Chemistry Reference
In-Depth Information
7 Electron Density in Embedding Schemes
Usually, quantum chemical calculations focus on the calculation of molecular
properties like the electron density for a single molecule. Experimental electron
densities, however, are obtained from a molecular crystal, in which crystal-packing
effects may play a nonnegligible role and have to be considered in the calculation
[
140
-
142
]. In order to ensure an adequate accuracy in such calculations, the system
under investigation might be “embedded” in a suitable environment to include such
environment effects [
143
-
148
]. This can be achieved by dividing a large molecule
of a molecular crystal into smaller pieces such that calculations on each subsystem
are feasible. In the optimization of the orbitals of a subsystem, the interaction with
the other subsystems of the “environment” can be modeled by an embedding
potential. Due to this partitioning, calculations on the full system are avoided,
which decreases the computational effort.
One example for such an embedding scheme being a subsystem version of
DFT is FDE, which was introduced by Cortona [
149
] to study properties of solids
within DFT. It was then further developed by Wesolowski and Warshel [
150
], who
extended the FDE formalism such that it can also be applied to molecules parti-
tioned into smaller building blocks. The density of the full
N
-electron system is then
partitioned into an “active” subsystem
r
1
, and a second subsystem
r
2
, representing
the environment (“frozen” subsystem):
r
tot
ðÞ¼r
1
ðÞþr
2
ðÞ:
(64)
The total energy
E
tot
[
r
tot
], appearing in (
53
), can then be rewritten as a bifunc-
tional [
151
] which depends on both subsystem electron densities
r
1
and
r
2
:
ð
d
3
r r
1
v
nuc
1
v
nuc
2
E
tot
r
1
; r
2
½
¼
E
NN
þ
ð
ðÞþr
2
ðÞ
Þ
ðÞþ
ðÞ
ðð
d
3
r
d
3
r
0
rðÞþr
2
rðÞ
1
2
ð
r
1
ðÞþr
2
ðÞ
Þ r
1
ð
Þ
þ
þ
E
xc
r
1
þ r
2
½
j
r r
0
j
þ
T
s
r
1
þ r
2
½
;
(65)
where
v
nuc
1
and
v
nuc
2
ðÞ
ðÞ
are the electrostatic potentials of the nuclei of the
subsystems.
Except for a few special cases, the subsystems cannot be expressed in terms of
canonical Kohn-Sham orbitals of the full system with the consequence that the
kinetic energy cannot be partitioned entirely. There always remains a term which
depends on both subsystem electron densities, the so-called nonadditive part of the
kinetic energy:
T
nadd
s
½
r
1
; r
2
¼
T
s
r
tot
½
T
s
r½
T
s
r½ :
(66)
