The Hartree (Eₕ or Ha) is the atomic unit of energy. It is the natural energy scale for calculations in atomic physics and quantum chemistry. The Hartree energy is defined as the magnitude of the electrostatic potential energy between an electron and a proton separated by one Bohr radius (a₀).
Physically, it is equal to twice the binding energy of the electron in the ground state of a hydrogen atom (which is approximately 13.6 eV). This system of units, known as atomic units, simplifies many fundamental equations of quantum mechanics by setting key constants (like the electron mass, elementary charge, and reduced Planck's constant) to unity.
The system of atomic units was developed starting in the 1920s by physicists like Douglas Hartree to simplify the complex self-consistent field calculations for multi-electron atoms. It has since become the standard in computational chemistry and condensed matter physics because it removes cumbersome constants and reduces numerical errors in simulations.
The Hartree (Eₕ), also known as the atomic unit of energy, is a fundamental physical constant. It is a scalar quantity that defines the natural energy scale for calculations in atomic physics and quantum chemistry, derived from other fundamental constants of nature.
| Property | Details |
|---|---|
| Nature | Scalar. Energy is a scalar quantity, and the Hartree is a specific unit of energy. |
| SI Units | Joule (J). It is also commonly expressed in electronvolts (eV). |
| Value | 1 Eₕ ≈ 4.359744722 × 10⁻¹⁸ J<br>1 Eₕ ≈ 27.211386246 eV |
| Dimensional Formula | M L² T⁻² |
| Physical Definition | It is defined as the magnitude of the electrostatic potential energy between a stationary proton and an electron separated by one Bohr radius (a₀). |
| Relation to Hydrogen Atom | The Hartree is exactly twice the absolute value of the binding energy (ionization energy) of the electron in the ground state of a hydrogen atom in the Bohr model. |
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| Eₕ | Hartree energy | Joule (J) | The atomic unit of energy. |
| e | Elementary charge | Coulomb (C) | The magnitude of the electric charge of a single proton or electron. |
| ε₀ | Vacuum permittivity | Farad per meter (F/m) | The permittivity of free space, a constant of proportionality in electrostatics. |
| a₀ | Bohr radius | meter (m) | The most probable distance between the electron and nucleus in a ground-state hydrogen atom. |
| ℏ | Reduced Planck constant | Joule-second (J·s) | Planck's constant divided by 2π. |
| mₑ | Electron mass | kilogram (kg) | The rest mass of an electron. |
The Hartree energy is defined from first principles using Coulomb's law for the electrostatic potential energy between two point charges.
Step 1: Start with Coulomb's Law
The potential energy \(U\) between two charges \(q_1\) and \(q_2\) separated by a distance \(r\) is given by:
Step 2: Apply to the Hydrogen Atom Model
We model the simplest atomic system: a single proton (charge \(q_1 = +e\)) and a single electron (charge \(q_2 = -e\)). The characteristic distance in this system is the Bohr radius, \(r = a_0\). The Hartree energy is defined as the magnitude of this interaction energy.
Step 3: Express in terms of other fundamental constants
The Bohr radius \(a_0\) can be expressed in terms of other constants as \(a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m_e e^2}\). Substituting this into the definition of \(E_h\) gives an alternative form:
As a defined constant, the Hartree itself does not have different types. However, it is the primary unit within a system of related atomic units of energy used in different contexts within physics and chemistry.
| Type / Case | Description | When to Use |
|---|---|---|
| Hartree (Eₕ) | The base unit of energy in the Hartree atomic units system. By definition, its value is exactly 1 in this system. | It is the standard unit for reporting energies from quantum chemistry calculations (e.g., in software like Gaussian or ORCA). |
| Rydberg (Ry) | A closely related unit of energy defined as exactly half a Hartree (1 Ry = 0.5 Eₕ). It represents the ionization energy of the hydrogen atom. | Often used in atomic spectroscopy and condensed matter physics for historical reasons and its direct connection to the Rydberg formula. |
| Electronvolt (eV) | A practical energy unit widely used in physics, representing the energy an electron gains when accelerated through a potential of one volt. 1 Eₕ ≈ 27.211 eV. | Used to connect theoretical calculations to experimental measurements, such as ionization potentials, electron affinities, and band gaps. |
Quantum Chemistry: The Hartree is the standard unit of energy in electronic structure calculations, such as Hartree-Fock theory and Density Functional Theory (DFT), used to predict molecular properties and reaction energies.
Computational Materials Science: Simulations of solids, surfaces, and nanomaterials rely on atomic units. The Hartree is used to express fundamental properties like band gaps, cohesive energies, and surface energies.
Atomic and Molecular Physics: It is used to describe ionization potentials, electron affinities, and the energies of electronic transitions observed in spectroscopy.
Spectroscopy: While experimental results are often reported in wavenumbers (cm⁻¹) or electronvolts (eV), theoretical calculations that predict these spectral lines are performed using the Hartree energy scale.
Drug Discovery: Pharmaceutical researchers use computational software to model how a potential drug molecule interacts with a protein target in the body. These programs calculate the binding energy in Hartrees to predict the drug's effectiveness, helping to screen thousands of candidates before expensive lab testing begins.
OLED Display Technology: The color of light emitted by an Organic Light Emitting Diode (OLED) depends on the energy gap between electronic orbitals in its organic molecules. Materials scientists perform quantum calculations, with results in Hartrees, to design new molecules with specific energy gaps that produce vibrant and efficient colors for phone and TV screens.
Solar Panel Development: In the search for more efficient solar cells, scientists simulate the electronic properties of novel semiconductor materials. The band gap, a critical property that determines how much of the sun's spectrum a material can absorb, is calculated in Hartrees to predict the material's potential performance before it is synthesized in a lab.
The dimension of energy in any system is \([M L^2 T^{-2}]\). We can verify this from the definition of the Hartree.
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Hartree Energy | Eₕ | Joule (J) | [M L² T⁻²] |
| Elementary Charge | e | Coulomb (C) | [I T] |
| Vacuum Permittivity | ε₀ | C²·N⁻¹·m⁻² | [M⁻¹ L⁻³ T⁴ I²] |
| Bohr Radius | a₀ | meter (m) | [L] |
Dimensional analysis of the defining formula: \( [E_h] = [\frac{e^2}{4\pi\epsilon_0 a_0}] \)
\( = \frac{[I T]^2}{[M^{-1} L^{-3} T^4 I^2] [L]} = \frac{[I^2 T^2]}{[M^{-1} L^{-2} T^4 I^2]} = [M L^2 T^{-2}] \)
This confirms that the Hartree has the correct dimensions of energy.
The Hartree (Eₕ) is the atomic unit of energy, defined by the formula Eₕ = ħ² / (mₑa₀²), where ħ is the reduced Planck constant, mₑ is the electron rest mass, and a₀ is the Bohr radius. It represents the magnitude of the potential energy of an electron in the first Bohr orbit of a hydrogen atom, which is twice its binding energy.
In the definition Eₕ = kₑe²/a₀, the variables represent fundamental physical quantities at the atomic scale. Here, kₑ is Coulomb's constant, e is the elementary charge of an electron, and a₀ is the Bohr radius, which is the most probable distance between the proton and electron in a hydrogen atom.
The Hartree is the standard unit of energy in atomic physics and computational quantum chemistry, particularly in methods like Hartree-Fock theory and Density Functional Theory (DFT). Using atomic units simplifies the Schrödinger equation by setting fundamental constants (ħ, mₑ, e, kₑ) to 1, making calculations more manageable.
A frequent error is confusing the Hartree energy (Eₕ ≈ 27.2 eV) with the Rydberg unit of energy (Ry). The Rydberg is exactly half a Hartree (Ry = Eₕ/2 ≈ 13.6 eV) and corresponds to the ionization energy of hydrogen. It is crucial to check which unit is required, as confusing them results in an error of a factor of two.
In computational materials science, researchers use the Hartree to calculate the total energy of crystal structures to predict their stability and electronic properties. For example, simulations using DFT calculate energies in Hartrees to determine whether a new material will be a conductor, semiconductor, or insulator before it is synthesized.
The Hartree energy is directly related to the ground-state energy of the hydrogen atom. The ground state energy is -13.606 eV, which is exactly -1/2 Eₕ (or -1 Rydberg). Therefore, the value of one Hartree (≈ 27.211 eV) represents twice the magnitude of the binding energy of the electron in a hydrogen atom.