Quantum Electronic Integrals

This dataset contains quantum interaction integrals between randomly sampled pairs/quadruples of Gaussian-Type Orbitals (GTOs).
The targets were computed in julia using GaussianBasis.jl.

Loading data from python

See qml/data/integrals.py. Loading a mono-electronic integral dataset should be as simple as:

from qml.data import MonoIntegral

I_2_1 = MonoIntegral.h5read("integrals/mono_20k/mono_2_1.h5")

The MonoIntegral class inherits its h5read method from the TensorDict mixin.

Each dataset contains its corresponding TensorDict dataclass, reading data from any compatible HDF5 storage (containing enough keys).

Mono-Electronic Integrals

See mono_20k and mono_100k for 2-electron integrals.

Each HDF5 file encodes an object of type:

# jqml/Data.jl
""" Object storing 1-electron integrals. """
struct MonoIntegral{T} <: ArrayFields
    l :: Vector{Int64}
    exp :: Union{SArray, Array{T}}
    xyz :: Union{SArray, Array{T}}
    overlap :: Array{T}
    kinetic :: Array{T}
    nuclear :: Array{T}
    Z :: Array{Int64}
end

Input wave functions (ψ1, ψ2) are primitive, spherical GTO-shells with unit coefficients, i.e.

ψ(C + r) = rˡ ⋅ Yₗₘ(r/|r|) ⋅ exp(-α |r|²)

where C is ψ.center, α is ψ.exp, and the magnetic quantum number m takes all possible values in {-l, ..., l} within each subshell.

Inputs:

• xyz : center of ψ2 (ψ1 is centered at 0)
• l : pair of angular momenta (l₁, l₂)
• exp : exponents (α₁, α₂)
• Z : atomic charges used to compute the nuclear integral.

Targets:

• overlap integrals S₁₂ = ∫ ψ1 ⋅ ψ2

• kinetic integrals T₁₂ = 1/2 * ∫ ∇ψ1 ⋅ ∇ψ2

• nuclear attraction integrals

  N₁₂ = ∫ ψ1 ⋅ [(Z₁ / |r|) + (Z₂ / |r - xyz|)] ⋅ ψ2

Note:

Mono-electronic integrals are square matrices of shape D × D with

D = (2 * l1 + 1) + (2 * l2 + 1)

Indices correspond to increasing values of m1 ∈ {-l1, …, l1} first,
then increasing values of m2 ∈ {-l2, …, l2}.

Bi-Electronic Integrals

Batches of 2-electron integrals are returned in the following sparse format:

"""Object for storing bi-electronic integrals"""
struct BiIntegral4c{T}  <: ArrayFields
    l :: Vector{Int64}
    exp :: Array{T}
    xyz :: Array{T}
    ijkl :: Array{Int16}
    Bijkl :: Array{Float64}
    index :: Vector{Int64}
end

The index field has the same length as ijkl and Bijkl, and maps each integral element to the index of the corresponding input GTOs.

See bi_200