Spin-dependence

This density mvolves a spin-dependent faetor whieh eouples only states (i,J) with the same spin eoordinates (.S ., Sj). It is not a true density in that it is dependent on r, r it has meaning only as defined below. 

Table Bl.12.3 Spin-dependent factor f(T) for the isotropic second-order quadnipole shift.

Fadley C S et al 1997 Photoelectron diffraction space, time and spin dependence of surface structures Surf. Rev. Left 4 421-40... 

If V is the total Coulombic potential between all the nuclei and electrons in the system, then, in the absence of any spin-dependent terms, the electronic Hamiltonian is given by... 

The electron—photon coupling that forms the microscopic basis of MOKE makes it possible, in principle, to determine the electron spin-dependent band structure of elements and alloys. This is done by examining the dependence of the Kerr response on the wavelength of the incident light. 

S. H. Vosko, L. Wilk and M. Nusair, Accurate spin-dependent electron liquid correlation energies for local spin density calculations a critical analysis, Canadian J. Phys., 58,1200 (1980). 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

Next we turn to our canonical band results. To do so we have used a generalized canonical band model that includes spin-polarization. Hence we introduce a spin dependent shift,... 

In the realistic case where the potential is spin-dependent, the spin-orbit method is in trouble (should the spin-orbit parameter be calculated with the up or the down spin potential ). The present formalism allows for the use of spin-dependent potential and wavefunctions. 

In order to include the spin of the two electrons in the wave function, it is assumed that the spin and spatial parts of the wave function can be separated so that the total wave function is the product of a spin and a spatial wave function F — iAspace sp n Since our Hamiltonian for the H2 molecule does not contain any spin-dependent terms, this is a good approximation (NB—the complete Hamiltonian does contain spin-dependent terms, but for hydrogen they are rather small and do not appreciably affect the energetics of chemical bonding). For a two-electron system it turns out that there are four possible spin wave functions they are ... 

Charle, K. P. Spin-Dependent Kinetics in Dye-Sensitized Charge-Carrier Injection into Organic Crystal Electrodes 19... 

The basis functions are most commonly chosen such that the spin-function is either a pure spin-up function a(cr) or a pure spin-down function )S(cr). They are defined such that a( ) = / ( j) = 1 and zero for any other argument cr. Since the BO and, consequently, the Fock operator do not contain any spin-dependent terms, the HF equations divide into spin-up and spin-down equations ... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

As a typical example we illustrate in Figure 2-1 the electron density of the water molecule in two different representations. In complete analogy, p(x) extends the electron density to the spin-dependent probability of finding any of the N electrons within the volume element dr, and having a spin defined by the spin coordinate s. 

Note that we must use i and not r, as variable here, because of the spin-dependence of the exchange integral, recall Section 1.3. 

Vosko, S. J., Wilk, L., Nusair, M., 1980, Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations A Critical Analysis , Can. J. Phys., 58, 1200. 

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