Physical Units and Conversions¶
This document provides detailed information about the atomic units system used in Schr and conversion factors for working with SI units.
Atomic Units System¶
Overview¶
Atomic units (also called Hartree atomic units) are a natural unit system for quantum mechanical calculations. In this system, fundamental constants are set to unity:
This simplifies the Schrödinger equation and eliminates unit-tracking overhead in numerical computations.
Fundamental Constants¶
Quantity |
Symbol |
SI Value |
Atomic Unit |
|---|---|---|---|
Reduced Planck constant |
\(\hbar\) |
\(1.054572 \times 10^{-34}\) J·s |
1 |
Electron mass |
\(m_e\) |
\(9.109384 \times 10^{-31}\) kg |
1 |
Elementary charge |
\(e\) |
\(1.602176 \times 10^{-19}\) C |
1 |
Coulomb constant |
\(k_e = 1/(4\pi\epsilon_0)\) |
\(8.987552 \times 10^9\) N·m²/C² |
1 |
Derived Units¶
Quantity |
Name |
SI Value |
Expression |
|---|---|---|---|
Length |
Bohr radius |
\(a_0 = 5.291772 \times 10^{-11}\) m |
\(\hbar^2/(m_e e^2 k_e)\) |
Energy |
Hartree |
\(E_h = 4.359744 \times 10^{-18}\) J = 27.211 eV |
\(e^2 k_e / a_0\) |
Time |
Atomic time unit |
\(\hbar/E_h = 2.418884 \times 10^{-17}\) s |
\(\hbar/E_h\) |
Velocity |
— |
\(2.187691 \times 10^6\) m/s |
\(a_0 E_h / \hbar\) |
Electric field |
Atomic unit of field |
\(5.142207 \times 10^{11}\) V/m |
\(E_h / (e a_0)\) |
Magnetic field |
— |
\(2.350518 \times 10^5\) T |
\(\hbar/(e a_0^2)\) |
Conversion Factors¶
Length Conversions¶
From |
To |
Multiply by |
|---|---|---|
Bohr (\(a_0\)) |
meters (m) |
\(5.291772 \times 10^{-11}\) |
Bohr (\(a_0\)) |
Ångström (Å) |
\(0.529177\) |
Bohr (\(a_0\)) |
nanometers (nm) |
\(0.0529177\) |
meters (m) |
Bohr (\(a_0\)) |
\(1.889726 \times 10^{10}\) |
Ångström (Å) |
Bohr (\(a_0\)) |
\(1.889726\) |
nanometers (nm) |
Bohr (\(a_0\)) |
\(18.89726\) |
Energy Conversions¶
From |
To |
Multiply by |
|---|---|---|
Hartree (\(E_h\)) |
Joules (J) |
\(4.359744 \times 10^{-18}\) |
Hartree (\(E_h\)) |
electron volts (eV) |
\(27.21138\) |
Hartree (\(E_h\)) |
wavenumbers (cm⁻¹) |
\(2.194746 \times 10^5\) |
Hartree (\(E_h\)) |
Kelvin (K) |
\(3.157746 \times 10^5\) |
electron volts (eV) |
Hartree (\(E_h\)) |
\(0.03674931\) |
Joules (J) |
Hartree (\(E_h\)) |
\(2.293712 \times 10^{17}\) |
Time Conversions¶
From |
To |
Multiply by |
|---|---|---|
Atomic time (\(\hbar/E_h\)) |
seconds (s) |
\(2.418884 \times 10^{-17}\) |
Atomic time (\(\hbar/E_h\)) |
femtoseconds (fs) |
\(0.02418884\) |
Atomic time (\(\hbar/E_h\)) |
attoseconds (as) |
\(24.18884\) |
seconds (s) |
Atomic time (\(\hbar/E_h\)) |
\(4.134137 \times 10^{16}\) |
femtoseconds (fs) |
Atomic time (\(\hbar/E_h\)) |
\(41.34137\) |
attoseconds (as) |
Atomic time (\(\hbar/E_h\)) |
\(0.04134137\) |