Interaction of Two Electrons

Consider two electrons, one on orbital /i, and another one on orbital /2. The repulsion energy of these two electrons is given by  

Renumbering electrons 1 and 2 in the last two terms, one can observe that the two positive and the two negative terms are identical, respectively  

Let us do the same derivation using second quantization. The interaction operator is a two-electron operator, thus it takes the form  

This result is quite general it is valid for any number of orbitals. In our particular case, the labels i and j are either 1 or 2, thus we get the same result as in Eq. (6.21)  

If the number of orbitals increases, the first quantized derivation presented above becomes more and more involved. As the size of the determinant increases, one should apply the general Slater-Condon rules, or rederive them similarly as was done for the Hiickel energy expression (Sect. 6.4). On the other hand, the general result of Eq. (6.23) is valid for any number of electrons and orbitals. 

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The first-order perturbation theory in this case does not give good results, since the mutual interaction of two electrons on one nucleus is so large as to greatly deform the eigenfunctions it leads... 

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

Fig. 5.1 Theoretical energy curves (a-J, f) for the hydrogen molecule. Hj. compared with the experimental curve (e). Curves o d show successive approximations m the wave function as discussed in the text. Curve /is the repulsive interaction of two electrons of like spin.

If two (or more adjacent) distinct chromophores exist in a chiral arrangement with respect to one another in a dendrimer molecule, the CD spectrum shows two intense Cotton effects of opposite signs which merge with each other [80]. This phenomenon, known as an exciton couplet, arises from the interaction of two electronic transition moments in a mutually chiral orientation, such as oc-... 

G. Breit, Dirac s equation and the spin-spin interactions of two electrons, Phys. Rev. 39 (1932) 616. 

It is possible to obtain the nuclear spin magnetic interaction terms by starting from the Breit equation. We recall that the Breit Hamiltonian describes the interaction of two electrons of spin 1 /2, each of which may be separately represented by a Dirac Hamiltonian ... 

A favourable interaction of two electrons with opposite m, values in the same orbital. 

The bond A— A, e.g. Na—Na in the molecule Nag, and the bond B—B, e.g. Cl—Cl in Clg are formed by the interaction of two electrons from different atoms, both electrons participating in the formation of the bond to the same degree. According to the method of Pauling, one half of the energy of the bond is apportioned to each electron, and each electron of the atoms A and B contribute the same energy to the molecule AB, as to the molecules AA and BB. 

Consider the interaction of two electrons, e and c2, that are located in the AOs and (frv. We do not exclude the possibility that the two electrons are in the same AO, pi = v, provided that they have opposite spin. The time-averaged distribution of electron 1 is given by 2(ci)dvi and that of electron 2 by 2(c2)dv2 (Bom interpretation). Therefore, the... 

To begin with, Schrbdinger attempted to interpret corpus(il( .s, and particularly electrons, as wave packets. Although his formuhn are entirely correct, his interpretation cannot be maintained, since on the one hand, as we have already explained above, the wave packiits must in course of time become dissipated, and on the other hand the description of the interaction of two electrons as a collision of two wave packets in ordinary three-dimensional space lands us in grave difficulties. 

The Hartree Fock determinant describes a situation where the electrons move independently of one another and where the probability of finding one electron at some point in space is independent of the positions of the other electrons. To introduce correlation among the electrons, we must allow the electrons to interact among one another beyond the mean field approximation. In the orbital picture, such interactions manifest themselves through virtual excitations from one set of orbitals to another. The most important class of interactions are the pairwise interactions of two electrons, resulting in the simultaneous excitations of two electrons from one pair of spin orbitals to another pair (consistent with the Pauli principle that no more than two electrons may occupy the same spatial orbital). Such virtual excitations are called double excitations. With each possible double excitation in the molecule, we associate a unique amplitude, which represents the probability of this virtual excitation happening. The final, correlated wave function is obtained by allowing all such virtual excitations to happen, in all possible combinations. 

Minimization of the interelectronic distance (Boys method) is in fact similar in concept to the maximization of the Coulomhic interaction of two electrons in the same orbital (Ruedenberg method). 

Auger decay The interaction of two electrons in higher shells by the electrostatic repulsion forces one of the two electrons to fill the inner-shell vacancy while the other electron is emitted into the continuum, thereby carrying the excess energy as kinetic energy. 

Presentations of the semi-classical theory are rare. The textbook by Mott and Sneddon [169] from 1948 is an example where this has been attempted in some detail, although — after having derived the quantum mechanical energy expression for the retarded electromagnetic interaction of two electrons — the authors come to the conclusion that... it loill he appreciated that the derivation does not depend on any very consecutive argument [169, p. 339]. In the following we will see that the situation is actually not that bad. 

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