Chemistry Reference
In-Depth Information
Equation (
11.21
) can be solved for the Laplacian of the chemical field with general
solutions:
4
ρ
2
v
2
(
2
ρ
−
v
·
−
j
ρ
2
4
2
mc
∂
∂t
±
−
−
∇
ρ
·
−
j
ρ
)
2
ρ
)
2
∇
(
∇
2
∇
ℵ
)
1,2
=
(
(11.23)
ρ
2
(
∇
ρ
)
2
4
e
mc
Equation (
11.23
), is a special propagation equation for the chemical field since it
links the spatial Laplacian
2
∇
ℵ=
Δ
ℵ
with temporal evolution of the chemical field
/∂t
)
1
/
2
; however, it may be considerable be simplified if assuming the stationary
chemical field, i.e. chemical field as not explicitly depend on time,
(
∂
ℵ
∂
∂t
=
0
(11.24)
in agreement with the fact that once established the chemical bonding should be
manifested stationary in order to preserve the stability of the structure it applies.
With condition (
11.24
) we may still have two solutions for the chemical field.
One corresponds with the
homogeneous
chemical bonding field
mc
e
Δ
ℵ=
0
⇒ℵ
h
=
v
−
j
X
bond
(11.25)
with the constant determined such that the field (
11.25
) to be of the same nature as
the Bohm phase action S in (
11.10
).
The second solution of (
11.23
) looks like
mc
e
v
·∇
ρ
Δ
ℵ=
(11.26)
ρ
Finally, Eq. (
11.26
) may be integrated to primarily give:
⎡
⎣
⎤
⎦
r
−
∇
·
−
j
ρ
0
−
∇
·
−
j
ρ
r
−
∇
·
−
j
ρ
mc
ρ
mc
ρ
ρ
−
∇ℵ =
e
−
v
e
−
v
dx
=
dx
+
dx
∞
∞
0
(11.27)
that can be projected on
bondonic
current direction
−
j
and then further integrated
as:
⎛
⎞
⎛
⎞
x
(
t
)
0
−
∇
·
−
j
ρ
r
−
∇
·
−
j
ρ
mc
e
ρ
mc
e
ρ
⎝
⎠
+
⎝
⎠
dr
ℵ−ℵ
0
=
v
−
j
X
bond
dl
v
−
j
dl
∞
0
0
(11.28)
from where there is identified both the so called
manifested
chemical bond field:
⎛
⎞
∞
−
∇
ρ
·
−
j
ρ
mc
e
⎝
⎠
ℵ
0
=
v
−
j
X
bond
dl
(11.29a)
0
