# 13.5: From Thomas-Fermi Theory to Density Functional Theory

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The analysis considered in this chapter is based on works from 1926 to 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 first principles while a calculation is called semi-empirical if some experimental data is used to find 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, \(T \approx 0\) 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 effects of a uniform cloud due to other electrons.
- The electrons of the atom do not have any spin or internal angular momentum.

Refined 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 functional theory calculations do not make as many or as severe of assumptions as were made above, especially for the \(E_{e \,e \,interact}\) term. These calculations have been used to calculate the angular dependence of the charge density, the allowed energy states of electrons that are part of molecules, the voltage felt by electrons at temperatures above absolute zero [136], and many other microscopic properties of atoms. Density functional theory is an active area of research. Often charge density is chosen as the generalized path instead of voltage [136].

Both Thomas [173] and Fermi [174] included numerical simulations. Amazingly, these calculations were performed way before computers were available! More recently, researchers have developed software packages for applying density functional theory to calculate the allowed energy levels, charge density, and so on of electrons around atoms and molecules [178] [179]. Because of the complexity of the calculations, parallel processing is used. Computers with multiple processors, supercomputers, and graphics cards with dozens of processors have all been used.