1 The Potential Energy Surface in Molecular Quantum Mechanics
27
with a particular configuration of the nuclei 16 that is deep enough for the lowest
energy eigenstates to be localized about x 0 . One can look at these states with a
‘microscope’ with a certain resolving power that depends on Planck’s constant.
The microscope transformation produces a translation to make x 0 the origin of
the coordinates, and a dilation (scale transformation)
ˆ
x λ = ˆ
x − (1 − λ)(ˆ x − x 0 ),
ˆ
p λ = ˆ
p +
(1 − λ)
λ
ˆ
p.
(1.44)
It is readily verified that the commutation relations are preserved for λ = 0
[ˆ x λ , ˆ
p λ ] = [ˆ x, ˆ
p].
Under this transformation a Hamiltonian of the form
ˆ
H =
g
ˆ
p 2
g
2m g
+ ˆ
V (x)
becomes
ˆ
H λ = ˆ
V (x 0 ) + λ
2 ˆ
N(λ)
where
ˆ
N(λ) = −
2
λ 4
g
∇ 2
g
2m g
+
1
λ 2
ˆ
V
x 0 + λ(x − x 0 )
− ˆ
V (x 0 )
.
We now put λ =
√
so as to eliminate λ from the kinetic energy term in ˆ
N(λ);
with this choice for λ, unitary equivalence of the spectrum implies that the eigenvalues of the original Hamiltonian ˆ
H are related to those of ˆ
N(λ) by
E n = V (x 0 ) + μ n (λ).
Provided ˆ
V is analytic in λ it can be expanded about λ = 0, and this puts ˆ
N(λ), in
lowest order, into the form of a sum of coupled oscillators so that the first approximation for the eigenvalue function μ n is
μ n =
k
n k +
1
2
.
In the Born-Oppenheimer calculation for the diatomic molecule the potential ˆ
V is
identified with the effective potential for the nuclei [63]; analyticity of ˆ
V in λ could
be proven, and the role of
√ was taken by the usual BO expansion parameter
κ = (m e /M N )
1
4 . In this way the molecular energy level formula (1.12) is recovered
as an asymptotic expansion.
16 The multiminima case can also be treated in this way.
27
with a particular configuration of the nuclei 16 that is deep enough for the lowest
energy eigenstates to be localized about x 0 . One can look at these states with a
‘microscope’ with a certain resolving power that depends on Planck’s constant.
The microscope transformation produces a translation to make x 0 the origin of
the coordinates, and a dilation (scale transformation)
ˆ
x λ = ˆ
x − (1 − λ)(ˆ x − x 0 ),
ˆ
p λ = ˆ
p +
(1 − λ)
λ
ˆ
p.
(1.44)
It is readily verified that the commutation relations are preserved for λ = 0
[ˆ x λ , ˆ
p λ ] = [ˆ x, ˆ
p].
Under this transformation a Hamiltonian of the form
ˆ
H =
g
ˆ
p 2
g
2m g
+ ˆ
V (x)
becomes
ˆ
H λ = ˆ
V (x 0 ) + λ
2 ˆ
N(λ)
where
ˆ
N(λ) = −
2
λ 4
g
∇ 2
g
2m g
+
1
λ 2
ˆ
V
x 0 + λ(x − x 0 )
− ˆ
V (x 0 )
.
We now put λ =
√
so as to eliminate λ from the kinetic energy term in ˆ
N(λ);
with this choice for λ, unitary equivalence of the spectrum implies that the eigenvalues of the original Hamiltonian ˆ
H are related to those of ˆ
N(λ) by
E n = V (x 0 ) + μ n (λ).
Provided ˆ
V is analytic in λ it can be expanded about λ = 0, and this puts ˆ
N(λ), in
lowest order, into the form of a sum of coupled oscillators so that the first approximation for the eigenvalue function μ n is
μ n =
k
n k +
1
2
.
In the Born-Oppenheimer calculation for the diatomic molecule the potential ˆ
V is
identified with the effective potential for the nuclei [63]; analyticity of ˆ
V in λ could
be proven, and the role of
√ was taken by the usual BO expansion parameter
κ = (m e /M N )
1
4 . In this way the molecular energy level formula (1.12) is recovered
as an asymptotic expansion.
16 The multiminima case can also be treated in this way.