Spin eigenfunction

SEMIEMPIRICAL CALCULATIONS ON LARGER MOLECULES The spin eigenfunctions are orthogonal... 

One consequence of the spin-polarized nature of the effective potential in F is that the optimal Isa and IsP spin-orbitals, which are themselves solutions of F ( )i = 8i d >i, do not have identical orbital energies (i.e., 8isa lsP) and are not spatially identical to one another (i.e., (l)isa and (l)isp do not have identical LCAO-MO expansion coefficients). This resultant spin polarization of the orbitals in P gives rise to spin impurities in P. That is, the determinant Isa 1 s P 2sa is not a pure doublet spin eigenfunction although it is an eigenfunction with Ms = 1/2 it contains both S = 1/2 and S = 3/2 components. If the Isa and Is P spin-orbitals were spatially identical, then Isa Is P 2sa would be a pure spin eigenfunction with S = 1/2. 

To consider the question in more detail, you need to consider spin eigenfunctions. If you have a Hamiltonian X and a many-electron spin operator A, then the wave function T for the system is ideally an eigenfunction of both operators ... 

In my discussion of pyridine, I took a combination of these determinants that had the correct singlet spin symmetry (that is, the combination that represented a singlet state). I could equally well have concentrated on the triplet states. In modem Cl calculations, we simply use all the raw Slater determinants. Such single determinants by themselves are not necessarily spin eigenfunctions, but provided we include them all we will get correct spin eigenfunctions on diago-nalization of the Hamiltonian matrix. 

In dealing with systems containing only two electrons we have not been troubled with the exclusion principle, but have accepted both symmetric and antisymmetric positional eigenfunctions for by multiplying by a spin eigenfunction of the proper symmetry character an antisymmetric total eigenfunction can always be obtained. In the case of two hydrogen atoms there are three... 

Substitution of this eigenfunction in an expression of the type of Equation 21 permits the evaluation of the perturbation energy W1, in the course of which use is made of the properties of orthogonality arid normalization of the spin eigenfunctions namely,... 

Triplet state. The spin eigenfunctions of the triplet state, with (z-component of the total spin) = 1, are written as [29]... 

For a particle that is not in a specific spin state, we denote the spin variable by o. A general state function fi (r, o, t) for a particle with spin 5 may be expanded in terms of the spin eigenfunctions l ms). 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

In the cases other than [case A] and [case B), so called "open-shell SCF methods are employed. The orbital concept becomes not quite certain. The methods are divided into classes which are "restricted 18> and "unrestricted 19> Hartree-Fock procedures. In the latter case the wave function obtained is no longer a spin eigenfunction. 

To distinguish between closed-shell and open-shell configurations (and determinants), one may generally include a prefix to specify whether the starting HF wavefunction is of restricted closed-shell (R), restricted open-shell (RO), or unrestricted (U) form. (The restricted forms are total S2 spin eigenfunctions, but the unrestricted form need not be.) Thus, the abbreviations RHF, ROHF, and UHF refer to the spin-restricted closed-shell, spin-restricted open-shell, and unrestricted HF methods, respectively. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

The conclusions from this rather elementary survey of the symmetry constraint problem all point in the same general direction. The imposition of symmetry constraints (other than the Pauli principle) on a variationally-based model is either unnecessary or harmful. Far from being necessary to ensure the physical reality of the wave function, these constraints often lead to absurd results or numerical instabilities in the implementation. The spin eigenfunction constraint is only realistic when the electrons are in close proximity and in such cases comes out of the UHF calculation automatically. The imposition of molecular spatial symmetry on the AO basis is not necessary if that basis has been chosen carefully — i.e. is near optimum. Further, any breakdowns in the spatial symmetry of the AO basis are a useful indication that the basis has been chosen badly or is redundant. 

Pauncz, R. "Spin Eigenfunctions Plenum Press New York,... 

Here a is an adjustable parameter, usually determined by comparing Hartree-Fock and Xa atomic calculations. In the spin-unrestricted version, the spin-up and spin-down orbitals are distinct, so that in general the resulting wavefunction is not a spin eigenfunction. 

For the constmction of spin eigenfunctions see, for example. Ref. [36]. The spin-coupled wavefunction may be extended by adding further configurations, in which case we may speak of a multiconfigurational spin-coupled (MCSC) description. In the... 

The dimensions of the spin spaces for the active electrons in Table 2, cf. Eq. (9)) are certainly not small. It proved difficult to find a spin basis in which very few of the coefficients were large and so we adopted instead a spin correlation scheme cf. Section 4.2). In the present work, we exploited the way in which expectation values of the two-electron spin operator evaluated over the total spin eigenfunction 4, depend on the coupling of the individual spins associated with orbitals ( )/ and j. Negative values indicate singlet character and positive values triplet character. Special cases of the expectation value are ... 

R. Pauncz Spin Eigenfunctions, Plenum Press, New York (1979). 

A complete set of spin eigenfunctions, e.g. oo i l = 1, 2,. .., 5) in the case of a six-electron singlet, can be constructed by means of one of several available algorithms. The most commonly used ones are those due to Kotani, Rumer and Serber [13]. Once the set of optimized values of the coefficients detining a spin-coupling pattern is available [see in EQ- (2)], it can be transformed easily [14] to a different spin basis, or to a modified set reflecting a change to the order in which the active orbitals appear in the SC wavefunction [see Eq. (1)]. 

Each of the five Rumer spin eigenfunctions for a six-electron singlet represents a product of three singlet two-electron spin ftmctions ... 

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