《从基础到高级：Ab Initio分子动力学理论与实践指南》

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"《第一原理分子动力学：基本理论与高级方法》是由Dominik Marx和Jürg Hutter共同编著的一本专著，由剑桥大学出版社出版。这本书是分子动力学与电子结构理论结合领域的一个里程碑，它革新了复杂分子系统和过程（如化学反应）的计算机模拟。书中对从头算（ab initio）分子动力学这一前沿技术进行了全面而深入的探讨。本书旨在为研究生和研究人员提供一个系统的介绍，涵盖了从基础理论到高级方法的广泛范围。作者精心设计了一系列循序渐进的讲解，让读者能够透彻理解并评估常用的ab initio分子动力学技术的优势和局限性。通过这种方法，研究者可以避免陷入目前科研文献中常见的误解，从而更准确地应用这些技术。特别值得一提的是，书中对广泛应用的Car-Parrinello方法进行了详尽剖析，澄清了该方法的一些常见误区。这不仅对于那些初次接触该领域的新人来说是宝贵的教育资源，也对现有程序开发者的优化工作提供了实用的指导。书中还包含了一些典型平面波电子结构代码的伪代码和程序布局，帮助读者熟悉常用软件包的工作原理，并激发开发者对代码进行改进和扩展的能力。《第一原理分子动力学：基本理论与高级方法》是一本不可或缺的参考书籍，不仅适用于深化理论研究，而且对于实际操作中的模拟和计算实践具有重要的指导意义，对于推动整个领域的发展起到了关键作用。"

4 Setting the stage: why ab initio molecular dynamics?

1970 1980 1990 2000 2010

Year n

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

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

molecular dynamics steps, i.e.

electronic structure calculations are needed to generate one trajectory.

Furthermore, it is assumed that 10

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

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

, 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.

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