Charged systems with density functional theory: Difference between revisions
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Surfaces and other two dimensional materials are treated with Coulomb kernel truncation methods. | Surfaces and other two dimensional materials are treated with Coulomb kernel truncation methods. | ||
These methods replace the Coulomb kernel <math display="inline"> v(\mathbf{G})</math>, i.e., the <math display="inline">\frac{4\pi}{\mathrm{G}^2}</math> in <math display="inline">V(\mathbf{G})</math> with a 2D kernel (<math display="inline"> v_{\text{2D}}(\mathbf{G})</math>) that removes electrostatic interactions between non-periodic replicas along the surface normal, | These methods replace the Coulomb kernel <math display="inline"> v(\mathbf{G})</math>, i.e., the <math display="inline">\frac{4\pi}{\mathrm{G}^2}</math> in <math display="inline">V(\mathbf{G})</math> with a 2D kernel (<math display="inline"> v_{\text{2D}}(\mathbf{G})</math>) that removes electrostatic interactions between non-periodic replicas along the surface normal,{{cite|rozzi:prb:2006}} | ||
<math display="block"> | <math display="block"> |
Revision as of 12:10, 16 October 2024
On this page, we describe technical issues with computing the energies of charged systems with periodic density functional theory (DFT) calculations. We first introduce the problem of divergence of the electrostatic energy in DFT calculations and illustrate how this divergence is removed for charge neutral computations. We then discuss why this divergence is present for charged systems. Finally, we present methods which have been implemented in VASP that allow for calculations of charged bulk, surface and molecular systems.
Treating divergence in charge neutral calculations
VASP makes use of efficient fast Fourier transforms (FFT) to compute the electrostatic potential from the charge density using the Poisson equation,
where and are all points in real space. Fourier transforming the Poisson equation to reciprocal space,
where is the reciprocal lattice vector and is its norm.
An obvious issue with computing is that it diverges for . This divergence is handled in charge neutral density DFT calculations by cancelling out individual divergences for the total electrostatic energy, which is the sum of electron-electron, ion-electron and ion-ion energies
which have the following functional forms,
where and are the electronic charge density and potential respectively.
where is the ion charge density. VASP does not explicitly compute , but the potential of the ion is reflected through the eigenvalues. The ion-ion interactions are treated by Ewald summation
where is the only component which sums over the vectors and has the form,
The summed terms of , and are given by,
which, can be expressed through the reciprocal space Poisson equation as,
Under the condition that , the individual divergences would cancel out and the total energy is convergent. Since the term of a Fourier transformed quantity is its average, the above condition is satisfied when the average electron and ion charge density are the same, i.e. the system is charge neutral.
Divergences in charged calculations
The total energy does not converge when , i.e. when the system is not charge neutral. Alternatively, we can think of this divergence as coming a compensating uniformly smeared out charge density, with energy
which cancels out the divergent terms in .
The consequences of adding a uniform compensating charge is system specific. For highly screened systems, the potential coming from a moderate compensating charge presents an acceptable amount of error. Conversely, systems containing vacuum (such as 2D materials, surfaces and molecules) have little screening and hence are more susceptible to artifacts caused by this compensating charge.
Treating charged systems in practice
VASP implements a few methods to deal with charged systems. These methods are specific to the dimensionality of the system that is being computed.
Bulk charged systems in orthorhombic cells can be treated by the applying analytical monopole-monolpole corrections (see LMONO for further practical details).
Surfaces and other two dimensional materials are treated with Coulomb kernel truncation methods. These methods replace the Coulomb kernel , i.e., the in with a 2D kernel () that removes electrostatic interactions between non-periodic replicas along the surface normal,[1]
where is the norm of the vector parallel to the surface and is the vector along the surface normal. A similar change can be made to the Coulomb kernel to treat charged molecules (so-called 0D systems with )) to remove interactions with periodic replicas in all directions,
See KERNEL_TRUNCATION/LTRUNCATE for further details on applying these Coulomb truncation methods in practice.