According to the molecular orbital bonding theory, what is the bond order of C2 ?

Chapter 8. Advanced Theories of Covalent Bonding

8.4 Molecular Orbital Theory

Learning Objectives

Past the end of this section, you will be able to:

• Outline the basic quantum-mechanical approach to deriving molecular orbitals from atomic orbitals
• Describe traits of bonding and antibonding molecular orbitals
• Calculate bail orders based on molecular electron configurations
• Write molecular electron configurations for beginning- and 2d-row diatomic molecules
• Relate these electron configurations to the molecules' stabilities and magnetic properties

For almost every covalent molecule that exists, nosotros can now draw the Lewis structure, predict the electron-pair geometry, predict the molecular geometry, and come up close to predicting bond angles. However, i of the most of import molecules we know, the oxygen molecule O₂, presents a problem with respect to its Lewis construction. Nosotros would write the following Lewis structure for O_two:

This electronic structure adheres to all the rules governing Lewis theory. There is an O=O double bond, and each oxygen atom has eight electrons around information technology. However, this picture show is at odds with the magnetic behavior of oxygen. By itself, O_two is not magnetic, but it is attracted to magnetic fields. Thus, when we pour liquid oxygen past a strong magnet, information technology collects between the poles of the magnet and defies gravity, equally in Figure 1 in Chapter 8 Introduction. Such attraction to a magnetic field is called paramagnetism, and it arises in molecules that have unpaired electrons. And all the same, the Lewis structure of O₂ indicates that all electrons are paired. How practise we business relationship for this discrepancy?

Magnetic susceptibility measures the force experienced by a substance in a magnetic field. When nosotros compare the weight of a sample to the weight measured in a magnetic field (Figure i), paramagnetic samples that are attracted to the magnet volition announced heavier because of the strength exerted by the magnetic field. We can calculate the number of unpaired electrons based on the increase in weight.

Figure i. A Gouy residual compares the mass of a sample in the presence of a magnetic field with the mass with the electromagnet turned off to determine the number of unpaired electrons in a sample.

Experiments show that each O_two molecule has two unpaired electrons. The Lewis-construction model does not predict the presence of these ii unpaired electrons. Unlike oxygen, the apparent weight of most molecules decreases slightly in the presence of an inhomogeneous magnetic field. Materials in which all of the electrons are paired are diamagnetic and weakly repel a magnetic field. Paramagnetic and diamagnetic materials practise not human action as permanent magnets. Only in the presence of an applied magnetic field do they demonstrate attraction or repulsion.

Water, like most molecules, contains all paired electrons. Living things comprise a big percentage of h2o, and then they demonstrate diamagnetic behavior. If you lot place a frog near a sufficiently big magnet, information technology will levitate. Y'all can see videos of diamagnetic floating frogs, strawberries, and more.

Molecular orbital theory (MO theory) provides an explanation of chemical bonding that accounts for the paramagnetism of the oxygen molecule. It as well explains the bonding in a number of other molecules, such equally violations of the octet rule and more molecules with more complicated bonding (beyond the scope of this text) 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 diminutive orbitals to yield molecular orbitals that are delocalized over the entire molecule rather than existence localized on its constituent atoms. MO theory also helps us understand why some substances are electrical conductors, others are semiconductors, and still others are insulators. Table 2 summarizes the primary points of the two complementary bonding theories. Both theories provide different, useful ways of describing molecular structure.

Valence Bond Theory	Molecular Orbital Theory
considers bonds as localized betwixt one pair of atoms	considers electrons delocalized throughout the entire molecule
creates bonds from overlap of atomic orbitals (due south, p, d…) and hybrid orbitals (sp, sp 2, sp 3…)	combines diminutive 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 organization of electrons in molecules
needs multiple structures to describe resonance
Table two. Comparison of Bonding Theories

Molecular orbital theory describes the distribution of electrons in molecules in much the aforementioned style that the distribution of electrons in atoms is described using atomic orbitals. Using quantum mechanics, the behavior of an electron in a molecule is withal described by a wave function, Ψ, coordinating to the behavior in an atom. Just similar electrons effectually isolated atoms, electrons effectually atoms in molecules are limited to detached (quantized) energies. The region of infinite in which a valence electron in a molecule is likely to be found is called a molecular orbital (Ψ ²). Similar an atomic orbital, a molecular orbital is total when it contains two electrons with opposite spin.

We will consider the molecular orbitals in molecules composed of two identical atoms (H₂ or Cl₂, for instance). Such molecules are chosen 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 chosen the linear combination of atomic orbitals (LCAO). The wave office describes the wavelike properties of an electron. Molecular orbitals are combinations of atomic orbital moving ridge functions. Combining waves can lead to constructive interference, in which peaks line upwardly with peaks, or subversive interference, in which peaks line up with troughs (Figure ii). 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.

Figure 2. (a) When in-phase waves combine, effective interference produces a wave with greater aamplitude. (b) When out-of-phase waves combine, destructive interference produces a moving ridge with less (or no) amplitude.

There are two types of molecular orbitals that can course from the overlap of ii atomic southward orbitals on next atoms. The two types are illustrated in Figure 3. The in-phase combination produces a lower free energy σ _south molecular orbital (read as "sigma-s") in which almost of the electron density is directly between the nuclei. The out-of-phase add-on (which can too exist thought of as subtracting the wave functions) produces a higher free energy [latex]\pmb \sigma^*_s[/latex]molecular orbital (read every bit "sigma-southward-star") molecular orbital in which at that place is a node between the nuclei. The asterisk signifies that the orbital is an antibonding orbital. Electrons in a σ _south orbital are attracted by both nuclei at the same time and are more stable (of lower energy) than they would be in the isolated atoms. Calculation electrons to these orbitals creates a force that holds the 2 nuclei together, so nosotros call these orbitals bonding orbitals. Electrons in the [latex]\sigma^*_s[/latex] orbitals are located well away from the region between the 2 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-free energy bonding orbital before the higher-free energy antibonding orbital, but every bit they make full lower-energy atomic orbitals before they fill higher-energy atomic orbitals.

Figure 3. Sigma (σ) and sigma-star (σ*) molecular orbitals are formed by the combination of two south atomic orbitals. The plus (+) signs indicate the locations of nuclei.

You tin can watch animations visualizing the calculated atomic orbitals combining to grade various molecular orbitals at the Orbitron website.

In p orbitals, the wave part gives ascension to two lobes with contrary phases, analogous to how a two-dimensional wave has both parts above and below the boilerplate. We betoken the phases by shading the orbital lobes different colors. When orbital lobes of the same phase overlap, constructive moving ridge 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 stop to end, they create σ and σ* orbitals (Figure 4). If two atoms are located along the ten-axis in a Cartesian coordinate organization, the two p_ten orbitals overlap end to end and form σ _px (bonding) and [latex]\sigma^*_{px}[/latex] (antibonding) (read equally "sigma-p-x" and "sigma-p-x star," respectively). Just as with s-orbital overlap, the asterisk indicates the orbital with a node betwixt the nuclei, which is a higher-energy, antibonding orbital.

Figure 4. Combining wave functions of two p atomic orbitals along the internuclear centrality creates ii molecular orbitals, σ _p and σ* _p .

The side-by-side overlap of 2 p orbitals gives ascent to a pi (π) bonding molecular orbital and a π* antibonding molecular orbital, equally shown in Figure 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, nosotros describe the π orbital past 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 ii atoms together, making it a bonding orbital. For the out-of-phase combination, there are two nodal planes created, one forth the internuclear axis and a perpendicular i between the nuclei.

Figure 5. Side-by-side overlap of each two p orbitals results in the formation of two π molecular orbitals. Combining the out-of-phase orbitals results in an antibonding molecular orbital with two nodes. I contains the internuclear axis, and ane is perpendicular to the axis. Combining the in-phase orbitals results in a bonding orbital. At that place is a node (bluish) containing the internuclear axis with the two lobes of the orbital located above and below this node.

In the molecular orbitals of diatomic molecules, each atom also has 2 sets of p orbitals oriented side past side (p_y and p_z ), so these iv atomic orbitals combine pairwise to create 2 π orbitals and two π* orbitals. The π _py and [latex]\pi^*_{py}[/latex] orbitals are oriented at right angles to the π _pz and [latex]\pi^*_{pz}[/latex] orbitals. Except for their orientation, the π _py and π _pz orbitals are identical and have the same energy; they are degenerate orbitals. The [latex]\pi^*_{py}[/latex] and [latex]\pi^*_{px}[/latex] 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 [latex]\sigma^*_{px}[/latex], π _py and [latex]\pi^*_{py}[/latex], π _pz and [latex]\pi^*_{pz}[/latex].

Case 1

Molecular Orbitals
Predict what type (if whatever) of molecular orbital would result from adding the moving ridge functions so each pair of orbitals shown overlap. The orbitals are all like in energy.

Solution
(a) is an in-phase combination, resulting in a σ_3p orbital

(b) volition non result in a new orbital considering the in-phase component (bottom) and out-of-phase component (acme) cancel out. Merely orbitals with the correct alignment tin combine.

(c) is an out-of-phase combination, resulting in a [latex]\pi^*_{3p}[/latex] orbital.

Check Your Learning
Label the molecular orbital shown as σ or π, bonding or antibonding and indicate where the node occurs.

Answer:

The orbital is located along the internuclear axis, so it is a σ orbital. There is a node bisecting the internuclear axis, then information technology is an antibonding orbital.

Walter Kohn: Nobel Laureate

Walter Kohn (Effigy half-dozen) is a theoretical physicist who studies the electronic structure of solids. His work combines the principles of breakthrough mechanics with advanced mathematical techniques. This technique, called density functional theory, makes it possible to compute properties of molecular orbitals, including their shape and energies. Kohn and mathematician John Pople were awarded the Nobel Prize in Chemistry in 1998 for their contributions to our understanding of electronic structure. Kohn also made significant contributions to the physics of semiconductors.

Effigy half-dozen. Walter Kohn developed methods to describe molecular orbitals. (credit: prototype courtesy of Walter Kohn)

Kohn'south biography has been remarkable outside the realm of concrete chemistry as well. He was built-in in Austria, and during World War Ii he was function of the Kindertransport program that rescued 10,000 children from the Nazi regime. His summer jobs included discovering gilded deposits in Canada and helping Polaroid explain how its instant motion-picture show worked. Although he is now an emeritus professor, he is still actively working on projects involving global warming and renewable energy.

Computational Chemical science in Drug Design

While the descriptions of bonding described in this chapter involve many theoretical concepts, they also have many applied, real-world applications. For example, drug design is an of import field that uses our understanding of chemical bonding to develop pharmaceuticals. This interdisciplinary surface 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 affliction pathway. By modeling the structures of the binding site and potential drugs, computational chemists tin predict which structures can fit together and how effectively they will demark (run across Effigy 7). Thousands of potential candidates tin be narrowed down to a few of the about promising candidates. These candidate molecules are then carefully tested to make up one's mind side effects, how finer 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 enquiry projects are underway.

Figure 7. The molecule shown, HIV-1 protease, is an of import target for pharmaceutical research. By designing molecules that demark to this protein, scientists are able to drastically inhibit the progress of the disease.

Molecular Orbital Energy Diagrams

The relative free energy levels of diminutive and molecular orbitals are typically shown in a molecular orbital diagram (Figure viii). 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 i orbital that can hold two electrons. The molecular orbitals formed by the combination of the atomic orbitals are shown in the heart. Dashed lines bear witness 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 consequence. Thus nosotros can see that combining the 6 2p atomic orbitals results in three bonding orbitals (one σ and ii π) and iii antibonding orbitals (one σ* and 2 π*).

Nosotros predict the distribution of electrons in these molecular orbitals by filling the orbitals in the aforementioned way that we make full atomic orbitals, past the Aufbau principle. Lower-energy orbitals fill first, electrons spread out amidst degenerate orbitals before pairing, and each orbital can hold a maximum of 2 electrons with opposite spins (Figure 8). Just as nosotros write electron configurations for atoms, we tin can write the molecular electronic configuration by listing the orbitals with superscripts indicating the number of electrons nowadays. For clarity, we place parentheses effectually molecular orbitals with the same energy. In this example, each orbital is at a unlike energy, so parentheses separate each orbital. Thus nosotros would expect a diatomic molecule or ion containing vii electrons (such as Be₂ ⁺) would have the molecular electron configuration [latex](\sigma_{1s})^ii[/latex] [latex](\sigma^*_{1s})^2[/latex] [latex](\sigma_{2s})^ii[/latex] [latex](\sigma^*_{2s})^i[/latex]. It is common to omit the core electrons from molecular orbital diagrams and configurations and include just the valence electrons.

Figure 8. This is the molecular orbital diagram for the homonuclear diatomic Exist₂ ⁺, showing the molecular orbitals of the valence shell but. The molecular orbitals are filled in the aforementioned manner every bit diminutive orbitals, using the Aufbau principle and Hund's dominion.

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 bail forcefulness of a molecule is identified past 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 ascertain bond order as the number of bonding pairs of electrons between two atoms. Thus a single bond has a bail order of 1, a double bail has a bond gild of 2, and a triple bond has a bail order of 3. We ascertain bail order differently when we utilise the molecular orbital description of the distribution of electrons, but the resulting bond order is usually the same. The MO technique is more accurate and can handle cases when the Lewis construction method fails, but both methods draw the same miracle.

In the molecular orbital model, an electron contributes to a bonding interaction if information technology occupies a bonding orbital and it contributes to an antibonding interaction if it occupies an antibonding orbital. The bond guild is calculated past subtracting the destabilizing (antibonding) electrons from the stabilizing (bonding) electrons. Since a bond consists of two electrons, we split by two to get the bond lodge. We tin make up one's mind bond guild with the following equation:

[latex]\text{bond society} = \frac{(\text{number of bonding electrons}) - (\text{number of antibonding electrons})}{two}[/latex]

The order of a covalent bail is a guide to its strength; a bond between two given atoms becomes stronger equally the bond order increases (Tabular array ane in Chapter viii.1 Valence Bail Theory). If the distribution of electrons in the molecular orbitals betwixt two atoms is such that the resulting bail would have a bond order of nil, a stable bail does non form. Nosotros next look at some specific examples of MO diagrams and bond orders.

Bonding in Diatomic Molecules

A dihydrogen molecule (H₂) forms from 2 hydrogen atoms. When the atomic orbitals of the two atoms combine, the electrons occupy the molecular orbital of everyman energy, the σ_1s bonding orbital. A dihydrogen molecule, H₂, readily forms because the energy of a H₂ molecule is lower than that of two H atoms. The σ_1s orbital that contains both electrons is lower in energy than either of the two 1due south atomic orbitals.

A molecular orbital can hold 2 electrons, and then both electrons in the H_two molecule are in the σ_1s bonding orbital; the electron configuration is [latex](\sigma_{1s})^2[/latex]. Nosotros represent this configuration past a molecular orbital energy diagram (Figure 9) in which a single upwards arrow indicates one electron in an orbital, and two (upward and downward) arrows indicate 2 electrons of opposite spin.

Figure ix. The molecular orbital energy diagram predicts that H₂ will be a stable molecule with lower energy than the separated atoms.

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

[latex]\text{bond social club in H}_2 = \frac{(two - 0)}{2} = 1[/latex]

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

A helium atom has two electrons, both of which are in its anes orbital. Ii helium atoms do non combine to form a dihelium molecule, He_two, with four electrons, considering the stabilizing effect of the two electrons in the lower-energy bonding orbital would be kickoff by the destabilizing effect of the two electrons in the higher-energy antibonding molecular orbital. Nosotros would write the hypothetical electron configuration of He_two as [latex](\sigma_{1s})^2[/latex] [latex](\sigma^*_{1s})^two[/latex] as in Figure 10. The net energy modify would be zero, so there is no driving strength for helium atoms to class the diatomic molecule. In fact, helium exists as discrete atoms rather than every bit diatomic molecules. The bond order in a hypothetical dihelium molecule would be nil.

[latex]\text{bond guild in He}_2 = \frac{(ii - 2)}{2} = 0[/latex]

A bond order of goose egg indicates that no bond is formed between ii atoms.

Figure 10. The molecular orbital free energy diagram predicts that He₂ volition not be a stable molecule, since it has equal numbers of bonding and antibonding electrons.

The Diatomic Molecules of the Second Menstruation

8 possible homonuclear diatomic molecules might be formed past the atoms of the second catamenia of the periodic table: Li_ii, Exist_two, B_ii, C₂, N₂, O₂, F₂, and Ne₂. Still, we tin can predict that the Be₂ molecule and the Ne₂ molecule would not be stable. We can see this past a consideration of the molecular electron configurations (Tabular array iii).

Nosotros predict valence molecular orbital electron configurations merely as we predict electron configurations of atoms. Valence electrons are assigned to valence molecular orbitals with the lowest possible energies. Consequent with Hund's rule, whenever in that location are two or more than degenerate molecular orbitals, electrons fill each orbital of that blazon singly before any pairing of electrons takes place.

As we saw in valence bond theory, σ bonds are generally more stable than π bonds formed from degenerate atomic orbitals. Similarly, in molecular orbital theory, σ orbitals are ordinarily more stable than π orbitals. Withal, this is not always the case. The MOs for the valence orbitals of the 2nd period are shown in Figure eleven. Looking at Ne_two molecular orbitals, nosotros see that the lodge is consequent with the generic diagram shown in the previous section. However, for atoms with three or fewer electrons in the p orbitals (Li through Due north) we observe a unlike pattern, in which the σ _p orbital is higher in free energy than the π _p set. Obtain the molecular orbital diagram for a homonuclear diatomic ion by adding or subtracting electrons from the diagram for the neutral molecule.

Figure 11. This shows the MO diagrams for each homonuclear diatomic molecule in the second menstruation. The orbital energies decrease across the menstruum as the effective nuclear accuse increases and atomic radius decreases. Between North₂ and O₂, the gild of the orbitals changes.

You tin can exercise labeling and filling molecular orbitals with this interactive tutorial from the University of Sydney.

This switch in orbital ordering occurs considering of a phenomenon called southward-p mixing. due south-p mixing does not create new orbitals; information technology merely influences the energies of the existing molecular orbitals. The σ_s wavefunction mathematically combines with the σ_p wavefunction, with the upshot that the σ_{due south} orbital becomes more stable, and the σ_p orbital becomes less stable (Effigy 12). Similarly, the antibonding orbitals also undergo s-p mixing, with the σ_s* condign more stable and the σ_p* becoming less stable.

Effigy 12. Without mixing, the MO pattern occurs every bit expected, with the σ_p orbital lower in energy than the σ_p orbitals. When southward-p mixing occurs, the orbitals shift as shown, with the σ_p orbital higher in energy than the π_p orbitals.

s-p mixing occurs when the s and p orbitals have similar energies. When a single p orbital contains a pair of electrons, the act of pairing the electrons raises the energy of the orbital. Thus the 2p orbitals for O, F, and Ne are college in energy than the 2p orbitals for Li, Exist, B, C, and N. Because of this, O_ii, F₂, and N₂ only have negligible due south-p mixing (not sufficient to modify the free energy ordering), and their MO diagrams follow the normal pattern, as shown in Figure 11. All of the other period 2 diatomic molecules do have s-p mixing, which leads to the design where the σ_p orbital is raised above the π_p ready.

Using the MO diagrams shown in Effigy 11, nosotros tin can add in the electrons and make up one's mind the molecular electron configuration and bond club for each of the diatomic molecules. As shown in Table 3, Be_two and Ne₂ molecules would accept a bond order of 0, and these molecules practise not exist.

Molecule	Electron Configuration	Bond Order
Li2	[latex](\sigma_{2s})^2[/latex]	ane
Exist2 (unstable)	[latex](\sigma_{2s})^two (\sigma^*_{2s})^2[/latex]	0
Bii	[latex](\sigma_{2s})^2 (\sigma^*_{2s})^2 (\pi_{2py}, \pi_{2pz})^two[/latex]	1
C2	[latex](\sigma_{2s})^two (\sigma^*_{2s})^2 (\pi_{2py}, \pi_{2pz})^4[/latex]	two
N2	[latex](\sigma_{2s})^2 (\sigma^*_{2s})^two (\pi_{2py}, \pi_{2pz})^four (\sigma_{2px})^2[/latex]	3
O2	[latex](\sigma_{2s})^2 (\sigma^*_{2s})^ii (\sigma_{2px})^2 (\pi_{2py}, \pi_{2pz})^4 (\pi^*_{2py}, \pi^*_{2pz})^2[/latex]	2
F2	[latex](\sigma_{2s})^2 (\sigma^*_{2s})^2 (\sigma_{2px})^2 (\pi_{2py}, \pi_{2pz})^iv (\pi^*_{2py}, \pi^*_{2pz})^iv[/latex]	1
Ne2 (unstable)	[latex](\sigma_{2s})^ii (\sigma^*_{2s})^ii (\sigma_{2px})^2 (\pi_{2py}, \pi_{2pz})^iv (\pi^*_{2py}, \pi^*_{2pz})^4 (\sigma^*_{2px})^ii[/latex]	0
Table 3. Electron Configuration and Bond Order for Molecular Orbitals in Homonuclear Diatomic Molecules of Flow Two Elements

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 iisouthward orbitals. Each of the two lithium atoms has one valence electron. Hence, we have two valence electrons available for the σ_{2due south} 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 appreciable concentration in lithium vapor at temperatures near the boiling signal of the element. All of the other molecules in Table 3 with a bond gild greater than nix are also known.

The O₂ molecule has enough electrons to half fill the ([latex]\pi^*_{2py}[/latex], [latex]\pi^*_{2pz}[/latex]) level. We expect the ii electrons that occupy these two degenerate orbitals to be unpaired, and this molecular electronic configuration for O₂ is in accordance with the fact that the oxygen molecule has two unpaired electrons (Figure 14). The presence of two unpaired electrons has proved to exist difficult to explain using Lewis structures, just the molecular orbital theory explains it quite well. In fact, the unpaired electrons of the oxygen molecule provide a potent piece of support for the molecular orbital theory.

Band Theory

When two identical atomic orbitals on dissimilar atoms combine, two molecular orbitals issue (see Effigy 3). 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 college in energy than the original atomic orbitals because the atomic orbitals are out-of-stage.

In a solid, similar things happen, simply on a much larger scale. Think that fifty-fifty in a minor sample in that location 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 aforementioned energy and each containing one (1) electron, are combined, Due north/2 (filled) bonding orbitals and North/2 (empty) antibonding orbitals volition result. Each bonding orbital volition show an free energy lowering equally the diminutive orbitals are mostly in-phase, but each of the bonding orbitals will be a little different and have slightly dissimilar energies. The antibonding orbitals will show an increment in energy as the atomic orbitals are mostly out-of-phase, just each of the antibonding orbitals will as well be a little different and accept slightly different energies. The immune free energy levels for all the bonding orbitals are then close together that they form a band, chosen the valence band. Likewise, all the antibonding orbitals are very shut together and form a band, chosen the conduction band. Figure 13 shows the bands for three important classes of materials: insulators, semiconductors, and conductors.

Figure 13. Molecular orbitals in solids are so closely spaced that they are described equally bands. The valence band is lower in energy and the conduction band is higher in free energy. The type of solid is determined by the size of the "ring gap" betwixt the valence and conduction bands. Simply a very small amount of energy is required to move electrons from the valance band to the conduction ring in a conductor, and and then they conduct electricity well. In an insulator, the ring gap is big, and so that very few electrons motility, and they are poor conductors of electricity. Semiconductors are in between: they conduct electricity ameliorate than insulators, but not as well as conductors.

In social club 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 ring gap, or the energy difference betwixt the height of the valence ring and the bottom of the conduction band, determines how piece of cake it is to motility electrons between the bands. Just a small amount of energy is required in a conductor because the band gap is very small. This small energy departure is "easy" to overcome, so they are good conductors of electricity. In an insulator, the band gap is so "large" that very few electrons movement into the conduction band; as a event, insulators are poor conductors of electricity. Semiconductors acquit electricity when "moderate" amounts of energy are provided to movement electrons out of the valence band and into the conduction ring. Semiconductors, such as silicon, are found in many electronics.

Semiconductors are used in devices such equally computers, smartphones, and solar cells. Solar cells produce electricity when light provides the energy to motion electrons out of the valence ring. The electricity that is generated may so be used to ability a light or tool, or it can be stored for afterwards use past charging a battery. As of December 2014, upward to 46% of the energy in sunlight could exist converted into electricity using solar cells.

Example 2

Molecular Orbital Diagrams, Bail Club, and Number of Unpaired Electrons
Describe the molecular orbital diagram for the oxygen molecule, O₂. From this diagram, calculate the bond order for O_two. How does this diagram account for the paramagnetism of O_two?

Solution

We draw a molecular orbital energy diagram similar to that shown in Effigy 11. Each oxygen cantlet contributes six electrons, so the diagram appears as shown in Figure 14.

Figure 14. The molecular orbital free energy diagram for O₂ predicts two unpaired electrons.

Nosotros summate the bail order as

[latex]\text{O}_2 = \frac{8 - 4}{2} = 2[/latex]

Oxygen's paramagnetism is explained past the presence of two unpaired electrons in the (π_2py , π_2pz )* molecular orbitals.

Check Your Learning

The principal component of air is North₂. From the molecular orbital diagram of N_two, predict its bond order and whether it is diamagnetic or paramagnetic.

Answer:

N₂ has a bond lodge of iii and is diamagnetic.

Example 3

Ion Predictions with MO Diagrams
Give the molecular orbital configuration for the valence electrons in C₂ ^two−. Will this ion be stable?

Solution

Looking at the appropriate MO diagram, we see that the π orbitals are lower in energy than the σ _p orbital. The valence electron configuration for C₂ is [latex](\sigma_{2s})^two (\sigma^*_{2s})^two (\pi_{2py}, \pi_{2pz})^4[/latex]. Adding 2 more than electrons to generate the C₂ ²⁻ anion volition give a valence electron configuration of [latex](\sigma_{2s})^2 (\sigma^*_{2s})^ii (\pi_{2py}, \pi_{2pz})^4 (\sigma_{2px})^ii[/latex]. Since this has six more than bonding electrons than antibonding, the bond gild will be three, and the ion should be stable.

Bank check Your Learning

How many unpaired electrons would exist nowadays on a Be₂ ²⁻ ion? Would information technology exist paramagnetic or diamagnetic?

Answer:

two, paramagnetic

Creating molecular orbital diagrams for molecules with more than two atoms relies on the aforementioned bones ideas as the diatomic examples presented hither. Withal, with more atoms, computers are required to calculate how the atomic orbitals combine. Come across three-dimensional drawings of the molecular orbitals for C₆H₆.

Key Concepts and Summary

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

We can draw the electronic structure of diatomic molecules past applying molecular orbital theory to the valence electrons of the atoms. Electrons fill up molecular orbitals following the same rules that apply to filling atomic orbitals; Hund's rule and the Aufbau principle tell u.s. that lower-energy orbitals will fill first, electrons volition spread out before they pair up, and each orbital can agree a maximum of two electrons with reverse 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.

Central Equations

• [latex]\text{bond order} = \frac{(\text{number of bonding electron}) - (\text{number of antibonding electrons})}{2}[/latex]

Chemistry End of Chapter Exercises

1. Sketch the distribution of electron density in the bonding and antibonding molecular orbitals formed from two south orbitals and from ii p orbitals.
2. How are the following like, and how do they differ?

   (a) σ molecular orbitals and π molecular orbitals

   (b) ψ for an diminutive orbital and ψ for a molecular orbital

   (c) bonding orbitals and antibonding orbitals

3. If molecular orbitals are created past combining five atomic orbitals from atom A and five atomic orbitals from atom B combine, how many molecular orbitals will event?
4. Can a molecule with an odd number of electrons ever be diamagnetic? Explain why or why not.
5. Can a molecule with an even number of electrons ever be paramagnetic? Explicate why or why not.
6. Why are bonding molecular orbitals lower in energy than the parent atomic orbitals?
7. Calculate the bond order for an ion with this configuration:

   [latex](\sigma_{2s})^two (\sigma^*_{2s})^2 (\sigma_{2px})^2 (\pi_{2py} , \pi_{2pz})^4 (\pi^*_{2py} , \pi^*_{2pz})^3[/latex]

8. Explain why an electron in the bonding molecular orbital in the H_ii molecule has a lower energy than an electron in the idue south atomic orbital of either of the separated hydrogen atoms.
9. Predict the valence electron molecular orbital configurations for the following, and state whether they will be stable or unstable ions.

   (a) Na_ii ²⁺

   (b) Mg_two ^two+

   (c) Al_ii ²⁺

   (d) Si₂ ²⁺

   (east) P₂ ²⁺

   (f) Southward₂ ^two+

   (k) F_ii ²⁺

   (h) Ar₂ ²⁺

10. Determine the bond order of each member of the following groups, and make up one's mind which member of each group is predicted past the molecular orbital model to have the strongest bond.

    (a) H₂, H₂ ⁺, H₂ ⁻

    (b) O₂, O₂ ²⁺, O₂ ²⁻

    (c) Li₂, Be₂ ⁺, Exist_two

    (d) F_ii, F_two ⁺, F₂ ⁻

    (eastward) N₂, N₂ ⁺, N₂ ⁻

11. For the first ionization energy for an N₂ molecule, what molecular orbital is the electron removed from?
12. Compare the diminutive and molecular orbital diagrams to identify the member of each of the following pairs that has the highest first ionization energy (the most tightly bound electron) in the gas phase:

    (a) H and H₂

    (b) N and N₂

    (c) O and O₂

    (d) C and C_two

    (eastward) B and B₂

13. Which of the flow 2 homonuclear diatomic molecules are predicted to be paramagnetic?
14. A friend tells you that the 2s orbital for fluorine starts off at a much lower energy than the iisouthward orbital for lithium, so the resulting σ_{2due south} molecular orbital in F_ii is more stable than in Li_two. Do you agree?
15. True or false: Boron contains twos ^two2p ¹ valence electrons, so just 1 p orbital is needed to form molecular orbitals.
16. What accuse would exist needed on F₂ to generate an ion with a bail guild of 2?
17. Predict whether the MO diagram for S_two would bear witness s-p mixing or not.
18. Explain why North_two ^two+ is diamagnetic, while O_ii ⁴⁺, which has the same number of valence electrons, is paramagnetic.
19. Using the MO diagrams, predict the bond guild for the stronger bond in each pair:

    (a) B₂ or B₂ ⁺

    (b) F_ii or F₂ ⁺

    (c) O_two or O₂ ²⁺

    (d) C₂ ⁺ or C₂ ⁻

Glossary

antibonding orbital

molecular orbital located outside of the region between 2 nuclei; electrons in an antibonding orbital destabilize the molecule

bond order

number of pairs of electrons between ii atoms; information technology tin be found by the number of bonds in a Lewis structure or by the difference betwixt the number of bonding and antibonding electrons divided by two

bonding orbital

molecular orbital located between two nuclei; electrons in a bonding orbital stabilize a molecule

degenerate orbitals

orbitals that have the same energy

diamagnetism

miracle in which a material is not magnetic itself but is repelled by a magnetic field; it occurs when in that location are merely paired electrons present

homonuclear diatomic molecule

molecule consisting of ii identical atoms

linear combination of atomic orbitals

technique for combining diminutive orbitals to create molecular orbitals

molecular orbital

region of space in which an electron has a high probability of being found in a molecule

molecular orbital diagram

visual representation of the relative energy levels of molecular orbitals

molecular orbital theory

model that describes the behavior of electrons delocalized throughout a molecule in terms of the combination of atomic wave functions

paramagnetism

phenomenon in which a material is not magnetic itself but is attracted to a magnetic field; it occurs when there are unpaired electrons present

π bonding orbital

molecular orbital formed by side-by-side overlap of atomic orbitals, in which the electron density is establish on opposite sides of the internuclear axis

π* bonding orbital

antibonding molecular orbital formed by out of phase side-past-side overlap of atomic orbitals, in which the electron density is found on both sides of the internuclear axis, and in that location is a node betwixt the nuclei

σ bonding orbital

molecular orbital in which the electron density is found forth the axis of the bond

σ* bonding orbital

antibonding molecular orbital formed by out-of-phase overlap of atomic orbital along the axis of the bond, generating a node between the nuclei

s-p mixing

change that causes σ _p orbitals to be less stable than π _p orbitals due to the mixing of s and p-based molecular orbitals of similar energies.

Solutions

Answers to Chemistry Terminate of Affiliate Exercises

2. (a) Similarities: Both are bonding orbitals that can contain a maximum of two electrons. Differences: σ orbitals are end-to-end combinations of atomic orbitals, whereas π orbitals are formed by side-by-side overlap of orbitals. (b) Similarities: Both are quantum-mechanical constructs that represent the probability of finding the electron about the atom or the molecule. Differences: ψ for an atomic orbital describes the behavior of but i electron at a time based on the cantlet. For a molecule, ψ represents a mathematical combination of atomic orbitals. (c) Similarities: Both are orbitals that can comprise two electrons. Differences: Bonding orbitals event in property two or more atoms together. Antibonding orbitals have the consequence of destabilizing any bonding that has occurred.

iv. An odd number of electrons tin can never be paired, regardless of the system of the molecular orbitals. It will always be paramagnetic.

6. Bonding orbitals take electron density in close proximity to more than i nucleus. The interaction between the bonding positively charged nuclei and negatively charged electrons stabilizes the organisation.

8. The pairing of the two bonding electrons lowers the energy of the organisation relative to the energy of the nonbonded electrons.

x. (a) H_ii bail order = 1, H_two ⁺ bond order = 0.5, H₂ ⁻ bond order = 0.5, strongest bail is H_ii; (b) O_ii bail order = ii, O_two ²⁺ bond order = 3; O₂ ²⁻ bond club = 1, strongest bail is O_ii ^two+; (c) Li_two bond social club = i, Be₂ ⁺ bond gild = 0.five, Be₂ bail order = 0, strongest bail is Li_two;(d) F_two bail order = 1, F₂ ⁺ bail order = i.5, F₂ ⁻ bail order = 0.5, strongest bond is F_ii ⁺; (e) Due north₂ bond order = 3, N_ii ⁺ bond order = ii.v, Due north₂ ⁻ bond club = 2.5, strongest bond is N_two

12. (a) H₂; (b) Northward_two; (c) O; (d) C_two; (e) B₂

xiv. Yeah, fluorine is a smaller cantlet than Li, so atoms in the iis orbital are closer to the nucleus and more stable.

16. 2+

eighteen. Due north₂ has s-p mixing, so the π orbitals are the last filled in Northward_two ⁱⁱ⁺. O₂ does non have due south-p mixing, so the σ _p orbital fills before the π orbitals.

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Source: https://opentextbc.ca/chemistry/chapter/8-4-molecular-orbital-theory/

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