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Based on the variational principle, one might expect that the ground state energy of a density functional theory (DFT) calculation will decrease as the basis set size increases. (As I understand it, this is the case in Hartree-Fock.)

Mathematically speaking, is it in fact true that the DFT energy decreases with basis set size? If so how would you prove it? It seems like the variational principle might not directly apply since DFT, unlike Hartree-Fock, does not explicitly construct a many-body wavefunction.

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    $\begingroup$ It is not only the size of the basis set, but also the form of the functions that comprise the set that make a difference. A larger basis set for a specific type of functions will allow a more accurate energy to be determined. Someone correct me if I'm wrong, but I believe with DFT, the lowest energy you can obtain will always be the most accurate one, since the actual energy would correspond to an exact solution. I don't know how to prove that though. $\endgroup$ – Nick Jul 3 '13 at 6:10
  • $\begingroup$ Yes, I had thought that the lowest energy is the most accurate one as well. But in some of my calculations, I've seen the energy increase with the number of plane waves. I'd like to know if this is a reasonable result or not. $\endgroup$ – Max Radin Jul 3 '13 at 14:28
  • $\begingroup$ @MaxRadin You mean the energy of a fixed structure increases with a larger basis set? That's bogus but potentially a natural consequence of some numerical issues in a particular code. Can you share details? $\endgroup$ – Jeff Jul 6 '13 at 20:09
  • $\begingroup$ I agree with Jeff^, if the basis set is purely planewaves and the volume of the cell does not change, the energy will go down as the basis set is increased. If it increases, something is wrong with either your input file or the software. @Max Radin, how large is the difference? if it is small it could just be a numerical convergence issue with the eigensolver. Try decreasing the tolerance for that. $\endgroup$ – Dr_bitz Jul 7 '13 at 8:44
  • $\begingroup$ I was doing a single-point PBE calculation (ion positions and lattice constants both fixed), and the energy increased by more than 0.1 eV per atom, which I think is quite large relative to the numerical accuracy of the calculation. I realize stackexchange isn't really meant to be a place for debugging, but here are the results and VASP input parameters. I also get qualitatively similar results using an entirely different code (QBox). $\endgroup$ – Max Radin Jul 9 '13 at 0:59
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In the standard form, KS-DFT is solved variationally, which means that additional degrees of freedom in the basis set must lead to a lower (or equal) energy. This is a very basic property of the variational method and the math is almost certainly explained on Wikipedia or equivalent.

I am assuming the same functional is used. Each DFT functional provides an effective one-body Schrodinger equation that is solved exactly using the SCF variational approach (assuming no spurious solutions are found). Hence, DFT is an approximate theory with an exact solution.

Hartree-Fock (HF), on the other hand, is the variational solution of the exact non-relativistic Born-Oppenheimer Schrodinger equation within the given basis set with the caveat that electron-electron interactions are approximated using a mean-field. In contrast to DFT, HF is an approximation solution to an exact^1 theory.

^1 For some definition of exact, i.e. the one where full configuration interface (FCI) in a finite basis set is considered the exact solution to a particular family of Hamiltonians.

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  • $\begingroup$ Thanks for your reply. After thinking about it some more, it makes sense to me that in principle the KS ground state energy will be monotonically decreasing with basis set size. However, computationally, I am wondering if there still may be a situation in which one might see the opposite behavior. Some codes calculate the KS kinetic contribution to the total energy based on the sum of the eigenvalues. If our basis is incomplete, then there will be some error in our eigenvalues. Maybe this could lead to an underestimate of the total energy. What do you think about this possibility? $\endgroup$ – Max Radin Jul 9 '13 at 3:38
  • $\begingroup$ I do not understand how one would compute the kinetic energy from eigenvalues. What eigenvalues are you referring to? $\endgroup$ – Jeff Jul 10 '13 at 19:56
  • $\begingroup$ I'm looking at equation 23 of these slides from the VASP tutorial. The total energy can be expressed as the sum of the KS eigenvalues, minus some additional terms. $\endgroup$ – Max Radin Jul 10 '13 at 22:31
  • $\begingroup$ Yeah, sorry, plane-wave basis functions are eigenstates of the kinetic energy operator. There is no approximation here. $\endgroup$ – Jeff Jul 14 '13 at 19:04
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Density functional theory optimizes the total energy with respect to the (possibly constrained) electronic density. The usual numerical procedure for this optimization is embodied by the KS method. The KS procedure will give you the optimal total energy and the ground-state (gs) density for which this optimal energy is achieved. This optimization procedure is independent of the explicit exchange-correlation (xc) density (or possibly orbital, where OEP-KS method applies) functional provided by you. Although, of course, the better the xc functional, the closer the total energy and gs density will be to the true one.(see Note added to the end)

The minimization procedure is done in the (in principle) infinite space of real valued functions that represent the density. However, by choosing a finite basis you constrain this search space. The more complete your basis set, the more flexibility you have to find the gs density, because the density is built from the single particle orbitals expanded in the given basis.

With this reasoning, it seems plausible to think that if you increase your basis set, you will have a bigger search space and the optimal gs density will be better than for a smaller basis set. This behavior would be in general the expected. However, it seems to me possible that you could add more basis functions that could increase the flexibility of the density in high energy regions, while deteriorating the description of the density in the low energy region (here I mean regions of the infinite dimensional space of densities), so one could somehow arrive at a higher energy by increasing the basis set in the wrong way. For example, think of a 1D quantum well in a real space representation of delta functions centered in a dicrete grid of points. If you increase your basis by adding many delta functions in low density regions but remove a few in places where the minimum of the potential occur, then for sure your total energy will increase, because your basis is incapable of describing accumulation of charge in the minimum of the potential.

Therefore, the take-home message is: if your energy is increasing upon increasing your basis set, then you are adding flexibility in the wrong place and deteriorating in the right place. If this is the behavior, then try to increase your basis in a different way, until you see a lowering of the total energy.

Note: All above is valid for a given fixed xc (explicit) density functional. Changing the functional will change your energy in a non-variational way. I mean, a given xc functional could give a total energy lower than the true one. However, this does not mean that there is a problem in the KS minimization procedure, but that your xc functional is not a very good one.

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