pyscf-vorticity 0.1.0

Λ³-DFT: Vorticity-based exchange-correlation analysis

pyscf-vorticity

Geometric correlation analysis for quantum chemistry

What it does

Computes an effective correlation dimension γ from first-principles two-particle reduced density matrices, helping you select the optimal DFT functional for your system.

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Quick Start

from pyscf import gto, scf, fci
from pyscf_vorticity import compute_vorticity

# Your molecule
mol = gto.M(atom='...', basis='cc-pvdz')
mf = scf.RHF(mol).run()
cisolver = fci.FCI(mf)
E_fci, fcivec = cisolver.kernel()

# Compute vorticity
rdm2 = cisolver.make_rdm12(fcivec, mol.nao, nelec)[1]
V, k = compute_vorticity(rdm2, mol.nao)
alpha = abs(E_fci - mf.e_tot) / V

# → Use γ to guide functional selection

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HOW TO: Choosing Your Functional

The following table provides practical guidelines based on physical interpretation of γ. These are recommendations, not absolute rules.

Your γ	Optimal a	Recommended Functional
γ ≈ 0	a ≈ 1.0	HF or high exact exchange hybrid
γ ≈ 1	a ≈ 0.5	M06-2X, BH&HLYP
γ ≈ 2	a ≈ 0.33	PBE0, B3LYP
γ ≈ 3-4	a ≈ 0.25	B3LYP
γ > 5	a < 0.15	TPSSh, pure GGA

Guideline formula: a ≈ 1 / (1 + γ)

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WHY: The Physics Behind It

The Problem

Standard DFT functionals contain empirical "magic numbers" - like B3LYP's 20% exact exchange - that work on average but fail systematically for:

• Strong correlation (Mott insulators, transition metals)
• Static correlation (bond breaking, diradicals)
• Weak correlation (metals)

The Root Cause

Most functionals implicitly assume a fixed correlation structure.

Reality: correlation strength varies across systems.

The Solution

The exchange-correlation energy scales with a vorticity measure V, allowing extraction of the correlation exponent γ through system-size dependence.

γ characterizes how correlated your system is:

γ value	Correlation type	Electron behavior
γ → 0	Strong (Mott)	Localized, atomic-like
γ ≈ 1	Intermediate	Partially localized
γ ≈ 2	Moderate	Molecular bonding
γ > 3	Weak (metallic)	Fully delocalized

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HOW TO: Strong Correlation Systems

When γ → 0, standard DFT typically fails.

What to do:

1. Use high exact exchange

   mf = dft.RKS(mol)
   mf.xc = 'HF*0.5 + PBE*0.5'  # 50% exact exchange

2. Consider range-separated hybrids

   mf.xc = 'camb3lyp'

3. Or go beyond DFT (CASSCF, DMFT, CC)

WHY this works:

Strong correlation (γ → 0) indicates localized electrons. Localized electrons require exact exchange to avoid self-interaction error. GGA/LDA assume delocalized electrons → systematic failure.

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HOW TO: Weak Correlation (Metals)

When γ > 3:

mf.xc = 'PBE'   # Pure GGA
mf.xc = 'TPSS'  # Meta-GGA

WHY:

Delocalized electrons screen effectively. Exact exchange over-localizes → wrong band structure.

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Validated Examples

H₂ Dissociation

R = 0.74Å: α = 0.062 (equilibrium)
R = 1.50Å: α = 0.039 (minimum)
R = 5.00Å: α = 0.097 (dissociated)

U-shaped α(R) captures weak → strong correlation transition.

Atoms

He (2e):  α = 0.090
Be (4e):  α = 0.003
Ne (10e): α = 0.004

The 30× drop from 2e to 4+e systems ("α-cliff") reflects the emergence of many-body correlation structure.

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Installation

pip install git+https://github.com/miosync-masa/pyscf-vorticity.git

GPU acceleration (recommended):

pip install jax jaxlib

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Citation

@article{iizumi2025geometric,
  title={Geometric Origin of Exchange-Correlation in Density Functional Theory},
  author={Iizumi, Masamichi},
  journal={arXiv preprint},
  year={2025}
}

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License

MIT