Orbital hybridisation

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In chemistry, orbital hybridisation (or hybridization) is the concept of mixing atomic orbitals into new hybrid orbitals (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds in valence bond theory. Hybrid orbitals are very useful in the explanation of molecular geometry and atomic bonding properties and are symmetrically disposed in space. Although sometimes taught together with the valence shell electron-pair repulsion (VSEPR) theory, valence bond and hybridisation are in fact not related to the VSEPR model.

History and uses

Chemist Linus Pauling first developed the hybridisation theory in 1931 to explain the structure of simple molecules such as methane (CH₄) using atomic orbitals. Pauling pointed out that a carbon atom forms four bonds by using one s and three p orbitals, so that "it might be inferred" that a carbon atom would form three bonds at right angles (using p orbitals) and a fourth weaker bond using the s orbital in some arbitrary direction. In reality, methane has four bonds of equivalent strength separated by the tetrahedral bond angle of 109.5°. Pauling explained this by supposing that in the presence of four hydrogen atoms, the s and p orbitals form four equivalent combinations or hybrid orbitals, each denoted by sp³ to indicate its composition, which are directed along the four C-H bonds. This concept was developed for such simple chemical systems, but the approach was later applied more widely, and today it is considered an effective heuristic for rationalising the structures of organic compounds. It gives a simple orbital picture equivalent to Lewis structures. 

Hybridisation theory is an integral part of organic chemistry, one of the most compelling examples being Baldwin's rules. For drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons. Hybridisation theory explains bonding in alkenes and methane. The amount of p character or s character, which is decided mainly by orbital hybridisation, can be used to reliably predict molecular properties such as acidity or basicity.

Overview

Orbitals are a model representation of the behaviour of electrons within molecules. In the case of simple hybridisation, this approximation is based on atomic orbitals, similar to those obtained for the hydrogen atom, the only neutral atom for which the Schrödinger equation can be solved exactly. In heavier atoms, such as carbon, nitrogen, and oxygen, the atomic orbitals used are the 2s and 2p orbitals, similar to excited state orbitals for hydrogen.

Hybrid orbitals are assumed to be mixtures of atomic orbitals, superimposed on each other in various proportions. For example, in methane, the C hybrid orbital which forms each carbon–hydrogen bond consists of 25% s character and 75% p character and is thus described as sp³ (read as s-p-three) hybridised. Quantum mechanics describes this hybrid as an sp³ wavefunction of the form N(s + √3pσ), where N is a normalisation constant (here 1/2) and pσ is a p orbital directed along the C-H axis to form a sigma bond. The ratio of coefficients (denoted λ in general) is √3 in this example. Since the electron density associated with an orbital is proportional to the square of the wavefunction, the ratio of p-character to s-character is λ² = 3. The p character or the weight of the p component is N²λ² = 3/4.

Types of hybridisation

sp³

Four sp³ orbitals.

Hybridisation describes the bonding atoms from an atom's point of view. For a tetrahedrally coordinated carbon (e.g., methane CH₄), the carbon should have 4 orbitals with the correct symmetry to bond to the 4 hydrogen atoms.

Carbon's ground state configuration is 1s² 2s² 2p² or more easily read: 

C	↑↓	↑↓	↑	↑
	1s	2s	2p	2p	2p

The carbon atom can use its two singly occupied p-type orbitals, to form two covalent bonds with two hydrogen atoms, yielding the singlet methylene CH₂, the simplest carbene. The carbon atom can also bond to four hydrogen atoms by an excitation (or promotion) of an electron from the doubly occupied 2s orbital to the empty 2p orbital, producing four singly occupied orbitals. 

C*	↑↓	↑	↑	↑	↑
	1s	2s	2p	2p	2p

The energy released by formation of two additional bonds more than compensates for the excitation energy required, energetically favouring the formation of four C-H bonds.

Quantum mechanically, the lowest energy is obtained if the four bonds are equivalent, which requires that they are formed from equivalent orbitals on the carbon. A set of four equivalent orbitals can be obtained that are linear combinations of the valence-shell (core orbitals are almost never involved in bonding) s and p wave functions, which are the four sp³ hybrids. 

C*	↑↓	↑	↑	↑	↑
	1s	sp3	sp3	sp3	sp3

In CH₄, four sp³ hybrid orbitals are overlapped by hydrogen 1s orbitals, yielding four σ (sigma) bonds (that is, four single covalent bonds) of equal length and strength. 

 

sp²

Three sp² orbitals.

 

Ethene structure

 

Other carbon compounds and other molecules may be explained in a similar way. For example, ethene (C₂H₄) has a double bond between the carbons. 

For this molecule, carbon sp² hybridises, because one π (pi) bond is required for the double bond between the carbons and only three σ bonds are formed per carbon atom. In sp² hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals, 

C*	↑↓	↑	↑	↑	↑
	1s	sp2	sp2	sp2	2p

forming a total of three sp² orbitals with one remaining p orbital. In ethylene (ethene) the two carbon atoms form a σ bond by overlapping one sp² orbital from each carbon atom. The π bond between the carbon atoms perpendicular to the molecular plane is formed by 2p–2p overlap. Each carbon atom forms covalent C–H bonds with two hydrogens by s–sp² overlap, all with 120° bond angles. The hydrogen–carbon bonds are all of equal strength and length, in agreement with experimental data.

sp

Two sp orbitals

 

The chemical bonding in compounds such as alkynes with triple bonds is explained by sp hybridisation. In this model, the 2s orbital is mixed with only one of the three p orbitals, 

C*	↑↓	↑	↑	↑	↑
	1s	sp	sp	2p	2p

resulting in two sp orbitals and two remaining p orbitals. The chemical bonding in acetylene (ethyne) (C₂H₂) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by p–p overlap. Each carbon also bonds to hydrogen in a σ s–sp overlap at 180° angles.

Hybridisation and molecule shape

Hybridisation helps to explain molecule shape, since the angles between bonds are (approximately) equal to the angles between hybrid orbitals, as explained above for the tetrahedral geometry of methane. As another example, the three sp² hybrid orbitals are at angles of 120° to each other, so this hybridisation favours trigonal planar molecular geometry with bond angles of 120°. Other examples are given in the table below. 

Classification	Main group	Transition metal
AX2	Linear (180°) sp hybridisation E.g., CO2	Bent (90°) sd hybridisation E.g., VO2+
AX3	Trigonal planar (120°) sp2 hybridisation E.g., BCl3	Trigonal pyramidal (90°) sd2 hybridisation E.g., CrO3
AX4	Tetrahedral (109.5°) sp3 hybridisation E.g., CCl4	Tetrahedral (109.5°) sd3 hybridisation E.g., TiCl4
AX6		C3v Trigonal prismatic (63.4°, 116.6°) sd5 hybridisation E.g., W(CH3)6

For two equivalent sp^x hybrids, the bond angle between them is given by ${\displaystyle \theta =\arccos(-{\frac {1}{x}})}$, while for two equivalent sd^x hybrids, the bond angle between them is given by ${\displaystyle \theta =\arccos \left(\pm {\sqrt {{\frac {1}{3}}(1-{\frac {2}{x}})}}\right)}$. There are no shapes with ideal bond angles corresponding to sd⁴ hybridisation as there is no regular 10-vertex polyhedron. Nonetheless, molecules such as Ta(CH₃)₅ with sd⁴ hybridisation adopt a square pyramidal shape.

Hybridisation of hypervalent molecules

Valence shell expansion

Hybridisation is often presented for main group AX₅ and above, as well as for many transition metal complexes, using the hybridisation scheme first proposed by Pauling. 

Classification Main group AX5 Trigonal bipyramidal (90°, 120°) sp3d hybridisation E.g., PF5 AX6 Octahedral (90°) sp3d2 hybridisation E.g., SF6 AX7 Pentagonal bipyramidal (90°, 72°) sp3d3 hybridisation E.g., IF7

Classification Transition metal AX4 (d8) Square planar (90°) dsp2 hybridisation E.g., PtCl42− AX6 (d6) Octahedral (90°) d2sp3 hybridisation E.g., Mo(CO)6 Note: AX5 and AX7 have variable molecular shapes

In this notation, d orbitals of main group atoms are listed after the s and p orbitals since they have the same principal quantum number (n), while d orbitals of transition metals are listed first since the s and p orbitals have a higher n. Thus for AX₆ molecules, sp³d² hybridisation in the S atom involves 3s, 3p and 3d orbitals, while d²sp³ for Mo involves 4d, 5s and 5p orbitals.

Contrary evidence

In 1990, Magnusson published a seminal work definitively excluding the role of d-orbital hybridisation in bonding in hypervalent compounds of second-row (period 3) elements, ending a point of contention and confusion. Part of the confusion originates from the fact that d-functions are essential in the basis sets used to describe these compounds (or else unreasonably high energies and distorted geometries result). Also, the contribution of the d-function to the molecular wavefunction is large. These facts were incorrectly interpreted to mean that d-orbitals must be involved in bonding.

For transition metal centres, the d and s orbitals are the primary valence orbitals, which are only weakly supplemented by the p orbitals. The question of whether the p orbitals actually participate in bonding has not been definitively resolved, but all studies indicate they play a minor role.

Resonance

In light of computational chemistry, a better treatment would be to invoke sigma bond resonance in addition to hybridisation, which implies that each resonance structure has its own hybridisation scheme. For main group compounds, all resonance structures must obey the octet (8-electron) rule. For transition metal compounds, the resonance structures that obey the duodectet (12-electron) rule suffice to describe bonding, with optional inclusion of d^mspⁿ resonance structures.

Isovalent hybridisation

Although ideal hybrid orbitals can be useful, in reality most bonds require orbitals of intermediate character. This requires an extension to include flexible weightings of atomic orbitals of each type (s, p, d) and allows for a quantitative depiction of bond formation when the molecular geometry deviates from ideal bond angles. The amount of p-character is not restricted to integer values; i.e., hybridisations like sp^2.5 are also readily described.

The hybridisation of bond orbitals is determined by Bent's rule: "Atomic s character concentrates in orbitals directed towards electropositive substituents".

Molecules with lone pairs

For molecules with lone pairs, the bonding orbitals are isovalent sp^x hybrids. For example, the two bond-forming hybrid orbitals of oxygen in water can be described as sp^4.0 to give the interorbital angle of 104.5°. This means that they have 20% s character and 80% p character and does not imply that a hybrid orbital is formed from one s and four p orbitals on oxygen since the 2p subshell of oxygen only contains three p orbitals. The shapes of molecules with lone pairs are:

• Trigonal pyramidal (>90°)
  • Three sp^x bond hybrids
  • E.g., NH₃
• Bent (>90°)
  • Two sp^x bond hybrids
  • E.g., SO₂, H₂O

In such cases, there are two mathematically equivalent ways of representing lone pairs. They can be represented by orbitals of sigma and pi symmetry similar to molecular orbital theory or by equivalent orbitals similar to VSEPR theory.

Hypervalent molecules

For hypervalent molecules with lone pairs, the bonding scheme can be split into a hypervalent component and a component consisting of isovalent sp^x bond hybrids. The hypervalent component consists of resonant bonds using p orbitals. The table below shows how each shape is related to the two components and their respective descriptions.

Hybridisation defects

Hybridisation of s and p orbitals to form effective sp^x hybrids requires that they have comparable radial extent. While 2p orbitals are on average less than 10% larger than 2s, in part attributable to the lack of a radial node in 2p orbitals, 3p orbitals which have one radial node, exceed the 3s orbitals by 20–33%. The difference in extent of s and p orbitals increases further down a group. The hybridisation of atoms in chemical bonds can be analysed by considering localised molecular orbitals, for example using natural localised molecular orbitals in a natural bond orbital (NBO) scheme. In methane, CH₄, the calculated p/s ratio is approximately 3 consistent with "ideal" sp³ hybridisation, whereas for silane, SiH₄, the p/s ratio is closer to 2. A similar trend is seen for the other 2p elements. Substitution of fluorine for hydrogen further decreases the p/s ratio. The 2p elements exhibit near ideal hybridisation with orthogonal hybrid orbitals. For heavier p block elements this assumption of orthogonality cannot be justified. These deviations from the ideal hybridisation were termed hybridisation defects by Kutzelnigg.

Photoelectron spectra

One misconception concerning orbital hybridisation is that it incorrectly predicts the ultraviolet photoelectron spectra of many molecules. While this is true if Koopmans' theorem is applied to localised hybrids, quantum mechanics requires that the (in this case ionised) wavefunction obey the symmetry of the molecule which implies resonance in valence bond theory. For example, in methane, the ionised states (CH₄⁺) can be constructed out of four resonance structures attributing the ejected electron to each of the four sp³ orbitals. A linear combination of these four structures, conserving the number of structures, leads to a triply degenerate T₂ state and a A₁ state. The difference in energy between each ionised state and the ground state would be an ionisation energy, which yields two values in agreement with experiment.

Localized MOs vs canonical MOs

Bonding orbitals formed from hybrid atomic orbitals may be considered as localized molecular orbitals, which can be formed from the delocalised orbitals of molecular orbital theory by an appropriate mathematical transformation. For molecules in the ground state, this transformation of the orbitals leaves the total many-electron wave function unchanged. The hybrid orbital description of the ground state is therefore equivalent to the delocalised orbital description for ground state total energy and electron density, as well as the molecular geometry that corresponds to the minimum total energy value.

Two localized representations

Molecules with multiple bonds or multiple lone pairs can have orbitals represented in terms of sigma and pi symmetry or equivalent orbitals. Different valence bond methods use either of the two representations, which have mathematically equivalent total many-electron wave functions and are related by a unitary transformation of the set of occupied molecular orbitals. 

For multiple bonds, the sigma-pi representation is the predominant one compared to the equivalent orbital (bent bond) representation. In contrast, for multiple lone pairs, most textbooks use the equivalent orbital representation. However, the sigma-pi representation is also used, such as by Weinhold and Landis within the context of natural bond orbitals, a localized orbital theory containing modernized analogs of classical (valence bond/Lewis structure) bonding pairs and lone pairs. For the hydrogen fluoride molecule, for example, two F lone pairs are essentially unhybridized p orbitals, while the other is an sp^x hydrid orbital. An analogous consideration applies to water (one O lone pair is in a pure p orbital, another is in an sp^x hybrid orbital).

Covalent bond

From Wikipedia, the free encyclopedia

A covalent bond forming H₂ (right) where two hydrogen atoms share the two electrons

 

A covalent bond, also called a molecular bond, is a chemical bond that involves the sharing of electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs, and the stable balance of attractive and repulsive forces between atoms, when they share electrons, is known as covalent bonding. For many molecules, the sharing of electrons allows each atom to attain the equivalent of a full outer shell, corresponding to a stable electronic configuration. 

Covalent bonding includes many kinds of interactions, including σ-bonding, π-bonding, metal-to-metal bonding, agostic interactions, bent bonds, and three-center two-electron bonds. The term covalent bond dates from 1939. The prefix co- means jointly, associated in action, partnered to a lesser degree, etc.; thus a "co-valent bond", in essence, means that the atoms share "valence", such as is discussed in valence bond theory. 

In the molecule H
₂, the hydrogen atoms share the two electrons via covalent bonding. Covalency is greatest between atoms of similar electronegativities. Thus, covalent bonding does not necessarily require that the two atoms be of the same elements, only that they be of comparable electronegativity. Covalent bonding that entails sharing of electrons over more than two atoms is said to be delocalized.

History

Early concepts in covalent bonding arose from this kind of image of the molecule of methane. Covalent bonding is implied in the Lewis structure by indicating electrons shared between atoms.

 

The term covalence in regard to bonding was first used in 1919 by Irving Langmuir in a Journal of the American Chemical Society article entitled "The Arrangement of Electrons in Atoms and Molecules". Langmuir wrote that "we shall denote by the term covalence the number of pairs of electrons that a given atom shares with its neighbors."

The idea of covalent bonding can be traced several years before 1919 to Gilbert N. Lewis, who in 1916 described the sharing of electron pairs between atoms. He introduced the Lewis notation or electron dot notation or Lewis dot structure, in which valence electrons (those in the outer shell) are represented as dots around the atomic symbols. Pairs of electrons located between atoms represent covalent bonds. Multiple pairs represent multiple bonds, such as double bonds and triple bonds. An alternative form of representation, not shown here, has bond-forming electron pairs represented as solid lines.

Lewis proposed that an atom forms enough covalent bonds to form a full (or closed) outer electron shell. In the diagram of methane shown here, the carbon atom has a valence of four and is, therefore, surrounded by eight electrons (the octet rule), four from the carbon itself and four from the hydrogens bonded to it. Each hydrogen has a valence of one and is surrounded by two electrons (a duet rule) – its own one electron plus one from the carbon. The numbers of electrons correspond to full shells in the quantum theory of the atom; the outer shell of a carbon atom is the n = 2 shell, which can hold eight electrons, whereas the outer (and only) shell of a hydrogen atom is the n = 1 shell, which can hold only two.

While the idea of shared electron pairs provides an effective qualitative picture of covalent bonding, quantum mechanics is needed to understand the nature of these bonds and predict the structures and properties of simple molecules. Walter Heitler and Fritz London are credited with the first successful quantum mechanical explanation of a chemical bond (molecular hydrogen) in 1927. Their work was based on the valence bond model, which assumes that a chemical bond is formed when there is good overlap between the atomic orbitals of participating atoms.

Types of covalent bonds

Atomic orbitals (except for s orbitals) have specific directional properties leading to different types of covalent bonds. Sigma (σ) bonds are the strongest covalent bonds and are due to head-on overlapping of orbitals on two different atoms. A single bond is usually a σ bond. Pi (π) bonds are weaker and are due to lateral overlap between p (or d) orbitals. A double bond between two given atoms consists of one σ and one π bond, and a triple bond is one σ and two π bonds. 

Covalent bonds are also affected by the electronegativity of the connected atoms which determines the chemical polarity of the bond. Two atoms with equal electronegativity will make nonpolar covalent bonds such as H–H. An unequal relationship creates a polar covalent bond such as with H−Cl. However polarity also requires geometric asymmetry, or else dipoles may cancel out resulting in a non-polar molecule.

Covalent structures

There are several types of structures for covalent substances, including individual molecules, molecular structures, macromolecular structures and giant covalent structures. Individual molecules have strong bonds that hold the atoms together, but there are negligible forces of attraction between molecules. Such covalent substances are usually gases, for example, HCl, SO₂, CO₂, and CH₄. In molecular structures, there are weak forces of attraction. Such covalent substances are low-boiling-temperature liquids (such as ethanol), and low-melting-temperature solids (such as iodine and solid CO₂). Macromolecular structures have large numbers of atoms linked by covalent bonds in chains, including synthetic polymers such as polyethylene and nylon, and biopolymers such as proteins and starch. Network covalent structures (or giant covalent structures) contain large numbers of atoms linked in sheets (such as graphite), or 3-dimensional structures (such as diamond and quartz). These substances have high melting and boiling points, are frequently brittle, and tend to have high electrical resistivity. Elements that have high electronegativity, and the ability to form three or four electron pair bonds, often form such large macromolecular structures.

One- and three-electron bonds

Lewis and MO diagrams of an individual 2e bond and 3e bond

 

Bonds with one or three electrons can be found in radical species, which have an odd number of electrons. The simplest example of a 1-electron bond is found in the dihydrogen cation, H⁺
₂. One-electron bonds often have about half the bond energy of a 2-electron bond, and are therefore called "half bonds". However, there are exceptions: in the case of dilithium, the bond is actually stronger for the 1-electron Li⁺
₂ than for the 2-electron Li₂. This exception can be explained in terms of hybridization and inner-shell effects.

The simplest example of three-electron bonding can be found in the helium dimer cation, He⁺
₂. It is considered a "half bond" because it consists of only one shared electron (rather than two); in molecular orbital terms, the third electron is in an anti-bonding orbital which cancels out half of the bond formed by the other two electrons. Another example of a molecule containing a 3-electron bond, in addition to two 2-electron bonds, is nitric oxide, NO. The oxygen molecule, O₂ can also be regarded as having two 3-electron bonds and one 2-electron bond, which accounts for its paramagnetism and its formal bond order of 2. Chlorine dioxide and its heavier analogues bromine dioxide and iodine dioxide also contain three-electron bonds.

Molecules with odd-electron bonds are usually highly reactive. These types of bond are only stable between atoms with similar electronegativities.

Modified Lewis structures with 3e bonds:

 

Nitric oxide

 

Dioxygen

Resonance

There are situations whereby a single Lewis structure is insufficient to explain the electron configuration in a molecule, hence a superposition of structures are needed. The same two atoms in such molecules can be bonded differently in different structures (a single bond in one, a double bond in another, or even none at all), resulting in a non-integer bond order. The nitrate ion is one such example with three equivalent structures. The bond between the nitrogen and each oxygen is a double bond in one structure and a single bond in the other two, so that the average bond order for each N–O interaction is 2 + 1 + 1/3 = 4/3. 

Aromaticity

In organic chemistry, when a molecule with a planar ring obeys Hückel's rule, where the number of π electrons fit the formula 4n + 2 (where n is an integer), it attains extra stability and symmetry. In benzene, the prototypical aromatic compound, there are 6 π bonding electrons (n = 1, 4n + 2 = 6). These occupy three delocalized π molecular orbitals (molecular orbital theory) or form conjugate π bonds in two resonance structures that linearly combine (valence bond theory), creating a regular hexagon exhibiting a greater stabilization than the hypothetical 1,3,5-cyclohexatriene.

In the case of heterocyclic aromatics and substituted benzenes, the electronegativity differences between different parts of the ring may dominate the chemical behaviour of aromatic ring bonds, which otherwise are equivalent.

Hypervalence

Certain molecules such as xenon difluoride and sulfur hexafluoride have higher co-ordination numbers than would be possible due to strictly covalent bonding according to the octet rule. This is explained by the three-center four-electron bond ("3c–4e") model which interprets the molecular wavefunction in terms of non-bonding highest occupied molecular orbitals in molecular orbital theory and resonance of sigma bonds in valence bond theory.

Electron deficiency

In three-center two-electron bonds ("3c–2e") three atoms share two electrons in bonding. This type of bonding occurs in electron deficient compounds like diborane. Each such bond (2 per molecule in diborane) contains a pair of electrons which connect the boron atoms to each other in a banana shape, with a proton (nucleus of a hydrogen atom) in the middle of the bond, sharing electrons with both boron atoms. In certain cluster compounds, so-called four-center two-electron bonds also have been postulated.

Quantum mechanical description

After the development of quantum mechanics, two basic theories were proposed to provide a quantum description of chemical bonding: valence bond (VB) theory and molecular orbital (MO) theory. A more recent quantum description is given in terms of atomic contributions to the electronic density of states.

Comparison of VB and MO theories

The two theories represent two ways to build up the electron configuration of the molecule. For valence bond theory, the atomic hybrid orbitals are filled with electrons first to produce a fully bonded valence configuration, followed by performing a linear combination of contributing structures (resonance) if there are several of them. In contrast, for molecular orbital theory a linear combination of atomic orbitals is performed first, followed by filling of the resulting molecular orbitals with electrons. 

The two approaches are regarded as complementary, and each provides its own insights into the problem of chemical bonding. As valence bond theory builds the molecular wavefunction out of localized bonds, it is more suited for the calculation of bond energies and the understanding of reaction mechanisms. As molecular orbital theory builds the molecular wavefunction out of delocalized orbitals, it is more suited for the calculation of ionization energies and the understanding of spectral absorption bands.

At the qualitative level, both theories contain incorrect predictions. Simple (Heitler–London) valence bond theory correctly predicts the dissociation of homonuclear diatomic molecules into separate atoms, while simple (Hartree–Fock) molecular orbital theory incorrectly predicts dissociation into a mixture of atoms and ions. On the other hand, simple molecular orbital theory correctly predicts Hückel's rule of aromaticity, while simple valence bond theory incorrectly predicts that cyclobutadiene has a larger resonance energy than benzene. 

Although the wavefunctions generated by both theories at the qualitative level do not agree and do not match the stabilization energy by experiment, they can be corrected by configuration interaction. This is done by combining the valence bond covalent function with the functions describing all possible ionic structures or by combining the molecular orbital ground state function with the functions describing all possible excited states using unoccupied orbitals. It can then be seen that the simple molecular orbital approach overestimates the weight of the ionic structures while the simple valence bond approach neglects them. This can also be described as saying that the simple molecular orbital approach neglects electron correlation while the simple valence bond approach overestimates it.

Modern calculations in quantum chemistry usually start from (but ultimately go far beyond) a molecular orbital rather than a valence bond approach, not because of any intrinsic superiority in the former but rather because the MO approach is more readily adapted to numerical computations. Molecular orbitals are orthogonal, which significantly increases feasibility and speed of computer calculations compared to nonorthogonal valence bond orbitals. However, better valence bond programs are now available.

Covalency from atomic contribution to the electronic density of states

In COOP, COHP and BCOOP, evaluation of bond covalency is dependent on the basis set. To overcome this issue, an alternative formulation of the bond covalency can be provided in this way. 

The center mass cm(n,l,m_l,m_s) of an atomic orbital |n,l,m_l,m_s⟩, with quantum numbers n, l, m_l, m_s, for atom A is defined as

${\displaystyle cm^{\mathrm {A} }(n,l,m_{l},m_{s})={\frac {\int \limits _{E_{0}}\limits ^{E_{1}}Eg_{|n,l,m_{l},m_{s}\rangle }^{\mathrm {A} }\left(E\right)dE}{\int \limits _{E_{0}}\limits ^{E_{1}}g_{|n,l,m_{l},m_{s}\rangle }^{\mathrm {A} }\left(E\right)dE}}}$

where g^A
_{|n,l,ml,ms⟩}(E) is the contribution of the atomic orbital |n,l,m_l,m_s⟩ of the atom A to the total electronic density of states g(E) of the solid

${\displaystyle g\left(E\right)=\sum _{\mathrm {A} }\sum _{n,l}\sum _{m_{l},m_{s}}{g_{|n,l,m_{l},m_{s}\rangle }^{\mathrm {A} }\left(E\right)}}$

where the outer sum runs over all atoms A of the unit cell. The energy window [E₀,E₁] is chosen in such a way that it encompasses all relevant bands participating in the bond. If the range to select is unclear, it can be identified in practice by examining the molecular orbitals that describe the electron density along the considered bond. 

The relative position C_nAlA,nBlB of the center mass of |n_A,l_A⟩ levels of atom A with respect to the center mass of |n_B,l_B⟩ levels of atom B is given as

${\displaystyle C_{n_{\mathrm {A} }l_{\mathrm {A} },n_{\mathrm {B} }l_{\mathrm {B} }}=-\left|cm^{\mathrm {A} }(n_{\mathrm {A} },l_{\mathrm {A} })-cm^{\mathrm {B} }(n_{\mathrm {B} },l_{\mathrm {B} })\right|}$

where the contributions of the magnetic and spin quantum numbers are summed. According to this definition, the relative position of the A levels with respect to the B levels is

${\displaystyle C_{\mathrm {A,B} }=-\left|cm^{\mathrm {A} }-cm^{\mathrm {B} }\right|}$

where, for simplicity, we may omit the dependence from the principal quantum number n in the notation referring to C_nAlA,nBlB.

In this formalism, the greater the value of C_A,B, the higher the overlap of the selected atomic bands, and thus the electron density described by those orbitals gives a more covalent A–B bond. The quantity C_A,B is denoted as the covalency of the A–B bond, which is specified in the same units of the energy E.