Spin eigenfunctions construction

Ruben Pauncz, Spin Eigenfunctions Construction and Use, Plenum, New York,... 

R. Pauncz, "Spin Eigenfunctions, Construction and Use", Plenum Press, New York (1979). 

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... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

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)]. 

These are quite useful for constructing spin eigenfunctions and are easily seen to be true, not only for three electrons, but for n. 

R. Pauncz. The Construction of Spin Eigenfunctions. Klnwer Academic/Plennm,... 

In summary, proper spin eigenfunctions must be constructed from antisymmetric (i.e., determinental) wavefunctions as demonstrated above because the total S2 and total Sz remain valid symmetry operators for many-electron systems. Doing so results in the spin-adapted wavefunctions being expressed as combinations of determinants with coefficients determined via spin angular momentum techniques as demonstrated above. In... 

Whatever method is used in practice to generate spin eigenfunctions, the construction of symmetry-adapted linear combinations, configuration state functions, or CSFs, is relatively straightforward. First, we note that all the methods we have considered involve 7V-particle functions that are products of one-particle functions, or, more strictly, linear combinations of such products. The application of a point-group operator G to such a product is... 

UHF Methods. A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this leads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,40 where two sets of singly occupied orbitals are constructed instead of the doubly occupied set. The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minimization,41 arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. 

The original Heitler-London calculation, being for two electrons, did not require any complicated spin and antisymmetrization considerations. It merely used the familiar rules that the spatial part of two-electron wave functions are symmetric in their coordinates for singlet states and antisymmetric for triplet states. Within a short time, however, Slater[10] had invented his determinantal method, and two approaches arose to deal with the twin problems of antisymmetrization and spin state generation. When one is constructing trial wave functions for variational calculations the question arises as to which of the two requirements is to be applied first, antisymmetrization or spin eigenfunction. 

Symmetric group methods. When using these we, in effect, first construct n-particle (spin only) eigenfunctions of the spin. From these we determine the functions of spatial orbitals that must be multiplied by the spin eigenfunctions in order for the overall function to be antisymmetric. It may be noted that this is precisely what is done in almost all treatments of two electron problems. Generating spatial functions... 

The HF wavefunction takes the form of a single Slater determinant, constructed of spin-orbitals, the spatial parts of which are molecular orbitals (MOs). Each MO is a linear combination of atomic orbitals (LCAOs), contributed by all atoms in the molecule. The wavefunction in classical VB theory is a linear combination of covalent and ionic configurations (or structures), each of which can be represented as an antis5nnmetrised product of a string of atomic orbitals (AOs) and a spin eigenfunction. The covalent structures recreate the different ways in which the electrons in the AOs on the atoms in the molecule can be engaged in bonding or lone pairs. An ionic structure contains one or more doubly-occupied AOs. Each of the structures within the classical VB wavefunction can be expanded in terms of several Slater determinants constructed from atomic spin orbitals. 

For the construction of spin eigenfunctions see, for example, Ref. [22], There are obviously many parallels to the multiconfiguration self-consistent field (MCSCF) methods of MO theory, such as the restriction to a relatively small active space describing the chemically most interesting features of the electronic structure. The core wavefunction for the inactive electrons, 4>core, may be taken from prior SCF or complete active space self-consistent field (CASSCF) calculations, or may be optimised simultaneously with the and cat. 

A further development was made by Knowles and Handy . They proposed to use Slater determinants instead of spin eigenfunctions as a basis (D,. In this case the coupling coefficients < y 7> take only the values 1 or 0, and can rapidly be recalculated each time they are required. With a suitable canonical addressing scheme for the determinants and Cl coefficients, the construction and use of the coupling coefficients can be vectorized. This makes it possible to use modern vector processors very efficiently. 

The unitary group approach attempts to solve this redundancy in two ways. First, the expansion terms are constructed in such a way that only spin eigenfunctions of the correct S value are considered. Secondly, the generator and generator product matrix elements in the CSF expansion space are computed in such a way that no reference to determinants is made, either implicitly or explicitly. This leads to computational schemes that depend only... 

Finally it is worth mentioning the Serber basis of spin functions in which pairs of electrons, 1 and 2, 3 and 4,...,etc., are coupled first to singlets or triplets, these pairs subsequently being coupled to one another to produce the required resultant spin. This differs from the Rumer basis in that triplet spin functions are used for the pairs as well as singlets, and the final set of spin functions is orthogonal. The construction of spin eigenfunctions is discussed in detail in the book by Pauncz. ... 

Though we never explicitly obtain (it could actually be constructed from the Ti coefficients), most of the relaxation and correlation efifeas are introduced by solving the CCSD equations. The coupling between relaxation and correlation is incomplete until all possible excitations (i.e., those due to T3, T4,. . . ) are also included. But CCSD is usually adequate because most correlation effects are found at just the T2 level. This is apparent from Table 17. The difference between using the QRHF reference (which is a spin eigenfunction) and the fully relaxed (but spin-contaminated) UHF reference for the ion is 0.03 to 0.07 eV for the valence levels. Even for the core, which is most sensitive to relaxation, the difference of 0.36 eV amounts to less than 0.1% ... 

R. Pauncz, The construction of spin eigenfunctions An exercise book, Kluwer Academic, New York, 2000, and references therein. 

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