Methods Module
This module provides methods for solving integrals and other calculations.
Main.SchrödingerSolver.Methods.build_matrices — Methodbuild_matrices(N, v, s, potential_gaussian_integral, potential_params)Builds the Hamiltonian matrix H and overlap matrix S for the variational method.
Arguments
N: Number of basis functions.v: Width parameters of the Gaussian basis functions.s: Centers of the Gaussian basis functions.potential_gaussian_integral: Function to compute the potential energy integral.potential_params: Additional parameters for the potential function.
Returns
(H, S): The Hamiltonian and overlap matrices.
Main.SchrödingerSolver.Methods.kinetic_integral — Methodkinetic_integral(v1, s1, v2, s2)Computes the kinetic energy integral between two Gaussian basis functions.
The kinetic energy integral is given by:
\[T_{ij} = \frac{v_1^{3/2} v_2^{3/2} \left( v_1 + v_2 - 2 v_1 v_2 (s_1 - s_2)^2 \right)}{\sqrt{\pi} (v_1 + v_2)^{5/2}} e^{- \frac{v_1 v_2 (s_1 - s_2)^2}{v_1 + v_2}}\]
Arguments
v1: Width parameter of the first Gaussian basis function.s1: Center of the first Gaussian basis function.v2: Width parameter of the second Gaussian basis function.s2: Center of the second Gaussian basis function.
Returns
T_ij: The kinetic energy integral value.
Main.SchrödingerSolver.Methods.overlap_integral — Methodoverlap_integral(v1, s1, v2, s2)Computes the overlap integral between two Gaussian basis functions with parameters (v1, s1) and (v2, s2).
The overlap integral is given by:
\[S_{ij} = \frac{\sqrt{v_1 v_2}}{\sqrt{\pi (v_1 + v_2)}} e^{- \frac{v_1 v_2 (s_1 - s_2)^2}{v_1 + v_2}}\]
Arguments
v1: Width parameter of the first Gaussian basis function.s1: Center of the first Gaussian basis function.v2: Width parameter of the second Gaussian basis function.s2: Center of the second Gaussian basis function.
Returns
S_ij: The overlap integral value.
Main.SchrödingerSolver.Methods.potential_integral_xn — Methodpotential_integral_xn(v1, s1, v2, s2, n)Computes the potential energy integral for ( V(x) = x^n ) between two Gaussian basis functions.
Arguments
v1,s1: Parameters of the first Gaussian basis function.v2,s2: Parameters of the second Gaussian basis function.n: The power of x in the potential function (integer from 0 to 4).
Returns
V_ij: The potential energy integral value.
Main.SchrödingerSolver.Methods.solve_schrodinger — Methodsolve_schrodinger(H, S, num_levels)Solves the generalized eigenvalue problem for the Hamiltonian H and overlap matrix S.
Arguments
H: Hamiltonian matrix.S: Overlap matrix.num_levels: Number of energy levels to compute.
Returns
(energies, states): The lowestnum_levelseigenvalues and eigenvectors.