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Perturbed Transition Density Matrices

The diagonalization of the effective Hamiltonian Heff yields  the model-space parts of the target relativistic wavefunctions (|m>, |n>, ...). Provided that efficient `diagrammatic' algorithms are employed, no explicit approximation for the remainder (outer-space) parts of the wavefunctions is obtained. A convenient way to incorporate the contributions from these parts to transition property values consists in a perturbative calculation of transition density matrices.

The first-order approximation for the m - n transition spin-free density matrix mnR compatible with the use of the present approximation for Heff is given by

mnRrs = <m | Esr | n> +  JJ'  <m | J><J' | m K ( < J |Esr  | K>HKJ'/ D(J',K)
                                                                                      + HJK <K | Esr | J'> / D(J,K))

where

J and J' are model-space determinant indices and the determinants |K> are out of the model space,
s,r are the spatial orbital indices,
Esr denotes the spin-free one-electron r -> s excitation operator,
the definition of energy denominators  D(I,K) is the same as used for Heff .
Provided that  the spin-orbit interactions outside of the model space are neglected, HJK  = HScJK and the summation over K is similar with that required in non-relativistic or scalar-relativistic calculations.

The  m -> n  transition value of an one-electron property (Anm) is readily obtained as

Anm  = tr (mnR A),

where A is the property matrix in the chosen spatial orbital basis.

References

  • Nikolayeva O., Klincare I., Auzinsh M., Tamanis M., Ferber R., Pazyuk E.A., Stolyarov A.V., Zaitsevskii A., Cimiraglia R., J Chem Phys 2000, 113, 4896
  • Zaitsevskii A., Ferber R., Teichteil Ch., Phys Rev A  2001, 63, 042511

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