Two-photon tensor

The two-photon tensor invariants proportional to two-photon absorptivities have been calculated for the transitions to the lowest singlet states, (BjJ and La+ (B J60 and for the prediction of the site of bromination, formylation, hydroxymethylation and nitration ofimida-zo[l,2-6][l,2,4]triazines.61... 

The CNDO/2 method has been used for the calculation of the sites of the protonation and alkylation of 1,2,4-triazines <88KGS525>. It was also applied to calculate the two-photon tensor invariants proportional to two-photon absorptivities for transitions to the lowest singlet states, Lb (B2 -) and L/ (B,/) <84CPL(107)125>. 

As noted in Section 2.1, the second-rank two-photon tensor Eq. 2.78, involving a sum over the manifold of excited electronic states, can be obtained from the single residue of the appropriate QRF,... 

It is evident from the form of Equation [17] that the two-photon tensor and hence the extent of two-photon absorption, is significantly enhanced if the molecule possesses an electronic excited state of an energy close to E + fuo, for then, in the sum over r, the term that state contributes has an absolute minimum value for the denominator. This situation, typical of multiphoton processes, is known as resonance enhancement, and its physical basis can be understood in terms of the time-energy uncertainty principle ... 

The two-photon transition operator is a tensor whose components may be nonzero. Thus an important reason for doing two-photon spectroscopy is that it allows us to observe the transitions directly as allowed transitions instead of indirectly as forbidden transitions as are all one-photon spectra of transition metal ions. 

Other commonly measured x interactions are those occurring with degenerate frequencies. Such interactions include phenomena such as four-wave-mixing, self-focusing, two-photon absorption, and CARS. If a single laser beam is focused into a material, the three applied fields interacting through the nonlinear tensor are coincident, and the interpretation is very similar to that for the... 

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. 

Based on the MCSCF/CM quadratic response method it is possible to calculate the hyperpolarizability tensor and the two-photon absorption cross-sections. The calculated MCSCF/CM properties exhibit for all the individual tensor components substantial shifts compared with the corresponding molecular properties of the molecule in vacuum. 

The basis of NLO-effects arising from susceptibilities of second order, is the interaction of three electric fields with a material. The practical implementation of optical devices requires strong, coherent and monochromatic radiation and hence, laser technology. Not all of the interacting fields need to be optical fields, however. In devices that make use of the Pockels effect, an externally applied electric field is used to alter reversibly the refractive index of a material. In a second harmonic generation (SHG) process two photons of circular frequency w can be transformed into one photon of frequency Iw. SHG is the NLO effect used most for the evaluation of /3-tensor elements in solution. 

So- r is fh two-photon transition moment tensor that can be visualized as a (3 X 3) matrix with elements (So r)a < which contain the sums of the products If the ground state % s of even (g) parity the inter-... 

The elements of transition moment tensors reflect the symmetry of the states involved, so even the allowed transitions can be observed only if the polarization of the photons in the molecular frame matches the nonzero components of the tensor. A major difference with respect to one-photon spectroscopy lies in the fact that this dependence on polarization does not vanish upon averaging over all orientations, and two-photon measurements on isotropic samples such as liquid solutions provide polarization information. 

The P tensor components are obtained from the slope of the scattered light intensity at harmonic frequency versus the square of the fundamental beam intensity, measured for different polarization configurations. The value of the g parameter intervening in Eqs. (9)-( 11) is obtained by calibration with a solvent of known /lvalue. These HRS measurements have to be done very carefully, as other processes such as two-photon-induced fluorescence, Raman, and higher order NLO processes can contribute to the measured HRS signal. Actually, the use of the time-re-solved technique with femtosecond pulses allows for the separation of photons coming from the harmonic generation of the others. 

In the present contribution the discussion of the NLO response is restricted to off-resonant case. The only exception is the purely resonant quantity, namely imaginary part of second-order hyperpolarizability in the resonant regime (Im-y(-tt> tu, —w, w)). This quantity describes the process of simultaneous absorption of two quanta. The two-photon absorption (TPA) process is much better understood than the three-photon absorption. The basic quantity associated with the two-photon absorption process is the two-photon absorption tensor (S ). In the most general case referring to two different photons (different polarizations and different energies is given by [75, 81] ... 

The averaging procedure of die two-photon absorption tensor over all orientations of the absorbing molecule leads to [75, 81] ... 

Table 6-1. C h molecular point group. The electronic states of the flat Tg molecule are classified according to the two-fold screw axis (C2), inversion (z), and glide plane reflection (ct/,) symmetry operations. The and excited states transform like translations (7) along the molecular axes and are optically allowed. The Ag and It, states are isomorphous with the polarizability tensor components (a), being therefore one-photon forbidden and two-photon allowed.

This molecular response tensor is a more general form of three apparently different tensors which have featured in previous work on multiphoton processes one is the Xy2 tensor arising in the single-frequency distributive two-photon absorption (Andrews and Harlow 1984a), and another is the tensor which appears in the theory of three-photon absorption (Andrews and Wilkes 1985). It is also exactly identical to the two-frequency hyper-Raman transition tensor (Andrews 1984). 

In this mechanism, two-photon transitions are forbidden and the excitation of the participating molecules occurs through one- and three-photon allowed transitions. Both the real (laser) photons are absorbed by one molecule, excitation of its partner resulting from the virtual photon coupling. Because of the difference in selection rules from the previous case, the first two terms of Eq. (5.13) are now zero, and contributions arise only from the third and fourth terms. It must also be noted that setting the two absorbed photon frequencies to be equal in Eq. (5.16) to produce zJy, (co,o>) introduces index symmetry into the tensor, as indicated by the brackets embracing the first two indices. A factor of j must then be introduced into the definition of this tensor in order to avoid over-counting contributions. The transition matrix... 

In the two-frequency distributive case, the molecular tensor y °(a)i,o)2) has resonance conditions similar to those for )- As the single-beam case, two of the proposed resonance conditions would be likely to allow the process to be masked by single-photon absorption a third leads to the possibility of conventional two-photon absorption, and a fourth cannot be satisfied if the centers involved are initially in their ground states. The remaining condition E x E — hu>i) if a>i t02, or E x(E — ho)2) if CO2 < ft)i) remains the only truly useful resonance. Naturally, since the energetics of the excitation process are constrained only by a condition on the sum of the photon frequencies, there is a wide scope for choosing laser frequencies specifically with the aim of exploit ng this type of resonance possibility. 

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