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BRILLOUIN– WIGNER METHODS FOR MANY-BODY SYSTEMS

Progress in Theoretical Chemistry and Physics
Editors-in-Chief:
Editorial Board:
Former Editors and Editorial Board Members:
K. Hirao (University of Tokyo, Japan)
R. Lefebvre (Université Pierre-et-Marie-Curie, Paris, France)
R. McWeeny (Università di Pisa, Italy)
M.A.C. Nascimento (Instituto de Química, Rio de Janeiro, Brazil)
P. Piecuch (Michigan State University, East Lansing, MI, U.S.A.)
S.D. Schwartz (Yeshiva University, Bronx, NY, U.S.A.)
A. Wang (University of British Columbia, Vancouver, BC, Canada)
R.G. Woolley (Nottingham Trent University, U.K.)
Y.G. Smeyers
R. Daudel
D. Avnir
J. Cioslowski
M.P. Levy
G.L. Malli
P.G. Mezey
N. Rahman
S. Suhai
O. Tapia
P.R. Taylor W.F. van Gunsteren
)
J. Rychlewski ()
()
()
()*
() *
() *
() *
() *
() *
() *
() *
() *
() *
R. Levine (Hebrew University of Jerusalem, Israel)

I. Prigogine (
*: deceased;
Honorary Editors:

K. Lindenberg (University of California at San Diego, CA, U.S.A.)
( )H. Ågren *
() *
G. Delgado-Barrio (Instituto de Matemáticas y Física Fundamental, Madrid, Spain)
W.N. Lipscomb (Harvard University, Cambridge, MA, U.S.A.)
E. Brändas (University of Uppsala, Sweden)
V. Aquilanti (Università di Perugia, Italy)
Yves Chauvin (Institut Français du Pétrole, Tours, France )
M. Mateev (Bulgarian Academy of Sciences and University of Sofia, Bulgaria)
: end of term
ˆ
E.K.U. Gross (Freie Universität, Berlin, Germany)
J. Maruani ( formerly Laboratoire de Chimie Physique, Paris, France)
S. Wilson (formerly Rutherford Appleton Laboratory, Oxfordshire, U.K.)
L. Cederbaum (Physikalisch-Chemisches Institut, Heidelberg, Germany)
For other titles published in this series go to www.springer.com/series/6464
I. Hubac
VOLUME 21

123

Comenius University
Bratislava
Slovakia
STEPHEN WILSON
CIVAN HUBA
Brillouin–Wigner Methods
for Many-Body Systems
and
Silesian University
Opava
Czech Republic
University of Oxford
UK
and
Comenius University
Bratislava
Slovakia

Ivan Hubac
Comenius University
Laboratory
United Kingdom
[email protected]
Springer Dordrecht Heidelberg London New York
c
No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by
any means, electronic, mechanical, photocopying, microﬁlming, recording or otherwise, without written
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Springer Science+Business Media B.V. 2010
Stephen Wilson
Physical & Theoretical ChemistryFaculty of Mathematics, Physics
Mlynska Dolina F 1
842 48 Bratislava
and Informatics
University of Oxford
South Parks Road
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Institute of Physics
Silesian University
[email protected]
74601 Opava
Czech Republic
Theoretical Chemistry Group
Library of Congress Control Number: 2009940111

PROGRESS IN THEORETICAL CHEMISTRY
AND PHYSICS
A series reporting advances in theoretical molecular and material sciences, including
theoretical, mathematical and computational chemistry, physical chemistry and
chemical physics
Aim and Scope
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The basic theories of physics – classical mechanics and electromagnetism,
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vi Progress in Theoretical Chemistry and Physics
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which involves the investigation of structure–property relationships aiming at the
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or genetic engineering. Relevant properties include conductivity (normal, semi- and
supra-), magnetism (ferro- or ferri-), optoelectronic effects (involving nonlinear
response), photochromism and photoreactivity, radiation and thermal resistance,
molecular recognition and information processing, and biological and pharmaceutical
activities; as well as properties favouring self-assembling mechanisms, and combina-
tion properties needed in multifunctional systems.
Progress in Theoretical Chemistry and Physics is made at different rates in these
various research ﬁelds. The aim of this book series is to provide timely and in-depth
coverage of selected topics and broad-ranging yet detailed analysis of contemporary
theories and their applications. The series will be of primary interest to those whose
research is directly concerned with the development and application of theoretical
approaches in the chemical sciences. It willprovide up-to-date reports on theoretical
methods for the chemist, thermodynamician or spectroscopist, the atomic, molec-
ular or cluster physicist, and the biochemist or molecular biologist who wishes to
employ techniques developed in theoretical, mathematical or computational chem-
istry in their research programmes. It is also intended to provide the graduate student
with a readily accessible documentation on various branches of theoretical chemistry,
physical chemistry and chemical physics.

This book is dedicated
to the late J. M´aˇsik
and to our children,
Michelle, Jonathan & James,
& grandchildren,
Nathael & Kierann.

PREFACE
The purpose of this book is to provide a detailed description of Brillouin–Wigner
methods and their application to the many-body problem in atomic and molecular
physics and quantum chemistry. Recently there has been a renewal of interest in
Brillouin–Wigner methods. This interest is fuelled by the need to develop robust,
yet efﬁcient multireference theoretical approaches to the electron correlation prob-
lem in molecules together with associated algorithms. Such theories are an essential
ingredient of the quantum mechanical description of most dissociative processes in
molecules, of excited states, and of ionization and electron attachment processes.
This volume contains a concise, systematic and self-contained account of the
Brillouin–Wigner methods. Chapter 1 is introductory giving the historical back-
ground to the Brillouin–Wigner methods and their application to atomic and
molecular structure. Chapter 2 uses the partitioning technique to develop both sin-
gle reference and multireference Brillouin–Wigner methods in a systematic fashion.
The corresponding Rayleigh–Schr¨odinger expansions are also considered since it has
been known for many years that they form the basis of a validmany-bodytheory
(in thepost-Brueckner sense). The many-body problem in atoms and molecules is
discussed in Chapter 3. The properties of a valid many-body theory are elaborated
and common approaches to the electron correlation problem are considered in some
detail. The linked diagram theorem of many-body perturbation theory is described
and the perturbative approach to the correlation problem considered alongside the
conﬁguration interaction and cluster expansion ansatz.
In Chapter 4, the application of Brillouin–Wigner methods to many-body systems
is described in some detail. This chapter deals with the central purpose of this mono-
graph – the development of many-body Brillouin–Wigner methods inparticular for
applications to the problem of describing molecular electronic structure using ab ini-
tio methods – methods which start from ﬁrst principles and can be systematically
reﬁned. (Semi-empirical methods and density functional theory, which in practice
involves parametrization, will not be considered.) The application of Brillouin–
Wigner theory to the conﬁguration interaction and cluster expansion techniques is
described as well as perturbation theory based methods. The use of Brillouin–Wigner
ix

x Preface
methods in obtaining many-body corrections to theoretical approaches which are not
valid many-body theories, such as limited conﬁguration interaction, is addressed.
Finally, Chapter 5 contains a summary and considers the prospects for future progress
in the application of Brillouin–Wigner methods to the correlation problem in atoms
and molecules and elsewhere.
A number of colleagues have participated in the development of many-body
Brillouin–Wigner methods over recent years. Without their dedicated efforts this
work would not have been possible. We single out the late J. M´aˇsik who contributed
substantially to the recent development of Brillouin–Wigner methods in electronic
structure theory before his untimely death at the age of only 33. We would also
like to mention(in alphabetical order): P. Babinec, P.ˇC´arsky, P. Mach, P. Neogr´ady,
P. Papp, J. Pittner, M. Pol´aˇsek, H.M. Quiney and J. Urban. We are most grateful for
their enthusiasm and their friendship.
We are grateful to our wives for their encouragement and support during the writ-
ing of this book, without which this book would not have been completed. We thank
Mr Radovan Javorˇc´ık, Charg´e dAffaires at the Embassy of the Slovak Republic, for
facilitating many of our meetings in London. We thank Mrs Kathryn Wilson who
undertook the labour of reading the whole book and corrected many careless slips
and helped us with points of style. Nevertheless any errors in the present volume are
ours alone and we would be grateful to any reader who takes the trouble to write to
us on such matters. We have established a webpage at
quantumsystems.googlepages.com/brillouin–wigner
where we can both collect errors in the present volume and details of further devel-
opments in Brillouin–Wigner many-body theory.
We are most grateful to the Royal Society for their permission to reproduce
material from J.E. Lennard-Jones, Proceedings of the Royal Society of London A
129, 598, 1930, and to Mrs. Eileen Hamilton Wigner for her permission to reproduce
material from E.P. Wigner, Math. u. Naturwiss. Anzeig. d. Ungar. Akad. Wiss LIII,
475, 1935.
IH acknowledges support from the VEGA Grant agency, Slovakia, under project
number 1/3040/06 and support by the Grant Agency of the Czech Republic under
project number MSM 4781305903. SW is a senioracademic visitor in the Physical
and Theoretical Chemistry Laboratory, University of Oxford, and is grateful to the
hospitality extended to him there.
Bratislava and Oxford, Ivan Hubaˇc
July, 2009 Stephen Wilson

CONTENTS
Preface............................................................ ix
Nomenclature...................................................... xv
Abbreviations...................................................... xvii
Atomic Units....................................................... xix
1 Introduction..................................................1
1.1 PREAMBLE ...................................................1
1.2 HISTORICAL BACKGROUND ...................................5
1.2.1 Lennard-Jones’ 1930 paper..................................5
1.2.2 Brillouin’s 1932 paper . . . ....................................7
1.2.3 Wigner’s 1935 paper........................................8
1.2.4 Studiesinperturbationtheory ................................10
1.3 PERTURBATIONTHEORY ...................................... 12
1.3.1 Brillouin–Wignerperturbationtheory..........................12
1.3.2 Rayleigh–Schr¨odingerperturbationtheory......................14
1.3.3 Comparison of Brillouin–Wigner and Rayleigh–Schr¨odinger
perturbationtheories ........................................17
1.3.3.1 AdvantagesoftheBrillouin–Wignertheory..............18
1.3.3.2 DisadvantagesoftheBrillouin–Wignertheory ...........22
1.4 THEMANY-BODYPROBLEM................................... 25
1.4.1 Linearscaling..............................................25
1.4.2 The re-emergence of Brillouin–Wigner methods . ................26
1.4.3 Anoverview...............................................32
References .....................................................33
2 Brillouin–Wigner perturbation theory............................37
2.1 THEPARTITIONINGTECHNIQUE............................... 37
2.1.1 The partitioning technique for a single-reference function . . .......38
2.1.1.1 Modelfunctionandprojectionoperators ................38
xi

xii Contents
2.1.1.2 AneffectiveHamiltonian .............................39
2.1.1.3 Thewaveoperator...................................40
2.1.1.4 Thereactionoperator ................................43
2.1.2 The partitioning technique for a multi-reference function . . .......44
2.1.2.1 Modelfunctionsandprojectionoperators................44
2.1.2.2 AneffectiveHamiltonian .............................45
2.1.2.3 Thewaveoperator...................................47
2.1.2.4 Thereactionoperator ................................48
2.2 PERTURBATIONEXPANSIONS ................................. 48
2.2.1 Single-referencefunctionperturbationexpansions ...............48
2.2.1.1 Expansionsoftheinverseoperator .....................48
2.2.1.2 Rayleigh–Schr¨odingerperturbationtheory...............50
2.2.1.3 Brillouin–Wignerperturbationtheory...................51
2.2.1.4 GeneralizedBrillouin–Wignerperturbationtheory ........53
2.2.1.5 Derivation of Rayleigh–Schr¨odinger perturbation theory
fromtheBrillouin–Wignerperturbationexpansion........56
2.2.2 Multi-referencefunctionperturbationexpansions................58
2.2.2.1 Multi-reference Rayleigh–Schr¨odinger perturbation theory .58
2.2.2.2 Multi-referenceBrillouin–Wignerperturbationtheory.....64
2.2.2.3 Ann-statesystem ...................................66
References .....................................................68
3 The many-body problem in atoms and molecules...................69
3.1 LINEARSCALINGINMANY-BODYSYSTEMS ................... 69
3.1.1 The exact electronic Schr¨odingerequation......................70
3.1.2 Independentparticlemodels..................................73
3.1.3 Many-body theories of electron correlation .....................76
3.2 MANY-BODYPERTURBATIONTHEORY......................... 77
3.2.1 Second-quantization formalism ...............................78
3.2.1.1 Second-quantization and the many-body problem . . .......78
3.2.1.2 Creation and annihilation operators and the occupation
numberrepresentation................................81
3.2.1.3 Normal products, contractions and Wick’s theorem .......87
3.2.1.4 Particle–hole formalism..............................90
3.2.1.5 TheHamiltonianoperatorinnormalform ...............94
3.2.2 Many-body Rayleigh–Schr¨odingerperturbationtheory ...........94
3.2.2.1 The many-body problem and quasiparticles . . . ...........94
3.2.2.2 Thealgebraicapproximation ..........................96
3.2.2.3 Many-body Rayleigh–Schr¨odinger perturbation theory . . . .98
3.2.2.4 Second-order contribution to the correlation energy .......101
3.2.2.5 Third-ordercontributionstothecorrelationenergy........108
3.2.2.6 Fourth-ordercontributionstothecorrelationenergy.......109
3.2.3 Many-body perturbation theory ...............................110

Contents xiii
3.3 MANY-BODYTHEORIESFORATOMSANDMOLECULES ........ 113
3.3.1 The full conﬁguration interaction method and limited conﬁguration
interaction.................................................114
3.3.2 Clusterexpansions..........................................119
References .....................................................129
4 Brillouin–Wigner methods for many-body systems.................133
4.1 INTRODUCTION . ..............................................133
4.2 BRILLOUIN–WIGNERCOUPLEDCLUSTERTHEORY............. 137
4.2.1 Single-reference Brillouin–Wigner coupled cluster theory . . .......137
4.2.2 Multi-reference Brillouin–Wigner coupled cluster theory . . .......143
4.2.2.1 Multi-root formulation of multi-reference
Brillouin–Wigner coupled cluster theory ................145
4.2.2.2 Multi-root multi-reference Brillouin–Wigner coupled
cluster:Hilbertspaceapproach ........................148
4.2.2.3 Basic approximations employed in the multi-reference
Brillouin–Wigner coupled cluster method...............152
4.2.3 Single-root formulation of the multi-reference Brillouin–Wigner
coupled-cluster theory . . . ....................................155
4.2.3.1 Single-root formulation of multi-reference
Brillouin–Wignerperturbationtheory...................156
4.2.3.2 Single-root multi-reference Brillouin–Wigner coupled
clustertheory:Hilbertspaceapproach ..................158
4.2.3.3 Single-root multi-reference Brillouin–Wigner coupled
cluster single- and double-excitations approximation......159
4.2.3.4 Linear scaling corrections in Brillouin–Wigner coupled
clustertheory.......................................162
4.3 BRILLOUIN–WIGNER CONFIGURATION INTERACTION THEORY 164
4.3.1 Single-reference Brillouin–Wigner conﬁguration interaction theory .167
4.3.1.1 Brillouin–Wigner perturbation theory and limited
conﬁgurationinteraction..............................167
4.3.1.2 Rayleigh–Schr¨odinger perturbation theory and limited
conﬁgurationinteraction..............................169
4.3.1.3 ‘Many-body’ correctionsfor Brillouin–Wigner limited
conﬁgurationinteraction..............................170
4.3.2 Multi-reference Brillouin–Wigner conﬁguration interaction theory .171
4.3.2.1 Multi-reference Brillouin–Wigner perturbation theory for
limitedconﬁgurationinteraction .......................171
4.3.2.2 Ap-statesystem ....................................174
4.3.2.3 Multi-reference conﬁguration interaction in
Brillouin–Wignerform...............................175
4.3.2.4 A posteriori Brillouin–Wigner correction to limited
multi-referenceconﬁgurationinteraction ................176

xiv Contents
4.4 BRILLOUIN–WIGNERPERTURBATIONTHEORY................. 177
4.4.1 Single-referenceBrillouin–Wignerperturbationtheory ...........177
4.4.2 Multi-referenceBrillouin–Wignerperturbationtheory............179
4.4.2.1 Multi-reference second-order Brillouin–Wigner
perturbationtheory ..................................179
4.4.3A posterioricorrection to multi-reference Brillouin–Wigner
perturbationtheory .........................................183
References .....................................................184
5 Summary and prospects........................................191
References .....................................................197
AL¨owdin’s studies in perturbation theory..........................201
B PROOF of the time-independent Wick’s theorem..................211
References .....................................................213
C Diagrammatic conventions.....................................215
D Rayleigh–Schr¨odinger perturbation theory.......................221
References .....................................................225
Index............................................................. 227

NOMENCLATURE
We present a summary of the nomenclature employed in this volume.
Hhamiltonian operator
H
0zero order hamiltonian operator
H
1perturbation operator
Ψexact wave function
Ψ
0exact ground state wave function
Ψ
αexact wave function for the stateα
Eexact energy
E
0exact ground state energy
E
αexact energy for stateα
Φzero order wave function
Φ
0zero order ground state wave function
Φ
αzero order wave function for stateα
Φ
mmodel space
Ψ
P
α
model function
Eenergy of the reference system
E
0ground state energy of the reference system
E
αenergy of the stateαof the reference system
λperturbation parameter
E
(p)
0
pth order energy coefﬁcient for the ground state
E
(p)
α
pth order energy coefﬁcient for the stateα
Ωwave operator
Ω
0wave operator for the ground state
Ω
αwave operator for the stateα
Vreaction operator
xv

xvi Nomenclature
V
0 reaction operator for the ground state
V
α reaction operator for the stateα
χ
p basis function
χ
(p)
0
pth order wave function for the ground state
χ
(p)
α
pth order wave function for the stateα
G general resolvent
R resolvent for Rayleigh–Schr¨odinger theory
R
0 resolvent for Rayleigh–Schr¨odinger theory
B resolvent for Brillouin–Wigner theory
B
0 resolvent for Brillouin–Wigner theory
B
α resolvent for Brillouin–Wigner theory
B
g resolvent for generalized Brillouin–Wigner theory
P reference or model space
Q orthogonal or complementary space
P projector onto the reference spaceP
Q projector onto the orthogonal or complementary spaceQ
H
effectiveeffective hamiltonian operator
X
+
i
creation operator in the particle formalism
X
i annihilation operator in the particle formalism
δ
pq Kronecker delta
[A, B]commutatorAB−BA
[A, B]
+anticommutatorAB+BA
n[...]normal product in the particle formalism
Y
+
i
creation operator in the particle–hole formalism
Y
i annihilation operator in the particle–hole formalism
N[...]normal product in the particle–hole formalism
T cluster operator
T
1 single excitation cluster operator
T
2 single excitation cluster operator
T
3 triple excitation cluster operator
T
4 quadruple excitation cluster operator
T
p p-fold excitation cluster operator

ABBREVIATIONS
We present here a deﬁnition of the abbreviations used in this volume.
CI conﬁguration interaction
BW Brillouin–Wigner
RS Rayleigh–Schr¨odinger
CC coupled cluster
PT perturbation theory
MR multi-reference
FCI full conﬁguration interaction
CISD single-reference conﬁguration interaction with single and double
replacements
MR-CI multi-reference conﬁguration interaction
BWPT Brillouin–Wigner perturbation theory
RSPT Rayleigh–Schr¨odinger perturbation theory
MBPT many-body perturbation theory
MPPT Møller–Plesset perturbation theory
MR-BWPTmulti-reference Brillouin–Wigner perturbation theory
MR-RSPTmulti-reference Rayleigh–Schr¨odinger perturbation theory
MR-MBPTmulti-reference many-body perturbation theory
MR-MPPTmulti-reference Møller–Plesset perturbation theory
CCSD single-reference coupled cluster with single and double replacements
CCSD(T) single-reference coupled cluster with single and double replacements
plus a perturbative estimate of the triple replacement component
xvii

xviii Abbreviations
CCSDT single-reference coupled cluster with single, double and
triple replacements
MR-CCSD multi-reference coupled cluster with single and double replace-
ments
MR-CCSD(T) multi-reference coupled cluster with single and double replace-
ments plus a perturbative estimate of the triple replacement
component
MR-CCSDT multi-reference coupled cluster with single, double and triple
replacements
BWCC Brillouin–Wigner coupled cluster
BWCCSD Brillouin–Wigner single-reference coupled cluster with single
and double replacements
BWCCSD(T) Brillouin–Wigner single-reference coupled cluster with single
and double replacements plus a perturbative estimate of the triple
replacement component
MR-BWCC multi-reference Brillouin–Wigner coupled cluster
MR-BWCCSD multi-reference Brillouin–Wigner coupled cluster with single and
double replacements
MR-BWCCSD(T) multi-reference Brillouin–Wigner coupled cluster with single and
double replacements plus a perturbative estimate of the triple
replacement component
CPA coupled-pair approximation
CPMET coupled-pair many-electron theory

ATOMIC UNITS
Atomic units are employed throughout this volume. In Table 1, we give the experi-
mentally determined values of the basic atomic units.
Table 1.Basic atomic units.
Quantity Symbol Value S.I. Units Relative
uncertainty (ppm)
Planck’s constantΨ=h/2π1.05457267×10
−34
Js 0.60
Elementary charge e 1 .60217733×10
−19
C0 .30
Electron mass m
e 9.1093897×10
−31
kg 0.59
The atomic unit of length is the ﬁrst Bohr radius
a
0=
Ψ
2
mee
2
The unit of energy is the Hartree
E
H=
e
2
a0
In Table 2, we give numerical values of these atomic units together with various other
derived atomic units.
xix

xx Atomic Units
Table 2.Some derived atomic units
Quantity Value S.I. Units
Bohr radius,a 0 5.2917726×10
−11
m
Energy,E
H 4.3598×10
−18
J
Time 2.4189×10
−17
s
Electric dipole moment 8 .4784×10
−30
Cm
Electric quadrupole moment 4.4866×10
−40
Cm
2
Electric octopole moment 2. 3742×10
−50
Cm
3
Electric ﬁeld 5.1423×10
11
Vm
−1
Electric ﬁeld gradient 9.7174×10
21
Vm
−2
Polarizability (dipole) 1.6488×10
41
C
2
m
2
J
−1
Hyperpolarizability 3.2063×10
−53
C
3
m
3
J
−2
Magnetic moment 1.8548×10
−23
JT
−1
Magnetizability 7.8910×10
−29
JT
−2
Magnetic vector potential 1 .2439×10
−5
mT
Force constant (harmonic) 1 .5569×10
3
Jm
−2
Force constant (cubic) 2.9421×10
13
Jm
−3
Force constant (quartic) 5. 5598×10
23
Jm
−4
Probability density 6.7483×10
30
m
−3

1
INTRODUCTION
AbstractAn overview of the many-body problem in atomic and molecular physics and in quantum
chemistry is given. Some historical background on the Brillouin–Wigner methodology is
presented together with basic derivations of the Rayleigh–Schr¨odinger and the Brillouin–
Wigner perturbation theories. The elementary formulations of the two theories are com-
pared. The recent resurgence of interest in Brillouin–Wigner methodology, particularly in
studies of the multi-reference correlation problem, is explained.
1.1. PREAMBLE
With its roots in astronomy and classical physics, perturbation theory is the most
general and systematic technique for theapproximation of the solutions of quantum
mechanical eigenvalue problems. The Schr¨odinger eigenvalue equation describes the
motion of the electrons and the nuclei which are the fundamental ingredients of atoms
and molecules. The solution of the appropriate Schr¨odinger equation is the key to
understanding atomic and molecular structure and properties, as well as the interac-
tions between atomic and molecular systems. The exact solution of the Schr¨odinger
equation is associated with considerable mathematical and computational difﬁculties.
Very few of the problems which occur in quantum mechanics can be solved exactly,
often because they involve many particles. The development of systematic methods of
approximation is, therefore, central to the theoretical models employed in the quan-
tum many-body problem in studies of molecular physics and quantum chemistry.
Perturbation theory relates the actual problem in hand to some simpliﬁed system
for which solutions are known – the difference being treated as a ‘perturbation’. In
quantum mechanics, this perturbation gives rise to an expansion in terms of a small
parameter: an idea which has been employed in classical mechanics and, in particu-
lar, in solving problems in celestial mechanics. For example, in the case of a many-
electron atom or molecule, the simpliﬁed system (or reference model) might be that
in which all electron–electron interactionsare completely neglected – each electron
moves in the ﬁeld of the nucleus or nuclei – and the perturbation is taken to be the
interactions between the electrons. An improved approximation scheme would take
into account the averaged interactions between the electrons in the reference model –
each electron moves in some mean ﬁeld, treating only the instantaneous interactions
1

2 I. Hubaˇc and S. Wilson
between the electrons by means of a perturbation expansion. These instantaneous
interactions are usually referred to as ‘electron correlation effects’. The component
of the total energy associated with electron correlation effects is but a small fraction
of the energy resulting from electron–electron interactions, if a suitably chosen mean
ﬁeld is employed in deﬁning the independent particle model used as a reference.
The accurate description of many-electron correlation effects remains one of the
central problems of molecular physics and quantum chemistry [1–4]. Although inde-
pendent particle models, such as the widely used Hartree–Fock theory, can account
for all but a fraction of the total energy of a molecular system, this fraction is often
as large or indeed larger than the energy changes which accompany chemical pro-
cesses: binding energies, interaction energies, activation energies, barriers to rotation,
and the like. The determination of the effects of electron correlation is therefore of
crucial importance
Molecular quantum mechanics was born in the late 1920s, but for many years the
use of perturbation theory in the study of molecules was rather limited. Perturbation
theory was used to describe the interactions between molecules at long range, but for
the description of the structure and properties of isolated molecular systems, meth-
ods based on the variation theorem found favour. The reason for this preference is
that many of the series encountered in atomic and molecular quantum mechanical
problems were found to be poorly convergent or even divergent. Although as early as
1930, Dirac [5] had recognized that
Even when the [perturbation] series does not converge, the ﬁrst approxima-
tion obtained by means of it is usually fairly accurate.
In early applications of non-relativistic quantum mechanics to atomic and molecular
systems, variational techniques were preferred for the majority of practical applica-
tions. The non-relativistic Hamiltonian operator is a semi-bounded operator and the
variation principle ensures that the energy expectation value corresponding to any
approximate wave function lies above the corresponding exact energy. Approximate
solutions of the Schr¨odinger equation could be constructed which depend on certain
arbitrary parameters, which can be reﬁned by invoking the variation principle. How-
ever, the theoretical description of heavy (and superheavy) atoms (and molecules
containing them) demands a fully relativistic formulation. The development of quan-
tum ﬁeld theory in the late 1940s and the early 1950s saw perturbation theory as
the only technique for obtaining information about the eigenvalues of the pertinent
Hamiltonians for relativistic problems.
The prospects for the approximate solution of the quantum mechanical equations
governing the properties of atoms and molecules changed radically with the advent
of the electronic digital computing machine around the middle of the twentieth cen-
tury. The systematic implementation of approximation techniques could be auto-
mated. However, it was soon recognized that these ﬁrst computers were too slow
and too short on memory to seriously consider applications to molecules containing
more than a few light atoms. Furthermore, theoretical progress was required both to

Brillouin–Wigner Methods for Many-Body Systems 3
cast existing theories in a form amenable to automatic computation and to tackle the
formidable electron correlation problem.
In 1951, Hall [6] and, independently, Roothaan [7] put the Hartree–Fock equations
– the ubiquitous independent particle model – in their matrix form. The Hartree–
Fock equations describe the motion of eachelectron in the mean ﬁeld of all the elec-
trons in the system. Hall and Roothaan invoked the algebraic approximation in which,
by expanding molecular orbitals in a ﬁnite analytic basis set, the integro-differential
Hartree–Fock equations become a set of algebraic equations for the expansion coef-
ﬁcients which are well-suited to computer implementation.
McWeeny [8] and, independently, Boys [9] introduced the Gaussian basis function,
which held the key to practical applications to arbitrary polyatomic molecules
because of the ease and accuracy with which theassociated molecular integrals could
be evaluated. The (cartesian) Gaussian basis functions have the form
χ
i=χi(ζi,i,mi,ni)
=x
Δi
y
mi
z
ni
exp
Ψ
ζ ir
2
Δ
(1.1)
r=(x, y, z)wherex,yandzare the cartesian coordinates.ζ
iis a screening con-
stant. The integers
i,miandn idetermine the nodal structure of the Gaussian basis
functions. The single particle state functions,φ
μ, of the independent particle model
are approximated as
φ
μ=
Φ
i
χicμ,i(1.2)
where the coefﬁcients are determined by iterative solution of the matrix Hartree–Fock
equations – the self-consistent ﬁeld method. However, it has to be noted [10] that the
use of Gaussian basis functions was not initially greeted with enthusiasm and initially
they were not used extensively – even by their proponents.
The second half of the twentieth century saw a sustained attack on the correlation
problem in atoms and molecules. By the mid-1950s the basic structure of correlated
wavefunctions was understood, thanks mainly to developments in solid state physics
and nuclear physics. The linked diagram theorem of the many-body perturbation the-
ory and the connected cluster structure of the exact wavefunctions were ﬁrmly estab-
lished. Goldstone [11] exploited diagrammatic techniques developed in quantum ﬁeld
theory [12–15], to complete Brueckner’s work [16] on the scaling of energies and
other expectation values with the number of electrons in the system. Hugenholtz [17]
provided an alternative approach. The exponential ansatz for the wave operator sug-
gested by Hubbard [18] in 1957 was ﬁrst exploited in nuclear physics by Coester
[19] and by Coester and K¨ummel [20].
In applications to the atomic and molecular electronic structure problem – the prob-
lem of describing the motion of the electrons in the ﬁeld of clamped nuclei – there
was, as Paldus [21] describes, an
initial hope that the conﬁguration interaction approach limited to doubly
excited conﬁgurations, originating from a single-reference state, [would]
provide a satisfactory description of correlation effects.

4 I. Hubaˇc and S. Wilson
However, this approach, which exploits the variation theorem to determine the cor-
related wavefunction for the non-relativistic problem, was soon thwarted by the slow
convergence of the conﬁguration interaction expansion.
The 1960s saw the applications of the many-body perturbation theory developed
during the 1950s by Brueckner [16], Goldstone [11] and others, to the atomic struc-
ture problem by Kelly [22–31]. These applications used the numerical solutions to the
Hartree–Fock equations which are available for atoms, because of the special coor-
dinate system. Kelly also reported applications to some simple hydrides in which the
hydrogen atom nucleus is treated as an additional perturbation. At about the same
time,ˇC´ıˇzek [32] developed the formalism of the coupled cluster approach for use in
the context of molecular electronic structure theory.
By the early 1970s, both the coupled cluster theory and the many-body perturba-
tion theory had been implemented in the algebraic approximation and applications to
arbitrary molecular systems became a reality. In 1972, Paldus,ˇC´ıˇzek and Shavitt [33]
initiated applications ofab initiocoupled cluster theory using ﬁnite basis set expan-
sions, whilst in the following year Kaldor [34] ﬁrst invoked the algebraic approxima-
tion in an application of the many-body perturbation theory to the hydrogen molecule
ground state. In 1976, Wilson and Silver [35] compared ﬁnite order many-body per-
turbation theory with limited conﬁguration interaction calculations when both meth-
ods are formulated in the algebraic approximation.
Progress was particularly rapid during the late 1970s and early 1980s with the
introduction of a new generation of ‘high performance’ computing machines which
enabled the realization of practical schemes of calculation which in turn gave new
levels of understanding of the nature of the electron correlation problem in atoms
and, more particularly, in molecules. Itbecame widely recognized that a successful
description of correlation effects in molecules must have two key ingredients:
1. It must be based either directly or indirectly on the linked diagram theorem of
many-body perturbation theory, so as to ensure that the calculated energies and
other expectation values scale linearly with particle number.
2. It must be based on a careful and systematic realization of the algebraic approx-
imation (i.e. the use of ﬁnite basis set expansions), since this can often be the
dominant source of error in calculations which aim to achieve high precision.
The past 20 years have witnessed a relentless increase in the power of computing
machines. It has been observed
1
that the processing power of computers seems to
double every eighteen months. As J.M. Roberts [36] points out
No other technology has ever improved so rapidly for so long.
This ever-increasing computing power has led to both higher accuracy in molecu-
lar electronic structure calculations, often because larger basis sets can be utilized,
and has opened up the possibility of applications to larger molecules and molecular
systems.
1
By Mr. Gordon Moore of Intel. This has been dubbed “Moore’s Law”.

Brillouin–Wigner Methods for Many-Body Systems 5
Today, there remain a number of problems in molecular electronic structure theory.
The most outstanding of these is undoubtedly the development of a robust theoret-
ical apparatus for the accurate description of dissociative processes which usually
demand the use of multi-reference functions. This requirement has recently kindled
a renewal of interest in the Brillouin–Wigner perturbation theory and its applica-
tion to such problems. This volume describes the application of Brillouin–Wigner
methods to many-body systems and, in particular, to molecular systems requiring a
multi-reference formalism.
This introductory chapter is organized as follows: In Section 1.2, we give some his-
torical background to the Brillouin–Wigner perturbation theory and its application in
the study of problems in molecular quantum mechanics. The Brillouin–Wigner and
the more familiar Rayleigh–Schr¨odinger perturbation theories are presented in ele-
mentary form in Section 1.3 and the advantages and disadvantages of the Brillouin–
Wigner expansion enumerated. In Section 1.4, we consider the re-emergence of
Brillouin–Wigner methods in recent years particularly in handling problems requir-
ing the use of a multi-reference formalism. Section 1.4 concludes with an outline of
the remaining chapters of this volume.
1.2. HISTORICAL BACKGROUND
The theory which is today called “Brillouin–Wigner perturbation theory” was intro-
duced in three seminal papers published in the 1930s. The ﬁrst of these, which
appears to be not so well known as the other two, was published in 1930 by
J.E. Lennard-Jones [37]. (Indeed, some authors [38, 39] refer to the method as
“Lennard-Jones–Brillouin–Wigner perturbation theory”.) Subsequently, L. Brillouin
published his famous paper in 1932 [40] whilst E.P. Wigner’s paper appeared some 2
years later [41]. In this section, we provide a brief synopsis of each of these important
papers.
During the period 1963–1971, P.-O. L¨owdin published
2
a series of papers [42–53]
with the general title “Studies in Perturbation Theory”, which afforded deep insight
into perturbation theory expansions, the relation between different expansions and
their application to quantum mechanical problems. We conclude this section with a
brief overview of L¨owdin’s work on perturbation theory.
1.2.1. Lennard-Jones’ 1930 paper
The seminal paper on what is nowadays called “Brillouin–Wigner perturbation the-
ory” by Lennard-Jones
3
was published in theProceedings of the Royal Society of
Londonin 1931. It was communicated by R.H. Fowler and received on 1stSeptem-
ber, 1930. Here we reproduce Lennard-Jones’ introduction:
2
The last paper in the series was coauthored by O. Goscinski.
3
Sir John Edward Lennard-Jones (1894–1954).

6 I. Hubaˇc and S. Wilson
Perturbation problems in quantum mechanics
J.E. Lennard-Jones
Department of Theoretical Physics, The University, Bristol
(Communicated by R.H. Fowler, F.R.S. – Received September 1, 1930)
Introduction
One of the great achievements of the Schr¨odinger wave-mechanics is the elegance
of its perturbation theory, which has brought many problems, formerly considered
intractable, within the range of highly-developed mathematical techniques. It is not
necessary at this stage to review the numerous applications which have been made of
this perturbation theory or to dwell upon its many advantages. The important advance
towards an understanding of chemical forces which it has made possible is in itself a
considerable achievement.
There are, however, certain disadvantages in the perturbation theory in its present
form, which limit the extent of its applications to complex problems of atomic and
molecular structure. If the interaction of atoms, for instance, is to be calculated, as is
most desirable, improved methods will have to be found.
One such improvement is considered in this paper. In its present form, it is easy to
calculate the ﬁrst approximation of the energy of a system, subject to small pertur-
bations, but difﬁcult to proceed further. This is a considerable disadvantage in those
problems where the ﬁrst approximation vanishes as in calculating the Stark effect
or the van der Waals attraction of two atoms at large distances apart. Moreover, the
theory expresses the perturbed eigenfunction in terms of all the unperturbed eigen-
functions of the system and these are not always known. This paper shows how, by a
slight modiﬁcation of the usual method, these difﬁculties may be overcome and the
energy and eigenfunction of a perturbed system calculated to higher approximations
with comparative ease.
As an illustration the method is applied to calculate the van der Waals ﬁelds of two
hydrogen atoms at large distances, and this is done with more ease and directness
than the usual form of the theory permits. It is possible, too, to extend the theory to
calculate the van der Waals ﬁelds of more complicated atoms, as it is hoped to show in
a later paper. The results of the Schr¨odinger perturbation theory are also obtained by
another method, which has the advantage ofexhibiting exactly what is neglected in
the usual successive approximations. This alternative method is not limited to small
perturbations; in fact, it is shown that certain perturbation problems can be solved,
however strong the perturbation. One such problem is that of rotating polar molecules
under the inﬂuence of an external electric ﬁeld, which is a necessary step in the theory
of gaseous dielectrics. A solution to this problem is given and the range of validity of
the usual dielectric theory is thus determined.
(Taken fromProceedings of the Royal Society of LondonA129, 598, 1930)

Brillouin–Wigner Methods for Many-Body Systems 7
We should emphasize the point made by Lennard-Jones that the
alternative method is not limited to small perturbations; in fact, it is shown
that certain perturbation problems canbe solved, however strong the pertur-
bation.
As we have noted already, some authors, such as Dalgarno [38] and Wilson [39], term
what is nowadays called “Brillouin–Wigner perturbation theory” “Lennard-Jones–
Brillouin–Wigner perturbation theory” in recognition of the seminal contribution of
Sir John Lennard-Jones.
1.2.2. Brillouin’s 1932 paper
Brillouin’s
4
seminal 1932 paper on perturbation theory was entitled“Les Probl`emes
de Perturbation et les Champs self-consistents”. It was published in 1932 inLe Jour-
nal de Physique et Le radium. It would appear that Brillouin was not aware of the
earlier work by Lennard-Jones, since the earlier work is not cited. Here we reproduce
the abstract of his paper, which is in French and which was submitted for publication
on 22nd July, 1932.
Les probl`emes de perturbations et les champs
self-consistents
L. Brillouin
Manuscript rec¸u le 22 juillet 1932
Sommaire
Le probl`eme des perturbations, en mecanique ondulatoire, peut ˆetre traite rigoureuse-
ment et aboutit `a une equation seculaire, ´ecrite sous forme de determinant; on
retrouve facilement les formules de Schr¨odinger pour un probl`eme non d´egen´er´eou
d´egen´er´e; mais cette equation permet aussi d’´etudier le cas d’une grosse perturbation
agissant sur un syst`eme qui poss`ede des niveaux d’´energie tr`es voisins. La m´ethode
des champs self-consistents est expos´ee, sous une forme tr`es directe qui donne non
seulement la premi`ere approximation, mais aussi toute la matrice de perturbation qui
subsiste apr`es cette premi`ere approximation. Deux types de champs self-consistents
sont compar´es, celui de Hartree et celui de Fock. Le second donne une matrice de per-
turbation plus faible, mais sans pouvoir annuler les termes complet`ement. Les formu-
las ´etablies dans cet article seront ult´erieurement utilis´ees pour l’etude des electron
libres dans les m´etaux.
(Taken fromLe Journal de Physique et le RadiumS´eries VII, TomeIII, 373,
1932)
4
L´eon Brillouin (1877–1972).

8 I. Hubaˇc and S. Wilson
Brillouin points out that the new perturbation series converges much more rapidly
than the power series of Schr¨odinger.
1.2.3. Wigner’s 1935 paper
Wigner’s
5
contribution to Brillouin–Wigner perturbation theory appeared some
2 years after Brillouin’s paper in 1935. It was published inMathematischer
und Naturwissenschaftlicher Anzeiger der Ungarischen Akademie der Wis-
senschaften. Wigner cites the earlier work of Brillouin but not that of Lennard-Jones.
The ﬁrst part of Wigner’s paper is in Hungarian:
A Rayleigh–Schr¨odinger-f´ele perturb´aci´o-elm´elet
egy m´odos´ıt´as´ar´ol
Wigner Jen˝o- t˝ol
Ismeretes, hogy m´ıg a Rayleigh–Schr¨odinger-f´ele perturb´aci´os elm´elet els˝o
k¨ozel´ıt´ese mindig t´ul magas eredm´enyt ad a legals´oenergian´ıv´ora, semmi hasonl´o
nem ´all´ıthat´oam´asodik ´es magasabb k¨ozel´ıt´esekre. A jelen dolgozat c´elja egy olyan
k¨ozelit˝oelj´ar´as kidolgoz´asa, mely a legals´o energianiv´ot mindig fel¨ulr˝ol k¨ozel´ıti meg,
´epp´ugy, mint a vari´aci´os m´odszer, melynek mindenik k¨ozel´ıt´ese a k´erd´esesH+V
operatornak egy k¨oz´ep´ert´eke eqy normaliz´alt hull´amf¨uggv´enyre.
HaH+Vk¨oz´ep´ert´ek´et a (4a) alatt adott f¨uggv´enyre kisz´am´ıtjuk (φ
kH-nakE k-
hoz tartoz´osaj´atf¨uggv´enye), a feladat nem m´as, mint az ´ıgy kapott (4b) kifejez´esnek
minimumm´aval´ot´etele. Ha a sz´aml´al´oban lev˝okett˝os ¨osszeget elhagyjuk, az ´ıgy
egyszer˝us´ıtett kifejez´es aza-knak (5a) alatt adott ´ert´eke mellett lesz legkisebb. Ebben
a kifejez´esbenF
(2)
1
azeg´eszminimaliz´aland´o kifejez´es ´ert´eke aza-k ezen ´ert´eke
mellett ´es meghat´aroz´as´ara a (6) implicit egyenlet szolg´al. Ezen implicit egyen-
let megold´asa teh´at egy - a Rayleigh–Sch¨odinger egyenlet harmadik k¨ozel´ıt´es´enek
megfelel˝o, - de mindig t´ul magas ´ert´eket ad az energi´ara.
A magasabb k¨ozel´ıt´eseket m´ar most ´ugy nyerj˝uk, hogy az eml´ıtett kett˝os ˝osszegben,
mely aza-knak biline´aris, kifejez´ese, az egyik faktort az el˝oz˝ok¨ozel´ıt´esnek meg-
felel˝o kifejez´essel helyettes´ıtj¨uk, mig a m´asik faktort valtozatlanul ismeretlennek tek-
inj¨uk. Az ´ıgy nyert kifejez´eseket a (7a) alatt adotta-k teszik legkisebb´e. A (7a)-ben
szerepl˝oF
(3)
1
megint az eg´esz minimaliz´aland´o kifejez´es ´ert´eke ezena-rendszer mel-
lett (meghat´aroz´as´ara a (7) egyenlet szolg´al) ´es ´ıgy ism´et egy fels˝ohat´ar´at k´epezi a
legkisebb energia´ert´eknek.
Az ´eppen le´ırt k¨ozelit˝o kifejez´esek a Rayleigh–Schr¨odinger-f´ele elm´elet megfelel˝o
kifejez´esein´el valamivel egyszer˝uebbek ´es ´ıgy k¨onnyen ´altal´anos´ıthat´okn=∞-re,
mely esetben a saj´at´ert´ekegyenletnek egy form´alis megold´as´at adj´ak egy v´egtelen
sor alakj´aban. Ezen v´egtelen sort el˝osz¨or L. Brillouin adta meg, a n´elk¨ul azon-
ban, hogy a r´eszlet¨osszegek viszony´at a val´odi megold´ashoz megvizsg´alta volna.
Bizonyos esetekben k¨onnyen ki lehet mutatni, hogy Brillouin v´egtelen sorai a val´odi
megold´ashoz knoverg´alnak, ´es vil´agos, hogy a jelen elj´ar´as sok esetben konverg´al,
amikor az eredeti nem haszn´alhat´o.
V´eg¨ul a nyert k¨ozelit˝o kifejez´es egy Mathieu egyenletre alkalmaztatik p´eldak´eppen.
5
Eugene P. Wigner (1902–1995), Nobel Prize in Physics, 1963.

Brillouin–Wigner Methods for Many-Body Systems 9
Wigner’s paper then continues in English:
On a modiﬁcation of the Rayleigh–Schr¨odinger
perturbation theory
Eugene Wigner
From the meeting of the IIIrd class of the Hungarian Academy of Sciences
on the 12th November 1934
1.The Rayleigh–Schr¨odinger perturbation theory
6
gives an explicit power series in
λfor the characteristic valuesF
nand the characteristic functionsφ nof a Hermitean
operatorH+λV
(1) (H+λV)φ
n=Fnφn
if the corresponding quantitiesE nandψ nfor the unperturbed operatorHare known
(1a)Hψ
n=E nψn.
If the so-called matrix elements ofVare denoted, as usual, by
(2) V
nm=(ψ n,Vψm)=V
∗
nm
the ﬁrst terms in the aforementioned series read
F
(2)
n
=E n+λVnn+λ
2
Λ
k
ﬀ
|Vnk|
2
En−Ek
(3a)
φ
(1)
n
=φn+λ
Λ
k
ﬀ
VknEn−Ek
ψ.(3b)
Generally only these ﬁrst terms of the series are used in actual calculations, the higher
order terms become increasingly complicated.
We shall ﬁx our attention on the lowest energy value,F
1. While it is evident that
the ﬁrst approximation for thisF
(1)
1
=E 1+λV11is always greater than the real
amount – since it is the expectation value of a normalized wave functionψ
1; noth-
ing like this holds for the second and higher approximations. It even happens quite
often that the last series in (3a) diverges in cases when the lowest energy value is
ﬁnite itself. In these cases, of course, Rayleigh–Schr¨odinger perturbation theory is
inapplicable to the problem. The aim of the present paper is to give an approxima-
tion formula forF
1which always yields values that are too high, and which can be
proved to converge at least in certain simple cases. Such an expression is naturally
provided by the variational method which had been used frequently indeed in cases
for which the general shape of the characteristic functions could be obtained by phys-
ical considerations.
6
J.W.S. Rayleigh,The Theory of Sound, London and New York, 1894, vol. 1,
p. 113; E. Schr¨odinger,Collected Papers on Wave Mechanics, London and Glasgow,
1928, p. 64.

10 I. Hubaˇc and S. Wilson
The ﬁnal result, the∞-approximation, will appear in the form of an inﬁnite series.
This inﬁnite series was ﬁrst found by L. Brillouin
7
who obtained it by an intuitive
consideration of the usual scheme. He has already pointed out in his important paper
that his series converges much more rapidly than the power series of Schr¨odinger. He
has not investigated, however, the successive approximations and their relations to
the actual problem.
(Taken fromMath.u.Naturwiss.Anzeig.d.Ungar.Akad.Wiss LIII, 475,
1935)
Wigner repeats Brillouin’s observation that the new
series converges much more rapidly than the power series of Schr¨odinger.
The papers by Lennard-Jones, by Brillouin and by Wigner represent the genesis of
what is now termed “Brillouin–Wigner perturbation theory”.
1.2.4. Studies in perturbation theory
During the 1960s, a series of papers with the general titleStudies in perturba-
tion theorywere published by P.-O. L¨owdin
8
[42–53]. The ﬁrst paper in the series
appeared in theJournal of Molecular Spectroscopyin 1963 [42]. A ﬁnal and 14th
paper in the series was co-authored by O. Goscinski and appeared in 1971 in the
International Journal of Quantum Chemistry[53].
This series of papers used the partitioning technique to develop perturbation series
in a very general way. The series provided a deep insight into the use of perturbation
methods in the study of quantum eigenproblems. We shall consider the partitioning
technique in detail in Chapter 2. Here we list the titles of the14papers with the gen-
eral title “Studies in perturbation theory” authored by L¨owdin and, in the case of the
fourteenth, Goscinski. The abstracts of these papers are reproduced in Appendix A.
For further details, the reader should consult the original publications.
1.Journal of Molecular Spectroscopy10, 12 (1963)
Studies in Perturbation Theory. I. An Elementary Iteration-Variation Procedure
for Solving the Schr¨odinger Equation by Partitioning technique
2.Journal of Molecular Spectroscopy13, 326 (1964)
Studies in Perturbation Theory. II. Generalization of the Brillouin–Wigner For-
malism
7
L. Brillouin,Le Journal de Physique et le RadiumS´eries VII, TomeIII, 373,
1932.
8
Per-Olov L¨owdin (1916–2000).

Brillouin–Wigner Methods for Many-Body Systems 11
3.Journal of Molecular Spectroscopy13, 326 (1964)
Studies in Perturbation Theory. III. Solution of the Schr¨odinger equation under a
variation of a parameter
4.Journal of Mathematical Physics3, 969 (1962)
Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem by Projection
Operator Formalism
5.Journal of Mathematical Physics3, 1171 (1962)
Studies in Perturbation Theory. V. Some Aspects on the Exact Self-Consistent
Field Theory
6.Journal of Molecular Spectroscopy14, 112 (1964)
Studies in Perturbation Theory. VI. Contraction of secular equations
7.Journal of Molecular Spectroscopy14, 119 (1964)
Studies in Perturbation Theory. VII. Localized perturbation
8.Journal of Molecular Spectroscopy14, 131 (1964)
Studies in Perturbation Theory. VIII. Separation of the Dirac equation and study
of the spin-orbit coupling and Fermi contact terms
9.Journal of Mathematical Physics6, 1341 (1965)
Studies in Perturbation Theory. IX. Connection Between Various Approaches in
the Recent Development – Evaluation of Upper Bounds to Energy Eigenvalues in
Schr¨odinger’s Perturbation Theory
10.The Physical Review139, A357 (1965)
Studies in Perturbation Theory. X. Lower Bounds to Energy Eigenvalues in
Perturbation-Theory Ground State
11.The Journal of Chemical Physics43, S 175 (1965)
Studies in Perturbation Theory. XI. Lower bounds to energy eigenvalues, ground
state, and excited state
12. inPerturbation theory and its applications in quantum mechanics,ed.C.H.
Wilcox, p.255, Wiley, New York (1966)
The Calculation of Upper and Lower Bounds of Energy Eigenvalues in Perturba-
tion Theory by means of Partitioning Technique
13.International Journal of Quantum Chemistry2, 867 (1968)
Studies in Perturbation Theory. XIII. Treatment of Constants of the Motion in
Resolvent Method, Partitioning Technique, and Pertubation Theory
14.International Journal of Quantum Chemistry5, 685 (1971)
Studies in Perturbation Theory. XIV. Treatment of Constants of the Motion,
Degeneracies and Symmetry Properties by Means of Multidimensional Parti-
tioning

12 I. Hubaˇc and S. Wilson
1.3. PERTURBATION THEORY
Brillouin–Wigner perturbation theory has a number of advantages over the more
widely used Rayleigh–Schr¨odinger theory. It also has signiﬁcant disadvantages which
have resulted in its neglect as the basis for many-body theories over the past 50 years.
Before describing these advantages and disadvantages, let us recall the basic structure
of the Brillouin–Wigner and Rayleigh–Schr¨odinger perturbation theories.
In this section, we shall ﬁrst provide an ‘elementary’ derivation of the Brillouin–
Wigner perturbation theory and then present a comparable introduction to the
Rayleigh–Schr¨odinger perturbation theory. The section concludes with a comparison
of the advantages and disadvantages of the two theories.
1.3.1. Brillouin–Wigner perturbation theory
We seek to develop approximate solutions of the time-independent Schr¨odinger
eigenvalue equation
ˆ
HΨ
μ=EμΨμ,μ=0,1,2,...(1.3)
where
ˆ
His the Hamiltonian operator,E
μis the energy eigenvalue, and the wave
function,Ψ
μis the eigenfunction for theμth state. For present purposes we shall not
concern ourselves with the precise form of the Hamiltonian. A perturbation theory is
developed by starting from some reference or model deﬁned by the equation
ˆ
H
0Φk=EkΦk,k=0,1,2,...(1.4)
for which the solutions are known. We write the total Hamiltonian operator in eq.
(1.3) as the sum of the zero-order Hamiltonian operator from eq. (1.4) and a pertur-
bation operator
ˆ
H=
ˆ
H
0+λ
ˆ
H 1(1.5)
where the parameterλis used to interpolate between the unperturbed problem(λ=0)
and the perturbed problem(λ=1). Perturbation theories are developed by making
expansions for the exact eigenvalue,E
μ, and the corresponding eigenfunction,Ψ μ.
Equation (1.5) allows the Schr¨odinger equation (1.3) to be written in the form
λ
E
μ−
ˆ
H0
Ω
Ψ
μ=λ
ˆ
H 1Ψμ(1.6)
which can then be rewritten in matrix form
(E
μ−Em)ΨΦm|ΨμΔ=λΨΦ m|
ˆ
H1|ΨμΔ(1.7)
or
ΨΦ
m|ΨμΔ=
λΨΦ
m|
ˆ
H1|ΨμΔ
(Eμ−Em)
(1.8)
In eqs. (1.7) and (1.8), the state|Φ
mΔis the unperturbed eigenket ofH 0with eigenen-
ergyE
m. We ﬁnd it convenient to put

Brillouin–Wigner Methods for Many-Body Systems 13
ΨΦ
μ|ΨμΔ=1(1.9)
in order to simplify the following formulae. Thus,|Ψ
μΔis not normalized to unity.
Equation (1.9) is termed the ‘intermediate normalization condition’. Settingm=μ
in eq. (1.7) and using eq. (1.9), the exact energy eigenvalue is given by
E
μ=Eμ+λΨΦ μ|
ˆ
H1|ΨμΔ(1.10)
The exact eigenstate for theμ
th
state is constructed using the completeness of the
unperturbed basis,|Φ
mΔ,sothat
|Ψ
μΔ=
∀
m
|ΦmΔΨΦm|ΨμΔ(1.11)
or
|Ψ
μΔ=|Φ μΔΨΦμ|ΨμΔ+
∀
m(Ψ=μ)
|ΦmΔΨΦm|ΨμΔ(1.12)
where we note that the summation excludes the casem=μ. Using eq. (1.8) for
ΨΦ
m|ΨμΔand eq. (1.9), we can write eq. (1.12) as
|Ψ
μΔ=|Φ μΔ+λ
∀
m(Ψ=μ)
|ΦmΔ
1
(Eμ−Em)
ΨΦ
m|
ˆ
H1|ΨμΔ(1.13)
We have included the parameterλin eq. (1.13) which is set equal to unity in order to
recover the perturbed problem. Equation (1.13) is the basic formula of the Brillouin–
Wigner perturbation theory fora single-reference function.
From eq. (1.13) we can develop a series expansion for the exact eigenfunction
|Ψ
μΔ,inpowersofλwith coefﬁcients depending on the perturbed energyE μ,rather
than the energy of the modelE
m, as would be the case in the more familiar Rayleigh–
Schr¨odinger perturbation series. Iterating this basic formula, we ﬁnd
|Ψ
μΔ=|Φ μΔ+λ
∀
m(Ψ=μ)
|ΦmΔ
1
(Eμ−Em)
ΨΦ
m|
ˆ
H1|ΦμΔ
+λ
2
∀
j(Ψ=μ),m(Ψ=μ)
|ΦjΔ
1
(Eμ−Ej)
ΨΦ
j|
ˆ
H1|ΦmΔ
1
(Eμ−Em)
?ΨΦ
m|
ˆ
H1|ΦμΔ+···(1.14)
The above procedure resembles the Lippmann–Schwinger method [106] for con-
structing the incoming state in scattering theory.
By introducing the Brillouin–Wigner type propagator (resolvent)
B=B(μ)=
∀
m(Ψ=μ)
|ΦmΔ
1
(Eμ−Em)
ΨΦ
m|,(1.15)
eq. (1.14) for the exact wave function for theμth state can be written in the more
compact form:
|Ψ
μΔ=
∼
1+λB
ˆ
H 1+λ
2
B
ˆ
H1B
ˆ
H1+···
√
|Φ μΔ(1.16)

14 I. Hubaˇc and S. Wilson
Substituting the expansion (1.14) for the exact wave function into eq. (1.10) for the
exact energy eigenvalue, the perturbed energy values are given by
E
μ=Eμ+λΨΦ μ|
ˆ
H1|ΦμΔ
+λ
2
λ
m(Ψ=μ)
ΨΦμ|
ˆ
H1|ΦmΔ
1
(Eμ−Em)
ΨΦ
m|
ˆ
H1|ΦμΔ
+λ
3
λ
j(Ψ=μ),m(Ψ=μ)
ΨΦμ|
ˆ
H1|ΦjΔ
1
(Eμ−Ej)
ΨΦ
j|
ˆ
H1|ΦmΔ
1
(Eμ−Em)
?ΨΦ
m|
ˆ
H1|ΦμΔ+···(1.17)
or, in terms of the Brillouin–Wigner-type resolvent deﬁned in eq. (1.15),
E
μ=Eμ+λΨΦ μ|
ˆ
H1|ΦμΔ+λ
2
ΨΦμ|
ˆ
H1B
ˆ
H1|ΦμΔ
+λ
3
ΨΦμ|
ˆ
H1B
ˆ
H1B
ˆ
H1|ΦμΔ+···(1.18)
It should be noted that the expansion (1.18) is not a simple power series expansion in
λ, since theE
μwhich appears in the denominators ofBalso depends onλ.However,
if the terms1/(E
μ−Em)are expanded in powers ofλ, then the usual Rayleigh–
Schr¨odinger series forE
μis recovered which is, as is well known, a simple power
series inλ.
In Brillouin–Wigner perturbation theory, we obtain the following expansion for the
ground state energy of the perturbed systems
E
0=E0+E
[1]
0
λ+E
[2]
0
λ
2
+E
[3]
0
λ
3
+···(1.19)
whereE
0is the ground state energy of the reference or model system and the energy
coefﬁcients,E
[1]
0
,E
[2]
0
,E
[3]
0
,...,aregivenby
E
[p]
0
=ΨΦ 0|
ˆ
H1
ν
B
ˆ
H
1
Φ
p−1
|Φ0Δ(1.20)
andB=B(0)is the Brillouin–Wigner resolvent for the ground state. We use square
brackets to distinguish the Brillouin–Wigner energy coefﬁcients from the Rayleigh–
Schr¨odinger coefﬁcients for which round brackets are used. We reiterate our obser-
vation that the expansion (1.19) is not a simple power series inλ, since each of the
energy coefﬁcients,E
(p)
0
[p>1]depends on the exact energyE.
1.3.2. Rayleigh–Schr¨odinger perturbation theory
In Rayleigh–Schr¨odinger perturbation theory, the expansion for the exact ground state
energy of the perturbed system can also be written in the form
E=E
0+E
(1)
0
λ+E
(2)
0
λ
2
+E
(3)
0
λ
3
+···(1.21)
whereλis again the perturbation parameter. An expansion in powers ofλis also
made for the exact ground state wave functionΨ
0,i.e.

Brillouin–Wigner Methods for Many-Body Systems 15
Ψ
0=Φ0+χ
(1)
0
λ+χ
(2)
0
λ
2
+χ
(3)
0
λ
3
+···(1.22)
whereχ
(p)
0
is thepth order wave function. The energy coefﬁcients,E
(p)
0
,aregiven
by
E
(p)
0
=
χ
d
p
E
dλ
p
δ
λ=0
(1.23)
whilst thepth order wave function,χ
(p)
0
,isgivenby
χ
(p)
0
=
χ
d
p
Ψ0
dλ
p
δ
λ=0
.(1.24)
The Rayleigh–Schr¨odinger perturbation theory is developed by substituting the
power series for the wave function (1.22) and that for the energy (1.21) into the
Schr¨odinger equation (1.3) in the form
λ
ˆ
H
0+λ
ˆ
H 1
Ω
Ψ
0=E0Ψ0(1.25)
and then equating powers ofλ. Explicitly, this equation then takes the form
λ
ˆ
H
0+λ
ˆ
H 1
Ωλ
Φ
0+χ
(1)
0
λ+χ
(2)
0
λ
2
+χ
(3)
0
λ
3
+···
Ω
=
λ
E
0+E
(1)
0
λ+E
(2)
0
λ
2
+E
(3)
0
λ
3
+···
Ω
λ
Φ
0+χ
(1)
0
λ+χ
(2)
0
λ
2
+χ
(3)
0
λ
3
+···
Ω
.(1.26)
Equating powers ofλleads to the basic equations of Rayleigh–Schr¨odinger perturba-
tion theory. The zero-order equation is simply the eigenvalue problem for the refer-
ence or model system with respect to which the perturbation expansion is developed:
ˆ
H
0Φ0=E0Φ0(1.27)
The ﬁrst-order equation takes the form
ˆ
H
0χ
(1)
0
+
ˆ
H1Φ0=E0χ
(1)
0
+E
(1)
0
Φ0(1.28)
the second-order equation is
ˆ
H
0χ
(2)
0
+
ˆ
H1χ
(1)
0
=E0χ
(2)
0
+E
(1)
0
χ
(1)
0
+E
(2)
0
Φ0(1.29)
and so on. In general, thepth order Rayleigh–Schr¨odinger perturbation equation takes
the form
ˆ
H
0χ
(p)
0
+
ˆ
H1χ
(p−1)
0
=
q=p−1
Φ
q=0
λ
E
(q)
0
χ
(p−q)
0
Ω
+E
(p)
0
Φ0.(1.30)
Without loss of generality, we can require that thep
th
order perturbed wave func-
tionsχ
(p)
0
be orthogonal to the reference function,Φ 0:
ΨΦ
0|χ
(p)
0
Δ=0,∀p.(1.31)

16 I. Hubaˇc and S. Wilson
Multiplying the zero-order equation (1.27) from the left byΦ
0and integrating gives
E
0=ΨΦ 0

ˆ
H
0

Φ
0Δ(1.32)
which deﬁnes the zero-order energy. Similarly, multiplying the ﬁrst-order equation
(1.28) from the left byΦ
0and integrating gives the ﬁrst-order energy
E
(1)
0
=ΨΦ 0

ˆ
H
1

Φ
0Δ.(1.33)
Thus, the ﬁrst-order energy is given by the matrix element of the perturbation oper-
ator for the reference wave function. An expression for the second-order energy can
be obtained by multiplying the second-order equation (1.29) byΦ
0and integrating
giving
E
(2)
0
=ΨΦ 0

ˆ
H
1

χ
(1)
0
Δ.(1.34)
The second-order energy therefore depends on the ﬁrst-order wave function.
An expression for the ﬁrst-order wave function may be obtained by rearranging the
ﬁrst-order perturbation equation (1.28) as
Ψ
E
0−
ˆ
H0
Δ
χ
(1)
0
=
Ψ
ˆ
H 1−E
(1)
0
Δ
Φ
0(1.35)
so that
χ
(1)
0
=
Ψ
E 0−
ˆ
H0
Δ
−1Ψ
ˆ
H
1−E
(1)
0
Δ
Φ
0(1.36)
or
χ
(1)
0
=R
Ψ
ˆ
H 1−E
(1)
0
Δ
Φ
0(1.37)
where
R=
Ψ
E
0−
ˆ
H0
Δ
−1
(1.38)
is the Rayleigh–Schr¨odinger resolvent.
Substituting eq. (1.37) for the ﬁrst-order wave function into expression (1.34),
gives the following expression for the second-order energy
E
(2)
0
=

Φ 0

ˆ
H
1R
ˆ
H1

Φ
0

.(1.39)
Thus the second-order energy is given by the matrix element of the operator
ˆ
H
1R
ˆ
H1
for the reference wave function.
Multiplying the third-order equation byΦ
0and integrating gives
E
(3)
0
=

Φ 0

ˆ
H
1

χ
(2)
0

(1.40)
and, in general,
E
(p)
0
=

Φ 0

ˆ
H
1

χ
(p−1)
0

.(1.41)
An expression for the second-order wave functionχ
(2)
0
, can be obtained from the
second-order perturbation equation (1.29) and then substituted in expression (1.40).

Brillouin–Wigner Methods for Many-Body Systems 17
In general, the(p−1)th order wave functionχ
(p−1)
0
can be obtained from the(p−1)th
order perturbation equation and then substituted in eq. (1.41).
The energy coefﬁcients in the Rayleigh–Schr¨odinger perturbation expansion have
a more complicated structure than those in the Brillouin–Wigner expansion and the
ﬁrst few orders take the form
(a) Zero-order energy coefﬁcient
E
0=ΨΦ 0|
ˆ
H0|Φ0Δ(1.42)
(b) First-order energy coefﬁcient
E
(1)
0
=ΨΦ 0|
ˆ
H1|Φ0Δ(1.43)
(c) Second-order energy coefﬁcient
E
(2)
0
=ΨΦ 0|
ˆ
H1R
ˆ
H1|Φ0Δ(1.44)
(d) Third-order energy coefﬁcient
E
(3)
0
=ΨΦ 0|
ˆ
H1R
ˆ
H1R
ˆ
H1|Φ0Δ?E
(1)
0
ΨΦ0|
ˆ
H1(R)
2ˆ
H
1|Φ0Δ(1.45)
(e) Fourth-order energy coefﬁcient
E
(4)
0
=ΨΦ 0|
ˆ
H1R
ˆ
H1R
ˆ
H1R
ˆ
H1|Φ0Δ?E
(1)
0
ΨΦ0|
ˆ
H1(R)
2ˆ
H
1R
ˆ
H1|Φ0Δ
−E
(1)
0
ΨΦ0|
ˆ
H1R
ˆ
H1(R)
2ˆ
H
1|Φ0Δ+(E
(1)
0
)
2
ΨΦ0|
ˆ
H1(R)
3ˆ
H
1|Φ0Δ
−E
(2)
0
ΨΦ0|
ˆ
H1(R)
2ˆ
H
1|Φ0Δ.(1.46)
In these expressions,Ris the Rayleigh–Schr¨odinger resolvent for the ground state,
which can be written in sum-over-states form as
R=R(0) =
Φ
r(Ψ=0)
|ΦrΔ
1
(E0−Er)
ΨΦ
r|.(1.47)
This should be compared with the Brillouin–Wigner resolvent (1.15) which for the
ground state takes the form
B(0) =
Φ
r(Ψ=0)
|ΦrΔ
1
(E0−Er)
ΨΦ
r|(1.48)
and which depends on the exact ground state energyE
0.
1.3.3. Comparison of Brillouin–Wigner and Rayleigh–Schr¨odinger
perturbation theories
In the preceding section, we have given elementary presentations of both the
Brillouin–Wigner perturbation theory and the Rayleigh–Schr¨odinger perturbation

18 I. Hubaˇc and S. Wilson
theory. We are now in a position to make an elementary comparison of the two meth-
ods. We approach this comparison by ﬁrst presenting the advantages of the Brillouin–
Wigner perturbation theory with respect to the Rayleigh–Schr¨odinger perturbation
theory, and then enumerating the disadvantages.
1.3.3.1. Advantages of the Brillouin–Wigner theory
We consider the advantages of the Brillouin–Wigner perturbation theory on the basis
of the elementary derivations given in Sections 1.3.1 and 1.3.2.
Brillouin–Wigner perturbation theory has ﬁve main advantages over the Rayleigh–
Schr¨odinger perturbation theory. These are:
(i) Rayleigh–Schr¨odinger theory can be regarded as an approximation to Brillouin–
Wigner theory.
This perspective on Brillouin–Wigner perturbation theory was mentioned by
Lennard-Jones in his seminal 1930 paper [37]. It was also described in the review
by Dalgarno [38] published in 1961.
Consider, for example, the second-order energy,E
[2]
0
, in the Brillouin–Wigner per-
turbation expansion for the ground state energy. From the third term on the right-hand
side of eq. (1.17), we see that this may be written in sum-over-states form as
E
[2]
0
=
Φ
rΨ=0
ΨΦ0|
ˆ
H1|ΦrΔΨΦr|
ˆ
H1|Φ0Δ
E0−Er
.(1.49)
If we make the approximation
E
0∼E0,(1.50)
then we are led immediately to the Rayleigh–Schr¨odinger second-order energy com-
ponent
E
(2)
0
=
Φ
rΨ=0
ΨΦ0|
ˆ
H1|ΦrΔΨΦr|
ˆ
H1|Φ0Δ
E0−Er
(1.51)
which is identical to eq. (1.44) upon substituting (1.47) for the resolvent. If we write
E
0=E0+ΔE 0(1.52)
or
ΔE
0=E0−E0(1.53)
then the denominator in (1.49) becomes
1
E0−Er
=
1
E0−Er+ΔE 0
=
1
E0−Er
+
ε
1
E0−Er
(−ΔE 0)
1
E0−Er+ΔE 0
∂
(1.54)

Brillouin–Wigner Methods for Many-Body Systems 19
and, ifΔE
0is small,
1
E0−Er
∼
1
E0−Er
(1.55)
and the second term on the right hand side of (1.54) is negligible.
(ii) Brillouin–Wigner theory is formally much simpler than the Rayleigh–
Schr¨odinger theory.
The relative simplicity of Brillouin–Wigner perturbation theory was noted by
Wigner in his original publication [41]. The simplicity of Brillouin–Wigner pertur-
bation theory was emphasized by Dalgarno in his review [38].
This simplicity is evident from a comparison of eqs. (1.15), (1.16) and (1.18) deﬁn-
ing Brillouin–Wigner perturbation theory with eqs. (1.27)–(1.30), (1.42)–(1.46) and
(1.47) for the Rayleigh–Schr¨odinger theory. In fact, eqs. (1.15), (1.16) and (1.18)
provide a complete deﬁnition of Brillouin–Wigner perturbation theory through all
orders.
The Brillouin–Wigner perturbation series is a simple geometric series, whereas in
every order beyond second-order in the energy, the Rayleigh–Schr¨odinger perturba-
tion theory gives rise to a very much more complicated structure. In every order of
the Rayleigh–Schr¨odinger perturbation theory, there is a principal term of the form
E
(p)
0
(principal)=ΨΦ 0|
ˆ
H1
λ
R
ˆ
H
1
Ω
p−1
|Φ0Δ,p=1,2,...(1.56)
which is analogous to the Brillouin–Wigner term deﬁned in (1.20)
E
[p]
0
=ΨΦ 0|
ˆ
H1
λ
B
ˆ
H
1
Ω
p−1
|Φ0Δ(1.57)
together with other terms in third and higher orders which are often called ‘renormal-
ization’ terms for reasons that will be elaborated below. There is only one ‘renormal-
ization’ term in third-order, four in fourth-order, thirteen in ﬁfth order, rising to, for
example, 4,861 in tenth order of perturbation [54].
(iii) Convergence of the Brillouin–Wigner perturbation theory is often more rapid
than that of the Rayleigh–Schr¨odinger theory for a given problem.
This advantage of the Brillouin–Wigner perturbation theory was recognized in the
original papers by Lennard-Jones [37] and of Brillouin [40]. It is also described in
the review by Dalgarno [38].
For example, it is well known that energy eigenvalue for a two-state problem is
given exactly by second-order Brillouin–Wigner perturbation theory. However, the
Rayleigh–Schr¨odinger perturbation expansion, in general, must be taken to inﬁnite
order to solve this simple problem. Speciﬁcally, taking a zero-order matrix

20 I. Hubaˇc and S. Wilson
H
0=
ε
00
0α
∂
,(1.58)
and a perturbation
H
1=
ε
0β
β0
∂
,(1.59)
and then solving the secular problem

−εβ
βα−ε

=0(1.60)
or
ε
2
−αε−β
2
=0,(1.61)
gives an exact solution
ε=
1
2
ϕ
α±
σ
α
2
+4β
2
τ
.(1.62)
The Brillouin–Wigner perturbation expansion through second-order for this prob-
lem is
ε=
β
2
ε−α
(1.63)
which is seen to be equivalent to eq. (1.61) and thus the exact solution, eq. (1.62).
For the two-state problem deﬁned above, second-order Brillouin–Wigner perturba-
tion theory provides an exact result. The exact energy is supported by second-order
Brillouin–Wigner perturbation theory no matter what the speciﬁc values ofαandβ
are assumed.
The Rayleigh–Schr¨odinger perturbation series for this problem can be obtained by
writing eq. (1.62) in the form
ε=
α
2
υ
1±
√
1+x
≈
,x=
4β
2
α
2
(1.64)
and using the identity
√
1+x =1+
1
2
x−
1
8
x
2
+
1
16
x
3
−
5
128
x
4
+
7
256
x
5
−
21
1026
x
6
...(1.65)
to obtain an expansion in powers ofβ. The Rayleigh–Schr¨odinger expansion there-
fore has the form
ε=−
β
2
α
+
β
4
α
3
−2
β
6
α
5
+···(1.66)
and summation to all orders is required to obtain the exact energy eigenvalue.

Brillouin–Wigner Methods for Many-Body Systems 21
(iv) Brillouin–Wigner perturbation theory may converge for problems for which
the Rayleigh–Schr¨odinger theory does not.
Wigner [41] noted this advantage of the Brillouin–Wigner perturbation theory in
his 1935 paper. Dalgarno [38] also identiﬁed this advantage of Brillouin–Wigner
perturbation theory in his 1961 review.
An example of this advantageous feature of Brillouin–Wigner perturbation the-
ory is again provided by the two-state problem deﬁned by eqs. (1.58) and (1.59).
The Brillouin–Wigner perturbation expansion leads to an exact solution by second-
order, irrespective of the particular values of the parameters deﬁning the zero-order
problem and the perturbation, i.e.αandβ. For this problem, the Brillouin–Wigner
perturbation expansion has an inﬁnite radius of convergence. Let us explicitly intro-
duce the perturbation parameter,λ, and determine the radius of convergence of the
Rayleigh–Schr¨odinger perturbation expansion. We write the Hamiltonian matrix for
the perturbed problem as
H
0+λH 1=
ε
00
0α
∂
+λ
ε
0β
β0
∂
(1.67)
which has exact solutions
ε=
1
2
ϕ
α±
σ
(α
2
+4β
2
λ
2
)
τ
.(1.68)
The Rayleigh–Schr¨odinger perturbation expansion now takes the form
ε=−
β
2
α
λ
2
+
β
4
α
3
λ
4
−2
β
6
α
5
λ
6
+... .(1.69)
The radius of convergence may be obtained directly by setting the discriminant in
eq. (1.68) to zero, i.e.α
2
+4β
2
λ
2
=0. We are thus led to the result that the Rayleigh–
Schr¨odinger perturbation expansion for the two-state model converges for values of
λ, satisfying
|λ|>
1
2

α
β

.(1.70)
For values ofλwhich do not satisfy the condition (1.70), the expansion (1.69) will
diverge.
It is important to note that, by forming certain approximants, it is usually possi-
ble to obtain valuable estimates of the energy eigenvalue. For example, by employ-
ing Pad´e approximants in cases where the Rayleigh–Schr¨odinger perturbation the-
ory diverges, useful energy values can beobtained, because such approximants are
usually able to handle a wider class of functions than the power series assumed in
perturbation theory.

22 I. Hubaˇc and S. Wilson
(v) Brillouin–Wigner perturbation theory is often more convenient to use for
degenerate problems.
This is one of the properties of the method which was emphasized by Wigner
[41] in his original paper. Indeed the Brillouin–Wigner perturbation theory can be
formally applied to degenerate problems without modiﬁcation.
For example, settingα=0in the two-state problem deﬁned by eqs. (1.58) and
(1.59) above, so as to give rise to a degenerate problem, leads immediately from
expressions (1.60) and (1.61) to the exact solutionε
2
=β
2
. The Brillouin–Wigner
second-order energy (1.64) becomesε=β
2
/εwhich clearly agrees with the exact
solution. On the other hand, the non-degenerate Rayleigh–Schr¨odinger perturbation
theory breaks down, because the radius of convergence, condition (1.70), is now zero.
1.3.3.2. Disadvantages of the Brillouin–Wigner theory
In spite of the ﬁve advantages listed in the preceding section, the Brillouin–Wigner
perturbation theory also has some disadvantages. Speciﬁcally, Brillouin–Wigner per-
turbation theory has two disadvantages in comparison with the Rayleigh–Schr¨odinger
perturbation theory. These disadvantages have prevented the widespread use of
Brillouin–Wigner methods over the past 50 years or so.
We will consider ﬁrst the least problematic of the disadvantages of the Brillouin–
Wigner perturbation theory and then the major negative feature of the theory in the
modelling of many-body systems.
(i) Brillouin–Wigner perturbation theory is iterative, since the exact energy is con-
tained in the denominators arising in the expressions for the energy components.
Consider, for example, the Brillouin–Wigner perturbation expansion for the ground
state energy truncated at second-order which takes the form
E
(2)
=ΨΦ 0|H0|Φ0∗+ΨΦ 0|H1|Φ0∗+
∞
rΨ=0
ΨΦ0|H1|ΦrΔΨΦr|H1|Φ0∗
E
(2)
−Er
.(1.71)
(We omit the subscript0to simplify the notation here, since we are concerned exclu-
sively with the ground state.) Equation (1.71) is solved iteratively. The right-hand-
side of this equation may be writtenf
(2)
(E
(2)
)and eq. (1.71) itself can then be put
in the form
E
(2)
i+1=f
(2)
(E
(2)
i).(1.72)
Taking an initial valueE
(2)
0
successive, application of the recursion (1.72) deﬁnes
a sequence of approximationsE
(2)
i
,i=0,1,2,..., which (if they are convergent)
converge to the second-order Brillouin–Wigner energy through second order, which
we denoteE
(2)
in the present discussion. In general, of course,E
(2)
is not necessarily
equal to the exact energy,E, but an approximation to it.
Through the third-order of the Brillouin–Wigner perturbation theory, we have for
the ground state energy:

Brillouin–Wigner Methods for Many-Body Systems 23
E
(3)
=ΨΦ 0|H0|Φ0Δ+ΨΦ 0|H1|Φ0Δ+
∀
rΨ=0
ΨΦ0|H1|ΦrΔΨΦr|H1|Φ0Δ
E
(3)
−Er
+
∀
rΨ=0
∀
sΨ=0
ΨΦ0|H1|ΦrΔΨΦr|H1|ΦsΔΨΦs|H1|Φ0Δ
(E
(3)
−Er)(E
(3)
−Es)
.(1.73)
The right-hand side of this equation may be written asf
(3)
(E)and eq. (1.73) itself
can then be put in the form
E
(3)
i+1
=f
(3)
(E
(3)
i
).(1.74)
As for the second-order energy, we take an initial valueE
(3)
0
and generate, by succes-
sive application of (1.74), a sequence of approximationsE
(3)
i
,i=0,1,2,...,which
(if they are convergent) converge to the Brillouin–Wigner energy through third-order,
which we denoteE
(3)
. Again, in general,E
(3)
is not necessarily equal to the exact
energy,E. When we substitute the converged value ofE
(3)
into (1.73), the third term
on the right-hand side is not necessarily equal to the second-order Brillouin–Wigner
energy obtained by iterative solution of (1.71).
In contrast to the Brillouin–Wigner approach, the Rayleigh–Schr¨odinger perturba-
tion theory is manifestly non-iterative. The denominators in Rayleigh–Schr¨odinger
perturbation theory depend on the energies obtained by solution of the reference
model equations. Furthermore, extending, say, a second-order Rayleigh–Schr¨odinger
perturbation theory to third-order or higher, does not change the second-order energy
component.
(ii) Brillouin–Wigner perturbation theory is not explicitly a many-body theory, in
that the energy expressions in each order do not scale linearly with particle number.
For example, the application of second-order Brillouin–Wigner perturbation theory
to an array ofnwell separated, i.e. non-interacting, ground state helium atoms, does
not yield a total energy which isntimes that of a single helium atom. This is a
direct consequence of the presence of the exact energy in the denominator in the
second-order energy expression. This property of the Brillouin–Wigner expansion
which was ﬁrst pointed out by Brueckner [16] in the mid-1950s, is the main reason
for the paucity of applications of the method to the atomic and molecular electronic
structure problem until recent years.
In their treatise on the many-body problem in quantum mechanics, March, Young
and Sampanthar [55] write that
the Brillouin–Wigner form of the [many-body] theory is completely inap-
propriate.

24 I. Hubaˇc and S. Wilson
They explicitly show that
...the ﬁrst term∝Nand that all succeeding terms [are] individually neg-
ligible compared to the ﬁrst term, whatever the strength of the perturbation.
However, it would be wrong to conclude that the above argument proves that
the Brillouin–Wigner series is convergent and that only the ﬁrst term need
be considered. Among other reasons, we know that [the ﬁrst-order term]
does not include dynamical correlations between particles and these must be
important (on physical grounds) for a strongly interacting system. The sit-
uation therefore must be that many(N)of the small terms beyond the ﬁrst
are of roughly comparable size, and add up to change the energy/particle by
a ﬁnite amount. Thus it will be completely misleading to apply many-body
perturbation theory in the Brillouin–Wigner form, short of considering an
inﬁnite number of terms in the limit of largeN.
In contrast to the Brillouin–Wigner perturbation theory, it is well known that
Rayleigh–Schr¨odinger perturbation theory in its “many-body” form does afford a
theoretical basis for the description of many-body systems. Again, March, Young
and Sampanthar [55] write:
the Rayleigh–Schr¨odinger perturbation theory...if carefully and system-
atically used, can yield an energy/particle proportional toNas required, in
spite of the appearance of spurious terms proportional toN
2
, etc., in any
given order. In fact, Brueckner showed that the non-physical terms cancel
up to fourth-order, and the generalization to all orders was effected by Gold-
stone.
They continue:
To third-order (and to any higher order) one can in fact verify that the
terms of the Rayleigh–Schr¨odinger perturbation expansion either vanish or
increase asNin the limitN, the non-physical terms proportional toN
2
,N
3
,
etc., all neatly cancelling each other.
and then conclude:
...the Rayleigh–Schr¨odinger series is more useful than the Brillouin–Wigner
form in that it yields an expansion leading to the energy per particle as inde-
pendent of the size of the system for largeN.
It is for these reasons, coupled with the efﬁciency of the corresponding computer
algorithms in low-order, that Rayleigh–Schr¨odinger perturbation theory, particularly
the Møller Plesset form [63], has been regarded as the method of choice in describing
electron correlation effects in atoms and molecules for more than 30 years [40, 58-
63].

Brillouin–Wigner Methods for Many-Body Systems 25
1.4. THE MANY-BODY PROBLEM
1.4.1. Linear scaling
The essential property of any true many-body theory of electronic structure is a linear
scaling of the energy components,E, with the number of electrons,N, in the system
[39, 55, 63, 64], that is,
E∝N.(1.75)
Any terms which scale non-linearly are unphysical and, therefore, should be dis-
carded. Equally, any theory which contains such unphysical terms is not acceptable
as a valid many-body method. Either the theory is abandoned or corrections are made
in an attempt to restore linear scaling, such as that of Davidson [65], which is used
in limited conﬁguration interaction studies.
With the development of many-body theories in the 1950s [16, 17], the shortcom-
ings of the Brillouin–Wigner perturbation expansion were widely recognized. The
presence of the exact energy in the Brillouin–Wigner denominators in the expres-
sions for the energy components ensures that the unphysical terms arise which scale
non linearly with the number of electrons. Theories which lead to a linear scaling
with the number of electrons are said to be ‘size-consistent’, ‘size-extensive’, or sim-
ply ‘extensive’ [2]. Brillouin–Wigner expansions are not ‘extensive’. They do not
scale linearly with the number of electrons in the system.
9
It was also recognized that
Brillouin–Wigner perturbation theory is not a simple power series in the perturbation
parameter,λ.
Brillouin–Wigner perturbation theory was, however, used as a step in the develop-
ment of an acceptable many-body perturbation theory most notably by Brandow [67]
in his pioneering work on multi-reference formalisms for the many-body problem.
In a review entitledLinked-Cluster Expansions for the Nuclear Many-Body Problem
and published in 1967, B.H. Brandow writes:
The Goldstone expansion is re-derived by elementary time-independent
methods, starting from Brillouin–Wigner (BW) perturbation theory. Inter-
action energy termsΔEare expanded out of the BW energy denominators,
and the series is then rearranged to obtain the linked-cluster result. Similar
algebraic methods lead to the linked expansions for the total wave function
(Hugenholtz) and the expectation value of a general operator (Thouless).
9
We would prefer to avoid the use of the terms ‘size-consistency’ and
‘size-extensivity’ and simply describe a theory as exhibiting correct scaling with
the number of particles considered. This point of view is also adopted by Nooijen,
Shamasundar and Mukherjee [66] who write: “The notions of size-extensivity and
size consistency are used very broadly in the literature and we prefer to speak more
speciﬁcally of a physical quantity that is to scale properly in the context of a partic-
ular type of physical system or state (e.g. open or closed shell).”

26 I. Hubaˇc and S. Wilson
As we have seen, in their 1967 text on many-body methods for quantum systems,
March, Young and Sampanthar [55] dismissBrillouin–Wigner perturbation theory as
a valid many-body technique. They write (p. 71):
...it will be completely misleading to apply many-body perturbation theory
in the Brillouin–Wigner form, short of considering an inﬁnite number of
terms in the limit of large N.
In a volume entitledAtomic Many-Body Theorypublished in 1982, Lindgren and
Morrison [63] state that:
The Brillouin–Wigner form of perturbation theory is formally very simple. It
has the disadvantage, however, that the operators depend on the exact energy
of the state considered. This requires a self-consistency procedure and limits
its application to one energy level at a time. There are also more fundamental
difﬁculties with the Brillouin–Wigner theory...The Rayleigh–Schr¨odinger
perturbation theory...does not have these shortcomings, and it is therefore
a more suitable basis for many-body calculations than the Brillouin–Wigner
form of the theory.
In his monographElectron correlation in molecules, Wilson [39] writes (p. 65):
The perturbation expansion of Lennard-Jones, Brillouin and Wigner does not
lead to expressions which are directly proportional to the number of electrons
in the system being studied.
As recently as 1992, in a book entitledAlgebraic and Diagrammatic Methods in
Many-Fermion Theory, Harris, Monkhorst and Freeman [64] write (p. 224):
A...fundamental difﬁculty with the Wigner–Brillouin expansion is its lack
of size consistency.
The above extracts serve to demonstrate what had until recently been the ‘standard
view’ of Brillouin–Wigner perturbation theory and its applicability to many-body
systems.
1.4.2. The re-emergence of Brillouin–Wigner methods
The use of Brillouin–Wigner perturbation theory in describing many-body systems
has been critically re-examined in recent years [69–96]. It has been shown that under
certain well-deﬁned circumstances, it can be regarded as a valid many-body theory.
The primary purpose of this volume is to provide a detailed and coherent account of
the Brillouin–Wigner methods in the study of the ‘many-body’ problem in atomic
and molecular quantum mechanics.
The renewal of interest in Brillouin–Wigner perturbation theory for many-body
systems seen in recent years, is driven by the need to develop a robust multi-reference
theory. As was mentioned in the ﬁrst section of the present chapter, multi-reference

Brillouin–Wigner Methods for Many-Body Systems 27
formalisms are an important prerequisite for theoretical descriptions of dissociative
phenomena. Brillouin–Wigner perturbation theory is seen as a remedy to a prob-
lem which plagues multi-reference Rayleigh–Schr¨odinger perturbation theory: the
so-called ‘intruder state’ problem.
Multi-reference Rayleigh–Schr¨odinger perturbation theory is designed to describe
a manifold of states. However, as the perturbation is ‘switched on’, the relative dispo-
sition of these states and states outside the reference space, may change in such a way
that convergence of the perturbation series is impaired or even destroyed. States from
outside the reference space, which assume an energy below that of any of the states
among the reference set when the perturbation is switched on, are termed ‘intruder
states’. The situation is illustrated schematically in Figure 1.1, which provides a
schematic representation of the intruder state problem. In this ﬁgure, the reference
spacePconsists of three states with energiesE
0,E1andE 2, which are represented
on the left-hand side. The exact energiesE
0,E1,...obtained when the perturbation
is turned on, are represented on the right-hand side of the ﬁgure. But the exact energy
E
2corresponds toE 3in the reference space. This is an intruder state.
unperturbed system perturbed system
λ=0 λ=1
intruder state
Ψ
ΨΨΔ
P
E0
E1
E2
E3
E4
E5
E6
E0
E1
E2
E3
E4
E5
E6
Φ
λ
Figure 1.1.In multi-reference Rayleigh–Schr¨odinger perturbation theory, states from outside the refer-
ence space,P, which assume an energy below that of any state among the reference set when the pertur-
bation is switched on, are termed ‘intruder states’.

28 I. Hubaˇc and S. Wilson
Intruder states arising when0<λ≤+1often have a physical origin. The
so-called ‘backdoor’ intruder states, which arise for−1≤λ<0, are frequently
unphysical. The occurrence of ‘backdoor’ intruder states is illustrated schematically
in Figure 1.2. In this ﬁgure, the reference spacePagain consists of three states with
energiesE
0,E1andE 2, which are represented in thecentral column. The perturbed
system, with energiesE
0,E1,..., is represented on the right-hand side and corre-
sponds toλ=+1. The ‘backdoor’ spectrum corresponding toλ=−1is shown on
the left-hand side. ‘Backdoor’ intruder states arise in this ‘backdoor’ spectrum.
Multi-reference Brillouin–Wigner theory overcomes the intruder state problem
because the energy is contained in the denominator factors. Calculations are there-
fore ‘state-speciﬁc’, that is, they are performed for one state at a time. This is in con-
trast to multi-reference Rayleigh–Schr¨odinger perturbation theory, which is applied
to a manifold of states simultaneously. Multi-reference Brillouin–Wigner perturba-
tion theory is applied to a single state. Wenzel and Steiner [77] write:
...the reference energy in Brillouin–Wigner perturbation theory is the fully
dressed energy...This feature guarantees the existence of a natural gap and
thereby rapid convergence of the perturbation series.
Multi-reference Brillouin–Wigner perturbation theory overcomes the intruder state
problem which has plagued multi-reference Rayleigh–Schr¨odinger perturbation the-
ory for many years. However, in general, Brillouin–Wigner expansions do not scale
linearly with the number of electrons; they are not extensive. These insights have led
to the critical re-examination of Brillouin–Wigner perturbation theory in describing
many-body systems in recent years.
When we consider the application of multi-reference Brillouin–Wigner methods to
many-body systems, two distinct approaches can be taken which we consider now in
turn:
(i) The Brillouin–Wigner perturbation theory can be employed to solve the equa-
tions associated with an explicit ‘many-body’ method.
For example, the full conﬁguration interaction problem can be solved by making
a Brillouin–Wigner perturbation theory expansion through inﬁnite order, as can the
equations of the coupled cluster theory. In the case of the full conﬁguration interac-
tion, it is obvious that provided the Brillouin–Wigner perturbation series is summed
through all orders, then a result will be obtained which is entirely equivalent, at
least as far as the results are concerned,
10
to any other technique for solving the full
conﬁguration interaction problem. We might designate the Brillouin–Wigner-based
approach to the Full Conﬁguration Interaction (
FCI) model “Brillouin–Wigner-Full
Conﬁguration Interaction” or (
BW-FCI).
Hubaˇc and Neogr´ady [68] have explored the use of Brillouin–Wigner perturbation
theory in solving the equations of coupled cluster theory. In a paper published in
The Physical Reviewin 1994, entitledSize-consistent Brillouin–Wigner perturbation
10
We are not concerned here with questions of computational eﬃciency.

Brillouin–Wigner Methods for Many-Body Systems 29
unperturbed systemperturbed system ‘backdoor’ spectrum
λ=0λ=1
λ=−1
‘backdoor’ intruder state
ΦΦΨ
P
E
0
E
1
E
2
E
3
E
4
E
5
E
6
E
0
E
1
E
2
E
3
E
4
E
5
E
6
Φ Ψ
Figure 1.2.In multi-reference Rayleigh–Schr¨ odinger perturbation theory, states from outside the reference space,P, which assume an energy below that of any
state among the reference set for −1≤λ<0are termed ‘backdoor’ intruder states. Un like the intruder states corresponding to 0<λ≤+1, which often have a
physical origin, ‘backdoor’ intrude r states are frequently unphysical.

30 I. Hubaˇc and S. Wilson
theory with an exponentially parametrized wave function: Brillouin–Wigner coupled
cluster theory, they write
Size-consistency of the Brillouin–Wigner perturbation theory is studied using
the Lippmann–Schwinger equation and an exponential ansatz for the wave
function. Relation of this theory to the coupled cluster method is studied and
a comparison through the effective Hamiltonian method is also provided.
By adopting an exponential expansion for the wave operator, they ensure that their
method is extensive. Hubaˇc and his co-workers, Neogr´ady and M´aˇsik, obtained the
“Brillouin–Wigner” coupled cluster theory [68–71] which is entirely equivalent to
other many-body formulations of coupled cluster theory for the case of a single-
reference function [104–107], since the Brillouin–Wigner perturbation expansion is
summed through all orders. We designate the Brillouin–Wigner-basedapproach to
the Coupled Cluster (
CC) model “Brillouin–Wigner Coupled Cluster” or (BW-CC).
If, for example, the equations of a limited coupled cluster expansion, such as that
usually designated
CCSD(in which all single and double excitation cluster operators
with respect to a single-reference function are considered), are solved by means of
Brillouin–Wigner perturbation theory, then a method, designated ‘Brillouin–Wigner-
Coupled Cluster Singles Doubles’ or (
BW-CCSD) theory, is obtained which is entirely
equivalent to ‘standard’
CCSDtheory.
(ii) Aposterioricorrections can be developed for calculations performed by using
the Brillouin–Wigner perturbation expansion. Thesea posterioricorrections can be
obtained for the Brillouin–Wigner perturbation theory itself and, more importantly,
for methods such as limited conﬁguration interaction or multi-reference coupled clus-
ter theory, which can be formulated within the framework of a Brillouin–Wigner per-
turbation expansion.
Thesea posterioricorrections are based on a very simple idea which is suggested
by the work of Brandow mentioned in the previous section, Section 1.4.1. Brandow
used the Brillouin–Wigner perturbation theory as a starting point for a derivation of
the Goldstone “linked diagram” expansion “by elementary time-independent meth-
ods”. At a NATO Advanced Study Institute held in 1991, Wilson wrote [107]:
The Rayleigh–Schr¨odinger perturbation theory can be derived from the
Lennard-Jones–Brillouin–Wigner perturbation theory by expanding the
energy-dependent denominators which occur in the latter.
In the work of Brandow [67], Brillouin–Wigner perturbation theory is used as a step
in the theoretical development of ﬁrst Rayleigh–Schr¨odinger perturbation theory and
then the many-body perturbation theory. In thea posterioricorrection developed by
the present authors in a paper [62] entitledOn the use of Brillouin–Wigner perturba-
tion theory for many-body systemsand published in theJournal of Physics B: Atomic,
Molecular and Optical Physicsin 2000, they write:
The use of Brillouin–Wigner perturbation theory in describing many-body
systems is critically re-examined.

Brillouin–Wigner Methods for Many-Body Systems 31
Brillouin–Wigner perturbation theory is employed as a computational technique – a
technique which avoids the intruder state problem – and then the relation between
the Brillouin–Wigner and Rayleigh–Schr¨odinger propagators is used to correct the
calculation for lack of extensivity.
If we compare the Brillouin–Wigner resolvent for the ground state given in
eq. (1.15)
(1.15)B=
λ
mΨ=0
|ΦmΔ
1
(E0−Em)
ΨΦ
m|
with the corresponding Rayleigh–Schr¨odinger resolvent given in eq. (1.47)
(1.47)R=
λ
mΨ=0
|ΦmΔ
1
E0−Em
ΨΦm|,
we then see that they differ only in the denominator factors. Using identity relation
[80]
(E−E
k)
−1
=(E 0−Ek)
−1
+(E 0−Ek)
−1
(−ΔE)(E−E k)
−1
(1.76)
where the exact ground state energy is written as
E
0=E0+ΔE 0,(1.77)
whereE
0is the ground state eigenvalue ofH 0andΔE 0is termed the level shift, we
can relate the Brillouin–Wigner resolvent to the Rayleigh–Schr¨odinger resolvent. In
this way, we can ﬁnda posterioriextensivity corrections to any Brillouin–Wigner
perturbation series.
We know that the Rayleigh–Schr¨odinger perturbation theory series leads directly
to the many-body perturbation theory by employing the linked diagram theorem. This
theory uses factors of the form(E
0−Ek)
−1
as denominators. Furthermore, this the-
ory is fully extensive; it scales linearly with electron number. The second term on the
right-hand side of eq. (1.76) can be viewed asan ‘extensivity correction term’ for the
Brillouin–Wigner series: a correction term which recovers the Rayleigh–Schr¨odinger
and many-body perturbation theoretic formulations. This simple idea has been used
to ﬁnda posterioricorrection for the limited conﬁguration interaction method [83],
as well as for state-speciﬁc multi-reference Brillouin–Wigner coupled cluster theory
[81,82]. Indeed,a posterioricorrections for a lack of extensivity can be obtained for
anyab initioquantum chemical method, provided that the method can be formulated
within Brillouin–Wigner perturbation theory.
Whereas the multi-reference Rayleigh–Schr¨odinger perturbation theory approxi-
mates a manifold of states simultaneously, the multi-reference Brillouin–Wigner per-
turbation theory approach is applied to a single state – it is said to be ‘state-speciﬁc’.
The multi-reference Brillouin–Wigner perturbation theory avoids the intruder state
problem. If a particular Brillouin–Wigner-based formulation is not a valid many-body
method, thena posterioricorrection can be applied. This correction is designed to
restore the extensivity of the method. This extensivity may be restored approximately

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move out to sea at once and take our own risk on this matter."
"Impossible," dissented Silverado, smiling sweetly, but with the
latent malice of triumph in his undertone. "Law of the nations--no right
to imperil the general safety. No, within two weeks we will give you
clearance if no new cases break out. Meantime----"
The officer coolly affixed the sealed document in his hand to the
mainmast.
Captain Broadbeam wriggled, fumed, groaned. He was too
thorough a seaman to mistake his predicament. His brow grew dark

and threatening.
"Bob, quick, come here."
With a violent jerk Dave Fearless pulled his startled chum to one
side.
"Quick as you can," he spoke rapidly, "rush to the purser. Tell him
to instantly send me up a rag that has been well saturated in
turpentine."
"Why, Dave----"
"No questions, no delay," ordered Dave peremptorily.
Bob shot away on his mission, Dave set his teeth, breathing hard.
In a flash a sinister suspicion had arisen in his mind. Like lightning
memory flew back to the overheard interview on the porch of the
native pilot between that crafty individual and the tricky Schmitt-
Schmitt.
"He said he could delay the Swallow, he hinted at spots, some
paint, at washing them off," mused Dave. "Good for you. Hold on."
Dave snatched the rag soaked with turpentine from Bob Vilett's
hands. He ran forward now to where his friends were depressedly
watching Tompkins arranging his shirt to replace it.
Dave made a dash at the man. He held him firmly by one shoulder.
With his free hand he slapped the rag briskly over his bare flesh to and
fro.
Dave's eyes sparkled immediately with the intensest satisfaction.
One by one the dark spots on the back of Tompkins began to
disappear.
"Captain Broadbeam," cried Dave, pulling the squirming Tompkins
around into full view, "a paint-trick. This man has got no more spotted
fever than I have myself."

CHAPTER VII
THE MYSTERIOUS JAR
Dave Fearless had saved the day. The young ocean diver knew this the
moment he glanced at the faces of those about him.
The wretch Tompkins shrank and cowered in a guilty manner. The
squeamish crew looked relieved. The governor's physician and his
military companion affected a profound astonishment, but secretly
were overwhelmed with confusion and chagrin.
Captain Broadbeam's eyes opened wide in amazement at the first.
Then as he guessed it out that a plot against him had been attempted
they blazed with wrath.
"Put that man in irons," he roared out.
"Pardon, captain," interrupted Silverado, stepping forward, "we will
do that. There is some grave mistake here."
"Mistake?" shouted Broadbeam. "Villainy, a conspiracy. Why----"
"The governor will investigate this matter thoroughly," said
Silverado.
Dave had glided to the captain's side. In a quick undertone he
advised him to smother his wrath for policy's sake. They allowed their
visitors to hustle Tompkins into their boat. To the last Silverado wore a
suave mask of forced politeness.
"You vile scum," broke out Broadbeam, shaking his fist after the
departing yawl. "It's hard to keep the bit between my teeth and say
nothing when I know that all hands from the governor down are in this
dirty plot."
The old salt bestowed an approving look on Dave and hustled to
the forecastle, calling the crew around him.

"Dave, how did you ever come to think of it?" marveled Bob Vilett.
"Why, it was simple--putting two and two together. I remembered
the pilot's talk about paint," replied Dave. "Hear that! Captain
Broadbeam is on his mettle."
Both boys listened to the sonorous voice of the commander of the
Swallow. He was greatly aroused. They heard him give orders to have
the entire armament of the Swallow put in active commission. A stand
of rifles was to be set ready for use. To Mr. Drake was delegated the
task of furbishing up two old brass ten-pounders from the hold.
"We sail to-morrow," announced the captain. "Look out for tricks
to-night. These villains won't let us go without meddling further if they
can help it. My men, I ask you all to stand by me if there's a
scrimmage, and there will be one if those fellows try to block my way."
Dave came in for a good deal of attention from the captain, Doctor
Barrell, and his father, when affairs had quieted down somewhat. They
all realized that his good memory and shrewd forethought had saved
them a vexatious delay and no end of further trouble from the
treacherous governor and his cohorts.
"I will be glad when we get clear of the island to-morrow," said
Dave, as Bob turned in for the night.
It had been a busy, exciting day, and Dave was glad to have a few
moments to himself to think over affairs in general.
He stretched himself on a heap of canvas in the shadow of the
rear cabin, overlooking the creek and the beautiful moonlit expanse
stretching out beyond it.
Dave mused, dozed, woke up, and stretched himself. He heard the
night-watch laughing and talking in low tones amidships.
"I'll join them, listen to one or two of their wild yarns, and then
turn in for the night myself," he decided.

Half-arising, however, Dave came to a rigid pose. He stared hard
beyond the rail and down into the still waters of the creek.
Everything was so calm and still that the least sound or movement
was vividly distinct to ear and eye.
Dave's eye had detected a ripple in the quiet waters. Then
momentarily a human head had protruded into view.
It bobbed down under water again. It came up ten feet nearer to
the Swallow. It disappeared once more, and this seemed to carry it
past the watcher's direct range of vision.
"Someone, and up to something," declared Dave to himself. "Hark,
now."
He bent his ear keenly. A soft drip-drip sounded just beyond the
rail. Then a black hand glistening with water clutched the rail itself.
Slowly, cautiously the body of a dusky native, attired only in
swimming garb, came into view. This was the person Dave had
detected swimming under water.
Straddling the rail, the intruder crouched, looking all about the
deck. Then he lifted both feet over onto the planking.
Dave now noticed that the man carried under one arm quite a
bulky package done up in black oilskin.
The intruder glanced sharply at the forecastle. Just abutting it was
a box-like section into which all kinds of odds and ends of canvas and
ropes were bundled. Its door was half-ajar. Dave saw the stranger glide
to this, thrust his package inside, glide back to the rail, slip over it, and
drop into the water.
A minute later the ripples in the creek showed where the fellow
was making his retreat under water. His head came up to the surface
once or twice. Then he arose at a distance down the stream and
disappeared among the dense shrubbery lining the creek.

"More mischief," instantly decided Dave Fearless.
Dave made a rush for the forecastle cubby hole. He pulled its door
wide open and groped about. His fingers closed about a dripping object
there.
"Hard and heavy," said Dave. "Wrapped in the oilskin to protect it.
What can it be?"
Dave arose to his feet. Suddenly a thrill passed through his frame.
"Put here for a purpose," he thought. "Can it be an explosive!"
Internally Dave became immensely excited. Coolly, however,
though carrying the dubious object as though it were an egg, he
proceeded to the ship's rail nearest the shore.
Dave set the object gently on the rail, climbed over, took it up
again, and, holding it above his head in one hand, dropped into the
water.
The splash, slight as it was, aroused the watch. Two men came
hurrying to the rail.
"Hold on, there," challenged one of them.
"It's only me--Dave Fearless," came the retort promptly, "cooling
off--a little swim, that's all."
"You pick a fine time for it."
Dave laughed. He liked water, and swam with one hand, came
ashore, and went past its fringe of brush to a clearing.
"Now then," said Dave, with a great sigh of relief, at a safe
distance from the ship, "burst, if you want to!"
Dave had set the object he carried down on the ground. He
stepped back a few feet and surveyed it suspiciously.
"A bomb?" he questioned himself. "How am I going to find out?
Perhaps it's some infernal machine loaded with phosphorus. Then those

villains intended to burn the Swallow. Certainly this means some black
mischief."
Dave roamed about till he found a stout long reed. Then he began
to poke at the object he had brought from the ship. He finally managed
to remove its oilskin covering.
"It's a jar, a stone jar," he said, "queer and foreign-looking, like we
get snuff or preserved ginger in. Labeled, too, and seals across the top.
It don't look very dangerous, for all the sinister way it came aboard."
Dave did not belie his name. He dallied with the situation no
longer and now took up the jar fearlessly.
Its label resembled the covering used on a package of firecrackers.
The seal was of tin-foil stamped with similar characters in red.
"Chinese, that's sure," thought Dave. "Shall I risk it?" he
questioned himself, his fingers surrounding the jar cover.
Dave snapped the seal and removed the cover. A layer of tissue
paper showed. He pulled this out. A dense stench was emitted by the
jar. He poked his finger down into the contents. They were solid and
sticky.
"Why," said Dave, a good deal puzzled, sniffing vigorously, "it's
opium."
CHAPTER VIII
OUTWITTING AN ENEMY
Dave Fearless stood looking over the queer jar and its contents very
thoughtfully.
"Well," he declared at length, "this is a puzzle."

Under ordinary circumstances Dave might have supposed that
some sailor addicted to the use of opium had hired some emissary to
smuggle some of the drug aboard ship.
This, however, did not look rational in the present case. In the first
place the contents of the jar represented over a year's pay of the
average sailor. In the next place it was too easy to get it aboard by
ordinary methods to occasion all this mystery.
Of course Dave at once decided that the placing of the opium in
the forecastle cubby-hole was part and parcel of the same plot that had
nearly wrecked the Swallow, that later just that day had developed the
unsuccessful attempt at quarantining the steamer.
"What's the motive in this latest trick?" mused Dave. "Aha!" he
exclaimed suddenly, "have I guessed it right?"
A quick suspicion, a prompt suggestion came to Dave's mind. He
was speedy to act.
"I think I've struck the clew," he said--"I think I'm acting right in
this matter."
Dave, carrying the jar with him, wandered about till he found a
decayed tree stump. He emptied the opium into a hole in the wood and
covered it over with bark.
Dave scraped the jar and made a little ball of the leavings, a
sample of the stuff he might need for later experience and evidence.
This he did up in a piece of paper, shoving it in a safe pocket. He
washed out the jar thoroughly. Then he wandered about studying the
branches of various trees under which he passed. Several of these
Dave ascended like a boy bird's-nesting.
He was quite a long time in one tree-top. When he descended to
the ground he had the cover firmly attached to the jar, which he carried
as if extremely careful of its contents.

"If I am guessing things out right," said Dave, with a kind of
satisfied chuckle, "I think we shall give our enemies quite a novel
surprise."
Dave swam back to the steamer. Arrived on deck he placed the jar
just where he had originally found it. Then he went to bed.
He overslept himself next morning. The ship was a scene of bustle
and activity. When he came up on deck, every member of the crew
proper was busy, even Bob Vilett.
So Dave found no opportunity to make a confidant of his special
chum, even had that been his desire or intention.
At nine o'clock Captain Broadbeam announced that all was ready
for their departure, and ordered steam up.
Within thirty minutes of getting under way the boatswain hurried
from the bow to where the captain was standing amidships.
"Coming again, sir," he announced, touching the peak of his cap
respectfully.
"Who's coming?" demanded Broadbeam.
"Those buzzards--same gang in the longboat that was here last
night."
"Humph!" growled the captain, gazing stormily at a yawl just
rounded from open water into the mouth of the creek.
The approaching craft was directed by the plausible Silverado.
Smiling as ever he came on board, three men with him.
"From his excellency the governor," he said.
"Yes, yes," answered Captain Broadbeam crossly; "I know all that
rigmarole. What do you want?"
"A complaint, captain."
"Who from?"
"I do not know."

"What about?"
"Contraband goods--smuggling."
Captain Broadbeam laughed in the officer's face outright.
"Guess not," he said. "I reckon, my friend, about all we will take
away from Minotaur Island will be a mighty poor opinion of its
inhabitants."
"Oh, I trust not," the polite official hastened to say, but added
tersely: "We must make a search."
"What for?"
"I have told you--contraband goods. We are having a good deal of
trouble in this line. Ships touching here make the island a sort of
clearing house for dutiable imports and exports. Our governor's high
sense of honor demands extreme vigilance and discipline. We are
authorized to make a search."
"Search away," cried Broadbeam indifferently, but with some show
of mental irritation.
Silverado and his aids went into the hold. They made a great
pretense of looking through the lockers in the cabins.
"Well?" demanded the captain of the Swallow as they came on
deck again, "found any smuggled goods?"
"None," reported Silverado promptly--"none, I am pleased to say."
"Then you give us a clean sheet on health and cargo, do you?"
said Broadbeam. "Reason I ask, is that we are going to swing out of
harbor soon as you get through with your tomfoolery."
Just here one of the officer's assistants came up and whispered in
the ear of his superior. He pointed at the forecastle.
"Yes, yes," nodded Silverado, "take a look there, and be thorough."
"Getting warm!" chuckled Dave to himself--"the precious
hypocrites!"

The man went into the forecastle and came out again. He looked
into the water barrel. He lifted some box covers. Just as Dave guessed
he would do, he kept up all this wise pretense until he landed up
against the forecastle cubby-hole.
"I have found something," he announced, after groping in the
hole. He had brought forth the stone jar.
"Ah, what is this?" spoke the officer. "Captain," he added,
assuming great sudden gravity as he inspected the jar, "this looks
pretty serious."
"Well, what's the mare's nest now?" petulantly demanded
Broadbeam.
The officer held up the jar in plain view.
"It is what we expected to find," he announced severely. "It is
opium. We know that last week a tramp steamer landed a lot of the
stuff on the island. The labels show that this is part of the same
contraband cargo. I declare this package and the Swallow under
confiscation, and arrest you. You must come to the governor."
"Oh, that so?" slowly spoke Captain Broadbeam, his shoulders
hunching dangerously. "I never saw that jar before, and, shiver my
timbers!" roared the incensed old captain, shaking his fist vigorously
under Silverado's nose, "I don't know the stuff is opium."
"Oh, yes, captain," insisted the officer. "The labels are
unmistakable. Look for yourself. Ough!"
With smart-Aleck readiness the suave Silverado untwisted the jar
cover. With a sharp cry he dropped it. In a cloud, a stream, there
instantly darted out from the receptacle an angry procession of
hornets.
They lit on those nearest to the jar, the officer and his assistants.
One of his aides was a special target. The poor fellow ran to the side to

escape them. He set up renewed yells as they stuck, pestered, and
stung. Then, splash! he took a reckless header into the waters of the
creek to escape his pertinacious tormentors.
Silverado lost all his usual calm dignity trying to evade the little
pests. He bit his lips and scowled as the captain faced him with a loud
derisive guffaw.
"Here, take away your contraband goods with you," shouted
Broadbeam, dropping jar and cover into the yawl, as the official hastily
descended into it, a crestfallen look on his face. "Ready, there," he
added to the boatswain. "Steam up."
"Aye, aye, sir."
Captain Broadbeam stepped to the little pilot house. He touched
an electric button.
Dave watched the maneuver with a glowing face. He was full of
the successful guess he had made concerning the planted opium, but
he did not try to explain that just then.
The jar of the starting steam below communicated a vibrating thrill
to his nerves. Dave ran up to Amos Fearless as the veteran diver
crossed the deck.
"Good news, father!" cried Dave gayly, "We've started."
"Hey and hallo for me paternal dominions--once more for the
Windjammers' Island and the stolen threasure!" shouted Pat Stoodles,
cutting a caper.
"Will we find it, I wonder?" sighed the old diver thoughtfully.
"I think we shall, father," answered Dave Fearless, with confidence.
CHAPTER IX
A BOLD PROJECT

The Swallow cleared her moorings in the creek on Minotaur Island, and
steamed out into the broad waters of the bay, a thing of life and
beauty.
"And what's that for now?" asked Pat Stoodles of Dave, who was
watching their progress and the coastline with great interest.
"I see," nodded Dave. "You mean the longboat from the
governor?"
"That same, lad. Luk at 'em, now. Ever since we came into open
wather they've been tearing along for the town like mad. Aha, there
goes one of those measly marines overboard."
Dave ran for a telescope. He viewed the government boat with a
good deal of curiosity.
The official, Silverado, stood up in the stern gesticulating with
energy, and evidently inciting his men to their best efforts at the oars.
"In a hurry to reach town, it seems," muttered Pat.
"In a tremendous hurry," said Dave. "So much so, that one of the
men has leaped overboard, waded ashore, and is making a lickety-
switch run across lots for the town."
Dave went at once to Captain Broadbeam and apprized him of the
maneuvers of their recent visitors.
"That's all right, lad," chuckled the old mariner. "Let 'em squirm.
We're safe out of their clutches."
"Not so safe," spoke Dave to his father, half an hour later. "Look
there."
The officer Silverado had seemingly got word to the governor of
the departure of the Swallow. A few minutes after the longboat had
disappeared around a neck of land, the ironclad gunboat hove into
view.

She was a saucy, spiteful little craft and a fast runner. She was
headed direct for the Swallow.
"Are they coming for us, captain?" inquired Amos Fearless,
somewhat anxiously.
"I hope not, for their own sakes," muttered Broadbeam quickly.
Then he shouted some orders down the tube and the Swallow made a
spurt.
"Running away?" said Pat Stoodles. "Shure, if I was in command
I'd sthand and give her one or two good welts."
"Captain Broadbeam knows his business, Mr. Stoodles," declared
Dave; "you can always count on that."
Far out in the bay were a group of sandbars and several small
wooded islands. The Swallow was headed for the largest of these islets.
The gunboat swung a challenge signal to which the Swallow made no
reply.
Then, just as the steamer, pursuant to her captain's orders, began
to slow up, the ironclad fired a gun.
"Give them their walking papers, Mr. Drake," rang out Broadbeam
to the boatswain.
The latter ran up a signal flag. This signified that the Swallow
announced herself two-and-one-half miles from shore, and therefore
out of the jurisdiction of Minotaur Island, claiming the freedom of
neutral waters.
"That'll hold her for a while," gloated Stoodles. "Aha! ye'll have to
take back wather now."
The gunboat reminded Dave of some spiteful being cheated out of
its prey. She circled, spit steam, and went more slowly back to port.
Captain Broadbeam now ordered the Swallow just without the
shoal line of a big sandy island they had neared. Here they came to

anchor.
Bob Vilett came up on deck reeking with the steam and grease of
the engine room.
"What's the programme, Bob?" asked Dave.
"Captain says we are going to stop here and take on ballast."
"For how long?"
"Till to-morrow, I reckon. I say, Dave, you've got your heart's
desire, eh?"
"I am the happiest boy living," answered the young diver.
"Something tells me we are going to get and enjoy that treasure after
all mishaps and disappointments."
In order to repair the Swallow in the creek, the ballast had been
taken out and the contents of the hold generally shifted about.
Now the captain set his men at work to take on new sand ballast
from the island and get things in the hold in regular order.
A pulley cable was run ashore. Dave and Bob were the first to take
an aerial spin along this, dangling from the big iron kettle that ran
down the incline.
Dave had told Captain Broadbeam and the others of his agency in
the matter of substituting the hornets for the opium. The recital had
made the captain good-natured, and he had given the boys permission
to rove over the sand island at will for the day.
Dave and Bob put in a pleasant hour or two talking, fishing, and
discussing the probable adventures that would greet them when they
again visited the Windjammers' Island.
At about five o'clock in the afternoon the work of securing ballast
was completed. The captain then announced that there was some work
still to do in the hold. They would make their real start with daylight.

Dave and Bob were taking a last swim in the cool of the day. A
clear sky and a fine breeze made the exercise delightful. Finally they
got daring one another. Dave swam to the little sand islet next to the
large one. Bob beat him in a race to the third of the group.
"Come on, if you've got the nerve," hailed Dave, making a quarter-
mile dash for a sand mound still beyond them.
Bob started, but turned back. Dave made port and threw himself
on the dry sand to rest. He got back his breath and sat up ready to
take the home course, when his eye was attracted to something on an
island about a furlong beyond the one he was on.
This was the nearest of the wooded islands. Dave had not noticed
it much before. What made him notice it now was that, half-hidden in a
great growth of bushes and vines, he noticed a small log hut.
In front of this a mast ran up into the air. At the moment that Dave
looked he saw a man fumbling at the lines along this mast. It was to
raise a blue bunting.
"Hello, hello," murmured Dave slowly, staring hard and thinking
desperately fast. "Why, that's easy to guess. That man is Schmitt-
Schmitt."
Dave could not precisely recognize the man at such a distance, but
felt sure that it was Schmitt-Schmitt. He thought this the more
positively as he saw that piece of blue bunting run up the mast.
"That was one of the signals I heard Schmitt-Schmitt tell the pilot
about," mused Dave. "Red for provisions, blue for sickness or help
wanted. Lantern at night, bunting by day. That's it, sure. He is signaling
the pilot. That island is Schmitt-Schmitt's place of hiding. Say, here's
something to think about."
Dave did not stay long to think about it. His eyes brightened and
he seemed moved by some inspiriting idea as he jumped into the water

and was soon back in the company of his chum, Bob Vilett.
Dave was quite silent and meditative till they had reached the big
sandy island. Arrived there, he slowly dressed himself.
"Come on, I'm hungry as a bear--don't want to miss a good
supper, Dave," hailed Bob, starting for the Swallow.
"Hold on!" challenged Dave. "I want to tell you something before
we go aboard."
"Fire away," directed Bob.
"Can you manage to get off duty about dusk?"
"There's nothing for me to do till we steam up again," replied Bob.
"Why?"
"Can we get one of the small boats for an hour or two, do you
think?"
Bob shook his head negatively.
"Heard the captain shut down on the chance of anybody sneaking
to town and making more trouble. No, it can't be done, unless the
captain gives special orders. Why?" pressed Bob curiously.
"I don't want to tell the captain what I am up to till I accomplish
something," explained Dave. "I'll tell you, though, for you've got to help
me."
"All right, Dave," piped Bob readily.
"We must rig up some kind of a craft to reach the first wooded
island."
"What for?"
"Schmitt-Schmitt is in hiding there."
"Aha, I see!" cried Bob excitedly.
"I propose," said Dave deliberately, "that we visit him, capture him,
and bring on board the Swallow--as a prisoner--the only man probably
who can guide us straight to that stolen treasure."

"Famous!" cried Bob Vilett enthusiastically--"but can we do it?"
"Let's try it, anyhow," answered Dave Fearless.
CHAPTER X
THE WOODED ISLAND
Captain Broadbeam gave pretty strict orders at dusk. A watch was set
with directions to allow no one to leave the Swallow. All the small boats
were chained stoutly.
"We'll have to defer going ashore, or report our plans to the
captain," said Bob Vilett about eight o'clock, coming up on deck with a
wry face. He was in overalls and his hands covered with oil. "No go,
Dave," he reported.
"You mean you can't join me?" asked Dave, in disappointment.
"That's it, Dave. There's work till twelve. I've got to stay. Say, why
don't you tell the captain your idea and have him send men and a boat
after Schmitt-Schmitt?"
"No," said Dave, "Captain Broadbeam wouldn't entertain the
project for a moment. He is a first-class captain, but hint at anything
outside of his ship, and he won't take the risk."
"What are you going to do, then?"
"Try it alone."
"Be careful, Dave. Don't undertake too much. You can never
manage Schmitt-Schmitt alone. Why don't you impress Stoodles into
service?"
"Mr. Stoodles is willing enough," answered Dave, "but he might
bungle. It will be all I can do to get off the Swallow alone."

Dave managed this, however, a little later, without discovery. Once
on the sand flat, he dragged some planks and ropes the ballast crew
had left there to the other side of the island. Dave constructed quite a
raft and pushed it into the water. Swimming, he propelled it before him.
Within half an hour he was on the wooded island.
The first thing that caught his eye was a blue light strung from a
tree at the end of the island nearer the town. Here there was a
favorable natural landing-place.
"The bunting signal didn't attract attention," reasoned Dave, "so
Schmitt-Schmitt has tried the lantern. Wonder if he is at the hut? I'll
work my way around that direction and find out."
Dave had the bold idea in mind of capturing this man. As he went
along he thought of plan after plan. If he could get Schmitt-Schmitt
helpless in his power, he could convey him to the Swallow on the raft.
"The very thing," said Dave gladly, as he neared the vicinity of the
hut. Lying across the top of some bushes was a fishing net. It had long
rope ends. Dave with his pocket knife cut these off and thrust them in
his pocket.
"Hey, what are you up to there?"
Dave thrilled at the sharp call, and turned quickly to face his
challenger.
It was Schmitt-Schmitt. He had abruptly emerged from the
greenery surrounding the hut. He carried a big cudgel, and as the clear
moonlight revealed the face of the intruder plainly he uttered a quick
gasp.
"Ha, I know you!" cried Schmitt-Schmitt, advancing with a scowling
face.
"It seems so," answered Dave coolly, cautiously retreating. "You
are Mr. Gerstein."

"No, you don't!" spoke the man, with a speedy leap forward.
Dave dodged, but not soon enough. The cudgel came down
directly on top of his head. He saw stars, sank flat, and knew no more
for fully five minutes.
Then, his lower limbs wound round and round with ropes, he
struggled upon the floor of a hut.
At a table on which burned a candle sat Schmitt-Schmitt. He had
just opened a bottle of lime juice and was about to pour some of its
contents into a glass to refresh himself.
He suspended operations, however, as Dave struggled to an
upright position, attracting his attention.
"Well," he spoke with a coarse chuckle, "how did that wallop suit
you?"
Dave rubbed his sore head and made a wry grimace.
"You don't treat visitors very politely, do you?" he said.
"You're a spy, you are," spoke Gerstein sullenly, "and don't you
deny it. I know you. Now then, what brought you here?"
"What brought you?" retorted Dave.
"Don't you get saucy," warned Schmitt-Schmitt. "All along you did
the big things that were done in baffling the Hankers. I hear, too, you
have been pretty smart with your tricks since you came to Minotaur
Island."
"Of course I've been trying to do all I could to protect my rights,"
said Dave. "I knew you were in hiding here."
"Ha! eh?" exclaimed Schmitt-Schmitt, pricking up his ears. "How
did you know that?"
"Oh, we have kept track of you," answered Dave lightly. "As soon
as we found you were back of the governor and the pilot in bothering
us, we naturally watched you."

Schmitt-Schmitt stared in stupefaction at Dave.
"Knew it, did you?" he muttered.
"Of course we did. We knew what you were up to. Now I can tell
you, Mr. Gerstein, you will never get that treasure away from the
Windjammers' Island, no matter how hard you try."
"Treasure! The Windjammers' Island!" gasped the man. "How--
when--where--the--the treasure was lost at sea."
"Not a bit of it, as you and I both know," asserted Dave blithely,
reading in the confusion and excitement of the man a confirmation of
his suspicions. "I say the Swallow, with or without me, sails in search of
that treasure at daylight. Come, sir, you have gone in with a measly
crowd who will only rob you in the end. Come to Captain Broadbeam,
save us the trouble of a long search, and my father will pay you all
right."
Schmitt-Schmitt got up and paced the floor. He seemed thinking
over what Dave had suggested. His face, however, gradually resumed
its customary ferocity and cunning.
"No," he said finally, striking the table with his fist and taking in his
captive's helpless situation with a good deal of satisfaction. "I have the
upper hand. I keep it."
"What upper hand?" asked Dave.
"You are my prisoner. Soon the pilot will be here in response to my
signal with his launch. I will take you to the island with me. I will hide
you. They will not get along so grandly without you. They will delay to
search for you, and delay is all I ask. Yes, yes, that is the programme."
Some whistles from craft in the bay echoed out. Schmitt-Schmitt
went outside, apparently to see if some answer was coming to his
signal.

"I am in it--deep," mused Dave. "Pshaw! I hate to think I shall
delay and bother Captain Broadbeam."
Dave found that the ropes securing him were not very tightly
arranged. They had been drawn to a loop about his waist and caught
with snap and hook behind.
"If I had time I could work loose," he thought. "I have not time, so
I suppose I must wait meekly and take what comes to me. Oh, by the
way--that's an idea!"
The "idea" in question was suggested by a glance at the bottle and
glass on the table. Dave's eyes sparkled. He fumbled under the ropes
and brought out wrapped up in a fragment of paper the sample of
opium he had discovered the night previous.
Frog-like he began hitching himself across the floor. Dave kept his
eye anxiously fixed on the open doorway. He got to the table, reached
up, dropped some grains of the drug into the glass there, and nimbly
as he could hitched his way back to his former position.
Two minutes later Schmitt-Schmitt reappeared. He went at once to
the table, poured out a drink, settled back in his chair, and said
complacently:
"My friend will soon be here. Do your friends also know I am
here?"
"Oh, dear, you mustn't expect me to tell any secrets to a fellow
who won't join in with us," said Dave.
"Maybe after a little solitude you will be willing to talk," observed
Schmitt-Schmitt meaningly.
"All right--we'll see," said Dave, with affected unconcern.
Dave's eyes sparkled as Schmitt-Schmitt began to blink. He was
delighted as the man fell back drowsily in the chair.

"Now's my chance," said Dave, as a prolonged snore announced
the complete subjugation of Schmitt-Schmitt to the influence of the
drug.
Dave did some brisk moving about. He managed to get to a
cupboard. He could not reach his own pocket knife. In the cupboard he
found a case knife and set at work sawing away the ropes that bound
him.
He laughed at his rare success, as stretching his cramped limbs he
went outside for a moment.
"I don't want to delay," he thought. "That signal may bring the
pilot at any moment, and that means two to handle instead of one.
This is just famous. Better than I planned out. How shall I get Schmitt-
Schmitt to the raft?"
Dave found an old wicker mattress on the rude porch of the hut. It
had rope ends to attach as a hammock. He took the precaution to tie
Schmitt-Schmitt's wrists and ankles together with ropes.
Then Dave dragged the insensible man from his chair across the
floor and let him down flat on the wicker mattress.
It required all his strength to pull this drag and its burden the two
hundred feet required down the beach.
"The mischief!" cried Dave, as, panting, he reached the spot where
he had left the rudely improvised raft.
It was nowhere in sight, and he readily surmised that he had
carelessly left it too near the surf, which had carried it away.
"Whatever am I to do now?" thought Dave. "I can't swim to the
Swallow with this man. I must find the material for a new raft. Pshaw!
there's a call to time."
Dave glanced keenly seawards. Then with due haste he dragged
mattress and burden back into the brush out of sight.

Peering thence, he watched a little launch making for the wooded
island at the point where the blue signal shone.
"The pilot, of course," said Dave. "He has come to see his friend.
What will he do when he fails to find him?"
With some anxiety Dave Fearless watched the little launch come
nearer and nearer to the wooded island.

CHAPTER XI
A RACE FOR LIFE
"Yes, it is the pilot," said Dave to himself, as the launch drove
directly into the little natural landing-place where the blue lantern
swung.
Dave peered from his bushy covert and closely watched the
maneuvers of its occupant.
The pilot ran the nose of the craft well into the sand, shut off
the power, and leaped ashore.
Dave saw him take up a basket and watched him depart for the
hut. As soon as some trees shut him out from view Dave leaped on
board of the launch.
A momentary inspection of the operating lever and steering
gear told Dave that he could easily navigate the boat.
"I must lose no time," he thought. "My only chance of getting
away with Schmitt-Schmitt is in taking the launch."
Dave forthwith dragged his unconscious captive to the launch. It
was no easy task to get that bulky individual aboard. Dave
accomplished it, however, and then paused to catch his breath and
wipe the perspiration from his face.
"Hi! hi! hi!"
A ringing yell, or rather three of them, uttered in rapid and
startling succession, made Dave turn with a shock.

Looking down the beach, he saw the pilot running towards him
at full speed. The latter had evidently visited the hut, had found it
vacated, and coming out to look for his missing friend, had
discovered the launch in the hands of a stranger.
Dave made no reply. He sprang to the little lever, reversing it,
and the launch slid promptly back into the water. Swinging the
steering gear south, Dave turned on full power.
"Stop. I'll shoot--stop! stop!" panted the pilot, gaining on Dave
with prodigious bounds of speed.
Dave kept his hand on the lever, his eyes fixed ahead. Suddenly-
---
Bang--ping! a shot whistled past his ear. Dave crouched and
darted a quick glance backward. The pilot, coming to a standstill,
was firing at him from a revolver.
Dave saw a point of refuge ahead. This was a broken irregular
wooded stretch, well-nigh impassable on foot. As a second shot
sounded out, Dave curved around this point of land.
He was now out of view of the pilot, who would find great
difficulty in crossing the stretch lying between them, as it was
marshy in spots. Dave lined the shore farther on, feeling pretty
proud of the success of his single-handed enterprise.
"Why," he mused, "we have the game in our own hands
completely now. I wonder what father and Captain Broadbeam will
say to all this. Of course they won't fancy such a guest as Schmitt-
Schmitt, but they must see how holding him a harmless captive
helps our plans."
Dave made a sweep with the launch to edge the rounding end
of the island. Here it narrowed to about two hundred feet. It would

now be a straight bolt past the same islets to where the Swallow
was.
"Won't do--the gunboat, sure as shingles!" spoke Dave
suddenly.
Almost directly in his course, and bearing down upon him, was
the ironclad. In that clear moonlight everything was plain as in
daylight. Dave could see the people on board the gunboat, and they
could see him--without doubt.
In fact, someone in uniform leaned over the bow of the ironclad
in his direction. Dave caught an indistinct hail. He paid no attention
to it.
He acted with the precipitancy of a school fugitive running away
from a truant officer. He saw just one chance to evade an
unpleasant overhauling by the ironclad, and took it.
This was to instantly steer to the north and shoot down the
narrow neck of water lying between the wooded island and the
nearest sand island.
Dave knew that this channel must be quite shallow. He doubted
if the cumbersome iron-clad could navigate it. Even if it tried to, it
would be some minutes before its crew could swing around into
position to make the chase.
The launch took the channel like an arrow. Dave's spirits rose
high, notwithstanding some loud and quite peremptory hails from
the direction of the gunboat.
"Better than before," soliloquized Dave. "I can swing around the
sandbars directly to the anchorage of the Swallow."
Glancing back, Dave saw that the gunboat did not intend to
follow the course he had taken. That craft had stopped and put

about.
"They must suspect that something's not exactly right,"
calculated Dave. "The mischief--that was close. Ouch! I'm hit."
Dave went keeling over from the bow seat. Very suddenly, from
some bushes on the wooded island, there were two sharp flashes
and reports. One bullet whizzed past his head, the second plowed a
furrow across his forearm. It was not deep, but the wound bled, and
the surprise and shock sent Dave over backwards.
The worst of it was that he jerked the lever, and this, turning
the launch, sent its nose directly into shore, and there the boat
stuck, vibrating with the impact of the still working machinery. The
pilot instantly ran from cover towards the boat, flourishing the
weapon in his hand. He had crossed the island, it seemed, to head
off the launch, and it looked as though Dave was doomed to disaster
in his present enterprise.
Dave scrambled to get back to the lever, and reverse the launch.
As he did so his hand touched something lying upon straps at the
side of the seat pit.
It was a rifle. Dave seized it, jerked it and its fastenings free,
and extended it directly at the running figure ashore.
"Get back," he shouted. "Drop that pistol, Mr. Pilot, or there will
be trouble."
The pilot, with a howl of rage, halted short. He flung the
revolver down. Dave guessed that it was now empty.
As Dave touched the lever and got out into the channel again,
he saw the pilot running back along the beach. He was headed for
the end of the island in the direction of the ironclad, and yelling out
some information to those aboard at the top of his bellowing voice.

"Now for a spurt," said Dave.
The channel was about a mile long. Dave came to its end in fine
spirits. It was a clear run now past the two outer sand islands, and a
half-mile turn would bring him to the Swallow.
He proceeded more leisurely now, for it did not seem possible
that the ironclad could make the opposite circuit in time to head him
off. Where the sand hills dropped, however, Dave had a view across
the two next islands.
"They are after me," he exclaimed. "The pilot has advised them
of the real state of affairs, and it's a sharp run. Full power--go!"
Dave had made out the gunboat whizzing down the channel
between the two outer sand islands. She was forcing full speed. It
was a question whether the gunboat would not emerge first into the
open sea and block his course.
Dave put on power that made the little launch strain and quiver
from stem to stern. He was terribly excited and anxious. His breath
came in quick jerks, his heart beat fast.
"Close shave," he panted, "but I've made it."
Two hundred feet down the channel was the gunboat, as Dave
crossed her outlet. The ironclad swung out after him not one minute
later.
The launch fairly skimmed the water. The ironclad loomed
portentously near, but Dave felt that, no mishap occurring, he would
win the race.
"They've got me, I guess," he gasped a second later.
A flash, a loud boom, and a terrific concussion plunged Dave
into a condition of extreme confusion and uncertainty.

The ironclad had fired a shot. It had struck the stern of the
launch, splintering it clear open. A great shower of water deluged
Dave and his insensible captive.
Dave regarded the damage done with grave dismay--the stern
had sunk and the launch was now on a slant.
In fact, the rear portion of the boat was under water to the rail.
Only by keeping up power could the launch be prevented from
filling and going down. Dave never let go his grasp on the lever. He
held firmly to the last notch in the indicator.
As he turned the end of the last sand island, the maneuver
made the launch wabble. Just here a second gun was fired from the
ironclad. The shot went far wide of its intended mark, but a vital
alarm urged Dave to change his course.
The launch went sideways, and a sudden inrush of water sunk
her to the middle. Dave headed for shore. There the launch struck, a
wreck.
Down the shore lay the Swallow. Active lights were bobbing
about her deck, so Dave knew that the crew had been aroused by
the firing at sea.
His first thought was to get Schmitt-Schmitt out of the half-
submerged launch. He dragged his captive to the beach, then he
took a look at the gunboat.
"Why," exclaimed Dave, in mingled astonishment and
satisfaction, "she's grounded."
Apparently the ironclad had struck some treacherous sandbar
over which the light swift launch had glided in safety. Loud orders,
quick bells, and whistles made a small babel aboard the craft in
distress.

Dave glanced down calculatingly at his helpless captive. He
must get him to the Swallow. But how?
The pit crate of the launch had floated up as the craft filled with
water. Dave waded to it, pulled it ashore, and rolled Schmitt-Schmitt
across it.
He was now quite hidden from the view of those aboard of the
gunboat, but he feared they might send a yawl on an investigating
expedition.
Dave swam, pushing the crate before him. Often he glanced
back. There was no pursuit. More hopefully and nearer and nearer
he approached the Swallow. With a kind of a faint cheer Dave hailed
her as he came within hearing distance.
"Ahoy, there!" rang back Captain Broadbeam's foghorn voice, as
he gazed down at crate, burden, and swimmer.
"It's me--Dave Fearless," began the latter.
"Bet it is! Had to have a rumpus, eh? What was the shooting?
Lower away there, men. Two of you, eh? What! that rascally
pawnbroker, Gerstein!" fairly yelled the captain, as by stages Dave
and his captive came nearer, were helped by the crew, and now
gained the deck of the Swallow.
"Yes, Captain Broadbeam," nodded the nearly exhausted Dave.
"The gunboat--after us--suggest you get away--at once--excuse--
weak and dizzy----"
And just then Dave Fearless sank flat to the deck of the
Swallow, overcome completely after the hardest work he had ever
done in his life.

CHAPTER XII
OVERBOARD
"What does he say, Captain Broadbeam?" asked Dave Fearless.
"Mum as an oyster, lad."
"Won't talk, eh?" remarked Dave's father. "Nothing come of
giving him free board, and after all the trouble you had, Dave, in
getting him onto the Swallow."
"You forget, father," reminded Dave, "it is one enemy the less to
worry about."
"The lad's right," declared Captain Broadbeam. "It means a
good deal to clip the wings of the main mover in this scheme against
us. If Gerstein, or Sehmitt-Schmitt as he calls himself, won't do us
any good, at least he can do us no harm as long as we hold him a
prisoner. I reckon those fellows back at Minotaur Island are a little
dazed at the slick way we disappeared,--ship, their crony, and all."
Bob Vilett, seated in the cabin with the others, laughed heartily.
"It was a big move and a good one, that of yours in capturing
this rascal," he declared to Dave. "Now we certainly have the field to
ourselves. The governor and the pilot can't follow us, for they don't
know where we have gone. No one is on this treasure search except
ourselves. It's a clear field, as I say."
"Until we reach the Windjammers' Island," suggested Dave. "I
wouldn't wonder if Gerstein had left Captain Nesik and the others
there, probably guarding the treasure while awaiting his return."
The Swallow had got away from the vicinity of Minotaur Island
two days previous. Just as soon as, after his exciting capture of
Gerstein, Dave had sufficiently recovered to explain matters to

Captain Broadbeam, the latter had ordered on full steam, leaving the
ironclad stuck on the sandbar.
Gerstein raved like a madman when the drug Dave had given
him began to lose its effect. He threatened all kinds of things--the
law, for one, for kidnapping--but Captain Broadbeam only laughed at
him.
"Just one word, my hearty," he observed spicily. "As long as you
behave yourself, outside of every man aboard having his eye on you
to look out for tricks, you'll have bed and food with the best of us.
Try any didos, though, and I clap you into irons--understand?"
Gerstein became at once sullen and silent. When he came on
deck after that he spoke to nobody. Most of the time he remained
shut up by himself in the little cabin apportioned to him.
The second day out Captain Broadbeam sought an interview
with him. It was after a talk with Amos Fearless.
He offered Gerstein a liberal share of the treasure if he would
divulge its whereabouts and tell what had become of the Raven and
her crew.
Gerstein declined to say a word. He simply regarded the captain
in a mocking, insolent way. It was evident that the fellow
appreciated the full value of his knowledge concerning the treasure.
"He's counting on getting away from us somehow, before the
cruise is over," reported Captain Broadbeam to his friends, "or he is
taking chances on our running into a nest of his friends when we
reach the Windjammers' Island."
The Swallow had a delightful run to Mercury Island. Before they
reached it Gerstein was placed in the hold, and there closely

guarded by two mariners until they had provisioned up and were
once more on their way.
Dave had little to do except to wait the end of their cruise, yet
he put in some busy hours. For three days he kept Stoodles at his
side at the table in the captain's cabin, questioning him on every
detail about the lay and outlines of the island they were sailing to.
Then he made a chart of the island, and as near as possible from
memory marked in the other island where they had recovered
possession of the Swallow after it had been stranded during a
cyclone.
The weather changed suddenly a day or two out from Mercury
Island. They rode into a fierce northeaster, and it rained nearly all
the time, with leaden skies and a choppy sea.
Dave was a good deal below. One afternoon, returning from a
brief visit to Bob Vilett, as he was making for the cabin passageway,
a chink of light attracted his attention.
It emanated from a crack in the paneling of the cabin occupied
by Gerstein. Dave drew nearer to the chink, and could look quite
clearly into the compartment that housed the person in whom he
was naturally very much interested at all times.
"H'm!" said Dave, with a bright flicker in his eye. "He's making a
chart, too, is he?"
The daylight was so dim that Gerstein had a lighted candle on
the table at which he sat. Spread out before him was a sheet of
heavy manila paper. It bore black outlines as if an irregular body of
land, and had crosses and dots all over it.
At this Gerstein was working, thoughtfully scanning it at times
and then making additions to it. Dave believed that it had something

to do with the treasure.
"Our treasure," he reflected, "and I'll play something else than
the spy if I get a chance to look over that chart, whatever it is."
He watched the man's movements for over half an hour. Then
Gerstein folded up the paper, placing it in a thin tin tobacco box.
This he secured in a pocket in the blue shirt he wore, buttoning the
pocket flap securely.
Dave got no further sight of the mysterious paper, if such it was,
during the next week. He felt himself justified in trying to get a
chance to secure the little tin box. Twice he visited Gerstein's cabin
secretly, while its occupant was on deck. Gerstein, however,
apparently carried the box with him wherever he went.
One night, when he slept, Dave crept into the cabin, the door of
which for a wonder had been left unlocked. He ransacked Gerstein's
clothing, but with no result.
"Got it somewhere in bed with him," thought Dave. "I don't dare
to try and find it, though. I would surely wake him up. I believe I will
tell Captain Broadbeam about the little tin box. If it in any way
concerns this treasure, why haven't we the right to take it away from
Gerstein, even by force?"
Before Dave had an opportunity to consult with Captain
Broadbeam, however, something transpired that changed all his
plans.
It was a dark and stormy night. The weather had been rough all
day. Dave came on deck about eight o'clock to find the captain on
duty. A few men were making things tidy about the stern deck.
The Swallow was plowing the water, slanted like a swordfish in
action. Dave held to a handle at the side of the cabin, peering into

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Brillouinwigner Methods For Manybody Systems 1st Edition Ivan Huba

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Brillouinwigner Methods For Manybody Systems 1st Edition Ivan Huba

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