Direct Energy, 2018a

308 13.5 From Thomas Fermi Theory to Density Functional Theory
13.5 From Thomas Fermi Theory to Density Functional
Theory
The analysis considered in this chapter is based on works from 1926to
1928 [173] [174]. They were early attempts at calculating the location of
electrons around an atom, and they were developed when the idea of an
atom itself was still quite new. Half of Fermi's work is in Italian, and half
is in German. However, it is clearer than most technical papers written in
English.
A calculation is called ab initio if it is from rst principles while a
calculation is called semi-empirical if some experimental data is used to
nd parameters of the solution [136, p. 13]. The Thomas Fermi method is
the simplest ab initio solution of the calculation of the charge density and
energy of electrons in an atom [136]. Since no experimental data is used,
the results of the calculation can be compared to experimental data from
spectroscopic experiments to verify the results.
We already know that the results are not very accurate because we made
a lot of rather extreme assumptions to make this problem manageable.
Assumptions include:
• There is no angular dependence to energy, charge density, voltage, or
other quantities.
• Temperature is near absolute zero, T ≈ 0 K, so that all electrons
occupy the lowest allowed energy states.
• There is only one isolated atom with no other charged particles around
it.
• The atom is not ionized and is not part of a molecule.
• The atom has many electrons, and one electron feels eects of a uniform
cloud due to other electrons.
• The electrons of the atom do not have any spin or internal angular
momentum.
Rened versions of this calculation are known as density functional theory.
A function is a quantity that takes in a scalar value and returns a scalar
value. A functional takes in a function and returns a scalar value. The
name density functional theory comes from the fact that the Lagrangian
and Hamiltonian are written as functionals of the charge density. Density