Computational Physics

9.4 Orthonormalisation; conjugate gradient and RM-DIIS techniques 287

feature ofEq. (9.81)for us is that the correction factor can be used as an error
estimate, which we call|j〉:

|j〉=

1

Hnn

(H−Hnn)|ψn(j)〉. (9.82)

Suppose we have a sequence of estimates|ψn(j)〉,j=1,...,Jfor the optimal
states, which are not necessarily constructed according to the recipe of Eq. (9.81).
From this sequence we construct a new state as a linear combination of the
previous ones

|ψnJ+^1 〉=

∑J

j= 1

dj|ψn(j)〉 (9.83)

such that it has a minimal error. We must, however, beware of the fact that the states
can be scaled at will – upon rescaling they still satisfy the same linear equations.
However, rescaling affects the norm of the error which we want to minimise. In
order to have an unambiguous measure of the error we require that

∑J

j= 1

dj=1, (9.84)

which turns out to be a convenient choice.
Substituting the linear expansion(9.83)in the expression for the error|J+ 1 〉
of the new state|ψnJ+^1 〉, we see that yields a linear combination of the individual
errors|j〉of the|ψn(j)〉:

|J+ 1 〉=

∑J

j= 1

dj|j〉. (9.85)

Now we shall be specific about the minimalisation of the norm of the error. This
is given by

|J+ 1 |^2 =

∑J

j,k= 1

djdk〈j|k〉. (9.86)

We abbreviate the matrix elements〈j|k〉byajk. Minimising the error norm,
respecting the appropriate constraint, which is included through a Lagrange
parameterλ, leads to







a 11 a 12 ... a 1 n 1

a 21 a 22 ... a 2 n 1

..

.

..

.

... ..

.

..

.

an 1 an 2 ... ann 1

11 ... 10

























d 1

d 2

..

.

dn

−λ













=













0

0

..

.

0

1













. (9.87)