Fock matrix spinor

The DIRAC package [55,56], devised by Saue and collaborators, rather than exploiting the group theoretical properties of Dirac spherical 4-spinors as in BERTHA, treats each component in terms of a conventional quantum chemical basis of real-valued Cartesian functions. The approach used in DIRAC, building on earlier work by Rosch [80] for semi-empirical models, uses a quaternion matrix representation of one electron operators in a basis of Kramers pairs. The transformation properties of these matrices, analysed in [55], are used to build point group transformation properties into the Fock matrix. 

We now have a collection of integrals over basis functions which must be evaluated in order to construct the Fock matrix. For S-spinors, these can be deduced from formulae given by Kim [95,37]. The main difference is that Kim was not aware of the importance of kinetic matching adaptation of his formulae is routine. The integrals involved are all related to the gamma function or the error function [83, Chapters 6 and 7]. We therefore concentrate here on G-spinors, which can be applied both to atomic and to molecular calculations. 

In deriving a set of relativistic one-particle functions, that is, spinors, from a mean-field approach, we typically start by making a simple guess at these functions (or the electron density), and then try to refine them iteratively. The refinement can be done by diagonalizing a suitable Hamiltonian (or Fock) matrix, which defines a rotation of the spinors in the entire function space available. Normally, this iterative process reaches a stage where further rotations do not change the spinors, that is, they are self-consistent. Provided we have chosen our sequence of rotations carefully, this should correspond to the optimal set of spinors from the mean field. For the present chapter our main concern is the rotation of the set of one-particle functions, and how this can be cast in a consistent theoretical framework that also accounts for the positron contributions. 

The terms in brackets may appear familiar, and are indeed nothing but the expression for the usual spinor or spin-orbital Fock matrix. If we insert the values of the density matrices for a single determinant, we get... 

The Fock matrix defined in terms of the densities in (8.30) is only nonzero if the second index is an occupied spinor index. A set of spinors that transforms the Fock matrix... 

The Fock matrix is identical in form to the electron Fock matrix given above, and the gradient is the same in form as the spinor gradient involving one unoccupied electron spinor. Thus, for the purposes of self-consistent field methods, we may treat the negative-energy spinors as simply an extension of the set of unoccupied spinors. The Hessian may similarly be defined using the principles and formulas set out above. 

Similar expressions follow for the other combinations of components, and for the Gaunt integrals. To arrive at a compact expression for the Fock matrix elements, we define the 2-spinor density matrices by... 

We would expect the open-open Fock matrix elements to correspond to redundant spinor rotation parameters because the energy should be invariant to rotations of equivalent spinors. However, if we construct the spinor rotation gradient, we see that there is a term that survives in the diagonal of the gradient ... 

Including the Breit term for the electron-electron interaction in a scalar basis requires extensive additions to a Dirac-Hartree-Fock-Coulomb scheme. It is not possible to achieve the same reductions as for the Coulomb term, and the derivation of the Fock matrix contributions requires considerable bookkeeping. We will not do this in detail, but will provide the development for the Gaunt interaction as we did for the 2-spinor case. 

The decision of whether to work with 2-spinors or a scalar spin-orbital basis must be made at an early stage of computer program construction because it affects all stages of the SCF process evaluation of the integrals, construction of the Fock matrix, and solution of the SCF equations. However, at each stage, the scalar spin-orbital basis can be transformed to the 2-spinor basis. Transformation of the integrals to a 2-spinor basis is not particularly difficult it is similar in principle to the transformation from Cartesians to spherical harmonics. Some efforts have been made to develop new algorithms in which these transformations are incorporated, and RKB is implemented from the start in the 2-spinor basis (Quiney et al. 1999, 2002, Yanai et al. 2002). 

Compared to the relatively simple matrix elements of the nonrelativistic Fock matrix, the expressions given in the sections above indicate the degree of increased complexity involved in a relativistic calculation. The critical issue is of course the logistics of the two-electron integrals. The following observations on the number of integrals can be made for a system that has no spatial symmetry using a 2-spinor RKB basis for the relativistic calculation. 

Thus, the scalar basis involves about 20% fewer real quantities than the 2-spinor basis, and therefore less work in the Fock matrix construction. This applies to an uncontracted basis set. 

In the first case, no approximation is made to the valence spinor. Then there is no approximation to the Fock matrix apart from the addition of the projector term. The gradient for mixing an arbitrary function into the valence spinor is deduced from (20.64)... 

This expression contains a core projector matrix element in addition to the model potential Fock matrix element. However, because it is off-diagonal the core projector term could be of either sign, depending on the contribution of the orthogonal component to the trial spinor. Whatever contributions to the Fock matrix elements come from core spinors, their signs are changed by the projector matrix element. In any case, the... 

Using the spinor form (2.2) we can go from the spin orbital form (2.5) of the Fock-Dirac matrix to the orbital form ... 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

By contrast to the numerical MCSCF method discussed in the last chapter, the basis-set approach has the convenient advantage that the virtual orbitals come for free by solution of the Roothaan equation. While the fully numerical approaches of chapter 8 do not produce virtual orbitals, as the SCF equations are solved directly for occupied orbitals only and smart b)q)asses must be devised, this problem does not show up in basis-set approaches. Out of the m basis functions, only N with N matrix Fock operator produces a full set of m orthogonal molecular spinor vectors that can be efficiently employed in the excitation process of any Cl-like method. 

Since the X matrix is directly evaluated from the electronic solutions of the four-component Fock operator, which must therefore be diagonalized, the same pathologies regarding the negative-energy states discussed in chapters 8 and 10 pose a caveat. However, if a four-component calculation must be carried out before the two-dimensional operator can be evaluated (as in the X2C case), the projection to electronic states by elimination of the small component in an exact two-component approach has no valid formal advantage (as the four-component variational solution for the 4-spinors already required (implicit) projection to the electronic solutions). 

Matrix Dirac-Hartree-Fock Equations in a 2-Spinor Basis... 

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