Description-A
Our approach to the development of first principles methods
A. Material agnostic Model Hamiltonian (MH) vs Material-Specific first principles approaches to condensed matter theory: The time-honored Model Hamiltonian approach to condensed matter theory forms the standard basis for teaching physics of solids. In this approach one explains generic physical effects as an outcome of generic, postulated, interactions. Examples include the explanation of magnetism from spin-spin interactions in the Ising/Heisenberg Hamiltonian, as well as the explanation of superconductivity from electron-phonon interactions; localization from electron-electron correlation (Mott) or from disorder (Anderson). The clear advantage of this historic MH approach is that it allows us to understand and teach recognizable effects in terms of well-posed effect-producing- interactions. Characteristic to such MH approaches is the fact that the chemical identity of the material--its Atomic numbers, Composition and Structure (ACS)-- is not explicitly encoded in the postulated interactions. Thus, the approach does not reveal in which material does the considered effect “live”, or how can we choose a material (i.e., an ACS) that will control the magnitude of the effect (the so-called, Inverse Problem). Since special physical effects (superconductivity, semiconductivity, topological insulation, catalytic activity, to name just a few) tend to “live” in specific materials and no others, the Model Hamiltonian philosophy did not prove to form a practical guide to material selection, design and engineering except as an explanation and description of known facts, rather than prediction and control. We note in passing that not all-formal work in condensed matter theory shares these properties of Model Hamiltonian approach. Certainly, formal work in areas that is intrinsically material-independent (such as group theory for topological insulators, general field theory for interacting particles, or renormalization group, to name a few) form a separate class of condensed matter theory.
B. Material-specific theories of matter: The alternative approach we focus on here is the predictive electronic structure theory of solids and molecules based on Hamiltonians that recognize the explicit identity of the system (its ACS). Whereas the mathematical foundations of the theories that necessitate the specification of the material identity (ACS) was known already from Hartree and Kohn for a long time, these approaches started having an impact on our understanding of solids many years after they were first published. Indeed, the initial optimistic expectations in the sixties, after the seminal works of L.D. Landau and W. Kohn was that solid state theory could guide practical efforts of systematic laboratory search for new materials and new functionalities. However, as exciting as were the previously developed ideas of Quantum Theory of Solids, encapsulated in the language of model Hamiltonians, they remained somewhat abstract concepts for a rather long time. The following section on Development of Theoretical and Computational First Principles Methods focuses on our contributions that helped transform the general ideas of material specific theories into a practical platform of computing predictively properties of solids given as input their characterization via their ACS.
C. The Four key areas of research (i- iv) that were instrumental enabling this approach. The ensuing “Standard Model” of predictive electronic structure theory is now forming the basis for unprecedented interaction between theorists and experimentalists in the fields of materials by design, surface science, high-pressure physics ,alloy theory and defect theory, arguably creating the basis for intelligent, theory- inspired material selection for a broad range of technologies, ranging from solar cells, LED’s, FET, flat panel displays.
(i) Development of universal exchange and correlation energy functionals. The seminal work of Hohenberg and Kohn and Kohn and Sham paved the way for all modern quantitative calculations on real materials. Yet, theirs was an “existence theorem” about a functional that would describe the exchange and correlation interactions response of the electron system. Without an actual functional, the use of the DFT machinery was simply impossible. The invention of a material-dependent realistic exchange and correlation functional (now called “ the Local Density Approximation (LDA) functional) is due to Perdew and Zunger 1981 (J.P. Perdew and A. Zunger, “Self-Interaction Correction to Density Functional Approximations for Many Electron Systems,” Phys. Rev. B23, 5048–5079 (1981); many subsequent functionals were invented by J. Perdew in the nineties. These have enabled the basic descriptor of inter electronic interaction in molecules and solids within the DFT. These functionals resulted in accurate description of Charge densities and bonding in solids compared to previous empirical pseudo potential generated charge densities that were qualitatively incorrect (even for Si).
(ii) Development of atomic pseudopotentials by inverting the density functional theory of atoms: It was known for a long time from traditional chemical intuition and from the more formal work (in theory of nuclear matter) of the 1930s due to Fermi and Amaldi that “core states” do not contribute much to binding, yet (because of Pauli’s principle) they must be carried in calculations along with “valence states” when calculating the quantum structure of many-particle systems. Thus, large systems could not be tackled because of the burden of treating so many (core+ valence) particles. The critical development steps, were (1) The concept of removal of “core states” by adding a repulsive (pseudo) potential to the conventional “all-electron” Hamiltonian (“pseudo potential theory”) was invented by Fermi and Amaldi (2) The application of the concept within non-structural theories (i.e., theories that can not predict structure because the potential describes an effect—the band structure—not the total energy) is due to Marvin Cohen and Bergstresser ( the Empirical Pseudo potential Method ,or EPM) .(3) The invention of atomic pseudo potentials constructed from a microscopic (DFT) theory ( “ first principles pseudo potentials” ) is due to Zunger ( S. Topiol, A. Zunger, and M. Ratner, “Local Density Pseudo potentials for the First Row Atoms,” Chem. Phys. Lett. 49, 367–373 (1977)) .This created a structural theory ,because the potentials had an external part ( “ionic pseudo potential”) ,as well as a density-dependent part ( the DFT screening) ,thus complied with the structure of DFT .This enabled rigorous DFT calculations for solids without the burden of having to deal with the many “core electrons” that would have made the problem intractable solids. (4) Improvements of such 1976-1978 first principles pseudo-potentials (making them “soft”, thus applicable within the plane-wave technology to transition metals) came later on by D. Hammon, L. Kleinman , and D. Vanderbilt.
(iii) Development of a non-divergent expression for the total energy of periodic solids: Until the late seventies, electronic structure calculations of poly-atomic solids generally gave only “band structures,”(eigenvalues of the Bloch equation) but not the total energies needed to predict the ground-state properties. Indeed, the individual terms in the total energy (electron-electron; ion- ion; electron -ion) were known to be individually divergent .The total energy formalism within the plane wave technology (by far, the currently most used approach in “first Principles theory of solids and molecules” was invented by Ihm , Zunger and Cohen in 1978/1979 (J. Ihm, A. Zunger, and M.L. Cohen, “A Momentum Space Formalism for the Total Energy of Solids Using Pseudopotentials,” J. Phys. C 12, 4409–4421 (1979) .Analogous total-energy approaches were invented in the context of all-electron theory by O.K.Anderson , Gunar Arbman and later A. J.Freeman in the eighties. These critical developments initiated the era of predicting ground-state properties of solids from pseudo potential or all-electron (such as LMTO and LAPW) DFT calculations. Properties included the structure and bond-lengths of solids and molecules, phonons, elastic properties; pressure effects ,etc. It would be indeed unimaginable to have today’s Material Theory without the total energy capability.
(iv) A paradigm-changing strategy to treat charge density self-consistently simultaneously with seeking the equilibrium position of the atoms in the solid. The application of DFT to molecules and solids requires both the solution of the electronic part at fixed atomic positions, and the optimization of the atomic positions. These two steps were traditionally carrier out consecutively, thus refining one’s knowledge on the electronic part for some geometry that were eventually found to be irrelevant. The invention of the technique of treating simultaneously the (i) atomic displacements towards equilibrium with (ii) attaining charge self-consistency is due to the 1983 Bendt-Zunger entitled “ Simultaneous Relaxation of Nuclear Geometries and Electronic Charge Densities in Electronic Structure Theories,” Phys. Rev. Lett.50, 1684–1688 (1983) .This work was the obvious precursor to the Car-Parrinelo method, published a year later, which coupled the previous idea Bendt-Zunger of simultaneous solution to Molecular Dynamics . These developments ushered in an era of rapid calculations of molecular and solid state structures, enabling numerous fundamental advances in material science.