We develop an alternative formulation in the energy-domain to calculate the second order Møller–Plesset (MP2) perturbation energies. The approach is based on repeatedly choosing four random energies using a nonseparable guiding function, filtering four random orbitals at these energies, and averaging the resulting Coulomb matrix elements to obtain a statistical estimate of the MP2 correlation energy. In contrast to our time-domain formulation, the present approach is useful for both quantum chemistry and real-space/plane wave basis sets. The scaling of the MP2 calculation is roughly linear with system size, providing a useful tool to study dispersion energies in large systems. This is demonstrated on a structure of 64 fullerenes within the SZ basis as well as on silicon nanocrystals using real-space grids.

We formulate the Kohn-Sham density functional theory (KS-DFT) as a statistical theory in which the electron density is determined from an average of correlated stochastic densities in a trace formula. The key idea is that it is sufficient to converge the total energy per electron to within a predefined statistical error in order to obtain reliable estimates of the electronic band structure, the forces on nuclei, the density and its moments, etc. The fluctuations in the total energy per electron are guaranteed to decay to zero as the system size increases. This facilitates “self-averaging” which leads to the first ever report of sublinear scaling KS-DFT electronic structure. The approach sidesteps calculation of the density matrix and thus, is insensitive to its evasive sparseness, as demonstrated here for silicon nanocrystals. The formalism is not only appealing in terms of its promise to far push the limits of application of KS-DFT, but also represents a cognitive change in the way we think of electronic structure calculations as this stochastic theory seamlessly converges to the thermodynamic limit.

An ab initio variational grand-canonical electronic structure mean-field method, based on the Gibbs–Peierls–Bogoliubov minimum principle for the Gibbs free energy, is applied to the di-lithium (Li+Li) system at temperatures around T \approx 10,000 K and electronic chemical potential of μ \approx -0.1Eh. The method is an extension of the Hartree–Fock approach to finite temperatures. We first study the Li2 molecule at a frozen inter-nuclear distance of R = 3 \AA as a function of temperature. The mean-field electronic structure changes smoothly as temperature increases, up to 104 K, where a sharp spontaneous spin-polarization emerges as the variational mean-field solution. Further increase in the temperature extinguishes this polarization. We analyze the mean-field behavior using a correlated single-site Hubbard model and show it arises from an attempt of the mean-field to mimic the polarization of the spin–spin correlation function of the exact solution. Next, we keep constant the temperature at 104 K and examine the electronic structure as a function of inter-nuclear distance R. At R = 3.7 \AA, a crossing between two free energy states occurs: One state is “spin-unpolarized” (becomes lower in energy when R \ge 3.7 \AA), while the other is “spin polarized”. This crossing causes near-discontinuous jumps in calculated properties of the system and is associated with using the noninteracting electron character of our mean-field approach. Such problems will likely plague FT-DFT calculations as well. We use second-order perturbation theory (PT2) to study effects of electron correlation on the potential of mean force between the two colliding Li atoms. We find that PT2 correlation free energy at  104 K is larger than at 0 K and tends to restore the spin-polarized state as the lowest free energy solution.

In recent generalized Kohn-Sham (GKS) schemes for density functional theory (DFT) Hartree-Fock type exchange is important. In plane waves and grid approaches the high cost of exchange energy calculations makes these GKS considerably more expensive than Kohn-Sham DFT calculations. We develop a stochastic approach for speeding up the calculation of exchange for large systems. We show that stochastic error per particle does not grow and can even decrease with system size (at a given number of iterations). We discuss several alternative approaches and explain how these ideas can be included in the GKS framework.

A stochastic method is developed to calculate the multiexciton generation (MEG) rates in semiconductor nanocrystals (NCs). The numerical effort scales near-linearly with system size allowing the study of MEG rates up to diameters and exciton energies previously unattainable using atomistic calculations. Illustrations are given for CdSe NCs of sizes and energies relevant to current experimental setups, where direct methods require treatment of over 1011 states. The approach is not limited to the study of MEG and can be applied to calculate other correlated electronic processes.

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

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.

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.

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.

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.

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.

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.