[Paper Review] Analytical solution of the Thomas-Fermi equation for atoms

M. Oulne|arXiv (Cornell University)|Nov 2, 2005

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper presents a new three-parameter analytical trial function for the Thomas-Fermi equation that significantly improves accuracy in modeling atomic potential and ionization energies. Using the Ritz variational method, the solution achieves a mean error of 0.28% in the range 0 ≤ x ≤ 10 and reproduces the derivative at x=0 with less than 2% error, yielding ionization energy predictions closer to Hartree-Fock results than any prior analytical approximation.

ABSTRACT

An approximate analytical solution of the Thomas-Fermi equation for neutral atoms is obtained, using the Ritz variational method, which reproduces accurately the numerical solution, in the range $0\leq x\leq50$, and its derivative at $x=0$. The proposed solution is used to calculate the total ionization energies of heavy atoms. The obtained results are in good agreement with the Hartree-Fock ones and better than those obtained from previously proposed trial functions by other authors.

Motivation & Objective

• To develop a more accurate analytical solution for the Thomas-Fermi equation that better matches numerical solutions and physical boundary conditions.
• To improve the prediction of total ionization energies for heavy atoms compared to existing trial functions.
• To ensure the solution accurately reproduces the derivative of the potential at the origin (x=0), a critical benchmark for physical consistency.
• To provide a variational trial function that outperforms previous one-, two-, and three-parameter forms in both potential shape and ionization energy accuracy.
• To validate the method using Hartree-Fock data as a reference standard for heavy atoms (Z ≥ 92).

Proposed method

• A new three-parameter trial function is proposed: φ(x) = (1 + α√x + βx e^{-γ√x})² e^{-2α√x}, designed to capture the correct behavior at x=0 and decay at infinity.
• The Ritz variational method is applied to minimize the Lagrangian functional L(φ) = ∫₀^∞ F(φ, φ', x) dx, with F defined as (1/2)(dφ/dx)² + (2/5)(φ^{5/2}/√x).
• The particle number constraint ∫ρ dv = Z is enforced via a Lagrange multiplier during minimization.
• The variational parameters α, β, and γ are optimized numerically using Maple Release 9 to minimize the energy functional while satisfying the normalization condition.
• The resulting solution is tested for accuracy in reproducing the numerical solution of the Thomas-Fermi equation and its derivative at x=0.
• Ionization energies are computed using the derived derivative at x=0 via the formula E = (12/7)(2/9π²)^{1/3} (dφ/dx)_{x=0} Z^{7/3}.

Experimental results

Research questions

• RQ1 Can a new analytical trial function be constructed that more accurately reproduces the numerical solution of the Thomas-Fermi equation across 0 ≤ x ≤ 50?
• RQ2 Does the proposed solution yield a more accurate derivative at x=0 than previous trial functions, especially given its importance for ionization energy calculations?
• RQ3 How does the accuracy of the new solution compare to Hartree-Fock results for total ionization energies of heavy atoms (Z ≥ 92)?
• RQ4 Can the new trial function maintain or improve accuracy as atomic number increases, unlike previous approximations?
• RQ5 Why do existing trial functions fail to reproduce both the solution and its derivative with sufficient precision, and how does the new form overcome these limitations?

Key findings

• The proposed solution achieves a mean error of 0.28% over 47 points in the range 0 ≤ x ≤ 10, significantly lower than the 1.13% error of the two-parameter function (Eq. 2) and over 2.5% for other prior functions.
• The derivative of the solution at x=0 is -1.61623647, with a relative error of less than 2% compared to the numerical value of -1.58807102.
• The ionization energy predictions from the new function (Eq. 21) show errors of only 4.0% at Z=100 and 2.0% at Z=106, improving steadily with increasing Z.
• The new solution outperforms all prior trial functions (Eqs. 1, 2, 3, 4, 5), with Eq. 5 being excluded due to an infinite derivative at x=0.
• At Z=109, the new method predicts 44,403 hartrees, only 0.8% above the Hartree-Fock value of 44,042, demonstrating high precision for superheavy atoms.
• The solution's accuracy in both potential shape and derivative at the origin makes it superior for calculating atomic properties in heavy elements.