
(the BLA, MEP
CT
, and MEP
DIR
paths) are shown by dashed lines on
the surface. Bottom left. A schematic magnification of the S
0
/S
1
CI
region. A loop centered around the CI and constructed using the
branching plane vectors parallel to the BLA and RC coordinates is
shown by the red dashed line on the S
0
surface. The angle α
corresponds to the 0−2π coordinate defining the position along the
circular loop. Bottom right. The S
0
and S
1
CASSCF energies (colored
according to the dominant electronic configuration) as well as the S
0
charge transfer character (gray area) along the angle α following the
loop around the CI. The charge-transfer character is determined by
summing the CASSCF Mulliken charges on the allyl (i.e., the C5H
2
C4H−C3H−) fragment of the PSB3. The energies and charge transfer
character are obtained from ref 10.
Journal of Chemical Theory and Computation Article
dx.doi.org/10.1021/ct300759z | J. Chem. Theory Comput. XXXX, XXX, XXX−XXX B
Figure 2.8: Potential energy surface in retinal showing the location of the conical intersec-
tion (CoIn or CI) between the charge transfer and diradical states. The two coordinates
are bond-alternation (BLA) and twisting reaction coordinate (RC). Relevant mechanistic
paths are shown by white dashed lines. MEP
CT
: minimum energy path on the ground
state that connects the cis and trans retinal equilibrium structures through a transition
state (TS
CT
). MEP
DIR
: connects cis, TS
DIR
and trans structures of the
DIR
state. The
BLA path connects the TS
CT
and TS
DIR
transition states and also intercepts a CoIn (CI)
point. Atomic charges of the two transition states are shown by bubble diagrams. Repro-
duced with permission from Ref. 62.
example to assess the ability of EOM-DEA-CCSD to describe conical intersections. Fig. 2.8
shows the PES of the lowest electronic state as function of the two key coordinates: bond-
length alternation (BLA) and twisting reaction coordinate (RC). As one can see, the character
of the lowest adiabatic state changes because the two lowest states exchange their character.
The location of conical intersection is very sensitive to the electronic structure method, as it
requires balanced description of the two electronic states. Previous benchmark studies
62–64
have
shown that EOM-SF-CCSD(dT) and and MR-CISD+Q results are in very good agreement,
while lower-level methods show large discrepancies.
Fig. 2.9 shows the PES scans along the BLA coordinate. The shape of the PES of the
diradical state along BLA coordinate computed by EOM-DEA-CCSD agrees well with the
MR-CISD+Q results, and is not affected by using different reference orbitals (+1 or +3). The
44
situation is different for the charge-transfer state. Overall shape agrees with MR-CISD+Q, but
the energy relative to the diradical state depends on the orbital choice, which results in large
differences in the location of the conical intersection. Using +3 Hartree-Fock orbitals in EOM-
DEA-CCSD, the conical intersection appears too early along the BLA coordinate, as compared
with MR-CISD. In contrast, when using +1 Hartree-Fock orbitals, conical intersection appears
too late along the BLA coordinate. Due to large positive charge in +3 Hartree-Fock reference,
the molecular orbital and their energy shows large deviations from +1 reference, and 3p1h part
of the EOM-DEA-CCSD ansatz is not sufficient to describe both orbital relaxation and the
correlation effects in the charge-transfer state. Consequently, the total energy of charge-transfer
state is higher, leading to the shift in the conical intersection position. This also leads to an oppo-
site trend in energy gap (S
0
and S
1
) at TS
CT
and TS
DIR
, when compared with MR-CISD+Q,
as shown in Table 2.15. We conclude by pointing out that both the errors in EOM-DEA-CCSD
and the dependence on the reference state are likely to be significantly reduced upon inclu-
sion of higher excitations. We remind that even for EOM-SF-CCSD, perturbative account of
triple excitations was necessary to exactly pin-point the location of the conical intersection in
this challenging system, and that MR-CISD results were considerably affected by the Davidson
correction (which also entails inclusion of higher excitations).
Table 2.15: The S
0
and S
1
energy gaps (kcal/mol) at TS
CT
, TS
DIR
, and cis-PSB3 geometry
of retinal; 6-31G
basis set.
Method TS
CT
TS
DIR
cis-PSB3
MRCISD+Q 10.2 0.6 101.4
MRCISD 8.8 1.6 104.8
CASSCF 4.5 7.4 110.3
EE-CCSD 16.6 6.9
SF-CCSD/UHF 2.4 7.5
SF-CCSD/ROHF 5.8 6.0 105.5
SF-CCSD(dT)/ROHF 11.1 0.6 102.1
SF-CCSD(fT)/ROHF 10.2 0.6
DEA-CCSD/+3 1.49 12.71 105.2
DEA-CCSD/+1 15.13 4.39 104.6
45
45
50
55
60
65
70
−0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04
ΔE(kcal/mol)
BLA coordinate (Å)
EOM−DEA−CCSD/+3 (CT)
EOM−DEA−CCSD/+1 (CT)
MRCISD+Q (CT)
EOM−DEA−CCSD/+3 (DIR)
EOM−DEA−CCSD/+1 (DIR)
MRCISD+Q (DIR)
Figure 2.9: The S
0
and S
1
energy profiles (retinal) along the BLA coordinate computed
with EOM-DEA-CCSD (with orbitals from the +3 and +1 charge Hartree-Fock reference)
and MRCISD+Q using the 6-31G
basis set. The energy values are relative to cis-PSB3.
2.3.9 Excited states in water and ammonia
Table 2.16: Ammonia. Vertical excitation energies (eV) for the four lowest singlets and the
lowest triplet states with different methods; aug-cc-pVQZ basis. Geometry from Ref. 87.
State EE-CC3
a
EE-CCSDT
a
exFCI
a
EE-CCSD
b
DEA-CCSD
b
Exp.
c
1
A
2
(n!3s) 6.61 6.61 6.64 6.67 6.14 6.39
1
E(n!3p) 8.18 8.17 8.22 8.21 7.66 7.93
1
A
1
(n!3p) 9.11 9.10 9.14 9.15 8.68 8.26
1
A
2
(n!4s) 9.96 9.77 9.77 9.81 9.27
3
A
2
(n!3s) 6.31 6.33 6.35 6.37 5.85 6.02
d
a
Ref. 87.
b
This work.
c
Electron impact experiment from Ref. 88.
d
Ref. 89.
In this section we consider two small molecules (water and ammonia) from Head-Gordon’s
data set and assess the performance of EOM-DEA-CCSD for singly excited states. In both
molecules, the low-lying states have predominantly Rydberg character. Table 2.16 compares
vertical excitation energies computed with EOM-DEA-CCSD and other methods using aug-
cc-pVQZ for ammonia; Table 2.17 shows relevant state and transition properties computed
with EOM-EE-CCSD and EOM-DEA-CCSD (aug-cc-pVQZ). The results for water are given
in Tables S3 and S4 in the SI.
46
In ammonia
87–89
, all lowest states are derived by the excitation from the HOMO (nirtogen’s
lone pair); therefore, EOM-DEA-CCSD should be able to describe these states reasonably well.
The results for ammonia show that EOM-DEA-CCSD excitation energies are generally below
extrapolated FCI (exFCI) ones and the errors can be as large as 0.5 eV , exceeding the errors of
EOM-EE-CCSD. This can be attributed to using the dicationic reference orbitals in EOM-DEA.
Despite differences in excitation energies, all properties are very close (featuringZ
HE
2 and
large electron-hole separation, as expected for pure Rydberg states), confirming that EOM-
DEA-CCSD and EOM-EE-CCSD yield wave functions of broadly similar quality for singly
excited states, despite using very different reference orbitals.
In water, some of the low-lying excited states are excitations from the HOMO-1; these states
are expected to show larger errors than the states derived from the excitations from the HOMO.
The problematic states can be identified by the large magnitude of the 3p1h amplitudes (these
values are reported in Table S4 in the SI). As one can see from Table S3, both EOM-EE-CCSD
and EOM-EE-CCSDT excitation energies are very close to exFCI (the largest deviation is 0.05
eV). EOM-DEA-CCSD errors for the “good” states (n
y
! Ry transitions) are large (0.6-0.7
eV), with the EOM-DEA excitation energies being too low. As in the previous examples, we
attribute this to the poor description of the ground state due to using the dicationic orbitals.
Indeed, by looking at EOM-DEA values computed using the neutral reference orbitals), we
observe increase in excitation energies and smaller errors (0.2 eV). These results illustrate,
once again, the effect of the reference orbitals on the computed excitation energies. The
1
A
1
and
3
A
1
states are derived by excitations from the HOMO-1 (n
z
); consequently, they have large
3p1h amplitudes. The errors for these “bad” states are around 4 eV and we observe only a
marginal improvement with neutral orbitals.
47
Table 2.17: Ammonia. One-particle state and transition properties
a
computed with EOM-
EE-CCSD and EOM-DEA-CCSD; aug-cc-pVQZ basis set.
State n
u;nl
f
l
jj
jj Z
HE
e-h sep
EOM-EE-CCSD
1
A
2
(n!3s) 2.00 0.08 0.91 2.02 3.02
1
E(n!3p) 2.00 0.00 0.92 2.01 3.79
1
A
1
(n!3p) 1.91 0.00 0.91 2.01 3.46
1
A
2
(n!4s) 2.00 0.00 0.91 2.18 4.57
3
A
2
(n!3s) 2.00 0.00 0.92 2.05 2.86
EOM-DEA-CCSD
1
A
2
(n!3s) 1.99 0.10 0.92 (0.07) 2.01 3.04
1
E(n!3p) 2.00 0.00 0.92 (0.06) 2.00 3.80
1
A
1
(n!3p) 1.82 0.00 0.91 (0.06) 2.01 3.52
1
A
2
(n!4s) 2.00 0.00 0.92 (0.07) 2.07 4.57
3
A
2
(n!3s) 2.00 0.00 0.92 (0.07) 2.03 2.89
a
n
u;nl
= Head-Gordon’s index,f
l
= oscillator strength,jj
jj= norm of 1PTDM (the numbers in
parenthesis are the squares of the 1h3p amplitudees of the DEA-CCSD wavefunction),Z
HE
=
hole-particle entanglement, e-h sep = electron hole separation (in
˚
A).
2.3.10 Conical intersection in HeH
2
It is well established that EOM-EE-CC methods have defects
62, 90–94
close to conical inter-
sections between excited states of the same symmetry. Yet, because the excited states are
described on the same footing, one can obtain a physically correct description by appropri-
ately modifying the EOM-CC method.
93, 95
Ground-state intersections present a more difficult
challenge: the intersecting states are described on an unequal footing and the CC ground-state
wave function does not adequately capture the required multi-configurational character. These
drawbacks can be removed by changing the reference wave function, as is done in EOM-SF,
EOM-DIP, and EOM-DEA. In these cases both states are associated with non-zero eigenvalues
of the Jacobian and the intersections are expected to resemble those between excited states in
EOM-EE-CC.
The HeH
2
system provides a simple test case for describing near-degeneracies between the
ground state and first excited state for which the FCI results are available. The states can be
48
Figure 2.10: HeH
2
molecule. One He-H bond length is varied (R).
characterized by inspecting the natural orbitals and their occupations (see Fig. 2.11). At short
bond lengths, the ground state has two electrons in the 1a
0
and 2a
0
orbitals. At the long bond
lengths, one electron has moved from 2a
0
to 3a
0
. The situation is reversed for the first excited
state.
1A’ 2A’ 3A’
X
1
A’
1
1
A’
1.998 1.941 0.028
1.996 1.086 0.909
1.996 1.112 0.882
1.997 1.939 0.029
Figure 2.11: Natural orbitals and their occupation numbers for ground and first excited
state. Black color corresponds to geometry at He-H = 0.6
˚
A. Red color corresponds to
geometry at He-H = 0.8
˚
A.
We consider a set of C
s
geometries where one of the He–H bond lengths is varied (see Fig.
2.10). Figure 2.12 shows potential energy curves obtained using EOM-EE-CCSD, EOM-DEA-
CCSD, and FCI. The EOM-EE-CCSD and EOM-DEA-CCSD curves are similar but shifted
relative to FCI. However, upon closer inspection, we see that EOM-EE-CCSD has defective
points and a region where we are not able to converge the equations, because EOM-CC roots
become complex (see the right panel of Fig. 2.12). In this particular scan, EOM-DEA-CCSD
gives an avoided crossing. Closer to the intersection, we expect to see complex pairs and defects
49
in EOM-DEA-CCSD—just as for excited states in EOM-EE-CCSD. However, EOM-DEA-
CCSD should give a generally more accurate description of the intersecting states because it
is able to describe the multi-configurational character in the two states. Furthermore, defects
encountered with EOM-DEA-CCSD can be removed by using the similarity constrained CC
approach.
93, 95
0.60 0.65 0.70 0.75 0.80
He-H length (Angstrom)
-3.75
-3.70
-3.65
-3.60
Energy (Hartree)
EE-CCSD
DEA-CCSD
FCI
0.695 0.700 0.705 0.710 0.715
He-H length (Angstrom)
-3.706
-3.704
-3.702
-3.700
-3.698
-3.696
-3.694
-3.692
Energy (Hartree)
EE-CCSD
DEA-CCSD
Figure 2.12: Conical intersection between X
1
A
0
and 1
1
A
0
computed using EOM-DEA-
CCSD, EOM-EE-CCSD, and full configuration interaction (FCI) methods with the aug-
cc-pVDZ basis set. The left pane shows all three limits for the full range of bond lengths
(0.6 to 0.8
˚
A). The right pane zooms in on the near-degeneracy for EOM-DEA-CCSD and
EOM-EE-CCSD.
50
2.4 Conclusions
In this contribution, we have documented a production-level implementation of EOM-DEA-
CCSD, including calculation of state and transition properties. This ansatz is suitable for treat-
ing electronic structure patterns that can be described as two-electrons-in-many-orbitals. Hence,
it further extends the scope of applicability of EOM-CC methods into the domain traditionally
described as multi-reference
22
. Following the pioneering work of Bartlett and coworkers
16, 17, 21
,
here we illustrate the advantages and limitations of EOM-DEA-CCSD relative to other EOM-
CCSD methods (EOM-EE-CCSD and EOM-SF-CCSD) by considering a diverse set of exam-
ples comprising well-behaved singly excited states, doubly excited states, diradicals, bond-
breaking, and conical intersections. The results can be summarized as follows:
1. Excited states in closed-shell molecules, which can be qualitatively characterized as exci-
tations from the HOMO, are described with similar accuracy as in EOM-EE-CCSD (the
errors of DEA are slightly larger, which is attributed to orbital relaxation effects). The
performance for excited states derived by excitations from lower occupied orbitals is poor,
as the method is clearly not designed for such applications.
2. Diradical states are described nearly as accurately as by EOM-SF-CCSD, but within spin-
adapted framework. The benefits are no spin-contamination and a smaller memory foot-
print of the calculations. Furthermore, EOM-DEA-CCSD allows access to higher excited
states, beyond the primary SF manifold (this has been recently illustrated for a set of
Rydberg diradicals
51, 61
).
3. Similarly to other EOM models, EOM-DEA-CCSD can describe states with mixed char-
acter, such as those featuring Rydberg-valence interactions.
51
4. EOM-DEA-CCSD can describe states with significant or even pure doubly excited char-
acter.
5. EOM-DEA-CCSD describes PES along bond breaking reasonably well, as illustrated in
the ethylene torsion example.
6. EOM-DEA-CCSD can also treat some problems involving conical intersections, where
standard EOM-EE-CCSD is inapplicable.
7. Orbital relaxation is somewhat of a problem. In most cases, using default dicationic
reference orbitals yields robust performance, but sometimes the charge of the reference
needs to be tweaked. Generally, we observed that using dicationic orbitals results in
the ground-state energies being too high. In many cases, using neutral orbitals results
in smaller errors in excitation energies. This dependence on the reference orbitals is
expected to be reduced upon inclusion of higher excitations in the EOM-DEA ansatz.
The results show that EOM-DEA-CCSD represents a useful addition to EOM-CC family of
methods. In addition to the examples considered in this paper, the EOM-DEA ansatz can be
used in combination with open-shell references —e.g., one can described
7
pattern by attach-
ing two electrons to a well-behaved high-spin d
5
reference (M
s
=
5
2
), in the same spirit
as EOM-EA was used to describe thed
6
pattern
96
. Future developments include performance
enhancements by exploiting resolution-of-identity and Cholesky decomposition of the electron-
repultion integrals
33, 34
, single-precision execution
97
, extenstions to compute additional proper-
ties (e.g., spin-orbit couplings
96
, polarizabilities
98
and two-photon absorption
99
, etc) and nuclear
gradients, as well as inclusion of higher excitations. We hope that this contribution provides a
useful guide for choosing the most appropriate EOM-CC method to tackle specific applications.
52
2.5 Appendix A
The left and right EOM-vectors are defined as:
1
= ([
H
SS
E
cc
]R
1
)
1
+ (
H
SD
R
2
)
1
;
2
= (
H
DS
R
1
)
2
+ ([
H
DD
E
cc
]R
2
)
2
;
~
1
= (L
1
[
H
SS
E
cc
])
1
+ (L
2
H
DS
)
1
;
~
2
= (L
1
H
SD
)
2
+ (L
2
[
H
DD
E
cc
])
2
:
Table 2.18: Programmable expressions for the right () and left (~ ) vectors in EOM-DEA-
CCSD.
ab
=P (ab)(
P
c
F
bc
r
ac
1
2
P
icd
I
7
ibcd
r
acd
i
)
+
1
2
P
cd
r
cd
I
5
abcd
+
P
ic
F
ic
r
abc
i
abc
i
=
1
2
P (abc)(
P
d
F
cd
r
abd
i
P
jd
I
1
idjc
r
abd
j
+
1
2
P
de
I
5
abde
r
cde
i
P
j
H
6
jc
t
ab
ij
P
d
I
3
idab
r
cd
)
P
j
F
ij
r
abc
j
~
ab
=P (ab)(
P
c
l
ac
F
ac
1
2
P
icd
l
acd
i
I
3
ibcd
)
P
ic
H
7
ic
I
7
icab
+
1
2
P
cd
l
cd
I
5
cdab
~
abc
i
=
1
2
P (abc)(
P
d
l
abd
i
F
dc
P
jd
l
abd
j
I
1
jcid
+
1
2
P
de
l
ade
i
I
5
debc
P
k
H
7
ka
hikjjbci +l
ab
F
ic
P
d
l
cd
I
7
idab
)
P
j
l
abc
j
F
ji
53
Table 2.19: Intermediates used in EOM-DEA-CCSD-vectors.
F
ia
=f
ia
+
P
jb
t
b
j
hijjjabi
F
ij
=f
ij
+
P
a
t
a
i
f
ja
+
P
ka
t
a
k
hjkjjiai +
P
kab
t
a
i
t
b
k
hjkjjabi +
1
2
P
kbc
t
bc
ik
hjkjjbci
F
ab
=f
ab
P
i
t
a
i
f
ib
P
ic
t
c
i
hiajjbci +
P
ijc
t
c
i
t
a
j
hijjjbci
1
2
P
jkc
t
ac
jk
hjkjjbci
I
1
iajb
=hiajjjbi
P
k
t
b
k
hjkjjiai
P
c
t
c
i
hjbjjaci +
P
kc
t
c
i
t
b
k
hjkjjaci
P
kc
t
bc
ik
hjkjjaci
I
3
icab
=hicjjabi + 2
P
d
t
d
i
I
5
bcad
+
P
j
(t
b
j
(hiajjjci
P
kd
t
cd
ik
hkjjjadi)t
c
j
(hiajjjbi
P
kd
t
bd
ik
hjkjjadi))
P
jk
t
b
j
t
c
k
hjkjjiai
P
jkd
t
d
j
t
bc
ik
hjkjjadi +
P
k
t
ab
ik
f
kc
1
2
P
kl
t
ab
kl
hicjjkli +
P
kd
(t
ad
ik
hkbjjcdit
bd
ik
hkajjcdi)
I
5
abcd
=habjjcdi +
1
2
P
ij
~
t
ab
ij
hijjjcdiP (ab)
P
i
t
a
i
hibjjcdi
I
7
iabc
=hiajjbci
P
j
t
a
j
hijjjbci
~
t
ab
ij
=t
ab
ij
+
1
2
P (ij)P (ab)t
a
i
t
b
j
H
6
ia
=
1
2
P
bc
r
bc
I
7
iabc
+
1
2
P
jbc
r
abc
j
hjijjbci
H
7
ia
=
1
2
P
jbc
l
abc
j
t
bc
ji
54
Table 2.20: Programmable expressions for EOM-DEA-CCSD density matrices.
ij
= ~
ij
+
ij
~
ij
=
~
l
ij
P
a
Y
1
ja
t
a
i
ai
=Y
1
ia
ab
=
~
l
ab
+Y
1
ia
t
b
i
ia
=t
a
i
+
P
jb
Y
1
jb
(t
ab
ij
t
a
j
t
b
i
) +
1
2
P
bc
l
bc
r
abc
i
P
k
~
l
ik
t
a
k
P
b
~
l
ab
t
b
i
+
P
b
Y
2
ib
r
ab
~
l
ij
=
1
6
P
abc
l
abc
j
r
abc
i
~
l
ab
=
P
c
l
ac
r
bc
+
1
2
P
icd
l
acd
i
r
bcd
i
Y
1
ia
=
1
2
P
bc
l
abc
i
r
bc
Y
2
ia
=
1
2
P
jbc
l
abc
j
t
bc
ij
Table 2.21: Ethylene torsion, DZP basis. Total energies (hartree) for the SF-
TDDFT(5050), MR-CI (TCSCF-CISD), CCSD, EOM-SF-CCSD, and EOM-DEA-CCSD
models.
SF-5050 MR-CI CCSD EOM-SF-CCSD/UHF EOM-DEA/+2 EOM-DEA/0
0 -78.53417 -78.36589 -78.35221 -78.35643 -78.34170 -78.37023
15 -78.52957 -78.36143 -78.34773 -78.35199 -78.33690 -78.36415
30 -78.51596 -78.34812 -78.33434 -78.33880 -78.32367 -78.34881
45 -78.49369 -78.32634 -78.31223 -78.31724 -78.30241 -78.32582
60 -78.46365 -78.29724 -78.28205 -78.28850 -78.27403 -78.29623
75 -78.42882 -78.26471 -78.24550 -78.25646 -78.24240 -78.26383
80 -78.41799 -78.25522 -78.23262 -78.24715 -78.23321 -78.25447
85 -78.40965 -78.24833 -78.22002 -78.24039 -78.22656 -78.24767
90 -78.40634 -78.24574 -78.20851 -78.23785 -78.22407 -78.22006
Geometry used:r
CC
= 1:330
˚
A,r
CH
= 1:076
˚
A, and
HCH
= 116
o
.
55
Neutral orbs:
SCF energy = -76.96046076
MP2 energy = -77.28705637
CCSD total energy = -77.30600968
EOMDEA-CCSD transition 1/A
Total energy = -78.22006171 a.u. Excitation energy = -24.8726 eV.
Dication orbs:
SCF energy = -77.09186812 higher, as it should be
MP2 energy = -77.28919723
CCSD total energy = -77.32414425 lower, meaning orbitals are better
EOMDEA-CCSD transition 1/A1
Total energy = -78.12325591 a.u. Excitation energy = -21.7449 eV.
R2^2 = 0.8959 R3^2 = 0.1041 Res^2 = 6.18e-06
HOMO
HOMO-1 HOMO
LUMO LUMO+1
LUMO LUMO+1
Neutral orbs:
SCF energy = -76.96046076
MP2 energy = -77.28705637
CCSD total energy = -77.30600968
EOMDEA-CCSD transition 1/A
Total energy = -78.22006171 a.u. Excitation energy = -24.8726 eV.
Dication orbs:
SCF energy = -77.09186812 higher, as it should be
MP2 energy = -77.28919723
CCSD total energy = -77.32414425 lower, meaning orbitals are better
EOMDEA-CCSD transition 1/A1
Total energy = -78.12325591 a.u. Excitation energy = -21.7449 eV.
R2^2 = 0.8959 R3^2 = 0.1041 Res^2 = 6.18e-06
HOMO
HOMO-1 HOMO
LUMO LUMO+1
LUMO LUMO+1
Neutral HF reference
Dication HF reference
CCSD Natural orbitals
Figure 2.13: Relevant MOs of ethylene at 90
twisted geometry.
Table 2.22: Comparing
^
R
2
2
(EOM,
^
R
2
amplitude contribution) for 2
1
A
g
excited state in
butadiene from EOM-EE-CCSD and EOM-EE-CCSDT
basis EE-CCSD EE-CCSDT
cc-pVDZ 0.2417 -
cc-pVTZ 0.1976 0.464
cc-pVQZ 0.1757 -
aug-cc-pVDZ 0.1323 -
aug-cc-pVTZ 0.1215 -
Table 2.23: Water. Vertical excitation energies (eV) for the 3 lowest singlet and 3 low-
est triplet states computed with different methods; aug-cc-pVQZ basis. Geometry from
Ref. 87.
State EE-CC3
a
EE-CCSDT
a
exFCI
a
EE-CCSD
b
DEA-CCSD/+2
b
DEA-CCSD/0
b
Exp.
c
1
B
1
(n
y
!3s) 7.65 7.64 7.68 7.68 6.97 7.92 7.41
1
A
2
(n
y
!3p) 9.43 9.41 9.46 9.44 8.72 9.70 9.20
1
A
1
(n
z
!3s) 10.00 9.98 10.02 10.02 13.94 13.03 9.67
3
B
1
(n
y
!3s) 7.28 7.26 7.30 7.29 6.62 7.53 7.20
3
A
2
(n
y
!3p) 9.26 9.25 9.28 9.27 8.57 9.53 8.90
3
A
1
(n
z
!3s) 9.56 9.54 9.58 9.55 13.55 13.48 9.46
a
Ref. 87.
b
This work.
c
Energy loss experiment from ref. ?.
56
Table 2.24: Water. One-particle state and transition properties
a
computed with EOM-
EE-CCSD and EOM-DEA-CCSD; aug-cc-pVQZ basis set.
State n
u;nl
f
l
jj
jj Z
HE
e-h sep
EOM-EE-CCSD
1
B
1
(n
y
!3s) 2.00 0.05 0.92 2.01 2.60
1
A
2
(n
y
!3p) 2.00 0.00 0.92 2.01 3.22
1
A
1
(n
z
!3s) 2.00 0.10 0.92 2.49 2.67
3
B
1
(n
y
!3s) 2.00 0.00 0.92 2.04 2.45
3
A
2
(n
y
!3p) 2.00 0.00 0.92 2.02 3.06
3
A
1
(n
z
!3s) 2.00 0.00 0.92 2.12 2.49
EOM-DEA-CCSD
1
B
1
(n
y
!3s) 2.00 0.06 0.92 (0.07) 2.01 2.63
1
A
2
(n
y
!3p) 2.00 0.00 0.92 (0.07) 2.00 3.23
1
A
1
(n
z
!3s) 1.98 0.12 0.85 (0.99) 2.21 2.83
3
B
1
(n
y
!3s) 2.00 0.00 0.93 (0.08) 2.02 2.48
3
A
2
(n
y
!3p) 2.00 0.00 0.92 (0.07) 2.01 3.09
3
A
1
(n
z
!3s) 2.00 0.00 0.86 (1.00) 2.12 3.10
a
n
u;nl
= Head-Gordon’s index,f
l
= oscillator strength,jj
jj= norm of 1PTDM(the numbers in
parenthesis are the squares of the 1h3p amplitudees of the DEA-CCSD wavefunction),Z
HE
=
hole-particle entanglement, e-h sep = electron hole separation (in
˚
A).
57
Basis set
TZ2P basis set for C and H
$basis
C 0
S 6 1.00
9471.0000000 0.0007760
1398.0000000 0.0062180
307.5000000 0.0335750
84.5400000 0.1342780
26.9100000 0.3936680
9.4090000 0.5441690
S 2 1.00
9.4090000 0.2480750
3.5000000 0.7828440
S 1 1.00
1.0680000 1.0000000
S 1 1.00
0.4002000 1.0000000
S 1 1.00
0.1351000 1.0000000
P 4 1.00
25.3700000 0.0162950
5.7760000 0.1020980
1.7870000 0.3402280
0.6577000 0.6682690
P 1 1.00
0.2480000 1.0000000
P 1 1.00
0.0910600 1.0000000
D 1 1.00
1.5000000 1.0000000
D 1 1.00
0.3750000 1.0000000
****
H 0
S 3 1.00
33.6400000 0.0253740
5.0580000 0.1896840
1.1470000 0.8529330
S 1 1.00
0.3211000 1.0000000
S 1 1.00
58
0.1013000 1.0000000
P 1 1.00
1.5000000 1.0000000
P 1 1.00
0.3750000 1.0000000
****
$end
59
Relevant Cartesian geometries
Methylene
Nuclear Repulsion Energy = 6.16086182 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C -0.0000000000 0.0000000000 0.1067875138
2 H -0.9892163971 -0.0000000000 -0.3203625414
3 H 0.9892163971 0.0000000000 -0.3203625414
----------------------------------------------------------------
Ortho-benzyne (singlet)
Nuclear Repulsion Energy = 189.10205245 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 H 2.5184660000 -0.0000000000 -0.1317769000
2 C 1.4433500000 -0.0000000000 -0.1297459000
3 C 0.6988000000 0.0000000000 1.0495961000
4 H 1.2189940000 0.0000000000 1.9935391000
5 C -0.6988000000 0.0000000000 1.0495961000
6 H -1.2189940000 0.0000000000 1.9935391000
7 C -1.4433500000 -0.0000000000 -0.1297459000
8 H -2.5184660000 -0.0000000000 -0.1317769000
9 C -0.6206040000 -0.0000000000 -1.2301439000
10 C 0.6206040000 -0.0000000000 -1.2301439000
----------------------------------------------------------------
Ortho-benzyne (triplet)
Nuclear Repulsion Energy = 186.77967739 hartrees
----------------------------------------------------------------
1 H 2.4760580000 0.0000000000 -0.1214619000
2 C 1.3978260000 -0.0000000000 -0.1159549000
3 C 0.6904280000 0.0000000000 1.0849821000
4 H 1.2299290000 0.0000000000 2.0172501000
5 C -0.6904280000 0.0000000000 1.0849821000
6 H -1.2299290000 0.0000000000 2.0172501000
7 C -1.3978260000 -0.0000000000 -0.1159549000
8 H -2.4760580000 -0.0000000000 -0.1214619000
9 C -0.6923260000 -0.0000000000 -1.2849919000
10 C 0.6923260000 -0.0000000000 -1.2849919000
----------------------------------------------------------------
60
Meta-benzyne (singlet)
Nuclear Repulsion Energy = 188.81019759 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 H -2.1449000000 -0.0000000000 -1.0915347500
2 C -1.1653200000 -0.0000000000 -0.6458647500
3 C -0.0000000000 0.0000000000 -1.4019747500
4 H -0.0000000000 0.0000000000 -2.4817347500
5 C 1.1653200000 0.0000000000 -0.6458647500
6 H 2.1449000000 0.0000000000 -1.0915347500
7 C 1.0083400000 -0.0000000000 0.7070852500
8 C 0.0000000000 -0.0000000000 1.6099052500
9 H 0.0000000000 -0.0000000000 2.6825752500
10 C -1.0083400000 -0.0000000000 0.7070852500
----------------------------------------------------------------
Meta-benzyne (triplet)
Nuclear Repulsion Energy = 187.20365817 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 H -2.1468900000 -0.0000000000 1.1685737500
2 C -1.2142300000 -0.0000000000 0.6317937500
3 C -0.0000000000 0.0000000000 1.3117837500
4 H -0.0000000000 0.0000000000 2.3900737500
5 C 1.2142300000 0.0000000000 0.6317937500
6 H 2.1468900000 0.0000000000 1.1685737500
7 C 1.1542500000 0.0000000000 -0.7333662500
8 C 0.0000000000 -0.0000000000 -1.4715662500
9 H 0.0000000000 -0.0000000000 -2.5496562500
10 C -1.1542500000 -0.0000000000 -0.7333662500
----------------------------------------------------------------
Para-benzyne (singlet)
Nuclear Repulsion Energy = 187.21381762 hartrees
----------------------------------------------------------------
1 H 2.1458100000 1.2252920000 -0.0000000000
2 C 1.2013820000 0.7092850000 -0.0000000000
3 C 1.2013820000 -0.7092850000 0.0000000000
4 H 2.1458100000 -1.2252920000 0.0000000000
5 C -0.0000000000 -1.3356640000 0.0000000000
6 C -1.2013820000 -0.7092850000 0.0000000000
61
7 H -2.1458100000 -1.2252920000 -0.0000000000
8 C -1.2013820000 0.7092850000 -0.0000000000
9 H -2.1458100000 1.2252920000 0.0000000000
10 C 0.0000000000 1.3356640000 -0.0000000000
----------------------------------------------------------------
Para-benzyne (triplet)
Nuclear Repulsion Energy = 187.10951165 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 H 2.1449940000 -1.2551650000 0.0000000000
2 C 1.2228020000 -0.6978500000 0.0000000000
3 C 1.2228020000 0.6978500000 -0.0000000000
4 H 2.1449940000 1.2551650000 -0.0000000000
5 C 0.0000000000 1.3088150000 -0.0000000000
6 C -1.2228020000 0.6978500000 -0.0000000000
7 H -2.1449940000 1.2551650000 -0.0000000000
8 C -1.2228020000 -0.6978500000 0.0000000000
9 H -2.1449940000 -1.2551650000 0.0000000000
10 C -0.0000000000 -1.3088150000 0.0000000000
----------------------------------------------------------------
Cyclobutadiene (D2h)
Nuclear Repulsion Energy = 98.88215601 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C 0.7830000000 0.6715000000 -0.0000000000
2 H 1.5412388325 1.4321246597 -0.0000000000
3 C 0.7830000000 -0.6715000000 0.0000000000
4 H 1.5412388325 -1.4321246597 0.0000000000
5 C -0.7830000000 -0.6715000000 0.0000000000
6 H -1.5412388325 -1.4321246597 0.0000000000
7 C -0.7830000000 0.6715000000 -0.0000000000
8 H -1.5412388325 1.4321246597 -0.0000000000
----------------------------------------------------------------
Cyclobutadiene (D4h)
Nuclear Repulsion Energy = 99.49319151 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
62
----------------------------------------------------------------
1 C 1.0175266581 -0.0000000000 -0.0000000000
2 H 2.0905266581 -0.0000000000 -0.0000000000
3 C 0.0000000000 1.0175266581 -0.0000000000
4 H 0.0000000000 2.0905266581 -0.0000000000
5 C 0.0000000000 -1.0175266581 0.0000000000
6 H 0.0000000000 -2.0905266581 0.0000000000
7 C -1.0175266581 0.0000000000 0.0000000000
8 H -2.0905266581 0.0000000000 0.0000000000
----------------------------------------------------------------
Butadiene
Nuclear Repulsion Energy = 103.42597579 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C 1.8422983928 -0.1219495418 0.0000000000
2 C -1.8422983928 0.1219495418 -0.0000000000
3 C 0.6087782883 0.4091598178 -0.0000000000
4 C -0.6087782883 -0.4091598178 0.0000000000
5 H -0.5076290212 -1.4984734630 0.0000000000
6 H 0.5076290212 1.4984734630 -0.0000000000
7 H 2.7136436771 0.5395583883 -0.0000000000
8 H -2.7136436771 -0.5395583883 0.0000000000
9 H 1.9605444464 -1.2095409730 0.0000000000
10 H -1.9605444464 1.2095409730 -0.0000000000
----------------------------------------------------------------
Ammonia
Nuclear Repulsion Energy = 11.95674636 hartrees
----------------------------------------------------------------
Standard Nuclear Orientation (Bohr)
I Atom X Y Z
----------------------------------------------------------------
1 N -0.0000000007 0.2163256500 0.0000000011
2 H 1.7716015767 -0.5047598488 0.0000000000
3 H -0.8858007865 -0.5047598525 1.5342519645
4 H -0.8858007856 -0.5047598488 -1.5342519724
----------------------------------------------------------------
Water
Nuclear Repulsion Energy = 9.17658047 hartrees
----------------------------------------------------------------
1 O 0.0000000000 0.0000000000 0.2223593500
63
2 H -1.4315287800 0.0000000000 -0.8894374000
3 H 1.4315287800 -0.0000000000 -0.8894374000
----------------------------------------------------------------
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Chapter 3: The quest to uncover the
nature of benzonitrile anion
3.1 Introduction
Benzonitrile (C
6
H
5
CN) has recently become the first aromatic molecule observed in the
interstellar medium using a radio telescope
1
. Its detection in the cold-core Tauras Molecular
Cloud 1 (TMC-1) has far-reaching implications as it provides a chemical link for unidentified
infrared bands in the interstellar medium. These emission features have long been thought
to be due to the polycyclic aromatic hydrocarbons (PAH)
2
and polycyclic aromatic nitrogen
heterocycles (PANH)
3
. However, proving the presence of these molecules in the interstellar
space has been a challenge for radio-astronomy due to the nonpolar or weakly polar nature of
the polycycles combined with the large density of states. Benzonitrile is one of the simplest
nitrogen-containing aromatic compounds. As a small molecule with a large dipole moment (>
4 Debye), it does not have the same detection limitations as the polycycles. The observation
of benzonitrile in TMC-1 provides the clearest indication to date that larger PAH and PANH
species are likely to be present there as well.
The molecular dipole moment also plays an important role in electron capture and anion
formation
4–9
. Among the negative ions previously detected in the interstellar medium
10
, many
are carbon-chain species
11–15
, such as C
2n1
N
and C
2n
H
. In these systems, electron cap-
ture by the neutral molecule may involve doorway dipole-bound states
16–21
or dipole-stabilized
resonances
22, 23
. The large dipole moment of benzonitrile is a predictor of the existence of a
70
dipole-bound state of its anion which may be a precursor for other chemical and photochemical
pathways involving negative ions in both the laboratory environments and interstellar space.
The goal of this work is to establish the key electronic properties of benzonitrile, such as its
electron affinity (EA) and the nature of electron binding in its anion. Despite several previous
investigations, including a direct photoelectron imaging measurement
24
, the debate about the
very nature of benzonitrile anion is still ongoing. The existence of both dipole-bound and
valence states of the anion has been predicted
24, 25
. Despite drastically different characters of
the respective wave-functions, both types of anions are expected to be weakly bound, with
similar electron attachment and detachment energies, making the assignment and interpretation
of experimental data a delicate task.
The first measurement of the EA of benzonitrile dates back to 1975
26
, when it was indi-
rectly determined to be 0.256(17) eV . Another, also indirect, measurement
27
based on electron
capture detection in 1983 yielded a value with a significantly larger uncertainty, 0.26(10) eV ,
nonetheless in perfect agreement with the earlier determination. Subsequent 1992 experiment
indicated that the ground-state anion of benzonitrile could not be observed by electron trans-
mission spectroscopy and hence it was concluded that the anion is bound by a few tenths of an
electron-volt
28
.
The 2015 photoelectron imaging experiment by Sanov and co-workers
24
yielded the photo-
electron spectra and angular distributions for the bare and microsolvated benzonitrile anion. Fig.
3.1 shows the photoelectron spectra of C
6
H
5
CN
and C
6
H
5
CN
H
2
O obtained at two different
photon energies. The authors reported vertical detachment energy (VDE) of bare C
6
H
5
CN
of 0.058(5) eV , while in the C
6
H
5
CN
H
2
O cluster the band was blue-shifted by 0.32 eV . The
spectra were assigned to the valence anion (V A) of benzonitrile, although a contribution of the
dipole-bound state (DBS) could not be ruled out. The authors also calculated the EAs and
VDEs for C
6
H
5
CN
and C
6
H
5
CN
H
2
O using equation-of-motion coupled-cluster (EOM-CC)
71
B n
B n
H
2
O
+ 0 . 3 2
+ 0 . 2 4
0 . 0 5 8
3 . 5 1
4 . 4 5
Electron Binding Energy / eV
B n
B n
H
2
O
+ 0 . 3 2
+ 0 . 2 4
0 . 0 5 8
3 . 5 1
4 . 4 5
Electron Binding Energy / eV
Figure 3.1: Photoelectron spectra of C
6
H
5
CN
(top) and C
6
H
5
CN
H
2
O (bottom)
obtained with two different energy photons. Red and blue lines correspond to the spectra
obtained with 1.165 eV and 4.661 eV photons, respectively. Reproduced with permission
from Ref. 24.
methods
29–31
. Of particular note, the computed VDE of C
6
H
5
CN
(0.047 eV) was in good
agreement with the 0.058(5) eV experimental value. The calculations were carried out for the
valence state of the benzonitrile anion in which the extra electron occupied a
-like (b
1
) orbital
and the target photodetachment transition corresponded to electron removal from the singly
occupied HOMO of the anion. The assignment of the experimental spectrum to the V A of ben-
zonitrile took into account the good agreement between the computed and experimental VDE
values. It was additionally supported by the analysis of the photoelectron angular distributions,
the Franck–Condon simulation of the low-energy photodetachment band, and the behavior of
the band under the microsolvation conditions. However, while concluding that the experimental
72
results could be completely explained by the V A structure of benzonitrile, the authors did not
explicitly analyze the DBS.
In a subsequent computational study
25
, Adamowicz and co-workers examined the structures
and energetics of both the V A and DBS of benzonitrile using the CCSD(T) method (CCSD
32
with perturbative account of triple excitations
33
). They reported
25
the VDE of the DBS of
benzonitrile to be 0.019 eV , while claiming that the V A is adiabatically unbound. Although the
above VDE did not agree well with the published photoelectron spectra, the authors nonetheless
suggested that the experimental spectra of Sanov and co-workers should be attributed to the
DBS, rather than the V A of benzonitrile.
In the present study, we resolve the ambiguity created by the above conflicting conclusions.
We report high-level calculations showing that the photodetachment transitions observed by
Sanov and co-workers are indeed attributed to the valence anion, but also discuss the possi-
bility that the DBS may serve as a doorway state in the formation of the V A. This route is
easily accessible in benzonitrile and therefore future detection of its anion in ISM should be
anticipated.
3.2 Theoretical methods
The EOM-CC approach
29–31, 34–41
provides an efficient and robust framework for computing
a variety of electronic states in closed- and open-shell species. Different variants of EOM-
CC enable access to different types of target electronic states, such as electronically excited,
electron attached or ionized states. The EOM-CC wave-function has the following form:
j i =Re
T
j
0
i; (3.1)
73
where the linear EOM operatorR acts on the reference CC wave-function,e
T
j
0
i. The operator
T is an excitation operator satisfying the CC equations for the reference state
h
j
Hj
0
i = 0; (3.2)
where
are the-tuply excited determinants and
H =e
T
He
T
. In EOM-CCSD, the CC and
EOM operator are truncated as follows
32, 38
:
TT
1
+T
2
; RR
0
+R
1
+R
2
; (3.3)
where T
1
and T
2
are single and double excitation operators, 1-hole-1-particle (1h1p) and 2-
holes-2-particles (2h2p):
T
1
=
X
ia
t
a
i
a
y
i; T
2
=
1
4
X
ijab
t
ab
ij
a
y
b
y
ji: (3.4)
Different variants
29–31
of EOM-CC are defined by different choices of the reference state and
the type of EOM operatorsR. For example, by choosing the reference as a neutral state andR
1
andR
2
as 1h and 2h1p operators, one can describe ground and excited states of the cation. This
strategy was used by Sanov and co-workers
24
, who computed VDE by EOM-IP-CCSD from the
open-shell V A state of benzonitrile. A more balanced treatment of open-shell anionic states is
afforded by the EOM-EA-CCSD variant
42
in which the reference is a neutral closed-shell state
andR
1
andR
2
are of 1p and 1h2p type, as illustrated in Fig. 3.2:
R
1
=
X
i
r
a
a
y
; R
2
=
1
2
X
iab
r
ab
i
a
y
b
y
i: (3.5)
Here we characterize the V A and DBS of benzonitrile by EOM-EA-CCSD starting from the
ground state of the neutral benzonitrile; in these calculations VDE is obtained as the energy
difference between the EOM-EA-CCSD and CCSD states. The accuracy of EOM-CC can be
systematically improved by including higher-level of excitations in T and R, up to the exact
74
limit. Here we account for the effect of triple excitations by using perturbative correction, i.e.,
the EOM-EA-CCSD(T)(a)
method
43, 44
.
EOM-EA-CCSD
R
1
+ R
2
Figure 3.2: EOM-EA target configurations generated from a closed-shell reference state.
The first two configurations are 1p ones and the third one is 2p1h.
The EOM amplitudes and the corresponding energies are found by diagonalizing the sim-
ilarity transformed Hamiltonian,
H. Since
H is a non Hermitian operator, its left and right
eigenstates are not identical but can be chosen to form a biorthonormal set.
HRj
0
i =ERj
0
i; (3.6)
h
0
jL
H =h
0
jLE; (3.7)
h
0
jL
M
R
N
j
0
i =
MN
; (3.8)
whereM andN denote theM
th
andN
th
EOM states and
L =L
1
+L
2
=
X
a
l
a
a +
1
2
X
iab
l
i
ab
i
y
ba: (3.9)
The left and right amplitudes are found by diagonalizing the corresponding matrix representa-
tion of
H. For energy calculations, right eigenstates are sufficient, but for property calculations,
such as Dyson orbitals
45
, both left and right eigenstates need to be computed.
75
Dyson orbitals are reduced quantities defined as the overlap between an initialN-electron
and finalN 1-electron states:
Dyson
if
(x
1
) =
p
N
Z
N
i
(x
1
;x
2
;:::;x
N
)
N1
f
(x
2
;:::;x
N
)dx
2
:::dx
N
: (3.10)
Because Dyson orbitals enter the expressions of various experimental observables, such as pho-
toionization/photodetachment cross sections
46, 47
, they can be interpreted as correlated states of
the ejected/attached electron. Thus, Dyson orbitals provide a basis for a rigorous extension
of molecular orbital theory to many-body correlated wave functions
45, 48, 49
. Here we compute
Dyson orbitals using the CCSD (as
N1
f
) and EOM-EA-CCSD (as
N
i
) wave functions of the
neutral and of the anion, respectively.
3.3 Computation details
As outlined above, we characterized valence and dipole-bound states of the benzonitrile
anion by EOM-EA-CCSD using the neutral CCSD reference. Core electrons were frozen in all
correlated calculations. Our primary focus is on the relative energetics of the neutral, V A, and
DBS. For negative ions, VDE is defined as energy gap between the ground-state energy of the
anion and the corresponding neutral molecule, both at the equilibrium geometry of the anion
(R
A
):
VDE =E
N
(R
A
)E
A
(R
A
) (3.11)
Analogously, vertical attachment energy (V AE) is computed at the equilibrium geometry of the
ground-state neutral molecule (R
N
):
V AE =E
A
(R
N
)E
N
(R
N
) (3.12)
76
The electronic part of the adiabatic electron affinity (AEA
ee
) is defined as the energy difference
between the ground-state energy of the anion at its equilibrium geometry and the ground-state
energy of the neutral at its equilibrium geometry:
AEA
ee
=E
N
(R
N
)E
A
(R
A
): (3.13)
To obtain the relative ordering of the anionic states and to compare with the experimental pho-
todetachment onset, we also computed AEA, which includes zero-point energy (ZPE) differ-
ences between the anion and the neutral states:
AEA = AEA
ee
+ ZPE: (3.14)
In calculations of energetics, we used the geometry of the neutral and V A states optimized
by CCSD/aug-cc-pVTZ and EOM-EA-CCSD/aug-cc-pVTZ, respectively. Both structures are
of C
2v
symmetry. As is well documented in the literature
4, 5
, the shape of potential energy sur-
faces of dipole-bound anions are very similar to the respective structures of the neutral species
because the extra electron resides largely outside the molecular core. Thus, the energies of DBS
were computed at the geometries of the neutral benzonitrile. Triples corrections to the VDE and
AEA of the V A were computed with EOM-EA-CCSD(T)(a)
method
43, 44
using aug-cc-pVTZ.
We assume that the effect of triples cancels out for the DBS, because the unpaired electron does
not participate in the bonding.
Because of their structural similarity, we also expect vibrational frequencies of the neutral
and DBS to be similar, giving rise to ZPE0. ZPEs of the neutral and the V A states were com-
puted with CCSD and EOM-EA-CCSD using aug-cc-pVDZ and resolution-of-the-identity (RI)
approximation
50, 51
with the matching basis set (ri-aug-cc-pVDZ), at the geometries optimized
at the same level of theory. The computed structures and normal modes were used to com-
pute the Franck–Condon factors within parallel-mode double-harmonic approximation using
77
the ezSpectrum software
52
. To further elucidate putative contributions of the DBS to the spec-
tra, we computed photoelectron cross sections using EOM-EA-CCSD Dyson orbitals and the
ezDyson software
53
.
To correctly describe DBS, large basis sets with additional sets of diffuse functions are
needed. We used the aug-cc-pVTZ basis augmented with several extra sets of diffuse functions
added to each atom, with the exponents obtained following the same procedure as in our pre-
vious studies
23, 54–57
; the details are provided in the SI. Our preliminary calculations monitoring
the convergence of the V AE of the DBS showed that the results converge with the aug-cc-
pVTZ+6s3p(3s) basis. Here, “6s3p” refers to the additional diffuse functions placed at the
heavy atoms and “(3s)” to those placed at the hydrogen atoms. Below we report the energetics
of the bare benzonitrile anion obtained with aug-cc-pVTZ+6s3p(3s). For the C
6
H
5
CN
H
2
O
complex, the VEA of the DBS converged with the aug-cc-pVTZ+4s4p(4s) basis. Thus, for the
benzonitrile-water complex, we report energetics obtained with aug-cc-pVTZ+4s4p(4s).
All electronic structure calculations were performed using the Q-Chem package
58, 59
. Below
we report symmetry labels using Mulliken’s convention
60
. Basis sets, relevant Cartesian coor-
dinates, and vibrational frequencies are given in the SI.
3.4 Results and discussion
3.4.1 The benzonitrile anion
Fig. 3.3 shows Dyson orbitals of the two lowest states of the benzonitrile anion,
2
A
1
and
2
B
1
. The shape of the orbitals identifies the former as DBS and the latter as V A. In the V A state,
the unpaired electron resides on a relatively compact
-like orbital ofb
1
symmetry, giving rise
to the
2
B
1
state. The Dyson orbital for the DBS is a diffuses-like orbital located on the opposite
end of the cyano-group, giving rise to the
2
A
1
state. The DBS is supported by the large dipole
moment of benzonitrile, 4.57 Debye (CCSD/aug-cc-pVTZ).
78
Neutral
VA
DBS
0.078 ev
0.024 ev
DBS
2
A1 VA
2
B2
DBS (
2
A
1
) VA (
2
B
1
)
Figure 3.3: Dyson orbitals for the two lowest electronic states of the benzonitrile anion,
2
A
1
and
2
B
1
, computed at the respective optimized geometries. Isovalue 0.007.
Neutral
VA
DBS
0.078 eV
0.024 eV
Figure 3.4: Schematic representation of the energy levels of the neutral, valence and
dipole-bound anionic states of benzonitrile (see text). Note that the V A adiabatically drops
below the neutral and the DBS due to zero-point energy.
Fig. 3.4 shows the schematic energy diagram of the neutral benzonitrile and the two anionic
states. The present EOM-EA-CCSD calculations reveal that at the neutral’s equilibrium geom-
etry (R
N
), the V A state is electronically unbound, and the only bound anionic state is the DBS,
with VEA of 0.0240 eV . This finding is consistent with the corresponding results of Adamow-
icz and co-workers
25
, who reported VDE of the DBS to be 0.019 eV with CCSD(T). However,
at the optimized geometry of the valence anion, both the DBS and V A are (vertically) bound
by 0.0264 eV and 0.0639 eV , respectively. The latter value is in excellent agreement with the
79
experimental VDE of 0.058(5) eV , previously assigned to the V A
24
. We note that this value is a
significant improvement over the corresponding EOM-IP-CCSD value of 0.047 eV , which was
obtained using a less balanced protocol based on the open-shell anionic reference and a smaller
basis set
24
.
Considering only the electronic energies (computed with EOM-EA-CCSD), the DBS min-
imum is 0.139 eV below the minimum of the V A and the V A is adiabatically unbound (as
indicated by the negative AEA
ee
). However, the ZPE correction makes the V A adiabatically
bound by 0.0106 eV . A relatively large effect of ZPE favoring the V A can be easily rational-
ized by the shapes of the respective Dyson orbitals (Fig. 3.3): because the electron is attached
to the
orbital, the vibrational modes of the anion become softer, thus lowering the magnitude
of ZPE relative to the neutral. The mode that is most affected by electron attachment is the
butterfly mode. The frequency of this mode softens by170 cm
1
upon electron attachment,
which is clearly illustrated in Fig. 3.5 by the reduction of the curvature of the potential energy
profile. The frequencies are given in Table SI in the SI.
0
0.2
0.4
0.6
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
Energy (eV)
Butterfly mode (Å)
Neutral
DBS
VA
Figure 3.5: Potential energy curves along the butterfly mode for V A, DBS, and the neu-
tral, showing the relaxation of V A. Energies along the scan are computed with EOM-EA-
CCSD/aug-cc-pVDZ+6s3p(3s). The scan was generated by taking the displacement along
the butterfly normal mode of the anion (mode #2 of 209.18 cm
1
).
80
The inclusion of perturbative triple excitations increases the attachment energy of the V A
by 0.0677 eV . Thus, the state ordering is reversed and the V A becomes adiabatically more
stable than the DBS when both triple excitations and ZPE are taken into account. Table 3.1
summarizes the key energetics and Fig. 3.4 presents the results graphically.
Table 3.1: Attachment and detachment energies (in eV) for valence (
2
B
1
) and dipole-bound
(
2
A
1
) anion of benzonitrile.
State V AE
a
VDE
a
AEA
a
ee
ZPE
b
(T)
c
AEA
d
2
A
1
-0.0240 - 0.0240 0 0 0.024
2
B
1
NB
e
0.0639 -0.1150 0.1256 0.0677 0.078
a
EOM-EA-CCSD/aug-cc-pVTZ+6s3p(3s).
b
RI-CCSD/RI-EOM-EA-CCSD and aug-cc-pVDZ for the V A.
c
EOM-EA-CCSD(T)(a)
/aug-cc-pVTZ for the V A.
d
AEA including ZPE and triples correction.
e
Electronically not bound.
Fig. 3.4, which shows the relevant energy levels of benzonitrile and its anion, highlights that
the V A is bound by 0.078 eV relative to the ground state of the neutral and is 0.054 eV more
stable adiabatically than the DBS. However, at the geometry of the neutral, the V A is above the
DBS and is electronically unbound. Thus, the DBS may act as a doorway for the V A formation.
In this mechanism, the electron is first captured by benzonitrile at its neutral geometry, forming
the DBS anion, followed by a non-adiabatic transition to the more stable V A state.
3.4.2 The C
6
H
5
CN
H
2
O complex
In the C
6
H
5
CN
H
2
O complex, the water molecule forms a hydrogen bond with the cyano
group, only weekly perturbing the electronic structure of the benzonitrile core. The electronic
states of the anion appear to be very similar to those of the bare benzonitrile, as one can clearly
see from the Dyson orbitals shown in Fig. 3.6. However, microsolvation affects the energetics
of the states. At the neutral’s geometry, the DBS(
1
A
0
) and V A(
1
A
00
) are vertically bound by
0.0661 eV and 0.0535 eV , respectively. The detachment energy for V A increases approximately
81
DBS
2
A’ VA
2
A’’
Figure 3.6: Dyson orbitals for the two lowest electronic states of the C
6
H
5
CN
H
2
O com-
plex,
2
A
0
and
2
A
00
, computed at the respective optimized geometries. Isovalue 0.007.
tenfold, up to 0.4566 eV at the optimized geometry of the V A. Adiabatically, the DBS and V A
are bound by 0.0661 and 0.169 eV , respectively.
Neutral
VA
DBS
0.457 ev
0.066 ev
Figure 3.7: Schematic representation of the neutral, valence anion, and dipole-bound
states of the benzonitrile-water complex.
Fig. 3.7 shows the relevant state ordering in this complex. Here, the ZPE also makes the V A
more bound, but the effect is small, relative to the VDE itself. The computed VDE agrees with
the experimental electron detachment energies, confirming that electron detachment happens
from the V A. One crucial experimental observation is that the detachment energy increases by
0.32 eV upon addition of water relative to the bare benzonitrile. This value is in good agreement
with the 0.39 eV increase in VDE predicted by theory.
82
Table 3.2: Attachment and detachment energies (in eV) for the valence (
1
A
00
) and dipole-
bound (
2
A
0
) states of the benzonitrile-water complex.
State V AE
a
VDE
a
AEA
a
ee
2
A
0
-0.0661 - 0.0661
2
A
00
-0.0535 0.4566 0.169
a
EOM-EA-CCSD/aug-cc-pVTZ+4s4p(4s)
3.4.3 Photodetachment spectra
0.7
0.8
0.9
1
1.1
0 100 200 300 400 500
%Population of VA
Temperature(K)
Boltzman Populations
N
VA
/N
Total
Figure 3.8: Ratio of the Boltzmann population of V A (N
VA
) relative to the total population
(N
total
) as a function of temperature.
The computed energetics of the bare benzonitrile suggest that the V A species are dominant
even at room temperature (300 K), as shown in Fig. 3.8. The much larger gap between the V A
and DBS in the microsolvated benzonitrile makes the presence of the DBS in C
6
H
5
CN
H
2
O
highly improbable. In this section, we analyze Franck–Condon factors and photodetachment
cross sections for the V A and DBS, to further confirm our assignment.
83
0
0.1
0.2
0.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Intensity (arb. unit)
Electron Binding Energy (eV)
Experiment
FCF VA
Convolution
00
0
1
1
2
1
1
26
1
0
Figure 3.9: Computed photoelectron spectrum for the V A and the experimental spectrum
obtained by Sanov and co-workers
24
. In the computed spectrum, the Franck-Condon
factors were convoluted with gaussians of width 0.05 eV .
Figure 3.10: Franck-Condon active modes: #0 (out-of-plane), #2 (butterfly), and #26 (ring
breathing).
Using the geometries of the anion and neutral forms of benzonitrile, we computed the
Franck–Condon factors for the photodetachment from the V A state and compared them with
the experimental spectrum in Fig. 3.9. The spectra show broad vibrational progression. The
analysis of the computed Franck-Condion factors (Table 3.3) reveals three dominant modes:
Butterfly mode (#2), ring-breathing mode (#26), and another soft out-of-plane mode (#0). The
frequencies are shown in Table SI of the SI and the modes are shown in Fig. 3.10. The character
of the Franck-Condon modes is consistent with the detachment from an antibonding
orbital
(Fig. 3.3, right panel).
84
As the formation of the DBS of the anion does not lead to a significant change in geometry,
the detachment from DBS would result in the sharp 00 transition with no vibrational structure.
Thus, the observed vibrational structure in the measured photodetachment spectrum also sup-
ports the conclusion that the observed photoelectron spectra must be attributed to the V A rather
than the DBS.
Table 3.3: Peak positions and assignments for the photoelectron spectrum of
C
6
H
5
CN
(
1
A
1
2
B
1
).
Energy (eV) Intensity (arb. units) Vibration
0.0634 0.1406 1
1
1
0.0639 0.3106 0
0
0
0.0684 0.1797 0
1
0
0.0847 0.1046 2
1
2
1.2701 0.1049 26
1
0
Extending the same argument to the C
6
H
5
CN
H
2
O complex, the V A is expected to show a
similar Franck–Condon progression in the photoelectron spectra. In the experiment with 1.165
eV photons, both C
6
H
5
CN
and C
6
H
5
CN
H
2
O indeed show similar Franck–Condon envelops.
This observation strongly suggests that the observed photoelectron spectra for both C
6
H
5
CN
and C
6
H
5
CN
H
2
O correspond to the detachment from V A.
To assess whether possible contributions from the DBS to photoelectron spectra are not seen
in the experiment due to potentially low cross section, we also computed absolute photodetach-
ment cross sections using Dyson orbitals and ezDyson software
53
. Different orbital characters
of DBS and V A result in different trends in the computed cross sections. The absolute cross
sections for the two anionic states are shown in Fig. S2 of SI. Fig. 3.11 shows the dependence
of ratio of the two cross sections as a function on the photon energy. The results reveal that at
low photon energy the DBS photodetachment cross section is 100 times higher than that of V A.
At the 1.165 eV photon energy, corresponding to the experimental spectrum shown in Fig. 3.9,
85
our calculations predict
DBS
=
VA
= 4.65 (see the inset Fig. 3.11). Taking this ratio value and
estimating the relative intensity of DBS photodetachment at this energy as
I
DBS
= I
VA
DBS
VA
(3.15)
one can compute the Franck–Condon spectra for an arbitrary mixture of DBS and V A. Fig. 3.11
shows such a spectrum for an equal mixture of the two anionic states. The sharp threshold peak
due to the DBS has a very high relative intensity, dwarfing the Franck-Condon envelope due
to the V A. This shows that if the population of the DBS were significant, the first sharp peak
in the photoelectron spectrum would be attributed to DBS. However, the very similar first peak
in the C
6
H
5
CN
H
2
O spectrum (Fig. 3.1, bottom) should definitely be attributed to V A, based
on the energetics shown in Fig. 3.7. Disregarding the 0.32 eV band shift, the C
6
H
5
CN
and
C
6
H
5
CN
H
2
O spectra in Fig. 3.1 are indeed very similar, strongly suggesting that the elec-
tronic character of the core anion should be the same in both cases. This comparison supports
the argument that the first peak observed in the experimental spectra of both C
6
H
5
CN
and
C
6
H
5
CN
H
2
O originates from the V A. If the DBS were involved, it would contributed more
prominently (with a sharp threshold peak) to the spectrum of the unsolvated anion, compared
to the micro-hydrated species, overpowering the Franck-Condon features from the V A.
A similar interplay between the V A and DBS was observed in uracil
61, 62
. An experimental
study by Bowenet.al
63
was able to distinguish between the DBS and V A in uracil. The photo-
electron spectrum of the uracil anion shows a strong narrow single peak at 93 meV . The absence
of the Franck–Condon progression indicated negligible structural changes from the anion to the
neutral, which is characteristic of DBS. Using argon and krypton as solvents did not affect
the narrow peak, but when xenon was used as solvent, the spectra changed dramatically. This
86
5
10
15
0.8 1 1.2 1.4
0
30
60
90
120
150
0 1 2 3 4 5
σ
DBS
/σ
VA
Photon energy (eV)
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8
Intensity (arb. unit)
Electron Binding Energy (eV)
VA
DBS
Figure 3.11: Top: Ratio of the cross sections (DBS versus V A) for electron detachment
from the benzonitrile anion. Bottom: Convolution of the Franck–Condon factors using
Eq. (3.15) with gaussian (width=0.05 eV) and assuming equal populations of the DBS and
V A.
behavior was attributed to the polarization effect of xenon, which resulted in breaking the sym-
metry of the molecule and stabilizing the V A. Using more polar solvent such as H
2
O resulted
in complete disappearance of the DBS peak.
87
The behavior of the benzonitrile anion differs from that of uracil in several respects. First,
the Franck–Condon features are observed in the photoelectron spectrum with no solvent present.
Second, no additional sharp peaks, which could be attributed to the DBS, are seen in the pho-
todetachment of the unsolvated anion, as compared to the micro-solvated species. Third, the V A
of benzonitrile is adiabatically more stable than the DBS. Thus, the DBS may act as a doorway
to capture the electron, subsequently transferring the population into the V A upon the collisional
relaxation of the anion. Since the cross sections for photodetachment are equal to the cross sec-
tions of the reverse process, radiative electron attachment, the large value of the cross section
for DBS further supports that electron capture directly into this state may be efficient. Given
the computed structures and frequencies, it is likely that the butterfly mode drives the relaxation
of the DBS into the V A.
3.5 Conclusion
CCSD and EOM-EA-CCSD calculations of the electronic states of benzonitrile and its anion
reveal that although, in terms of electronic energies the V A is adiabatically above the neutral
state, the respective ZPE-corrected levels reverse the state ordering, making the V A bound. The
inclusion of triples corrections further stabilizes the V A state relative to the neutral and DBS.
The computed energetics are in excellent agreement with the experimental values reported in
earlier work
24
. The computed Franck–Condon factors provide further support to the assignment
of the ground state of the benzonitrile anion as the valence-bound anion,
2
B
1
, rather than dipole-
bound state (
2
A
1
), as was claimed by Adamowicz and co-workers
25
. The DBS
2
A
1
, which is
0.054 eV above the V A (adiabatically) and is electronically bound at the equilibrium geome-
try of the neutral, is likely serving as a doorway state for electron capture. The calculations
on benzonitrile solvated with one water molecule show that the V A is stabilized further and
becomes bound even at the structure of the neutral complex. The calculations of photoelectron
88
spectra show similar Franck–Condon envelops, blue-shifted by 0.39 eV with respect to the bare
benzonitrile, also in excellent agreement with the experimental findings (0.32 eV) of Sanov and
co-workers
24
. As was pointed out in the original experimental study
24
, the similarity in the
vibrational structure of the photodetachment spectra of the bare and microsolvated benzonitrile
provides additional evidence that the ground state of benzonitrile is the valence
2
B
1
state. This
work highlights the importance of balanced and accurate treatment of electron correlation and
the need to consider nuclear motion.
3.6 Appendix A: Computational details
We used the standard aug-cc-pVTZ basis set was augmented by an additional sets of diffuse
s and p functions on heavy atoms and s function on H, with the following exponents of the
first additional basis functions: (N,s) = 0.0288,(N,p) = 0.0245,(C,s) = 0.02201,(C,p) =
0.017840, (H,s) = 0.01263; the exponents of the subsequent sets of the additional functions
were obtained according to
i+1
= 05
i
. The basis set printout is given below. For frequencies
calculations, we used aug-cc-pVDZ.
3.7 Appendix B: Normal modes
Table 3.4: Frequencies (cm
1
) for the neutral and valence anion computed with RI-CCSD
and RI-EOM-EA-CCSD using the aug-cc-pVDZ basis set.
Mode Neutral Anion Mode Neutral Anion Mode Neutral Anion
0 145.95 110.11 11 855.10 731.77 22 1332.71 1347.77
1 160.86 165.23 12 922.33 741.78 23 1463.68 1378.79
2 377.10 209.18 13 956.62 864.83 24 1513.42 1454.09
3 398.67 433.07 14 966.47 899.66 25 1638.76 1497.91
4 458.94 440.53 15 1007.98 973.11 26 1663.48 1649.02
5 544.92 478.20 16 1046.41 983.34 27 2311.96 2161.34
6 550.26 518.61 17 1098.78 1052.40 28 3199.53 3142.57
7 626.34 549.59 18 1169.55 1109.17 39 3209.82 3145.13
8 649.87 583.59 29 1193.67 1175.36 30 3217.54 3176.54
9 756.86 621.66 20 1217.71 1232.88 31 3226.14 3177.02
10 762.82 687.67 21 1298.16 1259.66 32 3231.12 3196.85
89
3.8 Appendix C: Absolute cross sections
0
2
4
6
8
0 1 2 3 4 5
Cross section (arb. unit)
Energy (eV)
DBS
0
0.2
0.4
0 1 2 3 4 5
Cross section (arb. unit)
Energy (eV)
VA
Figure 3.12: Absolute cross sections for detachment from the dipole bound (left) and
valence (right) states of C
6
H
5
CN
.
3.9 Appendix D: Basis set
aug-cc-pVTZ+6s6p(3s) for C6H5CN
$basis
H 0
S 3 1.00
33.8700000 0.0060680
5.0950000 0.0453080
1.1590000 0.2028220
S 1 1.00
0.3258000 1.0000000
S 1 1.00
0.1027000 1.0000000
S 1 1.00
0.0252600 1.0000000
S 1 1.00
0.0126300 1.0000000
S 1 1.00
0.0063150 1.0000000
S 1 1.00
0.0031570 1.0000000
P 1 1.00
1.4070000 1.0000000
P 1 1.00
90
0.3880000 1.0000000
P 1 1.00
0.1020000 1.0000000
D 1 1.00
1.0570000 1.0000000
D 1 1.00
0.2470000 1.0000000
****
C 0
S 8 1.00
8236.0000000 0.0005310
1235.0000000 0.0041080
280.8000000 0.0210870
79.2700000 0.0818530
25.5900000 0.2348170
8.9970000 0.4344010
3.3190000 0.3461290
0.3643000 -0.0089830
S 8 1.00
8236.0000000 -0.0001130
1235.0000000 -0.0008780
280.8000000 -0.0045400
79.2700000 -0.0181330
25.5900000 -0.0557600
8.9970000 -0.1268950
3.3190000 -0.1703520
0.3643000 0.5986840
S 1 1.00
0.9059000 1.0000000
S 1 1.00
0.1285000 1.0000000
S 1 1.00
0.0440200 1.0000000
S 1 1.00
0.0220100 1.0000000
S 1 1.00
0.0110000 1.0000000
S 1 1.00
0.0055000 1.0000000
S 1 1.00
0.0022500 1.0000000
S 1 1.00
0.0011250 1.0000000
S 1 1.00
91
0.0005625 1.0000000
P 3 1.00
18.7100000 0.0140310
4.1330000 0.0868660
1.2000000 0.2902160
P 1 1.00
0.3827000 1.0000000
P 1 1.00
0.1209000 1.0000000
P 1 1.00
0.0356900 1.0000000
P 1 1.00
0.0178400 1.0000000
P 1 1.00
0.0089200 1.0000000
P 1 1.00
0.0044600 1.0000000
D 1 1.00
1.0970000 1.0000000
D 1 1.00
0.3180000 1.0000000
D 1 1.00
0.1000000 1.0000000
F 1 1.00
0.7610000 1.0000000
F 1 1.00
0.2680000 1.0000000
****
N 0
S 8 1.00
11420.0000000 0.0005230
1712.0000000 0.0040450
389.3000000 0.0207750
110.0000000 0.0807270
35.5700000 0.2330740
12.5400000 0.4335010
4.6440000 0.3474720
0.5118000 -0.0085080
S 8 1.00
11420.0000000 -0.0001150
1712.0000000 -0.0008950
389.3000000 -0.0046240
110.0000000 -0.0185280
35.5700000 -0.0573390
92
12.5400000 -0.1320760
4.6440000 -0.1725100
0.5118000 0.5999440
S 1 1.00
1.2930000 1.0000000
S 1 1.00
0.1787000 1.0000000
S 1 1.00
0.0576000 1.0000000
S 1 1.00
0.0288000 1.0000000
S 1 1.00
0.0144000 1.0000000
S 1 1.00
0.0072000 1.0000000
S 1 1.00
0.0036000 1.0000000
S 1 1.00
0.0018000 1.0000000
S 1 1.00
0.0009000 1.0000000
P 3 1.00
26.6300000 0.0146700
5.9480000 0.0917640
1.7420000 0.2986830
P 1 1.00
0.5550000 1.0000000
P 1 1.00
0.1725000 1.0000000
P 1 1.00
0.0491000 1.0000000
P 1 1.00
0.0245500 1.0000000
P 1 1.00
0.0122700 1.0000000
P 1 1.00
0.0061350 1.0000000
D 1 1.00
1.6540000 1.0000000
D 1 1.00
0.4690000 1.0000000
D 1 1.00
0.1510000 1.0000000
F 1 1.00
93
1.0930000 1.0000000
F 1 1.00
0.3640000 1.0000000
****
$end
aug-cc-pVTZ+4s4p(4s) for C6H5CN + H2O
$basis
H 0
S 3 1.00
33.8700000 0.0060680
5.0950000 0.0453080
1.1590000 0.2028220
S 1 1.00
0.3258000 1.0000000
S 1 1.00
0.1027000 1.0000000
S 1 1.00
0.0252600 1.0000000
S 1 1.00
0.0126300 1.0000000
S 1 1.00
0.0063150 1.0000000
S 1 1.00
0.0031570 1.0000000
S 1 1.00
0.0015785 1.0000000
P 1 1.00
1.4070000 1.0000000
P 1 1.00
0.3880000 1.0000000
P 1 1.00
0.1020000 1.0000000
D 1 1.00
1.0570000 1.0000000
D 1 1.00
0.2470000 1.0000000
****
C 0
S 8 1.00
8236.0000000 0.0005310
1235.0000000 0.0041080
280.8000000 0.0210870
94
79.2700000 0.0818530
25.5900000 0.2348170
8.9970000 0.4344010
3.3190000 0.3461290
0.3643000 -0.0089830
S 8 1.00
8236.0000000 -0.0001130
1235.0000000 -0.0008780
280.8000000 -0.0045400
79.2700000 -0.0181330
25.5900000 -0.0557600
8.9970000 -0.1268950
3.3190000 -0.1703520
0.3643000 0.5986840
S 1 1.00
0.9059000 1.0000000
S 1 1.00
0.1285000 1.0000000
S 1 1.00
0.0440200 1.0000000
S 1 1.00
0.0220100 1.0000000
S 1 1.00
0.0110000 1.0000000
S 1 1.00
0.0055000 1.0000000
S 1 1.00
0.0027500 1.0000000
P 3 1.00
18.7100000 0.0140310
4.1330000 0.0868660
1.2000000 0.2902160
P 1 1.00
0.3827000 1.0000000
P 1 1.00
0.1209000 1.0000000
P 1 1.00
0.0356900 1.0000000
P 1 1.00
0.0178400 1.0000000
P 1 1.00
0.0089200 1.0000000
P 1 1.00
0.0044600 1.0000000
95
P 1 1.00
0.0022300 1.0000000
D 1 1.00
1.0970000 1.0000000
D 1 1.00
0.3180000 1.0000000
D 1 1.00
0.1000000 1.0000000
F 1 1.00
0.7610000 1.0000000
F 1 1.00
0.2680000 1.0000000
****
N 0
S 8 1.00
11420.0000000 0.0005230
1712.0000000 0.0040450
389.3000000 0.0207750
110.0000000 0.0807270
35.5700000 0.2330740
12.5400000 0.4335010
4.6440000 0.3474720
0.5118000 -0.0085080
S 8 1.00
11420.0000000 -0.0001150
1712.0000000 -0.0008950
389.3000000 -0.0046240
110.0000000 -0.0185280
35.5700000 -0.0573390
12.5400000 -0.1320760
4.6440000 -0.1725100
0.5118000 0.5999440
S 1 1.00
1.2930000 1.0000000
S 1 1.00
0.1787000 1.0000000
S 1 1.00
0.0576000 1.0000000
S 1 1.00
0.0288000 1.0000000
S 1 1.00
0.0144000 1.0000000
S 1 1.00
0.0072000 1.0000000
96
S 1 1.00
0.0036000 1.0000000
P 3 1.00
26.6300000 0.0146700
5.9480000 0.0917640
1.7420000 0.2986830
P 1 1.00
0.5550000 1.0000000
P 1 1.00
0.1725000 1.0000000
P 1 1.00
0.0491000 1.0000000
P 1 1.00
0.0245500 1.0000000
P 1 1.00
0.0122700 1.0000000
P 1 1.00
0.0061350 1.0000000
P 1 1.00
0.0030675 1.0000000
D 1 1.00
1.6540000 1.0000000
D 1 1.00
0.4690000 1.0000000
D 1 1.00
0.1510000 1.0000000
F 1 1.00
1.0930000 1.0000000
F 1 1.00
0.3640000 1.0000000
****
O 0
S 8 1.00
15330.0000000 0.0005080
2299.0000000 0.0039290
522.4000000 0.0202430
147.3000000 0.0791810
47.5500000 0.2306870
16.7600000 0.4331180
6.2070000 0.3502600
0.6882000 -0.0081540
S 8 1.00
15330.0000000 -0.0001150
2299.0000000 -0.0008950
97
522.4000000 -0.0046360
147.3000000 -0.0187240
47.5500000 -0.0584630
16.7600000 -0.1364630
6.2070000 -0.1757400
0.6882000 0.6034180
S 1 1.00
1.7520000 1.0000000
S 1 1.00
0.2384000 1.0000000
S 1 1.00
0.0737600 1.0000000
S 1 1.00
0.0368800 1.0000000
S 1 1.00
0.0184400 1.0000000
S 1 1.00
0.0092200 1.0000000
S 1 1.00
0.0046100 1.0000000
P 3 1.00
34.4600000 0.0159280
7.7490000 0.0997400
2.2800000 0.3104920
P 1 1.00
0.7156000 1.0000000
P 1 1.00
0.2140000 1.0000000
P 1 1.00
0.0597400 1.0000000
P 1 1.00
0.0298700 1.0000000
P 1 1.00
0.0149350 1.0000000
P 1 1.00
0.0074675 1.0000000
P 1 1.00
0.0037338 1.0000000
D 1 1.00
2.3140000 1.0000000
D 1 1.00
0.6450000 1.0000000
D 1 1.00
0.2140000 1.0000000
98
F 1 1.00
1.4280000 1.0000000
F 1 1.00
0.5000000 1.0000000
****
$end
3.10 Appendix E: Relevant Cartesian geometries
C6H5CN (anion) [EOM-EA-CCSD/aug-cc-pVTZ]
Nuclear Repulsion Energy = 297.4254488750 hartrees
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C 0.0000000000 0.0000000000 2.2198396566
2 C 1.2148152484 0.0000000000 1.4847353494
3 C 1.2286801655 0.0000000000 0.1090561197
4 C 0.0000000000 0.0000000000 -0.6418412548
5 C -1.2286801655 0.0000000000 0.1090561197
6 C -1.2148152484 0.0000000000 1.4847353494
7 H 0.0000000000 0.0000000000 3.3013966583
8 H 2.1614221562 0.0000000000 2.0155280474
9 H 2.1721584196 0.0000000000 -0.4237371437
10 H -2.1721584196 0.0000000000 -0.4237371437
11 H -2.1614221562 0.0000000000 2.0155280474
12 C 0.0000000000 0.0000000000 -2.0456075662
13 N 0.0000000000 0.0000000000 -3.2200270210
----------------------------------------------------------------
C6H5CN (neutral) [CCSD/aug-cc-pVTZ]
Nuclear Repulsion Energy = 300.0912167081 hartrees
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C 0.0000000000 0.0000000000 2.1806397523
2 C 1.2072838898 0.0000000000 1.4863451086
3 C 1.2122184965 0.0000000000 0.0965882257
4 C 0.0000000000 0.0000000000 -0.5963508767
5 C -1.2122184965 0.0000000000 0.0965882257
6 C -1.2072838898 0.0000000000 1.4863451086
7 H 0.0000000000 0.0000000000 3.2620055195
8 H 2.1441224787 0.0000000000 2.0255164220
99
9 H 2.1428929996 0.0000000000 -0.4529820599
10 H -2.1428929996 0.0000000000 -0.4529820599
11 H -2.1441224787 0.0000000000 2.0255164220
12 C 0.0000000000 0.0000000000 -2.0378191747
13 N 0.0000000000 0.0000000000 -3.1939971402
----------------------------------------------------------------
C6H5CN (anion) [RI-EOM-EA-CCSD/aug-cc-pVDZ]
Nuclear Repulsion Energy = 294.24941846 hartrees
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C 0.0000000000 0.0000000000 2.2385619165
2 C 1.2270100469 0.0000000000 1.4955660302
3 C 1.2414555922 0.0000000000 0.1048278944
4 C 0.0000000000 0.0000000000 -0.6517153293
5 C -1.2414555922 0.0000000000 0.1048278944
6 C -1.2270100469 0.0000000000 1.4955660302
7 H 0.0000000000 0.0000000000 3.3333378557
8 H 2.1849792477 0.0000000000 2.0326591595
9 H 2.1961421075 0.0000000000 -0.4345093168
10 H -2.1961421075 0.0000000000 -0.4345093168
11 H -2.1849792477 0.0000000000 2.0326591595
12 C 0.0000000000 0.0000000000 -2.0709831529
13 N 0.0000000000 0.0000000000 -3.2613636061
----------------------------------------------------------------
C6H5CN (neutral) [RI-CCSD/aug-cc-pVDZ]
Nuclear Repulsion Energy = 296.89544041 hartrees
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C 0.0000000000 0.0000000000 2.1973518112
2 C 1.2198603297 0.0000000000 1.4955590882
3 C 1.2256042525 0.0000000000 0.0911948716
4 C 0.0000000000 0.0000000000 -0.6074800981
5 C -1.2256042525 0.0000000000 0.0911948716
6 C -1.2198603297 0.0000000000 1.4955590882
7 H 0.0000000000 0.0000000000 3.2914713787
8 H 2.1676784048 0.0000000000 2.0411918025
9 H 2.1671350168 0.0000000000 -0.4645081830
10 H -2.1671350168 0.0000000000 -0.4645081830
11 H -2.1676784048 0.0000000000 2.0411918025
100
12 C 0.0000000000 0.0000000000 -2.0630362867
13 N 0.0000000000 0.0000000000 -3.2352712420
----------------------------------------------------------------
C6H5CN + H2O (anion) [EOM-EA-CCSD/aug-cc-pVDZ]
Nuclear Repulsion Energy = 356.60342912 hartrees
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C -1.0559591922 -1.2416910481 0.0000000000
2 C -0.2962128176 -0.0021660398 0.0000000000
3 C -1.0459008917 1.2434058339 0.0000000000
4 C -2.4367045617 1.2324629599 0.0000000000
5 C -3.1822450186 0.0092421753 0.0000000000
6 C -2.4462223691 -1.2203476331 0.0000000000
7 H -0.5186771746 -2.1970717339 0.0000000000
8 H -0.5018692828 2.1950706178 0.0000000000
9 H -2.9720373840 2.1909451057 0.0000000000
10 H -4.2767950617 0.0135752338 0.0000000000
11 H -2.9885843803 -2.1748283637 0.0000000000
12 C 1.1169746423 -0.0112036653 0.0000000000
13 N 2.3065654217 -0.0249614375 0.0000000000
14 O 5.2039805652 -0.0787786240 0.0000000000
15 H 5.3301318889 0.8760470161 0.0000000000
16 H 4.2276501721 -0.1569943179 0.0000000000
----------------------------------------------------------------
C6H5CN + H20 (neutral) [CCSD/aug-cc-pVDZ]
Nuclear Repulsion Energy = 357.0277249080 hartrees
Standard Nuclear Orientation (Angstroms)
I Atom X Y Z
----------------------------------------------------------------
1 C -1.0860467968 -1.2296201743 0.0000000000
2 C -0.3819739079 -0.0069710232 0.0000000000
3 C -1.0708767787 1.2244245809 0.0000000000
4 C -2.4750926197 1.2261974571 0.0000000000
5 C -3.1828604323 0.0096059813 0.0000000000
6 C -2.4898591578 -1.2153419764 0.0000000000
7 H -0.5357320682 -2.1741751448 0.0000000000
8 H -0.5114531691 2.1637519582 0.0000000000
9 H -3.0175160887 2.1761282806 0.0000000000
10 H -4.2767261424 0.0160758864 0.0000000000
101
11 H -3.0424983076 -2.1592565052 0.0000000000
12 C 1.0724883600 -0.0234315776 0.0000000000
13 N 2.2425517096 -0.0481918041 0.0000000000
14 O 5.3364158461 -0.0671302711 0.0000000000
15 H 5.5137281879 0.8800305480 0.0000000000
16 H 4.3695459215 -0.1193901367 0.0000000000
----------------------------------------------------------------
102
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47
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48
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51
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53
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54
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106
Chapter 4: EOM-CC guide to Fock-space
travel: The C
2
edition
4.1 Introduction
Ironically, the smallest form of neat carbon, the C
2
molecule, features the most complex
electronic structure. The complexity stems from the inability of the eight valence electrons
of the two carbons to form a quadruple bond (remarkably, the bonding in C
2
is still hotly
debated
1–8
). Because the optimal electron pairing cannot be attained, multiple electronic config-
urations have similar likelihood, leading to a dense manifold of low-lying electronic states. This
results in rich spectroscopy: C
2
features multiple low-lying electronic transitions, which have
been extensively studied experimentally
9–15
. Nevertheless, C
2
continues to generate interest.
For example, recently, new band systems have been identified
16–18
.
Besides its obvious fundamental importance, C
2
(and its anionic forms, C
2
and C
2
2
), play
a role in combustion
19
, plasma
20–22
, and astrochemistry
20, 23
. For example, C
2
and C
2
have been
observed in comet tails, protoplanetary nebulae, the atmospheres of stars, and in the diffuse
interstellar medium
23–28
. C
2
is responsible for the color of blue flames
19
. It is also a prominent
product of electrical discharges containing hydrocarbons
21
.
From the theoretical point of view, C
2
is arguably the most difficult molecule among
homonuclear diatomics from the first two rows of the periodic table. Electronic near-
degeneracies lead to multiconfigurational wave-functions. Small energy separations between
107
different electronic states also call for high accuracy. Because of its complex electronic struc-
ture, C
2
has often been described as the poster child of multi-reference methodology. The avail-
ability of high-quality spectroscopic data, complex electronic structure, and its small size make
C
2
a popular benchmark system for quantum chemistry studies
29–33
. Among recent theoretical
studies of the low-lying states of C
2
, the most comprehensive are the tour-de-force MR-CISD
(multi-reference configuration interaction with single and double excitations) calculations by
Schmidt and coworkers
34
and by Szalay and co-workers
35
. In both studies, the effect of basis
set and higher-order corrections have been carefully investigated. To correct MR-CISD ener-
gies for violation of size-extensivity, Davidson’s quadruple correction was used. Szalay and
co-workers have also reported results obtained with an alternative strategy, the so-called MR-
average quadratic coupled-cluster (AQCC) method. In both studies, the theoretical values of the
reported equilibrium distances (r
e
) and term energies (T
ee
) agreed well with the experimental
data.
The anionic forms of C
2
, C
2
and C
2
2
, have received less attention. C
2
is produced in
plasma discharge from acetylene
36, 37
. Electronically excited C
2
has been observed in a carbon-
rich plasma via fluorescence
22
. Recently, C
2
has been proposed as a candidate for laser cooling
of anions
38
; this makes these species interesting in the context of quantum information storage.
Ervin and Lineberger
39
have measured the photoelectron spectrum of C
2
using 3.53 eV photons;
they reported the adiabatic electron affinity (AEA) of C
2
to be 3.2690.006 eV . A similar value
(3.2730.008 eV) has been derived by Neumark and coworkers
40
, who reported vibrationally
resolved photodetachment spectra using 4.66 eV radiation. Feller has reported an AEA of 3.267
eV calculated using a composite method based on coupled-cluster (CC) methods
41
.
Because of its highly unsaturated character, C
2
has relatively large electron attachment
energy, so that even the two lowest excited states of C
2
are bound electronically. In contrast,
C
2
2
is metastable with respect to electron detachment. Its existence, which has been postulated
108
on the basis of features observed
42, 43
in electron scattering from C
2
, has been confirmed by
calculations
44, 45
.
In this contribution, we present detailed electronic structure calculations of C
2
, C
2
, and C
2
2
,
with an emphasis on spectroscopically relevant properties. We employ an alternative methodol-
ogy based on CC and equation-of-motion CC (EOM-CC) theory
46–50
. We show that electronic
states of C
2
and C
2
are well described by the double ionization potential (DIP)
51
and ioniza-
tion potential (IP)
52, 53
variants of EOM-CCSD (EOM-CC with single and double substitutions)
using a dianionic reference state. Formulated in a strictly single-reference fashion, the EOM-
CC family of methods provides an accurate, robust, and effective alternative to cumbersome
multi-reference calculations
46–50
. To describe metastable species, such as C
2
2
, we employ the
complex-variable extension of CCSD and EOM-CCSD via the complex absorbing potential
(CAP) approach
54–56
.
4.2 Molecular orbital framework and essential features of
electronic structure of C
2
species
Fig. 4.1 shows the molecular orbital diagram and describes orbital occupation patterns in C
2
,
C
2
, and C
2
2
. Due to orbital near-degeneracies, various electronic configurations of six electrons
over the upper four orbitals have similar energies, leading to closely lying electronic states
and multi-configurational wave-functions. In C
2
, there are four important configurations in
which the unpaired electron resides on one of the upper orbitals. In C
2
2
, which is isoelectronic
with N
2
, all four upper orbitals are doubly occupied, resulting in the electronic configuration
[core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
2
(
2pz
)
2
. Consequently, the ground state of C
2
2
is a well-behaved
closed-shell state dominated by a single Slater determinant; thus, it can be well described by
single-reference methods such as, for example, CCSD. From this reference state, EOM-IP and
109
!
"#
!
"#
∗
!
%#
!
%#
∗
&
%'
(
&
%'
)
*
%+
,
[core]
6
Figure 4.1: Molecular orbital diagram. The three lowest orbitals that remain doubly
occupied in the low-energy manifold of electronic states of C
2
and C
2
are denoted as
’core’. The electronic states of C
2
are derived by distributing six additional electrons over
four upper orbitals,
2s
,
2px
=
2py
, and
2pz
. Shown is the leading electronic configuration
of the ground state, X
1
g
. Low-lying states of C
2
are derived by distributing five electrons
over the four upper orbitals. In C
2
2
, all four upper orbitals are doubly occupied. Shown
are Dyson orbitals (isovalue = 0.05) computed with EOM-IP-CCSD/aug-cc-pVTZ from
the dianionic reference.
EOM-DIP operators can generate all important electronic configurations needed for describing
the electronic states of C
2
and C
2
, respectively, as illustrated in Fig. 4.2.
EOM-DIP
!
"
#$%
EOM-IP
!
"
#$
Reference
Figure 4.2: EOM-IP (left) and EOM-DIP (right) manifolds generated from the dianionic
reference (center). Only configurations generated by
^
R
1
from the top four orbitals from
Fig. 4.1 are shown. EOM-IP enables access to the ground and electronically excited states
of C
2
, whereas EOM-DIP describes the ground and excited states of C
2
.
110
Mathematically, the EOM-CCSD target states are described by the following ansatz
47–49
:
= (
^
R
1
+
^
R
2
)e
^
T
1
+
^
T
2
0
; (4.1)
wheree
^
T
1
+
^
T
2
0
is the reference CCSD wave function (the amplitudes of the excitation operator
^
T are determined by the CCSD equations for the reference state) and operator
^
R is a general
excitation operator. In EOM-IP-CCSD,
^
R comprises all 1h (one hole) and 2h1p (two hole one
particle) operators
52, 53
, whereas in EOM-DIP-CCSD it comprises all 2h and 3h1p operators. In
EOM-EE-CCSD (EOM-CCSD for excitation energies
57
) and EOM-SF-CCSD (spin-flip EOM-
CCSD
58, 59
),
^
R is particle-conserving and includes 1h1p and 2h2p operators (in the SF variant,
^
R changes the number of and electrons). In the EA (electron attachment) variant
60
, the
operatorR is of the 1p and 1h2p type. The amplitudes of
^
R are found by diagonalization of the
similarity-transformed Hamiltonian,
H:
H =e
T
He
T
; (4.2)
HR
k
=E
k
R
k
: (4.3)
Linear parameterization ensures that different configurations can mix and interact. There are
no assumptions about their relative importance—the relative weights of different configura-
tions are defined by the EOM eigen-problem and can span the entire range of situations, from
those dominated by a single electronic configuration to those of equal contributions from mul-
tiple determinants. The EOM-CC ansatz is capable of reproducing exact degeneracies (such
as between the two components of states in linear molecules or Jahn-Teller degeneracies),
which are violated by state-specific MR treatments. Since all important configurations appear
at the same excitation level, they are treated in a balanced way. As a multi-state method, EOM-
CC produces the entire manifold of electronic states without requiring user input regarding
state character. These features make EOM-CC very attractive for treating multiple electronic
111
states and extensive degeneracies
50
. Recent applications illustrating the power of EOM-CC
include calculations of electronic states of copper oxide anions
61
, Cvetanovi´ c diradicals
62
, and
molecules with several unpaired electrons
63, 64
.
The success of EOM-CC in treating a particular electronic structure depends on whether a
proper well-behaved reference can be found from which the manifold of target states can be
reached by an appropriately chosen
^
R
1
. As illustrated in Fig. 4.2, the electronic structure of C
2
is best described by EOM-DIP using the dianionic reference state. The DIP method is capable
of describing electronic degeneracies beyond two-electrons-in-two-orbitals or three-electrons-
in-three-orbitals patterns
51, 61, 62, 65–69
; however, its applications are limited by complications due
to the use of the dianionic reference.
Isolated dianions of small molecules are usually unstable with respect to electron detach-
ment and exist only as transient species (resonances).
70
In dianions, resonances emerge due to
the competition between the two factors: (i) long-range repulsion between the anionic core and
an extra electron and (ii) short-range stabilizing valence interactions. Together, these lead to a
repulsive Coulomb barrier. The extra electron is trapped behind this barrier but can escape the
system by tunneling. This is similar to metastable radical monoanions, where the extra electron
is trapped behind an angular-momentum barrier, which affords resonance character. In a com-
putational treatment using a sufficiently large basis, the wave function of a resonance becomes
more and more diffuse, approximating a continuum state corresponding to an electron-detached
system and a free electron
71–73
.
Resonances can be described by a non-Hermitian extension of quantum mechanics
74
by
using, for example, complex absorbing potential (CAP)
75, 76
. If one is interested in the dianionic
state itself, then the CAP-based extension of CC theory can be used
56
. However, in practical
calculations using EOM-DIP-CC, the dianionic state just serves as a reference for generating
target configurations. Thus, less sophisticated approaches can be used to mitigate complications
112
arising from its metastable character. The easiest and most commonly used approach is to use
a relatively small basis set, such that the reference state is artificially stabilized
51, 61, 62, 65–67, 77
.
Ku´ s and Krylov have investigated an alternative strategy: stabilization of the resonance using
an artificial Coulomb potential with a subsequent de-perturbative correction
72, 73
. Here we show
that in the case of C
2
, using the aug-cc-pVTZ basis provides a robust description of the dian-
ionic reference, which delivers accurate results for the target states. To further validate these
calculations, we carried out CAP-EOM-IP-CCSD calculations in which the dianionic reference
is stabilized by the CAP and compare the potential energy curves of C
2
2
and C
2
obtained by
these two calculations.
In the CAP approach
75, 76
, the Hamiltonian is augmented by a purely imaginary confining
potential iW (the parameter controls the strength of the potential). This transformation
converts the resonances intoL
2
-integrable wave functions with complex energies
E =E
res
i
2
; (4.4)
where the real and imaginary parts correspond to the resonance position (E
res
) and width ().
In a complete basis set, the exact resonance position and width can be recovered in the limit
of! 0. In finite bases, the resonance can only be stabilized at finite values of. The per-
turbation due to the finite-strength CAP can be removed by applying first-order de-perturbative
correction
54, 55
and identifying the special points of-trajectories at which the dependence of the
computed energy on is minimal. When combined with the EOM-CCSD ansatz, this approach
has been shown to yield accurate and internally consistent results for both bound and metastable
states
56
. For example, these calculations yield smooth potential energy curves
78–80
and in many
cases correctly identify the points where resonances become bound. We note, however, that in
some polyatomic molecules spurious widths of about 0.04 eV for bound states persist
80
. In our
previous calculations
54, 56, 79–83
, we used CAP-EOM-CCSD to describe metastable EOM states
113
from stable (bound) CCSD references. Here we present the first example of a calculation where
the CCSD reference is metastable, but the target EOM-CCSD states are bound.
4.3 Computational details
As explained above, we describe the electronic states of C
2
and C
2
by EOM-IP-CCSD and
EOM-DIP-CCSD, respectively, using the dianionic reference (see Fig. 4.2). In real-valued
EOM-CCSD calculations, we used the aug-cc-pVTZ basis. In the CAP-augmented CCSD
and EOM-IP-CCSD calculations, we used the aug-cc-pVTZ+3s3p and aug-cc-pCVTZ+6s6p6d
basis sets (the exponents of the additional diffuse sets were generated using the same proto-
col as in our previous studies
55, 82
). Two core orbitals,
1s
and
1s
, were frozen in correlated
calculations except when employing the aug-cc-pCVTZ basis. In the calculations using aug-cc-
pVTZ+3s3p, the CAP onset was set according to the expectation value ofR
2
of the triplet UHF
wave function of C
2
(atr
cc
=1.28
˚
A, the onsets were:x
0
=y
0
=1.6
˚
A,z
0
= 2.6
˚
A). In the calcula-
tions with aug-cc-pCVTZ+6s6p6d, the CAP onset was set according to the expectation value of
R
2
of the dianion computed using CCSD/aug-cc-pCVTZ (atr
cc
=1.2761
˚
A, this gavex
0
=y
0
=
2.4
˚
A,z
0
= 3.6
˚
A). First-order correction
54
was applied to the computed total energy and then
optimal values of were determined from these corrected trajectories using our standard proto-
col
54, 55
. All electronic structure calculations were carried out using the Q-Chem package
84, 85
.
The calculations of partial widths were carried out using ezDyson
86
.
114
0
20000
40000
60000
1 1.2 1.4 1.6 1.8
Energy (cm
−1
)
Bond length (Å)
X
1
Σ
g
+
A
1
Π
u
B
1
Δ
g
B
’1
Σ
g
+
C
1
Π
g
D
1
Σ
u
+
a
3
Π
u
b
3
Σ
g
−
c
3
Σ
u
+
d
3
Π
g
Figure 4.3: Potential energy curves of low-lying singlet and triplet states of C
2
.
Table 4.1: Equilibrium bond lengths (r
e
,
˚
A) and term energies (T
ee
, cm
1
) of the low-lying
states of C
2
.
State Configuration EOM-DIP-CCSD
a
MR-CISD+Q
b
Expt.
c
r
e
T
ee
r
e
T
ee
r
e
T
ee
X
1
+
g
[core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
2
1.224 - 1.2536 - 1.2425 -
A
1
u
[core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
1
(
2pz
)
1
1.316 8127 1.3294 8000 1.3184 8391
B
1
g
[core]
6
(
2s
)
2
(
2px
)
1
(
2py
)
1
(
2pz
)
2
1.404 10408 1.3972 11684 1.3855 12082
B
0
1
+
g
[core]
6
(
2s
)
2
(
2px
)
1
(
2py
)
1
(
2pz
)
2
1.377 15012 1.3897 15134 1.3774 15409
C
1
g
[core]
6
(
2s
)
1
(
2px
)
2
(
2py
)
1
(
2pz
)
2
1.246 36489 1.2682 34788 1.2552 34261
D
1
+
u
[core]
6
(
2s
)
1
(
2px
)
2
(
2py
)
2
(
2pz
)
1
1.208 45166 1.2521 43810 1.2380 43239
a
3
u
[core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
1
(
2pz
)
1
1.310 694 1.3228 256 1.3119 716
b
3
g
[core]
6
(
2s
)
2
(
2px
)
1
(
2py
)
1
(
2pz
)
2
1.390 4971 1.3786 5794 1.3692 6434
c
3
+
u
[core]
6
(
2s
)
1
(
2px
)
2
(
2py
)
2
(
2pz
)
1
1.185 10531 1.2170 9618 1.2090 9124
d
3
g
[core]
6
(
2s
)
1
(
2px
)
2
(
2py
)
1
(
2pz
)
2
1.258 23025 1.2777 20382 1.2661 20022
a
aug-cc-pVTZ basis set.
b
MR-CISD with Davidson correction using the cc-pVTZ basis set from Ref. 35.
c
From Refs. 9–14.
115
4.4 Results and discussion
4.4.1 C
2
Fig. 4.3 shows the potential energy curves of low-lying singlet and triplet states of C
2
com-
puted using EOM-DIP-CCSD/aug-cc-pVTZ. The respective electronic configurations, equilib-
rium distances, and term values are summarized in Table 4.1. Table 4.1 also presents MR-
CISD+Q/cc-pVTZ results from Ref. 35 and the experimental values. As one can see, C
2
fea-
tures 10 electronic states within24,000 cm
1
(about 3 eV).
The results illustrate that EOM-DIP-CCSD is capable of tackling the complexity of C
2
rather
well. It describes the entire manifold of the low-lying states with an accuracy comparable to that
of much more cumbersome and labor-intensive multi-reference calculations. When compared
to the experimental values, the root-mean-square (RMS) errors in the equilibrium bond lengths
and term energies computed with EOM-DIP-CCSD/aug-cc-pVTZ are 0.0165
˚
A and 1661 cm
1
.
The errors in bond lengths are only marginally bigger than those of MR-CISD+Q/cc-pVTZ
values (0.0114
˚
A). Remarkably, the errors in energy are consistently smaller than a conservative
estimate of the EOM-CCSD error bars, which is roughly 0.3 eV (2420 cm
1
). The relative state
ordering is also correctly described. MR-CISD+Q/cc-pVTZ yields, on average, smaller errors
in term energies (RMS of 469 cm
1
), however, for three out of nine states, the EOM-DIP-
CCSD/aug-cc-pVTZ values are closer to the experiment.
For a fair comparison, it is important to stress that the EOM-DIP-CCSD ansatz is very com-
pact and includes only 2h and 3h2p configurations, whereas in MR-CISD+Q and AQCC, the
size-extensivity corrections entail contributions of up to quadruply excited configurations. As
with other EOM-CCSD methods, perturbative or explicit inclusion of connected triple excita-
tions is expected to significantly reduce the errors. We note that higher excitations can also
describe orbital relaxation thus mitigating the effect of the unstable dianionic reference.
116
To put the results presented in Table 4.1 in a perspective, it is instructive to compare the
performance of various flavors of multireference methods and to discuss the effects of basis
set increase and higher-order corrections. Szalay et al. carried out
35
extensive comparisons
between MR-CISD, MR-CISD+Q, and MR-AQCC for thirteen states of C
2
. The effects of
higher-order corrections have also been investigated by Jiang and Wilson
32
in the framework of
the correlation-consistent composite approach (MR-ccCA) based on the complete active space
self-consistent field (CASSCF) theory with second-order perturbative corrections (CASPT2).
The size-extensivity correction is significant—the errors of MR-CISD decrease when either
Davidson’s correction or MR-AQCC is employed. Without size-extensivity corrections, the
RMS in the equilibrium bond lengths and term energies computed with MR-CISD/cc-pVTZ are
0.0117
˚
A and 623 cm
1
. The effect of the basis set on the term energies is less systematic
35
. The
RMS error in bond lengths within MR-AQCC/cc-pVTZ is 0.0115
˚
A (to be compared to 0.0114
˚
A of MR-CISD+Q). The errors in term energies were also comparable to MR-CISD+Q/cc-
pVTZ. We note that in the MR-AQCC(TQ) calculations, the largest errors in term energies
were observed for
1
u
ande
3
g
(999 cm
1
and 722 cm
1
). Both MR-AQCC and MR-CISD+Q
calculations were sensitive to the orbital choice and showed improved performance when using
state-averaged CASSCF orbitals. Extrapolation to the complete basis set based on the cc-pVTZ
and cc-pVQZ calculations results in a systematic decrease of equilibrium bond lengths by 0.01
˚
A.
Several studies have also investigated the magnitude of higher-order corrections, with an
aim to achieve spectroscopic accuracy
32, 87
. Schmidt and co-workers showed that the inclusion
of core-valence correlation combined with scalar relativistic corrections in the framework of
MR-CISD+Q brings the spectroscopic constants within 1% from the experimental values
34
.
Jiang and Wilson have reported similar trends
32
.
117
In addition to the states shown in Table 4.1, we also computed two elec-
tronic states,
1
u
and e
3
g
, which have been recently identified experimentally
16–18
.
The electronic configurations of these states are: [core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
1
(
2px
)
1
and
[core]
6
(
2s
)
2
(
2px
)
1
(
2py
)
1
(
2pz
)
1
(
2px
)
1
. Thus, they cannot be generated by the 2h operator
from the dianionic reference, so that the norm of the 3h1p EOM amplitudes becomes large (1).
Consequently, the computed term energies are too high. In order to describe these states with
the same accuracy as the states dominated by 2h configurations, the EOM-DIP ansatz needs to
be extended up to 4h2p operators.
We note that several lowest state of C
2
can also be described by EOM-SF-CCSD using
a high-spin triplet reference, e.g., [core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
1
(
2pz
)
1
. Using ROHF-EOM-SF-
CCSD/aug-cc-pVTZ, vertical excitation energy from
1
+
g
to a
3
u
atr
cc
=1.2425 of C
2
is 319
cm
1
, to be compared with 1924 cm
1
computed by EOM-DIP-CCSD/aug-cc-pVTZ. To quan-
tify the bonding pattern in C
2
, we also computed Head-Gordon’s index
88
, which characterizes
the number of effectively unpaired electrons. For the EOM-SF-CCSD wave function of the
ground state of C
2
at equilibrium,n
u;nl
=0.29. This value indicates that C
2
has substantial dirad-
ical character, comparable
64
to that of singlet methylene (0.25) ormeta-benzyne (0.26). In other
words, there is no support for a quadruple bond, which would be manifested byn
u;nl
0.
4.4.2 C
2
Table 4.2: Equilibrium bond lengths (r
e
,
˚
A) and term energies (T
ee
, cm
1
) of bound elec-
tronic states of C
2
. EOM-IP-CCSD vertical excitation energies (E
ex
, cm
1
) and oscillator
strengths (f
l
) are also shown.
State Configuration EOM-IP-CCSD/aug-cc-pVTZ Expt.
a
r
e
T
ee
E
ex
f
l
r
e
T
ee
2
+
g
[core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
2
(
2pz
)
1
1.260 1.268
2
u
[core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
1
(
2pz
)
2
1.310 3989 4575 0.004 1.308 3986
2
+
u
[core]
6
(
2s
)
1
(
2px
)
2
(
2py
)
2
(
2pz
)
2
1.219 19113 19801 0.085 1.223 18391
a
From Ref. 15.
118
0
20000
40000
60000
1 1.2 1.4 1.6 1.8
Energy (cm
−1
)
Bond length (Å)
2
Σ
g
+
2
Π
u
2
Σ
u
+
Figure 4.4: Potential energy curves of the three lowest states of C
2
.
Fig. 4.4 shows the potential energy curves of the three bound states of C
2
computed using
EOM-IP-CCSD/aug-cc-pVTZ. The respective electronic configurations, equilibrium distances,
and term values are given in Table 4.2. The Dyson orbitals
89
representing the unpaired electrons
in C
2
are shown in Fig. 4.1.
As one can see, the computed equilibrium distances and term energies are in excellent agree-
ment with the experimental data. The computed oscillator strengths show that transitions to both
excited states are optically allowed. The computed T
ee
of the
2
+
u
!
2
+
g
transition is 2.37 eV .
Vertically, at the equilibrium geometry of the
2
+
u
state, the energy gap between two states is
2.29 eV , which is exactly equal to the fluorescence signal observed in Ref. 22. Thus, our results
confirm that fluorescence observed in Ref. 22 can be attributed to the
2
+
u
!
2
+
g
transition of
C
2
.
We also computed AEA of C
2
. Using EOM-DIP-CCSD/aug-cc-pVTZ total energy ofX
1
+
g
and EOM-IP-CCSD/aug-cc-pVTZ total energy of theX
2
+
g
state at the respectiver
e
, the com-
puted value of AEA is 4.57 eV (without zero-point energy), which is more than 1 eV larger than
the experimental value
39, 40
of 3.27 eV and high-level ab initio estimates
41
. This suggests that
the current correlation level is insufficient to describe relative position of the two manifolds.
119
The two relevant states,X
1
+
g
and
2
+
g
, can also be computed using an alternative EOM-CC
scheme, via SF and EA using the high-spin triplet reference, [core]
6
(
2s
)
2
(
2px
)
2
(
2py
)
1
(
2pz
)
1
.
These calculations yield AEA of 3.44 eV when using UHF triplet reference and 3.42 eV eV
when using the ROHF reference. The analysis of the total energies shows that the EOM-EA
energy of the anion is very close to the corresponding EOM-IP energy whereas the EOM-SF
energy of the neutral state is significantly lower than the EOM-DIP energy. We attribute this to
orbital relaxation effects—while the dianionic orbitals are reasonably good for the anion, they
are too diffuse for the neutral and the EOM-DIP ansatz with only 2h and 3h1p operators is not
sufficiently flexible to account for that.
4.4.3 C
2
2
−75.9
−75.8
−75.7
−75.6
−75.5
1 1.2 1.4 1.6 1.8
Energy (hartree)
Bond length (Å)
2
Σ
g
+
2
Π
u
2
Σ
u
+
C
2
2−
(
1
Σ
g
+
)
CAP
C
2
2−
(
1
Σ
g
+
)
Figure 4.5: Potential energy curves of C
2
2
and C
2
. Total electronic energies are shown.
Solid lines show CCSD/aug-cc-pVTZ and EOM-IP-CCSD energies. Orange squares show
the results from CAP-CCSD/aug-cc-pvTZ+3s3p (first-order corrected energy).
Fig. 4.5, which shows potential energy curves of C
2
2
and C
2
, clearly illustrates the
metastable nature of C
2
2
. Adiabatically, C
2
2
is 3.41 eV (at the EOM-IP-CCSD/aug-cc-pVTZ
level) above the ground state of C
2
and can decay into any of the 3 states of the anion. The
120
squared norms of the respective Dyson orbitals
89
computed using the EOM-IP-CCSD/aug-cc-
pVTZ wave functions at the equilibrium bondlength of C
2
2
(1.28
˚
A) are 0.86, 0.80, and 0.86
for the
2
+
g
,
2
u
, and
2
+
u
states, respectively. These values indicate that each of these channels
corresponds to a one-electron detachment process. In the case of the autoionization, the shape
of the Dyson orbital represents the state of the outgoing electron. Thus, the lowest channel
(
2
g
) corresponds to a d-wave whereas the two other channels correspond to p-waves. This
qualitative analysis is supported by the calculations of partial waves usingezDyson.
Table 4.3: First-order corrected energies of C
2
2
at optimal values of the parameter
and the corresponding trajectory velocities (in a.u.) computed with CAP-CCSD/aug-cc-
pVTZ+3s3p.
r
CC
/
˚
A E
Re
opt
j
dE
d
j
1.1 -75.67858 0.02790 0.0176 8:324 10
5
1.2 -75.73037 0.02618 0.0164 2:061 10
5
1.28 -75.74059 0.02506 0.0156 1:250 10
4
1.3 -75.74022 0.02458 0.0148 2:355 10
4
1.4 -75.72646 0.02346 0.0140 2:016 10
4
1.5 -75.70046 0.02302 0.0128 1:104 10
4
1.6 -75.66865 0.02224 0.0120 1:827 10
4
1.7 -75.63523 0.02172 0.0116 1:427 10
4
To characterize lifetimes of the dianion and to quantify the effect of its resonance charac-
ter on the computed quantities of C
2
, we carried out CAP-CCSD and CAP-EOM-IP-CCSD
calculations. The results are summarized in Tables 4.3 and 4.4 and shown in Figs. 4.5 and 4.6.
As one can see from Fig. 4.5, the total energies of C
2
2
obtained from the CAP-augmented
calculations are nearly identical to the real-valued results. Moreover, the impact on the com-
puted term energies of C
2
is also small: atr
CC
=1.28
˚
A, the differences in excitation energies
of C
2
between the two calculations are0.03 eV . The adiabatic energy gap between C
2
2
and
C
2
is 3.16 eV computed with CAP-CCSD/aug-cc-pCVTZ+6s6p6d, only slightly smaller than
the value obtained in real-valued calculations (3.41 eV).
121
Table 4.4: First-order corrected energies of C
2
2
at optimal values of the parameter com-
puted with CAP-CCSD/aug-cc-pCVTZ+6s6p6d and CAP-HF/aug-cc-pCVTZ+6s6p6d.
CAP-CCSD CAP-HF
r
CC
/
˚
A E
Re
opt
E
Re
opt
1.0372 -75.730907 0.014406 0.0030 -75.298280 0.004573 0.0028
1.0901 -75.786473 0.012892 0.0030 -75.353147 0.003828 0.0026
1.1430 -75.821827 0.011584 0.0030 -75.387436 0.003267 0.0024
1.1959 -75.841880 0.010590 0.0028 -75.406232 0.002968 0.0020
1.2489 -75.850425 0.009711 0.0028 -75.413353 0.002924 0.0018
1.2761 -75.851410 0.009350 0.0028 -75.413483 0.003025 0.0018
1.3018 -75.850542 0.009089 0.0026 -75.411820 0.003178 0.0018
1.3547 -75.844523 0.008589 0.0026 -75.403959 0.003601 0.0018
1.4076 -75.834038 0.008303 0.0024 -75.391521 0.004147 0.0018
1.4605 -75.820500 0.007975 0.0024 -75.375802 0.004719 0.0018
1.5134 -75.804992 0.007837 0.0024 -75.357905 0.005367 0.0018
1.5664 -75.788195 0.007646 0.0024 -75.338755 0.005815 0.0020
1.6193 -75.770789 0.007618 0.0024 -75.318637 0.006414 0.0020
1.6722 -75.753150 0.007604 0.0022 -75.298126 0.007005 0.0020
1.7251 -75.735635 0.007578 0.0022 -75.277573 0.007564 0.0020
1.7780 -75.718455 0.007528 0.0022 -75.257580 0.007895 0.0020
1.8310 -75.701660 0.007486 0.0020 -75.237545 0.008336 0.0020
Previous calculations using the charge-stabilization method
44
estimated that the closed-shell
1
+
g
resonance of C
2
2
lies below 4 eV , roughly around 3.4 eV , above the ground state of C
2
.
Later, CAP-augmented MR-CISD calculations
45
yieldedE
res
= 3.52 eV andr
e
=1.285
˚
A. Thus,
our results confirm the findings of these earlier studies
44, 45
.
The resonance position and width are rather sensitive to the basis set employed, as Table 4.3
illustrates. For example, at the equilibrium bond length (r
CC
=1.28
˚
A), the aug-cc-pVTZ+3s3p
basis yields adiabatic E
res
=3.7 eV and =0.68 eV , whereas the aug-cc-pCVTZ+6s6p6d basis
produces E
res
=3.16 eV and =0.25 eV . A distinct stabilization point of the-trajectory is only
obtained using the larger basis set (see Fig. 4.6); in the small basis only first-order corrected
trajectory shows the stabilization point. Our best value for the resonance width (0.25 eV) is in
very good agreement with the CAP-MR-CISD value (0.30 eV)
45
and also agrees qualitatively
with the estimate from charge-stabilization calculations (0.26-0.55 eV)
44
. Compared to singlet
122
resonances with open-shell character, for example, those of CN
that have =0.48-0.56 eV
82
,
C
2
2
is a narrow resonance. On the other hand, C
2
2
resonance is rather broad compared to other
small dianions
70
, such as CO
2
3
or SO
2
4
.
A B
−0.008
−0.006
−0.004
−75.85 −75.84
Im(E)/a.u.
Re(E)/a.u.
without correction
including first−order correction
−0.03
−0.02
−0.01
0
−75.75 −75.74
Im(E)/a.u.
Re(E)/a.u.
without correction
including first−order correction
A B
−0.008
−0.006
−0.004
−75.85 −75.84
Im(E)/a.u.
Re(E)/a.u.
without correction
including first−order correction
−0.03
−0.02
−0.01
0
−75.75 −75.74
Im(E)/a.u.
Re(E)/a.u.
without correction
including first−order correction
Figure 4.6: Uncorrected and first-order corrected CAP-CCSD using aug-cc-pCVTZ+3s3p
(top) and aug-cc-pCVTZ+6s6p6d (bottom) -trajectories for C
2
2
at equilibrium
bondlengths.
We also estimated partial widths corresponding to the three decay channels. Within the
Feshbach formalism, partial widths of autodetachment can be approximated by the following
matrix element
83
:
c
=
2h
!c
j
^
Fj
d
c
i
2
; (4.5)
123
where
c
is the partial width corresponding to detachment channelc,!
c
and
d
c
are the respective
detachment energy and Dyson orbital,
!c
is the wave function of the free electron, and
^
F is the
Fock operator. Given the localized nature of
^
F , this matrix element is bound by the value of the
overlap between the Dyson orbital and the free-electron wave function. Thus, branching ratios
x
p
corresponding to different detachment channels can be estimated as follows:
x
p
=
h
!p
j
d
p
i
2
P
c
h
!c
j
d
c
i
2
; (4.6)
giving rise to
p
= x
p
. Note that the contributions from the degenerate channels (such as
u
) should be multiplied by the respective degeneracy number (2 for -states). The overlap
h
!p
j
d
p
i
2
is proportional to the norm of
d
c
and depends strongly on the energy of the detached
electron and the shape of the Dyson orbital. Fig. 4.7 shows the energy dependence of the com-
puted values of the squared overlap between the normalized Dyson orbitals and the free-electron
wave function approximated by the Coulomb wave. As one can see, the overlap values are zero
at low detachment energies and increase at higher energies. The trends for the
u
and
u
chan-
nels are very similar, which is not surprising given the similar shape of the respective Dyson
orbitals. Fig. 4.7 immediately suggest that the autodetachment process will be dominated by
the channels producing the two lowest states of the anion,
g
and
u
.
Table 4.5: Calculation of partial widths using Coulomb wave and Dyson orbitals from
real-valued and complex-valued EOM-IP-CCSD calculations.
EOM-IP-CCSD/aug-cc-pVTZ CAP-EOM-IP-CCSD/aug-cc-pVTZ+6s6p6d
Channel/DE
a
jj
d
jj
2
Overlap
b
x
p
p
jj
d
jj
2
Overlap
b
x
p
p
2
+
g
/3.41 0.86 1.12 0.69 0.17 0.73 3.61 0.83 0.21
2
u
/2.91 0.86 0.25 0.31 0.08 0.71 0.39 0.17 0.04
2
+
u
/1.04 0.80 2:13 10
4
1:4 10
4
0 0.62 1:6 10
4
0 0
a
Adiabatic EOM-IP-CCSD/aug-cc-pVTZ energies (eV).
b
Overlap (squared) is computed between normalized Dyson orbitals and the Coulomb wave
with charge=-1 and kinetic energy corresponding to adiabatic detachment energy.
124
0
0.5
1
1.5
0 1 2 3 4
0
1
2
3
4
5
Overlap
Overlap
Energy (eV)
2
Σ
g
+
2
Π
u
2
Σ
u
+
2
Σ
g
+
2
Π
u
2
Σ
u
+
Figure 4.7: Squared overlap between Dyson orbitals and a Coulomb wave with charge=-
1. Solid lines correspond to Dyson orbitals from EOM-IP-CCSD/aug-cc-pVTZ (scale
on the left). Dashed lines correspond to Dyson orbitals (real part) from CAP-EOM-IP-
CCSD/aug-cc-pVTZ+6s6p6d (scale on the right).
Table 4.5 lists the computed values using E
res
=3.41 eV (from EOM-IP-CCSD/aug-cc-
pVTZ). As one can see, the contribution of the
u
is negligible and the
g
channel is dominant.
When using lower energy value (3.16 eV , from CAP-EOM-IP-CCSD/aug-cc-pCVTZ+6s6p6d),
the contribution from the
u
channels drops even further while the ratio between the
g
and
u
channels remains unchanged. Using Dyson orbitals from the CAP-EOM-IP-CCSD/aug-
cc-pCVTZ+6s6p6d calculations leads to the increase of the relative weight of the
g
channel.
These simple estimates are in qualitative agreement with partial widths computed using
CAP-MR-CISD wave function and an approach based on the CAP projection
45
; their reported
values correspond tox
p
of 0.31, 0.66, and 0.02 for the
g
,
u
, and
u
channels. One important
difference is that our calculations predict that the dominant decay channel is
g
, producing the
ground-state of C
2
. We note that using plane wave to describe the state of the free electron
yields an entirely different picture: the overlaps are rather large around the threshold and
change much slower, resulting in comparable branching ratios for all three channels.
125
0
0.1
0.2
0.3
0.4
1.2 1.4 1.6 1.8
Resonance width (eV)
Bond length (Å)
CAP−CCSD
CAP−HF
Figure 4.8: First-order corrected resonance width of C
2
2
as a function of bond length
computed with CAP-CCSD and CAP-HF and the aug-cc-pCVTZ+6s6p6d basis set.
Finally, we investigate the dependence of the resonance width on the bond length. As illus-
trated by Figure 4.8, the CAP-CCSD resonance width shrinks with an increasing bond length
near the equilibrium distance while it is nearly constant beyond 1.6
˚
A. This is consistent with
the potential energy curves of C
2
2
and the
2
+
g
and
2
u
states of C
2
becoming nearly parallel
at elongated bond distances (see Figure 4.5). However, the behavior is different from that of
valence shape resonances in diatomic molecules (for example, H
2
or N
2
) that become bound
when the bond is stretched somewhat.
56
It is more reminiscent of dipole-stabilized resonances
whose width is also rather insensitive to the changes of the bond lengths.
81
Figure 4.8 also shows
that the resonance width behaves differently at the CAP-CCSD and CAP-HF levels. Within the
HF approximation, has a minimum around the equilibrium structure (0.08 eV) and grows
when the bond is stretched. This behavior is similar to the results reported in Ref. 45 where
CAP-CIS and CAP-MR-CISD also yielded increasing with bond length between 1.2 and 1.4
˚
A. A detailed investigation of these differences is beyond the scope of the present work, but we
note that the resonance width of C
2
2
has to vanish eventually, when the bond is stretched far
enough, because the
4
S ground state of C
obtained in the dissociation limit is stable towards
electron detachment.
126
The description of the decay channels reveals a shortcoming of the CAP-CCSD approach
based on a metastable reference. The CAP-EOM-IP-CCSD energies of the three bound states
of C
2
feature sizable positive imaginary parts of more than 0.3 eV (at the equilibrium bond
length and optimal values for the dianionic resonance). This is despite that real parts of abso-
lute CAP-EOM-IP-CCSD energies agree with the CAP-free values within0.1 eV . Also, it is
in stark contrast to the performance of CAP-EOM-CCSD based on bound reference states
54, 55
,
where the imaginary energies of bound states typically stay below 0.03 eV . We note that applica-
tion of the de-perturbative correction
54, 55
does not rectify this problem. This is not surprising as
the original analysis ofE() in terms of perturbation theory
76
was designed for resonances but
not bound states. Furthermore, the imaginary energies of the three bound states of C
2
differ by
more than a factor of two so that a single, not state-specific, correction is not realistic. However,
since a positive imaginary energy is unphysical and since no stabilization of the -trajectory
is observed for the CAP-EOM-IP-CCSD states, the problem is easily discernible. Importantly,
despite this shortcoming, CAP-EOM-CCSD calculations using an unstable reference clearly
distinguish bound and metastable states.
Experimentally
42, 43
, the C
2
2
resonance manifest itself as a broad feature around 10 eV in
electron scattering detachment spectra from C
2
, however, the interpretation of these spectra in
terms of the position of the resonance is not straightforward, as explained by Sommerfeld and
co-workers
44
. We hope that our results will stimulate further experimental efforts to characterize
electronic structure of C
2
2
.
4.5 Conclusion
We reported electronic structure calculations of C
2
, C
2
, and C
2
2
using the CC/EOM-CC
family of methods. Our results illustrate that EOM-CCSD provides an attractive alternative to
127
MR approaches. The low-lying states of C
2
and C
2
are well described by EOM-DIP-CCSD and
EOM-IP-CCSD using the dianionic closed-shell reference (C
2
2
), despite its metastable nature.
EOM-DIP-CCSD offers a much simpler computational approach based on a single-
reference formalism. In the EOM-DIP calculations, no active-space selection is required, and
the results of the calculations do not depend on the number of states computed, in contrast to
state-averaged MR schemes. One does not need to guess the electronic configurations of the
states to be computed—once the user specifies how many states in each irrep are desired, the
algorithm computes them.
The electronic structure of C
2
2
was characterized by CAP-augmented CCSD. The calcu-
lations placed the closed-shell C
2
2
resonance 3.16 eV adiabatically above the ground state of
C
2
. The computed resonance width is 0.25 eV , corresponding to a lifetime of 2.6 fs. The C
2
2
resonance can, in principle, decay into three open channels, producing the ground (X
2
g
) or
an excited (
2
u
or
2
u
) state of C
2
. Our calculations of the partial widths suggest that the
dominant decay channel (70-80%) corresponds to the ground state of the anion while the
2
u
channel is essentially blocked. The analysis of the respective Dyson orbitals reveals that the
main effect controlling the branching ratios in this system is the energy of the outgoing elec-
trons. Importantly, the CAP-augmented calculations yield detachment energies that are very
close to the real-valued EOM-CCSD calculations with the aug-cc-pVTZ basis set, thus con-
firming the validity of the results obtained with EOM-DIP-CCSD and EOM-IP-CCSD using
the dianion reference.
128
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Chapter 5: Dissociative Electron
Attachment in C
2
H
Investigation of microwave activated CH
4
/H
2
plasma in CVD revealed the formation of
electronically excited C
2
(B
2
+
u
). Here we investigate electronic structure of C
2
H
and for-
mation of C
2
in the ground (X
2
+
g
) and excited (B
2
+
u
) states via electronic resonances. We
employ equation-of-motion coupled-cluster methods augmented by complex absorbing poten-
tial to study the electronic resonances of C
2
H
. The resonance wavefunctions are analyzed
using natural transition orbitals. We identified several resonances, including the state that can
lead to bond dissociation and form C
2
(B
2
+
u
).
5.1 Introduction
Resonances
1–5
are electronic states that have finite lifetime and decay by ejecting an elec-
tron. Hence, they are located above the electron detachment (or ionization) threshold and belong
to the continuum part of the spectra. Resonances play an important role in plasma physics,
atmospheric chemistry, fusion reactors, interstellar medium, and in other high-energy environ-
ments. These states can be produced by electronic excitation, ionization, or electron attachment.
Dissociative electron attachment (DEA) is the process in which an electron attaches to a
molecule and causes its fragmentation. DEA processes play a crucial role in plasma physics.
134
A
3
Σ
+
(→
*
)
D
3
Σ
-
(→
*
)
E
1
Σ
-
(→
*
)
B
3
Π (
3
→
*
)
C
3
Δ (→
*
)
F
1
Δ (→
*
)
a
3
Σ
+
(
3
→
*
3
)
b
3
Π (→
*
3
)
X
2
Σ
+
g
A
2
Π
u .
(→
3
)
B
2
Σ
+
u
(
*
2
→
3
)
A
3
Σ
+
(
*
2
→
*
3
),(→
*
)
D
3
Σ
-
(→
*
)
E
1
Σ
-
(→
*
)
B
3
Π (
3
→
*
)
C
3
Δ (→
*
)
F
1
Δ (→
*
)
DE
X
1
Σ
+
X
1
Σ
+
DE
*
*
3
*
3
*
2
*
*
*
3
3
*
2
*
2
3
C C H
1.662 1.204
1.062 1.204
C C H
1.219
C C
Figure 5.1: Left panel: electronic states at C
2
H
at the ground state equilibrium geometry
of the of anion. Middle: electronic states of C
2
H
at elongated C-H bond geometry. Right
panel: electronic states of C
2
at equilibrium geometry of C
2
(X
2
+
g
).
It was shown that chemical vapor deposition (CVD) of diamond leads to the formation of elec-
tronically excited C
2
(B
2
+
u
) in a microwave activated plasma. The presence of excited C
2
was detected by its fluorescence. The proposed pathway invoked DEA:
C
2
H +e! C
2
+ H (5.1)
However, no explanation was given why electronically excited C
2
is formed. The goal of
our study is to clarify this issue. We report the calculations of bound and metastable states of
C
2
H
. We employ the equation-of-motion coupled-cluster (EOM-CC)
6–8
method augmented
with complex absorbing potential (CAP)
9
to describe the metastable states. EOM-CCSD
6–8, 10, 11
can treat closed-shell and open-shell species on an equal footing, which allows us to compare
the energy levels of the anion and the neutral. We also analyze the DEA pathway in C
2
H. The
135
main finding is that at the neutral ground-state geometry of C
2
H
, there are several resonances
of the!
type, but none can lead to the formation of excited C
2
(B
2
+
u
). The resonance
that can do both appears only when the C-H bond is elongated.
Fig.5.1 shows an overview of the low-lying electronic states at three different geometries.
On the left panel, the electronic states of the C
2
H
anion at the neutral geometry are shown.
C
2
H
is isolectronic to CN
and their low-lying states have similar character
12
. Even the order
of resonance states in C
2
H
is the same as in CN
. The middle panel shows the electronic states
of C
2
H
anion with a stretched C-H bond. The ground state of the stretched C
2
H
is higher than
the ground state of C
2
H
, which shifts all states to higher energy. Two extra features appear
at this geometry. First, two new bound states are formed. Second, the character of the A
3
+
resonance state changes: it develops a contribution from an orbital pair that has dissociative
character for the C-H bond, as in the excited C
2
(B
2
+
u
). The right panel of Fig.5.1 shows the
ground and excited states of C
2
relative to the ground state of C
2
H
anion (to compare energy
of C
2
with C
2
H
, the energy of hydrogen atom, -0.5 hartree, was added to the energy of C
2
).
The comparison of states allows us to retrace the path of formation for the ground-state of C
2
to the bound excited state of C
2
H
(a
3
+
). Similarly, the path of formation of the excited state
of C
2
(
2
+
u
) can be traced back to the resonance state of C
2
H
(A
3
+
).
5.2 Theoretical methods and computational details
EOM-CC framework enables treating diverse types of electronic structure
6–8, 11, 13–19
. It pro-
duces size-intensive transition energies for electronic excitation, ionization, or attachment. The
accuracy of EOM-CC is also systematically improvable up to the exact solution. The major
advantage of EOM-CC is, treating metastable and near-threshold bound states is the balanced
description of the target states because it allows to employ the same model Hamiltonian. We
136
investigated the electronic spectrum of C
2
H
below and above the electron detachment con-
tinuum onset by means of EOM-CCSD. In all calculations, the reference wave function cor-
responds to the closed-shell ground-state CCSD solution for the respective anion (N-electron
system). We determined the electron-detachment thresholds and properties of the parent neutral
radicals by the EOM-IP variant of the method in which target N-1-electron states are described
by the EOM operators that remove one electron (i.e., 1h and 2h1p). To study bound excited
states we employed the EOM-EE variant in which the target N-electron states are described
by the EOM operators that excite one or two electrons (i.e., 1h1p and 2h2p). To describe N-
electron metastable states embedded in the electron-detachment continuum, we augment the
standard EOM Hamiltonian by CAP (CAP-EOM-EE-CCSD).Fig. 5.2 illustrates the target-state
manifolds accessed by EOM-EE and EOM-IP. In both calculations the same closed-shell refer-
EOM-IP EOM-EE
Figure 5.2: Manifolds of target states accessed by EOM-IP and EOM-EE. In both cases,
the same closed-shell reference is used.
ence state is used. The onset of the continuum in EOM-EE exactly corresponds to the electron
detachment energy obtained from EOM-IP, which allows one to unambiguously distinguish
between the bound and continuum states.
In the CAP
9, 20–23
method the standard Hamiltonian
^
H
0
is augmented with an imaginary
potentiali
^
W .
^
H() =
^
H
0
i
^
W (r); (5.2)
where represents the strength of the CAP. The imaginary potential absorbs the diverging
tail of a metastable state making itL
2
-integrable. The eigen-energies of the CAP-augmented
137
Hamiltonian are complex. The real part is the position of the metastable state and the complex
part is the resonance width (which is inversely proportional to the lifetime of the resonance). In
our calculations, the CAP potential is quadratic with cuboid shape
^
W (r) =
^
W
x
(r
x
) +
^
W
y
(r
y
) +
^
W
y
(r
y
); (5.3)
with
^
W
(r
) =
8
>
>
<
>
>
:
0 ifjr
jr
0
(jr
jr
0
)
2
otherwise;
(5.4)
where the coordinates (r
0
x
;r
0
y
;r
0
z
) define the onset of the CAP in each dimension. Following the
same strategy as in previous calculations, we fixed the CAP onset at the spatial extent of the
wave function for the reference state,R =
p
h
CCSD
jRj
CCSD
i, where
CCSD
is the CCSD
solution (CAP-free) for the ground state of the anion. The optimal value of CAP’s strength
parameter is determined for each metastable state by calculating-trajectories and searching
for the minimum of the function:
j:
dE()
d
j =min; (5.5)
This procedure minimizes the error introduced by the incompleteness of the one-electron basis
set and the finite strength of CAP. More accurate resonance parameters can be obtained if,
instead of considering the raw -trajectories E(), one analyzes the -trajectories of deper-
turbed energiesU() from which the explicit dependence of CAP is removed in the first order
20
.
The deperturbed complex energiesU() are calculated by subtracting from raw energiesE()
the correctioniT [
W ], where
is the one-particle density matrix of the resonance state.
To analyze the transition properties between the EOM-CCSD states, we employ reduced
quantities such as one-particle transition density matrices. These objects are related to concrete
138
physical observables
24
and provide a way to derive a molecular orbital picture
24
of the many-
body wave functions. This is achived by using natural transition orbitals (NTOs). The one-
particle transition density matrix
25–30
(1PTDM) between two many-body wave functions
I
and
J
is
IJ
pq
=h
I
jp
y
qj
J
i: (5.6)
1PTDM can be used to compute one-electron transition properties such as oscillator strengths,
transition dipole moments, non-adiabatic
31
and mean-field spin-orbit couplings
32, 33
. The norm
of
IJ
(defined as
q
P
pq
(
IJ
pq
)
2
) provides a measure of one-electron character of the transition.
The 1PTDM provides a compact representation of the difference between the two states in terms
of hole-particle excitations. Singular-value decomposition of
IJ
yields a set of left and right
eigenvectors
IJ
=UV
T
; (5.7)
which define hole (
h
K
) and particle (
e
K
) orbitals corresponding to singular value
K
(elements
of the diagonal matrix )
h
K
=
X
q
U
qK
q
; (5.8)
e
K
=
X
q
V
qK
q
: (5.9)
Such pairs of hole and particle orbitals are called NTOs. The electronic resonances have com-
plex energy and complex densities whose real and imaginary component are
34
Re
IJ
pq
=h
Re
I
jp
y
qj
Re
J
ih
Im
I
jp
y
qj
Im
J
i; (5.10)
Im
IJ
pq
=h
Im
I
jp
y
qj
Re
J
i +h
Re
I
jp
y
qj
Im
J
i: (5.11)
139
The analysis of real part 1PTDM provides information about the orbitals involved in the tran-
sition, similar to bound state analysis. The imaginary part analysis provides insight into the
coupling of the resonance with the continuum and its decay channels
34
.
5.3 Computational details
All calculations were performed using the Q-Chem electronic structure program
35, 36
. The
geometry optimization of C
2
H and its anion was performed with EOM-IP-CCSD and CCSD,
respectively using the aug-cc-pVTZ basis set. Electron detachment for C
2
H
at the neutral
and anionic geometries were computed using EOM-IP-CCSD. The aug-cc-pVTZ basis set was
augmented by the 3s3p1d diffuse functions on carbon and 3s3p diffuse functions on hydrogen.
The exponents of the additional diffuse functions were generated in an even tempered manner
(with a spacing of 2.0), starting with the most diffuse function in the original basis set.
We computed electronic resonances for C
2
H
with CAP-EOM-EE-CCSD/aug-cc-
pVTZ+3s3p1d(3s3p) at the neutral geometry, stretched C
2
H
, and bend C
2
H
geometry.
Ground and excited states of C
2
were computed using EOM-IP-CCSD/aug-cc-pVTZ with dian-
ionic
37
reference (C
2
2
), as described in detail in Ref 37. Core orbitals were frozen in all calcu-
lations. All calculations used pure angular momentum polarization functions.
140
5.4 Results and discussion
5.4.1 Electronic structure of C
2
We begin by reviewing electronic states of C
2
. To describe this open-shell molecule, we
employ EOM-IP-CCSD with the dianion (C
2
2
) closed-shell reference. The dianionic reference
is stable
37
and have been used to describe the electronic structure of C
2
, C
2
, and C
2
2
. Table 5.1
shows the vertical excitation energy to A
2
u
and B
2
+
u
obtained using EOM-IP-CCSD/aug-cc-
pVTZ at the equilibrium geometry of C
2
(X
2
+
g
). The energy gap between X
2
+
g
and B
2
+
u
is
2.45 eV , which is in perfect agreement to the fluorescence signal observed in the CVD exper-
iment.
38
We also characterize the NTOs for excitation from the ground state of C
2
(X
2
+
u
) to
Table 5.1: Excitation energies (eV) and oscillator strength for C
2
at R
CC
= 1.219
˚
A using
EOM-IP-CCSD/aug-cc-pVTZ with the dianionic reference (C
2
).
State EE f
A
2
u
0.57 0.01
B
2
+
u
2.45 0.09
the A
2
u
and B
2
+
u
excited states. NTOs are shown in Fig. 5.3.
NTO
2
=0.92
2
=0.89
A
2
Π
u .
B
2
Σ
+
u
Figure 5.3: Natural transition orbitals for X
2
+
u
!A
2
u
and X
2
+
u
!B
2
+
u
in C
2
at
equilibrium geometry of C
2
using EOM-IP-CCSD/aug-cc-pVTZ with dianionic reference.
The experiment with microwave activated plasma of CVD reported the spectroscopic signal
at 541 nm (2.28 eV), which corresponds to emission from B
2
+
u
to X
2
+
u
. To form the excited
141
state (B
2
+
u
) of C
2
from an electronic resonance of C
2
H
, one should find a resonance that
has similar NTO structure. This means that the hole of NTO pair should have a node in the C-C
bond and the particle of the same NTO should not have node in the C-C bond. Because we want
hydrogen to dissociate from carbon, there should be a node in the C-H bond. Before looking
into electronic resonances and their NTOs we discuss the electronic structure of C
2
H
in next
section.
5.4.2 Electronic structure of C
2
H
The C
2
H
anion is a closed-shell species and isolectronic to CN
. Figure 5.4 shows cal-
culated ionization energies of the anion at different geometries. The continuum begins at the
X
1
Σ
+
X
1
Σ
+
2
Σ
+
2
Σ
+
2
Π
2
Π
1.068 1.246 1.062 1.204
C C C C H H
3.02
3.73
3.19
3.61
Anion Neutral
X
1
Σ
+
2
Σ
+
2
Π
1.668 1.246 1.662 1.204
C C H
3.28
3.606
Stretched neutral
1
X
2
X
2
A
3.27
3.85
3.87
2
B
1
Σ
+
1
Σ
+
2
Σ
+
2
Σ
+
2
Π
2
Π
1.068 1.246 1.062 1.204
C C C C H H
3.021
3.732
3.187
3.606
3.102
Anion Neutral
1
Σ
+
1
Σ
+
2
Σ
+
2
Σ
+
2
Π
2
Π
1.068 1.246 1.062 1.204
C C C C H H
3.021
3.732
3.187
3.606
3.102
Anion Neutral
Stretched and bend neutral
xi
TABLE VII: Vertical detachment energies (VDE) of the anions (in eV) at stretched and
bend geometry of neutral using EOM-IP-CCSD/aug-cc-pVTZ+3s3p1d(3s3p)
State VDE
2
X(A
0
)3.27
2
A(A
0
)3.87
2
B(A
00
)3.85
(a)
1
X(A
0
)!
2
X(A
0
)(b)
1
X(A
0
)!
2
A(A
0
)(c)
1
X(A
0
)!
2
B(A
00
)
FIG. 5: Dyson orbitals for stretched and bend neutral geometry of C2H
(C-H = 1.662
\ CCH = 170
o
)
TABLE VIII: Resonance position ER (eV) and widths (eV) at stretched and bend
neutral geometry of C2H
;CAP-EOM-EE-CCSD/aug-cc-pVTZ+3s3p1d(3s3p).
State E
(0)
R
(0)
E
(1)
R
(1)
a
3
A
0
(bound) 3.00 - - -
b
3
A
0
(bound) 3.14 - - -
c
3
A
00
(bound) 3.12 - - -
A
3
A
0
5.25 0.29 5.23 0.27
B
3
A
0
5.85 0.70 5.79 0.60
C
3
A
00
5.92 0.73 5.81 0.57
D
3
A
00
6.25 0.74 6.17 0.59
E
3
A
00
6.60 1.09 6.49 1.03
xi
TABLE VII: Vertical detachment energies (VDE) of the anions (in eV) at stretched and
bend geometry of neutral using EOM-IP-CCSD/aug-cc-pVTZ+3s3p1d(3s3p)
State VDE
2
X(A
0
)3.27
2
A(A
0
)3.87
2
B(A
00
)3.85
(a)
1
X(A
0
)!
2
X(A
0
)(b)
1
X(A
0
)!
2
A(A
0
)(c)
1
X(A
0
)!
2
B(A
00
)
FIG. 5: Dyson orbitals for stretched and bend neutral geometry of C2H
(C-H = 1.662
\ CCH = 170
o
)
TABLE VIII: Resonance position ER (eV) and widths (eV) at stretched and bend
neutral geometry of C2H
;CAP-EOM-EE-CCSD/aug-cc-pVTZ+3s3p1d(3s3p).
State E
(0)
R (0)
E
(1)
R (1)
a
3
A
0
(bound) 3.00 - - -
b
3
A
0
(bound) 3.14 - - -
c
3
A
00
(bound) 3.12 - - -
A
3
A
0
5.25 0.29 5.23 0.27
B
3
A
0
5.85 0.70 5.79 0.60
C
3
A
00
5.92 0.73 5.81 0.57
D
3
A
00
6.25 0.74 6.17 0.59
E
3
A
00
6.60 1.09 6.49 1.03
xi
TABLE VII: Vertical detachment energies (VDE) of the anions (in eV) at stretched and
bend geometry of neutral using EOM-IP-CCSD/aug-cc-pVTZ+3s3p1d(3s3p)
State VDE
2
X(A
0
)3.27
2
A(A
0
)3.87
2
B(A
00
)3.85
(a)
1
X(A
0
)!
2
X(A
0
)(b)
1
X(A
0
)!
2
A(A
0
)(c)
1
X(A
0
)!
2
B(A
00
)
FIG. 5: Dyson orbitals for stretched and bend neutral geometry of C2H
(C-H = 1.662
\ CCH = 170
o
)
TABLE VIII: Resonance position ER (eV) and widths (eV) at stretched and bend
neutral geometry of C2H
;CAP-EOM-EE-CCSD/aug-cc-pVTZ+3s3p1d(3s3p).
State E
(0)
R
(0)
E
(1)
R
(1)
a
3
A
0
(bound) 3.00 - - -
b
3
A
0
(bound) 3.14 - - -
c
3
A
00
(bound) 3.12 - - -
A
3
A
0
5.25 0.29 5.23 0.27
B
3
A
0
5.85 0.70 5.79 0.60
C
3
A
00
5.92 0.73 5.81 0.57
D
3
A
00
6.25 0.74 6.17 0.59
E
3
A
00
6.60 1.09 6.49 1.03
C C
H
Figure 5.4: Ionization energy and the respective Dyson orbitals for the C
2
H anion at dif-
ferent geometries.
detachment threshold and electronic states above the threshold are the resonances. Resonances
were computed with CAP-EOM-EE-CCSD at equilibrium geometry of the neutral. Table 5.2
list all resonances we found. The respective NTOs are shown in Table 5.3. There are multiple
142
Table 5.2: Resonance positions E
R
(eV) and widths (eV) at equilibrium geometry of
neutral C
2
H
using CAP-EOM-EE-CCSD/aug-cc-pVTZ+3s3p1d(3s3p).
State E
(0)
R
(0)
E
(1)
R
(1)
A
3
+
5.54 0.44 5.59 0.50
B
3
5.74 0.66 5.80 0.71
C
3
6.17 0.61 6.25 0.66
D
3
6.61 0.87 6.66 0.82
E
1
6.64 0.90 6.69 0.84
F
1
6.73 1.11 6.77 1.02
resonances, but there is no state whose hole of NTO pair has a node in the C-C bond and there is
no resonance in which particle NTO has a node in the C-H bond. Hence, there is no resonance
that can form excited C
2
(B
2
+
u
) and dissociate C-H bond.
Table 5.3: NTOs (real part) at equilibrium geometry of neutral (R
CC
= 1.204
˚
A, and R
CH
= 1.062
˚
A.)
State Type SVD(
2
) Hole Particle
A
3
+
!
0.454
0.454
B
3
!
0.894
C
3
(F
1
) !
0.451(0.498)
0.451(0.498)
D
3
(E
1
) !
0.468(0.473)
0.468(0.473)
There is no resonance corresponding to the dissociation of C-H bond and formation of
C
2
(B
2
+
u
) at the equilibrium geometry of neutral. To identify a resonance with a repul-
sive potential with respect to the C-H bond, we carried out CAP-EOM-EE-CCSD/aug-cc-
pVTZ+3s3p1d(3s3p) calculations at a geometry with the C-H bond stretched by 0.6
˚
A. At
143
R
CH
= 1:66
˚
A, states appeared corresponding to excitations in a
type orbital. We observe
two bound states, a
3
+
and b
3
, derived by excitation from and occupied orbitals to a
-
type orbital. Table 5.4 shows the resonance positions and widths for all computed resonances.
NTO analysis are summarized in Table 5.5. The A
3
+
resonance starts to show a contribution
Table 5.4: Resonance positions E
R
(eV) and widths (eV) at a stretched geometry (R
CC
=
1.204
˚
A, and R
CH
= 1.662
˚
A.); CAP-EOM-EE-CCSD/aug-cc-pVTZ+3s3p1d(3s3p).
State E
(0)
R
(0)
E
(1)
R
(1)
a
3
+
(bound) 3.06 - - -
b
3
(bound) 3.14 - - -
A
3
+
5.22 0.29 5.25 0.31
B
3
5.92 0.73 5.81 0.57
C
3
6.22 0.68 6.13 0.55
D
3
6.60 1.00 6.47 0.93
E
1
6.61 1.09 6.48 1.00
F
1
6.65 1.24 6.49 1.16
of an additional NTO pair, which has a node in the CC bond for hole, a node in the CH bond
for particle and, no node in the CC bond for the particle in the NTO pair.
Table 5.5: NTOs (real part) at stretched geometry (R
CC
= 1.204
˚
A, and R
CH
= 1.662
˚
A.)
State Type SVD(
2
) Hole Particle
a
3
+
(bound) !
0.807
b
3
(bound) !
0.787
A
3
+
!
0.380
0.261
0.261
To test the effect of bending vibration on the resonance, we carried out calculations at the
structure with a CCH bend at a stretched CH bond. The right panel in Fig. 5.4 shows the vertical
detachment energies and the corresponding Dyson orbitals at the bend and stretched geometry.
144
Table 5.6 shows the resonance positions and its widths and the respective NTOs are shown
in Table 5.7. Due to symmetry lowering, the states are split. Table 5.7 also compare each
resonance with resonances obtained from just stretched geometry.
Table 5.6: Resonance positions E
R
(eV) and widths (eV) at a stretched and bend geom-
etry (R
CC
= 1.204
˚
A, R
CH
= 1.662
˚
A, and\CCH = 170
o
); CAP-EOM-EE-CCSD/aug-cc-
pVTZ+3s3p1d(3s3p).
State E
(0)
R
(0)
E
(1)
R
(1)
a
3
A
0
(bound) 3.00 - - -
b
3
A
0
(bound) 3.14 - - -
c
3
A
00
(bound) 3.12 - - -
A
3
A
0
5.25 0.29 5.23 0.27
B
3
A
0
5.85 0.70 5.79 0.60
C
3
A
00
5.92 0.73 5.81 0.57
D
3
A
00
6.25 0.74 6.17 0.59
E
3
A
00
6.60 1.09 6.49 1.03
The A
3
+
resonance has all nodes features that can lead to a formation of excited C
2
(B
2
+
u
). We also observe that one of the bound excited state of C
2
H
(a
1
+
) can form the
ground state of C
2
(X
2
+
g
). Fig 5.1 shows the same connection. We note that the experi-
ment was carried out in high energy plasma, which was microwave activated. These conditions
suggest the molecule to dissociate via DEA mechanism discussed here. High energetic environ-
ment allows C
2
H radical to capture the highly energetic resonance state. Microwave activation
allows C
2
H to have access to longer bond lengths.
145
Table 5.7: NTOs (real part) at stretched and bend geometry (R
CC
= 1.204
˚
A, R
CH
= 1.662
˚
A, and\CCH = 170
o
).
State Type SVD(
2
) Hole Particle Similar
a
3
A
0
(bound) A
0
!A
0
0.790 a
3
+
3
A
0
(bound) A
0
!A
0
0.785 b
3
3
A
00
(bound) A
00
!A
0
0.789 b
3
3
A
0
A
0
!A
0
0.421 A
3
+
A
0
!A
0
0.209
A
00
!A
00
0.240
3
A
0
A
0
!A
0
0.714 B
3
A
0
!A
0
0.162
3
A
00
A
0
!A
00
0.876 B
3
3
A
00
A
0
!A
00
0.464 C
3
A
0
!A
00
0.444
3
A
00
A
0
!A
00
0.461 D
3
A
0
!A
00
0.493
a
Correlation between states at bend and stretched geometry with states obtained at linear
geometry.
146
5.5 Chapter 5 References
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23
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24
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149
Chapter 6: Future directions
Previous chapters presented the extension of EOM to double attachment (DEA) and illus-
trated the power of other EOM variants (EE, EA, IP, DIP) to treat the electronic structure of
open-shell species. However, there are many more features that one has to implement for both
DEA and DIP methods to exploit their full potential fully. This chapter presents future direc-
tions for method development, aiming to expand the scope of applications for DEA and DIP
methods.
6.1 Spin-orbit coupling for EOM-DEA and EOM-DIP
Relativistic effects such as spin-orbit couplings (SOC) play a crucial role in chemistry and
spectroscopy. Small in magnitude, the SOC facilitates the mixing of non-interacting states and
opens new reaction channels. For example, the reaction of O
2
(
3
g
) with unsaturated carbon
happens via triplet diradical formation
1–4
. SOC is crucial for open-shell species because it
allows degenerate electronic states to mix and brings changes in energetics and spectra. Because
SOC contributes to errors in bond dissociation energies by changing the energetics of open-shell
fragments, SOC is also essential for obtaining high-accuracy thermochemical data.
The SOC originates in the coupling of the angular momentum of an electron with its intrinsic
magnetic moment. For light atoms, this relativistic effect can be incorporated by using the Breit-
Pauli (BP) Hamiltonian
5, 6
. There are two ways in which spin-orbit effects can be incorporated.
The first one is fully variational: the spin-orbit part is included in the electronic Hamiltonian
and the wavefunctions are computed in the presence of SOC. In an other approximation, the
150
wavefunction are computed without BP Hamiltonian and then BP matrix elements are evalu-
ated for each state. Diagonalizing this perturbated Hamiltonian yields the wavefunctions and
energies including SOC. InQ-Chem, SOC can be computed for states obtained from the EOM-
EE- CCSD, EOM-IP-CCSD, EOM-EA-CCSD, EOM-SF-CCSD methods. To extend this to
EOM-DEA-CCSD and EOM-DIP-CCSD, one has to derive, implement, and test the equations.
In order to compute SOC, we need two-particle density matrix. For EOM-DIP- CCSD, this has
been done and implemented in Q-Chem. I am currently working on implementing parts of the
two-particle density matrix for EOM-DEA-CCSD.
6.2 Analytic gradients for EOM-DEA and EOM-DIP
Computing global potential energy surfaces is only feasible for small molecules. Comput-
ing spectroscopic properties of the large system requires finding stationary points on the surface
and, hence, requires gradients. One can compute the gradients numerically by just evaluating
energy and using a finite difference procedure. For the numerical evaluation of the gradient for
a system ofN degrees of freedom requires 2N single-point evaluations. The finite-difference
method often encounters numerical issues, like poor convergence, numerical noise, etc. In
contrast, a calculation using analytic gradients is free of numerical instabilities and can be per-
formed approximately at a single-point energy calculation cost. This provides significant time
savings for larger systems.
At present, in theQ-Chem software, we have only code for energy for the EOM-DEA-CCSD
and EOM-DIP-CCSD methods. In order to locate stationary points and perform normal mode
analysis, one has to derive and implement analytic gradients and Hessian for both methods.
151
6.3 CVS-EOM-DIP-CCSD
In recent years, double core hole (DCH) spectroscopy has been explored at synchrotrons and
free-electron lasers
7–9
When two core vacancies are created in a molecule, the chemical shift
is sensitive to both chemical environment and the molecular structure. There is an urgent need
for a theoretical tool to understand these experiments. One strategy to simulate core ionized
electronic states is to employ the core-valence separation (CVS) scheme proposed by Ceder-
baum et al. in 1980
10
. In a recent article by Lee et al.
11
, the authors simulate the energetics of
DCH states by employing CVS-CCSD and CVS-CCSD(T) methods. One can also compute
the energetics of DCH states using the CVS extension of EOM-DIP-CCSD. This will require
deriving the EOM-DIP-CCSD equations within the CVS scheme and then implementation in
theQ-Chem software to test it.
152
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Abstract (if available)
Abstract
This thesis focuses on the electronic structure and spectroscopy of open-shell and electronically excited species, which are crucial for understanding chemistry and difficult to simulate. This thesis provides several examples of challenging open-shell systems and discusses their roles in chemistry. To study open-shell species, one needs reliable methods to describe their physics accurately. The equation-of-motion coupled-cluster (EOM-CC) approach is a powerful tool for accurate calculations of electronically excited and open-shell molecules. This work shows how different variants of EOM-CC treat electronic structure of various open-shell species. ❧ Chapter 1 introduces open-shell species and explains their importance in chemistry. It defines different types of anions and explains the complexities in the simulation of their electronic structure. Chapter 1 also contains a brief introduction to the EOM-CC theory and its extensions to electronic resonances. Chapter 2 provides derivations and benchmark results for the double electron attachment (DEA) variant of EOM-CC. This chapter describes the implementation and benchmarking of the method in the Q-Chem electronic structure package. The EOM-DEA-CC method can treat diradicals, doubly excited states, some types of conical intersection and bond breaking. ❧ Chapter 3 reports our study of the electronic structure of the benzonitrile anion. It compares experimental electron attachment energies with calculations using the electron attachment (EA) variant of EOM-CC. In this study, we were able to resolve the long-standing controversy about the nature of the lowest bound anion state of benzonitrile. The dipole-bound and valence-bound anion states are close in energy, making it difficult to resolve the character of the ground state. The experiment suggested the formation of the valence-bound anion. EOM-EA-CCSD calculations with the inclusion of zero-point energy corrections supported the conclusion drawn from the experimental measurements. Chapter 4 shows the application of two variants, double ionization potential (DIP) and ionization potential (IP). The EOM-DIP method was used to study the electronic structure of the neutral carbon dimer and the EOM-IP method was used to study its anion. This chapter shows that EOM-DIP-CCSD offers a simple computational approach based on a single-reference formalism to study the electronic states of the carbon dimer. In the same study, complex-variable extension of EOM was employed to study the metastable dianion of the carbon dimer. ❧ Chapter 5 describes the electronic resonances in the C₂H⁻ anion. This study was inspired by an experimental finding of the formation of excited carbon dimer anion in plasma. The EOM-CC method augmented with the complex absorbing potential was used to study electronic resonances. The calculations explain the formation of excited C₂̄ via dissociative electron attachment to C₂H molecule. Chapter 6 discusses the future directions and possible extensions of the work presented in this thesis.
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Gulania, Sahil
(author)
Core Title
New electronic structure methods for electronically excited and open-shell species within the equation-of-motion coupled-cluster framework
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Chemistry
Publication Date
05/03/2021
Defense Date
03/10/2021
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Krylov, Anna (
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), Nakano, Aiichiro (
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), Prezhdo, Oleg (
committee member
)
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gulania@usc.edu,sgulania@gmail.com
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Tags
double electron attachment
double ionization potential
electronic structure
EOM-CCSD