1. Physics 250-06 “Advanced Electronic Structure” Pseudopotentials Contents: 1. Plane Wave Representation 2. Solution for Weak Periodic Potential 3. Solution for a single scattering problem 4. Pseudopotential Approximation

2. Solving Schroedingers Equation via Fourier Transoforms. Periodic array of potential wells placed at distance a between the wells. Basic property of the potential V(x+l)=V(x), and of the Hamiltonian H(x+l)=H(x), i.e periodicity. Periodic boundary condition is imposed Properties of solutions: The solutions are traveling (or Bloch waves). There are infinite number of solutions which can be labeled by wave vector K, i.e. where t=m*l, m is any integer. Vectors k can take the values following from the periodic boundary condition: when n is any integer.

3. Non trivial values of K are those for which n runs from 0 to N, where N is number of wells considered, i.e. L=N*l. Important properties of the Bloch waves: Orthogonality

4. Example of periodic potential This potential is periodic function as for x=l*m, m=0,…so that it can be a model to analyze the solutions of Schroedingers equation in periodic potential. For V1=0 the solutions can be written as plane waves The eigenvalues E are

5. Consider now the equation Let us use a single plane wave as a basis: The expansion coefficient is simply equal 1 (assuming plane waves are normalized to 1 within the volume). Let us find the eigenvalues. The matrix eigenvalue problem is collapsed to which means that there is no correction to the free electrons eigenstates linear to V1.

6. We need to improve our basis! With the plane waves it is pretty easy. For this recall first what is where n is any integer. Let us agree on n running from 0 to N and this will be called k. Then, where n is an integer from 0 to N and n’ is any integer. In other words, so that and note also that the latter is periodic function on the lattice We thus separated the space of all wave vectors K on the subspace of so called irreducible wave vectors k and the subspace of so called reciprocal lattice vectors G. (concept of Brillouin Zone) This can always be done as soon as we introduce a period set by length l, and the periodic boundary condition set by length L=Nl. Now we have a powerful basis set since we can expand In other words, for each given k, we use the subspace G which delivers us basis functions. We use those basis functions to represent the wave function for given wave vector k.

7. Let us return back to our example. Restrict the expansion by two nearest G, i.e. pick n’=-1,0,1, or G=-2pi/l,0,+2pi/l. We have where G0=2pi/l. Compute matrix eigenvalue problem in this basis Now we realize that plane waves with different K’s or different G’s are orthogonal and therefore We also realize that diagonal element does not depend on the oscillating part of the potential! We realize that off diagonal elements depend only on oscillating part of the potential.because

8. The matrix elements are trivially evaluated We finally obtain The roots can be found by looking at the determinant Opening the determinant produces This quation delivers us three roots E1(k), E2(k), E3(k).

9. Solution for Periodic Potential

10. Sphere Scattering by a single potential Consider scattering by a potential assumed to be spherically symmetric inside a sphere

11. Sphere Solve radial Schroedinger equation inside the sphere Solve Helmholtz equation outside the sphere

12. Solution of Helmholtz equation outside the sphere where coefficients provide smooth matching with

13. A single L-partial wave solves single scattering problem Any linear combination solves it as well

14. A potential information is hidden inside the coefficients providing smooth matching One can replace real potential by a pseudopotential which would provide the same scattering property!

15. Replace real potential for each L by its by its pseudopotential should produce such solution that In other words, (i) Solve Equation for, find (ii) Adjust so that it produces pseudo with the same scattering properties.

16. The Pseudopotential Approximation for Solids Unfortunately a plane wave basis set is usually very poorly suited to expanding the electronic wavefunctions because a very large number are required to accurately describe the rapidly oscillating wavefunctions of electrons in the core region. Also we have a divergency problem in 3D ionic potential Solving solid state problem using plane wave basis set

17. It is well known that most physical properties of solids are dependent on the valence electrons to a much greater degree than that of the tightly bound core electrons. It is for this reason that the pseudopotential approximation is introduced. This approximation uses this fact to remove the core electrons and the strong nuclear potential and replace them with a weaker pseudopotential which acts on a set of pseudowavefunctions rather than the true valence wavefunctions. So, for each atom in periodic table ionic potential can be replaced by pseudopotential which would produce the same scattering process:

18. A pseudopotential is not unique, therefore several methods of generation exist. However they must obey several criteria. The core charge produced by the pseudo wavefunctions must be the same as that produced by the atomic wavefunctions. This ensures that the pseudo atom produces the same scattering properties as the ionic core. Pseudo-electron eigenvalues must be the same as the valence eigenvalues obtained from the real wavefunctions. Pseudopotentials of this type are known as non-local norm-conserving pseudopotentials and are the most transferable since they are capable of describing the scattering properties of an ion in a variety of atomic environments.

20. Mathematically, the angular dependent pseudopotential can be written as: Here = is an angular momentum projection operator. In computation, this can be evaluated Analytical formula exists to write down the matrix elements. For large systems he Kleinman-Bylander implementation of the nonlocal potential is used. Basically, we first take one l (s, or, p, or d) as the local potential, Then we can define a nonlocal part as Then the above nonlocal pseudopotential is approximated as: are the atomic pseudowavefunctions for angular l. in G-space), or real-space. In the real-space representation, the projection function can be cut off beyond a radius rcut (which is used as a parameter in PEtot). As a result, for large system calculations, real-space implementation of the Kleinman-Bylander form is faster than G-space implementation

21. The sphere of Gc2 being inside the Fourier box of n1*n2*n3 grid. Plane Wave Grids

22. Gygi’s adaptive grid: ideas from the theory of gravity Francois Gygi (UC Davis)