Edited byPaolo Carloni and Frank Alber

Quantum Medicinal Chemistry

InnodataFile Attachment3527605304.jpg

Quantum Medicinal Chemistry

Edited byPaolo Carloni and Frank Alber

Edited byR. Mannhold, H. Kubinyi, G. Folkers

Editorial BoardH.-D. Höltje, H. Timmerman,J. Vacca, H. van de Waterbeemd, T. Wieland

Methods and Principles in Medicinal Chemistry

Edited byPaolo Carloni and Frank Alber

Quantum Medicinal Chemistry

Series Editors

Prof. Dr. Raimund MannholdBiomedical Research CenterMolecular Drug Research GroupHeinrich-Heine-UniversitätUniversitätsstraße 140225 DüsseldorfGermanyraimund.mannhold@uni-duesseldorf.de

Prof. Dr. Hugo KubinyiBASF AG Ludwigshafenc/o Donnersbergstraße 967256 Weisenheim am SandGermanykubinyi@t-online.de

Prof. Dr. Gerd FolkersDepartment of Applied BiosciencesETH ZürichWinterthurer Straße 1908057 ZürichSwitzerlandfolkers@pharma.ethz.ch

Volume Editors

Prof. Dr. Paolo CarloniInternational School for Advanced StudiesVia Beirut 434014 TriesteItalycarloni@sissa.it

Dr. Frank AlberLaboratories of Molecular BiophysicsThe Rockefeller University1230 York Avenue, Box 270New York, NY 10021-6399USAcurrent address:Dept. of Biopharmaceutical SciencesUniversity of CaliforniaSan Francisco, CA 94143USAfrank.alber@salilab.org

Cover illustrationElectron density map of morphine showingthe aromatic and part of the furanoid ring.Courtesy of C. Matta.

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Bibliographic information published byDie Deutsche BibliothekDie Deutsche Bibliothek lists this publicationin the Deutsche Nationalbibliografie; detailedbibliographic data is available in the Internetat http://dnb.ddb.de.

ISBN 3-527-30456-8

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Preface XI

Foreword XIII

List of Contributors XV

Outline of the Book 1

Density Functional Theory

1 Advances in Density-functional-based ModelingTechniques – Recent Extensions of the Car-ParrinelloApproach 5Daniel Sebastiani and Ursula Röthlisberger

1.1 Introduction 51.2 The Car-Parrinello Approach – Basic Ideas 61.2.1 How It Can be Done 81.2.2 Ab Initio Molecular Dynamics Programs 151.3 Mixed Quantum Mechanical/Molecular

Mechanical (QM/MM) Car-Parrinello Simulations 151.3.1 Gly-Ala Dipeptide in Aqueous Solution –

Do We Need a Polarizable Force Field? 191.4 Density-functional Perturbation Theory

and the Calculation of Response Properties 211.4.1 Introduction to Density-functional

Perturbation Theory 211.4.2 Basic Equations of Density-functional

Perturbation Theory 221.4.3 NMR Chemical Shieldings within DFPT 26

V

Contents

1.4.3.1 Introduction to Nuclear Magnetic ResonanceChemical Shifts 26

1.4.3.2 NMR Chemical Shielding 271.4.3.3 Calculation of NMR Chemical Shifts in QM/MM

Car-Parrinello Simulations 301.5 Introduction to Time-dependent Density-functional

Theory (TD-DFT) 321.5.1 Basic Equations of TD-DFT 331.5.2 Applications of TD-DFT within the QM/MM

Framework – Opsochromic Shift of Acetonein Water 35

1.6 Acknowledgments 361.7 References 36

2 Density-functional Theory Applicationsin Computational Medicinal Chemistry 41Andrea Cavalli, Gerd Folkers, Maurizio Recanatini,and Leonardo Scapozza

2.1 Introduction 412.2 Density-functional Theory and Related Methods 422.2.1 Density-functional Theory 422.2.2 Ab Initio Molecular Dynamics 452.3 SAR Studies of Ligand-Target Interactions 482.3.1 The Case Study: Herpes Simplex Virus Type 1

Thymidine Kinase Substrates and Inhibitors 482.3.1.1 Rationalizing Substrate Diversity –

SAR of HSV1 TK Ligands 512.3.1.2 What Can be Learned from this Case Study –

From SAR to Drug Design 562.4 Theoretical Studies of Enzymatic Catalysis 572.4.1 The Phosphoryl Transfer Reaction 582.4.1.1 Cdc42-catalyzed GTP Hydrolysis 582.4.1.2 HIV-1 Integrase 632.5 Studies on Transition Metal Complexes 642.5.1 Radiopharmaceuticals 652.6 Conclusions and Perspectives 672.7 References 68

ContentsVI

3 Applications of Car-Parrinello Molecular Dynamicsin Biochemistry – Binding of Ligands in Myoglobin 73Carme Rovira

3.1 Introduction 743.2 Computational Details 793.3 Myoglobin Active Center 813.3.1 Structure, Energy, and Electronic State 813.3.2 The Picket-fence-oxygen Biomimetic Complex 863.3.2.1 Interplay Structure/Electronic State 863.3.2.2 Optimized Structure and Energy of O2 Binding 903.3.3 Heme-Ligand Dynamics 933.4 Interaction of the Heme with the Protein 993.5 Conclusions 1063.6 Acknowledgments 1083.7 References 108

4 Density-functional Theory in Drug Design –the Chemistry of the Anti-tumor Drug Cisplatinand Photoactive Psoralen Compounds 113Johan Raber, Jorge Liano, and Leif A. Eriksson

4.1 Introduction 1134.2 Density-functional Theory 1144.2.1 Basic Equations 1154.2.2 Gradient Corrections and Hybrid Functionals 1174.2.3 Time-dependent Density-functional Response

Theory (TD-DFRT) 1204.2.4 Applicability and Applications 1224.3 Modes of Action of Anti-tumor Drug Cisplatin 1244.3.1 Activation Reactions 1274.3.2 Interactions Between DNA and Cisplatin 1344.4 Photochemistry of Psoralen Compounds 1414.4.1 Ionization Potentials 1434.4.2 Excitation Spectra 1464.5 Acknowledgments 1504.6 References 150

Contents VII

QM/MM Approaches

5 Ab Initio Methods in the Study of ReactionMechanisms – Their Role and Perspectivesin Medicinal Chemistry 157Mikael Peräkylä

5.1 Introduction 1575.2 Methods 1615.2.1 Hybrid QM/MM Potential 1615.2.2 QM/MM Boundary – The Link Atom Approach 1615.2.3 QM/MM Boundary – The Hybrid

Orbital Approach 1655.3 Thermodynamically Coupled QM/MM 1665.4 Selected Applications of QM/MM Methods 1685.4.1 Uracil-DNA Glycosylase 1685.4.2 QM/MM Simulations of Quantum Effects 1695.4.3 Miscellaneous Applications 1705.5 Conclusions 1735.6 References 173

6 Quantum-mechanical/Molecular-mechanicalMethods in Medicinal Chemistry 177Francesca Perruccio, Lars Ridder,and Adrian J. Mulholland

6.1 Introduction 1776.2 Theory 1786.2.1 Methodology 1786.2.2 Basic Theory 1796.2.3 QM/MM Partitioning Schemes 1806.3 Practical Aspects of Modeling Enzyme Reactions 1826.3.1 Choice and Preparation of the Starting Structure 1826.3.2 Definition of the QM Region 1836.3.3 Choice of the QM Method 1846.4 Techniques for Reaction Modeling 1856.4.1 Optimization of Transition Structures

and Reaction Pathways 1856.4.2 Dynamics and Free Energy Calculations 1866.5 Some Recent Applications 1896.5.1 Human Aldose Reductase 1896.5.2 Glutathione S-Transferases 1916.5.3 Influenza Neuraminidase 193

ContentsVIII

6.5.4 Human Thrombin 1936.5.5 Human Immunodeficiency Virus Protease 1946.6 Conclusions 1956.7 References 195

Molecular Properties

7 Atoms in Medicinal Chemistry 201Richard F. W. Bader, Cherif F. Matta,and Fernando J. Martin

7.1 Why Define Atoms in Molecules? 2017.2 Theory of Atoms in Molecules 2027.2.1 Definition of Atoms and Molecular Structure 2037.3 Definition of Atomic Properties 2087.3.1 Atomic Charges, Multipole Moments and Volumes 2097.4 QTAIM and Correlation of Physicochemical

Properties 2117.4.1 Use of Atomic Properties in QSAR 2117.4.2 Use of Bond Critical Point Properties in QSAR 2137.4.3 QTAIM and Molecular Similarity 2157.5 Use of QTAIM in Theoretical Synthesis

of Macromolecules 2187.5.1 Assumed Perfect Transferability in the Synthesis

of a Polypeptide 2197.5.2 The Assembly of Buffered Open Systems

in a Macrosynthesis 2227.6 The Laplacian of the Density and the Lewis Model 2247.6.1 The Laplacian and Acid-Base Reactivity 2257.6.2 Molecular Complementarity 2287.7 Conclusions 2297.8 References 230

8 The Use of the Molecular Electrostatic Potentialin Medicin Chemistry 233Jane S. Murray and Peter Politzer

8.1 Introduction 2338.2 Methodology 2358.3 An Example that Focuses on Vmin –

the Carcinogenicity of Halogenated Olefinsand their Epoxides 239

Contents IX

8.4 An Example Focusing on the General Patternsof Molecular Electrostatic Potentials – Toxicityof Dibenzo-p-dioxins and Analogs 244

8.5 Statistical Characterization of the MolecularSurface Electrostatic Potential – the GeneralInteraction Properties Function (GIPF) 246

8.6 Summary 2508.7 Acknowledgment 2508.8 References 250

9 Applications of Quantum Chemical Methodsin Drug Design 255Hans-Dieter Höltje and Monika Höltje

9.1 Introduction 2559.2 Application Examples 2569.2.1 Force Field Parameters from Ab Initio Calculations 2569.2.1.1 Equilibrium Geometry for a Dopamine-D3-Receptor

Agonist 2609.2.1.2 Searching for a Bioactive Conformation 2629.2.2 Atomic Point Charges 2649.2.3 Molecular Electrostatic Potentials 2669.2.4 Molecular Orbital Calculations 2689.3 Outlook 2739.4 References 274

Subject Index 275

ContentsX

Everyone relies on the power of computers, including chemicaland pharmaceutical laboratories. Increasingly faster and more ex-act simulation algorithms have made quantum chemistry a valu-able tool in the search for bioactive substances. The much largercomputational cost is more than compensated by a deeper under-standing of the physicochemical events taking place at the inter-action of ligands and proteins. Special interest in biomolecularsimulation is now given to catalytic centers in proteins whichcontain metals. Many of the DNA binding proteins involved inthe control of the transcription processes contain metallic cen-ters. Standard empirical methods, which have undeniable meritsin the field of structure-based design, nevertheless fail to de-scribe subtle chemical phenomena as partially covalent bonds ornon-rigid aromatic moieties. Another field of high interest inmedicinal chemistry are ligands that interfere with ion channels.Also here the presence of large electric fields demands a moresophisticated approach. Ab initio molecular dynamics which typi-cally make use of density functional theory add another piece tothe mosaic pattern of understanding ligand-protein interactions.Experience that has accumulated in recent years in the fields ofmaterial sciences and medicinal chemistry shows a unique roleof ab initio molecular dynamics in studying complex interactionphenomena with a close coupling to experimental, mostly spec-troscopical data.

These few remarks highlight that quantum-chemical methodshave adapted an important role in medicinal chemistry. It is theintention of the present volume to document this role in ade-quate detail. Accordingly, the book is divided into three main sec-tions. The first section is dedicated to density functional theory.

XI

Preface

A description of advances in density functional based modellingtechniques is followed by application examples in computationalmedicinal chemistry, biochemistry and drug design. The follow-ing section is focussed on QM/MM approaches and describese.g. the use of ab initio methods in the study of reaction mecha-nisms. The last section presents a survey of pharmaceutically rel-evant properties derived by quantum-chemical calculations suchas molecular electrostatic potentials. In a finalizing chapter appli-cations of quantum-chemical methods to systems of biologicaland pharmacological relevance are described.

The series editors would like to thank the authors and in par-ticular the volume editors, Paolo Carloni and Frank Alber, thatthey devoted their precious time to compiling and structuringthe comprehensive information on medicinal quantum chemis-try. Last, but not least we want to express our gratitude to FrankWeinreich and Gudrun Walter from Wiley-VCH publishers forthe fruitful collaboration.

December 2002 Raimund Mannhold, DüsseldorfHugo Kubinyi, LudwigshafenGerd Folkers, Zürich

PrefaceXII

Quantum-chemical (QC) calculations are a key element in biolog-ical research. When constantly tested for their range of validityQC methods provide a description of how molecules interact andform their three-dimensional shape, which in turn determinesmolecular function. They can aid the formulation of hypothesesthat provide the connecting link between experimentally deter-mined structures and biological function. QC calculations can beused to understand enzyme mechanisms, hydrogen bonding, po-larization effects, spectra, ligand binding and other fundamentalprocesses both in normal and aberrant biological contexts. Thepower of parallel computing and progress in computer algo-rithms are enlarging the domain of QC applications to ever morerealistic models of biological macromolecules. This book ismeant to serve as a reference for chemists, biochemists, andpharmacologists interested in learning about and using state-of-the-art QC techniques to investigate systems and processes ofpharmaceutical relevance. We are confident that the contribu-tions presented here will provide further support for the develop-ment and applications of quantum-mechanical methods in theapplied biosciences.

New York and Trieste Frank Alber and Paolo CarloniJuly 2002

XIII

Foreword

Editors

Dr. Frank AlberLaboratories of MolecularBiophysicsThe Rockefeller University1230 York Avenue, Box 270New York, NY 10021-6399USAfrank.alber@salilab.org

Prof. Dr. Paolo CarloniInternational School for AdvancedStudies (SISSA/ISAS)Via Beirut 434014 TriesteItalycarloni@sissa.it

Authors

Prof. Dr. Richard F. W. BaderDepartment of ChemistryMcMaster University1280 Main Street WestHamiltonOntario, L8S 4M1Canadabader@mcmail.cis.mcmaster.ca

Dr. Andrea CavalliDepartment of PharmaceuticalSciencesUniversity of BolognaVia Belmeloro 640126 BolognaItalycavan@alma.unibo.it

Prof. Dr. Leif A. ErikssonDivision of Structural andComputational BiophysicsDepartment of BiochemistryUppsala UniversityBox 57675123 UppsalaSwedenleif.eriksson@xray.bmc.uu.se

Prof. Dr. Gerd FolkersDepartment of Applied BiosciencesETH ZürichWinterthurerstraße 1908057 ZürichSwitzerlandfolkers@pharma.anbi.aethz.ch

XV

List of Contributors

Prof. Dr. Hans-Dieter HöltjeInstitute of PharmaceuticalChemistryHeinrich-Heine UniversitätUniversitätsstraße 140225 DüsseldorfGermanyhoeltje@pharm.uni-duesseldorf.de

Dr. Monika HöltjeInstitute of PharmaceuticalChemistryHeinrich-Heine-UniversitätUniversitätsstraße 140225 DüsseldorfGermanymhoeltje@pharm.uni-duesseldorf.de

Dr. Jorge LlanoDivision of Structural andComputational BiophysicsDepartment of Biochemistry andDepartment of Quantum ChemistryUppsala UniversityBox 57675123 UppsalaSwedenJorge.Llano@xray.bmc.uu.se

Dr. Fernando J. MartínDepartment of PharmaceuticalChemistryUniversity of CaliforniaSan Francisco513 Parnassus AvenueBox 0446San Francisco, CA 94143USA

Dr. Cherif F. MattaDepartment of ChemistryMcMaster UniversityHamilton1280 Main Street WestOntario, L8S 4M1Canadamattacf@mcmaster.cacurrent adress:Lash Miller Chemical LaboratoriesChemistry DepartmentUniversity of TorontoToronto, OntarioCanada, M5S 1A1

Prof. Dr. Adrian J. MulhollandSchool of ChemistryUniversity of BristolBristol BS8 1TSUKAdrian.Mulholland@bris.ac.uk

Prof. Dr. Jane S. MurrayDepartment of ChemistryUniversity of New OrleansElysian Fields Ave.New Orleans, LA 70148USAjsmurray@uno.edu

Dr. Francesca PerruccioSchool of ChemistryUniversity of BristolBristol BS8 1TSUKcurrent adress:Molecular InformaticsStructure and DesignPfizer Global Researchand DevelopmentRamsgate RoadSandwich, Kent CT13 9NJUK

List of ContributorsXVI

Prof. Dr. Mikael PeräkyläDepartment of ChemistryUniversity of KuopioSavilahdentie 9 FP.O. Box 162770211 KuopioFinlandmikael.perakyla@uku.fi

Prof. Dr. Peter PolitzerDepartment of ChemistryUniversity of New OrleansElysian Fields Ave.New Orleans, LA 70148USAppolitzer@uno.edu

Dr. Johan RaberDivision of Structural and Compu-tational BiophysicsDepartment of BiochemistryUppsala UniversitySavilahdentie 9 FBox 57675123 UppsalaSwedenJohan.Raber@xray.bmc.uu.se

Prof. Dr. Maurizio RecanatiniDepartment of PharmaceuticalSciencesUniversity of BolognaVia Belmeloro 640126 BolognaItalymreca@alma.unibo.it

Dr. Lars RidderSchool of ChemistryUniversity of BristolBristol BS8 1TSUKcurrent adress:Molecular Design&InformaticsN.V. Organon, 5340 BH OssThe Netherlands

Prof. Dr. Ursula RöthlisbergerInstitute of Molecular andBiological ChemistrySwiss Federal Institute ofTechnologyEPFL1015 LausanneSwitzerlandand Institute of InorganicChemistryETH ZürichUniversitätsstraße 68092 ZürichSwitzerlanduro@inorg.chem.ethz.ch

Dr. Carme RoviraCentre de Recerca en Química TeòricaParc Científic de Barcelona (PCB)Annex A, pta. 1Josep Samitier 1–508028 BarcelonaSpainand Departament de Química FísicaFacultat de QuímicaUniversitat de BarcelonaMartí i Franquès 108028 BarcelonaSpaincrovira@pcb.ub.es

List of Contributors XVII

Prof. Dr. Leonardo ScapozzaDepartment of Applied BiosciencesETH ZürichWinterthurerstraße 1908057 ZürichSwitzerlandleonardo.scapozza@pharma.anbi.ethz.ch

Dr. Daniel SebastianiInstitute of Molecularand Biological ChemistrySwiss Federal Instituteof TechnologyEPFL1015 LausanneSwitzerlandand Max-Planck-Instituteof Polymer ResearchAckermannweg 1055128 MainzGermanysebastia@mpip-mainz.mpg.de

List of ContributorsXVIII

The book is organized into three major parts which cover impor-tant and emerging methods and applications in biological andpharmacological research.

The first part focuses on density functional theory (DFT), oneof the most successful first-principle approaches to investigationof the electronic structure of relatively large model systems; itssuitability for tackling chemical problems was recognized by theNobel prize awarded to W. Kohn in 1998. In the first chapter, U.Röthlisberger and D. Sebastiani outline the principles of DFTand describe selected major advances, from the DFT-based mo-lecular dynamics (Car-Parrinello) method (and its extension tohybrid DFT/classical molecular dynamics), to time-dependentDFT methods which enable extension of DFT to investigation ofexcited states and DFT perturbation theory for calculation ofNMR chemical shifts. This rather methodological chapter servesas a reference for Chapters 2–4, which describe a wide spectrumof applications. A. Cavalli et al. (Chapter 2) discuss DFT studiesof ligand-target interactions and the mechanism of specific en-zyme systems. Subsequently, C. Rovira presents a detailed surveyof Car-Parrinello applications to iron porphyrin proteins (Chap-ter 3). Finally, J. Raber et al. describe applications of DFT to themode of action of the antitumor drug cisplatin and to descrip-tion of the excitation spectra of photochemotherapeutic com-pounds (Chapter 4).

The second part of the book provides a description of QM/MM approaches. Because of their large size, biological systemssuch as proteins and nucleic acids cannot be treated fully at the

1

Outline of the Book

quantum chemical (QC) level. QM/MM approaches combine aQC method (performed at semiempirical ab initio or density-functional levels) for the region of interest (for example, the ac-tive site of an enzyme) with molecular mechanics (MM) treat-ment of the environment. M. Peräkylä introduces the basic theo-ry of structurally and thermodynamically coupled QM/MMapproaches. He describes, in detail, treatment of the boundarybetween the QM and the MM regions (Chapter 5). Practicalaspects and selected applications are further discussed by F. Per-ruccio et al. (Chapter 6).

The final part is devoted to a survey of molecular properties ofspecial interest to the medicinal chemist. The Theory of Atomsin Molecules by R. F. W. Bader et al., presented in Chapter 7, en-ables the quantitative use of chemical concepts, for examplethose of the functional group in organic chemistry or molecularsimilarity in medicinal chemistry, for prediction and understand-ing of chemical processes. This contribution also discusses possi-ble applications of the theory to QSAR. Another important prop-erty that can be derived by use of QC calculations is the molecu-lar electrostatic potential. J. S. Murray and P. Politzer describethe use of this property for description of noncovalent interac-tions between ligand and receptor, and the design of new com-pounds with specific features (Chapter 8). In Chapter 9, H.D.and M. Höltje describe the use of QC methods to parameterizeforce-field parameters, and applications to a pharmacophoresearch of enzyme inhibitors. The authors also show the use ofQC methods for investigation of charge-transfer complexes.

Outline of the Book2

Density Functional Theory

1.1Introduction

During the last decade, density-functional theory (DFT)-basedapproaches [1, 2] have advanced to prominent first-principlesquantum chemical methods. As computationally affordable toolsapt to treat fairly extended systems at the correlated level, theyare also of special interest for applications in medicinal chemis-try (as demonstrated in the chapters by Rovira, Raber et al. andCavalli et al. in this book). Several excellent text books [3–5] andreviews [6] are available as introduction to the basic theory and tothe various flavors of its practical realization (in terms of differ-ent approximations for the exchange-correlation functional). Theactual performance of these different approximations for diversechemical [7] and biological systems [8] has been evaluated in anumber of contributions.

In this chapter we will focus on one particular, recently devel-oped DFT-based approach, namely on first-principles (Car-Parri-nello) molecular dynamics (CP-MD) [9] and its latest advance-ments into a mixed quantum mechanical/molecular mechanical(QM/MM) scheme [10–12] in combination with the calculation ofvarious response properties [13–18] within DFT perturbation the-ory (DFTPT) and time-dependent DFT theory (TDDFT) [19].

First-principles (or ‘ab initio’) molecular dynamics, the directcombination of DFT with classical molecular dynamics (MD),was introduced in 1985 in a seminal paper by Car and Parrinello[9]. This novel scheme has first been applied to the study of metalclusters [20] and amorphous and liquid silicon [21] but has sincemoved rapidly into chemistry [22] and biology [23, 24]. CP-MD of-

5

1Advances in Density-functional-basedModeling Techniques– Recent Extensions of the Car-Parrinello ApproachDaniel Sebastiani and Ursula Röthlisberger

fers the unique possibility of performing parameter-free MD sim-ulations in which all the interactions are calculated on-the-fly with-in the framework of DFT (or alternative electronic structure meth-ods [70–73]). In this way, finite temperature and entropic effects areincluded in a straightforward manner in the context of a quantumchemical electronic structure calculation and simulations can beperformed in realistic condensed-phase environments. Further-more, this approach is also highly amenable to parallelization sothat currently simulations of 100–1000 atoms can be performedat relative ease. CP-MD thus offers promising perspectives for ap-plications in medicinal chemistry, and a growing number of suchstudies has started to emerge in recent years [25].

Lately, the CP-MD approach has been combined with a mixedQM/MM scheme [10–12] which enables the treatment of chemi-cal reactions in biological systems comprising tens of thousandsof atoms [11, 26]. Furthermore, CP-MD and mixed QM/MM CP-MD simulations have also been extended to the treatment of ex-cited states within a restricted open-shell Kohn-Sham approach[16, 17, 27] or within a linear response formulation of TDDFT[16, 18], enabling the study of biological photoreceptors [28] andthe in situ design of optimal fluorescence probes with tailored op-tical properties [32]. Among the latest extensions of this methodare also the calculation of NMR chemical shifts [14].

Here, we will first give an introduction to the basic ideas un-derlying the Car-Parrinello method, especially addressed to com-plete newcomers in the field. We will then try to outline some ofthe recent methodological extensions, with particular emphasison aspects with potential interest for applications in medicinalchemistry. The power and limitations of these new modelingtools will be illustrated with few selected examples.

1.2The Car-Parrinello Approach – Basic Ideas

The Car-Parrinello approach combines an electronic structuremethod with a classical molecular dynamics scheme and thusunifies two major fields of computational chemistry, which havehitherto been essentially orthogonal. Through this unification a

1 Advances in Density-functional-based Modeling Techniques6

series of new features has become available that goes far beyondthe capabilities of each of the single parts on their own. A few ofthese special aspects are summarized schematically in Tab. 1.1and illustrated in Fig. 1.1a, b.

Quantum chemical (QC) methods have the advantage of highintrinsic accuracy but are essentially limited to the treatment ofsmall molecules in the gas phase at static nuclear configurations.In a typical QC calculation only few points of the potential en-ergy surface (PES) of the system are characterized by localizingthe stationary point (minimum or transition state) that lies clos-est to a given starting configuration. In this way only a limitedportion of the PES can be sampled and the system might betrapped in a local minimum far from the energetically most fa-vorable configurations.

1.2 The Car-Parrinello Approach – Basic Ideas 7

Tab. 1.1 Comparison of the properties of quantum chemical electronicstructure calculations (QC methods), classical molecular dynamics (Clas-sical MD) based on empirical force fields and first-principles moleculardynamics (ab initio MD) simulations.

QC Methods Classical MD ab initio MD

� High accuracy � Limited to accuracyof empirical forcefield

� QC accuracy

� First-principlesapproach

� Parameterization ef-fort

� limited transferabil-ity

� Parameter-free MDbased on first-prin-ciples

� Description ofchemical reactions

� MD simulation ofchemical reactionsdifficult

� MD simulations ofchemical reactionspossible

� Treatment of transi-tion metal ions pos-sible

� Treatment of transi-tion metal ions dif-ficult

� Treatment of transi-tion metal ions pos-sible

� Zero Kelvin � Finite temperature � Finite temperature� Mostly gas phase

only� Also condensed

phases� Also condensed

phases� Limited to charac-

terization of fewselected points ofthe PES

� Finite temperaturesampling of thePES

� Finite temperaturesampling of thePES

� Local geometryoptimization

� Simulated anneal-ing

� Simulated anneal-ing

In classical molecular dynamics, on the other hand, particlesmove according to the laws of classical mechanics over a PESthat has been empirically parameterized. By means of their ki-netic energy they can overcome energetic barriers and visit amuch more extended portion of phase space. Tools from statisti-cal mechanics can, moreover, be used to determine thermody-namic (e.g. relative free energies) and dynamic properties of thesystem from its temporal evolution. The quality of the results is,however, limited to the accuracy and reliability of the (empiri-cally) parameterized PES.

1.2.1

How It Can be Done

How can one join an electronic structure calculation with a clas-sical MD scheme? In principle, this is possible in a straightfor-ward manner – we can optimize the electronic wavefunction fora given initial atomic configuration (at time t =0) and calculatethe forces acting on the atoms via the Hellman-Feynman theo-rem:

1 Advances in Density-functional-based Modeling Techniques8

Fig. 1.1 (a) In traditional quantumchemical methods the potential en-ergy surface (PES) is characterizedin a pointwise fashion. Startingfrom an initial geometry, optimiza-tion routines are applied to localizethe nearest stationary point (mini-mum or transition state). Whichpoint of the PES results from thisprocedure mainly depends on thechoice of the initial configuration.The system can get trapped easilyin local minima without ever arriv-ing at the global minimum struc-

ture. On the other hand, the PEScan be described with high accu-racy.(b) In classical molecular dy-namics, the PES is approximatedvia an appropriately parameterizedempirical force field. The particlespossess kinetic energy with whichthey can overcome local barriersand access a wide portion of thePES. The reliability of the approachis limited by the accuracy of theunderlying empirical force field.

�FI � � � dHd�RI

���������

� �

These forces can then be plugged into the classical equations ofmotion (EOM) (Newton’s equations of motion) and the system canbe propagated to a new configuration at a time t+�t, at which wecan repeat the whole procedure again. Implementations which usethis direct procedure are usually referred to as ‘Born-Oppenheimer’first-principles MD schemes. With the powerful computers avail-able today this type of dynamics has become possible but it usuallyrequires special care in order to optimize the efficiency of the fullelectronic structure calculation that has to be performed at everytime step. An alternative way that does not require full blown elec-tronic structure calculation at every time step has been proposed byCar and Parrinello in 1985 [9]. They have suggested including theelectronic wavefunctions (i.e. in DFT as the underlying QC meth-od, the Kohn-Sham one-particle states) explicitly in the calculationand propagating them in parallel to the motions of the atoms. Thiscan be achieved by the elegant trick of considering the one-particleorbitals as fictitious classical degrees of freedom that evolve underthe laws of classical mechanics. Instead of using the familiar New-ton’s equation of motion, such a scheme is more conveniently for-mulated in terms of the equivalent Lagrangian formulation of clas-sical mechanics. In Lagrangian mechanics the system is describedin terms of generalized coordinates qi and their conjugate momen-ta pi =m�qi/� t. The use of generalized coordinates facilitates theintroduction of the electronic variables as additional classical de-grees of freedom and allows us to treat them on the same footingas the atomic motion. The central quantity that describes these dy-namics is the Lagrangian L:

L � K � Epotwhere K is the kinetic energy and Epot is the potential energy.For our combined system consisting of nuclear and electronic co-ordinates the extended Lagrangian, Lex, can be written as:

Lex � KN � Ke � Epotwhere KN is the kinetic energy of the nuclei, Ke is the analogousterm for the electronic degrees of freedom, and Epot is the poten-

1.2 The Car-Parrinello Approach – Basic Ideas 9

tial energy, which depends on both the nuclear positions ��RI�and the electronic variables ��i�. The Lagrangian determines thetime evolution of the classical system via EOM that are given bythe Euler-Lagrange equations:

ddt

�L� �qi

� �� �L

�qi

A further advantage of using Lagrangian dynamics is that wecan easily impose boundary conditions and constraints by apply-ing the method of Lagrangian multipliers. This is particularlyimportant for the dynamics of the electronic degrees of freedom,as we will have to impose that the one-electron wavefunctions re-main orthonormal during their time evolution. The Lex of our ex-tended system can then be written as:

Lex ��

I

12 MI

�R� 2I ��

i

�� ��i�2 � �0�H��0�

��

i� j

�ij�

�i ��r��j��r�d�r�� �ij�

where the �ij are Lagrange multipliers that ensure orthonormal-ity of the one-electron wavefunctions ��i�� � is a fictitious massassociated with the electronic degrees of freedom and the poten-tial energy is given by the expectation value of the total (groundstate) energy of the system E � �0�H��0�. Lex determines thetime evolution of a fictitious classical system in which nuclearpositions as well as electronic degrees of freedom are treated asdynamic variables and the equation of motion for both degreesof freedom can be derived via the Euler-Lagrange equations. TheEOM for the nuclear degrees of freedom become:

MI�R�

I � � �E��RI

and for the electronic ones:

���i � �H�i ��

j

�ij �j

where the term with the Lagrange multipliers �ij describes the con-straint forces needed to keep the wavefunctions orthonormal dur-

1 Advances in Density-functional-based Modeling Techniques10