Friday 13 September 2013

Density fitting and tensor hyper contraction

One of the main bottle necks in quantum chemistry is the computation of the two electron integrals and their transformation between the AO and MO bases. These integrals are defined as

where μ, ν, ρ and σ are one-electron orbitals.

A well established way to approximate the two electron integrals is density fitting (also called "resolution of the identity"). The idea is that the orbital products are one-electron functions (actually they are little densities) and may therefore be expressed using a new set of one-electron functions called "auxiliary basis set"

If the set of one electron functions χA is complete, then these would actually "resolve the identity" and the coefficients cμν,A could be chosen in a way to make this expression exact. For practical calculations with finite basis sets the optimal cμν,A can be obtained through density fitting.

The cμν,A are only three-index quantities, which requires less storage room as opposed to the full four-index MO integrals. The original two-electrons integrals can be obtained by using a double summation:

If this approach is combined with appropriate changes in the remaining algorithm, very efficient implementations of ab-initio excited state methods are possible (as this is for example done in Turbomole).

Going one step ahead, there is the tensor hyper contraction, as implemented by the Martinez group. In this case one only uses two-index quantities Xμ,P and ZPQ. In this approach the two-electron integrals are approximated according to

To define the Xμ,P they use a grid of points rP and simply set

Finally, the Z-matrix is fitted in order to allow the best possible approximation in the above defined form. And apparently, this is efficient and allows a fourth order scaling of CC2, as opposed to fifth order in the standard and RI implementations.

The idea is quite nice, I would say. Of course, only time will tell if "formal fourth order" is really faster than well implemented fifth order. But it looks like a nice step ahead.


Tim said...

As written in a basis of grid points, the THC factorization just looks like evaluation of the double integral by quadrature, in which case Z_{PQ} should just be 1/(rP-rQ) up to weighting factors. Is this a useful way to think about it, or is it more subtle than this?

Felix said...

I have not really thought it through in detail, but what your are saying makes sense ...

With an increasing number of sampling points this sum should resemble the exact integral more and more. And then Z_PQ should approach 1/(rP-rQ).