We introduce both rigorous and non-rigorous distance-dependent integral estimates for four-center two-electron integrals derived from a distance-including Schwarz-type inequality. The estimates are even easier to implement than our so far most efficient distance-dependent estimates [S. A. Maurer et al., J. Chem. Phys. 136, 144107 (2012)] and, in addition, do not require well-separated charge-distributions. They are also applicable to a wide range of two-electron operators such as those found in explicitly correlated theories and in short-range hybrid density functionals. For two such operators with exponential distance decay [er12 and erfc(0.11r12)/r12], the rigorous bound is shown to be much tighter than the standard Schwarz estimate with virtually no error penalty. The non-rigorous estimate gives results very close to an exact screening for these operators and for the long-range 1/r12 operator, with errors that are completely controllable through the integral screening threshold. In addition, we present an alternative form of our non-rigorous bound that is particularly well-suited for improving the PreLinK method [J. Kussmann and C. Ochsenfeld, J. Chem. Phys. 138, 134114 (2013)] in the context of short-range exchange calculations.

1.
J. L.
Whitten
,
J. Chem. Phys.
58
,
4496
(
1973
).
2.
M.
Häser
and
R.
Ahlrichs
,
J. Comput. Chem.
10
,
104
(
1989
).
3.
S. A.
Maurer
,
D. S.
Lambrecht
,
D.
Flaig
, and
C.
Ochsenfeld
,
J. Chem. Phys.
136
,
144107
(
2012
).
4.
D. S.
Hollman
,
H. F.
Schaefer
, and
E. F.
Valeev
,
J. Chem. Phys.
142
,
154106
(
2015
).
5.
M.
Beuerle
,
J.
Kussmann
, and
C.
Ochsenfeld
,
J. Chem. Phys.
146
,
144108
(
2017
).
6.
C.
Hättig
,
W.
Klopper
,
A.
Köhn
, and
D. P.
Tew
,
Chem. Rev.
112
,
4
(
2012
).
7.
L.
Kong
,
F. A.
Bischoff
, and
E. F.
Valeev
,
Chem. Rev.
112
,
75
(
2012
).
8.
J.
Heyd
,
G. E.
Scuseria
, and
M.
Ernzerhof
,
J. Chem. Phys.
118
,
8207
(
2003
).
9.
S.
Ten-no
,
Chem. Phys. Lett.
398
,
56
(
2004
).
10.
K. A.
Peterson
,
T. B.
Adler
, and
H. J.
Werner
,
J. Chem. Phys.
128
,
084102
(
2008
).
11.
A. V.
Krukau
,
O. A.
Vydrov
,
A. F.
Izmaylov
, and
G. E.
Scuseria
,
J. Chem. Phys.
125
,
224106
(
2006
).
12.
J. G.
Brandenburg
,
E.
Caldeweyher
, and
S.
Grimme
,
Phys. Chem. Chem. Phys.
18
,
15519
(
2016
).
13.
R.
Peverati
and
D. G.
Truhlar
,
Phys. Chem. Chem. Phys.
14
,
16187
(
2012
).
14.
J.
Kussmann
,
M.
Beer
, and
C.
Ochsenfeld
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
3
,
614
(
2013
).
15.
C. A.
White
,
B. G.
Johnson
,
P. M.
Gill
, and
M.
Head-Gordon
,
Chem. Phys. Lett.
230
,
8
(
1994
).
16.
E.
Schwegler
,
M.
Challacombe
, and
M.
Head-Gordon
,
J. Chem. Phys.
106
,
9708
(
1997
).
17.
C.
Ochsenfeld
,
C. A.
White
, and
M.
Head-Gordon
,
J. Chem. Phys.
109
,
1663
(
1998
).
18.
C.
Ochsenfeld
,
Chem. Phys. Lett.
327
,
216
(
2000
).
19.
H.-A.
Le
and
T.
Shiozaki
, e-print arXiv:1708.05353v1 (
2017
).
20.
J.
Kussmann
and
C.
Ochsenfeld
,
J. Chem. Phys.
138
,
134114
(
2013
).
21.
C. C. J.
Roothaan
,
Rev. Mod. Phys.
23
,
69
(
1951
).
22.
T.
Helgaker
,
P.
Jorgensen
, and
J.
Olsen
,
Molecular Electronic-Structure Theory
(
Wiley
,
2000
), pp.
431
432
.
23.
E. H.
Lieb
and
M.
Loss
,
Analysis
, 2nd ed. (
American Mathematical Society
,
2001
).
24.
E. H.
Lieb
and
R.
Seiringer
,
The Stability of Matter in Quantum Mechanics
(
Cambridge University Press
,
2010
).
25.
J.
Heyd
and
G. E.
Scuseria
,
J. Chem. Phys.
121
,
1187
(
2004
).
26.
T. B.
Adler
,
H.-J.
Werner
, and
F. R.
Manby
,
J. Chem. Phys.
130
,
054106
(
2009
).
27.
S.
Bochner
,
Lectures on Fourier Integrals
(
Princeton University Press
,
1959
).
28.
L.
Grafakos
and
G.
Teschl
,
J. Fourier Anal. Appl.
19
,
167
(
2013
).
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