Overlap amplitude

For atoms and molecules it is usual to make the weak coupling expansion for the one electron target-ion overlap amplitude [1,2] ... 

Eigenfunctions that accompany these eigenvalues have a clear physical meaning that corresponds to electron attachment or detachment. These functions are known as Dyson orbitals, Feynman-Dyson amplitudes, or generalized overlap amplitudes. For ionization energies, they are given by... 

Now assume that P = af af and Q — a a. Following the same reasoning as above we see that m> = Al + 2> if 0> = V>isan N-electron state. The poles and residues yield energy differences and overlap amplitudes of Auger spectra, respectively. Two-electron propagators of the type afia a yy have in... 

O. Goscinski and P. Lindner, Natural Spin Orbitals and Generalized Overlap Amplitudes, Preliminary Research Report No. 238, Quantum Chemistry Group, Uppsala University, 1969. 

O. Gosdnski and B. T. Pickup Generalized Overlap Amplitudes and Transition Spin Orbitals Chem. Phys. Lett. 33, 265 (1975). 

Figure Al.6.24. Schematic representation of a photon echo in an isolated, multilevel molecule, (a) The initial pulse prepares a superposition of ground- and excited-state amplitude, (b) The subsequent motion on the ground and excited electronic states. The ground-state amplitude is shown as stationary (which in general it will not be for strong pulses), while the excited-state amplitude is non-stationary. (c) The second pulse exchanges ground- and excited-state amplitude, (d) Subsequent evolution of the wavepackets on the ground and excited electronic states. Wlien they overlap, an echo occurs (after [40]).

If aos on one atom overlap aos on more than one neighboring atom, mos that involve amplitudes on three or more atomic centers can be formed. Such mos are termed delocalized or multicenter mos. 

In (a), two photon waves combine to give a new waveform, which has the same appearance and frequency as the initial separate waves. The photons are said to be coherent, and the amplitude of the waves (light intensity) is simply doubled. In (b), the two photon waves are shown out of step in time (incoherent). Addition of the two waveforms does not lead to a doubling of amplitude, and the new waveform is more complex, composed of a doubled overlapping frequency. In (c), the two waveforms are completely out of step (out of phase) and completely cancel each other, producing darkness rather than light (an interference phenomenon). 

For explicitness, let us assume that off-diagonal disorder is caused by chain twists, which randomly diminish the overlap between the n-orbitals of neighboring carbon atoms (see Fig. 3-7). The electron hopping amplitudes that depend both on the interatomic distances and on the relative orientation of the electronic orbitals on neighboring atoms can then be written in the form ... 

FIGURE U5 When two ls-orbitals overlap in the same region of space in such a way that their wavefunctions have the same signs in that region, their wavefunctions (red lines) interfere constructively and give rise to a region of enhanced amplitude between the two nuclei (blue line). 

The amplitude of the frontier orbitals determines the selectivity. The most reactive atom in a molecule has the largest amplitude of the frontier orbitals. The frontier orbitals overlap each other to the greatest extent at the sites with the largest amphtudes. Reactions occur on the atoms in the electron donors and acceptors, where the HOMO and LUMO amplitudes are largest, respectively. Electrophiles prefer the a position of naphthalene, an electron donor, with the larger HOMO amplitude (Scheme 21). Nucleophiles attack the carbons of the carbonyl groups, an electron acceptor, with the larger LUMO amplitude (Scheme 7). 

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