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Electronic correlation
Electronic correlation
Electronic correlation
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Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.

Atomic and molecular systems

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Electron correlation energy in terms of various levels of theory of solutions for the Schrödinger equation.

Within the Hartree–Fock method of quantum chemistry, the antisymmetric wave function is approximated by a single Slater determinant. Exact wave functions, however, cannot generally be expressed as single determinants. The single-determinant approximation does not take into account Coulomb correlation, leading to a total electronic energy different from the exact solution of the non-relativistic Schrödinger equation within the Born–Oppenheimer approximation. Therefore, the Hartree–Fock limit is always above this exact energy. The difference is called the correlation energy, a term coined by Löwdin.[1] The concept of the correlation energy was studied earlier by Wigner.[2]

A certain amount of electron correlation is already considered within the HF approximation, found in the electron exchange term describing the correlation between electrons with parallel spin. This basic correlation prevents two parallel-spin electrons from being found at the same point in space and is often called Pauli correlation. Coulomb correlation, on the other hand, describes the correlation between the spatial position of electrons due to their Coulomb repulsion, and is responsible for chemically important effects such as London dispersion. There is also a correlation related to the overall symmetry or total spin of the considered system.

The word correlation energy has to be used with caution. First it is usually defined as the energy difference of a correlated method relative to the Hartree–Fock energy. But this is not the full correlation energy because some correlation is already included in HF. Secondly the correlation energy is highly dependent on the basis set used. The "exact" energy is the energy with full correlation and complete basis set.

Electron correlation is sometimes divided into dynamical and non-dynamical (static) correlation. Dynamical correlation is the correlation of the movement of electrons and is described under electron correlation dynamics[3] and also with the configuration interaction (CI) method. Static correlation is important for molecules whose ground state is well described only with more than one (nearly) degenerate determinant. In this case the Hartree–Fock wavefunction (only one determinant) is qualitatively wrong. The multi-configurational self-consistent field (MCSCF) method takes account of this static correlation, but not dynamical correlation.

If one wants to calculate excitation energies (energy differences between the ground and excited states) one has to be careful that both states are equally balanced (e.g., Multireference configuration interaction).

Methods

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In simple terms, the molecular orbitals of the Hartree–Fock method are optimized by evaluating the energy of an electron in each molecular orbital moving in the mean field of all other electrons, rather than including the instantaneous repulsion between electrons.

To account for electron correlation, there are many post-Hartree–Fock methods, including:

One of the most important methods for correcting for the missing correlation is the configuration interaction (CI) method. Starting with the Hartree–Fock wavefunction as the ground determinant, one takes a linear combination of the ground and excited determinants as the correlated wavefunction and optimizes the weighting factors according to the Variational Principle. When taking all possible excited determinants, one speaks of Full-CI. In a Full-CI wavefunction all electrons are fully correlated. For non-small molecules, Full-CI is much too computationally expensive. One truncates the CI expansion and gets well-correlated wavefunctions and well-correlated energies according to the level of truncation.

Perturbation theory gives correlated energies, but no new wavefunctions. PT is not variational. This means the calculated energy is not an upper bound for the exact energy. It is possible to partition Møller–Plesset perturbation theory energies via Interacting Quantum Atoms (IQA) energy partitioning (although most commonly the correlation energy is not partitioned).[4] This is an extension to the theory of Atoms in Molecules. IQA energy partitioning enables one to look in detail at the correlation energy contributions from individual atoms and atomic interactions. IQA correlation energy partitioning has also been shown to be possible with coupled cluster methods.[5][6]

There are also combinations possible. E.g. one can have some nearly degenerate determinants for the multi-configurational self-consistent field method to account for static correlation and/or some truncated CI method for the biggest part of dynamical correlation and/or on top some perturbational ansatz for small perturbing (unimportant) determinants. Examples for those combinations are CASPT2 and SORCI.

  • Explicitly correlated wavefunction (R12 method)

This approach includes a term depending on interelectron distance into wavefunction. This leads to faster convergence in terms of basis set size than pure gaussian-type basis set, but requires calculation of more complex integrals. To simplify them, interelectron distances are expanded into a series making for simpler integrals. The idea of R12 methods is quite old, but practical implementations begun to appear only recently.

Crystalline systems

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In condensed matter physics, electrons are typically described with reference to a periodic lattice of atomic nuclei. Non-interacting electrons are therefore typically described by Bloch waves, which correspond to the delocalized, symmetry adapted molecular orbitals used in molecules (while Wannier functions correspond to localized molecular orbitals). A number of important theoretical approximations have been proposed to explain electron correlations in these crystalline systems.

The Fermi liquid model of correlated electrons in metals is able to explain the temperature dependence of resistivity by electron-electron interactions. It also forms the basis for the BCS theory of superconductivity, which is the result of phonon-mediated electron-electron interactions.

Systems that escape a Fermi liquid description are said to be strongly-correlated. In them, interactions plays such an important role that qualitatively new phenomena emerge.[7] This is the case, for example, when the electrons are close to a metal-insulator transition. The Hubbard model is based on the tight-binding approximation, and can explain conductor-insulator transitions in Mott insulators such as transition metal oxides by the presence of repulsive Coulombic interactions between electrons. Its one-dimensional version is considered an archetype of the strong-correlations problem and displays many dramatic manifestations such as quasi-particle fractionalization. However, there is no exact solution of the Hubbard model in more than one dimension.

The RKKY interaction can explain electron spin correlations between unpaired inner shell electrons in different atoms in a conducting crystal by a second-order interaction that is mediated by conduction electrons.

The Tomonaga–Luttinger liquid model approximates second order electron-electron interactions as bosonic interactions.

Mathematical viewpoint

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For two independent electrons a and b,

where ρ(ra,rb) represents the joint electronic density, or the probability density of finding electron a at ra and electron b at rb. Within this notation, ρ(ra,rb) dra drb represents the probability of finding the two electrons in their respective volume elements dra and drb.

If these two electrons are correlated, then the probability of finding electron a at a certain position in space depends on the position of electron b, and vice versa. In other words, the product of their independent density functions does not adequately describe the real situation. At small distances, the uncorrelated pair density is too large; at large distances, the uncorrelated pair density is too small (i.e. the electrons tend to "avoid each other").

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Electronic correlation

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Electronic correlation refers to the interactions among electrons in atoms, molecules, and solids that account for their correlated motions, reducing the probability of electrons occupying the same spatial and thereby lowering the total electronic beyond what is captured by the mean-field Hartree-Fock approximation. This correlation arises from the instantaneous Coulomb repulsion between electrons, which the Hartree-Fock method treats only on average, resulting in a correlation defined as the difference between the exact non-relativistic ground-state and the Hartree-Fock limit . In quantum chemistry and physics, electronic correlation is crucial for achieving chemical accuracy in computational predictions, as the Hartree-Fock approximation recovers about 99% of the total non-relativistic energy, but the remaining 1% due to correlation is on the order of typical bond energies and reaction barriers, influencing properties like bond lengths, dissociation energies, and spectroscopic constants. It manifests in two primary forms: dynamic correlation, which describes the short-range, instantaneous adjustments in electron positions to avoid mutual repulsion in systems with large energy gaps between occupied and virtual orbitals, and static correlation, which emerges from near-degeneracies in electronic configurations, such as in bond breaking or transition metal complexes, requiring multireference treatments. Historically, the concept was formalized by Per-Olov Löwdin in the 1950s, who emphasized its role in partitioning the total , and it has since driven the development of post-Hartree-Fock methods like configuration interaction (CI), Møller-Plesset (MP2 and beyond), and coupled-cluster , with the CCSD(T) approach—often called the "gold standard"—providing high accuracy for dynamic correlation in single-reference systems. (DFT) also incorporates correlation through exchange-correlation functionals, though it approximates rather than explicitly computes it, making it efficient for larger systems. Quantifying electronic correlation remains challenging, with measures like entanglement entropy proposed as alternatives to traditional energy-based definitions to better capture its quantum . Overall, understanding and modeling electronic correlation is fundamental to advancing computational materials , drug design, and quantum simulations, where neglecting it leads to qualitative errors in predicting molecular behavior.

Fundamentals

Definition and Correlation Energy

Electronic correlation refers to the interactions between electrons in multi-electron that cause their motions to be coupled beyond the assumptions of independent-particle models, such as the Hartree-Fock . This coupling primarily originates from the repulsion term in the many-electron Hamiltonian, which accounts for the instantaneous electrostatic interactions between electrons. Physically, electronic correlation arises because electrons tend to avoid each other more effectively than predicted by mean-field theories, due to the combined effects of the —which prevents electrons of the same spin from occupying the same spatial —and the repulsion that pushes like-charged electrons apart despite their overall attraction to the nucleus. This avoidance behavior creates a "correlation " in the pair probability , where the likelihood of finding two electrons in close proximity is significantly reduced compared to uncorrelated models. The energy, denoted EcE_c, quantifies this effect and is defined as the difference between the non-relativistic ground-state energy EE of the and the Hartree-Fock energy EHFE_{\mathrm{HF}}: Ec=E−EHFE_c = E - E_{\mathrm{HF}} Since the wavefunction incorporates these correlations, EcE_c is always negative, reflecting the additional lowering of the 's energy due to the stabilizing influence of correlated motions. The Hartree-Fock method provides the baseline independent-particle against which EcE_c is measured. The concept of correlation energy was first explicitly quantified in the late 1920s and early 1930s through variational wavefunction calculations by Egil Hylleraas on the , the simplest multi-electron system. Hylleraas's work yielded a correlation energy correction of approximately 42 millihartrees relative to the Hartree-Fock limit, demonstrating the significant role of interactions even in this two-electron case.

Types of Electronic Correlation

Electronic correlation in is broadly categorized into static (also known as nondynamic) and dynamic types, each arising from distinct physical mechanisms and contributing differently to the total , where the total is the sum of these two components. Static correlation originates from the near-degeneracy of multiple electronic configurations, which becomes prominent in situations such as transition states, bond dissociation, or systems with degenerate orbitals, necessitating a multi-reference description of the wavefunction to capture the proper mixing of these configurations. A classic example is the dissociation of the H₂ molecule, where the single-reference Hartree-Fock method fails to describe the correct limiting behavior of two separated hydrogen atoms, as it predicts an incorrect energy curve due to the degeneracy between bonding and antibonding orbitals at large internuclear distances. In contrast, dynamic correlation reflects the short-range, instantaneous adjustments in electron positions driven by Coulomb repulsion, leading to electrons avoiding each other on a timescale faster than nuclear motion; this type dominates in systems with well-separated orbitals and can often be adequately captured by single-reference post-Hartree-Fock methods that account for electron-electron interactions beyond the mean-field level. The distinction between static and dynamic correlation can be understood through the analysis of orbital occupancies in natural orbitals, where static correlation is associated with significant fractional occupations (deviating substantially from 0 or 2 electrons per orbital), indicating a multi-configurational character, whereas dynamic correlation manifests as small fluctuations around the mean-field occupancies in a predominantly single-determinant description. A quantitative measure to distinguish the prevalence of static involves the variance in pair densities or related diagnostics, such as the nondynamic derived from expansions of the two-particle , which quantifies deviations from independent particle behavior; specifically, the T1 diagnostic from coupled-cluster theory, defined as the norm of the singles amplitude vector divided by the of the number of electrons, provides a practical threshold, with values exceeding 0.02 (for main-group elements) signaling significant static that may compromise the reliability of single-reference approximations.

Manifestations in Systems

Atomic and Molecular Systems

In atomic systems, electronic correlation significantly refines the prediction of key properties such as ionization potentials (IPs) and electron affinities (EAs), which are often overestimated or underestimated by Hartree-Fock (HF) methods due to the neglect of electron-electron interactions. For instance, in the , the HF ground-state energy is -2.86168 , while the exact non-relativistic value is -2.903724 , meaning correlation lowers the energy by approximately 0.042 (or 1.14 eV), providing crucial stabilization beyond the independent-particle approximation. This correlation contribution enhances the accuracy of IPs, reducing the HF value of 0.917 for to the experimental 0.903 , and similarly improves EAs in systems like the by accounting for the avoidance of electron coalescence. In molecular systems, electronic correlation influences by allowing electrons to correlate their motions, which typically lengthens equilibrium bond lengths compared to HF predictions and alters curves, particularly near dissociation limits. For example, in diatomic molecules like N₂ and CO, post-HF methods incorporating correlation yield bond lengths that are 0.01–0.03 Å longer than HF values, reflecting increased accounting of electron repulsion and more accurate effective . This effect is pronounced in dissociation curves, where correlation prevents unphysical artifacts in HF, such as incorrect bond breaking in homonuclear diatomics. A specific illustration is the SN₂ reaction, like F⁻ + CH₃Cl → Cl⁻ + CH₃F, where electron correlation reduces classical barrier heights by about 50% (from ~50 kcal/mol in HF to ~25 kcal/mol), enabling accurate modeling of reaction kinetics that single-reference methods fail to capture. Spectroscopically, electronic induces shifts in excitation energies, particularly for states involving diffuse orbitals, by incorporating dynamic that adjusts densities in response to excitations. In Rydberg states of atoms and molecules, such as those in atoms or small hydrocarbons, dynamic is essential for correctly spacing energy levels and determining quantum defects, as HF alone overestimates excitation energies by 0.1–0.5 eV due to inadequate treatment of core polarization and screening. For Rydberg series in or , contributions lower vertical excitation energies and refine the convergence to the limit, ensuring alignment with experimental spectra. Open-shell atomic and molecular systems, such as radicals (e.g., CH₃•) or diradicals (e.g., methylene), present unique challenges due to strong static correlation arising from near-degeneracies in electronic configurations, which impart multireference character and render single-determinant methods inadequate. In these cases, static correlation stabilizes multiple Slater determinants of comparable energy, leading to symmetry breaking in unrestricted HF and errors in properties like spin densities or bond angles exceeding 10–20% compared to experiment. For diradicals like O₂ or carbenes, this multireference nature requires treating static correlation explicitly to avoid spurious barriers or incorrect ground-state multiplicities, as seen in the singlet-triplet splitting of methylene, where HF predicts a triplet ground state but correlation confirms it with accurate energy differences of ~9 kcal/mol.

Crystalline and Extended Systems

In crystalline and extended systems, electronic correlation profoundly influences the electronic structure and collective properties of periodic solids, such as metals, semiconductors, and insulators. Unlike weakly correlated systems where electrons behave as independent quasiparticles, strong correlations arise from the interplay of interactions and , leading to emergent phenomena like metal-insulator transitions and unconventional . These effects are particularly prominent in transition metal oxides and other materials with partially filled d- or f-shells, where correlation energies can rival bandwidths, necessitating beyond-mean-field descriptions. One key manifestation is the modification of band structures, where electronic correlations narrow bandwidths and open gaps that are absent in Hartree-Fock approximations. For instance, in the prototypical Mott-Hubbard insulator NiO, unrestricted Hartree-Fock predicts a metallic state due to partial d-band filling, but incorporating correlation via methods like the self-interaction-corrected local spin density approximation reveals an insulating gap of approximately 4 eV, driven by the localization of Ni 3d electrons. This correlation-induced gap arises from the Hubbard U parameter, which penalizes double occupancy and splits the d-band into upper and lower Hubbard bands, exemplifying the Mott transition in half-filled systems. Similar effects occur in other late monoxides like MnO and CoO, where band structure calculations incorporating correlation accurately reproduce experimental optical gaps and spectral features. Electronic correlation also governs magnetic properties through mechanisms like , which mediates antiferromagnetic interactions in insulating oxides. In these materials, virtual hopping of electrons between metal sites via oxygen 2p orbitals is suppressed by on-site repulsion, favoring antiparallel spin alignments to minimize kinetic energy loss; this Goodenough-Kanamori rules predict for 180° metal-oxygen-metal bonds. For example, in NiO, correlation stabilizes high-spin d^8 configurations (S=1), enabling type-II antiferromagnetic ordering with a Néel temperature of 523 K, as confirmed by simulations that capture the superexchange J ~ 20 meV. In broader families of perovskites like LaMnO3, correlation-driven orbital ordering further enhances these interactions, leading to complex spin structures. Regarding transport phenomena, correlations impact charge carrier mobility and enable exotic states like superconductivity. In high-Tc cuprates (e.g., up to ~90 K in YBa_2Cu_3O_7), such as La_{2-x}Sr_xCuO_4, strong correlations in the Cu d_{x^2-y^2} bands enhance electron pairing via spin-fluctuation-mediated mechanisms, where doping suppresses antiferromagnetism and promotes d-wave superconductivity. This correlation-enhanced pairing is evident near oxygen dopants, where local stripe-like orders amplify superconducting correlations, as probed by scanning tunneling microscopy. Additionally, in correlated metals, correlations scatter , reducing mobility and contributing to phenomena like the , though the provides quasiparticle corrections that partially restore delocalization. The tension between electron delocalization and localization is a hallmark of strongly correlated systems, contrasting free-electron-like behavior in simple metals with Mott localization in insulators. In weakly correlated metals like alkali metals, electrons delocalize over the lattice, forming broad bands, but strong correlations—quantified by the ratio U/W (where U is the on-site repulsion and W the bandwidth)—localize electrons when U > W, as in f-electron heavy-fermion compounds. This duality underlies bad-metal behavior in oxides, where static correlation in d-electron systems blurs the itinerant-localized boundary, leading to non-Fermi-liquid .

Mathematical Formalism

Pair Correlation Functions

Pair correlation functions provide a mathematical description of the of electrons in a many-body quantum , capturing deviations from independent particle behavior due to interactions and quantum statistics. The radial pair distribution function, often denoted as g(r1,r2)g(\mathbf{r}_1, \mathbf{r}_2), quantifies the probability of finding one at position r1\mathbf{r}_1 and another at r2\mathbf{r}_2, relative to the product of the one-electron densities. Specifically, it is defined as g(r1,r2)=Γ(r1,r2)ρ(r1)ρ(r2),g(\mathbf{r}_1, \mathbf{r}_2) = \frac{\Gamma(\mathbf{r}_1, \mathbf{r}_2)}{\rho(\mathbf{r}_1) \rho(\mathbf{r}_2)}, where Γ(r1,r2)\Gamma(\mathbf{r}_1, \mathbf{r}_2) is the diagonal element of the spin-traced two-particle reduced (2-RDM), representing the pair density, and ρ(r)\rho(\mathbf{r}) is the . This function arises from the expectation value of the pair density operator with respect to the many-electron wavefunction Ψ\Psi, given by Γ(r1,r2)=N(N−1)∫∣Ψ(r1,r2,r3,…,rN)∣2dr3⋯drN\Gamma(\mathbf{r}_1, \mathbf{r}_2) = N(N-1) \int |\Psi(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3, \dots, \mathbf{r}_N)|^2 d\mathbf{r}_3 \cdots d\mathbf{r}_N, where NN is the number of electrons. The exchange-correlation hole, nxc(r1,r2)n_{xc}(\mathbf{r}_1, \mathbf{r}_2), describes the depletion of around a reference at r1\mathbf{r}_1 due to both exchange (from the Pauli ) and correlation effects. It is defined as nxc(r1,r2)=ρ(r2)[g(r1,r2)−1],n_{xc}(\mathbf{r}_1, \mathbf{r}_2) = \rho(\mathbf{r}_2) [g(\mathbf{r}_1, \mathbf{r}_2) - 1], which ensures that the average number of electrons in the hole is exactly -1 when integrated over r2\mathbf{r}_2, maintaining overall charge neutrality: ∫nxc(r1,r2)dr2=−1\int n_{xc}(\mathbf{r}_1, \mathbf{r}_2) d\mathbf{r}_2 = -1. This property follows directly from the normalization of the 2-RDM and reflects the self-interaction cancellation inherent in exact many-electron theory. The pair correlation function and associated holes derive from the full many-electron wavefunction Ψ\Psi, through contraction of the 2-RDM obtained by integrating the NN-particle density matrix over all but two coordinates. Near points of electron-electron coalescence (∣r1−r2∣=0|\mathbf{r}_1 - \mathbf{r}_2| = 0), the wavefunction exhibits a cusp singularity to cancel the divergence in the Coulomb potential, as established by Kato's theorem. Specifically, for opposite-spin electrons, the wavefunction behaves as Ψ(r1,r2,… )≈Ψ(0)(1+12∣r1−r2∣+⋯ )\Psi(\mathbf{r}_1, \mathbf{r}_2, \dots) \approx \Psi(0) \left(1 + \frac{1}{2} |\mathbf{r}_1 - \mathbf{r}_2| + \cdots \right) at coalescence, leading to a linear short-range behavior in g(r1,r2)≈1+∣r1−r2∣+⋯g(\mathbf{r}_1, \mathbf{r}_2) \approx 1 + |\mathbf{r}_1 - \mathbf{r}_2| + \cdots. This cusp condition imposes a universal constraint on the 2-RDM and pair correlation at short distances, ensuring finite energies in the Hamiltonian. The total exchange-correlation can be decomposed into contributions from the Fermi (arising from exchange due to the antisymmetry of the wavefunction for same-spin electrons) and the Coulomb (arising from correlation effects, primarily for opposite-spin electrons due to repulsion). The Fermi , nx(r1,r2)n_x(\mathbf{r}_1, \mathbf{r}_2), captures the Pauli exclusion, reducing the probability of same-spin electrons occupying the same spatial region, while the Coulomb , nc(r1,r2)n_c(\mathbf{r}_1, \mathbf{r}_2), further depletes beyond mean-field effects to minimize repulsion. This decomposition is given by nxc=nx+ncn_{xc} = n_x + n_c, where the Fermi alone integrates to -1 for each spin, and the Coulomb integrates to zero, ensuring the total satisfies charge neutrality. Such separation highlights the distinct roles of quantum statistics and dynamic interactions in electronic .

Energy Decomposition

Electronic correlation energy, denoted as EcE_c, represents the difference between the exact non-relativistic ground-state energy of a many-electron system and its Hartree-Fock approximation, capturing the effects of electron-electron interactions beyond mean-field treatment. A common decomposition partitions EcE_c into static and dynamic components: Ec=Estatic+EdynamicE_c = E_{\text{static}} + E_{\text{dynamic}} The static correlation energy EstaticE_{\text{static}} arises from configuration interaction due to near-degeneracies in the electronic structure, such as in bond breaking or transition states, where multiple determinants contribute significantly to the wavefunction. This term accounts for the inadequacy of a single reference determinant and is often recovered variationally through multi-reference methods. In contrast, the dynamic correlation energy EdynamicE_{\text{dynamic}} stems from instantaneous fluctuations in electron pairs, avoiding Coulomb repulsion at short interelectronic distances, and is typically smaller in magnitude but pervasive across all systems. This decomposition is rooted in analyses of pair density deviations from the Hartree-Fock limit, where static effects dominate long-range correlations and dynamic effects short-range adjustments. In density functional theory, the Görling-Levy perturbation theory provides a systematic expansion of the correlation energy functional EcE_c with respect to the electron-electron coupling constant λ\lambda, treating the Kohn-Sham non-interacting system as the unperturbed reference. The second-order term Ec(2)E_c^{(2)} corresponds to the leading correlation correction, analogous to the second-order Møller-Plesset perturbation theory (MP2) in wavefunction-based approaches, capturing pairwise dynamic correlations in the high-density limit. Higher-order terms Ec(3)E_c^{(3)}, Ec(4)E_c^{(4)}, and beyond account for more complex interactions, with the full series converging to the exact EcE_c for finite systems. This formalism links density-based and many-body perturbation theories, enabling approximations like GL2 for practical computations. Size-consistency is a crucial property for correlation energy treatments, ensuring that the energy of a composite system scales correctly with size, particularly for non-interacting subsystems. For two separated molecules A and B at infinite distance, the total correlation energy must satisfy Ec(A+B)=Ec(A)+Ec(B)E_c(A+B) = E_c(A) + E_c(B), avoiding spurious interactions or overestimation that plagues non-size-consistent methods like configuration interaction singles and doubles (CISD). Methods such as coupled cluster theory achieve size-consistency by incorporating linked diagrams, while perturbation theories like MP2 inherently scale additively due to pairwise contributions. Violations lead to incorrect dissociation energies and poor extensivity for large systems, emphasizing the need for consistent formulations in intermediate Hamiltonian approaches. The Hellmann-Feynman theorem extends to correlated systems, dictating that atomic forces are given by the expectation value of the explicit derivative of the Hamiltonian with respect to nuclear coordinates, incorporating correlation contributions directly through the correlated wavefunction or . In exact or fully variational treatments, such as full configuration interaction, forces arise solely from these Hellmann-Feynman terms, with correlation effects manifesting as modifications to electrostatic interactions between nuclei and the correlated distribution. For approximate correlated methods, additional non-Hellmann-Feynman (Pulay) terms may appear due to basis set incompleteness, but the theorem ensures that correlation influences forces via integrated pair functions without direct computation of wavefunction responses. This relation is foundational for geometry optimizations and in correlated .

Computational Methods

Wavefunction-Based Approaches

Wavefunction-based approaches to electronic correlation explicitly construct multi-electron wavefunctions by incorporating contributions from multiple Slater determinants, thereby accounting for both static and dynamic correlation effects beyond the Hartree-Fock approximation. These methods are variational or perturbative in nature and are particularly suited for small to medium-sized molecular systems where high accuracy is required, such as in benchmark calculations for and . Unlike density-based methods, they provide a direct representation of the correlated wavefunction, enabling detailed analysis of electron pairing and entanglement, though at a high computational cost that scales steeply with basis set size and number of electrons. Configuration interaction (CI) methods form the foundation of many wavefunction-based treatments by expanding the total wavefunction as a of configuration state functions derived from Slater determinants: Ψ=∑IcIΦI\Psi = \sum_I c_I \Phi_I where ΦI\Phi_I are the determinants obtained by exciting electrons from occupied to virtual orbitals in a chosen basis, and the coefficients cIc_I are determined variationally by diagonalizing the electronic Hamiltonian. Full configuration interaction (full CI), which includes all possible excitations within the basis set, yields the exact non-relativistic energy and wavefunction for the given basis, serving as the gold standard for validating approximate methods in small systems like the or H2_2 . However, its factorial scaling with the number of orbitals limits full CI to systems with fewer than 20 electrons in minimal bases. Truncated approximations, such as configuration interaction singles and doubles (CISD), restrict the expansion to single and double excitations, capturing much of the dynamic while remaining computationally feasible for larger s, though they suffer from lack of size consistency. Møller-Plesset perturbation theory (MPn) treats correlation as a perturbation to the Hartree-Fock Hamiltonian, using the uncorrelated Hartree-Fock wavefunction as the zeroth-order reference. The second-order term, MP2, primarily accounts for dynamic correlation through pairwise interactions and is given by: E(2)=−14∑i,j,a,b∣⟨ij∣∣ab⟩∣2ϵi+ϵj−ϵa−ϵbE^{(2)} = -\frac{1}{4} \sum_{i,j,a,b} \frac{|\langle ij || ab \rangle|^2}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b} where i,ji,j index occupied orbitals, a,ba,b virtual orbitals, ⟨ij∣∣ab⟩\langle ij || ab \rangle are antisymmetrized two-electron integrals, and ϵ\epsilon are Hartree-Fock orbital energies. MP2 provides a size-extensive correction that improves upon Hartree-Fock energies for weakly correlated systems, achieving chemical accuracy (1 kcal/mol) for many organic molecules, and serves as a cost-effective starting point for higher-order methods. Higher orders like MP3 and MP4 include additional excitation classes but often exhibit oscillatory convergence, limiting practical use to MP2 or MP4 in most applications. Coupled cluster (CC) theory offers a systematically improvable framework by parameterizing the correlated wavefunction via an exponential cluster operator applied to the Hartree-Fock reference: Ψ=eT^Φ0\Psi = e^{\hat{T}} \Phi_0 where T^=T^1+T^2+⋯\hat{T} = \hat{T}_1 + \hat{T}_2 + \cdots sums excitation operators, with T^1\hat{T}_1 and T^2\hat{T}_2 denoting single and double excitations, respectively. The coupled cluster singles and doubles (CCSD) model solves the projected Schrödinger equations for these operators, providing a size-consistent and size-extensive treatment of correlation that surpasses CI in efficiency and accuracy for dynamic correlation-dominated systems. For near-quantitative results, CCSD is often augmented with a perturbative triples correction, CCSD(T), which adds fourth-order triples contributions approximately, achieving sub-kcal/mol accuracy for single-reference molecules like water or ammonia. CC methods are widely adopted for high-precision quantum chemistry due to their robust handling of electron correlation in closed-shell systems. Multi-reference methods address cases with significant static , such as bond breaking or transition metals, by optimizing a multi-determinantal reference function before adding dynamic . The complete active self-consistent field (CASSCF) approach variationally optimizes both orbitals and coefficients within a complete active (CAS) of mm electrons in nn active orbitals, yielding a compact wavefunction that captures non-dynamical exactly in the active . To include dynamic affordably, CASSCF is followed by second-order perturbation theory (CASPT2), which treats the difference between the full Hamiltonian and the zeroth-order CASSCF Hamiltonian perturbatively, providing balanced treatment of static and dynamic effects in systems like diradicals or excited states. These methods excel in challenging multi-reference scenarios, offering accuracies within 2-3 kcal/mol for surfaces when paired with adequate active spaces.

Density-Based Approaches

Density-based approaches to electronic correlation are fundamentally rooted in (DFT), a framework that maps the many-electron problem onto a non-interacting system governed by the ρ(r)\rho(\mathbf{r}). The Hohenberg-Kohn theorems establish that the ground-state energy is a unique functional of the density, and the exact energy functional can be variationally minimized to yield the correct density. In practice, the Kohn-Sham formulation introduces auxiliary orbitals {ϕi}\{\phi_i\} satisfying −12∇2ϕi(r)+veff(r)ϕi(r)=ϵiϕi(r),-\frac{1}{2}\nabla^2 \phi_i(\mathbf{r}) + v_{\text{eff}}(\mathbf{r}) \phi_i(\mathbf{r}) = \epsilon_i \phi_i(\mathbf{r}), where the effective potential veff=vext+vH+vxcv_{\text{eff}} = v_{\text{ext}} + v_{\text{H}} + v_{\text{xc}} includes the external, Hartree, and exchange-correlation components, with ρ(r)=∑i∣ϕi(r)∣2\rho(\mathbf{r}) = \sum_i |\phi_i(\mathbf{r})|^2. Electronic correlation enters via the exchange-correlation energy Exc[ρ]E_{\text{xc}}[\rho], which accounts for quantum effects beyond the classical Coulomb repulsion. The simplest approximations to ExcE_{\text{xc}} are local density approximations (LDAs), which assume the exchange-correlation energy density is that of a uniform electron gas: ExcLDA[ρ]=∫excUEG(ρ(r)) drE_{\text{xc}}^{\text{LDA}}[\rho] = \int e_{\text{xc}}^{\text{UEG}}(\rho(\mathbf{r})) \, d\mathbf{r}, parametrized from quantum Monte Carlo data for the uniform gas correlation energy. LDAs capture short-range dynamic correlation reasonably for atoms and simple molecules but suffer from self-interaction errors and poor description of non-uniform systems, leading to overbinding in molecules. Generalized gradient approximations (GGAs) improve upon LDAs by incorporating density gradients, as in the Perdew-Burke-Ernzerhof (PBE) functional: ExcPBE[ρ]=∫fxc(ρ,∇ρ) drE_{\text{xc}}^{\text{PBE}}[\rho] = \int f_{\text{xc}}(\rho, \nabla\rho) \, d\mathbf{r}, where fxcf_{\text{xc}} enhances correlation for slowly varying densities and reduces self-interaction. GGAs like PBE provide balanced performance for molecular geometries and reaction energies, though they underestimate long-range dispersion. Hybrid functionals address limitations in capturing exact exchange, mixing a fraction of Hartree-Fock exchange with DFT terms, as in B3LYP: ExcB3LYP=0.20ExHF+0.80ExLDA+0.72ΔExB88+0.19EcVWN+0.81EcLYPE_{\text{xc}}^{\text{B3LYP}} = 0.20 E_x^{\text{HF}} + 0.80 E_x^{\text{LDA}} + 0.72 \Delta E_x^{\text{B88}} + 0.19 E_c^{\text{VWN}} + 0.81 E_c^{\text{LYP}}. This admixture better describes static correlation in transition states and charge-transfer excitations, achieving chemical accuracy (mean absolute errors ~3-5 kcal/mol) for diverse thermochemistry benchmarks. Meta-GGA functionals further include the kinetic energy density τ(r)=12∑i∣∇ϕi∣2\tau(\mathbf{r}) = \frac{1}{2} \sum_i |\nabla \phi_i|^2, enabling orbital-dependent corrections that reduce delocalization errors, as in the TPSS functional, which improves cohesive energies in solids by 20-30% over GGAs. Range-separated hybrids, such as CAM-B3LYP, partition the Coulomb operator into short- and long-range parts using the error function erf(μr)/r\text{erf}(\mu r)/r, assigning exact exchange to long-range interactions to better handle dispersion and Rydberg states. Beyond approximate functionals, the random phase approximation (RPA) provides a parameter-free treatment of correlation energy by summing ring diagrams in the adiabatic connection fluctuation-dissipation theorem: EcRPA=12∫01dλTr[ln⁡(1−χ0(λ)v)+χ0(λ)v]E_c^{\text{RPA}} = \frac{1}{2} \int_0^1 d\lambda \text{Tr} \left[ \ln(1 - \chi_0(\lambda) v) + \chi_0(\lambda) v \right], where χ0\chi_0 is the non-interacting response function from Kohn-Sham orbitals and vv the Coulomb interaction. RPA excels in describing van der Waals forces and dispersion-dominated systems, with errors under 0.1 eV/atom for layered materials like graphite, outperforming GGAs without empirical damping. However, RPA often overestimates correlation in covalent bonds, necessitating inclusion of exchange (RPAx) or higher-order terms for balanced accuracy across bonding types. These methods collectively enable efficient computation of correlated electron densities for systems up to thousands of atoms, though challenges persist for strong correlation in Mott insulators or multireference scenarios. Recent developments as of 2025 include approaches to approximate exchange-correlation functionals or correlation energies directly from high-accuracy data, achieving near-CCSD(T) accuracy at DFT cost for larger systems. For example, models trained on datasets enhance the treatment of dynamic and static correlation in and materials simulations.

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