Infinite electron repulsion

At infinite electron repulsion, if the total numbers of electrons and lattice sites are coincident (half-filled band), each site is occupied by one electron only. 

The purpose of these notes is to show how some strongly correlated electron models like the one-band Hubbard model with infinite electron repulsion on rectangular and triangular lattices can be described in terms of spinless fermions and the operators of cyclic spin permutations. We will consider in detail the... 

The Hubbard model with infinite electron repulsion represents restricted hopping in a space with no doubly occupied sites. For the iV-electron system on the lattice formed by L sites the model Hamiltonian has the form [27]... 

For simplicity, let us first consider the case of two-site segments with 3 electrons and t3=0. Because of infinite electron repulsion there are only two... 

Quite recently the ferromagnetic ground state in a class of Hubbard systems on decorated lattices that contain flat or nearly flat lowest energy bands has been found [42,43]. Obviously such multiband systems have more degrees of freedom than the one-band model considered by Nagaoka, and we can expect the appearance of unusual effects even in the case of infinite electron repulsion. 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

The one electron operator h, describes the motion of electron i in the field of all the nuclei, and gy is a two electron operator giving the electron-electron repulsion. We note that the zero point of the energy corresponds to the particles being at rest (Tc = 0) and infinitely removed from each other (Vne = Vee = V n = 0). 

More recently, Caves and Karplus71 have used diagrammatic techniques to investigate Hartree-Fock perturbation theory. They developed a double perturbation expansion in the perturbing field and the difference between the true electron repulsion potential and the Hartree-Fock potential, V. This is compared with a solution of the coupled Hartree-Fock equations. In their interesting analysis they show that the CPHF equations include all terms first order in V and some types of terms up to infinite order. They propose an alternative iteration procedure which sums an additional set of diagrams and thus should give results more accurate than the CPHF scheme. Calculations on Ha and Be confirmed these conclusions. 

In the period 1940-1946, Ogg (132) developed the first quantitative theory for the solvated electron states in liquid ammonia. The Ogg description relied primarily on the picture of a particle in a box. A spherical cavity of radius R is assumed around the electron, and the ammonia molecules create an effective spherical potential well with an infinitely high repulsive barrier to the electron. It is this latter feature that does not satisfactorily represent the relatively weakly bound states of the excess electron (9,103). However, the idea of a potential cavity formed the basis of subsequent theoretical treatments. Indeed, as Brodsky and Tsarevsky (9) have recently pointed out, the simple approach used by Ogg for the excess electron in ammonia forms the basis of the modem theory (157) of localized excess-electron states in the nonpolar, rare-gas systems. [The similarities between the current treatments of trapped H atoms and excess electrons in the rare-gas solids has also recently been reviewed by Edwards (59).]... 

The electronic bands of an infinite crystal can cross as a function of some parameter (pressure, concentration etc.). If one treats the e /r,2 term of the electron repulsion correctly, one sees that the crossing transition of the two bands is a first-order phase transition, between the metallic and insulating states. This transition was predicted by Mott in 1946 and has carried his name ever since. In fact, the original Mott criterion does not predict such a transition for Hg, but the criterion was derived for monovalent atoms. For divalent mercury it should not be applicable. Also the semiempirical Herzfeld criterion, which was very successful in predicting the insulator to metal transition in compressed xenon, predicts bulk Hg to be non-metallic. All this seems to imply that Hg is a rather special case. 

This result is very typical of methods of integral evaluation by separation in polar coordinates the expansion of an operator in terms of products of spherical heirmonics has an infinite number of terms but the forms of the orbitals involved in any integrand ensures that the expansion cuts off after a finite number of terms. The atomic electron-repulsion integrals expanded as a sum of F and G Slater-Condon parameters is perhaps the prototype. 

Fig. 4. Spectrum of localized electrons for an Anderson model with an ionic one-particle level at Cj = -3A and an infinite Coulomb repulsion U, calculated at four different temperatures k T=A /3. Energies are given in units of the Anderson width A = TrV Nf. (a) NCA calculation for an impurity, Atji7 = 0.024. At the lowest temperature k T = A 90 < k Ty the ASR near o) = 0 is nearly saturated with a peak value of pI = (ird) . The original ionic resonance at j is broadened on the scale A. (b) LNCA calculation for the lattice on an extended scale, showing the formation of a coherence gap in the ASR.

In the expression for the total energy per unit cell of an infinite chain [equation (1.66)] the infinite sum of the nuclear attraction occurs twice [in hr and also in Fr g)] and this is compensated by the positive infinite sums (electron-repulsion and nuclear-repulsion terms). This compensation is not valid even approximately if we truncate the cell summation too early. This causes the total eneigy per unit cell to be very sensitive in the calculations to the number of neighbors taken into account explicitly. 

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