Quantum effects and dynamics in hydrogen-bonded systems [Elektronische Ressource] : a first-principles approach to spectroscopic experiments / Jochen Schmidt

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2007
• Nombre de lectures: 7
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

Quantum Effects and Dynamics in
Hydrogen-Bonded Systems: A First-Principles
Approach to Spectroscopic Experiments
Dissertation zur Erlangung des Grades
“Doktor der Naturwissenschaften”
am Fachbereich Physik
der Johannes Gutenberg-Universit¨at
in Mainz
Jochen Schmidt
geboren in Mainz
Mainz 2007Tag der mundlic¨ hen Prufung:¨ 14.09.2007Contents
1 Introduction 1
2 Basic Theory 5
2.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 The Born-Oppenheimer Approximation . . . . . . . . . . . 7
2.1.3 The Hohenberg-Kohn Theorem . . . . . . . . . . . . . . . 9
2.1.4 The Kohn-Sham Method . . . . . . . . . . . . . . . . . . . 14
2.1.5 The Local-Density Approximation . . . . . . . . . . . . . . 18
2.1.6 Gradient Corrected functionals . . . . . . . . . . . . . . . 19
2.2 Pseudopotentials and Basis Sets . . . . . . . . . . . . . . . . . . . 21
2.2.1 The Pseudopotential Approximation . . . . . . . . . . . . 21
2.2.2 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Computational Realizations . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Gaussian and Plane Waves Method (GPW) . . . . . . . . 27
2.3.2 Gaussian and Augmented Plane Waves Method (GAPW) . 30
2.4 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . 34
2.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2 Born-Oppenheimer MD . . . . . . . . . . . . . . . . . . . 35
2.4.3 Car-Parrinello MD . . . . . . . . . . . . . . . . . . . . . . 36
2.4.4 Realization of the NVT-Ensemble . . . . . . . . . . . . . . 38
3 Calculation of Spectroscopic Properties 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Ground State Properties . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 Nuclear Quadrupole Coupling Constants (NQCC) . . . . . 43
3.2.2 Calculation of Electric Field Gradients . . . . . . . . . . . 47
iii Contents
3.2.3 Relaxation via Quadrupole Couplings . . . . . . . . . . . . 49
3.3 Second-Order Properties . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.2 Chemical Shifts . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.3 The gauge origin problem . . . . . . . . . . . . . . . . . . 56
4 The Path Integral Formalism 59
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Formal Derivation of Path Integrals . . . . . . . . . . . . . . . . . 61
4.3 Path Integrals in MD Simulations . . . . . . . . . . . . . . . . . . 64
4.3.1 Representation with Ring Polymers . . . . . . . . . . . . . 64
4.3.2 The Staging Transformation . . . . . . . . . . . . . . . . . 65
4.3.3 Finite-Discretization Errors . . . . . . . . . . . . . . . . . 67
5 Nuclear Quantum Effects in Molecular Systems 69
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Staging Transformation and Spectroscopic Properties . . . . . . . 71
5.3 Tunneling Effects in Acetylacetone . . . . . . . . . . . . . . . . . 77
5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.2 Proton Density from a Path Integral Simulation . . . . . . 78
5.3.3 Electric Field Gradients: Classic vs. Quantum MD . . . . 83
5.3.4 NMR: Classical vs. Quantum MD . . . . . . . . . . . . . . 86
5.4 Nuclear quadrupole couplings in benzoic acid. . . . . . . . . . . . 91
5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4.2 Quantum effects on the NQCC . . . . . . . . . . . . . . . 93
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 Quadrupole Relaxation in Water 99
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 EFG in GAPW - Tests and Benchmarks . . . . . . . . . . . . . . 100
6.3 MD Simulation of Liquid Water . . . . . . . . . . . . . . . . . . . 103
6.4 Autocorrelation and Relaxation . . . . . . . . . . . . . . . . . . . 107
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Contents iii
7 Constant Pressure Simulations 119
7.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . 119
7.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1.2 Basic Thermodynamics . . . . . . . . . . . . . . . . . . . . 120
7.1.3 Stress-Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.1.4 Pressure and Periodic Boundary Conditions . . . . . . . . 127
7.2 Stress Tensor in the GPW framework . . . . . . . . . . . . . . . . 128
7.2.1 Forces in GPW . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2.2 Grid Independent Terms of the Stress Tensor . . . . . . . 132
7.2.3 Grid Dependent Terms of the Stress Tensor . . . . . . . . 133
7.2.4 Test of the Implementation . . . . . . . . . . . . . . . . . 138
7.3 Simulation of Liquid Water at Ambient Conditions . . . . . . . . 139
7.3.1 Prelude: The NPT Integrator . . . . . . . . . . . . . . . . 139
7.3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3.3 Technical Details . . . . . . . . . . . . . . . . . . . . . . . 142
7.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8 Summary and Outlook 157
A Derivation of the Stress Tensor 161
B Technical Details: Simulations 164
C Technical Details: NMR Measurements 165
D Atomic units 167
E Abbreviations 169
Bibliography 1701 Introduction
The year 1985 can be considered a milestone for the field of computational mate-
rials science. Car and Parrinello published their “Unified Approach for Molecular
Dynamics and Density-Functional Theory” [1]. This work provided a novel basis
formoleculardynamicssimulationsbasedonapotentialenergysurfacecomputed
from electronic structure, nowadays known as “Car-Parrinello molecular dynam-
ics”. Secondly, Cray Research released the Cray-2 supercomputer, yielding a
1performance of 3.9 GigaFLOPS , which was not outperformed by another ma-
chine until 1990. Starting in the late 80’s, the power of the new techniques, both
conceptual and technical, led to an explosion of the activity in this field. 22 years
later, the fastest supercomputer is BlueGene/L with 280 TeraFLOPS. Interest-
ingly, both Cray-2 and BlueGene/L have been built for the Lawrence Livermore
National Laboratory, where also a part of this work has been conducted.
While the Cray-2 cannot be found on 2007’s TOP500 list of the fastest super-
2computers , the Car-Parrinello method is still in widespread use. By July 2007,
the original paper has been cited in more than 4000 publications. The first appli-
cations of ab-initio molecular dynamics were limited to a few atoms and several
tensoffemtoseconds,bothusingtheCar-Parrinellomethodandotherapproaches,
such as Born-Oppenheimer molecular dynamics. Modern computer clusters, such
as BlueGene/L, which provides the impressive number of 131,000 processors, en-
able the ab-initio treatment of thousands of atoms and simulation times up to
severalnanoseconds. This has been achieved by pure computational poweron the
one hand, but also by developing efficient and sophisticated algorithms. “Linear
scaling techniques” that circumvent the cubic scaling with the system size ex-
hibited by many conventional schemes, are mentioned here as an example that
1Floating Point Operations Per Second
2see http://www.top500.org/
12 1 Introduction
opens the way for the modeling of biological systems. Before, these have been ac-
cessible only with the help of methods that use empirical and pre-parameterized
potentials.
Typically, a molecular dynamics simulation is carried out for investigating the
energetics and structure of a system under conditions that include physical pa-
rameterssuchastemperatureandpressure. Abinitioquantumchemicalmethods,
which form the basis of the dynamics scheme, have also proven to be capable of
predicting other experimentally accessible quantities, e.g. spectroscopic param-
eters. Still, the combination of these two features, dynamics simulations and
property calculations, is not commonly used, although it provides valuable infor-
mation, for instance the temperature dependence of the parameters. This is due
to the fact that both types of calculations are computationally very demanding,
leading to an immense effort for the combined scheme.
Furthermore, conventional molecular dynamics consider the nuclei as classical
particlesthataresubjecttotheforcescreatedbythepotentialduetothequantum
mechanically treated electrons. Not only motional effects, but also the quantum
natureofthenucleiareexpectedtoinfluencethepropertiesofamolecularsystem.
This has already been investigated using approximate methods, but a scheme
basedonpathintegralsprovidesacomputationalmethodfortherigorousinclusion
of such effects at a high level of accuracy.
In this work, the computational methods mentioned above have been combined,
aimingforamorerealisticandcompletedescriptionofpropertiesthatareaccessi-
3bleviaNMR experiments,suchastheNMRchemicalshiftorNuclearQuadrupole
Coupling Constants. Isotope effects, caused by the quantum mechanical behavior
of the involved particles, are well known and experimentally observed in NMR.
He