Static Polarizabilities and Hyperpolarizabilities

2 Static Polarizabilities and Hyperpolarizabilities. - In the case of the static polarizabilities the methods employed in semi-empirical and ab initio work are essentially the same. Naturally most of the calculations on small molecules are now based on ab initio methods while semi-empirical systems (often of the MOPAC/MNDO genre) come into play for larger molecules. 

There is a distinction between two groups of methods. The first is the finite field technique.35 In this case finite perturbations representing the external fields are added to the molecular hamiltonian and the calculation of the ground state wavefunction and energy is carried out as for the unperturbed molecule. The finite field method can be applied in conjunction with any quantum mechanical method that is available for molecular calculations. There are two principal subdivisions of the finite field method. In one of these terms of the form tff-Fi 

The second main group of methods analyse the response to successive powers of the perturbation separately using analytical re-arrangements of the perturbed equations. The procedure is typified in the Coupled Perturbed Hartree-Fock39,40,41 method (CPHF). which produces variationally optimized solutions in each order. Since the results represent a solution of the variational Hartree-Fock equations to each order they satisfy the energy derivative equations for the polarisabilities and the (2n+ 1) rule for the derivatives can be used to simplify the calculations. Corrections to the perturbed HF solutions can be made through MP2 or MP4 perturbation theory. 

Semi-empirical methods, usually of the MNDO/MOPAC genre, have also 

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Cammi, R., Cossi M. and Tomasi J., Analytical derivatives for molecular solutes. III. Hartree -Fock static polarizability and hyperpolarizabilities in the polarizable continuum model. J. Chem Phys. (1996) 104 4611-4620. 

Advances in Chemical Physics as have Bishop[18] and Luo et al.[ 19] for Advances in Quantum Chemistry. Two reviews which cover both theoretical and experimental developments have been written by Bonin and Kadar-Kallen[20] for polarizabilities and by Shelton and Rice[21] for hyperpolarizabilities. Finally, though it is not a review as such, attention is drawn to the paper by Stiehler and Hinze[22] on the calculation of static polarizabilities and hyperpolarizabilities for the atoms He through Kr. This paper is so thorough in its reference to other calculations on these atoms that it deserves to be specifically singled out as an excellent source for theoretical data. 

Calculations of the vibrational contributions to the static polarizability and hyperpolarizability have also been attempted. As far as the EFISH experiment is concerned, which depends on the square of an optical frequency field, it is assumed that there will be no direct contribution to (—2static contribution is comparable with the static electronic contribution to /1(0 0,0). An indirect vibrational effect through the linear polarizability of the solvent molecules is more important. Calculations of the vibrational effects in pNA cannot be carried out reliably even for the static case since the second term in the perturbation theory is much larger then the first and there is no evidence of convergence. 

In two previous papers [8,9] we have calculated the static polarizabilities and hyperpolarizabilities for ls3p Pj (J = 0, 2)-states of helium. The method was based on degenerate perturbation-theory expressions for these quantities. The necessary dipole matrix elements were found by using the high-precision wave function on framework of the configuration-interaction (Cl) method [10]. The perturbed wave functions are also expanded in a basis of accurate variational eigenstates [11]. These basis sets of the wave functions explicitly take account of electron correlation. To control the result we have also carried out similar calculations with Fues model potential method. 

Static Polarizabilities and Hyperpolarizabilities by ab initio Methods. -Using methods similar to those described in reference 125, Maroulis and... 

In this section we report a second extract of the study we have published on the Journal of the American Chemical Society about solvent effects on electronic and vibrational components of linear and nonlinear optical properties of Donor-Acceptor polyenes. In a previous section we have presented the analysis on geometries, here we report the results obtained for the electronic and vibrational (in the double harmonic approximation) static polarizability and hyperpolarizability for the two series of noncentrosym-metric polyenes NH2(CH=CH) R (n=l,2), with R=CHO (series I) and with R=N02 (series II) both in vacuo and in water. 

R. Cammi, M. Cossi, and J. Tomasi,/. Chem. Phys., 104, 4611 (1996). Analytical Derivatives for Molecular Solutes. III. Hartree-Fock Static Polarizability and Hyperpolarizabilities in the Polarizable Continuum Model. 

Reis et al have carried out DFT calculations on the static polarizabilities and hyperpolarizabilities of bare boron clusters incuding up to 10 boron atoms. They find that the y-hyperpolarizabdity saturates when the cluster size reaches approximately five atoms. A maximum in the hyperpolarizabilty per atom occurs for the cluster containing six atoms. 

G. Maroulis, Accurate electric multipole moment, static polarizability and hyperpolarizability derivatives for N2. J. Chem. Phys. 118(6), 2673-2687 (2003)... 

Many molecular properties may be calculated (static polarizability and hyperpolarizability frequency-dependent polarizability electric moments electric field and electric field gradient). 

Finite differences of the dipole moment can be used to give static polarizabilities and hyperpolarizabilities. Of course, these can be obtained from energies only, but greater numerical stability and CPU savings are possible if the dipole moment is calculated analytically. 

Static polarizabilities and hyperpolarizabilities were mentioned above. Frequency-dependent polarizabilities and hyperpolarizabilities can be calculated using two different approaches. First, the time-dependent Hartree-Fock (TDHF) approach, which is not correlated. This formalism enables a wide range of optical properties to be computed analytically and for any order. Correlated approaches to frequency-dependent polarizabilities and hyperpolarizabilities are available in the EOM-CCSD formalism. Obviously, this is significantly more expensive than TDHF, and it is sometimes useful to compute static properties with correlated methods and estimate the dynamic dispersion contribution from TDHF. 

This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field—induced polarization functions have been constructed from the first- and second-order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion. We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities. 

M. Spassova and B. Champagne, aP-SOS, a Code for Evaluating Static and D5mamic Polarizabilities and Hyperpolarizabilities (Namur Sofia, 1996). 

In order to provide accurate static values for this purpose, we have recently undertaken a series of calculations of the polarizability and hyperpolarizability of the rare gases. An extensive basis set investigation was performed for Ne, and we shall consider these results in detail. We shall also discuss aspects of the correlation treatment and computational methodology. We begin by considering methods for the calculation of the polarizability and hyperpolarizability. 

We have seen above how to calculate polarizabilities and hyperpolarizabilities for static electric fields. I will consider a simple oscillating electric field... 

Jensen L, Astrand P-O, Mikkelsen KV (2004) The static polarizability and the second hyperpolarizability of fullerenes and carbon nanotubes. J. Phys. Chem. A 108 8795-8800... 

Dipole moments, static averaged polarizabilities and hyperpolarizabilities of thiazole, benzothiazole, and their dipolar nitro and amino derivatives possessed... 

We will divide the survey into three parts (3.1) static dipole polarizabilities, (3.2) static dipole hyperpolarizabilities, and (3.3) dynamic dipole polarizabilities and hyperpolarizabilities. Within each part there will be sub-sections dealing with the three isoelectronic series He, Ne, and Ar. For (3.2) and (3.3) the hydrogen atom will also be included. 

Table 1 Various properties for a water molecule in the gas phase and in an water solvent. The FD results were obtained with a frozen-density approximation, and the QM with a polarizable-molecules approximation. Listed are the size of the dipole moment, the three independent components of the quadrupole moment, excitation energies, static polarizability, and static hyperpolarizability. All results are from ref. 28... 

The experimental measures of these molecular electric properties involve oscillating fields. Thus, the frequency-dependence effects should be considered when comparing the experimental results . Currently, there are fewer calculations of the frequency-dependent polarizabilities and hyperpolarizabilities than those of the static properties. Recent advances have enabled one to study the frequency dispersion effects of polyatomic molecules by ab initio methods In particular, the frequency-dependent polarizability a and hyperpolarizability y of short polyenes have been computed by using the time-dependent coupled perturbed Hartree-Fock method. The results obtained show that the dispersion of a increases with the increase in the optical frequency. At a given frequency, a and its relative dispersion increase with the chain length. Also, like a, the hyperpolarizability y values increase with the chain length. While the electronic static polarizability is smaller than the dynamic one, the vibrational contribution is smaller at optical frequencies. ... 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

Table 14 Electronic and nuclear relaxation contributions to the polarizabilities and hyperpolarizabilities of structures I, ref 96. Static and infinite frequency limit values are given for pnr(—co a>, 0)...

Table 4. The basis set dependence of the average static polarizability and the second hyperpolarizability of benzene. The lib, the DFT Potential/Kcrnel combinations and the CCSD methods were used. Reproduced from [42]...

Values for the static polarizabilities of, obtained without use of the Born-Oppenheimer approximation, are reported by Bhatia and Drachman.105 The agreement with recent measurements is good. Shigeta et a/.106 have formulated a non-Born-Oppenheimer density functional theory and have used it to calculate the polarizability and hyperpolarizabilities of the H2 molecule and its isotopes. 

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