[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-326533-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-326533-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","hartreefock-approximate-molecular-orbital-theory-3-lectures-on-theory-1-lecture-on-programming","Hartree–Fock Approximate - Molecular Orbital Theory - 3 lectures on theory - 1 lecture on programming","","Hartree–Fock approximate molecular orbital theory lectures outline the variational formulation of electronic energy starting from the Schrödinger equation ĤΨ=EΨ. The material motivates the need to understand the Hamiltonian operator, choose tractable wave functions, and examine variational parameters to obtain an upper bound. It derives the Hamiltonian split into kinetic and potential energy, applies the Born–Oppenheimer approximation to obtain the electronic Hamiltonian, then introduces one-electron molecular spin orbitals and constructs many-electron Hartree products. The discussion explains why Hartree products violate antisymmetry, then builds antisymmetric wave functions using determinants, leading to the Slater determinant framework used as the basis for Hartree–Fock equations and matrix approaches.",{"@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/hartreefock-approximate-molecular-orbital-theory-3-lectures-on-theory-1-lecture-on-programming/326533/",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/hartreefock-approximate-molecular-orbital-theory-3-lectures-on-theory-1-lecture-on-programming/326533.png","ImageObject",300,407,{"name":42,"@type":43},"Hazel","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-06","2026-09-21",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",9,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What is the central objective of the Hartree–Fock variational approach described here?","Question",{"text":63,"@type":64},"Derive an analytic energy expression that can be minimized with respect to variational parameters, providing an upper bound to the true energy.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How does the Born–Oppenheimer approximation simplify the problem in this lecture?",{"text":68,"@type":64},"It treats nuclei as fixed while electrons move quickly, separating nuclear and electronic motion and leading to an electronic Hamiltonian by removing nuclear-related terms.",{"name":70,"@type":61,"acceptedAnswer":71},"Why are Hartree products not physically realistic, and how is the issue fixed?",{"text":72,"@type":64},"Hartree products ignore electron antisymmetry required by the Pauli principle. The fix is to construct antisymmetric wave functions using determinants, culminating in the Slater determinant for N electrons.","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},326533,1790241184,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,119,122,126],{"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":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"research-report",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},"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":97,"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":112,"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,"preview_page_count":4},137441390410,"https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984","Hartree–Fock Approximate  \nMolecular Orbital Theory  \nJustin T. Fermann  \n3 lectures on theory  \n1 lecture on programming  \nPurpose  \nFirst, we define the problem, beginning with the Schr¨odinger equation  \nˆHΨ = EΨ . (1)  \nOur goal is to come up with an analytic equation for the energy which can be minimized with respect to some variational parameter(s) to give an upper bound on the energy. To do this, we must  \n1. Understand the hamiltonian operater ˆH .  \n2. Find an appropriate wave function Ψ which allows simple calculation of the electronic energy.  \n3. Examine potential variational parameters to figure out how to minimize the energy.  \nIt will likely be convenient to have an outline of where we’re going, so here it is:  \nI. Puropse (which we’ve been through)  \nII. The Hamiltonian, ˆH  \n• Kinetic energy operators  \n• Coulombic potential operators  \nIII. The Wave Function, Ψ  \n• 1 electron orbitals  \n• Hartree products  \n• Slater determinants  \nIV. Hamiltonian as Energy Operator, Schr¨odinger Equation  \n• E = hΨ|ˆH|Ψi  \n• Integrals over one electron operators  \n• Integrals over two electron operators  \n• Specific case energy expressions  \n• General form of HF energy  \nV. What Variational Parameter?  \n• LCAO–MO theory, energy in AO basis  \n• Density Matrices  \nVI. Hartree–Fock Equations  \n• Lagrangs’s Undetermined Multipliers  \n• A load ’o math  \nVII. Matrix Formalism  \nVIII. Program Outline  \nThe Hamiltonian, ˆH  \nThe Hamiltonian is the total energy operator for a system, and is written as the sum of the kinetic energy of all the components of the system and the internal potential energy. In an atom or molecule, comprised of positive nuclei and negative electrons, the potential energy is simply that due to the coulombic interactions present. Thus for the kinetic energy in a system of M nuclei and N electrons:  \nˆTN = − MX  1  ∇2A (2)  \nA 2MA  \nˆTe = − NX 1 ∇2i . (3)  \ni 2  \nAnd for the potential energy:  \nM Z Z  \nˆVNN = X  A B  (4)  \nA>B rAB  \nˆVee = NX  1  (5)  \ni>j rij  \nM N Z  \nˆVeN = −XX  A . (6)  \nA i riA  \nSince ˆH = ˆT + ˆV ,  \nˆH = − NXi 12∇2i − NXi ZAr~~ ~~iA +Xij r1ij −  21MA ∇2A +  ZAr~~ ~~AZBB~~ ~~ . (7)  \nWithin the Born–Oppenheimer approximation, we assume the nuclei are held fixed while the electrons move really fast around them. (note: Mp /Me ≈ 1840.) In this case, nuclear motion and electronic motion are seperated. The last two terms can be removed from the total hamiltonianto give the electronic hamiltonian, ˆHe , since ˆVNN = K, and ∇2A = 0 . The nuclear motion is handled in a rotational/vibrational analysis. We will be working within the B–O approximation, so realizing  \nˆHe = − NXi 12∇2i −  NXi ZAr~~ ~~iA +Xij r1ij~~ ~~ (8)  \nwe completly define the problem. Solving the electronic Schrdinger equation using this will give the electronic structure of a molecular system at a fixed nuclear geometry.  \nThe Wave Function, Ψ  \nWe’ve derived a complete many–electron Hamiltonian operator. Of course, the Schr¨odinger equation involving it is intractable, so let’s consider a simpler problem, involving the one–electron hamiltonian  \nˆh(i) = − ~~ ~~12∇2i +  r1iA~~ ~~ (9)  \nwhich involves no electron–electron interaction. This is soluable in the B–O approximation (recall the hydrogen atom by letting M = 1) . Call the solutions to the one–electron Schr¨odinger equation χi (x) . These will be molecular spin orbitals when we get around to it, but for now let it suffice to know they satisify the eigenequation  \nˆh(i)χj (xi ) = εj χj (xi) (10)  \nwith the interpretation that electron i occupies spin orbital χj with energy εj . If we ignore electron–electron interaction in ˆHe , we construct a simpler system with Hamiltonian  \nN  \nˆH =X ˆh(i) . (11)  \ni  \nIt will have eigenfunctions which are simple products of occupied spin orbitals, and thus an energy which is a sum of individual orbital energies, as  \nΨHP = χi (x1 )χj (x2 )χk (x3 ) ···χη (xN ) (12)  \nhΨ|ˆH|Ψi = εi + εj + εk + . . . + εη = E. (13)  \nThis kind of wavefunction is ","cbCaii86Gp2USo0V","https://ap.wps.com/l/cbCaii86Gp2USo0V","pdf",132724,21,"English","# Purpose\n## Variational goal and outline\n# Hamiltonian and energy operator\n## Kinetic energy operators\n## Coulombic potential operators\n## Born–Oppenheimer approximation\n# Wave function construction\n## One-electron Hamiltonian and orbitals\n## Hartree product (independent-electron model)\n## Antisymmetry and Pauli principle\n## Two-electron antisymmetrization\n## Slater determinant and notation","[{\"question\":\"What is the central objective of the Hartree–Fock variational approach described here?\",\"answer\":\"Derive an analytic energy expression that can be minimized with respect to variational parameters, providing an upper bound to the true energy.\"},{\"question\":\"How does the Born–Oppenheimer approximation simplify the problem in this lecture?\",\"answer\":\"It treats nuclei as fixed while electrons move quickly, separating nuclear and electronic motion and leading to an electronic Hamiltonian by removing nuclear-related terms.\"},{\"question\":\"Why are Hartree products not physically realistic, and how is the issue fixed?\",\"answer\":\"Hartree products ignore electron antisymmetry required by the Pauli principle. The fix is to construct antisymmetric wave functions using determinants, culminating in the Slater determinant for N electrons.\"}]","Hartree–Fock Approximate - Molecular Orbital Theory - 3 lectures on theory - 1 lecture on programming | PDF",1789974492,53]