Reduced matrix elements operators

The only difference is that a(0) is now an operator acting in jm) space of angular momentum eigenfunctions. This space consists of an infinite number of states, unlike those discussed above which had only four. This complication may be partly avoided if one takes into account that the scalar product in Eq. (4.55) does not depend on the projection index m. From spherical isotropy of space, Eq. (4.55) may be expressed via reduced matrix elements (/ a(0 /) as follows... 

A general development of matrix elements of an irreducible tensor operator leads to the Wigner-Eckart theorem (see, for example, Tinkham [2] or Chisholm [7]), which relates matrix elements between specific symmetry species to a single reduced matrix element that depends only on the irrep labels, but this is beyond the scope of the present course. 

The relation between the CFP with a detached electrons and the reduced matrix elements of operator q>(lNfiLS generating [see (15.4)] the <7-electron wave function is established in exactly the same way as in the derivation of (15.21). Only now in the appropriate determinants we have to apply the Laplace expansion in terms of a rows. The final expression takes the form... 

In [90] the relationship between eigenvalues of the Casimir operators of higher-rank groups and quantum numbers v, N, L, S is taken into account to work out algebraic expressions for some of the reduced matrix elements of operators (Uk Uk) and (Vkl Vkl). However, the above formulas directly relate the operators concerned, and some of these formulas are not defined by the Casimir operators of respective groups. 

Equations of this kind can also be derived for the special cases of reduced matrix elements of operators composed of irreducible tensors. Then, using the relation between CFP and the submatrix element of irreducible tensorial operators established in [105], we can obtain several algebraic expressions for two-electron CFP [92]. Unfortunately such algebraic expressions for CFP do not embrace all the required values even for the pN shell, which imposes constraints on their practical uses by preventing analytic summation of the matrix elements of operators of physical quantities. It has turned out, however, that there exist more general and effective methods to establish algebraic expressions for CFP, which do not feature the above-mentioned disadvantages. 

In special cases some of these terms may be identically equal to zero, for example, with the electric dipole transition operator (see (4.12) at k = 1) the intrashell terms are zero, and with the kinetic and potential energy operators the intershell terms are zero (at h h) -either case follows directly from the explicit form of relevant one-electron reduced matrix elements. 

The reduced matrix element of the one-electron tensor operator for n-electrons collapses to... 

As for any one-electron operator, the (orbital) unit tensor operator has the following reduced matrix elements ... 

Thus any one-electron operator has reduced matrix elements equal to those of the unit tensor operator times its one-electron reduced matrix element (Z 1 1 l). Other results are as follows ... 

Reduced matrix element of Racah operator (rationalized spherical harmonics)... 

In terms of the ITO approach, the reduced matrix elements of the electron repulsion operator (which gives rise to terms) are expressed as follows ... 

The reduced matrix elements of a double-tensor operator (formed as a scalar product of tensor operators of rank k and 1) become... 

The reduced matrix elements of a many-electron operator can be expressed through the unit tensor operator... 

Table 49 terms a Non-zero reduced matrix elements of the unit tensor operators between dn- ... 

The Wigner-Eckart theorem states that the matrix element of a tensor operator can be expressed through a more fundamental quantity - the reduced matrix element (which is free of projections of angular momenta) and a coupling coefficient... 

Having the reduced matrix element determined one can easily evaluate all the matrix elements of a tensor operator. 

The (orbital) unit tensor operator of rank k for one electron is introduced as having the reduced matrix element equal to unity, i.e. 

The unit tensor operators are irreducible-tensor operators with reduced matrix elements of unity. They are a valid choice to use as a basis in order to express any arbitrary tensor operator as a linear combination, since they are linearly independent. Attention is restricted to these for the sake of simplicity. Hence the definition of unit tensor single-particle operator U (aK, a L, r) is... 

In this section, we follow the symmetry-adapted approach put forward by Acevedo et al. [10], and introduce the vibronic crystal coupling constants Av y(i, t), the tensor operators 0 (Txr i, t) and the general symmetry-adapted coefficients to give a master formula to evaluate the relevant reduced matrix elements as given below ... 

Let us now consider bosonizing [4] the pair operators, based on the reduced matrix elements that appear on the rhs of (1). We then have an expansion which contains no s bosons in it. This BET, which we may call SR+BET, is exact (assuming that the boson expansion is carried out to a desired order). 

Table D.l. Reduced matrix elements of the angular momentum operator...

The disadvantages of the cogredient form of definition are connected with the presence of the factor (—1) in the normalizing coefficient 2Nk (D.32), which is a result of defining the reduced matrix element of the tensor operator (D.ll) into which this factor is introduced, in our opinion, without particular necessity. 

From the WET, Eq. [166], it is obvious that the reduced matrix element (RME) depends on the specific wave functions and the operator, whereas it is independent of magnetic quantum numbers m. The 3/ symbol depends only on rotational symmetry properties. It is related to the corresponding vector... 

So far we know the selection rules for spin-orbit coupling. Further, given a reduced matrix element (RME), we are able to calculate the matrix elements (MEs) of all multiplet components by means of the WET. What remains to be done is thus to compute RMEs. Technical procedures how this can be achieved for Cl wave functions are presented in the later section on Computational Aspects. Regarding symmetry, often a complication arises in this step Cl wave functions are usually determined only for a single spin component, mostly Ms = S. The Ms quantum numbers determine the component of the spin tensor operator for which the spin matrix element (S selection rules dictated by the spatial part of the ME. 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

The operators Op, and Op2 are used in the reduced matrix elements where, however, their tensor character without components fi has to be understood. Op, is a tensor of rank 1, Op2 a tensor of rank 0. 

It is a common occurrence that we wish to evaluate the reduced matrix element of an operator which acts on only one part of a coupled scheme. For example, the general formula for the reduced matrix element of an operator Tk(A ) which acts only on part 1 of a coupled scheme j + j2 = j is ... 

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