GaInt is abbreviation of Gaussian Intgral, it is a python package for the non-normalizecd molecular integrals over Primitive Cartesian Gaussian orbitals. This is a particularly naive implementation in python: little attempt is made to conserve memory or CPU time. Nevertheless, it is useful for small test calculations, in particular for investigating ideas quantum chemistry.
-
Prerequisties:
- Python 3.5 or above
- numpy 1.13.1
- scipy 0.19.1
-
Compile from source
git clone https://github.com/ZhaoYilin/gaint.git cd gaint python setup.py install --user -
Using pip to install python package on GitHub
pip install git+https://github.com/ZhaoYilin/gaint
The core mechanical quantities of a chemistry system is the Hamiltonian. Hamiltonian operator should include the kinetic energy and potential energy terms of all atomic nuclei and all electrons. It is generally assumed that the molecule is in a vacuum and adiabatic state in isolation. At this time, the interaction potential energy between the nucleus and the electron in the molecule is only related to distance from each other and time independent. Its expression is:
$$ \begin{aligned} \hat{H}{elec}= &-\sum^N{i=1}\frac{\hbar^2}{2m_i}{\nabla}i^2 -\sum^N{i=1}\sum^M_{\alpha=1} \frac{Z_a e^2}{\textbf{r}{ia}}\ &+\sum^N{i=1}\sum^N_{j>i} \frac{e^2}{\textbf{r}_{ij}} \end{aligned} $$
Where
| Name | Operators | Symbol | Shape |
|---|---|---|---|
| Overlap | 1 | S | (nbasis,nbasis) |
| Kinetic | T | (nbasis,nbasis) | |
| Nuclear Attraction | V | (nbasis,nbasis) | |
| Electron Repulsion | Eri | (nbasis,nbasis,nbasis,nbasis) |
Most quantum chemistry package supports Contracted Gaussian type orbtial(CGTO), it is an atomic orbital forming in linear combinations of primitive Gaussians type orbital(PGTO). Here in this note, only the integral over primitive Gaussian type orbital is to be considered which are defiend by an angular part which is homogeneous polynomial in the compoents x, y, and z of the position vector
- a>0 is the orbital exponent.
-
$r_A = r − A$ is the electronic coordinate. - i ≥ 0, j ≥ 0, k ≥ 0 is the quantum number.
- Total angular-momentum quantum number l = i + j + k ≥ 0
from gaint.gauss import PrimitiveGaussian
# Coordinate of H2O molecule
H2O = [[0., 1.43233673, -0.96104039],
[0., -1.43233673, -0.96104039],
[0., 0., 0.24026010]]
# Primitive contraction coefficients
PrimCoeff = np.array([[0.1543289673, 0.5353281423, 0.4446345422],
[0.1543289673, 0.5353281423, 0.4446345422],
[0.1543289673, 0.5353281423, 0.4446345422],
[-0.09996722919, 0.3995128261, 0.7001154689],
[0.155916275, 0.6076837186, 0.3919573931],
[0.155916275, 0.6076837186, 0.3919573931],
[0.155916275, 0.6076837186, 0.3919573931]])
# Orbital exponents
OrbCoeff = np.array([[3.425250914, 0.6239137298, 0.168855404],
[3.425250914, 0.6239137298, 0.168855404],
[130.7093214, 23.80886605, 6.443608313],
[5.033151319, 1.169596125, 0.38038896],
[5.033151319, 1.169596125, 0.38038896],
[5.033151319, 1.169596125, 0.38038896],
[5.033151319, 1.169596125, 0.38038896]])
# H1s, H2s, O1s, O2s, O2px , O2py, O2p
FCenter = [H2O[0], H2O[1], H2O[2], H2O[2], H2O[2], H2O[2], H2O[2]]
CartAng = [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0],
[1, 0, 0], [0, 1, 0], [0, 0, 1]]
pga = PrimitiveGaussian(PrimCoeff[0,0],FCenter[0],CartAng[0],OrbCoeff[0,0])
pgb = PrimitiveGaussian(PrimCoeff[6,0],FCenter[6],CartAng[6],OrbCoeff[6,0])# The non-normalizecd integral between two primitive Gaussian type orbital
from gaint.obara_saikai.overlap import Overlap
S = Overlap()
s17 = S(pga,pgb)
print(np.isclose(s_17,-0.0000888019))
from gaint.obara_saikai.kinetic import Kinetic
T = Kinetic()
t17 = T(pga,pgb)
print(np.isclose(t17,0.00167343))
from gaint.obara_saikai.nuclear_attraction import NuclearAttraction
V = NuclearAttraction()
v17 = V(pga,pgb,FCenter[0])
print(np.isclose(v17,-0.0000854386))
from gaint.obara_saikai.electron_repulsion import ElectronRepulsion
Eri = ElectronRepulsion()
eri1717 = Eri(pga,pgb,pga,pgb)
print(np.isclose(eri1717,1.9060888184873294e-08))# The non-normalizecd integral between two primitive Gaussian type orbital
from gaint.mcmurchie_davidson.overlap import Overlap
S = Overlap()
s17 = S(pga,pgb)
print(np.isclose(s_17,-0.0000888019))
from gaint.mcmurchie_davidson.kinetic import Kinetic
T = Kinetic()
t17 = T(pga,pgb)
print(np.isclose(t17,0.00167343))
from gaint.mcmurchie_davidson.nuclear_attraction import NuclearAttraction
V = NuclearAttraction()
v17 = V(pga,pgb,FCenter[0])
print(np.isclose(v17,-0.0000854386))
from gaint.mcmurchie_davidson.electron_repulsion import ElectronRepulsion
Eri = ElectronRepulsion()
eri1717 = Eri(pga,pgb,pga,pgb)
print(np.isclose(eri1717,1.9060888184873294e-08))
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