Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 437
The theory addresses the problem of giving an approximate solution for a target system
hard to solve directly, by exploiting the knowledge of a simpler (but similar) system.
The target system is represented by its Hamiltonian H and by the corresponding wave function
Ψ defined as solution of the corresponding Schrödinger equation (see eq. [8.1])
HΨ = EΨ
[8.30]
The simpler unperturbed system is described in terms of a similar equation
0
H Φ = E Φ
[8.31]
0 0 0
The following partition of H is then introduced
0
H = H + λ V
[8.32]
as well as the following expansion of the unknown wave function and energy as powers of
the parameter λ:
( 1) 2 ( 2) 3 ( 3)
Ψ = Φ + λΦ + λ Φ + λ Φ + � [8.33]
0
( 1) 2 ( 2) 3 ( 3)
E = E + λE + λ E + λ E + � [8.34]
0
The corrections to the wave function and to the energy are obtained introducing the
formal expressions [8.32]-[8.34] in the equation [8.30] and separating the terms according
to their order in the power of λ. In this way one obtains a set of integro-differential equations
to be separately solved. The first equation is merely the Schrödinger equation [8.31] of the
simple system, supposed to be completely known. The others give, order by order, the corrections
Φ (n) to the wave function, and E (n) to the energy. These equations may be solved by
exploiting the other solutions, Φ1, Φ2, Φ3, �, ΦK
, of the simpler problem [8.31], that constitute
a complete basis set and are supposed to be completely known. With this approach
every correction to Ψ is given as linear combination
=∑
( n)
n
Φ C Φ
K
K
k
[8.35]
The coefficients are immediately defined in terms of the integrals V LK = where
the indexes L and K span the whole set of the eigenfunctions of the unperturbed system (including
Φ 0 where necessary). The corrections to the energy follow immediately, order by
order. They only depend on the V LK integrals and on the energies E 1,E 2, ..., E K, ... of the simple
(unperturbed) system. The problem is so reduced to a simple summation of elements, all
derived from the simpler system with the addition of a matrix containing the V LK integrals.
This formulation exactly corresponds to the original problem, provided that the expansions
[8.33] and [8.34] of Ψ and E converge and that these expansions are computed until
convergence.
We are not interested here in examining other aspects of this theory, such as the convergence
criteria, the definitions to introduce in the case of interrupted (and so approximate)
expansions, or the problems of practical implementation of the method.