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. Solving Schroedingers Equation via Fourier Transoforms.
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Physics 250-06 “Advanced Electronic Structure”
1. Plane Wave Representation
2. Solution for Weak Periodic Potential
3. Solution for a single scattering problem
4. Pseudopotential Approximation
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
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.
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
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
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.
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,
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.
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
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
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).
Solution for Periodic Potential
Scattering by a single potential
Consider scattering by a potential assumed to be spherically
symmetric inside a sphere
Solve radial Schroedinger equation inside the sphere
Solve Helmholtz equation outside
Solution of Helmholtz equation outside
where coefficients provide smooth
A single L-partial wave solves single scattering problem
Any linear combination solves it as well
A potential information is hidden inside the coefficients providing
One can replace real potential by a pseudopotential
which would provide the same scattering property!
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.
The Pseudopotential Approximation for Solids
Solving solid state problem using plane wave basis set
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
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:
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.
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.
Pseudo-electron eigenvalues must be the same as the
valence eigenvalues obtained from the real wavefunctions.
Mathematically, the angular dependent pseudopotential can be written as:
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
Plane Wave Grids
The sphere of Gc2 being inside the Fourier box of n1*n2*n3 grid.
Gygi’s adaptive grid: ideas from the theory of gravity
Francois Gygi (UC Davis)