Momentum quasispin

Quasispin formalism can be used in describing the properties of the occupation number space for an arbitrary pairing state. For one shell of equivalent electrons, these pairing states can be chosen to be two one-particle states with the opposite values of angular momentum projections. 

The operator of total quasispin angular momentum of the shell can be obtained by the vectorial coupling of quasispin momenta of all the pairing states. For a shell of equivalent electrons, instead of (15.35) we have... 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

We noted in Chapter 6 that the seniority quantum number v in reduction chain (14.38) unambiguously classifies the irreducible representations of S P4i+2 group. Then one may well ask how can the earlier group-theoretical schemes include a rotation group defined by the operators of quasispin angular momentum ... 

We have established above that one-electron operators are expressible in terms of tensors W(kK) related to triple tensors W KkK by (15.59). Therefore, we shall find here the expansion in terms of irreducible tensors in quasispin space only for the two-particle operator that is a scalar in the total momentum. 

It follows from (14.5) that when Q operates on wave function (15.44), this changes only the z-projection of the quasispin momentum, and so we can obtain from the wave function of N electrons appropriate values for N + 2 electrons with the same quantum numbers... 

Since the number of electrons in the lN configuration is uniquely determined by the value of the z-projection of quasispin momentum, then the quasispin method provides us with a common approach to the spin-angular parts of the wave functions of partially and almost filled shells that differ only by the sign of that z-projection. The phase relations between various quantities will then uniquely follow from the symmetry... 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

For operators corresponding to physical quantities, we can also obtain an expansion in terms of irreducible tensors in quasispin space. Specifically, for two-particle operators (13.23) that are scalars with respect to the total momentum J... 

At some specific values of the ranks on the left side of (23.55) we can get a set of equations for sums with the same structure of the ranks of total angular momentum. For the operators that are scalars in the spaces of quasispin and total angular momenta (K" = k" = 0), we have... 

Thus, the use of wave function in the form (23.67) and operators in the form (23.62)—(23.66) makes it possible to separate the dependence of multi-configuration matrix elements on the total number of electrons using the Wigner-Eckart theorem, and to regard this form of superposition-of-configuration approximation itself as a one-configuration approximation in the space of total quasispin angular momentum. 

In (23.80) and (23.81) the rank sum y+k is an odd number, otherwise these operators are identically equal to zero. We shall separate sets of operators that are scalars in the space of total angular momentum but tensors in isospin space. If we go through a similar procedure for one subshell of equivalent electrons we shall end up with the quasispin classification of its states. It turns out that ten operators l/(00), U 0 vffl, F 0) are generators of a group of five-dimensional quasispin, wnich can be easily verified by comparing their commutation relations with the standard commutation relations for generators of that group. 

Now we shall have to express operators for physical quantities in terms of irreducible tensors in the spaces of total angular momentum and quasispin. One-electron terms of relativistic energy operator (2.1) (formulas (2.2)-(2.4)) are expressed in terms of operators (23.69), (23.71)-(23.73) in a trivial way. With two-electron operators the procedure of deriving the pertinent relations is more complex. The relativistic counterpart of (18.50)... 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

Ayl transforms as an irreducible tensor operator under operations of G, and as a rank-2 spinor in the angular momentum algebra generated by the quasispin operators. We form the quasispin generators as a coupled tensor in quasispin space Q(A) = i[AAAA]7V2, where [AB] = Y.qq lm q c/)AqBqi. In the Condon and Shortley spherical basis choice (with m = 1, 0, — 1) for the SO(3) Clebsch-Gordan coefficients [11-13,21-23] this takes the form [6,21] ... 

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 ... 

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