Quantum Monte Carlo: Atoms, Molecules, Clusters, Liquids, and Solids

Anderson, James B.

doi:10.1002/9780470125908.ch3

Cited by 45 publications

(15 citation statements)

References 128 publications

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“…(6) also has the eigenvalue λ = E (ρ) 0 and is clearly nonnegative, so it is the lowest eigenfunction of Eq. (6). Therefore, when the trial function is exact, E (ρ) 0 must be the lowest eigenvalue of Eq.…”

Section: Symmetry-constrained Variational Principlementioning

confidence: 97%

“…(6) to the corresponding wave function ψ, is now derived. Suppose that ψ = ξφ and f = ξQ are both differentiable to second order.…”

Section: E Fixed-ray Eigenvalue Equationmentioning

confidence: 99%

“…The FN-DMC method is reviewed in Refs. [5][6][7][8][9]. These reviews show that FN-DMC has been applied to a broad range of problems in quantum mechanical systems, ranging from the electronic structure of atoms, molecules, and crystalline solids to the exchange-correlation energy of the homogeneous electron gas (the initial step in a released-node calculation), 10 which is fundamental for the local-density approximation used in most density functional theory (DFT) calculations.…”

Section: Introductionmentioning

confidence: 99%

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Diffusion Monte Carlo method with symmetry-constrained variational principle for degenerate excited states

Hipes¹

2011

Phys. Rev. B

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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.

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Section: Symmetry-constrained Variational Principlementioning

confidence: 97%

“…(6) to the corresponding wave function ψ, is now derived. Suppose that ψ = ξφ and f = ξQ are both differentiable to second order.…”

Section: E Fixed-ray Eigenvalue Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Diffusion Monte Carlo method with symmetry-constrained variational principle for degenerate excited states

Hipes¹

2011

Phys. Rev. B

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“…A modification to the scaled distance function facilitates a linear scaling implementation of the Schmidt-Moskowitz-Boys-Handy (SMBH) correlation function that preserves the efficient matrix multiplication structure of the SMBH function. For the evaluation of the local energy, these two methods lead to asymptotic linear scaling with respect to the size of the molecule.In the quantum Monte Carlo (QMC) method [1][2][3][4][5][6][7], the expectation value of the Hamiltonian H is computed as a statistical average of the local energy of a trial wave-function, Ψ T (R), where R denotes the 3N coordinates of the N-particle system. …”

mentioning

confidence: 99%

“…In the quantum Monte Carlo (QMC) method [1][2][3][4][5][6][7], the expectation value of the Hamiltonian H is computed as a statistical average of the local energy of a trial wave-function, Ψ T (R), where R denotes the 3N coordinates of the N-particle system.…”

mentioning

confidence: 99%

Linear-Scaling Evaluation of the Local Energy in Quantum Monte Carlo

Austin

Aspuru‐Guzik

Salomón-Ferrer

et al. 2006

ACS Symposium Series

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For atomic and molecular quantum Monte Carlo calculations, most of the computational effort is spent in the evaluation of the local energy. We describe a scheme for reducing the computational cost of the evaluation of the Slater determinants and correlation function for the correlated molecular orbital (CMO) ansatz. A sparse representation of the Slater determinants makes possible efficient evaluation of molecular orbitals. A modification to the scaled distance function facilitates a linear scaling implementation of the Schmidt-Moskowitz-Boys-Handy (SMBH) correlation function that preserves the efficient matrix multiplication structure of the SMBH function. For the evaluation of the local energy, these two methods lead to asymptotic linear scaling with respect to the size of the molecule.In the quantum Monte Carlo (QMC) method [1][2][3][4][5][6][7], the expectation value of the Hamiltonian H is computed as a statistical average of the local energy of a trial wave-function, Ψ T (R), where R denotes the 3N coordinates of the N-particle system.

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Computing Quantum Phase Transitions

Vojta

2008

Reviews in Computational Chemistry

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A phase transition occurs when the thermodynamic properties of a material display a singularity as a function of the external parameters. Imagine, for instance, taking a piece of ice out of the freezer. Initially, its properties change only slowly with increasing temperature. However, at 0°C, a sudden and dramatic change occurs. The thermal motion of the water molecules becomes so strong that it destroys the crystal structure. The ice melts, and a new phase of water forms, the liquid phase. At the phase transition temperature of 0°C the solid (ice) and the liquid phases of water coexist. A finite amount of heat, the so-called latent heat, is required to transform the ice into liquid water. Phase transitions involving latent heat are called first-order transitions. Another well known example of a phase transition is the ferromagnetic transition of iron. At room temperature, iron is ferromagnetic, i.e., it displays a spontaneous magnetization. With rising temperature, the magnetization decreases continuously due to thermal fluctuations of the spins.At the transition temperature (the so-called Curie point) of 770°C, the magnetization vanishes, and iron is paramagnetic at higher temperatures. In contrast to the previous example, there is no phase coexistence at the transition temperature; the ferromagnetic and paramagnetic phases rather become indistinguishable. Consequently, there is no latent heat. This type of phase transition is called continuous (second-order) transition or critical point.

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Quantum Monte Carlo: Atoms, Molecules, Clusters, Liquids, and Solids

Cited by 45 publications

References 128 publications

Diffusion Monte Carlo method with symmetry-constrained variational principle for degenerate excited states

Diffusion Monte Carlo method with symmetry-constrained variational principle for degenerate excited states

Linear-Scaling Evaluation of the Local Energy in Quantum Monte Carlo

Computing Quantum Phase Transitions

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