Zero-field energy matrix

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

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... 

For the zero-field problem F remains a good quantum number, but J is not because of the hyperfine mixing. The spin spin, spin orbit and spin rotation energies have already been listed in table 8.6. The complete zero-field energy matrix, including the hyperfine terms, is as follows. 

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... 

In the case of the 5T%g) term the orbital kets are the same as for the 2T2(g) term. A higher spin multiplicity enlarges the dimension of the interaction matrix to N = 3-5=15. The secular equation can be split into a set of secular equations of lower dimension (Table 8.40). The zero-field energy levels are shown in Fig. 8.23. 

The zero-field energies of a cluster are obtained by constructing the spin Hamiltonian matrix, which should be diagonalised when non-zero off-diagonal matrix elements occur. 

These rules can be proved by diagonalization of the crystal-field energy matrix within a J = 1 multiplet. The matrix elements can be calculated using the method described above. The reduced matrix elements are left unspecified, because we want to consider a general case. The non-zero matrix elements are //n = oo ... 

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

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

It is now instructive to make use of the weak field results to obtain an energy level pattern for the J = 1 rotational level. The complete zero-field matrix is as follows. 

Each coupled spin level in the zero-field or correlation diagram had energy E(Si2, S34, S) for each wave function Si2, S34, S, M). The coupling scheme adopted was S12 = Sj + S2 S34 = S3 + S4 S = S12 + S34. Since the matrix of Eq. (14) cannot be diagonalized, matrix elements were worked out by tensor operator methods (50, 68). 

Vibrational spectra are not only good tests of a given theoretical model but also can aid the identification of unusual gas-phase or matrix isolated species. In addition, the complete vibrational force field is required to calculate zero point energies and important thermodynamic data such as enthalpies, entropies and hence Gibbs Free energies [10]. Moreover, the second derivatives are crucial to the calculation of Transition State geometries. 

In Table 1 we give the matrix elements Hy in terms of Dim and Gim-The position functions D ni and G are given in Table 2. These tables should enable the reader to calculate the relative one-electron energies of the crystal field potential. Note that in low symmetries (e.g. Dza, Cz , Dad) where different rf-orbitals (or sets of d-orbitals) are mixed by the ligand field and non-zero off-diagonal matrix elements Hy ( Vy) arise, it is necessary to solve the secular equations. 

Hzf and Hfield are respectively the zero-field and field-dependent parts of H and Mfii is the tuning rate for the energy of basis function iM) in the field F. The Hamiltonian matrix for this A J = 0 two-level problem is... 

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