Calculate an ionized hydrogen molecule

In this tutorial, we will calculate the properties of the dihydrogen cation ($ H_2^+ $). It is the simplest molecular ion, as it consists of two hydrogen nuclei and one electron. The simplicity makes it an ideal model system and is the reason why it has been extensively studied.

Exploring 3D Calculations

To analyze the properties of $ H_2^+ $, we must solve a 3D quantum mechanical problem. Let’s begin with a simple example calculation in 3D to understand how the computational library handles this scenario and to observe the differences in approach compared to simpler systems.

In [1]:
import sigspace as sg
import numpy as np

H = sg.Hamiltonian(sdims=3) + (lambda x: -2 / (1 + np.sum(x ** 2, axis=1)) * x[:,0]**2)

solver = sg.Solver(H)

solver.grid((-10, 10, 20))
solver.print_info = True

sol = solver.run()
psi = sol.state

sg.quickplot(sol)
Iter: 310, energy -1.544, residual 0.009867, duration 0.222112s