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A pretext to Density Functional Theory

By the 1960s, the tenets of quantum mechanics have been well established. The idea that every physical system is described by a wavefunction that abides by the Schrödinger equation and contains all the information about the system has been explored to benefit chemistry. Paul Dirac first presented the idea of a many-electron wavefunction in his 1929 paper “Quantum mechanics of many-electron systems” (Proc. R. Soc. Lond. A 1929, 123, 714–733). The Hartree-Fock method has been known since the 1930s, and scientists like Slater have contributed greatly to the refinement of it—the Hartree-Fock method remains the basis of the most important quantum chemistry methods today.

However, citing C. J. Cramer in his (very good) 2004 textbook Essentials of Computational Chemistry: Theories and Models, the wavefunction was a “strange and complicated beast.” It was a rather unintuitive mathematical object that has no obvious physical interpretation on its own: we cannot draw any observable physical properties of a system just by adoring its mathematical form with a naked pair of eyes, instead these properties have to be derived from the wavefunction by applying operators to it. The wavefunction essentially is a black box that screams “shut up and calculate”, leaving little room for suggestive intuition. The wavefunction is also practically unmanageable for systems with a large number of electrons (exactly how large is “large” depends on the specific method and available computational resources), since the wavefunction for a system with $N$ electrons is a function of $3N$ spatial coordinates and $N$ spin coordinates.

Two questions emerged: (1) is there a way to use, for chemical systems, something that sparks more intuition than the wavefunction, and (2) is there a way to reduce the complexity of the wavefunction so that it can be applied to larger systems? Motivated by these questions, in 1964, Pierre Hohenberg and Walter Kohn published their landmark paper “Inhomogeneous electron gas” (Phys. Rev. 1964, 136, B864). In this paper, they introduced the two theorems that would later be known as the Hohenberg-Kohn theorems, forming the foundation of density functional theory (DFT).

The Hohenberg-Kohn theorems presented some very elegant mathematical results that immediately reduced the $3N+N$ scaling of the wavefunction to a simple electron density function that only requires three spatial variables, but as we will see in a later post, we do not have a good idea of how to solve it exactly. But that means we should strive for the best approximations possible. In 1965, Walter Kohn (the same one!) and Lu Jeu Sham published the second landmark paper of DFT “Self-consistent equations including exchange and correlation effects” (Phys. Rev. 1965, 140, A1133), presenting what we now call the Kohn-Sham method, Kohn-Sham equations, or Kohn-Sham formulation. Essentially, they introduced a set of equations analogous to the Hartree-Fock equations; we will also go through the Kohn-Sham equations in a later post. Different flavors of DFT that we know today are essentially different approximations (that we call functionals) to solve the Kohn-Sham equations. We will also see later the reasons why there are different approximations available and why we need approximations first place for the Kohn-Sham equations first place are the same reasons why Kohn-Sham DFT is, very importantly, not improvable from a theoretical/mathematical standpoint. That being said, Kohn-Sham DFT remains one of the most important methods for computational applications for its many other advantages, and the two papers by Hohenberg-Kohn and Kohn-Sham remain two of the most cited papers in the history of chemistry.

References:

  1. Dirac, P. A. M. Quantum mechanics of many-electron systems. Proc. R. Soc. Lond. A 1929, 123, 714–733. DOI: https://doi.org/10.1098/rspa.1929.0024
  2. Cramer, C. J. Essentials of Computational Chemistry: Theories and Models, 2nd ed.; John Wiley & Sons, 2004.
  3. Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. 1964, 136, B864. DOI: https://doi.org/10.1103/PhysRev.136.B864
  4. Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133. DOI: https://doi.org/10.1103/PhysRev.140.A1133