**Appendix: Short necessary knowledge on the permutation group**

The permutation symmetry is classified according to the irreducible representations of the permutation group *πN*. <sup>1</sup> The latter are labeled by the Young diagrams

$$[\lambda] = [\lambda\_1 \lambda\_2 \dots \lambda\_k],$$

$$\lambda\_1 \ge \lambda\_2 \ge \dots \ge \lambda\_k, \sum\_{i=1}^k \lambda\_i = N. \tag{14}$$

where *λ<sup>i</sup>* is represented by a row of *λ<sup>i</sup>* cells. The presence of several rows of equal length *λ<sup>i</sup>* is convenient to indicate by a power of *λi*. For example,

At present, the apparatus of permutations groups cannot be described without using the Young diagrams, I would like to note here some unusual details of the biography of Alfred Young. He was a country clergyman and has not any mathematical education. Young published studies were extending from 1900 to 1935, and in total he published 8 papers. The keystone of his studies was the reduction of the permutation groups to its irreducible representations in an explicit form. It is quite remarkable the gap of 25 years between his second paper in 1902 and the third in 1927. This gap will not be surprising, if we take into account that Young was a clergyman with numerous clerical duties.

It is obvious that one can form from two cells only two Young diagrams:

For the permutation group of three elements, *π3*, one can form from three cells three Young diagrams:

<sup>1</sup> For a more detailed treatise see books by Rutherford [66], Chapter 2 in book [28] or Appendix B in book [5].

*Modern State of the Conventional DFT Method Studies and the Limits Following… DOI: http://dx.doi.org/10.5772/intechopen.102670*

The group *π<sup>4</sup>* has five Young diagrams:

As we mentioned in the beginning of this Appendix, the representation of the permutation group *π<sup>N</sup>* are labeled by the Young diagram ½ � *λ* . The rules how to find from a given Young diagram the matrices of representation of the permutation group were formulated by Young, but they were very complex. The Japanese mathematician Yamanouchi considerably simplified these rules. At present, it is called Young-Yamanouchi representation, which is described in detail in my book [28]. For each irreducible representation Γ½ � *<sup>λ</sup>* , the normalized basis functions can be easily constructed by the Young operator,

$$
\alpha\_{\boldsymbol{\pi}}^{[\boldsymbol{\lambda}]} = \sqrt{\frac{f\_{\boldsymbol{\lambda}}}{N!}} \sum\_{\boldsymbol{P}} \Gamma\_{\boldsymbol{\pi}}^{[\boldsymbol{\lambda}]}(\boldsymbol{P}) \boldsymbol{P}. \tag{15}
$$

In Eq. (15), Γ½ � *<sup>λ</sup> rt* ð Þ *<sup>P</sup>* are the matrix elements of the representation <sup>Γ</sup>½ � *<sup>λ</sup>* , *<sup>f</sup> <sup>λ</sup>* is the dimension of Γ½ � *<sup>λ</sup>* and the sum over operations *P* are taken for all *N*! permutations of the group **πN**. The normalized functions are obtained by acting the operator (15) on some non-symmetrized product of one-electron orbitals,

$$
\Phi\_0 = \wp\_1(\mathbf{1})\wp\_2(\mathbf{2})\dots\wp\_N(\mathbf{N}).\tag{16}
$$

The normalized functions

$$
\Phi\_{rt}^{[\boldsymbol{\lambda}]} = o\_{rt}^{[\boldsymbol{\lambda}]} \Phi\_0 = \sqrt{\frac{f\_{\boldsymbol{\lambda}}}{N!}} \sum\_{\boldsymbol{P}} \Gamma\_{rt}^{[\boldsymbol{\lambda}]}(\boldsymbol{P}) P \Phi\_0,\tag{17}
$$

are transformed in accordance with the representation Γ½ � *<sup>λ</sup>* . If in Φ½ � *<sup>λ</sup> rt* , Eq. (17), the second index *t* is fixed, then *f <sup>λ</sup>* function form a basis for the representation Γ½ � *<sup>λ</sup>* , each different index *k* enumerates different bases. Let us prove this statement applying an arbitrary permutation *Q* of the group *π<sup>N</sup>* to the function (17):

$$\mathbb{E}\,\mathbb{Q}\Phi\_{\boldsymbol{\pi}}^{[\boldsymbol{k}]} = \sqrt{\frac{f\_{\boldsymbol{k}}}{N!}}\sum\_{\boldsymbol{P}}\Gamma\_{\boldsymbol{\pi}}^{[\boldsymbol{k}]}(\boldsymbol{P})\mathbb{Q}\mathbb{P}\Phi\_{\boldsymbol{0}} = \sqrt{\frac{f\_{\boldsymbol{k}}}{N!}}\sum\_{\boldsymbol{P}}\Gamma\_{\boldsymbol{\pi}}^{[\boldsymbol{k}]}(\boldsymbol{P})\left(\boldsymbol{Q}^{-1}\mathbb{R}\right)\mathbb{R}\Phi\_{\boldsymbol{0}}.\tag{18}$$

Using the invariance properties of a sum over all group elements and the property of orthogonal matrices, we obtain the matrix element of the product of permutations as a product of matrix elements.

$$
\Gamma\_{\boldsymbol{m}}^{[\boldsymbol{\boldsymbol{\lambda}}]}(\boldsymbol{P})(\boldsymbol{Q}^{-1}\boldsymbol{R}) = \sum\_{\boldsymbol{u}} \Gamma\_{\boldsymbol{m}}^{[\boldsymbol{\boldsymbol{\lambda}}]}(\boldsymbol{Q}^{-1})\Gamma\_{\boldsymbol{u}}^{[\boldsymbol{\boldsymbol{\lambda}}]}(\boldsymbol{R}) = \sum\_{\boldsymbol{u}} \Gamma\_{\boldsymbol{u}\boldsymbol{r}}^{[\boldsymbol{\boldsymbol{\lambda}}]}(\boldsymbol{Q})\Gamma\_{\boldsymbol{u}\boldsymbol{t}}^{[\boldsymbol{\boldsymbol{\lambda}}]}(\boldsymbol{R}).\tag{19}
$$

In Eq. (19) we denoted the permutation *QP* by *R*. Substituting (19) in (18), we obtain finally

*Density Functional Theory - Recent Advances, New Perspectives and Applications*

$$Q\Phi\_{\boldsymbol{\upmu}}^{[\boldsymbol{\uplambda}]} = \sqrt{\frac{f\_{\boldsymbol{\uplambda}}}{N!}} \sum\_{\boldsymbol{\upmu}} \Gamma\_{\boldsymbol{\upmu}}^{[\boldsymbol{\upmu}]}(\boldsymbol{Q}) \left(\sum\_{\boldsymbol{\upR}} \Gamma\_{\boldsymbol{\upmu}}^{[\boldsymbol{\upmu}]}(\boldsymbol{R}) R \Phi\_{\boldsymbol{\upR}}\right) = \sum\_{\boldsymbol{\upmu}} \Gamma\_{\boldsymbol{\upmu}}^{[\boldsymbol{\upmu}]}(\boldsymbol{Q}) \Phi\_{\boldsymbol{\upmu}}^{[\boldsymbol{\upmu}]}.\tag{20}$$

Thus, if the second index *t* is fixed, then *f <sup>λ</sup>* functions form a basis for the representation Γ½ � *<sup>λ</sup>* , and each different index *k* enumerates different bases. It is also important to mention that in the function Φ½ � *<sup>λ</sup> rt* , index *r* characterizes the symmetry under permutation of the arguments, while index *t* characterizes the symmetry under permutation of the one-particle functions *φ*a.
