Energy & Thermodynamics

Scientific Unit Matrix

Select a domain and enter a value. The matrix instantly computes all corresponding unit equivalents using strict scientific constants. Supports standard and exponential notation (e.g., 1.5e-3).

Conversion logic

All calculations run locally in your browser. Factors are the 2018 CODATA recommended values and exact SI (2019) definitions; hover any unit's for its definition and source, or see the full method & references below.

Method & References

How the factors are defined, and where they come from

Every conversion in this tool reduces to a single reference constant per category. The numbers are not rounded conveniences, they are the 2018 CODATA recommended values and the exact SI (2019) definitions. Pick a domain below for the exact relations and the primary source of each constant.

Energy: the Hartree as the anchor

All energy units are expressed relative to the Hartree, Eh, the atomic unit of energy. Its SI value is fixed by the Rydberg constant through Eh = 2R∞hc.

The defining chain:

Eh = 4.359 744 722 2071 × 10-18 J = 27.211 386 246 eV = 2 Ry
1 Eh ≡ 2625.4996 kJ/mol = 627.5095 kcal/mol = 219474.63 cm-1

Molar units (kcal/mol, kJ/mol) multiply the per-particle energy by the Avogadro constant NA = 6.022 140 76×1023 mol-1, which is exact since the 2019 SI redefinition. The thermochemical calorie (1 cal = 4.184 J, exact by definition) links kcal/mol to kJ/mol, this is the calorie used by AMBER and CHARMM, not the 15°C or IT calorie. The wavenumber relation E = hcν̃ makes cm-1 an energy unit in spectroscopy.

Unit1 Eh equalsBasis
eV27.211386246CODATA 2018 (exact eV since 2019)
kJ/mol2625.499639CODATA 2018 × NA
kcal/mol627.509474+ thermochemical calorie (4.184 J)
cm-1219474.6314CODATA 2018 hartree–inverse-metre
K (E/kB)315775.02CODATA 2018 hartree–kelvin

Atomic units: length, force, dipole, charge, polarizability

Quantum-chemistry codes (Gaussian, ORCA, Psi4, Quantum ESPRESSO) work internally in atomic units, where ℏ = me = e = 1 and 4πε0 = 1. Converting their output to lab units needs the CODATA values below.

The atomic units used here:

a0 (length) = 5.291 772 109 03 × 10-11 m
e a0 (dipole) = 8.478 353 6255 × 10-30 C·m = 2.541746 D
e (charge) = 1.602 176 634 × 10-19 C (exact)
Eh/a0 (force) = 8.238 723 4983 × 10-8 N
Eh/a03 (pressure) = 2.942 101 5697 × 1013 Pa
e2a02/Eh (polariz.) = 1.648 777 274 36 × 10-41 C2m2J-1

The debye is fixed exactly by 1 D = 10-21/c C·m = 3.335 641×10-30 C·m (c is exact), giving 1 e·a0 = 2.541746 D. Force-field partial charges are always in units of e, so the charge tab lets you cross-check RESP/CHELPG output against C or statC. Volume polarizability (Å3, the number usually tabulated) follows the Gaussian-CGS convention α[Å3] = α[a.u.] × 0.148185.

Spectroscopy: one photon, four ways to write it

Wavenumber, frequency, photon energy and wavelength all describe the same excitation. Three of them scale linearly; wavelength is the reciprocal.

The relations (h and c are exact SI constants):

E = hν = hcν̃    λ = 1/ν̃ = c/ν
1 cm-1 ≡ 29.9792458 GHz ≡ 0.1239842 meV ≡ 1/λ[cm]

Because wavelength is inversely proportional to the others, the matrix inverts correctly only when you type into a single field, the tool normalises to a base wavenumber and reciprocates for λ. Constants used: c = 299 792 458 m/s and h = 6.626 070 15×10-34 J·s, both exact since 2019. The cm-1↔eV factor, 1.239842×10-4, is the reciprocal of 8065.54 cm-1/eV.

Temperature, kBT, and molar entropy

Temperature conversions include the two offset scales (Celsius, Fahrenheit) and the thermal-energy equivalents kBT that appear constantly in statistical mechanics and free-energy work.

Offset scales are affine, not multiplicative:

T[°C] = T[K] − 273.15    T[°F] = (T[K] − 273.15)×9/5 + 32
kBT at 300 K = 2.4943 kJ/mol = 0.5961 kcal/mol = 25.85 meV

The tool converts temperature through kelvin with explicit affine maps, so °C and °F are handled correctly (a plain ratio would be wrong). The kBT rows use kB = 1.380 649×10-23 J/K (exact) and, for molar values, the gas constant R = NAkB = 8.314 462 618 J mol-1 K-1. In the Heat Capacity tab, the entropy unit (e.u., “gibbs”) is 1 cal mol-1 K-1, and quantities are also given as dimensionless multiples of R (or kB per particle).

References & data sources

  1. Tiesinga, E., Mohr, P. J., Newell, D. B., & Taylor, B. N. (2021). CODATA recommended values of the fundamental physical constants: 2018. Rev. Mod. Phys. 93, 025010. doi:10.1103/RevModPhys.93.025010.
  2. NIST. Fundamental Physical Constants (2018 CODATA adjustment), Hartree energy, Bohr radius, atomic units of time, force, pressure, electric dipole moment and polarizability, elementary charge. physics.nist.gov/cuu/Constants.
  3. NIST. Non-SI units accepted for use with the SI, and units based on fundamental constants (2018 CODATA). nonsi_2018.pdf (a.u. of electric dipole moment = 8.4783536255×10-30 C·m; a.u. of polarizability = 1.64877727436×10-41 C2m2J-1).
  4. BIPM. The International System of Units (SI Brochure), 9th edition (2019). Exact defined values of c, h, e, kB and NA. bipm.org/en/publications/si-brochure.
  5. Thompson, A., & Taylor, B. N. (2008). Guide for the Use of the International System of Units (SI), NIST Special Publication 811. Definitions of the atmosphere, torr, psi, calorie and other non-SI units. nist.gov/pml/special-publication-811.
  6. Abraham, M. J., et al. (2015). GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers. SoftwareX 1–2, 19–25. See also the GROMACS Reference Manual, Definitions and Units. manual.gromacs.org.
  7. Case, D. A., et al. AMBER Reference Manual (unit system: kcal·mol-1, Å, thermochemical calorie). ambermd.org.
  8. Debye unit definition: 1 D = 10-18 statC·cm = (10-21/c) C·m ≈ 3.335641×10-30 C·m. See NIST Molecular Spectroscopy special-units notes. physics.nist.gov.
NIST CODATA constants SI Brochure GROMACS units

Conversion factors are the 2018 CODATA / exact SI (2019) values. All computation runs entirely in your browser, nothing you type leaves your device.

Frequently asked questions

Which calorie does the kcal/mol use?
The thermochemical calorie, 1 cal = 4.184 J exactly. This is the definition used by the AMBER and CHARMM force fields, so energies convert consistently with those programs. It is not the 15°C calorie (4.1855 J) or the international-table calorie (4.1868 J).
Why is the Hartree the base unit for energy?
Because it is the natural unit that quantum-chemistry codes report, and its SI value is one of the most precisely known conversions (fixed through the Rydberg constant, Eh = 2R∞hc). Anchoring on it keeps the whole energy matrix internally consistent to the full CODATA precision.
Why does the wavelength field behave differently in Spectroscopy?
Wavenumber, frequency and photon energy are all directly proportional to each other, but wavelength is their reciprocal (λ = 1/ν̃). The tool normalises everything to a base wavenumber and inverts for wavelength, which is exact as long as you type into one field at a time.
Why can't temperature use a single multiplication factor?
Celsius and Fahrenheit are offset (affine) scales, not ratio scales, 0 °C is not zero energy. The converter therefore routes temperature through kelvin with explicit T = K − 273.15 and T = (K−273.15)×9/5+32 formulas. The kBT rows convert the thermal energy at that temperature.
How precise are these numbers?
They are the 2018 CODATA recommended values and exact SI (2019) definitions, carried to the digits shown in each tooltip. Since the 2019 redefinition, c, h, e, kB and NA are exact, so any conversion depending only on those (e.g. eV↔J, charge, bar↔Pa) is exact; those depending on measured constants (Bohr radius, Hartree) inherit CODATA's relative uncertainty of order 10-10.
Is a.u. of polarizability the same as the Å3 I see in tables?
Not numerically. Molecular polarizabilities are usually tabulated as a volume in Å3 (Gaussian-CGS convention), whereas QM codes print the atomic-unit value. The conversion is α[Å3] = α[a.u.]×0.148185. Both are provided so you can move between a calculation and a reference table.
Is my data sent anywhere?
No. Every calculation runs locally in your browser using JavaScript. Nothing you enter is transmitted or stored.

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