[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-657132-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-657132-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","introduction-to-molecular-structure-lecture","Introduction to Molecular Structure - Lecture","","Lecture notes on molecular structure covering key frameworks from basic principles to practical applications. It introduces the Born–Oppenheimer approximation, explaining how separation of electronic and nuclear motion leads to electronic Schrödinger equations and a potential energy surface concept. It then develops Molecular Orbital Theory using the LCAO method, describing how molecular orbitals form as linear combinations of atomic orbitals and how coefficients follow from solving the Schrödinger equation. The outline previews extensions to diatomic molecules, polyatomic systems, conjugated π-systems, and the Hückel approximation.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/introduction-to-molecular-structure-lecture/657132/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/introduction-to-molecular-structure-lecture/657132.png","ImageObject",300,407,{"name":42,"@type":43},"fresh2021","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-29","2026-10-26",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What problem does the Born–Oppenheimer approximation address in molecular quantum mechanics?","Question",{"text":63,"@type":64},"It notes that solving the full Schrödinger equation for all particles together is intractable, then uses the large mass difference to treat nuclei as fixed while solving the electronic problem.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How does the Born–Oppenheimer approximation separate variables in the wave function?",{"text":68,"@type":64},"It proposes a product form Ψ(r, RA, RB)=ψ(r; RA, RB)χ(RA, RB) and substitutes it into the Schrödinger equation to obtain two coupled equations with electronic energy Ee and nuclear energy E.",{"name":70,"@type":61,"acceptedAnswer":71},"What is the key idea of Molecular Orbital Theory and LCAO?",{"text":72,"@type":64},"Molecular orbitals are constructed as linear combinations of atomic orbitals, with single-electron MOs written as Ψ(r)=Σ ci ϕ(r−Ri). The coefficients ci are determined by solving the Schrödinger equation for the molecular Hamiltonian.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},657132,1793105926,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":55,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145},962093912154,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Introduction to Molecular Structure  \nMolecular Orbital Theory, LCAO, Hydrogen Molecule, H¨uckel Method  \nDaniele Toffoli  \nDipartimento di Scienze Chimiche e Farmaceutiche, Universit`a di Trieste  \nOctober 16, 2024  \n Daniele Toffoli (UniTS)  Introduction to Molecular Structure  October 16, 2024 1 / 35   \nOutline  \n1 Born-Oppenheimer Approximation  \n2 Molecular Orbital Theory  \n3 MOT Applied to H2  \n4 MOT for Diatomic Molecules  \n5 MOT for Polyatomic Molecules  \n6 Conjugated π-Systems  \n7 The H¨uckel Approximation  \nDaniele Toffoli (UniTS)  Introduction to Molecular Structure  October 16, 2024 2 / 35   \nBorn-Oppenheimer Approximation  \nOutline  \n1 Born-Oppenheimer Approximation  \n2 Molecular Orbital Theory  \n3 MOT Applied to H2  \n4 MOT for Diatomic Molecules  \n5 MOT for Polyatomic Molecules  \n6 Conjugated π-Systems  \n7 The H¨uckel Approximation  \nDaniele Toffoli (UniTS)  Introduction to Molecular Structure  October 16, 2024 3 / 35   \nBorn-Oppenheimer Approximation  \nBorn-Oppenheimer Approximation Introduction  \n Even a simple system like the H+2 ion has a complex Hamiltonian:  \nˆH = − 2ℏ2me~~ ~~ ∇2 − 2ℏ2mp 􀀀 ∇2A + ∇2B􀀁 + V ( ⃗r; ⃗RA , ⃗RB )  \nwhere:  \n me is the electron mass, and mp is the proton mass.  \n ∇2 acts on electron’s coordinates  \n ∇2A and ∇2B act on the nuclei  \n V ( ⃗r; ⃗RA , ⃗RB ) = − 4~~e~~π2ϵ0 􀀐 ~~ ~~|~~ ~~⃗r⃗RA~~ ~~| + |~~ ~~⃗r~~ ~~−~~1~~⃗RB~~ ~~| + |~~ ~~⃗RA~~ ~~⃗RB~~ ~~|~~ ~~ 􀀑  \n Impossible to solve for the wave function if we are to consider all three particles together.  \n BO approximation: mp is at least 1000me → treat the nuclei as fixed  This decouples electronic and nuclear degrees  of freedom.     \nDaniele Toffoli (UniTS)  Introduction to Molecular Structure  October 16, 2024 4 / 35   \nBorn-Oppenheimer Approximation  \nBorn-Oppenheimer approximation Separation of Variables  \n Propose a solution  \nΨ( ⃗r, ⃗RA , ⃗RB ) = ψ( ⃗r; ⃗RA , ⃗RB )χ ( ⃗RA , ⃗RB )  \n Substituting this into the full Schr¨odinger equation yields two equations:  \n􀀔 − 2ℏ2me~~ ~~ ∇2 + V ( ⃗r; ⃗RA , ⃗RB )􀀕 ψ ( ⃗r; ⃗RA , ⃗RB ) = Eeψ( ⃗r; ⃗RA , ⃗RB )  \n􀀔 − 2ℏ2mp 􀀀 ∇2A + ∇2B􀀁+ Ee􀀕 χ ( ⃗RA , ⃗RB ) = Eχ( ⃗RA , ⃗RB )  \n After we solve the first equation for the fixed nuclei, we can solve the second equation with Ee .  \n Daniele Toffoli (UniTS)  Introduction to Molecular Structure  October 16, 2024 5 / 35   \nBorn-Oppenheimer Approximation  \nBorn-Oppenheimer approximation Potential Energy Surface  \n We notice that Ee = Ee( ⃗RA , ⃗RB ) for both the ground state and excited states.  \n Assuming that excited levels do not cross, we can view this function as a potential energy surface on which electrons move.  \nGround and 1st excited states of the molecule formaldimine for two angles.  \n Daniele Toffoli (UniTS)  Introduction to Molecular Structure  October 16, 2024 6 / 35   \nMolecular Orbital Theory  \nOutline  \n1 Born-Oppenheimer Approximation  \n2 Molecular Orbital Theory  \n3 MOT Applied to H2  \n4 MOT for Diatomic Molecules  \n5 MOT for Polyatomic Molecules  \n6 Conjugated π-Systems  \n7 The H¨uckel Approximation  \nDaniele Toffoli (UniTS)  Introduction to Molecular Structure  October 16, 2024 7 / 35   \nMolecular Orbital Theory  \nMolecular Orbital Theory  \nLinear Combination of Atomic Orbitals (LCAO)  \n Molecular Orbital Theory (MOT) describes the formation of molecular orbitals (MOs) from atomic orbitals (AOs) .  \nAn atomic orbital, in this context, is an orbital that was obtained by solving the Schr¨odinger equation for an isolated atom, localized on that atom.  \n Daniele Toffoli (UniTS)  Introduction to Molecular Structure  October 16, 2024 8 / 35   \nMolecular Orbital Theory  \nMolecular Orbital Theory  \nLinear Combination of Atomic Orbitals (LCAO)  \n In LCAO, we construct single-electron MO’s as superpositions of atomic orbitals on all atoms involved  \nΨ( ⃗r) =X ciϕ( ⃗r − ⃗Ri)  \ni  \nwhere i runs over all the atoms, and ϕ are the atomic orbitals.  \n By placing this wave function into a Schr¨odinger equation and solving for the coefficients, we obtain ","cbCaicstN7rpNpMf","https://ap.wps.com/l/cbCaicstN7rpNpMf","pdf",1779963,35,"English","# Outline\n## Born-Oppenheimer Approximation\n## Molecular Orbital Theory\n## MOT Applied to H2\n## MOT for Diatomic Molecules\n## MOT for Polyatomic Molecules\n## Conjugated π-Systems\n## The Hückel Approximation","[{\"question\":\"What problem does the Born–Oppenheimer approximation address in molecular quantum mechanics?\",\"answer\":\"It notes that solving the full Schrödinger equation for all particles together is intractable, then uses the large mass difference to treat nuclei as fixed while solving the electronic problem.\"},{\"question\":\"How does the Born–Oppenheimer approximation separate variables in the wave function?\",\"answer\":\"It proposes a product form Ψ(r, RA, RB)=ψ(r; RA, RB)χ(RA, RB) and substitutes it into the Schrödinger equation to obtain two coupled equations with electronic energy Ee and nuclear energy E.\"},{\"question\":\"What is the key idea of Molecular Orbital Theory and LCAO?\",\"answer\":\"Molecular orbitals are constructed as linear combinations of atomic orbitals, with single-electron MOs written as Ψ(r)=Σ ci ϕ(r−Ri). The coefficients ci are determined by solving the Schrödinger equation for the molecular Hamiltonian.\"}]","Introduction to Molecular Structure - Lecture | PDF",1792974610,88]