Many-body theories of electron

Shaul Mukamel, who is currently the C. E. Kenneth Mees Professor of Chemistry at the University of Rochester, received his Ph.D. in 1976 from Tel Aviv University, follot by postdoctoral appointments at MIT and the University of California at Berkeley and faculty positions at the Weizmann Institute and at Rice University. He has b n the recipient of the Sloan, Dreyfus, Guggenheim, and Alexander von Humboldt Senior Scientist awards. His research interests in theoretical chemical physics and biophysics include developing a density matrix Liouville-space approach to femtosecond spectroscopy and to many body theory of electronic and vibrational excitations of molecules and semiconductors multidimensional coherent spectroscopies of sbucture and folding dynamics of proteins nonlinear X-ray and single molecule spectroscopy electron transfer and energy ftrnneling in photosynthetic complexes and Dendrimers. He is the author of over 400 publications in scientific journals and of the textbook. Principles of Nonlinear OfMical Spectroscopy (Oxford University Press), 1995. 

Under certain conditions the Brillouin-Wigner pertiubation theory forms the basis for a many-body theory. Whether it can provide a robust multireference many-body theory of electron correlation effects is the subject of current research. 

The essential property of any true many-body theory of electronic structure is a linear scaling of the energy components, E, with the number of electrons, N, in the system [39,55,63,64], that is,... 

As we have seen in Chapter 3, that linear scaling of the energy with the number of electrons in a given system is the distinguishing feature of many-body theories of electronic stmcture. It is because of their lack of extensivity that Brillouin-Wigner methods have been largely dismissed as the basis for a viable many-body theory... 

L. M. Falicov and V. Heine, The Many-body theory of electrons in metal or has a metal really got a Fermi surface Adv. Phys. 10, 57-105 (1961). 

Freed, K. F. [1971] Many-Body Theories of the Electronic Structure of Atoms and Molecules , Annual Review of Physical Chemistry, 22, p. 313. 

Density Functional Theory (DFT) in the Kohn-Sham version can be considered as an improvement on HF theory, where the many-body effect of electron correlation is modelled by a function of the electron density. DFT is, analogously to HF, an... 

Freed, K. F., Many-body theories of the electronic structure of atoms and molecules, Ann. Rev. Phys. Chem. 22 313 (1971). An excellent review with almost three hundred references. It is written for the nonspecialist with a historical perspective. 

Schrodinger equation and the solutions supported by an independent electron model, scale linearly with electron number. Approximate treatments of electron correlation effects may not necessarily scale linearly with electron number. Approximations which display such a linear scaling are termed many-body methods. In the previous section, we described the many-body perturbation theory which is the basic ingredient of all many-body theories. It provided a fundamental tool for both the synthesis and the analysis of many-body methods. In this section, we consider some many-body theories of atomic and molecular structure and some theories which contain unphysical terms which scale nonlinearly with electron number. 

These limitations, most urgently felt in solid state theory, have stimulated the search for alternative approaches to the many-body problem of an interacting electron system as found in solids, surfaces, interfaces, and molecular systems. Today, local density functional (LDF) theory (3-4) and its generalization to spin polarized systems (5-6) are known to provide accurate descriptions of the electronic and magnetic structures as well as other ground state properties such as bond distances and force constants in bulk solids and surfaces. 

Let us recall that the Hohenberg-Kohn theorems allow us to construct a rigorous many-body theory using the electron density as the fundamental quantity. We showed in the previous chapter that in this framework the ground state energy of an atomic or molecular system can be written as... 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

The numerical determination of E grr by the use of many-body theory is a formidable task, and estimates of it based on E j and E p serve as important benchmarks for the development of methods for calculating electron correlation effects. The purpose of this work is to obtain improved estimates of Epp by combining the leading-order relativistic and many-body effects which have been omitted in Eq. (1) with experimentally determined values of the total electronic energy, and precise values of Epjp. We then obtain empirical estimates of E grr for the diatomic species N2, CO, BF, and NO using Epip and E p and the definition of E g in Eq. (1). 

We have chosen here to hint at the density matrix concept, typical of many-body theories, in order to stress that E c is still a two-electrons operator. (For the many body derivation of (20), see ). 

The BW form of PT is formally very simple. However, the operators in it depend on the exact energy of the state studied. This requires a self-consistency procedure and limits its application to one energy level at a time. The Rayleigh-Schrodinger (RS) PT does not have these shortcomings, and is, therefore, a more suitable basis for many-body calculations of many-electron systems than the BW form of the theory, it is applicable to a group of levels simultaneously. 

To account for the interchannel coupling, or, which is the same, electron correlation in calculations of photoionization parameters, various many-body theories exist. In this paper, following Refs. [20,29,30,33], the focus is on results obtained in the framework of both the nonrelativistic random phase approximation with exchange (RPAE) [55] and its relativistic analogy the relativistic random phase approximation (RRPA) [56]. RPAE makes use of a nonrelativistic HF approximation as the zero-order approximation. RRPA is based upon the relativistic Dirac HF approximation as the zero-order basis, so that relativistic effects are included not as perturbations but explicitly. Both RPAE and RRPA implicitly sum up certain electron-electron perturbations, including the interelectron interaction between electrons from... 

When tackling the task of developing a theory for the description of IETS with the STM, a many-body approach is thus unavoidable. The work by Davies [17] shows that one can actually get away with a simple many-body theory in which only the anti-symmetry of electron states is needed. The mandatory question is what a theory should account for. In order to answer this we need to re-consider what the IETS-STM technique is. 

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