Spin permutation

Dirac theory, 266—268 nonrelativistic states, 263- 265 Electron spin, permutational symmetry, 711-712 Electron transfer ... 

The product of bond wave functions in Equation 3.8, involves so-called perfect pairing, whereby we take the Lewis structure of the molecule, represent each bond by a HL bond, and finally express the full wave function as a product of all these pair-bond wave functions. As a rule, such a perfectly paired polyelectronic VB wave function having n bond pairs will be described by 2" determinants, displaying all the possible 2x2 spin permutations between the orbitals that are singlet coupled. 

As shown above in Scheme 4.2, the Rumer basis for butadiene is made of the VB structures 6 and 7. From Section 3.1.4, the VB function corresponding to a given bonding scheme is the one that involves singlet coupling between the AOs that are paired in this scheme. This, however, can be done in two ways. In the first way, the AOs are kept in the same order in the various determinants, and the determinants display all the possible 2x2 spin permutations between the orbitals that are singlet coupled. This is the convention used in Equation 4.16, which is similar to Equation 3.9. In this case, the determinant has a positive or... 

Since the spin permutations are successive, the 1B ig state will possess bonding across the C—C circumference (Fig. 7.5c). In contrast, the excited state is the positive Aig combination, which eliminates the QC determinants ... 

Remembering that the energy of the spin-alternant is taken as zero, and that determinants interact only if they differ by only one spin permutation between adjacent atoms (in which case their matrix element is X), the Hamiltonian matrix element between the two Rumer structure is... 

Here, grs is a parameter that is quantified either from experimental data, or is calculated by an ab initio method as one-half of the singlet-triplet excitation energy gap of the r—s bond. In terms of the qualitative theory in Chapter 3, grs is therefore identical to the key quantity —2(3 5 - This empirical quantity incorporates the effect of the ionic components of the bond, albeit in an implicit way. (c) The Hamiltonian matrix element between two determinants differing by one spin permutation between orbitals r and s is equal to grs. Only close neighbor grs elements are taken into account all other off-diagonal matrix elements are set to zero. An example of a Hamiltonian matrix is illustrated in Scheme 8.1 for 1,3-butadiene. 

Spin permutation technique in the theory of strongly correlated electron systems... 

The purpose of these notes is to show how some strongly correlated electron models like the one-band Hubbard model with infinite electron repulsion on rectangular and triangular lattices can be described in terms of spinless fermions and the operators of cyclic spin permutations. We will consider in detail the... 

CYCLIC SPIN PERMUTATION FORMALISM FOR HUBBARD MODEL WITH INFINITE REPULSION... 

The considered spin permutation can be written in the following form ... 

Let us enumerate all the electrons of the chain in succession along the cells of the chain. Making use of the spin permutations, one can obtain the Hamiltonian (8) with (/ = oo in the form... 

For several holes, spin permutations Q/ /+1occuring in (9) lead to the mixing... 

Rewriting the spin permutation with the help of the Dirac identity, one can obtain the following spin Hamiltonian ... 

When 1/2second order in t2 we can write the lattice Hamiltonian for this filling in the form... 

Assume that the rectangular lattice consists of two weakly interacting segments, one of them having a nonzero value of a. This lattice may be considered as a simple correlated electron model for stacked donor-acceptor crystals. For a t2 one can use the cyclic spin permutation technique within... 

So, to use the spin permutation technique we constructed the symmetry adapted lattice Hamiltonian in a compact operator form and essentially reduced the dimensionality of the corresponding eigenvalue problem. The effects of tpp 0 and the additional superexchange of copper holes are considered in [48]. 

The allowed transitions in the N and (N—l) quantum spectra of an. /V-spin system must belong to the highest symmetry class of the spin permutation group. By creating N or (N— 1) quantum coherence and then converting this into an observable IQ coherence, one obtains a symmetry-filtered lQ-spec-trum which contains only a subset of those transitions originating from the most symmetric class. The IQ spectra of solutes aligned in liquid crystalline... 

With Eqs. (70) or (100), the distinct pairs are easily found. For example, in Ne 2 instead of P, Z) and 5 2y> , the three unique pairs can he taken as p,.a.py ), p cup ) and p xp a). Other pairs differ from these only hy changes of axes or by spin permutations. Therefore, one needs to obtain the of these unique pairs only. 

To see that this is true it is only necessary to write out each full permutation operator in the definition of the antisymmetrising operator explicitly as a product of a spatial permutation and a spin permutation ... 

When a spin/space permutation is applied to a function of space and spin the Pauli principle requires that the product of the coefficient of the spatial part multiplied by the coefficient of the spin part must be (—1), where p is the parity of the permutation. So, if we know the coefficients which occur in the effect of a spin permutation on the chosen the coefficients of the associated spatial part are fixed. But, as we have seen, the effect of permutations of spin on a spin eigenfunction is simply to send it into a linear combination of other spin eigenfunctions with the same value of S and M the spin functions form a basis for the irreducible representations of the symmetric group of permutations of n objects. 

The formal group-theoretical result is that the representation matrices of the two spatial and spin permutations and P,) are linked by the Pauli principle through... 

The Hamiltonian matrix element between two determinants differing by one spin permutation between orbitals r and s is equal to Any other off-diagonal matrix elements are set to zero (see Scheme 12). 

Set up the irrep D5, dual to that obtained in Problem 4.8, and use it to form Wigner operators as in Problem 4.9. Apply a suitable pair of operators to a product a ri)b(r2)c(rx) to generate spatial functions that behave, under permutations of spatial variables, like the two basis vectors. Construct the function (4.3.5) and show that it is indeed emtisymmetric under space-spin permutations P12, P23, and hence satisfies the Pauli principle. 

To obtain the basic matrix element expressions, assuming orthonormal orbitals, we again start from (7.2.5), and systematically evaluate the matrix elements (7.2.7) of the spin permutations. For the moment, we restrict the discussion to the singlet structures of one orbital configuration. We write (7.2.5) as... 

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