A quantum Monte Carlo method that combines the second-order many-body perturbation theory and Monte Carlo (MC) integration has been developed for correlation and correlation-corrected (quasiparticle) energy bands of one-dimensional solids. The sum-of-product expressions of correlation energy and self-energy are transformed, with the aid of a Laplace transform, into high-dimensional integrals, which are subject to a highly scalable MC integration with the Metropolis algorithm for importance sampling. The method can compute correlation energies of polyacetylene and polyethylene within a few $\text{m}{E}_{\text{h}}$ and quasiparticle energy bands within a few tenths of an eV. It does not suffer from the fermion sign problem and its description can be systematically improved by raising the perturbation order.A quantum Monte Carlo method that combines the second-order many-body perturbation theory and Monte Carlo (MC) integration has been developed for correlation and correlation-corrected (quasiparticle) energy bands of one-dimensional solids. The sum-of-product expressions of correlation energy and self-energy are transformed, with the aid of a Laplace transform, into high-dimensional integrals, which are subject to a highly scalable MC integration with the Metropolis algorithm for importance sampling. The method can compute correlation energies of polyacetylene and polyethylene within a few $\text{m}{E}_{\text{h}}$ and quasiparticle energy bands within a few tenths of an eV. It does not suffer from the fermion sign problem and its description can be systematically improved by raising the perturbation order.

Physical Review BNov 18, 201420 citations

The effect of quantization on the full configuration interaction quantum Monte Carlo sign problem

Michael Kolodrubetz, James S. Spencer, Bryan K. Clark +1 more

The sign problem in full configuration interaction quantum Monte Carlo (FCIQMC) without annihilation can be understood as an instability of the psi-particle population to the ground state of the matrix obtained by making all off-diagonal elements of the Hamiltonian negative. Such a matrix, and hence the sign problem, is basis dependent. In this paper, we discuss the properties of a physically important basis choice: first versus second quantization. For a given choice of single-particle orbitals, we identify the conditions under which the fermion sign problem in the second quantized basis of antisymmetric Slater determinants is identical to the sign problem in the first quantized basis of unsymmetrized Hartree products. We also show that, when the two differ, the fermion sign problem is always less severe in the second quantized basis. This supports the idea that FCIQMC, even in the absence of annihilation, improves the sign problem relative to first quantized methods. Finally, we point out some theoretically interesting classes of Hamiltonians where first and second quantized sign problems differ, and others where they do not.The sign problem in full configuration interaction quantum Monte Carlo (FCIQMC) without annihilation can be understood as an instability of the psi-particle population to the ground state of the matrix obtained by making all off-diagonal elements of the Hamiltonian negative. Such a matrix, and hence the sign problem, is basis dependent. In this paper, we discuss the properties of a physically important basis choice: first versus second quantization. For a given choice of single-particle orbitals, we identify the conditions under which the fermion sign problem in the second quantized basis of antisymmetric Slater determinants is identical to the sign problem in the first quantized basis of unsymmetrized Hartree products. We also show that, when the two differ, the fermion sign problem is always less severe in the second quantized basis. This supports the idea that FCIQMC, even in the absence of annihilation, improves the sign problem relative to first quantized methods. Finally, we point out some theoretically interesting classes of Hamiltonians where first and second quantized sign problems differ, and others where they do not.

arXiv:1209.3044The Journal of Chemical PhysicsJan 11, 201330 citations

Monte Carlo methods for the many-fermion problem

Malvin H. Kalos

Physica A Statistical Mechanics and its ApplicationsMar 1, 19840 citations

Fermiyonik sistemler için Kuantum Monte Carlo yöntemleri : sabit düğüm yakınlaştırması ötesi

Dugan, Nazım

Developments are made on the quantum Monte Carlo methods towards increasing the precision and the stability of the non fixed-node projector calculations of fermions. In the first part of the developments, the wavefunction correction scheme, which was developed to increase the precision of the di usion Monte Carlo (DMC) method, is applied to non fixed-node DMC to increase the precision of such fermion calculations which do not have nodal error. The benchmark calculations indicate a significant decrease of statistical error due to the usage of the correction scheme in such non fixed-node calculations. The second part of the developments is about the modifications of the wavefunction correction scheme for having a stable non fixed-node DMC algorithm for fermions. The minus signed walkers of the non fixed-node calculations are avoided by these modifications in the developed stable algorithm. However, the accuracy of the method decreases, especially for larger systems, as a result of the discussed modifications to overcome the sign instability.Developments are made on the quantum Monte Carlo methods towards increasing the precision and the stability of the non fixed-node projector calculations of fermions. In the first part of the developments, the wavefunction correction scheme, which was developed to increase the precision of the di usion Monte Carlo (DMC) method, is applied to non fixed-node DMC to increase the precision of such fermion calculations which do not have nodal error. The benchmark calculations indicate a significant decrease of statistical error due to the usage of the correction scheme in such non fixed-node calculations. The second part of the developments is about the modifications of the wavefunction correction scheme for having a stable non fixed-node DMC algorithm for fermions. The minus signed walkers of the non fixed-node calculations are avoided by these modifications in the developed stable algorithm. However, the accuracy of the method decreases, especially for larger systems, as a result of the discussed modifications to overcome the sign instability.

Jan 1, 20100 citations

\text{m}{E}_{\text{h}}

and quasiparticle energy bands within a few tenths of an eV. It does not suffer from the fermion sign problem and its description can be systematically improved by raising the perturbation order.