We report an implementation of analytic second derivatives with respect to nuclear displacement of the Kohn–Sham energy for gradient‐corrected functionals. The second derivative equations are given in a form well‐suited for computer implementation, and the exchange‐correlation contributions are discussed in detail. The algorithms presented have favorable asymptotic exchange‐correlation cost scaling requirements relative to other aspects of the calculation. The results obtained show that analytic calculation of Kohn–Sham second derivatives is indeed a viable technique in practice.  

1.
J. A.
Pople
,
R.
Krishnan
,
H. B.
Schlegel
, and
J. S.
Binkley
,
Int. J. Quantum Chem. Symp.
13
,
225
(
1979
).
2.
J.
Gerratt
and
I. M.
Mills
,
J. Chem. Phys.
49
,
1719
(
1968
).
3.
J.
Gerratt
and
I. M.
Mills
,
J. Chem. Phys.
49
,
1730
(
1968
).
4.
P.
Hohenberg
and
W.
Kohn
,
Phys. Rev. B
136
,
864
(
1964
).
5.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev. A
140
,
1133
(
1965
).
6.
R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University, New York, 1989).
7.
R. O.
Jones
and
O.
Gunnarsson
,
Rev. Mod. Phys.
61
,
689
(
1989
).
8.
Density Functional Methods in Chemistry, edited by J. K. Labanowski and J. W. Andzelm (Springer, New York, 1991).
9.
T.
Ziegler
,
Chem. Rev.
91
,
651
(
1991
).
10.
P. A. M.
Dirac
,
Proc. Cam. Philos. Soc.
26
,
376
(
1930
).
11.
J. C. Slater, The Self-Consistent Field for Molecules and Solids, Quantum Theory of Molecules and Solids (McGraw-Hill, New York, 1974), Vol. 4.
12.
G.
Brual
and
S. M.
Rothstein
,
J. Chem. Phys.
69
,
1177
(
1978
).
13.
S. H.
Vosko
,
L.
Wilk
, and
M.
Nusair
,
Can. J. Phys.
58
,
1200
(
1980
).
14.
J. P.
Perdew
,
Phys. Rev. B
33
,
8822
(
1986
).
15.
A. D.
Becke
,
Phys. Rev. A
38
,
3098
(
1988
).
16.
A.
Cedillo
,
J.
Robles
, and
J. L.
Gásquez
,
Phys. Rev. A
38
,
1697
(
1988
).
17.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
,
Phys. Rev. B
37
,
785
(
1988
).
18.
J. P.
Perdew
,
J. A.
Chevary
,
S. H.
Vosko
,
K. A.
Jackson
,
M. R.
Pederson
,
D. J.
Singh
, and
C.
Fiolhais
,
Phys. Rev. B
46
,
6671
(
1992
).
19.
A. D.
Becke
,
J. Chem. Phys.
96
,
2155
(
1992
).
20.
A. D.
Becke
,
J. Chem. Phys.
97
,
9173
(
1992
).
21.
A. D.
Becke
,
J. Chem. Phys.
98
,
5648
(
1993
).
22.
J.
Andzelm
and
E.
Wimmer
,
J. Chem. Phys.
96
,
1280
(
1992
).
23.
C. W.
Murray
,
G. J.
Laming
,
N. C.
Handy
, and
R. D.
Amos
,
Chem. Phys. Lett.
199
,
551
(
1992
).
24.
P. M. W.
Gill
,
B. G.
Johnson
,
J. A.
Pople
, and
M. J.
Frisch
,
Int. J. Quantum Chem. Symp.
26
,
319
(
1992
).
25.
P. M. W.
Gill
,
B. G.
Johnson
,
J. A.
Pople
, and
M. J.
Frisch
,
Chem. Phys. Lett.
197
,
499
(
1992
).
26.
B. G.
Johnson
,
P. M. W.
Gill
, and
J. A.
Pople
,
J. Chem. Phys.
98
,
5612
(
1993
).
27.
J. M.
Seminario
,
Chem. Phys. Lett.
206
,
547
(
1993
).
28.
P. M. W. Gill, B. G. Johnson, C. A. Gonzalez, C. A. White, D. R. Maurice, and M. Head-Gordon, Q-CHEM, Q-Chem, Inc., Pittsburgh, Pennsylvania, 1993.
29.
M. J. Frisch, G. W. Trucks, M. Head-Gordon, P. M. W. Gill, M. W. Wong, J. B. Foresman, B. G. Johnson, H. B. Schlegel, M. A. Robb, E. S. Replogle, R. Gomperts, J. L. Andres, K. Raghavachari, J. S. Binkley, C. Gonzalez, R. L. Martin, D. J. Fox, D. J. Defrees, J. Baker, J. J. P. Stewart, and J. A. Pople, GAUSSIAN 92, Gaussian, Inc., Pittsburgh, Pennsylvania, 1992.
30.
R. D. Amos, I. L. Alberts, J. S. Andrews, S. M. Colwell, N. C. Handy, D. Jayatilaka, P. J. Knowles, R. Kobayashi, N. Koga, K. E. Laidig, P. E. Maslen, C. W. Murray, J. E. Rice, J. Sanz, E. D. Simandiras, A. J. Stone, and M.-D. Su, CADPAC, The Cambridge Analytic Derivatives Package, Issue 5.2, Cambridge, 1992.
31.
A. D.
Becke
,
Int. J. Quantum Chem. Symp.
23
,
1280
(
1989
).
32.
B.
Delley
,
J. Chem. Phys.
92
,
508
(
1990
).
33.
D. R. Salahub, R. Fournier, P. Mlynarski, I. Papai, A. S. Amant, and J. Ushio, in Density Functional Methods in Chemistry, edited by J. K. Labanowski and J. W. Andzelm (Springer, New York, 1991).
34.
J.
Andzelm
and
E.
Wimmer
,
Physica B
172
,
307
(
1991
).
35.
E. J.
Baerends
,
D. E.
Ellis
, and
P.
Ros
,
Chem. Phys.
2
,
41
(
1973
).
36.
R.
Fournier
,
J. Chem. Phys.
92
,
5422
(
1990
).
37.
B. I.
Dunlap
and
J.
Andzelm
,
Phys. Rev. A
45
,
81
(
1992
).
38.
A.
Komornicki
and
G.
Fitzgerald
,
J. Chem. Phys.
98
,
1398
(
1993
).
39.
N. C.
Handy
,
D. J.
Tozer
,
C. W.
Murray
,
G. J.
Laming
, and
R. D.
Amos
,
Isr. J. Chem.
33
,
331
(
1993
).
40.
J. A.
Pople
and
R. K.
Nesbet
,
J. Chem. Phys.
22
,
571
(
1954
).
41.
C. C. J.
Roothaan
,
Rev. Mod. Phys.
23
,
69
(
1951
).
42.
B. I.
Dunlap
,
J. W. D.
Connolly
, and
J. R.
Sabin
,
J. Chem. Phys.
71
,
3396
(
1979
).
43.
A. D.
Becke
,
J. Chem. Phys.
98
,
1372
(
1993
).
44.
P. M. W. Gill, Adv. Quantum Chem. (in press).
45.
J. A.
Pople
,
P. M. W.
Gill
, and
B. G.
Johnson
,
Chem. Phys. Lett.
199
,
557
(
1992
).
46.
K.
Kobayashi
,
N.
Kurita
,
H.
Kumahora
, and
K.
Tago
,
Phys. Rev. A
43
,
5810
(
1991
).
47.
M. J.
Frisch
,
M.
Head-Gordon
, and
J. A.
Pople
,
Chem. Phys.
141
,
189
(
1990
).
48.
P. M. W. Gill, B. G. Johnson, and J. A. Pople, Chem. Phys. Lett. (submitted).
49.
M. J.
Frisch
,
B. G.
Johnson
,
P. M. W.
Gill
,
D. J.
Fox
, and
R. H.
Nobes
,
Chem. Phys. Lett.
206
,
225
(
1993
).
50.
B. G.
Johnson
,
P. M. W.
Gill
, and
J. A.
Pople
,
Chem. Phys. Lett.
206
,
229
(
1993
).
51.
A. D.
Becke
,
J. Chem. Phys.
88
,
2547
(
1988
).
52.
C. W.
Murray
,
N. C.
Handy
, and
G. J.
Laming
,
Mol. Phys.
78
,
997
(
1993
).
53.
P. M. W.
Gill
,
B. G.
Johnson
, and
J. A.
Pople
,
Chem. Phys. Lett.
209
,
506
(
1993
).
54.
V. I.
Lebedev
,
Zy. Vychisl. Mat. Mat. Fiz.
16
,
293
(
1976
).
55.
V. I.
Lebedev
,
Sibirsk. Mat. Zh.
18
,
132
(
1977
).
56.
B. G.
Johnson
and
M. J.
Frisch
,
Chem. Phys. Lett.
216
,
133
(
1993
).
57.
M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. W. Wong, J. B. Foresman, M. A. Robb, M. Head-Gordon, E. S. Replogle, R. Gomperts, J. L. Andres, K. Raghavachari, J. S. Binkley, C. Gonzalez, R. L. Martin, D. J. Fox, D. J. Defrees, J. Baker, J. J. P. Stewart, and J. A. Pople, GAUSSIAN 92/DFT, Revision F.4, Gaussian, Inc., Pittsburgh, Pennsylvania, 1993.
58.
R. S.
Jones
,
J. W.
Mintmire
, and
B. I.
Dunlap
,
Int. J. Quantum Chem. Symp.
22
,
77
(
1988
).
59.
J. W. Andzelm, in Density Functional Methods in Chemistry, edited by J. K. Labanowski and J. W. Andzelm (Springer, New York, 1991).
60.
B. G. Johnson, P. M. W. Gill, and J. A. Pople, Chem. Phys. Lett. (submitted).
61.
G. W. Trucks and M. J. Frisch (to be published).
This content is only available via PDF.
You do not currently have access to this content.