Transition moment third order

Periodic oscillations in this dipole can act as a source term in the generation of new optical frequencies. Here a is the linear polarizability discussed in Exps. 29 and 35 on dipole moments and Raman spectra, while fi and x are the second- and third-order dielectric susceptibilities, respectively. The quantity fi is also called the hyperpolarizability and is the material property responsible for second-harmonic generation. Note that, since E cos cot, the S term can be expressed as -j(l + cos 2 wt). The next higher nonlinear term x is especially important in generating sum and difference frequencies when more than one laser frequency is incident on the sample. In the case of coherent anti-Stokes Raman scattering (CARS), X gives useful information about vibrational and rotational transitions in molecules. 

The single residue related to third-order transition moments is... 

The three-photon absorption cross section is related to the fifth order susceptibility which clearly forms a challenging computational task. Most often we are interested in the resonant absorption to a particular state, f, in which case the cross section can be expressed in terms of the third-order transition moment to this state, which in turn is much easier to compute. Hence, for 3PA it is sufficient to evaluate the third-order transition moment, T ", as... 

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 ... 

Even though the response approach represents a significant simplification, it is still computationally demanding for three-photon absorption, and the third order transition moments are therefore obtained at the RPA level, i.e. cubic response... 

Table 2. Excitation energies of the final state, /ioj, the third order transition moment, 7",. and the three-photon probability, calculated at SCF level with the basis sets 6-31G and 6-310. Geometry is optimized by DFT/B3LYP with 6-3IG ...

We recalculated the three-photon absorption coefficient, y, making use of the formulas which relate the third order transition dipole moment, three-photon transition probability, 8 and 7 ... 

We can deduce the symmetry of a response tensor by considering the operators that enter the numerator of its quantum mechanical expression. For example, the product of three electric-dipole transition moment operators in Eq. (14) render SFG a parity-odd and time-even process. It follows that a third-order process requires nonlocal magnetic-dipole contributions in order to be parity-odd and that a local fourth-order process is parity-odd within the electric-dipole approximation. Some pseudoscalars that arise at order n are tabulated below. 

In addition to expectation values and transition moments, we have a third class of important physical properties, which we refer to as second-order response properties. To develop some understanding for the meaning of and theoretical methods for studying these responses, let us investigate the response of a state 0> corresponding to H to an external time-independent one-cicetron perturbation (a//i)... 

The shapes of the orbitals for the other molecules of Fig. 2 are shown in Fig. 5. Molecules B and C of Fig. 2 have donors at the both ends. For B and C in Fig. 5, HOMO-1 spreads somewhat into extremities, and electron-hole symmetry does not hold either between HOMO and LUMO nor between HOMO-1 and LUMO-i-1. These shapes are beneficial for %(3). However, the orbital HOMO does not spread into the extremities. This means that transition from HOMO-1 to HOMO does not have so much moment as that of SBA. The orbital shape for molecules B and C is not so good as SBA. With respect to the orbitals of the molecules E and F, both HOMO and HOMO-1 spread, and the electron-hole symmetry does not hold. These are beneficial for Molecules D, E and F, which have amino at both ends and have nitrogen atoms at the X positions of Fig. 1, are expected to have large third order susceptibilities. These results correspond well with the experimental fact that their measured X is large and as good as conjugated polymers. The x values of SBA and SBAC, which is dichloro SBA, are 1.0 X 10 esu and 1.3 x lO" esu. The X( ) of PU-STAD is 1.5 X 10 esu, which is a derivative of molecule F in Fig. 2. 

A sufficient amount of oriented chiral molecules can be obtained in an induced cholesteric liquid crystal phase if the induced helical structure has been untwisted by an electric field. In the following description tensors are needed for the sake of simplicity (At least there are three tensors required the transition moment tensor (absorption tensor ,y), the rotational strength tensor (circular dichroism tensor A ,y), and the order tensor g,y33 (i,j= 1,2,3). If the molecules do not possess any symmetry, the principal axes of all of these tensors are differently oriented with respect to the molecular frame (the coordinate system in which only the three diagonal elements of a tensor are different from zero).) The only tensorial property, needed here explicitly, is the existence of three coordinates (components) of a tensor with respect to three specially chosen mutually perpendicular axes. This means that three information instead of one information about a molecule are needed instead of one CD value, namely Ae, three CD values, namely As, (i=l, 2, 3), have to be introduced. Ac is then one-third of a sum of the three so-called tensor coordinates of the CD tensor ... 

Linear optical processes give important information about the energies of the dipole allowed states. However, dark states - namely those with no dipole moment to the ground state - are inaccessible. Nonlinear optical processes, on the other hand, involve transitions between two or more states, so these access the dipole-forbidden states. In this chapter we explain how third order nonlinear process can be used to identify these forbidden states. 

Here, f, Fx and are local field factors, a is a molecular parameter including the dipole moment, second order and third order hyperpolarizabilities. Under condition F = Fx = the formula does not agree with the experiment for 8CB. Indeed, it follows that rji should increase at the transition from the nematic to smectic A phase according to the increase in (F2) However, in the experiment, Fn markedly decreases. The similar temperature behavior was earlier observed for dielectric permittivity of 8CB. In the latter case the decrease in is due to the antiparallel correlation of molecular dipoles in the smectic A phase, which results in a decrease in the effective dipole moment /x Thus, the decrease... 

---