COMPUTER SIMULATION METHODS IN CHEMISTRY AND PHYSICS

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AB INITIO MOLECULAR DYNAMICS:
BASIC THEORY AND ADVANCED METHODS
Ab initio molecular dynamics revolutionized the field of realistic computer
simulation of complex molecular systems and processes, including chemical
reactions, by unifying molecular dynamics and electronic structure theory. This
book provides the first coherent presentation of this rapidly growing field, covering
a vast range of methods and their applications, from basic theory to advanced
methods.
This fascinating text for graduate students and researchers contains systematic
derivations of various ab initio molecular dynamics techniques to enable readers
to understand and assess the merits and drawbacks of commonly used methods.
It also discusses the special features of the widely used Car–Parrinello approach,
correcting various misconceptions currently found in the research literature.
The book contains pseudo-code and program layout for typical plane wave
electronic structure codes, allowing newcomers to the field to understand commonly
used program packages, and enabling developers to improve and add new features
in their code.
D
ominik Marx is Chair of Theoretical Chemistry at Ruhr-Universität Bochum,
Germany. His main areas of research are in studying the dynamics and reactions
of complex molecular many-body systems and the development of novel ab initio
simulation techniques.
J
ürg Hutter is a Professor at the Physical Chemistry Institute at the University of
Zürich in Switzerland, where he researches problems in theoretical chemistry, in
particular, methods for large-scale density functional calculations.

AB INITIO MOLECULAR DYNAMICS:
BASIC THEORY AND ADVANCED

METHODS
DOMINIK MARX
Ruhr-Universität Bochum
and
JÜRG HUTTER
University of Zürich
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
First published in print format
ISBN-13 978-0-521-89863-8
ISBN-13 978-0-511-53333-4
© D. Marx and J. Hutter 2009
2009
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Contents
Preface page viii
1 Setting the stage: why ab initio molecular dynamics? 1
Part I Basic techniques 9
2 Getting started: unifying MD and electronic structure 11
2.1 Deriving classical molecular dynamics 11
2.2 Ehrenfest molecular dynamics 22
2.3 Born–Oppenheimer molecular dynamics 24
2.4 Car–Parrinello molecular dynamics 27
2.5 What about Hellmann–Feynman forces? 51
2.6 Which method to choose? 56
2.7 Electronic structure methods 67
2.8 Basis sets 75
3 Implementation: using the plane wave basis set 85
3.1 Introduction and basic definitions 85
3.2 Electrostatic energy 93
3.3 Exchange and correlation energy 99
3.4 Total energy, gradients, and stress tensor 104
3.5 Energy and force calculations in practice 109
3.6 Optimizing the Kohn–Sham orbitals 111
3.7 Molecular dynamics 119
3.8 Program organization and layout 128
4 Atoms with plane waves: accurate pseudopotentials 136
4.1 Why pseudopotentials? 137
4.2 Norm-conserving pseudopotentials 138
4.3 Pseudopotentials in the plane wave basis 152

4.4 Dual-space Gaussian pseudopotentials 157
v
vi Contents
4.5 Nonlinear core correction 160
4.6 Pseudopotential transferability 162
4.7 Example: pseudopotentials for carbon 167
Part II Advanced techniques 175
5 Beyond standard ab initio molecular dynamics 177
5.1 Introduction 177
5.2 Beyond microcanonics: thermostats, barostats, meta-
dynamics 178
5.3 Beyond ground states: ROKS, surface hopping, FEMD,
TDDFT 194
5.4 Beyond classical nuclei: path integrals and quantum
corrections 233
5.5 Hybrid QM/MM molecular dynamics 267
6 Beyond norm-conserving pseudopotentials 286
6.1 Introduction 286
6.2 The PAW transformation 287
6.3 Expectation values 290
6.4 Ultrasoft pseudopotentials 292
6.5 PAW energy expression 296
6.6 Integrating the Car–Parrinello equations 297
7 Computing properties 309
7.1 Perturbation theory: Hessian, polarizability, NMR 309
7.2 Wannier functions: dipole moments, IR spectra, atomic
charges 327
8 Parallel computing 350
8.1 Introduction 350
8.2 Data structures 352

8.3 Computational kernels 354
8.4 Massively parallel processing 359
Part III Applications 369
9 From materials to biomolecules 371
9.1 Introduction 371
9.2 Solids, minerals, materials, and polymers 372
9.3 Interfaces 376
9.4 Mechanochemistry and molecular electronics 380
9.5 Water and aqueous solutions 382
Contents vii
9.6 Non-aqueous liquids and solutions 385
9.7 Glasses and amorphous systems 389
9.8 Matter at extreme conditions 390
9.9 Clusters, fullerenes, and nanotubes 392
9.10 Complex and fluxional molecules 394
9.11 Chemical reactions and transformations 396
9.12 Homogeneous catalysis and zeolites 399
9.13 Photophysics and photochemistry 400
9.14 Biophysics and biochemistry 403
10 Properties from ab initio simulations 407
10.1 Introduction 407
10.2 Electronic structure analyses 407
10.3 Infrared spectroscopy 410
10.4 Magnetism, NMR and EPR spectroscopy 411
10.5 Electronic spectroscopy and redox properties 412
10.6 X-ray diffraction and Compton scattering 413
10.7 External electric fields 414
11 Outlook 416
Bibliography 419
Index 550

Preface
In this book we develop the rapidly growing field of ab initio molecular
dynamics computer simulations from the underlying basic ideas up to the
latest techniques, from the most straightforward implementation up to mul-
tilevel parallel algorithms. Since the seminal contributions of Roberto Car
and Michele Parrinello starting in the mid-1980s, the unification of molecu-
lar dynamics and electronic structure theory, often dubbed “Car–Parrinello
molecular dynamics” or just “CP”, widened the scope and power of both
approaches considerably. The forces are described at the level of the many-
body problem of interacting electrons and nuclei, which form atoms and
molecules as described in the framework of quantum mechanics, whereas the
dynamics is captured in terms of classical dynamics and statistical mechan-
ics. Due to its inherent virtues, ab initio molecular dynamics is currently
an extremely popular and ever-expanding computational tool employed to
study physical, chemical, and biological phenomena in a very broad sense.
In particular, it is the basis of what could be called a “virtual laboratory
approach” used to study complex processes at the molecular level, including
the difficult task of the breaking and making of chemical bonds, by means
of purely theoretical methods. In a nutshell, ab initio molecular dynam-
ics allows one to tackle vastly different systems such as amorphous silicon,
Ziegler-Natta heterogeneous catalysis, and wet DNA using the same compu-
tational approach, thus opening avenues to deal with molecular phenomena
in physics, chemistry, and biology in a unified framework.
We now feel that the time has come to summarize the impressive develop-
ments of the last 20 years in this field within a unified framework at the level
of an advanced text. Currently, any newcomer in the field has to face the
problem of first working through the many excellent and largely complemen-
tary review articles or Lecture Notes that are widespread. Even worse, much
of the significant development of the last few years is not even accessible at
viii

Preface ix
that level. Thus, our aim here is to provide not only an introduction to the
beginner such as graduate students, but also as far as possible a comprehen-
sive and up-to-date overview of the entire field including its prospects and
limitations. Both aspects are also of value to the increasing number of those
scientists who wish only to apply ab initio molecular dynamics as a pow-
erful problem-solving tool in their daily research, without having to bother
too much about the technical aspects, let alone about method development.
This is indeed possible, in principle, since several rather easy-to-use program
packages are now on the market, mostly for free or at low cost for academic
users.
In particular, different flavors of ab initio molecular dynamics methods
are explained and compared in the first part of this book at an introductory
level, the focus being on the efficient extended Lagrangian approach as in-
troduced by Car and Parrinello in 1985. But in the meantime, a wealth of
techniques that go far beyond what we call here the “standard approach”,
that is microcanonical molecular dynamics in the electronic ground state
using classical nuclei and norm-conserving pseudopotentials, have been de-
vised. These advanced techniques are outlined in Part II and include meth-
ods that allow us to work in other ensembles, to enhance sampling, to include
excited electronic states and nonadiabatic effects, to deal with quantum ef-
fects on nuclei, and to treat complex biomolecular systems in terms of mixed
quantum/classical approaches. Most important for the practitioner is the
computation of properties during the simulations, such as optical, IR, Ra-
man, or NMR properties, mostly in the context of linear response theory
or the analysis of the dynamical electronic structure in terms of fragment
dipole moments, localized orbitals, or effective atomic charges. Finally in
Part III, we provide a glimpse of the wide range of applications, which not
only demonstrate the enormous potential of ab initio molecular dynamics
for both explaining and predicting properties of matter, but also serve as a

compilation of pertinent literature for future reference and upcoming appli-
cations.
In addition to all these aspects we also want to provide a solid basis of
technical knowledge for the younger generation such as graduate students,
postdocs, and junior researchers beginning their career in a nowadays well-
established field. For this very reason we also decided to include, as far as
possible, specific references in the text to the original literature as well as
to review articles. To achieve this, the very popular approach of solving
the electronic structure problem in the framework of Kohn–Sham density
functional theory as formulated in terms of plane waves and pseudopoten-
tials is described in detail in Part I. Although a host of “tricks” can already
x Preface
be presented at that stage, specific aspects can only be made clear when
discussing them at the level of implementation. Here, the widely used and
ever-expanding program package CPMD serves as our main reference, but
we stress that the techniques and paradigms introduced apply analogously
to many other available codes that are in extensive use. This needs to
be supplemented with an introduction to the concept of norm-conserving
pseudopotentials, including definitions of various widely used pseudopoten-
tial types. In Part I, the norm-conserving pseudopotentials are explained,
whereas in Part II, the reader will be exposed to the powerful projector
augmented-wave transformation and ultrasoft pseudopotentials. A crucial
aspect for large-scale applications, given the current computer architectures
and the foreseeable future developments, is how to deal with parallel plat-
forms. We account for this sustainable trend by devoting special attention
in Part II to parallel programming, explaining a very powerful hierarchical
multilevel scheme. This paradigm allows one to use not only the ubiqui-
tous Beowulf clusters efficiently, but also the largest machines available, viz.
clustered shared-memory parallel servers and ultra-dense massively parallel
computers.

Overall, our hope is that this book will contribute not only to strengthen
applications of ab initio molecular dynamics in both academia and indus-
try, but also to foster further technical development of this family of com-
puter simulation methods. In the spirit of this idea, we will maintain the
site www.theochem.rub.de/go/aimd-book.html where corrections and ad-
ditions to this book will be collected and provided in an open access mode.
We thus encourage all readers to send us information about possible errors,
which are definitively hidden at many places despite our investment of much
care in preparing this manuscript.
Last but not least we would like to stress that our knowledge of ab initio
molecular dynamics has grown slowly within the realms of a fruitful and
longstanding collaboration with Michele Parrinello, initially at IBM Zurich
Research Laboratory in R¨uschlikon and later at the Max-Planck-Institut f¨ur
Festk¨orperforschung in Stuttgart, which we gratefully acknowledge on this
occasion. In addition, we profited enormously from pleasant cooperations
with too many friends and colleagues to be named here.
1
Setting the stage: why ab initio
molecular dynamics?
Classical molecular dynamics using predefined potentials, force fields, ei-
ther based on empirical data or on independent electronic structure calcu-
lations, is well established as a powerful tool serving to investigate many-
body condensed matter systems, including biomolecular assemblies. The
broadness, diversity, and level of sophistication of this technique are doc-
umented in several books as well as review articles, conference proceed-
ings, lecture notes, and special issues [25, 120, 136, 272, 398, 468, 577,
726, 1189, 1449, 1504, 1538, 1539]. At the very heart of any molecular
dynamics scheme is the question of how to describe – that is in prac-
tice how to approximate – the interatomic interactions. The traditional
route followed in molecular dynamics is to determine these potentials in

advance. Typically, the full interaction is broken up into two-body and
many-body contributions, long-range and short-range terms, electrostatic
and non-electrostatic interactions, etc., which have to be represented by
suitable functional forms, see Refs. [550, 1405] for detailed accounts. Af-
ter decades of intense research, very elaborate interaction models, including
the nontrivial aspect of representing these potentials analytically, were de-
vised [550, 1280, 1380, 1405, 1539].
Despite their overwhelming success – which will, however, not be praised
in this book – the need to devise a fixed predefined potential implies serious
drawbacks [1123, 1209]. Among the most significant are systems in which
(i) many different atom or molecule types give rise to a myriad of different
interatomic interactions that have to be parameterized and/or (ii) the elec-
tronic structure and thus the chemical bonding pattern changes qualitatively
during the course of the simulation. Such systems are termed here “chemi-
cally complex”. An additional aspect (iii) is of a more practical nature: once
a specific system is understood after elaborate development of satisfactory
potentials, changing a single species provokes typically enormous efforts to
1
2 Setting the stage: why ab initio molecular dynamics?
parameterize the new potentials needed. As a result, systematic studies are
a tour de force if no suitable set of consistent potentials is already available.
The reign of traditional molecular dynamics and electronic structure
methods was extended greatly by a family of techniques that is referred
to here as “ab initio molecular dynamics” (AIMD). Apart from the widely
used general notion of “Car–Parrinello” or just “CP simulations” as de-
fined in the Physics and Astronomy Classification Scheme, Pacs [1093],
other names including common abbreviations that are currently in use
for such methods are for instance first principles (FPMD), on-the-fly, di-
rect, extended Lagrangian (ELMD), density functional (DFMD), quantum
chemical, Hellmann–Feynman, Fock-matrix, potential-free, or just quantum

(QMD) molecular dynamics amongst others. The basic idea underlying ev-
ery ab initio molecular dynamics method is to compute the forces acting
on the nuclei from electronic structure calculations that are performed “on-
the-fly” as the molecular dynamics trajectory is generated, see Fig. 1.1 for a
simplifying scheme. In this way, the electronic variables are not integrated
out beforehand and represented by fixed interaction potentials, rather they
are considered to be active and explicit degrees of freedom in the course
of the simulation. This implies that, given a suitable approximate solution
of the many-electron problem, also “chemically complex” systems, or those
where the electronic structure changes drastically during the dynamics, can
be handled easily by molecular dynamics. But this also implies that the ap-
proximation is shifted from the level of devising an interaction potential to
the level of selecting a particular approximation for solving the Schr¨odinger
equation, since it cannot be solved exactly for the typical problems at hand.
Applications of ab initio molecular dynamics are particularly widespread
in physics, chemistry, and more recently also in biology, where the afore-
mentioned difficulties (i)-(iii) are particularly severe [39, 934]. A collection
of problems that have already been tackled by ab initio molecular dynamics,
including the pertinent references, can be found in Chapter 9 of Part III.
The power of this novel family of techniques led to an explosion of activity
in this field in terms of the number of published papers, see the squares in
Fig. 1.2 that can be interpreted as a measure of the activity in the area
of ab initio molecular dynamics. This rapid increase in activity started
in the mid to late 1980s. As a matter of fact the time evolution of the
number of citations of a particular paper, the one by Car and Parrinello
from 1985 entitled “Unified approach for molecular dynamics and density-
functional theory” [222, 1216], initially parallels the growth trend of the
entire field, see the circles in Fig. 1.2. Thus, the resonance evoked by this
publication and, at its very heart, the introduction of the Car–Parrinello
Setting the stage: why ab initio molecular dynamics? 3

Ab initio ES
Schr¨odinger Eq:
Hartree–Fock/
Kohn–Sham LDA
ˆ
HΨ=E
0
Ψ
Ab initio MD
Car–Parrinello
M
¨
R = −∇E
0
Classical MD
Newton Eq:
Fermi–Pasta–Ulam/
Alder–Wainwright
M
¨
R = −∇E
eff
⇓
Statics and Dynamics
Electrons and Nuclei
Molecules, Clusters, Complexes
Liquids, Solids, Surfaces
Composites
.
.

.
at T ≥ 0
Fig. 1.1. Ab initio molecular dynamics unifies approximate ab initio electronic
structure theory (i.e. solving Schr¨odinger’s wave equation numerically using, for in-
stance, Hartree–Fock theory or the local density approximation within Kohn–Sham
theory) and classical molecular dynamics (i.e. solving Newton’s equation of motion
numerically for a given interaction potential as reported by Fermi, Pasta, Ulam,
and Tsingou for a one-dimensional anharmonic chain model of solids [409] and
published by Alder and Wainwright for the three-dimensional hard-sphere model of
fluids [19]; see Refs. [33, 272, 308, 453, 652] for historic perspectives on these early
molecular dynamics studies).
“Lagrangean” [995], has gone hand in hand with the popularity of the en-
tire field over the last decade. Incidentally, the 1985 paper by Car and
Parrinello is the last one included in the section “Trends and Prospects”
in the reprint collection of “key papers” from the field of atomistic com-
puter simulations [272]. Evidence that the entire field of ab initio molecular
dynamics has matured is also provided by the separate Pacs classification
number (“71.15.Pd - Electronic Structure: Molecular dynamics calculations
4 Setting the stage: why ab initio molecular dynamics?
1970 1980 1990 2000 2010
Year n
0
1000
2000
3000
4000
5000
6000
7000
8000

9000
10000
11000
12000
Number N
CP PRL 1985
AIMD
Fig. 1.2. Publication and citation analysis up to the year 2007. Squares: number of
publications N which appeared up to the year n containing the keyword “ab initio
molecular dynamics” (or synonyms such as “first principles MD”, “Car–Parrinello
simulations” etc.) in title, abstract or keyword list. Circles: number of publications
N which appeared up to the year n citing the 1985 paper by Car and Parrinello [222]
(including misspellings of the bibliographic reference). Self-citations and self-papers
are excluded, i.e. citations of Ref. [222] in their own papers and papers coauthored
by R. Car and/or M. Parrinello are not considered in the respective statistics;
note that this, together with the correction for misspellings, is probably the main
reason for a slightly different citation number up to the year 2002 as given here
compared to that (2819 citations) reported in Ref. [1216]. The analysis is based
on Thomson/ISI Web of Science (WoS), literature file CAPLUS of the Chemical
Abstracts Service (CAS), and INSPEC file (Physics Abstracts) as accessible under
the database provider STN International. Earlier reports of these statistics [933,
934, 943] are updated as of March 13, 2008; the authors are most grateful to
Dr. Werner Marx (Information Service for the Institutes of the Chemical Physical
Technical Section of the Max Planck Society) for carrying out these analyses.
(Car–Parrinello) and other numerical simulations”) introduced in 1996 into
the Physics and Astronomy Classification Scheme [1093].
Despite its obvious advantages, it is evident that a price has to be payed
for putting molecular dynamics onto an ab initio foundation: the corre-
lation lengths and relaxation times that are accessible are much smaller
than what is affordable in the framework of standard molecular dynamics.

More recently, this discrepancy was counterbalanced by the ever-increasing
power of available computing resources, in particular massively parallel plat-
forms [661, 662], which shifted many problems in the physical sciences right
into the realm of ab initio molecular dynamics. Another appealing feature
of standard molecular dynamics is less evident, namely the experimental
Setting the stage: why ab initio molecular dynamics? 5
aspect of “playing with the potential”. Thus, tracing back the properties of
a given system to a simple physical picture or mechanism is much harder
in ab initio molecular dynamics, where certain interactions cannot easily be
“switched off” like in standard molecular dynamics. On the other hand,
ab initio molecular dynamics has the power to eventually map phenom-
ena onto a firm basis in terms of the underlying electronic structure and
chemical bonding patterns. Most importantly, however, is the fact that new
phenomena, which were not foreseen before starting the simulation, can sim-
ply happen if necessary. All this lends ab initio molecular dynamics a truly
predictive power.
Ab initio molecular dynamics can also be viewed from another perspec-
tive, namely from the field of classical trajectory calculations [1284, 1514].
In this approach, which has its origin in gas phase reaction dynamics, a
global potential energy surface is constructed in a first step either empir-
ically, semi-empirically or, more and more, based on high-level electronic
structure calculations. After fitting it to a suitable analytical form in a sec-
ond step (but without imposing additional approximations such as pairwise
additivity, etc.), the dynamical evolution of the nuclei is generated in a third
step by using classical mechanics, quantum mechanics, or semi/quasiclassical
approximations of various sorts. In the case of using classical mechanics to
describe the dynamics - which is the focus of the present book - the limiting
step for large systems is the first one, why should this be so? There are 3N−6
internal degrees of freedom that span the global potential energy surface of
an unconstrained N-body system. Using, for simplicity, 10 discretization

points per coordinate implies that of the order of 10
3N−6
electronic struc-
ture calculations are needed in order to map such a global potential energy
surface. Thus, the computational workload for the first step in the approach
outlined above grows roughly like ∼ 10
N
with increasing system size ∼ N.
This is what might be called the “curse of dimensionality” or “dimensional-
ity bottleneck” of calculations that rely on global potential energy surfaces,
see for instance the discussion on p. 420 in Ref. [551].
What is needed in ab initio molecular dynamics instead? Suppose that
a useful trajectory consists of about 10
M
molecular dynamics steps, i.e.
10
M
electronic structure calculations are needed to generate one trajectory.
Furthermore, it is assumed that 10
n
independent trajectories are necessary
in order to average over different initial conditions so that 10
M+n
ab initio
molecular dynamics steps are required in total. Finally, it is assumed that
each single-point electronic structure calculation needed to devise the global
potential energy surface and one ab initio molecular dynamics time step
require roughly the same amount of cpu time. Based on this truly simplistic
6 Setting the stage: why ab initio molecular dynamics?
order of magnitude estimate, the advantage of ab initio molecular dynamics

vs. calculations relying on the computation of a global potential energy
surface amounts to about 10
3N−6−M −n
. The crucial point is that for a
given statistical accuracy (that is for M and n fixed and independent of N)
and for a given electronic structure method, the computational advantage
of “on-the-fly” approaches grows like ∼ 10
N
with system size. Thus, Car–
Parrinello methods always outperform the traditional three-step approaches
if the system is sufficiently large and complex. Conversely, computing global
potential energy surfaces beforehand and running many classical trajectories
afterwards without much additional cost always pays off for a given system
size N like ∼ 10
M+n
if the system is small enough so that a global potential
energy surface can be computed and parameterized.
Of course, considerable progress has been achieved in accelerating the
computation of global potentials by carefully selecting the discretization
points and reducing their number, choosing sophisticated representations
and internal coordinates, exploiting symmetry and decoupling of irrelevant
modes, implementing efficient sampling and smart extrapolation techniques
and so forth. Still, these improvements mostly affect the prefactor but not
the overall scaling behavior, ∼ 10
N
, with the number of active degrees of
freedom. Other strategies consist of, for instance, reducing the number of
active degrees of freedom by constraining certain internal coordinates, rep-
resenting less important ones by a (harmonic) bath or by friction forces, or
building up the global potential energy surface in terms of few-body frag-

ments. All these approaches, however, invoke approximations beyond those
of the electronic structure method itself. Finally, it is evident that the com-
putational advantage of the “on-the-fly” approaches diminishes as more and
more trajectories are needed for a given (small) system. For instance, exten-
sive averaging over many different initial conditions is required in order to
calculate scattering or reactive cross-sections quantitatively. Summarizing
this discussion, it can be concluded that ab initio molecular dynamics is the
method of choice to investigate large and “chemically complex” systems.
Quite a few reviews, conference articles, lecture notes, and overviews
dealing with ab initio molecular dynamics have appeared since the early
1990s [38, 228, 338, 460, 485, 486, 510, 563, 564, 669, 784, 933, 934, 936–
938, 943, 1099, 1103, 1104, 1123, 1209, 1272, 1306, 1307, 1498, 1512, 1544]
and the interested reader is referred to them for various complementary view-
points. This book originates from the Lecture Notes [943] “Ab initio molec-
ular dynamics: Theory and implementation” written by the present authors
on the occasion of the NIC Winter School 2000 titled “Modern Methods and
Algorithms of Quantum Chemistry”. However, it incorporates in addition
Setting the stage: why ab initio molecular dynamics? 7
many recent developments as covered in a variety of lectures, courses, and
tutorials given by the authors as well as parts from our previous review and
overview articles. Here, emphasis is put on both the broad extent of the
approaches and the depth of the presentation as demanded from both the
practitioner’s and newcomer’s viewpoints.
With respect to the broadness of the approaches, the discussion starts in
Part I, “Basic techniques”, at the coupled Schr¨odinger equation for electrons
and nuclei. Classical, Ehrenfest, Born–Oppenheimer, and Car–Parrinello
molecular dynamics are derived in Chapter 2 from the time-dependent mean-
field approach that is obtained after separating the nuclear and electronic
degrees of freedom. The most extensive discussion is related to the fea-
tures of the standard Car–Parrinello approach, however, all three ab ini-

tio approaches to molecular dynamics - Car–Parrinello, Born–Oppenheimer,
and Ehrenfest - are contrasted and compared. The important issue of how
to obtain the correct forces in these schemes is discussed in some depth.
The two most popular electronic structure theories implemented within ab
initio molecular dynamics, Kohn–Sham density functional theory but also
the Hartree–Fock approach, are only touched upon since excellent text-
books [363, 397, 625, 760, 762, 913, 985, 1102, 1423] already exist in these
well-established fields. Some attention is also given to another important
ingredient in ab initio molecular dynamics, the choice of the basis set.
As for the depth of the presentation, the focus in Part I is clearly on
the implementation of the basic ab initio molecular dynamics schemes in
terms of the powerful and widely used plane wave/pseudopotential formu-
lation of Kohn–Sham density functional theory outlined in Chapter 3. The
explicit formulae for the energies, forces, stress, pseudopotentials, bound-
ary conditions, optimization procedures, etc. are noted for this choice of
method to solve the electronic structure problem, making particular ref-
erence to the CPMD software package [696]. One should, however, keep
in mind that an increasing number of other powerful codes able to per-
form ab initio molecular dynamics simulations are available today (for in-
stance ABINIT [2], CASTEP [234], CONQUEST [282], CP2k [287], CP-PAW [288],
DACAPO [303], FHI98md [421], NWChem [1069], ONETEP [1085], PINY [1153],
PWscf [1172], SIESTA [1343], S/PHI/nX [1377], or VASP [1559] amongst oth-
ers), which are partly based on very similar techniques. An important in-
gredient in any plane wave-based technique is the usage of pseudopotentials
to represent the core electrons, therefore enabling them not to be considered
explicitly. Thus, Chapter 4 of Part I introduces the norm-conserving pseu-
dopotentials up to the point of providing an overview about the different
generation schemes and functional forms that are commonly used.
8 Setting the stage: why ab initio molecular dynamics?
In Part II devoted to “Advanced techniques”, the standard ab initio molec-

ular dynamics approach as outlined in Part I is extended and generalized
in various directions. In Chapter 5, ensembles other than the microcanoni-
cal one are introduced and explained along with powerful techniques used to
deal with large energetic barriers and rare events, and methods to treat other
electronic states than the ground state, such as time-dependent density func-
tional theory in both the frequency and time domains. The approximation
of using classical nuclei is lifted by virtue of the path integral formulation
of quantum statistical mechanics, including a discussion of how to approxi-
mately correct classical time-correlation functions for quantum effects. Var-
ious techniques that allow us to represent only part of the entire system in
terms of an electronic structure treatment, the hybrid, quantum/classical, or
“QM/MM” molecular dynamics simulation methods, are outlined, including
continuum solvation models. Subsequently, advanced pseudopotential con-
cepts such as Vanderbilt’s ultrasoft pseudopotentials and Bl¨ochl’s projector
augmented-wave (PAW) transformation are introduced in Chapter 6.
Modern techniques to calculate properties directly from the available elec-
tronic structure information in ab initio molecular dynamics, such as in-
frared, Raman or NMR spectra, and methods to decompose and analyze
the electronic structure including its dynamical changes are discussed in
Chapter 7. Last but not least, the increasingly important aspect of writing
highly efficient parallel computer codes within the framework of ab initio
molecular dynamics, which take as much advantage as possible of the par-
allel platforms currently available and of those in the foreseeable future, is
the focus of the last section in Part II, Chapter 8.
Finally, Part III is devoted to the wealth of problems that can be addressed
using state-of-the-art ab initio molecular dynamics techniques by referring to
an extensive set of references. The problems treated are briefly outlined with
respect to the broad variety of systems in Chapter 9 and to specific properties
in Chapter 10. The book closes with a short outlook in Chapter 11. In
addition to this printed version of the book corrections and additions will be

provided at www.theochem.rub.de/go/aimd-book.html in an open access
mode.
Part I
Basic techniques

2
Getting started: unifying molecular dynamics and
electronic structure
2.1 Deriving classical molecular dynamics
The starting point of all that follows is non-relativistic quantum mechanics
as formalized via the time-dependent Schr¨odinger equation
i
∂
∂t
Φ({r
i
}, {R
I
}; t)=HΦ({r
i
}, {R
I
}; t) (2.1)
in its position representation in conjunction with the standard Hamiltonian
H = −

I

2
2M

I
∇
2
I
−

i

2
2m
e
∇
2
i
+
1
4πε
0

i<j
e
2
|r
i
− r
j
|
−
1
4πε

0

I,i
e
2
Z
I
|R
I
− r
i
|
+
1
4πε
0

I<J
e
2
Z
I
Z
J
|R
I
− R
J
|
= −


I

2
2M
I
∇
2
I
−

i

2
2m
e
∇
2
i
+ V
n−e
({r
i
}, {R
I
})
= −

I


2
2M
I
∇
2
I
+ H
e
({r
i
}, {R
I
}) (2.2)
for the electronic {r
i
} and nuclear {R
I
} degrees of freedom. Thus, only
the bare electron-electron, electron-nuclear, and nuclear-nuclear Coulomb
interactions are taken into account. Here, M
I
and Z
I
are mass and atomic
number of the Ith nucleus, the electron mass and charge are denoted by
m
e
and −e, and ε
0
is the vacuum permittivity. In order to keep the current

derivation as transparent as possible, the more convenient atomic units (a.u.)
will be introduced only at a later stage.
The goal of this section is to derive molecular dynamics of classical point
particles [25, 468, 577, 1189], that is essentially classical mechanics, starting
11
12 Getting started: unifying MD and electronic structure
from Schr¨odinger’s quantum-mechanical wave equation Eq. (2.1) for both
electrons and nuclei. As an intermediate step to molecular dynamics based
on force fields, two variants of ab initio molecular dynamics are derived in
passing. To achieve this, two complementary derivations will be presented,
both of which are not considered to constitute rigorous derivations in the
spirit of mathematical physics. In the first, more traditional route [355]
the starting point is to consider the electronic part of the Hamiltonian
for fixed nuclei, i.e. the clamped-nuclei part H
e
of the full Hamiltonian,
Eq. (2.2). Next, it is supposed that the exact solution of the corresponding
time-independent (stationary) electronic Schr¨odinger equation,
H
e
({r
i
}; {R
I
})Ψ
k
= E
k
({R
I

})Ψ
k
({r
i
}; {R
I
}) , (2.3)
is known for clamped nuclei at positions {R
I
}. Here, the spectrum of H
e
is
assumed to be discrete and the eigenfunctions to be orthonormalized

Ψ

k
({r
i
}; {R
I
})Ψ
l
({r
i
}; {R
I
}) dr = δ
kl
(2.4)

at all possible positions of the nuclei;

··· dr refers to integration over all
i =1, variables r = {r
i
}. Knowing all these adiabatic eigenfunctions at
all possible nuclear configurations, the total wave function in Eq. (2.1) can
be expanded
Φ({r
i
}, {R
I
}; t)=
∞

l=0
Ψ
l
({r
i
}; {R
I
})χ
l
({R
I
}; t) (2.5)
in terms of the complete set of eigenfunctions {Ψ
l
} of H

e
where the nuclear
wave functions {χ
l
} can be viewed to be time-dependent expansion coef-
ficients. This is an ansatz of the total wave function, introduced by Born
in 1951 [179, 811] for the time-independent problem, in order to separate
systematically the light electrons from the heavy nuclei [180, 771, 811] by
invoking a hierarchical viewpoint.
1
Insertion of this ansatz Eq. (2.5) into the time-dependent coupled Schr¨o-
dinger equation Eq. (2.1) followed by multiplication from the left by
1
“The terms of the molecular spectra comprise, as is known, contributions of varying orders of
magnitude; the largest contribution originates from the electron movement around the nuclei,
there then follows a contribution stemming from the nuclear vibrations, and, ultimately, the
contribution arising from the nuclear rotation. The justification of the existence of such a
hierarchy emanates from the magnitude of the mass of the nuclei, compared to that of the
electrons.” Translated by the authors from “Die Terme der Molekelspektren setzen sich bekan-
ntlich aus Anteilen verschiedener Gr¨oßenordnung zusammen; der gr¨oßte Beitrag r¨uhrt von der
Elektronenbewegung um die Kerne her, dann folgt ein Beitrag der Kernschwingungen, endlich
die von den Kernrotationen erzeugten Anteile. Der Grund f¨ur die M¨oglichkeit einer solchen
Ordnung liegt offensichtlich in der Gr¨oße der Masse der Kerne, verglichen mit der der Elektro-
nen.” Cited from the Introduction of the seminal paper [180] by Born and Oppenheimer from
1927.
2.1 Deriving classical molecular dynamics 13
Ψ

k
({r

i
}; {R
I
}) and integration over all electronic coordinates r leads to
a set of coupled differential equations

−

I

2
2M
I
∇
2
I
+ E
k
({R
I
})

χ
k
+

l
C
kl
χ

l
= i
∂
∂t
χ
k
(2.6)
where
C
kl
=

Ψ

k

−

I

2
2M
I
∇
2
I

Ψ
l
dr

+
1
M
I

I


Ψ

k
[−i∇
I
]Ψ
l
dr

[−i∇
I
] (2.7)
is the exact nonadiabatic coupling operator. The first term is a matrix
element of the kinetic energy operator of the nuclei, whereas the second
term depends on their momenta.
The diagonal contribution C
kk
depends only on a single adiabatic wave
function Ψ
k
and as such represents a correction to the adiabatic eigenvalue
E

k
of the electronic Schr¨odinger equation Eq. (2.3) in this kth state. As
a result, the “adiabatic approximation” to the fully nonadiabatic problem
Eq. (2.6) is obtained by considering only these diagonal terms,
C
kk
= −

I

2
2M
I

Ψ

k
∇
2
I
Ψ
k
dr , (2.8)
the second term of Eq. (2.7) being zero when the electronic wave function
is real, which leads to complete decoupling

−

I


2
2M
I
∇
2
I
+ E
k
({R
I
})+C
kk
({R
I
})

χ
k
= i
∂
∂t
χ
k
(2.9)
of the fully coupled original set of differential equations Eq. (2.6). This,
in turn, implies that the motion of the nuclei proceeds without changing
the quantum state, k, of the electronic subsystem during time evolution.
Correspondingly, the coupled wave function in Eq. (2.1) can be decoupled
simply
Φ({r

i
}, {R
I
}; t) ≈ Ψ
k
({r
i
}; {R
I
})χ
k
({R
I
}; t) (2.10)
into a direct product of an electronic and a nuclear wave function. Note
that this amounts to taking into account only a single term in the general
expansion Eq. (2.5).