Atomic units
Atomic units are a system of natural units of measurement particularly convenient for atomic physics and quantum chemistry, in which the reduced Planck constant (ħ), the electron rest mass (mₑ), and the elementary charge (e) are set equal to unity.[1][2] This choice eliminates these fundamental constants from many equations in non-relativistic quantum mechanics, simplifying theoretical calculations and numerical simulations of atomic and molecular systems.[2][3] The system was introduced by Douglas Hartree in a series of papers beginning in 1928 to facilitate self-consistent field calculations in atomic structure.[4] In the atomic unit system, derived quantities take on natural scales relevant to the hydrogen atom: the unit of length is the Bohr radius (a₀ ≈ 5.29 × 10⁻¹¹ m), the unit of energy is the hartree (Eₕ ≈ 27.211 eV), and the unit of time is ħ/Eₕ ≈ 2.42 × 10⁻¹⁷ s.[2][3] The hartree energy, named after Hartree, corresponds to twice the ionization energy of hydrogen, while the Bohr radius defines the characteristic size of the hydrogen ground state.[2] In SI units, the system implicitly sets the Coulomb constant 1/(4πε₀) = 1 as well, though variants exist in cgs-Gaussian units where e = 1 in electrostatic units (esu).[2][3] Atomic units are recognized for their utility in scientific communication by standards guides such as the NIST Guide to the SI, provided they are clearly identified, though they are not formally accepted by the CIPM for use with the SI, and they form a coherent set alongside other non-SI units like the electronvolt.[1] A related but distinct Rydberg atomic unit system sets the Rydberg energy (13.606 eV) as the energy unit instead of the hartree, halving certain scales for specific applications in spectroscopy.[2] These units are widely employed in computational chemistry software and theoretical models, enabling dimensionless formulations that highlight intrinsic physical behaviors at the atomic scale.[3]Background and Motivation
Historical Development
The origins of atomic units trace back to the foundational developments in quantum mechanics during the 1920s. Erwin Schrödinger's formulation of wave mechanics in 1926 provided the essential framework for describing electron behavior in atoms through the time-independent Schrödinger equation, which naturally lent itself to simplified unit systems in subsequent calculations. Shortly thereafter, Paul Dirac's 1928 derivation of the relativistic wave equation for the electron incorporated natural units to handle high-speed corrections, influencing the adoption of dimensionless conventions in atomic and relativistic contexts.[5] Douglas Hartree introduced the practical system of atomic units in 1928 while developing numerical methods for solving the Schrödinger equation in multi-electron atoms, particularly in his work on self-consistent fields for non-Coulomb potentials.[6] This approach set the electron mass, charge, and reduced Planck's constant to unity, facilitating the Hartree-Fock method in quantum chemistry by eliminating cumbersome constants from equations.[7] A key milestone came in 1957 with Hans Bethe and Edwin Salpeter's comprehensive treatment of one- and two-electron atoms, where they explicitly adopted and popularized Hartree's atomic units throughout their calculations in atomic physics.[7] In the 1950s, atomic units expanded beyond isolated atoms to molecular systems, enabling efficient quantum chemical computations for electron densities and bonding in simple molecules like H2.[8] Post-World War II efforts toward standardization accelerated their integration into mainstream physics, with widespread use in literature by the 1960s and formal recognition in recommendations from the National Institute of Standards and Technology (NIST) for fundamental constants and the International Union of Pure and Applied Physics (IUPAP) for nomenclature in quantum calculations.[9][10]Reasons for Use
Atomic units provide significant theoretical and practical advantages in atomic and molecular physics, primarily by setting fundamental constants such as the reduced Planck's constant \hbar, the electron mass m_e, and the elementary charge e to unity, which simplifies the form of key equations in quantum mechanics.[8] This approach reduces the Schrödinger equation for systems like the hydrogen atom or multi-electron atoms to a dimensionless form, eliminating extraneous constants and emphasizing the intrinsic physics of electron-nucleus interactions.[11] For instance, the time-independent Schrödinger equation in atomic units becomes \hat{H} \psi = E \psi, where the Hamiltonian \hat{H} lacks scaling factors, making it easier to identify eigenfunctions and verify solutions analytically or numerically.[8] In approximation methods such as perturbation theory and variational principles, atomic units eliminate scaling factors that would otherwise complicate the expansion of wave functions or energy corrections.[11] Perturbations, like those from electron-electron repulsion in multi-electron atoms, appear as pure numerical terms without dimensional multipliers, allowing for clearer identification of dominant contributions and more straightforward series expansions.[12] Similarly, in variational methods, trial wave functions can be optimized using dimensionless parameters, reducing the sensitivity to arbitrary unit choices and improving the convergence of approximations.[11] For computational physics, atomic units facilitate easier numerical integration and the use of dimensionless parameters, which align with the natural scales of atomic phenomena and enable efficient scaling laws in simulations.[8] This standardization minimizes unit conversion errors during algorithms for solving the Schrödinger equation or density functional theory calculations, leading to greater numerical accuracy and reduced computational overhead.[11] In practice, quantities like lengths in bohr radii or energies in hartrees avoid the cumbersome large or small numbers inherent in SI units, such as expressing the Bohr radius as approximately $5.29 \times 10^{-11} meters.[8] Compared to SI units, atomic units keep important dimensionless constants explicit and manageable; for example, the fine structure constant \alpha \approx 1/137 remains a simple fraction that directly influences relativistic corrections without additional scaling.[13] A concrete illustration is the hydrogen atom's energy levels, which simplify to E_n = -\frac{1}{2n^2} in atomic units, directly reflecting the Rydberg formula without multiplicative constants like $13.6 eV.[8] This inherent scaling also reduces errors in approximations, as perturbations or variational parameters operate on a unified numerical scale, enhancing the reliability of results in both theoretical derivations and computational implementations.[11]Core Definitions
Basis of the System
Atomic units are established by setting four fundamental quantities to unity: the electron rest mass m_e, the elementary charge e, the reduced Planck's constant \hbar, and $4\pi \epsilon_0 (where \epsilon_0 is the vacuum permittivity).[14][15] These choices define a natural scale tailored to the behavior of electrons in atoms, where the electron mass governs inertial effects, the charge dictates electrostatic interactions, and \hbar sets the quantum of action.[16] The Bohr radius a_0, given by a_0 = \frac{4\pi \epsilon_0 \hbar^2}{m_e e^2}, is thus set to 1, ensuring coherence in describing bound electron states and atomic sizes.[14] By normalizing these quantities, atomic units eliminate dimensional prefactors in the Schrödinger equation and Coulomb potential, facilitating calculations of electron dynamics within electromagnetic fields at atomic scales, such as orbital velocities on the order of the Bohr velocity (approximately 1 atomic unit).[16][15] The system draws heavily from the Gaussian cgs unit framework, where electromagnetic equations appear symmetric without explicit factors of $4\pi or c; in atomic units, \epsilon_0 = 1/(4\pi), making it compatible with SI units while retaining the simplicity of Gaussian forms, which predominate in theoretical atomic physics.[14][3] Formally, atomic units constitute a complete, coherent system for non-relativistic quantum electrodynamics in vacuum, encompassing the dynamics of electrons, nuclei, and quantized electromagnetic fields without relativistic corrections.[17] In this framework, the speed of light c emerges as a derived parameter rather than a base unit, with value c \approx 1/\alpha \approx 137 (where \alpha is the fine-structure constant), highlighting the non-relativistic approximation since electron velocities in atoms are much smaller than c.[16] This separation allows precise treatment of light-matter interactions at atomic energies while treating c as a large finite number.[17]Dimensionless Quantities
In atomic units, the system sets several fundamental constants to unity, leaving a set of dimensionless parameters that characterize deviations from the idealized non-relativistic, infinite-nuclear-mass electron model. These parameters arise naturally in expressions for atomic spectra, binding energies, and interactions, allowing precise quantification of effects like electromagnetic coupling strength and finite nuclear mass.[18] The fine-structure constant, denoted \alpha, is the primary dimensionless parameter governing the strength of electromagnetic interactions in atomic physics. Defined as \alpha = \frac{e^2}{4\pi\epsilon_0 \hbar c}, it measures the coupling between charged particles and the electromagnetic field.[13] Its numerical value (2022 CODATA) is \alpha \approx 7.2973525643 \times 10^{-3}, or equivalently \alpha^{-1} \approx 137.035999177.[19] In atomic units, where the Bohr velocity is \alpha c, this small value justifies the non-relativistic approximation for most atomic processes, as electron speeds are much less than the speed of light (v/c \sim \alpha \ll 1); relativistic effects become significant only when \alpha corrections are included, such as in fine-structure splittings.[18] Another key dimensionless parameter is the electron-to-proton mass ratio, m_e / m_p \approx 5.4461702149 \times 10^{-4} (2022 CODATA).[20] This ratio parameterizes corrections due to the finite nuclear mass, which deviate from the ideal atomic scale assuming infinite nuclear mass. For instance, the Rydberg constant for hydrogen, accounting for reduced mass effects, is R_H = R_\infty / (1 + m_e / m_p), where R_\infty is the infinite-mass Rydberg constant; this introduces a small shift of order m_e / m_p in spectral lines, reflecting nuclear recoil and other nuclear effects.[20] In atomic units, the non-relativistic Hamiltonian for a hydrogen-like atom takes the form H = -\frac{\nabla^2}{2} - \frac{Z}{r}, with relativistic corrections appearing as perturbations scaled by powers of \alpha. A leading-order relativistic kinetic energy correction is \delta H = -\frac{p^4}{8 c^2} = -\frac{\alpha^2 p^4}{8}, which contributes to the fine structure of energy levels proportional to \alpha^2 times the non-relativistic binding energy.[21] These dimensionless quantities ensure that physical laws in atomic units retain their standard mathematical forms, with \alpha serving as the sole coupling constant in quantum electrodynamics (QED) formulations for atomic systems. This structure simplifies theoretical calculations while explicitly highlighting the scale of electromagnetic and mass-ratio effects.[13]Unit Specifications
Base Physical Units
The atomic unit (au) system establishes a set of base physical units derived from fundamental properties of the electron and quantum mechanics, facilitating dimensionless formulations in atomic and molecular physics. These units are defined such that the electron rest mass m_e, elementary charge e, reduced Planck's constant \hbar, and Coulomb constant k_e = 1/(4\pi\epsilon_0) each equal 1 au, with the Bohr radius a_0 derived as 1 au; their SI equivalents are provided by CODATA recommendations.[22][23] The base unit of mass is the electron rest mass, m_e = 9.109\,383\,7139(28) \times 10^{-31} kg = 1 au.[22] The base unit of charge is the elementary charge, e = 1.602\,176\,634 \times 10^{-19} C = 1 au (exact since the 2019 SI redefinition).[22] The base unit of action is the reduced Planck's constant, \hbar = 1.054\,571\,817 \times 10^{-34} J s = 1 au (exact).[22] The base unit for electrostatics is the Coulomb constant, defined as k_e = \frac{1}{4\pi\epsilon_0} = 8.987\,551\,789 \times 10^{9}\ \text{N m}^2\ \text{C}^{-2} = 1\ \text{au}, (exact in form, but with uncertainty from fine-structure constant); this, combined with the other base units, yields the derived unit of length as the Bohr radius a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m_e e^2} \approx 5.291\,772\,105\,44(82) \times 10^{-11} \ \text{m} = 1\ \text{au}, with a conversion factor of 1 au length = 0.529,177,210,544(82) Å.[22]| Quantity | Atomic Unit Symbol | SI Value (CODATA 2022) | Unit in SI |
|---|---|---|---|
| Mass | m_e | $9.109\,383\,7139(28) \times 10^{-31} | kg |
| Charge | e | $1.602\,176\,634 \times 10^{-19} | C |
| Action | \hbar | $1.054\,571\,817 \times 10^{-34} | J s |
| Coulomb constant | k_e | $8.987\,551\,789 \times 10^{9} | N m² C⁻² |
Derived Units
In atomic units, derived units are constructed from combinations of the base units (including the implied vacuum permittivity via $4\pi\epsilon_0 = 1), such as the Bohr radius a_0 for length (derived as 1 au), electron mass m_e for mass (base), elementary charge e for charge (base), and reduced Planck's constant \hbar for action (base). These derived units simplify expressions in quantum mechanics and atomic physics by setting multiple fundamental constants to unity. The primary energy unit is the hartree E_h, defined as E_h = \frac{\hbar^2}{m_e a_0^2}, which is equivalently expressed as E_h = \frac{m_e e^4}{(4\pi\epsilon_0)^2 \hbar^2}. This corresponds to 27.211,386,245,981(30) eV or $4.359\,744\,722\,2060(48) \times 10^{-18} J (CODATA 2022).[24][25] In atomic units, energies are expressed in hartrees, with 1 au = 1 E_h. The time unit is derived as t_0 = \frac{\hbar}{E_h}, which simplifies to approximately $2.418\,884\,326\,5864 \times 10^{-17} s in SI units, or 1 au.[26] An alternative derivation yields t_0 = \frac{\hbar}{m_e \alpha^2 c^2}, where \alpha is the fine-structure constant and c is the speed of light, reflecting the timescale of electron motion in hydrogen-like atoms. The velocity unit is given by v_0 = \frac{e^2}{4\pi\epsilon_0 \hbar} = \alpha c, equivalent to the orbital speed in the Bohr model, with a value of approximately $2.187\,691\,262\,16 \times 10^6 m/s or 1 au.[27] The electric field unit arises as E_0 = \frac{E_h}{e a_0}, representing the field strength at the Bohr radius due to the nuclear charge, approximately $5.142\,206\,751\,12 \times 10^{11} V/m or 1 au.[28] Common conversions, such as 1 au of energy to joules, facilitate bridging atomic-scale calculations to macroscopic systems.[25]Conventions in Practice
Explicit Unit Representations
In explicit unit representations within atomic units (AU), physical quantities are expressed with explicit multipliers or symbols to denote the AU scale, facilitating clear traceability to conventional units like those in the International System of Quantities (SI). Common notations include appending "au" to symbols (e.g., energy in E_au) or using subscripts such as "at" for atomic (e.g., r = 1.5 a_0, where a_0 is the Bohr radius). This approach contrasts with fully unitless formalisms by preserving dimensional awareness, which aids in integrating AU-based theoretical results with experimental data measured in SI units. Computational chemistry software often incorporates explicit AU flags to control input and output formats, ensuring users can specify or retrieve values in atomic units while maintaining compatibility with SI conversions. For instance, in the Gaussian suite, the Units=Bohrs keyword can be used in the route section to specify atomic units for geometry input, with energies output in hartrees (E_h) alongside optional SI equivalents. Similarly, Psi4 supports explicit AU through its input syntax, such as setting "units au" in the molecule block, which scales coordinates to bohr and outputs properties like dipole moments in atomic units (e_au). These features allow practitioners to perform calculations in AU for numerical efficiency while explicitly labeling results to avoid ambiguity in interdisciplinary work. The primary advantage of explicit AU representations is the maintenance of traceability to SI units, enabling straightforward comparisons between theoretical predictions and experimental measurements, such as aligning computed ionization potentials with photoelectron spectroscopy data. For example, the non-relativistic Schrödinger equation for the hydrogen atom is commonly written as Ĥ ψ = E ψ, where Ĥ = - (1/2) ∇² - 1/r in AU, with the Laplacian ∇² in bohr⁻², the potential -1/r in hartree·bohr, and energies E labeled in hartrees (1 E_h ≈ 27.211386 eV). This notation highlights the unit dependencies without altering the numerical coefficients, promoting clarity in derivations. Quantum chemical results are typically reported in atomic units, often accompanied by SI conversions to facilitate comparison with experimental data, particularly for thermochemical and spectroscopic data, to ensure reproducibility and standardization across publications. In cases of mixed units, such as strong-field physics involving laser-matter interactions, intensities are often expressed in AU (e.g., I = 3.51 × 10^{16} W/cm² corresponding to 1 AU for electric field strength), while temporal parameters like pulse duration may retain SI units (femtoseconds) to interface with experimental laser specifications. This hybrid approach leverages AU for scaling atomic-scale phenomena while accommodating macroscopic experimental conditions.Unitless Formalism
In the unitless formalism of atomic units, physical quantities are expressed purely as numerical values by implicitly setting the base constants—electron mass m_e = 1, reduced Planck's constant \hbar = 1, elementary charge e = 1, and vacuum permittivity factor $4\pi\epsilon_0 = 1—to unity, thereby eliminating all explicit unit symbols from expressions and equations. This standard practice simplifies notation in atomic physics, where, for instance, the ground-state energy of the hydrogen atom is denoted simply as -0.5, corresponding to -0.5 hartrees.[29][30] The formalism arises from rescaling variables to render equations dimensionless; for the wave function, this involves transforming \psi(\mathbf{r}) \to \psi(\mathbf{r}/a_0) / a_0^{3/2}, where a_0 is the Bohr radius (now 1 in atomic units), which normalizes lengths, energies, and times to their characteristic atomic scales.[29] As a result, the time-dependent Schrödinger equation takes the compact, fully unitless form i \frac{\partial \psi}{\partial t} = \left[ -\frac{\nabla^2}{2} + V(\mathbf{r}) \right] \psi, where the Laplacian \nabla^2 operates in bohr units and V(\mathbf{r}) is the potential in hartrees.[30] This approach streamlines theoretical derivations and numerical implementations by reducing algebraic complexity and avoiding constant prefactors, a convention widely adopted in textbooks such as Physics of Atoms and Molecules by Bransden and Joachain.[31] However, it demands caution in multi-scale problems, such as those incorporating nuclear motion, where the large mass ratio between nuclei and electrons (typically ~1836 for protons) requires explicit rescaling to capture small but significant corrections.[32] In modern computational contexts, like density functional theory codes such as DFTK, atomic units serve as the default, with input quantities provided numerically without units for seamless integration.[33]Key Physical Constants
Fundamental Constants in AU
In atomic units, fundamental constants are expressed relative to the base units of mass (electron mass m_e = 1), charge (elementary charge e = 1), and action (reduced Planck constant \hbar = 1), with the unit of length defined as the Bohr radius a_0 = 1. This system simplifies many expressions in quantum mechanics and atomic physics by setting several constants to unity or simple numerical values. However, constants not incorporated into the base definitions, such as the speed of light and particle masses, retain measured numerical values with associated uncertainties from experimental data. The speed of light c is c = 1/\alpha, where \alpha is the fine-structure constant; the CODATA 2022 recommended value is \alpha^{-1} = 137.035999177(21), so c \approx 137.035999177 in atomic units of velocity.[22] The Planck constant h = 2\pi in atomic units of action, exactly, as it follows directly from \hbar = 1. The vacuum permittivity \varepsilon_0 = \frac{1}{4\pi} \approx 0.0795774715 in atomic units of permittivity, though it is often left implicit in calculations due to the definition of the charge unit. The gravitational constant G is extremely small in atomic units, with a value of approximately $2.40 \times 10^{-43} in units of mass^{-1} length^3 time^{-2}, rendering gravitational effects negligible compared to electromagnetic forces in atomic-scale phenomena; this value is derived from the 2022 CODATA recommended SI values for G, a_0, m_e, and the atomic unit of time.[22] The proton mass m_p \approx 1836.152673426(32) in atomic units of mass (relative to m_e). As an illustration of derived quantities, the Bohr magneton \mu_B = e \hbar / (2 m_e) = 1/2 exactly in atomic units of magnetic moment, highlighting how the system streamlines expressions for magnetic properties. The following table summarizes selected CODATA 2022 values of fundamental constants expressed in atomic units, including relative standard uncertainties (in parentheses as the last digits of the quoted value).| Constant | Symbol | Value in AU | Relative Uncertainty |
|---|---|---|---|
| Speed of light | c | 137.035999177(21) | $1.5 \times 10^{-10} |
| Proton-electron mass ratio | m_p / m_e | 1836.152673426(32) | $1.7 \times 10^{-11} |
| Gravitational constant | G | $2.40 \times 10^{-43} (calculated) | $2.2 \times 10^{-5} (from SI) |