Least Squares Tensor Hypercontraction (LS-THC) for ERIs (Part 1)

[andyj10224]

Description

In this PR, we implement a pilot version of least squares tensor hypercontraction (LS-THC)... factoring 4-index ERIs (O(N^4)) into two-index quantities (i.e. $(mn|rs) = x_{m}^{P}x_{n}^{P}Z^{PQ}x_{r}^{Q}x_{s}^{Q}$). In the LS-THC formulation, $P$ and $Q$ are derived from grid points, and

$x_{m}^{P} = w_{P}^{\frac{1}{4}} \phi_{m}(x_{P}, y_{P}, z_{P})$,
$E^{PQ} = x_{m}^{P}x_{n}^{P}(mn|rs)x_{r}^{Q}x_{s}^{Q}$
$S^{PP'} = (x_{m}^{P}x_{m}^{P'})^{2}$
$Z^{PQ} = (S^{PP'})^{-1} E^{P'Q'} (S^{QQ'})^{-1}$

This is based off of the work of Parrish et al. (2012)

These quantities can be constructed integral-direct and non-iteratively. No four or even three-index quantities are stored, thereby allowing for the future development of efficient, low-memory quantum chemistry algorithms in Psi.

This is a PR in a series of planned PRs that will involve:
1.) LS-THC-JK (integral-direct SCF without recomputing ERIs every iteration and faster K construction)
2.) LS-THC-MP2
3.) Local grid implementation of LS-THC to allow for use in local correlation methods (DLPNO-MP2, DLPNO-CCSD/(T))

User API & Changelog headlines

• Added Python interface (Pybind) for building LS-THC quantities that allow for rapid prototyping of new methods involving THC
• Developed infrastructure for the development of alternate ways to perform THC (i.e. PF-THC)

Dev notes & details

• Implemented LS-THC C/Py-side for future QC methods

Questions

• Is this the proper way to export this object Py-side?

Checklist

• Tests added for any new features
• All or relevant fraction of full tests run

Status

• Ready for review
• Ready for merge

[andyj10224]

Added some comments and addressed initial reviews. @loriab @jturney feel free to take a look

[andyj10224]

Pinging @loriab once again, requesting review

[PhillCli]

FYI: I've run 10 water molecules 240 AO basis, based on test_thc_eri.py, and it seems

Matrix->pseudoinverse bloats memory, more than I would expected.

   --------------------------------------------
       Least-Squares Tensor Hypercontraction   
                   by Andy Jiang               
              doi: 10.1063/1.4768233           
   --------------------------------------------

  Detailed LS-THC thresholds and cutoffs:
    LS_THC_SPHERICAL_POINTS   =    38 
    LS_THC_RADIAL_POINTS      =     8 
    LS_THC_BASIS_TOLERANCE    = 1.000e-12 
    LS_THC_WEIGHTS_TOLERANCE  = 1.000e-12 
    LS_THC_S_EPSILON          = 1.000e-10 
    Using DF?                     Yes 


    Tensor Hypercontraction Complete! 

    Number of Grid Points (Rank) :   6623 (220 per atom)

    Memory Required to Store Exact Integrals :  3.116 [GiB]
    Memory Required to Store DF Integrals :  0.181 [GiB]
    Memory Required in LS-THC factored form  :  0.339 [GiB]

THC completed: 
Z_PQ.shape=(6623, 6623)
x1.shape=(6623, 240)
Filename: thc_eri.py

mprof (a rough idea of process memory), after 100s mark in mprof memory is due to 3-idx AO and einsums.

memray flamegraph (.tar.gz), seems to point into Matrix.pseudoinverse and subsequent svd.

memray_flamegraph.tar.gz

I know it's a Part 1, nevertheless I wanted to share what I found.

Very good job anyway on adding THC into the Psi4 stack! Can't wait to try it out more, for prototyping!

[andyj10224]

@PhillCli Thank you for your tests and for going above and beyond! I will definitely look into improving the memory footprint! One possible solution is to use truncated SVD for the pseudoinverse (algorithm available in @jturney 's Einsums). I will look into this further once einsums is fully merged with Psi4.

[davpoolechem]

I think I can consider myself happy with this! Looking forward to seeing this in action for different methods!