The coupling matrix element

It follows from these relations that in the vicinity of resonance the exciton in a hybrid structure, roughly speaking, lives half-time as a Frenkel exciton and half-time as a Wannier-Mott exciton. Thus, in a hybrid structure we can expect a strong exciton-light interaction typical for Frenkel excitons as well as strong resonance optical nonlinearity typical for Wannier-Mott excitons. 

To evaluate the matrix element F(k) determining the resonance interaction between Frenkel and Wannier-Mott excitons we write down the interaction Hamiltonian as 

Since our system is translationally invariant in two dimensions, it is convenient to consider the Fourier transform  

The matrix element of the IQW polarization between the ground state 0) and W, k) for a ls-exciton with Bohr radius aF is equal to (18), (19) 

Now we can write the final expression for the coupling matrix element  

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The immediate question is where (in phase space) to place the newly spawned basis functions. The optimal choice will maximize the absolute value of the coupling matrix element between the existing basis function (i.e., the... 

Note that, if the donor and acceptor s and p orbitals refer to the same atomic center, the coupling matrix elements and /pp- are identically zero, and hybridization cannot lower the energy. Hence, atomic hybridization is intrinsically a bonding effect. 

Figure 15 Calculated total and state-to-state excitation transfer cross sections in the de-excitation of He(2 P)-Ne. (From Ref. 151.) Both electron exchange and dipole-dipole interactions are included in the coupling matrix elements. The threshold energy into each exit channel is shown on the upper axis.

In practice, if the lower state energy and the corresponding wave function are known accurately then the coupling matrix element (4>o H i) is small. Experience shows that, because the finite basis set approximation is more restrictive for than it is for o, the calculated excited state energy lies above the corresponding exact value. 

Table 2 Effect of an external electric field F on the coupling matrix element Vda between the G and A sites of the dimer [(GC),(AT)j, calculated at the HF/6-31G level with the GMH, SFCD, and FCD methods ...

Figure 4.7. Variations in the adsorption energy (from Figure 4.1) with the size of the coupling matrix element (from Table 4.1) for Cu, Ag and Au. Adapted from Ref. [4].

We will now consider a number of systems where the energy of the adsorbate state(s), sa, and the coupling matrix element, Vad, are essentially constant. We will... 

In what follows we study the coupling matrix elements between vibronic states following the treatment of Lin,88 and of Bixon and Jortner.8 To begin, we display the coupling matrix elements in terms of intramolecular normal coordinates (see eq. (4-10)) ... 

It can be safely assumed that the contribution of the term involving the nuclear kinetic energy operator is negligibly small. The coupling matrix element then assumes the form... 

Finally, we display the general expressions for the coupling matrix elements for the case of internal conversion... 

The final state of the system, corresponding to the ground state of the molecule plus a photon, is represented by ipB = 0 k, e>. and the set 0, are the eigenfunctions of Hth while vac ) and k, e) represent the zero-photon and the one-photon eigenstates of HR, respectively. The time evolution of the amplitudes a,(t) and CE(t) can be computed from time-dependent perturbation theory. The equations of motion are determined by the energy levels of the zero-order states of Hel + HR, by the coupling matrix elements... 

Another case which may be treated analytically is the case of exact resonance, WA = WB, if in addition, I aa - bb — 0 and VAB — VgAt i.e. the coupling matrix element is real. In this case Eqs. (14.16) are readily decoupled, leading to two identical uncoupled equations. The equation for CA(f) is... 

That the observed resonances for m 0 become narrower and more symmetric as the power is increased can be understood in the following way, using the one-photon process as an example. The coupling matrix element of Eq. (15.2) has two... 

The coupling matrix elements due to the kinetic energy operator,... 

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