Spin eigenvalues

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),... 

Hence, for a given total spin eigenvalue S there exist 2S + 1 states that all yield the same energy but may split when magnetic fields described as spin interactions are important in the Hamiltonian. The individual spin states are referred to as the S = 0 singlet state with 2S + 1 = 1, as doublet S — j with 2S + 1 = 2, as triplet S = 1 with 2S + 1 = 3, and so on. 

Next we discuss examples of the two types of excited states. To focus the treatment we deal only with the excited states of the smallest spin eigenvalue for a given system, that is, singlet states for a closed-shell molecule and doublet states for radicals. 

The nuclear spin eigenvalues are small compared to the Boltzmann energies (equation 9)... 

For example, a CSF written as 3300> indicates that the first two orbitals are both doubly occupied. This CSF is a four-electron singlet. The ket 3120> indicates that the first orbital is doubly occupied the second orbital is singly occupied and coupled to the lower-level orbitals to increase the spin, in this case giving a three-electron doublet. Finally the singly occupied third orbital is coupled to decrease the overall spin, in this case giving a four-electron singlet. The spin eigenvalue of any ket. denoted by S or by b , may be determined simply by the difference in the number of = 1 step vector entries and the number of dj = 2 step vector entries. In fact, the sum of the step vector entries is the same for all CSFs with the same total spin and number of electrons . The CSF expansion space may easily be limited to include only those terms that contribute to the correct overall spin. An important property of CSFs constructed in this manner is that they form an orthonormal expansion basis. 

So there is just one such product with Ua = n with spin eigenvalue M = nf 2 and Just one with spin eigenvalue M = —n/2 when all the electrons have (3 spin. If all electrons but one have a spin there arc n ways of realising this situation and so there are n independent spin products with spin eigenvalue M = n/2 — 1. Similarly, if two a factors are replaced by 0s we get n(n — l)/2 products with spin eigenvalue M = n/2 — 2 and so on. 

In general, a product containing na spin a factors and np spin 0 factors has spin eigenvalue... 

In addition to the operators di ussed above, it is often important in quantum-chemical applications to evaluate commutators of pairs of operators. For example, to show that the creation operator is of doublet spin character (i.e., has the potential to change the total spin eigenvalue of any function upon which it acts by h) it is sufficient to demonstrate that [5. = hr, and [S+, ] = 0. As an example of how to... 

The spin operator 5 commutes with the hJ Hamiltonian operator, so we can have molecular spin orbitals that are eigenfunctions of S. The spin eigenvalues are equal to nish = (1/2), the same as with the hydrogen atom. We can represent the spin orbitals by a space orbital multiplied by the spin function a or the spin function as with atomic orbitals. 

It is a trivial matter to verify these spin eigenvalues by operating with and S, and we may easily extend the second result to show that parallel coupling of ft spins may also be described by a single product ... 

Since, using (5.9.7), the cartesian components S , are linear combinations of S+, S i (which yield non-zero spin-density contributions Q i between functions 0, 0x with different spin eigenvalues), it is... 

Since Hi is spin-independent, the first matrix element vanishes unless aM) and bM") have identical spin eigenvalues, and then takes the A/-independent value... 

We conclude that T - lvac) - provided that it does not vanish - represents a tenscH- state with spin eigenvalues 5 and M. Because of the close relationship between spin tensw operators and spin eigenfunctions, the terminology for spin functions is often used for spin tensor operators as well. Thus, a spin tensor operator with 5 = 0 is referred to as a singlet operator, S — gives a doublet operator, 5 = 1 a triplet, and so on. 

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