Electron spin expectation value

A detailed analysis of (S2) in DFT can be found in Refs. (129,130).) Note the change of meaning in the summation indices in Eq. (91) In the second line i,j label electronic (spin) coordinates, while they denote the indices of spin orbitals in the third line. For the study of open-shell transition-metal clusters, it is necessary to obtain an expression for the total spin expectation value, where the summation rims over the number of a- and / -electrons rather than over the total number of electrons N. Thus, the sum in Eq. (91) may be split into four sums over the various spin combinations,... 

Note that we introduced the superscript CD in order to distinguish the expressions obtained by Clark and Davidson from those by Mayer, which will be given in the following marked by Ma. In a similar fashion, Mayer s partitioning of the total spin expectation value can be derived. Starting from Lowdin s expression for the total spin expectation value, Eq. (96), a one-electron basis set is introduced as in Eq. (102) and the numbers of a- and / -electrons, Na and N13, respectively, are replaced by sums over diagonal matrix elements Y (P"S)W and E (P S) w [cf. Eq. (104)], M... 

Selected Ideal Total and Local Spin Expectation Values for the [Fe2] Cluster 1, Where Each Metal Center Carries Four Unpaired Electrons Resulting in a Total Spin State of S=4 for the High-Spin State. Local Spin Values on the Iron Atoms are Displayed in the Last Column. Up-Pointing Arrows Indicate Excess of q-Spin and Down-Pointing Arrows Excess of /3-Spin... 

Table II up- and down-pointing arrows were chosen to illustrate the local spin distributions, where an up-pointing arrow represents an electron with -spin and a down-pointing arrow an electron with / -spin. The ideal total spin expectation values are given in terms of the eigenvalues and local spins on the metal centers are indicated by ideal Msxvalues.

Note that the second and third integrals on the r.h.s. are zero because of the orthonormality of the spatial orbitals a and b, whose products appear over the same electronic coordinate in those integrals. The spatial functions integrate to one in the first and fourth integrals, and the remaining spin expectation values are just diose of Eq. (C.17). Thus, the expectation value of Eq. (C.23) is (1 — 0 — 0+1) = 1. With additional work, it can be shown that 50 50q, not an eigenfunction of S-. 

In evaluating Eq. (C.23), we invoked orthonormality between the spatial orbitals a and b, each of which contains an electron of different spin. However, in a UHF wave function, the a and orbitals are not necessarily orthogonal to one another (only within each set, either (X or jS, are all of the orbitals mutually orthogonal to one another). In that case, the second and third terms on the r.h.s of Eq. (C.23) survive as — a b. In general, one can show that for a UHF wave function where the number of a electrons is greater than or equal to the number of electrons, the expectation value of may be computed as... 

In this chapter, we reviewed different quantum chemical approaches to determine local quantities from (multireference) wave functions in order to provide a qualitative interpretation of the chemical bond in open-shell molecules. Chemical bonding in open-shell systems can be described by covalent interactions and electron-spin coupling schemes. For different definitions of the (effective) bond order as well as various decomposition schemes of the total molecular spin expectation value into local contributions, advantages and shortcomings have been pointed out. For open-sheU systems, the spin density distribution is an essential ingredient in the... 

As a check for the presence of spin contamination, most ah initio programs will print out the expectation value of the total spin <(A >. If there is no spin contamination, this should equal. v(.v + 1), where s equals times the number of unpaired electrons. One rule of thumb, which was derived from experience with... 

For a quantum mechanical calculation, the single point calculation leads to a wave function for the molecular system and considerably more information than just the energy and gradient are available. In principle, any expectation value might be computed. You can get plots of the individual orbitals, the total (or spin) electron density and the electrostatic field around the molecule. You can see the orbital energies in the status line when you plot an orbital. Finally, the log file contains additional information including the dipole moment of the molecule. The level of detail may be controlled by the PrintLevel entry in the chem.ini file. 

In general, if there are p a-spin electrons and q 8-spin electrons, the UHF wave-function can contain spin contributions ranging from s = p — q) to j p + q). The expectation value... 

An important property of the dimerized Peierls stale is the existence of gaps in the spectra of spin and charge excitations. For free electrons (//ci-ci=0) both gaps are equal, while in the presence of Coulomb repulsion the spin gap is smaller than the charge gap [23, 24]. In what follows, we will assume the temperature to be much smaller than these two gaps, so that we can neglect electronic excitations and replace Hcl [ A (.v)] by its ground state expectation value. 

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