A general ansatz for the nuclear-electronic wavefunction that includes explicit dependence on the nuclear-electronic distances with Gaussian-type geminal functions is proposed. Based on this ansatz, expressions are derived for the total energy and the electronic and nuclear Fock operators for multielectron systems. The explicit electron-proton correlation is incorporated directly into the self-consistent-field procedure for optimizing the nuclear-electronic wavefunction. This approach is computationally practical for many-electron systems because only a relatively small number of nuclei are treated quantum mechanically, and only electron-proton correlation is treated explicitly. Electron-electron correlation can be included by combining the NEO-XCHF approach with perturbation theory, density functional theory, and multiconfigurational methods. Previous nuclear-electronic orbital-based methods produce nuclear densities that are too localized, resulting in abnormally high stretching frequencies and inaccuracies in other properties relying on these densities. The application of the NEO-XCHF approach to the [He-H-He](+) model system illustrates that this approach includes the significant electron-proton correlation, thereby leading to an accurate description of the nuclear density. The agreement between the proton densities obtained with the NEO-XCHF and grid-based methods validates the underlying theory and the implementation of the NEO-XCHF method.

The NEO approach is designed to treat a relatively small number of nuclei quantum mechanically, while the remaining nuclei are treated classically. In the NEO-DFT(ee) approach, the correlated electron density is used to obtain the nuclear molecular orbitals, and the resulting nuclear density is used to obtain the correlated electron density during an iterative procedure that continues until convergence of both the nuclear and electronic densities. This approach includes feedback between the correlated electron density and the nuclear wavefunction. The application of this approach to bihalides and acetylene indicates that the nuclear quantum effects do not significantly impact the electron correlation energy, but the quantum nuclear energy is enhanced in the NEO-DFT(ee) B3LYP method. The excellent agreement of the NEO-DFT(ee)-optimized bihalide structures with the vibrationally averaged geometries from grid-based quantum dynamical methods provides validation for the NEO-DFT(ee) approach. Electron-proton correlation could be included by the development of an electron-nucleus correlation functional. Alternatively, explicit electron-proton correlation could be included directly into the NEO self-consistent-field framework with Gaussian-type geminal functions.

The internal partition function of a polyatomic system is often computed by making a series of approximations. First, the Born-Oppenheimer approximation is invoked to treat the motion of the nuclei separately from that of the electrons. Second, rotation and vibration are assumed separable. Third, the rotational partition function is calculated by the rigid rotator approximation. Fourth, the harmonic oscillator approximation is used to calculate the vibrational partition function. These are very convenient approximations because they allow us to write the partition function in an analytical form that depends only on the temperature, equilibrium geometry, and the normal mode frequencies of vibration. The most serious approximation in this sequence is usually the harmonic oscillator approximation. An alternative to the harmonic oscillator approximation is to include the anharmonic effects in the partition function calculation, which is the objective of the present work. Converged vibrational eigenvalue calculations have been successfully carried out for small systems such as H2O and CH4; however, it is very difficult to obtain enough accurate eigenvalues to calculate the partition function for more complex systems.

Converged vibrational levels and converged quantum mechanical vibrational partition function of ethane in the temperature range of 200-600 K were studied. The Born-Oppenheimer approximation is used, and the calculations are carried out on the ground-state electronically adiabatic potential energy surface using a combined valence-bond molecular mechanics (CVBMM) potential energy surface.The calculations are carried out for zero total angular momentum (*J*=0), for which all rotation-vibrational coupling terms in the Hamiltonian are neglected. The vibrational energy levels are computed using the vibrational configuration interaction (VCI) schemethat has been successfully used on other smaller polyatomic systems. The VCI scheme is a variational method in which the matrix elements of the vibrational Hamiltonian are evaluated in a suitable basis set, and the resulting matrix is diagonalized to obtain the eigenvalues. After the vibrational energy levels of ethane are calculated, the partition function is obtained by summing over the Boltzmann factors associated with the energy levels at a given temperature.

12. Calculation of the positron annihilation rate in PsH with the positronic extension of the explicitly correlated nuclear-electronic orbital method, M. V. Pak, A. Chakraborty and S. Hammes-Schiffer, Journal of Physical Chemistry A, submitted (2008).

11. Density matrix formulation of the nuclear-electronic orbital approach with explicit electron-proton correlation, A. Chakraborty and S. Hammes-Schiffer, Journal of Chemical Physics, **129**, 204101 (2008).DOI

10. Development of electron-proton functionals for multicomponent density functional theory, A. Chakraborty, M. V. Pak, and S. Hammes-Schiffer, Physical Review Letters, **101**, 153001 (2008).DOI

9. Inclusion of explicit electron-proton correlation in the nuclear-electronic orbital approach using Gaussian-type geminal functions, A. Chakraborty, M. V. Pak, and S. Hammes-Schiffer, Journal of Chemical Physics **129**, 014101-13 (2008). DOI

8. Density functional theory treatment of electron correlation in the nuclear-electronic orbital approach, M. V. Pak, A. Chakraborty and S. Hammes-Schiffer, Journal of Physical Chemistry A **111**, 4522-4526 (2007).DOI

7. Analysis of nuclear quantum effects on hydrogen bonding, C. Swalina, Q. Wang, A. Chakraborty and S. Hammes-Schiffer, Journal of Physical Chemistry A **111**, 2206-2212 (2007).DOI

6. Explicit dynamical electron-proton correlation in the nuclear-electronic orbital framework, C. Swalina, M. V. Pak, A. Chakraborty and S. Hammes-Schiffer, Journal of Physical Chemistry A **110**, 9983-9987 (2006).DOI

5. Converged vibrational energy levels and quantum mechanical vibrational partition function of ethane, A. Chakraborty and D. G. Truhlar, Journal of Chemical Physics **124**, 184310-6 (2006).DOI

4. Combined valence bond-molecular mechanics potential-energy surface and direct dynamics study of rate constants and kinetic isotope effects for the H+C2H6 reaction, A. Chakraborty, Y. Zhao, H. Lin, and D. G. Truhlar, Journal of Chemical Physics **124**, 044315-14 (2006).DOI

3. Quantum mechanical reaction rate constants by vibrational configuration interaction. The OH + H2 -> H2O + H reaction as a function of temperature, A. Chakraborty and D. G. Truhlar, Proceedings of the National Academy of Sciences U.S.A. **102**, 6744-6749 (2005). (Special Feature issue on Chemical Theory and Computation) DOI

2. Calculation of converged rovibrational energies and partition function for methane using vibrational-rotational configuration interaction, A. Chakraborty, D. G. Truhlar, J. M. Bowman, and S. Carter, Journal of Chemical Physics **121**, 2071-2084 (2004).DOI

1.Photodissociation of LiFH and NaFH van der Waals complexes: A semiclassical trajectory study, A. W. Jasper, M. D. Hack, A. Chakraborty, D. G. Truhlar, and P. Piecuch, Journal of Chemical Physics **115**, 7945-7952 (2001); erratum: **119**, 9321 (2003).DOI