C6 Coefficients Research Articles

Static Dipole Polarizability of Ytterbium

The static dipole polarizability of the ground state ytterbium atom is calculated using full and approximate relativistic ab initio methods. Our recommended polarizability of 143 au is consistent with experimental atomic spectral data. The corresponding van der Waals coefficient C6 of Yb2, derived using Padé approximants, is 2062 (200) au.

Dipole oscillator strength distributions, properties, and dispersion energies for ethylene, propene, and 1-butene

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the ethylene molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength (DOS) data; the DOS data employed are recent experimental results not available at the time of the original constrained DOSD analysis of this molecule. The constraints are furnished by molar refractivity data and the Thomas–Reiche–Kuhn sum rule. The DOSD is used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for ethylene. Pseudo-DOSDs for this molecule, and for propene and 1–butene, which are based on an earlier constrained DOSD analysis for these molecules, are developed. They are used to obtain reliable results for the isotropic dipole–dipole dispersion-energy coefficients C6, for the interactions of the alkenes with each other and with 47 other species, and the triple-dipole dispersion-energy coefficients C9 for interactions involving any triple of molecules taken from ethylene, propene, and 1–butene.Key words: alkenes, dipole properties, pseudo-states, dipole–dipole and triple-dipole dispersion energies, long-range additive, non-additive interaction energies.

The Effect of the Basis-Set Superposition Error on the Calculation of Dispersion Interactions: A Test Study on the Neon Dimer.

The dispersion interactions of the Ne2 dimer were studied using both the long-range perturbative and supramolecular approaches: for the long-range approach, full CI or string-truncated CI methods were used, while for the supramolecular treatments, the energy curves were computed by using configuration interaction with single and double excitation (CISD), coupled cluster with single and double excitation, and coupled-cluster with single and double (and perturbative) triple excitations. From the interatomic potential-energy curves obtained by the supramolecular approach, the C6 and C8 dispersion coefficients were computed via an interpolation scheme, and they were compared with the corresponding values obtained within the long-range perturbative treatment. We found that the lack of size consistency of the CISD approach makes this method completely useless to compute dispersion coefficients even when the effect of the basis-set superposition error on the dimer curves is considered. The largest full-CI space we were able to use contains more than 1 billion symmetry-adapted Slater determinants, and it is, to our knowledge, the largest calculation of second-order properties ever done at the full-CI level so far. Finally, a new data format and libraries (Q5Cost) have been used in order to interface different codes used in the present study.

Trap loss collisions of 6Li and 7Li with 23Na in a combined magneto-optical trap

We investigated heteronuclear inelastic cold collisions in a mixed-species Li–Na magneto-optical trap, where the Li trap loss from collisions with Na was measured. Experimental results for the rate coefficient βNaLi for the isotopes 6Li as well as 7Li as a function of the Li-MOT laser intensity are presented. Furthermore, we discuss some unique characteristics of cold collisions of Na and Li which arise from (1) the very small spin–orbit splitting of Li, (2) the very small C6 coefficients of the Na(3 2S) + Li(2 2P) asymptotes and (3) the very small and inverted hyperfine structure of Li.

Electric dipole polarizabilities and C6 dipole-dipole dispersion coefficients for sodium clusters and C60

The frequency-dependent polarizabilities of closed-shell sodium clusters containing up to 20 atoms have been calculated using the linear complex polarization propagator approach in conjunction with Hartree-Fock and Kohn-Sham density functional theories. In combination with polarizabilities for C(60) from a previous work [J. Chem. Phys. 123, 124312 (2005)], the C(6) dipole-dipole dispersion coefficients for the metal-cluster-to-cluster and cluster-to-buckminster-fullerene interactions are obtained via the Casimir-Polder relation [Phys. Rev. 73, 360 (1948)]. The B3PW91 results for the polarizability of the sodium dimer and tetramer are benchmarked against coupled cluster calculations. The error bars of the reported theoretical results for the C(6) coefficients are estimated to be 5%, and the results are well within the error bars of the experiment.

A post-Hartree-Fock model of intermolecular interactions: Inclusion of higher-order corrections

We have previously demonstrated that the dipole moment of the exchange hole can be used to derive intermolecular C(6) dispersion coefficients [J. Chem. Phys. 122, 154104 (2005)]. This was subsequently the basis for a novel post-Hartree-Fock model of intermolecular interactions [J. Chem. Phys. 123, 024101 (2005)]. In the present work, the model is extended to include higher-order dispersion coefficients C(8) and C(10). The extended model performs very well for prediction of intermonomer separations and binding energies of 45 van der Waals complexes. In particular, it performs twice as well as basis-set extrapolated MP2 theory for dispersion-bound complexes, with minimal computational cost.

Exchange-hole dipole moment and the dispersion interaction: High-order dispersion coefficients

In recent publications [A. D. Becke and E. R. Johnson, J. Chem. Phys. 122, 154104 (2005); E. R. Johnson and A. D. Becke 123, 024101 (2005)] we have demonstrated that the position-dependent dipole moment of the exchange hole can be used to generate dispersion interactions between closed-shell systems. Remarkably accurate C6 coefficients and intermolecular potential-energy surfaces can be obtained from Hartree-Fock occupied orbitals and polarizability data alone. In the present work, our model is extended to predict C8 and C10 coefficients as well. These higher-order coefficients are obtained as easily as C6 and with comparable accuracy.

Rapid estimation of the electron correlation energy for van der Waals complexes RgX (Rg = Kr, Xe, X = Br, I)

We present the rules of electron correlation energies for RgX (Rg = Kr, Xe, X = Br, I) van der Waals (vdW) complex systems at CCSD(T) theoretical level with SDB-cc-pVQZ basis set by the Gaussian 98 program. A new method to derive the dispersion coefficient C6 by fitting the intermonomer electron correlation energies to C6R−6 function is introduced. The present C6 values are compared with the corresponding theoretical ones.

Complex polarization propagator method for calculation of dispersion coefficients of extended π-conjugated systems: The C6 coefficients of polyacenes and C60

The frequency-dependent polarizabilities and the C6 dipole-dipole dispersion coefficients for the first members of the polyacenes namely benzene, naphthalene, anthracene, and naphthacene as well as the fullerene C60 have been calculated at the time-dependent Hartree-Fock level and the time-dependent density-functional theory level with the hybrid B3LYP exchange-correlation functional. The dynamic polarizabilities at imaginary frequencies are obtained with use of the complex linear polarization propagator method and the C6 coefficients are subsequently determined from the Casimir-Polder relation. We report the first ab initio calculations of the C6 coefficients for the molecules under consideration, and our recommended value for the dispersion coefficient of the fullerene is 101.0 a.u.

The NaK 1Δ1,3 states: Theoretical and experimental studies of fine and hyperfine structure of rovibrational levels near the dissociation limit

Earlier high-resolution spectroscopic studies of the fine and hyperfine structure of rovibrational levels of the 1 3delta state of NaK have been extended to include high lying rovibrational levels with v < or = 59, of which the highest levels lie within approximately 4 cm(-1) of the dissociation limit. A potential curve is determined using the inverted perturbation approximation method that reproduces these levels to an accuracy of approximately 0.026 cm(-1). For the largest values of v, the outer turning points occur near R approximately 12.7 angstroms, which is sufficiently large to permit the estimation of the C6 coefficient for this state. The fine and hyperfine structure of the 1 3delta rovibrational levels has been fit using the matrix diagonalization method that has been applied to other states of NaK, leading to values of the spin-orbit coupling constant A(v) and the Fermi contact constant b(F). New values determined for v < or = 33 are consistent with values determined by a simpler method and reported earlier. The measured fine and hyperfine structure for v in the range 44 < or = v < or = 49 exhibits anomalous behavior whose origin is believed to be the mixing between the 1 3delta and 1 1delta states. The matrix diagonalization method has been extended to treat this interaction, and the results provide an accurate representation of the complicated patterns that arise. The analysis leads to accurate values for A(v) and b(F) for all values of v < or = 49. For higher v (50 < or = v < or = 59), several rovibrational levels have been assigned, but the pattern of fine and hyperfine structure is difficult to interpret. Some of the observed features may arise from effects not included in the current model.

Time-dependent density-functional theory based on optimized effective potentials for van der Waals forces

The C6 coefficients of van der Waals interactions have been evaluated by the dynamic polarizabilities at imaginary frequencies computed by time-dependent density-functional theory using the optimized effective potentials. The C6 coefficients thus obtained agree accurately with those from the exact-exchange counterpart in ab initio molecular-orbital theory.

A post-Hartree–Fock model of intermolecular interactions

Intermolecular interactions are of great importance in chemistry but are difficult to model accurately with computational methods. In particular, Hartree-Fock and standard density-functional approximations do not include the physics necessary to properly describe dispersion. These methods are sometimes corrected to account for dispersion by adding a pairwise C6R6 term, with C6 dispersion coefficients dependent on the atoms involved. We present a post-Hartree-Fock model in which C6 coefficients are generated by the instantaneous dipole moment of the exchange hole. This model relies on occupied orbitals only, and involves only one, universal, empirical parameter to limit the dispersion energy at small interatomic separations. The model is extensively tested on isotropic C6 coefficients of 178 intermolecular pairs. It is also applied to the calculation of the geometries and binding energies of 20 intermolecular complexes involving dispersion, dipole-induced dipole, dipole-dipole, and hydrogen-bonding interactions, with remarkably good results.

Dynamic polarizability, dispersion coefficient C6 and dispersion energy in the effective fragment potential method

The development of a fragment–fragment dispersion energy expression, for the general effective fragment potential (EFP2) method is presented. C6 dispersion coefficients, expressed in terms of the dynamic polarizabilties over the imaginary frequency range (α(iν)), were calculated for a set of homo and hetero dimers. Using these coefficients the dispersion energy has been calculated. The dispersion energy is expressed using a simple London series expansion terminated after the n=6 term and implemented using distributed localized molecular orbitals (LMOs). The EFP2 dispersion energy is compared to symmetry adapted perturbation theory (SAPT) values. From this comparison, it is apparent that one needs to include higher order terms in the dispersion energy. Adding an estimated C8 term to the C6 energy greatly improves the agreement with the benchmark SAPT energies.

Dipole Oscillator Strength Distributions and Properties for Methanol, Ethanol and Propan-1-ol and Related Dispersion Energies

Recommended isotropic dipole oscillator strength distributions (DOSDs) have been constructed for the methanol and ethanol molecules through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength (DOS) data; the DOS data employed are recent experimental results not available at the time of the original constrained DOSD analysis of these molecules. The constraints are furnished by molar refractivity data and the Thomas-Reiche-Kuhn sum rule. The DOSDs are used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for the molecules. Pseudo-DOSDs for these molecules, and for propan-1-ol based on an earlier constrained DOSD analysis for this molecule, are also presented. They are used to obtain reliable results for the isotropic dipole-dipole dispersion energy coefficients C6, for the interactions of the alcohols with each other and with 36 other species, and the triple-dipole dispersion energy coefficients C9for interactions involving any triple of molecules involving methanol, ethanol and propan-1-ol.

Focal switching of partially polarized Gaussian Schell-model beams passing through a system with the aperture and astigmatic lens separated

On the basis of the beam coherence?polarization (BCP) matrix approach, the focal switching of partially polarized Gaussian Schell-model (PGSM) beams is studied. Our main attention is focused on the effect of astigmatism of the lens, partial coherence and polarization on the behaviour of the focal switch. It is shown that the astigmatism and auto-correlation affect the focal switch. For the aberration-free case there is no focal switching of PGSM beams. Only when the astigmatic coefficient C6 is greater than a certain value can the focal switching take place. The influence of cross-correlation and polarization on the focal switch is demonstrated, after inserting a polarizer.

Accurate description of van der Waals complexes by density functional theory including empirical corrections.

An empirical method to account for van der Waals interactions in practical calculations with the density functional theory (termed DFT-D) is tested for a wide variety of molecular complexes. As in previous schemes, the dispersive energy is described by damped interatomic potentials of the form C6R(-6). The use of pure, gradient-corrected density functionals (BLYP and PBE), together with the resolution-of-the-identity (RI) approximation for the Coulomb operator, allows very efficient computations for large systems. Opposed to previous work, extended AO basis sets of polarized TZV or QZV quality are employed, which reduces the basis set superposition error to a negligible extend. By using a global scaling factor for the atomic C6 coefficients, the functional dependence of the results could be strongly reduced. The "double counting" of correlation effects for strongly bound complexes is found to be insignificant if steep damping functions are employed. The method is applied to a total of 29 complexes of atoms and small molecules (Ne, CH4, NH3, H2O, CH3F, N2, F2, formic acid, ethene, and ethine) with each other and with benzene, to benzene, naphthalene, pyrene, and coronene dimers, the naphthalene trimer, coronene. H2O and four H-bonded and stacked DNA base pairs (AT and GC). In almost all cases, very good agreement with reliable theoretical or experimental results for binding energies and intermolecular distances is obtained. For stacked aromatic systems and the important base pairs, the DFT-D-BLYP model seems to be even superior to standard MP2 treatments that systematically overbind. The good results obtained suggest the approach as a practical tool to describe the properties of many important van der Waals systems in chemistry. Furthermore, the DFT-D data may either be used to calibrate much simpler (e.g., force-field) potentials or the optimized structures can be used as input for more accurate ab initio calculations of the interaction energies.

Collisional properties of trapped cold chromium atoms

We report on calculations of the elastic cross section and thermalization rate for collision between two maximally spin-polarized chromium atoms in the cold and ultracold regimes, relevant to buffer-gas and magneto-optical cooling of chromium atoms. We calculate ab initio potential energy curves for Cr2 and the van der Waals coefficient C6, and construct interaction potentials between two colliding Cr atoms. We explore the effect of shape resonances on elastic cross section, and find that they dramatically affect the thermalization rate. Our calculated value for the s-wave scattering length is compared in magnitude with a recent measurement at ultracold temperatures.

Accurate asymptotic ground state potential curves of Cs $\mathsf{_2}$ from two-colour photoassociation

Over 100 high lying level energies of the lowest electronic states $X^1{\rm\Sigma}^ + _g$ and $a^3{\rm\Sigma}^ + _u$ in Cs2 are determined in a $\Lambda$ -like scheme two-colour photoassociation spectroscopy. The results are analyzed with a coupled channel model using an asymptotic approach, based on nodal lines. From this analysis we determine the long range dispersion coefficient C6 to 6846.2 $\pm$ 15.6 a.u. We also obtain the first experimental determination of the amplitude of the asymptotic exchange term.

Dispersion corrections to density functionals for water aromatic interactions.

We investigate recently published methods for extending density functional theory to the description of long-range dispersive interactions. In all schemes an empirical correction consisting of a C6r(-6) term is introduced that is damped at short range. The coefficient C6 is calculated either from average molecular or atomic polarizabilities. We calculate geometry-dependent interaction energy profiles for the water benzene cluster and compare the results with second-order Møller-Plesset calculations. Our results indicate that the use of the B3LYP functional in combination with an appropriate mixing rule and damping function is recommended for the interaction of water with aromatics.

Long-range interactions for hydrogen molecular ions

A general formalism for calculating long-range interaction energies between two ground-state atomic or molecular systems is presented. The static polarizabilities α1, α2, α3 and α4, as well as the dispersion coefficients C6, C8 and C10 are evaluated to high precision for the ground-state hydrogen molecular ions H+2, D+2, T+2, HD+, HT+ and DT+, using variational wavefunctions in Hylleraas coordinates.