The Thomas-Fermi theory of atoms, molecules and solids.(English)Zbl 0938.81568

The Thomas-Fermi (TF) model of the quantum theory of atoms, molecules and solids is put on a firm mathematical footing. A system of $$N$$ electrons moving about a number of fixed positive charges $$Z_i$$, where $$\sum Z_i=Z$$, is considered. If $$\rho(x)$$ denotes the electron charge density at any point $$x$$ and $$V(x)$$ the Coulomb potential due to positive charges at the point $$x$$, the energy $${\mathcal E}(\rho,V)$$ of the system can be written as a functional of $$\rho$$ only. For a suitable choice of units, $${\mathcal E}(\rho,V)=\tfrac 35\int\sigma(x)^{5/3} dx-\int V(x)\rho(x) dx+\tfrac 12\int\int(\rho(x)\rho(y)/|x-y|) dxdy$$. The Euler-Lagrange, equation for minimizing $${\mathcal E}(\rho,V)$$ with the subsidiary conditions $$\int\rho(x) dx=N$$ and $$\rho\geq 0$$ leads to the TF equation. It is shown that the TF equations have a unique solution for $$N\leq Z$$, that these solutions minimize the TF energy function and that no solution exists for $$N>Z$$. The chemical potential is monotone, strictly increasing and concave as a function of $$N$$. The Hamiltonian for the system of $$N$$ electrons and the nuclei gives rise to a quantum-mechanical operator that has the ground-state energy $$E_N{}^Q$$, taking into account the spins of the electrons and the Pauli principle. It is proved that, as the nuclear charges go to infinity, quantum mechanics and TF theory become identical, and the ratio of the ground-state energies tends to unity as $$N\rightarrow\infty$$. For the heavy atoms, far away from the nuclei, the TF density $$\rho(x)$$ is strictly positive for the neutral case and $$\rho(x)\rightarrow 27\pi^{-3}|x|^{-6}$$ as $$|x|\rightarrow\infty$$. For molecules one adds the intermolecular potential to the TF energy, and it is proved that theory does not lead to binding. The TF theory is also extended to solids, i.e., to a system of infinitely large, periodic molecules. The periodic TF equation is obtained and it is shown that $$\rho$$ exists and tends to a finite limit as the volume tends to infinity. The application of TF theory to solids under high pressures is justified. A TF theory of the screening of an impurity in a solid by the electrons in the solid is presented.

MSC:

 81V55 Molecular physics
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