H₂⁺ Molecule Overlap Integral, Coulomb Integral and Exchange Integral Solver.
The easiest possible molecule system is H₂⁺ ion consisting 2 protons and 1 electron. It can be investigated with Linear Combination of Atomic Orbitals (LCAO) method where the wave functions for the cases that the electron is bound to the nucleus A or to the nucleus B
are linearly combined. They are defined as:
,
where and
are relative distances of electron to respective proton;
Schematic of the ion[2].
Normalization of the systems wavefunction creates an overlap integral. Overlap Integral S is defined as:
Where is the triple integral in spherical coordinates;
Even though the limit of r is infinite, values bigger than 100 generally breaks the code. But this upper limit seems to be producing consistent results.
Exchange or Resonance integral K is defined as;
Coulomb integral J is defined as;
V(r) is Coulomb potential;
Bonding and anti-bonding molecular orbitals
are defined as;
Integration is done with scipy-integrate. Resulting graph;
