T-P isomorphism

For a single determinantal wave function k = 1 holds true (no reduction). 8.5.8 T-P isomorphism 

The angular momentum operators acting on the set of complex (pi,Po P-i) functions yield 

Except for the sign, equivalent results are obtained for the set of 12 functions forming the basis of the T2 term 

Thus it is legitimate to claim that the transformation matrix of / within the t2 representation of the -functions is equal to the matrix of (—/) for the p-functions (see Table 8.36). 

As far as the spin-orbit coupling for a one-electron system is concerned, one can use an equivalence 

---

The second alternative utilizes the T-p isomorphism according to which the angular momentum is substituted for... 

As the ground CFT is orbitally triply degenerate, it does not fulfill the conditions for a direct application of the SH formalism. By using the T-p isomorphism, the 9 x 9 interaction matrix in the basis set of transformed octahedral kets (which accounts for the Cl of the 3 -terms) is of the form that can be factored into secular equations of lower dimensions as already presented in Table 13 the eigenvalues can be written in a closed form. 

When dealing with the electronic ground state T2, the advantage of the T-p isomorphism can be employed the I0 Ly, and Lz components of the (fictitious) orbital angular momentum (1=1) fulfill a relationship... 

On the other hand, the T—P isomorphism says that the following correspondence of the functions JM) = 2,2) of the P-term and F (3Tl)) exists... 

Owing to the T—P isomorphism, we have to substitute L — 1 when dealing with T-term kets. 

Now, applying an equivalence coefficient that properly accounts for the T—P isomorphism... 

The perturbation Hamiltonian, in the form that properly accounts for the T— P isomorphism, is... 

Exercise 4.36 Show that the representation of SU(2) on (matrix multiplication) is isomorphic to the representation. Hint Define T —> P ... 

Now W is a finite-dimensional, unitary, irreducible representation, so by Proposition 6.16 there must be a nonnegative even integer h and an isomorphism T P W of representations. Because T is an isomorphism, the fist of weights for P must be the same as the fist of weights for W. Hence... 

We shall use the T-/ -isomorphism that allows us to consider the orbital triplet T2 as a state possessing the fictitious orbital angular momentum L = 1, keeping in mind that the matrix elements of the angular momentum operator L within T2 and P bases are of the opposite signs, L(T2) = —L(P) [2]. As it was shown in our recent paper [10] this approach provides both an efficient computational tool and a clear insight on the magnetic anisotropy of the system that appears due to the orbital contributions. Within T-P formalism the spin-orbital and Zeeman terms can be represented as ... 

---