Wednesday, March 14, 2012

Being more precise about Density Functional Theory (DFT)?

It seems to me many authors (particularly non-experts) are rather sloppy when they refer to electronic structure calculations based on Density Functional Theory (DFT). They say things like "our results are in good agreement with DFT calculations" or "we calculate the excited state energies using DFT". What don't I like about this?

DFT is an exact theory,.... provided one has the exact exchange-correlation functional and one can solve the non-linear variational functional equation the density must satisfy....
Hence, DFT should always agree with experiment!

But in reality, pure DFT is practically useless. One cannot do either of the above and so much use approximate functionals such as those based on the Local Density Approximation (LDA) or Generalised Gradient Approximation (GGA). These then lead to Kohn-Sham type equations which are straight-forward to solve.

Hence, it seems to me it is preferable to say, "our experimental results are comparable to values obtained from DFT-based calculations using the LDA".
But, maybe I am just pedantic...


  1. I totally agree!

    P.S. I think your blog is great. I'd comment more, but logging in is a slight hassle. My temperature is too low to surmount even small activation barriers.

  2. The statement that DFT is an exact theory is at least practically useless, and can't be used to justify most implementations of DFT. More specifically, DFT is exact when you use the density obtained by solving the Schrodinger equation. Of course, if you have to do that, why use DFT in the first place?

    Then there is always the small fact that the v-representability conditions are not known... and the assumption of the differentiability of the energy functional...

    So, when I hear "DFT is exact", I usually stop listening to whoever is talking and pay attention to someone more worth listening too - namely, the voice in my own head.

  3. While we are on this topic, I have never really understood the argument that the HK functional is universal. I understand how it doesn't depend on v(r), and so doesn't depend on the density... but none of this makes it obvious to me that it's universal - unless one makes assumptions about the topology of the set of possible functionals that are not clearly justified. So, I just don't believe that the assertion of universality (which is always being made) is true, or, if it is true, it may only be true in some limit which doesn't apply in any practical situation.

    If anyone has a better argument supporting universality of the HK functional, then I'd love to hear it...

  4. This is outside my expertise, but I wanted to highlight something that was brought up at Jaime Ferrer's APS talk this year.

    The point is that DFT is only in principle exact for the auxiliary problem it's meant to solve - i.e. it'll come up with the right density, and the right total energy, but the one-electron states that come out of a ground state calculation are a priori meaningless.

    In the literature, this DFT "band structure" from the ground state calculation is often compared with measured photoemission data or activation energies. Actually, there is no reason why these should correspond at all, even if the EXACT functional is known. The many-body spectrum is, generally speaking, richer than the DFT band structure suggests. This point is discussed in his paper here:

  5. After reading some more, I understand now that the universality of the exact HK functional depends on the ability to establish a global minimum by the Levy constrained search. Parr and Yang say that Lieb proved the minimum exists. I will try to understand Lieb's argument that the global minimum exists for the N-electron search, and why its existence implies that the functional is the same regardless of what N actually is. It is not clear to me that it should, because antisymmetry is a global constraint...

  6. BTW, I agree with Steve. It is almost unfortunate that the the KS orbital energies (for certain well-used functionals) agree well with electrochemistry experiments. This has spawned a whole generation of chemists who make statements like "I have experimentally measured the HOMO and LUMO energies through cyclic voltammetry".

  7. I agree with Steve as well. DFT is an exact *ground state* theory. The Kohn-Sham scheme is only useful to construct the ground-state density. So unless you can represent your observable as a functional of the density you are out of luck in terms of any statements of exactness, even with the exact XC functional. Lots of people try to use the KS levels (single particle energies whose only purpose is to construct the true many-body ground state density) as the poles of the real system. This is completely wrong and has no a priori hope of getting anything right (symmetries, trends or otherwise). For the interested reader, the history of DFT in quantum transport is especially illustrative of this fact.

  8. I think DFT is quite popular thanks to the B3LYP functional. Many people have been able to compare their organic molecules and "validate" them with
    B3LYP. They show that their IR spectrum agrees pretty well with the B3LYP spectrum. I totally agree there is a misuse of the word DFT. People should refer to it as Density Functional Approximation unless they are really studying density functional theory. Regarding the universality of the energy functional,
    nobody knows the exact form of the wave function for more than two electrons, and it is also true for energy functional. However, it is more useful to know a little piece of the functional because you can simulate many systems with it, while you cannot do that with the Helium wave-function.