基本计算过程 编辑
记双原子分子中两个原子的波函数分别为
φ
A
{\displaystyle \varphi _{A}}
φ
B
{\displaystyle \varphi _{B}}
Ψ
=
c
A
φ
A
+
c
B
φ
B
{\displaystyle \Psi ={c_{A}}{\varphi _{A}}+{c_{B}}{\varphi _{B}}}
代入到定态薛定谔方程
H
^
Ψ
=
E
Ψ
{\displaystyle {\hat {H}}\Psi =E\Psi }
H
^
(
c
A
φ
A
+
c
B
φ
B
)
=
E
(
c
A
φ
A
+
c
B
φ
B
)
{\displaystyle {\hat {H}}\left({{c_{A}}{\varphi _{A}}+{c_{B}}{\varphi _{B}}}\right)=E\left({{c_{A}}{\varphi _{A}}+{c_{B}}{\varphi _{B}}}\right)}
分别用两个原子波函数与上式做内积，
∫
d
τ
φ
A
∗
H
^
(
c
A
φ
A
+
c
B
φ
B
)
=
E
∫
d
τ
(
c
A
φ
A
∗
φ
A
+
c
B
φ
A
∗
φ
B
)
{\displaystyle \int {d\tau \;\varphi _{A}^{*}{\hat {H}}\left({{c_{A}}{\varphi _{A}}+{c_{B}}{\varphi _{B}}}\right)}=E\int {d\tau \;\left({{c_{A}}\varphi _{A}^{*}{\varphi _{A}}+{c_{B}}\varphi _{A}^{*}{\varphi _{B}}}\right)}}
∫
d
τ
φ
B
∗
H
^
(
c
A
φ
A
+
c
B
φ
B
)
=
E
∫
d
τ
(
c
A
φ
B
∗
φ
A
+
c
B
φ
B
∗
φ
B
)
{\displaystyle \int {d\tau \;\varphi _{B}^{*}{\hat {H}}\left({{c_{A}}{\varphi _{A}}+{c_{B}}{\varphi _{B}}}\right)}=E\int {d\tau \;\left({{c_{A}}\varphi _{B}^{*}{\varphi _{A}}+{c_{B}}\varphi _{B}^{*}{\varphi _{B}}}\right)}}
展开，
c
A
∫
d
τ
φ
A
∗
H
^
φ
A
⏟
H
A
A
+
c
B
∫
d
τ
φ
A
∗
H
^
φ
B
⏟
H
A
B
=
E
c
A
∫
d
τ
φ
A
∗
φ
A
⏟
1
+
E
c
B
∫
d
τ
φ
A
∗
φ
B
⏟
S
A
B
{\displaystyle {c_{A}}\underbrace {\int {d\tau \;\varphi _{A}^{*}{\hat {H}}{\varphi _{A}}}} _{H_{AA}}+{c_{B}}\underbrace {\int {d\tau \;\varphi _{A}^{*}{\hat {H}}{\varphi _{B}}}} _{H_{AB}}=E{c_{A}}\underbrace {\int {d\tau \;\varphi _{A}^{*}{\varphi _{A}}}} _{1}+E{c_{B}}\underbrace {\int {d\tau \;\varphi _{A}^{*}{\varphi _{B}}}} _{S_{AB}}}
c
A
∫
d
τ
φ
B
∗
H
^
φ
A
⏟
H
B
A
+
c
B
∫
d
τ
φ
B
∗
H
^
φ
B
⏟
H
B
B
=
E
c
A
∫
d
τ
φ
B
∗
φ
A
⏟
S
B
A
+
E
c
B
∫
d
τ
φ
B
∗
φ
B
⏟
1
{\displaystyle {c_{A}}\underbrace {\int {d\tau \;\varphi _{B}^{*}{\hat {H}}{\varphi _{A}}}} _{H_{BA}}+{c_{B}}\underbrace {\int {d\tau \;\varphi _{B}^{*}{\hat {H}}{\varphi _{B}}}} _{H_{BB}}=E{c_{A}}\underbrace {\int {d\tau \;\varphi _{B}^{*}{\varphi _{A}}}} _{S_{BA}}+E{c_{B}}\underbrace {\int {d\tau \;\varphi _{B}^{*}{\varphi _{B}}}} _{1}}
因此得到，
c
A
(
H
A
A
−
E
)
+
c
B
(
H
A
B
−
E
S
A
B
)
=
0
{\displaystyle {c_{A}}\left({{H_{AA}}-E}\right)+{c_{B}}\left({{H_{AB}}-E{S_{AB}}}\right)=0}
c
A
(
H
B
A
−
E
S
B
A
)
+
c
B
(
H
B
B
−
E
)
=
0
{\displaystyle {c_{A}}\left({{H_{BA}}-E{S_{BA}}}\right)+{c_{B}}\left({{H_{BB}}-E}\right)=0}
相应的久期方程矩阵形式为
[
H
A
A
−
E
H
A
B
−
E
S
A
B
H
B
A
−
E
S
B
A
H
B
B
−
E
]
[
c
A
c
B
]
=
0
{\displaystyle {\begin{bmatrix}{{H_{AA}}-E}&{{H_{AB}}-E{S_{AB}}}\\{{H_{BA}}-E{S_{BA}}}&{{H_{BB}}-E}\\\end{bmatrix}}{\begin{bmatrix}{c_{A}}\\{c_{B}}\\\end{bmatrix}}=0}
线性组合的系数由此可求得。
双原子分子体系的能量
E
{\displaystyle E}
H
A
A
−
E
H
B
A
−
E
S
B
A
=
H
A
B
−
E
S
A
B
H
B
B
−
E
{\displaystyle {\frac {{H_{AA}}-E}{{H_{BA}}-E{S_{BA}}}}={\frac {{H_{AB}}-E{S_{AB}}}{{H_{BB}}-E}}}
最简单的分子： H
2
+
{\displaystyle _{2}^{+}}
编辑 H
2
+
{\displaystyle _{2}^{+}}
2
+
{\displaystyle _{2}^{+}}
H
A
A
=
H
B
B
=
α
,
H
A
B
=
H
B
A
=
β
,
S
A
B
=
S
B
A
=
S
{\displaystyle {H_{AA}}={H_{BB}}=\alpha ,{H_{AB}}={H_{BA}}=\beta ,{S_{AB}}={S_{BA}}=S}
库仑积分 ，β为交换积分 ，S为重叠积分 。于是，代入用于求能量的比值式：
α
−
E
β
−
E
S
=
β
−
E
S
α
−
E
{\displaystyle {\frac {\alpha -E}{\beta -ES}}={\frac {\beta -ES}{\alpha -E}}}
可得到两个可能的能量值；回代入久期方程，可得到系数
c
A
{\displaystyle c_{A}}
c
B
{\displaystyle c_{B}}
E
+
=
α
+
β
1
+
S
{\displaystyle {E_{+}}={\frac {\alpha +\beta }{1+S}}}
c
A
=
c
B
{\displaystyle c_{A}=c_{B}}
E
−
=
α
−
β
1
−
S
{\displaystyle {E_{-}}={\frac {\alpha -\beta }{1-S}}}
c
A
=
−
c
B
{\displaystyle c_{A}=-c_{B}}
因此，令
c
A
=
c
B
=
c
{\displaystyle c_{A}=c_{B}=c}
Ψ
+
=
c
(
φ
A
+
φ
B
)
{\displaystyle {\Psi _{+}}=c\left({{\varphi _{A}}+{\varphi _{B}}}\right)}
Ψ
−
=
c
(
φ
A
−
φ
B
)
{\displaystyle {\Psi _{-}}=c\left({{\varphi _{A}}-{\varphi _{B}}}\right)}
c可由归一化条件最终确定。
已知氢原子基态波函数（1s）在空间中表示为
e
−
r
a
0
{\displaystyle e^{-{\frac {\mathbf {r} }{a_{0}}}}}
r
=
(
x
,
y
)
{\displaystyle \mathbf {r} =(x,y)}
x
=
0
{\displaystyle x=0}
φ
A
(
r
)
=
e
−
x
2
+
y
2
a
0
{\displaystyle \varphi _{A}(\mathbf {r} )=e^{-{\frac {\sqrt {x^{2}+y^{2}}}{a_{0}}}}}
x
=
x
0
{\displaystyle x=x_{0}}
φ
B
(
r
)
=
e
−
(
x
−
x
0
)
2
+
y
2
a
0
{\displaystyle \varphi _{B}(\mathbf {r} )=e^{-{\frac {\sqrt {(x-x_{0})^{2}+y^{2}}}{a_{0}}}}}
Ψ
+
(
x
,
y
)
=
c
(
e
−
x
2
+
y
2
a
0
+
e
−
(
x
−
x
0
)
2
+
y
2
a
0
)
{\displaystyle {\Psi _{+}}(x,y)=c\left(e^{-{\frac {\sqrt {x^{2}+y^{2}}}{a_{0}}}}+e^{-{\frac {\sqrt {(x-x_{0})^{2}+y^{2}}}{a_{0}}}}\right)}
Ψ
−
(
x
,
y
)
=
c
(
e
−
x
2
+
y
2
a
0
−
e
−
(
x
−
x
0
)
2
+
y
2
a
0
)
{\displaystyle {\Psi _{-}}(x,y)=c\left(e^{-{\frac {\sqrt {x^{2}+y^{2}}}{a_{0}}}}-e^{-{\frac {\sqrt {(x-x_{0})^{2}+y^{2}}}{a_{0}}}}\right)}
H2+分子的成键轨道
Ψ
+
(
x
,
y
)
{\displaystyle \Psi _{+}(x,y)}
的几率分布示意图
H2+分子的反键轨道
Ψ
−
(
x
,
y
)
{\displaystyle \Psi _{-}(x,y)}
的几率分布示意图
