/StationarySE

Julia language library for solving stationary schrodinger equation with Variation and Hartree Fock methods

Primary LanguageJuliaMIT LicenseMIT

StationarySE

Usage

The Library can be tested by running the following command:

julia           # activate the julia enviroment
]               # activate julia Pkg enviroment
activate .      # activate this project
instantiate     # install and precompile the dependencies for this project
test            # running the build-in test of this project

Running on multiple cores

Use this command in the shell to set the number of processors used in the calculation:

export  JULIA_NUM_THREADS=${THE_NUMBER_OF_THE_CORES}

It should be set before activating the Julia REPL environment.

Solver API

Single electron

1. eigen_solve(grid_size, potential, cell_length, hbar, mass; sparse)

Solves the eigenvalue problem for a given Hamiltonian matrix. The space will be devided into sparse grid.

Parameters:

  • grid_size::Int: The size of the grid for the problem.
  • potential::Array: The potential energy array.
  • cell_length::AbstractFloat: The length of each cell in the grid.
  • hbar::AbstractFloat: The reduced Planck's constant.
  • mass::AbstractFloat: The mass of the particle.
  • sparse::Bool (optional, default true): A flag indicating whether to use sparse matrix representation. (it will save memory, especially when thge grid_size is large)

Returns:

  • evals: The eigenvalues of the Hamiltonian matrix.
  • evecs: The corresponding eigenvectors of the Hamiltonian matrix.

2. variation_solve(basis_func, num_orbitals, start_bound, end_bound; potential, symmetric, dimension)

Find the ground state energy using the variational method.

Parameters:

  • basis_func::Function: The basis function for the problem.
  • num_orbitals::Int: The number of orbitals in the system.
  • start_bound::Vector: The starting boundary for the problem.
  • end_bound::Vector: The ending boundary for the problem.
  • potential::Function (optional, default (_) -> 0): The potential energy function.
  • symmetric::Symbol (optional, default :Nothing): A flag indicating whether the problem is symmetric.
  • dimension::Int (optional, default 1): The dimension of the problem.

Returns:

  • evals: The eigenvalues of the Hamiltonian matrix.
  • evecs: The corresponding eigenvectors of the Hamiltonian matrix.

Multiple electrons

3. hartree_fock_solve(nuclear_z, basis_func, num_orbitals; symmetric, dimension, iter_steps, torrlence, start_bound, end_bound, potential_func, mode, converge_alpha)

Solves the Hartree-Fock equations for a given system.

Parameters:

  • nuclear_z::Int: The atomic number of the nucleus.
  • basis_func::Function: The basis function for the problem.
  • num_orbitals::Int: The number of orbitals in the system.
  • symmetric::Symbol (optional, default :Spherical): The symmetry of the system.
  • dimension::Int (optional, default 3): The dimension of the problem.
  • iter_steps::Int (optional, default 50): The number of iteration steps.
  • torrlence::Float64 (optional, default 1e-2): The tolerance for convergence.
  • start_bound::Vector (optional, default [0.0]): The starting boundary for the problem.
  • end_bound::Vector (optional, default [4.0]): The ending boundary for the problem.
  • potential_func::Function (optional, default (r) -> -nuclear_z / r): The potential energy function.
  • mode::Symbol (optional, default :RHF): The mode of the calculation (only RHF, that is, restricted Hartree–Fock, is implemented).
  • converge_alpha::Float64 (optional, default 0.6): The convergence factor asigning the weight of the density matrix in last step and last-last step

Returns:

  • ground_state_energy: The energy of the ground state.

Example (Test)

Build-in tests are on the test/runtest.jl, they are:

println("")
println("=== Testing H atom ===")
potential_func = (r) -> -1/r
evals, _ = variation_solve(gaussian_hydrogen, 4, [0.0], [100.0]; potential=potential_func, symmetric=:Spherical, dimension=3)
ground_state_energy = evals[1]


println("")
println("=== Testing He atom ===")
ground_state_energy = hartree_fock_solve(2, gaussian_helium, 2; end_bound=[8.0])

println("")
println("=== Testing C atom ===")
ground_state_energy = hartree_fock_solve(6, gaussian_carbon, 6; end_bound=[8.0])

println("")
println("=== Testing O atom ===")
ground_state_energy_1 = hartree_fock_solve(8, gaussian_oxygen_constrain, 8, end_bound=[8.0])

The results of these atom should be:

-0.4992783898562091 hatree for H atom
-2.852043619183923 hatree for He atom
-38.39474787071248 hatree for C atom
-72.29553139062395 hatree for O atom

On progress

  1. Implement the open-shell system for Hartree-fock Method
  2. Adapt the Hartree-fock Method in moleculars using other symmetric option instead of Spherical