:orphan: Basis sets for relativistic calculations ======================================== Gaussian Type Orbitals (GTOs) ----------------------------- *Cartesian Gaussians* are defined as .. math:: G^{\alpha}_{ijk}=G^{\alpha}_i(x)G^{\alpha}_j(y)G^{\alpha}_k(z) with .. math:: 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] Alternatively one may express a Cartesian Gaussian as .. math:: 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 where the sum :math:i+j+k is associated with orbital angular momentum :math:l. The factor .. math:: F_{ijk}=\left(2i-1\right)!!\left(2j-1\right)!!\left(2k-1\right)!! 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 :math:F_{ijk} meaning that Cartesian Gaussians are normalized to .. math:: \left=F_{ijk} In practice this means that s- and p-functions are normalized to one. So are :math:d110, :math:d101 and :math:d011, whereas :math:d200, :math:d020 and :math:d002 are normalized to three. :math:f111 is normalized to one, :math:f210, :math:f201, :math:f120, :math:f102 and :math:f012 are normalized to three, and :math:f300, :math:f030 and :math:f003 are normalized to 15. *Spherical Gaussians* are defined by .. math:: G^{\alpha}_{lm} = R^{\alpha}_l\left(r\right)Y_{lm}\left(\theta,\phi\right);\quad where the angular part is given by spherical harmonics :math:Y_{lm} and the radial part by .. math :: 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) In passing we note that .. math:: N^{\alpha}_l=2\sqrt{\frac{\alpha}{2l+1}}N^{\alpha}_{l-1} For given :math:l there are :math:\frac{1}{2}\left(l+2\right)\left(l+1\right) :math:\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 .. math:: 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} We can now distinguish two cases .. math:: 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. The case :math:\kappa^L < 0 is straightforward, but a bit more care is needed for the implementation for the case :math:\kappa^L>0. The modified spherical Gaussian to be constructed is .. math:: 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)}} Constructing modified spherical harmonics for kinetic balance; the gritty details --------------------------------------------------------------------------------- We write the spherical harmonic :math:G^{\alpha}_{l-2,m} as a linear combination of Cartesian Gaussians .. math:: G^{\alpha}_{l-2,m} = \sum_{i+j+k=l-2}c^{l-2,m}_{ijk}G^{\alpha}_{ijk} In the HERMIT integral code we have selected a transformation such that the solid harmonics are normalized to unity. From this we obtain .. math:: \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} The ration of normalization constants in the above relation is given by .. math:: \frac{N^{\alpha}_{ijk}}{N^{\alpha}_{i+2,j,k}}=\frac{1}{4\alpha}\sqrt{\frac{F_{i+2,j,k}}{F_{ijk}}} However, the expression is much simplified by the fact that factors :math:F_{ijk} area set to one in HERMIT, such that .. math:: 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]