Matrix three-spin cases

Afterward, we have to find approximations for 2-RDM spin components D". Let us now focus on high-spin cases only, such as doublet, triplet, quartet... spins for one, two, three. .. unpaired electrons outside the closed shells. Accordingly, singly occupied orbitals will always have the same spin (Al p = 0 or = 0) so the trace of the one-matrix, Eq. (6), becomes... 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

If matrix elements off-diagonal in N are neglected, equation (8.254) yields the following simple results for the energies of the three spin components of a given N level in a case (b)3 state ... 

In order to calculate the spin-angular parts of matrix elements of the two-particle operator (1) with an arbitrary number of open shells, it is necessary to consider all possible distributions of shells upon which the second quantization operators are acting. In [2] they are found to be grouped into 42 different distributions, subdivided into 4 different classes. This also explains why operator (1) is written as the sum of four complex terms. The first term represents the case when all second-quantization operators act upon the same shell (distribution 1 in [2]), the second describes the situation when these operators act upon the two different shells (distributions 2-10), third and fourth are in charge of the interactions upon three and four shells respectively (distributions 11-18 and 19-42). Such expression is particularly convenient to take into account correlation effects, because it describes all possible superpositions of configurations for the case of two-electron operator. 

We are now in a position to examine the details of the Zeeman effect in the para-H2, TV = 2 level, and thereby to understand Lichten s magnetic resonance studies. For each Mj component we may set up an energy matrix, using equations (8.180) and (8.181) which describe the Zeeman interactions, and equations (8.201), (8.206) and (8.214) which give the zero-field energies. Since Mj = 3 components exist only for J = 3, diagonalisation in this case is not required. For Mj = 2 the J = 2 and 3 states are involved. For Mj = 0 and I, however, the matrices involve all three fine-structure states and take the form shown below in table 8.7. Note that /. is equal to a0 + 3 63-2/4 and the spin-rotation terms have been omitted. The diagonal Zeeman matrix elements are... 

We continue to use the case (b) hyperfine-coupled basis set used earlier. The matrix elements of the first three terms in the effective Hamiltonian (8.251) do not involve the nuclear spins and are therefore independent of, and diagonal in, the quantum number F. The required matrix elements are now tabulated. 

Note the separation of the terms into those involving J, with anomalous commution rules in the molecule-fixed system, and those which do not involve J Because both CIO and BrO are good case (a) molecules we choose to work in the case (a) basis set ij. A . S. X ./. Q. Mj) we will subsequently add nuclear spin to the basis set. If we separate the q = 0 components of (9.32) from the q = 1 it is readily apparent that the matrix elements within the case (a) basis have three different types of contribution,... 

For adequate simplicity, we return to consider the case when there is only a single Zeeman splitting term in the spin-Hamiltonian. For a given matrix g, much depends on the relations of its principal values. If all three such values are identical, then of course only one spectral line is observed, at all orientations of n = B/B, where B > 0. Except in that isotropic case, it becomes important what crystal symmetry is at hand there are 11 distinct cases (Laue groups).12... 

Though we can compare electron densities directly, there is often a need for more condensed information. The missing link in the experimental sequence are the steps from the electron density to the one-particle density matrix f(1,1 ) to the wavefunction. Essentially the difficulty is that the wavefunction is a function of the 3n space coordinates of the electrons (and the n spin coordinates), while the electron density is only a three-dimensional function. Drastic assumptions must be introduced, such as the description of the molecular orbitals by a limited basis set, and the representation of the density by a single Slater-determinant, in which case the idempotency constraint reduces the number of unknowns... 

As an application of the method, we consider the lowest excitation energies of the alkaline earth elements and the zinc series. Here, in addition to the degeneracy with respect to the spin index, the s -+ p transitions under consideration are threefold degenerate in the magnetic quantum number m of the final state. Hence, we have six degenerate poles and Eq. (354) is a (6x6) eigenvalue problem. In our case, however, the matrix in Eq. (354) consists of (three)... 

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