Gaussian Basis Functions

OpenQuantum uses Cartesian Gaussian-type orbitals (GTOs):

$χ (r) = N x^{l_{x}} y^{l_{y}} z^{l_{z}} exp (- α r^{2})$

where:

$N$ is the normalization constant
$l_{x} + l_{y} + l_{z} = L$ is the angular momentum
$α$ is the Gaussian exponent

Contracted Gaussians

Contracted Gaussians are linear combinations of primitives:

$χ_{contracted} = i \sum c_{i} χ_{i} (α_{i})$

The contraction coefficients $c_{i}$ and exponents $α_{i}$ are read from basis set definition files (.obs format, compatible with EMSL BSE).

Normalization

Primitives are normalized so that:

$\int ∣ χ (r) ∣^{2} d r = 1$

The normalization constant for a Cartesian GTO is:

$N = (\frac{2 α}{π})^{3/4} [\frac{( 2 α ) ^{L} ( 2 l _{x} - 1 )!! ( 2 l _{y} - 1 )!! ( 2 l _{z} - 1 )!!}{( 2 L + 1 )!!}]^{1/2}$

Pure Spherical Harmonics

By default, OpenQuantum transforms Cartesian basis functions to pure spherical harmonics (5d, 7f, etc.). This reduces the number of basis functions for higher angular momentum:

Shell	Cartesian	Pure	Reduction
d	6	5	1
f	10	7	3
g	15	9	6

Use 5d/7f (default) for pure harmonics, 6d/10f for Cartesian.

Embedded Basis Sets

OpenQuantum ships with over 100 basis sets via embedded .obs files:

Family	Examples	Coverage
Pople	STO-3G, 3-21G, 6-31G, 6-31G, 6-31G, 6-31+G, 6-311G	H–Xe
Dunning	cc-pVDZ, cc-pVTZ, cc-pVQZ, aug-cc-pVDZ, aug-cc-pVTZ	H–Kr
Ahlrichs	def2-SVP, def2-TZVP, def2-TZVPP, def2-QZVP	H–Rn
Other	D95, ANO-SEG, UGBS, CBSB7	varies

See Basis Sets Reference for the complete list and usage.