Supplementary materials for
The role of the exchange in the embedding electrostatic potential for
the fragment molecular orbital method
Dmitri G. Fedorov1, and Kazuo Kitaura (北浦和夫)1,2
1
RICS, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1
Umezono, Tsukuba, Ibaraki 305-8568, Japan.
2
Graduate School of Pharmaceutical Sciences, Kyoto University, Sakyo-ku, Kyoto
606-8501, Japan.
Fock matrix diagonalization
The Fock matrix obtained according to Eq. (1) is diagonalized once, and its
eigenvalues describe the orbital energies for the whole system. Because of the
nonorthogonality of the Gaussian basis sets and the spherical component transformation
(reducing, e.g., 6d into 5d), the actual set of equations is given by:
FC  SCε
(S1)
F  Q† FQ
(S2)
FC  Cε
(S3)
C  QC
(S4)
S   R † SR
(S5)
SV  Vs
(S6)
Q  RVs 1 / 2
(S7)
where C is the MO coefficient matrix, S is the overlap matrix,  and s are diagonal
matrices of eigenvalues of F and S, respectively. R is the transformation matrix
1
eliminating spherical contaminants from the basis set (e.g., s from Cartesian d
functions); it is block diagonal, and each block describes all components of an AO, e.g.,
for a d-function, the corresponding block is a 65 matrix.
As the matrix sizes can be huge, a careful implementation is needed for efficiency.
For instance, R is never constructed, instead the multiplications is performed on the fly
using the structure of R. Also, if the basis set has only s and p AOs, then the
transformation in Eq. (S5) are not needed. Obtaining MOs (C) is a more
resource-consuming part, requiring solving for C’ in Eq. (S3) and an extra
transformation in Eq. (S4).
Definition of the properties
N
N
I
I J
The FMO energy of N fragments is given by E FMO   E I   E IJ  E I  E J  ,
where EI and EIJ are the monomer (single fragment) and dimer (fragment pair) energies,
respectively, computed in the electrostatic field of other fragments.
The one and two-electron parts E mFMO of the energy (m=1,2) can be defined as,
E1FMO   E I1   E IJ1  E I1  E 1J 
N
N
I
I J
E2FMO  E FMO  E1FMO  E NR
(S8)
(S9)
where E NR is the nuclear repulsion energy (the same in FMO and ab initio).
No
†
~
E   ni CiX H X CiX
1
X
(S10)
i 1
where CiX are MOs of X, i runs over No occupied orbitals and ni are the occupation
numbers (equal to 2 for closed-shell methods).
2
~
H X  TX  v X  u X
(S11)
where T and v are the kinetic and Coulomb integrals (for charges in X), respectively,
and u describes the charges outside X, i.e., the first sum in U in Eq. (5). Similarly to Eq.
(S8), the kinetic energy T FMO can be defined.
Once the eigenvalues i and eigenvectors Ci of F are obtained, the total energy can
be defined as,
E FMO/F  E1FMO/F  E2FMO/F  E NR
No
(S12)
E1FMO/F   ni Ci HC i
†
(S13)
i 1
E
FMO/F
2
1
No
No
ni Ci FC i  1 E1FMO/F  1  ni  i  1 E1FMO/F
2
2
2 i 1
2
i 1
†
(S14)
where and H is given by the sum of the kinetic (T) and Coulomb (v) contributions,
similar to Eq. (S11) but describing the whole system. The same form of equations is
used for FMO/FX and FMO/XF (the difference being in the values of quantities
appearing in the equations, such as the Fock matrices and various energies).
Finally, we define the virial ratio R (ideally, it should be equal to 2) as:
R
E FMO  T FMO
T FMO
(S15)
(for FMO/F the kinetic energy T is obtained similarly to Eq. (S13)).
For the RESDIM approximation replacing SCF of far separated dimers IJ by the
electrostatic interaction, we used F IJ  F I  F J in Eq. (1). For consistency, the same
relation was used in H (Eq. S13) and T (i.e., the corresponding off-diagonal blocks for
such IJ were set to zero in these matrices).
The graphical representation of errors is shown in Figure SII.
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Table SI. Accuracy of the componentsa of the total energy for phenol+(H2O)16, RHF/6-31G*.
FMO
FMO
FMO/X FMO
FMO/F FMO/F FMO/F FMO/FX FMO/XF FMO/F FMO/FX
b
RESPPC
0/0
2/2
2/4
-1/-1
0/0
2/2
2/4
2/4
2/4
-1/-1
-1/4
14.10
35.46 124.74
-2.07
-4.59
30.76
0.87
-9.12
-55.89
27.87
-8.12
E1
2
-19.36 -40.69 -146.35
-1.32
-1.27 100.76
4.82
9.15
-3.66 -47.05
29.82
E
-34.26 -34.34 -79.19 -26.09
42.98
32.53
41.27
37.42
156.88
11.60
23.64
T
a
1
2
E and E are the errors in the one and two-electron energies (Eqs. (S8,S9)), respectively; T is the error in the kinetic energy, (all
energy values are in kcal/mol vs ab initio).
b
a/b notation means a in SCF and b in the final construction of F.
Table SII. Accuracy of the componentsa of the total energy for phenol+(H2O)16, RHF/6-311G**.
FMO
FMO/X
FMO
FMO/F
FMO/FX FMO/XF
FMO/F
a
RESPPC
2/4
2/4
-1/-1
2/4
2/4
2/4
-1/-1
88.31
942.06
35.51
145.12
-165.48
-743.47
22180.14
E1
-104.90
-1042.30
-37.58
-143.19
13.33
-405.60
-11117.70
E2
-112.78
-390.18
-47.93
35.33
155.35
822.34
-1333.74
T
a
See Table SI for the definitions and units.
E1
E2
T
FMO/FX
-1/4
-63.66
48.46
178.99
Table SIII. Accuracy of the componentsa of the total energy for phenol+(H2O)n, RHF. RESPPC=2/4 is used.
n=32, 6-31G*
n=64, 6-31G*
n=32, 6-311G**
n=64, 6-311G**
FMOb FMO/Fb FMO/FXb FMOb FMO/Fb FMO/FXb FMOb
FMO/Fb FMO/FXb FMOb
FMO/Fb FMO/FXb FMO/FXc
132.55 -111.12
-148.52 623.40 -101.16
-169.51 161.11 41235.18
-724.99
945.64 46256.75 -2680.84 -1084.23
-156.65
-34.85
-21.29 -683.83 609.88
631.82 -222.91 -21960.81
-14.20 -1102.64 -30677.71
360.97
-144.69
-155.32 180.36
151.69 -393.85 392.80
333.23 -447.68 -4629.86
999.40 -1189.12 -21977.66
1283.58
791.80
a
b
c
See Table SI for the definitions and units. One water molecule per fragment. Two water molecules per fragment.
4
Fig. S1. Structures of PhOH+(H2O)n systems, n=16, 32 and 64, from top to bottom.
5
a)
FMO1 errors in the orbital energies
2.0
1.0
0.0
error, eV
-1.0
-2.0
-3.0
-4.0
-5.0
-6.0
-7.0
-8.0
1
20 39 58 77 96 115 134 153 172 191 210 229 248 267 286 305 324 343 362 381
orbital number
FMO, 0
FMO, 2
FMO, -1
FMO/X, 2/4
b)
FMO2/F errors in the orbital energies
1.0
0.8
0.6
0.4
error, eV
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
1
20 39 58 77 96 115 134 153 172 191 210 229 248 267 286 305 324 343 362 381
orbital number
FMO/F, 0
FMO/FX, 2/4
FMO/F, 2
FMO/F, -1
FMO/F, 2/4
FMO/XF, 2/4
Fig. S2. The errors in the total energies, PhOH+(H2O)16, 6-31G* for a) FMO (union of
orbital energies) and b) FMO/F family. The numbers after comma in the legend are
RESPPC values; if the two values are the same, only one is shown. 105 occ. orbitals.
6