Elementary Molecular Quantum Mechanics: Mathematical Methods and Applications

927 Pages·2013·8.99 MB·English

by  Valerio Magnasco

Elementary Molecular
Quantum Mechanics:
Mathematical Methods
and Applications
Second Edition
Valerio Magnasco
University of Genoa, Genoa, Italy
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1314151617 10987654321
ISBN:978-0-444-62647-9
To Charles Alfred Coulson
To Paola, my wife
To my former Quantum Chemistry students
Giorgio Aliani, Mauro Amelio, Luca Arcesi, Paolo Armanino, Cristina Artini,
Roberto Austi, Fiorenza Azzurri, Monica Beggiato, Walter Bellini, Angelo Benazzo,
Maurizio Benzo, Anna Berti, Luca Berti, Federico Bianchi, Anja Bijedic Mitranic,
Sergio Bisio, Claudio Boffito, Vito Boido, Gabriella Borzone, Carlo Bottero, Andrea
Briatore, Stefano Brusco, Santino Bruzzone, Giuseppe Burgisi, Vincenzo Buscaglia,
Fabio Canepa, Maria Rosaria Carcassi, Francesca Carosi, Silvio Carrettin, Angela
Cartosio, Massimo Casanova, Danilo Cassaglia, Rosana Cavalli, Marino Cervetto,
RobertaCesari,SilviaCiuffini,FabrizioColace,BiancaCosma,CamillaCosta,Aldo
Curti, Marco Delfino, Giovanna Dellepiane, Fabrizio De Melas, Serena De Negri,
Maria Grazia Desirello, Silvia De Vito, Ilaria Devoti, Daniele Duce, Roberto
Eggenhoffner, Leandro Esposito, Sandro Ferraro, Alessandra Ferretti, Dino Romano
Ferro, Giuseppe Figari, Maria Luisa Fornasini, Giuseppe Francese, Ambra Frare,
Silvano Fuso, Massimo Gambetta, Silvia Garbarino, Paolo Gatti, Chiara Ghiron,
Michele Giardina, Claudia Grandi, Paola Guarnone, Marina Guenza, Roberto La
Ferla, Marco Leoncini, Serena Leporatti, Alessandra Leprini, Franco Lichene, Nara
Liessi, Andrea Lionello, Romeo Losso, Mattia Lucchini, Costanzo Luzzago, Filippo
Maido, Alberto Martinelli, Roberta Martolini, Bruno Matteazzi, Franco Merlo,
Valeria Merlo, Giovan Battista Minetto, Tomaso Munari, Riccardo Musenich, Gian
Franco Musso, Riccardo Narizzano, Marco Nencioni, Sergio Novelli, Riccardo
OlivieriColombo,MassimoOttonelli,DonatellaPaci,SilviaPalacioAzcona,Andrea
Palenzona, Christopoulos Panagiotis, Laura Pardo, Paolo Parodi, Lise Pascale,
Stefano Passalacqua, Elisa Pastore, Marta Patrone, Gianfranco Patrucco, Enrica
Perrotta,RobertoPeverati,MatteoPiccardo,GiulioPieretti,FrancescaPin,Arnaldo
Rapallo, Marcello Rasparini, Anna Ravina, Renzo Rizzo, Caterina Rocca, Giuseppe
Roncallo, Massimiliano Rossi, Marina Rui, Paolo Russo, Paolo Sacco, Carlo
Scapolla, Daniele Scavo, Giovanna Schmid,Andrea Sciutto, Andrea Siciliano, Lucia
Somma, Giuseppe Tassano, Anton Thumiger, Giuseppe Tonello, Giulio Andrea Tozzi,
Riccardo Tubino, Alessandro Valente, Vito Vece, Massimo Vizza, Gilberto Vlaich,
Ugo Zanelli, Maria Teresa Zannetti
Preface
The idea of a book presenting in a simple way most of the essential mathematical tools beyond
elementarycalculus,andpossiblyentitledMathematicalMethodsinMolecularQuantumMechanics,
grew for some time in the author’s mind during the many years of teaching and research at the
UniversityofGenoa.In2007,Elsevierpublishedateachingbookbytheauthorforgraduateuniversity
studentsofchemistryandphysicsentitledElementaryMethodsofMolecularQuantumMechanics,the
book being intended as a bridge between the classic elementary Coulson’s Valence and the more
advancedMcWeeny’sMethodsofMolecularQuantumMechanics.After5years,thisbooknecessarily
needed some revisionand restyling toeliminatevarious misprints.
After an intense correspondence with the Elsevier’s editorial board, to avoid the narrowness of
asinglemathematicaltext,itwasdecidedtomergethetwobooksintothepresentsinglebook,entitled
ElementaryMolecularQuantumMechanics:MathematicalMethodsandApplications.Thenewbook
ismostlybasedonthecontentoftheoldone,ofwhichitmaintainstheinterestingelementaryapproach
toquantumchemicalapplications,andcontainsanewintroductorypartmorespecificallydevotedto
mathematicalmethods.So,thenewbookcanbeconsideredasadeeplyrevisededitionoftheformer
Elsevier’sbookwithasensiblyenlargedmathematicalpart.Itishopedthat,inthisway,thenewbook
can meet better the students’ needs, enlarging its usability even for university students of applied
maths, at least in their first years.
Thefirstpartofthebookisnotintendedtocoverallmathematicaltopics,ratheritisintendedto
offer the reader in a plane way and more in detail many of the building stones of the advanced
mathematicsneededinworkingapplicationsofQuantumMechanicstoQuantumChemistry.Forthe
same reason, some important topics are omitted, such as functional, numerical, and computational
analysis, but these arguments are so wide that they would imply separate books in themselves. For
them, the reader should be addressed to specific treatises or to internet resources. The required
background for students of physics and chemistry is still that of a few semester courses in mathe-
matical analysis and physics.
The book might help as an auxiliary advanced mathematical and physical text of about 60 h for
universitystudents ofchemistry, physics,and applied maths.
ThemaindifferenceswiththeformerElsevier’sbookarethefollowing.(1)Theshiftingtothenew
Chapter 1 of both the fundamental approximation methods of Quantum Mechanics, the variational
methodandthevariousperturbativeapproaches(formerlypartlygiveninChapters5and11),(2)the
introduction of three new mathematical chapters devoted to the solution of the partial differential
equationsofQuantumMechanics(Chapter3),tothetheoryofspecialfunctions,includingFourierand
LaplacetransformsandGreen’sfunctions(Chapter4),tothefunctionsofacomplexvariableandthe
evaluationofintegralsofarealvariablebythemethodofresidues(Chapter5),(3)thesubdivisionof
the former unusually large Chapter 7 into the two new Chapters 14 and 16, the first devoted to the
theoryofmodelHamiltonians (Hartree–Fock, Hu¨ckel andapproximate MO methods),the second to
the correlation problem and to the Post-Hartree–Fock methods, (4) the accurate revision of the old
Chapter12onatomicandmolecularinteractions,whichwascompletelyre-writtenandupdatedinthe
newChapter17,(5)theadditionoftheGaussianGTOapproachbesidestheprevalentSTOapproachin
theevaluationofmulticentremolecularintegralsinthenewChapter18,(6)theadditionofChapter19,
xix
xx Preface
which introduces the reader to the still actually growing field of relativistic molecular quantum
mechanics,startingfromtheDirac’sequationfortherelativisticelectronuptoitsmajorapplicationsto
atomicandmolecularsystems,and,lastly,(7)ashortoutlooktotheproblemofmolecularvibrations
(Chapter 20) with particular emphasis to the vibronic interactions, a topic which was completely
omitted in the former book. The original feature of the former book which completed each chapter
with problems and solved problems was maintainedand enlargedinthe new book.
ThecontentsofthenewbookaregivenintheintroductorysectionContents,whereallitemsineach
chapter are specifiedindetail, sothat they will be omitted here.
Awide list of alphabetically ordered references, mostly taken from original research papers, and
authorand subjectindices complete the book.
Thanks are due to my son Mario and his daughter Laura, who prepared the drawings at the
computer, to Professor Giuseppe Figari, who helped much during the many years of research at the
DepartmentofChemistryoftheUniversityofGenoa,toDrMicheleBattezzatiforusefulsuggestions
and discussions on specific points, to Dr Tomas Marti(cid:1)sius of Vilnius (Lithuania), who provided the
softwarefortranslatingfromLatextoWord,whichmadethisworkpossible,andtoDrCamillaCosta,
who helped in the translation of the former published text from Latex to Word. Particularly warm
thanks are due to the Elsevier’ Senior Acquisition Editor, Mr Adrian Shell, PhD, for his patience in
discussing the best structure of the new book, and to Dr Egbert van Wezenbeek for his constant
encouragementandsupport.Finally,IacknowledgetheItalianMinistryforEducationUniversityand
Research (MIUR) and the University of Genoa for their financial support during the many years of
scientific research ofthe Genoa group.
MylastthanksareformywifePaola,whosupportedmestronglyduringthelongtimethiswork
was inprogress.
Valerio Magnasco
Genoa,30th November2012
CHAPTER
1
Mathematical foundations
and approximation methods
CHAPTER OUTLINE
1.1 MathematicalFoundations.................................................................................................................4
1.1.1 Regularfunctions..........................................................................................................4
1.1.2 Schmidtorthogonalization..............................................................................................5
1.1.3 Lo¨wdinorthogonalization................................................................................................8
1.1.4 Setoforthonormalfunctionsandbasisset......................................................................8
1.1.5 Linearoperators............................................................................................................9
1.1.6 Hermitianoperators.....................................................................................................10
1.1.7 Expansiontheorem......................................................................................................14
1.1.8 Basicprinciplesofquantummechanics........................................................................17
1.2 TheVariationalMethod...................................................................................................................19
1.2.1 Non-linearparameters.................................................................................................21
1.2.2 Linearparameters:theRitzmethod..............................................................................21
1.3 PerturbativeMethodsforStationaryStates......................................................................................25
1.3.1 RSperturbationtheory.................................................................................................25
1.3.2 Second-orderapproximationmethodsinRSperturbationtheory......................................32
1.3.2.1 TheUnso¨ldapproximation......................................................................................32
1.3.2.2 TheHylleraasmethod.............................................................................................33
1.3.2.3 TheKirkwoodmethod.............................................................................................34
1.3.2.4 TheRitzmethodforE~ :linearpseudostates............................................................35
2
1.3.3 BWperturbationtheory................................................................................................37
1.3.4 PerturbationmethodswithoutpartitioningoftheHamiltonian.........................................40
1.3.4.1 LSperturbationtheory............................................................................................41
1.3.4.2 ENperturbationtheory...........................................................................................42
1.3.5 Perturbationtheoriesincludingexchange(SAPTs)..........................................................45
1.3.6 Themomentmethod....................................................................................................53
1.4 The Wentzel–Kramers–BrillouinMethod..........................................................................................55
1.5 Problems1....................................................................................................................................59
1.6 SolvedProblems............................................................................................................................62
In this chapter, after a short review of the mathematical and physical foundations of quantum
mechanics, we shall be mostly concerned with the approximation methods on which all quantum
chemistryapplicationsarebased.ThevariationalmethodduetoRayleighwillbeexaminedfirst,next
3
ElementaryMolecularQuantumMechanics2E.http://dx.doi.org/10.1016/B978-0-444-62647-9.00001-4
(cid:1)2013,2007ElsevierB.V.Allrightsreserved
4 CHAPTER 1 Mathematical foundations and approximation methods
different perturbative methods for stationary states, starting from the Rayleigh–Schroedinger (RS)
perturbation theory, including some efficient methods of treating second-order quantities, up to the
Brillouin–Wigner(BW)andEpstein–Nesbet(EN)theories,thesymmetry-adaptedtheoriesincluding
exchange (SAPT), and the recently developed moment method. A glance at the Wentzel–Kramers–
Brillouin (WKB)method concludes this introductorychapter.
1.1 MATHEMATICAL FOUNDATIONS
We begin by giving a few mathematical definitions. We denote by x either a single independent
variable or a collective for many variables. Vectors and arguments of functions of more than one
variable will be written in bold type: thus, we use r to denote a three-dimensional vector, say
r¼ixþjyþkz in Cartesian coordinates, or the three variables (x,y,z) of a function f(x,y,z)¼f(r).
Generalized coordinates and transformation from Cartesian to other coordinate systems will be
examinedinChapter 2.
1.1.1 Regular functions
In what follows, we shall be concerned only with regular functions of the general variable x. A
function j(x), defined in a given domain D, is said regular if it satisfies the three constraints of
being there: (1) single valued, (2) piecewise continuous with its first derivatives, and (3)
quadratically integrable, i.e. it must be zero at infinity. So, not all functions that can exist
mathematically are physically acceptable. Such functions are usually obtained, at least in prin-
ciple, as solutions of differential equations of the second order like those examined in Chapter 3.
According to Dirac, such functions can be treated as vectors having infinite components in an
abstract linear space. Function j(x) can be complex, its complex conjugate being denoted by
j*(x). Complex numbers and functions of a complex variable will be treated to some extent in
Chapter 5.
Weoften use the Dirac1notation:
(cid:1)
Function jðxÞ¼jji 0 ket
(1)
Complex conjugate j(cid:2)ðxÞ¼hjj 0 bra
TheDirac jji vectorshaveall thewell-knownproperties of alinear vector space.
Thescalar(orinner)product(analogybetweenregularfunctionsandcomplexvectorsofinfinite
dimensions)of j*byj can then bewritten inthe bra-ket (‘bracket’) form:
Z
dxj(cid:2)ðxÞjðxÞ¼hjjji¼finite number>0 (2)
1DiracPaulAdrienMaurice1902–1984,Englishphysicist,ProfessorofmathematicsattheUniversityofCambridge(UK);
1933NobelPrizeforPhysics.
1.1 Mathematical foundations 5
If:
hjjji¼Nj (3)
wesaythatfunctionj(x)(theketjji)isnormalizedtoNj(thenormofj).Thefunctionjcanthenbe
normalized to1 by multiplyingit bythe normalization factor N ¼ðNjÞ(cid:3)1=2.
Thescalar productof two regularfunctions jand 4 satisfiesthe Schwarzinequality:
hjj4i2 (cid:4)hjjjih4j4i (4)
wherethe equality sign holds if andonly ifj and 4 are proportional.
If:
Z
hjj4i¼ dxj(cid:2)ðxÞ4ðxÞ¼0 (5)
we say that 4is orthogonalðtÞ toj. If:
hj0j40i¼Sðs0Þ (6)
40andj0arenotorthogonal,butcanbeorthogonalizedineitheroftwoways.Theprimednotationwill
bemostlyusedfornon-orthogonalfunctionsandtheunprimednotationfortheorthogonalset.Toput
inevidencethatwehavetwofunctions,itisconvenienttodenotej0byc0 and40byc0;jbyc and4
1 2 1
by c .
2
1.1.2 Schmidt orthogonalization
Thefirstorthogonalizationmethodistheso-calledunsymmetricalorthogonalizationorGram–Schmidt
orthogonalization,inshortSchmidtorthogonalization.Itconsistsinchoosingthelinearcombination:
(cid:3) (cid:4)
c ¼c0; c ¼N c0 (cid:3)Sc0 ; hc jc i¼0 (7)
1 1 2 2 1 1 2
where N ¼ð1(cid:3)S2Þ(cid:3)1=2 is the normalization factor. In fact, it is easily seen that if c0 and c0 are
1 2
normalized to1:
(cid:5) (cid:6) (cid:7) (cid:3) (cid:4)
hc jc i¼N c0(cid:6)c0 (cid:3)Sc0 ¼N S(cid:3)S ¼0 (8)
1 2 1 2 1
Thegeneral Schmidtorthogonalization processfor two functionsisconstructed as follows. If:
(cid:5) (cid:6) (cid:7) (cid:5) (cid:6) (cid:7) (cid:5) (cid:6) (cid:7) (cid:5) (cid:6) (cid:7)
c0(cid:6)c0 ¼ c0(cid:6)c0 ¼1; c0(cid:6)c0 ¼ c0(cid:6)c0 ¼Sðs0Þ (9)
1 1 2 2 1 2 2 1
we take the newunprimed set:
(
c ¼c0
1 1
(10)
c ¼Ac0 þBc0
2 2 1
whereweimposeonc theconditionsoforthogonalityandnormalization(inshort,theorthonormality
2
conditions). Then:
8
(cid:5) (cid:6) (cid:7)
<hc jc i¼ Ac0 þBc0(cid:6)c0 ¼ASþB¼0
2 1 (cid:5) 2 1(cid:6) 1 (cid:7) (11)
:hc jc i¼ Ac0 þBc0(cid:6)Ac0 þBc0 ¼A2þB2þ2ABS¼1
2 2 2 1 2 1
6 CHAPTER 1 Mathematical foundations and approximation methods
Weobtainthe system:
(cid:1)
ASþB¼0
(12)
A2þB2þ2ABS¼1
which issolvedinterms ofthe ratio (B/A):
8
>>><B¼(cid:3)S As0
A
(cid:8) (cid:9) (cid:10) (cid:9) (cid:10)(cid:11) (13)
>>>:A2 1þ B 2þ2S B ¼A2(cid:3)1(cid:3)S2(cid:4)¼1
A A
sothat:
(cid:3) (cid:4) (cid:3) (cid:4)
A¼ 1(cid:3)S2 (cid:3)1=2; B¼(cid:3)S 1(cid:3)S2 (cid:3)1=2 (14)
andthe Schmidt orthogonalized two-set will be:
8
><c1 ¼c01
>:c ¼cp02ffiffi(cid:3)ffiffiffiffiffiSffifficffiffiffi01ffi (15)
2
1(cid:3)S2
UsingthematrixformofChapter6,wecanwriteSchmidtorthogonalizationasthetransformation
between the two sets offunctions describedby (1 (cid:5)2)row vectors:
(cid:3) (cid:4) !
(cid:3) (cid:4) 1 (cid:3)S 1(cid:3)S2 (cid:3)1=2 (cid:3) (cid:4)
ðc c Þ¼ c0 c0 (cid:3) (cid:4) ¼ c0 c0 O (16)
1 2 1 2 0 1(cid:3)S2 (cid:3)1=2 1 2 S
where: (cid:3) (cid:4) !
1 (cid:3)S 1(cid:3)S2 (cid:3)1=2
O ¼ (cid:3) (cid:4) (17)
S 0 1(cid:3)S2 (cid:3)1=2
isthe unsymmetrical (2(cid:5)2) matrix doing Schmidt’s orthonormalization.
For a set of three normalized but non-orthogonal functions ðc0 c0 c0 Þ having non-orthogo-
1 2 3
nalitiesS ,S ,S ,respectively,itisbettertoorthogonalizeinafirststepthesecondfunctiontothe
12 13 23
first one, next the third function to the resulting orthogonalized ones. In terms of the original set,
Schmidt’sorthogonalization isgiveninthis case by the (3(cid:5) 3)transformation:
(cid:3) (cid:4)
ðc c c Þ¼ c0 c0 c0 O (18)
1 2 3 1 2 3 S
where:
0 1
1 (cid:3)N S (cid:3)N ðS (cid:3)S S Þ
2 21 3 31 32 21
B C
B C
OS ¼B@0 N2 (cid:3)N3ðS32(cid:3)S31S21ÞCA
(cid:3) (cid:4)
0 0 N 1(cid:3)S2
3 21
(cid:5) (cid:6) (cid:7)
Skl ¼ c0k(cid:6)c0l k;l¼1;2;3
(cid:3) (cid:4) (cid:13)(cid:3) (cid:4)(cid:3) (cid:4)(cid:14)
N ¼ 1(cid:3)S2 (cid:3)1=2; N ¼ 1(cid:3)S2 1(cid:3)S2 (cid:3)S2 (cid:3)S2 þ2S S S (cid:3)1=2 (19)
2 21 3 21 32 31 21 32 31 21