Decoupled representation

Nalewajski RF (1991) Normal (decoupled) representation of electronegativity equalization equations in a molecule. Int J Quant Chem 40(2) 265—285... 

The totally decoupled representation involves displacements in populations of PNM,... 

It is easy to show that the components of J obey the standard commutation relationships (5.13). The commuting operators J and J/ therefore have eigenfunctions j,m). In this section we describe how these eigenfunctions are related to those of j and j. We can describe the combined system by a simple product of the wavefunctions y i, m ) j2, m2) which we write as j mi j2m2) this is known as the uncoupled or decoupled representation. These products are still eigenfunctions of j, Jxz, J and J2z- They are also eigenfunctions of Jz with eigenvalue m = m +m2 ... 

Remembering that the J operator acts only on the jx, mi) part of the decoupled representation and J2 acts only on the j2, m2) part, we see that J2 connects states with different values of mi and m2 but with constant (mi + m2). 

Table 8.1. Energy matrix for Ih in the J = 1 level in the decoupled representation... 

As stated earlier, the decoupled representation used above is most appropriate for the high field measurements of the magnetic resonance spectrum. Before examining the field dependence of the energy levels and transition probabilities, we recalculate the matrix elements of Xusing the coupled basis, J, I, F, MF). We note that for J = 1 and / = 1, we can have F = 2 (with Mp = 2, 1, 0), F = 1 (with Mp = 1, 0), and F = 0 (Mf = 0). 

Finally in this section we turn to the heteronuclear species HD, which was studied in both the 1940 phase [9] and the post-war higher-resolution phase [11], The three coupled angular momenta J, 7D and 7h have space-fixed projections, which are appropriate for strong field studies, of Mj, Md and Mh. The experimental studies involved HD again in its J = 1 level, so that in the decoupled representation there are 18 possible levels (Mj = 0, 1 MD = 0, 1 Mh = 1/2). Consequently there are three types of transition involving these energy levels which occur in clusters at different... 

The required matrix elements in the decoupled representation are now calculated all of the scalar products which occur are expanded in the space-fixed coordinate system. 

We now apply these general results to the specific problem of HF. Apart from the Stark effect, we shall otherwise ignore the very small matrix elements which are off-diagonal in J, and evaluate the terms for J = 1, Mj = 0, 1, /H = h = 1 /2. As Weiss pointed out, the total magnetic component Mz = Mj + MF + Mh is a good quantum number and may be used to set up a decoupled representation in which states of different Mz value are diagonalised separately. For J = 1 there are twelve primitive basis states, which we write below in the form MF, Mh, Mj). 

The decoupled representation involves the normal population modes defined by the columns of the transformation matrix, U = f4, which diagonalizes the AIM hardness matrix, i.e., rotates the coordination system to the principal axes of the hardness tensor [33-36, 44] ... 

Of interest in the theory of chemical reactivity is also the intra-reactant decoupled representation [35,36] (see also Sect. 3.1). It corresponds to the choice of the (m -1- n) basis vectors consisting of eigenvectors of qA.A and Iib.b, (I a>5... 

As already mentioned in Sect. 2.2 the AIM and MO resolutions correspond to the strongly coupled EE equations (15), with the large off-diagonal elements of the respective hardness matrices. The normal representation, on the other hand, totally decouples the EE equations (23). For a reactive system = A—B, consisting of two reactants A (an acid) and B (a base), various partially decoupled representations are also of interest [35, 36]. 

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