Pseudo-Angular Momentum States

The familiar set of the three t2g orbitals in an octahedral complex constitutes a three-dimensional shell. Classical ligand field theory has drawn attention to the fact that the matrix representation of the angular momentum operator t in a p-orbital basis is equal to the matrix of — if in the basis of the three d-orbitals with t2g symmetry [2,3]. This correspondence implies that, under a d-only assumption, l2 g electrons can be treated as pseudo-p electrons, yielding an interesting isomorphism between (t2g) states and atomic (p) multiplets. We will discuss this relationship later on in more detail. 

Rotations in a vector space of three orbitals are described by the group SO(3) of orthogonal 3x3 matrices with determinant +1. To embed the octahedral rotation group in this covering group one needs a matrix representation of O which also consists of orthogonal and unimodular 3x3 matrices. Such a matrix representation is sometimes called the fundamental vector representation of the point group. In the case of O the fundamental vector representation is Ti and not T2. Indeed the 7] matrices are unimodular, i.e. have determinant +1, while the determinants of the T2 matrices are equal to the characters of the one-dimensional representation A2. 

The development of a pseudo-angular momentum theory of (t2g)n states, which is rooted in the point group under consideration can thus proceed in the following way one first converts the T2 basis into a T1 basis. The 7 functions can then be coupled to multiplet states using the coupling theory of the full 

For real components the coupling table corresponding to the A2 x T2 product has the following form [3]  

Here A2 symbolizes a pseudo-scalar of A2 symmetry, normalized to unity. The actual form of this pseudoscalar need not bother us. The only property we will have to use later on is that even powers of A2 are equal to +1. Now we can proceed by defining rotation generators f x, y,t 2 in the standard way, as indicated in Table 1 [10]. Note that primed symbols are used here to distinguish the pseudo-operators from their true counterparts in real coordinate space. Evidently the action of the true angular momentum operators t y, (z on the basis functions is ill defined since these functions contain small ligand terms. 

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A final point concerns the finite point group symmetry adaptation of the resulting states. Since the octahedral group has been rooted in the Lie group of the pseudo-angular momentum generators the standard SO (3) J. O j D4 sub-duction relations [13] can be used to obtain the symmetry adapted fMr combinations. The S, P, and D states thus match resp. Au Tt and E + T2... 

The quasispin classification of the ligand-field-split states was detailed in Ref. [19] following Judd s analysis for the rotation group. This problem turns out to have some subtleties, for example, the difficulty Ceulemans [10] discusses (and resolves) when bestowing a pseudo-angular momentum on his f2 subshell. From Judd [5] and Wyboume [19] we note the following. The total subshell state space is ... 

The levels structure of the EM acceptor centres is determined by the characteristics of the VB of their host crystal near from its absolute extremum. As mentioned before, this extremum is located at k = 0 in most semiconductors. The contribution of the atomic p states of the constituent semiconductor atoms is predominant in the VB (for the compound crystals, it is related to the most electronegative atom). When spin-orbit (s-o) coupling is included, the pseudo-angular momentum J associated with the upper VB is L + S where L = 1 corresponds to the p electrons of the host crystal. For this reason and since they correspond to the pseudo-angular momenta J = 3/2 and 1/2, in the description of the acceptor states in diamond-type semiconductors, the T8 and r7- VBs are often labelled the p3/2 and j> /2 bands, respectively. 

A pseudo potential approach was adopted by Hickman et al. [259] to calculate the excited metastable states of a He atom under liquid He. The density functional approach developed by Dupont-Roc et al. [260] was applied subsequently [261] for the description of the nature of the cavity formed around an alkali atom in the excited state of non-zero angular momentum. The resulting form of the cavity differs very much from the spherical shape. A similar approach was adopted by De Toffol et al. [262] to find qualitatively the first excited states of Na and Cs in liquid He. Earlier work in this direction was given in detail in Ref. [263]. 

The disadvantage of the pseudo-Hamiltonian is that one does not have very much flexibility in matching the core response to valence electrons with different angular momentum because the restrictions on the mass tensor are too severe, especially for first-row and transition metal atoms (i.e., for the cases with strong nonlocalities). In particular, for transition metals it is not possible to use an Ar core because the first electron must always go into an s state [48]. In fact, this is of secondary importance since for accurate calculations, which are the aim of QMC, one has to include 35 and 3p states in the valence space for the 3d transition elements. 

In the B state, the three sodium atoms perform a nearly free pseudorotational motion in the moat of a pseudo Jahn-Teller potential that is characterized by a vibronic angular momentum quantum number j. 

For 4Jp + 2Jf = 0, the matrix splits into two separate 2x2 blocks, which have the same eigenvalues. The splitting pattern is thus as in the central panel of Fig. 7.5. Such a case can occur for a state. The orbital part of this state has no angular momentum, since the corresponding operator is not included in the direct square Ti E x E. As result, the magnetic moment of such a state is due only to the doublet spin part. Such a state behaves as a pseudo-doublet. 

The two sheets of the Lis DMBE potential energy surface have been used for detailed zero total angular momentum calculations of the bound and pseudo-bound vibrational states of Lis without (NGP) and with (GP) consideration of the GP effect. 33,134 particular, for the lower sheet. 

The values for the wave functions at the origin are known exactly for hydrogen and can be substituted in. The 2p state can have total angular momentum J = 3/2 and J = 1/2. Only the J = 1/2 term contributes, as can be seen by evaluating directly the matrix elements of VpNc- This, of course, is expected because our potential is a (pseudo)-scalar. 

These results i.e. the great influence on the dissociation of the pseudo rotation angular momentum 1 about the Hg -N-N axis has also been demonstrated in theoretical calculations by Jouvet and Beswick (i ). The states of the excited Hg N2 complex are represented for the perturbed mercury by I J (= L + S), Q > as II, O" " > and II, 1 > and as 10,0 > for the Pq state. There the coupling between the 0" state ( Pi) and 0 ( Pq) is essentially 0 even through higher mercury states owing to the +, - parity difference. On the other hand, this interdiction is lifted when 1=1 vibrations become excited as we have observed. Nesbitt et al( ) observed the influence of the +, - parity upon the coupling of TT "/" and I" " vibrations in the NeHF predissociation, with similar conclusions. 

The two degrees of freedom associated with the ring puckering are, therefore, an ordinary vibration and a type of one-dimensional rotation in which the phase of the puckering moves around the ring the latter is not, however, a true rotation since there in no angular momentum about the axis of rotation, and so is described as a pseudo-rotation . This separation of the wave equation is not exact, but it has been stated that exact separation is possible and, on the assumptions of harmonic oscillations and small amplitudes of vibrations, leads to the same results as those given. 

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