Internally contracted multireference coupled-cluster (icMRCC) theory is extended to the computation of first-order properties (expectation values). We use the previously defined Lagrange formulation of the energy functional to derive the required equations for the Lagrange multipliers and arrive at an expression for first-order properties according to the generalized Hellmann-Feynman theorem, analogous to single-reference coupled-cluster theory. The present formulation does not include orbital relaxation, but in line with previous experience in coupled-cluster theory, the single-excitation cluster operator can recover a significant portion of orbital relaxation. Further aspects of the theory that arise from the internal contraction approach are discussed. Using automated derivation techniques, we have implemented a pilot code for icMRCCSD and icMRCCSDT for testing the method numerically. We find good agreement with full configuration interaction for several properties of boron monohydride and dipole moment curves of hydrogen fluoride and chromium hydride. A particular focus is given to spin-dependent properties: The hyperfine coupling tensors of Σ and Π radicals have been computed and compared to experiment and previous computations. We discuss the problem of describing spin polarization with properly spin-adapted wavefunctions, which requires either including pseudo-triple excitations or employing sufficiently flexible reference functions.

1.
J. A.
Pople
,
J. W.
McIver
, and
N. S.
Ostlund
,
J. Chem. Phys.
49
,
2965
(
1968
).
2.
R. J.
Bartlett
and
G. D.
Purvis
,
Phys. Rev. A
20
,
1313
(
1979
).
3.
G. D.
Purvis
and
R. J.
Bartlett
,
Phys. Rev. A
23
,
1594
(
1981
).
4.
G. H. F.
Diercksen
,
B. O.
Roos
, and
A. J.
Sadlej
,
Chem. Phys.
59
,
29
(
1981
).
6.
H. J.
Monkhorst
,
Int. J. Quantum Chem.
12
(
S11
),
421
(
1977
).
7.
E.
Dalgaard
and
H.
Monkhorst
,
Phys. Rev. A
28
,
1217
(
1983
).
8.
L.
Adamowicz
,
W.
Laidig
, and
R. J.
Bartlett
,
Int. J. Quantum Chem.
26
,
245
(
1984
).
9.
E. A.
Salter
,
G. W.
Trucks
, and
R. J.
Bartlett
,
J. Chem. Phys.
90
,
1752
(
1989
).
10.
N. C.
Handy
and
H. F.
Schaefer
 III
,
J. Chem. Phys.
81
,
5031
(
1984
).
11.
T.
Helgaker
and
P.
Jørgensen
,
Theor. Chim. Acta
75
,
111
(
1989
).
12.
H.
Koch
,
H.
Jensen
,
P.
Jørgensen
,
T.
Helgaker
,
G. E.
Scuseria
, and
H. F.
Schaefer
,
J. Chem. Phys.
92
,
4924
(
1990
).
13.
P.-O.
Löwdin
,
Adv. Chem. Phys.
2
,
207
(
1959
).
14.
T.
Helgaker
and
J.
Almlöf
,
Int. J. Quantum Chem.
26
,
275
(
1984
).
15.
O.
Christiansen
,
P.
Jørgensen
, and
C.
Hättig
,
Int. J. Quantum Chem.
68
,
1
(
1998
).
16.
H.
Koch
and
P.
Jørgensen
,
J. Chem. Phys.
93
,
3333
(
1990
).
17.
F.
Aiga
,
K.
Sasagane
, and
R.
Itoh
,
J. Chem. Phys.
99
,
3779
(
1993
).
18.
E. A.
Salter
,
H.
Sekino
, and
R. J.
Bartlett
,
J. Chem. Phys.
87
,
502
(
1987
).
19.
J.
Noga
and
M.
Urban
,
Theor. Chim. Acta
73
,
291
(
1988
).
20.
D. I.
Lyakh
,
M.
Musiał
,
V. F.
Lotrich
, and
R. J.
Bartlett
,
Chem. Rev.
112
,
182
(
2012
).
21.
A.
Köhn
,
M.
Hanauer
,
L. A.
Mück
,
T.-C.
Jagau
, and
J.
Gauss
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
3
,
176
(
2013
).
22.
D.
Mukherjee
,
R. K.
Moitra
, and
A.
Mukhopadhyay
,
Mol. Phys.
33
,
955
(
1977
).
23.
M. A.
Haque
and
D.
Mukherjee
,
J. Chem. Phys.
80
,
5058
(
1984
).
24.
B.
Jeziorski
and
H. J.
Monkhorst
,
Phys. Rev. A
24
,
1668
(
1981
).
25.
U. S.
Mahapatra
,
B.
Datta
,
B.
Bandyopadhyay
, and
D.
Mukherjee
,
Adv. Quantum Chem.
30
,
163
(
1998
).
26.
U. S.
Mahapatra
,
B.
Datta
, and
D.
Mukherjee
,
J. Chem. Phys.
110
,
6171
(
1999
).
27.
M.
Hanrath
,
J. Chem. Phys.
123
,
084102
(
2005
).
28.
A.
Banerjee
and
J.
Simons
,
Int. J. Quantum Chem.
19
,
207
(
1981
).
29.
F. A.
Evangelista
and
J.
Gauss
,
J. Chem. Phys.
134
,
114102
(
2011
).
30.
M.
Hanauer
and
A.
Köhn
,
J. Chem. Phys.
134
,
204111
(
2011
).
31.
P. G.
Szalay
,
Int. J. Quantum Chem.
55
,
151
(
1995
).
32.
J.
Pittner
and
J.
Šmydke
,
J. Chem. Phys.
127
,
114103
(
2007
).
33.
E.
Prochnow
,
F. A.
Evangelista
,
H. F.
Schaefer
 III
,
W. D.
Allen
, and
J.
Gauss
,
J. Chem. Phys.
131
,
064109
(
2009
).
34.
T.-C.
Jagau
,
E.
Prochnow
,
F. A.
Evangelista
, and
J.
Gauss
,
J. Chem. Phys.
132
,
144110
(
2010
).
35.
S.
Chattopadhyay
,
U. S.
Mahapatra
, and
D.
Mukherjee
,
J. Chem. Phys.
111
,
3820
(
1999
).
36.
T.-C.
Jagau
and
J.
Gauss
,
J. Chem. Phys.
137
,
044115
(
2012
).
37.
A.
Banerjee
and
J.
Simons
,
J. Chem. Phys.
76
,
4548
(
1982
).
38.
A.
Banerjee
and
J.
Simons
,
Chem. Phys.
87
,
215
(
1984
).
39.
D.
Mukherjee
,
Chem. Phys. Lett.
274
(
5-6
),
561
(
1997
).
40.
W.
Kutzelnigg
and
D.
Mukherjee
,
J. Chem. Phys.
107
(
2
),
432
(
1997
).
41.
D.
Mukherjee
, “
A coupled cluster approach to the electron correlation problem using a correlated reference state
,” in
Recent Progress in Many-Body Theories
, edited by
E.
Schachinger
,
H.
Mitter
, and
H.
Sormann
(
Plenum Press
,
New York
,
1995
), Vol. 4, pp.
127
133
.
42.
M.
Hanauer
and
A.
Köhn
,
J. Chem. Phys.
136
,
204107
(
2012
).
43.
P. K.
Samanta
,
D.
Mukherjee
,
M.
Hanauer
, and
A.
Köhn
,
J. Chem. Phys.
140
,
134108
(
2014
).
44.
F. A.
Evangelista
,
M.
Hanauer
,
A.
Köhn
, and
J.
Gauss
,
J. Chem. Phys.
136
,
204108
(
2012
).
45.
M.
Hanauer
and
A.
Köhn
,
J. Chem. Phys.
137
,
131103
(
2012
).
46.
M.
Hanauer
, “
Internally contracted multireference coupled-cluster methods
,” Ph.D. thesis,
Johannes Gutenberg University
,
Mainz, Germany
,
2013
, available at http://ubm.opus.hbz-nrw.de/volltexte/2013/3423.
47.
H.-J.
Werner
and
P. J.
Knowles
,
J. Chem. Phys.
89
,
5803
(
1988
).
48.
K.
Andersson
,
P.-Å.
Malmqvist
, and
B. O.
Roos
,
J. Chem. Phys.
96
,
1218
(
1992
).
49.
D.
Sinha
,
R.
Maitra
, and
D.
Mukherjee
,
Comput. Theor. Chem.
1003
,
62
(
2013
).
50.
R.
Fink
and
V.
Staemmler
,
Theor. Chim. Acta
87
,
129
(
1993
).
51.
H. J. J.
van Dam
,
J. H.
van Lenthe
, and
P.
Pulay
,
Mol. Phys.
93
,
431
(
1998
).
52.
53.
T.
Helgaker
and
P.
Jørgensen
,
Adv. Quantum Chem.
19
,
183
(
1988
).
54.
P.
Jørgensen
and
T.
Helgaker
,
J. Chem. Phys.
89
,
1560
(
1988
).
55.
R.
McWeeny
,
Spins in Chemistry
(
Academic Press
,
1970
).
56.
H. M.
McConnell
,
J. Chem. Phys.
28
,
1188
(
1958
).
57.
A. V.
Luzanov
,
Theor. Exp. Chem.
21
,
329
(
1985
).
58.
D.
Datta
and
J.
Gauss
,
J. Chem. Phys.
143
,
011101
(
2015
).
59.
R.
Shepard
,
H.
Lischka
,
P. G.
Szalay
,
T.
Kovar
, and
M.
Ernzerhof
,
J. Chem. Phys.
96
,
2085
(
1992
).
60.
P.
Celani
and
H.-J.
Werner
,
J. Chem. Phys.
119
,
5044
(
2003
).
61.
W.
Liu
,
M.
Hanauer
, and
A.
Köhn
,
Chem. Phys. Lett.
565
,
122
(
2013
).
62.
Y. A.
Aoto
and
A.
Köhn
,
J. Chem. Phys.
144
,
074103
(
2016
).
63.
F.
Lipparini
,
T.
Kirsch
,
A.
Köhn
, and
J.
Gauss
,
J. Chem. Theory Comput.
13
,
3171
(
2017
).
64.
Dalton, a molecular electronic structure program, release 2.0,
2005
, see http://www.kjemi.uio.no/software/dalton/dalton.html.
65.
J. F.
Stanton
,
J.
Gauss
,
M. E.
Harding
,
P. G.
Szalay
,
A. A.
Auer
,
R.
Bartlett
,
U.
Benedikt
,
C.
Berger
,
D. E.
Bernholdt
,
Y. J.
Bomble
,
O.
Christiansen
,
M.
Heckert
,
O.
Heun
,
C.
Huber
,
T.-C.
Jagau
,
D.
Jonsson
,
J.
Jusélius
,
K.
Klein
,
W. J.
Lauderdale
,
D. A.
Matthews
,
T.
Metzroth
,
D. P.
O’Neill
,
D. R.
Price
,
E.
Prochnow
,
K.
Ruud
,
F.
Schiffmann
,
S.
Stopkowicz
,
J.
Vázquez
,
F.
Wang
, and
J.
Watts
, Cfour, Coupled-Cluster techniques for Computational Chemistry, a quantum-chemical program package, with contributions from the integral packages Molecule (
J.
Almlöf
and
P. R.
Taylor
), Props (
P. R.
Taylor
), Abacus (
T.
Helgaker
,
H. J. A.
Jensen
,
P.
Jørgensen
, and
J.
Olsen
), and effective-core-potential routines by
A. V.
Mitin
and
C.
van Wüllen
for the current version, see http://www.cfour.de,
2009
.
66.
H.-J.
Werner
,
P.
Knowles
,
G.
Knizia
,
F.
Manby
, and
M.
Schütz
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
,
242
(
2012
).
67.
H.-J.
Werner
,
P. J.
Knowles
,
G.
Knizia
,
F. R.
Manby
,
M.
Schütz
 et al, molpro, version 2015.1, a package of ab initio programs, 2015, see http://www.molpro.net.
68.
T. H.
Dunning
,
J. Chem. Phys.
90
,
1007
(
1989
).
69.
D. E.
Woon
and
T. H.
Dunning
, Jr.
,
J. Chem. Phys.
100
,
2975
(
1994
).
70.
P. J.
Knowles
and
H.-J.
Werner
,
Chem. Phys. Lett.
145
,
514
(
1988
).
71.
72.
A.
Halkier
,
H.
Larsen
,
J.
Olsen
,
P.
Jørgensen
, and
J.
Gauss
,
J. Chem. Phys.
110
,
734
(
1999
).
73.
K. P.
Huber
and
G. H.
Herzberg
,
Molecular Spectra and Molecular Structure. Constants of Diatomic Molecules
(
Van Nostrand-Reinhold
,
New York
,
1979
).
74.
D. E.
Woon
and
T. H.
Dunning
, Jr.
,
J. Chem. Phys.
103
,
4572
(
1995
).
75.
P. J.
Knowles
and
N. C.
Handy
,
Chem. Phys. Lett.
111
(
4-5
),
315
(
1984
).
76.
E. R.
Davidson
and
D. W.
Silver
,
Chem. Phys. Lett.
52
,
403
(
1977
).
77.

There is some confusion in the literature regarding the occurrence of the fine structure constant in the expression for PK, as this depends on the chosen unit system. In true Gaussian units, one factor α is absorbed in each of the Bohr magneton and the nuclear magneton, respectively. Atomic units, however, are based on the electrostatic system of units (esu), so the α2 should appear explicitly.

78.
T. N.
Lan
,
Y.
Kurashige
, and
T.
Yanai
,
J. Chem. Theory Comput.
10
,
1953
(
2014
).
79.
T.
Shiozaki
and
T.
Yanai
,
J. Chem. Theory Comput.
12
,
4347
(
2016
).
80.
S.
Kossmann
and
F.
Neese
,
J. Phys. Chem. A
114
,
11768
(
2010
).
81.
H.
Sekino
and
R. J.
Bartlett
,
J. Chem. Phys.
82
,
4225
(
1985
).
82.
I.
Carmichael
,
J. Phys. Chem.
95
,
108
(
1991
).
83.
J.
Gauss
,
J. F.
Stanton
, and
R. J.
Bartlett
,
J. Chem. Phys.
95
,
2639
(
1991
).
84.
S. A.
Perera
,
J. D.
Watts
, and
R. J.
Bartlett
,
J. Chem. Phys.
100
,
1425
(
1994
).
85.
C.
Puzzarini
and
V.
Barone
,
J. Chem. Phys.
133
,
184301
(
2010
).
86.
V.
Barone
, in
Recent Advances in Density Functional Methods. Part 1
, edited by
D. P.
Chong
(
World Scientific
,
1995
), p.
287
.
87.
J. D.
Watts
,
J.
Gauss
, and
R. J.
Bartlett
,
J. Chem. Phys.
98
,
8718
(
1993
).
88.
J. S. M.
Harvey
,
L.
Evans
, and
H.
Lew
,
Can. J. Phys.
50
,
1719
(
1972
).
89.
T.
Kiljunen
,
J.
Eloranta
,
J.
Ahokas
, and
H.
Kunttu
,
J. Chem. Phys.
114
,
7144
(
2001
).
90.
W. W.
Holloway
, Jr.
,
E.
Lüscher
, and
R.
Novick
,
Phys. Rev.
126
,
2109
(
1962
).
91.
C. W.
Bauschlicher
, Jr.
,
S. R.
Langhoff
,
H.
Partridge
, and
D. P.
Chong
,
J. Chem. Phys.
89
,
2985
(
1988
).
92.
P. G.
Szalay
and
J.
Gauss
,
J. Chem. Phys.
112
,
4027
(
2000
).
93.

We follow the nomenclature of Szalay et al.,92 who also use the term “pseudo-triples.” Note that other authors, like Paldus and Li in Ref. 95 take a different viewpoint and speak about pseudo-singles and pseudo-doubles, where we use the terms pseudo-doubles and pseudo-triples.

94.
W.
Meyer
, in
Methods of Electronic Structure Theory
, edited by
H. F.
Schaefer
 III
(
Plenum
,
1977
).
95.
X.
Li
and
J.
Paldus
,
J. Chem. Phys.
102
,
8897
(
1995
).
96.
P. S.
Bagus
,
B.
Liu
, and
H. F.
Schaefer
,
Phys. Rev. A
2
,
555
(
1970
).
97.
J. F.
Stanton
,
J. Chem. Phys.
101
,
371
(
1994
).
98.
G. H.
Herzberg
,
Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules
(
Van Nostrand-Reinhold
,
New York
,
1950
).
99.
W.
Kutzelnigg
,
U.
Fleischer
, and
M.
Schindler
,
NMR Basis Principles and Progress
(
Springer-Verlag
,
Heidelberg, Germany
,
1990
).
100.

A subsequent discussion with one of the authors of Ref. 58 (D. D.) revealed that upon decontraction one should remove the extra s functions of the core-valence correlation set. This was not done in the present work (the impact on the results is small) but is recommended in general as it serves for numerical stability.

101.
L. B.
Knight
, Jr.
,
M. B.
Wise
,
E. R.
Davidson
, and
L. E.
McMurchie
,
J. Chem. Phys.
76
,
126
(
1982
).
102.
L. B.
Knight
, Jr.
,
J. O.
Herlong
,
T. J.
Kirk
, and
C. A.
Arrington
,
J. Chem. Phys.
96
,
5604
(
1992
).
103.
N. D.
Piltch
,
P. G.
Szanto
,
T. G.
Anderson
,
C. S.
Gudeman
,
T. A.
Dixon
, and
R. C.
Woods
,
J. Chem. Phys.
76
,
3385
(
1982
).
104.
L. B.
Knight
, Jr.
and
J.
Steadman
,
J. Chem. Phys.
77
,
1750
(
1982
).
105.
L. B.
Knight
and
J.
Steadman
,
J. Am. Chem. Soc.
106
,
900
(
1984
).
106.
T. A.
Dixon
,
J. Chem. Phys.
67
,
3956
(
1977
).
107.
W. C.
Easley
and
W.
Weltner
, Jr.
,
J. Chem. Phys.
52
,
197
(
1970
).
108.
C.
Yamada
,
E. A.
Cohen
,
M.
Fujitake
, and
E.
Hirota
,
J. Chem. Phys.
92
,
2146
(
1990
).
109.
L. B.
Knight
, Jr.
and
W.
Weltner
, Jr.
,
J. Chem. Phys.
55
,
5066
(
1971
).
110.
L. B.
Knight
, Jr.
,
T. J.
Kirk
,
J.
Herlong
,
J. G.
Kaup
, and
E. R.
Davidson
,
J. Chem. Phys.
107
,
7011
(
1997
).
111.
G. K.-L.
Chan
and
M.
Head-Gordon
,
J. Chem. Phys.
116
,
4462
(
2002
).
112.
G. K.-L.
Chan
and
M.
Head-Gordon
,
J. Chem. Phys.
118
,
8551
(
2003
).
113.
D.
Zgid
and
M.
Nooijen
,
J. Chem. Phys.
128
,
144116
(
2008
).
114.
D.
Ghosh
,
J.
Hachmann
,
T.
Yanai
, and
G. K.-L.
Chan
,
J. Chem. Phys.
128
,
144117
(
2008
).
115.
K.
Boguslawski
,
K. H.
Marti
,
Ö.
Legeza
, and
M.
Reiher
,
J. Chem. Theory Comput.
8
,
1970
(
2012
).
116.
M.
Yoshimine
,
A. D.
McLean
, and
B.
Liu
,
J. Chem. Phys.
58
,
4412
(
1973
).
117.
C.
Zenouda
,
P.
Blottiau
,
G.
Chambaud
, and
P.
Rosmus
,
J. Mol. Struct.: THEOCHEM
458
,
61
(
1999
).
118.
A. D.
Becke
,
J. Chem. Phys.
107
,
8554
(
1997
).
119.
P. J.
Stephens
,
F. J.
Devlin
,
C. F.
Chabalowski
, and
M. J.
Frisch
,
J. Phys. Chem. A
98
,
11623
(
1994
).
120.
J.
Tao
,
J. P.
Perdew
,
V. N.
Staroverov
, and
G. E.
Scuseria
,
Phys. Rev. Lett.
91
,
146401
(
2003
).
121.
T. D.
Varberg
,
F.
Stroh
, and
K. M.
Evenson
,
J. Mol. Spectrosc.
196
,
5
(
1999
).
122.
H. S. P.
Müller
,
K.
Kobayashi
,
K.
Takahashi
,
K.
Tomaru
, and
F.
Matsushima
,
J. Mol. Spectrosc.
310
,
92
(
2015
).
123.
R. W.
Fessenden
,
J. Phys. Chem.
71
,
74
(
1967
).
124.
D. M.
Chipman
,
J. Chem. Phys.
78
,
3112
(
1983
).
125.
J.
Janecka
,
H. M.
Vyas
, and
M.
Fujimoto
,
J. Chem. Phys.
54
,
3229
(
1971
).
126.
M. T.
Rogers
and
L. D.
Kispert
,
J. Chem. Phys.
46
,
221
(
1967
).
127.
T.
Shiga
,
H.
Yamaoka
, and
A.
Lund
,
Z. Naturforsch., A
29
,
653
(
1974
).

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