Wojciech Cencek

Sub-microhartree accuracy potential energy surface for H3+ including adiabatic and relativistic effects. I. Calculation of the potential points

Sixty-nine points of the Born–Oppenheimer (BO) potential energy surface (PES) for the ground state of H3+ have been computed using explicitly correlated Gaussian wave functions with optimized nonlinear parameters. The calculated points have an absolute error of about 0.02 cm−1 (0.1 microhartree), i.e., they are by at least one order of magnitude more accurate than ever reported. Similarly accurate adiabatic and relativistic corrections have also been evaluated by means of the Born–Handy formula and by direct perturbation theory (DPT), respectively.

Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium

The adiabatic, relativistic, and quantum electrodynamics (QED) contributions to the pair potential of helium were computed, fitted separately, and applied, together with the nonrelativistic Born-Oppenheimer (BO) potential, in calculations of thermophysical properties of helium and of the properties of the helium dimer. An analysis of the convergence patterns of the calculations with increasing basis set sizes allowed us to estimate the uncertainties of the total interaction energy to be below 50 ppm for interatomic separations R smaller than 4 bohrs and for the distance R = 5.6 bohrs. For other separations, the relative uncertainties are up to an order of magnitude larger (and obviously still larger near R = 4.8 bohrs where the potential crosses zero) and are dominated by the uncertainties of the nonrelativistic BO component. These estimates also include the contributions from the neglected relativistic and QED terms proportional to the fourth and higher powers of the fine-structure constant α. To obtain such high accuracy, it was necessary to employ explicitly correlated Gaussian expansions containing up to 2400 terms for smaller R (all R in the case of a QED component) and optimized orbital bases up to the cardinal number X = 7 for larger R. Near-exact asymptotic constants were used to describe the large-R behavior of all components. The fitted potential, exhibiting the minimum of -10.996 ± 0.004 K at R = 5.608 0 ± 0.000 1 bohr, was used to determine properties of the very weakly bound (4)He(2) dimer and thermophysical properties of gaseous helium. It is shown that the Casimir-Polder retardation effect, increasing the dimer size by about 2 Å relative to the nonrelativistic BO value, is almost completely accounted for by the inclusion of the Breit-interaction and the Araki-Sucher contributions to the potential, of the order α(2) and α(3), respectively. The remaining retardation effect, of the order of α(4) and higher, is practically negligible for the bound state, but is important for the thermophysical properties of helium. Such properties computed from our potential have uncertainties that are generally significantly smaller (sometimes by nearly two orders of magnitude) than those of the most accurate measurements and can be used to establish new metrology standards based on properties of low-density helium.

Many‐electron explicitly correlated Gaussian functions. I. General theory and test results

The Gaussian functions containing correlation factors of the type exp(−βrij2), employed so far in variational calculations of two‐electron atoms and molecules are generalized for many‐electron systems. Explicit formulas for necessary one‐, two‐, three‐, and four‐electron integrals over s‐type correlated Gaussians are given. Preliminary computations for the H3 and LiH molecules yield significantly lower energy values than all previously published variational results.

Pair potential for helium from symmetry-adapted perturbation theory calculations and from supermolecular data

Symmetry-adapted perturbation theory (SAPT) was applied to the helium dimer for interatomic separations R from 3 to 12 bohrs. The first-order interaction energy and the bulk of the second-order contribution were obtained using Gaussian geminal basis sets and are converged to about 0.1 mK near the minimum and for larger R. The remaining second-order contributions available in the SAPT suite of codes were computed using very large orbital basis sets, up to septuple-zeta quality, augmented by diffuse and midbond functions. The accuracy reached at this level was better than 1 mK in the same region. All the remaining components of the interaction energy were computed using the full configuration interaction method in bases up to sextuple-zeta quality. The latter components, although contributing only 1% near the minimum, have the largest uncertainty of about 10 mK in this region. The total interaction energy at R=5.6 bohrs is -11.000+/-0.011 K. For R< or =6.5 bohrs, the supermolecular (SM) interaction energies computed by us recently turned out to be slightly more accurate. Therefore, we have combined the SM results for R< or =6.5 bohrs with the SAPT results from 7.0 to 12 bohrs to fit analytic functions for the potential and for its error bars. The potential fit uses the best available van der Waals constants C(6) through C(16), including C(11), C(13), and C(15), and is believed to be the best current representation of the Born-Oppenheimer (BO) potential for helium. Using these fits, we found that the BO potential for the helium dimer exhibits the well depth D(e)=11.006+/-0.004 K, the equilibrium distance R(e)=5.608+/-0.012 bohrs, and supports one bound state for (4)He(2) with the dissociation energy D(0)=1.73+/-0.04 mK, and the average interatomic separation R=45.6+/-0.5 A.

Potential energy surface for interactions between two hydrogen molecules

Nonrelativistic clamped-nuclei energies of interaction between two ground-state hydrogen molecules with intramolecular distances fixed at their average value in the lowest rovibrational state have been computed. The calculations applied the supermolecular coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)] and very large orbital basis sets-up to augmented quintuple zeta size supplemented with bond functions. The same basis sets were used in symmetry-adapted perturbation theory calculations performed mainly for larger separations to provide an independent check of the supermolecular approach. The contributions beyond CCSD(T) were computed using the full configuration interaction method and basis sets up to augmented triple zeta plus midbond size. All the calculations were followed by extrapolations to complete basis set limits. For two representative points, calculations were also performed using basis sets with the cardinal number increased by one or two. For the same two points, we have also solved the Schrodinger equation directly using four-electron explicitly correlated Gaussian (ECG) functions. These additional calculations allowed us to estimate the uncertainty in the interaction energies used to fit the potential to be about 0.15 K or 0.3% at the minimum of the potential well. This accuracy is about an order of magnitude better than that achieved by earlier potentials for this system. For a near-minimum T-shaped configuration with the center-of-mass distance R=6.4 bohrs, the ECG calculations give the interaction energy of -56.91+/-0.06 K, whereas the orbital calculations in the basis set used for all the points give -56.96+/-0.16 K. The computed points were fitted by an analytic four-dimensional potential function. The uncertainties in the fit relative to the ab initio energies are almost always smaller than the estimated uncertainty in the latter energies. The global minimum of the fit is -57.12 K for the T-shaped configuration at R=6.34 bohrs. The fit was applied to compute the second virial coefficient using a path-integral Monte Carlo approach. The achieved agreement with experiment is substantially better than in any previous work.

Sub-microhartree accuracy potential energy surface for H3+ including adiabatic and relativistic effects. II. Rovibrational analysis for H3+ and D3+

The 69 potential energy points of H3+ computed by Cencek et al. [J. Chem. Phys., 108, 2831 (1998), preceding paper] have been fitted to an analytical potential energy surface (PES). Rovibrational frequencies have been derived for the symmetric H3+ and D3+ isotopomers. A comparison with experiment shows residual discrepancies of a few tenths of cm−1 which can be ascribed mainly to nonadiabatic effects.

Benchmark calculations for two-electron systems using explicitly correlated Gaussian functions

Abstract Explicitly correlated Gaussian functions and nonlinear optimization techniques have been used to calculate Born-Oppenheimer energies of the ground states of H 3 + and HeH + ions and several excited states of the hydrogen molecule at equilibrium nuclear configurations. In all the cases the results are more accurate than any previously reported.

Accurate relativistic energies of one‐ and two‐electron systems using Gaussian wave functions

Gaussian wave functions with optimized nonlinear variational parameters are applied to evaluate clamped‐nucleus nonrelativistic energies as well as the lowest order relativistic corrections for the ground states of the hydrogen atom, hydrogen molecular ion H+2, helium atom, and hydrogen molecule. The two‐electron functions used depend explicitly on the interelectronic distance r12 but do not describe the cusp properly. Despite this, for H+2 and H2 the results are more accurate than ever reported, and for He they are inferior only to the best calculations employing Hylleraas‐type expansions. It is demonstrated that, contrary to a common opinion, Gaussian wave functions are very well suited for high‐accuracy relativistic computations even in the Breit–Pauli approximation, provided that the nonlinear parameters are optimized with respect to the nonrelativistic energy.

The equivalence of explicitly correlated Slater and Gaussian functions in variational quantum chemistry computations: The ground state of H2

Abstract It is demonstrated that variational calculations based on explicitly correlated Gaussian functions for the hydrogen molecule in its ground state lead to energies of the same level of accuracy as those based on the Kolos-Wolniewicz functions. The energies obtained are the lowest reported so far, in contrast to the well-known bad asymptotic properties of Gaussian functions.

Benchmark calculations for He2+ and LiH molecules using explicitly correlated Gaussian functions

Abstract Explicitly correlated Gaussian (ECG) functions with carefully optimized non-linear parameters are used to calculate the electronic energies of He2+ and LiH at their equilibrium internuclear distances. The obtained variational upper bounds (−4.99464392 and −8.070538 hartree, respectively) are the lowest reported to date. By extrapolating results obtained with various expansion lengths, the estimations of the Born–Oppenheimer limits are made.