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arXiv:1609.02241 · Physical Review A

Nodal variational principle for excited states

Federico E. Zahariev, Mark S. Gordon, Mel Levy

Published Jul 31, 2018• 10 citations
arXivHTMLPDF

Abstract

It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. Simple numerical illustrations of the error functions are presented.

DOI
10.1103/physreva.98.012144

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It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. Simple numerical illustrations of the error functions are presented.
A recently developed self-healing diffusion Monte Carlo algorithm [F. A. Reboredo, R. Q. Hood, and P. R. C. Kent, Phys. Rev. B 79, 195117 (2009)] is extended to the calculation of excited states. The formalism is based on an excited-state fixed-node approximation and the mixed estimator of the excited-state probability density. The fixed-node ground-state wave-functions of inequivalent nodal pockets are found simultaneously using a recursive approach. The decay of the wave-function into lower-energy states is prevented using two methods: (i) the projection of the improved trial-wave function into previously calculated eigenstates is removed; and (ii) the reference energy for each nodal pocket is adjusted in order to create a kink in the global fixed-node wave-function, which, when locally smoothed, increases the volume of the higher-energy pockets at the expense of the lower-energy ones until the energies of every pocket become equal. This reference energy method is designed to find nodal structures that are local minima for arbitrary fluctuations of the nodes within a given nodal topology. It is demonstrated in a model system that the algorithm converges to many-body eigenstates in bosonic and fermionic cases.
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.
Fixed-node diffusion Monte Carlo (FN-DMC) is an accurate and useful method for estimating the wave function and the energy of the quantum ground state of a many-fermion system. However, it has been shown that difficulties with the method may occur when it is applied to a degenerate excited state because the nodal surface of the degenerate trial function is generally insufficient to impose the complete symmetry properties of the trial function on the FN-DMC wave function. As a result, the tiling theorem and the symmetry-constrained variational principle may be violated by FN-DMC when the excited state is degenerate. There are two practical consequences for the study of degenerate excited states: The FN-DMC energy may lie below the energy of the lowest stationary state that transforms according to the same degenerate irreducible representation as the trial function; and the convergence of the FN-DMC energy with improvements in the trial function may not be quadratic. In this paper a diffusion Monte Carlo method for degenerate excited states is presented. It provides a direct generalization of the FN-DMC method, and when applied to the study of degenerate excited states, it has the support of the tiling theorem and the symmetry-constrained variational principle. The method is applied to the lowest degenerate state of a simple test problem in which FN-DMC has been shown to violate both the tiling theorem and the symmetry-constrained variational principle. The numerical results support the assertion that this method for degenerate excited states satisfies both the tiling theorem and the symmetry-constrained variational principle.
We investigate the use of different variational principles in quantum Monte Carlo, namely, energy and variance minimization, prompted by the interest in the robust and accurate estimation of electronic excited states. For two prototypical, challenging molecules, we readily reach the accuracy of the best available reference excitation energies using energy minimization in a state-specific or state-average fashion for states of different or equal symmetry, respectively. On the other hand, in variance minimization, where the use of suitable functionals is expected to target specific states regardless of the symmetry, we encounter severe problems for a variety of wave functions: as the variance converges, the energy drifts away from that of the selected state. This unexpected behavior is sometimes observed even when the target is the ground state and generally prevents the robust estimation of total and excitation energies. We analyze this problem using a very simple wave function and infer that the optimization finds little or no barrier to escape from a local minimum or local plateau, eventually converging to a lower-variance state instead of the target state. For the increasingly complex systems becoming in reach of quantum Monte Carlo simulations, variance minimization with current functionals appears to be an impractical route.
The steepest descent perturbation theory is extended to the calculation of the energy and eigenfunction of the excited state of a quantum system. In case of the orthogonality of the trial function for the excited state to those for lower-energy state or ground state in the same symmetry class is preserved, the variational collapse to lower energy state can be avoided in this proposal. An iterative procedure is given for generating better eigenvalue and eigenfunction of the excited state without requiring an infinite summation over reference states as in conventional perturbation theory. This new perturbation method can be applied to calculate the excitation energy and wave function of excited states for any many-body quantum system to a high degree of accuracy without so much computational effort as in conventional method.
Recently developed neural network-based wave function methods are capable of achieving state-of-the-art results for finding the ground state in real space. In this work, a neural network-based method is used to compute excited states. We train our network via variational principle, along a further penalty term that imposes the orthogonality with lower-energy eigenfunctions. As a demonstration of the effectiveness of this approach, results from numerical calculations for one-dimensional and two-dimensional harmonic oscillators are presented.