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Work carried out from 9/2008 through 5/2009

• An analytical model has been developed to describe the interaction between rigid ammonia molecules including the explicit description of induction. The parameters of the model potential were chosen by fitting high quality ab initio data obtained using MP2 and extended basis sets. The description of polarization effects is introduced by using a non iterative form of the "charge on spring model". Global minima for (NH₃)_n (n=3-20) have been obtained.

• Several importance sampling strategies are developed and tested for Stereographic Projection Diffusion Monte Carlo in manifolds. We test a family of one parameter trial wavefunctions for Variational Monte Carlo in stereographically projected manifolds which can be used to produce importance sampling. We find that Diffusion Monte Carlo with importance sampling in manifolds is orders of magnitude more efficient compared to unguided Diffusion Monte Carlo. We obtain the ground state energy and the wavefunction for the Stokmayer trimer.

• Classical and quantum simulations of ammonia clusters in the dimer through the hendecamer range are performed using the Stereographic Projection Path Integral and a recently developed polarizable potential. We develop a first order finite difference algorithm to integrate the geodesic equations in the inertia ellipsoid generated by n rigid nonlinear bodies mapped with stereographic projections. We use the technique to optimize configurations and to explore the potential surface of the hendecamer.

• Several stochastic simulations of the TIP4P [J. Chem. Phys. 79, 926 (1983).] water octamer are performed. Use is made of the Stereographic Projection Path Integral, and the Green's Function Stereographic Projection Diffusion Monte Carlo techniques, recently developed. The quantum heat capacity of the TIP4P octamer contains a pronounced melting peak at 160 K, about 50 K lower than the classical melting peak. The zero point energy of the TIP4P water octamer is 0.034(8) hartree. By characterizing several large samples of configurations visited by both guided and unguided diffusion walks, we determine that the ground state wavefunction is predominantly contained within the D_2d basin of attraction, in spite of the fact that the global minimum for the TIP4P octamer is a S₄ cube.

Work in progress (summer 2009)

• We are performing high level ab-initio calculations on the H₂ - NH₃ dimer and the H₂ - (NH₃)₂ trimer in order develop analytical surfaces for the para-H₂ - (NH₃)_n and ortho-H₂ - (NH₃)_n systems. Radial scans with MP2 and MP4 and extended basis sets are currently being compared.

• We are computing the ground state energy for ammonia clusters in the n=3-20 size range using both guided and unguided DMC with stereographic projections to map the rigid body rotations of the ammonia molecules. Selected deuterated species such as ND₃-(NH₃)_n-1 for n = 2,8,12, and 16 are also being investigated to measure the quantum effects on the hydrogen bonding structures and stability of ammonia clusters.

• We are optimizing a number of two parameter trial wave functions for the evaluation of the ground state of clusters with Variational Monte Carlo, and eventually to improve the efficiency of Diffusion Monte Carlo for atomic and molecular clusters. We are exploring the inclusion of terms as part of the arguments of the trial wavefunction that account for the mass difference in the constituents of matter under investigation. We are testing the approach by simulating heterogeneous systems and extremely weekly bound systems. The latter exploration will be crucial for the successful completion of the investigation of hydrogen molecules bound to covalent ammonia clusters, as proposed. At the moment these theories are tested with mixed Ar-Ne, and Ar-He clusters.

• We are testing the sampling strength of Parallel Tempering against a set of increasingly complex multi-funneled potential energy surfaces over a wide range of temperatures. We determine the rate of successful sampling by direct comparison of the emerging Boltzmann distribution normalized numerically with the trapezoid rule. We have found a completely deterministic, yet sufficiently complicated system to cause parallel tempering employed with an arbitrarily chosen standard move block size of one million to fail. The characteristic feature that causes such failure is the presence of a narrow funnel containing a deep minimum inside. This work bares fundamental significance in a number of activities that the group engages in, from the variational Monte Carlo to the imaginary time Path Integral simulations of clusters. This investigation has helped elucidate one of the potential problems that are encountered when simulating sufficiently complex systems with Parallel Tempering such as (NH₃)₁₁ and Ar₃₈. We are using this system to establish the validity of the "seeding" procedure that has been used in these extreme cases.

• We are developing importance sampling functions for the stochastic evaluation of Feynman Path Integrals in real time. The importance sampling probability density function is obtained via a change of variables. The theories we are currently testing are developed by inserting the resolution of the identity for the derivation of the Feynman Path integral. The resulting short-time evolution operator is split into a potential and kinetic local action contributions. Then the exponential of the kinetic action is remapped with a single stereographic projection coordinate onto the curved, multiply connected space, S¹, with unit radius. We are presently testing code for the evaluation of the trace of the thermally averaged time evolution operator with a family of quartic potentials. We will use the code to produce data that can be readily compared with the results obtained employing vector spaces and explore the efficiency of the algorithms we have generated.

A total of four students have been trained and have been involved in the reseach during the 2008 - 2009 academic year. All four of these students have coauthored one article. Two articles have been published, and one has been submitted. Five students are working in the group this summer, of which two have completed their first level training.