The Ehrenfest field has already been examined in some test systems, but its
relation to E has been obscure due to the computational problems commented
above.
6.2.3 The Potential Acting on an Electron in a Molecule
Zhao and Yang [7, 8] define the potential acting on an electron in a molecule
(PAEM) as the interaction energy of a local electron that belongs to the molecule
(say electron 1) with all the nuclei and the remaining electrons. The instantaneous
Coulomb interaction energy of the first electron at r 1 with the rest of particles is
V 1 ¼ À
X
B
Z
B
jr 1 À R B j
þ
X N
i [ 1
1
jr 1 À r i j
:
ð6:14Þ
Averaging V r 1
ð Þ over the positions of all of the remaining electrons in the
system gives
V
0 r
ð Þ ¼
Z
dr 2 . . .dr N W
H V 1 W;
ð6:15Þ
which, in terms of q r
ð Þ and q 2 r; r 2
ð
Þ, reduces to
V
0 r
ð Þ ¼ À
q r
ð Þ
N
X
B
Z
B
jr À R B j
þ
1
N
Z q 2 r; r 2
ð
Þ
jr À r 2 j
dr 2 :
ð6:16Þ
Since q r
ð Þ=N measures the probability of finding the first electron at r and
N N À 1
ð
Þ
½
Š
À1 q 2 r; r 2
ð
Þ the probability of finding the first electron at r while the
second electron is at r 2 , the potential acting on an electron at r is expressed as
V PAEM r
ð Þ ¼
V
0 r
ð Þ
q r
ð Þ=N
¼ À
X
B
Z
B
jr À R B j
þ
1
q r
ð Þ
Z q 2 r; r 2
ð
Þ
jr À r 2 j
dr 2 :
ð6:17Þ
Using again Eq. 6.4 we have
V PAEM r
ð Þ ¼ ÀV mep r
ð Þ À
V xc r
ð Þ
q r
ð Þ
:
ð6:18Þ
As pointed out by Zhao et al., the potentials ÀV mep r
ð Þ and V PAEM r
ð Þ display
essential differences. First, V PAEM r
ð Þ describes the interaction energy of an internal
electron with the rest of the molecule, while ÀV mep r
ð Þ represents the interaction
6 Emergent Scalar and Vector Fields in Quantum Chemical Topology
137