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The Method of Linear Variations
In this section we show that the method of linear variations (also called
the Ritz method [11]) is equivalent to the matrix formulation
of the Schrödinger equation
.
Our treatment is similar to that of Szabo and Ostlund
[1], p. 116.
The linear variation method states that, given the linear expansion

(tex2html_deferredtex2html_deferred3.tex2html_deferred1  = _i c_i  _i
) 
we vary the coefficients c_{i} so that we minimize
.
We begin by requiring that the
wavefunction be normalized so that
.
Normalization means that we cannot minimize E simply by solving

(tex2html_deferredtex2html_deferred3.tex2html_deferred2 c_k  H  = 0
k=1,2,...,N
) 
because the c_{i}'s are not independent.
In this case we have a constrained
minimization, so we apply Lagrange's method of undetermined multipliers,
and we minimize the functional

(tex2html_deferredtex2html_deferred3.tex2html_deferred3 L =  H   E (   1)
) 
which has the same minimum as E when
is normalized.
When we substitute equation (3.1) into equation (3.3),
we obtain

(tex2html_deferredtex2html_deferred3.tex2html_deferred4 L = _ij c_i^*c_j _i  H  _j  E
( _ij c_i^*c_j _i  _j  1 )
) 
which we may rewrite as

(tex2html_deferredtex2html_deferred3.tex2html_deferred5 L = _ij c_i^* c_j H_ij 
E ( _ij c_i^* c_j S_ij  1 )
) 
where of course
and
.
Now set the first variation in
equal to zero:

(tex2html_deferredtex2html_deferred3.tex2html_deferred6 L = _ij c_i^* c_j H_ij 
E _ij c_i^* c_j S_ij +
_ij c_i^* c_j H_ij 
E _ij c_i^* c_j S_ij = 0
) 
Since the summations run over all i and j, and since
H_{ij} = H_{ji}^{*}
and
S_{ij} = S_{ji}^{*}, we can simplify to

(tex2html_deferredtex2html_deferred3.tex2html_deferred7 L = _i c_i^* [
_j H_ij c_j  E S_ij c_j ] +
complex conj. = 0
) 
Since each term is a sum of a number and its complex conjugate, the
imaginary parts will cancel. However, the real part will not necessarily
be zero; in fact, since all the
's are arbitrary (that is the
whole point of using Lagrange's method), then for
to be
zero, the term in brackets must be zero. We may rewrite this condition as
a matrix equation

(tex2html_deferredtex2html_deferred3.tex2html_deferred8 H c = E S c
) 
If the basis functions {
} are chosen orthonormal (as is
usually the case), then
the identity matrix, and we
have
.
Of course
is the columnvector
representation of
in the basis {
}. We thus
have two equivalent ways of viewing a CIeither as the matrix formulation
of the Schrödinger equation within the given linear vector space of
Nelectron basis functions, or as the minimization of the energy
with respect to the linear expansion coefficients c_{i} of
(3.1), subject to the constraint that the wavefunction remain
normalized. Another way of viewing the results of this section is to note
that only eigenvectors of the Hamiltonian matrix
are stable with
respect to variations in the linear expansion coefficients.
At this point it is reasonable to ask why we wish to minimize the
energy by varying the coefficients in equation (3.1).
How do we know that this will give us the best estimate of the wavefunction?
There are two answers to this. First, as we have just shown, minimizing the
energy by variation of the linear expansion coefficients gives the
Schrödinger equation in matrix form; thus the procedure is justified
a posteriori by the validity of its result. The other reason is that,
for the ground state, the linear expansion in equation (3.1)
gives an expectation value for the energy E which is always an
upper bound to the exact nonrelativistic ground state energy
.
We will prove this assertion in the next section; the
result is called the Variational Theorem.
The best estimate of E, then, is the minimum value which
can be obtained by varying the coefficients in equation (3.1)
(while also maintaining normalization). These arguments also hold for
excited states, so long as each excited state is made orthogonal to
all lower states.
Next: Variational Theorem for the
Up: The Variational Theorem
Previous: The Variational Theorem
C. David Sherrill
20000418