Solution of the stationary-value equations

Since the operator equations (8.2.8) are intractible, as they stand, we turn at once to finite-basis forms. There are two common choices of basis the n occupied orbitals may be expressed directly in terms of a completely arbitrary set Xr of m functions (e.g. AOs) or in terms of, say, first approximations to the pr (e g. simple SCF MOs). Both possibilities may be handled using the same equations (8.2.11)-(8.2.15), but in the second case the intermediate set should contain all 

It is convenient to start from (8.2.31), with defined in (8.2.10) or (8.2.14), according to the choice of basis. An obvious way to satisfy this equation would be to make a series of 2 x 2 rotations , as in the Jacobi method for matrix diagonalization, in order to reduce the elements of - to zero. This approach has been used by Hinze and coworkers (see e.g. Hinze, 1973 Hinze and Yurtsever, 1979), who choose the 0-basis and consider the reduction of 

A more satisfactory procedure (Golebiewski et ai, 1979) is to adopt the full transformation (8.3.2), choosing the unitary matrix V so that in each cycle the transformed matrix becomes more nearly Hermitian. Here we derive the required algorithm using the basis x and the transformation (8.3.3), with T in the partitioned form (8.3.1). 

First we note that the matrices T and X in (8.2.14) are rectangular, their columns referring only to the n occupied orbitals. With the notation of (8.3.1) the n X n matrix becomes TlXi, Xi being the first block of an extended X matrix in which Xj is identically zero. In terms of the full matrices, the quantity of interest is thus simply the leading diagonal block of 

Let us confine attention to the non-zero part of this matrix, namely 

---