Methodology

The usual approach in quantum chemistry for approximating the wave function is to build molecular wave functions from antisymmetrised products of atomic centred one-electron functions. This method is also applied to the Dirac-Coulomb-(Breit) equation. Some details of this Dirac-Fock-CI approach are discussed in this chapter. A more extended discussion of the methods that are used can be found in the thesis of Visscher.

NUCLEAR MODEL

There are two different models used in our program package. One can use the point-charge model (for non-relativistic or two-component calculations) or a finite nuclear model. Our finite nucleus model is based on a Gaussian distribution function of the nuclear charge [1]:




							     (8)

Here ZK is the charge of nucleus K and MK is its mass. In this formula ksi is related to the homogeneously charged sphere model by the formula S = R / 2, with R the radius of the sphere and S the standard deviation of the radial Gaussian distribution. The formula R = 2.27D-5 * MK1/3 is used to relate the radii and mass of the nuclei.

BASIS SET APPROACH

Our scalar basis set is formed by primitive cartesian Gaussian basis functions.



						    (9)

The basis is subdivided in a large(L) and small(S) component set, that will be used to describe the upper and lower two components of the 4-spinors respectively.
This basis set can be contracted using the general contraction scheme that was introduced by Raffenetti [2].



							    (10)

With these real scalar functions a set of 4-component functions is constructed:

				(11)

This basis set is used to expand the atomic or molecular spinors and defines the matrix representation of the one-electron Dirac or pseudo one-electron Dirac-Fock Hamiltonian. One can use the (point group) symmetry by a unitary transformation of the 4-component functions (forming a symmetry adapted basis).



						  (12)

KINETIC BALANCE

It is possible to establish an approximate relation between the large and small component parts of the 4-component spinors. This kinetic balance relation [3] becomes exact only in the non-relativistic limit:

						 (13)

Basis sets that fulfil this relation are hence called kinetically balanced. For primitive basis sets, this concept of kinetic balance is widely accepted as method for generating small component basisfunctions.

ATOMIC BALANCE

When one wants to use contracted basis sets one has to generate the small component basis functions but also the contraction coefficients for these functions. Straightforward application of the kinetic balance operator, which means that we use the contraction coefficients of the large component functions, gives erroneous results in the contracted functions that describe the core region. Now one needs a operator that resembles the exact operator more closely:

				  (14)

This has lead to the development of the atomic balance procedure [4]. In this procedure one first performs calculations on the atom(s) using kinetically balanced, uncontracted, basis sets. The new large and small component basis functions are then formed by contracting the primitive basis with the spinor coefficients of these calculations. One can construct a highly contracted basis in much the same way as in non-relativistic calculations. An important difference is caused by spin-orbit splitting. The difference in radial character of the l-1/2 and l+1/2 spinors in principle doubles the number of contracted Gaussian functions relative to a non-relativistic contraction scheme.

SYMMETRY

The program package can handle all double groups that are subgroups of the Oh double group. The full point group symmetry is used up to the Dirac-Fock level. At the relativistic CI level the highest abelian subgroup of the point group under consideration is used. Another symmetry that can be used is the Kramers' or Time-Reversal symmetry [5]. This symmetry is exploited in the 4-index transformation (ROTRAN) and the Dirac-Fock (MFDSCF) program but is at present not used in the CI codes.

References in Methodology

  1. O.Visser, P.J.C.Aerts, D.Hegarty and W.C.Nieuwpoort, Chem.Phys.Letters 134, 34 (1987).
  2. R.C.Raffenetti, J.Chem.Phys. 58, 4452 (1973).
  3. A.D.McLean and Y.S.Lee, in: "Current Aspects Of Quantum Chemistry 1981", ed.R.Carbó,(Elsevier: Amsterdam 1982).
  4. L.Visscher, O.Visser, P.J.C.Aerts and W.C.Nieuwpoort, Int.J.Quant.Chem.: Quant.Chem.Symp. 25, 131 (1991).
  5. H.A.Kramers, Proc.Acad.Amsterdam 33, 959 (1930). 
  MOLFDIR Contents