Total-spin operator

The wave funetion obtained eorresponds to the Unrestricted Hartree-Fock scheme and beeomes equivalent to the RHF ease if the orbitals (t>a and (()p are the same. In this UHF form, the UHF wave funetion obeys the Pauli prineiple but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-eleetron case, an alternative form of the wave funetion which has the same total energy, which is a pure singlet state, but whieh is no longer antisymmetric as required by thePauli principle, is ... 

Table 3 Unrestricted Hartree-Fock energies (Hartrees) for Ceo and C70 and their muon adducts, muon in atomic units (and MHz) after quartet spin state annihilation. The value of the total spin operator, < >, is also given after quartet spin state...

One of the pedagogically unfortunate aspects of quantum mechanics is the complexity that arises in the interaction of electron spin with the Pauli exclusion principle as soon as there are more than two electrons. In general, since the ESE does not even contain any spin operators, the total spin operator must commute with it, and, thus, the total spin of a system of any size is conserved at this level of approximation. The corresponding solution to the ESE must reflect this. In addition, the total electronic wave function must also be antisymmetric in the interchange of any pair of space-spin coordinates, and the interaction of these two requirements has a subtle influence on the energies that has no counterpart in classical systems. 

Then, the molecular total spin operator S defined as a sum over one-electron operators may be rewritten in terms of local spin operators SA,... 

In order to present the derivation of local spin expectation values, we shall briefly recall the foundations of spin eigenvalue equations in the non-relativistic framework. Information on the spin states of a molecule can be extracted from either the total spin operator S2 or its -component Sz (i.e., from its projection on the -axis),... 

In order to obtain expressions for the local spin expectation values, different decomposition schemes exist. One may either partition the total spin expectation value (S2) (122,124) as suggested by Mayer or the total spin operator S2 (113,114) as proposed by Clark and Davidson. The corresponding decomposition schemes for multi-determinant wave functions may be found in Refs. (125-127). 

We start with the partitioning of the total spin operator, introduce then the decomposition of the expectation value, and discuss the differences between these results. In accordance with Eq. (86) the decomposition of S2 yields,... 

Knowing both decomposition schemes yield the same squared total spin expectation value, we may write the expectation value of the total spin operator S2 in terms of Mayer s local expressions... 

The quantum numbers listed are for the eigenvalues of the total-spin operators S2 and Sz, where the total spin S is defined as S = S, + S2. Since electrons are fermions, the symmetric two-electron spatial function (1.249) must be multiplied by the antisymmetric spin function (1.254) to give an overall wave function that is antisymmetric the antisymmetric spatial function (1.250) must be multiplied by one of the symmetric spin functions (1.251H1.253). 

S denotes a vector operator comprising three components Sx, Sy, and Sz. Note that generally a spin Hamiltonian replaces all the orbital coordinates required to define the system by spin coordinates. With the total spin operator S = SA + SB and S2 = SA + Sg + 2SaSb, the Hamiltonian can be rewritten as... 

The Hamiltonian (1) is spin free, commutative with the spin operator S2 and its z-component Sz for one-electron and many-electron systems. The total spin operator of the hydrogen molecule relates to the constituent one-electron spin operators as... 

In order to evaluate integrals over spin variables let us note that Eq. (5) implies that the symmetric group operators form bases for representations of [Pg.614]

Starting from n)> we notice that it is neither an eigenstate of a Heisenberg Hamiltonian nor an eigenstate of the total spin operator S2. For instance, the Neel state is not invariant under the action of the nearest-neighbor XY spin terms of the Heisenberg Hamiltonian, which may produce a spin-flip on two nearest-neighbor sites at a time with respect to n). These XY excitation operators from the vacuum 4>n > can be written as... 

Since the Heisenberg Hamiltonian commutes with the total-spin operators, the 22iV-dimensional spin space can be separated in subspaces with... 

The operator commutes with S, the total spin operator, and so it cannot mix states of... 

It is very easily checked that these functions are orthogonal by construction. Their spin characters may be simply verified by applying to each of the previous collective states the convenient total spin operator. Then, introducing the expressions previously obtained for the molecular spin-orbital states (cf. 

The transition of one electron from the HOMO to the LUMO in a closed-shell system leads to four different excited wave functions (Fig. 7.2a). While two states f) and fj) correspond to energetically degenerate triplets, the mixed states mf) and mj) are not eigenfunctions of the total-spin operator. They can be combined to form another triplet state t3) and the singlet state s) (Fig. 7.2b). The total energy of the Sj state is then given by... 

The theory of symmetry-preserving Kramers pair creation operators is reviewed and formulas for applying these operators to configuration interaction calculations are derived. A new and more general type of symmetry-preserving pair creation operator is proposed and shown to commute with the total spin operator and with all of the symmetry operations which leave the core Hamiltonian of a many-electron system invariant. The theory is extended to cases where orthonormality of orbitals of different configurations cannot be assumed. 

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