Quantum number quasispin

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

The methods of theoretical description of many-electron atoms on the basis of tensorial properties of the orbital and spin angular momenta are well established [14, 18] and enable the spectral characteristics of these systems to be effectively found. The relation between the seniority quantum number and quasispin makes it possible to extend the mathematical tools to include the quasispin space and to work out new modifications of the mathematical techniques in the theory of spectra of many-electron atoms that take due account of the tensorial properties of the quasispin operator. 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

The concept of quasispin quantum number was discussed in the Introduction and Chapter 9 (see formulas (9.22) and (9.23)). Now let us consider it in the framework of the second-quantization technique. We can introduce the following bilinear combinations of creation and annihilation operators obeying commutation relations (14.2) - the quasispin operator ... 

Let us recall that the seniority quantum number, by definition, is the number of unpaired particles in a given state, and two electrons are called paired when their orbital and spin momenta are zero. Since it follows from definitions (15.39) and (15.40) that operators and Q operating on wave function lNaLS) respectively, create and annihilate two paired electrons, the seniority and quasispin quantum numbers v and Q must be somehow related. Let a certain iV-particle state have absent the paired electrons ... 

Consequently, the quantum numbers of quasispin Q and seniority v are related by expression (9.22) which is also valid for the wave function in the general case at N v. Operator Qj1, acting on wave function (15.44), increases the number of particles by two paired electrons, leaving, by definition, number v unchanged. 

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 ... 

In consequence, classification of antisymmetric wave functions of lN configurations using the characteristics of irreducible representations of Sp4i+2 and R21+1 groups is fully equivalent to classification by the eigenvalues of operators S2 and Q2. If now we take into account formula (9.22) relating the quasispin quantum number to the seniority quantum number, we can establish the equivalence of (15.80) and (5.38). 

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 wave functions with N > v can be found from the wave functions with N = v using (16.1), in the second-quantization representation it is necessary to construct in an explicit form only wave functions with the number of electrons minimal for given v, i.e. IolQLSMq = —Q). But even such wave functions cannot be found by generalizing directly relation (15.4) if operator cp is still defined so that it would be an irreducible tensor in quasispin space, then the wave function it produces in the general case will not be characterized by some value of quantum number Q v). This is because the vacuum state 0) in quasispin space of one shell is not a scalar, but a component of a tensor of rank Q = l + 1/2... 

Operator q>(lN)av LS in the general case consists of a linear combination of creation operators that provides a classification of states according to quantum numbers a, v. Since each term of this expansion contains N creation operators then from (15.49) the quasispin rank of operator (LS) an(j jts projectjon are equal to N/2 (regardless of the values of quantum numbers a, v). Accordingly, for a function with a certain... 

In general, the operator (p(lN) Ls can be expressed in terms of a linear combination of creation operators that provides a classification of the wave function produced in the additional quantum numbers a. For example, in relationship (16.8) there first appear the linear combinations of tensorial products of creation operators that provide classification of the wave functions produced from the vacuum in the seniority quantum number v = 3 (quasispin Q = l — 1). 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

Applying the Wigner-Eckart theorem in all three spaces, we establish the property of SCFP in relation to interchanges of the spin and quasispin quantum numbers... 

This expression for /-electrons can be derived using the phase relations established for isoscalar parts of factorized CFP with different parities of the seniority number [24]. It turned out [91] that phase (16.55) provides sign relations between the CFP in the tables for d- and /-electrons, but it is unsuitable for the p-electrons. In this connection, in what follows all the relationships derived using the symmetry properties under transposition of the quantum numbers of spin and quasispin are provided up to the sign. 

CFP (9.11) also have a simple algebraic form. In the previous paragraph we discussed the behaviour of coefficients of fractional parentage in quasispin space and their symmetry under transposition of spin and quasispin quantum numbers. The use of these properties allows one, from a single CFP, to find pertinent quantities in the interval of occupation numbers for a given shell for which a given state exists [92]. 

Using (16.16) and the above relations, we can work out algebraic expressions for SCFP, and hence for CFP, in the entire interval of the number of electrons in the shell existing for given v. Taking account of the symmetry of CFP under transposition of spin and quasispin quantum numbers further expands the number of such expressions. Formulas of this kind can be established also for larger v, but with v = 5 and above they become so unwieldy and difficult to handle that this limits their practical uses. They may be found in [108]. 

The concept of the vectorial coupling of quasispin momenta was first applied to the nucleus to study the short-range pairing nucleonic interaction [117]. For interactions of that type the quasispin of the system is a sufficiently good quantum number. In atoms there is no such interaction - the electrons are acted upon by electrostatic repulsion forces, for which the quasispin quantum number is not conserved. Therefore, in general, the Hamiltonian matrix defined in the basis of wave functions (17.56) is essentially non-diagonal. 

In certain special cases the approximate symmetries in atoms are sufficiently well explained using the quasispin formalism. In particular, the quasispin technique can be utilized to describe fairly accurately configuration mixing for doubly excited states of the two-electron atom. In the quasispin basis the energy matrix of the electrostatic interaction operator of such configurations is nearly diagonal, and hence the quantum number of total quasispin Q is approximately good . 

The algebra of the Sp4 group coincides with the algebra of the rotation group of five-dimensional Euclidean space R5, i.e. these groups are locally isomorphic. The irreducible representations of the Sp4 group can be characterized by a set of two parameters (v, t) where the seniority quantum number v for the five-dimensional quasispin group indicates the number... 

Calculations carried out with hydrogen radial orbitals indicate that the supermultiplet basis S U4 in many cases is more diagonal than the basis in which an additional classification of states is achieved using the quantum numbers v, t) of the five-dimensional quasispin group. 

Accounting for the properties of the seniority (quasispin) quantum numbers, we are able to express the matrix elements of the energy operator in terms of the corresponding quantities for the electronic configuration, for which this term has occurred for the first time (/f ajV L/S — ... 

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