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

Starting from our earlier model [J. Chem. Phys. 66, 1496 (1977)] a simple expression is derived for the radial dependent damping functions for the individual dispersion coefficients C2n for arbitrary even orders 2n. The damping functions are only a function of the Born–Mayer range parameter b and thus can be applied to all systems for which this is known or can be estimated. For H(1S)–H(1S) the results are in almost perfect agreement with the very accurate recent ab initio damping functions of Koide, Meath, and Allnatt. Comparisons with less accurate previous calculations for other systems also show a satisfactory agreement. By adding a Born–Mayer repulsive term [A exp(−bR)] to the damped dispersion potential, a simple universal expression is obtained for the well region of the atom–atom van der Waals potential with only five essential parameters A, b, C6, C8, and C10. The model has been tested for the following representative systems: H2 3Σ, He2, and Ar2 as well as NaK 3Σ and LiHg, which include four che...

An improved simple model for the van der Waals potential based on universal damping functions for the dispersion coefficients

Upper and lower bounds of the multipole van der Waals coefficients C6, C8, and C10 for hydrogen, noble gas, and alkali atoms are established. Nonadditive three‐body coefficients involving dipole, quadrupole, and octupole interactions are also determined. For the dipole polarizabilities a three‐term, two‐point Pade approximant is used to obtain the upper bound and a two‐term Pade approximant is used to obtain the lower bound. For the quadrupole and octupole polarizabilities a one‐term approximation of the dynamic polarizability is used, except for the helium quadrupole polarizability, where extended approximations are possible.

Upper and lower bounds of two‐ and three‐body dipole, quadrupole, and octupole van der Waals coefficients for hydrogen, noble gas, and alkali atom interactions

The interatomic van der Waals potentials for all the possible 21 homogeneous and heterogeneous pairs of rare gas atoms including radon are determined using the Tang–Toennies potential model and a set of previously derived combining rules. The three dispersion coefficients and the two Born–Mayer parameters needed for calculating the potential curves are listed.

The van der Waals potentials between all the rare gas atoms from He to Rn

A simple analytical expression for the entire van der Waals potential curve of the helium dimer is derived from perturbation theory. The repulsive part is essentially from Duman and Smirnov [Opt. Spectrosc. 29, 229 (1970)] who assume the exchange of only one pair of electrons at any one time. The potential depends only on the known dispersion coefficients and the amplitude of the asymptotic wave function and the ionization energy of the atoms. Without any adjustable parameters the potential is in excellent agreement with the latest experimental and theoretical results and predicts the existence of the He dimer. The same method can be applied to heavier rare gases.

Accurate analytical He-He van der Waals potential based on perturbation theory.

The Tang–Toennies model [J. Chem. Phys. 80, 3725 (1984)] has been modified to predict the potentials for ion–atom systems. First order SCF energies are used to describe the repulsive potential. The long range second order induction and dispersion potential terms up to R−10 are either taken from ab initio calculations or estimated and each term is appropriately damped. The potentials for Li+, Na+, K+, F−, and Cl− interacting with He, Ne, and Ar are found to agree well with both theoretical and experimental data within the expected errors. For comparison with the model new ab initio calculations have been performed for Na+–Ar and the results are in excellent agreement with the model predictions (<10%).

K. T. Tang

Papers

An improved simple model for the van der Waals potential based on universal damping functions for the dispersion coefficients

Upper and lower bounds of two‐ and three‐body dipole, quadrupole, and octupole van der Waals coefficients for hydrogen, noble gas, and alkali atom interactions

The van der Waals potentials between all the rare gas atoms from He to Rn

Accurate analytical He-He van der Waals potential based on perturbation theory.

Interaction potentials for alkali ion–rare gas and halogen ion–rare gas systems