Electrons mean-field approximation

To improve upon die mean-field picture of electronic structure, one must move beyond the singleconfiguration approximation. It is essential to do so to achieve higher accuracy, but it is also important to do so to achieve a conceptually correct view of the chemical electronic structure. Although the picture of configurations in which A electrons occupy A spin orbitals may be familiar and usefiil for systematizing the electronic states of atoms and molecules, these constructs are approximations to the true states of the system. They were introduced when the mean-field approximation was made, and neither orbitals nor configurations can be claimed to describe the proper eigenstates T, . It is thus inconsistent to insist that the carbon atom... 

In order to perform the calculation., of the conductivity shown here we first performed a calculation of the electronic structure of the material using first-principles techniques. The problem of many electrons interacting with each other was treated in a mean field approximation using the Local Spin Density Approximation (LSDA) which has been shown to be quite accurate for determining electronic densities and interatomic distances and forces. It is also known to reliably describe the magnetic structure of transition metal systems. 

Mean-field approximation of quasi-free electrons (the Hartree-Fock approximation). The total wave function is described, in this case, by a single Slater determinant. 

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

The case of intermediate and strong electron-vibron interaction at intermediate coupling to the leads is the most interesting, but also the most difficult. The existing approaches are mean-field [131-133], or start from the exact solution for the isolated system and then treat tunneling as a perturbation [134-140]. The fluctuations beyond mean-field approximations were considered in Refs. [141,142]... 

We use now the results of the foregoing section to discuss the electronic transport properties of our model in some limiting cases for which analytic expressions can be derived. We will discuss the mean-field approximation and the weak-coupling regime in the electron-bath interaction as well as to elaborate on the strong-coupling limit. Furthermore, the cases of ohmic (s = 1) and superohmic (s = 3) spectral densities are treated. 

Fig. 37 Electronic transmission and corresponding current in the mean-field approximation for two different temperatures. Parameters N = 20, Jo/oJc = 0.12, t /t = 0.5,rL/R/t =0.5.

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

QMSTAT is an effective quantum chemical solvent model with an explicit solvent representation. Effective here means that the quantum chemical electronic Hamiltonian only pertains to a small subset of the total system (typically the solute), with the solvent entering as a perturbation operator to the Hamiltonian explicit solvent means that the solvent is described with a set of spatial coordinates and parametrized physical features significantly simplified compared to a full quantum chemical description. The explicit solvent representation implies that it is possible to go beyond the mean-field approximation inherent in the often used continuum... 

In ideally one-dimensional systems, only intrachain electron-phonon and spin-phonon couplings are, within mean-field approximation, at the origin of electronic-Peierls and/or spin-Peierls transitions, respectively. In real systems, such as the TCNQ salts under concern here, it is clear, however, that one should take properly into account the coupling of the electrons to external potentials also and, in the first case, to the periodic electrostatic cation potential. 

Although a mean-field approximation will not provide the correct answer to the problem of interacting one-dimensional electrons, a brief survey using such an approach is still useful to show that the competition between superconducting and dielectric order is the basic problem of one-dimensional conductors [25a,b]. 

The second drawback of the mean-field approximation is neglect of electron-phonon interaction, which should contribute attractively to the gi contribution and lead to a lattice modulation (Peierls instability). However, a ground state of such a nature is not stabilized in the TM2X family. The absence of long-range order is deeply rooted in the one-dimensional problem and arises when the effect of interactions on the different response functions is calculated perturbatively [26]. 

The HE approach gives an exact treatment of the exchange interachon between electrons. However, because it uses the mean field approximation, it ignores the effect of the Coulomb interaction on the relative positions of the electrons at... 

In the mean-field approximation for electron-electron interactions, the Hartree U[n] and exchange Ex energies read ... 

The quantum chemist s traditional way to approximate solutions of the electronic Schrodinger equation is so-called ab initio, wave function-based electron correlation methods. These methods improve upon the HF mean-field approximation by adding many-body corrections in a systematic way [15]. As of the time of this writing, efforts to accelerate ab initio calculations with GPUs are scarce. However, it is expected that this will change in the near future because these methods are of critical importance whenever higher accuracy is required than what can be achieved by DFT or for types of interactions and properties for which DFT breaks down. 

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