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Trial wave function construction and the nodes of trial and exact wave functions in quantum Monte Carlo

Dario Bressanini, W Ageno, En Libby, Dam-Won Oh, Crowther Ma

Published Jan 1, 2003• 0 citations
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We develop a formalism and present an algorithm for optimization of the trial wave function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. The formalism is based on the DMC mixed estimator of the ground-state probability density. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project out a multideterminant expansion of the fixed-node ground-state wave function and (ii) to define a cost function that relates the fixed-node ground-state and the noninteracting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave function with better nodal structure and (b) we argue that the noise in the fixed-node wave function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust toward the ones of the exact many-body ground state in a simulated annealing-like process. Based on these principles, we propose a method to improve both single determinant and multideterminant expansions of the trial wave function. The method can be generalized to other wave-function forms such as pfaffians. We test the method in a model system where benchmark configuration-interaction calculations can be performed and most components of the Hamiltonian are evaluated analytically. Comparing the DMC calculations with the exact solutions, we find that the trial wave function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave functions orthogonal to the exact ground state. Similarly, the DMC total energy and density converges to the exact solutions for the model. In the optimization process we find an optimal noninteracting nodal potential of density-functional-like form whose existence was predicted in a previous publication [Phys. Rev. B 77, 245110 (2008)]. Tests of the method are extended to a model system with a conventional Coulomb interaction where we show we can obtain the exact Kohn-Sham effective potential from the DMC data.

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Trial wave function based quantum Monte Carlo is a promising family of methods for the solution of quantum mechanical Hamiltonians. These fundamentally non-perturbative methods can be used to treat bosons or fermions in weakly or strong interacting systems and in any phase. Properties computed for particles with bosonic statistics can be converged to their exact value at relatively moderate computational cost using diffusion quantum Monte Carlo. For fermions the problem is slightly more complicated. Exact methods, such as released-node diffusion quantum Monte Carlo, exist for computing unbiased fermion properties, however their computational expense increases rapidly with the Bose-Fermi energy gap and system size. It is possible to use a fixed-node version of diffusion quantum Monte Carlo, but it introduces a bias due to the nodes of the trial wave function. In this thesis we work towards reducing trial wave function bias in fixed-node calculations and then perform several benchmark studies. We begin with a pedagogical overview of the three methods used throughout the thesis, variational, diffusion, and reptation quantum Monte Carlo. Then we discuss the problem, trial wave function bias in the energy and other observables. Next we outline the Hamiltonians we are typically interested in for electronic systems and trial wave function forms used to solve them. The algorithm used to optimize the trial wave functions is presented along with special considerations for some particular cases. Next we present three studies of the pressure of the electron gas using improved estimators in variational, diffusion, and reptation quantum Monte Carlo. Benchmark results for a set of molecules are then presented for the massive multi-determinant expansion and optimization algorithm tailored to suit it. We conclude with two studies of the interacting Fermi liquid, one on the momentum properties of the electron gas in three dimensions and another on the entanglement properties of interacting Fermi liquids in two dimensions.

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In this paper we proposed a novel exact fixed-node quantum Monte Carlo (EFNQMC)algorithm,which is a self-optimizing and self-improving procedure,In contrast to the previous EFNQMC method,the trial function is optimized synchronistically in the diffusion procedure,but not before the beginning of EFNQMC computation.In order to optimize the trial function,the improved steepest descent technique is used,in which the step size is automatically adjustable.The procedure is quasi-Newton and converges super linearly.We also use a novel trial function,which has correct electron-electron and electron-nucleus cusp conditions.The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 1^1A1 state of CH2,^1Ag state of C8 and the ground-states of H2,LiH,Li2,H2O,respectively.The test results show that both the novel algorithm and the trial function proposed in the present paper are very excellent.

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EXPLICITLY CORRELATED WAVE FUNCTIONS IN QUANTUM MONTE CARLO*

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We develop a formalism and present an algorithm for optimization of the trial wave function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. The formalism is based on the DMC mixed estimator of the ground-state probability density. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project out a multideterminant expansion of the fixed-node ground-state wave function and (ii) to define a cost function that relates the fixed-node ground-state and the noninteracting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave function with better nodal structure and (b) we argue that the noise in the fixed-node wave function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust toward the ones of the exact many-body ground state in a simulated annealing-like process. Based on these principles, we propose a method to improve both single determinant and multideterminant expansions of the trial wave function. The method can be generalized to other wave-function forms such as pfaffians. We test the method in a model system where benchmark configuration-interaction calculations can be performed and most components of the Hamiltonian are evaluated analytically. Comparing the DMC calculations with the exact solutions, we find that the trial wave function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave functions orthogonal to the exact ground state. Similarly, the DMC total energy and density converges to the exact solutions for the model. In the optimization process we find an optimal noninteracting nodal potential of density-functional-like form whose existence was predicted in a previous publication [Phys. Rev. B 77, 245110 (2008)]. Tests of the method are extended to a model system with a conventional Coulomb interaction where we show we can obtain the exact Kohn-Sham effective potential from the DMC data.
Trial wave function based quantum Monte Carlo is a promising family of methods for the solution of quantum mechanical Hamiltonians. These fundamentally non-perturbative methods can be used to treat bosons or fermions in weakly or strong interacting systems and in any phase. Properties computed for particles with bosonic statistics can be converged to their exact value at relatively moderate computational cost using diffusion quantum Monte Carlo. For fermions the problem is slightly more complicated. Exact methods, such as released-node diffusion quantum Monte Carlo, exist for computing unbiased fermion properties, however their computational expense increases rapidly with the Bose-Fermi energy gap and system size. It is possible to use a fixed-node version of diffusion quantum Monte Carlo, but it introduces a bias due to the nodes of the trial wave function. In this thesis we work towards reducing trial wave function bias in fixed-node calculations and then perform several benchmark studies. We begin with a pedagogical overview of the three methods used throughout the thesis, variational, diffusion, and reptation quantum Monte Carlo. Then we discuss the problem, trial wave function bias in the energy and other observables. Next we outline the Hamiltonians we are typically interested in for electronic systems and trial wave function forms used to solve them. The algorithm used to optimize the trial wave functions is presented along with special considerations for some particular cases. Next we present three studies of the pressure of the electron gas using improved estimators in variational, diffusion, and reptation quantum Monte Carlo. Benchmark results for a set of molecules are then presented for the massive multi-determinant expansion and optimization algorithm tailored to suit it. We conclude with two studies of the interacting Fermi liquid, one on the momentum properties of the electron gas in three dimensions and another on the entanglement properties of interacting Fermi liquids in two dimensions.
In this paper we proposed a novel exact fixed-node quantum Monte Carlo (EFNQMC)algorithm,which is a self-optimizing and self-improving procedure,In contrast to the previous EFNQMC method,the trial function is optimized synchronistically in the diffusion procedure,but not before the beginning of EFNQMC computation.In order to optimize the trial function,the improved steepest descent technique is used,in which the step size is automatically adjustable.The procedure is quasi-Newton and converges super linearly.We also use a novel trial function,which has correct electron-electron and electron-nucleus cusp conditions.The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 1^1A1 state of CH2,^1Ag state of C8 and the ground-states of H2,LiH,Li2,H2O,respectively.The test results show that both the novel algorithm and the trial function proposed in the present paper are very excellent.