The talk will combine the two subjects indicated in the title. The first topic is a renewal of the use of the second order reduced density matrix (2-RDM) for electronic structure calculations. Previous explorations, going back to the early 1970's and culminating in recent work of M. Nakata et al [1], imposed the well-known P, Q, and G conditions of Coleman and Garrod and Percus. These conditions provide an accuracy that compares favourably with Hartree-Fock, but that is not impressive by the standards of other conventional ab initio methods. In our recent paper [2] we impose additional conditions towards N-representability. These conditions (which we call T1 and T2) are of semidefinite form just as are the P, Q, and G conditions, and they still involve only the 1-RDM and 2-RDM; there is no need for an approximate reconstruction of higher-order RDM's in our approach. We study a variety of molecules, extending the list of Nakata et al., and obtain in all cases an accuracy that is better than that of CISD or CCSD(T) on the same model space using full CI as the benchmark. In the talk I will review the reduced density matrix method and the T1 and T2 conditions, and discuss the results, including some more recent work.

The second topic is the construction of global fits for the potential energy surface (PES) of several molecular systems, among them CH5+, H3O2-, H4O2, H5O2+, and C3H3O [3,4]. Part of the novelty of our approach, and key to the success, is to use a functional form that is invariant under the complete symmetry group of permutations of like nuclei. This is technically quite difficult if one goes beyond the first few orders in an expansion, and we rely on the mathematical theory of invariants of finite groups and on a computational algebra system to help generate the codes. The accuracy of these global fits is superb. For example, for H5O2+ all the way up to dissociation into H2O and H3O+ (at energy around 120000/cm) we have an rms error in the fit less than 40/cm. The fitted potential is evaluated on a millisecond timescale, so we can do many long MD or QMC calculations at essentially ab initio accuracy without anywhere near the cost that is normally associated with ab initio MD, or even with a Car-Parrinello treatment. As shown by our treatment of C3H3O (including six different break-up and reaction channels) the work is immediately relevant to the evaluation of ab initio cross-sections for reactions in combustion. In the talk I will present the mathematics behind these fitting functions and discuss prospects for further applications.

[1] M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa

Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm

J. Chem. Phys. 114 (2001) 8282-8292.

[2] Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton and J. K. Percus

The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions

J. Chem. Phys. 120 (2004) 2095-2104.

[3] A. B. McCoy, B. J. Braams, A. Brown, X. C. Huang, Z. Jin, and J. M. Bowman

Ab initio diffusion Monte Carlo calculations of the quantum behavior of CH5+ in full dimensionality

J. Phys. Chem., in press.

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[4] X. Huang, B. J. Braams, S. Carter, and J. M. Bowman

Quantum Calculations of Vibrational Energies of H3O2- on an ab Initio Potential

J. Amer. Chem. Soc., in press.

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