Hartree atomic units

Hartree atomic units form a system of natural units that is especially convenient for atomic physics and quantum mechanics calculations. In this system, the reduced Planck constant (ħ), the elementary charge (e), the mass of the electron (m_e), and the Coulomb's constant (k_e or 1/4πε₀) are all normalized to 1. The system is named after the British physicist Douglas Hartree. It simplifies many of the physical equations by setting fundamental constants to unity, thereby making calculations more straightforward.

Definition

In Hartree atomic units, the fundamental constants are set as follows:

• The reduced Planck constant, ħ = 1.
• The elementary charge, e = 1.
• The mass of the electron, m_e = 1.
• The Coulomb's constant, k_e = 1.

This choice of units naturally leads to the unit of energy being the Hartree (Ha), which is equivalent to twice the ground state energy of the hydrogen atom. The unit of length is the Bohr radius (a₀), and the unit of time is derived from these quantities.

Advantages

The use of Hartree atomic units simplifies many equations in quantum mechanics and atomic physics. For example, the Schrödinger equation for the hydrogen atom becomes significantly simpler, and the expression for the energy levels of the hydrogen atom can be written in a more straightforward manner. This simplification arises because factors of 4πε₀, ħ, and e, which commonly appear in these equations, are set to 1.

Applications

Hartree atomic units are widely used in computational chemistry and quantum physics. They are particularly useful in the field of ab initio quantum chemistry calculations, where electronic structure problems are solved from first principles. By eliminating the need to carry physical constants through the calculations, the Hartree system allows for more direct comparison of results across different studies.

Conversion to SI Units

To convert quantities measured in Hartree atomic units to SI units, one must use the specific values of the physical constants that were set to 1 in the Hartree system. For example, the energy in joules (J) can be calculated from Hartrees (Ha) using the relation E_J = E_Ha × 4.359744650×10⁻¹⁸ J.

Limitations

While Hartree atomic units greatly simplify theoretical analyses, their use is mostly confined to the realm of atomic and molecular physics. In other areas of physics and engineering, the SI units are more commonly used. Additionally, the normalization of physical constants to 1 can sometimes obscure the physical significance of these constants in governing the dynamics of physical systems.

See Also

• Natural units
• Quantum mechanics
• Computational chemistry
• Douglas Hartree