Chemistry Reference
In-Depth Information
Fig. 14 Population analysis from the double atomic basin integration of the local source function
in the second-row diatomic hydrides H-
X
and comparison with corresponding terms in the
Mulliken's scheme. (a) Out-of-diagonal terms of the electron population matrix
M
as obtained
from the SF approach:
M
(H,
X
)+
M
(
X
,H)
ICP, the interchanged electron population between H
and
X
, given by the sum of the contribution to the electron population of H from
X
and of the
contribution to the electron population of
X
from H; (b) comparison between the ICP as obtained
from the SF approach and the corresponding overlap term [
M
(H,
X
)+
M
(
X
,H)]
M
in Mulliken's
analysis. Note that by definition
M
(H,
X
)
¼
M
(
X
,H) in Mulliken's analysis; (c) contributions of H
and
X
basins to determining their own electron populations in the SF approach; (d)asin(c) but for
formally corresponding terms in the Mulliken's scheme (this figure is adapted, with permission,
from material presented by the authors in [
16
,
33
])
¼
where
N
basis
is the number of basis set functions and P and S are the first-order
density and the overlap matrix representations over that basis, respectively. These
matrices are both symmetric, so that (15) may be also written as:
"
#
X
X
X
N
e
¼
P
mn
S
mn
þ
2
P
mn
S
mn
A
m2A;n2A
m2A;n2B6¼A
n
o
;
X
X
B6¼A
½ð
¼
M
ð
A
;
A
Þ
M
þ
M
ð
A
;
B
Þþ
M
ð
B
;
A
Þ
M
(16)
A
where the contribution to the number of electrons due to basis functions only
centered on A is named [
M
(A, A)]
M
, and that due to one basis function centered
on A and the other on B is called [
M
(A, B)
M
(B, A)]
M
. The subscript “M” denotes
that these quantities are defined within a Mulliken's scheme and a basis set
approach. Note that [
M
(A,B)
þ
þ
M
(B,A)]
M
corresponds to the Mulliken's total
