Wave function overlap

The quantity J dr is called the vibrational overlap integral, as it is a measure of the degree to which the two vibrational wave functions overlap. Its square is known as the Franck-Condon factor to which the intensity of the vibronic transition is proportional. In carrying out the integration the requirement that r remain constant during the transition is necessarily taken into account. 

As a result they form an almost free electron gas that spreads out over the entire metal. Hence, the atomic sp electron wave functions overlap to a great extent, and consequently the band they form is much broader (Fig. 6.10). 

Vibrational factors involve the operation of the vibrational wave-function overlap with the radiationless transitions. 

Here Psa is the transfer probability. Esa represents the resonance condition (in practise the spectral overlap of the emission of S and the relevant absorption of A) and occurs in both formulas. The quantity gsA comprises the optical strengths of the relevant transitions and a distance-dependence of the type rsA n=6,8, etc.). The quantity /sa, however, is proportional to the wave function overlap of S and A and comprises an exponential distance-dependence. 

Oces the presence or one or more nodes and maxima have any chemical effect The answer depends upon the aspect or bonding in which we are interested. We shall see later that covalent bonding depends critically upon the overlap of orbitals. Conceivably, ir one atom had a maximum in its radial wave function overlapping with a region with a node (minimum) in the wave function of a second atom, the overlap would be poor.4 However, in every case in which careful calculations liave been made, it has been found that the nodes lie too close to the nucleus to affect the bonding appreciably. 

The definite integral J dr is called the overlap integral, S, between the two wave functions A and The reason for this name is clear. For example, if both A and ip are s orbitals, we obtain the picture shown in Figure 2-2. We see that the two wave functions overlap each other in the region between the two nuclei. 

Factors of 10-100 indicate that spin projections of the initial and final states are parallel, but the wave function overlap is not favorable. 

For all the following considerations it is an important fact that within the CC of interest mutual chromophore wave function overlap and electron exchange effects among different chromophores do not take place (absence of the Dexter mechanism). Therefore, we may assume the orthogonality relation Pvnf Pnb) — A/ ,i A,.h to be valid, where A, / - /A,) denotes the electronic... 

If inserted into the time-dependent Schrodinger equation (49) an multiplication with 4>q and from the left results in equations of motion for the expansion coefficients. In doing so, one also produces overlap expressions like (o), 4>m d/dt o), and /dt n) (non-adiabatic couplings), which all can be neglected in line with the neglect of the mutual chromophore wave function overlap. Therefore, we obtain the expansion coefficient s equations of motion as... 

The metallic bond can be seen as a collection of molecular orbitals between a large number of atoms. As Figure A.6 illustrates, the molecular orbitals are very close and form an almost continuous band of levels. It is impossible to detect that the levels are actually separated from each other. Rather, the bands behave in many respects similarly to the orbitals of the molecule in Figure A. 5 if there is little overlap between the electrons, the interaction is weak and the band is narrow. This is the case for the d-electrons of the metal. Atomic d-orbitals have pronounced shapes and orientations that are largely retained in the metal. This in contrast to the s electrons, which are strongly delocalized that is, they are not restricted to well-defined regions between atoms, and form an almost free electron gas that spreads out over the whole metal. Hence, the atomic s-electron wave functions overlap to a great extent, and consequently the band they form is much broader. 

Since the two wave functions overlap, we can form from them two orthogonal wave functions using the Graham—Schmidt orthogonalization procedure,... 

Since no physisorption well is present, the question has to be considered how the inclusion of a physisorption state would alter the trapping dynamics. Physisorption wells are created by a combination of the attractive van der Waals interaction with Pauli repulsion caused by the overlap of molecular and substrate wave functions. While the former effect is not reproduced by the DFT calculation, the repulsion due to wave function overlap is well described by present DFT functionals. Hence, the calculated PES would only become more attractive if van der Waals forces were correctly included. For a more quantitative description of the trapping process at kinetic energies below 0.05 eV certainly the physisorption channel has to be included. 

If two conductors are so close that their leak out electron wave functions overlap, then an electron can tunnel back and forth through the vacuum or potential barrier between them, generating a tunneling current (Fig. 3). 

We have already noted that hole transport exhibiting an overall pattern of behavior similar to PMPS is observed (50) in poly(methylcyclohex-ylsilylene), the aliphatic system most closely resembling PMPS but distinguished by the absence of tt electrons. On the basis of simple chemical reasoning, one should expect no significant pendant wave function overlap in this system. From this and related comparative studies, the side groups in Si polymers do not seem to control transport critically. 

Dexter, following the classic work by Forster, considered energy transfer between a donor (or a sensitizer) S and an acceptor (or activator) A in a solid. This process occurs if the energy difference between the ground and excited states of S and A are equal (resonance condition) and if a suitable interaction between both systems exists. The interaction may be either an exchange interaction (if we have wave function overlap) or an electric or magnetic multipolar interaction. In practice the resonance condition can be tested by considering the spectral overlap of the S emission and the A absorption spectra. The Dexter result looks as follows ... 

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