Matrix element reduced

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... 

The second term of the product is a reduced matrix element which contains the /-state dependence of the f electrons since only 4f" configuration, for which 1 = 3,... 

The last term on the right in Equation 1.22 represents a doubly reduced matrix element, which can be calculated by recursive formula in terms of the coefficients of fractional parentage [4, 14], tabulated in the work of Nielson and Koster [27]. Finally, Equation 1.18 is rewritten as... 

The reduced matrix elements of the spherical harmonics can be written in terms of a Wigner 3 - j symbol... 

Relative weights of the three equivalent changes are given hy the coefficients and an appropriate reduced matrix element determines the magnitude of the perturbation. 

Infinitesimal changes of the atomic positions were analyzed above. An estimate of the magnitude of reduced matrix element (/ V /) will he obtained from calculations at small, finite distortions. Table 1 presents essential data from the calculations with GAMESS [10]. Basic displacements were introduced and bond length renormalizations were effected in order to avoid stretching contributions. This process will make it more difficult to ensure that the various forms have moved an equal amount. 

We assume a linear dependence of the reduced matrix element on the displacement and find that... 

By Wigner-Eckart s theorem [6] Eq. (2) can be expressed in terms of a reduced matrix element that is independent of M and M, ... 

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. 

Likewise, we can rewrite other equations that yield the totally reduced matrix elements of complex tensorial products. 

Here we need to carefully follow consistently one of the conventions found in the literature. The form used hereafter matches that of Racah [35-38], Judd [57], and Wybourne [58], which was accepted also by Slater [40]. Notice that the tabulated values of the reduced matrix elements can differ when taken from different sources as a consequence of different phase systems or different constant-factor conventions [41,52], Usually (lnvLS Tk lnVL S ) is used in place of (/ vL(S) Tfc( ) fVL (S )). 

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

The reduced matrix element for a two-electron system is given by the decoupling formula, which adopts a special form ... 

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 ... 

Values of the reduced matrix elements (Z vLS Uk lnVL S ) can be retrieved from Slater [40], which refers to Racah [35-38], Table 49. Note that tabulations of Jucys [41] differ by a factor of (2S + l)l/2. 

Reduced matrix elements connect the terms of different seniority v [lnvLS yee lnv L S ) = SLpSsp J2... 

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... 

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