Generally, the plan is to specify a finite number of basis functions, construct the matrix representation of the Dirac equation, and then diagonalize it to get the set of eigenfunctions and eigenvectors. We start by introducing radial basis sets for the large and the small components,
and
(for each
), which means that the radial functions
and
are expanded as
giving the matrix representation of the Dirac equation as
with
2.3.1. Gaussian-Type Functions
We shall work with Gaussian-type functions, since they play a central role in quantum chemical calculations. The large and the small component radial Gaussian functions are given by
with
the normalization constants. We choose the exponents
and
to reproduce the small
r behavior of the radial functions in the case of a finite nucleus [
21]; that is,
This also corresponds to the small
r behavior of the free-particle radial solutions (Equations (
10) and (
11)). Furthermore, Sun et al., in calculations on
using the dual kinetic balance prescription (discussed later in
Section 2.3.4), investigated the use of different integer powers
and
of
r for large and small Gaussian-type functions [
13] and concluded that optimal results, in particular, avoiding variational collapse and divergent integrals, were obtained using the powers given in Equation (
23).
2.3.2. C-Symmetry in The Finite Basis Approximation
We say that a basis set respects
C-symmetry, thereby leading to a
C-symmetric matrix representation, if the
C-conjugation of all the elements of the basis set belongs to the basis set itself
For simplicity’s sake, we shall set the phase factor in Equation (
8) to 1, since it does not contribute to expectation values.
is a map of
; the last condition is equivalent to say that the subspace
of
consisting of basis functions
, is preserved by the
-map. We have seen before that in the spherically symmetric case, the
C-operation replaces
,
and
, which means that the realization of the
C-symmetry at the basis set level implies
Under these conditions, we find that
and using the last equations, we get the connection between eigenvalues and eigenvectors
2.3.3. Kinetic Balance
Starting from the radial equation (Equation (
7)), we get two coupled equations that relate the large and small radial components of the Dirac equation.
The exact couplings are energy and potential-dependent, and therefore not appropriate for the construction of basis sets prior to the calculation of the energy. The energy-dependence can be eliminated by taking non-relativistic limit
. It is sometimes stated that the expressions in square brackets go to one provided
. However, the correct statement is rather that
should have a finite value as the limit is taken. For a point nucleus, the scalar potential
is singular at
, and so this condition is not satisfied. It can be restored by rather considering nuclei of finite charge distributions [
24,
25]. As it stands, the energy depends quadratically on the speed of light. This dependence can be eliminated by constant shifts, but implies taking different limits for the positive and negative energy branches.
For the positive energy branch, we introduce the shifted energy
, and from Equation (
29) obtain
For the negative energy branch, we introduce the shifted energy
, and from Equation (
30) obtain
Conventional atomic and molecular relativistic calculations in a finite basis focus on positive-energy solutions, and so bases are constructed according to the prescription of kinetic balance which imposes the non-relativistic coupling Equation (
31) at the level of individual basis functions; that is,
The numerical factor
, here set to half the reduced Compton wavelength in accordance with Equation (
31), is arbitrary. For instance, if one did not introduce an imaginary
i in the atomic spinor, Equation (
6), our present choice would be multiplied with this factor. This particular choice of basis functions provides a proper representation of the kinetic energy operator in the non-relativistic limit [
4], and therefore prevents the appearance of spurious states in the calculation. For calculations at finite values of the speed of light, it is necessary that the basis is sufficiently flexible so that the relativistic coupling can be realized through adjustment of basis expansion coefficients [
26].
The one-to-one correspondence between large and small component basis functions can only be realized in a two-component basis and is denoted
restricted kinetic balance (RKB). The terminology was introduced by Dyall and Fægri [
12] to contrast with the use of scalar basis functions, where the small component basis functions are taken as derivatives of the large component ones, in no particular linear combinations. This latter scheme is denoted
unrestricted kinetic balance (UKB).
Restricted kinetic balance (Equation (
33)) leads to the matrix eigenvalue equation (Equation (
16)) with elements given in (Equations (
A1) and (
A2)) of the
Appendix A. The matrix representation of the Dirac equation in an RKB basis is that of the modified Dirac equation [
27] (see also [
28]). From this, Sun et al. [
13] conclude that there is no "modified" Dirac equation. However, this is formally incorrect, since the modified Dirac equation has an independent existence at the operator level.
A corresponding prescription
that favors negative-energy solutions has been termed
inverse kinetic balance [
13] (IKB) and is based on Equation (
32). In this prescription, the small component basis functions are introduced first; then, the large ones are generated using the last equation. This prescription leads to the eigenvalue equation, whose elements are given in the
Appendix A (Equations (
A3) and (
A4)).
In order to respect the
C-symmetry, we impose the conditions of Equation (
25) on the RKB prescription, Equation (
33), which leads the following equation for large component basis functions
Its general solution is
and the small components are then
. The
are arbitrary coefficients;
and
are spherical Bessel functions of the first and second kind respectively. If, on the other hand, we impose
C-symmetry on the IKB prescription, Equation (
34), we obtain the same general solution, now with
. We also note that if we combine the RKB and IKB prescriptions, to describe both positive and negative energy solutions on the same footing, we again get the same general solution, with
. The problem with this specific choice of functions is that the boundary conditions
and
, are not obeyed simultaneously. Therefore, they are not useful for atomic and molecular calculations.
2.3.4. Dual Kinetic Balance
The kinetic balance prescription is widely employed in atomic and molecular calculations, but favors the positive-energy solutions. The
dual kinetic balance prescription (DKB) ensures the right coupling between the large and the small components (in the non-relativistic limit) for both positive and negative energy solutions. It was introduced by Shabaev et al. [
14] with the use of B-splines and tested by calculating the one-loop self-energy correction for a hydrogen-like ion. The radial function is expanded as
where the first and second set of basis functions have the non-relativistic coupling of positive and negative-energy solutions, respectively, as indicated by the
symbol on the coefficients. This particular expansion leads to a generalized eigenvalue problem whose elements are defined in Equations (
A9)–(
A11) of the
Appendix A. Contrary to the case of RKB/IKB the conditions for
C-symmetry, Equation (
25) can now be imposed without putting constraints on the choice of basis functions. The two matrix representations associated with
and
, become related by
leading to the
C-connection between the eigenvalues and the eigenvectors
Note, however, that the condition in Equation (
25) that ensures the
C-symmetry, implies that one has to use the same exponents for both
Gaussian type functions, as has also been pointed out by Dyall [
29]. This corresponds, in the terminology of Dyall to
j bases, where exponents are optimized for the total angular momentum
j quantum number [
30], contrary to conventional basis sets where functions are optimized for orbital angular momentum
ℓ.