Many-electron systems relations

The situation described here is based on a simple one-electron model which can hardly be expected to predict the behaviour of complex many-electron systems in quantitative detail. There can be no doubt however, that the qualitative picture is convincing and probably that the broad principles of electronic behaviour in solids have been identified. The most significant feature of the model is the band structure that makes no sense except in terms of the electron as a wave. Important, but largely unexplored aspects of solid-state reactions and heterogeneous catalysis must also relate to the nearly-free models of electrons in solids. 

A differential virial theorem represents an exact, local (at space point r) relation involving the external potential u(r), the (ee) interaction potential u r,r ), the diagonal elements of the 1st and 2nd order DMs, n(r) and n2(r,r ), and the 1st order DM p(ri r2) close to diagonal , for a particular system. As it will be shown, it is a very useful tool for establishing various exact relations for a many electron systems. The mentioned dependence on p may be written in terms of the kinetic energy density tensor, defined as... 

The horizontal axis relates the extent to which the motions of electrons in a many-electron system are independent of each other (uncorrelated). At the extreme left are found Hartree-Fock models. 

Primas, H. Separability in many-electron systems. In Modem quantum chemistry. Istanbul lectures. Sinanoglu, O. (ed.). New York Academic Press 1965 s°) Paldus, J., Ctzek, J. Relation of coupled-pair theory, Cl and some other many-body approaches. In Energy, Structure, and Reactivity. Proceedings of the 1972 Boulder Summer Research Conference on Theoretical Chemistry. Smith, D. W., McRae, W. B. (eds.). New York Wiley 1973, pp. 198-212... 

The Hamiltonian (1) is spin free, commutative with the spin operator S2 and its z-component Sz for one-electron and many-electron systems. The total spin operator of the hydrogen molecule relates to the constituent one-electron spin operators as... 

When this expression is extended to many-electron systems, two related problems arise. Firstly, what is the effective spin-orbit hamiltonian for the electron in open shells Secondly, what is the potential in which they move For a hydrogen-like atom the field would be written... 

Whereas, for non-Coulombic potentials, one can define nr and , n is then no longer related simply to the binding energy. Indeed, for a complex, many-electron atom, it is not at all obvious how one should set about quantising the system, since there is no guarantee that the orbits of individual electrons will close.6 In fact, conservation of the angular momentum for individual electrons is, at best, only an approximation. It would hold exactly for central fields. Even then, the same simple, precise relationship between n and as for H is not to be expected for many-electron atoms. As we shall see, the very meaning of n (the principal or most important quantum number) becomes less clear-cut for many-electron systems. In a nutshell the n and quantum numbers of... 

Quantum defect theory (QDT) was developed by Seaton [111] and his collaborators, from ideas which can be traced to the origins of quantum mechanics, through the work of Hartree and others. They relate to early attempts to extend the Bohr theory to many-electron systems (see e.g. [114]). 

Shakeup represents a fundamental many-body effect that takes place in optical transitions in many-electron systems. In such systems, an absorption or emission of light is accompanied by electronic excitations in the final state of the transition. The most notable shakeup effect is the Anderson orthogonality catastrophe [5] in the electron gas when the initial and final states of the transition have very small overlap due to the readjustment of the Fermi sea electrons in order to screen the Coulomb potential of pho-toexcited core hole. Shakeup is especially efficient when the optical hole is immobilized, and therefore it was widely studied in conjunction with the Fermi edge singularity (FES) in metals [6-8] and doped semiconductor quantum wells [9-15]. Comprehensive reviews of FES and related issues can be found in Refs. [16,17]. 

DFT aspires to predict exactly properties of many-electron systems without recourse to the wave function, using only the information contained (explicitly or implicitly) in the ground-state electron density. This section reviews the basic DFT formalism and introduces fundamental relations that will recur throughout this work. 

The relations of Eqs. 7.32 and 7.33 show that lx, ly, and It transform like arbitrary rotations about the x, y, and z axes, respectively. Each operator Ij is invariant to rotations about one axis (that is, Ry). In a many-electron system the total angular momentum operator is the sum of the one-electron operators Ixk, lyk, hk, where k labels the electron. Thus,... 

These results indicate that the correlation energy for U = 3 is closely related to that for the D oo limit. The possibility of exploiting this for many-electron systems is very inviting. The computation of AEoo only requires finding the minimum of the effective potential for the full problem and its value at a nearby point for the Hartree-Fock approximation. 

There is, however, one paper of Dirac [5] that keeps being cited, namely the one at which we want to have a look now. Most people who cite this paper hardly know that its title is Quantum mechanics of many-electron systems and are unaware of its scientific context. It deals mainly with the relation between permutation symmetry and spin and contains a formula which relates the expectation value of the operator of electron exchange to the total spin of the state. 

Eq. (8.94) represents an exact expression for the quantum mechanical state of a many-electron system. Note that the expansion coefficients Cj of the N-particle basis states are directly related to the expansion coefficients of the one-particle states. It is sufficient to know either of them (and this fact is related to the observation made below that a full configuration interaction wave function does not require the optimization of orbitals see the next section). 

This approximation inherent in the CI-RI2 method is important in view of the related methods for many-electron systems that will be discussed in Section 5.3 (these methods are denoted as linear RI2 methods). The approximation becomes exact in the limit where the corresponding one-electron problem (e.g., H2 or H3 ) is solved exactly. This is the fundamental idea on which the approximations in the linear R12 methods are based. It is much easier to converge to near-basis set completeness in one-electron than in two-electron space. It is therefore understood that atomic basis sets of near-Hartree-Fock limit quality are used as a starting point for linear R12 calculations. 

The Real Crux of the Matter. I have kept till last the most categorical reason for wanting to deny the possibility of ever observing an orbital. After all, the fact that orbitals might only provide an approximation to the motion of many-electron systems is not a sufficient reason for the complete denial that they or something related to orbitals can possibly exist. My final argument, which I claim to be the most decisive one, is that orbitals depict a quantity called probability amplitude, which has been known to be unobservable in principle since the birth of quantum mechanics as distinct from the old quantum theory. 

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