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QUANTUMCHEMISTRYINTHEAGEOF MACHINELEARNING QUANTUM CHEMISTRYINTHE AGEOFMACHINE LEARNING Editedby
PAVLO O.DRAL
Professor,StateKeyLaboratoryofPhysicalChemistryofSolidSurfaces,FujianProvincialKey LaboratoryofTheoreticalandComputationalChemistry,DepartmentofChemistry,CollegeofChemistryand ChemicalEngineering,XiamenUniversity,China
Elsevier
Radarweg29,POBox211,1000AEAmsterdam,Netherlands
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
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Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicor mechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem,without permissioninwritingfromthepublisher.Detailsonhowtoseekpermission,furtherinformationaboutthe Publisher’spermissionspoliciesandourarrangementswithorganizationssuchastheCopyrightClearance CenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions.
ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher(other thanasmaybenotedherein).
Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperiencebroadenourunderstanding, changesinresearchmethods,professionalpractices,ormedicaltreatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusinganyinformation, methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethodstheyshouldbemindfulof theirownsafetyandthesafetyofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeanyliabilityforany injuryand/ordamagetopersonsorpropertyasamatterofproductsliability,negligenceorotherwise,orfromanyuseor operationofanymethods,products,instructions,orideascontainedinthematerialherein.
ISBN:978-0-323-90049-2
ForinformationonallElsevierpublications visitourwebsiteat https://www.elsevier.com/books-and-journals
Publisher: SusanDennis
AcquisitionsEditor: CharlesBath
EditorialProjectManager: HowiDeRamois
ProductionProjectManager: PaulPrasadChandramohan
CoverDesigner: MatthewLimbert
TypesetbySTRAIVE,India
Companionwebsiteix
Contributorsxi
Prefacexv 1
Introduction
1.Verybriefintroductiontoquantum chemistry
XunWuandPeifengSu
Introduction—Thefoundationsofquantum chemistry3
Methodsofmolecularelectronicstructure computations8
Methodsofconceptionalinterpretationbasedon electronicstructurecalculations20
Casestudies21
Conclusionsandoutlook24
Acknowledgments24 References24
2.Density-functionaltheory
HongJiangandHuai-YangSun
Introduction27
TheoreticalfoundationsofDFT28
Density-functionalapproximations34
PracticalaspectsofDFTimplementations50
Casestudies53
Concludingremarks60
Acknowledgments61 References61
3.Semiempiricalquantummechanical methods
PavloO.DralandJan Řezac
Introduction67
Methods71
Casestudies87
Conclusionsandoutlook89
Acknowledgments90 References90
4.Fromsmallmoleculestosolid-state materials:Abriefdiscourseonanexample ofcarboncompounds
BiliChen,LeyuanCui,ShuaiWang,andGangFu
Introduction93 Methods95
Casestudies108
Conclusionsandoutlook114
Acknowledgments114 References114
5.Basicsofdynamics
XinxinZhongandYiZhao
Introduction117 Methods119
Casestudies125
Conclusionsandoutlook131
Acknowledgments131 References131
6.Machinelearning: Anoverview
EugenHruskaandFangLiu
Introduction135 Methods136
Basicconceptsofmachinelearning143
Casestudy147
Conclusionsandoutlook149
Acknowledgment149 References149
7.Unsupervisedlearning RoseK.CersonskyandSandipDe
Introduction153
Notationguide154 Methods155
Casestudies171
Conclusions178 References178
8.Neuralnetworks
PavloO.Dral,AlexeiA.Kananenka,FuchunGe, andBao-XinXue
Introduction183 Methods184
Casestudy200
Conclusionsandoutlook201
Acknowledgments202 References203
9.Kernelmethods
MaxPinheiroJrandPavloO.Dral
Introduction205 Methods207
Casestudies227
Conclusionsandoutlook229
Acknowledgments230
Authorcontributions230 References230
10.Bayesianinference
WeiLiangandHongshengDai
Introduction233
BasicconceptsofBayesianstatistics234
Bayesianregression239
Bayesianinferenceinmachinelearning:Bayesian neuralnetworks246
Casestudy246
Conclusionsandoutlook249
Acknowledgments249 References249
2 Machinelearningpotentials 11.Potentialsbasedonlinearmodels
GauthierTallec,GaetanLaurens,OwenFresse-Colson,and JulienLam
Introduction253 Methods255
Casestudies272 Conclusionandoutlook276
Acknowledgments276 References277
12.Neuralnetworkpotentials
JinzheZeng,LiqunCao,andTongZhu
Introduction279 Methods281
Casestudies287
Conclusionsandoutlook290 Acknowledgment290 References290
13.Kernelmethodpotentials
Yi-FanHouandPavloO.Dral
Introduction295 Methods296 Casestudy309
Conclusionsandoutlook310 Acknowledgments311 References311
14.Constructingmachinelearning potentialswithactivelearning
ChengShangandZhi-PanLiu
Introduction313 Methods317 Casestudy322
Conclusionandoutlook325 Acknowledgments325 References325
15.Excited-statedynamicswith machinelearning
LinaZhang,ArifUllah,MaxPinheiroJr,PavloO.Dral, andMarioBarbatti
Introduction329 Methods332 Casestudies344
Conclusionsandoutlook349 Acknowledgments350 References350
16.Machinelearningforvibrational spectroscopy
SergeiManzhos,ManabuIhara,andTuckerCarrington
Introduction355 Methods357 Casestudies372
Conclusionsandoutlook383 References384
17.Molecularstructureoptimizations withGaussianprocessregression
RolandLindhandIgnacioFdez.Galva ´ n
Introduction391 Methods392 Casestudies411 Conclusionsandoutlook423 Acknowledgments423 References424
3 Machinelearningofquantum chemicalproperties
18.Learningelectrondensities
BrunoCuevas-Zuvirı´a
Introduction431 Methods435 Casestudies445
Conclusionsandoutlook449 Acknowledgments450 References450
19.Learningdipolemomentsand polarizabilities
YaolongZhang,JunJiang,andBinJiang
Introduction453 Methods455 Casestudies461
Conclusionsandoutlook463 Acknowledgments463 References464
20.Learningexcited-stateproperties
JuliaWestermayr,PavloO.Dral,andPhilippMarquetand
Introduction467 Methods470
Casestudies477 Conclusionsandoutlook484 Acknowledgments486 References486
4 Machinelearning-improved quantumchemicalmethods
21.Learningfrommultiple quantumchemicalmethods: Δ-learning,transferlearning,co-kriging, andbeyond
PavloO.Dral,TetianaZubatiuk,andBao-XinXue
Introduction491 Methods493 Casestudies503 Conclusionsandoutlook505 Acknowledgments506 Authorcontributions506 References506
22.Data-drivenaccelerationof coupled-clusterandperturbationtheory methods
GrierM.Jones,P.D.VarunaS.Pathirage,and KonstantinosD.Vogiatzis
Introduction510 Methods511
Casestudies523
Conclusionsandoutlook527
Acknowledgment527
References528
23.Redesigningdensityfunctional theorywithmachinelearning
JiangWu,GuanhuaChen,JingchunWang,andXiaoZheng
Introduction531 Methods532
Casestudy552
Conclusionsandoutlook554
References554
24.Improvingsemiempiricalquantum mechanicalmethodswithmachinelearning
PavloO.DralandTetianaZubatiuk
Introduction559 Methods560
Casestudy572
Conclusionsandoutlook573
Acknowledgments574 References574
25.Machinelearningwavefunction
StefanoBattaglia
Introduction577 Methods581
Casestudies605
Conclusionsandoutlook612
Acknowledgments614
References614 5 Analysisofbigdata
26.Analysisofnonadiabaticmolecular dynamicstrajectories
YifeiZhu,JiaweiPeng,HongLiu,andZhenggangLan
Introduction619
Theoreticalmethods621
Examples630
Casestudies642
Conclusionsandoutlook648
References649
27.Designoforganicmaterialswith tailoredopticalproperties:Predicting quantum-chemicalpolarizabilitiesand derivedquantities
GauravVishwakarma,AdityaSonpal,AatishPradhan, MojtabaHaghighatlari,MohammadAtifFaizAfzal,and JohannesHachmann
Introduction654 Methods658
Casestudies:Implementingtherational designprotocol662
Conclusionsandoutlook669
References670
Index675
Pleasevisitthebelowcompanionsitetoviewrepositorieswithprograms,data,instructions, sampleinputandoutputfilesrequiredforcasestudiesaswellasanypost-publicationupdates.
https://www.elsevier.com/books-and-journals/book-companion/9780323900492
Contributors MohammadAtifFaizAfzal Departmentof ChemicalandBiologicalEngineering, UniversityatBuffalo,TheStateUniversityof NewYork,Buffalo,NY,UnitedStates
MarioBarbatti AixMarseilleUniversity,CNRS, ICR,Marseille;InstitutUniversitairedeFrance, Paris,France
StefanoBattaglia DepartmentofChemistry— BMC,UppsalaUniversity,Uppsala,Sweden
LiqunCao ShanghaiEngineeringResearch CenterofMolecularTherapeutics&New DrugDevelopment,SchoolofChemistryand MolecularEngineering,EastChinaNormal University;NYU-ECNUCenterfor ComputationalChemistryatNYUShanghai, Shanghai,China
TuckerCarrington ChemistryDepartment, Queen’sUniversity,Kingston,ON,Canada
RoseK.Cersonsky LaboratoryofComputationalScienceandModeling,SwissFederal InstituteofTechnology(EPFL),Lausanne, Switzerland
BiliChen StateKeyLaboratoryforPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,andCollegeof ChemistryandChemicalEngineering, XiamenUniversity,Xiamen,China
GuanhuaChen HongKongQuantumAILab andDepartmentofChemistry,TheUniversity ofHongKong,Pokfulam,HongKong
BrunoCuevas-Zuvirı ´ a CentrodeBiotecnologı ´ a yGeno ´ micadePlantas(CBGP,UPM-INIA), UniversidadPolitecnicadeMadrid(UPM), InstitutoNacionaldeInvestigacio ´ ny Tecnologı´aAgrariayAlimentaria(INIA), Madrid,Spain;NASACenterforEarlyLife andEvolution;DepartmentofBacteriology,
UniversityofWisconsin-Madison,Madison, WI,UnitedStates
LeyuanCui StateKeyLaboratoryforPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,andCollegeof ChemistryandChemicalEngineering, XiamenUniversity,Xiamen,China
HongshengDai DepartmentofMathematical Sciences,UniversityofEssex,Colchester, UnitedKingdom
SandipDe BASFSE,LudwigshafenamRhein, Germany
PavloO.Dral StateKeyLaboratoryofPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,Departmentof Chemistry,andCollegeofChemistryand ChemicalEngineering,XiamenUniversity, Xiamen,Fujian,China
IgnacioFdez.Galva ´ n Departmentof Chemistry—BMC,UppsalaUniversity, Uppsala,Sweden
OwenFresse-Colson CEMES,CNRSand UniversitedeToulouse,ToulouseCedex,France
GangFu StateKeyLaboratoryforPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,andCollegeof ChemistryandChemicalEngineering, XiamenUniversity,Xiamen,China
FuchunGe StateKeyLaboratoryofPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,Departmentof Chemistry,andCollegeofChemistryand ChemicalEngineering,XiamenUniversity, Xiamen,Fujian,China
JohannesHachmann DepartmentofChemical andBiologicalEngineering,Universityat Buffalo;ComputationalandData-Enabled ScienceandEngineeringGraduateProgram, UniversityatBuffalo,TheStateUniversityof NewYork;NewYorkStateCenterof ExcellenceinMaterialsInformatics,Buffalo, NY,UnitedStates
MojtabaHaghighatlari Departmentof ChemicalandBiologicalEngineering, UniversityatBuffalo,TheStateUniversity ofNewYork,Buffalo,NY,UnitedStates
Yi-FanHou StateKeyLaboratoryofPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,Departmentof Chemistry,andCollegeofChemistryand ChemicalEngineering,XiamenUniversity, Xiamen,Fujian,China
EugenHruska DepartmentofChemistry, EmoryUniversity,Atlanta,GA,UnitedStates
ManabuIhara DepartmentofChemicalScience andEngineering,TokyoInstituteof Technology,Tokyo,Japan
BinJiang HefeiNationalResearchCenterfor PhysicalSciencesattheMicroscale, DepartmentofChemicalPhysics,University ofScienceandTechnologyofChina,Hefei, Anhui,China
HongJiang BeijingNationalLaboratoryfor MolecularSciences,InstituteofTheoretical andComputationalChemistry,Collegeof ChemistryandMolecularEngineering,Peking University,Beijing,China
JunJiang HefeiNationalResearchCenterfor PhysicalSciencesattheMicroscale, DepartmentofChemicalPhysics,University ofScienceandTechnologyofChina,Hefei, Anhui,China
GrierM.Jones DepartmentofChemistry, UniversityofTennessee,Knoxville,TN, UnitedStates
AlexeiA.Kananenka DepartmentofPhysics andAstronomy,UniversityofDelaware, Newark,DE,UnitedStates
JulienLam CEMES,CNRSandUniversitede Toulouse,ToulouseCedex,France
ZhenggangLan SouthChinaNormal University,Guangzhou,China
GaetanLaurens InstitutLumie ` reMatie ` re, UMR5306UniversiteLyon1-CNRS,Universite deLyon,VilleurbanneCedex,France
WeiLiang SchoolofMathematicalSciences, XiamenUniversity,Xiamen,China
RolandLindh DepartmentofChemistry— BMC,UppsalaUniversity,Uppsala,Sweden
FangLiu DepartmentofChemistry,Emory University,Atlanta,GA,UnitedStates
HongLiu SouthChinaNormalUniversity, Guangzhou,China
Zhi-PanLiu CollaborativeInnovationCenterof ChemistryforEnergyMaterial,ShanghaiKey LaboratoryofMolecularCatalysisand InnovativeMaterials,KeyLaboratoryof ComputationalPhysicalScience(Ministryof Education),DepartmentofChemistry,Fudan University;ShanghaiQiZhiInstitute, Shanghai,China
SergeiManzhos DepartmentofChemical ScienceandEngineering,TokyoInstituteof Technology,Tokyo,Japan
PhilippMarquetand InstituteofTheoretical Chemistry,FacultyofChemistry,University ofVienna,Vienna,Austria
JiaweiPeng SouthChinaNormalUniversity, Guangzhou,China
MaxPinheiroJr AixMarseilleUniversity, CNRS,ICR,Marseille,France
AatishPradhan DepartmentofChemicaland BiologicalEngineering,UniversityatBuffalo, TheStateUniversityofNewYork,Buffalo, NY,UnitedStates
Jan Ř eza ´ c InstituteofOrganicChemistryand BiochemistryoftheCzechAcademyof Sciences,Prague,CzechRepublic
P.D.VarunaS.Pathirage Departmentof Chemistry,UniversityofTennessee, Knoxville,TN,UnitedStates
ChengShang CollaborativeInnovationCenter ofChemistryforEnergyMaterial,Shanghai KeyLaboratoryofMolecularCatalysisand InnovativeMaterials,KeyLaboratoryof ComputationalPhysicalScience(Ministryof
Education),DepartmentofChemistry,Fudan University;ShanghaiQiZhiInstitute, Shanghai,China
AdityaSonpal DepartmentofChemicaland BiologicalEngineering,UniversityatBuffalo, TheStateUniversityofNewYork,Buffalo, NY,UnitedStates
PeifengSu TheStateKeyLaboratoryof PhysicalChemistryofSolidSurfaces,Fujian ProvincialKeyLaboratoryofTheoreticaland ComputationalChemistry,andCollegeof ChemistryandChemicalEngineering, XiamenUniversity,Xiamen,Fujian,China
Huai-YangSun BeijingNationalLaboratoryfor MolecularSciences,InstituteofTheoreticaland ComputationalChemistry,Collegeof ChemistryandMolecularEngineering,Peking University,Beijing,China
GauthierTallec Institutdessyste ` mes intelligentsetderobotique(ISIR),Sorbonne Universite,Paris,France
ArifUllah StateKeyLaboratoryofPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,Departmentof Chemistry,andCollegeofChemistryand ChemicalEngineering,XiamenUniversity, Xiamen,Fujian,China
GauravVishwakarma DepartmentofChemical andBiologicalEngineering,Universityat Buffalo,TheStateUniversityofNewYork, Buffalo,NY,UnitedStates
KonstantinosD.Vogiatzis Departmentof Chemistry,UniversityofTennessee, Knoxville,TN,UnitedStates
JingchunWang DepartmentofChemical Physics,UniversityofScienceandTechnology ofChina,Hefei,China
ShuaiWang StateKeyLaboratoryforPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,andCollegeof ChemistryandChemicalEngineering, XiamenUniversity,Xiamen,China
JuliaWestermayr TheoreticalSurface ChemistryGroup,DepartmentofChemistry,
UniversityofWarwick,Coventry,United Kingdom
JiangWu HongKongQuantumAILaband DepartmentofChemistry,TheUniversityof HongKong,Pokfulam,HongKong
XunWu TheStateKeyLaboratoryofPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,andCollegeof ChemistryandChemicalEngineering, XiamenUniversity,Xiamen,Fujian,China
Bao-XinXue StateKeyLaboratoryofPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,Departmentof Chemistry,andCollegeofChemistryand ChemicalEngineering,XiamenUniversity, Xiamen,Fujian,China
JinzheZeng ShanghaiEngineering ResearchCenterofMolecularTherapeutics& NewDrugDevelopment,Schoolof ChemistryandMolecularEngineering, EastChinaNormalUniversity;NYU-ECNU CenterforComputationalChemistryatNYU Shanghai,Shanghai,China
LinaZhang StateKeyLaboratoryofPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,Departmentof Chemistry,andCollegeofChemistryand ChemicalEngineering,XiamenUniversity, Xiamen,Fujian,China
YaolongZhang HefeiNationalResearchCenter forPhysicalSciencesattheMicroscale, DepartmentofChemicalPhysics,University ofScienceandTechnologyofChina,Hefei, Anhui,China
YiZhao StateKeyLaboratoryofPhysical ChemistryofSolidSurfaces,FujianProvincial KeyLaboratoryofTheoreticaland ComputationalChemistry,Collegeof ChemistryandChemicalEngineering, XiamenUniversity,Xiamen,China
XiaoZheng DepartmentofChemicalPhysics, UniversityofScienceandTechnologyof China,Hefei,China
XinxinZhong CollaborativeInnovationCenter forAdvancedOrganicChemicalMaterials Co-constructedbytheProvinceandMinistry, MinistryofEducationKeyLaboratoryforthe SynthesisandApplicationofOrganic FunctionalMolecules,HubeiUniversity, Wuhan,China
TongZhu ShanghaiEngineeringResearch CenterofMolecularTherapeutics&New DrugDevelopment,SchoolofChemistryand
MolecularEngineering,EastChinaNormal University;NYU-ECNUCenterfor ComputationalChemistryatNYUShanghai, Shanghai,China
YifeiZhu SouthChinaNormalUniversity, Guangzhou,China
TetianaZubatiuk DepartmentofChemistry, CarnegieMellonUniversity,Pittsburgh,PA, UnitedStates
Preface Quantumchemistryisaverymaturefield ofscience—itsmethoddevelopmentstarted asearlyas1926,whentheSchrodinger equationwasproposedand,ayearlater,appliedtosolvingthefirstmolecularsystem H2.Nevertheless,thequickandastonishing progressinanotherdiscipline—artificialintelligenceandmachinelearning—hashada bigimpactonhowweperformquantum chemicalsimulations.Machinelearningin quantumchemistryhasemergedasavibrantandever-changingfieldrichinmany methodologicaladvantagesandhasrapidly becomeacommonlyusedtechniqueincomputationalchemistry.Thiscallsforabook thatcanbeusedasatextbookinadvanced coursesontheoreticalandcomputational chemistryaswellasareferencematerial forspecialistsandnonspecialistsalike.
Teachingmachinelearningmethodsin quantumchemistryhasbecomeamustin theuniversity-levelcoursesontheoretical andcomputationalchemistryasIhavearguedinmyPerspectivearticlewiththesame titleasthisbookinearly2020 [1].Ihave startedtoteachtheverybasicsofmachine learningtopicsmyselfinthespringsemester of2021forundergraduatechemistrystudentsatXiamenUniversity.Mylecturenotes turnedouttobeveryusefulwhencontributingtosomeofthechapters.TheaforementionedPerspectivehasalsoprettymuch formedtheblueprintforthisbook,largely definingthescopeofthetopicscovered, andthereadermayrefertothisPerspective foraveryconcise,bird’s-eyeviewofmachinelearninginquantumchemistry.
Part1ofthisbookisdedicatedtointroducingtheminimumoffundamentaltopicsin bothquantumchemistryandmachinelearning,whichareindispensabletolaythefoundationfortheotherparts;thus,Part1with10 chaptersisthelargestinthebook.Quantum chemistrytopicsstartwithabriefintroductiontoquantumchemistryin Chapter1, followedbydensityfunctionaltheoryin Chapter2,semiempiricalquantummechanicalmethodsin Chapter3,andabriefdiscoursedealingwithquantumchemical calculationsofsystemsfrommoleculesto solid-statematerialsin Chapter4. Chapter5 coversanimportantintroductiontodynamicssimulations.Machinelearningisintroducedin Chapter6 followedbychapters coveringspecifickeytopicssuchas unsupervisedlearningin Chapter7,neural networksin Chapter8,kernelmethodsin Chapter9,andBayesianinferencein Chapter10
Part2isdedicatedtooneofthemost abundantapplicationsofmachinelearning whenitisusedasasurrogateforquantum chemicalpotentials;bypotentialwemean theatomisticsystemtotalenergyasafunctionofnuclearcoordinates. Chapter11 coverspotentialsbasedonlinearmodels followedby Chapters12and13 introducing neuralnetworkpotentialsandkernel methodpotentials,respectively. Chapter14 coverstheimportanttopicofcreatingtrainingdataforsuchpotentialsviaactivelearning. Chapter15 focusesonthechallengesin creatingmachinelearningmodelsfor performingexcited-statedynamics.Machine
learningalsohelpsinvibrationalspectroscopyasdiscussedin Chapter16 andinacceleratingmolecularstructureoptimizations withquantumchemicalmethodsasshown inChapter17.
Part3providesatasteofhowmachine learningcanbeusedtolearnotherimportant quantumchemicalpropertiessuchaselectrondensities(Chapter18),dipolemoments, andpolarizabilities(Chapter19),and excited-stateproperties(Chapter20).
Part4describeshowmachinelearningbecomesanintegralpartofquantumchemical simulationsbyimprovingquantumchemical methods,e.g.,viageneralapproachessuch as Δ-learning(Chapter21,whichalsocovers relatedtopicsoflearningfrommultiplequantumchemicalmethodssuchastransferlearning,co-kriging,hierarchicalmachinelearning, andtheircombinations)ordedicatedapproachesforacceleratingcoupled-cluster andperturbationtheoryquantumchemical methods(Chapter22),redesigningdensity functionaltheorymethods(Chapter23),and improvingsemiempiricalquantumchemical methods(Chapter24).Part4concludes byoverviewinghowmachinelearningcan learntheessenceofquantumchemistry— wavefunction—inChapter25.
Part5concludesthebookwithexamples ofanalysisofbigdatainaquantumchemical context,showingspecificexamplesforanalyzingnonadiabaticmoleculardynamics trajectories(Chapter26)andsearchingfor organicmaterialswithtailoredopticalpropertiesinthewideseaofpossiblecompounds (Chapter27).
Oneofthemainmotivationsofthisbook hasbeentoofferpurelytheoreticalknowledgealongsidepracticalknowledgebyprovidingmathematicalexpositionsinthe Methods sectionandincorporatingcomputationaland,sometimes,pen-and-paperexercisesattheendofeachchapterinthe Case
studies section.Thecasestudiesinmany chaptersrevolvearoundoneofthesimplest chemicalsystems,theH2 molecule,whichis intentionallychosentodemonstratethebasic ideasofvariousmethodstoensurethatthey arewellunderstoodbeforemovingonto morecomplexexamples.Manychapters alsocontainexamplesfrommodernstudies sothatthereadercanbetterunderstand moreadvancedconcepts.Thereaderisalwaysencouragedtogobeyondtheinstructionsinthecasestudiesandexperimenton theirown.
Thisbroadnessofthetopicscouldnot becoveredbyasinglepersonwithina reasonablyshortamountoftimeadequate forsuchafast-pacedfield,especiallyas manycollegeshaveal readyintroduced machinelearning-relatedtopicsforchemistsandwehaveneededatextbookpreferablyyesterday.Thus,thisbookisa productofamassiveeffortby65authors fromverydifferentbackgroundsranging frommathematicstophysics,materials science,andchemistry.Manyoftheauthorsareamongthebestspecialistsin thetopicstheycovered,whichisevident fromthefactthatmanyofthemwere speakersatthefirstInternationalSymposiumonMachineLearninginQuantum ChemistrythatwasheldonlinefromNovember12to14,2021.
Thisisonlythetipoftheiceberg,asmany morepeoplewerepeerreviewingandallof mygroupmembers(LinaZhang,ArifUllah, Bao-XinXue,Yi-FanHou,FuchunGe, PeikunZheng,ShuangZhang,WudiYang, RanWang,LinqiangWei)wereheavilyinvolvedincreatingthisbookbybecoming co-authors,additionalpeerreviewers,and testersforthecasestudies.Thepublisher’s team(AnnekaHess,DevlinPerson,JaiMarie Jose,HowellAngeloM.DeRamos,Paul PrasadChandramohan,IndhumathiMani,
andmanymorebehindthescenes),wereinstrumentaltoaccomplishthisprojectfrom initiationtofinalizingandsupportingitall thewaythrough.Justtogiveashortperspectiveontheamountofeffortputintothis book:fromthesuggestiontowritethebook onMay13,2020(byAnnekaHessfrom Elsevier)untilwritingthisPreface,myemail folderforthisprojectcontainsaround2000 emails;thisisnotincludingallthediscussionsviamessagingappsandmeetingstocoordinateourefforts.Icordiallythankallthe authors,peerreviewers,mygroupmembers,
andpublisher’steamfortheireffortsthat madethisbookpossible.
Happyreadingandlearning!
PavloO.Dral Xiamen,China
Reference
[1] P.O.Dral,Quantumchemistryintheageofmachine learning,J.Phys.Chem.Lett.11(2020)2336–2347.
PART1 Introduction Verybriefintroductiontoquantum chemistry XunWuandPeifengSu TheStateKeyLaboratoryofPhysicalChemistryofSolidSurfaces,FujianProvincialKey LaboratoryofTheoreticalandComputationalChemistry,andCollegeofChemistryandChemical Engineering,XiamenUniversity,Xiamen,Fujian,China
Abstract
Thischapterprovidesabriefintroductiontoquantumchemistryincludingthebasicconceptsandapproximationsofquantummechanics,andaseriesofquantumchemicalmethods.
Keywords: Quantummechanics,Quantumchemistry,Hartree–Fockmethod,Configurationinteraction, Møller–Plessetperturbationtheory,Coupled-clustermethod,Multi-configurationalself-consistent-field, Time-dependentHartree–Fock
Introduction—Thefoundationsofquantumchemistry Quantumchemistry(QC)referstotheapplicationofquantummechanicstostudythe chemicalandphysicalpropertiesofatoms,molecules,andmaterials.Itisgenerallyaccepted thatthefirstQCcalculationwasperformedbytheGermanphysicistsWalterHeitlerandFritz Londononthehydrogen(H2)moleculein1927.Now,QChasbecomeanirreplaceabletoolfor chemiststoexplorefundamentalscientificquestionsinchemistryandrelatedfields, [1–4] withnumerousapplicationstoatomisticsystemssuchasatoms,moleculesorionicsolids. ThischapterintroducesbasicknowledgeofquantummechanicsandQCmethodsforsolving theSchr € odingerequation.
Briefintroductionofquantummechanics Basicconcepts
Thewavefunctionofasystemisamathematicalexpressionthatcontainsinformation aboutanythingthatmaybemeasuredforthissystem.Inquantummechanics,thewave
function Ψ(r1, r2 rn)isafunctionofthepositionsofalltheparticlesinthesystem,describing atime-independentquantummechanicalsystem,inwhich r1, r2 rn arethepositionsof N particles.Thesquareofthewavefunction, j Ψ(r1, r2 rn)j2,representstheprobabilityoffinding N particlesinagivenplace.
Everyobservablephysicalquantity,suchasposition,energy,ormomentum,isdescribed byan operator.AHermitianoperator ^ A satisfies Ð f ∗ ^ Agdτ ¼ Ð g ^ Af ∗ dτ ,where f and g are well-behavedfunctions.Measurementshowsapredictionasaresultofanoperatoracting onawavefunction.Themostimportantoperatoristhetotalenergyoperator ^ H or Hamiltonian. Themeasurementofthe Hamiltonian operatorisdescribedbyaneigen-equationcalledthe time-independentSchrodingerequation.Solutionsforthetime-independentSchrodinger equationonlyexistforcertainvaluesofenergy(energyisquantizedinboundstates).The resultingeigenvalueequationis:
InEq. (1), E istheeigenvalue,whichistheenergyofthesystemdescribedby Ψ(r1, r2 rn). Moreover,thewavefunctionmustgenerallydependontimetoreproduceinformationabout thesystem’sevolutionovertime.The time-dependent wavefunction,expressedas Ψ(r1, r2 rn, t), satisfyingthe time-dependent Schr€ odingerequation:
Born–Oppenheimerapproximation ThesolutionoftheSchrodingerequationformoleculessystemsistheprimarytaskofQC. However,anexactsolutioncannotbeobtainedformulti-electronsystems.Foramultielectronsystem(atomormolecule),the time-independent Hamiltonianinatomicunitsis:
TheHamiltonianinEq. (3) containstheelectronkineticoperators(thefirstterm),nuclear kineticoperators(thesecondterm),nucleus-electronattractivepotentialoperators(thethird term),electron–electronrepulsion(thefourthterm),andnucleus-nucleusrepulsion(thefinal term).TheBorn–Oppenheimer(BO)approximationisintroducedtosimplifytheHamiltonian.IntheBOapproximation,thecouplingbetweenthenuclear r2 andtheelectronicwave functionisneglected.Inmostcases,thecouplingbetweentwoelectronicstatesisnotstrong, i.e.,theenergydifferencebetweentwoelectronicstatesisbigenoughsothatasmalldisplacementofnucleiwillnotchangetheelectronicstate,makingtheBOapproximationvalid,which allowstheseparationofelectrons(lightparticles)fromnuclei(heavyparticles)ontimescales, resultinginaconsiderablesimplification.
Accordingtothisapproximation,thetotalHamiltonianoperatorcanbeexpressedasthe sumofelectronicHamiltonianandnuclearHamiltonian.TheelectronicHamiltonianiswrittenas:
Unlessotherwisespecified,theHamiltonianusedinthischapterreferstotheelectronic HamiltonianofEq. (4).
AccordingtothePauliprinciple,thewavefunctionofamulti-electronatomormolecule shouldbeantisymmetricwithrespecttotheexchangeoftwoelectrons.Antisymmetrization leadstoFermiholes,preventingthecloseapproachoftheelectronswiththesamespinand reducingtherepulsionenergybetweenthem.Thus,themany-electronfunctioncanbeapproximatelywrittenastheantisymmetrizedproductof N one-electronfunctionsintheform ofaSlaterdeterminant(SD):
InEq. (5), ϕ1…ϕN refertospinorbitals.
ThesolutionoftheSchr € odingerequationinvolvescalculationsofallpairwiseinteractions betweenelectrons(seethelasttermofelectronicHamiltonian,Eq. (4))andelectronsandnuclei,whichbecomesanalyticallyintractableformorethanatwo-bodyproblem(onenucleus andoneelectron).ThisleadstothefactthattheSchr € odingerequationcannotbesolvedexactly formulti-electronsystems.Twoapproximatingapproaches,variationalprinciple,andperturbationtheoryarewidelyusedtosolvetheSchr € odingerequationtoextractthenecessaryinformationaboutatomsormoleculesofinterest.
Variationalprinciple GivenasystemwhoseHamiltonianistime-independentandwhoselowest-energyeigenvaluesis E1,if ϕ isanynormalized,well-behavedfunctionofcoordinatesofthesystem’sparticlesthatsatisfiestheboundaryconditions,then
Ifthetrialfunctionhasaminimumofenergywithrespecttovariationalcoefficient c,we canfinditby
Thelowerenergy,thebetterthetrialwavefunction.Andifthetrialfunctionisaccurate enough,thefunctionwegetwillbecloseenoughtothetrueground-statewavefunction. Toimprovethequalityoftrialwavefunction,thelinearvariationalprincipleisapplied, bywhichthetrialwavefunctionisconstructedbyalinearcombinationof n linearlyindependentfunctions(χ 1, χ 2, ⋯ χ n):
where c1 cn arethevariationalcoefficientstobeoptimized.
Basedonthisprinciple,theminimizationroutineregulatestheparameterstoobtainthe wavefunctioncorrespondingtotheminimumenergy,whichisthewavefunctionthatclosely approximatesthegroundstate.
Fromthevariationalprinciple,thevalueofthelowestrootistheupperboundforthesystem’sground-stateenergy.Withthelinearvariationalprinciple,insteadofobtainingthetrial wavefunction,thecombinationcoefficientsofthebasisfunctionsshouldbesolved.
Perturbationtheory Perturbationtheoryprovidesaprocedureforfindingapproximatesolutionstothe Schr € odingerequationforasystemthatdiffersonlyslightlyfromasystemforwhichthe solutionsareknown.Thismethodisfrequentlyemployedtoestimatethefunctionsorvalues basedonpartialknowledgeaboutthesolutionsoftheinvestigatedproblem.
IftheSchrodingerequationfortheHamiltonianoperator ^ H
cannotbesolvedexactly,butweknowtheexactsolutionsoftheHamiltonianoperator
wherethesuperscript(0)denoteszero-order,whichmeansthatthewavefunctionandenergy arealreadyknown,thesubscript n denotesthe n th electronicstate.
Inperturbationtheory,weassumethat ^ H maybeexpandedas:
Theoperator ^ H 0 iscalledtheperturbation.Sincetheoperator ^ H differsonlyslightlyfrom ^ H 0 ðÞ ,theeigenfunctionsandeigenvaluesof ^ H willnotdiffersignificantlyfromthoseofthe unperturbedHamiltonianoperator ^ H 0 ðÞ .
Ifwesuggestthatthewavefunctionandenergycanbeexpressedasthelinearexpansion accordingto λ,whichcanbewrittenas
Herethesuperscript(k)denotestheorderofwavefunctionandenergyaccordingtotheexpansion.Inthelimit λ ! 0,theperturbedsystemreducestotheunperturbedsystem.
First,itisassumedthatthezero-orderwavefunctionisnotdegenerateforsimplicity.By equatingthecoefficientsofthesamepowerof λ,wecanobtainaseriesofequations,from whichwecanobtainthefirst-,second-,andhigher-ordercorrections.Thefirst-orderenergy correctionisanaverageoftheperturbationoperatorovertheunperturbedwavefunction
whilethesecond-orderenergycorrectionscanbeexpressedas:
Thefirst-orderwavefunctioncorrection ψ n (1) isorthogonaltotheunperturbedwave function.
Itcanbefoundthat ψ n (1) canbeexpressedasthelinearcombinationof m th (m ¼ n)electronic states.Thesmallertheenergydifferencebetween m th stateand n th stateis,orthelargerthe couplingterm ψ 0 ðÞ m ^ H 0 ψ 0 ðÞ n DE is,thelargerthecombinationcoefficientofthe m th electronic statewavefunctioncouldbe.
Second,ifthezero-orderwavefunctionisdegenerate,anyarbitrarycombinationofthedegeneratewavefunctionsistheeigenfunctionof ^ H 0 ðÞ.However,introducingtheperturbation operator ^ H 0 willcompletelyorpartiallyremovethedegeneracy.Thus,themostimportant issueistoderivethecombinationcoefficientsofthedegeneratewavefunction.Itiscalled adegenerateperturbativetheory,whichcanbeseenfromthetextbook.
Comparisonofthevariationprincipleandperturbationtheory Theperturbationmethodappliestoalltheboundstatesofasystem.Thevariationtheorem appliestotheloweststateofagivensymmetry.Also,wecanusethelinearvariationmethod totreattheexcitedboundstates.Intheperturbationmethod,onecancalculatetheenergy muchmoreaccurately(toanorderof2 k +1)thanthewavefunction(toanorderof k).The accuracyofperturbationcalculationsdependsonzero-orderHamiltonianandwavefunction, whilethevariationmethodcangetarelativelygoodresultwitharatherinaccuratewave function.
Fundamentalsofquantumchemistry Basedonthefundamentalpostulates,conceptsandapproximationsinquantummechanics,QCisdevotedtoexploringtheelectronicstructureofmulti-electronsystemsbyusingmolecularelectronicstructuremethodsthatsolvethemolecularSchr € odingerequationassociated withdifferentkindsofmolecularHamiltonian.Methodsthatdonotincludeanyempiricalor semiempiricalparametersarecalledabinitiomethods.Thisdoesnotindicatethattheirsolutionsareexact;theyareallapproximate.Itmeansthataparticularapproximationisrigorouslydefinedonfirstprinciples(quantumtheory)andthensolvedwithinanerrormargin thatisqualitativelyknownbeforehand.
