The “chain of spheres” (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix,

$\mathbf {Q}$
Q⁠, which leads the numerical and analytically evaluated overlap matrices to coincide. Using
$\mathbf {Q}$
Q
, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calculations and geometry optimizations. The fitting procedure results in a reduction of grid sizes necessary for achieving chemical accuracy. We demonstrate this by testing a number of grids and comparing results to the fully analytic and the earlier COS approximations. This turns out to be favourable for total and reaction energies, for which chemical accuracy can now be reached with a corresponding ∼30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favourable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theoretical framework.

1.
F.
Neese
,
F.
Wennmohs
,
A.
Hansen
, and
U.
Becker
,
Chem. Phys.
356
,
98
(
2009
).
2.
I.
Panas
and
J.
Almlöf
,
Int. J. Quantum Chem.
42
,
1073
(
1992
).
3.
G. R.
Ahmadi
and
J.
Almlöf
,
Chem. Phys. Lett.
246
,
364
(
1995
).
4.
J.
Almlof
,
K.
Faegri
 Jr.
, and
K.
Korsell
,
J. Comp. Chem.
3
,
385
(
1982
).
5.
I.
Panas
,
J.
Almlöf
, and
M. W.
Feyereisen
,
Int. J. Quantum Chem.
40
,
797
(
1991
).
6.
E. J.
Baerends
,
D. E.
Ellis
, and
P.
Ros
,
Chem. Phys.
2
,
41
(
1973
).
7.
J. L.
Whitten
,
J. Chem. Phys.
58
,
4496
(
1973
).
8.
N. H. F.
Beebe
and
J.
Linderberg
,
Int. J. Quantum. Chem.
12
,
683
(
1977
).
9.
F.
Aquilante
,
T. B.
Pedersen
, and
R.
Lindh
,
J. Chem. Phys.
127
,
194106
(
2007
).
10.
T. B.
Pedersen
,
F.
Aquilante
, and
R.
Lindh
,
Theor. Chem. Acc.
124
,
1
(
2009
).
11.
F.
Aquilante
,
L.
Boman
,
J.
Boström
,
H.
Koch
,
R.
Lindh
,
A. S.
de Merás
, and
T. B.
Pedersen
, in
Linear-Scaling Techniques in Computational Chemistry and Physics. Methods and Applications
, edited by
R.
Zaleśny
,
M. G.
Papadopoulos
,
P. G.
Mezey
, and
J.
Leszczynski
(
Springer
,
New York
,
2011
), Vol. 13, pp.
301
343
.
12.
O.
Vahtras
,
J.
Almlöf
, and
M. W.
Feyereisen
,
Chem. Phys. Lett.
213
,
514
(
1993
).
13.
K.
Eichkorn
,
O.
Treutler
,
H.
Ohm
,
M.
Haser
, and
R.
Ahlrichs
,
Chem. Phys. Lett.
242
,
652
(
1995
).
14.
K.
Eichkorn
,
F.
Weigend
,
O.
Treutler
, and
R.
Ahlrichs
,
Theor. Chem. Acc.
97
,
119
(
1997
).
15.
F.
Neese
,
J. Comput. Chem.
24
,
1740
(
2003
).
16.
J.
Almlöf
, in
Modern Electronic Structure Theory
, edited by
D. R.
Yarkony
(
World Scientific
,
Singapore
,
1995
), p.
110
.
17.
M.
Challacombe
and
E.
Schwegler
,
J. Chem. Phys.
106
,
5526
(
1997
).
18.
C.
Ochsenfeld
,
C. A.
White
, and
M.
Head-Gordon
,
J. Chem. Phys.
109
,
1663
(
1998
).
19.
R.
Polly
,
H.-J.
Werner
,
F. R.
Manby
, and
P. J.
Knowles
,
Mol. Phys.
102
,
2311
(
2004
).
20.
F.
Weigend
,
Phys. Chem. Chem. Phys.
4
,
4285
(
2002
).
21.
C. M.
Carlson
,
J. Chem. Phys.
47
,
862
(
1967
).
22.
D. K.
Harris
and
C. M.
Carlson
,
J. Chem. Phys.
51
,
5458
(
1969
).
23.
V.
Termath
and
N. C.
Handy
,
Chem. Phys. Lett.
230
,
17
(
1994
).
24.
C.
van Wüllen
,
Chem. Phys. Lett.
245
,
648
(
1995
).
25.
J. C.
Light
,
I. P.
Hamilton
, and
J. V.
Lill
,
J. Chem. Phys.
82
,
1400
(
1985
).
26.
J. C.
Light
and
Z.
Bacic
,
J. Chem. Phys.
85
,
4594
(
1986
).
27.
R. A.
Friesner
,
Chem. Phys. Lett.
116
,
39
(
1985
).
28.
R. A.
Friesner
,
J. Chem. Phys.
85
,
1462
(
1986
).
29.
R. A.
Friesner
,
J. Chem. Phys.
86
,
3522
(
1987
).
30.
T. J.
Martinez
and
E. A.
Carter
, in
Modern Electronic Structure Theory
, edited by
D. R.
Yarkony
(
World Scientific
,
Singapore
,
1995
), p.
1132
.
31.
W.
Yang
and
A. C.
Peet
,
Chem. Phys. Lett.
153
,
98
(
1988
).
32.
R. A.
Friesner
,
J. Phys. Chem.
92
,
3091
(
1988
).
33.
S.
Ten-no
, private communication (
2010
).
34.
S.
Ten-no
,
J. Chem. Phys.
121
,
117
(
2004
).
35.
C. C.
Tazartes
,
C. R.
Anderson
, and
E. A.
Carter
,
J. Comput. Chem.
19
,
1300
(
1998
).
36.
W. Y.
Yang
,
W.
Cao
,
T.-S.
Chung
, and
J.
Morris
,
Applied Numerical Methods Using MATLAB
(
Wiley
,
Hoboken, NJ
,
2005
), p.
72
.
37.
Y.
Won
,
J.-G.
Lee
,
M. N.
Ringalda
, and
R. A.
Friesner
,
J. Chem. Phys.
94
,
8152
(
1991
).
38.
G. C.
Corey
and
J. W.
Tromp
,
J. Chem. Phys.
103
,
1812
(
1995
).
39.
F.
Neese
, ORCA, an ab initio density functional and semiempirical program package, University of Bonn, Germany,
2011
.
40.
A.
Schäfer
,
H.
Horn
, and
R.
Ahlrichs
,
J. Chem. Phys.
97
,
2571
(
1992
).
41.
A.
Schäfer
,
C.
Huber
, and
R.
Ahlrichs
,
J. Chem. Phys.
100
,
5829
(
1994
).
42.
F.
Weigend
,
F.
Furche
, and
R.
Ahlrichs
,
J. Chem. Phys.
119
,
12753
(
2003
).
43.
TURBOMOLE basis set library, ftp://ftp.chemie.uni-karlsruhe.de/pub/basen.
44.
M.
Krack
and
A. M.
Köster
,
J. Chem. Phys.
108
,
3226
(
1998
).
45.
P. M.W.
Gill
,
B. G.
Johnson
, and
J. A.
Pople
,
Chem. Phys. Lett.
209
,
506
(
1993
).
46.
J.
Baker
,
J.
Andzelm
,
A.
Scheiner
, and
B.
Delley
,
J. Chem. Phys.
101
,
8894
(
1994
).
47.
S.
Grimme
,
M.
Steinmetz
, and
M.
Korth
,
J. Org. Chem.
72
,
2118
(
2007
).
48.
F.
Neese
,
A.
Hansen
,
F.
Wennmohs
, and
S.
Grimme
,
Acc. Chem. Res.
42
,
641
(
2009
b).
49.
See supplementary material at http://dx.doi.org/10.1063/1.3646921 for deviations from the analytic value for each species in the test set.
50.
S.
Kossmann
and
F.
Neese
,
J. Chem. Theory Comput.
6
,
2325
(
2010
).

Supplementary Material

You do not currently have access to this content.