Electric fields electronic hyperpolarizabilities

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 square of the molecular wave function 2, defines the molecular charge distribution. The wave function of state i, ( ), can be calculated in the presence or absence of an electric field. Details of the zero field occupied and unoccupied states determine the size of the hyperpolarizability. Summing the expectation value of the electronic position over occupied states (1 to M) gives the polarization [11]... 

In the simplest case of a donor-acceptor (D-A) molecule, the nonlinear optical activity arises from the electric-field-induced mixing of electronic states such as D-A and D+-A . This makes the response (polarizability) of the molecule different according to the sense of the electric field, and a second-order hyperpolarizability fi coefficient) is observed. If D and A are connected by some bridge, its role in promoting the electronic interaction will be quite similar to the bridge role in mixed-valence complexes. Metal complexes can play the role of donor or acceptor groups. Recent examples have been described with ferrocene or ruthenium(pentaammine) groups [48], but they are either monometallic or too short to be considered in this review. 

Apart from purely electronic effects, an asymmetric nuclear relaxation in the electric field can also contribute to the first hyperpolarizability in processes that are partly induced by a static field, such as the Pockels effect [55, 56], and much attention is currently devoted to the study of the vibrational hyperpolarizability, can be deduced from experimental data in two different ways [57, 58], and a review of the theoretical calculations of p, is given in Refs. [59] and [60]. The numerical value of the static P is often similar to that of static electronic hyperpolarizabilities, and this was rationalized with a two-state valence-bond charge transfer model. Recent ab-initio computational tests have shown, however, that this model is not always adequate and that a direct correlation between static electronic and vibrational hyperpolarizabilities does not exist [61]. 

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). 

Second, the development of methods and concrete numerical calculations of the constants (reduced matrix elements of the dipole and quadru-pole moments, polarizability, and hyperpolarizabilities, vibronic constant, etc.) determining the effects of electronic degeneracy on electric properties of molecules predicted in this paper seems to be one of the most up-to-date problem in the topics under consideration. Such calculations are quite possible, in principle, provided that the wave functions of the degenerate electronic term (for the calculation of the dipole moment), as well as the excited ones (for the calculation of the polarizabilities), are known. Considering the advances in quantum chemistry, the solution of the problem is quite possible from the practical point of view, especially if one takes into account that in the cases under consideration one can determine numerically the wave function of the system in the presence of an electric field instead of a calculation of excited states. 

Hyperpolarizabilities can be calculated in a number of different ways. The quantum chemical calculations may be based on a perturbation approach that directly evaluates sum-over-states (SOS) expressions such as Eq. (14), or on differentiation of the energy or induced moments for which (electric field) perturbed wavefunctions and/or electron densities are explicitly calculated. These techniques may be implemented at different levels of approximation ranging from semi-empirical to density functional methods that account for electron correlation through approximations to the exact exchange-correlation functionals to high-level ab initio calculations which systematically include electron correlation effects. 

The tensor component of the molecular polarizability (ajj) and hyperpolarizabilities (jSjjk and yyki) may be calculated by taking the appropriate derivatives of the total electronic energy or dipole moment with respect to the external electric field ... 

One of the hurdles in this field is the plethora of definitions and abbreviations in the next section I will attempt to tackle this problem. There then follows a review of calculations of non-linear-optical properties on small systems (He, H2, D2), where quantum chemistry has had a considerable success and to the degree that the results can be used to calibrate experimental equipment. The next section deals with the increasing number of papers on ab initio calculations of frequency-dependent first and second hyperpolarizabilities. This is followed by a sketch of the effect that electric fields have on the nuclear, as opposed to the electronic, motions in a molecule and which leads, in turn, to the vibrational hyperpolarizabilities (a detailed review of this subject has already been published [2]). Section 3.3. is a brief look at the dispersion formulas which aid in the comparison of hyperpolarizabilities obtained from different processes. 

To obtain hyperpolarizabilities of calibrational quality, a number of standards must be met. The wavefunctions used must be of the highest quality and include electronic correlation. The frequency dependence of the property must be taken into account from the start and not be simply treated as an ad hoc add-on quantity. Zero-point vibrational averaging coupled with consideration of the Maxwell-Boltzmann distribution of populations amongst the rotational states must also be included. The effects of the electric fields (static and dynamic) on nuclear motion must likewise be brought into play (the results given in this section include these effects, but exactly how will be left until Section 3.2.). All this is obviously a tall order and can (and has) only been achieved for the simplest of species He, H2, and D2. Comparison with dilute gas-phase dc-SHG experiments on H2 and D2 (with the helium theoretical values as the standard) shows the challenge to have been met. 

In general, the physical properties of an electron system are defined by referring to a specific perturbation problem and can be classified according to the order of the perturbation effect. For instance, the electric dipole moment is associated with the first-order response to an applied electric field (i.e. the perturbation), the electric polarizability with the second-order response, hyperpolarizabilities with higher-order terms. In addition to dipole moments, there is a number of properties which can be calculated as a first-order perturbation energy and identified with the expectation value... 

The polarizability of an atom or molecule describes the response of the electron cloud to an external field. The atomic or molecular energy shift KW due to an external electric field E is proportional to i for external fields that are weak compared to the internal electric fields between the nucleus and electron cloud. The electric dipole polarizability a is the constant of proportionality defined by KW = -0(i /2. The induced electric dipole moment is aE. Hyperpolarizabilities, coefficients of higher powers of , are less often required. Technically, the polarizability is a tensor quantity but for spherically symmetric charge distributions reduces to a single number. In any case, an average polarizability is usually adequate in calculations. Frequency-dependent or dynamic polarizabilities are needed for electric fields that vary in time, except for frequencies that are much lower than electron orbital frequencies, where static polarizabilities suffice. 

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