Classes, Fukutome

2 FUKUTOME CLASSES Symmetry dilemmas and the Fock operator 

We have derived the general Hartree-Fock method (GHF, p. 341) providing completely free variations for the spinorbitals taken from formula (8.1). As a result, the Fock equation of the form (8.26) was derived. 

We then decided to limit the spinorbital variations via our own condition of the double occupancy of the molecular orbitals as the real functions. This has led to the RHF method and to the Fock equation in the form (8.30). 

The Hartree-Fock method is a complex (nonlinear) procedure. Do the HF solutions have any symmetry features as compared to the Hamiltonian ones This question may be asked both for the GHF method, and also for any spinorbital constraints (e.g., the RFIF eonstraints). The following problems may be addressed  

The Fukutome classes may be characterized according to total spin as a function of position in space  

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The names and the properties of the Fukutome classes deserve some comments. Some of them are characterized by the expression current... 

Spinor Forms of the Spin Orbitals in the Eight Fukutome Classes... 

If we are seeking a GHF solution in a particular Fukutome class, symmetry will impose certain conditions on 0. 13 One finds, for example, that functions of the class ASDW must in fact have the form (3.1). In the ASCW class the pairing must be of the form... 

The existence of an instability means that it is possible to find solutions of symmetry lower than the original one. One speaks of broken symmetry solutions. In Fukutome s classification system (Table II), the new solutions can belong to the same or to a different class. In the former case it is the spatial symmetry which has been lowered. If the new solution belongs to a different Fukutome class, the symmetry with respect to spin and/or time reversal has been lowered. 

The forms (4.25) and (4.32-4.34) of the second variation refer to the spin orbital level In investigating possible instabilities we always choose a particular set - The corresponding set reference determinant or to a different one. In either case this type of information can be used to simplify the second variation further.17 The most well-known examples are the singlet and triplet instabilities of Paldus and Cizek.24 In the first case the set M is doubly filled and real and therefore belongs to the Fukutome class TICS like the set. As we have seen in (4.33), the fact that the spin orbitals are real reduces the matrix T from 2N x 2N to N x N. The fact that the orbitals are doubly filled reduces it further to (N/2) x (N/2). A similar reduction (but for another reason) occurs when we go from RHF to the Fukutome class ASDW.17 This provides an example of a triplet (or rather a nonsinglet) instability. 

The first explicit use of a gap equation in the type of problems discussed here was in a calculation of total and orbital energies for a linear chain of hydrogen atoms.30 Berggren and Martino arrived at their gap equation in a way other than the one described here. They studied the HN chain in the AMO approximation [cf. (3.1)], which means that their spm orbitals belong to the Fukutome class ASDW. The pairing partners i//M were taken from the unoccupied part of the band, the lower half of which is doubly filled in RHF. The pairing was vertical in the sense of a band... 

The fermion gas with S-function interactions has been treated in three dimensions by Berggren and Johansson34 and in one dimension by Sykja and Calais.35 Several Fukutome classes were investigated. 

In the case of a one-dimensional Coulomb gas the gap equation has been solved for the Fukutome class CCW.39 There the RHF orbital energy... 

Now we allow for a diversification of flie orbital part (keeping the functions reah i.e., staying within flie Fukutome class 4, usually called UHF in quantum chemistiy) for both spinorbitals. 

Is the resulting UHF energy (calculated for such afunction) lower than the corresponding RHF energy (calculated for jrRHF), i-e., is the RHF solution unstable toward UHF-type spinorbitals changes (no. 4 in flie table of Fukutome classes) ... 

The Fukutome classes allow some of the posed questions to be answered ... 

The probability densities of finding the o- = are different for the majority of Fukutome classes ""spin waves ). 

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