1 The Potential Energy Surface in Molecular Quantum Mechanics
13
The Coulomb energy is simply U(x, X). They then define the ‘unperturbed’ Hamiltonian
T e + U = H o
x,
∂
∂x
, X
(1.8)
and express the total Hamiltonian as
H = H o + κ
4 H 1
(1.9)
with Schrödinger equation
(H − E)ψ(x, X) = 0.
(1.10)
At this point in their argument they state
Setzt man in (12) [(1.10) above] κ = 0, so bekommt man eine Differentialgleichung für die
x k allein, in der die X l als Parameter vorkommen:
H o
x,
∂
∂x
; X
− W
ψ = 0.
Sie stellt offenbar die Bewegung der Elektronen bei festgehaltenen Kernen dar. 10
This splitting of the Hamiltonian into an ‘unperturbed’ part (κ = 0) and a ‘perturbation’ is essentially the same as in the earlier Old Quantum Theory version [36].
The difference here is that the action-angle perturbation theory of the Old Quantum
Theory is replaced by Schrödinger’s quantum mechanical perturbation theory. In
the following it is understood that the overall translational motion of the molecule
has been separated off by a suitable coordinate transformation; this is always possible. The initial step in setting up the perturbation expansion involves rewriting the
Hamiltonian H o as a series in increasing powers of κ. This is achieved by introducing new relative coordinates that depend on κ
X = X o + κζ
(1.11)
for some fixed X o , and using the {ζ } as the nuclear variables, in an expansion of
H o about X o .
Then as usual the eigenfunction and eigenvalue of (1.10) are presented as series
in κ
ψ = ψ
(0)
+ κψ
(1)
+ κ
2 ψ
(2)
+ · · · ,
E = E
(0)
+ κE
(1)
+ κ
2 E
(2)
+ · · · ,
the expansions are substituted into the Schrödinger equation (1.10), and the terms
separated by powers of κ. This gives a set of equations to be solved sequentially.
10 If one sets κ = 0. . . one obtains a differential equation in the x k alone, the X l appearing as
parameters:. . . . Evidently, this represents the electronic motion for stationary nuclei.
13
The Coulomb energy is simply U(x, X). They then define the ‘unperturbed’ Hamiltonian
T e + U = H o
x,
∂
∂x
, X
(1.8)
and express the total Hamiltonian as
H = H o + κ
4 H 1
(1.9)
with Schrödinger equation
(H − E)ψ(x, X) = 0.
(1.10)
At this point in their argument they state
Setzt man in (12) [(1.10) above] κ = 0, so bekommt man eine Differentialgleichung für die
x k allein, in der die X l als Parameter vorkommen:
H o
x,
∂
∂x
; X
− W
ψ = 0.
Sie stellt offenbar die Bewegung der Elektronen bei festgehaltenen Kernen dar. 10
This splitting of the Hamiltonian into an ‘unperturbed’ part (κ = 0) and a ‘perturbation’ is essentially the same as in the earlier Old Quantum Theory version [36].
The difference here is that the action-angle perturbation theory of the Old Quantum
Theory is replaced by Schrödinger’s quantum mechanical perturbation theory. In
the following it is understood that the overall translational motion of the molecule
has been separated off by a suitable coordinate transformation; this is always possible. The initial step in setting up the perturbation expansion involves rewriting the
Hamiltonian H o as a series in increasing powers of κ. This is achieved by introducing new relative coordinates that depend on κ
X = X o + κζ
(1.11)
for some fixed X o , and using the {ζ } as the nuclear variables, in an expansion of
H o about X o .
Then as usual the eigenfunction and eigenvalue of (1.10) are presented as series
in κ
ψ = ψ
(0)
+ κψ
(1)
+ κ
2 ψ
(2)
+ · · · ,
E = E
(0)
+ κE
(1)
+ κ
2 E
(2)
+ · · · ,
the expansions are substituted into the Schrödinger equation (1.10), and the terms
separated by powers of κ. This gives a set of equations to be solved sequentially.
10 If one sets κ = 0. . . one obtains a differential equation in the x k alone, the X l appearing as
parameters:. . . . Evidently, this represents the electronic motion for stationary nuclei.