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Relativistic State-Selective (Intermediate) Effective Hamiltonian Method

Andrei ZAITSEVSKII

A. Starting approximation

Model

We work within the relativistic effective pseudopotential (REP) model (see e.g. [1,2]), defining the quasirelativistic many-electron valence-shell molecular Hamiltonian as

where the total pseudopotential operator W consist of semilocal atomic contributions dependent on the one-electron total angular momentum j defined with respect to the corresponding atomic center:

Here  denotes the projector onto the subspace of two-component spinor spherical harmonics centered on the ath nucleus and the functionsdepend on the distance betwen an electron and the ath nucleus. Each atomic pseudopotential operator is split into the spin-averaged part (AREP) and effective spin-orbit pseudopotential. This splitting induces the partitioning of the total Hamiltonian into

  • the scalar relativistic Hamiltonian HSc, comprising kinetic energy, Coulomb electron-electron repulsion and AREP terms, and
  • the effective spin-orbit operator HSO.
  • The former operator is formally similar with conventional valence-shell Hamiltonian used within the semilocal pseudopotential approximation and commutes with point group transformations and total spin operations; the latter one is a sum of one-electron terms describing the spin-dependent deviation of the REP from the spin-averaged potential and can mix functions with different spin and point group symmetries. Being one-body operator, it implicitly incorporates the bulk of effects which manifest theirselves as two-electron spin-orbit interactions in all-electron two-component quasirelativistic theories (Douglas - Kroll etc.).

To convert the Hamiltonian H into a discretized form, we
    solve an appropriate SCF- or MCSCF-like problem for the scalar relativistic Hamiltonian HScyielding a set of spatial molecular orbitals,
    define a basis in the space of two-component one-electron spinors consisting of these molecular orbitals with spin factors "alpha" or "beta" (spinorbitals),
    construct a many-electron basis of Slater determinants {| J >}built from these spinorbitals (note that these basis should comprise Slater determinants with different spatial symmetries and total spin projections).
We are interested in M lowest eigenstates of the total relativistic Hamiltonian H (target states).
 

Restriction of the spin-orbit operator

To simplify the treatment of low- (double-group-) symmetry spin-orbit interactions, we try to construct a linear span of properly chosen Slater determinants (model space) in such a way that the spin-orbit interactions out of this space are insignificant in what concerns the target states. In other words, we want that the lowest M eigenstates of the operator

(where P is the projector onto the model space) provide a good approximation to those of (this obviously does not mean that the mentioned operators are globally similar!).
 

To this end, we
    define scalar auxiliary states, i. e.  HSc eigenstates which are supposed to have large overlap with the relativistic target states,
    choose a reference set comprising the leading configurations of the auxiliary (and therefore relativistic target)  states,
    starting from the reference set, the model space is built by means of a conventional (spin-orbit-free) numerical perturbative selection of configurations with potentially significant contributions to the auxiliary states [3].
 
We do realize that such selection in the absence of the spin-orbit operator is only a simple and rather reasonable but certainly not the best way to satisfy the abovementioned requirement.


Now the spin-orbit interactions outside of the model space are forgotten.
 

B. Second-order quasirelativistic state-selective effective Hamiltonian

We proceed via the calculation of a state-selective (intermediate) quasirelativistic effective Hamiltonian Heff [4,5] acting within the model space. The diagonalization of Heff should yield (approximate) eigenenergies and model space parts of wavefunctions for all target relativistic states simultaneously. In contradistinction with the  more commonly used state-universal effective Hamiltonian approach, the remainder (Tr(P)-M) eigenstates should not be necessarily meaningful.

Provided that we had chosen a scalar zero-order Hamiltonian approximation (or several approximations, if a multipartitioning version of the perturbation theory is used) which is diagonal in the determinantal basis, the approximation for  Hermitian intermediate effective Hamiltonian matrix up to second order is given by

where I and J are model-space determinant indices and the sum runs over all the determinants |K> out of model space. The expression for the energy denominator D(I,K) associated with a pair of determinants (|I>, |K>) depends on the particular form of  state-selective effective Hamiltonian perturbation theory (singe partitioning with shifts [5], multipartitioning [6] etc).

Since we neglect the spin-orbit interactions outside of the model space,

,,

i.e. the second-order contributions are reduced to that for the scalar relativistic problem and will have non-relativistic ((point group)x(spin)) symmetry. If we arrange the model-space determinants according to the values of the total spin projections Sz, the effective Hamiltonian matrix will have the form


represents the second-order scalar-relativistic intermediate Hamiltonian matrices for given Sz corresponds to the effective spin-orbit operator matrix

(even number of electrons is assumed). The procedure of computation of the blocks is identical to that for non-relativistic intermediate Hamiltonians and obviously can fully make use of non-relativistic symmetry. Provided that the system under study has a non-trivial point group, each fixed-Sz block has a subblock-diagonal structure (the subblocks are associated with definite non-relativistic spatial symmetries).  Moreover, only the Sz = 0 block (Sz = 1/2 for an odd number of electrons) should be computed directly; the remainder blocks are readily derived from the first one by means of S+ and S- operators.  The total Heff matrix is not block diagonal since non-zero spin-orbit matrix elements can be located outside of the blocks, but the evaluation of these matrix elements is extremely simple since they do not incorporate perturbative corrections.

The evaluation of few lowest eigenvalues of possibly large (though much smaller than in double-group CI treatment!) but rather sparse complex hermitian Heff matrix is eficiently  performed by an appropriate version of Davidson algorithm [7].

Bibliography

[1] M.Krauss, W.J.Stevens, Ann.Rev.Phys.Chem. 35, 357 (1984)
[2] W.Kuechle, M.Dolg, H.Stoll, H.Preuss, Mol.Phys. 74, 1245 (1991)
[3] B.Huron, P.Rancurel, J.P.Malrieu, J.Chem.Phys. 58, 5745 (1973)
[4] J.P.Malrieu, Ph.Durand, J.P.Daudey, J.Phys.A: Math.Gen. 18, 809 (1985)
[5] A.Zaitsevskii, J.L.Heully,  J.Phys.B: At.Mol.Opt.Phys. 25, 603 (1992)
[6] A.Zaitsevskii, J.P.Malrieu, Theor.Chem.Acc. 96, 269 (1997)
[7] V.Vallet, L.Maron, Ch.Teichteil, J.P.Flament, J.Chem.Phys.113, 1391 (2000)
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