16
B. Sutcliffe and R.G. Woolley
the role of parameters in the Schrödinger equation (1.13) for the electronic Hamiltonian; it differs from the earlier approach of Born and Oppenheimer because now
the values of X f range over the whole nuclear configuration space. Substituting this
expansion into (1.10), multiplying the result by ϕ(x, X) ∗
n and integrating over the
electronic coordinates x leads to an infinite dimensional system of coupled equations for the nuclear functions {Φ},
T N + E
o (X) n − E
Φ(X) n +
nn
C(X, P ) nn Φ(X) n = 0
(1.15)
where the coupling coefficients {C(X, P ) nn } have a well-known form which we
need not record here [44].
In this formulation the adiabatic approximation consists of retaining only the
diagonal terms in the coupling matrix C(X, P ), for then a state function can be
written as
ψ(x, X) ≈ ψ(x, X)
AD
n = ϕ(x, X) n Φ(X) n
(1.16)
and a product wavefunction corresponds to additive electronic and nuclear energies.
The special character of the electronic wavefunctions {ϕ(x, X) m } is, by (1.13), that
they diagonalize the electronic Hamiltonian H o ; they are said to define an ‘adiabatic’ basis (cf. the approximate form (1.16)) because the electronic state label n is
not altered as X varies. The Born approach does not really require the diagonalization of H o ; it is perfectly possible to define other representations of the electronic
expansion functions through unitary transformations of the {ϕ}, with concomitant
modification of the coupling matrix C. This leads to so-called ‘diabatic’ bases; the
freedom to choose the representation is very important in practical applications to
spectroscopy and atomic/molecular collisions [50, 51].
1.3.3 Formal Quantum Theory of the Molecular Hamiltonian
We now start again and develop the quantum theory of the Hamiltonian for a collection of n charged particles with Coulombic interactions. 12 We remind ourselves
again from Sect. 1.1 that for particles with classical Hamiltonian variables {q i , p i }
this is
H =
n
i
p 2
i
2m i
+
n
i
4πε 0 |q i − q j |
(1.17)
with the non-zero Poisson-bracket
{x i , p j } = δ ij .
12 The reader may find it helpful to refer to the Appendix which summarizes some mathematical
notions that are needed here, and illustrates them in a simple model of coupled oscillators with two
degrees of freedom.
B. Sutcliffe and R.G. Woolley
the role of parameters in the Schrödinger equation (1.13) for the electronic Hamiltonian; it differs from the earlier approach of Born and Oppenheimer because now
the values of X f range over the whole nuclear configuration space. Substituting this
expansion into (1.10), multiplying the result by ϕ(x, X) ∗
n and integrating over the
electronic coordinates x leads to an infinite dimensional system of coupled equations for the nuclear functions {Φ},
T N + E
o (X) n − E
Φ(X) n +
nn
C(X, P ) nn Φ(X) n = 0
(1.15)
where the coupling coefficients {C(X, P ) nn } have a well-known form which we
need not record here [44].
In this formulation the adiabatic approximation consists of retaining only the
diagonal terms in the coupling matrix C(X, P ), for then a state function can be
written as
ψ(x, X) ≈ ψ(x, X)
AD
n = ϕ(x, X) n Φ(X) n
(1.16)
and a product wavefunction corresponds to additive electronic and nuclear energies.
The special character of the electronic wavefunctions {ϕ(x, X) m } is, by (1.13), that
they diagonalize the electronic Hamiltonian H o ; they are said to define an ‘adiabatic’ basis (cf. the approximate form (1.16)) because the electronic state label n is
not altered as X varies. The Born approach does not really require the diagonalization of H o ; it is perfectly possible to define other representations of the electronic
expansion functions through unitary transformations of the {ϕ}, with concomitant
modification of the coupling matrix C. This leads to so-called ‘diabatic’ bases; the
freedom to choose the representation is very important in practical applications to
spectroscopy and atomic/molecular collisions [50, 51].
1.3.3 Formal Quantum Theory of the Molecular Hamiltonian
We now start again and develop the quantum theory of the Hamiltonian for a collection of n charged particles with Coulombic interactions. 12 We remind ourselves
again from Sect. 1.1 that for particles with classical Hamiltonian variables {q i , p i }
this is
H =
n
i
p 2
i
2m i
+
n
i
4πε 0 |q i − q j |
(1.17)
with the non-zero Poisson-bracket
{x i , p j } = δ ij .
12 The reader may find it helpful to refer to the Appendix which summarizes some mathematical
notions that are needed here, and illustrates them in a simple model of coupled oscillators with two
degrees of freedom.