API

plot_density(c₀, basis_fns; range=(-4.0, 4.0), N=200)

Plots the 2D probability density |ψ₀(x, y)|² in Jacobi space.

plot_wavefunction(c₀, basis_fns; axis=1, range=(-4.0, 4.0), N=400)

Plots a 1D slice of the wavefunction ψ₀(x, 0) or ψ₀(0, y).

ψ₀(r::Vector{Float64}, c₀::Vector{Float64}, basis_fns::Vector{<:GaussianBase})

Evaluates the ground state wavefunction ψ₀ at position r.

computegroundstate_energy(basis::BasisSet, ops::Vector{Operator})

Construct the Hamiltonian and overlap matrices and return the lowest eigenvalue.

corput(n::Int, b::Int=2)

Computes the n-th number in the van der Corput sequence in base b using digit reversal. Follows the standard definition: qₙ = ∑ dₖ(n) * b^(-k-1)

generate_basis(widths::Vector{Matrix{Float64}}, rank::Int=0)

Construct a BasisSet from a list of correlation matrices and optional rank.

generate_bij(method::Symbol, i::Int, n_terms::Int, b1::Float64) -> Vector{Float64}

Generate a bij vector using the specified sampling method. Supported methods: :quasirandom, :psudorandom

halton(n::Int, d::Int)

Generates the nth Halton sequence vector in d dimensions.

struct ParticleSystem

A structure representing a system of particles with associated masses and coordinate transformations.

Fields

• masses::Vector{Float64}: A vector containing the masses of the particles in the system. Must contain at least two elements.
• J::Matrix{Float64}: The Jacobi transformation matrix for the particle system, computed based on the masses.
• U::Matrix{Float64}: An auxiliary transformation matrix for the particle system, computed based on the masses.
• scale::Union{Symbol,Nothing}: An optional symbol indicating the scale of the system (e.g., :atomic, :molecular, :nuclear). Defaults to nothing.

Constructor

• ParticleSystem(masses::Vector{Float64}; scale::Union{Symbol,Nothing}=nothing): Creates a new ParticleSystem instance. The masses vector must contain at least two elements. The scale parameter is optional and can be used to specify the scale of the system. The Jacobi and auxiliary transformation matrices (J and U) are computed internally using the jacobi_transform function.

default_b0(scale::Union{Symbol,Nothing}) -> Float64

Returns a default value for the parameter b0 based on the provided scale.

Arguments

• scale::Union{Symbol,Nothing}: A symbol representing the scale type or nothing.
  • :atomic: Returns 1.0, corresponding to the Bohr radius in atomic units.
  • :molecular: Returns 3.0, representing a typical molecular bond length.
  • :nuclear: Returns 0.03, approximately 1 femtometer in atomic units.
  • nothing: Returns 1.0 as a fallback default.

jacobi_transform(masses::Vector{Float64})::Tuple{Matrix{Float64}, Matrix{Float64}}

Computes the Jacobi transformation matrix J and its pseudoinverse U for a given vector of masses.

Arguments

• masses::Vector{Float64}: A vector of masses for the system. Must contain at least two masses.

Returns

• Tuple{Matrix{Float64}, Matrix{Float64}}: A tuple containing:
  • J::Matrix{Float64}: The Jacobi transformation matrix.
  • U::Matrix{Float64}: The pseudoinverse of the Jacobi transformation matrix.

Details

The Jacobi transformation is used to convert the coordinates of a system of particles into a set of relative coordinates. The transformation matrix J is constructed based on the masses of the particles, and its pseudoinverse U is computed using the pinv function.

Constraints

• The input vector masses must have a length of at least 2. An assertion is raised if this condition is not met.

shift_vectors(a::Matrix{Float64}, b::Matrix{Float64}, mat::Union{Nothing, Matrix{Float64}}=nothing) -> Float64

Compute a weighted sum of dot products between columns of two matrices a and b, optionally using a weighting matrix mat.

Arguments

• a::Matrix{Float64}: A matrix where each column represents a vector.
• b::Matrix{Float64}: A matrix where each column represents a vector. Must have the same number of columns as a.
• mat::Union{Nothing, Matrix{Float64}}: An optional square weighting matrix. If nothing is provided, the identity matrix is used.

Returns

• Float64: The computed weighted sum of dot products.

Constraints

• The number of columns in a and b must be the same.
• If mat is provided, it must be a square matrix with dimensions equal to the number of columns in a and b.

transform_list(α::Vector{Float64})::Vector{Matrix{Float64}}

Transforms a vector of Float64 values into a vector of Matrix{Float64} objects. Each element of the input vector α is wrapped into a 1x1 matrix and returned as an element of the resulting vector.

Arguments

• α::Vector{Float64}: A vector of Float64 values to be transformed.

Returns

• Vector{Matrix{Float64}}: A vector where each element is a 1x1 matrix containing the corresponding value from the input vector α.

optimizegroundstateenergy(initwidths::Vector{Matrix{Float64}}, ops::Vector{Operator}; max_iter=100)

Uses local optimization (Nelder-Mead) on Gaussian widths to minimize ground state energy.