{"id":505,"date":"2017-12-14T21:39:00","date_gmt":"2017-12-14T21:39:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/chapter\/molecular-orbitals\/"},"modified":"2017-12-14T21:39:00","modified_gmt":"2017-12-14T21:39:00","slug":"molecular-orbitals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/chapter\/molecular-orbitals\/","title":{"raw":"Molecular Orbitals","rendered":"Molecular Orbitals"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul><li>Gain an \u00a0understanding of molecular orbital theory.<\/li>\n\t<li>Learn to calculate bond orders.<\/li>\n\t<li>Learn to draw molecular orbital electron configuration energy diagrams.<\/li>\n<\/ul><\/div>\n\u00a0\n\nValence bond theory is able to explain many aspects of bonding, but not all. To complement this theory, we use another called the\u00a0<a class=\"glossary\">molecular orbital (MO) theory<\/a>. Molecular orbital theory is a more sophisticated model for understanding the nature of chemical bonding.\n\nMO theory takes the idea of atomic orbitals overlapping to a new level, where new molecular orbitals are generated using a mathematical process called <em>linear combination of atomic orbitals<\/em> (LCAO).\n\nMolecular orbitals share many similarities with atomic orbitals:\n\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 They are filled from lowest energy to highest energy (Aufbau principle).\n\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 They can hold a maximum of two electrons of opposite spin per orbital\u00a0(Pauli exclusion principle).\n\nThe major difference between atomic and molecular orbitals is that atomic orbitals represent electron density in space associated with a particular atom. Molecular orbitals are associated with the entire molecule, meaning the electron density is delocalized (spread out) over more than one atom.\n\n\u00a0\n<h2>The Molecular Orbitals of the Hydrogen Molecule<\/h2>\nCombining the 1<em>s<\/em> orbitals of each hydrogen atom using LCAO, two molecular orbitals are generated \u03c3<sub>1<em>s<\/em><\/sub> (pronounced sigma one <em>s<\/em>) and \u03c3*<sub>1<em>s<\/em><\/sub> (pronounced sigma star one <em>s<\/em>).\n\nThe \u03c3<sub>1<em>s<\/em><\/sub> orbital is generated by a constructive combination (or interference), where the two atomic orbitals wave functions reinforce (add to) each other. This is the lower energy of the two molecular orbitals and is known as the <a class=\"glossary\">bonding molecular orbital<\/a>. Notice in Figure 9.19 \"<span class=\"Apple-style-span\">Hydrogen molecular orbital combination diagram\"\u00a0<\/span>that the electron density of this orbital is concentrated between the two nuclei. These electrons are stabilized by attractions to both nuclei, and they hold the atoms together with a covalent bond.\n\nThe \u03c3*<sub>1<em>s<\/em><\/sub> orbital is generated by a destructive combination (or interference), where the\u00a0wave functions of the two atomic orbitals cancel each other. This type of combination results in an area of zero electron density between the two nuclei, known as a <strong>nodal plane (or node)<\/strong>. This node of zero electron density is destabilizing toward the bond, making it higher energy, and subsequently this type of orbital is known as an <a class=\"glossary\">antibonding molecular orbital <\/a>(denoted by the asterisk in the orbital name).\n\n[caption id=\"attachment_2550\" align=\"alignnone\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/h2_mo_combination_diagram.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213843\/h2_mo_combination_diagram-1.png\" alt=\"Figure #.#. Hydrogen molecular orbital combination diagram.\" class=\"wp-image-2550\" height=\"197\" width=\"400\"\/><\/a> Figure 9.19. Hydrogen molecular orbital combination diagram.[\/caption]\n\n\u00a0\n\nSimilar to atomic orbitals, we can write electron configuration energy diagrams for molecular orbitals (Figure 9.20 \"<span class=\"Apple-style-span\">Hydrogen molecular orbital electron configuration energy diagram\"<\/span>). Notice that the atomic orbitals of each atom are written on either side, and the newly formed molecular orbitals are written in the centre of the diagram. The bonding molecular orbital is filled and is relatively lower in energy than the contributing atomic orbitals, supporting the fact that hydrogen molecules (H<sub>2<\/sub>) are more stable than lone hydrogen atoms.\n\n[caption id=\"attachment_2551\" align=\"alignnone\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/h2_MO_energy_diagram.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213845\/h2_MO_energy_diagram-1.png\" alt=\"Figure #.#. Hydrogen molecular orbital electron configuration energy diagram.\" class=\"wp-image-2551\" height=\"291\" width=\"400\"\/><\/a> Figure 9.20. Hydrogen molecular orbital electron configuration energy diagram.[\/caption]\n\n\u00a0\n<h2>Bond Order<\/h2>\nWe have just seen that the bonding molecular orbital is lower energy and promotes the formation of a covalent bond, while the antibonding molecular orbital is higher energy with a node of zero electron density between the atoms that destabilizes the formation of a covalent bond. We can evaluate the strength of a covalent bond by determining its <a class=\"glossary\">bond order<\/a>.\n\nBond order = 1\/2 (# of electrons in bonding MOs - # of electrons in antibonding MOs)\n\n\u00a0\n\nBond-order values can be whole numbers, fractions, or zero. These values correspond to the valence bond model, so a bond order of 1 is equal to a single bond, and 2 is equal to a double bond. A value of zero means that there is no bond present, and the atoms exist separately.\n<div class=\"textbox shaded\">\n\nExample 11\n\nDetermine the bond order of the hydrogen molecule.\n\nSolution\n\nBond order = 1\/2 (# of electrons in bonding MOs - # of electrons in antibonding MOs)\n\nBond order = 1\/2 (2 - 0) = 1\n\nTherefore there is a single bond in the hydrogen molecule.\n\n<\/div>\n\u00a0\n<h2>Molecular Orbitals of Li<sub>2<\/sub><\/h2>\nGenerating molecular orbitals of molecules more complex than hydrogen using the LCAO method requires following a few additional guidelines:\n\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The number of MOs generated is equal to the number of atomic orbitals combined.\n\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Combined atomic orbitals should be of similar energy levels.\n\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The effectiveness of atomic orbital combination depends on the amount of orbital overlap. Increased overlap lowers the energy of the bonding molecular orbital further, and raises the energy of the antibonding molecular orbital.\n\n\u00a0\n\nLet\u2019s follow these guidelines and generate a molecular orbital electron configuration diagram for Li<sub>2\u00a0<\/sub>(Figure 9.21 \"<span class=\"Apple-style-span\">Molecular orbital electron configuration energy diagram for dilithium\"<\/span>):\n\n\u00a0\n\n[caption id=\"attachment_2557\" align=\"alignnone\" width=\"481\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/Li2_MO_electron_config_diagram.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213847\/Li2_MO_electron_config_diagram-1.png\" alt=\"Figure #.#. Molecular orbital electron configuration energy diagram for dilithium.\" class=\"wp-image-2557 size-full\" height=\"607\" width=\"481\"\/><\/a> Figure 9.21. Molecular orbital electron configuration energy diagram for dilithium.[\/caption]\n\nNotice that we have combined the 1<em>s<\/em> atomic orbitals, as before in the H<sub>2<\/sub> example, to generate bonding and antibonding molecular orbitals that are completely filled by both atoms' 1<em>s<\/em> electrons. Similarly 2<em>s<\/em> atomic orbitals combine, giving a bonding orbital and an antibonding orbital, which are filled with the remaining valence electrons starting from the bottom up. The atomic orbitals that\u00a0combine are of similar energy levels; a 1s orbital <em>does not<\/em> combine with one of the 2<em>s<\/em> orbitals.\n\nThe bond order can be determined for this molecule to be:\n\nBond order = 1\/2 (4 - 2) = 1\n\nTherefore Li<sub>2<\/sub> would have a single bond.\n\n\u00a0\n<h2>Molecular Orbitals from <em>p<\/em>\u00a0Atomic Orbitals<\/h2>\nTo determine the molecular orbitals of many other molecules, we need to examine how <em>p<\/em> orbitals combine to give molecular orbitals. The\u00a0<em>p<\/em>\u00a0orbitals can overlap in two ways: head-to-head or sideways. Head-to-head overlap of <em>p<\/em> atomic orbitals results in a bonding and antibonding molecular orbital, where the electron density is centred along the internuclear axis, making them \u03c3 orbitals (Figure 9.22 \"<span class=\"Apple-style-span\">Head-to-head overlap of\u00a0<em>p<\/em>\u00a0orbitals\"<\/span>).\n\n\u00a0\n\n[caption id=\"attachment_2560\" align=\"alignnone\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/2p_head_to_head_overlap.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213849\/2p_head_to_head_overlap-1.png\" alt=\"Figure #.#. Head-to-head overlap of p orbitals.\" class=\"wp-image-2560\" height=\"141\" width=\"400\"\/><\/a> Figure 9.22. Head-to-head overlap of <em>p<\/em> orbitals.[\/caption]\n\nSideways overlap of the remaining four <em>p<\/em> atomic orbitals can occur along the two other axes, generating four \u03c0 molecular orbitals having electron density on opposite sides of the internuclear axis (Figure 9.23 \"<span class=\"Apple-style-span\">Sideways overlap of\u00a0<em>p<\/em>\u00a0orbitals\"<\/span>).\n\n\u00a0\n\n[caption id=\"attachment_2561\" align=\"alignnone\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/2p_sideways_overlap.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213851\/2p_sideways_overlap-1.png\" alt=\"Figure #.#. Sideways overlap of p orbitals.\" class=\"wp-image-2561\" height=\"420\" width=\"400\"\/><\/a> Figure 9.23. Sideways overlap of <em>p<\/em> orbitals.[\/caption]\n\nThe head-to-head overlap giving \u03c3 molecular orbitals results in greater overlap, making its bonding molecular orbital the most stable and lowest energy, while the \u03c3* antibonding is least stable and has the highest energy (Figure 9.24 \"<span class=\"Apple-style-span\">Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 8-10\"<\/span>). Sideways overlap gives four \u03c0 molecular orbitals, two lower-energy degenerate-bonding molecular orbitals, and two higher-energy antibonding orbitals.\n\n\u00a0\n\n[caption id=\"attachment_2562\" align=\"alignnone\" width=\"499\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/2p_molecular_orbitals_electron_config.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213853\/2p_molecular_orbitals_electron_config-1.png\" alt=\"Figure #.#. Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 8-10.\" class=\"wp-image-2562 size-full\" height=\"701\" width=\"499\"\/><\/a> Figure 9.24. Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 8-10.[\/caption]\n\n\u00a0\n\nThe energy diagram we have just generated fits experimentally with O<sub>2<\/sub>, F<sub>2<\/sub>, and Ne<sub>2<\/sub>, but does not fit for B<sub>2<\/sub>, C<sub>2<\/sub>, and N<sub>2<\/sub>. In the latter, homonuclear diatomic molecules (B<sub>2<\/sub>, C<sub>2<\/sub>, and N<sub>2<\/sub>), interactions take place between the 2<em>s<\/em> and 2<em>p<\/em> atomic orbitals that are strong enough to swap the ordering of the \u03c3<sub>2<em>p<\/em><\/sub> and \u03c0<sub>2<em>p<\/em><\/sub> molecular orbitals (Figure 9.25).\n\n[caption id=\"attachment_2563\" align=\"alignnone\" width=\"496\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/2p_molecular_orbital_energy_diagram_swapped.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213855\/2p_molecular_orbital_energy_diagram_swapped-1.png\" alt=\"Figure #.#. Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 5-7.\" class=\"size-full wp-image-2563\" height=\"710\" width=\"496\"\/><\/a> Figure 9.25. Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 5-7.[\/caption]\n<h2>Heteronuclear Diatomic Molecules<\/h2>\nIn heteronuclear diatomic molecules, where two different molecules are bonded, the energy levels of the individual atoms' atomic orbitals may differ. However, the molecular orbital diagram we\u00a0see in\u00a0Figure 9.25 (\"<span class=\"Apple-style-span\">Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 5-7\")<\/span> can be used to estimate the electron configuration and bond order.\n\n\u00a0\n<h2>Frontier Molecular Orbitals<\/h2>\nWe can focus further on two very important types of molecular orbitals: the <a class=\"glossary\">highest occupied molecular orbital (HOMO)<\/a> and the <a class=\"glossary\">lowest unoccupied molecular orbital (LUMO)<\/a>, also referred to collectively as the <a class=\"glossary\">frontier molecular orbitals<\/a> (Figure 9.26 \"<span class=\"Apple-style-span\">Frontier molecular orbitals HOMO and LUMO\"<\/span>). As their names imply, the HOMO is the molecular orbital that has the highest energy and contains electrons, while the LUMO is the lowest energy molecular orbital that does not contain electrons.\n\n[caption id=\"attachment_2567\" align=\"alignnone\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/Frontier_Molecular_Orbitals_1.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213857\/Frontier_Molecular_Orbitals_1-1.png\" alt=\"Figure #.#. Frontier molecular orbitals HOMO and LUMO.\" class=\"wp-image-2567\" height=\"307\" width=\"400\"\/><\/a> Figure 9.26. Frontier molecular orbitals HOMO and LUMO.[\/caption]\n\n\u00a0\n\nWhen molecules absorb energy, it is typical for a HOMO electron to use this energy to transition from the ground HOMO orbital to the LUMO excited-state orbital. This type of transition can be observed in ultraviolet-visible (UV-Vis) radiation spectroscopy experiments. As well, in many chemical reactions, one reactant molecule may donate HOMO electrons to the LUMO of another reactant (Figure 9.27 \"<span class=\"Apple-style-span\">Formation of a new bonding molecular orbital by combining reactant HOMO and LUMO\"<\/span>). Therefore, understanding frontier molecular orbital energy levels can provide chemists with a great deal of insight in the areas of molecular spectroscopy and reactivity.\n\n[caption id=\"attachment_2568\" align=\"alignnone\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/HOMO_and_LUMO_reaction.png\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213859\/HOMO_and_LUMO_reaction-1.png\" alt=\"Figure #.#. Formation of a new bonding molecular orbital by combining reactant HOMO and LUMO.\" class=\"wp-image-2568\" height=\"179\" width=\"400\"\/><\/a> Figure 9.27. Formation of a new bonding molecular orbital by combining reactant HOMO and LUMO.[\/caption]\n\n\u00a0\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul><li>Atomic orbitals can combine to make bonding and antibonding molecular orbitals.<\/li>\n\t<li>Bonding orbitals are lower in energy than antibonding orbitals.<\/li>\n\t<li>Molecular orbitals are filled using similar principles to atomic orbitals.<\/li>\n\t<li>Bond order can be used to evaluate bond strength.<\/li>\n\t<li>Frontier molecular orbitals are of particular importance in molecular\u00a0spectroscopy and reactivity.<\/li>\n<\/ul><\/div>\n\u00a0","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Gain an \u00a0understanding of molecular orbital theory.<\/li>\n<li>Learn to calculate bond orders.<\/li>\n<li>Learn to draw molecular orbital electron configuration energy diagrams.<\/li>\n<\/ul>\n<\/div>\n<p>\u00a0<\/p>\n<p>Valence bond theory is able to explain many aspects of bonding, but not all. To complement this theory, we use another called the\u00a0<a class=\"glossary\">molecular orbital (MO) theory<\/a>. Molecular orbital theory is a more sophisticated model for understanding the nature of chemical bonding.<\/p>\n<p>MO theory takes the idea of atomic orbitals overlapping to a new level, where new molecular orbitals are generated using a mathematical process called <em>linear combination of atomic orbitals<\/em> (LCAO).<\/p>\n<p>Molecular orbitals share many similarities with atomic orbitals:<\/p>\n<p>&#8211;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 They are filled from lowest energy to highest energy (Aufbau principle).<\/p>\n<p>&#8211;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 They can hold a maximum of two electrons of opposite spin per orbital\u00a0(Pauli exclusion principle).<\/p>\n<p>The major difference between atomic and molecular orbitals is that atomic orbitals represent electron density in space associated with a particular atom. Molecular orbitals are associated with the entire molecule, meaning the electron density is delocalized (spread out) over more than one atom.<\/p>\n<p>\u00a0<\/p>\n<h2>The Molecular Orbitals of the Hydrogen Molecule<\/h2>\n<p>Combining the 1<em>s<\/em> orbitals of each hydrogen atom using LCAO, two molecular orbitals are generated \u03c3<sub>1<em>s<\/em><\/sub> (pronounced sigma one <em>s<\/em>) and \u03c3*<sub>1<em>s<\/em><\/sub> (pronounced sigma star one <em>s<\/em>).<\/p>\n<p>The \u03c3<sub>1<em>s<\/em><\/sub> orbital is generated by a constructive combination (or interference), where the two atomic orbitals wave functions reinforce (add to) each other. This is the lower energy of the two molecular orbitals and is known as the <a class=\"glossary\">bonding molecular orbital<\/a>. Notice in Figure 9.19 &#8220;<span class=\"Apple-style-span\">Hydrogen molecular orbital combination diagram&#8221;\u00a0<\/span>that the electron density of this orbital is concentrated between the two nuclei. These electrons are stabilized by attractions to both nuclei, and they hold the atoms together with a covalent bond.<\/p>\n<p>The \u03c3*<sub>1<em>s<\/em><\/sub> orbital is generated by a destructive combination (or interference), where the\u00a0wave functions of the two atomic orbitals cancel each other. This type of combination results in an area of zero electron density between the two nuclei, known as a <strong>nodal plane (or node)<\/strong>. This node of zero electron density is destabilizing toward the bond, making it higher energy, and subsequently this type of orbital is known as an <a class=\"glossary\">antibonding molecular orbital <\/a>(denoted by the asterisk in the orbital name).<\/p>\n<div id=\"attachment_2550\" style=\"width: 410px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/h2_mo_combination_diagram.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2550\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213843\/h2_mo_combination_diagram-1.png\" alt=\"Figure #.#. Hydrogen molecular orbital combination diagram.\" class=\"wp-image-2550\" height=\"197\" width=\"400\" \/><\/a><\/p>\n<p id=\"caption-attachment-2550\" class=\"wp-caption-text\">Figure 9.19. Hydrogen molecular orbital combination diagram.<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>Similar to atomic orbitals, we can write electron configuration energy diagrams for molecular orbitals (Figure 9.20 &#8220;<span class=\"Apple-style-span\">Hydrogen molecular orbital electron configuration energy diagram&#8221;<\/span>). Notice that the atomic orbitals of each atom are written on either side, and the newly formed molecular orbitals are written in the centre of the diagram. The bonding molecular orbital is filled and is relatively lower in energy than the contributing atomic orbitals, supporting the fact that hydrogen molecules (H<sub>2<\/sub>) are more stable than lone hydrogen atoms.<\/p>\n<div id=\"attachment_2551\" style=\"width: 410px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/h2_MO_energy_diagram.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2551\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213845\/h2_MO_energy_diagram-1.png\" alt=\"Figure #.#. Hydrogen molecular orbital electron configuration energy diagram.\" class=\"wp-image-2551\" height=\"291\" width=\"400\" \/><\/a><\/p>\n<p id=\"caption-attachment-2551\" class=\"wp-caption-text\">Figure 9.20. Hydrogen molecular orbital electron configuration energy diagram.<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<h2>Bond Order<\/h2>\n<p>We have just seen that the bonding molecular orbital is lower energy and promotes the formation of a covalent bond, while the antibonding molecular orbital is higher energy with a node of zero electron density between the atoms that destabilizes the formation of a covalent bond. We can evaluate the strength of a covalent bond by determining its <a class=\"glossary\">bond order<\/a>.<\/p>\n<p>Bond order = 1\/2 (# of electrons in bonding MOs &#8211; # of electrons in antibonding MOs)<\/p>\n<p>\u00a0<\/p>\n<p>Bond-order values can be whole numbers, fractions, or zero. These values correspond to the valence bond model, so a bond order of 1 is equal to a single bond, and 2 is equal to a double bond. A value of zero means that there is no bond present, and the atoms exist separately.<\/p>\n<div class=\"textbox shaded\">\n<p>Example 11<\/p>\n<p>Determine the bond order of the hydrogen molecule.<\/p>\n<p>Solution<\/p>\n<p>Bond order = 1\/2 (# of electrons in bonding MOs &#8211; # of electrons in antibonding MOs)<\/p>\n<p>Bond order = 1\/2 (2 &#8211; 0) = 1<\/p>\n<p>Therefore there is a single bond in the hydrogen molecule.<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<h2>Molecular Orbitals of Li<sub>2<\/sub><\/h2>\n<p>Generating molecular orbitals of molecules more complex than hydrogen using the LCAO method requires following a few additional guidelines:<\/p>\n<p>&#8211;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The number of MOs generated is equal to the number of atomic orbitals combined.<\/p>\n<p>&#8211;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Combined atomic orbitals should be of similar energy levels.<\/p>\n<p>&#8211;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The effectiveness of atomic orbital combination depends on the amount of orbital overlap. Increased overlap lowers the energy of the bonding molecular orbital further, and raises the energy of the antibonding molecular orbital.<\/p>\n<p>\u00a0<\/p>\n<p>Let\u2019s follow these guidelines and generate a molecular orbital electron configuration diagram for Li<sub>2\u00a0<\/sub>(Figure 9.21 &#8220;<span class=\"Apple-style-span\">Molecular orbital electron configuration energy diagram for dilithium&#8221;<\/span>):<\/p>\n<p>\u00a0<\/p>\n<div id=\"attachment_2557\" style=\"width: 491px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/Li2_MO_electron_config_diagram.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2557\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213847\/Li2_MO_electron_config_diagram-1.png\" alt=\"Figure #.#. Molecular orbital electron configuration energy diagram for dilithium.\" class=\"wp-image-2557 size-full\" height=\"607\" width=\"481\" \/><\/a><\/p>\n<p id=\"caption-attachment-2557\" class=\"wp-caption-text\">Figure 9.21. Molecular orbital electron configuration energy diagram for dilithium.<\/p>\n<\/div>\n<p>Notice that we have combined the 1<em>s<\/em> atomic orbitals, as before in the H<sub>2<\/sub> example, to generate bonding and antibonding molecular orbitals that are completely filled by both atoms&#8217; 1<em>s<\/em> electrons. Similarly 2<em>s<\/em> atomic orbitals combine, giving a bonding orbital and an antibonding orbital, which are filled with the remaining valence electrons starting from the bottom up. The atomic orbitals that\u00a0combine are of similar energy levels; a 1s orbital <em>does not<\/em> combine with one of the 2<em>s<\/em> orbitals.<\/p>\n<p>The bond order can be determined for this molecule to be:<\/p>\n<p>Bond order = 1\/2 (4 &#8211; 2) = 1<\/p>\n<p>Therefore Li<sub>2<\/sub> would have a single bond.<\/p>\n<p>\u00a0<\/p>\n<h2>Molecular Orbitals from <em>p<\/em>\u00a0Atomic Orbitals<\/h2>\n<p>To determine the molecular orbitals of many other molecules, we need to examine how <em>p<\/em> orbitals combine to give molecular orbitals. The\u00a0<em>p<\/em>\u00a0orbitals can overlap in two ways: head-to-head or sideways. Head-to-head overlap of <em>p<\/em> atomic orbitals results in a bonding and antibonding molecular orbital, where the electron density is centred along the internuclear axis, making them \u03c3 orbitals (Figure 9.22 &#8220;<span class=\"Apple-style-span\">Head-to-head overlap of\u00a0<em>p<\/em>\u00a0orbitals&#8221;<\/span>).<\/p>\n<p>\u00a0<\/p>\n<div id=\"attachment_2560\" style=\"width: 410px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/2p_head_to_head_overlap.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2560\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213849\/2p_head_to_head_overlap-1.png\" alt=\"Figure #.#. Head-to-head overlap of p orbitals.\" class=\"wp-image-2560\" height=\"141\" width=\"400\" \/><\/a><\/p>\n<p id=\"caption-attachment-2560\" class=\"wp-caption-text\">Figure 9.22. Head-to-head overlap of <em>p<\/em> orbitals.<\/p>\n<\/div>\n<p>Sideways overlap of the remaining four <em>p<\/em> atomic orbitals can occur along the two other axes, generating four \u03c0 molecular orbitals having electron density on opposite sides of the internuclear axis (Figure 9.23 &#8220;<span class=\"Apple-style-span\">Sideways overlap of\u00a0<em>p<\/em>\u00a0orbitals&#8221;<\/span>).<\/p>\n<p>\u00a0<\/p>\n<div id=\"attachment_2561\" style=\"width: 410px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/2p_sideways_overlap.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2561\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213851\/2p_sideways_overlap-1.png\" alt=\"Figure #.#. Sideways overlap of p orbitals.\" class=\"wp-image-2561\" height=\"420\" width=\"400\" \/><\/a><\/p>\n<p id=\"caption-attachment-2561\" class=\"wp-caption-text\">Figure 9.23. Sideways overlap of <em>p<\/em> orbitals.<\/p>\n<\/div>\n<p>The head-to-head overlap giving \u03c3 molecular orbitals results in greater overlap, making its bonding molecular orbital the most stable and lowest energy, while the \u03c3* antibonding is least stable and has the highest energy (Figure 9.24 &#8220;<span class=\"Apple-style-span\">Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 8-10&#8243;<\/span>). Sideways overlap gives four \u03c0 molecular orbitals, two lower-energy degenerate-bonding molecular orbitals, and two higher-energy antibonding orbitals.<\/p>\n<p>\u00a0<\/p>\n<div id=\"attachment_2562\" style=\"width: 509px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/2p_molecular_orbitals_electron_config.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2562\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213853\/2p_molecular_orbitals_electron_config-1.png\" alt=\"Figure #.#. Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 8-10.\" class=\"wp-image-2562 size-full\" height=\"701\" width=\"499\" \/><\/a><\/p>\n<p id=\"caption-attachment-2562\" class=\"wp-caption-text\">Figure 9.24. Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 8-10.<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>The energy diagram we have just generated fits experimentally with O<sub>2<\/sub>, F<sub>2<\/sub>, and Ne<sub>2<\/sub>, but does not fit for B<sub>2<\/sub>, C<sub>2<\/sub>, and N<sub>2<\/sub>. In the latter, homonuclear diatomic molecules (B<sub>2<\/sub>, C<sub>2<\/sub>, and N<sub>2<\/sub>), interactions take place between the 2<em>s<\/em> and 2<em>p<\/em> atomic orbitals that are strong enough to swap the ordering of the \u03c3<sub>2<em>p<\/em><\/sub> and \u03c0<sub>2<em>p<\/em><\/sub> molecular orbitals (Figure 9.25).<\/p>\n<div id=\"attachment_2563\" style=\"width: 506px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/2p_molecular_orbital_energy_diagram_swapped.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2563\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213855\/2p_molecular_orbital_energy_diagram_swapped-1.png\" alt=\"Figure #.#. Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 5-7.\" class=\"size-full wp-image-2563\" height=\"710\" width=\"496\" \/><\/a><\/p>\n<p id=\"caption-attachment-2563\" class=\"wp-caption-text\">Figure 9.25. Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 5-7.<\/p>\n<\/div>\n<h2>Heteronuclear Diatomic Molecules<\/h2>\n<p>In heteronuclear diatomic molecules, where two different molecules are bonded, the energy levels of the individual atoms&#8217; atomic orbitals may differ. However, the molecular orbital diagram we\u00a0see in\u00a0Figure 9.25 (&#8220;<span class=\"Apple-style-span\">Molecular orbital energy diagram for homonuclear diatomic molecules made from atoms of atomic number 5-7&#8243;)<\/span> can be used to estimate the electron configuration and bond order.<\/p>\n<p>\u00a0<\/p>\n<h2>Frontier Molecular Orbitals<\/h2>\n<p>We can focus further on two very important types of molecular orbitals: the <a class=\"glossary\">highest occupied molecular orbital (HOMO)<\/a> and the <a class=\"glossary\">lowest unoccupied molecular orbital (LUMO)<\/a>, also referred to collectively as the <a class=\"glossary\">frontier molecular orbitals<\/a> (Figure 9.26 &#8220;<span class=\"Apple-style-span\">Frontier molecular orbitals HOMO and LUMO&#8221;<\/span>). As their names imply, the HOMO is the molecular orbital that has the highest energy and contains electrons, while the LUMO is the lowest energy molecular orbital that does not contain electrons.<\/p>\n<div id=\"attachment_2567\" style=\"width: 410px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/Frontier_Molecular_Orbitals_1.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2567\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213857\/Frontier_Molecular_Orbitals_1-1.png\" alt=\"Figure #.#. Frontier molecular orbitals HOMO and LUMO.\" class=\"wp-image-2567\" height=\"307\" width=\"400\" \/><\/a><\/p>\n<p id=\"caption-attachment-2567\" class=\"wp-caption-text\">Figure 9.26. Frontier molecular orbitals HOMO and LUMO.<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p>When molecules absorb energy, it is typical for a HOMO electron to use this energy to transition from the ground HOMO orbital to the LUMO excited-state orbital. This type of transition can be observed in ultraviolet-visible (UV-Vis) radiation spectroscopy experiments. As well, in many chemical reactions, one reactant molecule may donate HOMO electrons to the LUMO of another reactant (Figure 9.27 &#8220;<span class=\"Apple-style-span\">Formation of a new bonding molecular orbital by combining reactant HOMO and LUMO&#8221;<\/span>). Therefore, understanding frontier molecular orbital energy levels can provide chemists with a great deal of insight in the areas of molecular spectroscopy and reactivity.<\/p>\n<div id=\"attachment_2568\" style=\"width: 410px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/06\/HOMO_and_LUMO_reaction.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2568\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14213859\/HOMO_and_LUMO_reaction-1.png\" alt=\"Figure #.#. Formation of a new bonding molecular orbital by combining reactant HOMO and LUMO.\" class=\"wp-image-2568\" height=\"179\" width=\"400\" \/><\/a><\/p>\n<p id=\"caption-attachment-2568\" class=\"wp-caption-text\">Figure 9.27. Formation of a new bonding molecular orbital by combining reactant HOMO and LUMO.<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul>\n<li>Atomic orbitals can combine to make bonding and antibonding molecular orbitals.<\/li>\n<li>Bonding orbitals are lower in energy than antibonding orbitals.<\/li>\n<li>Molecular orbitals are filled using similar principles to atomic orbitals.<\/li>\n<li>Bond order can be used to evaluate bond strength.<\/li>\n<li>Frontier molecular orbitals are of particular importance in molecular\u00a0spectroscopy and reactivity.<\/li>\n<\/ul>\n<\/div>\n<p>\u00a0<\/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-505\">\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>Introductory Chemistry- 1st Canadian Edition . <strong>Authored by<\/strong>: Jessie A. Key and David W. Ball. <strong>Provided by<\/strong>: BCCampus. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/opentextbc.ca\/introductorychemistry\/\">https:\/\/opentextbc.ca\/introductorychemistry\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Download this book for free at http:\/\/open.bccampus.ca<\/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":23485,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Chemistry- 1st Canadian Edition \",\"author\":\"Jessie A. Key and David W. Ball\",\"organization\":\"BCCampus\",\"url\":\"https:\/\/opentextbc.ca\/introductorychemistry\/\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Download this book for free at http:\/\/open.bccampus.ca\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["jessie-a-key"],"pb_section_license":""},"chapter-type":[],"contributor":[59],"license":[],"class_list":["post-505","chapter","type-chapter","status-publish","hentry","contributor-jessie-a-key"],"part":352,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/pressbooks\/v2\/chapters\/505","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/pressbooks\/v2\/chapters\/505\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/pressbooks\/v2\/parts\/352"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/pressbooks\/v2\/chapters\/505\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/wp\/v2\/media?parent=505"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/pressbooks\/v2\/chapter-type?post=505"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/wp\/v2\/contributor?post=505"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-introductory-chemistry\/wp-json\/wp\/v2\/license?post=505"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}