CCD for the pairing Hamiltonian
You learned about the pairing Hamiltonian earlier in this
school. Convince yourself that this Hamiltonian does not induce any
1p-1h excitations. Let us solve the CCD equations for this
problem. This consists of the following steps
- Write a function that compute the potential, i.e. it returns a four-indexed array (or tensor). We need \( \langle ab\vert V\vert cd\rangle \), \( \langle ij\vert V\vert kl\rangle \), and \( \langle ab\vert V\vert ij\rangle \). Why is there no \( \langle ab\vert V\vert id\rangle \) or \( \langle ai\vert V\vert jb\rangle \) ?
- Write a function that computes the Fock matrix, i.e. a two-indexed array. We only need \( f_a^b \) and \( f_i^j \). Why?
- Initialize the cluster amplitudes according to Eq. (37), and solve Eq. (36) by iteration. The cluster amplitudes \( T_1 \) and \( T_2 \) are two- and four-indexed arrays, respectively.
Please note that the contraction of tensors (i.e. the summation over
common indices in products of tensors) is very user friendly and
elegant in python when numpy.einsum is used.