Molecule-Environment Embedding with Quantum Monte Carlo: Electrons Interacting with Drude Oscillators

Abstract

We present a comprehensive investigation of the El-QDO embedding method [Phys. Rev. Lett. 131, 228001 (2023)], where molecular systems described through the electronic Hamiltonian are immersed in a bath of charged quantum harmonic oscillators, i.e., quantum Drude oscillators (QDOs). In the El-QDO model, the entire system of electrons and drudons─the quantum particles in the QDOs─is modeled through a single Hamiltonian which is solved through quantum Monte Carlo (QMC) methods. We first describe the details of the El-QDO Hamiltonian, of the proposed El-QDO ansatz, and of the QMC algorithms implemented to integrate both electronic and drudonic degrees of freedom. Then we analyze short-range regularization functions for the interacting potential between electrons and QDOs in order to accurately treat equilibrium and repulsive regions, resolving the overpolarization error that occurs between the electronic system and the environment. After benchmarking various regularization (damping) functions on the cases of argon and water dimers, the El-QDO method is applied to study the solvation energies of the benzene and water dimers, verifying the accuracy of the El-QDO approach compared to accurate fully electronic ab initio calculations. Furthermore, through the comparison of the El-QDO interaction energies with the components of Symmetry-Adapted Perturbation Theory calculations, we illustrate the El-QDO's explicit many-body treatment of electrostatic, polarization, and dispersion interactions between the electronic subsystem and the environment.

Figure 1

Schematic of the Hamiltonian in eq 4 in the case of an electronic system interacting with two interacting QDOs, where QDO₂ has one additional point charge (only selected interactions between the QDOs and the electronic system are shown).

Figure 2

Damped Coulomb potentials from eqs 7–10 compared to bare Coulomb for σ = 0.1 Bohr and σ = 0.5 Bohr. The values of σ are indicated via the dashed vertical lines.

Figure 3

Interaction energies of the El-QDO argon dimer for erf, exp₂, exp₄, and swave damping functions as a function of the distance between the two subsystems. The results are compared to two SAPT2 + 3(CCD) curves explained in the main text. The vertical dashed line is the equilibrium geometry of the argon dimer.

Figure 4

Interaction energies of the El-QDO water dimer for the error function damping as a function of the distance between the two subsystems. The results are compared to two SAPT2 + 3(CCD) curves explained in the main text. The vertical dashed line is the equilibrium geometry of the water dimer.

Figure 5

Monomer₁ (green) and monomer₂ (blue) in the environment composed of 50 water molecules for three values of the expansion of the water cage from its center. R_min are the minimal atom–atom distances between the monomer of the corresponding color and the environment. The original geometry for dR = 0.0 Å is taken from ref (126).

Figure 6

Solvation energies of benzene (monomer₁, monomer₂, and dimer) in an environment composed of 50 water molecules as a function of the expansion of the water cage from its center. El-QDO results are compared to the SAPT0 decomposition explained in the main text.

Figure 7

Monomer₁ (green) and monomer₂ (blue) in the environment composed of 28 water molecules for three values of the expansion of the water cage from its center. R_min are the minimal atom–atom distances between the monomer of the corresponding color and the environment. The original geometry for dR = 0.0 Å is taken from ref (128).

Figure 8

Solvation energies of water dimer and its monomers in an environment composed of 28 water molecules as a function of the expansion of the water cage from its center. El-QDO results are compared to the SAPT0 decomposition explained in the main text.

Figure 9

Change of the binding energy (eq 28) of the central water dimer due to the presence of the environment (Figure 7) as a function of the QDOs cage deformation. El-QDO results are compared to the SAPT0 decomposition explained in the main text.