**1. Introduction**

The DFT method developed by Walter Kohn with his collaborators more than 55 years ago is widely used for study molecular systems and solids. First, Kohn published with Hohenberg [1] their famous theorem on which the DFT theory is based. Then Kohn with Sham [2] obtained their well-known Kohn-Sham equation. The number of citations on these first Kohn and collaborators publications is

increased every year [3]. In 2010 papers [1, 2] were cited 11,000 times and in 2012 14,000 times. So, we can expect that at present the number of citations will be larger.

Last years, the number of Congresses on Nanosciences and Nanotechnologies is considerably increased. In one of the reports made in Las Vegas on October 2016 by Miyazaki, it was claimed that using the modern computational facilities they were able to apply the DFT molecular dynamic simulation to systems with million atoms, see Ref. [4]. Certainly, it can be done only using the traditional Kohn-Sham (KS) formalism based on the electron density *ρ*ð Þ*r* and its modifications by gradients.

The electron density is the diagonal element of the spinless one-particle reduced density matrix,

$$\rho(\mathbf{r}\_1) = N \sum\_{\sigma\_1, \dots, \sigma\_N} \int |\Psi(\mathbf{r}\_1 \sigma\_1, \dots, \mathbf{r}\_N \sigma\_N)|^2 dV^{(1)} \tag{1}$$

In Eq. (1), the spin projections *σ<sup>i</sup>* span over the whole spin space and the spatial coordinates are integrated over the N � 1 electrons excepting the first. If Ψ is defined in 4N-dimensional Gilbert space, *ρ*ð Þ*r* is defined in a three-dimensional space. Evidently, the calculations, in which only *ρ*ð Þ*r* -formalism is used, will be considerably faster than Ψ-formalism is used, and the *ρ*ð Þ*r* -formalism can be applied to larger systems.

Let us consider another point: what we lose, if we use ρ-formalism with the modulus of j j <sup>Ψ</sup> <sup>2</sup> squared, as in Eq. (1), instead of the wave function formalism. It is evident that in the transition from the wave function formalism to the probability density j j <sup>Ψ</sup> <sup>2</sup> , we lose the phase of the wave function. Due to the insensitivity of the probability density to the symmetry of the state (we will discuss it in the following sections), we also lose the symmetry characteristic of the wave function and cannot determine the Pauli permitted states, on which molecular spectroscopy is based. The diagonal element of the full and all reduced density matrices, as it was proved in my studies and discussed in book [5], does not depend on the symmetry of the state and its dimensionality.

It can be expected that after integration we lose some information. In the case of electron system, the one-particle reduced density matrix must be used. This leads to loss of information connected with the two-particle correlations, which are described by the two-particle reduced density matrix.

What is not evident and deserves a special discussion: as it was proved by the author for an arbitrary many-electron system, the total spin S of the system in principle cannot be introduced in the DFT studies. This can be done at the two-particle reduced density matrix level. In the Section 2, we will discuss this problem in detail.

On the other hand, even at the framework of the two-particle reduced density matrix formalism, one cannot study the non-additive many-body effects, which determined by many-body forces, In this connection, I would like to mention that when I arrived from Moscow to Mexico by invitation of the Director of IF-UNAM Octavio Novaro and was working in his laboratory, we obtained a closed formula for the energy of N-body interactions [6].

Later on, in the author book [7], Chapter 4, different general cases for the manybody forces have been considered, see also paper [8]. In several publications, e.g., in Refs. [9, 10], it was conclusively demonstrated that the clusters built from closedshell atoms (atoms without valence electrons) are stabilized by the three-body forces. The alkaline-earth clusters: Ben, Mgn, Can, and etc. are the typical example of such clusters. They are stabilized by the three-body forces, It is also important to mention that for the stability of rare-gas clusters the three-body dispersion forces, which are known as the Axilrod-Teller-Muto forces, play a decisive role, see Section 4.3.3 in Ref. [7] and recent review by Johnson and co-authors [11].

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

It is instructive to discuss shortly the physical sense of the non-additivity and many-body forces concepts. As is well known, the interaction of charge particles is described by the Coulomb law

$$V = \sum\_{a$$

where charges *qi* are considered as points. Eq. (2) contains only two-particle interactions, so, it is additive. However, in quantum mechanics, the charge particles are not points and they are not rigid. Atoms and molecules obey quantummechanical laws. If we consider the Coulomb interactions between charged atoms (or charged molecules), the additivity is lost. The interaction will depend on surrounding. In the case of three atoms, the third atom can polarize the electronic structure of two others, and this leads to three-body forces, since the interaction depends on three interatomic distances.

It should be noted that the possibility of application of DFT approaches to large systems, which were not available to be studied before, induced a euphoria in the DFT community. This euphoria led to wide using DFT methods without an analysis of the limitations following from quantum mechanics.

In many publications it was revealed that the applications of DFT method in some cases lead to incorrect results. First, it was recognized in the DFT studies of intermolecular interactions. The potential curves obtained by the early created DFT functionals for many stable in experiments dimers were repulsive, since in these DFT functionals the dispersion energy was not taken into account.

Then it became clear that DFT methods meet serious difficulties in studies of transition metals with nd electrons. These problems were analyzed by many authors, e.g., by Cramer and Truhlar [12]. I would like to stress that most of difficulties discussed in their review [12] are connected with the problem of spin in DFT approach and in principle cannot be resolved in the framework of the electron density, *ρ*(*r*), which belong to the one-particle reduced density matrix, see Eq. (1). The spin problem in DFT approaches will be analyzed in detail in Sections 2 and 3.

Last years, many comparative studies of the relative precision of exchangecorrelation (XC) functionals are published. Below I will discuss some important, from my viewpoint, DFT papers published on this topic in the last years. Certainly, the list of selected papers is only a little part of thousands DFT papers that are published each year.

Gillan et al. [13] analyzed different kind of XC functionals for liquid and ice water and water clusters. The conclusion was that many functionals are not satisfactory because do not describe correctly the dispersion. Let us stress that this situation takes place for such widely studied substance as water. The authors mentioned that after they included in XC functionals the non-local dispersion, the results still cannot be admitted as completely satisfactory.

In the Taylor et al. [14] paper, the precision of DFT calculations of intermolecular interactions with respect to highly accurate benchmarks for 10 dimers was analyzed. Their review is comprised 17 authors, among them are the well-known creators of XC functionals Angyán, Hirao, Scuseria, Truhlar and others.

The perspectives of DFT theory were discussed in the paper by Truhlar and collaborators [15]. The authors also analyzed recent Minnesota functionals. On the other hand, Mardirossian and Head-Gordon [16] benchmarked the Minnesota functionals using a very comprehensive database and came to conclusion that none of them are state-of-the-art for non-covalent interactions and isomerization energy.

In the article published by Medvedev et al. in collaboration with Perdue [17], 128 XC functionals created in period 1974–2015 were analyzed. The authors [17] made

the comparison of the normalized error for different functionals for atoms and its ions, see **Figure 1**, using the CCSD approximation as a reference level. It is important to stress that according to their results the normalized error of *ρ*ð Þ*r* up to 2000 decreased and then increased due to the introduction of semiempirical functionals.

Kepp [18] made a comment on the conclusion by the authors [17] that after the early 2000 the functionals strayed from the path toward exactness. Kepp indicated that the "straying" is not chemically relevant to the studied systems. In their response [19], the authors performed a special calculation, from which follows that their results are valid also for heavier system, including molecules.

In the next paper published by Perdew with Medvedev et al. [20], the authors discuss the possibilities and pitfalls of statistical error analysis, taking as an example the ranking of approximate functionals by the accuracy of their self-consistent electron densities.

As mentioned above in our discussion of last publications, the DFT community is concerned on the still existing problems in some applications of DFT approaches and most of these problems are connected with the quality of XC functionals. I like and completely agree with the witty comment made by Savin [21]:

*"The existing great number of different expressions for the XC functionals can be considered as evidence that we still have not satisfactory correct XC functionals".*

Among new publications, it is important to mention the very recent paper published by Perdue with collaborates [22], in which the problem of the symmetry breaking in DFT was discussed. The interesting paper was recently published by Bartlett [23]. Even the title of the paper "Adventures in DFT by a wavefunction theorist" looks quite intriguing. I also recommend readers the quite comprehensive and interesting review by Johnson and co-authors [11] published in 2021.

In my following discussions, I will analyze in detail the two problems:

The first problem is connected with the definition of spin in the KS-DFT framework. It will be shown that the concept of the total spin S of the state, in principle, cannot be defined in the frame of electron density formalism.

**Figure 1.** *The precision of calculating* ρ *using different functionals.*

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

The second problem is related to the degenerate states in DFT. As follows from the general conceptions of quantum mechanics, at Born-Oppenheimer approximation in the case of degenerate states, the electronic and nuclear motions cannot be separated, they are mixt by so-called vibronic interactions. This problem will be discussed in Section 3.

In this chapter I will not discuss Ψ-versions of DFT that solves many problems in DFT but lost the simplicity of KS-DFT formulation. The Ψ*-*formalism was successfully combined with DFT approach by Gőrling, Trickey, and some other investigators. Nevertheless, the existing problems in the applications of DFT approach are still remaining actual. I will discuss them in the next two sections. The discussion will be based on two theorems proved by the author in 2007 [24].
