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Large scale CI calculations of frequency-dependent dipole polarizabilities (FDPs) of ground state BH at the experimental distance of R = 2.329a0 have been performed using two sets of contracted GTO functions involving up to 316 × 106 symmetry-adapted Slater determinants, comparing recently developed interpolation and variational techniques. Final calculation of BH–BH dispersion coefficients was done using the larger basis set of 268 AOs in the frozen-core FCI approximation and the lower bounds resulting from the variational approach.

The solution of the asymptotic quantum mechanical differential equations in the neighbourhood of singular points is suggested to overcome some complexities occurring with the technique of the indicial equations. After separation of the variables, the differential equations are solved in their asymptotic regions, the general regular solutions in the whole interval being then found by replacing integration constants by functions of the variables and finding the differential equations determining these functions. The resulting differential equations are now free from singularities and can be solved by the usual series expansion, giving simple polynomial solutions for the harmonic oscillator and the hydrogen atom and more complicated solutions for H 2 + .

Quantum corrections in the moderately low-temperature regime to the interaction potential energy of two permanent molecular dipoles are evaluated through the Wigner–Uhlenbeck–Gropper (WUG) semiclassical expansion of the density matrix, whose leading term is proportional to ℏ 2 , the dipoles being assumed in oscillatory motion around their position of minimum energy. Application of the resulting formulae to the ground states of a few heteropolar diatomic molecules and radicals shows that the quantum correction is not negligible, being particularly relevant for BH and OH.

In the elementary Huckel theory of ground state HF at its simplest valence level the two unknown parameters ∣αp − αh∣ and ∣β∣ can be determined in terms of experimental bond dissociation energy and electric dipole moment. If some sp hybridization is admitted on F in a bond orbital approximation, optimization of the resulting Huckel energy against the hybridization parameter ω using 1-term SCF/STOs shows that little hybridization (ω ≈ 8°) gives a 3% increase in the bond energy parameter ∣β∣ and a 14% reduction in the value of the formal charges ∣δ∣ on the interacting atoms.

Huckel transformation theory is applied to the study of the hybridization problem in ground state NH 3 . Under assumptions similar to those recently introduced for H 2 O, minimization of the model Huckel energy at the valence level yields a trigonometric relation between bond angle and hybridization parameter showing that hybridization opens the interbond angle beyond 90°. A further orthogonal transformation mixing in AOs belonging to A 1 and E symmetry yields four orthogonal sp 3 hybrids of C 3v symmetry on nitrogen that can be engaged in the axially symmetric lone pair and three equivalent straight N–H bonds making the experimental interbond angle.

An orthogonal transformation of the oxygen valence AOs belonging to A 1 symmetry to an ω -hybridized sp basis set allows for useful simplifications reducing Huckel transformed matrix for ground state H 2 O to nearly block-diagonal form. Neglecting delocalization of the hybrid lone pair into the bond, minimization of the resulting Huckel energy shows that for ω ≠ 0 the interbond angle opens beyond 90°. A further orthogonal transformation involving the B 2 y AO yields three orthogonal sp 2 hybrids of C 2v symmetry on oxygen which can be engaged in two equivalent straight O–H bonds making the experimental interbond angle and locally minimizing the Huckel energy of the valence electron configuration.

A self-consistent theory of the linear response previously developed by one of us is applied to the dipole–dipole dispersion interaction of two ground state H atoms in the united atom (He) limit, using a Sellmeier representation of the polarizability in terms of pseudostates. Numerical calculations show that a truncated 12-term expansion out of the N = 22 H(1s) dipole pseudospectrum built from STOs with orbital exponent c = 0.985 gives a result exceeding by no more than 5% the variational result for the He dispersion energy.

A recently proposed technique for interpolating numerical evaluations performed on frequency-dependent polarizabilities and obtaining analytically the related dispersion constants is improved in order to deal with larger sets of available data.

T-dependent long-range Keesom coefficients are evaluated up to the R-10 term for small values of the dimensionless parameter vertical bar a(ll ')vertical bar For large values of vertical bar all 'vertical bar, I corrections must be introduced mostly for the dipole-dipole term, the correct values of Q being best obtained from a recently derived asymptotic formula. The corresponding attractive energies are the isotropic electrostatic contributions to the interaction energy and are temperature-dependent. Comparison with long-range induction and dispersion energy results for some simple polar axially symmetric molecules in the gas phase shows that at R = 10a(0) and T = 293 K the electrostatic dipole-dipole component is dominant for vertical bar a(11)vertical bar > 0.5. For centrosymmetric molecules the corresponding electrostatic contribution is usually negligible with respect to dispersion.

Explicit analytical formulae are presented for the exact R n -expansion of the short-range electronic interaction energy for ground state H 2 + up to n = 10. The corresponding calculated values for small values of the internuclear distance are in excellent agreement with Peek’s accurate results up to R = 0.30 a 0 .

It is shown in this Letter that the erroneous bond length resulting for the bound 1 Σ u + state of H 2 described by the ionic valence bond structure over the minimum set of 1s STO functions is corrected by the variational inclusion with a substantial weight of the singly excited covalent structures involving 2s and 2pσ STOs onto each H atom. The corresponding value for the molecular energy resulting from the complete optimization of the orbital exponents gives now over 96% of the accurate Kolos–Wolniewicz results.

Frozen-core full-CI calculations of frequency-dependent dipole polarizabilities (FDPs) of ground state BeH 2 at the experimental distance of R = 2.506 a 0 have been performed using an extended set of 208 contracted GTO functions [9s9p5d3f] on Be and [9s8p6d] on H involving about 58 × 10 6 symmetry-adapted Slater determinants. The Casimir–Polder integral was then evaluated analytically using eight optimized imaginary frequencies chosen according to a recently developed interpolation technique which allows for the evaluation of the three dipole dispersion constants for the BeH 2 –BeH 2 homodimer, from which isotropic C 6 and anisotropy γ 6 coefficients are derived for the first time.

Making the usual assumptions typical of elementary Huckel theory including overlap it is shown that the Huckel energy of the two-electron bond is minimized for orbitals having maximum overlap, which yields maximum strength for the straight bond. Analysis of the bond energy for the general A–B bond shows for the molecular orbital description a more complicated dependence on the one-electron exchange-overlap densities of A and B than that recently derived by us in a corresponding valence bond treatment.

Revisitation of the Heitler–London theory for H2 shows that the exchange-overlap component of the bond energy has a strength which is maximum at the bond length. Making the usual assumptions typical of elementary Huckel theory including overlap it is similarly found that in the valence bond description of the chemical bond covalent bonding is strongest for atomic orbitals maximizing the strength of the complete exchange-overlap component of the bond energy.

A two-state model of van der Waals (VdW) interactions is presented which completes the elementary description of the interactions occurring between atoms and molecules. A weak VdW bond is formed between closed-shell atoms or molecules as soon as the small Pauli repulsion is offset in long range by attractive VdW interactions. These are distortion and dispersion interactions, related to the electric properties of the interacting molecules, and described as small second-order effects in terms of long range interaction integrals β s. The explicit forms of these β s are derived from classical electrostatics for atom–atom, atom–linear molecule dispersion, and atom–linear molecule induction. Comparison between C 6 dispersion and induction coefficients for simple homodimers shows that dispersion is, in general, the most important attractive VdW interaction in long range.

The simple Huckel model of the chemical bond, proposed earlier and applied mostly to homonuclear diatomics, is extended to the case of the heteropolar bond. The filling of electrons into the MO levels has now more possibilities than in the previous homonuclear case. For orthogonal AOs, the fundamental quantities entering the model, the atomic energy difference α2−α1 and the bond integral |β| can be determined using the experimental values of the atomization energy De and the electric dipole moment μ. The results obtained for the two-electron σ-bonds occurring in the first-row series of ground state diatomic hydrides are analyzed and shortly discussed.

Fluctuation theory techniques are used to evaluate short-range interactions in ground state H 2 + by calculating in a direct way small energy differences taken with respect to the ground state of the united atom He + . Appropriate coordinate transformation allows to obtain inner and outer Hermitean eigenvalue equations, which are then solved perturbatively in positive powers of the internuclear distance R choosing He + as unperturbed problem. The separation constants are deduced from the regularity conditions on the functions. Next, the function endowed with the right symmetry is constructed by solving an integral equation in a perturbative way. A second-order approximation gives short-range interaction coefficients which are exact up to R 5 .

An elementary technique has been developed to intercept available values of a given atomic or molecular multipole dynamic polarizability through an expression providing a faithful representation for its dependence on the imaginary frequency. The explicitation of the implied interpolative parameters is obtained by imposing that the available data ought to be exactly intercepted. Numerical tests concerning a ground state hydrogen atom have confirmed the reliability of such interpolative procedure for extracting continuous trends from some initial estimates performed on multipole polarizabilities and for evaluating dispersion constants.

Ordinary LCAO-MO theory allows for the ab-initio definition of α and β parameters occurring in the Huckel theory including overlap of simple σ homonuclear systems. One- and two-electron terms not included in Huckel theory are explicitly considered. When such correction terms are neglected in the very spirit of Huckel theory, simple calculation in the hydrogenic approximation for the ground states of H2+, H2, He2+, He2 at the corresponding bond lengths shows semiquantitative agreement with experimental or accurate theoretical results for the bond energies, providing a simple and clear physical picture of the nature of the chemical bond.

Small well-shaped basis sets have been used to perform highly accurate variational-perturbative evaluations of dynamic (frequency-dependent) multipole polarizabilities for a hydrogen atom in its ground state. Their structure has been suggested by the ab initio polynomial power expansion we have recently developed to get correspondingly exact results within a finite range covering the region of low frequencies. The frequency-dependent coefficients of the optimized linear combinations of the excited pseudostate orbitals were obtained by diagonalizing the representative matrix of the excitation operator from the ground state. High levels of accuracy have been observed over a set of frequencies now covering the complete real and imaginary ranges.

A recently proposed exact power expansion for the dynamic (frequency-dependent) multipole polarizabilities of a ground-state hydrogen atom has been improved keeping its simple polynomial structure substantially unaffected and enlarging its range of convergence. The improvement is performed by adding unexpanded frequency terms taken from the formal expression of the polarizability as a sum-over-states and subtracting their corresponding power expansions. The enlarged range of convergence obtained in this way covers all real frequencies of physical importance and the related imaginary frequencies.

The partition function of coupled molecular dipoles in a static electric field parallel to the supermolecular axis is evaluated by expanding the interaction energy function around the equilibrium position of minimum potential energy. The resulting expansion of the configurational integral is, as was proved by these same authors, an asymptotic representation in the limit of the strong interaction parameter a. This allows for the computation of one-sided averages of the supermolecular dipole moment not attainable through symmetrical procedures.

Long-range dispersion and induction coefficients for the homodimers of Li2, Na2, K2 are evaluated in the LALBM scheme using explicit formulae presented previously and numerical values of moments, polarizabilities and dispersion constants obtained from 2-term reduced spectra of the Kaiserslautern group. While induction coefficients are 5–8 times smaller than dispersion, induction anisotropies are always larger than the corresponding dispersion anisotropies for all homodimers.

The C 8 and C 10 long-range induction coefficients for two like centrosymmetric linear molecules are derived explicitly using the irreducible properties of spherical tensor operators. An application to the H 2 –H 2 system shows that whereas the isotropic induction coefficients are 50–100 times smaller than dispersion, the induction anisotropies γ n are always larger than the corresponding dispersion anisotropies.

Formulae and computer programs for evaluating two-centre molecular integrals over an STO basis in spheroidal coordinates are presented. A unified formula is obtained for the two-electron integrals. New formulae are suggested for the auxiliary functions that remarkably improve the accuracy and speed of calculation for high values of the quantum numbers n and l.

An accurate evaluation of C 6 dispersion coefficients for long-range interactions involving H, He, H 2 + and H 2 has been performed using the London formula in terms of ab initio pseudostate decomposition of the static dipole polarizabilities of the individual atoms and molecules.

The pseudostate decomposition of accurate Kolos-Wolniewicz static dipole polarizabilities for ground-state H 2 allows an extremely accurate evaluation of C 6 dispersion coefficients for (H 2 ) 2 .

Lebeda—Schrader perturbation calculations are performed for ground and excited states of H2 using an Inui ψ0. The first-order function is expanded in a two-centre basis set of linear pseudostates involving powers of spheroidal coordinates and explicitly including the interelectronic distance. Third-order results give errors well below 1% with respect to the corresponding variational values.

The two-parameter Inui wavefunction for the ground and excited states of H 2 is improved variationally through an expansion in powers of confocal spheroidal coordinates including the interelectronic distance. A single 69-term expansion gives, for both states, fairly accurate interaction energies in the region of the chemical and van der Waals bond.

The Guillemin-Zener wavefunction for the ground and excited states of H + 2 is improved in terms of a separate double expansion in powers of confocal spheroidal coordinates. Variational optimization by the Ostrowski method shows that no more than 15 linear coefficients are needed to yield for both states practically exact interaction energies over the whole range of nuclear separations.