A truncated Davidson method for the efficient "chemically accurate" calculation of full configuration interaction wavefunctions without any large matrix diagonalization

TL;DR

This paper introduces a truncated Davidson method that efficiently computes chemically accurate full CI wavefunctions without large matrix diagonalization by incrementally building a small, highly truncated subspace for Hamiltonian diagonalization.

Contribution

A novel truncated Davidson approach that avoids large matrix diagonalization by incrementally constructing a small subspace for accurate full CI calculations.

Findings

Achieves chemical accuracy with small subspace sizes

Demonstrates efficiency on large Hilbert spaces (~10^18)

Provides a scalable alternative to standard CI methods

Abstract

This work develops and illustrates a new method of calculating "chemically accurate" electronic wavefunctions (and energies) via a truncated full configuration interaction (CI) procedure which arguably circumvents the large matrix diagonalization that is the core problem of full CI and is also central to modern selective CI approaches. This is accomplished simply by following the standard/ubiquitous Davidson method in its "direct" form -- wherein, in each iteration, the electronic Hamiltonian operator is applied directly in second quantization to the Ritz vector/wavefunction from the prior iteration -- except that (in this work) only a small portion of the resultant expansion vector is actually even computed (through application of only a similarly small portion of the Hamiltonian). Specifically, at each iteration of this truncated Davidson approach, the new expansion vector is taken to be twice as large as that from the prior iteration. In this manner, a small set of highly truncated expansion vectors (say 10--30) of increasing precision is incrementally constructed, forming a small subspace within which diagonalization of the Hamiltonian yields clear, consistent, and monotonically variational convergence to the approximate full CI limit. The good efficiency in which convergence to the level of chemical accuracy (1.6 mHartree) is achieved suggests, at least for the demonstrated problem sizes -- Hilbert spaces of $10^{18}$ and wavefunctions of $10^8$ determinants -- that this truncated Davidson methodology can serve as a replacement of standard CI and complete-active space (CAS) approaches, in circumstances where only a few chemically-significant digits of accuracy are required and/or meaningful in view of ever-present basis set limitations.