COMPUTER SIMULATION METHODS IN CHEMISTRY AND PHYSICS

Finally, it is assumed thateach single-point electronic structure calculation needed to devise the global potential energy surface and one ab initio molecular dynamics time step require

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AB INITIO M O L E C U L A R D Y N A M I C S :

B A S I C T H E O RY A N D A D VA N C E D M E T H O D S

Ab initio molecular dynamics revolutionized the field of realistic computer

simulation of complex molecular systems and processes, including chemicalreactions, by unifying molecular dynamics and electronic structure theory Thisbook provides the first coherent presentation of this rapidly growing field, covering

a vast range of methods and their applications, from basic theory to advancedmethods

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 waveelectronic structure codes, allowing newcomers to the field to understand commonlyused program packages, and enabling developers to improve and add new features

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São PauloCambridge 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

Information on this title: www.cambridge.org/9780521898638

This publication is in copyright Subject to statutory exception and to the

provision of relevant collective licensing agreements, no reproduction of any partmay take place without the written permission of Cambridge University Press

Cambridge University Press has no responsibility for the persistence or accuracy

of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain,

accurate or appropriate

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

eBook (EBL)hardback

3.4 Total energy, gradients, and stress tensor 1043.5 Energy and force calculations in practice 109

4.3 Pseudopotentials in the plane wave basis 152

v

vi Contents

6.6 Integrating the Car–Parrinello equations 297

9.2 Solids, minerals, materials, and polymers 372

9.4 Mechanochemistry and molecular electronics 380

Contents vii

9.11 Chemical reactions and transformations 396

10.5 Electronic spectroscopy and redox properties 41210.6 X-ray diffraction and Compton scattering 413

In this book we develop the rapidly growing field of ab initio molecular

dynamics computer simulations from the underlying basic ideas up to thelatest techniques, from the most straightforward implementation up to mul-tilevel parallel algorithms Since the seminal contributions of Roberto Carand 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 body problem of interacting electrons and nuclei, which form atoms andmolecules as described in the framework of quantum mechanics, whereas thedynamics is captured in terms of classical dynamics and statistical mechan-

many-ics Due to its inherent virtues, ab initio molecular dynamics is currently

an extremely popular and ever-expanding computational tool employed tostudy physical, chemical, and biological phenomena in a very broad sense

In particular, it is the basis of what could be called a “virtual laboratoryapproach” used to study complex processes at the molecular level, includingthe 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 ments of the last 20 years in this field within a unified framework at the level

develop-of an advanced text Currently, any newcomer in the field has to face theproblem 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 thebeginner 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 andlimitations 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 bothertoo much about the technical aspects, let alone about method development.This is indeed possible, in principle, since several rather easy-to-use programpackages are now on the market, mostly for free or at low cost for academicusers

In particular, different flavors of ab initio molecular dynamics methods

are explained and compared in the first part of this book at an introductorylevel, the focus being on the efficient extended Lagrangian approach as in-troduced by Car and Parrinello in 1985 But in the meantime, a wealth oftechniques that go far beyond what we call here the “standard approach”,that is microcanonical molecular dynamics in the electronic ground stateusing 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 includeexcited electronic states and nonadiabatic effects, to deal with quantum ef-fects on nuclei, and to treat complex biomolecular systems in terms of mixedquantum/classical approaches Most important for the practitioner is thecomputation 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 fragmentdipole moments, localized orbitals, or effective atomic charges Finally inPart 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 acompilation of pertinent literature for future reference and upcoming appli-cations

In addition to all these aspects we also want to provide a solid basis oftechnical 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 aspossible, specific references in the text to the original literature as well as

to review articles To achieve this, the very popular approach of solvingthe electronic structure problem in the framework of Kohn–Sham densityfunctional 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 whendiscussing them at the level of implementation Here, the widely used andever-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-conservingpseudopotentials, 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 projectoraugmented-wave transformation and ultrasoft pseudopotentials A crucialaspect for large-scale applications, given the current computer architecturesand 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 hierarchicalmultilevel 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 parallelcomputers

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 puter simulation methods In the spirit of this idea, we will maintain thesite 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

com-We thus encourage all readers to send us information about possible errors,which are definitively hidden at many places despite our investment of muchcare 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 andlongstanding collaboration with Michele Parrinello, initially at IBM ZurichResearch Laboratory in R¨uschlikon and later at the Max-Planck-Institut f¨urFestk¨orperforschung in Stuttgart, which we gratefully acknowledge on thisoccasion In addition, we profited enormously from pleasant cooperationswith too many friends and colleagues to be named here

ei-726, 1189, 1449, 1504, 1538, 1539] At the very heart of any moleculardynamics scheme is the question of how to describe – that is in prac-tice how to approximate – the interatomic interactions The traditionalroute followed in molecular dynamics is to determine these potentials inadvance Typically, the full interaction is broken up into two-body andmany-body contributions, long-range and short-range terms, electrostaticand non-electrostatic interactions, etc., which have to be represented bysuitable functional forms, see Refs [550, 1405] for detailed accounts Af-ter decades of intense research, very elaborate interaction models, includingthe 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 seriousdrawbacks [1123, 1209] Among the most significant are systems in which(i) many different atom or molecule types give rise to a myriad of differentinteratomic interactions that have to be parameterized and/or (ii) the elec-tronic structure and thus the chemical bonding pattern changes qualitativelyduring 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 satisfactorypotentials, 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 usefor such methods are for instance first principles (FPMD), on-the-fly, di-rect, extended Lagrangian (ELMD), density functional (DFMD), quantumchemical, 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 the-fly” as the molecular dynamics trajectory is generated, see Fig 1.1 for asimplifying scheme In this way, the electronic variables are not integratedout beforehand and represented by fixed interaction potentials, rather theyare considered to be active and explicit degrees of freedom in the course

“on-of the simulation This implies that, given a suitable approximate solution

of the many-electron problem, also “chemically complex” systems, or thosewhere the electronic structure changes drastically during the dynamics, can

be handled easily by molecular dynamics But this also implies that the proximation is shifted from the level of devising an interaction potential tothe level of selecting a particular approximation for solving the Schr¨odingerequation, since it cannot be solved exactly for the typical problems at hand

ap-Applications of ab initio molecular dynamics are particularly widespread

in physics, chemistry, and more recently also in biology, where the mentioned difficulties (i)-(iii) are particularly severe [39, 934] A collection

afore-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 inFig 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 thenumber of citations of a particular paper, the one by Car and Parrinellofrom 1985 entitled “Unified approach for molecular dynamics and density-functional theory” [222, 1216], initially parallels the growth trend of theentire field, see the circles in Fig 1.2 Thus, the resonance evoked by thispublication and, at its very heart, the introduction of the Car–Parrinello

Setting the stage: why ab initio molecular dynamics? 3

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 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).

in-“Lagrangean” [995], has gone hand in hand with the popularity of the tire field over the last decade Incidentally, the 1985 paper by Car andParrinello is the last one included in the section “Trends and Prospects”

en-in the repren-int 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 classificationnumber (“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

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 smallerthan what is affordable in the framework of standard molecular dynamics.More recently, this discrepancy was counterbalanced by the ever-increasingpower 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 andchemical bonding patterns Most importantly, however, is the fact that newphenomena, 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 electronicstructure calculations After fitting it to a suitable analytical form in a sec-ond step (but without imposing additional approximations such as pairwiseadditivity, etc.), the dynamical evolution of the nuclei is generated in a thirdstep by using classical mechanics, quantum mechanics, or semi/quasiclassicalapproximations of various sorts In the case of using classical mechanics todescribe 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 103N −6 electronic

struc-ture calculations are needed in order to map such a global potential energysurface Thus, the computational workload for the first step in the approachoutlined 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 10M molecular dynamics steps, i.e

10M electronic structure calculations are needed to generate one trajectory.Furthermore, it is assumed that 10n independent trajectories are necessary

in order to average over different initial conditions so that 10M +n ab initio

molecular dynamics steps are required in total Finally, it is assumed thateach 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 energysurface amounts to about 103N −6−M−n The crucial point is that for agiven 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 trajectoriesafterwards 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 thecomputation of global potentials by carefully selecting the discretizationpoints and reducing their number, choosing sophisticated representationsand internal coordinates, exploiting symmetry and decoupling of irrelevantmodes, implementing efficient sampling and smart extrapolation techniquesand so forth Still, these improvements mostly affect the prefactor but notthe overall scaling behavior, ∼ 10 N, with the number of active degrees offreedom Other strategies consist of, for instance, reducing the number ofactive degrees of freedom by constraining certain internal coordinates, rep-resenting less important ones by a (harmonic) bath or by friction forces, orbuilding 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 putational advantage of the “on-the-fly” approaches diminishes as more andmore trajectories are needed for a given (small) system For instance, exten-sive averaging over many different initial conditions is required in order tocalculate scattering or reactive cross-sections quantitatively Summarizing

com-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, andtutorials given by the authors as well as parts from our previous review andoverview articles Here, emphasis is put on both the broad extent of theapproaches and the depth of the presentation as demanded from both thepractitioner’s and newcomer’s viewpoints

With respect to the broadness of the approaches, the discussion starts inPart I, “Basic techniques”, at the coupled Schr¨odinger equation for electronsand nuclei Classical, Ehrenfest, Born–Oppenheimer, and Car–Parrinellomolecular dynamics are derived in Chapter 2 from the time-dependent mean-field approach that is obtained after separating the nuclear and electronicdegrees of freedom The most extensive discussion is related to the fea-

tures of the standard Car–Parrinello approach, however, all three ab tio approaches to molecular dynamics - Car–Parrinello, Born–Oppenheimer,

ini-and Ehrenfest - are contrasted ini-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 books [363, 397, 625, 760, 762, 913, 985, 1102, 1423] already exist in thesewell-established fields Some attention is also given to another important

text-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 lation of Kohn–Sham density functional theory outlined in Chapter 3 Theexplicit formulae for the energies, forces, stress, pseudopotentials, bound-ary conditions, optimization procedures, etc are noted for this choice ofmethod to solve the electronic structure problem, making particular ref-erence to the CPMD software package [696] One should, however, keep

formu-in mformu-ind that an formu-increasformu-ing number of other powerful codes able to form ab initio molecular dynamics simulations are available today (for in-

per-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 consideredexplicitly Thus, Chapter 4 of Part I introduces the norm-conserving pseu-dopotentials up to the point of providing an overview about the differentgeneration 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 cal one are introduced and explained along with powerful techniques used todeal with large energetic barriers and rare events, and methods to treat otherelectronic states than the ground state, such as time-dependent density func-tional theory in both the frequency and time domains The approximation

microcanoni-of using classical nuclei is lifted by virtue microcanoni-of the path integral formulation

of quantum statistical mechanics, including a discussion of how to mately correct classical time-correlation functions for quantum effects Var-ious techniques that allow us to represent only part of the entire system interms of an electronic structure treatment, the hybrid, quantum/classical, or

approxi-“QM/MM” molecular dynamics simulation methods, are outlined, includingcontinuum solvation models Subsequently, advanced pseudopotential con-cepts such as Vanderbilt’s ultrasoft pseudopotentials and Bl¨ochl’s projectoraugmented-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 analyzethe electronic structure including its dynamical changes are discussed inChapter 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 allel platforms currently available and of those in the foreseeable future, isthe focus of the last section in Part II, Chapter 8

par-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 withrespect 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 Inaddition to this printed version of the book corrections and additions will beprovided at www.theochem.rub.de/go/aimd-book.html in an open accessmode

Part I

Basic techniques

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

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

meand −e, and ε0 is the vacuum permittivity In order to keep the currentderivation 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 pointparticles [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 bothelectrons 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 thespirit of mathematical physics In the first, more traditional route [355]the starting point is to consider the electronic part of the Hamiltonianfor fixed nuclei, i.e the clamped-nuclei part He 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,

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

in 1951 [179, 811] for the time-independent problem, in order to separatesystematically the light electrons from the heavy nuclei [180, 771, 811] byinvoking a hierarchical viewpoint.1

Insertion of this ansatz Eq (2.5) into the time-dependent coupled Schr¨dinger equation Eq (2.1) followed by multiplication from the left by

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-

Elektronenbewegung um die Kerne her, dann folgt ein Beitrag der Kernschwingungen, endlich

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

is the exact nonadiabatic coupling operator The first term is a matrixelement of the kinetic energy operator of the nuclei, whereas the secondterm depends on their momenta

The diagonal contribution C kk depends only on a single adiabatic wavefunction Ψk 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,

the second term of Eq (2.7) being zero when the electronic wave function

is real, which leads to complete decoupling

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 decoupledsimply

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 Notethat this amounts to taking into account only a single term in the generalexpansion Eq (2.5)

14 Getting started: unifying MD and electronic structure

The ultimate simplification consists in neglecting also the diagonal pling terms

of the mass ratio (me/M I)1/4, see also§ 14 and Appendix VII in Ref [179])

are readily derived as special cases based on the particular functional ansatz

Eq (2.5) of the total wave function In the above simplified presentationsubtleties due to Berry’s geometric phase [1329] have been ignored, but theinterested reader is referred to excellent reviews [168, 986, 1642] that coverthis general phenomenon with a focus on molecular systems

The next step in the derivation of molecular dynamics is the task of proximating the nuclei as classical point particles How can this be achieved

ap-in the framework where a full quantum-mechanical wave equation, χ k, scribes the motion of all nuclei in a selected electronic state Ψk? In order

de-to proceed, it is first noted that for a great number of physical situationsthe Born–Oppenheimer approximation can safely be applied, but see Sec-tion 5.3 for a discussion of cases where this is not the case Based on thisassumption, the following derivation will be built on Eq (2.11) being theBorn–Oppenheimer approximation to the fully coupled solution, Eq (2.6).Secondly, a well-known route to extract semiclassical mechanics from quan-tum mechanics in general starts with rewriting the corresponding wave func-tion

χ k({RI }; t) = A k({RI }; t) exp [iS k({RI }; t)/
] (2.12)

in terms of an amplitude factor A k and a phase S kwhich are both considered

to be real and A k > 0 in this polar representation, see for instance Refs [345,

996, 1268] After transforming the nuclear wave function in Eq (2.11) for

a chosen electronic state k accordingly and after separating the real and

imaginary parts, the equations for the nuclei

2.1 Deriving classical molecular dynamics 15

using Reχ k and Imχ k It is noted in passing that this quantum fluid dynamic(or hydrodynamic, Bohmian) representation [169, 1636], Eqs (2.13)-(2.14),can actually be used to solve the time-dependent Schr¨odinger equation [340,878]

The relation for the amplitude, Eq (2.14), may be rewritten after

multi-plying by 2A k from the left as a continuity equation [345, 996, 1268]

k, obtained directly from the definition Eq (2.12), and with the

associated current density defined as Jk,I = A2k(∇I S k )/M I This continuityequation Eq (2.16) is independent of
and ensures locally the conservation

of the particle probability density |χ k |2 of the nuclei in the presence of aflux

More important for the present purpose is a detailed discussion of the

relation for the phase S k, Eq (2.13), of the nuclear wave function that is

associated with the kth electronic state This equation contains one term

that depends explicitly on
, a contribution that vanishes

if the classical limit is taken as
→ 0 Note that a systematic expansion in

terms of
would, instead, lead to a hierarchy of semiclassical methods [562,996] The resulting equation Eq (2.17) is now isomorphic to the equation ofmotion in the Hamilton–Jacobi formulation [528, 1282] of classical mechanics

∂S k

∂t + H k({R I }, {∇ I S k }) = 0 (2.18)with the classical Hamilton function

16 Getting started: unifying MD and electronic structure

defined in terms of (generalized) coordinates{R I } and their conjugate

canon-ical momenta {P I } With the help of the connecting transformation

PI ≡ ∇ I S k = M IJk,I

ρ k

(2.21)

the Newtonian equations of motion, ˙ PI =−∇ I V k({R I }), corresponding to

the Hamilton–Jacobi form Eq (2.17) can be read off

ac-by the Born–Oppenheimer potential energy surface E k obtained by solving

simultaneously the time-independent electronic Schr¨odinger equation for the

kth state, Eq (2.3), at the given nuclear configuration {R I (t) } In other

words, this time-local many-body interaction potential due to the quantumelectrons is a function of the set of all classical nuclear positions at time

t Since the Born–Oppenheimer total energies in a specific adiabatic tronic state yield directly the forces used in this variant of ab initio molec-

elec-ular dynamics, this particelec-ular approach is often called “Born–Oppenheimermolecular dynamics”, to be discussed in more detail later in Section 2.3

In order to present an alternative derivation, which does maintain a

quantum-mechanical time evolution of the electrons and thus does not voke solving the time-independent electronic Schr¨odinger equation Eq (2.3)

in-as before, the elegant route taken in Refs [1516, 1517] is followed; see alsoRef [943] To this end, the nuclear and electronic contributions to the totalwave function Φ({r i }, {R I }; t) are separated directly such that, ultimately,

the classical limit can be imposed for the nuclei only The simplest possibleform is a product ansatz

where the nuclear and electronic wave functions are separately normalized

to unity at every instant of time, i.e χ; t|χ; t = 1 and Ψ; t|Ψ; t = 1,

respectively In addition, a phase factor

˜

Ee=



Ψ
({ri }; t) χ
({RI }; t) HeΨ({ri }; t) χ({R I }; t) drdR (2.24)was introduced at this stage for convenience such that the final equations

2.1 Deriving classical molecular dynamics 17

will look simpler; again

· · · drdR refers to the integration over all i = 1, and I = 1, electronic and nuclear variables r = {r i } and R = {R I }, re-

spectively It is mentioned in passing that this approximation is called a

one-determinant or single-configuration ansatz for the total wave function,

which at the end must lead to a mean-field description of the coupled namics Note in addition that this product ansatz differs, independentlyfrom the issue of phase factor, from Born’s ansatz Eq (2.5) used above in

dy-terms of adiabatic electronic states {Ψ k } even if only a single electronic state

k is considered according to Eq (2.10).

Inserting this particular separation ansatz Eq (2.23) into Eqs (2.1)–(2.2)yields (after multiplying from the left by Ψ
and χ
, integrating over nu-clear and electronic coordinates, respectively, and imposing conservation

d H /dt ≡ 0 of the total energy) the following relations

ba-as early ba-as 1930 by Dirac [344], see also Ref [338] Both electrons and

nuclei move quantum-mechanically in time-dependent effective potentials,

i.e self-consistently obtained average fields, given by the expressions in thebraces These potentials are obtained from appropriate averages (defined asquantum-mechanical expectation values ) over the other class of degrees

of freedom by using the nuclear and electronic wave functions, respectively.Thus, the single-determinant ansatz Eq (2.23) produces, as already antici-

pated, a mean-field description of the coupled nuclear–electronic quantum dynamics This is the price to pay for the simplest possible separation of electronic and nuclear variables in terms of dynamics.

At this stage the nuclei must again be approximated as classical pointparticles, however this time in the presence of electrons which do movequantum-mechanically in time according to Eq (2.25) Invoking the same

18 Getting started: unifying MD and electronic structure

trick as before, see Eq (2.12), but using χ as defined in Eq (2.23) instead,

one now arrives at the TDSCF equations

if the classical limit is taken again as
→ 0 Correspondingly, the Newtonian

equations of motion of the classical nuclei equivalent to Eq (2.29) are givenby

time-freedom, VE

e =Ψ|He|Ψ

However, the very TDSCF equation that describes the time evolution ofthe electrons, Eq (2.25), still contains the full quantum-mechanical nuclear

wave function χ( {R I }; t) instead of just the classical-mechanical nuclear

po-sitions {R I (t) } In this case the classical reduction can be achieved simply

by replacing the nuclear density|χ({R I }; t)|2in Eq (2.25) in the limit
→ 0

by a product of delta functions

I δ(R I − R I (t)) centered at the

instanta-neous positions {R I (t) } of the classical nuclei as given by Eq (2.30) This

naive approach yields, e.g for the position operator,



χ
({R I }; t) R I χ( {R I }; t) dR −→ R
→0 I (t) (2.31)the required expectation value This classical limit leads to a time-dependent

2.1 Deriving classical molecular dynamics 19

wave equation for the electrons

which evolves self-consistently as the classical nuclei are propagated via

Eq (2.30) Note that now He depends parametrically on the classical clear positions {R I (t) } at time t through Vn −e({r i }, {R I (t) }) This means

nu-that feedback between the classical and quantum degrees of freedom is corporated in both directions, although in a mean-field sense only This is

in-at variance with the much simpler “classical pin-ath” or Mott non-SCF proach to dynamics [1516, 1517] where the quantum subsystem moves along

ap-a predefined clap-assicap-al trap-ajectory

The approach to ab initio molecular dynamics that relies on solving

New-ton’s equation for the nuclei, Eq (2.30), simultaneously with Schr¨odinger’sequation for the electrons, Eq (2.32), is often called “Ehrenfest moleculardynamics” in honor of Paul Ehrenfest who was the first to address the es-sential question2of how Newtonian classical dynamics of point particles can

be derived from Schr¨odinger’s time-dependent wave equation [382] In thepresent case this leads to a hybrid or mixed quantum–classical approach be-cause only the nuclei are forced to behave like classical particles, whereas theelectrons are still treated as quantum objects At variance with the Born–Oppenheimer molecular dynamics approach relying on solving a stationarySchr¨odinger equation, however, the electronic subsystem evolves explicitly

in time according to a time-dependent Schr¨odinger equation, see Section 2.2for more detail

Although the TDSCF approach underlying Ehrenfest molecular dynamics

is clearly a mean-field theory concerning the dynamical evolution, tions between electronic states are included in this scheme at variance withthe Born–Oppenheimer molecular dynamics technique This can be made

transi-transparent by expanding the electronic wave function Ψ in Eq (2.23) (as opposed to the total wave function Φ according to Eq (2.5)) in a basis of

to be able to answer the following question as elementarily as possible: which retrospective view results when considering Newton’s fundamental equations of classical mechanics from the

vom Standpunkt der Quantenmechanik auf die Newtonschen Grundgleichungen der klassischen Mechanik?”

20 Getting started: unifying MD and electronic structure

electronic states Ψl

Ψ({r i }, {R I }; t) =
∞

l=0

c l (t)Ψ l({r i }; {R I }) (2.33)

with complex time-dependent coefficients {c l (t) } In this case, the

coeffi-cients|c l (t) |2 satisfying l |c l (t) |2≡ 1 describe explicitly the time evolution

of the populations (occupations) of the different states l whereas the

nec-essary interferences between any two such states are included via the

off-diagonal term, c
k c l =k One possible choice for the basis functions{Ψ k } is the

instantaneous adiabatic basis obtained from solving the time-independentelectronic Schr¨odinger equation

He({r i }; {R I })Ψ k = E k({R I })Ψ k({r i }; {R I }) , (2.34)where {R I } are the instantaneous nuclear positions at time t that are de-

termined according to Eq (2.30) The actual equations of motion in terms

of expansion coefficients {c k }, adiabatic energies {E k }, and nonadiabatic

couplings are presented in Section 2.2

Here, instead, a further simplification is invoked in order to reduce fest molecular dynamics to Born–Oppenheimer molecular dynamics Toachieve this, the electronic wave function Ψ is restricted to be the groundstate adiabatic wave function Ψ0 ofHe at each instant of time according to

Ehren-Eq (2.34), which implies|c0(t) |2≡ 1 and thus a single term in the expansion

Eq (2.33) This should be a good approximation if the energy differencebetween Ψ0 and the first excited state Ψ1 is large everywhere compared to

the thermal energy kBT , roughly speaking In this limit the nuclei move

by solving the time-independent electronic Schr¨odinger equation Eq (2.34)

for k = 0 at each nuclear configuration {R I } generated during molecular dynamics This leads to the identification VeE ≡ E0 = V0BO and thus to

Eq (2.22), i.e in this limit the Ehrenfest potential is identical to the groundstate Born–Oppenheimer (or “clamped nuclei”) potential

As a consequence of this observation, it is conceivable to fully decouple thetask of generating the classical nuclear dynamics from the task of computingthe quantum potential energy surface In a first step, the global potential

energy surface E0, which depends on all nuclear degrees of freedom {R I },

2.1 Deriving classical molecular dynamics 21

is computed for many different nuclear configurations by solving the tionary Schr¨odinger equation separately for all these situations In a secondstep, these data points are fitted to an analytical functional form to yield

sta-a globsta-al potentista-al energy surfsta-ace [1280], from which the grsta-adients csta-an beobtained analytically In a third step, the Newtonian equations of motionare solved on this surface for many different initial conditions, producing a

“swarm” of classical trajectories{R I (t) } This is, in a nutshell, the basis of classical trajectory calculations on global potential energy surfaces as used

very successfully to understand scattering and chemical reaction dynamics

of small systems in vacuum [1284, 1514]

As already explained in the general introduction, Chapter 1, such proaches suffer severely from the “dimensionality bottleneck” as the number

ap-of active nuclear degrees ap-of freedom increases One traditional way out ap-ofthis dilemma, making possible calculations of large systems, is to approxi-mate the global potential energy surface

N

I<J v2(R I , R J)

when the nuclei are propagated Thus, the mixed quantum/classical lem is reduced to purely classical mechanics, once the{v n } are determined.

prob-“Standard” molecular dynamics

In force field-based molecular dynamics, typically only two-body v2 or

three-body v3 interactions are taken into account [25, 468, 577, 1189], though very sophisticated models exist that include nonadditive many-body

al-22 Getting started: unifying MD and electronic structure

interactions This amounts to a dramatic simplification and removes, inparticular, the dimensionality bottleneck since the global potential surface

is reconstructed from a manageable sum of additive few-body contributions.The flipside of the medal is the introduction of the drastic approximationembodied in Eq (2.36), which basically excludes the study of chemical re-actions from the realm of computer simulation

As a result of the derivation presented above, the essential assumptionsunderlying standard force field-based molecular dynamics become very trans-parent The electrons follow adiabatically the classical nuclear motion andcan be integrated out so that the nuclei evolve on a single Born–Oppenheimerpotential energy surface (typically, but not necessarily, given by the elec-tronic ground state), which is generally approximated in terms of few-bodyinteractions

Actually, force field-based molecular dynamics for many-body systems is

only made possible by somehow decomposing the global potential energy

In order to illustrate this point, consider the simulation of N = 500 argon

atoms in the liquid phase [395] where the interactions can be described

faith-fully by additive two-body terms, i.e VFF

I<J v2(|R I − R J |) Thus, the determination of the pair potential v2 from ab initio electronic

structure calculations consists in computing and fitting a one-dimensionalfunction only The corresponding task of determining a global potential en-ergy surface, however, amounts to doing that in about 101500 dimensions,which is simply impossible In the case of neat argon this is obviously notnecessary, but this assessment changes drastically if the 500 atoms are notall identical and, for instance, are those that form a small enzyme catalyzing

a particular biochemical reaction

2.2 Ehrenfest molecular dynamics

A systematic and general way out of the dimensionality bottleneck, otherthan to approximate the global potential energy surface Eq (2.36) or reducethe number of active degrees of freedom, is to take seriously the classicalnuclei approximation to the TDSCF equations, Eqs (2.30) and (2.32) Thisimplies computing the Ehrenfest force by actually solving numerically

M IR ¨I (t) = −∇ I



Ψ
He Ψ dr

2.2 Ehrenfest molecular dynamics 23

the coupled set of quantum/classical equations simultaneously Thereby, the

a priori construction of any type of potential energy surface is avoided from the outset by solving the time-dependent electronic Schr¨odinger equation

“on-the-fly” as the nuclei are propagated using classical mechanics This lows one to compute the force from−∇ I He for each configuration {R I (t) }

al-generated by molecular dynamics; see Section 2.5 for the issue of using theHellmann–Feynman forces, i.e −Ψ|∇ I He|Ψ instead of −∇ I Ψ|He|Ψ

The equations of motion corresponding to Eqs (2.38)-(2.39) can be pressed conveniently by representing the electronic wave function Ψ in terms

ex-of the instantaneous adiabatic electronic states Eq (2.34) and the dependent expansion coefficients Eq (2.33) This particular representation

time-of Ehrenfest molecular dynamics reads [355, 1516, 1517]

with the property dkk I ≡ 0 of the nonadiabatic coupling vectors for real

wave functions The Ehrenfest approach to ab initio molecular dynamics

is thus seen to include rigorously nonadiabatic transitions between differentelectronic states Ψk and Ψl within the framework of classical nuclear mo-

tion and the mean-field (TDSCF) approximation Eq (2.23) to the coupled

problem, see e.g Refs [355, 1516, 1517] for reviews and Section 5.3.5 forimplementations in terms of time-dependent density functional theory.The restriction to one electronic state in the expansion Eq (2.33), which

24 Getting started: unifying MD and electronic structure

is in most cases the ground state Ψ0, leads to

in the context of condensed matter problems More recently, however, itsuse in conjunction with time-dependent density functional theory to describethe electronic subsystem gained a lot of attention, see Section 5.3.5 for anoutline of these methods and, for instance, Refs [77, 369, 370, 451, 1258,

1316, 1400, 1463, 1637, 1638] for some of the implementations, but there

is also progress within the realm of Hartree–Fock-based electronic structuremethods [862]

2.3 Born–Oppenheimer molecular dynamics

An alternative approach to include the electronic structure in molecular

dy-namics simulations consists in straightforwardly solving the static electronic structure problem in each molecular dynamics step given the set of fixed

nuclear positions at that instant of time Thus, the electronic structure part

is reduced to solving a time-independent quantum problem, e.g by ing the time-independent, stationary Schr¨odinger equation, concurrently topropagating the nuclei according to classical mechanics This implies thatthe time dependence of the electronic structure is imposed and dictated byits parametric dependence on the classical dynamics of the nuclei which it

solv-just follows Thus, it is not an intrinsic dynamics as in Ehrenfest molecular

dynamics The resulting Born–Oppenheimer molecular dynamics method

2.3 Born–Oppenheimer molecular dynamics 25

can be written down readily and is defined by

de-of electrons and nuclei

Concerning the nuclear equation of motion, a profound difference withrespect to Ehrenfest dynamics is that the minimum ofHe has to be reached

in each time step of a Born–Oppenheimer molecular dynamics propagationaccording to Eq (2.46), for instance by diagonalizing the Hamiltonian InEhrenfest dynamics, on the other hand, a wave function that minimizedHe

initially will stay, in the absence of external perturbations, in its respectiveground state minimum as the nuclei move according to Eq (2.44) by virtue

of the unitarity of the wave function propagation according to Eq (2.45).Within the framework of Born–Oppenheimer dynamics it is easily possi-ble to apply the scheme to some specific excited electronic state Ψk , k > 0,

but without considering any interferences with other states Ψl =k nor with

it-self Thus, both the nondiagonal and diagonal corrections are neglected and,

hence, Born–Oppenheimer molecular dynamics should not be called

“adia-batic molecular dynamics” as is sometimes done, since the latter methodwould require including the diagonal corrections C kk as defined in Eq (2.8)and thus solving Eq (2.9)

For the sake of later reference, it is useful at this stage to formulate theelectronic part of the Born–Oppenheimer molecular dynamics equations ofmotion, i.e the stationary Schr¨odinger equation Eq (2.47), for the specialcase of effective one-particle Hamiltonians such as Hartree–Fock theory (seeSection 2.7.3 for a concise introduction to this electronic structure method).The Hartree–Fock approximation [625, 762, 985, 1423] is obtained fromthe variational minimum of the energy expectation valueΨ0 |He| Ψ0 using

a single Slater determinant Ψ0 = 1/ √

N ! det{φ i } to represent the exact

electronic wave function subject to the constraint that the one-particle wave

functions (i.e the orbitals) φ i are orthonormal, i.e φ i |φ j = δ ij Thecorresponding constrained minimization of the total energy with respect tothe orbitals

26 Getting started: unifying MD and electronic structure

can be cast into Lagrange’s formalism

L = − Ψ0 |He| Ψ0 +

i,j

Λij(φi |φ j − δ ij) (2.49)

where Λij are the associated Lagrange multipliers that are necessary in order

to impose the constraints Unconstrained variation of this Lagrangian withrespect to the orbitals

δ L

δφ
i

as discussed amply in standard textbooks [625, 762, 985, 1423] The more

familiar diagonal canonical form HeHFφ i =  i φ i is obtained after a unitary

transformation and HeHFdenotes the effective one-particle Hamiltonian, seeSection 2.7 for more details The equations of motion resulting from thegeneral formulas Eqs (2.46)-(2.47) read

e has to be replaced by the Kohn–Sham effective one-particle

Hamiltonian HeKS, see Section 2.7.2 for more details on Kohn–Sham theory

Instead of diagonalizing a one-particle Hamiltonian such as HeHFor HeKS, analternative but equivalent approach consists in performing the constrainedminimization directly according to Eq (2.48), using nonlinear optimizationtechniques explicitly

Early applications of the iterative Born–Oppenheimer molecular dynamicsscheme have been performed in the framework of semiempirical approxima-tions to the electronic structure problem [1590, 1596] But only a few years

2.4 Car–Parrinello molecular dynamics 27

later an ab initio approach was implemented within the Hartree–Fock

ap-proximation [853] Most notably, combining the Born–Oppenheimer agation scheme with density functional theory [1122] in conjunction withefficient orbital prediction schemes [45, 1123] and parallelization strate-gies [274] in the late 1980s to early 1990s was particularly successful in

prop-the realm of ab initio condensed matter and also cluster simulations [75,

234, 785, 786, 1123, 1307, 1419, 1559] In addition to these efforts, Born–Oppenheimer dynamics started to become popular in more general terms inthe early 1990s [423, 551, 605, 607, 620, 896], including molecular dynamics

in electronically excited states [605], with the availability of more efficientquantum chemistry electronic structure codes in conjunction with sufficientcomputer power to solve “interesting problems”, see for instance the compi-lation of such studies in Table 1 of an overview article [174] More recently,

a revival of these activities with greatly improved algorithms to performBorn–Oppenheimer simulations is observed [634, 795, 1203, 1552]

Undoubtedly, the breakthrough of Kohn–Sham density functional theory

in the realm of chemistry - which took place around the same time in theearly 1990s - also helped by improving greatly the “price/performance ratio”

of all ab initio molecular dynamics methods, see e.g Refs [766, 1390, 1622].

A third, and possibly the crucial reason that established and boosted the

field of ab initio molecular dynamics enormously [39] was the seminal tribution of the Car–Parrinello approach [222, 1216] to ab initio molecular

con-dynamics The conclusion that this particular paper was highly influential

is demonstrated by the time evolution of its citation response as depicted inFig 1.2 At that time, this numerically highly efficient technique not only

opened novel avenues to treat large-scale problems via ab initio molecular

dynamics and catalyzed the entire field by making “interesting calculations”possible [38], but it also united the two quite distinct electronic structurecommunities (i.e “Quantum Chemistry” and “Total Energy”) with thecomputer simulation (“Computational Physics”) community [39, 934] This

particularly efficient approach to performing ab initio simulations is

pre-sented in detail in the next section

2.4 Car–Parrinello molecular dynamics

2.4.1 Motivation

A non-obvious approach to cut down the computational expenses of ular dynamics, which includes the electrons as active degrees of freedom,was proposed by Roberto Car and Michele Parrinello in 1985 [222, 1216] Inretrospect it can be considered to combine the advantages of both Ehrenfest

molec-28 Getting started: unifying MD and electronic structure

and Born–Oppenheimer molecular dynamics in an optimal way In Ehrenfestdynamics the time scale and thus the time step to integrate Eqs (2.44) and(2.45) simultaneously is dictated by the intrinsic dynamics of the electrons

as described by the time-dependent Schr¨odinger equation Since electronicmotion is typically much faster than nuclear motion - being the physicalbasis of the adiabatic and Born–Oppenheimer approximations - the largestpossible time step in Ehrenfest dynamics is that which allows us to integratethe electronic equations of motion properly Contrary to that, there is noelectron dynamics whatsoever involved in solving the Born–Oppenheimerequations of motion, Eqs (2.46)-(2.47), because the electronic problem is

treated within the time-independent, stationary Schr¨odinger equation Thisimplies that these equations of motion can be integrated on the time scalegiven by nuclear motion, which is much slower and thus allows us to use

a larger molecular dynamics time step However, this means that the tronic structure problem has to be solved self-consistently at each moleculardynamics step, whereas this is avoided in Ehrenfest dynamics due to the pos-sibility of propagating the wave function simply by applying the Hamiltonian

elec-to an initial wave function (obtained by a single self-consistent optimization

at the very beginning of such a simulation)

From an algorithmic point of view the main task achieved in ground stateEhrenfest dynamics is simply to keep the wave function automatically min-imized as the nuclei are propagated, in addition to keeping the orbitals or-thonormal in effective one-particle theories such as Hartree–Fock or Kohn–Sham approaches This, however, might be achieved - in principle - byanother sort of deterministic dynamics than first-order Schr¨odinger dynam-ics In summary, the “Best of all Worlds Method” should (i) integrate theequations of motion on the (long) time scale set by the nuclear motion butnevertheless (ii) intrinsically take advantage of the smooth time evolution ofthe dynamically evolving electronic subsystem as much as possible The sec-ond point allows us to circumvent explicit diagonalization or minimization

to solve the electronic structure problem iteratively before the next ular dynamics step can be made Car–Parrinello molecular dynamics is anefficient method to satisfy requirement (ii) automatically in a numericallystable and efficient fashion, and makes an acceptable compromise concerningthe length of the time step (i)

molec-2.4.2 Car–Parrinello Lagrangian and equations of motion

The basic idea of the Car–Parrinello approach can be viewed as taking most

direct advantage of the quantum-mechanical adiabatic time scale separation

particularly efficient approach to performing ab initio simulations...

is reduced to solving a time-independent quantum problem, e.g by ing the time-independent, stationary Schrăodinger equation, concurrently topropagating the nuclei according to classical mechanics... efficientquantum chemistry electronic structure codes in conjunction with sufficientcomputer power to solve “interesting problems”, see for instance the compi-lation of such studies in Table of an