Three-photon matrix element

This expression is closely related to the three-photon matrix element describing an excitation from die molecular ground state to the excited state /) by simultaneous absorption of three photons. When the residue above is evaluated at ftjj = = -Wy o/3 it will provide the matrix element corresponding to absorp-... 

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. 

The residue analysis of the CRF yields different types of excited-state quantities such as three-photon transition matrix elements (three-photon absorption) [27], the two-photon matrix elements between excited states (the cross section for second-order transitions), and the excited-state polarizability (dynamic second-order property). 

Dipole matrix elements, one- vs. three-photon excitation, coherence spectroscopy, 163—166... 

Here the matrix elements Hi , = (e Hul e ) are operators with respect to nuclear = wave functions in the ground and excited electronic states and At, g = e dE g, and so forth. The two-and three-photon absorption operators D and T are defined by the r bove, identities. 

Carried out to yet a higher order in the perturbation, we will be able to obtain an expression for the three-photon absorption matrix element. The third-order amplitude, taken from Eq. (40), is... 

The three-photon absorption matrix element, symmetrized in the dummy indices, can thus be written as... 

For molecules or crystals having a symmetry center, the total eigenfunctions are classified into symmetric g states and antisymmetric u states. In the equation for a three-photon process in Table 1.5, if a symmetric molecule has a g state at i), m) will be the u state and n) will be the g state, so that the matrix elements have non-zero values. But this results in an u state for f )m, which is in contradiction with the fact that f) and i) should be in the same state for three-photon processes. When a molecule or crystal has no symmetry center, there is no distinction between g and u states. So, three-photon processes can proceed only in molecules and crystals without symmetry center. The intuitive correspondence between phenomena (schemes) and equations is emphasized in Table 1.5. 

The second factor in (2.66) describes quite generally the transition probability for all possible two-photon transitions such as Raman scattering or two-photon absorption and emission. Figure 2.30 illustrates schematically three different two-photon processes. The important point is that the same selection rules are valid for all these two-photon processes. Equation (2.66) reveals that both matrix elements D,- and Dkf must be nonzero to give a nonvanishing transition probability A,/. This means that two-photon transitions can only be observed between two states i) and I/) that are both connected to intermediate levels fe) by allowed single-photon optical transitions. Because the selection rule for single-photon transitions demands that the levels i) and A ) or A ) and /) have opposite parity, the two levels i) and I/) connected by a two-photon transition must have the same parity. In atomic two-photon spectroscopy s s or s d transitions are allowed, and in diatomic homonuclear molecules Eg Eg transitions are allowed. 

For molecular transitions the matrix elements Dim and Dmf are composed of three factors the electronic transition dipole element, the Franck-Condon factor, and the Honl-London factor (Sect. 1.7). Within the two-photon dipole approximation a co) becomes zero if one of these factors is zero. The calculation of linestrengths for two- or three-photon transitions in diatomic molecules can be found in [243, 244]. 

An important aspect in control schemes based on the Stark effect is the choice of the frequency of the control field. It should be chosen such as to ensure a non-resonant interaction with the molecule. In our case, as mentioned in Sect.7.2.1, all the elements of the dipole moment matrix along the z direction are zero by symmetry. However, two-photon transitions between the Si and S2 states can be mediated by the non-zero af2(Q) matrix element. A value of hujc = 1-8 eV, which is high with respect to any two-photon transition between the Si and S2 states was chosen. Calculations with peak intensities of 0,10, 20,30,40 and 50 TW/cm for the control field were performed. The peak intensity of the pump pulse was set to 0.2TW/cm. In order to address various parts of the spectmm, three different photon energies (4.6,4.7 and 4.8 eV) were considered. The TDSE of Eq. (7.27), for each set of parameters was solved using the MCTDH method in the multi-set formalism. In each... 

Fig. 3. Construction of an electron energy distribution curve (EDC) from a density of states. The top panel depicts a parabolic density of states with structure centered around E-,. For simpUcity, we show photoexcitation of three initial state levels by a photon energy hv with no account taken of dipole matrix element effects. p is the Fermi energy, Vo is the inner potential,

From the discussed residues, we can determine properties explicitly dependent on different electronic states, for instance aU the transition moments coupling the reference state 0) and other states e) (from linear response residues) and aU the one-photon transition moments coupling other pairs of states (from quadratic response double residues). In addition, from single residues of the second- and third-order response functions we can identify two- and three-photon transition matrix elements. 

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