Improving half-projected spin-contaminated wave functions by multi-configuration perturbation theory.

Allowing triplet components of individual geminals, spin-contaminated strongly orthogonal geminal wave functions may emerge, which can be ameliorated by spin-projection techniques. Of the latter, half-projection was previously shown to be useful, offering a compromise between the amount of remaining spin-contamination and the violation of size consistency generated by projection. This paper investigates how a half-projected spin-contaminated geminal wave function can be improved by multi-configuration perturbation theory to incorporate dynamical correlation effects.

Bilinear Constraints upon the Correlation Contribution to the Electron-Electron Repulsion Energy as a Functional of the One-Electron Reduced Density Matrix.

Perturbative analysis of the functional U[n, ψ] that yields the correlation component U of the electron-electron repulsion energy in terms of the vectors ψ(1) and n of the natural spinorbitals and their occupation numbers (the 1-matrix functional) facilitates examination of the flaws inherent to the present implementations of the density matrix functional theory. Recognizing that the practical usefulness of any approximate 1-matrix functional hinges upon its capability of exactly reproducing the leading contribution to U at the limit of vanishing electron-electron interactions gives rise to asymptotic bilinear constraints for the (exact or model) 2-cumulant G2 that enters the expression for U. The asymptotic behavior of certain blocks of G2 is found to be equally important. These identities, which are obtained for both the single-determinantal and a model multideterminantal cases, take precedence over the linear constraints commonly enforced in the course of approximate construction of such functionals. This observation reveals the futility of designing sophisticated approximations tailored for the second-order contribution to G2 while neglecting proper formulation of the respective first-order contribution that in the case of the so-called JKL-only functionals requires abandoning the JK-dependence altogether. It has its repercussions not only for the functionals of the PNOF family but also for the expressions involving only the L-type two-electron repulsion integrals (in the guise of their exchange counterparts) that account only for the correlation effects due to electrons with antiparallel spins and are well-defined only for spin-unpolarized and high-spin systems (yielding vanishing U for the latter).

Mean first passage times in variational coarse graining using Markov state models.

Markov state models (MSMs) provide some of the simplest mathematical and physical descriptions of dynamical and thermodynamical properties of complex systems. However, typically, the large dimensionality of biological systems studied makes them prohibitively expensive to work in fully Markovian regimes. In this case, coarse graining can be introduced to capture the key dynamical processes-slow degrees of the system-and reduce the dimension of the problem. Here, we introduce several possible options for such Markovian coarse graining, including previously commonly used choices: the local equilibrium and the Hummer Szabo approaches. We prove that the coarse grained lower dimensional MSM satisfies a variational principle with respect to its slowest relaxation time scale. This provides an excellent framework for optimal coarse graining, as previously demonstrated. Here, we show that such optimal coarse graining to two or three states has a simple physical interpretation in terms of mean first passage times and fluxes between the coarse grained states. The results are verified numerically using both analytic test potentials and data from explicit solvent molecular dynamics simulations of pentalanine. This approach of optimizing and interpreting clustering protocols has broad applicability and can be used in time series analysis of large data.

Application of the Cauchy integral formula as a tool of analytic continuation for the resummation of divergent perturbation series.

Previous attempts to the resummation of divergent power series by means of analytic continuation are improved applying the Cauchy integral formula for complex functions. The idea is tested on divergent Møller-Plesset perturbation expansions of the electron correlation energy. In particular, the potential curve of the LiH molecule is computed from single reference MPn results which are divergent for bond distances larger than 3.6 Å. Preliminary results for the Hartree-Fock molecule are also tabulated.

Effect of partitioning on the convergence properties of the Rayleigh-Schrödinger perturbation series.

Convergence features of the Rayleigh-Schrödinger perturbation theory (PT) strongly depend on the partitioning applied. We investigate the large order behavior of the Møller-Plesset and Epstein Nesbet partitionings in comparison with a less known partitioning obtained by level shift parameters minimizing the norm of operator Q^W^, with W^ being the perturbation operator while Q standing for the reduced resolvent of the zero order Hamiltonian H^0. Numerical results, presented for molecular systems for the first time, indicate that it is possible to find level shift parameters in this way which convert divergent perturbation expansions to convergent ones in some cases. Besides numerical calculations of high-order PT terms, convergence radii of the corresponding perturbation expansions are also estimated using quadratic Padé approximants.