The method of dispersion correction as an add-on to standard Kohn-Sham density functional theory (DFT-D) has been refined regarding higher accuracy, broader range of applicability, and less empiricism. The main new ingredients are atom-pairwise specific dispersion coefficients and cutoff radii that are both computed from first principles. The coefficients for new eighth-order dispersion terms are computed using established recursion relations. System (geometry) dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination numbers (CN). They are used to interpolate between dispersion coefficients of atoms in different chemical environments. The method only requires adjustment of two global parameters for each density functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of atomic forces. Three-body nonadditivity terms are considered. The method has been assessed on standard benchmark sets for inter- and intramolecular noncovalent interactions with a particular emphasis on a consistent description of light and heavy element systems. The mean absolute deviations for the S22 benchmark set of noncovalent interactions for 11 standard density functionals decrease by 15%-40% compared to the previous (already accurate) DFT-D version. Spectacular improvements are found for a tripeptide-folding model and all tested metallic systems. The rectification of the long-range behavior and the use of more accurate C(6) coefficients also lead to a much better description of large (infinite) systems as shown for graphene sheets and the adsorption of benzene on an Ag(111) surface. For graphene it is found that the inclusion of three-body terms substantially (by about 10%) weakens the interlayer binding. We propose the revised DFT-D method as a general tool for the computation of the dispersion energy in molecules and solids of any kind with DFT and related (low-cost) electronic structure methods for large systems.

http://dns2.asia.edu.tw/~ysho/YSHO-English/1000%20WC/PDF/J%20Che%20Phy132,%20154104.pdf

A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu

A new density functional (DF) of the generalized gradient approximation (GGA) type for general chemistry applications termed B97-D is proposed. It is based on Becke's power-series ansatz from 1997 and is explicitly parameterized by including damped atom-pairwise dispersion corrections of the form C(6) x R(-6). A general computational scheme for the parameters used in this correction has been established and parameters for elements up to xenon and a scaling factor for the dispersion part for several common density functionals (BLYP, PBE, TPSS, B3LYP) are reported. The new functional is tested in comparison with other GGAs and the B3LYP hybrid functional on standard thermochemical benchmark sets, for 40 noncovalently bound complexes, including large stacked aromatic molecules and group II element clusters, and for the computation of molecular geometries. Further cross-validation tests were performed for organometallic reactions and other difficult problems for standard functionals. In summary, it is found that B97-D belongs to one of the most accurate general purpose GGAs, reaching, for example for the G97/2 set of heat of formations, a mean absolute deviation of only 3.8 kcal mol(-1). The performance for noncovalently bound systems including many pure van der Waals complexes is exceptionally good, reaching on the average CCSD(T) accuracy. The basic strategy in the development to restrict the density functional description to shorter electron correlation lengths scales and to describe situations with medium to large interatomic distances by damped C(6) x R(-6) terms seems to be very successful, as demonstrated for some notoriously difficult reactions. As an example, for the isomerization of larger branched to linear alkanes, B97-D is the only DF available that yields the right sign for the energy difference. From a practical point of view, the new functional seems to be quite robust and it is thus suggested as an efficient and accurate quantum chemical method for large systems where dispersion forces are of general importance.

Semiempirical GGA-type density functional constructed with a long-range dispersion correction.

It is shown by an extensive benchmark on molecular energy data that the mathematical form of the damping function in DFT-D methods has only a minor impact on the quality of the results. For 12 different functionals, a standard "zero-damping" formula and rational damping to finite values for small interatomic distances according to Becke and Johnson (BJ-damping) has been tested. The same (DFT-D3) scheme for the computation of the dispersion coefficients is used. The BJ-damping requires one fit parameter more for each functional (three instead of two) but has the advantage of avoiding repulsive interatomic forces at shorter distances. With BJ-damping better results for nonbonded distances and more clear effects of intramolecular dispersion in four representative molecular structures are found. For the noncovalently-bonded structures in the S22 set, both schemes lead to very similar intermolecular distances. For noncovalent interaction energies BJ-damping performs slightly better but both variants can be recommended in general. The exception to this is Hartree-Fock that can be recommended only in the BJ-variant and which is then close to the accuracy of corrected GGAs for non-covalent interactions. According to the thermodynamic benchmarks BJ-damping is more accurate especially for medium-range electron correlation problems and only small and practically insignificant double-counting effects are observed. It seems to provide a physically correct short-range behavior of correlation/dispersion even with unmodified standard functionals. In any case, the differences between the two methods are much smaller than the overall dispersion effect and often also smaller than the influence of the underlying density functional.

Effect of the damping function in dispersion corrected density functional theory

We present a parameter-free method for an accurate determination of long-range van der Waals interactions from mean-field electronic structure calculations. Our method relies on the summation of interatomic C6 coefficients, derived from the electron density of a molecule or solid and accurate reference data for the free atoms. The mean absolute error in the C6 coefficients is 5.5% when compared to accurate experimental values for 1225 intermolecular pairs, irrespective of the employed exchangecorrelation functional. We show that the effective atomic C6 coefficients depend strongly on the bonding environment of an atom in a molecule. Finally, we analyze the van der Waals radii and the damping function in the C6R � 6 correction method for density-functional theory calculations.

/pdf/accurate-molecular-van-der-waals-interactions-from-ground-4sgen01nnb.pdf

Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data

Feshbach resonances are the essential tool to control the interaction between atoms in ultracold quantum gases. They have found numerous experimental applications, opening up the way to important breakthroughs. This review broadly covers the phenomenon of Feshbach resonances in ultracold gases and their main applications. This includes the theoretical background and models for the description of Feshbach resonances, the experimental methods to find and characterize the resonances, a discussion of the main properties of resonances in various atomic species and mixed atomic species systems, and an overview of key experiments with atomic Bose-Einstein condensates, degenerate Fermi gases, and ultracold molecules.

Feshbach resonances in ultracold gases

An atom–diatomic molecule collision is simulated by considering an idealized potential energy surface which is a two‐dimensional duct with an adjustable potential in the corner region. This potential is symmetric with respect to an interchange of the x and y Cartesian coordinates. Explicit expressions for the wavefunctions are obtained which make use of this symmetry. Also analytical relations are obtained between the transmission and reflection coefficients and their phases. Quantum mechanical streamlines are computer graphed for a large number of energies and positive, negative, and zero values of the potential energy in the corner region. Special attention is given to the quantized vortices (surrounding wavefunction nodes) which appear in the streamlines. When only one energy channel is open, the streamlines are symmetric and the flux is antisymmetric. This occurs because the wavefunction is a linear combination (with complex coefficients) of two real solutions, one symmetric, the other antisymmetric. ...

Quantum mechanical streamlines. III. Idealized reactive atom–diatomic molecule collision

Potentials for some rare gas and alkali-helium systems calculated from the surface integral method

A combining rule calculation of the van der Waals potentials of the rare-gas hydrides

A quantum-mechanical study is made of reactive scattering in the (H, ${\mathrm{H}}_{2}$) system. The problem is formulated in terms of a form of the distorted-wave Born approximation (DWBA) suitable for collisions in which all particles have finite mass. For certain incident energies, differential and total cross sections, as well as other attributes of the reactive collisions, (e.g., the reaction configuration) are determined. Two limiting models in the DWBA formulation are compared; in one, the molecule is unperturbed by the incoming atom, and in the other, the molecule adiabatically follows the incoming atom. For thermal incident energies and the semiempirical interaction potential employed, the adiabatic model seems to be more appropriate. Since the DWBA method is too complicated for a general study of the (H, ${\mathrm{H}}_{2}$) reaction, a much simpler approximation method, the "linear model," is developed. This model is very different in concept from treatments in which the three atoms are constrained to move on a line throughout the collision. The present model includes the full three-dimensional aspect of the collision, and it is only the evaluation of the transition matrix element itself that is simplified. It is found that the linear model, when appropriately normalized, gives results in good agreement with that of the DWBA method. By application of this model, the energy dependence, rotational-state dependence, and other properties of the total and differential reaction cross sections are determined. These results of the quantum-mechanical treatment are compared with the classical calculation for the same potential surface. The most important result is that, in agreement with the classical treatment, the differential cross sections are strongly backward peaked at low energies and shift toward the forward direction as the energy increases. Finally, the implications of the present calculations for a theory of chemical kinetics are discussed.

Quantum Theory of (H, H 2 ) Scattering: Approximate Treatments of Reactive Scattering

A simple theory for the van der Waals potential in the region of the well minimum, which previously has been successfully applied to the prediction of the isotropic atom–atom [J. Chem. Phys. 66, 1496 (1977)] and the anisotropic atom–diatom potentials [J. Chem. Phys. 68, 5501 (1978); 74, 1148 (1981)], has been extended to calculate the full potential hypersurface including the H2 bond distance dependence for He–H2 and Ne–H2. By taking advantage of the known potential parameters in the united atom limits He–He and Ne–He, respectively, the potential hypersurface is predicted over a wide range of bond distances. The model is modified to also provide a good estimate of the true potential in the repulsive region (V≃1 eV). The results for He–H2 are compared with a recently calculated CI type hypersurface [Meyer, Hariharan, and Kutzelnigg, J. Chem. Phys. 73, 1880 (1980)] and found to be in good agreement in the region of intermediate distances of the center of masses (R≈3.0 A), and for H2 bond distances r in the ...

A simple theoretical model for the van der Waals potential at intermediate distances. IV. The bond distance dependence of the potential hypersurfaces for He–H2 and Ne–H2 also for the repulsive region

K. T. Tang

Papers

Quantum mechanical streamlines. III. Idealized reactive atom–diatomic molecule collision

Potentials for some rare gas and alkali-helium systems calculated from the surface integral method

A combining rule calculation of the van der Waals potentials of the rare-gas hydrides

Quantum Theory of (H, H 2 ) Scattering: Approximate Treatments of Reactive Scattering

A simple theoretical model for the van der Waals potential at intermediate distances. IV. The bond distance dependence of the potential hypersurfaces for He–H2 and Ne–H2 also for the repulsive region