Spin-orbit matrix

The required spin-orbit matrix elements are given in Table 8, whence the eigenvalues of the matrix of the first and second order contributions are easily found to be 0, and 2 — ((3/A i) + (2/A 2)), where AEl and A E2 represent respectively the energies of the 3II(o7r) and 3Il(7r6) states above the ground level. Introducing the appropriate metal mixing coefficient, c, for the a, ir, and 5 levels the required g value then becomes... 

Table 4. Spin-orbit matrix elements in formaldehyde...

The space parts of (n,7i ) states are of Ai or Si those of [a,n ) or n,a ) states are of 2 or S 2 symmetry. A closer inspection of the magnitude of the spin-orbit matrix elements, retaining only one-center contributions, shows that Hso Pnn -y = 0. ... 

Table 4. Calculated spin-orbit matrix elements (using Eq. (3)), calculated (using Eq. (2)) and observed Zero Field Splittings...

The phenomenological spin-orbit Hamiltonian ought not to be used for computing spin-orbit matrix elements, though. An example for a failure of such a procedure will be discussed in detail in the later subsection on a word of caution. 

Equation [172] or related expressions (Table 10) are applied extensively when evaluating the spin part of spin-orbit matrix elements, for configuration interaction (Cl) wave functions. The latter are usually provided for a single Ms component only. 

Silver,71 we generally do not employ the WET for the spatial part of a spin-orbit matrix element. In the Russell-Saunders (LS) coupling scheme, we still assume, however, that S is a (fairly) good quantum number 1 and that we... 

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

Because of the spin-orbit selection rules, only triplet zeroth-order states contribute to the first-order perturbation correction of a singlet wave function. In Rayleigh-Schro dinger perturbation theory, the expansion coefficient a of a triplet zeroth-order state (3spin-orbit matrix element with the electronic ground state (in the numerator) and its energy difference with respect to the latter (in the denominator). 

Spin-Orbit Matrix Elements for Internally Contracted Multireference Configuration Interaction Wavefunctions. 

Note Added in Proofs For the evaluation of spin-orbit matrix elements, standard basis sets of... 

The HF calculations predict a 3II ground state, which is the same as found for CaO, but as noted above, Cl might well reverse this ordering, and it is clear that very extensive calculations are needed before this question is definitely settled. A very recent paper reporting PE curves for BeO, MgO, and CaO, using DZ + P basis sets, has also appeared,310 and the authors also discuss the dissociation behaviour of the ground state. Calculation of the spin-orbit matrix elements shows that 3I1 and 12+ states are not significantly mixed. 

Hence the spin-orbit matrix element is given by... 

Thus, from a single calculation we may calculate a complete spin-orbit matrix over a manifold of eigenstates of Hq (see section 4.II). 

Equation (54) is the basis for the semiempirical methods to be discussed below. Several approximations have been used for (rXi). In all of the methods, the matrix elements of Hso are evaluated with respect to a wave function variationally optimized in A-S coupling. The spin-orbit matrix is then diagonalized for the various values of the total angular momentum of the system. 

The spin—orbit matrix element The spin—orbit potential (3.174) is... 

Table 3. Definition of the spin-orbit matrix elements involving the S2 (l4Ag) state, together with our schematic representation given in Figure 3.

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