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:

.. math::

   \hbar = m_e = e = \frac{1}{4\pi\epsilon_0} = 1

This simplifies the Schrödinger equation and eliminates unit-tracking overhead in numerical computations.

Fundamental Constants
~~~~~~~~~~~~~~~~~~~~~

.. list-table::
   :header-rows: 1
   :widths: 25 20 35 20

   * - Quantity
     - Symbol
     - SI Value
     - Atomic Unit
   * - Reduced Planck constant
     - :math:`\hbar`
     - :math:`1.054572 \times 10^{-34}` J·s
     - 1
   * - Electron mass
     - :math:`m_e`
     - :math:`9.109384 \times 10^{-31}` kg
     - 1
   * - Elementary charge
     - :math:`e`
     - :math:`1.602176 \times 10^{-19}` C
     - 1
   * - Coulomb constant
     - :math:`k_e = 1/(4\pi\epsilon_0)`
     - :math:`8.987552 \times 10^9` N·m²/C²
     - 1

Derived Units
~~~~~~~~~~~~~

.. list-table::
   :header-rows: 1
   :widths: 25 20 35 20

   * - Quantity
     - Name
     - SI Value
     - Expression
   * - Length
     - Bohr radius
     - :math:`a_0 = 5.291772 \times 10^{-11}` m
     - :math:`\hbar^2/(m_e e^2 k_e)`
   * - Energy
     - Hartree
     - :math:`E_h = 4.359744 \times 10^{-18}` J = 27.211 eV
     - :math:`e^2 k_e / a_0`
   * - Time
     - Atomic time unit
     - :math:`\hbar/E_h = 2.418884 \times 10^{-17}` s
     - :math:`\hbar/E_h`
   * - Velocity
     - —
     - :math:`2.187691 \times 10^6` m/s
     - :math:`a_0 E_h / \hbar`
   * - Electric field
     - Atomic unit of field
     - :math:`5.142207 \times 10^{11}` V/m
     - :math:`E_h / (e a_0)`
   * - Magnetic field
     - —
     - :math:`2.350518 \times 10^5` T
     - :math:`\hbar/(e a_0^2)`

Conversion Factors
------------------

Length Conversions
~~~~~~~~~~~~~~~~~~

.. list-table::
   :header-rows: 1
   :widths: 30 40 30

   * - From
     - To
     - Multiply by
   * - Bohr (:math:`a_0`)
     - meters (m)
     - :math:`5.291772 \times 10^{-11}`
   * - Bohr (:math:`a_0`)
     - Ångström (Å)
     - :math:`0.529177`
   * - Bohr (:math:`a_0`)
     - nanometers (nm)
     - :math:`0.0529177`
   * - meters (m)
     - Bohr (:math:`a_0`)
     - :math:`1.889726 \times 10^{10}`
   * - Ångström (Å)
     - Bohr (:math:`a_0`)
     - :math:`1.889726`
   * - nanometers (nm)
     - Bohr (:math:`a_0`)
     - :math:`18.89726`

Energy Conversions
~~~~~~~~~~~~~~~~~~

.. list-table::
   :header-rows: 1
   :widths: 30 40 30

   * - From
     - To
     - Multiply by
   * - Hartree (:math:`E_h`)
     - Joules (J)
     - :math:`4.359744 \times 10^{-18}`
   * - Hartree (:math:`E_h`)
     - electron volts (eV)
     - :math:`27.21138`
   * - Hartree (:math:`E_h`)
     - wavenumbers (cm⁻¹)
     - :math:`2.194746 \times 10^5`
   * - Hartree (:math:`E_h`)
     - Kelvin (K)
     - :math:`3.157746 \times 10^5`
   * - electron volts (eV)
     - Hartree (:math:`E_h`)
     - :math:`0.03674931`
   * - Joules (J)
     - Hartree (:math:`E_h`)
     - :math:`2.293712 \times 10^{17}`

Time Conversions
~~~~~~~~~~~~~~~~

.. list-table::
   :header-rows: 1
   :widths: 30 40 30

   * - From
     - To
     - Multiply by
   * - Atomic time (:math:`\hbar/E_h`)
     - seconds (s)
     - :math:`2.418884 \times 10^{-17}`
   * - Atomic time (:math:`\hbar/E_h`)
     - femtoseconds (fs)
     - :math:`0.02418884`
   * - Atomic time (:math:`\hbar/E_h`)
     - attoseconds (as)
     - :math:`24.18884`
   * - seconds (s)
     - Atomic time (:math:`\hbar/E_h`)
     - :math:`4.134137 \times 10^{16}`
   * - femtoseconds (fs)
     - Atomic time (:math:`\hbar/E_h`)
     - :math:`41.34137`
   * - attoseconds (as)
     - Atomic time (:math:`\hbar/E_h`)
     - :math:`0.04134137`