qml/integrals

---
license: mit
---

# 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](https://github.com/FermiQC/GaussianBasis.jl).

## Loading data from python

See [qml/data/integrals.py](https://github.com/aklipf/qml).
Loading a mono-electronic integral dataset should be as simple as:

```py
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](https://huggingface.co/datasets/qml/integrals/tree/main/mono_20k)
and [mono_100k](https://huggingface.co/datasets/qml/integrals/tree/main/mono_100k)
for 2-electron integrals.

Each HDF5 file encodes an object of type:

```julia
# 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:

```julia
"""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](https://huggingface.co/datasets/qml/integrals/tree/main/bi_200)

原始信息汇总

Quantum Electronic Integrals 数据集概述

数据集内容

单电子积分（Mono-Electronic Integrals）

• 数据结构：每个HDF5文件包含一个MonoIntegral类型的对象，该对象包含以下字段：
  • l：角动量向量
  • exp：指数
  • xyz：位置坐标
  • overlap：重叠积分
  • kinetic：动能积分
  • nuclear：核吸引积分
  • Z：原子电荷
• 输入波函数：原始的球形GTO壳层，系数为单位系数。
• 目标积分：
  • 重叠积分 S₁₂ = ∫ ψ1 ⋅ ψ2
  • 动能积分 T₁₂ = 1/2 * ∫ ∇ψ1 ⋅ ∇ψ2
  • 核吸引积分 N₁₂ = ∫ ψ1 ⋅ [(Z₁ / |r|) + (Z₂ / |r - xyz|)] ⋅ ψ2
• 矩阵形状：D × D，其中 D = (2 * l1 + 1) + (2 * l2 + 1)

双电子积分（Bi-Electronic Integrals）

• 数据结构：BiIntegral4c类型的对象，包含以下字段：
  • l：角动量向量
  • exp：指数
  • xyz：位置坐标
  • ijkl：整数数组
  • Bijkl：浮点数数组
  • index：索引向量
• 存储格式：稀疏格式

数据集使用

单电子积分加载示例（Python）

py from qml.data import MonoIntegral

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

双电子积分参考

• 数据集 bi_200