When dealing with approximate wave functions, molecular properties can be computed either as expectation values or as derivatives of the energy with respect to a corresponding perturbation. In this work, the intermediate state representation (ISR) formalism for the computation of expectation values is compared to the Lagrange formalism following a derivative ansatz, which are two alternative approaches of which neither one can be considered superior in general. Within the ISR formalism, terms are included up to a given order of perturbation theory only, while in the Lagrange formalism, all terms are accounted for arising through the differentiation. Similarities and differences of the Lagrange and ISR formalism are illustrated using explicit working equations for selected methods and analyzing numerical results for a range of coupled-cluster as well as algebraic-diagrammatic construction (ADC) methods for excited states. The analysis explains why the ADC(3/2) method is able to yield a large amount of the orbital-relaxation effects for p-h states in contrast to ADC(2) although the same second-order ISR is used to represent the corresponding operator.
REFERENCES
1.
A. C.
Hurley
and J. E.
Lennard-Jones
, Proc. R. Soc. London, Ser. A
226
, 179
(1954
).2.
J.
Gerratt
and I. M.
Mills
, J. Chem. Phys.
49
, 1719
(1968
).3.
G. H.
Diercksen
, B. O.
Roos
, and A. J.
Sadlej
, Chem. Phys.
59
, 29
(1981
).4.
P.
Pulay
, Mol. Phys.
17
, 197
(1969
).5.
V.
Bakken
, T.
Helgaker
, W.
Klopper
, and K.
Ruud
, Mol. Phys.
96
, 653
(1999
).6.
G. W.
Trucks
, E. A.
Salter
, C.
Sosa
, and R. J.
Bartlett
, Chem. Phys. Lett.
147
, 359
(1988
).7.
O.
Christiansen
, P.
Jørgensen
, and C.
Hättig
, Int. J. Quantum Chem.
68
, 1
(1998
).8.
9.
R. P.
Feynman
, Phys. Rev.
56
, 340
(1939
).10.
J. F.
Stanton
and R. J.
Bartlett
, J. Chem. Phys.
98
, 7029
(1993
).11.
J.
Schirmer
, Phys. Rev. A
26
, 2395
(1982
).12.
A.
Dreuw
and M.
Wormit
, Wiley Interdiscip. Rev.: Comput. Mol. Sci.
5
, 82
(2015
).13.
J.
Schirmer
, Phys. Rev. A
43
, 4647
(1991
).14.
J.
Schirmer
and A. B.
Trofimov
, J. Chem. Phys.
120
, 11449
(2004
).15.
J.
Schirmer
and F.
Mertins
, Theor. Chem. Acc.
125
, 145
(2010
).16.
C. M.
Aikens
, S. P.
Webb
, R. L.
Bell
, G. D.
Fletcher
, M. W.
Schmidt
, and M. S.
Gordon
, Theor. Chem. Acc.
110
, 233
(2003
).17.
T.
Helgaker
, P.
Jørgensen
, and N. C.
Handy
, Theor. Chim. Acta
76
, 227
(1989
).18.
H.
Koch
, H. J. A.
Jensen
, P.
Jørgensen
, T.
Helgaker
, G. E.
Scuseria
, and H. F.
Schaefer
, J. Chem. Phys.
92
, 4924
(1990
).19.
P.
Jørgensen
and T.
Helgaker
, J. Chem. Phys.
89
, 1560
(1988
).20.
T.
Helgaker
and P.
Jørgensen
, Theor. Chim. Acta
75
, 111
(1989
).21.
K.
Hald
, A.
Halkier
, P.
Jørgensen
, S.
Coriani
, C.
Hättig
, and T.
Helgaker
, J. Chem. Phys.
118
, 2985
(2003
).22.
F.
Furche
and R.
Ahlrichs
, J. Chem. Phys.
117
, 7433
(2002
).23.
C.
Hättig
, J. Chem. Phys.
118
, 7751
(2003
).24.
J.
Heuser
and S.
Höfener
, J. Comput. Chem.
38
, 2316
(2017
).25.
E. A.
Salter
, H.
Sekino
, and R. J.
Bartlett
, J. Chem. Phys.
87
, 502
(1987
).26.
A. A.
Auer
and J.
Gauss
, J. Chem. Phys.
115
, 1619
(2001
).27.
C. Y.
Cheng
, M. S.
Ryley
, M. J.
Peach
, D. J.
Tozer
, T.
Helgaker
, and A. M.
Teale
, Mol. Phys.
113
, 1937
(2015
).28.
T. C.
Caves
and M.
Karplus
, J. Chem. Phys.
50
, 3649
(1969
).29.
A. J.
Sadlej
, J. Chem. Phys.
75
, 320
(1981
).30.
T.
Helgaker
, P.
Jørgensen
, and J.
Olsen
, Molecular Electronic-Structure Theory
(John Wiley & Sons
, 2000
).31.
C.
Møller
and M. S.
Plesset
, Phys. Rev.
46
, 618
(1934
).32.
A.
Szabo
and N. S.
Ostlund
, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory
(Dover
, Mineola, New York
, 1996
).33.
N. C.
Handy
and H. F.
Schaefer
, J. Chem. Phys.
81
, 5031
(1984
).34.
R. D.
Amos
, Chem. Phys. Lett.
73
, 602
(1980
).35.
M.
Wormit
, D. R.
Rehn
, P. H. P.
Harbach
, J.
Wenzel
, C. M.
Krauter
, E.
Epifanovsky
, and A.
Dreuw
, Mol. Phys.
112
, 774
(2014
).36.
A. B.
Trofimov
, G.
Stelter
, and J.
Schirmer
, J. Chem. Phys.
111
, 9982
(1999
).37.
A. B.
Trofimov
, G.
Stelter
, and J.
Schirmer
, J. Chem. Phys.
117
, 6402
(2002
).38.
P. H. P.
Harbach
, M.
Wormit
, and A.
Dreuw
, J. Chem. Phys.
141
, 064113
(2014
).39.
D. R.
Rehn
and A.
Dreuw
, J. Chem. Phys.
150
, 164117
(2019
).40.
C.
Hättig
and A.
Köhn
, J. Chem. Phys.
117
, 6939
(2002
).41.
C.
Hättig
, “Response theory and molecular properties (A tribute to Jan Linderberg and Poul Jørgensen)
,” in Advances in Quantum Chemistry
, edited by H.
Jensen
(Academic Press
, 2005
), Vol. 50, pp. 37
–60
.42.
Y.
Shao
, Z.
Gan
, E.
Epifanovsky
, A. T.
Gilbert
, M.
Wormit
, J.
Kussmann
, A. W.
Lange
, A.
Behn
, J.
Deng
, X.
Feng
, D.
Ghosh
, M.
Goldey
, P. R.
Horn
, L. D.
Jacobson
, I.
Kaliman
, R. Z.
Khaliullin
, T.
Kuś
, A.
Landau
, J.
Liu
, E. I.
Proynov
, Y. M.
Rhee
, R. M.
Richard
, M. A.
Rohrdanz
, R. P.
Steele
, E. J.
Sundstrom
, H.
Lee Woodcock
, III, P. M.
Zimmerman
, D.
Zuev
, B.
Albrecht
, E.
Alguire
, B.
Austin
, G. J. O.
Beran
, Y. A.
Bernard
, E.
Berquist
, K.
Brandhorst
, K. B.
Bravaya
, S. T.
Brown
, D.
Casanova
, C.-M.
Chang
, Y.
Chen
, S. H.
Chien
, K. D.
Closser
, D. L.
Crittenden
, M.
Diedenhofen
, R. A.
DiStasio
, Jr., H.
Do
, A. D.
Dutoi
, R. G.
Edgar
, S.
Fatehi
, L.
Fusti-Molnar
, A.
Ghysels
, A.
Golubeva-Zadorozhnaya
, J.
Gomes
, M. W.
Hanson-Heine
, P. H.
Harbach
, A. W.
Hauser
, E. G.
Hohenstein
, Z. C.
Holden
, T.-C.
Jagau
, H.
Ji
, B.
Kaduk
, K.
Khistyaev
, J.
Kim
, J.
Kim
, R. A.
King
, P.
Klunzinger
, D.
Kosenkov
, T.
Kowalczyk
, C. M.
Krauter
, K. U.
Lao
, A. D.
Laurent
, K. V.
Lawler
, S. V.
Levchenko
, C. Y.
Lin
, F.
Liu
, E.
Livshits
, R. C.
Lochan
, A.
Luenser
, P.
Manohar
, S. F.
Manzer
, S.-P.
Mao
, N.
Mardirossian
, A. V.
Marenich
, S. A.
Maurer
, N. J.
Mayhall
, E.
Neuscamman
, C. M.
Oana
, R.
Olivares-Amaya
, D. P.
O’Neill
, J. A.
Parkhill
, T. M.
Perrine
, R.
Peverati
, A.
Prociuk
, D. R.
Rehn
, E.
Rosta
, N. J.
Russ
, S. M.
Sharada
, S.
Sharma
, D. W.
Small
, A.
Sodt
, T.
Stein
, D.
Stück
, Y.-C.
Su
, A. J.
Thom
, T.
Tsuchimochi
, V.
Vanovschi
, L.
Vogt
, O.
Vydrov
, T.
Wang
, M. A.
Watson
, J.
Wenzel
, A.
White
, C. F.
Williams
, J.
Yang
, S.
Yeganeh
, S. R.
Yost
, Z.-Q.
You
, I. Y.
Zhang
, X.
Zhang
, Y.
Zhao
, B. R.
Brooks
, G. K.
Chan
, D. M.
Chipman
, C. J.
Cramer
, W. A. G. M. S.
Gordon
III, W. J.
Hehre
, A.
Klamt
, H. F.
Schaefer
III, M. W.
Schmidt
, C. D.
Sherrill
, D. G.
Truhlar
, A.
Warshel
, X.
Xu
, A.
Aspuru-Guzik
, R.
Baer
, A. T.
Bell
, N. A.
Besley
, J.-D.
Chai
, A.
Dreuw
, B. D.
Dunietz
, T. R.
Furlani
, S. R.
Gwaltney
, C.-P.
Hsu
, Y.
Jung
, J.
Kong
, D. S.
Lambrecht
, W.
Liang
, C.
Ochsenfeld
, V. A.
Rassolov
, L. V.
Slipchenko
, J. E.
Subotnik
, T. V.
Voorhis
, J. M.
Herbert
, A. I.
Krylov
, P. M.
Gill
, and M.
Head-Gordon
, Mol. Phys.
113
, 184
(2015
).43.
J. F.
Stanton
, J.
Gauss
, L.
Cheng
, M. E.
Harding
, D. A.
Matthews
, and P. G.
Szalay
, “Cfour, coupled-cluster techniques for computational chemistry, a quantum-chemical program package,” With contributions from A. A.
Auer
, R. J.
Bartlett
, U.
Benedikt
, C.
Berger
, D. E.
Bernholdt
, Y. J.
Bomble
, O.
Christiansen
, F.
Engel
, R.
Faber
, M.
Heckert
, O.
Heun
, M.
Hilgenberg
, C.
Huber
, T.-C.
Jagau
, D.
Jonsson
, J.
Jusélius
, T.
Kirsch
, K.
Klein
, W. J.
Lauderdale
, F.
Lipparini
, T.
Metzroth
, L. A.
Mück
, D. P.
O’Neill
, D. R.
Price
, E.
Prochnow
, C.
Puzzarini
, K.
Ruud
, F.
Schiffmann
, W.
Schwalbach
, C.
Simmons
, S.
Stopkowicz
, A.
Tajti
, J.
Vázquez
, F.
Wang
, J. D.
Watts
and The integral packages MOLECULE (J.
Almlöf
and P. R.
Taylor
), PROPS (P. R.
Taylor
), ABACUS (T.
Helgaker
, H. J. Aa.
Jensen
, P.
Jørgensen
, and J.
Olsen
), and ECP routines by A. V.
Mitin
and C.
van Wüllen
, For the current version, see www.cfour.de.44.
S.
Höfener
, J. Comput. Chem.
35
, 1716
(2014
).45.
T. H.
Dunning
, Jr., J. Chem. Phys.
90
, 1007
(1989
).46.
A. K.
Wilson
, T.
van Mourik
, and T. H.
Dunning
, Jr., J. Mol. Struct.: THEOCHEM
388
, 339
(1996
).47.
T. D.
Crawford
and H.
Sekino
, “On the performance of a size-extensive variant of equation-of-motion coupled cluster theory for optical rotation in chiral molecules
,” in Advances in the Theory of Atomic and Molecular Systems: Conceptual and Computational Advances in Quantum Chemistry
, edited by P.
Piecuch
, J.
Maruani
, G.
Delgado-Barrio
, and S.
Wilson
(Springer Netherlands
, Dordrecht
, 2009
), pp. 225
–239
.48.
T. D.
Crawford
, “Reduced-scaling coupled-cluster theory for response properties of large molecules
,” in Recent Progress in Coupled Cluster Methods: Theory and Applications
, edited by P.
Cársky
, J.
Paldus
, and J.
Pittner
(Springer Netherlands
, Dordrecht
, 2010
), Chap. 2, pp. 37
–55
.49.
J. D.
Watts
, “An introduction to equation-of-motion and linear-response coupled-cluster methods for electronically excited states of molecules
,” in Radiation Induced Molecular Phenomena in Nucleic Acids: A Comprehensive Theoretical and Experimental Analysis
, edited by M. K.
Shukla
and J.
Leszczynski
(Springer Netherlands
, Dordrecht
, 2008
), pp. 65
–92
.50.
J. H.
Starcke
, M.
Wormit
, J.
Schirmer
, and A.
Dreuw
, Chem. Phys.
329
, 39
(2006
).51.
M.
Schreiber
, M. R.
Silva-Junior
, S. P. A.
Sauer
, and W.
Thiel
, J. Chem. Phys.
128
, 134110
(2008
).52.
C. S.
Page
and M.
Olivucci
, J. Comput. Chem.
24
, 298
(2003
).53.
S.
Gozem
, F.
Melaccio
, A.
Valentini
, M.
Filatov
, M.
Huix-Rotllant
, N.
Ferr
, L. M.
Frutos
, C.
Angeli
, A. I.
Krylov
, A. A.
Granovsky
, R.
Lindh
, and M.
Olivucci
, J. Chem. Theory Comput.
10
, 3074
(2014
).© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
