Screening in the Heitler-London Model: Revisiting the Bonding and Antibonding States of the Hydrogen Molecule

TL;DR

This work revisits the Heitler–London description of the hydrogen molecule, first deriving the bonding and antibonding HL states with the canonical HL wave function and energy $E^{HL}_{T\pm}(R)$. It then introduces a variational parameter $\alpha$ to model electronic screening within the HL ansatz and benchmarks the approach using variational quantum Monte Carlo (VQMC) to optimize $\alpha$ as a function of inter-nuclear distance $R$, yielding improved bond lengths and vibrational frequencies. Building on this, the authors propose an analytically simple screening form with $\alpha_{0\pm}(R)=\beta_\pm+(\alpha_{\mathrm{He}}-\gamma_\pm) e^{-\lambda_\pm R}$, obtaining an $\alpha_0$-HL model that better reproduces the experimental bond length $R_0$ (e.g., $R_0\approx1.40$ a.u.) and exhibits vibrational frequencies closer to experiment than the original HL estimate. Overall, the paper demonstrates that a minimal screening modification to the HL wave function can capture essential physics of molecular bonding and dissociation, providing a useful, analytically tractable input for constructing simple yet physically meaningful variational wave functions.

Abstract

The present manuscript revisits one of the earliest approaches to treating molecular systems within the Schrödinger formalism of quantum mechanics: the Heitler-London (HL) model. Originally proposed in 1927 and based on a linear combination of atomic orbitals, the HL model provided a foundational description of covalent bonds and has served as the basis for numerous variational methods. Focusing on the hydrogen molecule, we begin by revisiting the analytical calculations of the original HL model, from which the qualitative physics of bonding and antibonding states can be obtained. Subsequently, we propose including electronic screening effects directly in the original HL wave function. We then compare our proposal with variational quantum Monte Carlo (VQMC) calculations, whose trial wave function allows us to optimize the electronic screening potential as a function of the inter-proton distance. We obtain the bond length, binding energy, and vibrational frequency of the H$_2$ molecule. Beyond revisiting this foundational approach in quantum mechanics, our proposal can serve as improved input for constructing new, but still analytically simple, variational wave functions to describe dissociation or bond formation.

Figures (5)

• Figure 1: Schematic of the hydrogen molecule showing electrons $i=1,\ 2$ and protons $j=A,\ B$. The electron-proton distances are $r_{ij}$. The electron-electron and proton-proton separations are $r_{12}$ and $R$, respectively.
• Figure 2: Bonding (solid lines) and antibonding (dashed lines) screened wave functions for $\alpha=0.5$ (black), $1.0$ (red), and $1.5$ (blue), with the nuclear separation fixed at $R=2.0$. The functions are defined in Eq. \ref{['HLwfT']} and constructed using the atomic orbitals given in Eq. \ref{['HLwfTalpha']}. Note that $\alpha=1.0$ corresponds to the original HL wave function.
• Figure 3: Exponential fits of $\alpha_{0\pm}(R)$ as defined in Eq. (\ref{['Qeff']}): $\alpha_{0+}(R)=0.970(5)+0.826(13)e^{-1.01(3)R}$ for the bonding state (red line) and $\alpha_{0-}(R)=1.01(0)-0.473(7)e^{-1.30(3)R}$ for the antibonding state (black line). In both cases, the fitted functions yield $\alpha_{0\pm}(R)\approx 1$ as $R\to \infty$, recovering the original HL model without screening. For $R\to 0$, the effective charges approach $1.80$ and $0.54$ for the bonding and antibonding states, respectively. The red solid and black open square points were obtained from VQMC calculations with a sampling size of $N_s=10^8$.
• Figure 4: Total energy of the hydrogen molecule as a function of the inter-proton distance $R$. The solid black and red lines correspond to the HL curves for the antibonding and bonding states, respectively, as given by Eq. \ref{['H']}. The black open and red solid square points were obtained from VQMC calculations with a sampling size of $N_s=10^8$. The dotted black and red lines represent the curves from our $\alpha_0$-HL proposal, constructed by incorporating the exponential fits of $\alpha_{0\pm}(R)$ into the HL model. The inset shows the bonding-state VQMC curve $E(R)$ near the energy minimum, based on 14 sampled points, along with the corresponding quadratic fit: $E(R)=-0.760(24)-0.536(35)R+0.189(12)R^2$.
• Figure 5: Bond length (black) and vibrational frequency (red) of the hydrogen molecule as functions of the parameter $\lambda_+$ in Eq. \ref{['Qeff']}. The blue line indicates the corresponding experimental values.