Instantaneous Coulomb interaction

Corrections to the mean-field model are needed to describe the instantaneous Coulombic interactions among the electrons. This is achieved by including more than one Slater determinant in the wavefunction. 

Note that the Breit-type operators are often neglected in quantum chemistry because they yield small energy contributions in comparison to the instantaneous Coulomb interaction. However, the effects may not be negligible in highly accurate quantum chemical calculations or for spin- or magnetic-field-dependent properties such as those measured by magnetic resonance spectroscopies. 

In their considerations on the field generated by a single moving charged particle, Chubykalo and Smimov-Rueda [2,56,57] have claimed the Lienard-Wiechert potentials to be incomplete. These potentials are then not able to describe long-range instantaneous Coulomb interaction. However, in a modified theory by Chubykalo and Smimov-Rueda such interaction is included. The applicability of these potentials is, however, still under discussion [9]. 

To arrive at Eq. (180) we have used the definitions (145), (148), (171) and (175) of the density response functions. Furthermore, we have abbreviated the kernel of the (instantaneous) Coulomb interaction by w(x, x ) = 3(t — t )/ r — r. Finally, by inserting Eq. (180) into (168) one arrives at the time-dependent Kohn-Sham equations for the second-order density response ... 

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 Hamiltonian (2.2) contains only the instantaneous Coulomb interaction between the carriers forming the crystal. Therefore, as in (12), the excitons corresponding to the operator (2.2) will be called Coulomb excitons. 

Note, however, that the dependence of the polariton energy on the wavevec-tor, which arose when only retardation is taken into account, is correct only if we can neglect the dependence of the energy of the Coulomb exciton Etl on k, arising from instantaneous Coulomb interaction. For example, if we apply this theory for 2D quantum well polaritons, the linear term in the dispersion of po-laritons will be cancelled because in this case the linear term as a function of the energy of the quantum well exciton on the wavevector has the same value with opposite sign. 

From this presentation it is clear that any expansion in Za is useless even in the region of intermediate Z and the calculation has to be performed to all orders of Za. Methods for these calculations were developed by decomposing the intermediate bound state into several terms but keeping all orders of Za. In the following discussion we employ the usual bound interaction ( Furry ) picture of QED. The external field is considered as instantaneous Coulomb interaction according to an infinitely heavy atomic nucleus. Deviations from this assumption result only in minor corrections and will be discussed later on. The Furry picture results in a possible separation of time and space variables contrary to the covariant formulation of eqs. (6-8). 

The 00-component simplifies to give the instantaneous Coulomb interaction between charges... 

The spectrum of the single-electron Dirac operator Hd and its eigenspinors (/> for Coulombic potentials are known in analytical form since the early days of relativistic quantum mechanics. However, this is no longer true for a many-electron system like an atom or a molecule being described by a many-particle Hamiltonian H, which is the sum of one-electron Dirac Hamiltonians of the above kind and suitably chosen interaction terms. One of the simplest choices for the electron interaction yields the Dirac-Coulomb-Breit (DCB) Hamiltonian, where only the frequency-independent first-order correction to the instantaneous Coulomb interaction is included. 

Our decision in favor of combining nonrelativistic quantum mechanics with the nrl of electrodynamics becomes very important, when we consider the interaction between moving electrons (section 7). In the nrl there is only a nonretarded (instantaneous) Coulomb interaction, while both the magnetic interaction and the retardation of the Coulomb interaction are relativistic corrections and are therefore neglected. One needs to consider them only if one also includes relativistic corrections to the kinematics. 

It contains the one-electron Dirac Hamiltonian hp plus the nuclear potential, V, and the operator Vy = 1/ry for the instantaneous Coulomb interaction between electrons... 

In non-relativistic theory only the first term of Eq. (13), which comes from the instantaneous Coulomb interaction, is present. All remaining contributions to the total energy are summarised in the exchange-correlation energy, Exc- actual calculations one has to use (rather crude) approximations to calculate Exc the four-current. 

The non-relativistic limit of the electron-electron interaction is the instantaneous Coulomb repulsion, also called the longitudinal part of the electron-electron interaction. The remainder is called the transversal part and mainly covers retardation and magnetic interactions (this separation is somewhat gauge dependent). The longitudinal and transversal part of the relativistic local exchange functional read... 

Fig. 3 Graphical representation of the second-order wave operator (17). The solid lines represent single-electron orbitals and the dashed lines instantaneous Coulomb interactions. The second folded diagram represents the part with the intermediate state in the model space (P)...

We use here the Coulomb gauge, where the interaction can be separated in an instantaneous Coulomb part and a Breit interaction that can be retarded. The Breit part is represented by two interactions of the type (27) with the /(k) function in Eq. (42) given by... 

The precise form of this correction depends upon the gauge condition used to describe the electromagnetic field. In the Coulomb gauge, which has been employed more often in relativistic atomic structure, the electron-electron interactions come from one-photon exchange process and is sum of instantaneous Coulomb interaction and the transverse photon interaction. 

The generic term van-der-Waals forces refers to any interaction between electrically neutral atoms and molecules which results from the non-uniform distribution of electrons around the atomic nuclei. This non-uniformity may be observable as a permanent polarisation of molecules (e.g. for water), yet it primarily exist on an instantaneous level due to the electron oscillations. Atoms and molecules are, therefore, fluctuating dipoles that interact via Coulombic forces, which eventually produces a net attraction. The strength of these forces diminishes very rapidly with the intermolecular distance r, ,- the corresponding energy potential uy obeys a power-law decay (Kralchevsky et al. 2002, Chap. 5.4.2) ... 

---