Journal of Computational Methods in Sciences and Engineering

A semiempirical method is discussed to construct the polarizability functions for a diatomic homonuclear molecule as a piecewise-continuous function that exhibits physically correct behavior at small and large internuclear distances and agrees with the polarizability function near the nuclear equilibrium position of the molecule. The method is applied to calculate the polarizability functions of N₂ and O₂ molecules in the range of internuclear distances (0, ∞).

Ab initio calculations of the static dipole polarizability and the static and dynamic second hyperpolarizabilities, including the electron correlation effects via the standard Coupled Cluster method with single and double excitations (CCSD) and non-iterative triple excitations (CCSD(T)) as well as the vibrational motion (nuclear relaxation) effects by means of the Finite Field method formulated by Kirtman and co-workers, have been carried out for difluoroacetylene (C₂F₂). The basis set dependence of the geometry parameters and molecular properties using various types of basis sets (ANO-L, POL, HyPOL, aug-cc-pVTZ) is analysed. A particular attention is devoted to the evaluation of the electron correlation effects on the vibrational contributions to the molecular properties.

Elevated magnitudes of the first and second static hyperpolarizabilities (AM1/TDHF), respectively, of organic molecules having electron-donation (D) and electron-accepting (A) groups can be obtained through of selection of molecular systems with appropriate structural and electronic parameters. The Homo-Lumo energy gap, the total number of π-electrons of the molecule, the Homo energy, and the ground state dipole moment were the parameters considered in this work. The designed D-A organic molecules investigated have polyenic/mesoionic bridge, which afford large β and γ hyperpolarizabilities, associated for different strength of donor-acceptor pairs. Large data sets of these parameters for the molecules, obtained from quantum chemistry semiempirical calculations with AM1 hamiltonian, were submitted to a principal components analysis (PCA). In addition, a model was proposed based in a principal components regression (PCR), having in mind the prevision and selection of organic molecules with potential applications in nonlinear optics.

A theory of the spectral distribution of the molecular time correlation function of the collision induced polarizabilities active in the collision induced light scattering has been developed. Within the framework of this approach a new concept of the averaging procedure is proposed, which – unlike the usually applied assumptions of total separation of the angle dependent and the translational contributions – partially includes the coupling between the rotational and translational degrees of freedom due to anisotropic intermolecular interactions. Introduction of the rotational spectral functions, valid for different frequency ranges, in order to produce a numerically treatable method of deriving collisional spectral profiles is discussed. The final results of the theory are formulated in a fashion providing a solid base for numerical calculations capable of interpreting data obtained in various fields of physical, environmental and astrophysical sciences.

The optical rotations (OR) of five chiral organic molecules have been calculated by time-dependent density functional response theory (TDDFT) employing four different density functionals. The results are compared with experimental gas phase data in order to explore the inherent accuracy of the functionals. The theoretical results obtained with the BHLYP hybrid functional including 50% “exact” Hartree-Fock exchange compare very well with the experimental data and also better with those derived from the gas-phase than from solution. Non-hybrid functionals show a tendency to overestimate the OR. The anomalous behavior of the methyloxirane molecule which shows sign changes with solvent and excitation frequency is discussed.

The static dipole moment (μ₀), polarizability (α₀) and first hyperpolarizability (β₀) of the hydrogen bond complex NH₃-HF with a equivalent triple π-type hydrogen bond are investigated by means of ab initio methods at the MP2 level based on the QCISD potential energy surface. The full counterpoise (CP) method is applied in the studies of the intermolecular interaction contributions to the above properties. The first hyperpolarizability obtained is 31.76 a.u. The results of the intermolecular interaction contributions to the above properties are 26% for μ₀, -4.2% for α₀ and 10.0% for β₀.

Pseudo – spectral dipole oscillator strengths and excitation energies, which are discrete representations of previously developed recommended continuous dipole oscillator strength distributions(DOSDs), are presented for the ground state formaldehyde, acetaldehyde, acetone, and mono - , di - , and tri – methylamine molecules. These pseudo – DOSDs, together with previously published pseudo – DOSDs for other atoms and molecules, are used to evaluate the dipole – dipole and the triple – dipole dispersion energy coefficients for all the two – body and three - body interactions between H₂CO, CH₃CHO, (CH₃)₂CO, CH₃NH₂, (CH₃)₂ NH, and (CH₃)₃N, and between these molecules and forty – four other species, namely Cl₂, SiH₄, SiF₄, CCl₄, H, Li, He, Ne, Ar, Kr, Xe, SF6, HF, HCl, HBr, SO₂, CS₂, OCS, H₂, N₂, O₂, NO, N₂O, H₂O, H₂S, NH₃, CO, CO₂, the normal alkanes CH₄, C₂H₆, C₃H₈, C₄H₁₀, C₅H₁₂, C₆H₁₄, C₇H₁₆ and C₈H₁₈, the 1-alkenes C₂H₄, C₃H₆ and C₄H₈, C₂H₂, C₆H₆, and the primary alcohols CH₃OH, C₂H₅OH and C₃H₇OH. Results are presented explicitly for all the dipole – dipole dispersion energy coefficients and for the triple – dipole coefficients for all three -body interactions involving the H₂CO, CH₃CHO, (CH₃)₂CO, CH₃NH₂, (CH₃)₂ NH, and (CH₃)₃N molecules. The estimated errors in the two - body and three – coefficients are 1% and 1–2%

The complex polarization propagator method has been applied to the calculation of dipole-dipole dispersion coefficients (also known as C 6 coefficients) of pyridine, pyrazine, and s-tetrazine. These calculations refer to the electronic ground states as well as the first excited states of π → π ∗ character. It is argued that accurate ground state dispersion coefficients are obtained with density functional theory using the B3LYP exchange-correlation functional. The proposed values for the C 6 coefficients of pyridine, pyrazine, and s-tetrazine in their ground states are 1543 a.u., 1398 a.u., and 1014 a.u., respectively. Multi-configurational complete active space calculations are performed on these compounds in their respective π → π ∗ excited state. The isotropic averages of the frequency-dependent polarizabilities are smaller in the excited states, but the effective frequencies defined in the London – van der Waals dispersion relation – are on the other hand larger.

Static vibrational and electronic contributions to the longitudinal polarizability (α) and first hyperpolarizability (β) have been calculated, within the double harmonic oscillator approximation, at the HF/6-31G for a set of polyconjugated, D-linker-A, compounds. Effects due to the nature of the linker and of the D/A-pair type, on the geometrical structures and the electrical properties, are investigated. It turns out that, for a given linker, in almost all cases, the static β L e (0) and β L v (0) vary in the same order, with the type of the acceptor group A (A # NO₂, COH, CF₃). As well, the pair HO/NO₂ was associated the greatest β L e ( 0 ) and β L v ( 0 ) values. The analysis upon the SOM expressions, due to Bishop and Kirtman, demonstrates that, more than 50% of β L v owes to three, two or even to only one vibrational normal mode contribution. Moreover, the vibrational modes with frequencies at, or more than, 3000 cm⁻¹ present a null individual vibrational first hyperpolarizability.

Components of the static electric dipole polarizabilty (α), and second hyperpolarizability (γ); tensors have been determined by accurate ab initio calculations for three molecules containing two conjugated carbon – nitrogen and/or carbon – phosphorus triple bonds. For the NCCP molecule the dipole moment (μ) and non-vanishing components of the first hyperpolarizability (β) tensor have also been calculated. Electron correlation effects have been taken into account by the second-order Many Body Perturbation Theory (MBPT(2)), and coupled cluster (CCSD and CCSD(T)) calculations. The basis set developed by Sadlej (Pol) and designed to be used in electric properties calculations together with its extension (HyPol) have been applied and compared to the results obtained with the standard aug-cc-PVTZ sets of Dunning.

The polarizability and first hyperpolarizability of a crystal surface are calculated from a sum over states method using crystalline orbitals of the CRYSTAL program. The linear and nonlinear optical properties of the (001) LiF surfaces are evaluated layer by layer, showing that the response of inner layers to an electric field tends effectively to the bulk response value while the χ⁽²⁾ susceptibility components responsible of second harmonic generation (SHG) are no longer equal to zero at the surface of the crystal.

An ab initio investigation of the molecular properties rationalizing the electric-field-gradient induced birefringence (Buckingham effect) for Cl₂ is presented. The quadrupole moment is determined using hierarchies of basis sets and wavefunction models. The electric dipole polarizability, the dipole – dipole – quadrupole and dipole – dipole – magnetic dipole hyperpolarizabilities are determined exploiting a Coupled Cluster Singles and Doubles (CCSD) response approach. The properties are zero-point vibrationally averaged, and the contribution of excited ro-vibrational states accounted for. To this end, the interatomic 1 Σ g + ground state potential has been computed at CCSD plus perturbative triples – CCSD(T) – level employing a large augmented correlation consistent basis set. The effect of relativity is estimated at the Dirac-Hartree-Fock level. Our best value for the quadrupole moment of Cl₂ is (2.327 ± 0.010) au and it is in excellent agreement with experiment which, after revision and dependent on the procedure employed for correcting the original estimate of (2.24 ± 0.04) au of Graham et al., [Mol. Phys., 93, 49, (1998)], ranges from (2.31 ± 0.04) au to (2.36 ± 0.04) au.

We briefly review the different ab initio methods that have been introduced for the calculation of excited-state polarizabilities of molecules in solution. Emphasis is put on the conceptual differences between the methods, emphasizing the strengths and weaknesses of the different approaches. A general discussion of the use of dielectric continuum methods in the modeling of linear and non-linear electric properties of ground- and excited state polarizabilities is given. Particular attention is given to the notion of equilibrium and non-equilibrium solvation models. We discuss the results of the few theoretical calculations that have been presented in the literature at the time of this review, and also give a few new results.

We report a study of the static dipole polarizability (α) of the o-, m- and p-benzyne isomers in their singlet ground state. Due to the biradical character of benzynes, calculations were performed at ab initio unrestricted Hartree-Fock (UHF) level of theory. The correlation effects were accounted for with second order Møller Plesset (MP2) method, Density Functional Theory (DFT) with the BLYP and B3LYP hybrid approaches and the coupled cluster CCSD method with 6-31+G(d,p) standard basis set with optimized geometries. C₂v symmetry restriction was employed for the geometry of o- and m-benzynes (closed and open conformation) and D₂h for p-benzynes (open and closed conformation). Because the high quality of the basis set is a requirement for accurate α determination, MP2 and DFT calculations were performed with the 6-31+G(d,p), the extended 6-311++G(3d,3p) and the specialized Sadlej basis sets. The CCSD calculations were only carried out with the 6-31+G(d,p) and the Sadlej basis sets. The results indicate that the average α polarizability values for o- and m-benzynes are lower than the corresponding to the experimental value of benzene, while this property for p-benzyne is slightly higher than the benzene one. The anisotropy Δα values of benzyne isomers are always higher than the anisotropy of benzene molecule. Finally, was found that the conformations of p-C₆H₄ are the less stable isomers, and are the most polarizables and anisotropic compounds from the benzyne series.

The structure and molecular first hyperpolarizability (β) of the nitro-amino-substituted 1,2-dicarba-closo-hexaboranes were investigated at B3LYP/6-31+G(d) level of theory. Conformations with different orientations of the substituents with respect to the carborane cage were considered. It is shown that two factors significantly influence the value of β: orientation of the amino- and nitro-group with respect to the cage, and their mutual orientation. The latter factor appears to be more important. The results obtained have revealed that a carborane substituted at boron atoms is characterized by higher value of the molecular first hyperpolarizability.

Over the last couple of years, it has been shown that Time Dependent Density Functional Theory (TD-DFT) is able to predict accurately and efficiently the polarizability of molecules, when using appropriate exchange-correlation potentials and (large) basis sets. In a previous paper, we compared the accuracy of the predicted mean polarizabilities of 15 organic molecules with experiment, and with two other computational methods: the Restricted Hartree-Fock (RHF) method and the Direct Reaction Field (DRF) approach, the first of which is ignored in this paper. The (empirical) DRF approach however was shown to give comparable accuracies to TD-DFT with the values computed in just a few seconds. In this paper, we use TD-DFT to compute molecular polarizabilities of the twenty amino acid residues, and compare them with the results obtained with the DRF approach. Although the mean absolute deviation of the DRF values from the TD-DFT values is reasonable (7%), it is more than two times the accuracy normally found with the DRF approach. Therefore we decided to optimize the atomic parameters for these systems, and found after optimization, a good agreement with the TD-DFT values (mean absolute deviation 1.0%). As the TD-DFT calculations were necessarily obtained with two additional hydrogens to saturate the backbone bonds, the molecular value of the polarizability of the amino acid residues is overestimated by the TD-DFT calculations. Therefore, the DRF approach (with the newly optimized atomic parameters) has been used to get the actual polarizabilities of the amino acid residues.

Recent interest in quantitative aromaticity indices has focussed on structural, magnetic, and energetic criteria. In this work, aromaticity indices based on polarizabilities are compared with indices based upon bond orders for benzene and 12 azines, pyrrole and 9 azoles, furan and 9 oxazoles, and thiophene and 8 thiazoles. The best polarizability-based index of aromaticity we find is the polarizability anisotropy of the π-electrons. However, none of the indices constructed from polarizabilities seem to be entirely suitable as measures of aromaticity. The comparison of the structural and polarizability scales enables us to formulate three critical tests that can be used to eliminate quickly unsuitable aromaticity scales for this set of heterocycles.

The interacting induced dipole polarization model implemented in our program for the calculation of molecular polarizabilities (POLAR) is used for the calculation of the molecular dipole-dipole polarizability α ‾ ‾ . POLAR is tested with Si n , Ge n and Ga n As m small clusters. The polarizability is an important quantity for the identification of clusters with different numbers of atoms and even for the separation of isomers. The results for the polarizability are in agreement with reference calculations performed with our version of the program PAPID (polarisabilités atomiques par interactions dipolaires) and with reference computations from Dr. J.R. Chelikowsky. The bulk limit for the polarizability is estimated from the Clausius-Mossotti relationship. The polarizability trend for the clusters as a function of size is different from what one might have expected. The clusters are more polarizable than what one might have inferred from the bulk polarizability. Previous experimental work yielded the opposite trend for somewhat larger clusters. At present, the origin of this difference is problematic. One might argue that smaller clusters need not behave like those of intermediate size. The high polarizability of small clusters is attributed to dangling bonds at the surface of the cluster. Indeed, most of the atoms within small clusters reside on the surface. In this respect, semiconductor clusters resemble metallic clusters.

We reported the first-principle study of linear and nonlinear optical polarizability of a di-nuclear transition metal cluster, [Mo₂S₃(CO)₆(C₆H₁₁)₃]·N(C₂H₃)₄, by using finite-field method. The calculations were performed at an HF/Lanl2dz level. The spatial average value of the first hyperpolarizability along z direction was very large, β z = 13.8 × 10 − 30 esu, indicating the potential applications of this cluster for IR nonlinear optical processes. Our results showed that the charge transfer inside the metal core cluster, [Mo₂S₃(CO)₆], played an important role in nonlinear optical response while the joint effect of metal core and organic ligands (C₆H₁₁)₃ also contributed to nonlinear optical polarizability. In addition, the organic ionic group, [N(C₂H₃)₄]⁺, situated in z direction enhanced the β z value. The study may help us to understand the nonlinear optical nature of the transition metal clusters and provided an example to show the computational efficiency of finite-field method in the first-principle theoretical study of the complex system containing transition-metal element.

We present a real-space method for computing the linear and nonlinear polarizabilities in the time-dependent density-functional theory. In our method, the three-dimensional Cartesian coordinate is discretized on a uniform grid and the wave functions are represented on the grid points. The dynamic polarizabilities may be calculated perturbatively by expanding the dipole moment in a power series of the external field strength or directly by integrating the real-time equations of motion under the time-varying external field. We describe the equations and computational techniques for the both methods. Besides calculating second-order and third-order hyperpolarizabilities, the method can be applied to photon-absorption cross sections, and we show examples of this application as well. We also present here for the first time calculations of the nonlinear polarizabilities using the real-time method.

The rational design rules for atomic Gaussian basis sets, tailored for the calculations of molecular polarizabilities α i j and hyperpolarizabilities β i j k at the SCF level, are elaborated. The sequence of the optimized polarization sets (2d), (3d), (4d), (3d1f), (4d3f) for the first-row atoms from B to F is constructed, the optimization technique is supposed to provide a rapid convergence of both α i j and β i j k with the size of the polarization set. Adding of these polarization sets to the [6s4p] and [6s5p] substrates, derived from the energy-optimized (10s6p) primitive set of Huzinaga, gives the [6s4pNdMf] and [6s5pNdMf] bases families. They are combined with the corresponding [4sNpMd] and [5sNpMd] bases for Hydrogen built up in the same manner. The convergence of SCF results with the size of the basis set has been thoroughly investigated by the examples of H₂CO, CH₃CN, and HCN molecules. It has been established that both α i j and β i j k values converge to their saturation limits at the [6s4p3d/4s3p] level. This basis set provides the results, which are very close to near HF limit values, estimated with the most sophisticated polarized Gaussian basis sets used up to now.

The design rules, elaborated earlier in the first paper of this series for the construction of atomic Gaussian basis sets, tailored for the calculations of molecular polarizabilities α i j and hyperpolarizabilities β i j k at the SCF level, are tested here by the example of the basis set families, generated from the energy-optimized (9s5p) parent set. Adding the polarization sets to contracted sp substrates of various flexibility, we generate the [4s3pNdMf], [5s3pNdMf] and [6s4pNdMf] bases families, which are combined with the corresponding [3sNpMd] and [4sNpMd] bases for Hydrogen. The SCF results for H₂CO, CH₃CN, and HCN molecules, obtained with (11s7p3d/6s3p)/[6s4p3d/4s3p], are used as a benchmark here. The contraction flexibility is shown to affect the convergence of both α i j and β i j k values only insignificantly, slightly increasing the uncertainties of the calculated data. The smallest [4s3p3d/3s3p] basis is recommended for the calculation of (hyper)polarizabilities of large molecules within the SCF approximation. For the conjugated organic molecules with dominant ÏĂ-electron polarization rather reliable SCF results are obtained at the [4s3p2d/3s] level. So, for p-nitroaniline this basis set reproduces the results obtained with [6s4p3d/4s3p] basis with the error 1% for α i j and 3% for dominant longitudinal hyperpolarizability β z z z .

A comparison of ab initio calculations of the static dipole polarizability components for the water molecule and hydroxide anion is reported. The discussion is focused on the difference between the computational needs for an accurate treatment of these two systems. A particular attention is paid to the danger of extrapolating to the anion some approximate techniques that are valid for the water molecule. Four important points are considered in this analysis: the analytical/numerical calculation of the components, the effect of the electronic correlation, the characteristics of the basis set, and the vibrational effects. In the finite-field approach, the interdependence of some control parameters, such as the energy convergence threshold and the associated field strength, are recalled within the frame of parabolic or quartic polynomial fits. The correlation effect is examined within the coupled-cluster method restricted to single, double, and approximate triple excitations, CCSD(T), and the nth-order Møller-Plesset perturbation theory, MPn. The oscillating behaviour of the perturbation series is very pronounced for the anion case, and its convergence is doubtful. It is crucial to include much more diffuse functions (valence and polarization) in the basis set for the anion than for the water molecule. A comparison of many basis sets is presented. The role of the vibrational effect is briefly addressed. Because ab initio calculations on supersystems generally cannot be performed at the best level reachable for the monomer polarizabilities, lower (but reasonnable) accuracy of the polarizability components is also discussed.