Spin projections

To overcome some of the problems inlierent in the UFIF method, it is possible to derive SCF equations based on minimizing the energy of a wavefiinction fomied by spin projecting a single Slater detemiinant starting... 

This wavefunetion is a pure S = 1/2 state. This preseription for avoiding spin eontamination (i.e., earrying out the UHF ealeulation and then forming a new spin-pure P) is referred to as spin-projection. 

Another approach is to run an unrestricted calculation and then project out the spin contamination after the wave function has been obtained (PUHF, PMP2). This gives a correction to the energy but does not affect the wave function. Spin projection nearly always improves ah initio results, but may seriously harm the accuracy of DFT results. 

A spin projected result does not give the energy obtained by using a restricted open-shell calculation. This is because the unrestricted orbitals were optimized to describe the contaminated state, rather than the spin-projected state. In cases of very-high-spin contamination, the spin projection may fail, resulting in an increase in spin contamination. 

If spin contamination is small, continue to use unrestricted methods, preferably with spin-annihilated wave functions and spin projected energies. Do not use spin projection with DFT methods. When the amount of spin contamination is more significant, use restricted open-shell methods. If all else fails, use highly correlated methods. 

Many transition metal systems are open-shell systems. Due to the presence of low-energy excited states, it is very common to experience problems with spin contamination of unrestricted wave functions. Quite often, spin projection and annihilation techniques are not sufficient to correct the large amount of spin contamination. Because of this, restricted open-shell calculations are more reliable than unrestricted calculations for metal system. Spin contamination is discussed in Chapter 27. 

The spatial localization of H atoms in H2 and HD crystals found from analysis of the hyperfine structure of the EPR spectrum, is caused by the interaction of the uncoupled electron with the matrix protons [Miyazaki 1991 Miyazaki et al. 1991]. The mean distance between an H atom and protons of the nearest molecules was inferred from the ratio of line intensities for the allowed (without change in the nuclear spin projections. Am = 0) and forbidden (Am = 1) transitions. It equals 3.6-4.0 A and 2.3 A for the H2 and HD crystals respectively. It follows from comparison of these distances with the parameters of the hep lattice of H2 that the H atoms in the H2 crystal replace the molecules in the lattice nodes, while in the HD crystal they occupy the octahedral positions. 

In the context of fra/u-polyacetylene cjia and c are, respectively, the creation and annihilation operators of an electron with spin projection a in the n-orbital of the nth carbon atom (n= l,...,N) that is perpendicular to the chain plane (see Fig. 3-3). Furthermore, u is the displacement along the chain of the nth CH unit from its position in the undimerized chain, P denotes the momentum of this unit, and M is its mass. 

Spin projection coefficients as given in (4.84) can be obtained in a general and mathematically proper manner by using the Wigner-Eckhart theorem. A detailed description of this topic is found in the textbook by Bencini and Gatteschi [106]. 

Wittbrodt, J. M., Schlegel, H. B., 1996, Some Reasons Not to Use Spin Projected Density Functional Theory , J. Chem. Phys., 105, 6574. 

C. Elsasser, M. Brecht and R. Bittl, Pulsed electron-electron double resonance on multinuclear metal clusters Assignment of spin projection factors bsed on the dipolar interaction, J. Am. Chem. Soc., 2002, 124, 12606. 

Apart from the selection of basis set and correlation procedure, an additional consideration arises in open-shell systems because of the presence of one or more unpaired electrons. This leads to treatments that are referred to as spin-restricted (R), spin-unrestricted (U), and spin-projected (P). 

A further alternative is to remove the higher-spin states from the unrestricted wavefunction by means of a spin-projection operator. Spin-projected energies are designated by a P prefix (e.g. PHF, PMP). 

It is obvious that the projection operators for the different species have different numbers of terms in them. The HON species have 12 terms (3 x 2 ) while the A2B-type species have four terms, and the HDT+ isotopomer has only two terms. This results in different sizes of the spin-projected basis sets, and for this reason the properties obtained in this work are not precisely comparable between the A3, A2B, and ABC systems, although a very good idea of the trends may be obtained from the data in Table XVI. While all of the above are given in terms of the original particles, it should be noted that the permutations used in the internal particle basis functions are pseudo -permutations induced by the permutations on real particles. 

Here nig denotes the sign of the spin projection (it takes two values, +1 and — 1). By taking into account the cumulant properties, Eqs. (10)-(13) with replaced by F, it can be shown that A must be a real symmetric matrix (A " = A j) with no unique diagonal elements, whereas 11 is a spin-independent (11 = =... 

In a recent work, Marx and collaborators presented a protocol to calculate the dynamics of the coupling constant KAB(t) on the basis of an improved Yamaguchi expression that incorporated spin projection in a Hubbard-corrected two-determinant approach to obtain more accurate KAB values (86,87). 

Once a description of the electronic structure has been obtained in these terms, it is possible to proceed with the evaluation of spectroscopic properties. Specifically, the hyperfine coupling constants for oligonuclear systems can be calculated through spin projection of site-specific expectation values. A full derivation of the method has been reported recently (105) and a general outline will only be presented here. For the calculation of the hyperfine coupling constants, the total system of IV transition metal centers is viewed as composed of IV subsystems, each of which is assumed to have definite properties. Here the isotropic hyperfine is considered, but similar considerations apply for the anisotropic hyperfine coupling constants. For the nucleus in subsystem A, it can be... 

The method described above is of general validity and can be applied to transition metal clusters of arbitrary shape, size, and nucleanty. It should be noted that in the specific case of a system comprising only two interacting exchanged coupled centers, our general treatment yields the same result as that of Bencini and Gatteschi (121), which was specifically formulated for dimers. In this case, the relation between the spin-projection coefficient and the on-site spin expectation value is simply given by... 

Calculated (Scaled Tpssh) 66mn Hyperfine parameters (Mhz) Site Values and Spin-Projected Isotropic and Anisotropic Hfcs For [MNmMNIV3(BpY)6]3+ , Compared with Experiment... 

Ca2+ depletion has been shown to change the binding affinity of Mn, to modulate the exchange coupling within the Mn cluster and to mediate photooxidation of the Mn ions (see reference 431).Based on the spin-projection model for clusters with localized spins coupled by two-centre spin-exchange interaction468 possible structural models have been evaluated as candidates for the OEC. Three model were found to satisfy the constraints of the EPR and most other spectroscopic data, furthermore possible locations of Ca2+ were proposed (for a thorough discussion see reference 431). 

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