Density Functional Theory from past and recent developments to applications to large (bio)molecules: dedicated to John Pople, Walter Kohn, and John Perdew

 Fig 1. John Pople (1998 Nobel Prize in Chemistry).

Fig 2. Walter Kohn (1998 Nobel Prize in Chemistry).

"This entry is dedicated to John Pople (1998 Nobel Prize in Chemistry), Walter Kohn (1998 Nobel Prize in Chemistry), and John Perdew for their contributions to the DFT; the first and second for its computational and theoretical first-principle development, and the third for his excellent understanding of the DFT. There are also that contribute in many ways to DFT, for both matter states, that we do not forget".

Fig 3. DFT densities.

Actually, the Density Functional Theory (DFT) is a most applicable and used method in both condensed matter physics and quantum chemistry. During years, the DFT has been on the scope of quantum chemists, physicist, and other related fields of the quantum mechanics. The DFT can ne devided into to well-stablished theories, that are represented by rigorous theorems, the well-known ground state, Kohn-Sham DFT (KS-DFT), and the excited state representation of the density, Time-Dependent DFT (TD-DFT). Both theories show the most actually methodological representations -for determining ground-state and excited-state properties, respectively- of the DFT. Several codes have been developed by, e.g. GAUSSIAN, ADF, et al., John Pople (died), E. J. Baerends (Vrije University), and other that they helped to develop the codes of DFT.

Fig 4. E. J. Baerends.

First of all, there are excellent books (see 1,2 for more details), paper reviews (3), articles, and others manuscripts related to the field of DFT. Also, John Perdew wrote some words in relation to develop 14th Lessons in KS-DFT in the International Journal of Quantum Chemistry (IJQC), see Ref. 4 to read the complete KS-DFT Lessons. Here a short phrase taken from the 14th Lessons: "It presents 14 easy lessons in nonrelativistic density functional theory (DFT) at a qualitative level. The selection of lessons is partial in both senses of the word: it is incomplete, and it reflects our own biases about what is most important and interesting (or at least most familiar)".

The DFT is based on two simple, but relenvant, theorems A and B.

(A) The external potential , and hence the total energy, is a unique functional of the electron density .

 (B) The groundstate energy can be obtained variationally: the density that minimises the total energy is the exact groundstate density.

More information concerning these theorems can be found in Reference 5 (their mathematical foundations and more). The Hohenberg-Kohn theorems (A and B) are key steps into the historical theoretical development of the DFT. There, of course, others authors that develop theory, at the begining of the DFT story, to include the correlation and exchange potentials in the DFT (see the introduction of the 14th Lessons of John Perdew and co-workers). 

"Orbital-free DFT began in the 1920s with the Thomas–Fermi theory, which expresses the total energy E approximately in terms of the electron density , where is the average number of electrons in volume element d³r at position , using the simplest density functional that makes sense. The density is then varied at fixed electron number to minimize the energy functional. This approach gives simple and useful estimates for the density and total energy of an atom (defined as minus the minimum work to strip all the electrons from the nucleus) but it is far too crude for chemistry. In fact, Teller proved that in Thomas–Fermi theory, atoms do not bind together to form molecules and solids. Without the exchange–correlation energy, “nature's glue”, chemical bonds are either absent or far too long and weak. However, orbital-free methods continue to improve; see recent work by Trickey and others".

 Fig 5. John Perdew.

These key figures -developing computationally and theoretically the DFT- configurate a qualitative step further in Quantum Chemistry theory. Again, we mention that others develop key theorems for matter states for DFT, and help to develop the theory and its applications. The TD-DFT is used to study excited-state properties of systems, and it is based on Runge-Gross theorem that provides (taken from revised Wikipedia information):

Employing the Schrödinger equation as its starting point, the Runge-Gross theorem shows that at any time, the density uniquely determines the external potential. This is done in two steps:

1. Assuming that the external potential can be expanded in a Taylor series about a given time, it is shown that two external potentials differing by more than an additive constant generate different current densities.
2. Employing the continuity equation, it is then shown that for finite systems, different current densities correspond to different electron densities.

Several books have appeared related to TD-DFT (see Ref. 2) for more details of books. The applicability of DFT have been widely studied by several authors, in particular the one that programme the DFT formulation, .g. J. A. Pople, E. J. Baerends, M. Head-Gordon, A. Dreuw et al. These authors (M. Head-Gordon and A. Dreuw) published an interesting review related to TD-DFT and its applications to large molecules (6). The relations within size and applicability of the TD-DFT have been intensively shown that the size of the system influences much on the computational cost. 

Fig 6. Nanomaterial.

The applications of DFT are significantly large to consider DFT as the most used quantum chemical method. There are also others methodologies -e.g. semiempirical and ab-initio schemes- that are relevant for their application to large (bio)system.

From our point of view, the DFT is most used method in the quantum chemistry. This scheme provides accurate and reliable estimation of electronic properties of large (bio)systems in a economical time. These (bio)systems can be proteins, nanoscale materials, etc. For these and others reasons, the DFT -for the ground excited states formulation- is recommended by mostly all community of quantum chemists.

 Video 1. KS-DFT and TD-DFT at Perdue's Lecture.

 Video 2. TD-DFT Lecture.

UPDATE: The ERC has awarded Dr. Eduard Matito (Institut de Química Computacional, Universitat de Girona) with a grant, related to non-empirical development of DFT functionals. Link to the grant description: here.