Eigenvalues many-body equations

However, we are concerned with a many-body system that is described by a particular one- and two-body interaction. For each eigenvalue the many-particle eigenfunction obeying the Schrbdinger equation... 

Equations (70) or (76) and (73) are the basic equations of the new EOMXCC theory. In order to solve an eigenvalue problem (70) we must decide about the source of information about the cluster operator T that defines H. We find the cluster amplitudes defining T by projecting Eq. (73) against the excited configurations included in the many-body expansion of... 

Most band-structure calculations in solid-state physics are actually calculations of the KS eigenvalues q.39 This simplification has proved enormously successful, but when one uses it one must be aware of the fact that one is taking the auxiliary single-body equation (71) literally as an approximation to the many-body Schrodinger equation. DFT, practiced in this mode, is not a rigorous many-body theory anymore, but a mean-held theory (albeit one with a very sophisticated mean held i>s(r)). 

A partial justihcation for the interpretation of the KS eigenvalues as starting point for approximations to quasi-particle energies, common in band-structure calculations, can be given by comparing the KS equation with other self-consistent equations of many-body physics. Among the simplest such equations are the Hartree equation... 

There are basically three distinct types of approximations involved in a DFT calculation. One is conceptual, and concerns the interpretation of KS eigenvalues and orbitals as physical energies and wave functions. This approximation is optional — if one does not want to make it one simply does not attach meaning to the eigenvalues of Eq. (71). The pros and cons of this procedure were discussed in Secs. 4.2.2 and 4.2.3. The second type of approximation is numerical, and concerns methods for actually solving the differential equation (71). A main aspect here is the selection of suitable basis functions, briefly discussed in Sec. 4.3. The third type of approximation involves constructing an expression for the unknown xc functional Exc[n, which contains all many-body aspects of the problem [cf. Eq. (55)]. It is with this type of approximation that we are concerned in the present section. 

The random phase approximation (RPA) was first introduced into many-body theory by Pines and Bohm.This approximation was shown to be equivalent to the TDHF for the linear opticcd response of many-electron systems by Lindhard. ° (See, for example, Chapter 8.5 in ref 83. The electronic modes are identical to the transition densities of the RPA eigenvalue equation.) The textbook of D. J. Thouless contains a good overview of Hailree—Fock and TDHF theory. 

Here, the e, value is the orbital s energy eigenvalue. Equation (6.9) is remarkably similar to the original Schrodinger equation. Equation (6.2), but the wave functions have been replaced with the KS orbitals and the exchange and correlation terms have been isolated. Thus, we have replaced the iV-body coupled electronic wave function with a collection of uncorrelated wave fimctions while at the same time defining precisely what the uncertain many-body terms in need of approximation are. 

Yaspatial positions rj of the N molecules yields a set of energy eigenvalues ( rj ), which can be interpreted as the effective Al-particle potential in the single-channel many-body Hamiltonian (Equation 12.1). The dependence of Vgl ( r ) on the electric fields E provides the basis for the engineering of the many body interactions in (Equation 12.2). The validity of this adiabatic approximation and of the associated decoupling of the Born-Oppenheimer channels will be discussed below. 

As it will be shown in Section 3.4, Eq. (89) is also called as Optimized Effective Potential (OEP) equation. Exchange-correlation energy functional that depends on all orbitals and eigenvalues are, e.g., the ones based on the second-order many-body perturbation-theory (MBPT) and the ones which uses virtual orbitals to model the static-correlation. ... 

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