[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-650804-105":59,"doc-detail-650804-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","crystal-ligand-field-theory-solution-and-orbitalspin-formalism","Crystal (ligand) field theory - Solution and orbital/spin formalism","","Crystal (ligand) field theory notes on atomic electronic structure, starting from the Schrödinger equation and hydrogenic solutions in spherical coordinates using radial functions and spherical harmonics Yl,m. The document derives Hamiltonian components (kinetic and potential), introduces quantum numbers n, l, ml, and evaluates orbital angular momentum and projection. It extends to effective nuclear charge and Slater’s rules for screening, then treats angular orbitals as linear combinations of spherical harmonics. Finally, it covers spin angular momentum, the Dirac equation, relativistic energy corrections, Pauli matrices, Darwin’s term, and spin–orbit coupling with qualitative filling rules.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/crystal-ligand-field-theory-solution-and-orbitalspin-formalism/650804/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/crystal-ligand-field-theory-solution-and-orbitalspin-formalism/650804.png","ImageObject",300,407,{"name":92,"@type":93},"Valentina","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-24",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the role of the Schrödinger equation in these notes?","Question",{"text":112,"@type":113},"It is used to set up and solve for hydrogen-like wave functions using kinetic and potential energy terms. The solutions are expressed with radial functions and spherical harmonics Yl,m.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are orbital angular momentum and its projection defined?",{"text":117,"@type":113},"The notes state that l can take values from 0 to n-1. The magnetic quantum number ml is the projection into the z-axis and ranges from -l to l.",{"name":119,"@type":110,"acceptedAnswer":120},"What relativistic effects are included beyond the Schrödinger treatment?",{"text":121,"@type":113},"The document introduces Dirac’s equation and relativistic corrections to electron energies. It specifically mentions spin–orbit coupling and Darwin’s contact interaction (Darwin’s term) for s orbitals.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},650804,1792846092,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},13056703020460,"https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923","Crystal (ligand) field theory  \nLiterature  \n• C.E. Housecroft, A.G. Sharpe: Inorganic chemistry  \nCrystal (ligand) field theory  \n• Sphere symmetry  \n• jj-coupling, LS-coupling  \nSchrödinger equation Crystal field  \nSchrödinger equation  \n􀀐 ~~􀀡~~22m 􀀧 􀀼(r) 􀀎  \n􀀉 􀀋  \n􀀊  \nˆV(r) 􀀼 (r) 􀀠 E􀀼(r)  \n􀀬  \nkinetic energy potential energy  \n􀀧 􀀠 ~~􀁷~~2􀁷x2~~ ~~ 􀀎 ~~􀁷~~2􀁷y2~~ ~~ 􀀎 ~~􀁷~~2􀁷z~~ ~~2~~  ~~Hydrogen atom: ˆV 􀀠 4~~e~~2􀁓􀁈or  \n􀀧 in spherical coordinates: 􀀼n ,l ,m 􀀠 Rn ,l (r) 􀂘Yl ,m (􀁔,􀁍)  \n􀀧 􀁼 ~~􀁷~~2􀁷r~~ ~~2~~ ~~ 􀀎 ~~􀁷~~2􀁷􀁔2~~ ~~ 􀀎 ~~􀁷~~2􀁷􀁍2~~ ~~  \nm: electron mass  \n􀁈o: permitivity of vacuum 􀀼: wave functions  \ne: electron charge  \nE: energy  \nħ: Planck’s constant  \nR: radial function  \nY: angular function  \nˆH􀀼n ,l ,m 􀀠 En 􀀼n ,l ,m 2 Yl ,m 􀀠 l (l 􀀎1)􀀡 2 Yl ,m ˆLzYl ,m 􀀠 ml 􀀡Yl ,m  \nˆH 􀀠 ˆT 􀀎 ˆV  \nn: principal quantum number  \nl: orbital quantum number  \ndetermine the orbital angular momentum l = 0 ... n-1  \nml: magnetic quantum number  \nprojection of the angular momentum into z-axis ml = -l … l  \nRadial function Crystal field  \n8  \n7 4􀁓r2·Rnl (r) 2  \n6  \n5  \n4  \n1s  \n2s  \n3  \n2p  \n3s  \n2  \n3p  \n1  \n0  \n0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  \nr / a0  \nBohr’s radius a0 = 0.52918 Å The distance of the electron (maximum of the orbital 1s density) from nuclei in hydrogen atom  \na0 􀀠 4􀁓􀁈m~~0~~e􀀡22  \nRadial function – the dependence on Z Crystal field  \nThe radius of maximal electron density  \nEffective nuclear charge  \na0* = a0 / Z* Z* = Z- σ  \nσ = screening constant, a sum over all electrons The electrons are merged into groups ()(1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)(5d)(5f) ...  \nSlater’s rules:  \nelectron on the right does not contribute to σ Electrons inside the group screen 0.35 (1s only 0.30) n–1 (s,p) screen 0.85  \nn–2 and lower screen 1.00  \nIf the electron is in d or f, all electron on the left screen 1.0  \nAn example for Fe (26):  \nAtomic orbitals – spherical harmonic functions (angular parts) Crystal field  \nSolution of Schrödinger equation (complex function):   \nYl ml  \nLinear combination:(real function):   \ns 􀀠 Y00  \npz 􀀠 Y10  \npx 􀀠 ~~1~~2~~ ~~ 􀀋Y11 􀀎 Y1􀀐1􀀌 py 􀀠 􀀐~~i~~2~~ ~~ 􀀋Y11 􀀐Y1􀀐1 􀀌  \ndz 2 􀀠 Y20  \ndx 2 􀀐 y2 􀀠 ~~1~~2~~ ~~ 􀀋Y22 􀀎 Y2􀀐2  \n􀀌  \nd xy  \ndxz  \ndyz  \n􀀠  \n􀀠  \n􀀠  \n 􀀐i   \n~~ ~~2  \n 􀀐1   \n~~ ~~2  \n i   \n~~ ~~2  \n􀀋Y22 􀀐 Y2􀀐2 􀀌  \n􀀋Y21 􀀐 Y2􀀐1 􀀌  \n􀀋Y21 􀀎 Y2􀀐1 􀀌  \nY 􀀐3􀀕 3  \n􀀠    \n6  \nAtomic orbitals Crystal field  \nOrbital and spin angular momentun Crystal field  \nˆH􀀼n ,l ,m 􀀠 En 􀀼n ,l ,m  \n2 Yl ,m  \nˆL Yz l ,m  \n􀀠 l (l 􀀎1)􀀡 2 Yl ,m  \n􀀠 ml 􀀡Yl ,m  \nˆH 􀀠 ˆT 􀀎 ˆV  \nˆs2 􀀠 􀀡 2 s (s􀀎1) s 􀀠ˆsz 􀀠 􀀡ms 􀀠 􀁲 ~~ ~~􀀡2  \nm: electron mass  \n􀁈o: permitivity of vacuum  1  􀀼: eigenfunctions  \n2 e: electron charge  \nE: energy  \nħ: Planck’s constant  \nR: radial function  \nY: angular function  \nn: principal quantum number  \nl: orbital quantum number  \ndetermine the orbital angular momentum l = 0 ... n-1  \nml: magnetic quantum number projection of the angular momentum into z-axis ml = -l … l  \ns: spin angular momentum  \nms: projection into z-axis  \nms = -1/2 … 1/2  \nDirac’s equation Crystal field  \n􀜧 = 􀜿 2 + 􀝉 20􀜿 2  \nRelativistic formula for the total energy of a free particle  \n􀜧 = 􀜿 −ℏ2 ∆ + 􀝉 20􀜿 2 = −􀝅ℏ􀜿􀟘 + 􀟚􀝉0 􀜿 2 where 􀟚 = 10 1  =   Pauli matrixes: 􀟪􀯫 = 01 10 􀟪􀯬 = 0􀝅 −0􀝅 􀟪􀯭 = 10 1  \n|  |􀀠 s (s 􀀎1)􀀡 s 􀀠 12  \nsz 􀀠 ms 􀀡 ms 􀀠 􀀐 2 , 2  \ns: spin quantum number  \ndetermines angular momentum of electron ms: projection into z-axis  \nms = -s …s  \nSpin-orbit coupling Magnetism  \n􀀈 􀀦 􀀦  \nHso 􀀠 􀁏SL  \n􀁏 > 0 : less than ½ occupied orbital,  \nJ=|L-S|, direction of L and S opposite 􀁏 \u003C 0 : more than ½ occupied orbital,  \nJ=L+S, direction of L and S identical  \n3d \u003C 4d \u003C 5d M2+ \u003C M3+\u003C …  \nApproximate solution of Dirac’s equation:  \nSpin orbit interaction is known in atomic physics as relativistic correction of the electron energies  \n(, , and 􀜮 mean vector operators of spin, momentum and angular momentum)  \n􀜸􀯌􀯅 = 4ℏ􀝉~~ ~~0􀜿~~ ~~2  ∙ ∇􀜸 × /􀝉0  \n􀜸􀯌􀯅,􀯦􀯣ℎ =  ℏ   1 􀝀􀜸 􀜮 ∙  4􀝉0 􀜿 2 􀝎 􀝀􀝎  \nD 􀀠 ˆHSchr 􀀎 ˆVm 􀀎 ˆVD 􀀎 ˆVso  \nVm : Relat","cbCaigXEHFX74IRB","https://ap.wps.com/l/cbCaigXEHFX74IRB","pdf",4908359,71,"English","# Crystal (ligand) field theory\n## Schrödinger equation and hydrogenic solutions\n## Quantum numbers and orbital angular momentum\n## Radial dependence, effective nuclear charge, Slater’s rules\n## Atomic orbitals as spherical harmonic linear combinations\n## Spin, Dirac equation, relativistic corrections\n## Darwin’s term and spin–orbit coupling","[{\"question\":\"What is the role of the Schrödinger equation in these notes?\",\"answer\":\"It is used to set up and solve for hydrogen-like wave functions using kinetic and potential energy terms. The solutions are expressed with radial functions and spherical harmonics Yl,m.\"},{\"question\":\"How are orbital angular momentum and its projection defined?\",\"answer\":\"The notes state that l can take values from 0 to n-1. The magnetic quantum number ml is the projection into the z-axis and ranges from -l to l.\"},{\"question\":\"What relativistic effects are included beyond the Schrödinger treatment?\",\"answer\":\"The document introduces Dirac’s equation and relativistic corrections to electron energies. It specifically mentions spin–orbit coupling and Darwin’s contact interaction (Darwin’s term) for s orbitals.\"}]","Crystal (ligand) field theory - Solution and orbital/spin formalism | PDF",179]