{"id":2057,"date":"2015-04-22T20:20:44","date_gmt":"2015-04-22T20:20:44","guid":{"rendered":"https:\/\/courses.candelalearning.com\/oschemtemp\/?post_type=chapter&#038;p=2057"},"modified":"2016-10-27T15:55:30","modified_gmt":"2016-10-27T15:55:30","slug":"molecular-orbital-theory","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/chapter\/molecular-orbital-theory\/","title":{"raw":"Molecular Orbital Theory","rendered":"Molecular Orbital Theory"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Outline the basic quantum-mechanical approach to deriving molecular orbitals from atomic orbitals<\/li>\r\n \t<li>Describe traits of bonding and anti-bonding molecular orbitals<\/li>\r\n \t<li>Calculate bond orders based on molecular electron configurations<\/li>\r\n \t<li>Write molecular electron configurations for first- and second-row diatomic molecules<\/li>\r\n \t<li>Relate these electron configurations to the molecules\u2019 stabilities and magnetic properties<\/li>\r\n<\/ul>\r\n<\/div>\r\nFor almost every covalent molecule that exists, we can now draw the Lewis structure, predict the electron-pair geometry, predict the molecular geometry, and come close to predicting bond angles. However, one of the most important molecules we know, the oxygen molecule O<sub>2<\/sub>, presents a problem with respect to its Lewis structure. We would write the following Lewis structure for O<sub>2<\/sub>:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211927\/CNX_Chem_08_04_O2_img1.jpg\" alt=\"A Lewis structure is shown. It is made up of two oxygen atoms, each with two lone pairs of electrons, bonded together with a double bond.\" data-media-type=\"image\/jpeg\" \/>\r\n\r\nThis 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 it. However, this picture is at odds with the magnetic behavior of oxygen. By itself, O<sub>2<\/sub> is not magnetic, but it is attracted to magnetic fields. Thus, when we pour liquid oxygen past a strong magnet, it collects between the poles of the magnet and defies gravity, as in Figure 1. Such attraction to a magnetic field is called <strong>paramagnetism<\/strong>, and it arises in molecules that have unpaired electrons. And yet, the Lewis structure of O<sub>2<\/sub> indicates that all electrons are paired. How do we account for this discrepancy?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"750\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211747\/CNX_Chem_08_00_LiqO21.jpg\" alt=\"A pitcher is shown pouring liquid oxygen through the gap between two magnets, where it has formed a solid disk. A call out box near the stream of liquid oxygen shows an image of six pairs of spheres, spread apart from one another. Another call out box near the solid disk shows ten pairs of spheres much closer together.\" width=\"750\" height=\"346\" data-media-type=\"image\/jpeg\" \/> Figure 1. Oxygen molecules orient randomly most of the time, as shown in the top magnified view. However, when we pour liquid oxygen through a magnet, the molecules line up with the magnetic field, and the attraction allows them to stay suspended between the poles of the magnet where the magnetic field is strongest. Other diatomic molecules (like N<sub>2<\/sub>) flow past the magnet. The detailed explanation of bonding described in this chapter allows us to understand this phenomenon. (credit: modification of work by Jefferson Lab)[\/caption]\r\n\r\nMagnetic susceptibility measures the force experienced by a substance in a magnetic field. When we compare the weight of a sample to the weight measured in a magnetic field (Figure 2), paramagnetic samples that are attracted to the magnet will appear heavier because of the force exerted by the magnetic field. We can calculate the number of unpaired electrons based on the increase in weight.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"800\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211928\/CNX_Chem_08_04_Gouy1.jpg\" alt=\"A diagram depicts a stand supporting two objects that are held in balance by a horizontal bar. On the right, the bar supports a dish that is holding two weights. On the left there is a line attached to a test tube labeled, \u201cSample tube.\u201d The test tube has been lowered into the space labeled, \u201cMagnetic field,\u201d between two structures labeled, \u201cElectromagnets.\u201d\" width=\"800\" height=\"601\" data-media-type=\"image\/jpeg\" \/> Figure 2. A Gouy balance 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.[\/caption]\r\n\r\nExperiments show that each O<sub>2<\/sub> molecule has two unpaired electrons. The Lewis-structure model does not predict the presence of these two 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 <strong>diamagnetic<\/strong> and weakly repel a magnetic field. Paramagnetic and diamagnetic materials do not act as permanent magnets. Only in the presence of an applied magnetic field do they demonstrate attraction or repulsion.\r\n<div class=\"textbox\">Water, like most molecules, contains all paired electrons. Living things contain a large percentage of water, so they demonstrate diamagnetic behavior. If you place a frog near a sufficiently large magnet, it will levitate. You can see videos of diamagnetic floating frogs, strawberries, and more at the <a href=\"http:\/\/www.ru.nl\/hfml\/research\/levitation\/diamagnetic\/\" target=\"_blank\">Radboud University Diamagnetic Levitation website<\/a>.<\/div>\r\nMolecular 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, such as 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 atomic orbitals to yield molecular orbitals that are <em>delocalized<\/em> over the entire molecule rather than being 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 1 summarizes the main points of the two complementary bonding theories. Both theories provide different, useful ways of describing molecular structure.\r\n<table summary=\"A table is shown that is composed of two columns and six rows. The header row reads, \u201cValence Bond Theory,\u201d and, \u201cMolecular Orbital Theory.\u201d The first column contains the phrases: \u201cconsiders bonds as localized between one pair of atoms,\u201d \u201ccreates bonds from overlap of atomic orbitals ( s, p, d\u2026) and hybrid orbitals ( s p , s p superscript 2, s p superscript 3 \u2026 ) ,\u201d \u201cforms sigma or pi bonds,\u201d \u201cpredicts molecular shape based on the number of regions of electron density,\u201d and, \u201cneeds multiple structures are needed to describe resonance.\u201d The second column reads, \u201cconsiders electrons delocalized throughout the entire molecule,\u201d \u201ccombines atomic orbitals to form molecular orbitals ( sigma, sigma superscript asterik, pi, pi superscript asterisk ),\u201d \u201ccreates bonding and antibonding interactions based on which orbitals are filled,\u201d \u201cpredicts the arrangement of electrons in molecules.\u201d\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\">Table 1. Comparison of Bonding Theories<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<th>Valence Bond Theory<\/th>\r\n<th>Molecular Orbital Theory<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>considers bonds as localized between one pair of atoms<\/td>\r\n<td>considers electrons delocalized throughout the entire molecule<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>creates bonds from overlap of atomic orbitals (<em>s, p, d<\/em>\u2026) and hybrid orbitals (<em>sp, sp<\/em><sup>2<\/sup>, <em>sp<\/em><sup>3<\/sup>\u2026)<\/td>\r\n<td>combines atomic orbitals to form molecular orbitals (\u03c3, \u03c3*, \u03c0, \u03c0*)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>forms \u03c3 or \u03c0 bonds<\/td>\r\n<td>creates bonding and antibonding interactions based on which orbitals are filled<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>predicts molecular shape based on the number of regions of electron density<\/td>\r\n<td>predicts the arrangement of electrons in molecules<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>needs multiple structures to describe resonance<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Molecular orbital theory <\/strong>describes the distribution of electrons in molecules in much the same way that the distribution of electrons in atoms is described using atomic orbitals. Using quantum mechanics, the behavior of an electron in a molecule is still described by a wave function, <em>\u03a8<\/em>, analogous to the behavior in an atom. Just like electrons around isolated atoms, electrons around atoms in molecules are limited to discrete (quantized) energies. The region of space in which a valence electron in a molecule is likely to be found is called a<strong> molecular orbital (<em>\u03a8<\/em><sup>2<\/sup>)<\/strong>. Like an atomic orbital, a molecular orbital is full when it contains two electrons with opposite spin.\r\n\r\nWe will consider the molecular orbitals in molecules composed of two identical atoms (H<sub>2<\/sub> or Cl<sub>2<\/sub>, for example). Such molecules are called <strong>homonuclear diatomic molecules<\/strong>. In these diatomic molecules, several types of molecular orbitals occur.\r\n\r\nThe mathematical process of combining atomic orbitals to generate molecular orbitals is called the <strong>linear combination of atomic orbitals (LCAO)<\/strong>. The wave function describes the wavelike properties of an electron. Molecular orbitals are combinations of atomic orbital wave functions. Combining waves can lead to constructive interference, in which peaks line up with peaks, or destructive interference, in which peaks line up with troughs (Figure 3). 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"882\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211931\/CNX_Chem_08_04_waveadd1.jpg\" alt=\"A pair of diagrams are shown and labeled, \u201ca\u201d and \u201cb.\u201d Diagram a shows two identical waves with two crests and two troughs. They are drawn one above the other with a plus sign in between and an equal sign to the right. To the right of the equal sign is a much taller wave with a same number of troughs and crests. Diagram b shows two waves with two crests and two troughs, but they are mirror images of one another rotated over a horizontal axis. They are drawn one above the other with a plus sign in between and an equal sign to the right. To the right of the equal sign is a flat line.\" width=\"882\" height=\"186\" data-media-type=\"image\/jpeg\" \/> Figure 3. (a) When in-phase waves combine, constructive interference produces a wave with greater amplitude. (b) When out-of-phase waves combine, destructive interference produces a wave with less (or no) amplitude.[\/caption]\r\n\r\nThere are two types of molecular orbitals that can form from the overlap of two atomic <em>s<\/em> orbitals on adjacent atoms. The two types are illustrated in Figure 4. The in-phase combination produces a lower energy <strong>\u03c3<sub><em>s<\/em><\/sub> molecular orbital<\/strong> (read as \"sigma-s\") in which most of the electron density is directly between the nuclei. The out-of-phase addition (which can also be thought of as subtracting the wave functions) produces a higher energy [latex]{\\sigma}_{s}^\\ast[\/latex] <strong>molecular orbital<\/strong> (read as \"sigma-s-star\") molecular orbital in which there is a node between the nuclei. The asterisk signifies that the orbital is an antibonding orbital. Electrons in a \u03c3<em><sub>s<\/sub><\/em> 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. Adding electrons to these orbitals creates a force that holds the two nuclei together, so we call these orbitals <strong>bonding orbitals<\/strong>. Electrons in the [latex]{\\sigma}_{s}^\\ast[\/latex] 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<strong> antibonding orbitals<\/strong>. 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"879\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211933\/CNX_Chem_08_04_ssigma1.jpg\" alt=\"A diagram is shown that depicts a vertical upward-facing arrow that lies to the left of all the other portions of the diagram and is labeled, \u201cE.\u201d To the immediate right of the midpoint of the arrow are two circles each labeled with a positive sign, the letter S, and the phrase, \u201cAtomic orbitals.\u201d These are followed by a right-facing horizontal arrow that points to the same two circles labeled with plus signs, but they are now touching and are labeled, \u201cCombine atomic orbitals.\u201d Two right-facing arrows lead to the last portion of the diagram, one facing upward and one facing downward. The upper arrow is labeled, \u201cSubtract,\u201d and points to two oblong ovals labeled with plus signs, and the phrase, \u201cAntibonding orbitals sigma subscript s superscript asterisk.\u201d The lower arrow is labeled, \u201cAdd,\u201d and points to an elongated oval with two plus signs that is labeled, \u201cBonding orbital sigma subscript s.\u201d The heading over the last section of the diagram are the words, \u201cMolecular orbitals.\u201d\" width=\"879\" height=\"358\" data-media-type=\"image\/jpeg\" \/> Figure 4. Sigma (\u03c3) and sigma-star (\u03c3*) molecular orbitals are formed by the combination of two s atomic orbitals. The plus (+) signs indicate the locations of nuclei.[\/caption]\r\n\r\n<div class=\"textbox\">You can watch animations visualizing the calculated atomic orbitals combining to form various molecular orbitals at the <a href=\"https:\/\/winter.group.shef.ac.uk\/orbitron\/MOs\/N2\/2s2s-sigma\/index.html\" target=\"_blank\">University of Sheffield's Orbitron website<\/a>.<\/div>\r\nIn <em>p<\/em> 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 <em>p<\/em> orbitals overlap end to end, they create \u03c3 and \u03c3* orbitals (Figure 5). If two atoms are located along the <em>x<\/em>-axis in a Cartesian coordinate system, the two <em>p<sub>x<\/sub><\/em> orbitals overlap end to end and form \u03c3<em><sub>px<\/sub><\/em> (bonding) and [latex]{\\sigma}_{px}^\\ast[\/latex] (antibonding) (read as \"sigma-p-x\" and \"sigma-p-x star,\" respectively). Just as with <em>s<\/em>-orbital overlap, the asterisk indicates the orbital with a node between the nuclei, which is a higher-energy, antibonding orbital.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"881\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211934\/CNX_Chem_08_04_pMOsigma1.jpg\" alt=\"Two horizontal rows of diagrams are shown. The upper diagram shows two equally-sized peanut-shaped orbitals with a plus sign in between them connected to a merged orbital diagram by a right facing arrow. The merged diagram has a much larger oval at the center and much smaller ovular orbitals on the edge. It is labeled, \u201csigma subscript p x.\u201d The lower diagram shows two equally-sized peanut-shaped orbitals with a plus sign in between them connected to a split orbital diagram by a right facing arrow. The split diagram has a much larger oval at the outer ends and much smaller ovular orbitals on the inner edges. It is labeled, \u201csigma subscript p x superscript asterisk\u201d.\" width=\"881\" height=\"235\" data-media-type=\"image\/jpeg\" \/> Figure 5. Combining wave functions of two p atomic orbitals along the internuclear axis creates two molecular orbitals, \u03c3<sub>p<\/sub> and \u03c3<sup>\u2217<\/sup><sub>p<\/sub>.[\/caption]\r\n\r\nThe side-by-side overlap of two <em>p<\/em> orbitals gives rise to a <strong>pi (\u03c0) bonding molecular orbital <\/strong>and a <strong>\u03c0* antibonding molecular orbital<\/strong>, as shown in Figure 6. In valence bond theory, we describe \u03c0 bonds as containing a nodal plane containing the internuclear axis and perpendicular to the lobes of the p\u2013 \u03c0 orbitals, with electron density on either side of the node. In molecular orbital theory, we describe the \u03c0 orbital by this same shape, and a \u03c0 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"650\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211935\/CNX_Chem_08_04_pMOpi1.jpg\" alt=\"Two horizontal rows of diagrams are shown. The upper and lower diagrams both begin with two vertical peanut-shaped orbitals with a plus sign in between followed by a right-facing arrow. The upper diagram shows the same vertical peanut orbitals bending slightly away from one another and separated by a dotted line. It is labeled, \u201cpi subscript p superscript asterisk.\u201d The lower diagram shows the horizontal overlap of the two orbitals and is labeled, \u201cpi subscript p.\u201d\" width=\"650\" height=\"511\" data-media-type=\"image\/jpeg\" \/> Figure 6. Side-by-side overlap of each two p orbitals results in the formation of two \u03c0 molecular orbitals. Combining the out-of-phase orbitals results in an antibonding molecular orbital with two nodes. One contains the axis, and one contains the perpendicular. Combining the in-phase orbitals results in a bonding orbital. There is a node (blue line) directly along the internuclear axis, but the orbital is located between the nuclei (red dots) above and below this node.[\/caption]\r\n\r\nIn the molecular orbitals of diatomic molecules, each atom also has two sets of <em>p<\/em> orbitals oriented side by side (<em>p<sub>y<\/sub><\/em> and <em>p<sub>z<\/sub><\/em>), so these four atomic orbitals combine pairwise to create two \u03c0 orbitals and two \u03c0* orbitals. The \u03c0<em><sub>py<\/sub><\/em> and [latex]{\\pi}_{py}^\\ast[\/latex] orbitals are oriented at right angles to the \u03c0<em><sub>pz<\/sub><\/em> and [latex]{\\pi}_{pz}^\\ast[\/latex] orbitals. Except for their orientation, the \u03c0<em><sub>py<\/sub><\/em> and \u03c0<em><sub>pz<\/sub><\/em> orbitals are identical and have the same energy; they are <strong>degenerate orbitals<\/strong>. The [latex]{\\pi}_{py}^\\ast[\/latex] and [latex]{\\pi}_{pz}^\\ast[\/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 <em>p<\/em> orbitals in two atoms: \u03c3<em><sub>px<\/sub><\/em> and [latex]{\\sigma}_{px}^\\ast[\/latex], \u03c0<em><sub>py<\/sub><\/em> and [latex]{\\pi}_{py}^\\ast,[\/latex] \u03c0<em><sub>pz<\/sub><\/em> and [latex]{\\pi}_{pz}^\\ast[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 1:\u00a0Molecular Orbitals<\/h3>\r\nPredict 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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211937\/CNX_Chem_08_04_AOtype_img1.jpg\" alt=\"Three diagrams are shown and labeled \u201ca,\u201d \u201cb,\u201d and \u201cc.\u201d Diagram a shows two horizontal peanut-shaped orbitals laying side-by-side. They are labeled, \u201c3 p subscript x and 3 p subscript x.\u201d Diagram b shows one vertical and one horizontal peanut-shaped orbital which are at right angles to one another. They are labeled, \u201c3 p subscript x and 3 p subscript y.\u201d Diagram c shows two vertical peanut-shaped orbitals laying side-by-side and labeled, \u201c3 p subscript y and 3 p subscript y.\u201d\" width=\"799\" height=\"247\" data-media-type=\"image\/jpeg\" \/>\r\n[reveal-answer q=\"150712\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"150712\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>is an in-phase combination, resulting in a \u03c3<sub>3<em>p<\/em><\/sub> orbital<\/li>\r\n \t<li>will not result in a new orbital, because the in-phase component (bottom) and out-of-phase component (top) cancel out. Only orbitals with the correct alignment can combine.<\/li>\r\n \t<li>is an out-of-phase combination, resulting in a [latex]{\\pi}_{3p}^\\ast[\/latex] orbital.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nLabel the molecular orbital shown as <em>s<\/em> or \u03c0, bonding or antibonding. Indicate where the nuclei and nodes occur.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211938\/CNX_Chem_08_04_siganti_img1.jpg\" alt=\"Two orbitals are shown lying end-to-end. Each has one enlarged and one small side. The small sides are facing one another\" data-media-type=\"image\/jpeg\" \/>\r\n[reveal-answer q=\"606234\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"606234\"]\r\nNuclei are shown by plus signs. The orbital is along the internuclear axis, so it is a \u03c3 orbital. There is a node bisecting the internuclear axis, so it is an antibonding orbital.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211940\/CNX_Chem_08_04_salabel_img1.jpg\" alt=\"Two orbitals are shown lying end-to-end. Each has one enlarged and one small side. The small sides are facing one another and are separated by a vertical dotted line.\" data-media-type=\"image\/jpeg\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Portrait of a Chemist: Walter Kohn: Nobel Laureate<\/h3>\r\n[caption id=\"\" align=\"alignright\" width=\"200\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211941\/CNX_Chem_08_04_Kohn1.jpg\" alt=\"A photograph of Walter Kohn is shown.\" width=\"200\" height=\"233\" data-media-type=\"image\/jpeg\" \/> Figure 7. Walter Kohn developed methods to describe molecular orbitals. (credit: image courtesy of Walter Kohn)[\/caption]\r\n\r\nWalter Kohn (Figure 7) is a theoretical physicist who studies the electronic structure of solids. His work combines the principles of quantum 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.\r\n\r\nKohn 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.Kohn\u2019s biography has been remarkable outside the realm of physical chemistry as well. He was born in Austria, and during World War II he was part of the Kindertransport program that rescued 10,000 children from the Nazi regime. His summer jobs included discovering gold deposits in Canada and helping Polaroid explain how its instant film worked. Although he is now an emeritus professor, he is still actively working on projects involving global warming and renewable energy.\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>How Sciences Interconnect: Computational Chemistry in Drug Design<\/h3>\r\nWhile 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 (see Figure 8). 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"650\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211943\/CNX_Chem_08_04_HIVProteas1.jpg\" alt=\"A diagram of a molecule is shown. The image shows a tangle of ribbon-like, intertwined, pink and green curling lines with a complex ball and stick model in the center.\" width=\"650\" height=\"353\" data-media-type=\"image\/jpeg\" \/> Figure 8. The molecule shown, HIV-1 protease, is an important target for pharmaceutical research. By designing molecules that bind to this protein, scientists are able to drastically inhibit the progress of the disease.[\/caption]\r\n\r\n<\/div>\r\n<h2>Molecular Orbital Energy Diagrams<\/h2>\r\nThe relative energy levels of atomic and molecular orbitals are typically shown in a <strong>molecular orbital diagram <\/strong>(Figure 9). 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 2<em>p<\/em> atomic orbitals results in three bonding orbitals (one \u03c3 and two \u03c0) and three antibonding orbitals (one \u03c3* and two \u03c0*).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 9). 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 [latex]{\\text{Be}}_{2}^{+}[\/latex] ) would have the molecular electron configuration [latex]{\\left({\\sigma}_{1s}\\right)}^{2}{\\left({\\sigma}_{1s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{1}.[\/latex] It is common to omit the core electrons from molecular orbital diagrams and configurations and include only the valence electrons.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"800\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211945\/CNX_Chem_08_04_FillMo1.jpg\" alt=\"A diagram is shown that has an upward-facing vertical arrow running along the left side labeled, \u201cE.\u201d At the bottom center of the diagram is a horizontal line labeled, \u201csigma subscript 2 s,\u201d that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, \u201c2 s.\u201d The line on the left has two vertical half arrows drawn on it, one facing up and one facing down while the line of the right has one half arrow facing up drawn on it. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but further up from the first. It is labeled, \u201csigma subscript 2 s superscript asterisk.\u201d This horizontal line has one upward-facing vertical half-arrow drawn on it. Moving farther up the center of the diagram is a long horizontal line labeled, \u201csigma subscript 2 p subscript x,\u201d which lies below two horizontal lines. These two horizontal lines lie side-by-side, and labeled, \u201cpi subscript 2 p subscript y,\u201d and, \u201cpi subscript 2 p subscript z.\u201d Both the bottom and top lines are connected to the right and left by upward-facing, dotted lines to three more horizontal lines, each labeled, \u201c2 p.\u201d These sets of lines are connected by upward-facing dotted lines to another single line and then pair of double lines in the center of the diagram, but farther up from the lower lines. They are labeled, \u201csigma subscript 2 p subscript x superscript asterisk,\u201d and, \u201c\u201cpi subscript 2 p subscript y superscript asterisk,\u201d and, \u201cpi subscript 2 p subscript z superscript asterisk,\u201d respectively. The left and right sides of the diagram have headers that read, \u201dAtomic orbitals,\u201d while the center is header reads \u201cMolecular orbitals\u201d.\" width=\"800\" height=\"524\" data-media-type=\"image\/jpeg\" \/> Figure 9. This is the molecular orbital diagram for the homonuclear diatomic Be<sub>2<\/sub><sup>+<\/sup>, showing the molecular orbitals of the valence shell only. The molecular orbitals are filled in the same manner as atomic orbitals, using the Aufbau principle and Hund\u2019s rule.[\/caption]\r\n<h2 data-type=\"title\">Bond Order<\/h2>\r\nThe 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 <strong>bond order<\/strong> that results from the filling of the molecular orbitals by electrons.\r\n\r\nWhen 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. The MO technique is more accurate and can handle cases when the Lewis structure method fails, but both methods describe the same phenomenon.\r\n\r\nIn 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:\r\n<p style=\"text-align: center;\">[latex]\\text{bond order}=\\frac{\\left(\\text{number of bonding electrons}\\right)-\\left(\\text{number of antibonding electrons}\\right)}{2f}[\/latex]<\/p>\r\nThe order of a covalent bond is a guide to its strength; a bond between two given atoms becomes stronger as the bond order increases (Table 1). 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.\r\n<h2 data-type=\"title\">Bonding in Diatomic Molecules<\/h2>\r\nA dihydrogen molecule (H<sub>2<\/sub>) forms from two hydrogen atoms. When the atomic orbitals of the two atoms combine, the electrons occupy the molecular orbital of lowest energy, the \u03c3<sub>1<em>s<\/em><\/sub> bonding orbital. A dihydrogen molecule, H<sub>2<\/sub>, readily forms, because the energy of a H<sub>2<\/sub> molecule is lower than that of two H atoms. The \u03c3<sub>1<em>s<\/em><\/sub> orbital that contains both electrons is lower in energy than either of the two 1<em>s<\/em> atomic orbitals.\r\n\r\nA molecular orbital can hold two electrons, so both electrons in the H<sub>2<\/sub> molecule are in the \u03c3<sub>1<em>s<\/em><\/sub> bonding orbital; the electron configuration is [latex]{\\left({\\sigma}_{1s}\\right)}^{2}.[\/latex] We represent this configuration by a molecular orbital energy diagram (Figure 10) in which a single upward arrow indicates one electron in an orbital, and two (upward and downward) arrows indicate two electrons of opposite spin.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"879\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211946\/CNX_Chem_08_04_H2MO1.jpg\" alt=\"A diagram is shown that has an upward-facing vertical arrow running along the left side labeled \u201cE.\u201d At the bottom center of the diagram is a horizontal line labeled, \u201csigma subscript 1 s,\u201d that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, \u201c1 s,\u201d and each with one vertical half-arrow facing up drawn on it. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first, and labeled, \u201csigma subscript 1 s superscript asterisk.\u201d The left and right sides of the diagram have headers that read, \u201dAtomic orbitals,\u201d while the center header reads, \u201cMolecular orbitals.\u201d The bottom left and right are labeled \u201cH\u201d while the center is labeled \u201cH subscript 2.\u201d\" width=\"879\" height=\"253\" data-media-type=\"image\/jpeg\" \/> Figure 10. The molecular orbital energy diagram predicts that H<sub>2<\/sub> will be a stable molecule with lower energy than the separated atoms.[\/caption]\r\n\r\nA dihydrogen molecule contains two bonding electrons and no antibonding electrons so we have\r\n<p style=\"text-align: center;\">[latex]{\\text{bond order in H}}_{2}=\\frac{\\left(2-0\\right)}{2}=1[\/latex]<\/p>\r\nBecause the bond order for the H\u2013H bond is equal to 1, the bond is a single bond.\r\n\r\nA helium atom has two electrons, both of which are in its 1<em>s<\/em> orbital. Two helium atoms do not combine to form a dihelium molecule, He<sub>2<\/sub>, 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 He<sub>2<\/sub> as [latex]{\\left({\\sigma}_{1s}\\right)}^{2}{\\left({\\sigma}_{1s}^\\ast\\right)}^{2}[\/latex] as in Figure 11.\u00a0The 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.\r\n<p style=\"text-align: center;\">[latex]{\\text{bond order in He}}_{2}=\\frac{\\left(2-2\\right)}{2}=0[\/latex]<\/p>\r\nA bond order of zero indicates that no bond is formed between two atoms.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"879\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211948\/CNX_Chem_08_04_He2MO1.jpg\" alt=\"A diagram is shown that has an upward-facing vertical arrow running along the left side labeled, \u201cE.\u201d At the bottom center of the diagram is a horizontal line labeled, \u201csigma subscript 1 s,\u201d that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, \u201c1 s,\u201d and each with one vertical half-arrow facing up and one facing down drawn on it. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first, and labeled, \u201csigma subscript 1 s superscript asterisk.\u201d This line has one upward-facing and one downward-facing vertical arrow drawn on it. The left and right sides of the diagram have headers that read, \u201cAtomic orbitals,\u201d while the center header reads, \u201cMolecular orbitals.\u201d The bottom left and right are labeled, \u201cH e,\u201d while the center is labeled, \u201cH e subscript 2.\u201d\" width=\"879\" height=\"297\" data-media-type=\"image\/jpeg\" \/> Figure 11. The molecular orbital energy diagram predicts that He<sub>2<\/sub> will not be a stable molecule, since it has equal numbers of bonding and antibonding electrons.[\/caption]\r\n<h3 data-type=\"title\">The Diatomic Molecules of the Second Period<\/h3>\r\nEight possible homonuclear diatomic molecules might be formed by the atoms of the second period of the periodic table: Li<sub>2<\/sub>, Be<sub>2<\/sub>, B<sub>2<\/sub>, C<sub>2<\/sub>, N<sub>2<\/sub>, O<sub>2<\/sub>, F<sub>2<\/sub>, and Ne<sub>2<\/sub>. However, we can predict that the Be<sub>2<\/sub> molecule and the Ne<sub>2<\/sub> molecule would not be stable. We can see this by a consideration of the molecular electron configurations (Table 2).\r\n\r\nWe 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\u2019s 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.\r\n\r\nAs we saw in valence bond theory, \u03c3 bonds are generally more stable than \u03c0 bonds formed from degenerate atomic orbitals. Similarly, in molecular orbital theory, \u03c3 orbitals are usually more stable than \u03c0 orbitals. However, this is not always the case. The MOs for the valence orbitals of the second period are shown in\u00a0Figure 12.\u00a0Looking at Ne<sub>2<\/sub> molecular orbitals, we see that the order is consistent with the generic diagram shown in the previous section. However, for atoms with three or fewer electrons in the <em>p<\/em> orbitals (Li through N) we observe a different pattern, in which the \u03c3<em><sub>p<\/sub><\/em> orbital is higher in energy than the \u03c0<em><sub>p<\/sub><\/em> set. Obtain the molecular orbital diagram for a homonuclear diatomic ion by adding or subtracting electrons from the diagram for the neutral molecule.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"881\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211950\/CNX_Chem_08_04_X2MOs1.jpg\" alt=\"A graph is shown in which the y-axis is labeled, \u201cE,\u201d and appears as a vertical, upward-facing arrow. Across the top, the graph reads, \u201cL i subscript 2,\u201d \u201cB e subscript 2,\u201d \u201cB subscript 2,\u201d \u201cC subscript 2,\u201d \u201cN subscript 2,\u201d \u201cO subscript 2,\u201d \u201cF subscript 2,\u201d and \u201cNe subscript 2.\u201d Directly below each of these element terms is a single pink line, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled, \u201csigma subscript 2 p x superscript asterisk\u201d. Directly below each of these lines is a set of two pink lines, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. It is consistently lower than the first line. This line is labeled, \u201cpi subscript 2 p y superscript asterisk,\u201d and, \u201cpi subscript 2 p z superscript asterisk.\u201d Directly below each of these double lines is a single pink line, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. It has a distinctive drop at the label, \u201cO subscript 2.\u201d This line is labeled, \u201csigma subscript 2 p x.\u201d Directly below each of these lines is a set of two pink lines, and all lines are connected to one another by a dashed line to create an overall line that decreases very slightly in height as it moves from left to right across the graph. It is consistently lower than the third line until it reaches the point labeled, \u201cO subscript 2.\u201d This line is labeled, \u201cpi subscript 2 p y,\u201d and, \u201cpi subscript 2 p z.\u201d Directly below each of these lines is a single blue line, and all lines are connected to one another by a dashed line to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled, \u201csigma subscript 2 s superscript asterisk.\u201d Finally, directly below each of these lines is a single blue line, and all lines are connected to one another by a dashed line to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled. \u201csigma subscript 2 s.\u201d\" width=\"881\" height=\"358\" data-media-type=\"image\/jpeg\" \/> Figure 12. This shows the MO diagrams for each homonuclear diatomic molecule in the second period. The orbital energies decrease across the period as the effective nuclear charge increases and atomic radius decreases. Between N2 and O2, the order of the orbitals changes.[\/caption]\r\n\r\n<div class=\"textbox\">You can practice labeling and filling molecular orbitals with this <a href=\"https:\/\/scilearn.sydney.edu.au\/fychemistry\/calculators\/mo_diagrams.shtml\" target=\"_blank\">interactive tutorial from the University of Sydney<\/a>.<\/div>\r\nThis switch in orbital ordering occurs because of a phenomenon called <strong>s-p mixing<\/strong>. s-p mixing does not create new orbitals; it merely influences the energies of the existing molecular orbitals. The \u03c3<sub>s<\/sub> wavefunction mathematically combines with the \u03c3<sub>p<\/sub> wavefunction, with the result that the \u03c3<sub>s<\/sub> orbital becomes more stable, and the \u03c3<sub>p<\/sub> orbital becomes less stable (Figure 13). Similarly, the antibonding orbitals also undergo s-p mixing, with the \u03c3<sub>s*<\/sub> becoming more stable and the \u03c3<sub>p*<\/sub> becoming less stable.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"880\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211952\/CNX_Chem_08_04_spmix1.jpg\" alt=\"A diagram is shown. At the bottom left of the diagram is a horizontal line that is connected to the right and left by upward-facing, dotted lines to two more horizontal lines. Those two lines are connected by upward-facing dotted lines to another line in the center of the diagram but farther up from the first. Each of the bottom two central lines has a vertical downward-facing arrow. Above this structure is a horizontal line that is connected to the right and left by upward-facing, dotted lines to two sets of three horizontal lines and those two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but further up from the first. In between the horizontal lines of this structure are two pairs of horizontal lines that are above the first line but below the second and connected by dotted lines to the side horizontal lines. The bottom and top central lines each have an upward-facing vertical arrow. These two structures are redrawn on the right side of the diagram, but this time, the central lines of the bottom structure are moved downward in relation to the side lines. The upper portion of the structure has its central lines shifted upward in relation to the side lines. This structure also shows the bottom line appearing above the set of two lines.\" width=\"880\" height=\"590\" data-media-type=\"image\/jpeg\" \/> Figure 13. Without mixing, the MO pattern occurs as expected, with the \u03c3<sub>p<\/sub> orbital lower in energy than the \u03c3<sub>p<\/sub> orbitals. When s-p mixing occurs, the orbitals shift as shown, with the \u03c3<sub>p<\/sub> orbital higher in energy than the \u03c0<sub>p<\/sub> orbitals.[\/caption]\r\n\r\ns-p mixing occurs when the <em>s<\/em> and <em>p<\/em> orbitals have similar energies. When a single <em>p<\/em> orbital contains a pair of electrons, the act of pairing the electrons raises the energy of the orbital. Thus the 2<em>p<\/em> orbitals for O, F, and Ne are higher in energy than the 2<em>p<\/em> orbitals for Li, Be, B, C, and N. Because of this, O<sub>2<\/sub>, F<sub>2<\/sub>, and N<sub>2<\/sub> only have negligible s-p mixing (not sufficient to change the energy ordering), and their MO diagrams follow the normal pattern, as shown in Figure 12. All of the other period 2 diatomic molecules do have s-p mixing, which leads to the pattern where the \u03c3<sub>p<\/sub> orbital is raised above the \u03c0<sub>p<\/sub> set.\r\n\r\nUsing the MO diagrams shown in Figure 12, 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 2, Be<sub>2<\/sub> and Ne<sub>2<\/sub> molecules would have a bond order of 0, and these molecules do not exist.\r\n<table summary=\"A table is shown that has three columns and nine rows. The header row reads: \u201cMolecule,\u201d \u201cElectron Configuration,\u201d and, \u201cBond Order.\u201d The first column contains the symbols \u201cL i subscript 2,\u201d \u201cB e subscript 2 ( unstable ),\u201d \u201cB e subscript 2,\u201d \u201cC subscript 2,\u201d \u201cN subscript 2,\u201d \u201cO subscript 2,\u201d \u201cF subscript 2,\u201d and, \u201cNe subscript 2 ( unstable ).\u201d The second column contains the symbols \u201c( sigma subscript 2 s ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4 ( sigma subscript 2 p x ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( sigma subscript 2 p x ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4 ( pi superscript asterisk subscript 2 p y, pi superscript asterisk subscript 2 p z ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( sigma subscript 2 p x ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4 ( pi superscript asterisk subscript 2 p y, pi superscript asterisk subscript 2 p z ) superscript 4,\u201d and \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( sigma subscript 2 p x ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4 ( pi superscript asterisk subscript 2 p y, pi superscript asterisk subscript 2 p z ) superscript 4 ( sigma superscript asterisk subscript 2 p x ) superscript 2.\u201d The third column contains the numbers: \u201c1,\u201d \u201c0,\u201d \u201c1,\u201d \u201c2,\u201d \u201c3,\u201d \u201c2,\u201d \u201c1,\u201d \u201c0.\u201d\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"3\">Table 2. Electron Configuration and Bond Order for Molecular Orbitals in Homonuclear Diatomic Molecules of Period Two Elements<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<th>Molecule<\/th>\r\n<th>Electron Configuration<\/th>\r\n<th>Bond Order<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>Li<sub>2<\/sub><\/td>\r\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Be<sub>2<\/sub> (unstable)<\/td>\r\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B<sub>2<\/sub><\/td>\r\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{2}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C<sub>2<\/sub><\/td>\r\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}[\/latex]<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>N<sub>2<\/sub><\/td>\r\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\sigma}_{2px}\\right)}^{2}[\/latex]<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>O<sub>2<\/sub><\/td>\r\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2px}\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)}^{2}[\/latex]<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<sub>2<\/sub><\/td>\r\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2px}\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)}^{4}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Ne<sub>2<\/sub> (unstable)<\/td>\r\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2px}\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)}^{4}{\\left({\\sigma}_{2px}^\\ast\\right)}^{2}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe combination of two lithium atoms to form a lithium molecule, Li<sub>2<\/sub>, is analogous to the formation of H<sub>2<\/sub>, but the atomic orbitals involved are the valence 2<em>s<\/em> orbitals. Each of the two lithium atoms has one valence electron. Hence, we have two valence electrons available for the \u03c3<sub>2<em>s<\/em><\/sub> bonding molecular orbital. Because both valence electrons would be in a bonding orbital, we would predict the Li<sub>2<\/sub> molecule to be stable. The molecule is, in fact, present in appreciable concentration in lithium vapor at temperatures near the boiling point of the element. All of the other molecules in Table 2 with a bond order greater than zero are also known.\r\n\r\nThe O<sub>2<\/sub> molecule has enough electrons to half fill the [latex]\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)[\/latex] level. We expect the two electrons that occupy these two degenerate orbitals to be unpaired, and this molecular electronic configuration for O<sub>2<\/sub> is in accord with the fact that the oxygen molecule has two unpaired electrons (Figure 15). 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.\r\n<div class=\"textbox shaded\">\r\n<h3>How Sciences Intersect: Band Theory<\/h3>\r\nWhen two identical atomic orbitals on different atoms combine, two molecular orbitals result (see Figure 14). 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.\r\n\r\nIn 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 &gt; 10<sup>23<\/sup> atoms), and therefore a huge number of atomic orbitals that may be combined into molecular orbitals. When <em>N<\/em> valence atomic orbitals, all of the same energy and each containing one (1) electron, are combined, <em>N<\/em>\/2 (filled) bonding orbitals and <em>N<\/em>\/2 (empty) antibonding orbitals will result. Each bonding orbital will show an energy lowering as the atomic orbitals are <em>mostly<\/em> 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 <em>mostly<\/em> 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 14 shows the bands for three important classes of materials: insulators, semiconductors, and conductors.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"748\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211953\/CNX_Chem_08_04_Band1.jpg\" alt=\"This figure shows three diagrams. The first is labeled, \u201cInsulator,\u201d and it consists of two boxes. The \u201cconduction\u201d box is above and the \u201cvalence\u201d box is below. A large gap marked by 4 dashed lines contains a double-headed arrow. One head pointing towards the \u201cconduction box\u201d and the other towards the \u201cvalence\u201d box. The arrow is labeled, \u201cBand gap.\u201d The second diagram is similar to the first, but the band gap is about half as large. This diagram is labeled, \u201cSemiconductor.\u201d The third diagram is similar to the other two, but the band gap is about a fifth that of the \u201cSemiconductor\u201d diagram. This diagram is labeled, \u201cConductor.\u201d\" width=\"748\" height=\"290\" data-media-type=\"image\/jpeg\" \/> Figure 14. Molecular orbitals in solids are so closely spaced that they are described as bands. The valence band is lower in energy and the conduction band is higher in energy. The type of solid is determined by the size of the \u201cband gap\u201d between the valence and conduction bands. Only a very small amount of energy is required to move electrons from the valance band to the conduction band in a conductor, and so they conduct electricity well. In an insulator, the band gap is large, so that very few electrons move, and they are poor conductors of electricity. Semiconductors are in between: they conduct electricity better than insulators, but not as well as conductors.[\/caption]\r\n\r\nIn 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 \u201ceasy\u201d to overcome, so they are good conductors of electricity. In an insulator, the band gap is so \u201clarge\u201d that very few electrons move into the conduction band; as a result, insulators are poor conductors of electricity. Semiconductors conduct electricity when \u201cmoderate\u201d 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.\r\n\r\nSemiconductors 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.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2:\u00a0Molecular Orbital Diagrams, Bond Order, and Number of Unpaired Electrons<\/h3>\r\nDraw the molecular orbital diagram for the oxygen molecule, O<sub>2<\/sub>. From this diagram, calculate the bond order for O<sub>2<\/sub>. How does this diagram account for the paramagnetism of O<sub>2<\/sub>?\r\n\r\n[reveal-answer q=\"292725\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"292725\"]\r\n\r\nWe draw a molecular orbital energy diagram similar to that shown in Figure 12. Each oxygen atom contributes six electrons, so the diagram appears as shown in Figure 15.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"708\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211955\/CNX_Chem_08_04_O2MO1.jpg\" alt=\"A diagram is shown that has an upward-facing vertical arrow running along the left side labeled, \u201cE.\u201d At the bottom center of the diagram is a horizontal line labeled, \u201csigma subscript 2 s,\u201d that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, \u201c2 s,\u201d and with two vertical half arrows drawn on them, one facing up and one facing down. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first and labeled, \u201csigma subscript 2 s superscript asterisk.\u201d This horizontal line has two vertical half-arrow drawn on it, one facing up and one facing down. Moving further up the center of the diagram is a horizontal line labeled, \u201csigma subscript 2 p subscript x,\u201d which lies below two horizontal lines, lying side-by-side, and labeled \u201cpi subscript 2 p subscript y,\u201d and \u201cpi subscript 2 p subscript z.\u201d Both the bottom and top lines are connected to the right and left by upward-facing, dotted lines to three more horizontal lines, each labeled, \u201c2 p,\u201d on either side. These sets of lines each hold three upward-facing and one downward-facing half-arrow. They are connected by upward-facing dotted lines to another single line and then pair of double lines in the center of the diagram, but farther up from the lower lines. They are labeled, \u201csigma subscript 2 p subscript x superscript asterisk,\u201d \u201cpi subscript 2 p subscript y superscript asterisk,\u201d and \u201cpi subscript 2 p subscript z superscript asterisk,\u201d respectively. The lower of these two central, horizontal lines each contain one upward-facing half-arrow. The left and right sides of the diagram have headers that read, \u201dAtomic orbitals,\u201d while the center header reads, \u201cMolecular orbitals.\u201d\" width=\"708\" height=\"529\" data-media-type=\"image\/jpeg\" \/> Figure 15. The molecular orbital energy diagram for O<sub>2<\/sub> predicts two unpaired electrons.[\/caption]\r\n\r\nWe calculate the bond order as\r\n<p style=\"text-align: center;\">[latex]{\\text{O}}_{2}=\\frac{\\left(10-6\\right)}{2}=2[\/latex]<\/p>\r\nOxygen's paramagnetism is explained by the presence of two unpaired electrons in the (\u03c0<sub>2<em>py<\/em><\/sub>, \u03c0<sub>2<em>pz<\/em><\/sub>)* molecular orbitals.\r\n\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nThe main component of air is N<sub>2<\/sub>. From the molecular orbital diagram of N<sub>2<\/sub>, predict its bond order and whether it is diamagnetic or paramagnetic.\r\n\r\n[reveal-answer q=\"768732\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"768732\"]N<sub>2<\/sub> has a bond order of 3 and is diamagnetic.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3:\u00a0Ion Predictions with MO Diagrams<\/h3>\r\nGive the molecular orbital configuration for the valence electrons in [latex]{\\text{C}}_{2}^{2-}.[\/latex] Will this ion be stable?\r\n\r\n[reveal-answer q=\"174387\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"174387\"]Looking at the appropriate MO diagram, we see that the \u03c0 orbitals are lower in energy than the \u03c3<em><sub>p<\/sub><\/em> orbital. The valence electron configuration for C<sub>2<\/sub> is [latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{\\text{2}s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}.[\/latex] Adding two more electrons to generate the [latex]{\\text{C}}_{2}^{2-}[\/latex] anion will give a valence electron configuration of [latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{\\text{2}s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\sigma}_{2px}\\right)}^{2}.[\/latex] Since this has six more bonding electrons than antibonding, the bond order will be 3, and the ion should be stable.[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nHow many unpaired electrons would be present on a [latex]{\\text{Be}}_{2}^{2-}[\/latex] ion? Would it be paramagnetic or diamagnetic?\r\n\r\n[reveal-answer q=\"421881\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"421881\"]\r\n\r\ntwo, paramagnetic\r\n\r\nCreating molecular orbital diagrams for molecules with more than two atoms relies on the same basic ideas as the diatomic examples presented here. However, with more atoms, computers are required to calculate how the atomic orbitals combine. See <a href=\"http:\/\/mw.concord.org\/modeler\/showcase\/simulation.html?s=http:\/\/mw2.concord.org\/public\/student\/quantumchemistry\/benzene.html\" target=\"_blank\">Molecular Workshop's\u00a0three-dimensional drawings of the molecular orbitals for C<sub>6<\/sub>H<\/a><sub><a href=\"http:\/\/mw.concord.org\/modeler\/showcase\/simulation.html?s=http:\/\/mw2.concord.org\/public\/student\/quantumchemistry\/benzene.html\" target=\"_blank\">6<\/a><\/sub>.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Concepts and Summary<\/h3>\r\nMolecular 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 \u03c3 MOs. They can be formed from <em>s<\/em> orbitals or from <em>p<\/em> orbitals oriented in an end-to-end fashion. Molecular orbitals formed from <em>p<\/em> orbitals oriented in a side-by-side fashion have electron density on opposite sides of the internuclear axis and are called \u03c0 orbitals.\r\n\r\nWe 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\u2019s 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.\r\n<h4>Key Equations<\/h4>\r\n<ul>\r\n \t<li>[latex]\\text{bond order}=\\frac{\\left(\\text{number of bonding electron}\\right)-\\left(\\text{number of antibonding electrons}\\right)}{2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\n<ol>\r\n \t<li>Sketch the distribution of electron density in the bonding and antibonding molecular orbitals formed from two <em>s<\/em> orbitals and from two <em>p<\/em> orbitals.<\/li>\r\n \t<li>How are the following similar, and how do they differ?\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>\u03c3 molecular orbitals and \u03c0 molecular orbitals<\/li>\r\n \t<li>\u03c8 for an atomic orbital and \u03c8 for a molecular orbital<\/li>\r\n \t<li>bonding orbitals and antibonding orbitals<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>If molecular orbitals are created by combining five atomic orbitals from atom A and five atomic orbitals from atom B combine, how many molecular orbitals will result?<\/li>\r\n \t<li>Can a molecule with an odd number of electrons ever be diamagnetic? Explain why or why not.<\/li>\r\n \t<li>Can a molecule with an even number of electrons ever be paramagnetic? Explain why or why not.<\/li>\r\n \t<li>Why are bonding molecular orbitals lower in energy than the parent atomic orbitals?<\/li>\r\n \t<li>Calculate the bond order for an ion with this configuration: [latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2px}\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)}^{3}[\/latex]<\/li>\r\n \t<li>Explain why an electron in the bonding molecular orbital in the H<sub>2<\/sub> molecule has a lower energy than an electron in the 1<em>s<\/em> atomic orbital of either of the separated hydrogen atoms.<\/li>\r\n \t<li>Predict the valence electron molecular orbital configurations for the following, and state whether they will be stable or unstable ions.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{\\text{Na}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{Mg}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{Al}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{Si}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{P}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{S}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{F}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{Ar}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Determine the bond order of each member of the following groups, and determine which member of each group is predicted by the molecular orbital model to have the strongest bond.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>H<sub>2<\/sub>, [latex]{\\text{H}}_{2}^{\\text{+}},[\/latex] [latex]{\\text{H}}_{2}^{-}[\/latex]<\/li>\r\n \t<li>O<sub>2<\/sub>, [latex]{\\text{O}}_{2}^{\\text{2+}},[\/latex] [latex]{\\text{O}}_{2}^{2-}[\/latex]<\/li>\r\n \t<li>Li<sub>2<\/sub>, [latex]{\\text{Be}}_{2}^{\\text{+}},[\/latex] Be<sub>2\u00a0<\/sub><\/li>\r\n \t<li>F<sub>2<\/sub>, [latex]{\\text{F}}_{2}^{\\text{+}},[\/latex] [latex]{\\text{F}}_{2}^{-}[\/latex]<\/li>\r\n \t<li>N<sub>2<\/sub>, [latex]{\\text{N}}_{2}^{\\text{+}},[\/latex] [latex]{\\text{N}}_{2}^{-}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>For the first ionization energy for an N<sub>2<\/sub> molecule, what molecular orbital is the electron removed from?<\/li>\r\n \t<li>Compare the atomic 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:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>H and H<sub>2\u00a0<\/sub><\/li>\r\n \t<li>N and N<sub>2\u00a0<\/sub><\/li>\r\n \t<li>O and O<sub>2\u00a0<\/sub><\/li>\r\n \t<li>C and C<sub>2\u00a0<\/sub><\/li>\r\n \t<li>B and B<sub>2<\/sub><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Which of the period 2 homonuclear diatomic molecules are predicted to be paramagnetic?<\/li>\r\n \t<li>A friend tells you that the 2<em>s<\/em> orbital for fluorine starts off at a much lower energy than the 2<em>s<\/em> orbital for lithium, so the resulting \u03c3<sub>2<em>s<\/em><\/sub> molecular orbital in F<sub>2<\/sub> is more stable than in Li<sub>2<\/sub>. Do you agree?<\/li>\r\n \t<li>True or false: Boron contains 2<em>s<\/em><sup>2<\/sup>2<em>p<\/em><sup>1<\/sup> valence electrons, so only one <em>p<\/em> orbital is needed to form molecular orbitals.<\/li>\r\n \t<li>What charge would be needed on F<sub>2<\/sub> to generate an ion with a bond order of 2?<\/li>\r\n \t<li>Predict whether the MO diagram for S<sub>2<\/sub> would show s-p mixing or not.<\/li>\r\n \t<li>Explain why [latex]{\\text{N}}_{2}^{\\text{2+}}[\/latex] is diamagnetic, while [latex]{\\text{O}}_{2}^{\\text{4+}},[\/latex] which has the same number of valence electrons, is paramagnetic.<\/li>\r\n \t<li>Using the MO diagrams, predict the bond order for the stronger bond in each pair:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>B<sub>2<\/sub> or [latex]{\\text{B}}_{2}^{+}[\/latex]<\/li>\r\n \t<li>F<sub>2<\/sub> or [latex]{\\text{F}}_{2}^{+}[\/latex]<\/li>\r\n \t<li>O<sub>2<\/sub> or [latex]{\\text{O}}_{2}^{\\text{2+}}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{C}}_{2}^{+}[\/latex] or [latex]{\\text{C}}_{2}^{-}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"503798\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"503798\"]\r\n\r\n2. The similarities and differences are as follows:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Similarities: Both are bonding orbitals that can contain a maximum of two electrons. Differences: <em>\u03c3<\/em> orbitals are end-to-end combinations of atomic orbitals, whereas <em>\u03c0<\/em> orbitals are formed by side-by-side overlap of orbitals.<\/li>\r\n \t<li>Similarities: Both are quantum-mechanical constructs that represent the probability of finding the electron about the atom or the molecule. Differences: <em>\u03c8<\/em> for an atomic orbital describes the behavior of only one electron at a time based on the atom. For a molecule, <em>\u03c8<\/em> represents either a mathematical combination of atomic orbitals.<\/li>\r\n \t<li>Similarities: Both are orbitals that can contain two electrons. Differences: Bonding orbitals result in holding two or more atoms together. Antibonding orbitals have the effect of destabilizing any bonding that has occurred.<\/li>\r\n<\/ol>\r\n4.\u00a0An odd number of electrons can never be paired, regardless of the arrangement of the molecular orbitals. It will always be paramagnetic.\r\n\r\n6.\u00a0Bonding orbitals have electron density in close proximity to more than one nucleus. The interaction between the bonding positively charged nuclei and negatively charged electrons stabilizes the system.\r\n\r\n8.\u00a0The pairing of the two bonding electrons lowers the energy of the system relative to the energy of the nonbonded electrons.\r\n\r\n10.\u00a0The bond order is equal to half the difference between the number of bonding electrons and the number of antibonding electrons. The bond with the greatest bond order is predicted to be the strongest.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>H<sub>2<\/sub> bond order = 1, [latex]{\\text{H}}_{2}^{+}[\/latex] bond order = 0.5, [latex]{\\text{H}}_{2}^{-}[\/latex] bond order = 0.5, strongest bond is H<sub>2<\/sub>;<\/li>\r\n \t<li>O<sub>2<\/sub> bond order = 2, [latex]{\\text{O}}_{2}^{\\text{2+}}[\/latex] bond order = 3; [latex]{\\text{O}}_{2}^{2-}[\/latex] bond order = 1, strongest bond is [latex]{\\text{O}}_{2}^{\\text{2+}};[\/latex]<\/li>\r\n \t<li>Li<sub>2<\/sub> bond order = 1, [latex]{\\text{Be}}_{2}^{+}[\/latex] bond order = 0.5, Be<sub>2<\/sub> bond order = 0, Li<sub>2<\/sub> [latex]{\\text{Be}}_{2}^{+}[\/latex] have the same strength bond;<\/li>\r\n \t<li>F<sub>2<\/sub> bond order = 1, [latex]{\\text{F}}_{2}^{+}[\/latex] bond order = 1.5, [latex]{\\text{F}}_{2}^{-}[\/latex] bond order = 0.5, strongest bond is [latex]{\\text{F}}_{2}^{\\text{+}};[\/latex]<\/li>\r\n \t<li>N<sub>2<\/sub> bond order = 3, [latex]{\\text{N}}_{2}^{+}[\/latex] bond order = 2.5, [latex]{\\text{N}}_{2}^{-}[\/latex] bond order = 2.5, strongest bond is N<sub>2<\/sub>.<\/li>\r\n<\/ol>\r\n12. The substance with the\u00a0highest first ionization energy in each pair is as follows:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>H<sub>2<\/sub><\/li>\r\n \t<li>N<sub>2<\/sub><\/li>\r\n \t<li>O<\/li>\r\n \t<li>C<sub>2<\/sub><\/li>\r\n \t<li>B<sub>2<\/sub><\/li>\r\n<\/ol>\r\n14.\u00a0Yes, fluorine is a smaller atom than Li, so atoms in the 2<em>s<\/em> orbital are closer to the nucleus and more stable.\r\n\r\n16.\u00a02+\r\n\r\n18.\u00a0N<sub>2<\/sub> has s-p mixing, so the \u03c0 orbitals are the last filled in [latex]{\\text{N}}_{2}^{\\text{2+}}.[\/latex] O<sub>2<\/sub> does not have s-p mixing, so the \u03c3<em><sub>p<\/sub><\/em> orbital fills before the \u03c0 orbitals.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<strong>antibonding orbital:<\/strong> molecular orbital located outside of the region between two nuclei; electrons in an antibonding orbital destabilize the molecule\r\n\r\n<strong>bond order:<\/strong> number of pairs of electrons between two atoms; it can be found by the number of bonds in a Lewis structure or by the difference between the number of bonding and antibonding electrons divided by two\r\n\r\n<strong>bonding orbital:<\/strong> molecular orbital located between two nuclei; electrons in a bonding orbital stabilize a molecule\r\n\r\n<strong>degenerate orbitals:<\/strong> orbitals that have the same energy\r\n\r\n<strong>diamagnetism<strong>:<\/strong><\/strong> phenomenon in which a material is not magnetic itself but is repelled by a magnetic field; it occurs when there are only paired electrons present\r\n\r\n<strong>homonuclear diatomic molecule<strong>:<\/strong><\/strong> molecule consisting of two identical atoms\r\n\r\n<strong>linear combination of atomic orbitals<strong>:<\/strong><\/strong> technique for combining atomic orbitals to create molecular orbitals\r\n\r\n<strong>molecular orbital<strong>:<\/strong><\/strong> region of space in which an electron has a high probability of being found in a molecule\r\n\r\n<strong>molecular orbital diagram<strong>:<\/strong><\/strong> visual representation of the relative energy levels of molecular orbitals\r\n\r\n<strong>molecular orbital theory<strong>:<\/strong><\/strong> model that describes the behavior of electrons delocalized throughout a molecule in terms of the combination of atomic wave functions\r\n\r\n<strong>paramagnetism<strong>:<\/strong><\/strong> phenomenon in which a material is not magnetic itself but is attracted to a magnetic field; it occurs when there are unpaired electrons present\r\n\r\n<strong>\u03c0 bonding orbital<strong>:<\/strong><\/strong> molecular orbital formed by side-by-side overlap of atomic orbitals, in which the electron density is found on opposite sides of the internuclear axis\r\n\r\n<strong>\u03c0* bonding orbital<strong>:<\/strong><\/strong> antibonding molecular orbital formed by out of phase side-by-side overlap of atomic orbitals, in which the electron density is found on both sides of the internuclear axis, and there is a node between the nuclei\r\n\r\n<strong>\u03c3 bonding orbital<strong>:<\/strong><\/strong> molecular orbital in which the electron density is found along the axis of the bond\r\n\r\n<strong>\u03c3* bonding orbital<strong>:<\/strong><\/strong> antibonding molecular orbital formed by out-of-phase overlap of atomic orbital along the axis of the bond, generating a node between the nuclei\r\n\r\n<strong>s-p mixing:<\/strong> change that causes \u03c3<em><sub>p<\/sub><\/em> orbitals to be less stable than \u03c0<em><sub>p<\/sub><\/em> orbitals due to the mixing of <em>s<\/em> and <em>p<\/em>-based molecular orbitals of similar energies.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Outline the basic quantum-mechanical approach to deriving molecular orbitals from atomic orbitals<\/li>\n<li>Describe traits of bonding and anti-bonding molecular orbitals<\/li>\n<li>Calculate bond orders based on molecular electron configurations<\/li>\n<li>Write molecular electron configurations for first- and second-row diatomic molecules<\/li>\n<li>Relate these electron configurations to the molecules\u2019 stabilities and magnetic properties<\/li>\n<\/ul>\n<\/div>\n<p>For almost every covalent molecule that exists, we can now draw the Lewis structure, predict the electron-pair geometry, predict the molecular geometry, and come close to predicting bond angles. However, one of the most important molecules we know, the oxygen molecule O<sub>2<\/sub>, presents a problem with respect to its Lewis structure. We would write the following Lewis structure for O<sub>2<\/sub>:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211927\/CNX_Chem_08_04_O2_img1.jpg\" alt=\"A Lewis structure is shown. It is made up of two oxygen atoms, each with two lone pairs of electrons, bonded together with a double bond.\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p>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 it. However, this picture is at odds with the magnetic behavior of oxygen. By itself, O<sub>2<\/sub> is not magnetic, but it is attracted to magnetic fields. Thus, when we pour liquid oxygen past a strong magnet, it collects between the poles of the magnet and defies gravity, as in Figure 1. Such attraction to a magnetic field is called <strong>paramagnetism<\/strong>, and it arises in molecules that have unpaired electrons. And yet, the Lewis structure of O<sub>2<\/sub> indicates that all electrons are paired. How do we account for this discrepancy?<\/p>\n<div style=\"width: 760px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211747\/CNX_Chem_08_00_LiqO21.jpg\" alt=\"A pitcher is shown pouring liquid oxygen through the gap between two magnets, where it has formed a solid disk. A call out box near the stream of liquid oxygen shows an image of six pairs of spheres, spread apart from one another. Another call out box near the solid disk shows ten pairs of spheres much closer together.\" width=\"750\" height=\"346\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. Oxygen molecules orient randomly most of the time, as shown in the top magnified view. However, when we pour liquid oxygen through a magnet, the molecules line up with the magnetic field, and the attraction allows them to stay suspended between the poles of the magnet where the magnetic field is strongest. Other diatomic molecules (like N<sub>2<\/sub>) flow past the magnet. The detailed explanation of bonding described in this chapter allows us to understand this phenomenon. (credit: modification of work by Jefferson Lab)<\/p>\n<\/div>\n<p>Magnetic susceptibility measures the force experienced by a substance in a magnetic field. When we compare the weight of a sample to the weight measured in a magnetic field (Figure 2), paramagnetic samples that are attracted to the magnet will appear heavier because of the force exerted by the magnetic field. We can calculate the number of unpaired electrons based on the increase in weight.<\/p>\n<div style=\"width: 810px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211928\/CNX_Chem_08_04_Gouy1.jpg\" alt=\"A diagram depicts a stand supporting two objects that are held in balance by a horizontal bar. On the right, the bar supports a dish that is holding two weights. On the left there is a line attached to a test tube labeled, \u201cSample tube.\u201d The test tube has been lowered into the space labeled, \u201cMagnetic field,\u201d between two structures labeled, \u201cElectromagnets.\u201d\" width=\"800\" height=\"601\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. A Gouy balance 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.<\/p>\n<\/div>\n<p>Experiments show that each O<sub>2<\/sub> molecule has two unpaired electrons. The Lewis-structure model does not predict the presence of these two 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 <strong>diamagnetic<\/strong> and weakly repel a magnetic field. Paramagnetic and diamagnetic materials do not act as permanent magnets. Only in the presence of an applied magnetic field do they demonstrate attraction or repulsion.<\/p>\n<div class=\"textbox\">Water, like most molecules, contains all paired electrons. Living things contain a large percentage of water, so they demonstrate diamagnetic behavior. If you place a frog near a sufficiently large magnet, it will levitate. You can see videos of diamagnetic floating frogs, strawberries, and more at the <a href=\"http:\/\/www.ru.nl\/hfml\/research\/levitation\/diamagnetic\/\" target=\"_blank\">Radboud University Diamagnetic Levitation website<\/a>.<\/div>\n<p>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, such as 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 atomic orbitals to yield molecular orbitals that are <em>delocalized<\/em> over the entire molecule rather than being 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 1 summarizes the main points of the two complementary bonding theories. Both theories provide different, useful ways of describing molecular structure.<\/p>\n<table summary=\"A table is shown that is composed of two columns and six rows. The header row reads, \u201cValence Bond Theory,\u201d and, \u201cMolecular Orbital Theory.\u201d The first column contains the phrases: \u201cconsiders bonds as localized between one pair of atoms,\u201d \u201ccreates bonds from overlap of atomic orbitals ( s, p, d\u2026) and hybrid orbitals ( s p , s p superscript 2, s p superscript 3 \u2026 ) ,\u201d \u201cforms sigma or pi bonds,\u201d \u201cpredicts molecular shape based on the number of regions of electron density,\u201d and, \u201cneeds multiple structures are needed to describe resonance.\u201d The second column reads, \u201cconsiders electrons delocalized throughout the entire molecule,\u201d \u201ccombines atomic orbitals to form molecular orbitals ( sigma, sigma superscript asterik, pi, pi superscript asterisk ),\u201d \u201ccreates bonding and antibonding interactions based on which orbitals are filled,\u201d \u201cpredicts the arrangement of electrons in molecules.\u201d\">\n<thead>\n<tr>\n<th colspan=\"2\">Table 1. Comparison of Bonding Theories<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>Valence Bond Theory<\/th>\n<th>Molecular Orbital Theory<\/th>\n<\/tr>\n<tr>\n<td>considers bonds as localized between one pair of atoms<\/td>\n<td>considers electrons delocalized throughout the entire molecule<\/td>\n<\/tr>\n<tr>\n<td>creates bonds from overlap of atomic orbitals (<em>s, p, d<\/em>\u2026) and hybrid orbitals (<em>sp, sp<\/em><sup>2<\/sup>, <em>sp<\/em><sup>3<\/sup>\u2026)<\/td>\n<td>combines atomic orbitals to form molecular orbitals (\u03c3, \u03c3*, \u03c0, \u03c0*)<\/td>\n<\/tr>\n<tr>\n<td>forms \u03c3 or \u03c0 bonds<\/td>\n<td>creates bonding and antibonding interactions based on which orbitals are filled<\/td>\n<\/tr>\n<tr>\n<td>predicts molecular shape based on the number of regions of electron density<\/td>\n<td>predicts the arrangement of electrons in molecules<\/td>\n<\/tr>\n<tr>\n<td>needs multiple structures to describe resonance<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Molecular orbital theory <\/strong>describes the distribution of electrons in molecules in much the same way that the distribution of electrons in atoms is described using atomic orbitals. Using quantum mechanics, the behavior of an electron in a molecule is still described by a wave function, <em>\u03a8<\/em>, analogous to the behavior in an atom. Just like electrons around isolated atoms, electrons around atoms in molecules are limited to discrete (quantized) energies. The region of space in which a valence electron in a molecule is likely to be found is called a<strong> molecular orbital (<em>\u03a8<\/em><sup>2<\/sup>)<\/strong>. Like an atomic orbital, a molecular orbital is full when it contains two electrons with opposite spin.<\/p>\n<p>We will consider the molecular orbitals in molecules composed of two identical atoms (H<sub>2<\/sub> or Cl<sub>2<\/sub>, for example). Such molecules are called <strong>homonuclear diatomic molecules<\/strong>. In these diatomic molecules, several types of molecular orbitals occur.<\/p>\n<p>The mathematical process of combining atomic orbitals to generate molecular orbitals is called the <strong>linear combination of atomic orbitals (LCAO)<\/strong>. The wave function describes the wavelike properties of an electron. Molecular orbitals are combinations of atomic orbital wave functions. Combining waves can lead to constructive interference, in which peaks line up with peaks, or destructive interference, in which peaks line up with troughs (Figure 3). 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.<\/p>\n<div style=\"width: 892px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211931\/CNX_Chem_08_04_waveadd1.jpg\" alt=\"A pair of diagrams are shown and labeled, \u201ca\u201d and \u201cb.\u201d Diagram a shows two identical waves with two crests and two troughs. They are drawn one above the other with a plus sign in between and an equal sign to the right. To the right of the equal sign is a much taller wave with a same number of troughs and crests. Diagram b shows two waves with two crests and two troughs, but they are mirror images of one another rotated over a horizontal axis. They are drawn one above the other with a plus sign in between and an equal sign to the right. To the right of the equal sign is a flat line.\" width=\"882\" height=\"186\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. (a) When in-phase waves combine, constructive interference produces a wave with greater amplitude. (b) When out-of-phase waves combine, destructive interference produces a wave with less (or no) amplitude.<\/p>\n<\/div>\n<p>There are two types of molecular orbitals that can form from the overlap of two atomic <em>s<\/em> orbitals on adjacent atoms. The two types are illustrated in Figure 4. The in-phase combination produces a lower energy <strong>\u03c3<sub><em>s<\/em><\/sub> molecular orbital<\/strong> (read as &#8220;sigma-s&#8221;) in which most of the electron density is directly between the nuclei. The out-of-phase addition (which can also be thought of as subtracting the wave functions) produces a higher energy [latex]{\\sigma}_{s}^\\ast[\/latex] <strong>molecular orbital<\/strong> (read as &#8220;sigma-s-star&#8221;) molecular orbital in which there is a node between the nuclei. The asterisk signifies that the orbital is an antibonding orbital. Electrons in a \u03c3<em><sub>s<\/sub><\/em> 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. Adding electrons to these orbitals creates a force that holds the two nuclei together, so we call these orbitals <strong>bonding orbitals<\/strong>. Electrons in the [latex]{\\sigma}_{s}^\\ast[\/latex] 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<strong> antibonding orbitals<\/strong>. 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.<\/p>\n<div style=\"width: 889px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211933\/CNX_Chem_08_04_ssigma1.jpg\" alt=\"A diagram is shown that depicts a vertical upward-facing arrow that lies to the left of all the other portions of the diagram and is labeled, \u201cE.\u201d To the immediate right of the midpoint of the arrow are two circles each labeled with a positive sign, the letter S, and the phrase, \u201cAtomic orbitals.\u201d These are followed by a right-facing horizontal arrow that points to the same two circles labeled with plus signs, but they are now touching and are labeled, \u201cCombine atomic orbitals.\u201d Two right-facing arrows lead to the last portion of the diagram, one facing upward and one facing downward. The upper arrow is labeled, \u201cSubtract,\u201d and points to two oblong ovals labeled with plus signs, and the phrase, \u201cAntibonding orbitals sigma subscript s superscript asterisk.\u201d The lower arrow is labeled, \u201cAdd,\u201d and points to an elongated oval with two plus signs that is labeled, \u201cBonding orbital sigma subscript s.\u201d The heading over the last section of the diagram are the words, \u201cMolecular orbitals.\u201d\" width=\"879\" height=\"358\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. Sigma (\u03c3) and sigma-star (\u03c3*) molecular orbitals are formed by the combination of two s atomic orbitals. The plus (+) signs indicate the locations of nuclei.<\/p>\n<\/div>\n<div class=\"textbox\">You can watch animations visualizing the calculated atomic orbitals combining to form various molecular orbitals at the <a href=\"https:\/\/winter.group.shef.ac.uk\/orbitron\/MOs\/N2\/2s2s-sigma\/index.html\" target=\"_blank\">University of Sheffield&#8217;s Orbitron website<\/a>.<\/div>\n<p>In <em>p<\/em> 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 <em>p<\/em> orbitals overlap end to end, they create \u03c3 and \u03c3* orbitals (Figure 5). If two atoms are located along the <em>x<\/em>-axis in a Cartesian coordinate system, the two <em>p<sub>x<\/sub><\/em> orbitals overlap end to end and form \u03c3<em><sub>px<\/sub><\/em> (bonding) and [latex]{\\sigma}_{px}^\\ast[\/latex] (antibonding) (read as &#8220;sigma-p-x&#8221; and &#8220;sigma-p-x star,&#8221; respectively). Just as with <em>s<\/em>-orbital overlap, the asterisk indicates the orbital with a node between the nuclei, which is a higher-energy, antibonding orbital.<\/p>\n<div style=\"width: 891px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211934\/CNX_Chem_08_04_pMOsigma1.jpg\" alt=\"Two horizontal rows of diagrams are shown. The upper diagram shows two equally-sized peanut-shaped orbitals with a plus sign in between them connected to a merged orbital diagram by a right facing arrow. The merged diagram has a much larger oval at the center and much smaller ovular orbitals on the edge. It is labeled, \u201csigma subscript p x.\u201d The lower diagram shows two equally-sized peanut-shaped orbitals with a plus sign in between them connected to a split orbital diagram by a right facing arrow. The split diagram has a much larger oval at the outer ends and much smaller ovular orbitals on the inner edges. It is labeled, \u201csigma subscript p x superscript asterisk\u201d.\" width=\"881\" height=\"235\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. Combining wave functions of two p atomic orbitals along the internuclear axis creates two molecular orbitals, \u03c3<sub>p<\/sub> and \u03c3<sup>\u2217<\/sup><sub>p<\/sub>.<\/p>\n<\/div>\n<p>The side-by-side overlap of two <em>p<\/em> orbitals gives rise to a <strong>pi (\u03c0) bonding molecular orbital <\/strong>and a <strong>\u03c0* antibonding molecular orbital<\/strong>, as shown in Figure 6. In valence bond theory, we describe \u03c0 bonds as containing a nodal plane containing the internuclear axis and perpendicular to the lobes of the p\u2013 \u03c0 orbitals, with electron density on either side of the node. In molecular orbital theory, we describe the \u03c0 orbital by this same shape, and a \u03c0 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.<\/p>\n<div style=\"width: 660px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211935\/CNX_Chem_08_04_pMOpi1.jpg\" alt=\"Two horizontal rows of diagrams are shown. The upper and lower diagrams both begin with two vertical peanut-shaped orbitals with a plus sign in between followed by a right-facing arrow. The upper diagram shows the same vertical peanut orbitals bending slightly away from one another and separated by a dotted line. It is labeled, \u201cpi subscript p superscript asterisk.\u201d The lower diagram shows the horizontal overlap of the two orbitals and is labeled, \u201cpi subscript p.\u201d\" width=\"650\" height=\"511\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. Side-by-side overlap of each two p orbitals results in the formation of two \u03c0 molecular orbitals. Combining the out-of-phase orbitals results in an antibonding molecular orbital with two nodes. One contains the axis, and one contains the perpendicular. Combining the in-phase orbitals results in a bonding orbital. There is a node (blue line) directly along the internuclear axis, but the orbital is located between the nuclei (red dots) above and below this node.<\/p>\n<\/div>\n<p>In the molecular orbitals of diatomic molecules, each atom also has two sets of <em>p<\/em> orbitals oriented side by side (<em>p<sub>y<\/sub><\/em> and <em>p<sub>z<\/sub><\/em>), so these four atomic orbitals combine pairwise to create two \u03c0 orbitals and two \u03c0* orbitals. The \u03c0<em><sub>py<\/sub><\/em> and [latex]{\\pi}_{py}^\\ast[\/latex] orbitals are oriented at right angles to the \u03c0<em><sub>pz<\/sub><\/em> and [latex]{\\pi}_{pz}^\\ast[\/latex] orbitals. Except for their orientation, the \u03c0<em><sub>py<\/sub><\/em> and \u03c0<em><sub>pz<\/sub><\/em> orbitals are identical and have the same energy; they are <strong>degenerate orbitals<\/strong>. The [latex]{\\pi}_{py}^\\ast[\/latex] and [latex]{\\pi}_{pz}^\\ast[\/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 <em>p<\/em> orbitals in two atoms: \u03c3<em><sub>px<\/sub><\/em> and [latex]{\\sigma}_{px}^\\ast[\/latex], \u03c0<em><sub>py<\/sub><\/em> and [latex]{\\pi}_{py}^\\ast,[\/latex] \u03c0<em><sub>pz<\/sub><\/em> and [latex]{\\pi}_{pz}^\\ast[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1:\u00a0Molecular Orbitals<\/h3>\n<p>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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211937\/CNX_Chem_08_04_AOtype_img1.jpg\" alt=\"Three diagrams are shown and labeled \u201ca,\u201d \u201cb,\u201d and \u201cc.\u201d Diagram a shows two horizontal peanut-shaped orbitals laying side-by-side. They are labeled, \u201c3 p subscript x and 3 p subscript x.\u201d Diagram b shows one vertical and one horizontal peanut-shaped orbital which are at right angles to one another. They are labeled, \u201c3 p subscript x and 3 p subscript y.\u201d Diagram c shows two vertical peanut-shaped orbitals laying side-by-side and labeled, \u201c3 p subscript y and 3 p subscript y.\u201d\" width=\"799\" height=\"247\" data-media-type=\"image\/jpeg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q150712\">Show Answer<\/span><\/p>\n<div id=\"q150712\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>is an in-phase combination, resulting in a \u03c3<sub>3<em>p<\/em><\/sub> orbital<\/li>\n<li>will not result in a new orbital, because the in-phase component (bottom) and out-of-phase component (top) cancel out. Only orbitals with the correct alignment can combine.<\/li>\n<li>is an out-of-phase combination, resulting in a [latex]{\\pi}_{3p}^\\ast[\/latex] orbital.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>Label the molecular orbital shown as <em>s<\/em> or \u03c0, bonding or antibonding. Indicate where the nuclei and nodes occur.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211938\/CNX_Chem_08_04_siganti_img1.jpg\" alt=\"Two orbitals are shown lying end-to-end. Each has one enlarged and one small side. The small sides are facing one another\" data-media-type=\"image\/jpeg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q606234\">Show Answer<\/span><\/p>\n<div id=\"q606234\" class=\"hidden-answer\" style=\"display: none\">\nNuclei are shown by plus signs. The orbital is along the internuclear axis, so it is a \u03c3 orbital. There is a node bisecting the internuclear axis, so it is an antibonding orbital.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211940\/CNX_Chem_08_04_salabel_img1.jpg\" alt=\"Two orbitals are shown lying end-to-end. Each has one enlarged and one small side. The small sides are facing one another and are separated by a vertical dotted line.\" data-media-type=\"image\/jpeg\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Portrait of a Chemist: Walter Kohn: Nobel Laureate<\/h3>\n<div style=\"width: 210px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211941\/CNX_Chem_08_04_Kohn1.jpg\" alt=\"A photograph of Walter Kohn is shown.\" width=\"200\" height=\"233\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7. Walter Kohn developed methods to describe molecular orbitals. (credit: image courtesy of Walter Kohn)<\/p>\n<\/div>\n<p>Walter Kohn (Figure 7) is a theoretical physicist who studies the electronic structure of solids. His work combines the principles of quantum 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.<\/p>\n<p>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.Kohn\u2019s biography has been remarkable outside the realm of physical chemistry as well. He was born in Austria, and during World War II he was part of the Kindertransport program that rescued 10,000 children from the Nazi regime. His summer jobs included discovering gold deposits in Canada and helping Polaroid explain how its instant film worked. Although he is now an emeritus professor, he is still actively working on projects involving global warming and renewable energy.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>How Sciences Interconnect: Computational Chemistry in Drug Design<\/h3>\n<p>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 (see Figure 8). 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.<\/p>\n<div style=\"width: 660px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211943\/CNX_Chem_08_04_HIVProteas1.jpg\" alt=\"A diagram of a molecule is shown. The image shows a tangle of ribbon-like, intertwined, pink and green curling lines with a complex ball and stick model in the center.\" width=\"650\" height=\"353\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8. The molecule shown, HIV-1 protease, is an important target for pharmaceutical research. By designing molecules that bind to this protein, scientists are able to drastically inhibit the progress of the disease.<\/p>\n<\/div>\n<\/div>\n<h2>Molecular Orbital Energy Diagrams<\/h2>\n<p>The relative energy levels of atomic and molecular orbitals are typically shown in a <strong>molecular orbital diagram <\/strong>(Figure 9). 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 2<em>p<\/em> atomic orbitals results in three bonding orbitals (one \u03c3 and two \u03c0) and three antibonding orbitals (one \u03c3* and two \u03c0*).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 9). 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 [latex]{\\text{Be}}_{2}^{+}[\/latex] ) would have the molecular electron configuration [latex]{\\left({\\sigma}_{1s}\\right)}^{2}{\\left({\\sigma}_{1s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{1}.[\/latex] It is common to omit the core electrons from molecular orbital diagrams and configurations and include only the valence electrons.<\/p>\n<div style=\"width: 810px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211945\/CNX_Chem_08_04_FillMo1.jpg\" alt=\"A diagram is shown that has an upward-facing vertical arrow running along the left side labeled, \u201cE.\u201d At the bottom center of the diagram is a horizontal line labeled, \u201csigma subscript 2 s,\u201d that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, \u201c2 s.\u201d The line on the left has two vertical half arrows drawn on it, one facing up and one facing down while the line of the right has one half arrow facing up drawn on it. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but further up from the first. It is labeled, \u201csigma subscript 2 s superscript asterisk.\u201d This horizontal line has one upward-facing vertical half-arrow drawn on it. Moving farther up the center of the diagram is a long horizontal line labeled, \u201csigma subscript 2 p subscript x,\u201d which lies below two horizontal lines. These two horizontal lines lie side-by-side, and labeled, \u201cpi subscript 2 p subscript y,\u201d and, \u201cpi subscript 2 p subscript z.\u201d Both the bottom and top lines are connected to the right and left by upward-facing, dotted lines to three more horizontal lines, each labeled, \u201c2 p.\u201d These sets of lines are connected by upward-facing dotted lines to another single line and then pair of double lines in the center of the diagram, but farther up from the lower lines. They are labeled, \u201csigma subscript 2 p subscript x superscript asterisk,\u201d and, \u201c\u201cpi subscript 2 p subscript y superscript asterisk,\u201d and, \u201cpi subscript 2 p subscript z superscript asterisk,\u201d respectively. The left and right sides of the diagram have headers that read, \u201dAtomic orbitals,\u201d while the center is header reads \u201cMolecular orbitals\u201d.\" width=\"800\" height=\"524\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9. This is the molecular orbital diagram for the homonuclear diatomic Be<sub>2<\/sub><sup>+<\/sup>, showing the molecular orbitals of the valence shell only. The molecular orbitals are filled in the same manner as atomic orbitals, using the Aufbau principle and Hund\u2019s rule.<\/p>\n<\/div>\n<h2 data-type=\"title\">Bond Order<\/h2>\n<p>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 <strong>bond order<\/strong> that results from the filling of the molecular orbitals by electrons.<\/p>\n<p>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. The MO technique is more accurate and can handle cases when the Lewis structure method fails, but both methods describe the same phenomenon.<\/p>\n<p>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:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{bond order}=\\frac{\\left(\\text{number of bonding electrons}\\right)-\\left(\\text{number of antibonding electrons}\\right)}{2f}[\/latex]<\/p>\n<p>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 (Table 1). 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.<\/p>\n<h2 data-type=\"title\">Bonding in Diatomic Molecules<\/h2>\n<p>A dihydrogen molecule (H<sub>2<\/sub>) forms from two hydrogen atoms. When the atomic orbitals of the two atoms combine, the electrons occupy the molecular orbital of lowest energy, the \u03c3<sub>1<em>s<\/em><\/sub> bonding orbital. A dihydrogen molecule, H<sub>2<\/sub>, readily forms, because the energy of a H<sub>2<\/sub> molecule is lower than that of two H atoms. The \u03c3<sub>1<em>s<\/em><\/sub> orbital that contains both electrons is lower in energy than either of the two 1<em>s<\/em> atomic orbitals.<\/p>\n<p>A molecular orbital can hold two electrons, so both electrons in the H<sub>2<\/sub> molecule are in the \u03c3<sub>1<em>s<\/em><\/sub> bonding orbital; the electron configuration is [latex]{\\left({\\sigma}_{1s}\\right)}^{2}.[\/latex] We represent this configuration by a molecular orbital energy diagram (Figure 10) in which a single upward arrow indicates one electron in an orbital, and two (upward and downward) arrows indicate two electrons of opposite spin.<\/p>\n<div style=\"width: 889px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211946\/CNX_Chem_08_04_H2MO1.jpg\" alt=\"A diagram is shown that has an upward-facing vertical arrow running along the left side labeled \u201cE.\u201d At the bottom center of the diagram is a horizontal line labeled, \u201csigma subscript 1 s,\u201d that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, \u201c1 s,\u201d and each with one vertical half-arrow facing up drawn on it. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first, and labeled, \u201csigma subscript 1 s superscript asterisk.\u201d The left and right sides of the diagram have headers that read, \u201dAtomic orbitals,\u201d while the center header reads, \u201cMolecular orbitals.\u201d The bottom left and right are labeled \u201cH\u201d while the center is labeled \u201cH subscript 2.\u201d\" width=\"879\" height=\"253\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 10. The molecular orbital energy diagram predicts that H<sub>2<\/sub> will be a stable molecule with lower energy than the separated atoms.<\/p>\n<\/div>\n<p>A dihydrogen molecule contains two bonding electrons and no antibonding electrons so we have<\/p>\n<p style=\"text-align: center;\">[latex]{\\text{bond order in H}}_{2}=\\frac{\\left(2-0\\right)}{2}=1[\/latex]<\/p>\n<p>Because the bond order for the H\u2013H bond is equal to 1, the bond is a single bond.<\/p>\n<p>A helium atom has two electrons, both of which are in its 1<em>s<\/em> orbital. Two helium atoms do not combine to form a dihelium molecule, He<sub>2<\/sub>, 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 He<sub>2<\/sub> as [latex]{\\left({\\sigma}_{1s}\\right)}^{2}{\\left({\\sigma}_{1s}^\\ast\\right)}^{2}[\/latex] as in Figure 11.\u00a0The 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.<\/p>\n<p style=\"text-align: center;\">[latex]{\\text{bond order in He}}_{2}=\\frac{\\left(2-2\\right)}{2}=0[\/latex]<\/p>\n<p>A bond order of zero indicates that no bond is formed between two atoms.<\/p>\n<div style=\"width: 889px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211948\/CNX_Chem_08_04_He2MO1.jpg\" alt=\"A diagram is shown that has an upward-facing vertical arrow running along the left side labeled, \u201cE.\u201d At the bottom center of the diagram is a horizontal line labeled, \u201csigma subscript 1 s,\u201d that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, \u201c1 s,\u201d and each with one vertical half-arrow facing up and one facing down drawn on it. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first, and labeled, \u201csigma subscript 1 s superscript asterisk.\u201d This line has one upward-facing and one downward-facing vertical arrow drawn on it. The left and right sides of the diagram have headers that read, \u201cAtomic orbitals,\u201d while the center header reads, \u201cMolecular orbitals.\u201d The bottom left and right are labeled, \u201cH e,\u201d while the center is labeled, \u201cH e subscript 2.\u201d\" width=\"879\" height=\"297\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 11. The molecular orbital energy diagram predicts that He<sub>2<\/sub> will not be a stable molecule, since it has equal numbers of bonding and antibonding electrons.<\/p>\n<\/div>\n<h3 data-type=\"title\">The Diatomic Molecules of the Second Period<\/h3>\n<p>Eight possible homonuclear diatomic molecules might be formed by the atoms of the second period of the periodic table: Li<sub>2<\/sub>, Be<sub>2<\/sub>, B<sub>2<\/sub>, C<sub>2<\/sub>, N<sub>2<\/sub>, O<sub>2<\/sub>, F<sub>2<\/sub>, and Ne<sub>2<\/sub>. However, we can predict that the Be<sub>2<\/sub> molecule and the Ne<sub>2<\/sub> molecule would not be stable. We can see this by a consideration of the molecular electron configurations (Table 2).<\/p>\n<p>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\u2019s 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.<\/p>\n<p>As we saw in valence bond theory, \u03c3 bonds are generally more stable than \u03c0 bonds formed from degenerate atomic orbitals. Similarly, in molecular orbital theory, \u03c3 orbitals are usually more stable than \u03c0 orbitals. However, this is not always the case. The MOs for the valence orbitals of the second period are shown in\u00a0Figure 12.\u00a0Looking at Ne<sub>2<\/sub> molecular orbitals, we see that the order is consistent with the generic diagram shown in the previous section. However, for atoms with three or fewer electrons in the <em>p<\/em> orbitals (Li through N) we observe a different pattern, in which the \u03c3<em><sub>p<\/sub><\/em> orbital is higher in energy than the \u03c0<em><sub>p<\/sub><\/em> set. Obtain the molecular orbital diagram for a homonuclear diatomic ion by adding or subtracting electrons from the diagram for the neutral molecule.<\/p>\n<div style=\"width: 891px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211950\/CNX_Chem_08_04_X2MOs1.jpg\" alt=\"A graph is shown in which the y-axis is labeled, \u201cE,\u201d and appears as a vertical, upward-facing arrow. Across the top, the graph reads, \u201cL i subscript 2,\u201d \u201cB e subscript 2,\u201d \u201cB subscript 2,\u201d \u201cC subscript 2,\u201d \u201cN subscript 2,\u201d \u201cO subscript 2,\u201d \u201cF subscript 2,\u201d and \u201cNe subscript 2.\u201d Directly below each of these element terms is a single pink line, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled, \u201csigma subscript 2 p x superscript asterisk\u201d. Directly below each of these lines is a set of two pink lines, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. It is consistently lower than the first line. This line is labeled, \u201cpi subscript 2 p y superscript asterisk,\u201d and, \u201cpi subscript 2 p z superscript asterisk.\u201d Directly below each of these double lines is a single pink line, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. It has a distinctive drop at the label, \u201cO subscript 2.\u201d This line is labeled, \u201csigma subscript 2 p x.\u201d Directly below each of these lines is a set of two pink lines, and all lines are connected to one another by a dashed line to create an overall line that decreases very slightly in height as it moves from left to right across the graph. It is consistently lower than the third line until it reaches the point labeled, \u201cO subscript 2.\u201d This line is labeled, \u201cpi subscript 2 p y,\u201d and, \u201cpi subscript 2 p z.\u201d Directly below each of these lines is a single blue line, and all lines are connected to one another by a dashed line to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled, \u201csigma subscript 2 s superscript asterisk.\u201d Finally, directly below each of these lines is a single blue line, and all lines are connected to one another by a dashed line to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled. \u201csigma subscript 2 s.\u201d\" width=\"881\" height=\"358\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 12. This shows the MO diagrams for each homonuclear diatomic molecule in the second period. The orbital energies decrease across the period as the effective nuclear charge increases and atomic radius decreases. Between N2 and O2, the order of the orbitals changes.<\/p>\n<\/div>\n<div class=\"textbox\">You can practice labeling and filling molecular orbitals with this <a href=\"https:\/\/scilearn.sydney.edu.au\/fychemistry\/calculators\/mo_diagrams.shtml\" target=\"_blank\">interactive tutorial from the University of Sydney<\/a>.<\/div>\n<p>This switch in orbital ordering occurs because of a phenomenon called <strong>s-p mixing<\/strong>. s-p mixing does not create new orbitals; it merely influences the energies of the existing molecular orbitals. The \u03c3<sub>s<\/sub> wavefunction mathematically combines with the \u03c3<sub>p<\/sub> wavefunction, with the result that the \u03c3<sub>s<\/sub> orbital becomes more stable, and the \u03c3<sub>p<\/sub> orbital becomes less stable (Figure 13). Similarly, the antibonding orbitals also undergo s-p mixing, with the \u03c3<sub>s*<\/sub> becoming more stable and the \u03c3<sub>p*<\/sub> becoming less stable.<\/p>\n<div style=\"width: 890px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211952\/CNX_Chem_08_04_spmix1.jpg\" alt=\"A diagram is shown. At the bottom left of the diagram is a horizontal line that is connected to the right and left by upward-facing, dotted lines to two more horizontal lines. Those two lines are connected by upward-facing dotted lines to another line in the center of the diagram but farther up from the first. Each of the bottom two central lines has a vertical downward-facing arrow. Above this structure is a horizontal line that is connected to the right and left by upward-facing, dotted lines to two sets of three horizontal lines and those two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but further up from the first. In between the horizontal lines of this structure are two pairs of horizontal lines that are above the first line but below the second and connected by dotted lines to the side horizontal lines. The bottom and top central lines each have an upward-facing vertical arrow. These two structures are redrawn on the right side of the diagram, but this time, the central lines of the bottom structure are moved downward in relation to the side lines. The upper portion of the structure has its central lines shifted upward in relation to the side lines. This structure also shows the bottom line appearing above the set of two lines.\" width=\"880\" height=\"590\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 13. Without mixing, the MO pattern occurs as expected, with the \u03c3<sub>p<\/sub> orbital lower in energy than the \u03c3<sub>p<\/sub> orbitals. When s-p mixing occurs, the orbitals shift as shown, with the \u03c3<sub>p<\/sub> orbital higher in energy than the \u03c0<sub>p<\/sub> orbitals.<\/p>\n<\/div>\n<p>s-p mixing occurs when the <em>s<\/em> and <em>p<\/em> orbitals have similar energies. When a single <em>p<\/em> orbital contains a pair of electrons, the act of pairing the electrons raises the energy of the orbital. Thus the 2<em>p<\/em> orbitals for O, F, and Ne are higher in energy than the 2<em>p<\/em> orbitals for Li, Be, B, C, and N. Because of this, O<sub>2<\/sub>, F<sub>2<\/sub>, and N<sub>2<\/sub> only have negligible s-p mixing (not sufficient to change the energy ordering), and their MO diagrams follow the normal pattern, as shown in Figure 12. All of the other period 2 diatomic molecules do have s-p mixing, which leads to the pattern where the \u03c3<sub>p<\/sub> orbital is raised above the \u03c0<sub>p<\/sub> set.<\/p>\n<p>Using the MO diagrams shown in Figure 12, 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 2, Be<sub>2<\/sub> and Ne<sub>2<\/sub> molecules would have a bond order of 0, and these molecules do not exist.<\/p>\n<table summary=\"A table is shown that has three columns and nine rows. The header row reads: \u201cMolecule,\u201d \u201cElectron Configuration,\u201d and, \u201cBond Order.\u201d The first column contains the symbols \u201cL i subscript 2,\u201d \u201cB e subscript 2 ( unstable ),\u201d \u201cB e subscript 2,\u201d \u201cC subscript 2,\u201d \u201cN subscript 2,\u201d \u201cO subscript 2,\u201d \u201cF subscript 2,\u201d and, \u201cNe subscript 2 ( unstable ).\u201d The second column contains the symbols \u201c( sigma subscript 2 s ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4 ( sigma subscript 2 p x ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( sigma subscript 2 p x ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4 ( pi superscript asterisk subscript 2 p y, pi superscript asterisk subscript 2 p z ) superscript 2,\u201d \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( sigma subscript 2 p x ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4 ( pi superscript asterisk subscript 2 p y, pi superscript asterisk subscript 2 p z ) superscript 4,\u201d and \u201c( sigma subscript 2 s ) superscript 2 ( sigma superscript asterisk subscript 2 s ) superscript 2 ( sigma subscript 2 p x ) superscript 2 ( pi subscript 2 p y, pi subscript 2 p z ) superscript 4 ( pi superscript asterisk subscript 2 p y, pi superscript asterisk subscript 2 p z ) superscript 4 ( sigma superscript asterisk subscript 2 p x ) superscript 2.\u201d The third column contains the numbers: \u201c1,\u201d \u201c0,\u201d \u201c1,\u201d \u201c2,\u201d \u201c3,\u201d \u201c2,\u201d \u201c1,\u201d \u201c0.\u201d\">\n<thead>\n<tr>\n<th colspan=\"3\">Table 2. Electron Configuration and Bond Order for Molecular Orbitals in Homonuclear Diatomic Molecules of Period Two Elements<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>Molecule<\/th>\n<th>Electron Configuration<\/th>\n<th>Bond Order<\/th>\n<\/tr>\n<tr>\n<td>Li<sub>2<\/sub><\/td>\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>Be<sub>2<\/sub> (unstable)<\/td>\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>B<sub>2<\/sub><\/td>\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{2}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>C<sub>2<\/sub><\/td>\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}[\/latex]<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>N<sub>2<\/sub><\/td>\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\sigma}_{2px}\\right)}^{2}[\/latex]<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>O<sub>2<\/sub><\/td>\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2px}\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)}^{2}[\/latex]<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>F<sub>2<\/sub><\/td>\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2px}\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)}^{4}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>Ne<sub>2<\/sub> (unstable)<\/td>\n<td>[latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2px}\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)}^{4}{\\left({\\sigma}_{2px}^\\ast\\right)}^{2}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The combination of two lithium atoms to form a lithium molecule, Li<sub>2<\/sub>, is analogous to the formation of H<sub>2<\/sub>, but the atomic orbitals involved are the valence 2<em>s<\/em> orbitals. Each of the two lithium atoms has one valence electron. Hence, we have two valence electrons available for the \u03c3<sub>2<em>s<\/em><\/sub> bonding molecular orbital. Because both valence electrons would be in a bonding orbital, we would predict the Li<sub>2<\/sub> molecule to be stable. The molecule is, in fact, present in appreciable concentration in lithium vapor at temperatures near the boiling point of the element. All of the other molecules in Table 2 with a bond order greater than zero are also known.<\/p>\n<p>The O<sub>2<\/sub> molecule has enough electrons to half fill the [latex]\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)[\/latex] level. We expect the two electrons that occupy these two degenerate orbitals to be unpaired, and this molecular electronic configuration for O<sub>2<\/sub> is in accord with the fact that the oxygen molecule has two unpaired electrons (Figure 15). 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.<\/p>\n<div class=\"textbox shaded\">\n<h3>How Sciences Intersect: Band Theory<\/h3>\n<p>When two identical atomic orbitals on different atoms combine, two molecular orbitals result (see Figure 14). 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.<\/p>\n<p>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 &gt; 10<sup>23<\/sup> atoms), and therefore a huge number of atomic orbitals that may be combined into molecular orbitals. When <em>N<\/em> valence atomic orbitals, all of the same energy and each containing one (1) electron, are combined, <em>N<\/em>\/2 (filled) bonding orbitals and <em>N<\/em>\/2 (empty) antibonding orbitals will result. Each bonding orbital will show an energy lowering as the atomic orbitals are <em>mostly<\/em> 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 <em>mostly<\/em> 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 14 shows the bands for three important classes of materials: insulators, semiconductors, and conductors.<\/p>\n<div style=\"width: 758px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211953\/CNX_Chem_08_04_Band1.jpg\" alt=\"This figure shows three diagrams. The first is labeled, \u201cInsulator,\u201d and it consists of two boxes. The \u201cconduction\u201d box is above and the \u201cvalence\u201d box is below. A large gap marked by 4 dashed lines contains a double-headed arrow. One head pointing towards the \u201cconduction box\u201d and the other towards the \u201cvalence\u201d box. The arrow is labeled, \u201cBand gap.\u201d The second diagram is similar to the first, but the band gap is about half as large. This diagram is labeled, \u201cSemiconductor.\u201d The third diagram is similar to the other two, but the band gap is about a fifth that of the \u201cSemiconductor\u201d diagram. This diagram is labeled, \u201cConductor.\u201d\" width=\"748\" height=\"290\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 14. Molecular orbitals in solids are so closely spaced that they are described as bands. The valence band is lower in energy and the conduction band is higher in energy. The type of solid is determined by the size of the \u201cband gap\u201d between the valence and conduction bands. Only a very small amount of energy is required to move electrons from the valance band to the conduction band in a conductor, and so they conduct electricity well. In an insulator, the band gap is large, so that very few electrons move, and they are poor conductors of electricity. Semiconductors are in between: they conduct electricity better than insulators, but not as well as conductors.<\/p>\n<\/div>\n<p>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 \u201ceasy\u201d to overcome, so they are good conductors of electricity. In an insulator, the band gap is so \u201clarge\u201d that very few electrons move into the conduction band; as a result, insulators are poor conductors of electricity. Semiconductors conduct electricity when \u201cmoderate\u201d 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.<\/p>\n<p>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.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2:\u00a0Molecular Orbital Diagrams, Bond Order, and Number of Unpaired Electrons<\/h3>\n<p>Draw the molecular orbital diagram for the oxygen molecule, O<sub>2<\/sub>. From this diagram, calculate the bond order for O<sub>2<\/sub>. How does this diagram account for the paramagnetism of O<sub>2<\/sub>?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q292725\">Show Answer<\/span><\/p>\n<div id=\"q292725\" class=\"hidden-answer\" style=\"display: none\">\n<p>We draw a molecular orbital energy diagram similar to that shown in Figure 12. Each oxygen atom contributes six electrons, so the diagram appears as shown in Figure 15.<\/p>\n<div style=\"width: 718px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23211955\/CNX_Chem_08_04_O2MO1.jpg\" alt=\"A diagram is shown that has an upward-facing vertical arrow running along the left side labeled, \u201cE.\u201d At the bottom center of the diagram is a horizontal line labeled, \u201csigma subscript 2 s,\u201d that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, \u201c2 s,\u201d and with two vertical half arrows drawn on them, one facing up and one facing down. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first and labeled, \u201csigma subscript 2 s superscript asterisk.\u201d This horizontal line has two vertical half-arrow drawn on it, one facing up and one facing down. Moving further up the center of the diagram is a horizontal line labeled, \u201csigma subscript 2 p subscript x,\u201d which lies below two horizontal lines, lying side-by-side, and labeled \u201cpi subscript 2 p subscript y,\u201d and \u201cpi subscript 2 p subscript z.\u201d Both the bottom and top lines are connected to the right and left by upward-facing, dotted lines to three more horizontal lines, each labeled, \u201c2 p,\u201d on either side. These sets of lines each hold three upward-facing and one downward-facing half-arrow. They are connected by upward-facing dotted lines to another single line and then pair of double lines in the center of the diagram, but farther up from the lower lines. They are labeled, \u201csigma subscript 2 p subscript x superscript asterisk,\u201d \u201cpi subscript 2 p subscript y superscript asterisk,\u201d and \u201cpi subscript 2 p subscript z superscript asterisk,\u201d respectively. The lower of these two central, horizontal lines each contain one upward-facing half-arrow. The left and right sides of the diagram have headers that read, \u201dAtomic orbitals,\u201d while the center header reads, \u201cMolecular orbitals.\u201d\" width=\"708\" height=\"529\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 15. The molecular orbital energy diagram for O<sub>2<\/sub> predicts two unpaired electrons.<\/p>\n<\/div>\n<p>We calculate the bond order as<\/p>\n<p style=\"text-align: center;\">[latex]{\\text{O}}_{2}=\\frac{\\left(10-6\\right)}{2}=2[\/latex]<\/p>\n<p>Oxygen&#8217;s paramagnetism is explained by the presence of two unpaired electrons in the (\u03c0<sub>2<em>py<\/em><\/sub>, \u03c0<sub>2<em>pz<\/em><\/sub>)* molecular orbitals.<\/p>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>The main component of air is N<sub>2<\/sub>. From the molecular orbital diagram of N<sub>2<\/sub>, predict its bond order and whether it is diamagnetic or paramagnetic.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q768732\">Show Answer<\/span><\/p>\n<div id=\"q768732\" class=\"hidden-answer\" style=\"display: none\">N<sub>2<\/sub> has a bond order of 3 and is diamagnetic.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3:\u00a0Ion Predictions with MO Diagrams<\/h3>\n<p>Give the molecular orbital configuration for the valence electrons in [latex]{\\text{C}}_{2}^{2-}.[\/latex] Will this ion be stable?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q174387\">Show Answer<\/span><\/p>\n<div id=\"q174387\" class=\"hidden-answer\" style=\"display: none\">Looking at the appropriate MO diagram, we see that the \u03c0 orbitals are lower in energy than the \u03c3<em><sub>p<\/sub><\/em> orbital. The valence electron configuration for C<sub>2<\/sub> is [latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{\\text{2}s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}.[\/latex] Adding two more electrons to generate the [latex]{\\text{C}}_{2}^{2-}[\/latex] anion will give a valence electron configuration of [latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{\\text{2}s}^\\ast\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\sigma}_{2px}\\right)}^{2}.[\/latex] Since this has six more bonding electrons than antibonding, the bond order will be 3, and the ion should be stable.<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>How many unpaired electrons would be present on a [latex]{\\text{Be}}_{2}^{2-}[\/latex] ion? Would it be paramagnetic or diamagnetic?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q421881\">Show Answer<\/span><\/p>\n<div id=\"q421881\" class=\"hidden-answer\" style=\"display: none\">\n<p>two, paramagnetic<\/p>\n<p>Creating molecular orbital diagrams for molecules with more than two atoms relies on the same basic ideas as the diatomic examples presented here. However, with more atoms, computers are required to calculate how the atomic orbitals combine. See <a href=\"http:\/\/mw.concord.org\/modeler\/showcase\/simulation.html?s=http:\/\/mw2.concord.org\/public\/student\/quantumchemistry\/benzene.html\" target=\"_blank\">Molecular Workshop&#8217;s\u00a0three-dimensional drawings of the molecular orbitals for C<sub>6<\/sub>H<\/a><sub><a href=\"http:\/\/mw.concord.org\/modeler\/showcase\/simulation.html?s=http:\/\/mw2.concord.org\/public\/student\/quantumchemistry\/benzene.html\" target=\"_blank\">6<\/a><\/sub>.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts and Summary<\/h3>\n<p>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 \u03c3 MOs. They can be formed from <em>s<\/em> orbitals or from <em>p<\/em> orbitals oriented in an end-to-end fashion. Molecular orbitals formed from <em>p<\/em> orbitals oriented in a side-by-side fashion have electron density on opposite sides of the internuclear axis and are called \u03c0 orbitals.<\/p>\n<p>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\u2019s 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.<\/p>\n<h4>Key Equations<\/h4>\n<ul>\n<li>[latex]\\text{bond order}=\\frac{\\left(\\text{number of bonding electron}\\right)-\\left(\\text{number of antibonding electrons}\\right)}{2}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<ol>\n<li>Sketch the distribution of electron density in the bonding and antibonding molecular orbitals formed from two <em>s<\/em> orbitals and from two <em>p<\/em> orbitals.<\/li>\n<li>How are the following similar, and how do they differ?\n<ol style=\"list-style-type: lower-alpha;\">\n<li>\u03c3 molecular orbitals and \u03c0 molecular orbitals<\/li>\n<li>\u03c8 for an atomic orbital and \u03c8 for a molecular orbital<\/li>\n<li>bonding orbitals and antibonding orbitals<\/li>\n<\/ol>\n<\/li>\n<li>If molecular orbitals are created by combining five atomic orbitals from atom A and five atomic orbitals from atom B combine, how many molecular orbitals will result?<\/li>\n<li>Can a molecule with an odd number of electrons ever be diamagnetic? Explain why or why not.<\/li>\n<li>Can a molecule with an even number of electrons ever be paramagnetic? Explain why or why not.<\/li>\n<li>Why are bonding molecular orbitals lower in energy than the parent atomic orbitals?<\/li>\n<li>Calculate the bond order for an ion with this configuration: [latex]{\\left({\\sigma}_{2s}\\right)}^{2}{\\left({\\sigma}_{2s}^\\ast\\right)}^{2}{\\left({\\sigma}_{2px}\\right)}^{2}{\\left({\\pi}_{2py},{\\pi}_{2pz}\\right)}^{4}{\\left({\\pi}_{2py}^\\ast,{\\pi}_{2pz}^\\ast\\right)}^{3}[\/latex]<\/li>\n<li>Explain why an electron in the bonding molecular orbital in the H<sub>2<\/sub> molecule has a lower energy than an electron in the 1<em>s<\/em> atomic orbital of either of the separated hydrogen atoms.<\/li>\n<li>Predict the valence electron molecular orbital configurations for the following, and state whether they will be stable or unstable ions.\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{\\text{Na}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<li>[latex]{\\text{Mg}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<li>[latex]{\\text{Al}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<li>[latex]{\\text{Si}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<li>[latex]{\\text{P}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<li>[latex]{\\text{S}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<li>[latex]{\\text{F}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<li>[latex]{\\text{Ar}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Determine the bond order of each member of the following groups, and determine which member of each group is predicted by the molecular orbital model to have the strongest bond.\n<ol style=\"list-style-type: lower-alpha;\">\n<li>H<sub>2<\/sub>, [latex]{\\text{H}}_{2}^{\\text{+}},[\/latex] [latex]{\\text{H}}_{2}^{-}[\/latex]<\/li>\n<li>O<sub>2<\/sub>, [latex]{\\text{O}}_{2}^{\\text{2+}},[\/latex] [latex]{\\text{O}}_{2}^{2-}[\/latex]<\/li>\n<li>Li<sub>2<\/sub>, [latex]{\\text{Be}}_{2}^{\\text{+}},[\/latex] Be<sub>2\u00a0<\/sub><\/li>\n<li>F<sub>2<\/sub>, [latex]{\\text{F}}_{2}^{\\text{+}},[\/latex] [latex]{\\text{F}}_{2}^{-}[\/latex]<\/li>\n<li>N<sub>2<\/sub>, [latex]{\\text{N}}_{2}^{\\text{+}},[\/latex] [latex]{\\text{N}}_{2}^{-}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>For the first ionization energy for an N<sub>2<\/sub> molecule, what molecular orbital is the electron removed from?<\/li>\n<li>Compare the atomic 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:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>H and H<sub>2\u00a0<\/sub><\/li>\n<li>N and N<sub>2\u00a0<\/sub><\/li>\n<li>O and O<sub>2\u00a0<\/sub><\/li>\n<li>C and C<sub>2\u00a0<\/sub><\/li>\n<li>B and B<sub>2<\/sub><\/li>\n<\/ol>\n<\/li>\n<li>Which of the period 2 homonuclear diatomic molecules are predicted to be paramagnetic?<\/li>\n<li>A friend tells you that the 2<em>s<\/em> orbital for fluorine starts off at a much lower energy than the 2<em>s<\/em> orbital for lithium, so the resulting \u03c3<sub>2<em>s<\/em><\/sub> molecular orbital in F<sub>2<\/sub> is more stable than in Li<sub>2<\/sub>. Do you agree?<\/li>\n<li>True or false: Boron contains 2<em>s<\/em><sup>2<\/sup>2<em>p<\/em><sup>1<\/sup> valence electrons, so only one <em>p<\/em> orbital is needed to form molecular orbitals.<\/li>\n<li>What charge would be needed on F<sub>2<\/sub> to generate an ion with a bond order of 2?<\/li>\n<li>Predict whether the MO diagram for S<sub>2<\/sub> would show s-p mixing or not.<\/li>\n<li>Explain why [latex]{\\text{N}}_{2}^{\\text{2+}}[\/latex] is diamagnetic, while [latex]{\\text{O}}_{2}^{\\text{4+}},[\/latex] which has the same number of valence electrons, is paramagnetic.<\/li>\n<li>Using the MO diagrams, predict the bond order for the stronger bond in each pair:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>B<sub>2<\/sub> or [latex]{\\text{B}}_{2}^{+}[\/latex]<\/li>\n<li>F<sub>2<\/sub> or [latex]{\\text{F}}_{2}^{+}[\/latex]<\/li>\n<li>O<sub>2<\/sub> or [latex]{\\text{O}}_{2}^{\\text{2+}}[\/latex]<\/li>\n<li>[latex]{\\text{C}}_{2}^{+}[\/latex] or [latex]{\\text{C}}_{2}^{-}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q503798\">Show Answer<\/span><\/p>\n<div id=\"q503798\" class=\"hidden-answer\" style=\"display: none\">\n<p>2. The similarities and differences are as follows:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Similarities: Both are bonding orbitals that can contain a maximum of two electrons. Differences: <em>\u03c3<\/em> orbitals are end-to-end combinations of atomic orbitals, whereas <em>\u03c0<\/em> orbitals are formed by side-by-side overlap of orbitals.<\/li>\n<li>Similarities: Both are quantum-mechanical constructs that represent the probability of finding the electron about the atom or the molecule. Differences: <em>\u03c8<\/em> for an atomic orbital describes the behavior of only one electron at a time based on the atom. For a molecule, <em>\u03c8<\/em> represents either a mathematical combination of atomic orbitals.<\/li>\n<li>Similarities: Both are orbitals that can contain two electrons. Differences: Bonding orbitals result in holding two or more atoms together. Antibonding orbitals have the effect of destabilizing any bonding that has occurred.<\/li>\n<\/ol>\n<p>4.\u00a0An odd number of electrons can never be paired, regardless of the arrangement of the molecular orbitals. It will always be paramagnetic.<\/p>\n<p>6.\u00a0Bonding orbitals have electron density in close proximity to more than one nucleus. The interaction between the bonding positively charged nuclei and negatively charged electrons stabilizes the system.<\/p>\n<p>8.\u00a0The pairing of the two bonding electrons lowers the energy of the system relative to the energy of the nonbonded electrons.<\/p>\n<p>10.\u00a0The bond order is equal to half the difference between the number of bonding electrons and the number of antibonding electrons. The bond with the greatest bond order is predicted to be the strongest.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>H<sub>2<\/sub> bond order = 1, [latex]{\\text{H}}_{2}^{+}[\/latex] bond order = 0.5, [latex]{\\text{H}}_{2}^{-}[\/latex] bond order = 0.5, strongest bond is H<sub>2<\/sub>;<\/li>\n<li>O<sub>2<\/sub> bond order = 2, [latex]{\\text{O}}_{2}^{\\text{2+}}[\/latex] bond order = 3; [latex]{\\text{O}}_{2}^{2-}[\/latex] bond order = 1, strongest bond is [latex]{\\text{O}}_{2}^{\\text{2+}};[\/latex]<\/li>\n<li>Li<sub>2<\/sub> bond order = 1, [latex]{\\text{Be}}_{2}^{+}[\/latex] bond order = 0.5, Be<sub>2<\/sub> bond order = 0, Li<sub>2<\/sub> [latex]{\\text{Be}}_{2}^{+}[\/latex] have the same strength bond;<\/li>\n<li>F<sub>2<\/sub> bond order = 1, [latex]{\\text{F}}_{2}^{+}[\/latex] bond order = 1.5, [latex]{\\text{F}}_{2}^{-}[\/latex] bond order = 0.5, strongest bond is [latex]{\\text{F}}_{2}^{\\text{+}};[\/latex]<\/li>\n<li>N<sub>2<\/sub> bond order = 3, [latex]{\\text{N}}_{2}^{+}[\/latex] bond order = 2.5, [latex]{\\text{N}}_{2}^{-}[\/latex] bond order = 2.5, strongest bond is N<sub>2<\/sub>.<\/li>\n<\/ol>\n<p>12. The substance with the\u00a0highest first ionization energy in each pair is as follows:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>H<sub>2<\/sub><\/li>\n<li>N<sub>2<\/sub><\/li>\n<li>O<\/li>\n<li>C<sub>2<\/sub><\/li>\n<li>B<sub>2<\/sub><\/li>\n<\/ol>\n<p>14.\u00a0Yes, fluorine is a smaller atom than Li, so atoms in the 2<em>s<\/em> orbital are closer to the nucleus and more stable.<\/p>\n<p>16.\u00a02+<\/p>\n<p>18.\u00a0N<sub>2<\/sub> has s-p mixing, so the \u03c0 orbitals are the last filled in [latex]{\\text{N}}_{2}^{\\text{2+}}.[\/latex] O<sub>2<\/sub> does not have s-p mixing, so the \u03c3<em><sub>p<\/sub><\/em> orbital fills before the \u03c0 orbitals.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Glossary<\/h2>\n<p><strong>antibonding orbital:<\/strong> molecular orbital located outside of the region between two nuclei; electrons in an antibonding orbital destabilize the molecule<\/p>\n<p><strong>bond order:<\/strong> number of pairs of electrons between two atoms; it can be found by the number of bonds in a Lewis structure or by the difference between the number of bonding and antibonding electrons divided by two<\/p>\n<p><strong>bonding orbital:<\/strong> molecular orbital located between two nuclei; electrons in a bonding orbital stabilize a molecule<\/p>\n<p><strong>degenerate orbitals:<\/strong> orbitals that have the same energy<\/p>\n<p><strong>diamagnetism<strong>:<\/strong><\/strong> phenomenon in which a material is not magnetic itself but is repelled by a magnetic field; it occurs when there are only paired electrons present<\/p>\n<p><strong>homonuclear diatomic molecule<strong>:<\/strong><\/strong> molecule consisting of two identical atoms<\/p>\n<p><strong>linear combination of atomic orbitals<strong>:<\/strong><\/strong> technique for combining atomic orbitals to create molecular orbitals<\/p>\n<p><strong>molecular orbital<strong>:<\/strong><\/strong> region of space in which an electron has a high probability of being found in a molecule<\/p>\n<p><strong>molecular orbital diagram<strong>:<\/strong><\/strong> visual representation of the relative energy levels of molecular orbitals<\/p>\n<p><strong>molecular orbital theory<strong>:<\/strong><\/strong> model that describes the behavior of electrons delocalized throughout a molecule in terms of the combination of atomic wave functions<\/p>\n<p><strong>paramagnetism<strong>:<\/strong><\/strong> phenomenon in which a material is not magnetic itself but is attracted to a magnetic field; it occurs when there are unpaired electrons present<\/p>\n<p><strong>\u03c0 bonding orbital<strong>:<\/strong><\/strong> molecular orbital formed by side-by-side overlap of atomic orbitals, in which the electron density is found on opposite sides of the internuclear axis<\/p>\n<p><strong>\u03c0* bonding orbital<strong>:<\/strong><\/strong> antibonding molecular orbital formed by out of phase side-by-side overlap of atomic orbitals, in which the electron density is found on both sides of the internuclear axis, and there is a node between the nuclei<\/p>\n<p><strong>\u03c3 bonding orbital<strong>:<\/strong><\/strong> molecular orbital in which the electron density is found along the axis of the bond<\/p>\n<p><strong>\u03c3* bonding orbital<strong>:<\/strong><\/strong> antibonding molecular orbital formed by out-of-phase overlap of atomic orbital along the axis of the bond, generating a node between the nuclei<\/p>\n<p><strong>s-p mixing:<\/strong> change that causes \u03c3<em><sub>p<\/sub><\/em> orbitals to be less stable than \u03c0<em><sub>p<\/sub><\/em> orbitals due to the mixing of <em>s<\/em> and <em>p<\/em>-based molecular orbitals of similar energies.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2057\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Chemistry. <strong>Provided by<\/strong>: OpenStax College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/openstaxcollege.org\">http:\/\/openstaxcollege.org<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Chemistry\",\"author\":\"\",\"organization\":\"OpenStax College\",\"url\":\"http:\/\/openstaxcollege.org\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2057","chapter","type-chapter","status-publish","hentry"],"part":3005,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/users\/17"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions"}],"predecessor-version":[{"id":6009,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters\/2057\/revisions\/6009"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/parts\/3005"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters\/2057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapter-type?post=2057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/contributor?post=2057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/license?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}