Wednesday, May 3, 2017

Computational density functional theory (DFT) in a nutshell

My recent post, Computational Quantum Chemistry in a nutshell, was quite popular. There are two distinct approaches to computational approaches: those based on calculating the wavefunction, which I described in that post, and those based on calculating the local charge density [one particle density matrix of the many-body system]. Here I describe the latter which is based on density functional theory (DFT). Here are the steps and choices one makes.

First, as for wave-function based methods, one assumes the Born-Oppenheimer approximation, where the atomic nuclei are treated classically and the electrons quantum mechanically.

Next, one makes use of the famous (and profound) Hohenberg-Kohn theorem which says that the total energy of the ground state of a many-body system is a unique functional of the local electronic charge density, E[n(r)]. This means that if one can calculate the local density n(r) one can calculate the total energy of the ground state of the system. Although this is an exact result, the problem is that one needs to know the exchange-correlational functional, and one does not. One has to approximate it.

The next step is to choose a particular exchange-correlation functional. The simplest one is the local density approximation [LDA] where one writes E_xc[n(r)] = f(n(r)), where f(x) is the corresponding energy for a uniform electron gas with constant density x. Kohn and Sham showed that if one minimises the total energy as a function of n(r) then one ends up with a set of eigenvalue equations for some functions phi_i(r) which have the identical mathematical structure to the Schrodinger equation for the molecular orbitals that one calculates in a wave-function based approach with the Hartree-Fock approximation. However, it should be stressed that the phi_i(r) are just a mathematical convenience and are not wave functions. The similarity to the Hartree-Fock equations means the problem is not just computationally tractable but also relatively cheap.

When one solves the Kohn-Sham equations on the computer one has to choose a finite basis set. Often they are similar to the atomic-centred basis sets used in wave-function based calculations. For crystals, one sometimes uses plane waves. Generally, the bigger and the more sophisticated and chemical appropriate the basis set, the better the results.

With the above uncontrolled approximations, one might not necessarily expect to get anything that proximates reality (i.e. experiment). Nevertheless, I would say the results are often surprisingly good. If you pick a random molecule LDA can give a reasonable answer (say within 20 per cent) of the geometry, bond lengths, heats of formation, and vibrational frequencies... However, it does have spectacular failures, both qualitative and quantitative, for many systems, particularly those involving strong electron correlations.

Over the past two decades, there have been two significant improvements to LDA.
First, the generalised gradient approximation (GGA) which has an exchange-correlation functional that allows for the spatial variations in the density that are neglected in LDA.
Second, hybrid functionals (such as B3LYP) which contain a linear combination of the Hartree- Fock exchange functional and other functionals that have been parametrised to increase agreement with experimental properties.
It should be stressed that this means that the calculation is no longer ab initio, i.e. one where you start from just Schrodinger's equation and Coulomb's law and attempts to calculate properties.

It should be stressed that for interesting systems the results can depend significantly on the choice of exchange-correlational functional. Thus, it is important to calculate results for a range of functionals and basis sets and not just report results that are close to experiment.

DFT-based calculations have the significant advantage over wave-function based approaches that they are computationally cheaper (and so are widely used). However, they cannot be systematically improved [the dream of Jacob's ladder is more like a nightmare], and become problematic for charge transfer and the description of excited states.

3 comments:

  1. The Born-Oppenheimer approximation, in general, does NOT treat the nuclei classically. In the case of electronic structure theory, it treats them as infinitely heavy (i.e. they don't have dynamics at all, neither classical nor quantum.) Its quite normal (for molecules) to then treat the nuclear rotations and vibrations quantum mechanically on the resulting potential surface, and then, less commonly, to use perturbation theory to correct for the Born-Oppenheimer approximation.

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  2. BO: I have also arrived to gripe about your use of 'Born-Oppenheimer approximation'. The Wikipedia article quite correctly states that the BO approx is that the nuclear and electronic wavefunctions are separable, that there are no correlation terms. So you can calculate the electronic and nuclear wavefunctions separately, and then combine them. B+O show this by considering the mismatch in mass of electrons and the nucleus, lowering the coupling (cross) terms, and meaning you can calculate your electronic wavefunction in a static snapshot of the nucleus, and the nucleus move in a potential generated by the electrons (the BO energy surface).

    But what people take the BO approximation to mean is the further approximation is made that the nuclear charges are motionless points, and never bother even considering nuclear motion (zero point or thermal). Some of the path-integral literature explicitly calls this the 'coarse Born-Oppenheimer approximation', which I think is pretty accurate!

    LDA: The LDA functional is simply that Ex[n(r)] = C * n(r)^1/3 . From a classical electrons in phase space argument (the Slater Exchange energy), C=-(3/4)(3/pi)^1/3 = -0.738...
    However, as far as I'm aware, all 1980s onwards codes use the Ceperley (1980) MC result, as parametrised by Perdew and Zunger (1981). Quite literally this is a set of 7 parameters for each of the polarised + unpolarised case, used as a set of n, sqrt(n), ln(n).
    So are LDA calculations ab-initio? I would argue not, though they are fitted to reproduce the HEG results.

    Incidentally I definitely recommend Thijssen's Computational Physics (2nd edition) for learning about the nuts and bolts of electronic structure.

    Not only is DFT difficult to systematically improve, but it is not variational. So with HF + post-HF quantum-chemistry methods, anything you do to try and decrease the energy of the system takes it towards the ground state. This includes basis-set expansion. Whereas with DFT, you can be above or below the true answer, and oscillating wrt. basis size.

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