Quantum number seniority

Here wi, W2, W3 are parameters characterizing the representations of group Rj u, U2 stand for the corresponding quantities of group G2 v is the seniority quantum number, defined in a simpler way in Chapter 9. On the other hand, the eigenvalues of the Casimir operator of group i 2(+i may be expressed in the following way by v and S quantum numbers... 

However, this is not the case for the dN shell. For d3 there are two of the same 2D terms. This problem with the dN shell and, partially, the fN shell was solved by Racah in his paper [23] introducing the seniority quantum number. In accordance with formula (9.7) we can build the antisymmetric wave function of shell lN with the help of the CFP with one detached electron. However, we could also use the CFP with two detached electrons... 

Thus, the repeating terms can be divided into two groups those occurring for the first time in the configuration under consideration and those already existing in the configurations lN 2, lN 4, etc. Adopting this method we can find the minimal number of electrons, usually marked by v, for which the term LS occurs for the first time. It is called the seniority quantum number and is usually denoted as pre-subscript 2S+ L. 

The seniority quantum number can also have a group-theoretical interpretation. If we define... 

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. 

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 ... 

Out of the set of additional quantum numbers a, the seniority number v is separated in an explicit form. 

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). 

For repeating terms of the fN configuration to be classified, the seniority quantum number is no longer sufficient in this case (see (14.39)) the characteristics of irreducible representations of the G2 group [24] are used. The generators of this group are the fourteen operators... 

In (16.4) and (16.5) ranks K,k,K are arbitrary save for the fact that they must obey the appropriate triangle conditions. Relation (16.5) enables us to construct wave functions characterized by seniority quantum number v, if the quantities with v = v — 2 are known. To establish a similar formula relating wave functions to v and v = v — 1, we must in (16.4) substitute operator for tensor TiKkK Reasoning along the same lines as in... 

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). 

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. 

The term LjS can be chosen in an arbitrary manner, and the normalization factor is found from the normalization"condition for coefficients of fractional parentage at fixed momenta L SJ and L,S. Equation (16.66) holds for repeating terms that are uniquely classified by the seniority quantum number v, but for non-repeating terms (when 5(L2S2,LS) = 0) that equation becomes the conventional Redmond formula [109]. 

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... 

Index k runs here through odd values 1,3,...,2 . Expression (19.79) is diagonal with respect to seniority quantum number v, i.e. 

Making use of the formulas presented in this section we are in a position to write the expression for the energy of the electrostatic interaction between any number of shells. In this case only a number of terms items increase in the corresponding formulas. However, we have to notice that these terms must be diagonal with respect to the quantum numbers of those shells on which they do not act. Let us also recall that in all formulas containing at least one almost filled shell, it is meant that the seniority quantum number v is excluded from a. This is necessary for the reconciliation of phase conditions between the corresponding quantities. 

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 — ... 

Relationships between coefficients gk for different degrees of occupation of the subshells (the cases of almost and completely filled subshells) are described by equalities (20.35), (20.36) and (20.38). Therefore, for magnetic interactions one has additionally to consider such conditions only in the case of coefficient dk. Bearing in mind that k acquires only odd values and that the submatrix elements of operator Tk are diagonal with respect to seniority quantum number v, we find... 

Unlike LS coupling, additional classification of states of the subshell of equivalent electrons in jj coupling in the cases of practical interest causes no problems. In fact, it is well known [18] that for the degenerate states of the jN configuration, at j < 7/2, to be classified it is sufficient only to use additionally the seniority quantum number v. 

Actually, only transitions described by one term in (25.23), take place. Formulas (25.22) and (25.23) are valid for the first and second forms of the Efc-transition operator (formulas (4.12) and (4.13)). The corresponding one-electron submatrix elements are given by (25.5) and (25.6). Analogous expressions for the third form of the /c-radiation operator are established in [77]. The appropriate selection and sum rules may be found in a similar way as was done for transitions between different configurations. It is interesting to mention that the non-zero conditions for submatrix elements for the operator Uk with regard to a seniority quantum number suggest new selection rules for the transitions in the shell of equivalent electrons v = v at odd and v = v, v 2 at even k values. 

Selection rules for electronic transitions with the participation of core electrons are similar to those for transitions when the core is left unchanged, with the exception of the selection rules following from the CFP with one detached electron. Then seniority quantum numbers t ,- and v- of the subshells, between which the electron is jumping , must be changed by unity, i.e. At ,- = 1, At - = 1. 

The seniority number (seniority for short), v, is a quantum number related to eigenvalues of the Racah seniority operator . The seniority matches the electron configuration ln in which the particular term first appears. We will see later that some matrix elements of the operators included in the Hamiltonian are diagonal in v so that their off-diagonal counterparts vanish. For some operators, however, there are crossing terms in v. 

The lj-SM is characterised by the fact that it is very easy to construct in it a basis states in the seniority scheme. Let a (basis) state n,v a> contain n particles, v of which contributing to make this state have seniority v. (a stands for additional quantum numbers.) This state then has S pairs (of Cooper type) of which number equals k=(n-v)/2. The states with n v (and hence k=0) are called highest senioriy (HS) states. 

The examination of the role of the two-electron term in the Hamiltonian shows that elements of the many-electron basis with occupied Kramers pairs of spin orbitals generally will have a larger energy than others and that the ground state configuration conforms with Hund s rule. Kramers pairs are related to the Racah seniority approximate quantum number. The rotationally invariant geminal creator... 

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