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arXiv:2607.27377

Excited state optimization for strongly correlated quantum defects using ensemble variational Monte Carlo

Kevin G. Kleiner, Lucas K. Wagner

Published Jul 29, 2026• 0 citations
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Abstract

Using the recently introduced ensemble variational Monte Carlo (VMC), we study optimized wave functions for strongly correlated point defects, including nitrogen-vacancy and silicon-vacancy centers in diamond and substitutional iron and chromium impurities in aluminum nitride. We study the effects of fully optimized determinant expansion parameters, orbitals, and Jastrow correlation factors on these systems. We find that orbitals from the hybrid functional PBE0 are much better (have lower objective functional) than semilocal PBE, which results in changes in the excitation energies up to 0.5 eV. Further improvements can be made by directly optimizing the objective functional, resulting in changes in excitation energies up to 0.2 eV. The most important parameter varies from defect to defect, reinforcing the necessity of optimizing all parameters to obtain accurate excited states in strongly correlated defects.

DOI
10.48550/arxiv.2607.27377

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Using the recently introduced ensemble variational Monte Carlo (VMC), we study optimized wave functions for strongly correlated point defects, including nitrogen-vacancy and silicon-vacancy centers in diamond and substitutional iron and chromium impurities in aluminum nitride. We study the effects of fully optimized determinant expansion parameters, orbitals, and Jastrow correlation factors on these systems. We find that orbitals from the hybrid functional PBE0 are much better (have lower objective functional) than semilocal PBE, which results in changes in the excitation energies up to 0.5 eV. Further improvements can be made by directly optimizing the objective functional, resulting in changes in excitation energies up to 0.2 eV. The most important parameter varies from defect to defect, reinforcing the necessity of optimizing all parameters to obtain accurate excited states in strongly correlated defects.
Certain point defects in solids can efficiently be used as qubits for applications in quantum technology. They have spin states that are initializable, readable, robust, and can be manipulated optically. New theoretical methods are needed to find the best host materials and defect configurations. Most methods proposed so far rely either on cluster models or restrict the many-body treatment of the defects to a subspace of single-particle orbitals. We explore best practices and theory for the use of quantum Monte Carlo to predict the excitation spectra for spin defects, by using the negatively charged nitrogen vacancy (${\mathrm{NV}}^{\ensuremath{-}}$) center in diamond as a test system. Quantum Monte Carlo can be used to explicitly simulate electronic correlations with larger systems and sets of orbitals than previous methods due to favorable scaling with respect to system size and computing power. We consider different trial wave functions for variational and diffusion Monte Carlo methods, explore the nodal surface errors of the ground and excited state wave functions, and study whether the variational principle holds for the excited states. We compute the vertical excitation energies in different simulation cell sizes and extrapolate to infinite system size, and include backflow corrections to the extrapolated energies. The final results for vertical excitation energies are found to overestimate the experimental estimates, but the triplet-to-triplet and singlet-to-singlet transitions are accurate against experiment. Finally, we list further developments for QMC needed to address the problem of accurately predicting structural and spin properties of the solid-state defects.
Point defects are of interest for many applications, from quantum sensing to modifying bulk properties of materials. Because of their localized orbitals, the electronic states are often strongly correlated, which has led to a proliferation of quantum embedding techniques to treat this correlation. In these techniques, most of the one-body states are treated with a weakly correlated theory such as density functional theory, and certain one-body states are singled out as an active space to be treated using an effective interaction. We assess these techniques for iron and chromium defects in aluminum nitride using quantum Monte Carlo (QMC) calculations on identical Hamiltonians. For these systems, we find the dominant errors in the embedding arise from the one-body crystal-field splitting in the d orbitals inherited from density functional theory (DFT), rather than double counting corrections, with the screened interactions also affected by the DFT orbitals. Strikingly, the best double counting recipe is opposite in these two cases. Because excitation energies can agree while the underlying wave functions do not, diagnosing these errors requires detailed information about the many-body wave functions, which QMC provides.
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Abstract Variational Monte Carlo methods have recently been applied to the calculation of excited states; however, it is still an open question what objective function is most effective. A promising approach is to optimize excited states using a penalty to minimize overlap with lower eigenstates, which has the drawback that states must be computed one at a time. We derive a general framework for constructing objective functions with minima at the the lowest N eigenstates of a many-body Hamiltonian. The objective function uses a weighted average of the energies and an overlap penalty, which must satisfy several conditions. We show this objective function has a minimum at the exact eigenstates for a finite penalty, and provide a few strategies to minimize the objective function. The method is demonstrated using ab initio variational Monte Carlo to calculate the degenerate first excited state of a CO molecule.
Modern quantum Monte Carlo (QMC) methods often capture electron correlation through both explicitly correlating Jastrow factors and small to mid-sized configuration interaction (CI) expansions. Here, we study the additional optimization difficulty created by including increasing numbers of CI parameters. We find evidence that the quality of Variational Mone Carlo (VMC) optimization can be limited by the ability to statistically resolve the CI parameters in the presence of a Jastrow factor. Although using larger statistical samples can mitigate this issue and bring an optimization closer to its true minimum, this approach is computationally intensive. We present evidence that similar gains to optimization quality can be had without increasing the sample size by avoiding CI parameters that are statistically the most difficult to resolve. Our findings suggest that, in addition to the value of using traditional selected configuration interaction (sCI) methods to prepare VMC wave functions, sCI-like methods can play an important role within VMC by improving the effectiveness of stochastic energy minimization.
We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.