Spin-polarized wavefunction

Both objects are much less complicated than the total A -particle wavefunction itself, since they only depend on three spatial variables. The electron density is manifestly positive (or zero) everywhere in space while the spin-density can be positive or negative. If, by convention, there are more spin-up than spin-down electrons, the positive part of the spin-density will prevail and there will usually be only small regions of negative spin-density that arise from spin-polarization. This spin-polarization is physically important and is already included in the UHF method but not in the ROHF method that, by construction, can only describe the... 

The impurity interacts with the band structure of the host crystal, modifying it, and often introducing new levels. An analysis of the band structure provides information about the electronic states of the system. Charge densities, and spin densities in the case of spin-polarized calculations, provide additional insight into the electronic structure of the defect, bonding mechansims, the degree of localization, etc. Spin densities also provide a direct link with quantities measured in EPR or pSR, which probe the interaction between electronic wavefunctions and nuclear spins. First-principles spin-density-functional calculations have recently been shown to yield reliable values for isotropic and anisotropic hyperfine parameters for hydrogen or muonium in Si (Van de Walle, 1990) results will be discussed in Section IV.2. 

One can dissociate the NO dimer simply by increasing the N-N bond distance to infinity. One can also require that during that process the molecule remain on the singlet surface, which by definition has a wavefunction and thus density that has equal spin-up and spin-down components everywhere in space. We are not interested in spin-restricted dynamics. We are interested in the much more balanced chemical dynamics that treats each half of the dissociated dimer correctly in DFT via a spin-polarized calculation. This decision must be made independent of whether or not one wants to use spatial symmetry to reduce the cost of the calculation. Spin-unrestricted DFT chemical dynamics will be called balanced in the following. 

The task of finding the single particle-like wavefunctions is now in principle equivalent to that within non-relativistic SIC-LSD theory. The four-component nature of the wavefunctions and the fact that neither spin nor orbital angular momentum are conserved separately presents some added technical difficulty, but this can be overcome using well-known techniques (Strange et al., 1984). The formal first-principles theory of MXRS, for materials with translational periodicity, is based on the fully relativistic spin-polarized SIC-LSD method in conjunction with second-order time-dependent perturbation theory (Arola et al., 2004). 

I> (Phi) spaced-fixed polar angle X (chi) spin wavefunction i/i (psi) spatial wavefunction P (Psi) spin-spatial wavefunction (o (omega) electron spin coordinate, angular frequency, vibrational constant (oig)... 

By (1.95), the LSD on-top exchange hole rix (r, r) is exact, at least when the Kohn-Sham wavefunction is a single Slater determinant. The LSD on-top correlation hole n (r,r) is nonexact [63] (except in the high-density, low-density, fully spin-polarized, or slowly-varying limit), but it is often quite realistic [64]. By (1.85), its cusp is then also realistic. 

Sometimes the term restricted Hartree-Fock (RHF) is used to emphasize that the wavefunction is restricted to be a single determinantal function for a configuration wherein electrons of a spin occupy the same space orbitals as do the electrons of P spin. When this restriction is relaxed, and different orbitals are allowed for electrons with different spins, we have an unrestricted Hartree-Fock (UHF) calculation. This refinement is most likely to be important when the numbers of a- and -spin electrons differ. We encountered this concept in Section 8-13, where we noted that the unpaired electron in a radical causes spin polarization of other electrons, possibly leading to negative spin density. 

A paradigmatic application has been reported for ethane [16]. It shows that the Fermi contact contributions to experimental nuclear spin-spin coupling constant are easy to explain in terms of current densities (7.25), which transport spin polarization along the coupling pathway, and associated plots of property density, Eq. (7.57). Same-spin electron correlation, the only kind of correlation recovered by the Hartree Fock wavefunction considered in Ref. [16], determines the alignment of the nuclear dipoles at its ends, as shown in the current-density maps reported for ethane. Fig. 7.44. 

As discussed above, UHF wavefunctions mirror the spin polarization that is seen experimentally, but they do so at the cost of introducing contamination from higher spin states. We now use allyl radical to illustrate how this occurs. Employing Dirac s ket notation for Slater determinants (introduced in... 

Since (S ) is close to 1 in allyl, has an uncomfortably high level of spin contamination. If one compares the observed hyperfin. couplings in allyl radical to those computed from a UHF wavefunction, one finds that spin contamination causes the amount of spin polarization to be exaggerated. In longer odd-alternant hydrocarbon radicals, spin contamination can become quite spectacular As shown in Figure 3, (S ) for the UHF wavefunctions of odd-alternant polyenyl radicals increases by 0.38 units for every pair of CH groups added, instead of remaining constant at (S ) = 0.75, as it would for a pure doublet wavefunction. 

However, even if one is not interested in modeling ESR spectra, preventing electrons of opposite spin from having different spatial distributions imposes a constraint on ROHF wavefunctions that has energetic consequences. For example, if one compares the ROHF energy of the planar allyl radical in C2J, symmetry (cf. below), where spin polarization is quite important, to that of the twisted species, where spin polarization of the electrons in the n bond is almost absent, one obtains a rotational barrier that is far too low (see Table 1), compared to the experimental value of 15 kcal/mol. 

We now return to the allyl radical to illustrate how a CASSCF wavefunction incorporates spin polarization, which is absent from ROHF wavefunctions, without introducing spin contamination, which is present in UHF wave-functions. Shown schematically in Figure 7 is the lowest energy configuration,... 

Equation [12] demonstrates that the terms in Eq. [7] that introduce spin polarization into the UHF wavefunction for the k electrons in the allyl radical simultaneously contaminate it with the quartet wavefunction. In fact, the greater the amount of spin polarization in q UHF coeffi-... 

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