Laplace transformation density matrix

To gain orientation, let us first neglect the self-consistent field (see below for its inclusion). The TF solution for the diagonal element of the canonical density matrix or the Slater sum S(r, fi) can be written (compare Eq. (20)) as SoB(P)exp(— PF(r)) with F(r) = — Ze /r. This is readily shown to yield, via the ground-state electron density p(r, B)=L (5(r, p, B)/P where L denotes the inverse Laplace transform discussed earlier ... 

The stochastic Liouville equations are readily solved for the time-dependent density matrix elements pu (e.g., through Laplace transforms) the latter may then be used in turn to develop expressions for the polarized fluorescence or absorption difference signals. The initial values of the density matrix elements under 5-function pulsed excitation are given by... 

D can project the density matrix elements onto the diagonal matrix elements of the bath. Apply Laplace transformation to density operator a ... 

Using this Laplace MP2 formalism, Suij(in [14] derived the explicit MP2 energy functional of the HF density matrix, while Ayala and Scuseria [17] suggested previously. Using the Roothaan Fq. (4), one can transform Fqs. (11) and (12) to... 

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