Polarizabilities calculated properties

The molecular electronic polarizability is one of the most important descriptors used in QSPR models. Paradoxically, although it is an electronic property, it is often easier to calculate the polarizability by an additive method (see Section 7.1) than quantum mechanically. Ah-initio and DFT methods need very large basis sets before they give accurate polarizabilities. Accurate molecular polarizabilities are available from semi-empirical MO calculations very easily using a modified version of a simple variational technique proposed by Rivail and co-workers [41]. The molecular electronic polarizability correlates quite strongly with the molecular volume, although there are many cases where both descriptors are useful in QSPR models. 

Ah initio calculations of polymer properties are either simulations of oligomers or band-structure calculations. Properties often computed with ah initio methods are conformational energies, polarizability, hyperpolarizability, optical properties, dielectric properties, and charge distributions. Ah initio calculations are also used as a spot check to verify the accuracy of molecular mechanics methods for the polymer of interest. Such calculations are used to parameterize molecular mechanics force fields when existing methods are insulficient, which does not happen too often. 

The calculation of polarizabilities is one of the research topics Jens Oddershede is working on since the beginning of his career [1-21], Already in one of his first papers he discussed the dipole polarizability of HF [1] and returned to it several times later [3,6,13,14,18]. Therefore, we decided to contribute to this special issue with a study of static dipole and quadrupole polarizabilities which are still one of the most studied electromagnetic properties. 

A Consequence of the Instability in First-order Properties.—Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places does an error in the fourth matter To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability a. In this method a perturbation term Ai—— fix(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wave-function calculated as usual. For small uniform fields,... 

Zgaevskii VE (1977) Theoretical description of the elastic properties of micro-inhomogeneous polymers. Polym Mech 13(4) 624-625 Zgaevskii VE, Pokrovskii VN (1970) Calculation of polarizability of macromolecule as elastic thread. Zh Prikl Spektrosk 12(2) 312-317 (in Russian)... 

We do not here attempt to review the developments of methods for calculating molecular polarizabilities and magnetizabilities, nor will we attempt any review of theoretical results obtained for these properties. We will instead outline the approach for calculating molecular properties in general, and the polarizability and the magnetizability of closed-shell systems in particular, using ab-initio methods. We will also devote some time to discuss... 

Theoretical knowledge of molecular electrical moments and polarizabilities has reached an important threshold. Methods have advanced considerably in the last several years and have reached the point that elusive electrical properties of molecules are obtainable from calculation. At the same time, there has been significant advancement in understanding the role of these properties in certain chemical phenomena, and application of electrical interaction ideas is likely to become even more widespread. A survey of calculated properties is presented here. 

General Theory. In addition to the work already quoted, a few papers considering various aspects of the general theory of polarizabilities are also included. We mention the efforts to consider the effects of the polarizability of relatively crude models (such as metallic spheres) [101, 103]. The tensor representations of molecular polarizabilities have been carefully discussed [84, 141, 362]. More general procedures applicable to a variety of systems (including big molecules and molecular complexes) that are often based on classical electrostatics have been proposed [4, 21, 118, 119, 127, 135, 136]. Various books and articles are concerned with the definition of the terms in common use [12, 13, 20, 91, 109]. Other articles deal with relevant general properties of matter [22, 113] and molecules in electric and magnetic fields [31,239], triplet polarizabilities [10,61, 102,126,208], and the calculations of polarizabilities of molecules in the liquid. 

We show how the response of a molecule to an external oscillating electric field can be described in terms of intrinsic properties of the molecules, namely the (hyper)polarizabilities. We outline how these properties are described in the case of exact states by considering the time-development of the exact state in the presence of a time-dependent electric field. Approximations introduced in theoretical studies of nonlinear optical properties are introduced, in particular the separation of electronic and nuclear degrees of freedom which gives rise to the partitioning of the (hyper)polarizabilities into electronic and vibrational contributions. Different approaches for calculating (hyper)polarizabilities are discussed, with a special focus on the electronic contributions in most cases. We end with a brief discussion of the connection between the microscopic responses of an individual molecule to the experimentally observed responses from a molecular ensemble... 

The toxicity of compounds has often been related to the polarizability of compounds. This descriptor is related to the intermolecular interactions in biological environments and can be ascribed both to the drug-receptor interactions as well as to the properties determining the bioavailability of a compound [112], Thus it was shown that even the CNDO/2 calculated molecular polarizability (a) can be successfully correlated with the acute toxicity in a series of 20 nitriles [113] ... 

The finite field method is the simplest method for obtaining nonlinear optical properties of molecules. This method was first used by Cohen and Roothaan to calculate atomic polarizabilities at the Hartree-Foclc level. The basic idea is to truncate the expansion of the energy (Eq. [6]) and solve for the desired coefficients by numerical differentiation. For example, if the expression is truncated after the quadratic term, the result is E(P) = E[0) — — iot yF,Fy. 

By suitable choices for the P, Q, R,... operators a whole variety of properties may be calculated. The polarizability corresponding to imaginary frequencies, for example, provides the van der Waals dispersion coefficients, with the leading term depending on the inverse sixth power of the interatomic distance between atoms A and... 

Cmtempa-ary numerical methods allow routine calculation of polarizability. It is difficult with the hyperpolariz-abdities that are much more sraisitive to fhe quality of fhe atomic basis set used. The hyperpolarizabilities relate to nonlinear properties, which are in high demand in new materials for technological applications. 

As noted in the Sect. 3.3.2, the polyacenes are characterized by a more complex structure of the wave function, and therefore, to adequately describe this structure a higher level of theory is needed. It is expected that for the polycyclic aromatic hydrocarbons discussed in this section the selection of an appropriate correlation radius is a very important aspect of the calculation. To study the effect of the level of accounting for the electron correlation effects for polyacenes, we have calculated the polarizability and 2nd hyperpolarizabUity values for different levels of the cue-CCSD theory. In Figs. 3.14 and 3.15 the dependencies of the specific values of these properties on the number of the r-electrons are shown. 

It should be pointed out that there are alternative methods to calculate properties other than direct differentiation. For instance, the second-order perturbation theory correction to the energy for a perturbing electric field yields an expression for the dipole polarizability tensor, a. 

Thus, having necessary properties of free molecules, it is not difficult to calculate the interaction-induced dipole moment using suggested formulas. As the components of the properties (polarizabilities, multipole moments, etc.) are dependent, in general case, on the orientation of molecules (Appendix A) we have a multidimensional surface of the dipole moment for a complex. 

On the other hand, linear and nonlinear responses are evaluated in a strategy of designing new systems with remarkable electric, magnetic, and optical properties. In this second motivation for calculating the polarizabilities and h q)erpolarizabilities, the emphasis is also put on the interpretation and the deduction of structure-property relationships while these investigations are parts of multidisciplinary approaches including synthetic and experimental characterizations. 

For second energy derivatives that appear in the calculation of polarizabilities, chemical hardness, van der Waals coefficients, vibrational frequencies and other second order properties, the perturbed density matrix is required. McWeeny s self-consistent perturbation (SCP) theory (Diercksen and McWeeny 1966 Dodds et al. 1977 McWeeny 1962, 2001 McWeeny and Dier-cksen 1968 McWeeny et al. 1977) represents a direct approach for the calculation of this matrix. For the clarity of the presentation we assume perturbation-independent basis and auxiliary functions and restrict ourselves to closed-shell systems. Under these conditions the elements of the perturbed density matrix are given by the SCP formalism of McWeeny et al. (1977) ... 

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

The quantum mechanical approach to the scattering process is quite different from the classic model the wave-particle duality of a light beam is incorporated by considering that the beam is made up of packets or quanta of light particles known as photons. Moreover, the quantization of molecular energy levels is taken into account and a means is provided for calculating the polarizability a, and thus Raman intensities, in terms of the electronic properties of a molecule. 

The discussion presented above outlines some definite trends of changes in the effective induced bond chaiges evaluated in analyzing oh in///o calculated atonuc polarizability tensors. are closely related with the polarizability properties of the respective bonds and depend strongly on bond lengths, atomic polarizabilities, bond multiplicity and conjugation with other bonds. 

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