Variational solution of the Schrödinger equation using plane waves in adaptive coordinates: The radial case

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• The Schrödinger equation is fundamental to quantum mechanics.
• Accurate solutions are crucial for understanding atomic and molecular properties.
• Existing methods may face challenges with complex systems or desired accuracy levels.

Purpose of the Study:

• To introduce a new, efficient, and accurate method for solving the Schrödinger equation.
• To demonstrate the method's applicability to atomic systems.
• To achieve high-precision energy calculations for atomic orbitals.

Main Methods:

• A coordinate transformation from Cartesian (r) to a new system (u) is employed.
• The wave function psi(r) is expressed using the Jacobian determinant and a function U(u).
• U(u) is expanded as a linear combination of plane waves, with coefficients optimized variationally.

Main Results:

• The method was tested on the radial Schrödinger equation for the hydrogen atom.
• Micro-Hartree accuracy or better was achieved for the energies of ns and np orbitals (n up to 5).
• Moderate-length expansions were sufficient to obtain high accuracy.

Conclusions:

• The proposed coordinate transformation method offers a viable and accurate approach to solving the Schrödinger equation.
• The technique shows promise for high-precision quantum mechanical calculations.
• Generalization of algorithms for Coulomb potentials facilitates this new method.