Complete-graph tensor network states: a new fermionic wave function ansatz for molecules

Konrad H Marti,Matthias Troyer,Frank Verstraete,Bela Bauer,Markus Reiher

doi:10.1088/1367-2630/12/10/103008

Abstract

We present a class of tensor network states specifically designed to capture the electron correlation within a molecule of arbitrary structure. In this ansatz, the electronic wave function is represented by a complete-graph tensor network (CGTN) ansatz, which implements an efficient reduction of the number of variational parameters by breaking down the complexity of the high-dimensional coefficient tensor of a full-configuration-interaction (FCI) wave function. This ansatz applied to molecules is new and based on a tensor network wave function recently studied in lattice problems. We demonstrate that CGTN states approximate ground states of molecules accurately by comparison of the CGTN and FCI expansion coefficients. The CGTN parametrization is not biased towards any reference configuration, in contrast to many standard quantum chemical methods. This feature allows one to obtain accurate relative energies between CGTN states, which is central to molecular physics and chemistry. We discuss the implications for quantum chemistry and focus on the spin-state problem. Our CGTN approach is applied to the energy splitting of states of different spins for methylene and the strongly correlated ozone molecule at a transition state structure. The parameters of the tensor network ansatz are variationally optimized by means of a parallel-tempering Monte Carlo algorithm.

Highlights

IntroductionThe electronic Hamiltonian in second quantization reads in Hartree atomic units (‘h = me = e = 4π 0 = 1’)
The electronic wave function is represented by a complete-graph tensor network (CGTN) ansatz, which implements an efficient reduction of the number of variational parameters by breaking down the complexity of the highdimensional coefficient tensor of a full-configuration-interaction (FCI) wave function
We investigate three active spaces that are successively enlarged, starting with a CAS(4,4) of four spatial orbitals comprising four electrons that is increased in each step by an occupied and a virtual orbital around the Fermi level yielding in total CAS(4,4), CAS(6,6) and CAS(8,8)

Summary

Introduction

The electronic Hamiltonian in second quantization reads in Hartree atomic units (‘h = me = e = 4π 0 = 1’). I, j i, j,k,l σ σ,σ which contains one-electron integrals hi j over spatial orbitals φi (r) given in non-relativistic theory by [16]. The nucleus–nucleus repulsion term is suppressed for the sake of brevity. The two-electron integrals Vi jkl are defined as. The Hamiltonian and its ingredients may be written in terms of spin orbitals φi (x) = φi (r)σ , where σ is a spin-up or spin-down spin eigenfunction. The coordinate x denotes both spatial and spin variables

Methods

Results

Conclusion