Many-Body Methods for Atoms, Molecules and Clusters

330 Pages·2018·7.68 MB·English

by  Jochen Schirmer

Lecture Notes in Chemistry 94
Jochen Schirmer
Many-Body
Methods
for Atoms,
Molecules and
Clusters
Lecture Notes in Chemistry
Volume 94
Series editors
Barry Carpenter, Cardiff, UK
Paola Ceroni, Bologna, Italy
Barbara Kirchner, Institut für Physikalische und Theo, Leipzig, Germany
KatharinaLandfester,Max-Planck-InstitutfürPolymerforschung,Mainz,Germany
Jerzy Leszczynski, Department of Chemistry and Biochemistry, Jackson State
University, Jackson, MS, USA
Tien-Yau Luh, Department of Chemistry, National Taiwan University, Taipei,
Taiwan
Eva Perlt, Institute for Physical and Theoretical Chemistry, University of Bonn,
Mulliken Center for Theoretical Chemistry, Bonn, Germany
NicolasC.Polfer,DepartmentofChemistry,UniversityofFlorida,Gainesville,FL,
USA
Reiner Salzer, Dresden, Germany
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Jochen Schirmer
Many-Body Methods
for Atoms, Molecules
and Clusters
123
JochenSchirmer
Institute of Physical Chemistry
Heidelberg University
Heidelberg, Germany
ISSN 0342-4901 ISSN 2192-6603 (electronic)
Lecture Notesin Chemistry
ISBN978-3-319-93601-7 ISBN978-3-319-93602-4 (eBook)
https://doi.org/10.1007/978-3-319-93602-4
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Preface
The present book is based on a one-year course of lectures given intermittently
during the years from 1990 to 2010 at the University of Heidelberg. The lectures
weredevisedasanin-depthintroductionintomany-bodytheoryforfiniteelectronic
systems, that is, molecules, atoms, and clusters, addressing graduate, doctoral, and
postdoctoralstudents,whoweregenerallyinterestedinquantum-chemicalmethods
and computations. The original course is essentially covered by the first 10 chap-
ters, while 7 additional chapters address further elaborations and extensions.
Many-body methods, or more accurately, field-theoretical many-body methods,
have originated in quantum field theory where they were developed as a means to
treatthephysicsofelementaryparticles.Aswassoonrealized,thesemethodscould
be transferred to the treatment of quantum many-body systems in solid-state phy-
sicsandstatistics,notconveyingnovelphysicsherebutsupplyingapowerfulnew
formalismandaroutetowardalternativecomputationalmethods.Shortlyafterward,
this formalism was taken up in the treatment of finite particle systems, first in
nuclear physics, and finally in quantum chemistry as well. It is now almost half a
century since computational schemes based on field-theoretical many-body theory
were developed and successfully applied to finite electronic systems.
In the field-theoretical approach, the many-particle problem is formulated in
termsofmany-bodyGreen’sfunctionsorpropagators.Theseentitiesaredefinedas
ground-state expectation values of time-dependent operator products, which, in
energyrepresentation,takeontheformofmatrixelementsofmany-bodyresolvent
operators.Theyallowfor adirectaccess totheenergiesandtransitionmomentsof
generalized excitation processes in the considered system, such as ionization
(electronremoval),affinities(electronattachment),andneutralelectronicexcitation.
What is the advantage inherent to these methods and what can they actually do
better than the conventional procedures in dealing with small, medium size, and
large molecules? An apparent advantage is a direct access to physically relevant
quantitiessuchasexcitationenergiesandtransitionmoments,whichotherwisehave
to be assembled from independent computations for the initial ground and final
excited states. But there is another, deeper justification, related to characteristic
shortcomings in the conventional approach.
v
vi Preface
In the conventional quantum-theoretical treatment of finite many-electron sys-
tems,therearetwobasictools:firstly,perturbationtheory(PT),which,however,as
acomputationalschemeappliesonlytotheN-electrongroundstate;and,secondly,
the standard numerical procedure of solving the time-independent Schrödinger
equation, that is, using suitable basis set expansions for the states of interest and
transforming the Schrödinger equation into the secular problem of the corre-
sponding matrix representation of the hamiltonian. The general problem of con-
figuration interaction (CI), as the standard procedure is referred to in quantum
chemistry, is the exponentially increasing dimension of the secular matrix,
(cid:1) (cid:3)
d ¼ M ,bothwiththesizeofthesystems,reflectedinthenumberofelectrons,N,
N
and the demand for accuracy, reflected in the number M of one-particle states
underlyingthemany-electronbasisstates(CIconfigurations).Thismeansthatafull
CI treatment is not viable except for very small systems and limited one-particle
basissets,andonehastoresorttoapproximateCIschemes obtained bytruncating
the configuration manifold in suitable ways.
Here, however, an unsuspected problem arises which disqualifies the CI as a
meansoftreatingextendedelectronsystems.IntheCIsecularequations,thereisan
interaction (mixing) of configurations that differ exactly by a double excitation,
such as in a singly (S) excited configuration (relative to the reference state) and a
triply (T) excited configuration comprising the former single excitation.
A corresponding S-T secular matrix element is potentially “non-local”; that is, its
magnitude does not decrease or vanish when the involved single and double
excitations can be assigned to distant parts of the system or even to separate
fragmentsofacompositesystem.IntruncatedCIexpansions,thepresenceofthese
potentially non-local admixtures causes an uncontrollable error which grows with
the spatial extension of the system and, accordingly, is referred to as
size-consistency error.
The propagator methods, by contrast, do not suffer from this deficiency. As a
commonfeature,approximationschemesderivingfromfield-theoreticalmany-body
theory combine perturbation expansions (of the ground-state type) and eigenvalue
algebra within a generalized secular problem where in particular any potentially
non-localcouplingcontributionsaretakencareofinthePTpart.Asaconsequence,
the propagator methods are inherently size-consistent and, moreover, more eco-
nomical,requiringdistinctlysmaller explicitconfigurationmanifoldsinthesecular
problem than in CI expansions of comparable accuracy.
Abriefguidetothetourthroughthefivepartsofthisbookmaybehelpful.The
first two chapters of Part I lay the groundwork for the quantum theory of
many-electron systems, addressing states, operators, the evaluation of matrix ele-
ments, and, finally, the use of second quantization. Thereupon, the prototypical
one-particle Green’s function or electron propagator is presented and discussed in
Chap. 3.
Preface vii
InthefourchaptersofPartII,theformalismofdiagrammaticperturbationtheory
is developed, based on three central theorems, the Gell-Mann and Low theorem,
Wick’s theorem, and the linked-cluster theorem. At the end of that part, the reader
should be able to draw and evaluate Feynman diagrams.
However,thediagrammaticartsdonotyetestablishaproceduretocomputethe
electron propagator or thephysical information conveyed therein.So with Chap. 8
inPartIII,thefocusshiftstotheissueofdevelopingcomputationalschemes.Here,
theprominentstartingpointistheDysonequation,relatingtheelectronpropagator
to the so-called self-energy. The latter quantity is itself subject to a diagrammatic
perturbation expansion, where the diagrams are simpler than those for the electron
propagator. The subject of Chap. 9 is the algebraic–diagrammatic construction
(ADC), a general procedure to generate systematic higher-order approximations
(ADC(n) schemes) to the self-energy, being consistent through order n, and, cru-
cially, reproducing the correct analytical structure of the self-energy. The ADC
procedure is quite versatile and can directly be applied to the electron propagator,
or, more accurately, to its ðN(cid:2)1Þ-electron parts, as is demonstrated in Chap. 10.
Then, in Chaps. 11 and 12, our tour takes a remarkable turn: The direct ADC
approximations can be derived via a radically different route, namely a
wave-function-based approach referred to as intermediate state representation
(ISR). (An impetuous reader, already familiar with the topics of Chaps. 1 and 2,
might take a shortcut directly to Chaps. 11 and 12). The ISR concept bridges the
gap between propagator and wave-function methods, lifts certain limitations
inherent to the diagrammatic propagator approach, and allows for a rigorous
foundation (Chap. 12) of the defining many-body features.
In Part IV, we turn toward the physics of N-electron excitations and the
polarization propagator relevant here. Chapter 13 discusses how diagrammatic
perturbation theory can be adapted to the polarization propagator. The ADC and
ISR concepts for N-electron excitations are presented in Chap. 14, while Chap. 15
reviews the prominent random-phase approximation (RPA), being a paradigmatic
modelinmany-bodytheory.ThefinalpartVtakesalookattworelatedapproaches,
which may be seen as ISR variants: The equation-of-motion (EOM) methods
(Chap. 16) and methods based on the coupled-cluster (CC) ansatz (Chap. 17).
Altogether 9 appendices supplement the main text: Appendix A.1 reviews
many-body perturbation theory and recollects some useful algebraic techniques;
some more lengthy proofs are deferred to Appendices A.2, A.3, A.4, and A.6;
extensions to Chaps. 8, 13, and 16 are given in Appendices A.5, A.7, and A.8,
respectively; the final Appendix A.9 compiles various explicit ADC expressions.
As may be permissible in a textbook, perhaps even advisable, the bibliography
has been kept relatively short and selective. In topics that are well documented in
the literature, only a few key papers or books are quoted. More comprehensive
reference is made to subjects or issues that are less familiar or amenable. And, of
course,Ihavetriedtoindicatethesourceswhereverthetextdrawsuponexemplary
previous presentations.
viii Preface
Many persons have contributed in various ways to the genesis of this book.
Foremost, it should be gratefully acknowledged that the students attending the
original lectures did, to paraphrase J. A. Wheeler, a great job in educating their
lecturer.AndparticularthanksgotoVitaliAverbukh,oneofthestudentsthen,now
holdingafacultypositionattheImperialCollegeLondon,whoencouraged—well,
more accurately—pressured me to make those lectures publicly available in the
form of a book. I am greatly indebted to Lukas Wirz for unremitting technical
assistanceduringtheentireproject,fortranslatingthebulkofhandwrittenformulae
intoLaTex,andfordevisingnumerousdiagramsandfigures.Overtime,quiteafew
people participated in the critical reading of selected chapters, contributing helpful
suggestionsandcomments.Iwouldliketothankallofthem,and,inparticularand
two for all, Tsveta Miteva, who intrepidly and skillfully shouldered the burden of
correcting the very first version of each chapter, and Bridgette Cooper for diligent
revisions in the earlier stages of the project. Sincere thanks are due to my former
collaborators Alexander B. Trofimov and Frank Mertins, who, perhaps in a less
direct but all the more profound way, have shaped various themes of the present
book.Andfinally,IwouldliketoexpressmygratitudetoLorenzS.Cederbaumfor
the initiation and longtime common commitment to the challenges of many-body
theory.
Heidelberg, Germany Jochen Schirmer
Contents
Part I Many-Electron Systems and the Electron Propagator
1 Systems of Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Many-Electron Wave Functions. . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Matrix Elements for Many-Electron States . . . . . . . . . . . . . . . . 9
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Definition of Creation and Destruction Operators . . . . . . . . . . . 19
2.2 Anticommutation Relations for Creation and Destruction
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Operators in Second Quantization . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Combining Second Quantization and Slater–Condon Rules . . . . 26
2.5 Change of the One-Particle Representation. . . . . . . . . . . . . . . . 27
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 One-Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Definition and Relation to Physical Quantities . . . . . . . . . . . . . 31
3.2 Ground-State Expectation Values. . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Ground-State Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Free One-Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Part II Formalism of Diagrammatic Perturbation Theory
4 Perturbation Theory for the Electron Propagator. . . . . . . . . . . . . . 45
4.1 Time-Development Operator in the Interaction Picture . . . . . . . 46
4.2 The Gell-Mann and Low Theorem. . . . . . . . . . . . . . . . . . . . . . 50
4.3 Expectation Values of Heisenberg Operators . . . . . . . . . . . . . . 53
4.4 Comparison with Rayleigh–Schrödinger Perturbation Theory. . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
ix