Atomic Units

Introduction

Write out the Schrödinger equation for hydrogen in SI units and it is cluttered with constants: ħ, the electron mass, the elementary charge and the permittivity of free space. They appear in every line of a derivation and hide the structure of the mathematics. Quantum chemists therefore work in atomic units , a system chosen so that these constants all equal one. Equations become shorter, numbers become sizes that are natural for atoms, and every major computational chemistry program reports its results in these units.

Core explanation

The definition. In atomic units (a.u.) four fundamental quantities are set to one:

- the reduced Planck constant, ħ = 1; - the electron mass, mₑ = 1; - the elementary charge, e = 1; - the Coulomb constant factor, 4πε₀ = 1.

Every other unit follows from combinations of these. The two that matter most to chemists are the units of length and energy.

The bohr. The atomic unit of length is the Bohr radius,

a₀ = 4πε₀ħ²/(mₑe²) = 5.292 × 10⁻¹¹ m = 52.92 pm = 0.5292 Å.

It is the most probable electron–nucleus distance in the hydrogen 1s orbital (using the electron mass rather than the reduced mass). A length of 1 bohr is simply written "1 a₀" or "1 bohr".

The hartree. The atomic unit of energy is the hartree,

Eₕ = e²/(4πε₀a₀) = 4.360 × 10⁻¹⁸ J = 27.211 eV = 2625.5 kJ mol⁻¹.

It is the magnitude of the Coulomb potential energy between an electron and a proton separated by one bohr. The ground-state energy of hydrogen is exactly −½ Eₕ, which is the familiar −13.606 eV.

The hydrogen Hamiltonian in atomic units. In SI units the Hamiltonian for a hydrogen-like atom of nuclear charge Z is

Ĥ = −(ħ²/2mₑ)∇² − Ze²/(4πε₀r).

Setting ħ = mₑ = e = 4πε₀ = 1 gives

Ĥ = −½∇² − Z/r,

where r is now measured in bohr and energies come out in hartree. The energy levels become E n = −Z²/(2n²) Eₕ, and the normalised 1s wavefunction is ψ₁ₛ = (Z³/π)^½ e^(−Zr). Every constant has disappeared into the choice of units.

Other derived units. The atomic unit of time is ħ/Eₕ ≈ 2.419 × 10⁻¹⁷ s, and the atomic unit of velocity is the speed of the electron in the Bohr model's first orbit, about 2.188 × 10⁶ m s⁻¹. In these units the speed of light is about 137.036, which is the reciprocal of the fine-structure constant α. The large value shows that electrons in light atoms move at a small fraction of c, so relativistic effects are small for hydrogen but grow as roughly (Zα)² for heavy elements.

Formulae

1 a₀ = 52.92 pm; 1 Eₕ = 27.211 eV = 2625.5 kJ mol⁻¹ = 219 475 cm⁻¹. Hamiltonian: Ĥ = −½∇² − Z/r. Levels: E n = −Z²/(2n²).

Step-by-step reasoning

To convert an SI equation into atomic units:

1. Replace ħ, mₑ, e and 4πε₀ by 1 wherever they appear. 2. Interpret every remaining length as a multiple of a₀ and every energy as a multiple of Eₕ. 3. Solve the simpler equation. 4. Convert the answer back by multiplying by the appropriate unit, for example 27.211 eV per hartree.

Visual explanation

Imagine a ruler whose marks are spaced by 52.9 pm and an energy scale whose ticks are spaced by 27.2 eV. A hydrogen atom drawn on this grid has its 1s peak at the first mark and its ground level at half a tick below zero. Atoms fit naturally onto a scale built for them.

Real-world analogy

Astronomers measure distances within the solar system in astronomical units rather than metres: Earth is at 1 AU, Jupiter at about 5.2 AU. The numbers are easy to compare and remember. Atomic units do the same for electrons, choosing the size of hydrogen as the natural yardstick.

Real-world example

Quantum chemistry programs typically print total energies in hartree and read or write geometries in bohr or ångström. A calculated energy for the water molecule of around −76 Eₕ is routine output; chemists then take differences between such numbers and convert them to kJ mol⁻¹ to discuss reaction energies.

Why?

Why bother? In SI units the numbers involved in atomic calculations are awkward powers of ten, such as 10⁻¹⁸ J and 10⁻¹¹ m, and constants multiply rounding errors. In atomic units typical quantities are of order one, the equations display their pure mathematical form, and scaling laws with Z become immediately visible.

Common misconception

"Atomic units are only an approximation." They are not; they are an exact change of units, like switching from metres to kilometres. No physics is lost. The only care needed is to convert back correctly when quoting a final answer in SI or chemical units.

Worked example

Question: The accurate non-relativistic ground-state energy of the helium atom is −2.9037 Eₕ. Express it in eV and in kJ mol⁻¹.

Reasoning: Multiply by the conversion factors. In eV: −2.9037 × 27.211 = −79.01 eV. In kJ mol⁻¹: −2.9037 × 2625.5 = −7624 kJ mol⁻¹.

Answer: −79.0 eV, or about −7620 kJ mol⁻¹. This is the energy needed to remove both electrons from helium.

Quick check

1. What is the ground-state energy of He⁺ (Z = 2) in atomic units and in electronvolts? Answer: E₁ = −Z²/2 = −2 Eₕ, which is −2 × 27.211 = −54.4 eV.

Exam focus

Memorise the defining conditions ħ = mₑ = e = 4πε₀ = 1 and the conversions 1 a₀ ≈ 52.9 pm and 1 Eₕ ≈ 27.2 eV. Be able to rewrite the hydrogen Hamiltonian as −½∇² − Z/r and to convert computed energies back to eV or kJ mol⁻¹.

Advanced insight

Strictly, the hydrogen atom's own energy involves the reduced mass μ = mₑmₚ/(mₑ + mₚ), which is about 0.9995 mₑ. In atomic units the true hydrogen ground-state energy is therefore −0.49973 Eₕ, not exactly −0.5. The Rydberg constant for infinite nuclear mass corresponds to ½ Eₕ, and the small reduced-mass correction is what shifts deuterium lines slightly from hydrogen lines.

Summary

Atomic units set ħ, mₑ, e and 4πε₀ equal to one. The unit of length is the bohr, a₀ = 52.92 pm, and the unit of energy is the hartree, Eₕ = 27.211 eV. In these units the hydrogen-like Hamiltonian is −½∇² − Z/r and the levels are −Z²/(2n²). Results are converted back to SI or chemical units at the end.

Practice questions

1. Express the bond length of H₂, 74.1 pm, in bohr. Answer: 74.1 ÷ 52.92 = 1.40 bohr. 2. A calculation gives an energy difference of 0.0100 Eₕ between two conformers. Convert this to kJ mol⁻¹. Answer: 0.0100 × 2625.5 = 26.3 kJ mol⁻¹. 3. Write the energy of the n = 3 level of Li²⁺ (Z = 3) in atomic units and in eV. Answer: E₃ = −9/(2 × 9) = −0.5 Eₕ, which equals −13.6 eV. 4. Which four constants are set equal to one in atomic units? Answer: The reduced Planck constant ħ, the electron mass mₑ, the elementary charge e and 4πε₀. 5. Why does the speed of light have the value of about 137 in atomic units, and what does this suggest about hydrogen? Answer: The atomic unit of velocity is αc, so c = 1/α ≈ 137. The electron in hydrogen moves at about 1/137 of c, so relativistic effects are small.