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.
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 :
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
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 .
With these real scalar functions a
set of 4-component functions is constructed:
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).
It is possible to establish an approximate relation between the
large and small component parts of the 4-component spinors. This kinetic balance
relation  becomes exact only in the non-relativistic limit:
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.
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:
This has lead to the development of the atomic
balance procedure . 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
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 . 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
- O.Visser, P.J.C.Aerts, D.Hegarty and W.C.Nieuwpoort, Chem.Phys.Letters 134, 34 (1987).
- R.C.Raffenetti, J.Chem.Phys. 58, 4452 (1973).
- A.D.McLean and Y.S.Lee, in: "Current Aspects Of Quantum Chemistry 1981", ed.R.Carbó,(Elsevier: Amsterdam 1982).
- L.Visscher, O.Visser, P.J.C.Aerts and W.C.Nieuwpoort, Int.J.Quant.Chem.: Quant.Chem.Symp. 25, 131 (1991).
- H.A.Kramers, Proc.Acad.Amsterdam 33, 959 (1930).