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Computational Physics (Lecture 25)

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Computational Physics(Lecture 25)

PHY4370

- The form of the coupling constant integration for the exchange-correlation energy is the basis for constructing a class of functionals called hybrid.
- Because they are a combination of orbital-dependent Hartree-Fock and an explicit density functional.
- The most accurate functionals available as far as energetics is concerned and the method of choice in the chemistry community.

- The hybrid formulation arises by approximating the integral in terms of information at the end points and the dependence as a form of the coupling constant.
- When the coupling constant is zero, it is Hartree-Fock exchange energy.
- The potential part of the LDA and GGA functional is most appropriate at full coupling when the coupling constant is 1.

- Therefore, it is possible to approximate the functional by assuming a linear dependence on the coupling constant
- Leading to a half and half form:
- Exc = ½(E_xHF + E_xcDFA)
- DFA denotes an LDA or GGA functional.
- Later Beck presented parameterized forms that are accurate for many molecules, such as B3P91, a three parameter functional that mixes HF exchange, the exchange functional of Becke and correlation from PW.

- Perdew, Ernzerhof and Burke proposed the form:
- Exc=E_xcLDA+1/4(E_xHF-E_xcDFA)
- The ¼ was found by fitting.

- Hybrid can be used in different ways.
- Not strictly within the usual K-S approach if the H_F equations are solved with a non-local exchange operator.

- A straightforward way to solve the Kohn-Sham equations is to follow the condition that the effective potential and the density are consistent
- i.e. to obtain a self-consistent solution.

- The first step is to make an initial guess of the density first;
- then calculate the corresponding effective potential and solve the Kohn-Sham equations;
- from the results, the new electron density can be calculated;
- if the density and the effective potential are consistent, the energy, force, stresses, eigenvalues etc can be achieved and the calculation is done;
- otherwise, the effective potential can be recalculated based on a linear combination of previous densities and the process is repeated.
- In this step, a so called steepest decent method can be applied to find the minimized energy. (Also, CG is widely used in many packages.)

- To solve the Kohn-Sham equations, a basis set need to be chosen.
- There are two types of basis sets:
- one is based on plane waves;
- the other is based on localized orbits.

- Plane waves and grids are two major methodologies to solve differential equations.
- Plane waves are especially appropriate for periodic crystals calculation
- and localized orbits (grids) are suitable for finite systems. In modern electronic structure calculations, both methods are extensively applied with fast Fourier transformation.

- To achieve accurate electronic calculation results, the Coulomb potential of the nucleus and the effects of the core electrons need to be included.
- In this sense, an all electron calculated potential is necessary.
- However, the Coulomb potential of the nucleus is usually strong and the core electrons are usually tightly bound, which usually results in a high kinetic energy and a lot of terms in Fourier transform which affect the calculation efficiency.
- Also, the core electrons have a small effect in valence electron bonding.

- One way to improve the calculation efficiency is to replace all electron calculated potential with a so-called “pseudopotential”.
- Instead of solving the difficult problem of the Coulomb potential of the nucleus, an effective ionic potential acting on the valence electrons can be calculated.

- This potential is called pseudopotential.
- A pseudopotential can be generated in an atomic calculation and the calculation result can be the input of calculations of properties of valence electrons in solids or molecules, because the core states almost remain unchanged in these situations
- except for some extreme conditions, such as very high pressures.

- Pseudo-potentials can usually be constructed in two parts, the local (l-independent) part plus the non-local (l-dependent) part (l is the orbit quantum number)
- To preserve the right physics of the all electron case, pseudo-potentials usually satisfy a so-called norm conserving condition: the eigenvalues and the orbits are required to be the same for the pseudo and the all electron case for r>Rc,
- each potential Vl(r) equals to the local (l-independent) all electron potential, and for r->.

Schematic illustration of a pseudo-potential.

- The ab-initio norm-conserving and ultra-soft pseudopotentials are the basis of the accurate electronic calculations.
- One goal of creating potential is to make the potential as smooth as possible, because in plane wave calculations, the potential is expanded in Fourier components and the performance of the calculation is to the power of the number of Fourier components.

- One way to achieve the smoothness is to introduce the concept of “ultra-soft pseudopotentials”. “Ultra-soft pseudopotential” is a practical approach for solving equations beyond the applicability of those formulations

- The concept of ultra-soft pseudopotential was proposed by Blochl and Vanderbilt
- They rewrote a non-local potential in a form involving a smooth function that is not norm conserving

Schematic illustration of an all electron calculated wave function (solid line), a norm conserving pseudo-potential calculated one (dotted) and an ultra-soft pseudo-potential calculated one(dashed)

- As we can see, the ultra-soft pseudo-potential yields the smoothest wave function among the three and is the most efficient in calculation.

- The projector augmented wave method is a general approach to solution of the electronic structure problem
- That formulates the orthogonalized plane wave method.

- Like the ultrasoft PP mthod, it introduces projectors and auxiliary locallized functions.
- The difference is that PAW approach keeps the full all-electron wave function in a form similar to the general OPW expression.

- It is usually more difficult to simulate continuous systems than discrete ones,
- especially when the properties under study are governed by nonlinear equations.

- The systems can be so complex
- that the length scale at the atomic level can be as important as the length scale at the macroscopic level.

- The basic idea in dealing with complicated systems is similar to a divide-and-conquer concept,
- that is, dividing the systems with an understanding of the length scales involved and then solving the problem with an appropriate method at each length scale.

- In order to show how a typical finite element method works for a specific problem,
- let us take the one-dimensional Poisson equation as an example.

- Assume that the charge distribution is ρ(x) and the equation for the electrostatic potential is
- In order to simplify our discussion here, let us assume that the boundary conditionisgiven as φ(0) = φ(1) = 0.

- we can always express the solution in terms of a complete set of orthogonal basis functions as

and ui(0) = ui(1) = 0.

- We have assumed that ui(x) is a real function and that the summation is over all the basis functions.
- One of the choices for ui(x) is to have ui(x) = √2 sin iπx.
- In order to obtain the actual value of φ(x), we need to solve all the coefficients aiby applying the differential equation and the orthogonal condition
- This becomes quite a difficult task if the system has a higher dimensionality and an irregular boundary.

- The finite element method is designed to find a good approximation of the solution for irregular boundary conditions or in situations of rapid change of the solution.
- Typically, the approximation is still written as a series with a finite number of terms.
- Here u_ii(x) is a set of linearly independent local functions, each defined in a small region around a node x_ii.

- If we use this approximation in the differential equation, we have a nonzero value:
- which would be zero if φn(x) were the exact solution.
- The goal now is to set up a scheme that would make r_n(x) small over the whole region during the process of determining all a_i.

- The selection of u_i(x) and the procedure for optimizing r_n(x) determine how accurate the solution is for a given n.
- We can always improve the accuracy with a higher n.
- One scheme for performing optimization is to introduce a weighted integral

which is then forced to be zero during the determination of ai. Here wi(x) is a

selected weight.

- The procedure just described is the essence of the finite element method.
- Theregion of interest is divided into many small pieces, usually of the same topology but not the same size, for example, different triangles for a two-dimensional domain.
- Then a set of linearly independent local functions ui(x) is selected, with each defined around a node and its neighborhood.
- We then select the weight wi(x),which is usually chosen as a local function as well.

- For the one-dimensional Poisson equation, the weighted integral becomes

- The advantage of choosing ui(x) and wi(x) as local functions lies in the solution of the linear equation set.
- Basically, we need to solve only a tridiagonalor a band matrix problem, which can be done quite quickly.
- The idea is to make the problem computationally simple but numerically accurate.
- Thatmeans the choice of the weight wi(x) is rather an art.
- A common choice is that wi(x) = ui(x), which is called the Galerkin method.