Self-consistent field spinors

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

Cnrrelafion tSACl models 169 Spinors, 213 Two Configuration Self-consistent Field reactions, 356, 363... 

We can also formulate this in a different manner and say that the self-consistent field procedure plays a crucial role in 4-component theory because it serves to define the spinors that isolate the n-electron subspaces from the rest of the Fock space. In this manner it determines in effect the precise form of the electron-electron interaction used in the calculations. Both aspects are a consequence of the renormalization procedure that was followed when fixing the energy scale and interpretation of the vacuum. The experience with different realizations of the no-pair procedure has learned that the differences in calculated chemical properties (that depend on energy differences and not on absolute energies) are usually small and that other sources of errors (truncation errors in the basis set expansion, approximations in the evaluation of the integrals) prevail in actual calculations. 

This equation is the so-called self-consistent field (SCF) equation which we may rearrange for spinor ipi to become... 

The derivation has been general so far, i.e., we have derived the most general form of the self-consistent field equations that one may use to optimize a set of one-electron spinors for truncated Cl expansions in the MCSCF approach. 

General Self-Consistent-Field Equations and Atomic Spinors... 

General Self-Consistent-Field Equations and Atomic Spinors 359 configuration wave function these equations turn out to read [351]... 

Y. S. Kim, Y. S. Lee. The Kramers restricted complete active space self-consistent-field method for two-component molecular spinors and relativistic effective core potentials including spin-orbit interactions. /. Chem. Phys., 119 (2003) 12169. 

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. 

What we would like to do is to remove all explicit reference to core spinors and to core basis functions from the Hamiltonian, incorporating their effect into some kind of local or nonlocal potential. The foundation for all such approaches is the Philips-Kleinman (1959) procedure, which was generalized by Weeks and Rice (1968). It starts from the all-electron self-consistent field (SCF) equation, which we will assume here is converged. 

Malli, G.L. Thirty years of relativistic self-consistent field theory for molecules Relativistic and electron correlation effects for atomic and molecular systems of transactinide superheavy elements up to ekaplutonium E126 with g-atomic spinors in the ground state configuration. Theor. Chem. Ace. 118, 473 82 (2007)... 

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. 

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