Hi all,
Is there anyone possessing the experience of coding DFT scripts (by solving Kohn-Sham equation)? Recently I try to code one for a hydrogen atom array which is periodic in space (one hydrogen atom per unit cell).
The question is that how to handle the non-localized operator, i.e. Hartree potential (electron-electron interact) and external potential (nuclei-electron interact). Taken Hartree potential as an example, the operator is defined as:
V(x)=∫(n(x')/|x-x'|)dx'
Obviously, if we integrate over the infinite space, the integral is not expected to convergence in most cases since n(x') is periodic. Therefore, a suitable selection of range might be important. The same question also arises in external potential. How to design the range get involved?
Or, alternatively, in some studies the researchers solved another equation, which is the derivation of the first one:
-ΔH(x)=4πn(x)
However, it seems that the equation is applied to an finite space with wave functions on boundaries equaling to zero (Dirichlet boundary condition). I don't know whether it is still valid in periodic system.
How you treat this operator in your case?
Thank you very much!
I have done some research and the equation -ΔH(x)=4πn(x) is in fact Poisson equation for static electronic field. This equation is derived from Maxwell equations and it is thus valid for any conditions only if the dielectric of space considered is a constant.
Indeed, the potential of a infinite space with periodic density of electrons are infinite, but the key is not the absolute value of potential but relative value in the system. Therefore the derivation of potential is suitable to our condition. However, some necessary correction is required (like normalization or other process) to make the results of different systems fit some "standard".
For periodic systems, it's often more convenient to solve the Kohn-Sham equations in 'momentum space', also called 'k-space', which is the Fourier transform of real space. In this space, the form of the Hartree potential is simpler to handle because it transforms from a convolution integral in real space to a straightforward multiplication. Specifically, the Hartree potential in k-space is given by the electron density's Fourier transform divided by the square of the wave vector k.
Yixian Liu Matthew J. P. Hodgson
It is quite challenging. The treatment of the non-localized operators, such as the Hartree potential and the external potential, is a critical aspect of this problem.
Hartree Potential:
The Hartree potential is defined as the potential experienced by an electron due to the charge distribution of all the other electrons in the system. As you mentioned, the integration over the infinite space poses a problem, as the integral may not converge due to the periodic nature of the electron density n(x').
One common approach to handle this issue is to use the Ewald summation method. The Ewald method splits the Hartree potential into two parts: a short-range part that is summed in real space, and a long-range part that is summed in reciprocal space. This allows for a efficient and convergent calculation of the Hartree potential, even in periodic systems.
The Ewald method involves introducing a convergence parameter that controls the split between the real-space and reciprocal-space contributions. The optimal choice of this parameter depends on the system and the desired accuracy. You can find more details about the Ewald method in the literature on computational DFT.
External Potential:
The external potential, which represents the interaction between the electrons and the nuclei, can also be treated using the Ewald summation method. In the case of a periodic system, the external potential is also periodic, and the Ewald method can be applied to ensure convergence.
Poisson Equation:
As you mentioned, an alternative approach is to solve the Poisson equation, which relates the electron density to the Hartree potential:
-∇^2 H(x) = 4π n(x)
This equation can be solved in a finite domain, with appropriate boundary conditions. In the case of a periodic system, the Poisson equation can be solved using periodic boundary conditions, which ensures that the solution is also periodic.
One common approach is to use a fast Fourier transform (FFT) method to solve the Poisson equation in reciprocal space. This allows for an efficient and accurate solution, even for large systems.
In summary, when coding DFT scripts for a periodic system like a hydrogen atom array, you can consider the following approaches:
1. Use the Ewald summation method to handle the non-localized Hartree and external potentials.
2. Solve the Poisson equation using periodic boundary conditions and an FFT-based method.
Hope it helps. partial credit AI
Matthew J. P. Hodgson Mohammad Imam
Thank you for your help, and your suggestions are very inspiring. I may need to do some more research on Hartree potential in k-space and Ewald summation methods.
However, when I tried to solve Poisson equation under periodic boundary conditions, it seems that the Laplace operator (-Δ) is singular and irreversible. Though zero is replaced by a very small value in practical calculation, I don't know whether it will make a big difference between different systems and thus make the comparison between these systems unconvincing. The eigenvalue (Hartree potential) obtained by this method is exceptionally high in absolute value.
Maybe k-space or Ewald summations are better solutions.
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