Probabilistic algorithms

Jacobi, S. ; Baer, R. The well-tempered auxiliary-field Monte Carlo. The Journal of chemical physics 2004, 120, 43–50.Abstract

The auxiliary-field Monte Carlo (AFMC) is a method for computing ground-state and excited-state energies and other properties of electrons in molecules. For a given basis set, AFMC is an approximation to full-configuration interaction and the accuracy is determined predominantly by an inverse temperature "\beta" parameter. A considerable amount of the dynamical correlation energy is recovered even at small values of \beta. Yet, nondynamical correlation energy is inefficiently treated by AFMC. This is because the statistical error grows with \beta, warranting increasing amount of Monte Carlo sampling. A recently introduced multideterminant variant of AFMC is studied, and the method can be tuned by balancing the sizes of the determinantal space and the \beta parameter with respect to a predefined target accuracy. The well tempered AFMC is considerably more efficient than a naive AFMC. We demonstrate the principles on dissociating hydrogen molecule and torsion of ethylene where we calculate the (unoptimized) torsional barrier and the vertical singlet-triplet

Baer, R. ; Neuhauser, D. Shifted Contour Auxiliary Field Monte Carlo. In Recent Advances in Quantum Monte Carlo Methods II; W.A. Lester Jr., S. R. ; Tanaka, S., Ed.; World Scientific: Singapore, 2002; Vol. 3, pp. 279. baer2002f.pdf
Baer, R. ; Neuhauser, D. Molecular electronic structure using auxiliary field Monte Carlo, plane-waves, and pseudopotentials. J. Chem. Phys. 2000, 112, 1679–1684.Abstract

Shifted contour auxiliary field Monte Carlo is implemented for molecular electronic structure using a plane-waves basis and norm conserving pseudopotentials. The merits of the method are studied by computing atomization energies of H2,H2, BeH2,BeH2, and Be2.Be2. By comparing with high correlation methods, DFT-based norm conserving pseudopotentials are evaluated for performance in fully correlated molecular computations. Pseudopotentials based on generalized gradient approximation lead to consistently better atomization energies than those based on the local density approximation, and we find there is room for designing pseudopotentials better suited for full valence correlation.

Baer, R. Ab initio computation of forces and molecular spectroscopic constants using plane waves based auxiliary field Monte Carlo with application to N-2. J. Chem. Phys. 2000, 113, 473–476.Abstract

Correlated sampling within the shifted contour auxiliary field Monte Carlo method, implemented using plane waves and pseudopotentials, allows computation of electronic forces on nuclei, potential energy differences, geometric and vibrotational spectroscopic constants. This is exemplified on the N2 molecule, where it is demonstrated that it is possible to accurately compute forces, dissociation energies, bond length parameters, and harmonic frequencies.

Baer, R. Ab-initio molecular deformation barriers using auxiliary-field quantum Monte Carlo with application to the inversion barrier of water. Chem. Phys. Lett. 2000, 324, 101–107.Abstract

The shifted-contour auxiliary field Monte Carlo method applied within a plane waves and pseudopotential framework is shown capable of computing accurate molecular deformation barriers. The inversion barrier of water is used as a test case. A method of correlated sampling is extremely useful for deriving highly accurate barriers. The inversion barrier height is determined to be 1.37 eV with a statistical error bar of "0.01 eV. Recent high-level ab initio results are within the error bars. Several theoretical and methodological issues are discussed.

Baer, R. Accurate and efficient evolution of nonlinear Schrödinger equations. Phys. Rev. A 2000, 62, 063810.Abstract

A numerical method is given for affecting nonlinear Schro¨dinger evolution on an initial wave function, applicable to a wide range of problems, such as time-dependent Hartree, Hartree-Fock, density-functional, and Gross-Pitaevskii theories. The method samples the evolving wave function at Chebyshev quadrature points of a given time interval. This achieves an optimal degree of representation. At these sampling points, an implicit equation, representing an integral Schro¨dinger equation, is given for the sampled wave function. Principles and application details are described, and several examples and demonstrations of the method and its numerical evaluation on the Gross-Pitaevskii equation for a Bose-Einstein condensate are shown.

Baer, R. ; Zeiri, Y. ; Kosloff, R. Quantum diffusion of hydrogen and deuterium on nickel (100). Surf. Sci. 1998, 411, L783–L788.Abstract

The diffusion constants of hydrogen and deuterium at low temperature were calculated using the surrogate Hamiltonian method and an embedded atom potential. A comparison with previous experimental and theoretical results is made. A crossover to temperature-independent tunneling occurs at 69 K for hydrogen and at 46 K for deuterium. An inverse isotope effect at intermediate temperatures is found, consistent with experiment. Deviations are found at low temperature where a large isotope effect is calculated.

Baer, R. ; Head-Gordon, M. Electronic structure of large systems: Coping with small gaps using the energy renormalization group method. J. Chem. Phys. 1998, 109, 10159–10168. baer1998electronic.pdf
Baer, R. ; Head-Gordon, M. Energy renormalization-group method for electronic structure of large systems. Physical Review B-Condensed Matter 1998, 58, 15296–15299.Abstract

A newly developed energy renormalization-group method for electronic structure of large systems with small Fermi gaps within a tight-binding framework is presented in detail. A telescopic series of nested Hilbert spaces is constructed, having exponentially decreasing dimensions and electrons, for which the Hamiltonian matrices have exponentially converging energy ranges focusing to the Fermi level and in which the contribution to the density matrix is a sparse contribution. The computational effort scales near linearly with system size even when the density matrix is highly nonlocal. This is illustrated by calculations on a model metal, a small radius carbon-nanotube and a two-dimensional puckered sheet polysilane semiconductor.

Baer, R. ; Head-Gordon, M. P. ; Neuhauser, D. Shifted-contour auxiliary field Monte Carlo for ab initio electronic structure: Straddling the sign problem. J. Chem. Phys. 1998, 109, 6219–6226. baer1998b.pdf