A comprehensive overview of the equation of motion coupled‐cluster (EOM‐CC) method and its application to molecular systems is presented. By exploiting the biorthogonal nature of the theory, it is shown that excited state properties and transition strengths can be evaluated via a generalized expectation value approach that incorporates both the bra and ket state wave functions. Reduced density matrices defined by this procedure are given by closed form expressions. For the root of the EOM‐CC effective Hamiltonian that corresponds to the ground state, the resulting equations are equivalent to the usual expressions for normal single‐reference CC density matrices. Thus, the method described in this paper provides a universal definition of coupled‐cluster density matrices, providing a link between EOM‐CC and traditional ground state CC theory. Excitation energy, oscillator strength, and property calculations are illustrated by means of several numerical examples, including comparisons with full configuration interaction calculations and a detailed study of the ten lowest electronically excited states of the cyclic isomer of C4.
REFERENCES
1.
J.
Gold-stone
, Proc. R. Soc. London Ser. A
239
, 267
(1957
). Molecular applications of MBPT were performed by2.
F.
Coester
and H.
Kiimmel
, Nucl. Phys.
17
, 477
(1960
). Systematic CC methods for quantum chemistry were first developed by, Nucl. Phys.
3.
See, for example,
R. J.
Bartlett
, Annu. Rev. Phys. Chem.
32
, 359
(1981
);4.
J. A.
Pople
, R.
Krishnan
, H. B.
Schlegel
, and J. S.
Binkley
, Int. J. Quantum Chem. Sympos.
13
, 225
(1979
).5.
G.
Fitzgerald
, R.
Harrison
, W. D.
Laidig
, and R. J.
Bartlett
, Chem. Phys. Lett.
82
, 4379
(1985
).6.
7.
E. A.
Salter
, G. W.
Trucks
, G.
Fitzgerald
, and R. J.
Bartlett
, Chem. Phys. Lett.
141
, 61
(1987
).8.
L.
Adamowicz
, W. D.
Laidig
, and R. J.
Bartlett
, Int. J. Quantum Chem. Sympos.
18
, 245
(1984
).9.
R. J. Bartlett, in Geometrical Derivatives of Energy Surfaces and Molecular Properties, edited by P. Jo/rgensen and J. Simons (Reidel, Dordrecht, 1986).
10.
A. C.
Scheiner
, G. E.
Scuseria
, J. E.
Rice
, T. J.
Lee
, and H. F.
Schaefer
, J. Chem. Phys.
87
, 5361
(1987
).11.
E. A.
Salter
, G. W.
Trucks
, and R. J.
Bartlett
, J. Chem. Phys.
90
, 1752
(1989
).12.
H.
Koch
, H. J. Aa.
Jensen
, P.
Jo/rgensen
, T.
Helgaker
, G. E.
Scuseria
, and H. F.
Schaefer
, J. Chem. Phys.
92
, 4924
(1990
).13.
J.
Gauss
, J. F.
Stanton
, and R. J.
Bartlett
, J. Chem. Phys.
95
, 2623
(1991
).14.
J.
Gauss
, J. F.
Stanton
, and R. J.
Bartlett
, J. Chem. Phys.
95
, 2639
(1991
). Explicit equations for the reduced one-particle density matrix for ground state CC theory are given in this paper, although they are presented in a symmetrized form.15.
J.
Gauss
, W. J.
Lauderdale
, J. F.
Stanton
, J. D.
Watts
, and R. J.
Bartlett
, Chem. Phys. Lett.
182
, 207
(1991
).16.
See, for example,
E. A.
Salter
, H.
Sekino
, and R. J.
Bartlett
, J. Chem. Phys.
87
, 502
(1987
).17.
18.
J. F.
Stanton
, J.
Gauss
, and R. J.
Bartlett
, J. Chem. Phys.
94
, 4084
(1991
);J. D.
Watts
, J. F.
Stanton
, J.
Gauss
, and R. J.
Bartlett
, J. Chem. Phys.
94
, 4320
(1991
)., J. Chem. Phys.
19.
See, for example,
L.
Adamowicz
and R. J.
Bartlett
, Int. J. Quantum Chem. Sympos.
19
, 217
(1986
).20.
21.
22.
For general reviews of the Fock space approach,see the articles in
Theor. Chim. Acta
80
, 427
(1991
).23.
24.
25.
J.
Geertsen
, M.
Rittby
, and R. J.
Bartlett
, Chem. Phys. Lett.
164
, 57
(1989
).26.
Excitation energies obtained in EOM-CC calculations are equivalent to those calculated using coupled-cluster linear response theory[
H.
Koch
and P.
Jo/rgensen
, J. Chem. Phys.
93
, 3333
(1990
);H.
Koch
, H. J. Aa.
Jensen
, T.
Helgaker
, and P.
Jo/rgensen
, J. Chem. Phys.
93
, 3345
(1990
)]. In addition, the EOM-CC approach is similar to Nakatsuji’s symmetry adapted cluster method[ , J. Chem. Phys.
27.
28.
In cases where nondynamical correlation effects are important, direct methods such as EOM-CCSD and the multireference Fock space CC approach should be more reliable than the single-configuration QRHF-CC and TD-CC methods since the latter approaches are intrinsically biased towards the reference configuration. In direct methods, there is no such bias and these methods therefore provide a more balanced treatment for problems of this type. This point has been discussed in some detail in a recent paper on the Fock space approach[
J. F.
Stanton
, R. J.
Bartlett
, and C. M. L.
Rittby
, J. Chem. Phys.
95
, 6224
(1992
)]. 29.
Strictly speaking, it is only the truncation of the vector which causes the method to be inexact.
30.
J. A.
Pople
, M.
Head-Gordon
, and K.
Raghavachari
, J. Chem. Phys.
87
, 5968
(1987
).31.
In order to retain consistency with the normal CC treatment of the ground state, the ℜ; vectors should be normalized to unity.
32.
P. S. Zalay and R. J. Bartlett, J. Chem. Phys, (in press). The functional form of the ground state CC energy and its potential advantage for property calculations had previously been noted by Arponen [
J. S.
Arponen
, Ann. Phys. (N.Y.)
151
, 311
(1983
)]. 33.
This expression applies equally well to transitions between two excited states.
34.
Here, we make an implicit distinction between the actual reduced one-particle density matrix of ground state CC theory and the “relaxed” or “effective” one-density (see Ref. 11, for example), which includes all effects of orbital relaxation. Only the former type of density matrix is considered in this paper.
35.
J. F.
Stanton
, J.
Gauss
, J. D.
Watts
, and R. J.
Bartlett
, J. Chem. Phys.
94
, 4334
(1991
).36.
In Ref. 13, the H matrix elements are designated as while the intermediates used in solving the CCSD equations are denoted as
37.
To save disk space, the terms involving may be handled differently from the rest. As discussed in Ref. 13, the contractions between and the trial vectors may be broken down into three terms, each of which involves one contribution to the amplitudes. If carried out in this fashion, no non-Hermitian type quantities need to be stored on disk, and the ordered lists may exploit the intrinsic permutational symmetry of the bare Hamiltonian integrals. This results in negligible computational overhead and a significant reduction in the demand for disk storage.
38.
We have not mentioned the and matrix elements in this context because the former vanish when the T amplitudes obey the CC equations and consequently need not be stored.
39.
40.
41.
Although H contains a number of three- and higher-body operators, only a few of the three-body terms contribute to the EOM-CCSD equations. These are not listed in Table I, but their explicit spin-orbital representation can be extracted from Eqs. (31) and (32) by considering only the t amplitudes and antisymmetrized two-electron integrals.
42.
43.
44.
ACES H, an ab initio program system, authored by J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett. The package also contains modified versions of the MOLECULE Gaussian integral program of J. Almlof and P. R. Taylor, the ABACUS integral derivative program of T. U. Helgaker, H. J. A. Jensen, P. Jo/rgensen, and P. R. Taylor, and the PROPS property integral package of P. R. Taylor.
45.
46.
J.
Olsen
, A. M.
Sanchez de Meras
, H. J. Aa.
Jensen
, and P.
Jo/rgensen
, Chem. Phys. Lett.
154
, 380
(1989
).47.
It should be pointed out that partitional EOM (Ref. 25) and the coupled-cluster linear response theory (Ref. 26) has previously been applied to calculate transition energies for these systems. Since the latter gives excitation energies identical to those obtained in the EOM-CC approximation, the energies listed in the first column of Table III have been previously presented in the literature but not the dipole strengths or AEL values.
48.
The ground state CCSD density matrix is of course non-Hermitian, and the approximate natural orbital basis is therefore biorthogonal. Nevertheless, to calculate the AEL values listed in this paper we have simplified matters by symmetrizing the ground and excited state density matrices. This should make essentially no difference to the numerical values of the AEL.
49.
50.
P. O.
Widmark
, P. A.
Malmqvist
, and B. O.
Roos
, Theor. Chim. Acta
77
, 291
(1990
). 51.
A. Balkova and R. J. Bartlett, J. Chem. Phys, (submitted).
52.
D. H.
Magers
, R. J.
Harrison
, and R. J.
Bartlett
, J. Chem. Phys.
84
, 3284
(1986
).53.
R. A.
Whiteside
, R.
Krishnan
, D. J.
DeFrees
, J. A.
Pople
, and P. v. R.
Schleyer
, Chem. Phys. Lett.
78
, 538
(1981
);J. P.
Ritchie
, H. F.
King
, and W. S.
Young
, J. Chem. Phys.
85
, 5175
(1986
);K.
Raghavachari
and J. S.
Binkley
, J. Chem. Phys.
87
, 2191
(1987
); , J. Chem. Phys.
D.
Michalska
, H.
Chojnacki
, B. A.
Hess
, and L. J.
Schaad
, Chem. Phys. Lett.
141
, 376
(1987
);D. E.
Bernholdt
, D. H.
Magers
, and R. J.
Bartlett
, J. Chem. Phys.
89
, 3612
(1988
);J. M. L.
Martin
, J. P.
Francois
, and R.
Gijbels
, J. Chem. Phys.
94
, 3753
(1991
);K.
Lammertsma
, O. F.
Giiner
, and P. V.
Sudhakar
, J. Chem. Phys.
94
, 8105
(1991
); , J. Chem. Phys.
V.
Parasuk
and J.
Almlöf
, J. Chem. Phys.
94
, 8172
(1991
)., J. Chem. Phys.
54.
55.
M. A.
Nygren
and L. G. M.
Petersson
, Chem. Phys. Lett.
191
, 3096
(1992
).56.
J. D.
Watts
, J.
Gauss
, J. F.
Stanton
, and R. J.
Bartlett
, J. Chem. Phys.
97
, 8372
(1992
).57.
M.
Algranati
, H.
Feldman
, D.
Kella
, E.
Malkin
, E.
Miklazky
, R.
Naaman
, Z.
Vager
, and J.
Zajfman
, J. Chem. Phys.
90
, 4617
(1989
).58.
For a review, see
W.
Weltner
and R. J.
Van Zee
, Chem. Rev.
89
, 1713
(1989
).59.
Z. Z. Wang, R. N. Diffendorfer, and I. Shavitt, paper presented at the 39th Symposium on Molecular Spectroscopy, Ohio State University, 1984, as cited in Ref. 54.
60.
See, for example,
J.
Noga
and M.
Urban
, Theor. Chim. Acta
73
, 291
(1988
);S. A. Blundell, in Applied Many-Body Methods in Spectroscopy and Electronic Structure, edited by D. Mukherjee (Plenum, New York, 1992), pp. 163–192.
61.
D. J. Thouless, The Quantum Mechanics of Many-Body Systems (Academic, New York, 1961), p. 126.
62.
In the limit of an exact calculation, the two approaches are of course equivalent, but the symmetric strategy still requires the summation of an infinite series of terms.
63.
J. F. Stanton and R. J. Bartlett, J. Chem. Phys, (in press).
64.
D. C. Comeau and R. J. Bartlett, Chem. Phys. Lett, (in press).
This content is only available via PDF.
© 1993 American Institute of Physics.
1993
American Institute of Physics
You do not currently have access to this content.
