Analysis of the Interlayer Force in Graphite by the Thomas-Fermi-Dirac Method and Kirzhnits Correction

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Atwater, Henry
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Scholarship Sewanee 2021 , Quantum Physics , Chemistry , Density Functional Theory
Density functional theory is an approach to mathematically model and interpret complex many-atom systems by expressing the energy of the system as a functional of electron density. There are many variations of density functional theory, along with many corrections made to each. I have implored a specific model in density functional theory called the Thomas-Fermi-Dirac Method, along with a correction for the inhomogeneity of kinetic energy that was calculated by Kirzhnits, to predict the Van der Waals forces between layers in a graphite crystal. In my model, I assumed that each carbon atom had the electron configuration: 1s^22s2p^3 (4 valence electrons). I then determined wave functions for each orbital in the structure, and wrote the electron density as an appropriate linear combination of their magnitudes. After obtaining the density function, I normalized it across an arbitrary distance between two parallel planes equidistant from the nucleus so that the planes enclosed 5 of the 6 electrons. The purpose of this normalization is to isolate the electron in the 〖2p〗_z orbital, as it is the farthest away and chiefly responsible for the Van der Waals forces. This effectively modeled the graphite layers as stacked positive ionic slabs containing the nuclei and 5 electrons for each carbon in the lattice, in a sea of 〖2p〗_z electrons. This allowed me to use Gauss’s law to calculate the electric field from a slab. I then solved the Thomas-Fermi-Dirac equation to acquire the electric potential as a function of a perpendicular distance from a sheet, applying boundary conditions necessary for the model. I then related the potential to the density by Poisson’s equation, and calculated the energy per atom using the Thomas-Fermi-Dirac functional. Adding the Kirszhnits correction, I evaluated the integrals numerically, varying the interlayer distance, to approximate the equilibrium distance and the force constant and compared them to empirical values. Lastly, using these values, I approximated the exfoliation energy of graphite.