18
B. Sutcliffe and R.G. Woolley
commutes with ˆ
H. It follows that the molecular Hamiltonian may be written as a
direct integral
ˆ
H =
⊕
R 3
ˆ
H (P )dP
(1.22)
where [52]
ˆ
H (P ) =
P 2
2M T
+ ˆ
H
(1.23)
is the Hamiltonian at fixed total momentum P and M T is the molecular mass. The
internal Hamiltonian ˆ
H is independent of the centre-of-mass variables and is explicitly translation invariant. The form of ˆ
H is not uniquely fixed but whatever coordinates are chosen the essential point is that it is always the same operator specified
in (1.23) acting on a Hilbert space H that may be parameterized by functions of the
electron and nuclear coordinates.
The separation of the centre-of-mass terms from the internal Hamiltonian is the
same in quantum mechanics as in classical mechanics so we need not distinguish
operators from classical variables in this step. It is convenient to choose the centreof-nuclear mass for the definition of suitable internal coordinates. 13 Let t e be a set
of internal electronic coordinates defined as the original electronic coordinates x
referred to the centre-of-nuclear mass, and let t n be a set of internal nuclear coordinates constructed purely from the original nuclear coordinates X. If there are s
electrons and M nuclei, there are s internal electronic coordinates, and M − 1 internal nuclear coordinates. There are corresponding canonically conjugate internal
momentum variables. In terms of these variables the total kinetic energy of all the
particles can be decomposed into the form
T 0 = T CM + T N + T e
(1.24)
where T CM is the kinetic energy for the centre-of-mass, T N is the kinetic energy for
the nuclei expressed purely in terms of the internal nuclear momentum variables,
and T e is the kinetic energy for the electrons expressed purely in terms of the internal electronic momentum variables. The Coulomb energy can be expressed purely
in terms of the internal coordinates, U = U(t e , t n ). These relations are true both
classically and in quantum mechanics with a suitable operator interpretation.
In parallel with the decomposition in (1.18), we define the quantum mechanical
‘electronic’ Hamiltonian as
ˆ
H
elec
= ˆ
T e + ˆ
U
ˆ t
e , ˆ t
n
(1.25)
13 It is always possible to split off the kinetic energy of the centre-of-mass without any approximation; with this choice we retain the separation of the electronic and nuclear kinetic energies as
well, as in (1.24). Explicit formulae are given in e.g. [3] where it is shown that the nuclear kinetic
energy terms involve reciprocals of the nuclear masses, so that overall, the nuclear kinetic energy
is proportional to κ 4 .
B. Sutcliffe and R.G. Woolley
commutes with ˆ
H. It follows that the molecular Hamiltonian may be written as a
direct integral
ˆ
H =
⊕
R 3
ˆ
H (P )dP
(1.22)
where [52]
ˆ
H (P ) =
P 2
2M T
+ ˆ
H
(1.23)
is the Hamiltonian at fixed total momentum P and M T is the molecular mass. The
internal Hamiltonian ˆ
H is independent of the centre-of-mass variables and is explicitly translation invariant. The form of ˆ
H is not uniquely fixed but whatever coordinates are chosen the essential point is that it is always the same operator specified
in (1.23) acting on a Hilbert space H that may be parameterized by functions of the
electron and nuclear coordinates.
The separation of the centre-of-mass terms from the internal Hamiltonian is the
same in quantum mechanics as in classical mechanics so we need not distinguish
operators from classical variables in this step. It is convenient to choose the centreof-nuclear mass for the definition of suitable internal coordinates. 13 Let t e be a set
of internal electronic coordinates defined as the original electronic coordinates x
referred to the centre-of-nuclear mass, and let t n be a set of internal nuclear coordinates constructed purely from the original nuclear coordinates X. If there are s
electrons and M nuclei, there are s internal electronic coordinates, and M − 1 internal nuclear coordinates. There are corresponding canonically conjugate internal
momentum variables. In terms of these variables the total kinetic energy of all the
particles can be decomposed into the form
T 0 = T CM + T N + T e
(1.24)
where T CM is the kinetic energy for the centre-of-mass, T N is the kinetic energy for
the nuclei expressed purely in terms of the internal nuclear momentum variables,
and T e is the kinetic energy for the electrons expressed purely in terms of the internal electronic momentum variables. The Coulomb energy can be expressed purely
in terms of the internal coordinates, U = U(t e , t n ). These relations are true both
classically and in quantum mechanics with a suitable operator interpretation.
In parallel with the decomposition in (1.18), we define the quantum mechanical
‘electronic’ Hamiltonian as
ˆ
H
elec
= ˆ
T e + ˆ
U
ˆ t
e , ˆ t
n
(1.25)
13 It is always possible to split off the kinetic energy of the centre-of-mass without any approximation; with this choice we retain the separation of the electronic and nuclear kinetic energies as
well, as in (1.24). Explicit formulae are given in e.g. [3] where it is shown that the nuclear kinetic
energy terms involve reciprocals of the nuclear masses, so that overall, the nuclear kinetic energy
is proportional to κ 4 .