**3. Methods allowing take into account the spin multiplet structure within the DFT approach**

It should be mentioned that the methods, taking into account the spin multiplet structure, are beyond the KS formalism. In most of these methods the Ψ-formalism is used. I will consider two widely used approaches:

1.One of the first publications, in which the spin multiplet structure was taken into account in the frame of DFT, was the paper by Ziegler et al. [32]. For each value of the total spin S, they built the appropriate combinations of the Slater determinants. The factor giving the value of spin was obtained by correcting the exchange energy, *EX*. In publications [33–35], in which scientists applied the Ziegler et al. approach, only the exchange energy was considered. Mineva et al. [36] are stressed that the scientists [33–35] developing after Ziegler et al. the methods that allows to take into account the conception of spin in the DFT studies, are considered only the exchange energy, *EX*, and did not consider the correlation energy, *EC*. This led to the incorrect multiplet structure. To the best of my knowledge, this drawback has not been discussed in the DFT community.

In some applications, the Ziegler et al. method was named as Multiplet Structure Method, or shortly MSM. In the following text, I will use this abbreviation. Usually the exchange-correlation functional, *EXC*, is presented as a sum of exchange, *EX*, and correlation *EC*, energies:

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

$$E\_{\infty} = E\_{\infty} + E\_{c}.\tag{9}$$

In the studies based or developing MSM approach [32–35], the value of the total spin S was found using only the exchange energy *EX*, and then applied to the total *Exc*. It is evident that the contribution of the correlation energy *EX*, is not the same as the exchange energy. Therefore, if *Ec* is multiplied by the same factor as it was found for *Ex* then it will give a wrong multiplet structure, because the exchange and correlations functionals should have a different dependence on S.

2. In the second group of methods [37–41], designated as restricted open-shell Kohn-Sham (ROKS) method, the open-shell theory of Roothaan [42] was used. In the first publication by Russo et al. [37], the Hamiltonian of Roothaan [42] was used, but the exchange term was replaced by the exchangecorrelation functional. The authors [38–41], combined the ROKS methods with the MSM approach. As a result, the methods elaborated in publications [38–41] carried the same mistakes as the first ROKS method created by Russo et al. [37]. They do not provide the correct spin value for the correlation functional.

As stated above, both approaches MSM and ROKS, do not provide the correct total spin S for the correlation functionals.

According to calculations by Illas et al. [43], the ferromagnetic coupling is exaggerated, if the DFT method approaches are used. This agreed with our precise Mn2 calculations performed with Mavridis group [44]. In most of calculations, Mn2 had the ferromagnetic ground state with maximum value of the total spin, *S* ¼ 5. I would like also to mention that in the following article [45] Illas and collaborators, using the Filatov-Shaik ROKS method [39, 40], failed to improve the agreement with experiment. The reason is that, as we noted above, in the ROKS calculation, the correlation energy was not considered.

The definition of the correlation energy was given by Löwdin [46] many years ago. According to it:

$$E\_{corr} = E\_{\text{exact}} + E\_{\text{HF}}.\tag{10}$$

The exact quantum-mechanical calculations can be performed only for small electron systems, for larger systems, the "exact" energy will depend upon the method used for its calculation. Thus, the correlation energy is method dependent. It should be also mentioned that the correlation energy has not an analytical expression, which leads to some problems in its applications, see Ref. [24].

As it was noted in Introduction, the total spin S of the system can be introduced only at the two-particle reduced density matrix level. The modern state of the development of the two-particle reduced density matrix formalism was discussed in a large number of articles [47–55]. Unfortunately, the spin problem still has not been considered by the DFT community.

### **4. Symmetry properties of the density matrix; degenerate states**

In an elegant proof, Hohenberg and Kohn [1] laid down the theoretical foundation of the DFT theory. In their fundamental paper, the degeneracy was not treated, since they considered the ground state, which very rare is degenerated, as it is in the case of O2 molecule.

Very soon, in the DFT community it was accepted that the Levy-Lieb [56, 57] constraint search procedure allows to study the degenerate states in the DFT calculations. First, it was shown by Levy [56] in 1969 and then in 1983 by Lieb [57], who applied more abstract mathematical approach. I would like to mention that Bersuker [58] was the first who criticized the possibility of application of the DFT approach to degenerate states. Bersuker considered it on the special case of the Jahn-Teller effect. Let us mention that according to the following from quantum mechanics the Born-Oppenheimer approximation (the molecules can be calculated only at this approximation) the vibronic interaction mixed the electronic and nuclear motions, and the electron and nuclear densities may not be constructed. Thus, the Levy-Lieb [56, 57] constraint search procedure contradicts quantum mechanics.

In Section 2, we already discussed that the author proved, see Ref. [24], the theorem that the electron density of the arbitrary N-electron system, defined in Eq. (1), does not depend upon the total spin S and always preserves the same form as it is for a single-determinantal wave function. From this theorem follows that the wave function of N-electron system does not depend on the degeneracy of the state and on its symmetry as well. It was proved using the permutation group apparatus, described in detail in chapter 2 of my book [28], see also Appendix to this chapter.

In general, it can be two types of degenerate states, the spatial and the spin degeneracy. In the case of the spatial degeneracy, the particles are described by spatial wave functions, although when they are degenerated in the spin space, they are described by spin wave functions. For constructing the degenerate in space wave function, the point group formalism should be used. Let us consider a point group **G** with *g* elements. In the book [28], the author constructed the wave functions belonging to the *f <sup>α</sup>*-dimensional representation *Г*ð Þ *<sup>α</sup>* of an arbitrary point group *G* as:

$$
\Psi\_{ik}^{(a)} = \frac{f\_a}{\mathcal{g}} \sum\_R \Gamma\_{ik}^{(a)}(R) \,^\* R \Psi\_0,\tag{11}
$$

where Γð Þ *<sup>α</sup> ik* ð Þ *<sup>R</sup>* are the matrix elements of the representation *<sup>Г</sup>*ð Þ *<sup>α</sup>* and the sum in Eq. (11) is taken over all *g* elements of the group **G**. The operations R of the group **G** are acting on some non-symmetrized product Ψ<sup>0</sup> of one-electron orbitals. If in Ψð Þ *<sup>α</sup> ik* the second index *k* is fixed, then *f <sup>α</sup>* function form a basis for the representation *Г*ð Þ *<sup>α</sup>* , each different indices *k* enumerates different bases.

If we have a *f <sup>α</sup>* degenerate state, each of its *f <sup>α</sup>* functions describe the system with the same probability and a pure state (the state described by wave function) cannot be selected. Therefore, the degenerate state must be considered as a mixed state, in which its basis functions enter the density with the same weight factors. The diagonal element of the density matrix in the case of degenerate state is written as:

$$D\_k^{(a)} = \frac{1}{f\_a} \sum\_{i=1}^{f\_a} \left| \Psi\_{ik}^{(a)} \right|^2. \tag{12}$$

Using expression (12), I proved the theorem [24], according to which, the diagonal elements of the full density matrix is invariant for all operations of the group symmetry of the state, that is, it is a group invariant. It was proved that for every operation *R* of group *G* and all its irreducible representations *Г*ð Þ *<sup>α</sup>*

$$RD\_k^a = D\_k^a. \tag{13}$$

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

This means that the diagonal element of the full density matrix (and all reduced density matrices as well) transforms according to the totally symmetric onedimensional representation *A*<sup>1</sup> of *G* regardless of the dimension of representation *Г*ð Þ *<sup>α</sup>* . It was proved for an arbitrary point group, but it is correct for any finite group. For the permutation group, this result was used in my publications [24, 59, 60] in analysis of the foundations of PEP. In these articles, I analyzed the case when PEP is not fulfilled and except of symmetrical and antisymmetrical states, an arbitrary permutation symmetry, including degenerate permutation states, are permitted. I have showed that if PEP is not fulfilled, this leads to contradictions with the concepts of particle independence and their identity. It was rigorously proved that the particles, described by wave functions with the permutation symmetry not allowed by PEP, may not exist in our Nature.

The arguments presented in Refs. [24, 59, 60], see also book [5], can be considered as a theoretical substantiation of PEP. They explained why in our Nature only completely symmetric or antisymmetric states, corresponding to onedimensional representations of the permutation group, are realized. From this result, the important consequence follows

*We may not expect that in future some unknown elementary particles can be discovered that are not fermions or bosons.*

On the other hand, according to the so-called *fractional* statistics, which is valid in the 2D-space, a continuum of intermedium cases between boson and fermion particles can exist, see subsection 5.4 in book [5]. As was showed by Leinaas and Myrheim [61] in their pioneer paper, in 2D-space can exist a continuum of states between boson and fermion symmetry. After Leinaas and Myrheim [61], Wilczek [62] introduced in 2D-space the *anyons*, which obey any statistics. However, we should take into account that anyons are quasiparticles defined in 2D-space. The real particles can exist only in 3D-space, and according to PEP, formulated for all elementary particles, see Eq. (7), the elementary particles can obey only the boson or fermion symmetry. It is important to stress that the discovery of the fractional statistics does not contradict PEP.

All experimental data, see my recent review [63], confirm the Pauli Exclusion Principle. Different very precise experiments did not show any Pauli-forbidden transitions.

This is confirmed also by very precise calculations of H2 molecule [7], in which, certainly, PEP was taken into account. The quantum mechanical calculations of the H2 dissociation energy and its first ionization potential [64, 65] are in a complete agreement with very precise experimental values, see Table 1.1 in [7]. From this follows not only an additional confirmation of PEP, but also a rather general conclusion that molecules obey the same quantum-mechanical laws that obey traditionally physical objects: atoms and solids; at nanoscale we should not distinguish between chemical and physical systems.

In the end, I would like to note that in some papers the authors claimed that they developed the non-Born-Oppenheimer DFT in the frame of the electron density approach. These publications were analyzed in my first paper on DFT limits [24], where it was shown that in spite of the authors claims, their formalisms must be attributed to the Born-Oppenheimer approximation.

### **Acknowledgements**

I am grateful to Maestro Ronald Columbié-Leyva for new references and for the technical and software support.
