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The Journal of Chemical Physics

The calculation of excited state properties with quantum Monte Carlo

David M. Ceperley, B. Bernu

Published Nov 15, 1988• 151 citations
Publisher

Abstract

A new Monte Carlo method for computing excited state properties of quantum systems is introduced. It is a generalization of the transient estimate method used for fermion Green’s function Monte Carlo and of subspace projection methods used for computing eigenstates of matrices. The time dependent autocorrelation function of a vector of trial functions is calculated for a random walk generated by the imaginary-time Schrödinger equation and estimates of energy levels are determined by the eigenvalues of the matrix of correlation functions. This method is especially useful for treating states with the same symmetry as it automatically keeps higher states orthogonal to lower states. The estimated energy converges to the exact eigenvalue with a rate which decreases with increasing excitation energy, thus limiting the method to relatively low-lying states. The method is zero variance in the sense that as better trial functions are introduced, the statistical error decreases to zero. The method has a nontrivial bias which is analyzed. As an illustration, the eigenvalues of a double well are computed.

DOI
10.1063/1.455398

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A new Monte Carlo method for computing excited state properties of quantum systems is introduced. It is a generalization of the transient estimate method used for fermion Green’s function Monte Carlo and of subspace projection methods used for computing eigenstates of matrices. The time dependent autocorrelation function of a vector of trial functions is calculated for a random walk generated by the imaginary-time Schrödinger equation and estimates of energy levels are determined by the eigenvalues of the matrix of correlation functions. This method is especially useful for treating states with the same symmetry as it automatically keeps higher states orthogonal to lower states. The estimated energy converges to the exact eigenvalue with a rate which decreases with increasing excitation energy, thus limiting the method to relatively low-lying states. The method is zero variance in the sense that as better trial functions are introduced, the statistical error decreases to zero. The method has a nontrivial bias which is analyzed. As an illustration, the eigenvalues of a double well are computed.
We introduce a new stochastic method for calculating ground-state properties of quantum systems. Segments of a Langevin random walk guided by a trial wave function are subject to a Metropolis rejection test performed on the time integral of the local energy. The algorithm---which is as simple as variational Monte Carlo---for bosons provides exact expectation values of local observables, as well as their static and dynamic (in imaginary time) response functions, without mixed-estimate nor population-control biases. Our method is demonstrated with a few case applications to 4He{}^{4}\mathrm{He}4He.
Solving the Schrodinger equation and finding excited states for quantum mechanical many-body systems is a fundamental problem. If this problem is formulated in terms of integrals, the dimensionality of the configuration space is often too high to perform the integrations required directly. This is because the volume of configuration space increases exponentially with dimension quickly, making its complete exploration impossible. Monte Carlo methods provide a way to estimate these integrals by statistically sampling a subset of configuration space, and these methods provide 1/√ N convergence regardless of dimension. For many problems involving the quantum mechanics of molecules there exists a large time scale separation between the high-frequency internal vibrations, and low-frequency intermolecular motions. This separation motivates the rigid-body approximation which freezes internal degrees of freedom in order to study intermolecular effects. The reduction in the dimensionality of configuration space further increases the range of accessible problems. This thesis is devoted to the construction and implementation of algorithms which incorporate the rigid-body approximation into existing Monte Carlo methods for solving the quantum mechanical many-body problem. Monte Carlo estimators are constructed as averages over samples drawn from some probability distribution. If this distribution is known in closed form the samples can be generated by application of the Metropolis algorithm. In most cases the distribution is not known in closed form or even representable with a finite number of variables. If this distribution is the dominant eigenstate of some known operator then a stochastic implementation of the power method can be used to generate the required samples. For the quantum mechanical problem this operator can be taken to be the imaginary-time evolution operator and its application can be represented in terms of random walks. For rigid bodies, this method involves the implementation of rotational Brownian motion. The use of quaternion algebra to implement this rotational motion enhances simplicity, performance, and numerical stability. The method described above can be generalized to investigate excited state properties using correlation function Monte Carlo which is a Monte Carlo implementation of the Rayleigh-Ritz variational method. This generalization requires the construction of a trial subspace which is then subjected to the variational method resulting in a generalized eigenvalue problem. It is important to represent optimized trial states accurately and efficiently. For this purpose, we write trial states as functions of invariant polynomials of the interparticle distances. By reducing the number of free interparticle distances, the rigid-body approximation greatly simplifies
… as suggested by Fermi, the time-independent Schrödinger equation … can be interpreted as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the value of the point function V. N. Metropolis and S. Ulam, 1949 Summary In the projector quantum Monte Carlo method, one uses a function of the hamiltonian to sample a distribution proportional to the exact ground-state wavefunction, and thereby computes exact matrix elements of it. An importance sampling transformation makes the algorithm much more efficient. In this chapter we introduce and develop the diffusion Monte Carlo method, which involves drifting, branching random walks. For any excited state, including any system with more than two electrons, one encounters the sign problem, limiting the direct application of these algorithms for most fermion systems. Instead, by using approximate fixed-node or fixed-phase boundary conditions, one can achieve efficiency similar to variational Monte Carlo. We also discuss the application of the projector method in a basis of Slater determinants. In this chapter, we discuss a different quantum Monte Carlo method, projector Monte Carlo (PMC). This general method was first suggested by Fermi [1049]; see the quote at the start of this chapter by two of the inventors of the Monte Carlo method. An implementation of PMC was tried out in the early days of computing [1050]. Advances in methodology, in particular importance sampling, resulted in a significant large-scale application: the exact calculation of the ground-state properties of 256 hard-sphere bosons by Kalos, Levesque, and Verlet [1051] in 1974. Calculations for electronic systems and the fixednode approximation were introduced by Anderson [1052, 1053]. One of the most important projector MC algorithms, the diffusion Monte Carlo algorithm with importance sampling for fermions, was used to compute the correlation energy of homogeneous electron gas by Ceperley and Alder [109] in 1980; the resulting HEG correlation energy was crucial in the development of density functional calculations. Types and properties of projectors In this method, a many-body projector G ( R , R ) = Ĝ is repeatedly applied to filter out the exact many-body ground state from an initial state; the operation of the projector is carried out with a random walk, hence the name of this class of methods.
We present in detail the recently developed multiconfigurational symmetrized projector quantum Monte Carlo method for excited states of the Hubbard model. We describe the implementation of the Monte Carlo method for a multiconfigurational trial wave function. We give a detailed discussion of issues related to the symmetry of the projection procedure that validates our Monte Carlo procedure for excited states. In this context we discuss various averaging procedures for the Green function and present an analysis of the errors incurred in these procedures. We study the ground-state energy and correlation functions of the one-dimensional Hubbard model at half-filling to confirm these analyses. We then study the energies and correlation functions of excited states of Hubbard chains. Hubbard rings away from half-filling are also studied and the pair binding energies for holes of 4n4n4n and 4n+24n+24n+2 systems are compared with the Bethe ansatz results of Fye, Martins, and Scalettar [Phys. Rev. B 42, 6809 (1990)]. Our study of the two-dimensional Hubbard model includes the $4\ifmmode\times\else\texttimes\fi{}2$ ladder and the $3\ifmmode\times\else\texttimes\fi{}4$ lattice with periodic boundary conditions. The $3\ifmmode\times\else\texttimes\fi{}4$ lattice is nonbipartite and amenable to exact diagonalization studies and is, therefore, a good candidate for checks on the method. We are able to reproduce accurately the energies of ground and excited states, both at and away from half-filling. We study the properties of the $4\ifmmode\times\else\texttimes\fi{}2$ Hubbard ladder with bond alternation as the correlation strength and filling are varied. The method reproduces the correlation functions accurately. We also examine the severity of sign problem for one- and two-dimensional systems.
A novel scheme to solve the quantum eigenvalue problem through the imaginary-time Green function Monte Carlo method is presented. This method is applicable not only to the ground states, but also to the excited states of generic system. We demonstrate the actual utility of the method with the numerical examples on three simple systems. Typeset using REVTEX 1 The stochastic quantization has been successfully applied in recent years to the various problems of physics ranging from quantum chromo dynamics to the plasmon oscillation [1-3]. While it can handle very complex systems which is usually beyond the approach either by the direct diagonalization or by the perturbation, it has been only applicable to the calculation of the ground state properties of the system. In this paper, we propose a new method based on the imaginary-time Green function Monte Carlo approach, which is extendable to the excited states of Hamiltonian system. We have primarily in mind the non-relativistic quantum few body problem such as encountered in nuclear physics [4-6].
As proposed by Ceperley and Bernu, the excited-state energies can be obtained from imaginary-time cross-correlation functions (rather than autocorrelation functions) generated by quantum Monte Carlo (QMC) simulations. We show that, when processed by the filter diagonalization method (FDM), the same cross-correlation functions yield excited-state energies of higher accuracy and greater stability. The reason is that unlike the other methods FDM uses all the time domain information available. The superior performance of the cross-correlation FDM is demonstrated for a two-dimensional harmonic oscillator with closely lying eigenvalues. Because QMC does not take advantage of the separability in the Hamiltonian, this model system provides a challenging and generic test case.