The hartree (symbol: E_h), also known as the Hartree energy, is the atomic unit of energy. It is defined as the absolute value of the ground-state energy of the hydrogen atom,[ E_h = \frac{\hbar^2}{m_e a_0^2} = \frac{e^2}{4 \pi \epsilon_0 a_0} ]where \hbar is the reduced Planck constant, m_e is the electron mass, a_0 is the Bohr radius, e is the elementary charge, and \epsilon_0 is the vacuum permittivity.[1] This corresponds to approximately 4.359744722 \times 10^{-18} J or 27.21138602 eV (2018 CODATA values).[2]The unit is named after the English physicist and mathematician Douglas Hartree (1897–1958), who made significant contributions to quantum mechanics and early computing.[3]
Definition
Role in atomic units
In the atomic unit system, commonly employed in non-relativistic quantum mechanics for atomic and molecular calculations, the reduced Planck's constant ħ, the elementary charge e, and the electron mass m_e are set to unity (ħ = e = m_e = 1).[4] This choice defines a natural scale for physical quantities, with the hartree (E_h) emerging as the fundamental unit of energy derived directly from these constants.[4] The system also includes the Bohr radius a_0 as the unit of length, the electron mass m_e as the unit of mass, and the elementary charge e as the unit of charge, providing a cohesive framework that eliminates explicit constants in many equations.[5]The hartree energy is explicitly derived as E_h = \frac{\hbar^2}{m_e a_0^2}, where a_0 represents the Bohr radius, the characteristic length scale in the hydrogen atom.[5] This expression arises from balancing the kinetic and potential energy terms in the Schrödinger equation under the atomic unit conventions, yielding a consistent energy scale tied to the electron's dynamics near the nucleus.[5]By adopting the hartree as the energy unit, equations in non-relativistic quantum mechanics are greatly simplified, as fundamental constants like ħ, m_e, and e disappear from the Hamiltonian and related operators.[4] For instance, the time-independent Schrödinger equation for a hydrogen-like atom reduces to a dimensionless form, facilitating analytical solutions and numerical computations without dimensional overhead.[5] This systemic role underscores the hartree's utility in establishing the energy benchmark equivalent to twice the binding energy of the hydrogenground state.[5]
Formal definition
The hartree energy, denoted E_\mathrm{h}, is the atomic unit of energy in the Hartree atomic unit system, with its 2022 CODATA recommended value given by E_\mathrm{h} = 4.359\,744\,722\,2060(48) \times 10^{-18} J.[6] This value results from a least-squares adjustment of fundamental physical constants and auxiliary quantities.[6]Formally, E_\mathrm{h} is expressed in terms of fundamental constants as E_\mathrm{h} = \frac{[e](/page/Elementary_charge)^2}{4\pi\epsilon_0 [a_0](/page/Bohr_radius)}, where e is the elementary charge, \epsilon_0 is the vacuum permittivity, and a_0 is the Bohr radius defined by a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m_e [e](/page/Elementary_charge)^2}, with m_e the electron mass and \hbar the reduced Planck constant.[6] Substituting the expression for a_0 yields the equivalent form E_\mathrm{h} = \frac{[m_e](/page/Electron_mass) [e](/page/Elementary_charge)^4}{(4\pi\epsilon_0)^2 \hbar^2}.[6]Following the 2019 revision of the SI, which fixed the numerical values of e, h (hence \hbar), and the speed of light c, the uncertainty in E_\mathrm{h} arises primarily from measurements of m_e and the fine-structure constant \alpha (since E_\mathrm{h} = \alpha^2 m_e c^2).[6] The relative standard uncertainty of $1.1 \times 10^{-9} in the 2022 CODATA value reflects these measurement precisions in the global adjustment.[6]An operational definition ties E_\mathrm{h} to spectroscopic data via the Rydberg constant: E_\mathrm{h} = 2 h c R_\infty, where R_\infty is the Rydberg constant for infinite nuclear mass.[6] This relation connects the atomic unit directly to measured atomic spectra without requiring reduced-mass corrections for finite nuclear mass.[6]
Numerical values
SI and base units
The hartree, denoted E_\mathrm{h}, has a CODATA-recommended value of $4.359\,744\,722\,2060(48) \times 10^{-18} J in SI units.[7]This energy unit is derived from the base physical constants including the electron mass m_\mathrm{e}, elementary charge e, reduced Planck's constant \hbar = h/(2\pi), speed of light c, and vacuum permittivity \epsilon_0, via the relation E_\mathrm{h} = m_\mathrm{e} c^2 \alpha^2, where \alpha = e^2/(4\pi \epsilon_0 \hbar c) is the fine-structure constant.[8]Equivalently, E_\mathrm{h} = \frac{m_\mathrm{e} e^4}{(4\pi \epsilon_0)^2 \hbar^2}.[8]In the context of energy-time relations, the inverse hartree corresponds to the atomic unit of time \tau_\mathrm{h} = \hbar / E_\mathrm{h} \approx 2.418\,884\,326\,5864(26) \times 10^{-17} s.[9]To illustrate its scale, the hartree represents energies on the order of $10^{-18} J, which is characteristic of electronic binding and transition energies in atomic systems.
Common conversions
In practical applications within physics and chemistry, the hartree (E_h) is frequently converted to electronvolts for electronic structure discussions, where 1 E_h = 27.211386245981(30) eV.[2]Spectroscopic measurements often express energies in wavenumbers, with 1 E_h = 219474.63136314(24) cm^{-1}.[10]In thermochemistry, molar energies are relevant, such that 1 E_h per molecule corresponds to 2625.4996394791(29) kJ/mol or 627.5094743374(69) kcal/mol.[11]The hartree also relates directly to the Rydberg energy (Ry) as 1 E_h = 2 Ry, reflecting its definition as twice the binding energy of the hydrogen atomground state.The following table summarizes these conversions based on CODATA 2022 recommendations, including uncertainties for precision in computations:
The hartree energy unit is intrinsically linked to the hydrogen atom, serving as twice the absolute value of its ground-state energy in the limit of infinite nuclear mass. In this fundamental system, the ground-state energy E_1 is given by E_1 = -\frac{1}{2} E_h, where E_h denotes one hartree. This relation arises because the hartree is defined through atomic units, where the hydrogen atom's binding energy sets the scale for electronic energies. Experimentally, the ground-state energy of the actual hydrogen atom, accounting for the finite proton mass via the reduced mass correction, is E_1 = -13.598433 \, \mathrm{eV}, which approximates -0.5 E_h since E_h \approx 27.211386 \, \mathrm{eV}.[12][2]The precise definition ties E_h = 2 |E_1(\mathrm{H})|, where E_1(\mathrm{H}) is the ground-state energy for hydrogen with infinite nuclear mass, yielding |E_1(\mathrm{H})| = 13.605693122994(26) \, \mathrm{eV}. This infinite-mass limit eliminates recoil effects, providing an idealized reference for the unit's physical basis. In practice, the reduced mass \mu = m_e (1 - m_e/M_p) slightly lowers the energy magnitude for real hydrogen, but the hartree remains anchored to the infinite-mass value for consistency in theoretical calculations.[13]Within the Bohr model of the hydrogen atom, the hartree emerges naturally from the balance of kinetic and potential energies in the electron's circular ground-state orbit. The model posits quantized orbits where the centripetal force equals the Coulomb attraction, leading to a Bohr radius a_0 such that the potential energy at this distance is -E_h. By the virial theorem for inverse-square forces, the time-averaged kinetic energy \langle T \rangle equals -\frac{1}{2} \langle V \rangle, so the total energy E = \langle T \rangle + \langle V \rangle = \frac{1}{2} \langle V \rangle = -\langle T \rangle. For the ground state, this yields E_1 = -\frac{1}{2} E_h, with the hartree scaling the magnitudes directly.[14]This connection extends to higher energy levels, where the hydrogen atom's energies form the simple infinite series E_n = -\frac{E_h}{2 n^2} for principal quantum number n = 1, 2, [3, \dots](/page/3_Dots). In atomic units, this simplifies to E_n = -\frac{1}{2 n^2}, making transitions between levels yield wavenumbers that are straightforward fractions, which greatly facilitates the analysis of hydrogen's spectral lines. The unit's design thus provides intuitive numerical simplicity for these exact solutions, underscoring its utility in atomic spectroscopy.[15]
Implications in quantum mechanics
In atomic units, the time-independent Schrödinger equation for multi-electron atoms takes a parameter-free form, where the electron mass, charge, and reduced Planck's constant are set to unity, and energies are expressed in hartrees (E_h). This simplification yields-\frac{1}{2} \sum_{i=1}^N \nabla_i^2 \psi - Z \sum_{i=1}^N \frac{1}{r_i} \psi + \sum_{i<j} \frac{1}{r_{ij}} \psi = E \psi,with E as the dimensionless energy eigenvalue in units of E_h, Z the nuclear charge, and the coordinates scaled by the Bohr radius. This dimensionless structure facilitates the numerical solution and analytical approximation of wavefunctions for systems beyond the hydrogen atom, enabling direct comparison of energy scales across atomic and molecular species.[16][17]The hartree provides a natural scale for electron correlation effects, which arise from the instantaneous Coulomb interactions neglected in mean-field approximations like Hartree-Fock. In molecules, these correlation energies—the difference between the exact non-relativistic ground-state energy and the Hartree-Fock limit—typically range from 0.04 to 0.4 E_h, depending on the number of electrons and bonding characteristics; for example, in small closed-shell atoms like neon, the value is approximately 0.38 E_h.[18] This scale underscores the small but crucial role of correlation in achieving chemical accuracy, often requiring post-Hartree-Fock methods for quantitative predictions.The hartree energy is intrinsically tied to the fine structure constant \alpha \approx 1/137, expressed as E_h = \alpha^2 m_e c^2, where m_e is the electron rest mass and c the speed of light. This relation highlights the non-relativistic nature of atomic units while linking to relativistic corrections, which scale as \alpha^2 relative to the leading-order energies; for instance, fine structure splittings in hydrogen-like atoms are of order (Z \alpha)^2 E_h / n^3, with higher-order terms like the Lamb shift further modulated by \alpha. Such scaling aids in assessing the validity of non-relativistic approximations for heavier elements.[19][20]In perturbation theory applied to multi-electron systems, the hartree unit normalizes the unperturbed Hamiltonian (typically the hydrogenic or Hartree-Fock operator) to order 1, allowing perturbation terms like electron-electron repulsions to be treated as expansions in fractions of E_h. This facilitates convergence analysis, where second-order corrections for correlation in helium, for example, contribute about 0.03 E_h, with higher orders ensuring rapid convergence for light atoms when expressed relative to E_h. The unit thus provides a benchmark for evaluating the perturbative series' reliability across quantum mechanical models.[21]
History
Origins of atomic units
The concept of atomic units emerged from Niels Bohr's 1913 model of the hydrogen atom, where the energy scale was defined by the magnitude of the ground-state binding energy, expressed as $2\pi^2 \frac{m_e e^4}{h^2}. This quantity, approximately 13.6 eV, served as a natural reference for the discrete energy levels in the model, E_n = -\frac{2\pi^2 m_e e^4}{h^2 n^2}, reflecting the quantized orbits of the electron around the nucleus.[22] Bohr's approach laid the groundwork for scaling atomic phenomena in terms of fundamental constants like the electron mass m_e, charge e, and Planck's constant h, though it predated a fully systematic unit system.[23]With the development of quantum mechanics in the mid-1920s, early wavefunction calculations for multi-electron atoms employed similar dimensionless forms by normalizing lengths to the Bohr radius and energies to the Rydberg scale. These practices facilitated numerical computations by reducing the equations to forms where key constants were set to unity, emphasizing the hydrogen-like structure as a baseline for more complex systems. In 1928, Douglas Hartree proposed the atomic unit system in his work on self-consistent field methods for solving the Schrödinger equation for multi-electron atoms, setting the electron mass, charge, and reduced Planck's constant to unity.[24] The framework evolved further with Paul Dirac's 1928 formulation of a relativistic wave equation for the electron, incorporating special relativity into quantum mechanics and yielding relativistic corrections to atomic energy levels.[25] Dirac's theory accounted for high velocities near the nucleus, with the speed of light c becoming a key parameter in relativistic extensions of atomic units; however, the non-relativistic hartree energy remained the standard for most atomic calculations due to its sufficiency for lighter elements.[26] Hartree's contributions in applying self-consistent field methods further refined these units for practical use in atomic structure computations.Prior to 1959, the energy unit—equivalent to twice the Rydberg energy, or about 27.2 eV—was commonly referred to simply as the "atomic unit of energy" in theoretical and computational work, without a eponymous name.[3] This terminology underscored its role as a universal scale derived from atomic constants, facilitating comparisons across quantum mechanical models.
Naming and adoption
The hartree unit of energy was formally named in 1959 by H. Shull and G. G. Hall in their paper "Atomic Units," published in Nature, to honor Douglas Hartree's foundational work on self-consistent field methods for multi-electron atomic systems.[27]Douglas Hartree (1897–1958) was a British physicist and mathematician renowned for developing numerical techniques to solve quantum mechanical problems during the 1920s and 1940s, including the Hartree method, which approximates electron densities through iterative self-consistent fields to compute atomic wavefunctions.[28]At the University of Manchester, Hartree pioneered the adaptation of the differential analyzer—an early analog computing device—for performing atomic structure calculations, enabling more accurate numerical solutions to differential equations in quantum theory.[28][29]The unit saw widespread adoption in the 1960s as computational quantum chemistry expanded with the advent of electronic computers, facilitating efficient simulations of molecular systems in a natural, parameter-free framework.In 1993, the International Union of Pure and Applied Chemistry (IUPAC) recommended the symbol E_h for the hartree in its Quantities, Units and Symbols in Physical Chemistry (Green Book), standardizing its notation in physical and chemical literature.[30]
Applications
Computational chemistry
In computational chemistry, the hartree (E_h) serves as the fundamental unit of energy for quantum mechanical calculations, enabling precise handling of electronic structure problems due to its alignment with atomic-scale phenomena. This unit facilitates the computation of molecular energies, where total electronic energies are typically on the order of tens to hundreds of E_h, reflecting the binding of electrons to nuclei. The use of E_h ensures numerical stability and consistency in algorithms, as it avoids scaling factors that could introduce errors in iterative solvers.In Hartree-Fock (HF) theory, the total energy is expressed as the sum of one-electron integrals (kinetic and nuclear attraction terms) and two-electron Coulomb and exchange integrals, with all contributions evaluated in E_h. For instance, the HF energy for the water molecule converges to approximately -76 E_h using minimal basis sets. Basis set convergence in HF calculations is typically targeted to within 10^{-5} E_h to achieve chemical accuracy, balancing computational cost with precision in molecular properties. This approach underpins self-consistent field methods, where the energy is minimized variationally over molecular orbitals expanded in Gaussian basis functions.Density functional theory (DFT), an extension of HF that incorporates electron correlation via the exchange-correlation functional, also scales all energy components in E_h. The exchange-correlation energy, a key term in the Kohn-Sham equations, is approximated using functionals like LDA or GGA, contributing fractions of E_h that capture binding effects beyond mean-field approximations. Typical binding energies for small molecules, such as atomization energies, range from ~0.1 to 2 E_h, as seen in DFT benchmarks for hydrocarbons and hydrides, highlighting the unit's suitability for quantifying molecular stability.Ab initio methods, including post-HF approaches like MP2 or CCSD(T), perform geometry optimizations and vibrational frequency calculations with internal energies computed in E_h for high precision, though results are often reported in converted units like kcal/mol for interpretability. Software packages such as Gaussian and ORCA default to E_h for energy outputs, ensuring seamless integration in workflows for molecular simulations. For example, Gaussian reports SCF energies directly in atomic units, while ORCA provides total energies like -75.56 E_h for the H2O+ ion in single-point calculations.
Atomic and solid-state physics
In atomic physics, Hartree atomic units provide a natural framework for solving the Schrödinger equation for multi-electron atoms by setting fundamental constants such as the electron mass m_e = 1, the elementary charge e = 1, and the reduced Planck's constant \hbar = 1, with the unit of energy defined as the Hartree, approximately 27.211 eV. This system simplifies the non-relativistic Hamiltonian for an N-electron atom to H = \sum_{i=1}^N \left( \frac{p_i^2}{2} - \frac{Z}{r_i} \right) + \sum_{i<j} \frac{1}{r_{ij}}, where Z is the nuclear charge, p_i the momentum operator, and r_{ij} the inter-electron distance, eliminating explicit constants and focusing on electron-nucleus and electron-electron interactions.[31]The Hartree-Fock method, foundational to atomic structure calculations, employs these units to approximate the many-body wavefunction as a Slater determinant of single-particle orbitals, minimizing the energy functional E[\Phi] = \sum_{\lambda=1}^N I_\lambda + \frac{1}{2} \sum_{\lambda,\mu} (J_{\lambda\mu} - K_{\lambda\mu}), where I_\lambda is the one-electron integral, J_{\lambda\mu} the Coulomb integral, and K_{\lambda\mu} the exchangeintegral. This approach yields ground-state energies and orbital shapes for atoms like helium and neon with errors typically under 1% for total energies, enabling predictions of ionization potentials and excitation spectra that align closely with experimental values. For instance, Hartree-Fock calculations in atomic units reproduce the helium ground-state energy as -2.8617 Hartrees, compared to the exact -2.9037 Hartrees, highlighting the method's utility despite neglecting electron correlation.[31]In solid-state physics, Hartree units extend to periodic systems through ab initio Hartree-Fock and density functional theory (DFT) methods, facilitating band structure computations in crystals by normalizing lattice parameters and energies relative to atomic scales. The Hartree potential, representing the classical electrostatic interaction of electrons with the nuclear and electronicdensity, is central to the Kohn-Sham equations in DFT, expressed in atomic units as v_H(\mathbf{r}) = \int \frac{n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d\mathbf{r}', where n(\mathbf{r}) is the electron density; this term accounts for mean-field electron-electron repulsion in solids without explicit many-body effects.[32]Applications include linear combination of atomic orbitals (LCAO) Hartree-Fock calculations for insulators, such as brucite (Mg(OH)_2), where basis sets like 8-6G for Mg yield band gaps of about 8.5 eV and bulk moduli of 68 GPa, closely matching experiments and revealing ionic Mg-O bonding alongside covalent O-H bonds under pressures up to 45 GPa. Similarly, for NaCl, near-Hartree-Fock-limit basis sets produce valence band widths within 2 mH (milliHartrees) of Bloch-orbital results, with total energies around -621.5 Hartrees per formula unit, demonstrating the method's accuracy for localized Wannier orbitals in ionic crystals. These computations in Hartree units enable quantitative predictions of electronic properties, such as pressure-induced phase transitions, by integrating over Brillouin zones while maintaining computational tractability for unit cells up to 20 atoms.[33][34]