Electronic spin operators

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 the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

For every electronic wavefunction that is an eigenfunction of the electron spin operator S, the one-electron density function always comprises an spin part... 

Our next goal is to transform this expression into one based on the total electron spin operator, S = si + s2. The first three terms can be simplified by making use of the identity (derived using raising and lowering operators) ... 

Here, /3 and / are constants known as the Bohr magneton and nuclear magneton, respectively g and gn are the electron and nuclear g factors a is the hyperfine coupling constant H is the external magnetic field while I and S are the nuclear and electron spin operators. The electronic g factor and the hyperfine constant are actually tensors, but for the hydrogen atom they may be treated, to a good approximation, as scalar quantities. 

The spin Hamiltonian contains electron spin operators that are completely defined as follows... 

Sg denotes the adjoint electron spin operator. One should notice that the expression [exp(—iLtx)Sp Do i p(f2ML)] results in the S-operators and the ml being (implicitly) time-dependent. In order to continue any further, we need to specify the lattice and its Liouvillian. 

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 ... 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

Then we obtain with the definition of the electron spin operator... 

Whereas Si and s2 are true one-electron spin operators, Ky is the exchange integral of electrons and in one-electron states i and j (independent particle picture of Hartree-Fock theory assumed). It should be stressed here that in the original work by Van Vleck (80) in 1932 the integral was denoted as Jy but as it is an exchange integral we write it as Ky in order to be in accordance with the notation in quantum chemistry, where Jy denotes a Coulomb integral. 

Though the true electron spin operators were employed here as well as in the Breit-Pauli Hamiltonian, the phenomenological Spin Hamiltonian, in which the spin coupling is an exchange effect, is in sharp contrast to the Breit-Pauli Hamiltonian, that is including the (magnetic) spin-spin interactions. Since the exchange effect is an effect introduced by the Pauli principle imposed on the wave function, we may write the electron-electron interaction as an expectation value,... 

A description in terms of local spins, however, raises the question of how to define these localized surrogate spins, where we have to move from one-electron spin operators Sj to multi-electron spin operators Sa that are located at one center and summarize all electronic contributions attributed to this center. This can be solved by the introduction of projection operators Pa (33,113,114,122-124). The projection operators do not alter any property of the molecule but divide it into local basins A. They add up to the identity operator,... 

The calculation of magnetic parameters such as the hyperfine coupling constants and g-factors for oligonuclear clusters is of fundamental importance as a tool for the evaluation of spectroscopic data from EPR and ENDOR experiments. The hyperfine interaction is experimentally interpreted with the spin Hamiltonian (SH) H = S - A-1, where S is the fictitious, electron spin operator related to the ground state of the cluster, A is the hyperfine tensor, and I is the nuclear spin operator. Consequently, it is... 

As an example, suppose we want the eigenfunctions and eigenvalues of the electron spin operator Sx. We shall use the basis functions a and ( for the calculation. Equation (2.56) gives the S matrix in this basis. The... 

Further, so may be expressed formally as = J2i where the s, are the usual one-electron spin operators and the first rank tensor operators denote the spatial part of >so related to electron i. (In the BP Hamiltonian (Eq. [104]), for example, 2 corresponds to the terms in braces.) One then obtains... 

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... 

Since the electric quadrupole interaction does not involve the electron spin operators, its form remains the same as in equation (7.146) with the operator T2(V )... 

Likewise the electronic spin operators are expanded over the (25 + 1) matrices which, in turn, are defined in the same way as the j [Eq. (2.29)]. 

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