Chemistry Reference
In-Depth Information
while the second Eq. (
11.6b
)in3D,
h
2
2
R
R
∂S
∂t
−
2
m
∇
1
2
m
(
¯
S
)
2
+
∇
+
V
=
0
(11.7b)
extends the basic Schrödinger Eq. (
11.4
) to include further quantum complexity. It
may be clearly seen since recognizing that:
p
2
2
m
=
1
2
m
(
∂S
∂t
=−
S
)
2
p
2
S
)
2
(
∇
=
⇒
∇
=
T
;
E
(11.8)
one gets from (
11.7b
) the total energy expression:
E
=
T
+
V
+
V
qua
(11.9a)
in terms of newly appeared so called quantum (or Bohm) potential
h
2
2
R
R
2
m
∇
V
qua
=−
¯
(11.9b)
Exploring the consequences of the existence of the Bohm potential (
11.9b
) reveals
most interesting features of the fundamental nature of electronic quantum behavior.
We will survey some of them in what follows.
Since the chemical bonding is carried by electrons only, one can see the basic de
Broglie-Bohm wavefunction (
11.1
) as belonging to gauge U(1) group transformation:
BB
(
x
,
t
)exp
i
¯
(
x
,
t
)
e
c
ℵ
U
(1)
(
x
,
t
)
=
h
(11.10)
R
(
x
,
t
)exp
i
¯
(
x
,
t
)
,
e
S
(
x
,
t
)
e
0
4
πε
0
e
c
ℵ
=
+
=
h
where
should account through of variational principle
(Schrödinger equation here) by the electronic bond, eventually being quantified by
associate corpuscle.
As such, one employs the gauge wavefunction (
11.10
) to compute the actual
Schrödinger partial derivative terms as:
the
chemical
field
ℵ
∂R
∂x
+
h
R
∂S
exp
i
¯
S
∂
U
(1)
∂x
i
¯
e
c
∂
∂x
e
c
ℵ
=
∂x
+
+
(11.11a)
h
∂
2
U
(1)
∂x
2
⎧
⎨
⎫
⎬
∂S
∂x
+
h
R
∂
2
S
∂
2
R
∂x
2
∂
2
2
i
¯
∂R
∂x
e
c
∂
∂x
i
¯
e
c
ℵ
∂x
2
+
+
+
h
∂x
2
∂S
∂x
2
=
(11.11b)
2
e
c
⎩
⎭
R
¯
∂
∂x
h
2
c
R
∂S
e
∂
∂x
−
+
−
2
h
2
∂x
¯
exp
i
¯
S
e
c
ℵ
×
+
h
