Page 241 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9

Approximation methods

È
In the preceding chapters we solved the time-independent Schrodinger equation
for a few one-particle and pseudo-one-particle systems: the particle in a box,
the harmonic oscillator, the particle with orbital angular momentum, and the
hydrogen-like atom. There are other one-particle systems, however, for which
È
the Schrodinger equation cannot be solved exactly. Moreover, exact solutions
of the Schrodinger equation cannot be obtained for any system consisting of
È
two or more particles if there is a potential energy of interaction between the
particles. Such systems include all atoms except hydrogen, all molecules, non-
ideal gases, liquids, and solids. For this reason we need to develop approxima-
È
tion methods to solve the Schrodinger equation with suf®cient accuracy to
explain and predict the properties of these more complicated systems. Two of
these approximation methods are the variation method and perturbation
theory. These two methods are developed and illustrated in this chapter.

9.1 Variation method
Variation theorem

The variation method gives an approximation to the ground-state energy E 0
^
(the lowest eigenvalue of the Hamiltonian operator H) for a system whose
È
time-independent Schrodinger equation is
^
Hø n ˆ E n ø n , n ˆ 0, 1, 2, ... (9:1)
In many applications of quantum mechanics to chemical systems, a knowledge
of the ground-state energy is suf®cient. The method is based on the variation
theorem:if ö is any normalized, well-behaved function of the same variables
as ø n and satis®es the same boundary conditions as ø n , then the quantity
^
E ˆhöjHjöi is always greater than or equal to the ground-state energy E 0
^
E höjHjöi > E 0 (9:2)

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