Hyperpolarizability fourth-order

The pointwise given MCSCF results for zero point vibrational corrections for the ESHG hyperpolarizability were added to the CCSD results obtained from the dispersion coefficients and than fitted to a fourth-order polynomial in... 

G. Maroulis and A. J. Thakkar, J. Chem. Phys., 88, 7623 (1988). Multipole Moments, Polarizabilities and Hyperpolarizabilities for N2 from Fourth-Order Many-Body Perturbation Theory Calculations. [Erratum, ibid., 89, 6558 (1988). 

Abstract. We have calculated the scalar and tensor dipole polarizabilities (/3) and hyperpolarizabilities (7) of excited ls2p Po, ls2p P2- states of helium. Our theory includes fine structure of triplet sublevels. Semiempirical and accurate electron-correlated wave functions have been used to determine the static values of j3 and 7. Numerical calculations are carried out using sums of oscillator strengths and, alternatively, with the Green function for the excited valence electron. Specifically, we present results for the integral over the continuum, for second- and fourth-order matrix elements. The corresponding estimations indicate that these corrections are of the order of 23% for the scalar part of polarizability and only of the order of 3% for the tensor part... 

In the perturbation theory for degenerate states the resonant hyperpolarizability is determined by the tensor part of polarizability [9] and may be extracted out of the fourth-order terms self-consistently in the case of nondegenerate perturbation theory the resonant part appears for separate sublevels of an atomic multiplet. The numerical results are listed in Table 2. 

The secular divergences in Eq. (65), i.e. the singularities due to terms in which one or more of the states equal 0), can be removed in a similar manner as we did for the lower-order properties. In fact, explicit nondivergent formulas have been derived for molecular hyperpolarizabilities up to fourth order [5], and for the second-order hyperpolarizability, the resulting formula can be written as (66) 7a/37s(-"<7 l t 2> 3)= E -<7,1,2,3... 

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... 

The microscopic origin of the three-photon absorption can be traced to the fourth order hyperpolarizability. Despite the high order of this property, it is still attainable due to two decisive steps of simplification. First, at resonance, i.e. under excitation with a frequency equal to one-third to the excitation energy of the final state, Wy/3, a fraction of the terms will dominate the summation completely, which leads to a formulation where the fourth order hyperpolarizability can be expressed as a product of third order transition dipole moments ... 

The derivatives of energy in the above equations can be obtained by numerical differentiation (Finite Field approach, FF). For instance, to calculate the second- and fourth-order derivatives which correspond to polarizability and 2nd hyperpolarizability the following seven point formulaes can be used ... 

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)... 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

Coherent Raman effects originate in the third-order susceptibility fourth-rank tensor) which is the bulk version of the second molecular hyperpolarizability gufifi in (4.1). For the general non-... 

Finally, fhe application of higher order nonlinear optical processes such as third-harmonic generation (Berkovic 1995 Tsang 1995) or fourth-harmonic generation (Lee et al. 1997) could provide even more detailed interface information. For example, in the case of fourth-harmonic generation the induced polarization is given by Pj 4u>) = j k,l,m,nx]tL Ek u>)Ei co)Em u>)E io), i.e., the hyperpolarizability is a tensor of rank 5. Hence if is possible fo resolve up to five-fold surface symmetries. However, the absolute values are... 

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