Hyperpolarizabilities expansion

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 ... 

The generality of a simple power series ansatz and an open-ended formulation of the dispersion formulas facilitate an alternative approach to the calculation of dispersion curves for hyperpolarizabilities complementary to the point-wise calculation of the frequency-dependent property. In particular, if dispersion curves are needed over a wide range of frequencies and for several optical proccesses, the calculation of the dispersion coefficients can provide a cost-efficient alternative to repeated calculations for different optical proccesses and different frequencies. The open-ended formulation allows to investigate the convergence of the dispersion expansion and to reduce the truncation error to what is considered tolerable. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

An alternative compact expansion with coefficients which are independent of the optical process can be derived for the isotropic parallel average of the second hyperpolarizability 7 defined as [13]... 

The much better performance of the diagonal Fade approximants compared to the Taylor expansion is not unexpected since similar convergence trends were found previously for the expansion of the first hyperpolarizabilities and u>) of ammonia. The good... 

A crude experimental estimate for the A s coefficient is obtained from Sheltons fit of ESHG measurements for the hyperpolarizability ratio 7 /7i to the expression 3(1 +A a 22)- Using the dispersion expansions (79) and (89), we obtain for the ratio 7 /7x up to second order in the frequencies the expansion... 

The expansion (Equation 24.12) does not contain even powers of the field because of the spherical symmetry of an isolated atom. Indeed for an atom, the even derivatives in Equation 24.10 are zero as well as for any molecule having an inversion center. Note that a3 and a5 are, in fact, the components of tensors, respectively of the so-called second and fourth hyperpolarizabilities [4]. 

The values for the dipoles, polarizabilities, and hyperpolarizabilities of the H2 series were obtained using (a) a 16-term basis with a fourfold symmetry projection for the homonuclear species and (b) a 32-term basis with a twofold symmetry projection for the heteronuclear species. These different expansion lengths were used so that when combined with the symmetry projections the resulting wave functions were of about the same quality, and the properties calculated would be comparable. A crude analysis shows that basis set size for an n particle system must scale as k", where k is a constant. In our previous work [64, 65] we used a 244-term wave function for the five-internal-particle system LiH to obtain experimental quality results. This gives a value of... 

The conjugation length expansion increases the second-order hyperpolarizability as good as donor-acceptor substitution with the advantage of better transparency in the visible and infrared spectra. 

The induced molecular dipole replaces the polarization and the local field, E, acting on the molecule is introduced in place of the macroscopic field. There are two conventions in use for defining the hyperpolarizability series one is the exact analogue of the macroscopic method (B convention), the other uses a Taylor series expansion (T convention) where a factor (l/ ) is introduced into wth order terms. The notation introduced by WRBS as been used. For a noncentros5unmetric molecule subjected to an internal field,... 

Table 2 lists the results of calculations of the static first hyperpolarizability. As the perturbation theory expansion shows, for w = 0, Kleinman symmetry holds exactly and the components of the tensor are invariant under permutations of the three co-ordinate indices so that Pz as defined in eqn (4.12) reduces to,... 

The polarization can now be expanded in several different ways. For example, if the expansion is in terms of powers of the first two terms on the right of eqn (8.2) which together comprise the effective fields as defined by the conventional field factors then effective polarizabilities and hyperpolarizabilities will be defined which incorporate the effects of the discrete terms that have been omitted from the fields in the power expansion. If the expansion is in terms of all contributions except then the polarizabilities and hyperpolarizabilities so defined will... 

It is possible to differentiate the quantum-mechanical electronic energy beyond first order, and means for doing this are discussed in Section III. The second derivatives are the usual polarizabilities, the third derivatives are the hyperpolarizabilities, and so on. These properties are associated with a power series expansion of the energy in terms of the elements of V. A second-degree polytensor is introduced for handling all the polarizabilities [7]. It is a square matrix whose rows and columns are labeled, in anticanonical order, by the same indices that label the elements of the column array M. For example. 

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). 

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