Convergence study of a Schrödinger-equation algorithm and structure-factor determination from the wavefunction

Area of Science:

• Crystallography
• Quantum Mechanics
• Computational Chemistry

Background:

• Solving the Schrödinger equation is crucial in quantum mechanics.
• Crystallography requires accurate determination of wavefunctions for structure analysis.
• Previous algorithms for crystallographic Schrödinger equation solutions needed refinement.

Purpose of the Study:

• To upgrade and test an existing algorithm for solving the Schrödinger equation within a crystallographic context.
• To assess the convergence properties and uniqueness of the determined wavefunction.
• To evaluate the accuracy of the algorithm in reproducing crystallographic structure factors.

Main Methods:

• An iterative, reciprocal-space algorithm was employed.
• The algorithm determines a wavefunction Phi(H) that satisfies the Schrödinger equation and minimizes differences between calculated and observed structure factors.
• Calculations were performed on three molecules of varying complexity (11, 41, and 110 non-H atoms).

Main Results:

• The upgraded algorithm demonstrated fast convergence towards a unique wavefunction, independent of initial phase values.
• The method showed high accuracy in determining wavefunctions, with mean phase errors of 3, 26, and 28 degrees for the tested structures.
• The combined conditions of fulfilling the Schrödinger equation and least-squares minimization proved critical for larger structures.

Conclusions:

• The upgraded algorithm is a reliable tool for determining wavefunctions in crystallography.
• The convergence and uniqueness of the wavefunction are established, paving the way for practical applications.
• The algorithm's accuracy is sufficient for precise crystallographic structure determination.