Konrad Patkowski

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.

Intermolecular Interactions via Perturbation Theory: From Diatoms to Biomolecules

This article is devoted to the most recent, i.e. taking place within the last few years, theoretical developments in the field of intermolecular interactions. The most important advancement during this time period was the creation of a new version of symmetry-adapted perturbation theory (SAPT) which is based on the density-functional theory (DFT) description of monomers. This method, which will be described in Sect. 5.2, allows SAPT calculations to be performed for much larger molecules than before. In fact, many molecules of biological importance can now be investigated. Another important theoretical advancement was made in understanding the convergence properties of SAPT. It has been possible to investigate such properties on a realistic example of a Li atom interaction with an H atom. This is the simplest system for which the coupling of physical states to the unphysical, Pauli forbidden continuum causes the divergence of the conventional polarization expansion and of several variants of SAPT. This development will be described in some detail in Sects. 2–4, where, in addition to a review of published work, we shall present several original results on this subject. In an unrelated way, one of the most interesting recent applications of ab initio methods concerns the helium dimer and allows first-principle predictions for helium that are in many cases more accurate than experimental results. Therefore, theoretical input can be used to create new measurement standards. This broad range of systems that were the subject of theoretical investigations in recent years made us choose the title of the current review. With a few exceptions, the investigations of individual systems discussed here utilized SAPT. The calculations for helium are described in Sect. 6, recent wave-function based applications in Sect. 7, the performance of SAPT(DFT) on model systems in Sect. 8, and applications of SAPT(DFT) in Sect. 9. Section 10 summarizes work on biosystems.

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.

Argon pair potential at basis set and excitation limits

A new ab initio interaction potential for the electronic ground state of argon dimer has been developed. The potential is a sum of contributions corresponding to various levels of the coupled-cluster theory up to the full coupled-cluster method with single, double, triple, and quadruple excitations. All contributions have been calculated in larger basis sets than used in the development of previous Ar(2) potentials, including basis sets optimized by us up to the septuple(sextuple)-zeta level for the frozen-core (all-electron) energy. The diffuse augmentation functions have also been optimized. The effects of the frozen-core approximation and the relativistic effects have been computed at the CCSD(T) level. We show that some basis sets used in literature to compute these corrections may give qualitatively wrong results. Our calculations also show that the effects of high excitations do not necessarily converge significantly faster (in absolute values) in basis set size than the effects of lower excitations, as often assumed in literature. Extrapolations to the complete basis set limits have been used for most terms. Careful examination of the basis set convergence patterns enabled us to determine uncertainties of the ab initio potential. The interaction energy at the near-minimum interatomic distance of 3.75 Å amounts to -99.291±0.32 cm(-1). The ab initio energies were fitted to an analytic potential which predicts a minimum at 3.762 Å with a depth of 99.351 cm(-1). Comparisons with literature potentials indicate that the present one is the most accurate representation of the argon-argon interaction to date.

Third-order interactions in symmetry-adapted perturbation theory

We present an extension of many-body symmetry-adapted perturbation theory (SAPT) by including all third-order polarization and exchange contributions obtained with the neglect of intramonomer correlation effects. The third-order polarization energy, which naturally decomposes into the induction, dispersion, and mixed, induction-dispersion components, is significantly quenched at short range by electron exchange effects. We propose a decomposition of the total third-order exchange energy into the exchange-induction, exchange-dispersion, and exchange-induction-dispersion contributions which provide the quenching for the corresponding individual polarization contributions. All components of the third-order energy have been expressed in terms of molecular integrals and orbital energies. The obtained formulas, valid for both dimer- and monomer-centered basis sets, have been implemented within the general closed-shell many-electron SAPT program. Test calculations for several small dimers have been performed and their results are presented. For dispersion-bound dimers, the inclusion of the third-order effects eliminates the need for a hybrid SAPT approach, involving supermolecular Hartree-Fock calculations. For dimers consisting of strongly polar monomers, the hybrid approach remains more accurate. It is shown that, due to the extent of the quenching, the third-order polarization effects should be included only together with their exchange counterparts. Furthermore, the latter have to be calculated exactly, rather than estimated by scaling the second-order values.

Accurate ab initio potential for argon dimer including highly repulsive region

An ab initio potential has been developed for the argon dimer. This potential is based on coupled-cluster calculations with single, double, and non-iterated triple excitations in a sequence of very large basis sets, up to augmented sextuple-zeta quality and containing bond functions, followed by extrapolations to the complete basis set limit. The calculations included intermolecular distances as small as 0.25 Å, where the interaction potential is of the order of 4 keV. The computed points were fitted by an analytic expression. The new potential has the minimum at 3.767 Å with a depth of 99.27 cm−1 respectively, very close to experimental values of 3.761 ± 0.003 Å and 99.2 ± 1.0 cm−1 respectively. The potential was used to compute the spectra of the argon dimer and the virial coefficients. The latter calculations suggest a possible revision of the established experimental reference results. From the agreement achieved with experimental values and from comparisons of the fit with available piecewise information on specific regions of the argon–argon interaction, one can assume that the present work provides the best overall representation of the true argon–argon potential to date.

On the Elusive Twelfth Vibrational State of Beryllium Dimer

One More Vibration The beryllium dimer has puzzled chemists for the better part of a century because its bond, though weak by molecular standards, is much stronger than standard bonding frameworks would predict. Recent spectroscopic measurements have characterized the molecule in great detail, but leave open the question of whether a high-energy vibrational state might lie just below the threshold for scission of the bond. Patkowski et al. (p. 1382) have now derived a potential energy function, based on high-level theoretical calculations and, by slightly modifying it through fitting to the experimental data, obtained strong evidence for the presence of the upper vibrational state. Taken together with the measured spectra, these results offer a well-grounded basis for understanding the unusual molecule. Theoretical calculations support a previous spectroscopic assignment of the highest vibrational level of the beryllium dimer. The beryllium dimer has puzzled chemists for roughly 80 years on account of its unexpectedly strong bonding interaction between two nominally closed-shell atoms. Recent spectroscopic measurements characterized the molecule’s ground electronic state with sufficient resolution to distinguish 11 vibrational levels; the possibility that a twelfth level lay just below the dissociation threshold remained unresolved. Here we present a potential function, based on ab initio calculations at the full configuration interaction level, that definitively supports the presence of this twelfth vibrational state. “Morphed” versions of this potential, fitted to experimental data, closely reproduce the observed spectra to within 0.1 cm−1, bolstering the strength of the assignment.

Revised Damping Parameters for the D3 Dispersion Correction to Density Functional Theory

Since the original fitting of Grimmes DFT-D3 damping parameters, the number and quality of benchmark interaction energies has increased significantly. Here, conventional benchmark sets, which focus on minimum-orientation radial curves at the expense of angular diversity, are augmented by new databases such as side chain-side chain interactions (SSI), which are composed of interactions gleaned from crystal data and contain no such minima-focused bias. Moreover, some existing databases such as S22×5 are extended to shorter intermolecular separations. This improved DFT-D3 training set provides a balanced description of distances, covers the entire range of interaction types, and at 1526 data points is far larger than the original training set of 130. The results are validated against a new collection of 6773 data points and demonstrate that the effect of refitting the damping parameters ranges from no change in accuracy (LC-ωPBE-D3) to an almost 2-fold decrease in average error (PBE-D3).

On the accuracy of explicitly correlated coupled-cluster interaction energies--have orbital results been beaten yet?

The basis set convergence of weak interaction energies for dimers of noble gases helium through krypton is studied for six variants of the explicitly correlated, frozen geminal coupled-cluster singles, doubles, and noniterative triples [CCSD(T)-F12] approach: the CCSD(T)-F12a, CCSD(T)-F12b, and CCSD(T)(F12*) methods with scaled and unscaled triples. These dimers were chosen because CCSD(T) complete-basis-set (CBS) limit benchmarks are available for them to a particularly high precision. The dependence of interaction energies on the auxiliary basis sets has been investigated and it was found that the default resolution-of-identity sets cc-pVXZ/JKFIT are far from adequate in this case. Overall, employing the explicitly correlated approach clearly speeds up the basis set convergence of CCSD(T) interaction energies, however, quite surprisingly, the improvement is not as large as the one achieved by a simple addition of bond functions to the orbital basis set. Bond functions substantially improve the CCSD(T)-F12 interaction energies as well. For small and moderate bases with bond functions, the accuracy delivered by the CCSD(T)-F12 approach cannot be matched by conventional CCSD(T). However, the latter method in the largest available bases still delivers the CBS limit to a better precision than CCSD(T)-F12 in the largest bases available for that approach. Our calculations suggest that the primary reason for the limited accuracy of the large-basis CCSD(T)-F12 treatment are the approximations made at the CCSD-F12 level and the non-explicitly correlated treatment of triples. In contrast, the explicitly correlated second-order Mo̸ller-Plesset perturbation theory (MP2-F12) approach is able to pinpoint the complete-basis-set limit MP2 interaction energies of rare gas dimers to a better precision than conventional MP2. Finally, we report and analyze an unexpected failure of the CCSD(T)-F12 method to deliver the core-core and core-valence correlation corrections to interaction energies consistently and accurately.

Unified treatment of chemical and van der Waals forces via symmetry-adapted perturbation expansion

We propose a symmetry-adapted perturbation theory (SAPT) expansion of the intermolecular interaction energy which in a finite order provides the correct values of the constants determining the asymptotics of the interaction energy (the van der Waals constants) and is convergent when the energy of the interacting system is submerged in the continuum of Pauli-forbidden states-the situation common when at least one of the monomers has more than two electrons. These desirable features are achieved by splitting the intermolecular electron-nucleus attraction terms of the Hamiltonian into regular (long-range) and singular (short-range) parts. In the perturbation theory development, the regular part is treated as in the conventional polarization theory, which guarantees the correct asymptotics of the interaction energy, while the singular part is weakened sufficiently by an application of permutational symmetry projectors so that a convergent perturbation series is obtained. The convergence is demonstrated numerically, for both the chemical and van der Waals minima, by performing high-order calculations of the interaction energy of the ground-state lithium and hydrogen atoms-the simplest system for which the physical ground state is submerged in the Pauli-forbidden continuum. The obtained expansion enables a systematic extension of SAPT calculations beyond second order with respect to the intermolecular interaction operator.