Extraction of a One-Particle Reduced Density Matrix from a Quantum Monte Carlo Electronic Density: A New Tool for Studying Nondynamic Correlation

Carmelo Naim,Claudio Amovilli

doi:10.3390/computation9120135

Abstract

In this work, we present a method to build a first order reduced density matrix (1-RDM) of a molecule from variational Quantum Monte Carlo (VMC) computations by means of a given correlated mapping wave function. Such a wave function is modeled on a Generalized Valence Bond plus Complete Active Space Configuration Interaction form and fits at best the density resulting from the Slater-Jastrow wave function of VMC. The accuracy of the method proposed has been proved by comparing the resulting kinetic energy with the corresponding VMC value. This 1-RDM is used to analyze the amount of correlation eventually captured in Kohn-Sham calculations performed in an unrestricted approach (UKS-DFT) and with different energy functionals. We performed test calculations on a selected set of molecules that show a significant multireference character. In this analysis, we compared both local and global indicators of nondynamic and dynamic correlation. Moreover, following the natural orbital decomposition of the 1-RDM, we also compared the effective temperatures of the corresponding Fermi-like distributions. Although there is a general agreement between UKS-DFT and VMC, we found the best match with the functional LC-BLYP.

Highlights

The first measure of electron correlation energy was introduced by Löwdin as the energy difference between the exact solution of the non-relativistic Schrödinger equation and the Hartree-Fock (HF) approximation [1]
density functional theory (DFT) usually is considered able to recover most of dynamic correlation, common density functional approximations (DFA) are generally unable to take into account nondynamic correlation effects or they can reproduce them in an unspecified manner [4]
The objective of this study has been to present a method able to recover a 1-RDM derived from an accurate wave function and studying the nondynamic correlation effects included by unrestricted KS-DFT calculations

Summary

Introduction

The first measure of electron correlation energy was introduced by Löwdin as the energy difference between the exact solution of the non-relativistic Schrödinger equation and the Hartree-Fock (HF) approximation [1]. Among the various multiconfiguration techniques, the Complete Active Space SCF (CASSCF) is the most appropriate to fix a reference for the estimate of the nondynamical correlation energy In such a case, the full valence active space should be taken into account in order to consider all possible arrangements of electrons on the different molecular orbitals. In order to preserve the spin symmetry of the electronic Hamiltonian, the wave function employed in HF or DFT computations has to be constructed in a restricted framework This formulation results inadequate for the accurate reproduction of ground state energy of systems with a multiconfigurational character. In the case of spin contamination we speak of symmetry breaking and in KS theory we can interpret the difference between restricted and unrestricted energy as an estimate of nondynamic correlation energy, namely

Methods

Results

Conclusion