The Non-Ground Case

The groundness assumption on E(r) is very strong. So we extend the MSG Method to sets Q(r) of not necessarily ground examples, called general examples. The idea is to introduce alternatives in examples. Alternatives are naturally characterized by disjunctions and existential quantifiers. 

Definition 10-11 A general example of predicate r/n is the existential closure of a disjunction of atoms of predicate r/n. 

Note that a single-disjunct general example is different from an atomic property (a property with no body), because properties have universal quantifiers. If there is no ambiguity whether we are referring to an atomic property or a general example, we omit quantifiers. 

Definition 10-12 Given a general example g, the set of admissible alternatives of g, denoted adm(g), is defined as follows  

Note that the admissible alternatives of a general example (set) are (sets of) atoms, not (sets of) ground examples. Also note that the adm relations may be infinite if variables occur in their arguments. 

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Note that the method of Task T is non-deterministic, always succeeds, but may be infinite, because of the non-ground case of the MSG Method. Moreover, the synthesized instances are non-determinislic and finite, for the same reason. The logic algorithm synthesized at Step 6 may thus be non-deterministic, but is certainly finite. All other tasks are fully deterministic. Hence Step 6 never fails. 

As in the TFD method, this equation has to be solved with the boundary conditions (t)(0) = 1 andXcCt) (Xc) = < Xc) where = bXc is the cutoff point where the pressure of the electron gas becomes zero. The value of ( )(Xc)/Xcnecessary for this differs from the non-relativistic case and can be found from similar grounds [27], just by making zero the pressure of a homogeneous electron gas, given by... 

The energy denominators in SOS-PT contain one or more factors of the general form (iT i.f / + to) where is the excitation frequency from the ground vibronic state to K,k>, l), - is the population decay rate for K,k>, and m is an optical frequency or a sum of Hie optical frequencies that characterize the particular process. Resonant, or near-resonant processes occur when or 0. Otherwise, the process is non-resonant and we neglect The resonant case is discussed in Section 1, while the remainder of this review focuses on the non-resonant case. 

The one-dimensional cases discussed above illustrate many of die qualitative features of quantum mechanics, and their relative simplicity makes them quite easy to study. Motion in more than one dimension and (especially) that of more than one particle is considerably more complicated, but many of the general features of these systems can be understood from simple considerations. Wliile one relatively connnon feature of multidimensional problems in quantum mechanics is degeneracy, it turns out that the ground state must be non-degenerate. To prove this, simply assume the opposite to be true, i.e. 

It is clear that an ah initio calculation of the ground state of AF Cr, based on actual experimental data on the magnetic structure, would be at the moment absolutely unfeasible. That is why most calculations are performed for a vector Q = 2ir/a (1,0,0). In this case Cr has a CsCl unit cell. The local magnetic moments at different atoms are equal in magnitude but opposite in direction. Such an approach is used, in particular, in papers [2, 3, 4], in which the electronic structure of Cr is calculated within the framework of spin density functional theory. Our paper [6] is devoted to the study of the influence of relativistic effects on the electronic structure of chromium. The results of calculations demonstrate that the relativistic effects completely change the structure of the Or electron spectrum, which leads to its anisotropy for the directions being identical in the non-relativistic approach. 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

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