Spin compensation

This result should be, of course, consistent with the result for the conventional DFT of a spin-compensated (paramagnetic) system characterized by n (ir) = n ) -) = in(r). From Eq. (135) its exchange energy can be written as... 

So far in this chapter we have discussed the NOE theory in terms of general set of SOs 4>i x) or in terms of restricted SOs, which are constrained to have the same spatial function spin functions. In this section we are concerned only with closed-shell systems. Our molecules are thus allowed to have only an even number of electrons, with all electrons paired such that the spatial orbitals are doubly occupied. In this case of spin-compensated systems, the two nonzero blocks of the 1-RDM are the same ( D = D ) that is. 

The spin-compensated version of Eq. (8) has a prefactor of 2, due to the double occupancy of occupied orbitals,... 

High-Explosive-Antitank (HEAT).. This is a special HE type, the effect of which derives from its shaped charge. The metal cone which shapes the charge, standoff (provided by projectile design), fuze action, and rotation affect depth of penetration. In high-velocity HEAT rounds, greater penetration is achieved with fin-stabilization and spin-compensation... 

Although the effects of rotation were not known with certainty, it was theorized that when the liner collapses in a rotating shaped chge, he jet formed from the liner also rotates and spreads or splits. To counteract these effects of rotation, tangential components of velocity are necesssry. These must be equal in magnitude but opposite in direction to those set up by the initial spin upon all the elements of the liner. The purpose of this counteracting force, called "spin-compensation , is to stop rotation of the jet (Ref 1). Both.theory and initial test results indicated that a fluted liner... 

Just as for the simple, spin compensated situation where pa(r) = pp(r) = j p(r), there are related expressions for the exchange and correlation energies per particle of the uniform electron gas characterized by p (r) pp(r), the so-called spin polarized case. The degree of spin polarization is often measured through the spin-polarization parameter... 

Atomic units are used in all equations and all considerations concern non-relativistic quantum mechanics in Born-Oppenheimer approximation. Square brackets, as in E[p] for instance, are used to indicate that the relevant quantity is a functional i.e. the correspondence between a function in real space p = p(r) and a real number (energy in this example). Abbreviations or acronyms denoting different approximate exchange-correlation functionals reflect their common usage in the literature. They are collected in Appendix. Unless specified, the equations are given for the spin-compensated case. 

Different fits were obtained by Vosko, Wilk, and Nusair for the spin-compensated and spin-... 

Unless specified, the formulas are given for the spin-compensated case. 

Tgad[pA, Pb] can be expressed analytically for a pair of electron densities (pa 1 and p ) obtained from the following partitioning of a noninteracting pure-state v-representable electron density (p 4 ) in a spin-compensated four-electron system ... 

GEA was tested. But the case of a nAS-representable pair of electron densities is much more general. It is striking to note that, the second-order GEA contribution to T ad pA, Pb is non-positive for all pairs of electron densities (for uAB-representable and not uAB-representable pairs alike). This follows from its explicit analytic form given here in the second line of Eq. 71. We recall now that the second-order GEA term (T2 in Eq. 69) is closely related to the von Weizsacker functional T [p 75 (T [p = 9T2[p]), which is the exact kinetic energy functional for one- and two-spin-compensated electron systems. Using Tsw p] in approximating T ad pA, Pb]... 

For any considered kinetic-energy functional defined for spin-compensated systems, the Oliver-Perdew spin-density generalization can be applied195 ... 

Similar to the spin-compensated case, the solution of the unrestricted Kohn-Sham equations starts with the external potential and the number of spin a and spin ji electrons in the state of interest (denoted Na and Np, respectively) then Eqs. (48) and (49) are solved until consistency is achieved. Using the Kohn-Sham orbitals and orbital energies, one then computes the total energy of the system using the spin-dependent generalization of Eq. (47),... 

Por spin-compensated systems, substitution of Eq. (37) into Eqs. (9) and (35) yields the following exact expression for the exchange energy... 

The exchange functionals appropriate to spin-compensated and to spin-polarized systems are related to each other by the spin-scaling relation [86]... 

Correlation is a much more difficult problem than exchange, so exact analytic forms of e r° (p) are known only for two hmiting cases. The first is the high-density (weak correlation) limit of a spin-compensated UEG... 

Perdew and Zunger [77] (PZ81) suggested the following parametrization of the Ceperley-Alder data for the spin-compensated and spin-polarized cases... 

Particular interest has been expressed in the Co/copper system because of the existence of a bound state formed between the localised magnetic impurity moments and the conduction-electron spins below a critical temperature [88]. The impurity moment is significantly quenched by spin compensation from the conduction electrons below the critical temperature, but this bound state can be perturbed by application of large external magnetic fields. 

Throughout this chapter, equations are written in atomic units. For the sake of simplicity, equations are given for spin-compensated electron densities hence the factor 2 in Eq. (3). The acronym KSCED stands for the Kohn-Sham equations with constrained electron density and is used to distinguish the two effective potentials expressed as density functionals the one in the considered one-electron equations, which involves an additional constraint (see Eq. (5) below), from that in the Kohn-Sham equations. 

Less stable blue solutions form when alkali metals are added to tetrahydrofuran, 1,2-dimethoxyethane, and other polyethers,and transient blue colours appear in alcohols and water. The ether solutions are mainly diamagnetic and the existence of a diamagnetic negative charge carrier seems necessary to account for their conductances. Both spin-compensated electron pairs, e ", and triple ions, e M e y have been suggested, and there is recent photolytic evidence for the Na ion in amine solvents. In amines a variety of ionic (M, e , M ion-pair and dimeric (M2, el ) species appears... 

Only for spin-compensated two-electron systems, we have Ex[p = — T-f l/ool therefore, Vx = can be... 

If two atoms in states with spins 8 and 82 are brought together, the lowest molecular state will be one in which the spins compensate each other as far as possible, in other words a molecular state with resultant spin /S = 5 — >821. If, and only if, such compensation actually takes place, i.e. if both 8 and 82 are different from zero, then will this molecular state have a function V with a minimum. 28 and 282 are to be identified... 

---