**2. Electron density** *ρ* **and the total spin S**

It was recognized sufficiently long ago that the concept of the total spin of the many-electron quantum state is quite difficult to base in the frame of DFT approaches. Studying the two-electron system, McWeeny [25] came to conclusion the electron density does not allow to identify the spin state. McWeeny formulated it as the following statement:

*"Electron spin is in a certain sense extraneous to the DFT".*

In their analysis of DFT foundation, Weiner and Trickey [26] came to conclusion that

*" … the way that the ρ-based XC potential takes account of spin is very obscure except in the simplest configurations".*

The statements of McWeeny and Weiner-Trickey are quite cautious. I would like to formulate it more definite:

*"The conception of spin in principle cannot be defined in DFT at the level of the first reduced density matrix".*

To the best of my knowledge, the concept of spin was discussed in DFT community only for two-electron systems. In my article [24], I proved the theorem that the electron density does not depend upon the total spin of the state for N-electron system. This proof was done applying the formalism of the permutation group.

I would like to mention that in the Ψ-formalism used by Slater [27], he presented the wave function as a linear combination of determinants, corresponding to a given value of the total spin S. It does not allow to obtain any conclusions about spin. On the other hand, as I noted above, in my articles, firstly in Ref. [24], I obtained the proof for N-particle system for the independency of the electron density from the total spin S of the state. It was achieved applying the permutation group apparatus. For understanding the following text, I presented in this section the short description of the formalism of permutation groups. For more detail description, see Appendix in the end of this chapter or chapter 2 in my book [28].

The permutation group is characterized by Young diagrams ½ � *λ* :

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

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$$
\lambda\_1 \ge \lambda\_2 \ge \dots \ge \lambda\_k,\\
\sum\_{i=1}^k \lambda\_i = N. \tag{3}
$$

The rows in the Young diagram are represented in decreasing order. The presence of several rows with equal length *λ<sup>i</sup>* is indicated by a power of *λi*. For example, [*λ*] = [22 12 ] is depicted graphically as

I would like to mention that in Appendix, I describe the unusual biography of Alfred Young who was not a professional mathematician but was a country clergyman.

Though the concept of spin has enabled to explain the nature of chemical bond, the electron spins are not involved directly in the formation of the latter. The interactions responsible for chemical bonding have a purely electrostatic nature. In nonrelativistic approximation, the Hamiltonian does not depend on the spin, that means that the spin is saved, and we can operate with the value of the total spin S in the considered state.

According to PEP, the total electron wave function can be constructed as a sum of product of the spatial and spin wave functions symmetrized in respect to the irreducible representations *Г*½ � *<sup>λ</sup>* [28].

$$
\Psi^{\left[1^{N}\right]} = \frac{1}{\sqrt{f\_{\lambda}}} \sum\_{r} \Phi\_{r}^{\left[\lambda\right]} \Omega\_{\vec{r}}^{\left[\lambda\right]}.\tag{4}
$$

In Eq. (4), ½ � *<sup>λ</sup>* is the Young diagram and *<sup>Г</sup>*½ �<sup>~</sup>*<sup>λ</sup>* denotes the representation conjugate to *Г*½ � *<sup>λ</sup>* . Its matrix elements are

$$
\Gamma\_{\vec{r}\vec{t}}^{[\vec{\lambda}]}(P) = (-\mathbf{1})^p \Gamma\_{rt}^{[\vec{\lambda}]}(P). \tag{5}
$$

where *p* is the parity of permutation *P*. The spin Young diagram ~*λ* � � is dual to ½ � *<sup>λ</sup>* , i.e., it is obtained from the latter by replacing rows by columns. For example,

Let us return to Eq. (4), where the sum is taken over all basis functions of the representation. The normalization of the total wave function is provided by the factor 1*=* ffiffiffiffiffiffi *f λ* p . It should be mentioned that the electron spin has only two projections *sz* ¼ �*½*, therefore the spin Young diagram <sup>~</sup>*<sup>λ</sup>* � � must have no more than

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two boxes per columns. In one box, the projection of spin *sz* ¼ ½ in the other box *sz* ¼ �*½*. It means that the total spin of this column equals 0. It is evident that the contribution to the total spin of the system of electrons will come only from uncoupled electron spins, that is, from the length of rows. The value of the total spin corresponding to spin Young diagram ~*λ* is equal to

$$\mathcal{S} = \frac{1}{2} \left( \tilde{\boldsymbol{\lambda}}^{(1)} - \tilde{\boldsymbol{\lambda}}^{(2)} \right). \tag{6}$$

Eq. (6) enables one to find easily the values of the spin S for each spin Young diagram. For example, the spin corresponding to the spin Young diagram ~*λ* <sup>¼</sup> ½ � 3 1 , is equal to *S* ¼ 1.

Let us mention that in the case of particles with *s* > *½*, for a given Young diagram can correspond several values of *S:*If spin of the particle *s* ¼ 1, to ½ �¼ *λ* ½ � 31 can be attributed three values of *S* ¼ 1, 2, and 3, see Table 2 in Section C4 in Appendix C of my book on PEP [5].

As follows from experiment, the wave function of elementary particles can be only completely symmetric or antisymmetric. It allowed to generalize the PEP, primary formulated by Pauli only for electrons, for all elementary particles:

*The only possible states of a system of identical particles possessing spin s are those for which the total wave function transforms upon interchange of any two particles as*

$$P\_{\vec{\eta}}\Psi(\mathbf{1},\ldots,i,\ldots j,\ldots,N) = (-\mathbf{1})^{\mathfrak{Z}}\Psi(\mathbf{1},\ldots,i,\ldots j,\ldots,N).\tag{7}$$

*That is, it is symmetric for integer values of s (the Bose-Einstein statistics) and antisymmetric for half-integer values of s (the Fermi-Dirac statistics).*

According to Ehrenfest and Oppenheimer [29], this formulation is valid not only for elementary particles, but it is valid for different composite particles as well. As examples of composite particles, the authors considered atoms, molecules, and nuclei composed by electrons and protons (at that time the neutron had not been discovered). According to the presented above the general formulation of PEP for elementary particles, the wave-functions that described them can have only two types of symmetry: completely symmetric or antisymmetric, depending on their intrinsic value of spins.

The composite particles considered by Ehrenfest and Oppenheimer [29], were composed by fermions, that is, from particles with spin ½. So, the even number of particles leads to the Bose-Einstein statistics and odd number to the Fermi-Dirac statistics,

To the best of my knowledge, the scientists that had developed methods allowing to use the conception of spin in DFT calculations considered only twoparticle systems. The general case of the N-electron system was considered firstly by the author in Ref. [24] where it was proved the theorem named by some authors as the Kaplan Theorem 2. This theorem was formulated in the following manner:

*"The electron density of an arbitrary N-electron system, characterized by the N-electron wave function corresponding to the total spin S and constructed on some orthonormal orbital set, does not depend upon the total spin S and always preserves the same form as it is for a single-determinantal wave function."*

According to this theorem, for any permutation symmetry of the spatial wave function described by the Young diagram [*λ*] that correspond to a definite value of spin S, the electron density is equal

$$\rho\_t^{[\boldsymbol{\mathbb{A}}]}(\boldsymbol{r}) = \sum\_{n=1}^{N} |\rho\_n|^2. \tag{8}$$

It is a well-known expression of the electron density for the state described by the one-determinantal function with single-occupied orbitals. It can be shown that in the case of orbital configuration with arbitrary occupation numbers, the final expression (6) has not changed and will also correspond to the electron density for the one-determinantal function.

As follows from the discussion in the beginning of this section, at the first reduced density matrix approximation the concept of spin in principle cannot be introduced in the frame of traditional KS approach and at the gradient correction level as well. In more detail see the discussion based on the theory of permutation group in my paper, Ref. [24] or in some earlier papers.

From the analysis of the discussed above theorem follows that for different values of the total spin S, the expression for obtained electron density does not changed and have the same value as for wave function presented as a single Slater determinant. In this connection, it should be mentioned that about the ambiguity of the description by the electron density was known many years ago. I will cite two publications: more than 40 years ago Harriman [30] demonstrated that for each electron density *ρ* it can be constructed an arbitrary number of orthonormal orbitals, while in 2001 Cappelle and Vignale [31] showed that at the LSDA approximation it can be constructed different sets of potentials having the same ground state density.

Mean-while, different methods allowing taking into account the spin multiplet structure were developed, see Ref. [24] and references therein. In next section, I will discuss two groups of these methods.
