11.3: Molecular Orbital Theory (2023)

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    Learning Objectives
    • Outline the basic quantum-mechanical approach to deriving molecular orbitals from atomic orbitals
    • Describe traits of bonding and antibonding molecular orbitals
    • Calculate bond orders based on molecular electron configurations
    • Write molecular electron configurations for first- and second-row diatomic molecules
    • Relate these electron configurations to the molecules’ stabilities and magnetic properties

    Introduction to MO Theory

    For almost any covalent molecule we can now draw the Lewis structure, predict the electron-pair geometry, predict the molecular geometry, and predict bond angles, and use the hybrid orbital model to explain the bonding and structure. However, the behavior of the oxygen molecule O2 is not consistent with predictions based on these models. We would write the following Lewis structure for O2:

    11.3: Molecular Orbital Theory (2)

    This electronic structure adheres to all the rules for Lewis theory and we can readily explain the structure and bonding. There is an O=O double bond, and each oxygen atom has eight electrons around it. However, this model is at odds with some experimental results. Although O2 is not magnetic, it is attracted to magnetic fields. When we pour liquid oxygen past a strong magnet, it collects between the poles of the magnet. This attraction to a magnetic field is called paramagnetism, and it arises in molecules with unpaired electrons. The Lewis structure of O2 shows all electrons are paired and is not consistent with this experimental observation.

    The Lewis dot structure for nitrogen gas, N2, also shows all the electrons are paired. Both N2 and O2 have similar Lewis structures with bonds and lone pair electrons that are all paired.

    11.3: Molecular Orbital Theory (3)

    Nitrogen demonstrates very different magnetic behavior than oxygen. We can pour liquid nitrogen through a magnetic field with no visible interactions, while liquid oxygen is attracted to the magnet and floats in the magnetic field. This behavior is not consistent with the Lewis structures and hybrid orbital models. A new model, molecular orbital theory, is needed to explain these observations.

    11.3: Molecular Orbital Theory (4)

    Molecular orbital theory (MO theory) provides an explanation of chemical bonding that accounts for the paramagnetism of the oxygen molecule. It also explains the bonding in a number of other molecules and more molecules with more complicated bonding that are difficult to describe with Lewis structures. Additionally, it provides a model for describing the energies of electrons in a molecule and the probable location of these electrons.

    Unlike valence bond theory, which uses hybrid orbitals that are assigned to one specific atom, MO theory uses the combination of atomic orbitals to yield molecular orbitals that are delocalized over the entire molecule. MO theory also helps us understand why some substances are electrical conductors, others are semiconductors, and still others are insulators. Table \(\PageIndex{1}\) summarizes the main points of the two complementary bonding theories. Both theories provide different, useful ways of describing molecular structure.

    Table \(\PageIndex{1}\) : Comparison of Bonding Theories
    Valence Bond Theory Molecular Orbital Theory
    considers bonds as localized between one pair of atoms considers electrons delocalized throughout the entire molecule
    creates bonds from overlap of atomic orbitals (s, p, d…) and hybrid orbitals (sp, sp2, sp3…) combines atomic orbitals to form molecular orbitals (σ, σ*, π, π*)
    forms σ or π bonds creates bonding and antibonding interactions based on which orbitals are filled
    predicts molecular shape based on the number of regions of electron density predicts the arrangement of electrons in molecules
    needs multiple structures to describe resonance

    Generating Molecular Sigma Orbitals

    We will consider the molecular orbitals in molecules composed of two identical atoms (H2 or Cl2, for example). Such molecules are called homonuclear diatomic molecules. In these diatomic molecules, several types of molecular orbitals occur.

    The mathematical process of combining atomic orbitals to generate molecular orbitals is called the linear combination of atomic orbitals (LCAO). This is similar to the mathematics for creating hybrid orbitals. The wave function describes the wavelike properties of an electron. Molecular orbitals are combinations of atomic orbital wave functions. Figure \(\PageIndex{2}\) shows how combining waves leads to constructive interference, in which peaks line up with peaks, or destructive interference, in which peaks line up with troughs. In orbitals, the waves are three dimensional, and they combine with in-phase waves producing regions with a higher probability of electron density and out-of-phase waves producing nodes, or regions of no electron density.

    11.3: Molecular Orbital Theory (5)

    There are two types of molecular orbitals that can form from the overlap of two atomic s orbitals on adjacent atoms. The two types are illustrated in Figure \(\PageIndex{3}\). The in-phase combination produces a lower energy σs molecular orbital (read as "sigma-s") in which most of the electron density is directly between the nuclei. This is similar to the overlap between hybrid orbitals to create a sigma bond. The new part in molecular orbital theory is recognizing that the out-of-phase addition (which can also be thought of as subtracting the wave functions) produces a higher energy \(σ^∗_s\) molecular orbital (read as "sigma-s-star"). This molecular orbital has a node between the nuclei. Since there is very little electron density between the two nuclei, their positive charge will push the two atoms apart - so this is called an anti-bond. The asterisk signifies that the orbital is an antibonding orbital.

    Electrons in a σs orbital are attracted by both nuclei at the same time and are more stable (of lower energy) than isolated atoms. Adding electrons to these orbitals creates a force that holds the two nuclei together, so we call these orbitals bonding orbitals. Electrons in the \(σ^∗_s\) orbitals are located well away from the region between the two nuclei. The attractive force between the nuclei and these electrons pulls the two nuclei apart. Hence, these orbitals are called antibonding orbitals. Electrons fill the lower-energy bonding orbital before the higher-energy antibonding orbital, just as they fill lower-energy atomic orbitals before they fill higher-energy atomic orbitals.

    11.3: Molecular Orbital Theory (6)

    In p orbitals, the wave function gives rise to two lobes with opposite phases, analogous to how a two-dimensional wave has both parts above and below the average. We indicate the phases by shading the orbital lobes different colors. When orbital lobes of the same phase overlap, constructive wave interference increases the electron density. When regions of opposite phase overlap, the destructive wave interference decreases electron density and creates nodes. When p orbitals overlap end to end, they create σ and σ* orbitals (Figure \(\PageIndex{4}\)). If two atoms are located along the x-axis in a Cartesian coordinate system, the two px orbitals overlap end to end and form σpx (bonding) and \(σ^∗_{px}\) (antibonding) (read as "sigma-p-x" and "sigma-p-x star," respectively). Just as with s-orbital overlap, the asterisk indicates the orbital with a node between the nuclei, which is a higher-energy, antibonding orbital.

    11.3: Molecular Orbital Theory (7)

    Generating MolecularPi Orbitals

    The side-by-side overlap of two p orbitals gives rise to a pi (\(π\)) bonding molecular orbital and a \(π\)* antibonding molecular orbital, as shown in Figure \(\PageIndex{5}\). In valence bond theory, we describe π bonds as containing a nodal plane containing the internuclear axis and perpendicular to the lobes of the p orbitals, with electron density on either side of the node. In molecular orbital theory, we describe the π orbital by this same shape, and a π bond exists when this orbital contains electrons. Electrons in this orbital interact with both nuclei and help hold the two atoms together, making it a bonding orbital. For the out-of-phase combination, there are two nodal planes created, one along the internuclear axis and a perpendicular one between the nuclei.

    11.3: Molecular Orbital Theory (8)

    In the molecular orbitals of diatomic molecules, each atom also has two sets of p orbitals oriented side by side (py and pz), so these four atomic orbitals combine pairwise to create two π orbitals and two \(π^*\) orbitals. The \(π_{py}\) and \(π^∗_{py}\) orbitals are oriented at right angles to the \(π_{pz}\) and \(π^∗_{pz}\) orbitals. Except for their orientation, the πpy and πpz orbitals are identical and have the same energy; they are degenerate orbitals. The \(π^∗_{py}\) and \(π^∗_{pz}\) antibonding orbitals are also degenerate and identical except for their orientation. A total of six molecular orbitals results from the combination of the six atomic p orbitals in two atoms: \(σ_{px}\) and \(σ^∗_{px}\), \(π_{py}\) and \(π^∗_{py}\), \(π_{pz}\) and \(π^∗_{pz}\).

    Example \(\PageIndex{1}\): Molecular Orbitals

    Predict what type (if any) of molecular orbital would result from adding the wave functions so each pair of orbitals shown overlap. The orbitals are all similar in energy.

    11.3: Molecular Orbital Theory (9)


    1. This is an in-phase combination, resulting in a σ3p orbital
    2. This will not result in a new orbital because the orbitals do not overlap. Only orbitals with the correct alignment can combine.
    3. This is an out-of-phase combination, resulting in a \(π^∗_{3p}\) orbital.
    Exercise \(\PageIndex{1}\)

    Label the molecular orbital shown as \(σ\) or \(π\), bonding or antibonding and indicate where the node occurs.

    11.3: Molecular Orbital Theory (10)


    The orbital is located along the internuclear axis, so it is a \(σ\) orbital. There is a node bisecting the internuclear axis, so it is an antibonding orbital.

    11.3: Molecular Orbital Theory (11)

    Application: Computational Chemistry in Drug Design

    While the descriptions of bonding described in this chapter involve many theoretical concepts, they also have many practical, real-world applications. For example, drug design is an important field that uses our understanding of chemical bonding to develop pharmaceuticals. This interdisciplinary area of study uses biology (understanding diseases and how they operate) to identify specific targets, such as a binding site that is involved in a disease pathway. By modeling the structures of the binding site and potential drugs, computational chemists can predict which structures can fit together and how effectively they will bind (Figure \(\PageIndex{6}\)). Thousands of potential candidates can be narrowed down to a few of the most promising candidates. These candidate molecules are then carefully tested to determine side effects, how effectively they can be transported through the body, and other factors. Dozens of important new pharmaceuticals have been discovered with the aid of computational chemistry, and new research projects are underway.

    11.3: Molecular Orbital Theory (12)

    Molecular Orbital Energy Diagrams

    The relative energy levels of atomic and molecular orbitals are typically shown in a molecular orbital diagram (Figure \(\PageIndex{7}\)). For a diatomic molecule, the atomic orbitals of one atom are shown on the left, and those of the other atom are shown on the right. Each horizontal line represents one orbital that can hold two electrons. The molecular orbitals formed by the combination of the atomic orbitals are shown in the center. Dashed lines show which of the atomic orbitals combine to form the molecular orbitals. For each pair of atomic orbitals that combine, one lower-energy (bonding) molecular orbital and one higher-energy (antibonding) orbital result. Thus we can see that combining the six 2p atomic orbitals results in three bonding orbitals (one σ and two π) and three antibonding orbitals (one σ* and two π*).

    11.3: Molecular Orbital Theory (13)

    We predict the distribution of electrons in these molecular orbitals by filling the orbitals in the same way that we fill atomic orbitals, by the Aufbau principle. Lower-energy orbitals fill first, electrons spread out among degenerate orbitals before pairing, and each orbital can hold a maximum of two electrons with opposite spins (Figure \(\PageIndex{7}\)). Just as we write electron configurations for atoms, we can write the molecular electronic configuration by listing the orbitals with superscripts indicating the number of electrons present. For clarity, we place parentheses around molecular orbitals with the same energy. In this case, each orbital is at a different energy, so parentheses separate each orbital. Thus we would expect a diatomic molecule or ion containing seven electrons (such as \(\ce{Be2+}\)) would have the molecular electron configuration \((σ_{1s})^2(σ^∗_{1s})^2(σ_{2s})^2(σ^∗_{2s})^1\). It is common to omit the core electrons from molecular orbital diagrams and configurations and include only the valence electrons.

    Bond Order

    The filled molecular orbital diagram shows the number of electrons in both bonding and antibonding molecular orbitals. The net contribution of the electrons to the bond strength of a molecule is identified by determining the bond order that results from the filling of the molecular orbitals by electrons.

    When using Lewis structures to describe the distribution of electrons in molecules, we define bond order as the number of bonding pairs of electrons between two atoms. Thus a single bond has a bond order of 1, a double bond has a bond order of 2, and a triple bond has a bond order of 3. We define bond order differently when we use the molecular orbital description of the distribution of electrons, but the resulting bond order is usually the same.

    In the molecular orbital model, an electron contributes to a bonding interaction if it occupies a bonding orbital and it contributes to an antibonding interaction if it occupies an antibonding orbital. The bond order is calculated by subtracting the destabilizing (antibonding) electrons from the stabilizing (bonding) electrons. Since a bond consists of two electrons, we divide by two to get the bond order. We can determine bond order with the following equation:

    \[\textrm{bond order}=\dfrac{(\textrm{number of bonding electrons})−(\textrm{number of antibonding electrons})}{2}\]

    The order of a covalent bond is a guide to its strength; a bond between two given atoms becomes stronger as the bond order increases. If the distribution of electrons in the molecular orbitals between two atoms is such that the resulting bond would have a bond order of zero, a stable bond does not form. We next look at some specific examples of MO diagrams and bond orders.

    Bonding in Diatomic Molecules

    A dihydrogen molecule (H2) forms from two hydrogen atoms. When the atomic orbitals of the two atoms combine, the electrons occupy the molecular orbital of lowest energy, the σ1s bonding orbital. A dihydrogen molecule, H2, readily forms because the energy of a H2 molecule is lower than that of two H atoms. The σ1s orbital that contains both electrons is lower in energy than either of the two 1s atomic orbitals.

    A molecular orbital can hold two electrons, so both electrons in the H2 molecule are in the σ1s bonding orbital; the electron configuration is \((σ_{1s})^2\). We represent this configuration by a molecular orbital energy diagram (Figure \(\PageIndex{8}\)) in which a single upward arrow indicates one electron in an orbital, and two (upward and downward) arrows indicate two electrons of opposite spin.

    11.3: Molecular Orbital Theory (14)

    A dihydrogen molecule contains two bonding electrons and no antibonding electrons so we have

    \[\ce{bond\: order\: in\: H2}=\dfrac{(2−0)}{2}=1\]

    Because the bond order for the H–H bond is equal to 1, the bond is a single bond.

    A helium atom has two electrons, both of which are in its 1s orbital. Two helium atoms do not combine to form a dihelium molecule, He2, with four electrons, because the stabilizing effect of the two electrons in the lower-energy bonding orbital would be offset by the destabilizing effect of the two electrons in the higher-energy antibonding molecular orbital. We would write the hypothetical electron configuration of He2 as \((σ_{1s})^2(σ^∗_{1s})^2\) as in Figure \(\PageIndex{9}\). The net energy change would be zero, so there is no driving force for helium atoms to form the diatomic molecule. In fact, helium exists as discrete atoms rather than as diatomic molecules. The bond order in a hypothetical dihelium molecule would be zero.

    \[\ce{bond\: order\: in\: He2}=\dfrac{(2−2)}{2}=0\]

    A bond order of zero indicates that no bond is formed between two atoms.

    11.3: Molecular Orbital Theory (15)

    The Diatomic Molecules of the Second Period

    Eight possible homonuclear diatomic molecules might be formed by the atoms of the second period of the periodic table: Li2, Be2, B2, C2, N2, O2, F2, and Ne2. However, we can predict that the Be2 molecule and the Ne2 molecule would not be stable. We can see this by a consideration of the molecular electron configurations (Table \(\PageIndex{1}\)).

    Table \(\PageIndex{1}\): Electron Configuration and Bond Order for Molecular Orbitals in Homonuclear Diatomic Molecules of Period Two Elements
    Molecule Electron Configuration Bond Order
    Li2 \((σ_{2s})^2\) 1
    Be2 (unstable) \((σ_{2s})^2(σ^∗_{2s})^2\) 0
    B2 \((σ_{2s})^2(σ^∗_{2s})^2(π_{2py},\:π_{2pz})^2\) 1
    C2 \((σ_{2s})^2(σ^∗_{2s})^2(π_{2py},\:π_{2pz})^4\) 2
    N2 \((σ_{2s})^2(σ^∗_{2s})^2(π_{2py},\:π_{2pz})^4(σ_{2px})^2\) 3
    O2 \((σ_{2s})^2(σ^∗_{2s})^2(σ_{2px})^2(π_{2py},\:π_{2pz})^4(π^∗_{2py},π^∗_{2pz})^2\) 2
    F2 \((σ_{2s})^2(σ^∗_{2s})^2(σ_{2px})^2(π_{2py},\:π_{2pz})^4(π^∗_{2py},\:π^∗_{2pz})^4\) 1
    Ne2 (unstable) \((σ_{2s})^2(σ^∗_{2s})^2(σ_{2px})^2(π_{2py},\:π_{2pz})^4(π^∗_{2py},π^∗_{2pz})^4(σ^∗_{2px})^2\) 0

    We predict valence molecular orbital electron configurations just as we predict electron configurations of atoms. Valence electrons are assigned to valence molecular orbitals with the lowest possible energies. Consistent with Hund’s rule, whenever there are two or more degenerate molecular orbitals, electrons fill each orbital of that type singly before any pairing of electrons takes place.

    11.3: Molecular Orbital Theory (16)

    Using the MO diagrams shown in Figure \(\PageIndex{10}\), we can add in the electrons and determine the molecular electron configuration and bond order for each of the diatomic molecules. As shown in Table \(\PageIndex{1}\), Be2 and Ne2 molecules would have a bond order of 0, and these molecules do not exist.

    The combination of two lithium atoms to form a lithium molecule, Li2, is analogous to the formation of H2, but the atomic orbitals involved are the valence 2s orbitals. Each of the two lithium atoms has one valence electron. Hence, we have two valence electrons available for the σ2s bonding molecular orbital. Because both valence electrons would be in a bonding orbital, we would predict the Li2 molecule to be stable. The molecule is, in fact, present in an appreciable concentration in lithium vapor at temperatures near the boiling point of the element. All of the other molecules in Table \(\PageIndex{1}\) with a bond order greater than zero are also known.

    The O2 molecule has enough electrons to half fill the \((π^∗_{2py},\:π^∗_{2pz})\) level. We expect the two electrons that occupy these two degenerate orbitals to be unpaired, and this molecular electronic configuration for O2 is in accord with the fact that the oxygen molecule has two unpaired electrons ( Figure \(\PageIndex{10}\)). The presence of two unpaired electrons has proved to be difficult to explain using Lewis structures, but the molecular orbital theory explains it quite well. In fact, the unpaired electrons of the oxygen molecule provide a strong piece of support for the molecular orbital theory.

    Application: Band Theory in Extended Systems

    When two identical atomic orbitals on different atoms combine, two molecular orbitals result (e.g., \(H_2\) in Figure \(\PageIndex{8}\)). The bonding orbital is lower in energy than the original atomic orbitals because the atomic orbitals are in-phase in the molecular orbital. The antibonding orbital is higher in energy than the original atomic orbitals because the atomic orbitals are out-of-phase.

    In a solid, similar things happen, but on a much larger scale. Remember that even in a small sample there are a huge number of atoms (typically > 1023 atoms), and therefore a huge number of atomic orbitals that may be combined into molecular orbitals. When N valence atomic orbitals, all of the same energy and each containing one (1) electron, are combined, N/2 (filled) bonding orbitals and N/2 (empty) antibonding orbitals will result. Each bonding orbital will show an energy lowering as the atomic orbitals are mostly in-phase, but each of the bonding orbitals will be a little different and have slightly different energies. The antibonding orbitals will show an increase in energy as the atomic orbitals are mostly out-of-phase, but each of the antibonding orbitals will also be a little different and have slightly different energies. The allowed energy levels for all the bonding orbitals are so close together that they form a band, called the valence band. Likewise, all the antibonding orbitals are very close together and form a band, called the conduction band. Figure \(\PageIndex{11}\)) shows the bands for three important classes of materials: insulators, semiconductors, and conductors.

    11.3: Molecular Orbital Theory (17)

    In order to conduct electricity, electrons must move from the filled valence band to the empty conduction band where they can move throughout the solid. The size of the band gap, or the energy difference between the top of the valence band and the bottom of the conduction band, determines how easy it is to move electrons between the bands. Only a small amount of energy is required in a conductor because the band gap is very small. This small energy difference is “easy” to overcome, so they are good conductors of electricity. In an insulator, the band gap is so “large” that very few electrons move into the conduction band; as a result, insulators are poor conductors of electricity. Semiconductors conduct electricity when “moderate” amounts of energy are provided to move electrons out of the valence band and into the conduction band. Semiconductors, such as silicon, are found in many electronics.

    Semiconductors are used in devices such as computers, smartphones, and solar cells. Solar cells produce electricity when light provides the energy to move electrons out of the valence band. The electricity that is generated may then be used to power a light or tool, or it can be stored for later use by charging a battery. As of December 2014, up to 46% of the energy in sunlight could be converted into electricity using solar cells.

    Example \(\PageIndex{2}\): Molecular Orbital Diagrams, Bond Order, and Number of Unpaired Electrons

    Draw the molecular orbital diagram for the oxygen molecule, O2. From this diagram, calculate the bond order for O2. How does this diagram account for the paramagnetism of O2?


    We draw a molecular orbital energy diagram similar to that shown in Figure \(\PageIndex{7}\). Each oxygen atom contributes six electrons, so the diagram appears as shown in Figure \(\PageIndex{7}\).

    11.3: Molecular Orbital Theory (18)

    We calculate the bond order as

    \[\ce{O2}=\dfrac{(8−4)}{2}=2 \nonumber\]

    Oxygen's paramagnetism is explained by the presence of two unpaired electrons in the (π2py, π2pz)* molecular orbitals.

    Exercise \(\PageIndex{2}\)

    The main component of air is N2. From the molecular orbital diagram of N2, predict its bond order and whether it is diamagnetic or paramagnetic.


    N2 has a bond order of 3 and is diamagnetic.

    Example \(\PageIndex{3}\): Ion Predictions with MO Diagrams

    Give the molecular orbital configuration for the valence electrons in \(\ce{C2^2-}\). Will this ion be stable?


    Looking at the appropriate MO diagram, we see that the π orbitals are lower in energy than the σp orbital. The valence electron configuration for C2 is


    Adding two more electrons to generate the \(\ce{C2^2-}\) anion will give a valence electron configuration of


    Since this has six more bonding electrons than antibonding, the bond order will be 3, and the ion should be stable.

    Exercise \(\PageIndex{3}\)

    How many unpaired electrons would be present on a \(\ce{Be2^2-}\) ion? Would it be paramagnetic or diamagnetic?


    two, paramagnetic

    Key Concepts and Summary

    Molecular orbital (MO) theory describes the behavior of electrons in a molecule in terms of combinations of the atomic wave functions. The resulting molecular orbitals may extend over all the atoms in the molecule. Bonding molecular orbitals are formed by in-phase combinations of atomic wave functions, and electrons in these orbitals stabilize a molecule. Antibonding molecular orbitals result from out-of-phase combinations of atomic wave functions and electrons in these orbitals make a molecule less stable. Molecular orbitals located along an internuclear axis are called σ MOs. They can be formed from s orbitals or from p orbitals oriented in an end-to-end fashion. Molecular orbitals formed from p orbitals oriented in a side-by-side fashion have electron density on opposite sides of the internuclear axis and are called π orbitals.

    We can describe the electronic structure of diatomic molecules by applying molecular orbital theory to the valence electrons of the atoms. Electrons fill molecular orbitals following the same rules that apply to filling atomic orbitals; Hund’s rule and the Aufbau principle tell us that lower-energy orbitals will fill first, electrons will spread out before they pair up, and each orbital can hold a maximum of two electrons with opposite spins. Materials with unpaired electrons are paramagnetic and attracted to a magnetic field, while those with all-paired electrons are diamagnetic and repelled by a magnetic field. Correctly predicting the magnetic properties of molecules is in advantage of molecular orbital theory over Lewis structures and valence bond theory.

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