Sergiy Bubin

Variational calculations of excited states with zero total angular momentum (vibrational spectrum) of H2 without use of the Born–Oppenheimer approximation

Very accurate, rigorous and fully variational, all-particle, non-Born–Oppenheimer calculations of the vibrational spectrum of the H2 molecule have been performed. Very high accuracy has been achieved by expanding the wave functions in terms of explicitly correlated Gaussian functions with preexponential powers of the internuclear distance. An indicator of the high accuracy of the calculations is the new upper bound for the H2 nonrelativistic nonadiabatic ground state energy equal to −1.164 025 030 0 hartree.

Born-Oppenheimer and non-Born-Oppenheimer, atomic and molecular calculations with explicitly correlated Gaussians.

This paper is part of the 2012 Quantum Chemistry thematic issue. Sergiy Bubin,*,† Michele Pavanello,*,‡ Wei-Cheng Tung, Keeper L. Sharkey, and Ludwik Adamowicz* †Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, United States ‡Department of Chemistry, Rutgers University Newark, Newark, New Jersey 07102, United States Department of Chemistry and Biochemistry and Department of Physics, University of Arizona, Tucson, Arizona 85721, United States

Non-Born–Oppenheimer calculations of atoms and molecules

We review a recent development in high-accuracy non-Born–Oppenheimer calculations of atomic and molecular systems in a basis of explicitly correlated Gaussian functions. Much of the recent progress in this area is due to the derivation and implementation of analytical gradients of the energy functional with respect to variational linear and non-linear parameters of the basis functions. This method has been used to obtain atomic and molecular ground and excited state energies and the corresponding wave functions with accuracy that exceeds previous calculations. Further, we have performed the first calculations of non-linear electrical properties of molecules without the Born–Oppenheimer approximation for systems with more than one electron. The results for the dipole moments of such systems as HD and LiH agree very well with experiment. After reviewing our non-Born–Oppenheimer results we will discuss ways this method can be extended to deal with larger molecular systems with and without an external perturbation.

Non-Born–Oppenheimer study of positronic molecular systems: e+LiH

Very accurate non-Born-Oppenheimer variational calculations of the ground state of e(+)LiH have been performed using explicitly correlated Gaussian functions with preexponential factors dependent on powers of the internuclear distance. In order to determine the positron detachment energy of e(+)LiH and the dissociation energy corresponding to the e(+)LiH fragmentation into HPs and Li(+) we also calculated non-BO energies of HPs, LiH, and Li(+). For all the systems the calculations provided the lowest ever-reported variational upper-bounds to the ground state energies. Annihilation rates of HPs and e(+)LiH were also computed. The dissociation energy of e(+)LiH into HPs and Li(+) was determined to be 0.036 548 hartree.

Nonrelativistic molecular quantum mechanics without approximations: electron affinities of LiH and LiD.

We took the complete nonrelativistic Hamiltonians for the LiH and LiH- systems, as well as their deuterated isotopomers, we separated the kinetic energy of the center of mass motion from the Hamiltonians, and with the use of the variational method we optimized the ground-state nonadiabatic wave functions for the systems expanding them in terms of n-particle explicitly correlated Gaussian functions. With 3600 functions in the expansions we obtained the lowest ever ground-state energies of LiH, LiD, LiH-, and LiD- and these values were used to determine LiH and LiD electrons affinities (EAs) yielding 0.330 30 and 0.327 13 eV, respectively. The present are the first high-accuracy ab initio quantum mechanical calculations of the LiH and LiD EAs that do not assume the Born-Oppenheimer approximation. The obtained EAs fall within the uncertainty brackets of the experimental results.

Non-Born–Oppenheimer calculations of the BH molecule

Variational calculations employing explicitly correlated Gaussian basis functions have been performed for the ground state of the boron monohydride molecule (BH) and for the boron atom (B). Up to 2000 Gaussians were used for each system. The calculations did not assume the Born-Oppenheimer (BO) approximation. In the optimization of the wave function, we employed the analytical energy gradient with respect to the Gaussian exponential parameters. In addition to the total nonrelativistic energies, we computed scalar relativistic corrections (mass-velocity and Darwin). With those added to the total energies, we estimated the dissociation energy of BH. The non-BO wave functions were also used to compute some expectation values involving operators dependent on the interparticle distances.

Charge asymmetry in HD

Expanding the wave functions of the ground and excited states of HD(+) (or pde) in terms of spherically symmetric explicitly correlated Gaussian functions with preexponential multipliers consisting of powers of the internuclear distance, and using the variational method, we performed very accurate nonadiabatic calculations of all bound states of this system corresponding to the zero total angular momentum quantum number (vibrational states; v=0-22). The total and the transition energies obtained agree with the best available calculations. For each state we computed the expectation values of the d-p, d-e, and p-e interparticle distances. This is the first time these quantities were computed for HD(+) using rigorous nonadiabatic wave functions. While up to the v=20 state some asymmetry is showing in the d-e and p-e distances, for v=21 and v=22 we observe a complete breakdown of the Born-Oppenheimer approximation and localization of the electron almost entirely at the deuteron.

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.

Non-Born-Oppenheimer calculations of the pure vibrational spectrum of HeH +

Very accurate calculations of the pure vibrational spectrum of the HeH(+) ion are reported. The method used does not assume the Born-Oppenheimer approximation, and the motion of both the electrons and the nuclei are treated on equal footing. In such an approach the vibrational motion cannot be decoupled from the motion of electrons, and thus the pure vibrational states are calculated as the states of the system with zero total angular momentum. The wave functions of the states are expanded in terms of explicitly correlated Gaussian basis functions multipled by even powers of the internuclear distance. The calculations yielded twelve bound states and corresponding eleven transition energies. Those are compared with the pure vibrational transition energies extracted from the experimental rovibrational spectrum.

Orbit-orbit relativistic corrections to the pure vibrational non-Born-Oppenheimer energies of H2

We report the derivation of the orbit-orbit relativistic correction for calculating pure vibrational states of diatomic molecular systems with sigma electrons within the framework that does not assume the Born-Oppenheimer (BO) approximation. The correction is calculated as the expectation value of the orbit-orbit interaction operator with the non-BO wave function expressed in terms of explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. With that we can now calculate the complete relativistic correction of the order of alpha(2) (where alpha=1/c). The new algorithm is applied to determine the full set of the rotationless vibrational levels and the corresponding transition frequencies of the H(2) molecule. The results are compared with the previous calculations, as well as with the frequencies obtained from the experimental spectra. The comparison shows the need to include corrections higher than second order in alpha to further improve the agreement between the theory and the experiment.