{"history":[],"article":{"id":"147fdeb7-1076-4bac-bbfe-ca1e6e81ba34","title":"Localized molecular orbitals","slug":"localized-molecular-orbitals","content":":::summary\n:::\n\n# \n**Localized [molecular orbitals](https://bliptext.com/articles/molecular-orbital)** are [molecular orbitals](https://bliptext.com/articles/molecular-orbital) which are concentrated in a limited spatial region of a molecule, such as a specific bond or lone pair on a specific atom.\nThey can be used to relate molecular orbital calculations to simple bonding theories, and also to speed up [post-Hartree–Fock](https://bliptext.com/articles/post-hartree-fock) electronic structure calculations by taking advantage of the local nature of [electron correlation](https://bliptext.com/articles/electronic-correlation).\nLocalized orbitals in systems with [periodic boundary conditions](https://bliptext.com/articles/periodic-boundary-conditions) are known as [Wannier functions](https://bliptext.com/articles/wannier-functions).\nStandard [ab initio quantum chemistry methods](https://bliptext.com/articles/ab-initio-quantum-chemistry-methods) lead to delocalized orbitals that, in general, extend over an entire molecule and have the symmetry of the molecule.\nLocalized orbitals may then be found as [linear combinations](https://bliptext.com/articles/linear-combination) of the delocalized orbitals, given by an appropriate [unitary transformation](https://bliptext.com/articles/unitary-transformation).\nIn the water molecule for example, ab initio calculations show bonding character primarily in two molecular orbitals, each with electron density equally distributed among the two O-H bonds.\nThe localized orbital corresponding to one O-H bond is the sum of these two delocalized orbitals, and the localized orbital for the other O-H bond is their difference; as per [Valence bond theory](https://bliptext.com/articles/valence-bond-theory).\nFor multiple bonds and lone pairs, different localization procedures [give different orbitals](https://bliptext.com/articles/sigma-pi-and-equivalent-orbital-models).\nThe Boys and Edmiston-Ruedenberg localization methods mix these orbitals to give equivalent [bent bonds](https://bliptext.com/articles/bent-bond) in ethylene and rabbit ear [lone pairs](https://bliptext.com/articles/lone-pair) in water, while the Pipek-Mezey method preserves their respective [σ and π symmetry](https://bliptext.com/articles/molecular-orbital).\n\n# Equivalence of localized and delocalized orbital descriptions\nFor molecules with a closed electron shell, in which each molecular orbital is doubly occupied, the localized and delocalized orbital descriptions are in fact equivalent and represent the same physical state.\nIt might seem, again using the example of water, that placing two electrons in the first bond and two other electrons in the second bond is not the same as having four electrons free to move over both bonds.\nHowever, in quantum mechanics all electrons are identical and cannot be distinguished as same or other.\nThe total [wavefunction](https://bliptext.com/articles/wavefunction) must have a form which satisfies the [Pauli exclusion principle](https://bliptext.com/articles/pauli-exclusion-principle) such as a [Slater determinant](https://bliptext.com/articles/slater-determinant) (or linear combination of [Slater determinant](https://bliptext.com/articles/slater-determinant)s), and it can be shown that if two electrons are exchanged, such a function is unchanged by any unitary transformation of the doubly occupied orbitals.\nFor molecules with an open electron shell, in which some molecular orbitals are singly occupied, the electrons of alpha and beta spin must be localized separately.\nThis applies to radical species such as nitric oxide and dioxygen.\nAgain, in this case the localized and delocalized orbital descriptions are equivalent and represent the same physical state.\n\n# Computation methods\nLocalized [molecular orbitals](https://bliptext.com/articles/molecular-orbitals) (LMO) are obtained by [unitary transformation](https://bliptext.com/articles/unitary-transformation) upon a set of canonical [molecular orbitals](https://bliptext.com/articles/molecular-orbitals) (CMO).\nThe transformation usually involves the optimization (either minimization or maximization) of the expectation value of a specific operator.\nThe generic form of the localization potential is:\n,\nwhere \\hat{L} is the localization operator and \\phi_i is a molecular spatial orbital.\nMany methodologies have been developed during the past decades, differing in the form of \\hat{L}.\nThe optimization of the objective function is usually performed using pairwise Jacobi rotations.\nHowever, this approach is prone to saddle point convergence (if it even converges), and thus other approaches have also been developed, from simple conjugate gradient methods with exact line searches, to Newton-Raphson and trust-region methods.\n\n## Foster-Boys\nThe Foster-[Boys](https://bliptext.com/articles/s-francis-boys) (also known as [Boys](https://bliptext.com/articles/s-francis-boys)) localization method minimizes the spatial extent of the orbitals by minimizing, where.\nThis turns out to be equivalent to the easier task of maximizing.\nIn one dimension, the Foster-Boys (FB) objective function can also be written as\n.\n\n## Fourth moment\nThe fourth moment (FM) procedure is analogous to Foster-Boys scheme, however the orbital fourth moment is used instead of the orbital second moment.\nThe objective function to be minimized is\n.\nThe fourth moment method produces more localized virtual orbitals than Foster-Boys method, since it implies a larger penalty on the delocalized tails.\nFor graphene (a delocalized system), the fourth moment method produces more localized occupied orbitals than Foster-Boys and Pipek-Mezey schemes.\n\n## Edmiston-Ruedenberg\nEdmiston-Ruedenberg localization maximizes the electronic self-repulsion energy by maximizing, where.\n\n## Pipek-Mezey\nPipek-Mezey localization takes a slightly different approach, maximizing the sum of orbital-dependent partial charges on the nuclei:\n.\nPipek and Mezey originally used [Mulliken charges](https://bliptext.com/articles/mulliken-population-analysis), which are [mathematically ill defined](https://bliptext.com/articles/mulliken-population-analysis).\nRecently, Pipek-Mezey style schemes based on a variety of mathematically well-defined partial charge estimates have been discussed.\nSome notable choices are Voronoi charges, Becke charges, Hirshfeld or Stockholder charges, intrinsic atomic orbital charges (see [intrinsic bond orbitals](https://bliptext.com/articles/intrinsic-bond-orbitals))\", Bader charges, or \"fuzzy atom\" charges. Rather surprisingly, despite the wide variation in the (total) partial charges reproduced by the different estimates, analysis of the resulting Pipek-Mezey orbitals has shown that the localized orbitals are rather insensitive to the partial charge estimation scheme used in the localization process. However, due to the ill-defined mathematical nature of Mulliken charges (and Löwdin charges, which have also been used in some works ), as better alternatives are nowadays available it is advisable to use them in favor of the original version.\nThe most important quality of the Pipek-Mezey scheme is that it preserves σ-π separation in planar systems, which sets it apart from the Foster-Boys and Edmiston-Ruedenberg schemes that mix σ and π bonds.\nThis property holds independent of the partial charge estimate used.\nWhile the usual formulation of the Pipek-Mezey method invokes an iterative procedure to localize the orbitals, a non-iterative method has also been recently suggested.\n\n# In organic chemistry\nOrganic chemistry is often discussed in terms of localized molecular orbitals in a qualitative and informal sense.\nHistorically, much of classical organic chemistry was built on the older [valence bond](https://bliptext.com/articles/valence-bond-theory) / [orbital hybridization](https://bliptext.com/articles/orbital-hybridisation) models of bonding.\nTo account for phenomena like [aromaticity](https://bliptext.com/articles/aromaticity), this simple model of bonding is supplemented by semi-quantitative results from [Hückel molecular orbital theory](https://bliptext.com/articles/h-ckel-molecular-orbital-theory).\nHowever, the understanding of [stereoelectronic effects](https://bliptext.com/articles/stereoelectronic-effect) requires the analysis of interactions between donor and acceptor orbitals between two molecules or different regions within the same molecule, and molecular orbitals must be considered.\nBecause proper (symmetry-adapted) molecular orbitals are fully delocalized and do not admit a ready correspondence with the \"bonds\" of the molecule, as visualized by the practicing chemist, the most common approach is to instead consider the interaction between filled and unfilled localized molecular orbitals that correspond to σ bonds, π bonds, lone pairs, and their unoccupied counterparts.\nThese orbitals and typically given the notation σ (sigma bonding), π (pi bonding), n (occupied nonbonding orbital, \"lone pair\"), p (unoccupied nonbonding orbital, \"empty p orbital\"; the symbol n* for unoccupied nonbonding orbital is seldom used), π* (pi antibonding), and σ* (sigma antibonding).\n(Woodward and Hoffmann use ω for nonbonding orbitals in general, occupied or unoccupied.) When comparing localized molecular orbitals derived from the same atomic orbitals, these classes generally follow the order σ < π < n < p (n*) < π* < σ* when ranked by increasing energy.\nThe localized molecular orbitals that organic chemists often depict can be thought of as qualitative renderings of orbitals generated by the computational methods described above.\nHowever, they do not map onto any single approach, nor are they used consistently.\nFor instance, the [lone pairs](https://bliptext.com/articles/lone-pair) of water are usually treated as two equivalent spx hybrid orbitals, while the corresponding \"nonbonding\" orbitals of [carbenes](https://bliptext.com/articles/carbene) are generally treated as a filled σ(out) orbital and an unfilled pure p orbital, even though the [lone pairs](https://bliptext.com/articles/lone-pair) of water could be described analogously by filled σ(out) and p orbitals (for further discussion, see the article on [lone pair](https://bliptext.com/articles/lone-pair) and the discussion above on [sigma-pi and equivalent-orbital models](https://bliptext.com/articles/sigma-pi-and-equivalent-orbital-models)).\nIn other words, the type of localized orbital invoked depends on context and considerations of convenience and utility.\n# [A list of localized molecular orbitals considered in organic chemistry, showing component atomic orbitals and all shapes of the MOs they constitute. In reality, AOs and MOs, as obtained from computations, are much \"fatter\" than depicted in these cartoons. | upload.wikimedia.org/wikipedia/commons/2/2d/Localized///MOs.png]","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T09:24:51.976Z","updated_at":"2025-01-26T09:24:51.976Z","search_vector":"'local':1 'molecular':2 'orbit':3","search_content":" \n \n\n# \n**Localized [molecular orbitals](https://bliptext.com/articles/molecular-orbital)** are [molecular orbitals](https://bliptext.com/articles/molecular-orbital) which are concentrated in a limited spatial region of a molecule, such as a specific bond or lone pair on a specific atom.\nThey can be used to relate molecular orbital calculations to simple bonding theories, and also to speed up [post-Hartree–Fock](https://bliptext.com/articles/post-hartree-fock) electronic structure calculations by taking advantage of the local nature of [electron correlation](https://bliptext.com/articles/electronic-correlation).\nLocalized orbitals in systems with [periodic boundary conditions](https://bliptext.com/articles/periodic-boundary-conditions) are known as [Wannier functions](https://bliptext.com/articles/wannier-functions).\nStandard [ab initio quantum chemistry methods](https://bliptext.com/articles/ab-initio-quantum-chemistry-methods) lead to delocalized orbitals that, in general, extend over an entire molecule and have the symmetry of the molecule.\nLocalized orbitals may then be found as [linear combinations](https://bliptext.com/articles/linear-combination) of the delocalized orbitals, given by an appropriate [unitary transformation](https://bliptext.com/articles/unitary-transformation).\nIn the water molecule for example, ab initio calculations show bonding character primarily in two molecular orbitals, each with electron density equally distributed among the two O-H bonds.\nThe localized orbital corresponding to one O-H bond is the sum of these two delocalized orbitals, and the localized orbital for the other O-H bond is their difference; as per [Valence bond theory](https://bliptext.com/articles/valence-bond-theory).\nFor multiple bonds and lone pairs, different localization procedures [give different orbitals](https://bliptext.com/articles/sigma-pi-and-equivalent-orbital-models).\nThe Boys and Edmiston-Ruedenberg localization methods mix these orbitals to give equivalent [bent bonds](https://bliptext.com/articles/bent-bond) in ethylene and rabbit ear [lone pairs](https://bliptext.com/articles/lone-pair) in water, while the Pipek-Mezey method preserves their respective [σ and π symmetry](https://bliptext.com/articles/molecular-orbital).\n\n# Equivalence of localized and delocalized orbital descriptions\nFor molecules with a closed electron shell, in which each molecular orbital is doubly occupied, the localized and delocalized orbital descriptions are in fact equivalent and represent the same physical state.\nIt might seem, again using the example of water, that placing two electrons in the first bond and two other electrons in the second bond is not the same as having four electrons free to move over both bonds.\nHowever, in quantum mechanics all electrons are identical and cannot be distinguished as same or other.\nThe total [wavefunction](https://bliptext.com/articles/wavefunction) must have a form which satisfies the [Pauli exclusion principle](https://bliptext.com/articles/pauli-exclusion-principle) such as a [Slater determinant](https://bliptext.com/articles/slater-determinant) (or linear combination of [Slater determinant](https://bliptext.com/articles/slater-determinant)s), and it can be shown that if two electrons are exchanged, such a function is unchanged by any unitary transformation of the doubly occupied orbitals.\nFor molecules with an open electron shell, in which some molecular orbitals are singly occupied, the electrons of alpha and beta spin must be localized separately.\nThis applies to radical species such as nitric oxide and dioxygen.\nAgain, in this case the localized and delocalized orbital descriptions are equivalent and represent the same physical state.\n\n# Computation methods\nLocalized [molecular orbitals](https://bliptext.com/articles/molecular-orbitals) (LMO) are obtained by [unitary transformation](https://bliptext.com/articles/unitary-transformation) upon a set of canonical [molecular orbitals](https://bliptext.com/articles/molecular-orbitals) (CMO).\nThe transformation usually involves the optimization (either minimization or maximization) of the expectation value of a specific operator.\nThe generic form of the localization potential is:\n,\nwhere \\hat{L} is the localization operator and \\phi_i is a molecular spatial orbital.\nMany methodologies have been developed during the past decades, differing in the form of \\hat{L}.\nThe optimization of the objective function is usually performed using pairwise Jacobi rotations.\nHowever, this approach is prone to saddle point convergence (if it even converges), and thus other approaches have also been developed, from simple conjugate gradient methods with exact line searches, to Newton-Raphson and trust-region methods.\n\n## Foster-Boys\nThe Foster-[Boys](https://bliptext.com/articles/s-francis-boys) (also known as [Boys](https://bliptext.com/articles/s-francis-boys)) localization method minimizes the spatial extent of the orbitals by minimizing, where.\nThis turns out to be equivalent to the easier task of maximizing.\nIn one dimension, the Foster-Boys (FB) objective function can also be written as\n.\n\n## Fourth moment\nThe fourth moment (FM) procedure is analogous to Foster-Boys scheme, however the orbital fourth moment is used instead of the orbital second moment.\nThe objective function to be minimized is\n.\nThe fourth moment method produces more localized virtual orbitals than Foster-Boys method, since it implies a larger penalty on the delocalized tails.\nFor graphene (a delocalized system), the fourth moment method produces more localized occupied orbitals than Foster-Boys and Pipek-Mezey schemes.\n\n## Edmiston-Ruedenberg\nEdmiston-Ruedenberg localization maximizes the electronic self-repulsion energy by maximizing, where.\n\n## Pipek-Mezey\nPipek-Mezey localization takes a slightly different approach, maximizing the sum of orbital-dependent partial charges on the nuclei:\n.\nPipek and Mezey originally used [Mulliken charges](https://bliptext.com/articles/mulliken-population-analysis), which are [mathematically ill defined](https://bliptext.com/articles/mulliken-population-analysis).\nRecently, Pipek-Mezey style schemes based on a variety of mathematically well-defined partial charge estimates have been discussed.\nSome notable choices are Voronoi charges, Becke charges, Hirshfeld or Stockholder charges, intrinsic atomic orbital charges (see [intrinsic bond orbitals](https://bliptext.com/articles/intrinsic-bond-orbitals))\", Bader charges, or \"fuzzy atom\" charges. Rather surprisingly, despite the wide variation in the (total) partial charges reproduced by the different estimates, analysis of the resulting Pipek-Mezey orbitals has shown that the localized orbitals are rather insensitive to the partial charge estimation scheme used in the localization process. However, due to the ill-defined mathematical nature of Mulliken charges (and Löwdin charges, which have also been used in some works ), as better alternatives are nowadays available it is advisable to use them in favor of the original version.\nThe most important quality of the Pipek-Mezey scheme is that it preserves σ-π separation in planar systems, which sets it apart from the Foster-Boys and Edmiston-Ruedenberg schemes that mix σ and π bonds.\nThis property holds independent of the partial charge estimate used.\nWhile the usual formulation of the Pipek-Mezey method invokes an iterative procedure to localize the orbitals, a non-iterative method has also been recently suggested.\n\n# In organic chemistry\nOrganic chemistry is often discussed in terms of localized molecular orbitals in a qualitative and informal sense.\nHistorically, much of classical organic chemistry was built on the older [valence bond](https://bliptext.com/articles/valence-bond-theory) / [orbital hybridization](https://bliptext.com/articles/orbital-hybridisation) models of bonding.\nTo account for phenomena like [aromaticity](https://bliptext.com/articles/aromaticity), this simple model of bonding is supplemented by semi-quantitative results from [Hückel molecular orbital theory](https://bliptext.com/articles/h-ckel-molecular-orbital-theory).\nHowever, the understanding of [stereoelectronic effects](https://bliptext.com/articles/stereoelectronic-effect) requires the analysis of interactions between donor and acceptor orbitals between two molecules or different regions within the same molecule, and molecular orbitals must be considered.\nBecause proper (symmetry-adapted) molecular orbitals are fully delocalized and do not admit a ready correspondence with the \"bonds\" of the molecule, as visualized by the practicing chemist, the most common approach is to instead consider the interaction between filled and unfilled localized molecular orbitals that correspond to σ bonds, π bonds, lone pairs, and their unoccupied counterparts.\nThese orbitals and typically given the notation σ (sigma bonding), π (pi bonding), n (occupied nonbonding orbital, \"lone pair\"), p (unoccupied nonbonding orbital, \"empty p orbital\"; the symbol n* for unoccupied nonbonding orbital is seldom used), π* (pi antibonding), and σ* (sigma antibonding).\n(Woodward and Hoffmann use ω for nonbonding orbitals in general, occupied or unoccupied.) When comparing localized molecular orbitals derived from the same atomic orbitals, these classes generally follow the order σ < π < n < p (n*) < π* < σ* when ranked by increasing energy.\nThe localized molecular orbitals that organic chemists often depict can be thought of as qualitative renderings of orbitals generated by the computational methods described above.\nHowever, they do not map onto any single approach, nor are they used consistently.\nFor instance, the [[lone pair](https://bliptext.com/articles/lone-pair)s](https://bliptext.com/articles/lone-pair) of water are usually treated as two equivalent spx hybrid orbitals, while the corresponding \"nonbonding\" orbitals of [carbenes](https://bliptext.com/articles/carbene) are generally treated as a filled σ(out) orbital and an unfilled pure p orbital, even though the [[lone pair](https://bliptext.com/articles/lone-pair)s](https://bliptext.com/articles/lone-pair) of water could be described analogously by filled σ(out) and p orbitals (for further discussion, see the article on [lone pair](https://bliptext.com/articles/lone-pair) and the discussion above on [sigma-pi and equivalent-orbital models](https://bliptext.com/articles/sigma-pi-and-equivalent-orbital-models)).\nIn other words, the type of localized orbital invoked depends on context and considerations of convenience and utility.\n# [A list of localized molecular orbitals considered in organic chemistry, showing component atomic orbitals and all shapes of the MOs they constitute. In reality, AOs and MOs, as obtained from computations, are much \"fatter\" than depicted in these cartoons. | upload.wikimedia.org/wikipedia/commons/2/2d/Localized///MOs.png]","revisionCount":0},"hasMore":false,"nextCursor":null}

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