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Extension and Parametrization of an Approximate Density Functional Method for Organic and Biomolecules [Elektronische Ressource] / Michael Gaus. Betreuer: M. Elstner

141 pages
Extension and Parametrization of an ApproximateDensity Functional Method for Organic andBiomoleculesZur Erlangung des akademischen Grades einesDOKTORS DER NATURWISSENSCHAFTEN(Dr. rer. nat.)Von derFakult at fur Chemie und BiowissenschaftenKarlsruher Institut fur Technologie (KIT) { Universit atsbereichgenehmigteDISSERTATIONvonMichael GausausWormsDekan: Prof. Dr. Stefan Br aseReferent: Prof. Dr. Marcus ElstnerKorreferent: Prof. Dr. Willem KlopperTag der mundlic hen Prufung: 15.07.2011Copyrights and PermissionsChapter 4 is reproduced in part with permission from Gaus, M.; Cui, Q.; Elstner, M. J.Chem. Theory Comput. 2011, 7, 931. Copyright 2011 American Chemical Society.Chapter 5 is reproduced in part with permission from Gaus, M.; Chou, C.-P.; Witek, H.;Elstner, M. J. Phys. Chem. A 2009, 113, 11866. Copyright 2009 American Chemical Society.Individual paragraphs of chapter 3 are reproduced with permission from Gaus, M.; Chou,C.-P.; Witek, H.; Elstner, M. J. Phys. Chem. A 2009, 113, 11866. Copyright 2009 AmericanChemical Society and Gaus, M.; Cui, Q.; Elstner, M. J. Chem. Theory Comput. 2011, 7,931. Copyright 2011 American Chemical Society.Druckjahr: 2011MICHAEL GAUSExtension and Parametrization of an Approximate Density Func-tional Method for Organic and BiomoleculesZusammenfassungDiese Arbeit befasst sich mit der Weiterentwicklung eines genaherten quantenchemi-schen Rechenverfahrens namens density functional tight binding\(DFTB).
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Extension and Parametrization of an Approximate
Density Functional Method for Organic and
Biomolecules
Zur Erlangung des akademischen Grades eines
DOKTORS DER NATURWISSENSCHAFTEN
(Dr. rer. nat.)
Von der
Fakult at fur Chemie und Biowissenschaften
Karlsruher Institut fur Technologie (KIT) { Universit atsbereich
genehmigte
DISSERTATION
von
Michael Gaus
aus
Worms
Dekan: Prof. Dr. Stefan Br ase
Referent: Prof. Dr. Marcus Elstner
Korreferent: Prof. Dr. Willem Klopper
Tag der mundlic hen Prufung: 15.07.2011Copyrights and Permissions
Chapter 4 is reproduced in part with permission from Gaus, M.; Cui, Q.; Elstner, M. J.
Chem. Theory Comput. 2011, 7, 931. Copyright 2011 American Chemical Society.
Chapter 5 is reproduced in part with permission from Gaus, M.; Chou, C.-P.; Witek, H.;
Elstner, M. J. Phys. Chem. A 2009, 113, 11866. Copyright 2009 American Chemical Society.
Individual paragraphs of chapter 3 are reproduced with permission from Gaus, M.; Chou,
C.-P.; Witek, H.; Elstner, M. J. Phys. Chem. A 2009, 113, 11866. Copyright 2009 American
Chemical Society and Gaus, M.; Cui, Q.; Elstner, M. J. Chem. Theory Comput. 2011, 7,
931. Copyright 2011 American Chemical Society.
Druckjahr: 2011MICHAEL GAUS
Extension and Parametrization of an Approximate Density Func-
tional Method for Organic and Biomolecules
Zusammenfassung
Diese Arbeit befasst sich mit der Weiterentwicklung eines genaherten quantenchemi-
schen Rechenverfahrens namens density functional tight binding\(DFTB). Der Fokus liegt
"
auf einer Erweiterung dieser Methode, die eine genauere Beschreibung gro er molekularer
Systeme fur ein breites Spektrum chemischer Umgebungen ermoglic hen soll, ohne dabei den
Rechenaufwand bedeutend zu erhohen.
Zunachst werden die Dichtefunktionaltheorie, ihre heutigen Implementierungen und de-
ren Stark en und Schwachen in Kurze dargestellt. Es folgt eine Beschreibung der Naherungen,
die zum DFTB-Formalismus fuhren. Daran anknupfend wird eine Erweiterung vorgeschlagen,
die konsistent mit dieser Herleitung ist. Damit wird die Anderung der Elektron-Elektron-
Wechselwirkung berucksichtigt, die durch die Umverteilung der Elektronendichte bei der
Bildung eines Molekuls verursacht wird. Die Einbeziehung dieses vorher noch nicht beruc k-
sichtigten E ektes und einiger weiterer Verbesserungen f uhrt zu einer verbesserten Beschrei-
bung von Protonena nit aten, Bindungsenergien von Wassersto br uc ken und Barrieren von
Protonentransferreaktionen. Au erdem wird auch die Phosphatchemie mit dieser Methode
zuganglich, wie an einigen Beispielen gezeigt wird.
In vielen Anwendungen ist eine quantitative Genauigkeit von au erster Bedeutung. Diese
kann nur durch eine geeignete Parametrisierung von DFTB erreicht werden. Bislang konn-
te eine solche nur durch aufwendige manuelle Arbeit erzeugt werden. In der vorliegenden
Arbeit wird ein Schema vorgestellt, das diesen Prozess zum gro ten Teil automatisiert. Ver-
schiedene Bedingungen werden in einem linearen Gleichungssystem zusammengefasst, das
dann im Sinne der Methode der kleinsten Fehlerquadrate gelost werden kann. So ist es
moglic h, optimale Parameter zu nden und zugleich Zielkon ikte zu entdecken, die in den
Unzulanglic hkeiten der DFTB-Methode begrundet sind. Mit diesem Schema wird ein Para-
metersatz fur Kohlenwassersto e erzeugt, der die Genauigkeit f ur Bildungsenthalpien und
Schwingungsfrequenzen gegenub er dem alten Parametersatz erhoh t. Fur spezi sche Anwen-
dungen wird ein alternativer Parametersatz vorgeschlagen. Dieser produziert etwas gro ere
Fehler fur Bildungsenthalpien, erreicht aber Genauigkeiten fur Schwingungsfrequenzen, die
sonst nur mit wesentlich rechenzeitaufwendigeren ab-initio-Methoden moglich sind.
In einem folgenden Kapitel wird eine allgemeine Parametrisierung fur die Elemente
Kohlensto , Wassersto , Sticksto und Sauersto vorgestellt, die auf die neuen metho-
dischen Erweiterungen abgestimmt ist. Eine nachfolgende Evaluierung an Atomisierungs-,
Isomerisierungs- und Reaktionsenergien, Gleichgewichtsgeometrien und Schwingungsfrequen-
zen fur einen gro en Satz organischer Molek ule zeigt eine Verbesserung gegenuber den bereits
vorhandenen Parametern. Besonders von Bedeutung bei dieser Parametrisierung ist die ein-
fache Einbeziehbarkeit weiterer Elemente.MICHAEL GAUS
Extension and Parametrization of an Approximate Density Func-
tional Method for Organic and Biomolecules
Abstract
This work deals with further developments of an approximate quantum chemical method
called \density functional tight binding" (DFTB). It focusses on extensions that allow a more
accurate description of large molecular systems for a broader range of chemical environments
while maintaining the computational speed of that method. After an outline of density
functional theory including its today’s implementations, their strengths, and limitations, the
appoximations made to derive the DFTB formalism are reviewed.
An extension is suggested that is formally consistent within this derivation. It considers
the change of electron-electron interaction due to the reorganization of the electron density
when forming a molecule, an e ect that had not been included before. Together with further
methodological re nements this leads to an improved description of hydrogen binding ener-
gies, proton a nities, and proton transfer barriers. Moreover, phosphate chemistry becomes
feasible as demonstrated on a few examples.
Quantitative accuracy is very important for many applications, but can only be achieved
by proper parametrization of such an approximative method. A scheme is presented that
automatizes large parts of this formerly manual process. Several constraints can be arranged
in a system of linear equations which can then be solved in a least squares sense. In this
manner, parameters for optimal performance are deduced, and also con icts of objectives
are detected which illustrate the limitations of the underlying formalism. Along these lines
a parameter set for hydrocarbons is produced that improves the accuracy for heats of for-
mation and vibrational frequencies. For speci c applications, an alternative parameter set
is presented that increases the error for heats of formation on the one hand, but yields
vibrational frequencies with an accuracy of high-level calculations on the other.
A more general parametrization speci cally designed for the new methodological ex-
tensions including the elements carbon, hydrogen, nitrogen and oxygen is suggested in a
following chapter. An evaluation on energies of atomization, reactions, and isomerizations,
equilibrium geometries, and vibrational frequencies for a large set of organic molecules shows
an improvement over yet existent parameters. Most important is the simplicity of the for-
malism to extend the parameter set to further elements.Contents
1 Introduction 9
2 Density Functional Theory 13
2.1 Basics of Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Exchange Correlation Functionals and Their Performance . . . . . . . . . . . 15
2.3 Current Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Density Functional Tight Binding 21
3.1 DFTB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 DFTB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 The MIO Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Electronic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.2 Repulsive P . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Performance and De ciencies of DFTB2 . . . . . . . . . . . . . . . . . . . . 35
4 DFTB3 37
4.1 Third Order Taylor Series Expansion . . . . . . . . . . . . . . . . . . . . . . 38
4.1.1 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1.2 Kohn-Sham Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.3 Atomic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
h4.2 The -Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Computational Details for Benchmarking . . . . . . . . . . . . . . . . . . . . 42
4.3.1 DFTB Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.2 Parameters for DFTB3 . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.3 Calculation of Proton A nities Using DFTB . . . . . . . . . . . . . . 46
4.4 Benchmarks and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.1 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.2 Binding Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4.3 Proton A nities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4.4 Transfer Barriers . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.5 Phosphorus Containing Molecules . . . . . . . . . . . . . . . . . . . . 56
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Partially Automatized Scheme for Parametrizing DFTB 63
5.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.1 Analytical Representation of Repulsive Potentials . . . . . . . . . . . 64
5.1.2 De ning a Linear Equation System . . . . . . . . . . . . . . . . . . . 65
5.1.3 Solving the Linear System . . . . . . . . . . . . . . . . . . . 68
78 CONTENTS
rep
5.1.4 Illustrative Example of Fitting V . . . . . . . . . . . . . . . . . . . 70HO
5.1.5 Further Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Application to Hydrocarbons and DFTB2 . . . . . . . . . . . . . . . . . . . 74
5.2.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.2 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Parametrization of DFTB3 for Elements C, H, N, and O 91
6.1 Parametrization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1.1 Initial Guess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1.2 Compression Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1.3 Repulsive Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1.4 Tuning the Hubbard Derivatives . . . . . . . . . . . . . . . . . . . . . 100
6.1.5 Summary of New Parameters . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Performance of the New Parameters . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.1 Atomization Energies, Geometries, Vibrational Frequencies . . . . . . 102
6.2.2 Proton A nities, Hydrogen Binding Energies, and Proton Transfer
Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.3 Reaction Energies and Isomerization Enthalpies . . . . . . . . . . . . 104
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7 Summary and Outlook 109
A Illustration of Repulsive Potentials 111
B Derivation of DFTB3 Expressions 113
@abB.1 Functional Form of and its Derivative . . . . . . . . . . . . . . . . . . 113
@Ua
B.2 Derivation of the Kohn-Sham Equations . . . . . . . . . . . . . . . . . . . . 114
B.3 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.4 Atomic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
hB.5 The -Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
C Molecule Sets and Benchmarks 121
D Self-Interaction Correction for DFTB3 129
Bibliography 131Chapter 1
Introduction
Quantum mechanics is a physical theory describing occurrences on the atomic and subatomic
level. One mathematical formulation of quantum mechanics is given by the Schr odinger
equation which turns out to be soluable only for very simple problems. Approximations are
necessary in order to determine the chemical behavior of molecules. Two main quantum
chemical approaches are established, wave function based ab initio methods and density
functional theory (DFT).
The Hartree-Fock (HF) method is a simple wave function based approach. It considers
the Pauli principle in so called exchange integrals, i.e. two electrons of same spin symmetry
cannot occupy the same quantum state (orbital) simultaneously. However, HF principally
allows two electrons of opposite spin to \move" independently from each other. To pre-
vent this unphysical behavior one has to include electron correlation. Therefore, post-HF
methods such as con guration interaction and perturbation theoretical methodologies are
necessary (for review see e.g. ref [1]). Unfortunately, these methods require tremendously
increased computer resources. While HF is often used for determining structural and vibra-
tional properties, post-HF methods are necessary for reliably describing chemical reactions,
energetics and structural properties of hydrogen bonding, and van der Waals complexes.
For these reasons, DFT has become very popular [2]. Even though an exact functional
that correctly describes all properties is not known, already simple approximations contain
the physics of electron exchange and correlation. This leads to results similar in quality
of those from post-HF methods for many molecular properties, but in remarkably reduced
computational time. Despite the success, DFT implementations reveal several problems, e.g.
in describing van der Waals complexes, transition states, and charge transfer excitations.
Therefore, further development of density functionals remains an important issue of current
investigation.
For many applications a further reduction of computational costs is required for two
reasons: First, to be able to handle large molecular systems, and second to investigate
dynamic properties such as conformational changes in biological systems. Even with rapidly
increasing computational speed it is not foreseeable that HF or DFT based methods will
meet the demand of the scientists for \su ciently" large systems and \su ciently" long
simulation times.
For accessing experiments that lie beyond these limitations of HF and DFT, semiempirical
methods are well established. These methods treat electronic systems in a quantum mechan-
ical sense, but explicit calculation of integrals is avoided by either neglect or parametrization.
That way, computational time is reduced by a factor of about 1000 as illustrated for a few
examples in Figure 1.1. For a molecular dynamics simulation, this means that a 1000 times
910 Chapter 1 Introduction
Figure 1.1: A popular DFT implementation B3LYP in comparison to the semiempirical SCC-DFTB
method with respect to computational time for a single point energy calculation. The calculations were
carried out on a standard desktop PC, B3LYP with the 6-31G(d) basis set utilizing the program package
TURBOMOLE [3], SCC-DFTB using DFTB+ [4].
longer simulation time is possible in comparison to HF and DFT based methods. The sim-
pli cations in these schemes usually lead to a reduction of reliability and accuracy. Thus, the
essence of the development of semiempirical methods is to provide schemes that maintain
the accuracy of HF and DFT based methods for as many properties as possible.
Again, two main developmental directions can be distinguished. The rst class consists of
HF based semiempirical methods which originate from work of the Nobel laureate John Pople
back in the 1950s and 60s. As in HF theory, electron correlation is not explicitly considered
in the formalism. However, it is captured to some degree implicitly via parameters that
are tted to experimental data. The strengths and weaknesses of such methods are well
documented (see e.g. ref [5]). Modern and popular representatives are PDDG/PM3 [6],
PM6 [7], and the OMx methods [8, 9] to name only a few.
The second class of semiempirical methods is comprised of tight binding approaches which
can be understood as approximate density functional theory [10]. Particularly the density
functional tight binding (DFTB) method has become very successful in recent years. Almost
all parameters are calculated from DFT and only few empirical ones remain. Like DFT, the
conceptually very simple DFTB method contains electron correlation explicitly.
The di erence between the rst and second class of semiempirical methods is its ori-
gin. However, computational costs and applications are rather similar. While methods like
PDDG/PM3 and PM6 have proven to give excellent results for heats of formation of small
molecules [7, 11], the strengths of DFTB are applications to large biological systems, e.g.
structure and relative energies of peptides are described very well [12, 13, 14]; furthermore,
DFTB turns out to be very reliable and robust in molecular dynamics simulations.
The aim of this thesis is a further development of DFTB. This method can be derived by a
Taylor series expansion of the DFT total energy around a given reference density. Originally,
all terms higher than rst order were neglected [15]. Therefore, reasonable results were only
1found for systems with small charge transfer between the atoms. Typical applications
concerned the chemistry of hydrocarbons. For describing systems with signi cant charge
transfer the method was extended by including also the second order terms of the Taylor
1Later it was found that also systems with very large intramolecular charge transfer, as e.g. in KCl and
NaCl molten salts, can be well described by this version of DFTB [16].