## Gab Gab Gab 830

! C H — Eo where |Qi is the ground state of H'. The above expression is used as the starting point of the Green function's numerical computation (see Sects. 8.4.3 and 8.4.4).

By inserting completeness relations in (8.30), one finds the Lehmann representation:

The two sums are over different sets of eigenstates, in the spaces with one more and one less electron, respectively. Let us introduce the notation eam = \ca\m) Qih = {Q |ct|n) (8.34)

as well as = Em — E0 > 0 and = — En + E0 < 0 to write

The Qaimm forms a L x N(e) matrix, where N(e) is the number of states \m) that give a non-zero contribution to the first sum above. Likewise, the QS forms a L x N(h matrix. Let M = N(e) + N(h) and let us introduce a L x M matrix Q by joining vertically the matrix Q(h) below the matrix Q(e), and let !r denote the elements of the concatenated sets {«If} and {&>nh)}. Then we can write

If we introduce the diagonal matrix Ars = Srs!r and g (!) = —(8.37) ! — A

then we have the matrix expression

This is a very general representation of the exact cluster Green function. If we restore spin and band indices, the matrix Q becomes NaL x M instead of L x M.

The band Lanczos method (Sect. 8.4.4) will provide a truncated Lehmann representation of the Green function, in which the number M of columns of the matrix Q is small, but with essentially the same properties as the exact matrix.

8.3.1 The Lehmann Representation and the CPT Green Function

The CPT Green function (8.4) also has a Lehmann representation of the form (8.38). Following [8], we write