Structure factors calculated from molecular wavefunctions. II. Analysis of the molecular charge distribution in diborane

The electron density distribution in the crystal of diborane has been determined using structure factors based on molecular densities. The libration-correction and third-cumulant terms have also been included in the expression for the temperature factor. The equilibrium r_e values for BH bond lengths obtained by this treatment are about 0.02Åshorter than the spectroscopic ones. A total librational motion of the diborane molecule is determined: the root-mean-square oscillations about inertial axis are 11.0° (8), 9.6° (5) and 7.2° (5), respectively. The dynamic theoretical deformation density shows a three-center BHB bond picture for the bridge structure.     

On the euler equation: Bi-Hamiltonian structure and integrals in involution

We propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the physical phase space so(n), and is different from the bi-Hamiltonian formulation on the extended phase space sl(n), considered previously in the literature. Using the bi-Hamiltonian structure on so(n), we construct new recursion schemes for the Mishchenko and Manakov integrals of motion.     

Reduction techniques for infinite-dimensional Hamiltonian systems: Some ideas and applications

In the language of tensor analysis on differentiable manifolds, we present a reduction method of integrability structures, and apply it to recover some well-known hierarchies of integrable nonlinear evolution equations.This research has been partially supported by the Italian Ministry of Public Education     

Stability of 3He2 4He M and 3He3 4He M L = 0,1 Clusters

We have studied the stability of mixed ³He/⁴He clusters in L = 0 and L = 1 states by the diffusion Monte Carlo method, employing the Tang-Toennies-Yiu (TTY) He-He potential. The clusters ³He⁴He_M ( ) and ³He₂⁴He_M (L = 0, S = 0) are stable for M > 1, while to bind two ³He in a triplet state the minimum number of ⁴He is four. Considering clusters with three ³He, ³He₃⁴He₄ is the smallest stable system in the L = 1 state, while ³He₃⁴He₈ is the smallest stable system in the L = 0 state.     

A variational principle for the dynamic problem of linear coupled thermoelasticity

Summary A functional is constructed by whose stationarity the mixed problem for linear and dynamic thermoelasticity is obtained. In contrast with previous results[2], [4], [8], neither constrained variations nor previous transformations upon the equations of the problem are used. The unrestricted variational formulation is made possible by means of a convolutive bilinear form.

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Sommario Si costruisce un funzionale dalla cui stazionarietà si ricava il problema misto della termoelasticità lineare accoppiata. A differenza di precedenti risultati[2], [4], [8] non si ricorre né a variazioni bloccate né a preliminari trasformazioni del problema. La formulazione variazionale non ristretta è resa possibile dall'uso di una forma bilineare convolutiva.

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Work done in the sphere of activity of the Group for Mathematical Research of the (Italian) C.N.R.     

On the nodal structure of single-particle approximation based atomic wave functions

The nodal structures of atomic wave functions based on a product of spatial orbitals, namely, restricted, unrestricted, and generalized valence bond wave functions, are shown to be equivalent. This result is verified by fixed node-diffusion Monte Carlo simulations for atoms up to Ne. Also for a molecular system, Li(2) at the equilibrium geometry, a multideterminantal generalized valence bond wave function does not improve the nodal surfaces of a restricted Hartree-Fock wave function.     

Improved diffusion Monte Carlo propagators for bosonic systems using Itô calculus

The construction of importance sampled diffusion Monte Carlo (DMC) schemes accurate to second order in the time step is discussed. A central aspect in obtaining efficient second order schemes is the numerical solution of the stochastic differential equation (SDE) associated with the Fokker-Plank equation responsible for the importance sampling procedure. In this work, stochastic predictor-corrector schemes solving the SDE and consistent with It? calculus are used in DMC simulations of helium clusters. These schemes are numerically compared with alternative algorithms obtained by splitting the Fokker-Plank operator, an approach that we analyze using the analytical tools provided by Ito; calculus. The numerical results show that predictor-corrector methods are indeed accurate to second order in the time step and that they present a smaller time step bias and a better efficiency than second order split-operator derived schemes when computing ensemble averages for bosonic systems. The possible extension of the predictor-corrector methods to higher orders is also discussed.     

An investigation of nodal structures and the construction of trial wave functions

The factors influencing the quality of the nodal surfaces, namely, the atomic basis set, the single-particle orbitals, and the configurations included in the wave-function expansion, are examined for a few atomic and molecular systems. The following empirical rules are found: the atomic basis set must be fairly large, complete active space and natural orbitals are usually better than Hartree-Fock orbitals, multiconfiguration expansions perform better than single-determinant wave functions, but only few configurations are effective and their choice is suggested by symmetry considerations, while too long determinantal expansions spoil the nodal surfaces. These rules allow us to reduce the nodal error and to compute the best fixed node-diffusion Monte Carlo energies for a series of dimers of first-row atoms.     

Many-electron correlated exponential wavefunctions. A quantum Monte Carlo application to H2 and He2

We propose to expand the solution of the Schrödinger equation for an atomic or molecular system as a linear combination of many-electron explicitly correlated exponentials. A series of trial wavefunctions has been optimized, minimizing the variance of the local energy for H₂ and He₂⁺ in their ground state at the equilibrium distance, and their variational energy has been computed using the variational Monte Carlo method. The He₂⁺ wavefunctions have been used in a series of fixed node diffusion Monte Carlo simulations, showing that, using a small number of terms, one can obtain a good estimate of the exact energy.