Computational Physics (Lecture 25). PHY4370. Hybrid functionals. The form of the coupling constant integration for the exchange-correlation energy is the basis for constructing a class of functionals called hybrid.
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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.
Therefore, it is possible to approximate the functional by assuming a linear dependence on the coupling constant
One way to improve the calculation efficiency is to replace all electron calculated potential with a so-called “pseudopotential”.
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)
The ab-initio norm-conserving and ultra-soft pseudopotentials are the basis of the accurate electronic calculations.
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
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)
In order to show how a typical finite element method works for a specific problem,
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.
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.
If we use this approximation in the differential equation, we have a nonzero value:
The selection of u_i(x) and the procedure for optimizing r_n(x) determine how accurate the solution is for a given n.
which is then forced to be zero during the determination of ai. Here wi(x) is a
The procedure just described is the essence of the finite element method.
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.