Dipole hyperpolarizability, electrical

Among the molecular properties introduced above are the permanent electric dipole moment /xa and traceless electric quadrupole moment a(8, the electric dipole polarizability aajg(—w to) [aiso(to) = aaa(—or, o>)], the magnetizability a(8, the dc Kerr first electric dipole hyperpolarizability jBapy(—(o a>, 0) and the dc Kerr second electric-dipole hyperpolarizability yapys(— ( >, 0,0). The more exotic mixed hypersusceptibilities are defined, with the formalism of modern response theory [9]... 

To understand the complete role of vibration in determining electrical properties, it is useful to consider a diatomic molecule in the harmonic oscillator approximation, where the stretching potential is taken to be quadratic in the displacement coordinate. The doubly harmonic model takes the various electrical properties to be linear functions of the coordinate. This turns out to be most reasonable in the vicinity of an equilibrium structure, but it breaks down at long separations. Letting x be a coordinate giving the displacement from equilibrium of a one-dimensional harmonic oscillator, the dipole moment, dipole polarizability, and dipole hyperpolarizability, within the doubly harmonic (dh) model, may be written in the following way ... 

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]. 

The conceptually simplest NLO property is the electric first dipole hyperpolarizability 13. Nevertheless, it is a challenging property from both the theoretical and experimental side, which is related to the fact that, as third-rank tensor, it is a purely anisotropic property. Experimentally this means that (3 in isotropic media (gas or liquid phase) cannot be measured directly as such, but only extracted from the temperature dependence of the third-order susceptibilities In calculations anisotropic properties are often subject to subtle cancellations between different contributions and accurate final results are only obtained with a carefully balanced treatment of all important contributions. 

In the previous section we discussed pure electric-dipole hyperpolarizabilities, in particular second harmonic generation. Another important class of NLO processes includes birefringences and dichroisms which can be rationalized (at least to lowest orders in perturbation theory) in terms of response functions involving, besides the electric-dipole, also magnetic-dipole and electric-quadrupole operators. Prominent examples related to quadratic response functions are ... 

David Pugh remarked that there seemed to be very much more to write about electric and magnetic properties than when David Bounds and I wrote our own Theoretical Chemistry SPR contribution all those years ago. New techniques in non-linear optics and non-linear spectroscopy have given a new impetus to the accurate calculation of quantities such as the dipole hyperpolarizability. 

Next, we would obtain higher-order dipole hyperpolarizabilities (y,... )< which wUl contribute to the characteristics of the way the molecule is polarized when subject to a weak electric field. 

The above calculation represents an example of the application to an atom of what is called the finite field method. In this method we solve the Schrodinger equation for the system in a given homogeneous (weak) electric field. Say, we are interested in the approximate values of Uqq/ for a molecule. First, we choose a coordinate system, fix the positions of the nuelei in space (the Born-Oppenheimer approximation) and ealeulate the number of electrons in the molecule. These are the data needed for the input into the reliable method we choose to calculate E S). Then, using eqs. (12.38) and (12.24) we calculate the permanent dipole moment, the dipole polarizability, the dipole hyperpolarizabilities, etc. by approximating E(S) by a power series of Sq A. 

G. Maroulis, Electric dipole hyperpolarizability and quadrupole polarizability of methane from finite-field coupled cluster and fourth-order many-body peituibation theory calculations. Chem. Phys. Lett. 226, 420 (1994)... 

G. Maroulis, A study of basis set and election correlation effects in the ab initio calculation of the electric dipole hyperpolarizability of ethene (H2C=CH2). J. Chem. Phys. 97(6), 4188-4194 (1992)... 

Gora et al reported systematic study of interaction-induced electric properties in linear HCN oligomer chains. The authors reported electric dipole moments, polarizabilities and hyperpolarizabilities for the sequence HCN, (HCN)2 and (HCN)3 at the HF, MP2, CCSD and CCSD(T) levels of theory with the aug-cc-pVQZ basis set. Excess electric properties were subsequently calculated at the same levels of theory. The excess mean second dipole hyperpolarizability for the dimer were found to be Ay X 10 = 0.2(HF), 2.0 (MP2), 1.0 (CCSD) and 1.2 (CCSD(T)) e ao Eh. For the trimer, the respective values are Ay x 10 = 2.8(HF), 4.6 (MP2), 2.8 (CCSD) and 3.6 (CCSD(T)) In addition, the... 

The perturbation operator in the calculation of electric dipole moments, electric dipole polarizabilities and hyperpolarizabilities, i.e. the electric dipole moment operator Eq. (4.30), contains the position vector r of the electrons, which implies that the tail of the wavefunction becomes important. However, this is not well described in GTOs as discussed before and it is therefore essential to include additional valence functions with very small exponents C - so-called diffuse basis functions. In the Pople-style basis sets this is done in the 6-31G-I- and 6-31G- -- - basis sets, where in the -h basis set one diffuse function is added only for second- and third-row atoms, while in the - -+ basis set one diffuse function is also added for hydrogen (Clark et al., 1983). In the series of correlation consistent and polarization consistent basis sets one set of diffuse functions of each type present in the basis set is added in the aug-cc-pVXZ (Kendall et at, 1992 Woon and Dunning Jr., 1993, 1994 Balabanov and Peterson, 2005) and aug-pc-n (Jensen, 2002c) version of these basis sets. In the series of correlation consistent basis sets it is also possible to add two or more sets of diffuse functions in the d-aug , t-aug and so forth versions. 

The quadratic response function describing the second-order induced electric dipole moment due to a uniform time-dependent electric field is related to the frequency-dependent electric dipole hyperpolarizability as... 

The first electric dipole hyperpolarizability, given by the quadratic response function... 

Pluta, X, Sadlej, A. J. (1998). HyPol basis sets for high-level-correlated calculations of electric dipole hyperpolarizabilities. Chemical Physics Letters, 297, 391. 

The perturbation V = H-H appropriate to the particular property is identified. For dipole moments ( i), polarizabilities (a), and hyperpolarizabilities (P), V is the interaction of the nuclei and electrons with the external electric field... 

The molecular quantities can be best understood as a Taylor series expansion. For example, the energy of the molecule E would be the sum of the energy without an electric field present, Eq, and corrections for the dipole, polarizability, hyperpolarizability, and the like ... 

If neither cOyis nor cOsfg is in resonance with an electric dipole transition in the material and only electric dipole transitions are considered, the hyperpolarizability. 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

Based on the fundamental dipole moment concepts of mesomeric moment and interaction moment, models to explain the enhanced optical nonlinearities of polarized conjugated molecules have been devised. The equivalent internal field (EIF) model of Oudar and Chemla relates the j8 of a molecule to an equivalent electric field ER due to substituent R which biases the hyperpolarizabilities (28). In the case of donor-acceptor systems anomalously large nonlinearities result as a consequence of contributions from intramolecular charge-transfer interaction (related to /xjnt) and expressions to quantify this contribution have been obtained (29). Related treatments dealing with this problem have appeared one due to Levine and Bethea bearing directly on the EIF model (30), another due to Levine using spectroscopically derived substituent perturbations rather than dipole moment based data (31.) and yet another more empirical treatment by Dulcic and Sauteret involving reinforcement of substituent effects (32). 

Experimental and theoretical results are presented for four nonlinear electrooptic and dielectric effects, as they pertain to flexible polymers. They are the Kerr effect, electric field induced light scattering, dielectric saturation and electric field induced second harmonic generation. We show the relationship between the dipole moment, polarizability, hyperpolarizability, the conformation of the polymer and these electrooptic and dielectric effects. We find that these effects are very sensitive to the details of polymer structure such as the rotational isomeric states, tacticity, and in the case of a copolymer, the comonomer composition. 

We have shown in this paper the relationships between the fundamental electrical parameters, such as the dipole moment, polarizability and hyperpolarizability, and the conformations of flexible polymers which are manifested in a number of their electrooptic and dielectric properties. These include the Kerr effect, dielectric polarization and saturation, electric field induced light scattering and second harmonic generation. Our experimental and theoretical studies of the Kerr effect show that it is very useful for the characterization of polymer microstructure. Our theoretical studies of the NLDE, EFLS and EFSHG also show that these effects are potentially useful, but there are very few experimental results reported in the literature with which to test the calculations. More experimental studies are needed to further our understanding of the nonlinear electrooptic and dielectric properties of flexible polymers. 

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