Reference: https://content.wolfram.com/sites/19/2012/02/Ho.pdf
I reproduced most of the computations in the MATHMATICA reference. The water molecule is a planar molecule that lies in the YZ-plane.
The optimization methods that were programmed are the Simplex Algorithm, the Steepest Descent method, and Newton’s Method. The Simplex Algorithm was given to me by Microsoft Copilot. The other two methods are from Elementary Numerical Analysis: An Algorithmic Approach 3rd Edition (c) 1980 by S.D. Conte and Carl de Boor.
Source code for the solutions to Exercise 1.18 and 1.19.
I modified my translation of a FORTRAN program mentioned in a couple of my recent blog entries. The hybrid C/C++ source code is 1,291 lines. I find the basis set of Gaussian Type 1s Orbitals (GTO-NG) using my evolutionary hill-climber, where the GTO1s-NGs curve fit a Slater Type 1s Orbital (STO1s-NG), where N = 4 and 5 in the cation case and N = 4 in the molecule calculation. The percent errors in both cases are considerably less than 1%.
Using my STO-4G curve fit for N = 4 basis Gaussian type 1s orbitals I was able to get better results than found using the N = 3 basis wavefunctions in the graduate-level textbook Modern Quantum Chemistry Introduction to Advanced Electronic Structure Theory by Attila Szabo and Neil S. Ostlund. My recreation for N = 3 discovered -2.97867 atomic units ground state energy for the helium-hydrogen ion and -1.11651 atomic units for the hydrogen molecule using the textbook’s basis wavefunctions. The percentage errors were 3.98002% and 4.15928% respectively. My STO-4G basis wavefunctions found a ground state energy for the helium-hydrogen ion of -2.94937 atomic units and for the hydrogen molecule -1.14344 atomic units, which have percentage errors of 0.98349% and 2.86607% respectively.
Numerical Explorations 