Determining Molecular Ground State with Quantum Imaginary Time Evolution using Broken-Symmetry Wave Function

Abstract

The Hartree-Fock (HF) wave function, commonly used for approximating molecular ground states, becomes nonideal in open shell systems due to the inherent multi-configurational nature of the wave function, limiting accuracy in Quantum Imaginary Time Evolution (QITE). We propose replacing the HF wave function with a spin- and spatial-symmetry broken wave function, enhancing convergence by adding a spin operator, $S^2$ as a penalty term to the original molecular Hamiltonian. We verify that this approach provides good convergence behavior towards the lowest energy eigenstate using direct matrix exponentiation for Imaginary Time Evolution (ITE). Numerical simulations were performed on the hydrogen molecule and a square tetrahydrogen cluster using measurement-assisted unitary approximation in QITE. QITE demonstrates faster convergence to the ground state with broken symmetry (BS) compared to HF, particularly after the molecule exhibits a diradical character of 0.56 for hydrogen. Prior to this point, HF remains more effective, suggesting a transition threshold of diradical character for wave function selection. Additionally, the overlap analysis with Complete Active Space Configuration Interaction (CAS-CI) wave function shows that BS has a larger initial overlap than HF in higher-spin, multi-configurational systems like triple bond dissociation in nitrogen molecule. This method provides a pathway for improved energy simulations in open shell systems, where wave function accuracy significantly impacts downstream quantum algorithms and practical applications in quantum chemistry.

Figures (7)

• Figure 1: Active orbitals of the N2 molecule. The blue dots represent N atoms. (a) RHF canonical orbitals. Arrows specify the RHF electronic configuration. (b) Localized orbitals constructed from the natural orbitals computed at the BS-UHF level. Arrows specify the electron occupancies of the BS3.
• Figure 2: (a) Energy gap between the singlet ground state and singlet excited state corresponding to the delocalized HF wave function. (b) Energy gap between the singlet ground state and the excited triplet state corresponding to the localized BS wave function after introducing penalty.
• Figure 3: (a) Comparison of the convergence behavior of ITE in N$_2$ molecule with the RHF, BS2, and BS3 wave functions as the starting wave function. Top panel shows the energy difference $\Delta E$ during the evolution. Fidelity of ITE evolved wave function with respect to the CAS-CI wave function in the middle panel, and contribution of the penalty term $\langle \hat{\mathbf{S}}^2 \rangle$ in the bottom panel. (b) Time of convergence to attain the chemical precision $\Delta E \le$ 1.0 kcal mol$^{-1}$ for bond length from 1.0 Å to 3.0 Å. Background colors show the recommended region of wave function as starting wave function depending on the N--N bond length. (c) Diradical character $y$ at the region of bond dissociation. The points of intersection of red and blue lines with corresponding red and blue curves indicate the recommended points to change the initial wave functions described in (b).
• Figure 4: (a) Comparison of the convergence behavior of QITE in H$_2$ molecule with the RHF and BS wave functions as the starting wave function. Top panel shows the energy difference between the ground state energy calculated using CAS-CI and QITE with unitary approximation. Dashed line represents chemical precision given by $1.59\times10^{-3}$ Hartree. Bottom panel shows fidelity of QITE evolved wave function with respect to the CAS-CI wave function. Dashed line marks unit fidelity. (b) Recommended region of wave function as starting wave function depending on the bond length between hydrogen atoms. (c) Diradical character $y$ at the region of bond dissociation. The point of intersection of two straight lines with the curve indicate the recommended point to change the initial wave function described in (b).
• Figure 5: Comparison of the energy difference between the ground state energy calculated using CAS-CI and using HF and BS wave function inspired QITE with unitary approximation in the top panel. Bottom panel shows the fidelity of QITE evolved wave function with respect to the CAS-CI wave function for tetrahydrogen cluster with H--H interatomic distance of 2.0 Bohr.