Density Functional Theory Energy Estimator

Calculator Inputs

Choose an input mode, set units, and enable optional energy terms.

Reset to defaults

Input mode

Inactive fields can be left blank.

Electrons (N)

Total electron count in the model region.

Density (n)

Volume (V)

Wigner–Seitz radius (r_s)

Used to derive density when mode is N + r_s.

Sphere radius (R)

Functional preset

Correlation uses a standard LDA fit (unpolarized).

Spin polarization (ζ)

Applies spin-scaling to exchange term only.

External energy term (optional)

Use this to add a known offset term.

Include kinetic Ts (Thomas–Fermi)

Ts term depends strongly on density.

Include correlation Ec

Turns correlation on/off within LDA.

Include Hartree self-energy (approx.)

Uniform sphere model: very approximate.

Clear Results

Example Data Table

These rows show typical inputs and resulting totals for the same model equations.

Case	N	V (bohr³)	n (e/bohr³)	r_s (bohr)	Functional	Total (Ha)	Total (eV)
A	10	500	0.020000	2.285391	LDA_XC	-1.540420	-41.916958
B	50	1200	0.041667	1.789400	LDA_XC	-5.750422	-156.476954
C	10	500	0.020000	2.285391	LDA_X	-1.116109	-30.370866

Formulas Used

This tool uses a uniform-electron-density model to illustrate common DFT energy components in atomic units.

Density from N and V: n = N / V
Wigner–Seitz radius: r_s = (3 / (4πn))^1/3
Fermi wavevector: k_F = (3π²n)^1/3
Fermi energy: ε_F = k_F² / 2
Thomas–Fermi kinetic energy (uniform): T_s = (3/10)(3π²)^2/3 n^5/3 V
LDA exchange per electron (Dirac): ε_x = −(3/4)(3/π)^1/3(1/r_s)
Spin scaling for exchange: f(ζ) = ½[(1+ζ)^4/3 + (1−ζ)^4/3]
Correlation (LDA fit): PZ81 piecewise function of r_s (unpolarized).
Hartree self-energy (rough): uniform sphere E_H ≈ (3/5)N²/R
Total energy: E = T_s + E_x + E_c + E_H + E_ext

Units: 1 Hartree = 27.211386 eV = 2625.500 kJ/mol.

How to Use This Calculator

Select an input mode (for example, N + V).
Enter the required values and choose the matching units.
Pick a functional preset and optional terms (Ts, Ec, EH).
Click Compute to view results above the form.
Use Download CSV or Download PDF after computation.

FAQs

1) Is this a full quantum chemistry DFT solver?

No. It is a uniform-density energy estimator for learning and quick checks, not a geometry-based electronic-structure calculation.

2) What does r_s represent?

r_s is the radius of a sphere that contains one electron on average at density n. It is widely used for electron-gas models.

3) Why do results change a lot with density?

Kinetic energy scales roughly with n^5/3, and exchange scales roughly with 1/r_s. Small density changes can strongly affect totals.

4) What does the spin polarization ζ do here?

ζ modifies the exchange term using a standard spin-scaling factor. Correlation spin-dependence is not included in this simplified model.

5) Should I enable the Hartree self-energy term?

Only for rough intuition. The uniform-sphere Hartree energy is a crude approximation and does not represent real molecular charge distributions.

6) Can I compare two materials or densities?

Yes. Use the same options for both cases, then compare total energy or individual terms. Consistent settings matter more than absolute values.

7) What units should I use for typical chemistry inputs?

If you have values in Å or Å³, select those units. The calculator converts to atomic units internally for the formulas and outputs.

8) Why is an external energy term included?

It lets you add a known constant offset, such as a reference energy, when comparing scenarios. Leave it at zero if unsure.