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arXiv:2312.00693 · Electronic Structure

Ensemble variational Monte Carlo for optimization of correlated excited state wave functions

William A Wheeler, Kevin G. Kleiner, Lucas K. Wagner

Published Mar 28, 2024• 9 citations
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Abstract

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.

DOI
10.1088/2516-1075/ad38f8

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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.
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.
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.
Determining quantum excited states is crucial across physics and chemistry but presents significant challenges for variational methods, primarily due to the need to enforce orthogonality to lower-energy states, often requiring state-specific optimization, penalty terms, or specialized ansatz constructions. We introduce a novel variational principle that overcomes these limitations, enabling the \textit{simultaneous} determination of multiple low-energy excited states. The principle is based on minimizing the trace of the inverse overlap matrix multiplied by the Hamiltonian matrix, Tr(S−1H)\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})Tr(S−1H), constructed from a set of \textit{non-orthogonal} variational states {∣ψi⟩}\{|ψ_i\rangle\}{∣ψi​⟩}. Here, Hij=⟨ψi∣H∣ψj⟩\mathbf{H}_{ij} = \langleψ_i | H | ψ_j\rangleHij​=⟨ψi​∣H∣ψj​⟩ and Sij=⟨ψi∣ψj⟩\mathbf{S}_{ij} = \langleψ_i | ψ_j\rangleSij​=⟨ψi​∣ψj​⟩ are the elements of the Hamiltonian and overlap matrices, respectively. This approach variationally optimizes the entire low-energy subspace spanned by {∣ψi⟩}\{|ψ_i\rangle\}{∣ψi​⟩} without explicit orthogonality constraints or penalty functions. We demonstrate the power and generality of this method across diverse physical systems and variational ansatzes: calculating the low-energy spectrum of 1D Heisenberg spin chains using matrix product states, finding vibrational spectrum of Morse potential using quantics tensor trains for real-space wavefunctions, and determining excited states for 2D fermionic Hubbard model with variational quantum circuits. In all applications, the method accurately and simultaneously obtains multiple lowest-lying energy levels and their corresponding states, showcasing its potential as a unified and flexible framework for calculating excited states on both classical and quantum computational platforms.
Optimization of wave functions in quantum Monte Carlo is a difficult task because the statistical uncertainty inherent to the technique makes the absolute determination of the global minimum difficult. To optimize these wave functions we generate a large number of possible minima using many independently generated Monte Carlo ensembles and perform a conjugate gradient optimization. Then we construct histograms of the resulting nominally optimal parameter sets and "filter" them to identify which parameter sets "go together" to generate a local minimum. We follow with correlated-sampling verification runs to find the global minimum. We illustrate this technique for variance and variational energy optimization for a variety of wave functions for small systellls. For such optimized wave functions we calculate the variational energy and variance as well as various non-differential properties. The optimizations are either on par with or superior to determinations in the literature. Furthermore, we show that this technique is sufficiently robust that for molecules one may determine the optimal geometry at tIle same time as one optimizes the variational energy.
We present a simple and efficient method to optimize within energy minimization the determinantal component of the many-body wave functions commonly used in quantum Monte Carlo calculations. The approach obtains the optimal wave function as an approximate perturbative solution of an effective Hamiltonian iteratively constructed via Monte Carlo sampling. The effectiveness of the method, as well as its ability to substantially improve the accuracy of quantum Monte Carlo calculations, are demonstrated by optimizing a large number of parameters for the ground state of acetone and the difficult case of the 11Bu1^{1}B_{u}11Bu​ state of hexatriene.
We present a technique for optimizing hundreds of thousands of variational parameters in variational quantum Monte Carlo. By introducing iterative Krylov subspace solvers and by multiplying by the Hamiltonian and overlap matrices as they are sampled, we remove the need to construct and store these matrices and thus bypass the most expensive steps of the stochastic reconfiguration and linear method optimization techniques. We demonstrate the effectiveness of this approach by using stochastic reconfiguration to optimize a correlator product state wave function with a Pfaffian reference for four example systems. In two examples on the two dimensional Fermionic Hubbard model, we study 16 and 64 site lattices, recovering energies accurate to 1% in the smaller lattice and predicting particle-hole phase separation in the larger. In two examples involving an ab initio Hamiltonian, we investigate the potential energy curve of a symmetrically dissociated $4\ifmmode\times\else\texttimes\fi{}4$ hydrogen lattice as well as the singlet-triplet gap in free base porphin. In the hydrogen system we recover 98% or more of the correlation energy at all geometries, while for porphin we compute the gap in a 24 orbital active space to within 0.02 eV of the exact result. The number of variational parameters in these examples ranges from $4\ifmmode\times\else\texttimes\fi{}{10}^{3}$ to $5\ifmmode\times\else\texttimes\fi{}{10}^{5}$.