Thursday, October 31, 2019

The minimum parameterization of the wave function for the many-body electronic Schrödinger equation. I. Theory and ansatz

Lasse Kragh Sørensen (2019)
Highlighted by Jan Jensen

It is well known that Full CI (FCI) scales exponentially with the basis set size. However, Sørensen claims that the "whole notion of the exponentially scaling nature of quantum mechanics simply stems from expanding the wavefunction in a sub-optimal basis",  i.e. one-electron functions. Sørensen goes on to argue that of two-ele tron functions (geminals) are used instead, the scaling is reduced to m(m−1)/2 where m is the number of basis functions.  Furthermore, because "the number of parameters is independent of the electronic structure the multi-configurational problem is a mere phantom summoned by a poor choice of basis for the wave function". 

I don't have the background to tell whether the arguments in this paper are correct and the main point os this post is to see if I can get some feedback from people who do have the background. 

In principle one could simply compare the new approach to FCI calculations but the new method isn't quite there yet:
A straight forward minimization of Eq. 84 unfortunately gives the solution for the two-electron problem of a +(N−2) charged system N/2 times so additional constraints must be introduced. These constraints can be found by the property of the wave function in limiting cases. The problem of finding the constraints for the geminals in the AGP ansatz is therefore closely related to finding the N-representability constraints for the two-body reduced density matrix (2-RDM). For the N-representability Mazziotti have showed a constructive solution[74] though the exact conditions are still illusive.[75, 76]  


  1. I have some doubts. m(m−1)/2 is O(N^2), so way cheaper than CCSD, or even just HF. I have difficulty believing FCI can be so cheap. Also, I think there is a conjecture that a method cannot be variational and size consistent at the same time without being FCI (or equivalent).
    With that said, geminals have done some wonderful things for basis set convergence, F12 methods are usually great.

  2. The excerpt is trivial. The 2-RDM is parameterized by $m*(m+1)/2$ variables. But parameterization of the 2-RDM is not the same as computational complexity of the solution. Or in other words, the computational complexity is not affected by this argument and remains exponential in the number of one-electron basis functions.