Supermatrix notation

The supermatrix notation emphasizes the structure of the problem. Each diagonal operator drives a wavepacket, just as in the adiabatic case of Eq. (10), but here the motion of the wavepackets in different adiabatic states is mixed by the off-diagonal non-adiabatic operators. In practice, a single matrix is built for the operator, and a single vector for the wavepacket. The operator matrix elements in the basis set [Pg.384]

In order to take into account the two-electron interaction terms we define a supermatrix notation that we illustrate for The components of a supervector t are the matrix elements t ij in dictionary order. Supervectors are transformed by supermatrices. We need the supermatrix V, defined by... 

Here a supermatrix notation is employed, each entry corresponding to a two by two matrix. The a matrices are the familiar Pauli spin matrices... 

Here we introduce the notation ( . ..) for the scalar product of vectors whose components are numbered by the Cartesian shifts of the nuclei). Next, let h" be the supermatrix of the second derivatives of the matrix of the Fock operator with respect to the same shifts. As previously, we refer here to the supermatrix indexed by the pairs of nuclear shifts in order to stress that the elements of this matrix are themselves the 10 x 10 matrices of the corresponding second derivatives of the Fock operator with respect to the shifts. The contribution of the second order in the nuclear shifts can be given the form of the (super)matrix average over the vector of the nuclear shifts ... 

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