relativistic electronic structure calculations for molecules and clusters 

Relativistic StateSelective (Intermediate) Effective Hamiltonian MethodAndrei ZAITSEVSKII
ModelWe work within the relativistic effective pseudopotential (REP) model (see e.g. [1,2]), defining the quasirelativistic manyelectron valenceshell molecular Hamiltonian aswhere the total pseudopotential operator W consist of semilocal atomic contributions dependent on the oneelectron total angular momentum j defined with respect to the corresponding atomic center: Here denotes the projector onto the subspace of twocomponent spinor spherical harmonics centered on the a^{th} nucleus and the functionsdepend on the distance betwen an electron and the a^{th} nucleus. Each atomic pseudopotential operator is split into the spinaveraged part (AREP) and effective spinorbit pseudopotential. This splitting induces the partitioning of the total Hamiltonian into
The former operator is formally similar with conventional valenceshell Hamiltonian used within the semilocal pseudopotential approximation and commutes with point group transformations and total spin operations; the latter one is a sum of oneelectron terms describing the spindependent deviation of the REP from the spinaveraged potential and can mix functions with different spin and point group symmetries. Being onebody operator, it implicitly incorporates the bulk of effects which manifest theirselves as twoelectron spinorbit interactions in allelectron twocomponent quasirelativistic theories (Douglas  Kroll etc.). To convert the Hamiltonian
H
into a discretized form, we
solve an appropriate SCF
or MCSCFlike problem for the scalar relativistic Hamiltonian H^{Sc}yielding
a set of spatial molecular orbitals,
define a basis in the space
of twocomponent oneelectron spinors consisting of these molecular orbitals
with spin factors "alpha" or "beta" (spinorbitals),
Restriction of the spinorbit operatorTo simplify the treatment of low (doublegroup) symmetry spinorbit interactions, we try to construct a linear span of properly chosen Slater determinants (model space) in such a way that the spinorbit 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. H^{Sc} 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 (spinorbitfree) numerical perturbative selection of configurations with potentially significant contributions to the auxiliary states [3].
We proceed via the calculation of a stateselective (intermediate) quasirelativistic effective Hamiltonian H^{eff} [4,5] acting within the model space. The diagonalization of H^{eff} should yield (approximate) eigenenergies and model space parts of wavefunctions for all target relativistic states simultaneously. In contradistinction with the more commonly used stateuniversal effective Hamiltonian approach, the remainder (Tr(P)M) eigenstates should not be necessarily meaningful. Provided that we had chosen a scalar zeroorder 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 modelspace 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 stateselective effective Hamiltonian perturbation theory (singe partitioning with shifts [5], multipartitioning [6] etc). Since we neglect the spinorbit interactions outside of the model space, ,, i.e. the secondorder contributions are reduced to that for the scalar relativistic problem and will have nonrelativistic ((point group)x(spin)) symmetry. If we arrange the modelspace determinants according to the values of the total spin projections S_{z}, the effective Hamiltonian matrix will have the form
(even number of electrons is assumed). The procedure of computation of the blocks is identical to that for nonrelativistic intermediate Hamiltonians and obviously can fully make use of nonrelativistic symmetry. Provided that the system under study has a nontrivial point group, each fixedS_{z} block has a subblockdiagonal structure (the subblocks are associated with definite nonrelativistic spatial symmetries). Moreover, only the S_{z} = 0 block (S_{z} = 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 H^{}eff matrix is not block diagonal since nonzero spinorbit 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 doublegroup CI treatment!) but rather sparse complex hermitian H^{eff} 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)
