Extension to Slater-type orbitals

[susilehtola]

Currently GauXC is limited to Gaussian-type basis functions

$$ R_i(r) \propto r^{l_i} \sum_i c_i \exp(-\alpha_i r^2) $$

but it shouldn't be too difficult to extend support to Slater-type orbitals

$$ R_i(r) \propto r^{n_i-1} \exp(-\zeta_i r) $$

where $n_i$ may be non-integer.

[wavefunction91]

Can you expand on this "non-integer" principal AM number criterion? I understand how this works for scaling the angular bits for integer AM (i.e. I don't treat $$r^l$$ explicitly, it's wrapped into the angular function). Can you give an example of how this would look for e.g. a cartesian and spherical $$d$$ function with non-integer $$n_i$$?

[susilehtola]

I think the handling would be similar for NAOs and STOs: since it is probably easiest for the basis function gradient and Laplacian to expand $r^l Y_{lm}$ in the Cartesian basis, one then needs to use the radial function $r^{-l} R(r)$.

For STOs, the new radial part is then $r^{n-1-l} \exp(-\zeta r)$. For traditional integer-$n$ STOs, $n\ge l+1$ so this part is well-behaved.

For non-integer STOs, the story is different: according to the pioneering paper by Parr and Joy, the best result is obtained for helium with an s-function with $\zeta=1.61162$ and $n=0.955$. The fits for the Kr example in my review paper shows that the optimal $n$ value can lead to $n-1 < l$ at least for $d$ functions

Fitting am = 0 orbital
Optimal n=1 STO exponent 3.529000e+01, projection 9.999872e-01
Optimal n=1.000000e+00 STO exponent 3.529000e+01, projection 9.999872e-01

Fitting am = 1 orbital
Optimal n=2 STO exponent 1.544000e+01, projection 9.994710e-01
Optimal n=1.800000e+00 STO exponent 1.420000e+01, projection 9.999334e-01

Fitting am = 2 orbital
Optimal n=3 STO exponent 6.390000e+00, projection 9.879646e-01
Optimal n=1.840000e+00 STO exponent 4.220000e+00, projection 9.988193e-01

One can thus expect the non-integer approach to lead to some potentially difficult numerics close to the nucleus: the adjusted radial part $r^{n-1-l} \exp (-\zeta r)$ blows up in the $r\to 0$ limit, but should be killed by the $r^l$ factor included in the angular part. The same challenges will also affect NAOs.

[susilehtola]

Of course, one can evaluate the spherical harmonics in a different way: just normalize the relative $x$, $y$, and $z$ coordinates, that way you don't get these issues.