We have developed a combined quantum mechanics/molecular mechanics (QM/MM) method with periodic boundary condition (PBC) treatment of explicit electron-charge interactions in a theoretically rigorous manner, for an accurate description of electronic structures for molecules in the condensed phase. The Ewald summation technique is employed for the calculation of the one-electron Hamiltonian in an ab initio framework. We decompose the Coulomb interactions into two components: those within the same cell and those between different cells. The former is calculated in the same way as the conventional QM/MM calculation for isolated systems; this article focuses on our novel method for calculating the latter type of Coulomb interactions. The detailed formulation of the Hamiltonian of this new QM/MM-PBC method, as well as the necessary one-electron integrals and their gradients, is given. The novel method is assessed by applying it to the dilute water system and a system with a coumarin molecule in water solvent; it successfully reproduces the electronic energies, frontier orbital energies, and Mulliken population charge of the real-space limit calculated by QM/MM using large isolated systems. We investigated the contribution from each term of the Hamiltonian and found that the surface-dipole term in the Ewald summation technique is indispensable for QM/MM-PBC calculations. The newly developed QM/MM-PBC method is promising for tackling chemical reactions and excited states of molecules in the condensed phase.

1.
A.
Warshel
and
M. J.
Levitt
,
J. Mol. Biol.
103
,
227
(
1976
).
2.
J.
Gao
and
D. G.
Truhlar
,
Annu. Rev. Phys. Chem.
53
,
467
(
2002
).
3.
A.
Warshel
,
Annu. Rev. Biophys. Biomol. Struct.
32
,
425
(
2003
).
4.
B.
McCann
and
O.
Acevedo
,
J. Chem. Theory Comput.
9
,
944
(
2013
).
5.
O.
Acevedo
,
J. Phys. Chem. A
118
,
11653
(
2014
).
6.
O.
Acevedo
and
W. L.
Jorgensen
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
4
,
422
(
2014
).
7.
D.
Fang
,
R. E.
Duke
, and
G. A.
Cisneros
,
J. Chem. Phys.
143
,
044103
(
2015
).
8.
R. G.
Alden
,
W. W.
Parson
,
Z. T.
Chu
, and
A.
Warshel
,
J. Am. Chem. Soc.
117
,
12284
(
1995
).
9.
A.
Warshel
and
A.
Papazyan
,
Curr. Opin. Struct. Biol.
8
,
211
(
1998
).
10.
W.
Im
,
S.
Bernèche
, and
B. T.
Roux
,
J. Chem. Phys.
114
,
2924
(
2001
).
11.
P.
Schaefer
,
D.
Riccardi
, and
Q.
Cui
,
J. Chem. Phys.
123
,
014905
(
2005
).
12.
T.
Benighaus
and
W.
Thiel
,
J. Chem. Theory Comput.
5
,
3114
(
2009
).
13.
T.
Benighaus
and
W.
Thiel
,
J. Chem. Theory Comput.
7
,
238
(
2011
).
14.
A.
Aleksandrov
and
M.
Field
,
Phys. Chem. Chem. Phys.
13
,
10503
(
2011
).
15.
S. W.
de Leeuw
,
J. W.
Perram
, and
E. R.
Smith
,
Proc. R. Soc. London, Ser. A
373
,
27
(
1980
).
16.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulation: From Algorithms to Applications
(
Academic Press
,
San Diego
,
2002
).
17.
J.
Gao
,
J. Phys. Chem.
96
,
6432
(
1992
).
18.
J.
Gao
and
C.
Alhambra
,
J. Chem. Phys.
107
,
1212
(
1997
).
19.
D.
Riccardi
,
P.
Schaefer
, and
Q.
Cui
,
J. Phys. Chem. B
109
,
17715
(
2005
).
20.
F.
Dehez
,
M. T. C.
Martins-Costa
,
D.
Rinaldi
, and
C.
Millot
,
J. Chem. Phys.
122
,
234503
(
2005
).
21.
G. d. M.
Seabra
,
R. C.
Walker
,
M.
Elstner
,
D. A.
Case
, and
A. E.
Roitberg
,
J. Phys. Chem. A
111
,
5655
(
2007
).
22.
H.
Nishizawa
and
H.
Okumura
,
J. Comput. Chem.
37
,
2701
(
2016
).
23.
K.
Nam
,
J.
Gao
, and
D. M.
York
,
J. Chem. Theory Comput.
1
,
2
(
2005
).
24.
T.
Laino
,
F.
Mohamed
,
A.
Laio
, and
M.
Parrinello
,
J. Chem. Theory Comput.
1
,
1176
(
2005
).
25.
T.
Laino
,
F.
Mohamed
,
A.
Laio
, and
M.
Parrinello
,
J. Chem. Theory Comput.
2
,
1370
(
2006
).
26.
R. C.
Walker
,
M. F.
Crowley
, and
D. A.
Case
,
J. Comput. Chem.
29
,
1019
(
2008
).
27.
W.
Xie
,
L.
Song
,
D. G.
Truhlar
, and
J.
Gao
,
J. Chem. Phys.
128
,
234108
(
2008
).
28.
L. D.
Jacobson
and
J. M.
Herbert
,
J. Chem. Phys.
134
,
094118
(
2011
).
29.
C. F.
Sanz-Navarro
,
R.
Grima
,
A.
García
,
E. A.
Bea
,
A.
Soba
,
J. M.
Cela
, and
P.
Ordejón
,
Theor. Chem. Acc.
128
,
825
(
2011
).
30.
J. M.
Herbert
,
L. D.
Jacobson
,
K. U.
Lao
, and
M. A.
Rohrdanz
,
Phys. Chem. Chem. Phys.
14
,
7679
(
2012
).
31.
P.
Zhang
,
D. G.
Truhlar
, and
J.
Gao
,
Phys. Chem. Chem. Phys.
14
,
7821
(
2012
).
32.
Z. C.
Holden
,
R. M.
Richard
, and
J. M.
Herbert
,
J. Chem. Phys.
139
,
244108
(
2013
).
33.
Z. C.
Holden
,
R. M.
Richard
, and
J. M.
Herbert
,
J. Chem. Phys.
142
,
059901
(
2015
).
34.
K.
Nam
,
J. Chem. Theory Comput.
10
,
4175
(
2014
).
35.
K.
Doll
and
T.
Jacob
,
J. Theor. Comput. Chem.
14
,
1550054
(
2015
).
36.
T.
Vasilevskaya
and
W.
Thiel
,
J. Chem. Theory Comput.
12
,
3561
(
2016
).
37.
M.
Matsui
and
S.
Sakaki
,
J. Phys. Chem. C
121
,
20242
(
2017
).
38.
L. R.
McMurchie
and
E. R.
Davidson
,
J. Chem. Phys.
26
,
218
(
1978
).
39.
H. F.
King
and
M.
Dupuis
,
J. Comput. Phys.
21
,
144
(
1976
).
40.
M.
Dupuis
,
J.
Rys
, and
H. F.
King
,
J. Chem. Phys.
65
,
111
(
1976
).
41.
J.
Rys
,
M.
Dupuis
, and
H. F.
King
,
J. Comput. Chem.
4
,
154
(
1983
).
42.
S.
Obara
and
A.
Saika
,
J. Chem. Phys.
84
,
3963
(
1986
).
43.
S.
Obara
and
A.
Saika
,
J. Chem. Phys.
89
,
1540
(
1988
).
44.
M.
Tachikawa
and
M.
Shiga
,
Phys. Rev. E
64
,
056706
(
2001
).
45.
K.
Ishimura
, SMASH Program 2.2.0, http://smash-qc.sourceforge.net,
2017
.
46.
47.
W. L.
Jorgensen
,
J.
Chandrasekhar
,
J. D.
Madura
,
R. W.
Impey
, and
M. L.
Klein
,
J. Chem. Phys.
79
,
926
(
1983
).
48.
T.
Sakata
,
Y.
Kawashima
, and
H.
Nakano
,
J. Chem. Phys.
134
,
014501
(
2011
).
49.
T. H.
Dunning
,
J. Chem. Phys.
90
,
1007
(
1989
).
51.
W. L.
Slattery
,
G. D.
Doolen
, and
H. E.
DeWitt
,
Phys. Rev. A
21
,
2087
(
1980
).
52.
B.
Cichocki
,
B. U.
Felderhof
, and
K.
Hinsen
,
Phys. Rev. A
39
,
5350
(
1980
).
53.
G.
Hummer
and
D. M.
Soumpasis
,
J. Chem. Phys.
98
,
581
(
1993
).
54.
J.-M.
Caillol
,
J. Chem. Phys.
101
,
6080
(
1994
).
55.
G.
Hummer
,
L. R.
Pratt
, and
A. E.
Garcia
,
J. Phys. Chem.
99
,
14188
(
1995
).
56.
V.
Ballenegger
and
J.-P.
Hansen
,
J. Chem. Phys.
122
,
114711
(
2005
).
57.
H. D.
Herce
,
A. E.
Garcia
, and
T.
Darden
,
J. Chem. Phys.
126
,
124106
(
2007
).
58.
V.
Ballenegger
,
J. J.
Cerda
,
O.
Lenz
, and
Ch.
Holm
,
J. Chem. Phys.
128
,
034109
(
2008
).
59.
V.
Ballenegger
,
J. Chem. Phys.
140
,
161102
(
2014
).
60.
Y.
Kawashima
,
M.
Dupuis
, and
K.
Hirao
,
J. Chem. Phys.
117
,
248
(
2002
).
61.
M.
Shiga
and
M.
Tachikawa
,
Mol. Simul.
33
,
171
(
2007
).
62.
M.
Shiga
and
M.
Masia
,
J. Chem. Phys.
139
,
144103
(
2013
).

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