1 The Potential Energy Surface in Molecular Quantum Mechanics
21
Hilbert space of a fixed configuration, H (X), in which X can be assumed to be a
‘parameter’ in the position space wavefunction ψ(x, X), whereas if they had continued with quantum mechanics they would have been working in the ‘big’ Hilbert
space H with ˆ
x and ˆ
X treated on an equal footing as operators, and all possible
nuclear configurations being treated simultaneously, rather than one at a time.
The unusual properties of the (‘electronic’) Hamiltonian ˆ
H o (ˆ x, ˆ
p, ˆ
X) = ˆ
H elec
in (1.32) 15 considered as a quantum-mechanical operator on the whole space H,
are of exactly the kind to be expected from the work of Kato [54]. In Lemma 4 of
his paper he showed that for a Coulomb potential U and for any function f in the
domain D 0 of the full kinetic energy operator ˆ
T 0 , the domain, D U , of the internal
Hamiltonian ˆ
H contains D 0 and there are two constants a, b such that
Uf ≤ a ˆ
T 0 f + bf
where a can be taken as small as is liked. This result is often summarised by saying that the Coulomb potential is small compared to the kinetic energy. Given this
result he proved in Lemma 5 (the Kato-Rellich theorem) that the usual Coulomb
Hamiltonian operator is essentially self-adjoint and so is guaranteed a complete set
of eigenfunctions, and is bounded from below.
In the present context the important point to note is that the Coulomb term is
small only in comparison with the kinetic energy term involving the same set of
variables. So the absence of one or more kinetic energy terms from the Hamiltonian
may mean that the Coulomb potential term cannot be treated as small. It is evident
that one can’t use the Kato-Rellich argument to guarantee self-adjointness for the
customary representation of H elec in a position representation as a differential and
multiplicative operator because it contains the nuclear positions {X} in Coulomb
terms that are not dominated by corresponding kinetic energy operators involving
the conjugate momentum operators {−i∇} since they have been separated off into
the ‘perturbation’ term ∝ κ 4 . As a quite separate matter, the abstract direct integral
operator (1.32) is self-adjoint since the resolvent of the clamped-nuclei Hamiltonian
is integrable. This is demonstrated in Theorem XIII.85 in the book by Reed and
Simon [53]. It is in this form that the operator is used in the mathematically rigorous
accounts (to be discussed later) of the Born-Oppenheimer approximation in [64]
and [70]. The operator used in the standard account of Born and Huang [44] is
however simply the usual one which, as discussed above, is not self-adjoint in the
Kato sense.
1.3.4 Approximate Calculations
It might have been hoped, in the light of the claim in the original paper by Born
and Oppenheimer quoted in Sect. 1.3.1, that the eigensolutions of the κ → 0 limit
15 We assume that the centre-of-mass contributions are eliminated as usual.
21
Hilbert space of a fixed configuration, H (X), in which X can be assumed to be a
‘parameter’ in the position space wavefunction ψ(x, X), whereas if they had continued with quantum mechanics they would have been working in the ‘big’ Hilbert
space H with ˆ
x and ˆ
X treated on an equal footing as operators, and all possible
nuclear configurations being treated simultaneously, rather than one at a time.
The unusual properties of the (‘electronic’) Hamiltonian ˆ
H o (ˆ x, ˆ
p, ˆ
X) = ˆ
H elec
in (1.32) 15 considered as a quantum-mechanical operator on the whole space H,
are of exactly the kind to be expected from the work of Kato [54]. In Lemma 4 of
his paper he showed that for a Coulomb potential U and for any function f in the
domain D 0 of the full kinetic energy operator ˆ
T 0 , the domain, D U , of the internal
Hamiltonian ˆ
H contains D 0 and there are two constants a, b such that
Uf ≤ a ˆ
T 0 f + bf
where a can be taken as small as is liked. This result is often summarised by saying that the Coulomb potential is small compared to the kinetic energy. Given this
result he proved in Lemma 5 (the Kato-Rellich theorem) that the usual Coulomb
Hamiltonian operator is essentially self-adjoint and so is guaranteed a complete set
of eigenfunctions, and is bounded from below.
In the present context the important point to note is that the Coulomb term is
small only in comparison with the kinetic energy term involving the same set of
variables. So the absence of one or more kinetic energy terms from the Hamiltonian
may mean that the Coulomb potential term cannot be treated as small. It is evident
that one can’t use the Kato-Rellich argument to guarantee self-adjointness for the
customary representation of H elec in a position representation as a differential and
multiplicative operator because it contains the nuclear positions {X} in Coulomb
terms that are not dominated by corresponding kinetic energy operators involving
the conjugate momentum operators {−i∇} since they have been separated off into
the ‘perturbation’ term ∝ κ 4 . As a quite separate matter, the abstract direct integral
operator (1.32) is self-adjoint since the resolvent of the clamped-nuclei Hamiltonian
is integrable. This is demonstrated in Theorem XIII.85 in the book by Reed and
Simon [53]. It is in this form that the operator is used in the mathematically rigorous
accounts (to be discussed later) of the Born-Oppenheimer approximation in [64]
and [70]. The operator used in the standard account of Born and Huang [44] is
however simply the usual one which, as discussed above, is not self-adjoint in the
Kato sense.
1.3.4 Approximate Calculations
It might have been hoped, in the light of the claim in the original paper by Born
and Oppenheimer quoted in Sect. 1.3.1, that the eigensolutions of the κ → 0 limit
15 We assume that the centre-of-mass contributions are eliminated as usual.