Efficient diagonalization of the interaction matrix

This operation count can be reduced greatly considering the special form the interaction matrix takes in the present problem. Significant improvement of computation time comes from noting that nonzero elements of the matrix appear only in one row and one column, as given in Eq. (3.72). It can be shown that such a matrix is diagonalized analytically to result in just two nonzero eigenvalues, A, where 

Short-time propagation with the interaction matrix may be written as 

Instead we split the transformation matrix into two parts, one analytically known and the other part computationally nontrivial, 

D22 is a 2 X 2 diagonal matrix of the exponentiated nonzero eigenvalues, and Dnn is the N -2) x N - 2) unit matrix. Noting that 

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