Irreducible/irreducibility, generally tensor components

Here, y is the unique symmetric and traceless (irreducible) part of Q, whereas D(i) are symmetric and traceless second-order (irreducible) tensors, and v(i) and S are vectors and a scalar. It may be noted that, if we write a general third-order tensor in the form (8 33), there is no loss of generality in assuming that the second-order tensor components are symmetric. The antisymmetric part of any second-order tensor can always be represented by a vector. For example, if D = D,s + D" then the antisymmetric part can always be written as D"= e d where d = — e D" and included in the vector terms of (8-33). 

These components transform under rotation like the spherical harmonic functions T]m. In general, an irreducible spherical tensor Tim transforms like the function Ylm. The spherical components of a second-rank tensor are again collected in Table 1.13. 

Although we have introduced irreducible spherical tensors, we do not yet have a formalism which admits ready generalization to D dimensions. This can be accomplished by transforming the spherical tensor to a Cartesian tensor. The spherical components of a tensor are related by a unitary transformation to Cartesian components [11,12,13]. For example consider a spherical harmonic of / = 1 (a spherical tensor of rank 1) written as a three component vector... 

The general problem is now clear the quantities i /,. p are tensor components, with respect to the group U(m), and we want to find linear combinations of these components that will display particular symmetries under electron permutations and hence under index permutations. Each set of symmetrized products, with a particular index symmetry, will provide a basis for constructing spin-free CFs (as in Section 7.6) for states of given spin multiplicity and in this way the full-CI secular equations will be reduced into the desired block form, each block corresponding to an irreducible representation of U(m). It is therefore necessary to study both groups U(m), which describes possible orbital transformations, and which provides a route (via the Young tableaux of Chapter 4) to the construction of rank-N tensors of particular symmetry type with reject to index permutations. 

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... 

In a tensor product 0(k) each component of is to be multiplied with each component of The resulting 2(k + f) -2( + -dimensional tensor is in general not irreducible. Reduction yields irreducible tensor operators with ranks ranging from k + j, k + j — 1, , k—j. ... 

The task of expressing K in terms of irreducible tensors is somewhat more involved. The general problem of reducing tensors of any order into irreducible components is discussed by Coope et al.,3 which was referred to earlier. These authors show that a general third-order tensor can always be expressed in the form... 

One mates the connection by counting the independent components of the irreducible tensor. For the completely symmetric r = [71,0,0] states (see the discussion above equation (15) for an explanation of this notation) this is given by equation (18) of Reference [16]. In three dimensions the F = [71,1,0] tableau, ie either the two box antisymmetric or the mixed symmetry tableaux, is associate (see Reference [5] p. 396 and Reference [6] p. 164) to the F = [71,0,0] tableau, i.e. a completely symmetric tableaux. Thus a second rank antisymmetric or a mixed symmetry tensor has the same dimension as the associate completely symmetric tensor. The three box completely antisymmetric F = [1,1,1] tableau is the associate tableau to the tableau with no boxes and is therefore a scalar. There aie more general formulas for the number of independent components of an irreducible tensor in D dimensions, however they are not required to achieve the above identification, see D.E. Littlewood, The Theory of Group Characters (Oxford University Press, Oxford, 1950), equation (11.8 6), p. 236. 

It follows that the spherical components of the electric dipole operator will transform like spherical harmonics of order unity under any arbitrary rotation of the coordinate system. A quantity which transforms under rotations like the spherical harmonic Y (0,( i) is said to be an irreducible tensor operator T of rank k and projection q where the projection quantum number can take any integer value from -k to +k. Since any arbitrary function of 0 and < can generally be expanded as a sum of spherical harmonics, it is usually possible to express any physical operator in terms of irreducible tensor operators. For instance, the electric quad-rupole moment operator defined by equation (4.45) can be shown to be a tensor of rank 2 (Problem 5.6). 

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