Matrix zero-field

In general, no simple, consistent set of analytical expressions for the resonance condition of all intradoublet transitions and all possible rhombicities can be derived with the perturbation theory for these systems. Therefore, the rather different approach is taken to numerically compute all effective g-values using quantum mechanics and matrix diagonalization techniques (Chapters 7-9) and to tabulate the results in the form of graphs of geff,s versus the rhombicity r = E/D. This is a useful approach because it turns out that if the zero-field interaction is sufficiently dominant over... 

It is informative to look at this matrix in zero field (B = 0) ... 

For axial symmetry (i.e., / = 0) this is a diagonal matrix with only two eigenvalues D and -D and a zero-field splitting AE = D —D = 2D as we have previously noted in... 

Figure 5.12. But even for systems with finite rhombicity the zero-field energy matrix can be reduced to 2 x 2 dimensionality...

We use these relations to write out the energy matrix initially in zero field (i.e., ignoring the Zeeman interaction) ... 

In Equation 7.33 we have written out both the g-value and the zero-field coefficient of the basic S2 interaction term in the form of diagonal 3x3 matrices in which all off-diagonal elements are equal to zero. The diagonal elements were indexed with subscripts x, y, z, corresponding to the Cartesian axes of the molecular axes system. But how do we define a molecular axis system in a (bio)coordination complex that lacks symmetry The answer is that if we would have made a wrong choice, then the matrices would not be diagonal with zeros elsewhere. In other words, if the spin Hamiltonian would have been written out for a different axes system, then, for example, the g-matrix would not have three, but rather six, independent elements ... 

This matrix is diagonal in the zero-field interaction, so the zeroth-order energy levels can be directly seen to be... 

The label IAmsl = 2 does not mean that two quanta hv are absorbed it is simply a somewhat unfortunate but widely divulged notation to indicate a transition (A E = hv) between two levels that we happen to have labeled as 1+1) and 1-1). Strictly speaking, these labels should only apply to the strong-field situation of (S S) B S) as we discussed in Chapter 4. In the present example of Figure 11.1 we are in the weak-to-intermediate field regime (S S > B S), which means that the actual wavefunctions are linear combinations of the ones in Equation 11.5. In particular, a rhombic E-term mixes the 1+1) and 1-1) levels as can be seen from its appearance in nondiagonal positions in the zero-field energy matrix... 

The arylcarbenes are also bent, the angle being about 150—155° for phenyl-and diphenylcarbenes 63,64) jj g zero-field splitting parameters were shown to be appreciably dependent on the host matrix used >. However diphenylcarbene prepared from different precursors proved to be identical and is believed to be the triplet species of diphenylcarbene. 

Using the zero field n(m states we calculate the matrix elements (/i m Ez n m ) of the Stark perturbation to the zero field Hamiltonian. Writing the matrix element in spherical coordinates and choosing the z axis as the axis of quantization,... 

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