# 15th Anniversary of the Nobel Prize in Chemistry

November 04, 2013

*It's been 15 years since the Nobel Prize in Chemistry was awarded to Walter Kohn for his development of density-functional theory (DFT) and John Pople for his development of computational methods in quantum chemistry. In this edition of the ARL Fellows Corner, Dr. Jan Andzelm discusses the discovery of quantum mechanics and density-functional theory (DFT), and how, the DFT became the most commonly used method of electronic structure calculations for many applications in chemistry, biophysics, nanotechnology and materials science, all important for the mission of Army Research Laboratory.*

On October 13, 2013 we will commemorate the 15th anniversary of the Nobel Prize in Chemistry awarded to Walter Kohn for his development of density-functional theory (DFT) and John Pople for his development of computational methods in quantum chemistry. The press release from the Royal Swedish Academy of Sciences reminds us that soon after quantum mechanics was discovered in the early 1900s, Paul A.M. Dirac stated in 1929: "The fundamental laws necessary for the mathematical treatment of large parts of physics and the whole of chemistry are thus fully known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved". It took about 60 years of theory, computational methods and computer technology development to challenge this pessimistic statement. The development of quantum mechanics relied on two fundamental quantities describing the motion of electrons in a molecule or solid, the wave function (Ψ) and the electron density (ρ). These quantities can be obtained by solving quantum-mechanical equations and they are related as the square of the wave function defines the probability of finding an electron at a particular position in a molecule.

The quest to solve quantum-mechanical equations using the wave function as a fundamental quantity is often called an "ab initio" approach, which originated from a pivotal publication by Erwin Schrödinger in 1926. Since then, numerous attempts have been made to find approximate solutions of Schrödinger's wave equations and this is where John Pople was most influential. His lasting contribution is a computer program called Gaussian used by thousands of chemists every day. This program can calculate many molecular properties such as structure or vibrations through a series of computational methods with varying degrees of sophistication.

A rudimentary form of DFT was first proposed by Llewellyn H. Thomas and Enrico Fermi in 1927. John Slater contributed significantly to its development in the early 1950s but it was the work by Walter Kohn, Pierre Hohenberg, and Lu Jeu Sham in 1964 and 1965 that set the stage for an explosion of DFT simulations in physics and chemistry. Walter Kohn showed that quantum-mechanical equations can be efficiently approximated by containing all of the many-body interactions in an exchange-correlation energy (E_{xc}), which depends on the electron density of the simulated system. Unlike the wave function Ψ, which is a function of the position of all electrons in the system, the electron density ρ is only a function of three positional coordinates. Therefore the DFT equations can be programmed much more efficiently and applied to more complex solid and molecular systems of practical importance for materials science. The DFT approach is often called a "first principles" method since in contrast to the "ab initio" approach it contains a component, the exchange correlation functional, whose exact form is not known. The ever increasing popularity of DFT methods can be approximately assessed by searching Web of Science for the keywords "density functional theory" as a function of publication year. This approximate measure indicates that after the Nobel Prize was awarded in 1997, the number of DFT citations per year increased almost nine-fold.

Kohn and Hohenberg showed that there is a one-to-one correspondence between the energy of a system and its electron density, yet they did not provide a systematic recipe for how to find the exact E_{xc}. In practical calculations, one of the many approximate forms of the DFT E_{xc} is selected, with different forms for different classes of systems or even for various properties within a given system.

In 2001, John Perdew described the hierarchy of DFT functional approximations as the "Jacob's ladder of DFT". The local density approximation (LDA) based on the total energy of a homogeneous electron gas occupies the bottom rung of the ladder and is primarily used for calculations on metallic systems. The next rung of the ladder is the generalized gradient approximation (GGA) which introduces a local gradient in the electron density. Typically, GGA calculations improve upon the LDA method with respect to the energetics of chemical reactions and structures. Meta-GGA exchange-correlation functionals, which include a dependence on the kinetic energy density, form the third rung. At the top of the ladder are the hybrid-GGA methods, which augment the density-based DFT exchange with the exact (i.e., Hartree Fock) exchange. The hybrid methods accurately predict chemical reactions and also improve the band gap of solids. One of the most commonly used hybrid DFT functionals, B3LYP was proposed by Axel Becke in 1993 and is now the standard for most quantum chemistry calculations.

For selected systems and properties, the most accurate DFT calculations can be performed with semi-empirical meta-hybrid functionals that depend on variables such as spin density, gradient density, kinetic energy density and exact exchange arranged in formulas with dozens of fitting parameters. The fitting procedure can involve probing hundreds of small molecular complexes for various thermodynamical, electronic and structural properties. B3LYP is the best known example of such semi-empirical finctionals. The transferability of such functionals to different classes of molecules and properties is problematic and one reason for continuing active research in the development and validation of new DFT functionals. The selection of the best DFT functional for a particular project requires significant experience in the field and often extensive benchmarking of several functionals. Despite many efforts to rationalize development of DFT functionals purely on theretical grounds, it is unlikely that semi-empirical meta-hybrid forms will be abandoned soon. The meta-hybrid forms are computationally more demanding than simple LDA or GGA functionals; however, these DFT computations are still much more efficient than the alternative "ab initio" wave function-based calculations beyond the Hartree-Fock level.

DFT calculations typically proceed through iteratively solving a set of one electron, nonlinear equations by using a linear combination of a set of basis functions representing the electron's electronic wave functions, a procedure proposed by Kohn and Sham (KS) in 1965. The choice of basis, functions and other computational parameters such as a numerical grid or strategies to solve self consistent (SCF) KS equations gives rise to numerous computational approaches and computer programs, many of which are commercialized and well optimized for modern parallel supercomputers. Solid state simulations are typically performed with plane-wave basis sets while calculations on molecules or clusters are more efficiently performed by using basis sets localized on atoms. There are efficient computer programs which use hybrid plane wave-localized orbitals, thus combining the advantages of both approaches. Since valence electrons determine chemical bonding and interactions, the most efficient DFT calculations replace the effect of core electrons by using an approximation of their field called "pseudo potentials". The local basis sets enable the implementation of order-N DFT algorithms, allowing computational effort to increase linearly (O (N)) with the number of atoms, N. For molecular and non-metallic solids, the O (N) technique allows the study of systems containing many thousands of atoms. In the case of metals, it is also possible to perform accurate DFT calculations without the Kohn-Sham scheme by using orbital-free DFT approach. This method can be applied to large systems; however its extension to non-metallic systems is limited.

The most recent survey of state of the art methods in DFT development and applications can be gleaned from the several included references [1]. Quantum mechanical calculations on molecules or solids are performed every day at ARL to elucidate properties of materials for variety of Army-relevant applications. Specifically, quantum mechanical calculations are used to predict molecular or solid structures, energies, reaction pathways as well as a variety of electronic, magnetic or mechanical properties. We will briefly review some of ARL's recent efforts in this field.

An accurate meta-hybrid DFT method for kinetics as well as sophisticated ab inito calculations were used to study reaction mechanisms and thermal decomposition of small energetic molecules [2]. A similar meta-hybrid DFT method was applied to study metal-ligand complexes and their dissociation pathways [3] as well as reactions of common electrolyte solvents for batteries and electrical double layer capacitors [4]. Highly accurate "ab initio" calculations for molecules are often used to validate DFT functionals or to provide benchmark data for fitting empirical force fields [5]. DFT simulations for crystals are very computationally intensive and often DFT calculations at the GGA-level are performed to study the band structure of semiconductors [6], mechanical properties of ceramic materials [7], and charge distribution in molecular crystals [8].

Simulations of energetic crystals as well as polymer models or biomolecules require special improvements to DFT functionals because properties of these materials are largely governed by weak intermolecular, dispersion forces that are poorly described by standard DFT. A recent assessment of various dispersion corrected DFT approaches [9] found that the GGA method, when augmented with dispersion-corrected atom-centered pseudo potentials, described well the structures of energetic crystals. A similar approach led to accurate prediction of shock Hugoniot properties of polymers [10]. Dispersion forces can be calculated directly by using symmetry-adapted perturbation theory based on the DFT description of interacting molecular dimers. Properties of molecular crystals [11] were studied using force fields fitted to results from this method and predictions were in excellent agreement with available experimental data.

Standard DFT fails to properly describe excited states and polarizability of dyes, in particular the charge-transfer chromophores. These shortcomings can be rectified by developing range-separated hybrid functionals. The short range of such a functional is described by a standard GGA approximation while the long range, affecting mainly excited states, is corrected using the exact exchange term. These functionals were found to be advantageous for predicting the excited states and color of chromophores relevant to the Army [12].

As part of its ballistic sciences research, ARL has a strong interest in modeling materials under extreme conditions such as high pressure, temperature and strain rates as well as in studying shock wave propagation in materials due to a blast or ballistic impact. These simulations require a molecular dynamics (MD-DFT) approach applied to large atomistic models. MD-DFT forces have to be calculated using an accurate DFT functional that can describe bond fracture in the material, which is particularly important in this case as the reactive classical potentials are less reliable. Recently, large scale simulations using the CP2K DFT program were performed to explore the impact of dynamic shock on polymeric nitrogen [13]. CP2K O (N) algorithms that scale linearly with the number of atoms are being actively developed at ARL to enable highly efficient simulations of large molecular or solid systems including polymers, biopolymers, ceramics and energetic materials. In the case of metallic systems, ARL is also developing finite element methods for solving DFT equations [14].

DFT is undoubtedly the most commonly used method of electronic structure calculation for many applications in chemistry, biophysics, nanotechnology and materials science, all important for the mission of Army Research Laboratory. In the years to come, the dominance of DFT will likely be maintained due to the favorable computational performance of the method coupled with the ever increasing accuracy and capabilities of DFT approach [15]. In six years, we should be able to perform simulations on much more realistic systems as the current generation of petaflop-supercomputers is replaced by exaflop (1000 petaflop) machines and more efficient O (N) DFT algorithms are developed and validated. Therefore, future DFT programs will expand the role of quantum mechanical simulations as a first step in multiscale modeling which is critical to rational design of materials at ARL.

Dr. Jan Andzelm, ARL Fellow, Multiscale Modeling of Macromolecules & Polymers

## References

- [1] http://www.chemistryviews.org/details/ezine/5039611/DFT_2013_Virtual_Issiously ue.html, http://th.fhi-berlin.mpg.de/sitesub/meetings/DFT-workshop-2013/index.php?n=Meeting.Program
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- [15] http://www.cecam.org/workshop-2-882.html