As a result, if the Anderson or Pulay scheme converges, and when the pseudo-inverse of Y k is computed in exact arithmetic, heuristic reasoning suggests that the convergence rate may be approximately bounded by. When a good initial approximation to the inverse of the Jacobian e. A natural question that arises when we apply a preconditioned fixed point iteration to a large atomistic system is whether the convergence rate depends on the size of the system.

## The Kohn-Sham Equation for Deformed Crystals

For periodic systems, the size of the system is often characterized by the number of unit cells in the computational domain. To simplify our discussion, we assume the unit cell to be a simple cubic cell with a lattice constant L. For non-periodic systems such as molecules, we can construct a fictitious cubic supercell that encloses the molecule and periodically extend the supercell so that properties of the system can be analyzed through Fourier analysis. In both cases, we assume the number of atoms in each supercell is proportional to L 3.

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The Lindhard function satisfies. Therefore, the Kerker preconditioner is an ideal preconditioner for simple metals. However, the Kerker preconditioner is not an appropriate preconditioner for insulating systems. When such a preconditioner is used the convergence the fixed point iteration becomes independent of the system size.

As we have seen above, simple insulating and metallic systems call for different types of preconditioners to accelerate the convergence of a fixed point iteration for solving the Kohn-Sham problem. A natural question one may ask is how we should construct a preconditioner for a complex material that may contain both insulating and metallic components or metal surfaces.

Although this assumption is generally acceptable for simple materials, it may not hold for more complex systems. In many cases, we can choose the approximation to be a local diagonal operator defined by a function b r , although other type of more sophisticated operators are possible. This additional change yields the following elliptic partial differential equation PDE. Because our construction of the preconditioner involves solving an elliptic equation, we call such a preconditioner an elliptic preconditioner.

Although the new framework we use to construct a preconditioner for the fixed point iteration is based on heuristics and certain simplifications of the Jacobian, it is consistent with the existing preconditioners that are known to work well with simple metals or insulators. The solution of the above equation is exactly the same as what is produced by the Kerker preconditioner.

For a complex material that consists of both insulating and metallic components, it is desirable to choose approximation of a r and b r that are spatially dependent. In this case, the operator defined on the left hand side of 39 is a strongly elliptic operator. Such an operator is symmetric positive semi-definite. The implementation of the elliptic preconditioner only requires solving an elliptic equation.

In general a r , b r are spatially dependent, and solving the elliptic preconditioner requires more than just a Fourier transform and scaling operation as is the case for the Kerker preconditioner. Even if we cannot achieve O N complexity, Eq. Our numerical experience suggests that simple choices of a r and b r can produce satisfactory convergence result for complicated systems.

The resulting piecewise constant function can be smoothed by convolving it with a Gaussian kernel.

Similarly, a r can be chosen to be 1 in the metallic region, and a constant larger than 1 in the vacuum region. This is the approach we take in the examples that we will show in the next section. In this section, we demonstrate the performance of the elliptic preconditioner proposed in the previous section, and compare it with other acceleration schemes through two examples.

The first example consists of a one-dimensional 1D reduced Hartree-Fock model problem that can be tuned to exhibit both metallic and insulating features. The simplified 1D model neglects the contribution of the exchange-correlation term. Nonetheless, typical behaviors of an SCF iteration observed for 3D problems can be exemplified by this 1D model. In addition to neglecting the exchange-correlation potential, we also use a pseudopotential to represent the electron-ion interaction.

This makes our 1D model slightly different from that presented in [ 46 ]. Each function m i x takes the form.

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The corresponding m i x is called a pseudo charge density for the i -th nucleus. We refer to the function m x as the total pseudo-charge density of the nuclei. The system satisfies charge neutrality condition, i. To simplify discussion, we omit the spin contribution here.

Instead of using a bare Coulomb interaction, which diverges in 1D, we adopt a Yukawa kernel. The parameters used in the reduced Hartree-Fock model are chosen as follows. Atomic units are used throughout the discussion unless otherwise mentioned. For all the systems tested below, the distance between each atom and its nearest neighbor is set to 10 a. The nuclear charge Z i is set to 2 for all atoms.

The Hamiltonian operator is represented in a planewave basis set. The temperature of the system is set to K, which is usually considered to be very low, especially for the simulation of metallic systems. We apply the elliptic preconditioner with different choices of a x and b x to all three cases. In the case of an insulator and a metal, both a x and b x are chosen to be constant functions.

## Frontiers | The Basics of Electronic Structure Theory for Periodic Systems | Chemistry

For the hybrid case, we partition the entire domain [ 0 , ] into two subdomains: [ 0 , ] and [ , ]. The first 64 eigenvalues correspond to occupied states, and the rest correspond to the first 10 unoccupied states.

For the insulator case, the electron density fluctuates between 0. The electron density associated with the metallic case is relatively uniform in the entire domain. The corresponding eigenvalues lie on a parabola which is the correct distribution for uniform electron gas. In this case, there is no gap between the occupied eigenvalues and the unoccupied eigenvalues. For the hybrid case, the electron density is uniformly close to a constant in the metallic region except at the boundary , and fluctuates in the insulating region.

There is no gap between the occupied and unoccupied states. In each one of the subfigures, the blue line with circles, the red line with triangles and the black line with triangles correspond to tests performed on a atom, atom and atom system respectively. In particular, the number of SCF iterations required to reach convergence is more or less independent from the type of system and system size.

We can clearly see that the use of the Kerker preconditioner leads to deterioration in convergence speed when the system size increases for insulating and hybrid systems. All these observed behaviors are consistent with the analysis we presented in the previous section. The model problem we construct consists of a chain of sodium atoms placed in a vacuum region that extends on both ends of the chain. The sodium chain contains a number of body-centered cubic BCC unit cells. The dimension of the unit cell along each direction is 8.

Each unit cell contains two sodium atoms. To examine the size dependency of the preconditioning techniques, we tested both a unit cell atoms model and a larger unit cell 64 atoms model. Furthermore, the number of iterations around 30 required to reach convergence does not change significantly as we move from the atom problem to the atom problem. However, it fails to reach convergence within iterations for the atom case. We discussed techniques for accelerating the convergence of the self-consistent iteration for solving the Kohn-Sham problem.

These techniques make use of the spectral properties of the Jacobian operator associated with the Kohn-Sham fixed point map. They can also be viewed as preconditioners for a fixed point iteration. We pointed out the crucial difference between insulating and metallic systems and different strategies for constructing preconditioners for these two types of systems. A desirable property of the preconditioner is that the number of fixed point iterations is independent of the size of the system.

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Sign up for new issue notifications. This article is concerned with the mathematical analysis of the perturbation method for extended Kohn—Sham models, in which fractional occupation numbers are allowed.

## The Kohn-Sham equation for deformed crystals.

All our results are established in the framework of the reduced Hartree—Fock rHF model, but our approach can be used to study other kinds of Kohn—Sham models, under some assumptions on the mathematical structure of the exchange—correlation functional. We then focus on the situation when the Fermi level is a degenerate eigenvalue of the rHF Hamiltonian, which has not been considered so far.

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