# Basis sets for relativistic calculations¶

## Gaussian Type Orbitals (GTOs)¶

Cartesian Gaussians are defined as

\begin{equation*} G^{\alpha}_{ijk}=G^{\alpha}_i(x)G^{\alpha}_j(y)G^{\alpha}_k(z) \end{equation*}

with

\begin{equation*} G^{\alpha}(x)=\left(\frac{2\alpha}{\pi}\right)^{1/4}\sqrt{\frac{\left(4\alpha\right)^i}{\left(2i-1\right)!!}} x^i\exp\left[-\alpha x^2\right] \end{equation*}

Alternatively one may express a Cartesian Gaussian as

\begin{equation*} G^{\alpha}_{ijk}=N^{\alpha}_{ijk}x^iy^jz^k\exp\left[-\alpha r^2\right];\quad N^{\alpha}_{ijk}=\left(\frac{2\alpha}{\pi}\right)^{3/4}\sqrt{\frac{2^l}{F_{ijk}}}\left(\sqrt{2\alpha}\right)^l \end{equation*}

where the sum $$i+j+k$$ is associated with orbital angular momentum $$l$$ . The factor

\begin{equation*} F_{ijk}=\left(2i-1\right)!!\left(2j-1\right)!!\left(2k-1\right)!! \end{equation*}

shows that the Cartesian components of a given shell may have different normalization constants. In the HERMIT integral code a single normalization is chosen for each shell by ignoring $$F_{ijk}$$ meaning that Cartesian Gaussians are normalized to

\begin{equation*} \left<G^{\alpha}_{ijk}|G^{\alpha}_{ijk}\right>=F_{ijk} \end{equation*}

In practice this means that s- and p-functions are normalized to one. So are $$d110$$ , $$d101$$ and $$d011$$ , whereas $$d200$$ , $$d020$$ and $$d002$$ are normalized to three. $$f111$$ is normalized to one, $$f210$$ , $$f201$$ , $$f120$$ , $$f102$$ and $$f012$$ are normalized to three, and $$f300$$ , $$f030$$ and $$f003$$ are normalized to 15.

Spherical Gaussians are defined by

\begin{equation*} G^{\alpha}_{lm} = R^{\alpha}_l\left(r\right)Y_{lm}\left(\theta,\phi\right);\quad \end{equation*}

where the angular part is given by spherical harmonics $$Y_{lm}$$ and the radial part by

\begin{equation*} R^{\alpha}_l=N^{\alpha}_lr^l\exp\left[-\alpha r^2\right];\quad N^{\alpha}_l = \frac{2\left(2\alpha\right)^{3/4}}{\pi^{1/4}}\sqrt{\frac{2^l}{\left(2l+1\right)!!}}\left(\sqrt{2\alpha}\right) \end{equation*}

In passing we note that

\begin{equation*} N^{\alpha}_l=2\sqrt{\frac{\alpha}{2l+1}}N^{\alpha}_{l-1} \end{equation*}

For given $$l$$ there are $$\frac{1}{2}\left(l+2\right)\left(l+1\right)$$ $$\left(2l+1\right)$$ spherical Gaussians. The latter basis functions therefore provide more compact basis set expansions. However, in 4-component relativistic calculations the use of spherical Gaussians is somewhat more complicated since the coupling of large and small component basis functions needs to be taken into account.

## Kinetic balance¶

Kinetic balance corresponds to the non-relativistic limit of the exact coupling of large and small component basis functions for positive-energy orbitals. Starting from the radial function of a large component spherical Gaussian basis function one obtains

\begin{equation*} R^S\propto -\left[\frac{\partial}{\partial r}+\frac{\left(1+\kappa^L\right)}{r}\right]R^{\alpha}_l = \sqrt{\alpha\left(2l+3\right)}R^{\alpha}_{l+1}-2\sqrt{\frac{\alpha}{2l+1}}R^{\alpha}_{l-1} \end{equation*}

We can now distinguish two cases

\begin{equation*} R^S \propto \left\{ \begin{array}{ll} \sqrt{\left(2l+3\right)}R^{\alpha}_{l+1}-2\sqrt{\left(2l+1\right)}R^{\alpha}_{l-1} & \mbox{if $\kappa^L = l$};\\ R^{\alpha}_{l+1} & \mbox{if $\kappa^L = -\left(l+1\right)$}.\end{array} \right. \end{equation*}

The case $$\kappa^L < 0$$ is straightforward, but a bit more care is needed for the implementation for the case $$\kappa^L>0$$ . The modified spherical Gaussian to be constructed is

\begin{equation*} G^*_{lm} = N\left\{\sqrt{\left(2l+1\right)\left(2l-1\right)}R^{\alpha}_l-2\left(2l-1\right)R^{\alpha}_{l-2}\right\}Y_{l-2,m};\quad N = \frac{1}{\sqrt{\left(2l+1\right)\left(2l-1\right)}} \end{equation*}

## Constructing modified spherical harmonics for kinetic balance; the gritty details¶

We write the spherical harmonic $$G^{\alpha}_{l-2,m}$$ as a linear combination of Cartesian Gaussians

\begin{equation*} G^{\alpha}_{l-2,m} = \sum_{i+j+k=l-2}c^{l-2,m}_{ijk}G^{\alpha}_{ijk} \end{equation*}

In the HERMIT integral code we have selected a transformation such that the solid harmonics are normalized to unity. From this we obtain

\begin{equation*} \begin{array}{lcl} G^*_{lm}&=&N\left[4\alpha r^2-2\left(2l-1\right)\right]\sum_{i+j+k=l-2}c^{l-2,m}_{ijk}G^{\alpha}_{ijk}\\ &=&N\sum_{i+j+k=l-2}c^{l-2,m}_{ijk} \left[4\alpha \left( \frac{N^{\alpha}_{ijk}}{N^{\alpha}_{i+2,j,k}}G^{\alpha}_{i+2,j,k} +\frac{N^{\alpha}_{ijk}}{N^{\alpha}_{i,j+2,k}}G^{\alpha}_{i,j+2,k} +\frac{N^{\alpha}_{ijk}}{N^{\alpha}_{i,j,k+2}}G^{\alpha}_{i,j,k+2} \right) -2\left(2l-1\right)G^{\alpha}_{ijk} \right] \end{array} \end{equation*}

The ration of normalization constants in the above relation is given by

\begin{equation*} \frac{N^{\alpha}_{ijk}}{N^{\alpha}_{i+2,j,k}}=\frac{1}{4\alpha}\sqrt{\frac{F_{i+2,j,k}}{F_{ijk}}} \end{equation*}

However, the expression is much simplified by the fact that factors $$F_{ijk}$$ area set to one in HERMIT, such that

\begin{equation*} G^*_{lm}=N\sum_{i+j+k=l-2}c^{l-2,m}_{ijk} \left[ \left(G^{\alpha}_{i+2,j,k}+G^{\alpha}_{i,j+2,k}+G^{\alpha}_{i,j,k+2}\right) -2\left(2l-1\right)G^{\alpha}_{ijk} \right] \end{equation*}