Electrons coulomb interaction

Such an equation differs from Hartree s equation only by virtue of the extra exchange term, TBx. Whereas the electronic Coulomb interaction of the Hartree scheme is formulated as... 

The existence of two coupling schemes for a shell of equivalent electrons is conditioned by the relative values of intra-atomic interactions. If the non-spherical part of electronic Coulomb interactions prevails over the spin-orbit, then LS coupling takes place, otherwise the jj coupling is valid. As we shall see later on, for the overwhelming majority of atoms and ions, including fairly highly ionized ones, LS coupling is valid in a shell of equivalent electrons, that is why we shall pay the main attention to it. 

The electronic Coulomb interaction u(r 12) = greatly complicates the task of formulating and carrying out accurate computations of iV-electron wave functions and their physical properties. Variational methods using fixed basis functions can only with great difficulty include functions expressed in relative coordinates. Unless such functions are present in a variational basis, there is an irreconcilable conflict with Coulomb cusp conditions at the singular points ri2 - 0 [23, 196], No finite sum of product functions or Slater determinants can satisfy these conditions. Thus no practical restricted Hilbert space of variational trial functions has the correct structure of the true V-electron Hilbert space. The consequence is that the full effect of electronic interaction cannot be represented in simplified calculations. 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

Of these, the pure electron-electron Coulomb interaction (4.14a) appears to be the obvious choice and, indeed, has been widely used [12,14,16]. The electron-electron contact interaction (4.14b), which only acts if both electrons are at the position of the ion (in effect, a three-body contact interaction), has also been frequently employed [15], Both interactions have been compared in various regards in [17,18,40]. More recently, the Coulomb interaction (4.14c), which is only effective if the second (bound) electron is located at the position of the ion, and the electron-electron contact interaction (4.14d), which is not restricted to the position of the ion, have also been studied [27]. The interactions (4.14b) and (4.14c) are effective three-body interactions, which attempt to take into account that the effective electron-electron interaction will depend on the positions of the electrons relative to the ion. An alternative interpretation, which formally leads to the same results, is to consider a two-body interaction Vi2 in (4.17) and a wave function (rlip ) in (4.18) that is extremely strongly localized at the position of the ion for details, see [27]. 

We are now in a position to present the total electronic Hamiltonian by summing over all possible electrons i. We must be very careful, however, not to count the various interactions twice. Thus on summing over i, we modify all terms which are symmetric in i and j by a factor of 1 /2. These terms are the electron-electron Coulomb interaction (3.141), the orbit-orbit interaction (3.145), the spin-spin interaction (3.151) and the spin-other-orbit interaction from (3.144) and (3.153). 

Table 3. Core-valence electron coulomb interaction terms, evaluated from relativistic Hartree-Fock-Slater wave functions...

Table 8. Effective core-valence electron Coulomb interaction energies and Eiatt term energies for Ti and C. (Values in eV)a)...

Most organic conductors behave as quasi-one-dimensional systems, at least at high temperatures. It is generally accepted that due to the narrow band-widths, the strongest interactions are the electron-electron Coulomb interaction, U for two electrons on the same site, and V for electrons in nearest-neighbor sites, in agreement with the extended Hubbard Hamiltonian, which is usually taken as a good approximation. 

However, as discussed extensively in review articles by Pouget [9] and by Barisic and Bjelis [10], the presence of both 2kF and 4kF anomalies, where kF is the Fermi wave vector of the quasi-one-dimensional electron gas, the fact that phonon softening at 2kF is relatively small, combined with theoretical considerations, have lead to the present-day viewpoint that electron-electron Coulomb interactions play an important role. 

It is conventional that the ligand field problem for systems with Na> d electrons requires the diagonalization of an effective Hamiltonian operator composed for the electronic kinetic energy T, and both one-electron ligand field terms, and two-electron Coulomb interactions ... 

When magnitude of the electron-electron Coulomb interaction increases, the system is expected to show a ehange from the SDW dominated state to the spin-Peierls one. This was verified in another system, (TMTTF(tetramethyltetra-thiafulvalene))2X and (TMTSF)2X [74]. The former materials have the narrower band width than the latter. This means that the role of Coulomb interaction is more important in the former system than in the latter. The former system is expected to have the spin-Peierls state and the latter one the SDW state. Systematic studies have proposed the phase diagram shown in Fig. 23 [74]. 

If we denote the electron-electron Coulombic interaction e2/ r — r by u(r,r )> then it is not difficult to show [41] that the electron-electron contribution Fee(t) in Eq. (29) involves the pair correlation function, say n2(r,r ), which is the diagonal element of the second order density matrix y2(r,r/,r,r/) through... 

However, the interaction potential between two charged particles, nucleus-electron or electron-electron, is not just the Coulomb interaction, since in the relativistic description a retarded, velocity-dependent interaction must be considered. The full and general derivation of these interaction potentials is involved and approximate relativistic corrections to the Coulomb interaction are used in general. The frequency-dependent correction to the electron-electron Coulomb interaction, the Breit operator... 

The electron-electron interaction is usually supposed to be well described by the instantaneous Coulomb interaction operator l/rn. Also, all interactions with the nuclei whose internal structure is not resolved, like electron-nucleus attraction and nucleus-nucleus repulsion, are supposed to be of this type. Of course, corrections to these approximations become important in certain cases where a high accuracy is sought, especially in computing the term values and transition probabilities of atomic spectroscopy. For example, the Breit correction to the electron-electron Coulomb interaction should not be neglected in fine-structure calculations and in the case of highly charged ions. However, in general, and particularly for standard chemical purposes, these corrections become less important. 

The effect of Z exchange between electrons can be taken into account by adding a weak correction to the electron-electron Coulomb interaction. This correction takes the form of a contact interaction... 

Although the proper point of departure for relativistic atomic structure calculations is quantum electrodynamics (QED), very few atomic structure calculations have been carried out entirely within the QED framework. Indeed, almost all relativistic calculations of the structure of many-electron atoms are based on some variant of the Hamiltonian introduced a half century ago by Brown and Ravenhall [1] to understand the helium fine structure. By decoupling the electron and radiation fields in QED to order a (the fine-structure constant) using a contact transformation. Brown and Ravenhall obtained a relativistic momentum-space Hamiltonian in which the electron-electron Coulomb interaction was surrounded by positive-energy projection operators. Owing to the fact that contributions from virtual electron-positron pairs are automatically projected out of... 

The Hartree-Fock approximation. The independent electron approximation, which includes the electron-electron Coulomb interaction and the exchange interaction between electrons with parallel spins. 

Notice that this is based not on the ab initio method but on the original Hartree method. With the ab initio method, the double-counted electron-electron Coulomb interaction must be removed from the total energy. In this way, the electronic states of molecules can be calculated using the LCAO-MO approximation. 

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