Approximate spin-projection

From all this one must conclude that the determinantal and second-quantized formulations should be regarded as a poor man s group theory which, while convenient, hides the basic freeon dynamics. These fermion methods have the additional disadvantage that their antisymmetric fermion functions are not normally pure spin (freeon) states so that spin-projection may be required. A method for avoiding (approximately) spin projection is the employment of the variation principle to approximate the ground state e. g., unrestricted Hartree-Fock theory. Finally the use of the fermion formulations has lead to the spin paradigm as a replacement for the more fundamental freeon dynamics. 

Approximate Spin Projection for Geometry Optimization of Biradical Systems Case Studies of Through-Space and Through-Bond Systems... 

Approximate Spin Projection for Geometry Optimization of Biradictil... 

A second term in Eq. (18.10) is the SCE in the S-T gap, and, consequently, a second term in a denominator of Eq. (18.8) projects the spin contamination in the BS singlet solution. In this way, Eq. (18.8) gives approximately spin-projected (AP) Jab values. Equation (18.8) can be easily expanded into any spin dimers, namely, the lowest spin state (LS) and the highest spin state (HS), e.g., singlet-quintet for S = Sh = 2/2 pairs, singlet-sextet for 5a = Sb = 3/2 pairs, and so on, as follows ... 

Approximate Spin Projection for BS Energy and Energy Derivatives... 

Because Jab calculated by Eq. (18.8) is a value that the spin contamination error is approximately eliminated, it should be equal to Jab value calculated by the approximately spin-projected LS energy (i p) as... 

The approximation techniques described in the earlier sections apply to any (non-relativistic) quantum system and can be universally used. On the other hand, the specific methods necessary for modeling molecular PES that refer explicitly to electronic wave function (or other possible tools mentioned above adjusted to describe electronic structure) are united under the name of quantum chemistry (QC).15 Quantum chemistry is different from other branches of theoretical physics in that it deals with systems of intermediate numbers of fermions - electrons, which preclude on the one hand the use of the infinite number limit - the number of electrons in a system is a sensitive parameter. This brings one to the position where it is necessary to consider wave functions dependent on spatial r and spin s variables of all N electrons entering the system. In other words, the wave functions sought by either version of the variational method or meant in the frame of either perturbational technique - the eigenfunctions of the electronic Hamiltonian in eq. (1.27) are the functions D(xi,..., xN) where. r, stands for the pair of the spatial radius vector of i-th electron and its spin projection s to a fixed axis. These latter, along with the... 

The coefficient one-half at the diagonal interaction element in the above expression reflects the fact that in the HFR approximation for the closed electron shell system, only that half of the electron density residing at the a-th AO contributes to the energy shift at the same AO, which corresponds to the opposite electron spin projection. Then the expression for the renormalized mutual atomic polarizability matrix IIA can be obtained ... 

A more appropriate spin-orbit coupling Hamiltonian can be derived if electron-positron pair creation processes are excluded right from the beginning (no-pair approximation). After projection on the positive energy states, a variationally stable Hamiltonian is obtained if one avoids expansion in reciprocal powers of c. Instead the Hamiltonian is transformed by properly chosen... 

The broken symmetry wavefunction is not itself a pure spin state. However, spin projection techniques allow the approximate energies and properties of the correct spin states to be calculated. 

In this context, it is pertinent to recall that in many cases one can obtain the so-called best overlap orbitals [64] of DODS type which are produced by the given many-electron wave function. These orbitals were considered in [65] where they were identified with spin-polarized Brueckner orbitals. However, they exist if and only if the so-called nonsinglet Brueckner instabihty conditions are satisfied. At last, if the correct spin-projected determinant is involved in the consideration, then it is always possible to construct the best overlap orbitals of DODS type for the given exact or approximate state vector ). These orbitals were recently introduced [62] and named the spin-polarized extended Bmeckner (SPEB) orbitals. By construction, they maximize T). 

The results obtained in post-HF methods for solids refer mainly to the energy of the ground state but do not provide the correlated density matrix. The latter is calculated for sohds in the one-determinant approximation. The density matrix calculated for crystals in RHF or ROHF one-determinant methods describes the many-electron state with the fixed total spin (zero in RHF or defined by the maximal possible spin projection in ROHF). Meanwhile, the UHF one-determinant approximation formally corresponds to the mixture of many-electron states with the different total spin allowed for the fixed total spin projection. Therefore, one can expect that the UHF approach partly takes into account the electron correlation. In particular, of interest is the question to what extent UHF method may account for correlation effects on the chemical bonding in transition-metal oxides. An answer to this question can be obtained in the framework of the molecular-crystalline approach, proposed in [577] to evaluate the correlation corrections in the study of chemical bonding in crystals. 

The simplest way to avoid the neglect of spin polarization is to use a spin-unrestricted approach, such as UHF. However, at the UHF level of theory, the spin polarization effects are usually overestimated considerably in magnitude. In the absence of a theoretical framework that includes electron correlation (whereby a significant fraction of the problems discussed above vanish), the excess spin polarization can be reduced considerably by applying spin projection (PUHF) or spin annihilation (UHF-AA) techniques. Still, significant and seemingly unpredictable errors do occur also at this level of approximation, due to spin contamination and lack of electron correlation. These will be discussed in Section 4 below. 

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