Hyperpolarizability definition

It should be noted that polarizabilities of various orders can be defined in an alternative way in the SI system of units to that discussed previously. A quantity having the dimension of volume a = a/47re0 can be considered to be an SI analogue of the cgs polarizability. Analogously, y = -y/47re0 (or y = y/eo) can be used as the third-order hyperpolarizability in the SI system, with y having the units of m5 V 3. The presence or absence of the factor of An in the definition of the hyperpolarizability is, unfortunately, not always obvious in literature data. 

Unfortunately, however, there are several alternative definitions of ft and y (and higher hyperpolarizabilities) and many studies do not explicitly state which convention is being followed. Therefore, one should be careful in comparing values from different studies. These are discussed in detail, along with conversion factors, in Ref. [1], For example, the (n - l)th-order hyperpolarizability is commonly equated to (1/n ) X (dn i/dEn), rather than simply to the differential. 

Definition of the Nonlinear Optical Susceptibilities and Molecular Hyperpolarizabilities... 

The factors 2 and 4 in the denominators are due to the definition of the field amplitudes (Eq.(14) and Eq.(15)). In order to prevent these factors some authors drop the factor 1/2 in the definitions of the amplitudes. The disadvantage of this convention is the unusual convergence behaviour of the electric field as the frequency to approaches zero. This different field definition of course additionally complicates the comparison of different hyperpolarizability values. The factors in the numerator arise from the different possibilities to permute the input frequencies. As an example, in self-phase modulation the three input electric fields each provide a factor 1/2 which, with the factor 1/2 from the polarization, results in a denominator of 4. The negative frequency for SPM allows three permutations yielding finally a prefactor of 3/4. 

While gas phase work on the h5q5erpolarizability of small molecules has been relatively free of problems concerned with the definitions of measured quantities and their formal relationship to computed quantities, the same cannot be said about solution studies of rather larger organic species. It is the latter that possess the very large nonlinear response functions that are of greatest interest. The prototype system for such studies has been 4-nitroaniline (pNA) and this review is mainly concerned with the relation between the measurements, in vacuo and in solution, of the hyperpolarizabilities of pNA and the closely related molecule, MNA (2-methyl, 4-nitroaniline) to ab initio and DFT calculations of these quantities. 

Whatever the validity of these formulae and the underlying cavity field theories it is very desirable that, before applying them, the definition and consistency of the macroscopic quantities as measured by different groups should be assessed. Much of the discussion in the literature deals directly with the final values of the hyperpolarizability presented in the various experimental papers. In calculating these values different versions of the above procedures may have been employed and, in particular, different values of the molecular dipole moment inserted into y to extract the y value. It is relatively easy to identify how the microscopic parameter has been obtained and which molecular convention is being used provided the identity of the macroscopic quantity is clearly established. The symbol, F), (and its derivative, (9F i/9H )o) is introduced here to denote provisionally a reported macroscopic nonlinearity before assessing its precise definition. The unprimed symbols are defined in accordance with eqn (4.16). The most troublesome ambiguity in the... 

The value attributed in column A to Bosshard et al. is quoted in their paper and has required only a conversion of the units. Their definitions make it clear that the quantity concerned is intended to be Tso that their hyperpolarizability is in the X convention. 

For group-theoretical selection of nonzero matrix elements of the hyperpolarizability components, one has to know the symmetry properties of the operator / < ( w). Owing to the definition (153), /3y (w) is symmetric with respect to the permutation of the indices./ and k, but it has no definite symmetry with respect to the permutation of all the indices and has no definite parity with respect to the operation of time reversal ... 

We have seen how the molecular properties in nonlinear optics are defined by the expansion of the molecular polarization in orders of the external electric field, see Eq. (5) beyond the linear polarization this definition introduces the so-called nonlinear hyperpolarizabilities as coupling coefficients between the two quantities. The same equation also expresses an expansion in terms of the number of photons involved in simultaneous quantum-mechanical processes a, j3, y, and so on involve emission or absorption of two, three, four, etc. photons. The cross section for multiphoton absorption or emission, which takes place in nonlinear optical processes, is in typical cases relatively small and a high density of photons is required for these to occur. 

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. 

Notation. The symbols a, f, y are used throughout to denote the electric field polarizability, first and second hyperpolarizabilities respectively, suitably qualified by frequency factors where necessary. The magnetizability is denoted by y and the nuclear screening tensor by a. The numerous but well-known acronyms specifying the computational procedures are used without definition. The possibly rather less well-known acronyms for the principal gauge invariant procedures are given in Table 1. 

In general case the multipole moments depend on the origin of the coordinate system (except for the charge of a molecule). It follows from the definition (2.1.1) if the vector r is shifted on any vector a. For uncharged molecules the origin dependence appears only for the mullipole moments of the rank n > 2. As a result, because in the book only the uncharged molecules and complexes are considered, their dipole moments and connected with them polarizabilities and first hyperpolarizabilities do not depend on the origin of the coordinate system. 

In practice, the calculation involves three steps the calculation of the property for three systems A- B, A and B. If the desired goal is the calculation of the interaction-induced mean dipole polarizability a) or second dipole hyperpolarizability (y), one must first obtain the Cartesian components of both tensors for all three systems. The general definition of the mean is... 

These expansions serve mainly as definitions of the polarizabilities and hyperpolarizabilities as proportionality constants in the correction terms to the permanent moments. The dipole polarizability a is a second-rank tensor with nine cartesian components the dipole-quadrupole polarizability and first dipole... 

Table B.l Definitions of tensor components of the electric polarizabilities and hyperpolarizabilities as derivatives of components of the field-dependent electric dipole Ha S,S) and quadrupole Qji(S, ) moments or of the field-dependent energy E , ). All derivatives have to be evaluated at zero field and field gradient.

The general definition of the quadratic response function, O Eq. 11.68, indicates its symmetry with respect to permutation of operators. Thus, for all the dipole hyperpolarizabilities we have... 

Electric Polarizabilities and Hyperpolarizabilities of Clusters Definitions and Theory... 

There are two approaches for the calculation of molecular second order hyperpolarizabilities. One is the finite field difference method, where a small external electric field is applied to the molecule via a modified Hamiltonian in order to obtain the induced dipole moment. The other approach is the sum-over-states (sos) method. " The sum-over-states method has been applied in a variety of semiempirical programs, such as CND0/S, 2 PPP, INDO/S, and AMI. There are currently different conventions used for the definition of hyperpolarizabilities. The one that will be discussed here is based on a Taylor series expansion and is commonly used. The first order hyperpolarizability corresponds to the third order energy and can be,expressed in the form of equation (11) if the Taylor series expansion is used. ... 

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