We present an analytical approach to treat higher order derivatives of Hartree-Fock (HF) and Kohn-Sham (KS) density functional theory energy in the Born-Oppenheimer approximation with respect to the nuclear charge distribution (so-called alchemical derivatives). Modified coupled perturbed self-consistent field theory is used to calculate molecular systems response to the applied perturbation. Working equations for the second and the third derivatives of HF/KS energy are derived. Similarly, analytical forms of the first and second derivatives of orbital energies are reported. The second derivative of Kohn-Sham energy and up to the third derivative of Hartree-Fock energy with respect to the nuclear charge distribution were calculated. Some issues of practical calculations, in particular the dependence of the basis set and Becke weighting functions on the perturbation, are considered. For selected series of isoelectronic molecules values of available alchemical derivatives were computed and Taylor series expansion was used to predict energies of the “surrounding” molecules. Predicted values of energies are in unexpectedly good agreement with the ones computed using HF/KS methods. Presented method allows one to predict orbital energies with the error less than 1% or even smaller for valence orbitals.

1.
C.
Wolverton
and
A.
Zunger
,
Phys. Rev. Lett.
81
,
606
(
1998
).
2.
M.
Wang
,
X.
Hu
,
D. N.
Beratan
, and
W.
Yang
,
J. Am. Chem. Soc.
128
,
3228
(
2006
).
3.
P.
Kirkpatrick
and
C.
Ellis
,
Nature (London)
432
,
823
(
2004
).
4.
R.
Lahana
,
Drug Discov. Today
4
,
447
(
1999
).
5.
A.
Franceschetti
and
A.
Zunger
,
Nature (London)
402
,
60
(
1999
).
6.
G. H.
Jóhannesson
,
T.
Bligaard
,
A. V.
Ruban
,
H. L.
Skriver
,
K. W.
Jacobsen
, and
J. K.
Nørskov
,
Phys. Rev. Lett.
88
,
255506
(
2002
).
7.
O. A. von
Lilienfeld
and
M. E.
Tuckerman
,
J. Chem. Phys.
125
,
154104
(
2006
).
8.
R. G.
Parr
,
R. A.
Donnely
,
M.
Levy
, and
W. E.
Palke
,
J. Chem. Phys.
68
,
3801
(
1978
).
9.
R. G.
Parr
and
R. G.
Pearson
,
J. Am. Chem. Soc.
105
,
7512
(
1983
).
10.
P.
Geerlings
,
F.
De Proft
, and
W.
Langenaeker
,
Chem. Rev.
103
,
1793
(
2003
).
11.
D.
Sheppard
,
G.
Henkelman
, and
O. A. von
Lilienfeld
,
J. Chem. Phys.
133
,
084104
(
2010
).
12.
O. A. von
Lilienfeld
,
J. Chem. Phys.
131
,
164102
(
2009
).
13.
R. P.
Feynman
,
Phys. Rev.
56
,
340
(
1939
).
14.
R. M.
Stevens
,
W. N.
Lipscomb
, and
R. M.
Pitzer
,
J. Chem. Phys.
38
,
550
(
1963
).
15.
J.
Gerratt
and
I. M.
Mills
,
J. Chem. Phys.
49
,
1719
(
1968
).
16.
J.
Gerratt
and
I. M.
Mills
,
J. Chem. Phys.
49
,
1730
(
1968
).
17.
P.
Pulay
,
Mol. Phys.
17
,
197
(
1969
).
18.
P.
Pulay
,
Mol. Phys.
18
,
473
(
1970
).
19.
J. A.
Pople
,
R.
Krishnan
,
H. B.
Schlegel
, and
J. S.
Binkley
,
Int. J. Quantum Chem.
16
,
225
(
1979
).
20.
Y.
Osamura
,
Y.
Yamaguchi
,
P.
Saxe
,
D. J.
Foxa
,
M. A.
Vincent
, and
H. F.
Schaefer
 III
,
J. Mol. Struct.: THEOCHEM
12
,
183
(
1983
).
21.
M.
Frisch
,
M.
Headgordon
, and
J.
Pople
,
Chem. Phys.
141
,
189
(
1990
).
22.
N. C.
Handy
,
D. J.
Tozer
,
G. J.
Laming
,
C. W.
Murray
, and
R. D.
Amos
,
Isr. J. Chem.
33
,
331
(
1993
).
23.
B. G.
Johnson
,
P. M. W.
Gill
, and
J. A.
Pople
,
J. Chem. Phys.
98
,
5612
(
1993
).
24.
B. G.
Johnson
and
M. J.
Frisch
,
J. Chem. Phys.
100
,
7429
(
1994
).
25.
Y.
Yamaguchi
,
Y.
Osamura
,
J. D.
Goddard
, and
H. F.
Schaefer
 III
,
A New Dimension to Quantum Chemistry: Analytic Derivative Methods in Ab Initio Molecular Electronic Structure
(
Oxford University Press
,
New York
,
1994
).
26.
J.
Baker
,
K.
Wolinski
,
M.
Malagoli
, and
P.
Pulay
,
Mol. Phys.
102
,
2475
(
2004
).
27.
S. M.
Colwell
,
C. W.
Murray
,
N. C.
Handy
, and
R. D.
Amos
,
Chem. Phys. Lett.
210
,
261
(
1993
).
28.
S. J. A. van
Gisbergen
,
J. G.
Snijders
, and
E. J.
Baerends
,
J. Chem. Phys.
109
,
10644
(
1998
).
29.
S. J. A. van
Gisbergen
,
J. G.
Snijders
, and
E. J.
Baerends
,
J. Chem. Phys.
103
,
9347
(
1995
).
30.
P.
Geerlings
and
F. De
Proft
,
Phys. Chem. Chem. Phys.
10
,
3028
(
2008
).
31.
R.
Balawender
and
L.
Komorowski
,
J. Chem. Phys.
109
,
5203
(
1998
).
32.
R.
Balawender
,
L.
Komorowski
,
F. De
Proft
, and
P.
Geerlings
,
J. Chem. Phys.
102
,
9912
(
1998
).
33.
R.
Balawender
and
P.
Geerlings
,
J. Chem. Phys.
114
,
682
(
2001
).
34.
R.
Balawender
and
P.
Geerlings
,
J. Chem. Phys.
114
,
4441
(
2001
).
35.
R.
Balawender
and
P.
Geerlings
,
J. Chem. Phys.
123
,
124103
(
2005
).
36.
P. E.
Maslen
,
N. C.
Handy
,
R. D.
Amos
, and
D.
Jayatilaka
,
J. Chem. Phys.
97
,
4233
(
1992
).
37.
P. E.
Maslen
,
D.
Jayatilaka
,
S. M.
Colwell
,
R. D.
Amos
, and
N. C.
Handy
,
J. Chem. Phys.
95
,
7409
(
1991
).
38.
S. M.
Colwell
,
D.
Jayatilaka
,
P. E.
Maslen
,
R. D.
Amos
, and
N. C.
Handy
,
Int. J. Quantum Chem.
40
,
179
(
1991
).
39.
R.
Bast
,
U.
Ekström
,
B.
Gao
,
T.
Helgaker
,
K.
Ruud
, and
A. J.
Thorvaldsen
,
Phys. Chem. Chem. Phys.
13
,
2627
(
2011
).
40.
A. O. von
Lilienfeld
,
R. D.
Lins
, and
U.
Rothlisberger
,
Phys. Rev. Lett.
95
,
153002
(
2005
).
41.
C. C. J.
Roothaan
,
Rev. Mod. Phys.
23
,
69
(
1951
).
42.
J. A.
Pople
and
R. K.
Nesbet
,
J. Chem. Phys.
22
,
571
(
1954
).
43.
P.
Hohenberg
and
W.
Kohn
,
Phys. Rev.
136
,
B864
(
1964
).
44.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
140
,
A1133
(
1965
).
45.
A.
D.
Becke
,
J. Chem. Phys.
98
,
1372
(
1993
).
46.
A.
D.
Becke
,
J. Chem. Phys.
98
,
5648
(
1993
).
47.
S.
Grimme
,
J. Chem. Phys.
124
,
034108
(
2006
).
48.
F.
Liu
,
Z.
Gan
,
Y.
Shao
,
C.-P.
Hsu
,
A.
Dreuw
,
M.
Head-Gordon
,
B. T.
Miller
,
B. R.
Brooks
,
J.-G.
Yu
,
T. R.
Furlani
, and
J.
Kong
,
Mol. Phys.
108
,
2791
(
2010
).
49.
U.
Ekström
,
L.
Visscher
,
R.
Bast
,
A. J.
Thorvaldsen
, and
K.
Ruud
,
J. Chem. Theory Comput.
6
,
1971
(
2010
).
50.
P. A. M.
Dirac
,
Proc. Cambridge Philos. R. Soc.
26
,
376
(
1930
).
51.
S. H.
Vosko
,
L.
Wilk
and
M.
Nusair
,
Can. J. Phys.
58
,
1200
(
1980
).
52.
J. P.
Perdew
and
Y.
Wang
,
Phys. Rev. B
45
,
13244
(
1992
).
53.
A.
D.
Becke
,
J. Chem. Phys.
88
,
2547
(
1987
).
54.
V. I.
Lebedev
,
Zh. Vychisl. Mat. Mat. Fiz.
15
,
48
(
1975
).
55.
V. I.
Lebedev
,
Zh. Vychisl. Mat. Mat. Fiz.
16
,
293
(
1976
).
56.
V. I.
Lebedev
,
Sibirsk. Mat. Zh.
18
,
132
(
1977
).
57.
A.
W.
Murray
,
N. C.
Handy
, and
G. J.
Laming
,
Mol. Phys.
78
,
997
(
1993
).
58.
J. C.
Slater
,
J. Chem. Phys.
41
,
3199
(
1964
).
59.
A.
G.
Johnson
,
P. M. W.
Gill
, and
J. A.
Pople
,
J. Chem. Phys.
98
,
5612
(
1992
).
60.
B. G.
Johnson
and
M. J.
Fisch
,
J. Chem. Phys.
100
,
7429
(
1994
).
61.
M.
Malagoli
and
J.
Baker
,
J. Chem. Phys.
119
,
12763
(
2003
).
62.
B. G.
Johnson
and
M. J.
Frisch
,
Chem. Phys. Lett.
216
,
133
(
1993
).
63.
O.
Treutler
and
R.
Ahlrichs
,
J. Chem. Phys.
102
,
346
(
1994
).
64.
T. H.
Dunning
 Jr.
,
J. Chem. Phys.
90
,
1007
(
1989
).
65.
R. A.
Kendall
,
T. H.
Dunning
 Jr.
, and
R. J.
Harrison
,
J. Chem. Phys.
96
,
6796
(
1992
).
66.
M. W.
Schmidt
,
K. K.
Baldridge
,
J. A.
Boatz
,
S. T.
Elbert
,
M. S.
Gordon
,
J. H.
Jensen
,
S.
Koseki
,
N.
Matsunaga
,
K. A.
Nguyen
,
S.
Su
,
T. L.
Windus
,
M.
Dupuis
, and
J. A.
Montgomery
,
J. Comput. Chem.
14
,
1347
(
1993
).
67.
M. S.
Gordon
and
M. W.
Schmidt
Advances in electronic structure theory: GAMESS a decade later
,” in
Theory and Applications of Computational Chemistry: The First Forty Years
, edited by
C. E.
Dykstra
,
G.
Frenking
,
K. S.
Kim
, and
G. E.
Scuseria
(
Elsevier
,
Amsterdam
2005
), pp.
1167
1189
.
68.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
[PubMed]
[Erratum
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
78
,
1396
(
1997
)].
69.
C.
Adamo
and
V.
Barone
,
J. Chem. Phys.
110
,
6158
(
1999
).
You do not currently have access to this content.