Partition Sum Calculator

Example data table

Use these sample values to test your setup. Enter them as discrete levels, choose eV, and set temperature near 300 K.

Formula used

The canonical partition sum for a set of energy levels is:

Z(T) = sum_i g_i * exp( -E_i / (kB * T) )

• E_i is the energy of level i.
• g_i is the degeneracy of level i.
• kB is the Boltzmann constant.
• T is absolute temperature in kelvin.

From Z the calculator estimates:

• Probabilities: p_i = (g_i e^{-E_i/(kB T)}) / Z
• Mean energy: U = sum_i p_i E_i
• Free energy: F = -kB T ln Z
• Entropy: S = (U - F) / T
• Heat capacity: Cv = ( - ^2 ) / (kB T^2)

A small energy shift is used internally to improve numerical stability without changing probabilities or thermodynamic results.

How to use this calculator

1. Select a mode: discrete levels, oscillator, or rotor.
2. Enter temperature and choose its unit.
3. Select the energy unit you will use for inputs.
4. Provide model parameters or paste your energy list.
5. Press Calculate to show results above the form.
6. Use the CSV/PDF buttons to save outputs.

Unit note: kJ/mol is converted to per-particle energy internally. cm^-1 uses E = h*c*nu_bar.

Professional article

1) Canonical partition sum as a bridge

The partition sum Z(T) links microscopic energy levels to macroscopic thermal behavior. In the canonical ensemble, each level contributes a Boltzmann weight g_ie^-E_i/(k_BT). When Z is known, you can derive probabilities, free energy, and response functions without fitting extra parameters.

2) Thermal scale and level spacing

A quick reality check is the thermal energy scale k_BT. At 300 K, k_BT ≈ 4.14×10^-21 J ≈ 0.0259 eV. If most level spacings are far larger than k_BT, the ground state dominates and Z stays close to the ground-state degeneracy.

3) Degeneracy and symmetry counting

Degeneracy g_i multiplies the statistical weight directly. For example, two states with the same energy double the contribution to Z and increase entropy through additional accessible microstates. In molecular applications, g often represents rotational, spin, or symmetry multiplicities and should be included explicitly.

4) Discrete level lists for spectroscopy

Discrete mode is useful when you have measured or computed energies from spectroscopy or electronic structure calculations. You can paste energies with optional degeneracies and compare populations across temperatures. The probability table helps identify which states meaningfully contribute, so you can justify truncating very high levels.

5) Truncated harmonic oscillator model

For vibrational motion, the oscillator option uses E_n = (n + 1/2) * epsilon. Because the sum is infinite, the calculator truncates at n_max. Increase n_max until Z and U stop changing. Higher temperature or smaller epsilon typically requires a larger n_max to capture the tail.

6) Rigid rotor model for rotations

The rotor option uses E_J=B J(J+1) with degeneracy g_J=2J+1. This captures the rapid growth of rotational state counts. As with the oscillator, the sum is truncated at J_max. If probabilities remain significant near J_max, increase it until results converge.

7) Thermodynamic outputs and interpretation

From ln Z the calculator reports F=-k_BT ln Z, then U from the probability-weighted mean energy, and S=(U-F)/T. Heat capacity is computed from energy fluctuations, C_v = (<E²> - <E>²)/(k_BT²). Peaks in C_v often indicate a crossover where additional levels become thermally accessible.

8) Practical numerical accuracy checks

Large energies can underflow e^-E/(k_BT). To stabilize the calculation, the code subtracts the minimum energy before exponentiating, which preserves probabilities and thermodynamic quantities. Validate your setup by changing units, increasing truncation limits, and confirming that Z, U, and C_v remain consistent.

FAQs

1) What does the partition sum represent?

It is the weighted count of thermally accessible energy levels at temperature T. Each level contributes g_ie^-E_i/(k_BT), which determines probabilities and thermodynamic functions.

2) Why do I need degeneracy?

Degeneracy counts how many distinct states share the same energy. Higher degeneracy increases the level's statistical weight, raising its probability and typically increasing entropy and Z.

3) How do I choose n_max or J_max?

Increase the cutoff until Z, U, and C_v stop changing within your desired tolerance. If the highest listed levels still have noticeable probability, the cutoff is too small.

4) What energy units are supported?

You can enter energies in J, eV, kJ/mol (converted to per-particle energy), or cm^-1 (converted using E = h*c*nu_bar). Choose the unit that matches your source data.

5) Why does the calculator show ln Z?

ln Z is numerically stable and directly feeds free energy F=-k_BT ln Z. It also avoids overflow/underflow when Z is extremely large or small.

6) What does C_v/k_B mean?

It is heat capacity normalized by the Boltzmann constant, giving a dimensionless measure. Multiply by k_B to obtain C_v in J/K per particle.

7) How can I sanity-check my inputs?

Compare level spacings to k_BT (0.0259 eV at 300 K). If energies are far larger, probabilities should concentrate near the lowest energy levels. Use the example table to verify behavior.