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Register a new userProof for an upper bound in fixed-node Monte Carlo for lattice fermions
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We justify a recently proposed prescription for performing Green Function Monte Carlo calculations on systems of lattice fermions, by which one is able to avoid the sign problem. We generalize the prescription such that it can also be used for problems with hopping terms of different signs. We prove that the effective Hamiltonian, used in this method, leads to an upper bound for the ground-state energy of the real Hamiltonian, and we illustrate the effectiveness of the method on small systems.
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The effectiveness of the recently developed Fixed-Node Quantum Monte Carlo method for lattice fermions, developed by van Leeuwen and co-workers, is tested by applying it to the 1D Kondo lattice, an example of a one-dimensional model with a sign problem. The principles of this method and its implementation for the Kondo Lattice Model are discussed in detail. We compare the fixed-node upper bound for the ground state energy at half filling with exact-diagonalization results from the literature, and determine several spin correlation functions. Our `best estimates for the ground state correlation functions do not depend sensitively on the input trial wave function of the fixed-node projection, and are reasonably close to the exact values. We also calculate the spin gap of the model with the Fixed-Node Monte Carlo method. For this it is necessary to use a many-Slater-determinant trial state. The lowest-energy spin excitation is a running spin soliton with wave number pi, in agreement with earlier calculations.
By combining density-functional theory (DFT) and wave function theory (WFT) via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multi-determinant trial wave functions. In particular, we combine here short-range exchange-correlation functionals with a flavor of selected configuration interaction (SCI) known as emph{configuration interaction using a perturbative selection made iteratively} (CIPSI), a scheme that we label RS-DFT-CIPSI. One of the take-home messages of the present study is that RS-DFT-CIPSI trial wave functions yield lower fixed-node energies with more compact multi-determinant expansions than CIPSI, especially for small basis sets. Indeed, as the CIPSI component of RS-DFT-CIPSI is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional CIPSI calculation. Importantly, by performing various numerical experiments, we evidence that the RS-DFT scheme essentially plays the role of a simple Jastrow factor by mimicking short-range correlation effects, hence avoiding the burden of performing a stochastic optimization. Considering the 55 atomization energies of the Gaussian-1 benchmark set of molecules, we show that using a fixed value of $mu=0.5$~bohr$^{-1}$ provides effective error cancellations as well as compact trial wave functions, making the present method a good candidate for the accurate description of large chemical systems.
The accuracy and efficiency of ab-initio quantum Monte Carlo (QMC) algorithms benefits greatly from compact variational trial wave functions that accurately reproduce ground state properties of a system. We investigate the possibility of using multi-Slater-Jastrow trial wave functions with non-orthogonal determinants by optimizing identical single particle orbitals independently in separate determinants. As a test case, we compute variational and fixed-node diffusion Monte Carlo (FN-DMC) energies of a C$_2$ molecule. For a given multi-determinant expansion, we find that this non-orthogonal orbital optimization results in a consistent improvement in the variational energy and the FN-DMC energy on the order of a few tenths of an eV. Our calculations indicate that trial wave functions with non-orthogonal determinants can improve computed energies in a QMC calculation when compared to their orthogonal counterparts.
Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multi-determinant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis set full configuration interaction (CBS-FCI) energies estimated with CIPSI are used as a reference in this comparative study between DMC and AFQMC. By focusing on a set of canonical finite size solid state systems, we show that both QMC methods can be made to systematically converge towards the same energy once basis set effects and systematic biases have been removed. AFQMC shows a much smaller dependence on the trial wavefunction than DMC while simultaneously exhibiting a much larger basis set dependence. We outline some of the remaining challenges and opportunities for improving these approaches.
Fixed-node Greens function Monte Carlo calculations have been performed for very large 16x6 2D Hubbard lattices, large interaction strengths U=10,20, and 40, and many (15-20) densities between empty and half filling. The nodes were fixed by a simple Slater-Gutzwiller trial wavefunction. For each value of U we obtained a sequence of ground-state energies which is consistent with the possibility of a phase separation close to half-filling, with a hole density in the hole-rich phase which is a decreasing function of U. The energies suffer, however, from a fixed-node bias: more accurate nodes are needed to confirm this picture. Our extensive numerical results and their test against size, shell, shape and boundary condition effects also suggest that phase separation is quite a delicate issue, on which simulations based on smaller lattices than considered here are unlikely to give reliable predictions.
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