12.7: Post-Hartree-Fock Methods - Density Functional Theory

We mentioned before that there are other model Hamiltonian methods that are applicable to larger systems. One of the most widely used methods in chemistry for these types of systems is Density Functional Theory (DFT). The conceptual picture is based on the Hohenberg-Kohn theorem, which effectively shows that the electron density (itself a function of space and time) can be used to uniquely specify the ground state of a system in the Born-Oppenheimer approximation. In other words, the ground state electronic wavefunction really is just telling us the distribution of electrons, so maybe instead of solving for the many-body wavefunction (whose dimensionality is 3 times the number of electrons), we can instead solve just for the electron density in 3D space. In principle, if we know the electron density, we should be able to back out the total energy.

Mathematically, it can be shown that

\[E(\rho) = E_K + E_{P;e,N} + E_{P;e,e} + E_{XC}(\rho) \nonumber \]

where \(\rho\) is the electron density, \(E_K\) is the kinetic energy of the electons, \(E_{P;e,N}\) and \(E_{P;e,e}\) are the potential energies from electron-nucleus and electron-electron interaction, and \(E_{XC}\) is the exchange-correlation energy, which takes into account effects arising from electron spin and correlation. The first three terms are all classical in nature, and can easily be calculated. The problem comes from the exchange-correlation energy: that is non-classical, but theoretically there exists a functional (a functional is a function that takes another function as its argument) of the electron density function that should allow us to get the exchange-correlation energy from the electron density. As you might imagine, though, that functional is unknown, if indeed it even exists. Instead, certain model systems are used to approximate the exchange-correlation functional; in chemistry, these model systems are molecules for which energies and geometries have been measured experimentally. The most popular functional in chemistry is called B3LYP; like all DFT functionals, it is just one way of trying to approximate the exchange-correlation energy from an electron density.

To actually perform a DFT calculation, the electron density is expressed in terms of a basis of normalized orbitals (typically GTOs). Thus, the notes we made earlier on basis sets in ab initio calculations still apply to DFT calcuations, but all of the integrals are just three-dimensional, so the time required for a DFT calculation does not scale as severely with molecular size (DFT calculations are routinely performed on proteins containing over a thousand atoms, which is not possible at present with wavefunction-based methods). By applying the variational principle to the energy and electron density, we arrive at the Kohn-Sham equations, which are numerically solved:

\[\left\{ -\frac{1}{2}\nabla^2 - \sum_j^N \frac{Z_j}{r_{j1}} + \int \frac{\rho(r_2)}{r_{12}} + V_{XC}(r_1) \right\}\psi(r_1) = \epsilon \psi(r_1). \nonumber \]

You can see this is analogous to the Hartree-Fock equations (it is another way of converting the Hamiltonian into a noninteracting problem), except for the exchange-correlation potential. \(V_{XC}\) is in turn expressed as the functional derivative of the exchange-correlation energy; i.e., like in the SCF calculation, the potential is defined in terms of the electron density that gives rise to it. However, since the true functional is unknown, we must use one of the approximate functionals out there like B3LYP. In the end, given a density functional like B3LYP, the Kohn-Sham equation is solved iteratively in a manner quite similar to the SCF method.

To reiterate, because the electron density can be expressed as a function of three spatial coordinates, the dimensionality of a DFT calculation is much lower than for a Hartree-Fock (or higher-level) ab initio calculation (whose dimensionality is 3\(N\)), and it can therefore be applied to much larger systems. DFT can be and is often used to calculate the properties of proteins, solid state materials, and other large molecules, though it has trouble with certain types of intermolecular interactions and dispersion. Compared with high-level ab initio methods, the accuracy of DFT is relatively poor, but it does calculate bond lengths to within a few percent, vibrational energies to within 10-20%, and reaction energetics that are at least qualitatively reasonable. While the initial formalism for DFT could only be applied to ground states of nondegenerate systems, it has been extended to cover degenerate systems and, with the advent of Time-Dependent DFT (TDDFT), it can even be used to calculate excited state properties.

The main limitation of DFT is the same as for other model-based systems: although they often get the right answer, they get the right answer for the wrong reasons. This has two consequences: first, it is impossible to know with any certainty how accurate a DFT calculation on any particular system is if the experimental value is unknown. Second, there is no way in which DFT methods can be systematically improved. This means that calculations involving larger and larger basis sets or more and more sophisticated exchange-correlation functionals are not guaranteed to be more accurate than more primitive calculations. Nevertheless, experienced quantum chemists have found certain functionals and basis sets perform better than others for certain classes of molecules or systems containing particular types of interactions. As cautioned before a DFT calculation will always give an answer, for better or for worse, and unlike ab initio methods, it can be very difficult to assess how reliable a DFT calculation is.

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