Thomas Fermi Method

The Thomas-Fermi model is a semiclassical approach used in quantum mechanics to describe the electronic structure of atoms. It was developed independently by Llewellyn Thomas and Enrico Fermi in the 1920s. The Thomas-Fermi model provides a simple and intuitive way to estimate the electron distribution within an atom without explicitly solving the Schrödinger equation.

In the Thomas-Fermi model:

1. Electron Density: Instead of determining the wave function of individual electrons, the model focuses on the average electron density within an atom.

2. Kinetic Energy: The kinetic energy of electrons is approximated using classical mechanics rather than quantum mechanics. The model assumes that electrons move in an average effective potential.

3. Electron-Electron Interaction: The electron-electron interaction is treated as a mean-field approximation. Electrons experience an average electrostatic potential due to the presence of other electrons.

The Thomas-Fermi model provides a set of equations that relate the electron density to the effective potential. While it is a relatively simple model and doesn't capture all the quantum effects accurately, it was an important step in the development of quantum mechanics and density functional theory.

It's essential to note that the Thomas-Fermi model has limitations, especially in describing the details of the electronic structure and predicting fine details of atomic spectra. Modern density functional theory (DFT) builds upon the ideas of the Thomas-Fermi model but incorporates more sophisticated mathematical formulations and considers exchange-correlation effects, providing a more accurate description of electronic structures.

𝜙(r) is the potential energy of electron in the atom.

𝜙 = - ze²/ 4𝜋𝜖or when r -> 0

𝜙(r) = 1 / 4𝜋𝜖o total charge/ r x (-e)

The field distribution in the normal state of the atom is determined by the symmetric solution 𝜙(r) of the equation that satisfies the following boundary condition for r -> 0 the field is due to only positively charged nucleus.

while r -> ∞ (outside the atom)

𝜙(r) r -> 0

Since 𝜙(r) = - 1 / 4𝜋𝜖o total charge / r (-e)

r𝜙(r) = - 1 / 4𝜋𝜖o total charge x (-e)

Since the atom is electrically neutral total charge is zero. So r𝜙(r) = 0. Now we introduce new variable x such that r = xbz^-1/3.

where b is constant and is

And a function X(x) in place of 𝜙(r) as

𝜙(r) = 1 / 4𝜋𝜖o ze²/ r X(x)

As r-> 0 𝜙(r) = - 1 / 4𝜋𝜖o ze²/ r

so as x-> 0 X(x) = 1

Also As r-> ∞, 𝜙(r) r -> 0

So, as r-> ∞, X(x) = 0

Then the above equation turns into new form

Numerical method of this differential equation gives X(x). For small x it can be represented by X(x) = 1 - 1.588 x + 4/3 x^3/2

Now,

The first term is potential energy due to positively charged nucleus and second term represents the potential energy of the electron due to all other electrons.

If we plot D(r) = n(r) 4𝜋r² versus r

The Thomas-Fermi model, despite its simplicity and limitations, has several advantages and applications in certain contexts:

1. Computational Simplicity: The Thomas-Fermi model is computationally less demanding than solving the full Schrödinger equation for the electronic structure of atoms. This makes it a practical and efficient tool for quick estimations of electronic properties.

2. Insight into Average Behaviour: The model provides insight into the average behaviour of electrons in an atom, particularly the average electron density. This can be useful for understanding trends and general features in atomic properties.

3. Qualitative Predictions: While the Thomas-Fermi model doesn't provide highly accurate quantitative predictions for atomic properties, it can yield qualitatively correct results. It is often used to estimate the order of magnitude of quantities like ionization energies and atomic sizes.

4. Density Functional Theory Basis: The Thomas-Fermi model laid the foundation for the development of density functional theory (DFT), which is a more advanced and widely used quantum mechanical method in modern computational chemistry. DFT retains the concept of electron density from the Thomas-Fermi model but incorporates more sophisticated approximations.

5. Educational Tool: The Thomas-Fermi model is often used in educational settings to introduce students to the concept of semiclassical methods and to provide a starting point for understanding quantum mechanics without delving into complex mathematical formulations.

While the Thomas-Fermi model has these advantages, it is important to recognize its limitations. It does not capture the detailed quantum mechanical behaviour of electrons and may not provide accurate results for certain atomic properties. For more precise calculations, especially in the study of electronic structures and properties, more advanced quantum mechanical methods such as Hartree-Fock or density functional theory are typically employed.

This note is a part of the Physics Repository.