3.4 Perturbation theory and variational calculus 37Then we can rewrite(3.13)as
V†HVV−^1 C=EV†SVV−^1 C (3.31)and, defining
C′=V−^1 C (3.32)
and
H′=V†HV, (3.33)
we obtain
H′C′=EC′. (3.34)
This is an ordinary eigenvalue problem which we can solve forC′andE, and then
we can find the eigenvectorCof the original problem asVC′.
The problem remains of finding a matrixVwhich bringsSto unit form accord-
ing to(3.30). This matrix can be found if we have a unitary matrixUwhich
diagonalisesS:
U†SU=s (3.35)
withsthe diagonalised form ofS. In fact, the matrixUis automatically gener-
ated when diagonalisingSby a Givens–HouseholderQRprocedure (seeAppendix
A8.2). From the fact thatSis an overlap matrix, defined by (3.12), it follows directly
that the eigenvalues ofSare positive (see Problem 3.2). Therefore, it is possible
to define the inverse square root ofs: it is the matrix containing the inverse of
the square root of the eigenvalues ofSon the diagonal. Choosing the matrixVas
Us−^1 /^2 , we obtain
V†SV=s−^1 /^2 U†SUs−^1 /^2 =I (3.36)
so the matrixVindeed has the desired property.
*3.4 Perturbation theory and variational calculusIn 1951, Löwdin[6]devised a method in which, in addition to a standard basis set
A, a number of extra basis states (B) is taken into account in a perturbative manner,
thus allowing for huge basis sets to be used without excessive demands on computer
time and memory. The size of the matrix to be diagonalised in this method is equal
to the number of basis states in the restricted setA; the remaining states are taken
into account in constructing this matrix. A disadvantage is that the latter depends
on the energy (which is obviously not known at the beginning), but, as we shall see,
this does not prevent the method from being useful in many cases.
We start with an orthonormal basis, which could be a set of plane waves. The
basis is partitioned into the two setsAandB, and for the plane wave example,A
will contain the slowly varying waves andBthose with shorter wavelength. We