Hartree spin polarization

A UHF wave function may also be a necessary description when the effects of spin polarization are required. As discussed in Differences Between INDO and UNDO, a Restricted Hartree-Fock description will not properly describe a situation such as the methyl radical. The unpaired electron in this molecule occupies a p-orbital with a node in the plane of the molecule. When an RHF description is used (all the s orbitals have paired electrons), then no spin density exists anywhere in the s system. With a UHF description, however, the spin-up electron in the p-orbital interacts differently with spin-up and spin-down electrons in the s system and the s-orbitals become spatially separate for spin-up and spin-down electrons with resultant spin density in the s system. 

Optimize the structure of acetyl radical using the 6-31G(d) basis set at the HF, MP2, B3LYP and QCISD levels of theory. We chose to perform an Opt Freq calculation at the Flartree-Fock level in order to produce initial force constants for the later optimizations (retrieved from the checkpoint file via OptsReadFC). Compare the predicted spin polarizations (listed as part of the population analysis output) for the carbon and oxygen atoms for the various methods to one another and to the experimental values of 0.7 for the C2 carbon atom and 0.2 for the oxygen atom. Note that for the MP2 and QCISD calculations you will need to include the keyword Density=Current in the job s route section, which specifies that the population analysis be performed using the electron density computed by the current theoretical method (the default is to use the Hartree-Fock density). 

Spin polarization can have important effects on systems with unpaired electrons. Theoretical techniques such as Hartree-Fock or density-... 

The intraatomic d-d electron-electron interaction includes Coulomb and exchange interactions, and it is responsible for orbital and spin polarization. To account for orbital polarization, the idea of the LDA + U method was followed.70 A generalized Hartree-Fock approximation including all possible pairings was then used to write... 

Table II The essentialiy-exact PW92 exchange-correlation energy per electron (in hartree) in a spin-polarized = 1) uniform electron gas of density parameter Tj(in bohr), and the deviation (in hartree) of other approximations from PW92. (1 hartree = 27.21 eV = 627.5 kcal/mol.]...

MOs can be different, and this permits spin polarization. Equations (6.8) and (6.9) define umestricted Hartree-Fock (UHF) theory. 

The only atomic wave-functions that do not have a node at the nucleus are s-functions. The isotropic coupling constant is thus a measure of the s-character of the wave-function of the unpaired electron at the nucleus in question. The coupling constant for an atomic s-electron can be either measured experimentally or calculated from Hartree-Fock atomic wave-functions so that, to a first approximation, the s-electron density may be calculated from the ratio of the experimental and atomic coupling constants. Should the first-order s-character of the wave-function of the unpaired electron be zero, as for example in the planar methyl radical, then a small isotropic coupling usually arises from second-order spin-polarization effects. The ESR spectra of solutions show only isotropic hyperfine coupling. 

There have been few first principles calculations for ions and no form factor calculations. Brown and Burton s spin-unrestricted Hartree-Fock SCFMO calculations (93) for K2NaCrFegave /< = -2.19%, f =2.SS% and/<,-/ = -4.74%, in quite good agreement with the experimental estimates (Table 4), but a somewhat small value of fa — fn —6.5%) was estimated for MnFe"" (cf. Table 5). On the other hand the Xa multiple scattering calculation of Larsson and Connolly (94) indicates that the contribution to /<, — f from Cg spin polarization is dominant... 

Hyperfine couplings, in particular the isotropic part which measures the spin density at the nuclei, puts special demands on spin-restricted wave-functions. For example, complete active space (CAS) approaches are designed for a correlated treatment of the valence orbitals, while the core orbitals are doubly occupied. This leaves little flexibility in the wave function for calculating properties of this kind that depend on the spin polarization near the nucleus. This is equally true for self-consistent field methods, like restricted open-shell Hartree-Fock (ROHF) or Kohn-Sham (ROKS) methods. On the other hand, unrestricted methods introduce spin contamination in the reference (ground) state resulting in overestimation of the spin-polarization. 

The LDA or its spin-polarized generalization, the local spin density (LSD) approximation provides a means of folding exchange and correlation effects, calculated on the basis of the local behavior of a uniform electron gas, into a set of self-consistent Hartree-like equations which contain only local operators for the potential. This procedure is represented schematically in Fig. 1. 

Transition metals are important materials with intriguing properties and they have been studied with ever improved methods. A major difficulty is posed by the standard one-electron models where the tight-binding model seems appropriate for the narrow, so-called d-bands while near-plane-wave crystal orbitals are adequate for the conduction bands. Canonical Hartree-Fock solutions are awkward starting points for the description of magnetic structures and the use of spin-polarized versions destroys basic symmetry properties. 

Although C does indeed depend only on the pair distribution function at zero separation, this quantity is nevertheless a non-local functional of the density, invalidating the short wavelength hypothesis. (We show this later with an explicit example.) On the other hand, for high densities, the Hartree-Fock approximation is valid, and g (r, r) = (1 - 2(r))/2, where C(r) = ( t(r) - nj(r))/n(r) is the relative spin-polarization at r, so that... 

Table 1. Stability, size, and magnetic moments for selected central atoms in icosahedral Fe12X clusters, that have all electrons in either filled or half-filled and maximally spin-polarized icosahedral electronic subshells. The binding energy (relative to Fe12 + X) is in Hartrees and radial bond distance in Bohr.

As a matter of fact, the choice of the best IPM for open-shell systems raises a number of problems there are many possible extended forms of the Hartree-Fock theory in addition to the standard RHF method. All these schemes include both spin polarization and correlation to some extent, but in an ill-defined way. The best way to include spin polarization is probably to take RHF as a starting point and to admit single excitations afterwards. 

Larsson [95] who subsequently lifted all restrictions to the wave function ansatz when performing atomic calculations. As a first step he allowed the radial functions to be different for different mg values of the participating orbitals (the spin-polarized Hartree-Fock method SPHF) leading to a core orbital contribution to magnetic effects. Furthermore the radial parts of functions possessing different mi values are... 

The broken-symmetry approach was developed within the framework of unrestricted Hartree-Fock theory or spin-polarized density functional theory, i.e., Xa theory. " It is based on the assumption that the energy difference between the singlet and triplet states resulting from the interaction between two 5 = 1/2 spin centers is given by ... 

Unpaired electron spin density p. Negative values of p for each H nucleus of PH2 (and thus for the related isotropic H magnetic hf coupling constant ap( H) see p. 65) are to be expected in view of a spin-polarization effect in the P-H o bond [5 to 7]. Numerical data were derived from ap( H) from two ESR spectra p = (-)0.035 (Kr matrix) [5] and -0.046 (cancrinite matrix) [6]. The following spin densities were theoretically calculated by ab initio (unrestricted Hartree-Fock, UHF) and semiempirical (INDO) methods [8] ... 

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