Transition hyperpolarizability

It is possible to develop expressions for hyper-Rayleigh and Raman intensity components in terms of sixth-rank tensor invariants, analogous to the familiar fourth-rank invariants given above, together with quantum-mechanical expressions for transition hyperpolarizability tensors. However, these expressions are too complicated to give here the articles by D.A. Long in reference 4] should be consulted for further details. 

These expressions include averaging over all orientations of the chromo-phore with respect to the field and the polarization of the measuring light The factors Uij and bijj in Eqs. (B4.15.2) and (B4.15.3) are elements of the transition polarizability tensor (a) and the transition hyperpolarizability (b), which describe the effects of the external field on the dipole strength of the absorption band. These effects probably are relatively minor in most cases, and are neglected in Eq. (4.64). The transition polarizabifity tensor... 

If neither cOyis nor cOsfg is in resonance with an electric dipole transition in the material and only electric dipole transitions are considered, the hyperpolarizability. 

Methods that are known to calculate transition matrix elements reliably for the systems of interest (e.g., 7r-electron systems) have been used extensively [13,17]. Especially for /3 calculations, where relatively few electronic states often dominate the hyperpolarizability, numerical methods are reliable. However, 7 calculations are more complicated because of the larger number of contributing terms and the possibility of subtle cancellations that can occur only when the full series is summed. General aspects of / and 7 calculations are discussed in the next section. 

This paper is a more extensive survey of the influence of the metal on the hyperpolarizability of a series of the transition metal tetrakis(cumylphenoxy)-phthalocyanines (MPcCP4). The compounds chosen were those most closely related to PtPcCP4, the compound which showed the largest hyperpolarizibility in the previous study. Specifically, phthalocyanines substituted with the last four members of the first row transition metal series (Co, Ni, Cu, and Zn) and also with the Ni, Pd, Pt triad were prepared and studied. The near IR spectra of these tetrakis(cumylphenoxy)-phthalocyanines are briefly discussed. Speculation on how metal substitution can influence the third order susceptibility of a near centrosymmetric structure, like that of the phthalocyanines, is presented. 

The data in Table 1 reveal systematic variations in the measured third order susceptibilities of these phthalocyanines with the metal. There is a monotonic variation of y in the series Co, Ni, Cu, Zn. The nonlinear susceptibility decreases as the d orbitals of the metal become filled. There is also a qualitative correlation between a large hyperpolarizibility and the presence of a weak, near IR transition. However, the variation of the figure of merit, x(3)/ , shows that the correlation between y and the absorption coefficient is not linear. No clear trend is seen in the triad Ni, Pd, Pt although PtPcCP4 does have a larger hyperpolarizibility as might be expected for a larger, more polarizable metal ion. 

The contributions of optically forbidden electronic states to the x(3) of centrosymmetric structures are of particular interest. (18) Each of the terms in a sum-over-states calculation of x(3) involves the product of transition moments between a sequence of four states. There are symmetry selection rules that govern which states which can contribute to the individual terms. In a centrosymmetric molecule the symmetry of the contributing states must be in a sequence g -> u --> g --> u --> g.(19) This means that all the non-zero terms in the summation which determines the hyperpolarizibility must include an excited electronic state of g symmetry (or the ground state) as an intermediate state. The tetrakis(cumylphenoxy)phthalocyanines are approximately centrosymmetric and many of the new electronic states in a metal phthalocyanine will be of g symmetry. Such states may well contribute to the dependence of the hyperpolarizibility on metal substitution. 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

Similar to the average hyperpolarizability, the two-photon absorption cross-sections are also affected by the interactions with the structured environment. For forbidden transitions we have observed that the structured environment perturbs these transitions significantly. Generally, the results from the MCSCF/CM model including polarization contributions compare very well with the available experimental data on two-photon cross-sections of liquid water. 

Fig. 11.3 Variation of fl for frequency doubling (solid lines) and the electrooptic effect (dotted line) with incident light frequency, m, according to the two-level Eqs. (17) and (18), respectively. /J0 is the static hyperpolarizability and a>ge is the transition energy from the ground state to the excited state responsible for / . Note that for frequency doubling there is an additional resonance seen when...

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