We present an approach for obtaining a molecular orbital picture of the first dipole hyperpolarizability (β) from correlated many-body electronic structure methods. Ab initio calculations of β rely on quadratic response theory, which recasts the sum-over-all-states expression of β into a closed-form expression by calculating a handful of first- and second-order response states; for resonantly enhanced β, damped response theory is used. These response states are then used to construct second-order response reduced one-particle density matrices (1PDMs), which, upon visualization in terms of natural orbitals (NOs), facilitate a rigorous and black-box mapping of the underlying electronic structure with β. We explain the interpretation of different components of the response 1PDMs and the corresponding NOs within both the undamped and damped response theory framework. We illustrate the utility of this new tool by deconstructing β for cis-difluoroethene, para-nitroaniline, and hemibonded OH· + H2O complex, computed within the framework of coupled-cluster singles and doubles response theory, in terms of the underlying response 1PDMs and NOs for a range of frequencies.
REFERENCES
1.
S.
Yamaguchi
and T.
Tahara
, “Two-photon absorption spectrum of all-trans retinal
,” Chem. Phys. Lett.
376
, 237
(2003
).2.
B.
Dick
and G.
Hohlneicher
, “Two-photon excitation spectroscopy of phenanthrene singlet states below 50000 cm−1
,” Chem. Phys. Lett.
97
, 324
(1983
).3.
A. M.
Rizzuto
, S.
Irgen-Gioro
, A.
Eftekhari-Bafrooei
, and R. J.
Saykally
, “Broadband deep UV spectra of interfacial aqueous iodide
,” J. Phys. Chem. Lett.
7
, 3882
(2016
).4.
H.
Mizuno
, A. M.
Rizzuto
, and R. J.
Saykally
, “Charge-transfer-to-solvent spectrum of thiocyanate at the air/water interface measured by broadband deep ultraviolet electronic sum frequency generation spectroscopy
,” J. Phys. Chem. Lett.
9
, 4753
(2018
).5.
O.
Fuchs
, M.
Zharnikov
, L.
Weinhardt
, M.
Blum
, M.
Weigand
, Y.
Zubavichus
, M.
Bär
, F.
Maier
, J. D.
Denlinger
, C.
Heske
, M.
Grunze
, and E.
Umbach
, “Isotope and temperature effects in liquid water probed by x-ray absorption and resonant x-ray emission spectroscopy
,” Phys. Rev. Lett.
100
, 027801
(2008
).6.
L.
Kjellsson
, K. D.
Nanda
, J.-E.
Rubensson
, G.
Doumy
, S. H.
Southworth
, P. J.
Ho
, A. M.
March
, A.
Al Haddad
, Y.
Kumagai
, M.-F.
Tu
, T.
Debnath
, M. S.
Bin Mohd Yusof
, C.
Arnold
, W. F.
Schlotter
, S.
Moeller
, G.
Coslovich
, J. D.
Koralek
, M. P.
Minitti
, M. L.
Vidal
, M.
Simon
, R.
Santra
, Z.-H.
Loh
, S.
Coriani
, A. I.
Krylov
, and L.
Young
, “Resonant inelastic x-ray scattering reveals hidden local transitions of the aqueous OH radical
,” Phys. Rev. Lett.
124
, 236001
(2020
).7.
E. C.
Carroll
, S.
Berlin
, J.
Levitz
, M. A.
Kienzler
, Z.
Yuan
, D.
Madsen
, D. S.
Larsen
, and E. Y.
Isacoff
, “Two-photon brightness of azobenzene photoswitches designed for glutamate receptor optogenetics
,” Proc. Natl. Acad. Sci. U. S. A
112
, E776
(2015
).8.
A. V.
Kachynski
, A.
Pliss
, A. N.
Kuzmin
, T. Y.
Ohulchanskyy
, A.
Baev
, J.
Qu
, and P. N.
Prasad
, “Photodynamic therapy by in situ nonlinear photon conversion
,” Nat. Photonics
8
, 455
(2014
).9.
H.-Q.
Peng
, L.-Y.
Niu
, Y.-Z.
Chen
, L.-Z.
Wu
, C.-H.
Tung
, and Q.-Z.
Yang
, “Biological applications of supramolecular assemblies designed for excitation energy transfer
,” Chem. Rev.
115
, 7502
(2015
).10.
W. A.
Velema
, W.
Szymanski
, and B. L.
Feringa
, “Photopharmacology: Beyond proof of principle
,” J. Am. Chem. Soc.
136
, 2178
(2014
).11.
T. Z.
Teisseyre
, A. C.
Millard
, P.
Yan
, J. P.
Wuskell
, M.-d.
Wei
, A.
Lewis
, and L. M.
Loew
, “Nonlinear optical potentiometric dyes optimized for imaging with 1064-nm light
,” J. Biomed. Opt.
12
, 044001
(2007
).12.
S.
Hunter
, F.
Kiamilev
, S.
Esener
, D. A.
Parthenopoulos
, and P. M.
Rentzepis
, “Potentials of two-photon based 3-D optical memories for high performance computing
,” Appl. Opt.
29
, 2058
(1990
).13.
S.
Kawata
and Y.
Kawata
, “Three-dimensional optical data storage using photochromic materials
,” Chem. Rev.
100
, 1777
(2000
).14.
C. C.
Corredor
, Z.-L.
Huang
, K. D.
Belfield
, A. R.
Morales
, and M. V.
Bondar
, “Photochromic polymer composites for two-photon 3D optical data storage
,” Chem. Mater.
19
, 5165
(2007
).15.
B. J.
Orr
and J. F.
Ward
, “Perturbation theory of the non-linear optical polarization of an isolated system
,” Mol. Phys.
20
, 513
(1971
).16.
J.
Olsen
and P.
Jørgensen
, “Linear and non-linear response functions for an exact state and for MCSCF state
,” J. Chem. Phys.
82
, 3235
(1985
).17.
T.
Helgaker
, S.
Coriani
, P.
Jørgensen
, K.
Kristensen
, J.
Olsen
, and K.
Ruud
, “Recent advances in wave function-based methods of molecular-property calculations
,” Chem. Rev.
112
, 543
(2012
).18.
19.
M.
Bass
, P. A.
Franken
, J. F.
Ward
, and G.
Weinreich
, “Optical rectification
,” Phys. Rev. Lett.
9
, 446
(1962
).20.
P. A.
Franken
, A. E.
Hill
, C. W.
Peters
, and G.
Weinreich
, “Generation of optical harmonics
,” Phys. Rev. Lett.
7
, 118
(1961
).21.
M.
Nakano
and B.
Champagne
, “Diradical character dependences of the first and second hyperpolarizabilities of asymmetric open-shell singlet systems
,” J. Chem. Phys.
138
, 244306
(2013
).22.
P. C.
Ray
, “The effects of π-conjugation on first hyperpolarizabilities of charged NLO chromophores
,” Chem. Phys. Lett.
394
, 354
(2004
).23.
R.
Andreu
, M. J.
Blesa
, L.
Carrasquer
, J.
Garín
, J.
Orduna
, B.
Villacampa
, R.
Alcalá
, J.
Casado
, M. C.
Ruiz Delgado
, J. T.
López Navarrete
, and M.
Allain
, “Tuning first molecular hyperpolarizabilities through the use of proaromatic spacers
,” J. Chem. Phys.
127
, 008835
(2005
).24.
T.
Woller
, P.
Geerlings
, F.
De Proft
, B.
Champagne
, and M.
Alonso
, “Aromaticity as a guiding concept for spectroscopic features and nonlinear optical properties of porphyrinoids
,” Molecules
23
, 1333
(2018
).25.
J. N.
Woodford
, M. A.
Pauley
, and C. H.
Wang
, “Solvent dependence of the first molecular hyperpolarizability of p-nitroaniline revisited
,” J. Phys. Chem. A
101
, 1989
(1997
).26.
M. M.
Alam
, V.
Kundi
, and P. P.
Thankachan
, “Solvent effects on static polarizability, static first hyperpolarizability and one- and two-photon absorption properties of functionalized triply twisted Möbius annulenes: A DFT study
,” Phys. Chem. Chem. Phys.
18
, 21833
(2016
).27.
K. D.
Nanda
and A. I.
Krylov
, “Visualizing the contributions of virtual states to two-photon absorption cross-sections by natural transition orbitals of response transition density matrices
,” J. Phys. Chem. Lett.
8
, 3256
(2017
).28.
K. D.
Nanda
and A. I.
Krylov
, “A simple molecular orbital picture of RIXS distilled from many-body damped response theory
,” J. Chem. Phys.
152
, 244118
(2020
).29.
A. I.
Krylov
, “From orbitals to observables and back
,” J. Chem. Phys.
153
, 080901
(2020
).30.
S. A.
Mewes
, F.
Plasser
, A.
Krylov
, and A.
Dreuw
, “Benchmarking excited-state calculations using exciton properties
,” J. Chem. Theory Comput.
14
, 710
(2018
).31.
F.
Plasser
, M.
Wormit
, and A.
Dreuw
, “New tools for the systematic analysis and visualization of electronic excitations. I. Formalism
,” J. Chem. Phys.
141
, 024106
(2014
).32.
A. V.
Luzanov
, A. A.
Sukhorukov
, and V. E.
Umanskii
, “Application of transition density matrix for analysis of excited states
,” Theory Exp. Chem.
10
, 354
(1976
)A. V.
Luzanov
, A. A.
Sukhorukov
, and V. E.
Umanskii
, [Teor. Eksp. Khim.
10
, 456
(1974
) (in Russian)].33.
A. V.
Luzanov
and V. F.
Pedash
, “Interpretation of excited states using charge-transfer number
,” Theory Exp. Chem.
15
, 338
(1979
).34.
M.
Head-Gordon
, A. M.
Grana
, D.
Maurice
, and C. A.
White
, “Analysis of electronic transitions as the difference of electron attachment and detachment densities
,” J. Phys. Chem.
99
, 14261
(1995
).35.
F.
Plasser
, S. A.
Bäppler
, M.
Wormit
, and A.
Dreuw
, “New tools for the systematic analysis and visualization of electronic excitations. II. Applications
,” J. Chem. Phys.
141
, 024107
(2014
).36.
S. A.
Bäppler
, F.
Plasser
, M.
Wormit
, and A.
Dreuw
, “Exciton analysis of many-body wave functions: Bridging the gap between the quasiparticle and molecular orbital pictures
,” Phys. Rev. A
90
, 052521
(2014
).37.
P.
Kimber
and F.
Plasser
, “Toward an understanding of electronic excitation energies beyond the molecular orbital picture
,” Phys. Chem. Chem. Phys.
22
, 6058
(2020
).38.
M.
de Wergifosse
, C. G.
Elles
, and A. I.
Krylov
, “Two-photon absorption spectroscopy of stilbene and phenanthrene: Excited-state analysis and comparison with ethylene and toluene
,” J. Chem. Phys.
146
, 174102
(2017
).39.
K. D.
Nanda
and A. I.
Krylov
, “The effect of polarizable environment on two-photon absorption cross sections characterized by the equation-of-motion coupled-cluster singles and doubles method combined with the effective fragment potential approach
,” J. Chem. Phys.
149
, 164109
(2018
).40.
P.
Norman
, D. M.
Bishop
, H. J. A.
Jensen
, and J.
Oddershede
, “Nonlinear response theory with relaxation: The first-order hyperpolarizability
,” J. Chem. Phys.
123
, 194103
(2005
).41.
D. C.
Comeau
and R. J.
Bartlett
, “The equation-of-motion coupled-cluster method. Applications to open- and closed-shell reference states
,” Chem. Phys. Lett.
207
, 414
(1993
).42.
J. F.
Stanton
and R. J.
Bartlett
, “The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties
,” J. Chem. Phys.
98
, 7029
(1993
).43.
Y.
Shao
, Z.
Gan
, E.
Epifanovsky
, A. T. B.
Gilbert
, M.
Wormit
, J.
Kussmann
, A. W.
Lange
, A.
Behn
, J.
Deng
, X.
Feng
et al., “Advances in molecular quantum chemistry contained in the Q-Chem 4 program package
,” Mol. Phys.
113
, 184
(2015
).44.
A. I.
Krylov
and P. M. W.
Gill
, “Q-Chem: An engine for innovation
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
3
, 317
(2013
).45.
K.
Kristensen
, J.
Kauczor
, A. J.
Thorvaldsen
, P.
Jørgensen
, T.
Kjærgaard
, and A.
Rizzo
, “Damped response theory description of two-photon absorption
,” J. Chem. Phys.
134
, 214104
(2011
).46.
J.
Kauczor
, P.
Norman
, O.
Christiansen
, and S.
Coriani
, “Communication: A reduced-space algorithm for the solution of the complex linear response equations used in coupled cluster damped response theory
,” J. Chem. Phys.
139
, 211102
(2013
).47.
K. D.
Nanda
, M. L.
Vidal
, R.
Faber
, S.
Coriani
, and A. I.
Krylov
, “How to stay out of trouble in RIXS calculations within the equation-of-motion coupled-cluster damped response theory framework? Safe hitchhiking in the excitation manifold by means of core-valence separation
,” Phys. Chem. Chem. Phys.
22
, 2629
(2020
).48.
K. D.
Nanda
and A. I.
Krylov
, “Two-photon absorption cross sections within equation-of-motion coupled-cluster formalism using resolution-of-the-identity and Cholesky decomposition representations: Theory, implementation, and benchmarks
,” J. Chem. Phys.
142
, 064118
(2015
).49.
K. D.
Nanda
and A. I.
Krylov
, “Static polarizabilities for excited states within the spin-conserving and spin-flipping equation-of-motion coupled-cluster singles and doubles formalism: Theory, implementation, and benchmarks
,” J. Chem. Phys.
145
, 204116
(2016
).50.
H.
Sekino
and R. J.
Bartlett
, “A linear response, coupled-cluster theory for excitation energy
,” Int. J. Quantum Chem.
26
, 255
(1984
).51.
H.
Koch
and P.
Jørgensen
, “Coupled cluster response functions
,” J. Chem. Phys.
93
, 3333
(1990
).52.
H.
Koch
, H. J. Aa.
Jensen
, P.
Jørgensen
, and T.
Helgaker
, “Excitation energies from the coupled clusters singles and doubles linear response functions (CCSDLR). Applications to Be, CH+, CO, and H2O
,” J. Chem. Phys.
93
, 3345
(1990
).53.
P. B.
Rozyczko
, S. A.
Perera
, M.
Nooijen
, and R. J.
Bartlett
, “Correlated calculations of molecular dynamic polarizabilities
,” J. Chem. Phys.
107
, 6736
(1997
).54.
P.
Rozyczko
and R. J.
Bartlett
, “Frequency dependent equation-of-motion coupled cluster hyperpolarizabilities: Resolution of the discrepancy between theory and experiment for HF?
,” J. Chem. Phys.
107
, 10823
(1997
).55.
C.
Hättig
, H.
Koch
, and P.
Jørgensen
, “Comment on ‘Frequency-dependent equation-of-motion coupled cluster hyperpolarizabilities: Resolution of the discrepancy between theory and experiment for HF?’ [J. Chem. Phys. 107, 10823 (1997)]
,” J. Chem. Phys.
109
, 3293
(1998
).56.
K. D.
Nanda
, A. I.
Krylov
, and J.
Gauss
, “Communication: The pole structure of the dynamical polarizability tensor in equation-of-motion coupled-cluster theory
,” J. Chem. Phys.
149
, 141101
(2018
).57.
K. D.
Nanda
and A. I.
Krylov
, “Cherry-picking resolvents: A general strategy for convergent coupled-cluster damped response calculations of core-level spectra
,” J. Chem. Phys.
153
, 141104
(2020
).58.
S.
Coriani
, F.
Pawłowski
, J.
Olsen
, and P.
Jørgensen
, “Molecular response properties in equation of motion coupled cluster theory: A time-dependent perspective
,” J. Chem. Phys.
144
, 024102
(2016
).59.
F.
Plasser
, “TheoDORE: A toolbox for a detailed and automated analysis of electronic excited state computations
,” J. Chem. Phys.
152
, 084108
(2020
).© 2021 Author(s).
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