Electric properties polarizabilities dynamic

Molecular dynamic studies used in the interpretation of experiments, such as collision processes, require reliable potential energy surfaces (PES) of polyatomic molecules. Ab initio calculations are often not able to provide such PES, at least not for the whole range of nuclear configurations. On the other hand, these surfaces can be constructed to sufficiently good accuracy with semi-empirical models built from carefully chosen diatomic quantities. The electric dipole polarizability tensor is one of the crucial parameters for the construction of such potential energy curves (PEC) or surfaces [23-25]. The dependence of static dipole properties on the internuclear distance in diatomic molecules can be predicted from semi-empirical models [25,26]. However, the results of ab initio calculations for selected values of the internuclear distance are still needed in order to test and justify the reliability of the models. Actually, this work was initiated by F. Pirani, who pointed out the need for ab initio curves of the static dipole polarizability of diatomic molecules for a wide range of internuclear distances. 

Molecular electric properties give the response of a molecule to the presence of an applied field E. Dynamic properties are defined for time-oscillating fields, whereas static properties are obtained if the electric field is time-independent. The electronic contribution to the response properties can be calculated using finite field calculations , which are based upon the expansion of the energy in a Taylor series in powers of the field strength. If the molecular properties are defined from Taylor series of the dipole moment /x, the linear response is given by the polarizability a, and the nonlinear terms of the series are given by the nth-order hyperpolarizabilities ()6 and y). 

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. ... 

Electron correlation plays a role in electrical response properties and where nondynamical correlation is important for the potential surface, it is likely to be important for electrical properties. It is also the case that correlation tends to be more important for higher-order derivatives. However, a deficient basis can exaggerate the correlation effect. For small, fight molecules that are covalently bonded and near their equilibrium structure, correlation tends to have an effect of 1 5% on the first derivative properties (electrical moments) [92] and around 5 15% on the second derivative properties (polarizabilities) [93 99]. A still greater correlation effect is possible, if not typical, for third derivative properties (hyperpolarizabilities). Ionic bonding can exhibit a sizable correlation effect on hyperpolarizabilities. For instance, the dipole hyperpolarizability p of LiH at equilibrium is about half its size with the neglect of correlation effects [100]. For the many cases in which dynamical correlation is not significant, the nondynamical correlation effect on properties is fairly well determined with MP2. For example, in five small covalent molecules chosen as a test set, the mean deviation of a elements obtained with MP2 from those obtained with a coupled cluster level of treatment was 2% [101]. 

Electric moments, polarizabilities, and hyperpolarizabilities for BH were calculated for the first time [23], as were field and field gradient polarizabilities [24]. Spectroscopic properties were calculated for BH using the coupled electron pair approximation. The potential curve for BH was calculated at 22 points and Rq was found to be 1.23115 A and p to be 1.244 D [21]. The radiative lifetime of the A state of BH was calculated from second-order polarization propagator calculations [25], and the singlet-triplet separation in BH was calculated using ab initio MO methods. The latter, described as the singlet-triplet separation, was found to be 31.9 kcal/mol [26]. Finally, the possible dynamical pathways in the system BH + H+ were probed [27]. 

Choosing a non-zero value for uj corresponds to a time-dependent field with a frequency u, i.e. the ((r r)) propagator determines the frequency-dependent polarizability corresponding to an electric field described by the perturbation operator QW = r cos (cut). Propagator methods are therefore well suited for calculating dynamical properties, and by suitable choices for the P and Q operators, a whole variety of properties may be calculated. " ... 

S. Kielich. The Determination of Molecular Electric Multipoles and Their Polarizabilities by Methods of Nonlinear Intermolecular Spectroscopy of Scattered Laser Light. In J. van Kranendonk (ed.), Intermolecular Spectroscopy and Dynamical Properties of Dense Systems—Proceedings of the International School of Physics Enrico Fermi," Course LXXV, North-Holland, Amsterdam, 1980, pp. 146-155. 

The molecular response tensors are characterized by peculiar properties and satisfy a series of very general quantum-mechanical relations. First we observe that the dynamic properties can be rewritten as a sum of the corresponding static property and a function multiplying the square of the angular frequency. Thus, for instance, in the case of dipole electric polarizability, using Eqn. (117),... 

Let us return to the problem of solving the response of the quantum mechanical system to an external electric field. The zeroth-order wave function of the quantum mechanical system is obtained by use of any of the standard approximate methods in quantum chemistry and the coupling to the field is described by the electtic dipole operator. There exist a number of ways to determine the response functions, some of which differ in formulation only, whereas others will be inherently different. We will give a short review of the characteristics of tire most common formulations used for the calculation of molecular polarizabilities and hyperpolarizabilities. The survey begins with the assumption that the external perturbing fields arc non-oscillatory, in which case we may determine molecular properties at zero frequencies, and then continues with the general situation of time-dependent fields and dynamic properties. 

A natural way to introduce equations for excited states into a quantum chemical approach is to consider stimulating the molecule by a time-varying electric field to which the molecule can respond by excitation, and derive solutions from the time-dependent Schroedinger equation. Analysis then leads to equations for the excitation energies and properties of the excited state eigensolutions like transition moments. In particular, such an approach, after a Fourier transformation from time to frequency, will yield the dynamic polarizability whose spectral expansion is... 

The (monochromatic) electric fields are characterized by Cartesian directions indicated by the Greek letters and by circular optical frequencies, coi, a>2, and The induced dipole moment oscillates at a> = EjO,. and are such that the jS and y values associated with different NLO processes converge towards the same static value. The 0 superscript indicates that the properties are evaluated at zero electric fields. Eqn (2) is not the unique phenomenological expression defining the (hyper)polarizabilities. Another widely-applied expression is the analogous power series expansion where the 1/2 and 1/6 factors in front of the second- and third-order terms are absent. The static and dynamic linear responses, o(0 0) and a(—correspond to the so-called static and dynamic polarizabilities, respectively. At second order in the fields, the responses are named first hyperpolarizabilities whereas second hyperpolarizabilities correspond to the third-order responses. Different phenomena can be distinguished as a function of the combination of optical frequencies. So, (0 0,0), a>,a>)... 

In the above discussion of the shell-model, we have assumed that the atoms or ions are not statically polarized. This is only the case for structures with sufficiently high symmetry. However, in general structures, the ions are often located at sites with low symmetry and therefore carry static induced electronic dipoles. Examples are layer structures such as Pbl. Because of the large polarizabilities of the nonmetals, we expect that their electric static dipole moments are important for the understanding of static and dynamic properties of these layer compounds. An extended shell model for... 

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