Very accurate, rigorous and fully variational, all-particle, non-Born–Oppenheimer calculations of the vibrational spectrum of the ${\mathrm{H}}_2$ 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 ${\mathrm{H}}_2$ nonrelativistic nonadiabatic ground state energy equal to $\text{−1.164\hspace{0.05em}025\hspace{0.05em}030\hspace{0.05em}0}$ hartree.

This work reviews the state-specific multireference coupled-cluster (CC) approaches which have been developed as approximate methods for performing high-level quantum mechanical calculations on quasidegenerate ground and excited states of atomic and molecular systems. The term “quasidegenerate” refers to a state that cannot be described even in the first approximation by a single-determinant wavefunction (a Slater determinant), but requires two or more determinants for this purpose. The main challenge with applying the coupled-cluster theory to such states is in describing the electron correlation effects in the wavefunctions representing these states in a manner that is size-extensive, yet accurate and simple enough so the method can be routinely applied to small and medium-size molecular systems. We are describing how this can be accomplished within a theory that focuses on only one state of the system in a single CC calculation (the state-specific theory).

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.

Using the core-mass approach, we have generated a vibrational-mass surface for the triatomic ${\mathrm{H}}_{3}{}^{+}$. The coordinate-dependent masses account for the off-resonance nonadiabatic coupling and permit a very accurate determination of the rovibrational states using a single potential energy surface. The new, high-precision measurements of 12 rovibrational transitions in the ${$\nu${}}_{2}$ bending fundamental of ${\mathrm{H}}_{3}{}^{+}$ by Wu et al. [Phys. Rev. A 88, 032507 (2013)] are used to scale this surface empirically and to derive state-dependent vibrational and rotational masses that reproduce the experimental transition energies to ${10}^{$-${}3}\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{$-${}1}$. Rotational term values for $J$\le${}10$ are presented for the two lowest vibrational states and equivalent transitions in D${}_{3}{}^{+}$ considered.

Benchmark calculations of the total and transition energies of the four lowest ${}^{1}S$ states of the beryllium atom are performed. The computational approach is based on variational calculations with finite mass of the nucleus. All-particle explicitly correlated Gaussian (ECG) functions are used to expand the total non-Born-Oppenheimer nonrelativistic wave functions and the ECG exponential parameters are optimized using the standard variational method. The leading relativistic and quantum electrodynamics energy corrections are calculated using the first-order perturbation theory. A comparison of the experimental transition frequencies with the ones calculated in this work shows excellent agreement. The deviations of $0.02–0.09{\mathrm{cm}}^{-1}$ are well within the estimated error limits for the experimental values.