**Synonym**: Rayleigh-Ritz method

Schroedinger's differential equation can always be solved by the variation method -- even when analytical methods fail. Many-electron atoms and molecules, for which "exact" solutions are almost never possible, are commonly solved on modern computers using the variation method. In its commonest application, the variation method uses an expansion in a basis to transform the DE into a matrix equation or linear algebra problem.

Suppose the system hamiltonian operator is *H* and the problem is to solve the
Schroedinger equation, *Hy _{n} = E_{n}y_{n}*, for eigenvalues and eigenfunctions. None of the
usual methods for analytical solutions may work if the system is complicated. Yet, we will
see that the Rayleigh quotient provides the means to approximately solve the problem.

Let *F* be an **arbitrary** state
function of the system -- not necessarily an energy eigenfunction. Assume only that F belongs to the space spanned by the (unknown) eigenfunctions; for
instance*F* is continuous, finite, differentiable,
and satisfies the boundary conditions. Define the Rayleigh quotient *W(F)* as

.

According to a postulate of quantum mechanics, *W(F)* is
the average value of a great many energy measurements on the system prepared in the state *F*.

__Theorem__: If H is a self-adjoint (Hermitian) operator with a purely discrete
spectrum and if *E _{0}* is its smallest eigenvalue, then the minimum value
of

The idea for using this result is to allow the function F to vary through some extended subspace
of possible state functions and to select the state with lowest Rayleigh quotient. Then
this lowest *W* is a more or less good approximation to *E _{0}*, and
the corresponding

A powerful way to use the variation theorem is to expand the trial function in a basis
of possible trial functions and minimize *W* with respect to the expansion
coefficients. That is, let . Here there are *M* basis functions and *M* unknown linear
coefficients, *a _{n}*. Trial basis functions are taken to be linearly
independent but not necessarily orthonormal; instead their inner products form the
elements of a

or, in matrix notation

where **H** and **S** are the
Hamiltonian and overlap matrices and **a** is the column of
expansion coefficients. This linear algebra problem is an approximation for the original
Schroedinger differential equation.

The secular equation has M eigenvalues and eigenvectors. The lowest eigenvalue is an
approximation to the ground state system energy and the corresponding eigenfunction
provides an approximation to the ground state wave function (when substituted into the
expansion for *F* in the basis of *f _{n}*). When the secular equation is rewritten as (

The variation method is not limited to the ground state. It can be shown that the *M*
roots of the secular determinant are sequentially upper bounds for excited states. [J.K.L.
MacDonald, *Phys. Rev.*, **43**, 830 (1933)] Denote the roots of the
secular equation by *W _{k}* for

Solve the particle in a box (PIB) problem using linear variation and for trial function use a linear combination of polynomials. Let the center of the box be placed at the origin: V=0 when -L/2 < x < L/2. Solution

Another way to vary the trial function is to scale the distances. Suppose *F(x)* is a trial function; then introduce the family of functions
*f(a,x)= F(ax)*. As *a* varies the trial
function expands and contracts. The trial energy becomes a function of *a*: *W(a) = <f(a,x)|H|f(a,x)>/<f(a,x)|f(a,x)>*. The objective is to minimize the Rayleigh quotient
with respect to *a*.

- P.W. Atkins and R.S. Friedman,
*Molecular Quantum Mechanics*[Oxford University Press, 1997] - F. L. Pilar,
*Elementary Quantum Chemistry*[McGraw-Hill, 1990]

Created or up-dated 08/03/99
by R.D. Poshusta