SOLUTIONS OF THE THOMAS–FERMI–DIRAC EQUATION

A large computational work*) has been carried out, using an electronic counting machine, to solve the Thomas–Fermi–Dirac equation. The equation considered is

\[ \frac{\partial^{2}\psi}{\partial x^{2}} = x\left(\varepsilon+\sqrt{\frac{\psi}{x}}\right)^{3}, \]

where

\[ \varepsilon=\left(\frac{3}{32\pi^{3}}\right)^{\frac13} Z^{-\frac23} =0.211783\,Z^{-\frac23},\qquad x=\frac{r}{\mu}, \]

\[ r\text{ is the coordinate along the radius},\qquad \mu=0.88534\,a_{0}Z^{-\frac13},\qquad a_{0}\text{ is the Bohr radius}. \]

For \(x\) close to zero, the function \(\psi\) can be represented by the series

\[ \psi=1+a_{2}x+a_{3}x^{\frac32}+a_{4}x^{2}+\cdots . \]

The coefficients \(a_k\) have been computed up to \(k=11\).

Only the coefficient \(a_2\) (the initial slope of \(\psi(x)\)) was specified to the calculating machine. The computation was carried out over equal intervals \(\Delta w=0.04\) of the variable \(w=\sqrt{2x}\). Up to \(w=0.48\) the calculation was performed by the series-expansion formula; for larger values the machine performed numerical integration.

The equation was solved for atoms of 24 elements (\(Z=6, 10, 14, 16, 18, 22, 26, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 84, 88,\) and \(92\)).

For each element, 6–10 different initial slopes were tried. Thus the article presents about 200 solutions of the Thomas–Fermi–Dirac equation.

The condition for a neutral atom is that the electric field vanish at the boundary. This corresponds to the requirement

\[ \left(\frac{d\psi}{dx}\right)_{x=x_0} = \left(\frac{\psi}{x}\right)_{x=x_0}, \]

which, depending on the initial slope, is satisfied at different \(x_0\). Thus the solutions obtained correspond to atoms of different sizes.

If the slope chosen is too large, then \(\psi\) may become zero before the condition written above is satisfied. In this case the solution must be “cut off” at that value of the argument \(x_1\) at which \(\psi\) becomes zero. Such solutions represent positive ions; the charge of the ions can be computed.

The tables given in the work are arranged as follows: for each \(a_2\), a table of values of \(\psi(w)\) is given at intervals of 0.01 unit; if the solution corresponds to a neutral atom, its size is given; if the solution corresponds to an ion, its size and charge are given.

All intermediate values not appearing in the tables can be obtained with sufficient accuracy by quadratic interpolation.

A. K.

) N. Metropolis and J. R. Reitz, J. Chem. Phys. 19*, 555 (1951).