Exchange Correlation Energy Calculator

Calculator

Enter density, volume, and optional spin polarization.

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Electron density (n)

Positive number. Interpreted as uniform density.

Density unit

Conversion uses 1 Bohr = 0.529177 Å.

Volume (V)

Choose any representative volume.

Volume unit

The model assumes a uniform region of volume V.

Spin polarization (ζ)

0 = unpolarized, 1 = fully polarized.

Output energy unit

Conversions are applied after computing in Hartree.

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Example data table

Case	n (e/Å³)	V (Å³)	ζ	What you will observe
A	0.02	1000	0.0	Lower density gives smaller magnitude energies.
B	0.05	1000	0.0	Exchange becomes more negative as n increases.
C	0.05	1000	0.7	Exchange magnitude rises due to spin scaling.

These examples assume a uniform region; real DFT uses spatially varying density n(r).

Formula used

1) Exchange energy (Dirac LDA)

C_x = 3/4 · (3/π)^1/3
E_x = −C_x · V · n^4/3 · f(ζ)
f(ζ) = [ (1+ζ)^4/3 + (1−ζ)^4/3 ] / 2

This is the uniform electron gas exchange contribution in atomic units.

2) Correlation energy (PZ81, unpolarized)

r_s = (3 / (4πn))^1/3
E_c = N · ε_c(r_s) , N = nV

ε_c(r_s) is computed with a common piecewise fit to QMC data.

Important: This calculator uses a uniform-density approximation. For real molecules/solids, E_xc is computed from n(r) over space, often with gradient corrections.

How to use this calculator

Enter a positive electron density and select its unit.
Enter the volume of the uniform region and select its unit.
Optionally set spin polarization ζ between 0 and 1.
Pick your preferred output unit (Hartree, eV, or kJ/mol).
Click Calculate to show results above the form.
Use Download CSV or Download PDF for exports.

Density inputs and unit handling

Electron density n drives every term in this calculator. Typical solid-state valence densities fall near 0.01–0.10 e/Å³, while compressed matter can exceed 0.2 e/Å³. The page converts n into atomic units (e/Bohr³) using 1 Bohr = 0.529177 Å, ensuring consistent DFT-style arithmetic.

Uniform volume and electron count

The model treats your region as uniformly filled, so the electron count is N = nV. For example, n = 0.05 e/Å³ and V = 1000 Å³ gives N ≈ 50 electrons. Keeping V fixed helps you see how exchange–correlation changes with density alone.

Exchange energy trend and spin effect

Exchange uses the Dirac LDA form Ex = −Cx V n^(4/3) f(ζ), where Cx = 3/4(3/π)^(1/3). Because of the 4/3 power, doubling density increases |Ex| by about 2^(4/3) ≈ 2.52. Spin polarization raises |Ex| through f(ζ); at ζ = 0.7, f(ζ) is noticeably larger than 1.

Correlation energy through rs

Correlation is parameterized by the Wigner–Seitz radius rs = (3/4πn)^(1/3). Larger n means smaller rs and typically more negative correlation per electron. In many materials, rs often sits around 1–5 Bohr; the calculator reports rs so you can judge whether you are in a high- or low-density regime.

Total Exc and normalized outputs

Total Exc = Ex + Ec is reported in Hartree, eV, or kJ/mol. The tool also reports Exc per electron, useful for comparing cases with different V. For reference, 1 Hartree ≈ 27.2114 eV and 1 eV ≈ 96.4853 kJ/mol, so energy-unit changes preserve trends while scaling magnitudes.

Graph-driven sensitivity checks

The Plotly chart sweeps density around your input and plots Ex, Ec, and Exc on the same axes. Use it to identify nonlinear regions: exchange curves steepen with density, while correlation varies more gently with rs. If curves look flat, increase the sweep span or check that n and V are not extremely small.

After calculation, the CSV export writes a tidy key–value summary that can be pasted into notebooks or lab logs. The PDF export produces a one-page snapshot for sharing results. When documenting runs, store the input unit choices, ζ, and rs alongside Exc, because changing units or polarization can otherwise hide meaningful differences in the underlying electronic regime clearly securely.

FAQs

1) What does exchange–correlation energy represent here?

It approximates the many-electron effects beyond classical Coulomb energy using a uniform electron gas model, reported as Ex, Ec, and their sum Exc for your chosen density, volume, and ζ.

2) Why does exchange become more negative with density?

In LDA, Ex scales as n^(4/3). Increasing density strengthens the exchange hole and lowers energy, so the magnitude grows faster than linear when density rises.

3) What is rs and why is it shown?

rs is the Wigner–Seitz radius in Bohr, derived from density. It is a standard density parameter for uniform electron gas fits and indicates whether you are in a high- or low-density regime.

4) How should I choose the volume input?

Use any representative region size. Since N = nV, changing volume mainly scales total energy. For comparisons across densities, keep V constant and focus on Exc per electron.

5) Does ζ affect correlation in this version?

This build applies spin scaling to exchange and uses an unpolarized correlation fit for simplicity. It is useful for learning trends, but spin-dependent correlation requires additional parameterization.

6) Is this suitable for publishing-grade DFT results?

No. It is an educational LDA-style calculator for uniform densities. Production work integrates n(r) on a grid and uses well-tested functionals and numerical settings within established electronic-structure codes.