Dipole hyperpolarizability

The next terms in the series, denoted. .. in equation 17.1 above, are called the dipole hyperpolarizabilities. The first one is and this also is a tensor. It has three indices, and the corresponding formula for the induced dipole, equation 17.3, becomes... 

Thus coefficients with an even total order I + m + n are real and coefficients with an odd total order I m + n are pure imaginary. In the following we consider only dipole hyperpolarizabilities. In this case the four operators A, B, C and D are cartesian components of the dipole operator and the odd dispersion coefficients vanish. 

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]... 

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. 

Dykstra and Jasien [9] used the general equations given in the preceding to implement an approach for the calculation of derivatives of the Hartree-Fock or SCF energy. Unique to the DHF approach is its open-endedness. The computer program that was written was immediately able to compute a tenth-dipole hyperpolarizability and beyond, if desired. Derivatives involving geometric parameters could be obtained, too, but then there exists the... 

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. 

The situation is somewhat different for the convergence with the wavefunction model, i.e. the treatment of electron correlation. As an anisotropic and nonlinear property the first dipole hyperpolarizability is considerably more sensitive to the correlation treatment than linear dipole polarizabilities. Uncorrelated methods like HF-SCF or CCS yield for /3 results which are for small molecules at most qualitatively correct. Also CC2 is for higher-order properties not accurate enough to allow for detailed quantitative studies. Thus the CCSD model is the lowest level which provides a consistent and accurate treatment of dynamic electron correlation effects for frequency-dependent properties. With the CC3 model which also includes the effects of connected triples the electronic structure problem for j8 seems to be solved with an accuracy that surpasses that of the latest experiments (vide infra). 

Naively, one would expect that second hyperpolarizabilities y are theoretically and experimentally more difficult to obtain than first hyperpolarizabilities (3. From a computational point of view the calculation of fourth-order properties requires, according to the 2n + 1-rule, second-order responses of the wavefunction and thus the solution of considerably more equations than needed for j3 (cf. Section 2.3). However, unlike (3 the second dipole hyperpolarizability y has two isotropic tensor... 

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 Homogeneous Field Dipole Polarizability and Dipole Hyperpolarizabilities... 

Table 5.10 Vibrational contributions to the first dipole hyperpolarizability of cyclopropenone,...

Table 5.11 Contributions to vibrational corrections to the parallel component (Pzzz) of the first dipole hyperpolarizability ...

The homogeneous field dipole polarizability and dipole hyperpolarizabilities... 

Similarly, we may obtain the perturbational expressions for the dipole, quadru-pole, octupole hyperpolarizabilities, etc. For example, the ground-state dipole hyperpolarizability /3o has the form (the qci q" component, where the prime means that the ground state is omitted - we skip the derivation) ... 

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. 

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