I. Harczuk — Atomic decomposition of molecular …929105/FULLTEXT02.pdfThe calculated molecular...

Atomic decomposition ofmolecular properties

Ignat Harczuk

Theoretical Chemistry and BiologySchool of Biotechnology

Royal Institute of TechnologyStockholm, 2016

Abstract

In this thesis, new methodology of computing properties aimed for multipleapplications is developed. We use quantum mechanics to compute propertiesof molecules, and having these properties as a basis, we set up equations basedon a classical reasoning. These approximations are shown to be quite good inmany cases, and makes it possible to calculate linear and non-linear propertiesof large systems.

The calculated molecular properties are decomposed into atomic propertiesusing the LoProp algorithm, which is a method only dependent on the overlapmatrix. This enables the expression of the molecular properties in the two-site atomic basis, giving atomic, and bond-centric force-fields in terms of themolecular multi-pole moments and polarizabilities. Since the original LoProptransformation was formulated for static fields, theory is developed which makesit possible to extract the frequency-dependent atomic properties as well. Fromthe second-order perturbation of the electron density with respect to an externalfield, LoProp is formulated to encompass the first order hyperpolarizability.

The original Applequist formulation is extended into a quadratic formula-tion, which produces the second-order shift in the induced dipole moments of thepoint-dipoles from the hyperpolarizability. This enables the calculation of a to-tal hyperpolarizability in systems consisting of interacting atoms and molecules.The first polarizability α and the first hyperpolarizability β obtained via theLoProp transformation are used to calculate this response with respect to anexternal field using the quadratic Applequist equations.

In the last part, the implemented analytical response LoProp procedureand the quadratic Applequist formalism is applied to various model systems.The polarizable force-field that is obtained from the decomposition of the staticmolecular polarizability α is tested by studying the one-photon absorption spec-trum of the green fluorescent protein. From the frequency dispersion of thepolarizability α(ω), the effect of field perturbations is evaluated in classicaland QM/MM applications. Using the dynamical polarizabilities, the Rayleigh-scattering of aerosol clusters consisting of water and cis–pinonic acid moleculesis studied. The LoProp hyperpolarizability in combination with the quadraticApplequist equations is used to test the validity of the model on sample wa-ter clusters of varying sizes. Using the modified point-dipole model developedby Thole, the hyper-Rayleigh scattering intensity of a model collagen triple-helix is calculated. The atomic dispersion coefficients are calculated from thedecomposition of the real molecular polarizability at imaginary frequencies. Fi-nally, using LoProp and a capping procedure we demonstrate how the QM/MMmethodology can be used to compute x-ray photoelectron spectra of a polymer.

i

Sammanfattning

I denna avhandling utvecklas ny metodik for berakningar av egenskaper medolika tillampningar. Vi anvander kvantmekanik for att berakna egenskaper hosmolekyler, och anvander sedan dessa egenskaper som bas i klassiska ekvationer.Dessa approximationer visas vara bra i flera sammanhang, vilket gor det direktmojligt att berakna linjara och icke-linjara egenskaper i storre system.

De beraknade molekylara egenskaperna delas upp i atomara bidrag genomLoProp transformationen, en metod endast beroende av den atomara overlapps-matrisen. Detta ger mojligheten att representera en molekyls egenskaper i entvaatomsbasis, vilket ger atomara, och bindningscentrerade kraftfalt tagna frande molekylara multipoler och polarisabiliteter.

Eftersom att den originella LoProp transformationen var formulerad medstatiska falt, sa utvecklas och implementeras i denna avhandling LoProp meto-den ytterligare for frekvensberoende egenskaper. Genom den andra ordnin-gens storning med avseende pa externa falt, sa formuleras LoProp sa att di-rekt bestamning av forsta ordningens hyperpolariserbarhet for atomara po-sitioner blir mojlig. De ursprungliga Applequist ekvationerna skrivs om tillen kvadratisk representation for att gora det mojligt att berakna den andraordningens induktion av dipolmomenten for punktdipoler med hjalp av denforsta hyperpolariserbarheten. Detta gor det mojligt att berakna den totalahyperpolariserbarheten for storre system. Har anvands den statiska polariser-barheten och hyperpolariserbarheten framtagna via LoProp transformationenfor att berakna ett systems egenskaper da det utsatts av ett externt elektrisktfalt via Applequists ekvationer till andra ordningen.

Tillampningar presenteras av den implementerade LoProp metodiken medden utvecklade andra ordnings Applequist ekvationer for olika system. Detpolariserbara kraftfaltet som fas av lokalisering av α testas genom studier avabsorptionsspektrat for det grona fluorescerande proteinet. Via berakningar avden lokala frekvensavhangande polariserbarheten α(ω), testas effekten av de ex-terna storningar pa klassiska och blandade kvant-klassiska egenskaper. Genomden linjara frekvensberoende polariserbarheten sa studeras aven Rayleigh sprid-ning av atmosfars partiklar. Via LoProp transformationen av hyperpolariser-barheten i kombination med de kvadratiska Applequist ekvationerna sa un-dersoks modellens rimlighet for vattenkluster av varierande storlek. Genom attanvanda Tholes exponentiella dampningsschema sa beraknas hyper-Rayleighspridningen for kollagen. Den atomara dispersionskoefficienten beraknas via delokala bidragen till den imaginara delen av den linjara polariserbarheten. Slutli-gen visar vi hur LoProp tekniken tillsammans med en s.k. inkapslingsmetod kananvandas i QM/MM berakningar av Rontgenfotoelektron spektra av polymerer.

iii

Abbreviations

QM — Quantum mechanicsMM — Molecular mechanicsQM/MM — Quantum mechanics molecular mechanicsMD — Molecular DynamicsLoProp — Localized PropertiesSCF — Self-consistent fieldOPA — One—photon absorptionTPA — Two—photon absorptionNLO — Non—linear opticsGFP — Green fluorescent proteinMFCC — Molecular fractionation with conjugate capsHF — Hartree—FockDFT — Density functional theoryCI — Configuration interactionTDHF — Time-dependent Hartree-Fock TheoryKS — Kohn-ShamTD-DFT — Time-dependent Density functional theorySHG — Second—harmonic generationEFISHG — Electric—field induced second—harmonic generationesu — Electrostatic unit conventionPDA — Point dipole approximationAMOEBA — Atomic Multipole Optimized Energetics for Biomolecular ApplicationsONIOM — Our own N—layered Integrated molecular Orbital and molecular Mechanics

PES — Potential energy surface

iv

List of Papers

Paper I. I. Harczuk, N. Arul Murugan, O. Vahtras, and H. Agren,“Studies of pH-sensitive optical properties of the deGFP1 green flu-orescent protein using a unique polarizable force field”, J. Chem.Theory. Comput., 10, (8), pp 3492–3502, 2014

Published

Paper II. I. Harczuk, O. Vahtras, and H. Agren, “Frequency-dependentforce fields for QM/MM calculations”, Phys. Chem. Chem. Phys.,17, 12, pp 7800–7812, 2015

Published

Paper III. I. Harczuk, O. Vahtras, and H. Agren, “Hyperpolariz-abilities of extended molecular mechanical systems”, Phys. Chem.Chem. Phys. , 18, 12, pp. 8710–8722, 2016

Published

Paper IV. I. Harczuk, O. Vahtras, and H. Agren, “Modeling RayleighScattering of Aerosol Particles”, J. Phys. Chem. B , 120, 18, pp.4296–4301, 2016

Published

Paper V. I. Harczuk, O. Vahtras, and H. Agren, “First hyperpo-larizability of collagen using the point dipole approximation”

Submitted to J. Phys. Chem. Lett.

Paper VI. I. Harczuk, B. Balazs, F. Jensen, O. Vahtras, H. Agren,“Local decomposition of imaginary polarizabilities and dispersioncoefficients”

Manuscript

Paper VII. T. Loytynoja, I. Harczuk, K. Jankala, O. Vahtras,and H. Agren, “Quantum-classical calculations of X-ray photoelec-tron spectra of polymers — polymethyl methacrylate revisited”

Manuscript

v

Own contributionPaper I: I computed the force-field, did the gas-phase and PCM

calculation of the GFP-chromophore states, and wrote the manuscriptdraft.

Papers II-III: I performed all of the calculations, did the analysis,and wrote the manuscript, with the exemption of the theory part.

Paper IV: I performed all of the calculations, did the analysis,and wrote the manuscript

Paper V: I performed all of the calculations, did the analysis,and wrote the manuscript.

Paper VI: I performed the calculations with exemption of theSAPT2(HF) calculations, and wrote most of the manuscript.

Paper VII: I performed the calculations of the force-field for theQM/MM calculations.

vi

AcknowledgmentsI firstly acknowledge the supervision provided for me by Hans Agren, my mainsupervisor, and Olav Vahtras, my co-supervisor.

After finished my Master thesis defense, I had no plans for academia, but withHans suggestion and encouragement to apply for the excellence position, I wasable to stay and continue research, something that has always been close tomy heart. Hans experience in the field of computational chemistry furthermoremade it possible for our project to be propelled in a clear direction, with all theaspects of our work inter-connected. The field of response theory has before myPhD studies seemed mystical and far too advanced for me, however, with Olav’sexpertise in the subject it became possible for me to learn details regardingproperty calculations I would never grasp on my own.

I want to give a special thanks to my collaborators in the department, namelyNatarajan Arul Murugan, Tuomas Loytynoja and Xin Li for helping me under-stand and analyze our computational results. Murugan helped me understandconcepts from MD, and as it is not my area of expertise, this is greatly appreci-ated. During work on a project with Tuomas, we struggled together to figure outhow to model PMMA, which was a fun process and which ultimately he solvedusing some rotations. Xin Li introduced me to the theory of light scattering inthe atmosphere, which helped me immensely for our aerosol project.

I furthermore want the express my gratitude to all the people working in ourdepartment, but also the people who I meet but finished their position duringmy research time. All of the people in our wonderful department made the officea very pleasant environment to work in, and contributed to a great atmosphere.

Others who deserve a special mention are Nina Bauer and Caroline Bramstang,who helped me with non-scientific but still important issues. I also want to givea huge thanks and acknowledgment to Per-Ake Nygren, who woke me up onemorning for a phone interview of the excellence position, which I luckily passed.

I would also give a special thanks to Wei Hu, Bogdan Frecuz, Lu Sun, JaimeAxel Rosal Sandsberg, Guanglin Kuang, Vinıcius Vaz da Cruz, Rafael Car-valho Couto, Zhen Xie, Rongfeng Zou, Shaoqi Zhan, Nina Ignatova, Yongfei Ji,Zhengzhong Kang, and Lijun Liang for the badminton games and all the greatparties we had in Roslagstull and Lappis.

I finally acknowledge KTH and the school of biotechnology, for providing me

with the excellence position. I would also like to thank SNIC for computa-

tional resources under the project names “SNIC-2013–26–31”, “SNIC-023/07–

18”, “SNIC-2014–1–179”, and “SNIC-2015–1–230”.

Thank you!

Ignat Harczuk, Stockholm, June 2016

vii

Contents

1 Prologue 1

2 Introduction 3

2.1 Quantum Mechanics Molecular Mechanics . . . . . . 3

2.2 Force-fields . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Molecular Dynamics . . . . . . . . . . . . . . 5

2.3 The Point-Dipole approximation . . . . . . . . . . . . 7

3 Theory 11

3.1 The Schrodinger equation . . . . . . . . . . . . . . . 12

3.2 Quantum Chemical Methods . . . . . . . . . . . . . . 13

3.2.1 Hartree-Fock theory . . . . . . . . . . . . . . 13

3.2.2 Density Functional theory . . . . . . . . . . . 17

3.3 Response theory . . . . . . . . . . . . . . . . . . . . . 18

3.4 Single reference TD-DFT . . . . . . . . . . . . . . . . 20

3.5 The LoProp Algorithm . . . . . . . . . . . . . . . . . 21

3.5.1 Local Polarizabilities from analytical responsetheory . . . . . . . . . . . . . . . . . . . . . . 23

3.5.2 Second-order perturbation . . . . . . . . . . . 24

3.6 The Applequist Equations . . . . . . . . . . . . . . . 25

3.6.1 First order . . . . . . . . . . . . . . . . . . . . 25

3.6.2 Second order . . . . . . . . . . . . . . . . . . 26

3.7 Tholes damping model . . . . . . . . . . . . . . . . . 28

3.8 Non-linear optics . . . . . . . . . . . . . . . . . . . . 29

ix

CONTENTS

4 Applications 31

4.1 The Green Fluorescent Protein . . . . . . . . . . . . 31

4.2 Frequency dependent force-fields . . . . . . . . . . . . 33

4.3 Hyperpolarizability of water . . . . . . . . . . . . . . 36

4.4 Rayleigh-scattering of aerosol particles . . . . . . . . 36

4.5 Hyperpolarizability of Collagen . . . . . . . . . . . . 38

4.6 C6 coefficients from complex polarizabilities . . . . . 41

4.7 Calculations of X-ray photoelectron spectra of polymers 42

5 Conclusions and Outlook 45

x

Chapter 1

Prologue

Before my PhD, the view I had of research was too idealized. Iassumed that all research was consistent and impeccably produced,which at all times led to new advances and technologies. This flaw-less notion of research turned out to be a little simplistic, as researchis produced, and reinforced, by humans. Even though researchersstudy repeatable phenomena with the purpose of either buildingmore efficient machines, or to learn more about the universe, re-search in itself is a human process, which comes with everythingelse of being a human. Furthermore, the collective research progresswe make as a humanity is only possible due to work produced byour previous generations. Even the tiniest discoveries should thusbe respected, as long as they have been achieved with the rigor thatscience requires.

In doing my PhD, I have learned many nuances of doing research,and science in general. These nuances has taught me to be even morecareful, and more critical, of the data and interpretations which I,or others, make and obtain. The peer-review system set in placeprovides good protection against false positive discoveries, and alsoagainst sub-par research methodologies which are not reproducible,but it is not always perfect. During my PhD I thus learned toalways be skeptical when reading about new findings, as these areprone to error at times, even with the current systems set in place.I learned that sometimes, my own results were wrong, and that I

1

CHAPTER 1. PROLOGUE

should always strive to keep careful procedural information loggedfor future references. This is where my appreciation of the IPythonNotebook came from.

During my research I was exposed to a lot of programming, and morespecifically, to the internal design of software architectures via mysmall programming implementations for Dalton, and my own codes.This made me aware of the importance of open-source software inscience, and the transparent research methodologies that goes withit.

Without droning on too long, I will say that the largest lesson Ilearned during my research is the appreciation of knowledge, andthe hard work that lies behind scientific discoveries. Where theknowledge comes from and what it is applicable to will of courseimpact its importance in terms of its intrinsic value, and whetherit should be disclosed at all, such as in the case of moral gray areasthat could cause devastation in the long haul. Aside from ethicallydubious discoveries, I truly do believe that scientific knowledge, ir-respective of what research area it comes from, such as for e.g. cos-mology, engineering, or genetics, is the foundation for prosperity inour world.

Therefore, I believe that as a researcher, I now hold a higher re-sponsibility towards our society as a whole to act scrupulously inthe eye of new data and findings, and that I am indebted to giveback to our society in forms of open science and open source.

2

Chapter 2

Introduction

In this part, an overview of core topics is given to set the contextof the work presented in the thesis. First there will be a historicaland broad presentation of QM/MM, which is the most importantapplication for this work. Then a section about force-fields used inMD and QM/MM will be given. Lastly, an overview of the point-dipole model is presented.

2.1 Quantum Mechanics Molecular Me-

chanics

In recent years, the advance in computer power has allowed forincreasingly larger chemical and biological systems to be studied,which has led to a wide window of physical phenomena being un-derstood at larger scales1. However, at the most fundamental levelof computational chemistry, there is an inherent bottleneck whichmakes the computational time of quantum mechanical methods scalenon-linearly with respect to the problem size.

For very accurate predictions, which require the most time-consumingmethods in computational chemistry, this means that the increasein raw computational power alone is not enough in order to be ableto describe large interesting systems at an accuracy higher than, orequivalent to that of, the corresponding experiments. These con-

3

CHAPTER 2. INTRODUCTION

Figure 2.1: Separation of a large system into a subset of two interact-ing systems, the quantum mechanical (QM) part, and the classicalmolecular mechanical (MM) part.

cerns are what led to the formulation of QM/MM by Warshel andLevitt back in the 70s2.

The basic idea of QM/MM is that the energy of a system can bepartitioned by the energy of different sub-spaces of the system, cal-culated at different levels of theory, see Figure 2.1 for an illustra-tion. The interesting subspace of a system can thus be modeledby a higher level of theory, by e.g. HF, DFT, or CC2, than therest of the system, modeled by classical MM, which is assumed toprovide only a minor perturbation to the system of interest. Thisintroduces an additional degree of freedom in the compromise be-tween computational power and accuracy in determined energies. Alarger QM region thus gives a more accurate picture, but at the ex-pense of computational resources, whereas a larger MM region canbe cheaply incorporated in calculations without too much increasein computation time, but with the advantage of better descriptionof the whole system. This methodology leads to a modified Hamil-tonian which contains additive contributions corresponding to theQM, MM, and the mixed QM/MM interaction parts.

H = HQM +HMM +HQM/MM (2.1)

4

2.2. FORCE-FIELDS

One implementation of Eq. (2.1) is to calculate the HQM/MM termby representing the external MM atoms using static point charges.These charges can be obtained in numerous ways, correspondingto different force-fields. For larger corrections to the HQM/MM term,higher order multi-poles can be included of the external MM-region,which gives an overall better description of the external potential.When charges and dipoles are included in the external region, thecore QM region will feel the external field caused by those chargesand dipoles. If the MM force-field also includes polarization, theexternal dipoles can be further polarized by the QM region, andthe response of the QM-region to the environment itself. This pro-cess can be iteratively solved and converged self-consistently. Themethod above is denoted as having a polarizable embedding3.

Another implementation of QM/MM is the ONIOM4,5 model, whereinseveral layers labeled are described by different Hamiltonians. Thiscan be separated into many different layers of systems, but is fornow implemented as highest for 3 layers. The energy evaluated ofa 2 layer system consisting of a “High” level for the region A repre-senting the core QM part, and a “Low” level semi-empirical methodon the outer region B, will be obtained as the sum

Etot = EHighA + ELow

A+B − ELowA (2.2)

Where A and B denote how large region is included for the micro-iteration of method defined by “High” and “Low”.

2.2 Force-fields

2.2.1 Molecular Dynamics

In molecular dynamics, the motion of all particles in a system canbe obtained by integrating Newton’s second law of motion

Fi = miai (2.3)

The forces acting on each particle are obtained from the derivativeof the potential energy

5


Fi = −∇Vi (2.4)

How the construction of this potential energy is made, is, howeverforce-field dependent. One example is the CHARMM force-field6,for which the total potential V of a system is

V =∑bonds

Kb(b− b0)2 +∑

angles

Kθ(θ − θ0)2 +∑

dihedrals

Kφ(1 + cos(nφ− δ))

+∑

improperdihedrals

Kϕ(ϕ− ϕ0)2 +

∑Urey−Bradtey

KUB(r1,3 − r1,3;0)2 (2.5)

+∑

nonbonded

qiqj4πDrij

+ εij

[(Rmin,ij

rij)12

− 2(Rmin,ij

rij)6]

The parameters in Eq. (2.5) are unique to a set of atom types, orpairs of atom types. The parameters of a force-field can subse-quently be fitted in order to reproduce common experimental datasuch as the density, isothermal compression, heat capacity, andmany other quantities. After satisfactory agreement is found fora set of parameters, the force-field is ready for use in production fornew systems.

The most commonly used force-fields today6–8, are additive and de-scribe many generic systems fairly well. However, these force fieldsare designed to be general and extensible to many different systems,which leads to its failure to describe important phenomena suchas bond-breaking. One special type of reactive force-fields solvesthis problem with bond-breaking by adding terms that describe theinter-molecular atom displacements9.

The other difficulty with traditional additive force-fields can be seenwhen polarizable molecules such as water or for e.g. phenoles bondedto proteins are in close proximity, as the charges and the net dipole-moments of the systems will not reflect the induction energy. Inreality, the electron clouds will overlap and sophisticated interac-tions will occur between the species. This is important, as evensubtle interaction can give rise to stability in biological systems10.

6

2.3. THE POINT-DIPOLE APPROXIMATION

Several polarizable force-fields such as the polarizable AMBER ff0211,and the newly proposed AMOEBA12 try to include this inducedenergy, but are still not mature enough to completely overtake theclassical fixed-charge force-fields for production. Additionally, force-fields based on the fluctuation of charges via parametrization of thecapacitance have been developed to study the more complex elec-tron dynamics involving organic ligands interacting with metallicnano-particles13,14.

The decomposition of the interaction energy in proteins was inves-tigated further by the MFCC procedure15,16, which is the process ofcutting a protein into distinct residues across the peptide back-bone,and capping their co-valency with so-called concap groups. By usingLoProp in combination with MFCC, choosing the cutting to occurover peptide bonds, and utilizing the first-order non-redundant cap-ping group HCONH2, an illustration is made in Figure 2.2, whichexplicitly shows how the protein properties are obtained for the MMpart for the use in QM/MM calculations. These atomic polarizableforce-fields have been used in the QM/MM calculations of the ab-sorption spectra17,18 of ligands bound to proteins. The idea hereis to describe the charges and polarizabilities of protein residues asthe MM part in QM/MM, while the QM region would be used forproperty calculations of photoactive probes.

2.3 The Point-Dipole approximation

Silberstein introduced the notion19,20 of light-matter interaction be-ing a result of polarizable atoms. In his theory, atomic polarizabili-ties where additive and summed up to the molecular polarizability.Applequist21 took Silbersteins idea one step further by describingthe instant induced atomic dipole moments of atoms in molecules asfunctions of their localized atomic polarizabilities, and the externalfield. This is the point-dipole model. In the Applequist picture,the molecular polarizability is thus obtained from the interaction ofatomic polarizabilities. The subsequent atomic polarizabilities werefit to a test set of molecules, for which the experimental structuresand polarizabilities were known. An approximate model of atomic

7


Figure 2.2: Illustration of how the properties of the N and the CAatoms of a tri-peptide are obtained by cutting it into smaller frag-ments. This example utilizes the HCONH2 capping group.

polarizabilities was constructed, which, however, had some draw-backs. One issue stemmed from the fact that the induced dipoleswere only dependent on the inter-atomic distances, and could insome cases lead to over-polarization and infinite polarizabilities ofmolecules. This consequence was by Applequist attributed to reso-nance conditions in the model.

Following Applequist, Thole22 modified the original equations witha damping term, which would depend on a damping parameter, andthe atomic polarizabilities themselves. Out of the various dampingforms introduced by Thole, the exponential one was the most useful,and is the one commonly used today in applications23. The dampingterm would serve as to decrease the field between atoms which werevery close, completely avoiding the over-polarization in the originalApplequist model. New sets of parameters where derived using thenew damping model24.

The polarizable AMOEBA force-field mentioned in Section 2.1 usesthe Thole damping scheme, but with a different parameters for the

8

2.3. THE POINT-DIPOLE APPROXIMATION

damping factor and polarizabilities25,26.

The point-dipole model has been applied27,28 to larger test sets ofmolecules. Moreover, with additional parametrizations to QM ob-tainable molecular polarizabilities29, the polarizability of large pro-teins could be studied30. The higher order point-dipole scheme31

also included the possibility of parametrization of atomic secondhyperpolarizabilities γ, but showed inadequacies for the predictionof the out-of-plane second hyperpolarizability tensor components insystems such as benzene.

9

Chapter 3

Theory

In this chapter, theories related to the thesis work will be presented.Since the work in this thesis concerns applications in various fieldsfrom polarizable embeddings in QM/MM and force-fields in molec-ular dynamics to semi-classical evaluation of optical properties, gen-eral considerations of the underlying theories will be presented.

First there will be a presentation of the most fundamental equationof quantum chemistry, namely the Schrodinger equation, which isthe basis of all the evaluated properties in the quantum mechanicalframework of theoretical predictions.

The basic theory of Hartree-Fock will be presented, along with itslimitations, followed by Density Functional Theory, as these are themain theories used for computations in the papers included in thisthesis.

Response theory is an integral part of the thesis work as it is used todetermine the static and dynamic polarizabilities and hyperpolariz-abilities for the atomic decomposition using the LoProp algorithm.Firstly, there will be a general discussion about response theory,then, the application of response theory to molecular properties ispresented, and in particular the equations for the single-referenceTD-DFT formalism.

Subsequently the LoProp method is presented, using the perturbeddensities obtained from the response calculations. Here we gener-alize the LoProp method, which originally was designed for static

11

CHAPTER 3. THEORY

polarizabilities obtained with the finite-field approach. Since webase our calculations on the analytical response formalism, equa-tions are provided which show how the localized polarizabilities areobtained in the LoProp basis. The second-order perturbation ofthe electron density is incorporated, which gives the possibility toextract the atomic hyperpolarizabilities via the LoProp approach.

Finally, the theory developed by Applequist et al. is presented in itsoriginal form, and the second-order perturbation of the point-dipolesis included in order to express the induced hyperpolarizability ofinteracting atoms and molecules, obtained with properties using theabove presented theories.

A summary of Tholes exponential damping equations and generalconsiderations of non-linear optics is shown in the last part of theTheory section.

3.1 The Schrodinger equation

The most fundamental equation in quantum chemistry is the Schrodingerequation. In its general time-dependent form, it reads

i~δΨ

δt= HΨ (3.1)

where the Hamiltonian H contains operators of the kinetic and po-tential energy of all the particles associated with the total wavefunction Ψ of the system, explicitly written

H =∑i

− ~2

2mi

∇2i + Vi (3.2)

where the first term is the kinetic energy operator, and the secondis the potential energy operator for particle i. Eq. (3.1) is applicableto any dynamic quantum system, but is in quantum chemistry mostoften used in its time-independent eigenvalue formulation

HΨ = EΨ (3.3)

12

3.2. QUANTUM CHEMICAL METHODS

as one is often only interested in stationary states of chemical sys-tems, which give the energetics of the electronic ground state. Thisgives the power to predict reaction rates32, water oxidation in vari-ous systems33,34, reaction mechanisms on surfaces35 etc.

The eigenvalue E of the Hamiltonian in Eq. (3.3) gives the energy ofeach respective eigenstate in Ψ . If one would study a molecule, thenΨ from Eq. (3.3) would contain all information about the molecularsystem in question, at this level of approximation.

If the theory of special relativity was included, which postulates thatno particle with mass can travel as fast as light, the Schrodingerequation would be extended to the relativistic equation denoted asthe Klein-Gordon equation

− ~2δ2ψ

δt2+ (~c)2∆2ψ = (mc2)

2ψ (3.4)

In the 1930s, Dirac generalized the relativistic equation proposed byKlein and Gordon by the inclusion of the Pauli matrices to describeparticles as having 4 components. For the work presented in thisthesis, the properties of interest will have negligible contributionsfrom relativistic effects such as spin-orbit or nuclear coupling, andthe form used will thus be restricted to the non-relativistic equa-tion (3.3).

3.2 Quantum Chemical Methods

Here an overview of the two most used methods in the thesis ispresented, Hartree-Fock, and Density Functional Theory.

3.2.1 Hartree-Fock theory

In quantum chemistry, one is mostly interested in the energy statesof the electrons in molecules, and the approach is therefore to finda practical form of Eq. (3.3) for the electronic wave-function only.This is done by using the Born-Oppenheimer approximation. Asthe nuclei have a mass of a factor approximately 103 larger than

13

CHAPTER 3. THEORY

the electrons, they will move much slower than the electrons. Theelectrons thus approximately move in a frame where the nuclei posi-tions are stationary. This enables the separation of the nuclear, andelectronic wave-function of Eq. (3.3), respectively, from the totalwave-function

Ψ(rel, Rnuc) = ΨN(Rnuc)ΨE(rel, Rfixednuc ) (3.5)

Now the electronic wave-function depends on the nuclei coordinatesparametrically, which are fixed when solving the Schrodinger equa-tion for the electrons only

Rnuc = Rfixednuc (3.6)

The problem is now to compute the wave-function of the electrons,and the subscript E in Ψ = ΨE is omitted to always refer to theelectronic wave-function for simplicity. For very heavy nuclei andexcited state PES, the Born-Oppenheimer approximation can seizeto be valid, and the nuclei has to be considered in some way. Forsmall systems and for applications related to this thesis, however,the BO-framework is sufficient.

The Hamiltonian from Eq. (3.2) can now be written by only consid-ering the kinetic operator for the electrons, in a potential createdby the fixed position of the atomic nuclei (here in atomic units,~ = mi = 1)

H = −1

2

∑i

∇2i −

N∑i=1

M∑A=1

ZAriA

+M∑B>A

ZAZBrAB

+N∑j>i

1

rij(3.7)

where Zj are the nuclei charge, N and M are the number of elec-trons, and nuclei, respectively. From quantum mechanics, the wave-function is mathematically a point in Hilbert space, which consistsof the linear combination of basis vectors representing complex func-tions. In this infinite basis space, the analytical solution to the elec-tronic Schrodinger equation is only known for one-electron systems.This is due to the electron repulsion term in Eq. (3.7). To numer-ically compute the electronic wave-function for a molecule, a finitebasis may be introduced.

14


The first step is to express the total wave function using a Slaterdeterminant. For a total of N electrons, the Slater determinant is

Ψ = |χ1χ2, . . . , χn >=1√N !

∣∣∣∣∣∣∣∣∣χ1(r1) χ2(r1) · · · χn(r1)

χ1(r2) χ2(r2). . .

......

.... . .

...χ1(rN) χ2(rN) · · · χn(rN)

∣∣∣∣∣∣∣∣∣ (3.8)

where χ are the molecular orbitals. As electrons are fermions, theirwave-function must be anti-symmetric with respect to interchangeof two fermion coordinates, due to the Pauli exclusion principle. TheSlater determinant is thus a convenient representation of an anti-symmetric wave-function, as swapping two electron-coordinates isequivalent to permuting two rows in the determinant, which changesits sign.

Each orbital χi can be expanded as a linear combination of atomicorbital functions |φ >

χi = cijφj (3.9)

introducing the finite basis set φj. Multiplying from the left and in-tegrating over all coordinates, the Schrodinger equation (3.3) can betransformed to a system of non-linear equations called the Roothan-Hall equations, that can be solved iteratively through the self-consistentfield (SCF) cycle. This is the standard Hartree-Fock (HF) method.For explicit details considering all of the aspects of HF, see36.

Static correlation

The SCF solutions obtained from restricted closed-shell HF calcula-tions via the Roothan-Hall equations give very crude energies, andfails to describe the static correlation energy.

The issue of static correlation can be visualized by pulling apart thetwo hydrogens in a H2 molecule, as seen in Figure 3.1. In HF theory,the α, and β electron of each respective hydrogen in H2 will forma σH1−H2 bond, independent of the inter-atomic displacement. Acalculation of the energy for the individual atoms will thus not bethe same as the sum of the individual energy calculations at large

15

CHAPTER 3. THEORY

Figure 3.1: The dissociation of the H2 molecule. The left energydiagram demonstrates the problem of static correlation in restrictedHF-theory.

distances. The above problem is partially solved by constructing thespin-unrestricted Hartree-Fock, UHF, where each spatial molecularorbital has it’s own unique spin coordinate which is not integratedout.

UHF leads to the coupled system of equations called the Pople-Nesbet equations, where each spatial molecular orbital contains itsown spin α, or spin β-electron. The drawback of the UHF is how-ever, that the electronic wave-function is not a pure eigenstate ofthe square of the total spin operator, and small contamination ofthe spin-quantum number occurs, which has contributions from thehigher order spin-states.

The problem of static correlation energy can be resolved by full-CIor by constructing a multi-configurational wave-function, which isobtained by the linear combination of configuration state functions.In this way, contributions from different kinds of electron popula-tions for a fixed nuclei-wavefunction can be found, and the staticcorrelation problem be remedied.

The above methods are in the post-HF ab initio category, and re-

16


quire much more computational resources such as memory and pro-cessing power, as compared to HF, and can only be used for thesmallest systems, if large configurational space is considered.

For the applications considered collectively in this thesis, the sys-tems are closed-shell which are well described using a single referencestate, and thus the closed-shell single determinant formalism will beutilized.

Dynamic correlation

The second issue stemming from the single-determinant nature ofHF-theory is that each electron only feels the average field of allthe other electrons. This is why HF is also denoted as a mean-fieldtheory. The problems this leads to is that the coulomb repulsionbetween two electrons is overestimated.

The use of multi-configurational methods solve the issue of dynamiccorrelation as well, but require many configuration states in orderto capture the dynamic correlation, which is a very local effect. Analternative solution to this problem is to use Density FunctionalTheory, DFT.

3.2.2 Density Functional theory

Hohenberg and Kohn37 described early on how the energy of anelectron gas would only depend on its electron density for a givenexternal potential. Later, Kohn and Sham38 formulated DFT, whichwhen using one-electron orbitals, gave the energy of molecules as thesum of functionals of the electronic density,

E [ρ] = T [ρ] + Vext[ρ] + J [ρ] + Exc + VNN (3.10)

where T is the kinetic operator T [ρ] = 12

∑i < φ|∇2|φ >, Vext is the

potential of electrons with respect to the stationary nuclei and theelectrons themselves

Vext =

∫ρ(r)

(∑l

Zlr−Rl

+ v(r)

)dτ (3.11)

17

CHAPTER 3. THEORY

J [ρ] is the coulomb integral and VNN is the constant nuclear repulsionterm. The term Exc is the exchange-correlation energy, which is acorrection deriving from expressing DFT as an independent-particletheory.

The DFT formalism reduces the problem of computing the 4N elec-tronic coordinates (3 spatial and 1 spin) of all N electrons to con-structing a 4 dimensional electronic spin density, which would anal-ogously describe the energetics of a system. The problems withDFT methods comes from the fact that the shape of the exchange-correlation functional Exc is not well-defined, and thus has to beparameterized in some ad-hoc fashion.

DFT is thus not an ab initio theory per se, but a semi-empiricalmethod. The advantage of the DFT formalism is that the computa-tional speed is comparable to that of Hartree-Fock, but the accuracycan sometimes be within the experimental fault tolerance for sometype of calculation. As a result, DFT has enjoyed a widespreadusage39,40 till this very day.

Due to the parametric nature of the DFT functionals, the disad-vantage can be that DFT in a way becomes a reverse lottery, i.e. itworks most of the time, but in a few cases gives nonsensical results.

3.3 Response theory

Response theory is a mathematical tool, which is used in multi-ple disciplines, e.g. for the design of electronic circuits, quantumchemistry, and economics. It is a formalism, which describes howa system of interest reacts to some kind of input, and how the re-action in the system occurs with respect to some time-delay, i.e.“response”.

In the most general way, this can be written

R(t) =

∫ t

−∞φ(t− t′)f(t′)dt′ (3.12)

Where the functionR(t) has been affected by some stimuli function,or force f(t′). φ(t− t′) is the delayed response in the time domain,

18

3.3. RESPONSE THEORY

Figure 3.2: The retarded response function φ. The green lines showexamples of how the function should look. The red line shows anexample of diverging response. Note that it can oscillate.

as it is a function which creates a change with respect to an exter-nal perturbation. The shape of φ is that it does not diverge withincreasing time, i.e. it can initially be large, but then must decreasewith time, as seen in Figure 3.2.

If we make the variable transformation

τ = t− t′ (3.13)

dτ = −dt′ (3.14)

and rewrite R(t) and f(t′) in Eq. (3.12) in terms of their Fouriertransforms

R(t) =

∫ ∞−∞R(ω)e−iωtdω (3.15)

f(t′) =

∫ ∞−∞

f(ω)e−iω(t′)dω =

∫ ∞−∞

f(ω)e−iω(t−τ)dω (3.16)

inserting (3.15) and (3.16) into Eq. (3.12) gives

∫ ∞−∞R(ω)e−iωtdω =

∫ ∞−∞

f(ω)e−iωtdω

∫ ∞0

φ(τ)eiωτdτ (3.17)

19

CHAPTER 3. THEORY

where the integration limits have been changed due to (3.13).

Now, Eqn. (3.17) can be written∫ ∞−∞

e−iωtdω

(R(ω)− (

∫ ∞0

φ(τ)eiωτdτ)f(ω)

)= 0 (3.18)

taking the integral over all frequencies, this corresponds to a Fourierseries decomposition where each amplitude corresponding to the fre-quencies taken has to match individually forR(ω) and (

∫∞0φ(τ)eiωτdτ)f(ω),

respectively. Defining the susceptibility

χ(ω) =

∫ ∞0

φ(τ)eiωτdτ (3.19)

we get the response in the frequency picture

R(ω) = χ(ω)f(ω) (3.20)

Response theory comes in handy for the calculation of propertiesof molecules and atoms subjected to external or internal perturba-tions. Choosing the interaction operator between a molecule and anexternal field, which adds the interaction energy to the Hamiltonianas

Hinteract = −µE(ω) (3.21)

and choosing the response to be the expectation of the dipole-moment operator as well, µ, we obtain the linear response functionas the polarizability of a molecule.

3.4 Single reference TD-DFT

In the work by Sa lek et al.41, the generalization of MCSCF re-sponse42 was applied to the KS equations, and the explicit formof the necessary functional derivatives was presented. An operatorapplied to a time-independent reference state |0〉, gives the time-evolution ∣∣0⟩ = e−κ(t) |0〉 (3.22)

20

3.5. THE LOPROP ALGORITHM

where the matrix representation of the operator κ(t) is

κ(t) =∑pq

κpq(t)Epq (3.23)

and the excitation operator Epq is written

Epq = a†pσaqσ (3.24)

The matrix-elements κpq form the parameters of the theory, which,given a set of chosen operators, provides a system of linear-equationsthat when solved gives the time-evolution for said operator. Basedon the Ehrenfest theorem, the time-evolution of any time-independentoperator can be obtained

δ

⟨0

∣∣∣∣[O, eκ(t)(H0 + V (t)− i ∂∂t

)e−κ(t)

]∣∣∣∣ 0⟩ = 0 (3.25)

Choosing the perturbation V (t), and expectation operator O to bethe dipole-moment gives the polarizability as the linear responsefunction, and the hyperpolarizability as the quadratic response func-tion.

In the first-order perturbed density this will read

δ⟨eκ~re−κ

⟩(ω) =

⟨[δκ, ~r

]⟩(ω) =

∑pq

~rpqδDpq(ω) (3.26)

δDpq(ω) =[δκT (ω), δD

]pq(3.27)

3.5 The LoProp Algorithm

The motivation behind the LoProp transformation43 was to find aconvenient method to study the property dependence of atoms inmolecules in a way that could be comparable across atoms in dif-ferent molecules and even with different basis sets. This would re-sult in transferable properties which would give origin-independent

21

CHAPTER 3. THEORY

Figure 3.3: Visual diagram of the diagonalization procedure in theLoProp transformation. A and B are arbitrary atoms and o, vdenote the occupied, and virtual subspace of orbitals belonging toA or B, respectively. The red color denotes non-zero elements.

electrostatics of groups in molecules which closely resemble the to-tal molecular electrostatics, with the idea of using these localizedcharges, dipole moments, and polarizabilities as force fields in MD.

The basic idea is to obtain the transformation matrix which or-thonormalizes the atomic overlap matrix. The transformation isdone in 4 orthonormalization (Gram-Schmidt (GS) or Lowdin) andprojection procedures, which lead to a total transformation matrixthat can be written as the product

T = T1T2T3T4 (3.28)

Schematically, the overlap matrix between each step is changed inthe manner shown in Figure 3.3.

The resulting matrix T can now be used to express the molecularproperties as contributions from two-atomic sites only, by summingup the contributions in the transformed basis.

Since the general property of the operator O is defined by the ex-

22

3.5. THE LOPROP ALGORITHM

pectation value

〈O〉 = Tr(DO) =∑µ,ν

Dµν

⟨µ∣∣∣O∣∣∣ ν⟩ , (3.29)

restricting the summation of properties between pairs of atoms

〈OAB〉 = Tr(DO) =∑µ∈Aν∈B

DLoPropµν

⟨µ∣∣∣O∣∣∣ ν⟩LoProp

, (3.30)

where⟨µ∣∣∣O∣∣∣ ν⟩LoProp

is the transformed integral in the LoProp ba-

sis, the atomic properties are obtained.

The LoProp procedure is similar to the orthonormalization of theone-particle density matrix, with the key difference in that the local-ized properties obtained with the LoProp transformation does notdepend on the electronic configuration, only on the overlap matrixgiven a basis set and nuclear configuration.

3.5.1 Local Polarizabilities from analytical re-sponse theory

In their original work, Gagliardi et al.43 showed how the molecularpolarizability could be written in terms of localized charges anddipoles.

α(AB)ij =

µ(AB)i (F + δj)− µ(AB)

i (F− δj)2δj

+(∆Q(AB)(F + δj)−∆Q(AB)(F− δj))(R(A)

i −R(B)i )

2δj

The above formalism used finite-field which gives static polarizabil-ities, i.e. those which occur without any direct influence of externaloscillating fields. Below is the equivalent atomic decomposition pre-sented in the formalism of analytical response theory, which presents

23

CHAPTER 3. THEORY

the ability to extract static as well as dynamic localized polarizabil-ities, and later on hyperpolarizabilities, using one single formalism.

The first-order change of an expectation value can be expressed interms of the first-order density matrix

δ〈eκAe−κ〉∣∣∣κ=0

= 〈[δκ, A]〉 =∑pq

ApqδDpq (3.31)

The total molecular dipole moment is

〈−~rC〉 = −∑AB

~rABDAB +

∑A

QA(~RA − ~RC) (3.32)

and the first-order variation of the dipole moment can be written

δ〈−~rC〉 =∑AB

−~rABδDAB +∑A

δQA~RA , (3.33)

where Eq. (3.32) is origin-dependent.

By adding a charge-transfer term ∆QAB analogously to the deriva-tion in Ref.43

δQA =∑B

∆QAB = −∑l∈A

δDll (3.34)

the origin-dependence in Eq. (3.32) is removed and the localizeddynamic polarizability can be written as

αAB = δ〈−~r〉AB = −~rABδDAB + ∆QAB(~RA − ~RB) (3.35)

3.5.2 Second-order perturbation

Since the expansion of the electron density holds to each order of theperturbation of an external field, collecting the second-order termsbecomes straightforward, and thus the localized hyperpolarizabilitycontributions in the LoProp basis can be obtained analogously tothe LoProp polarizability.

The second-order perturbed density is

δ2Dpq(ω1, ω2) =[δ2κT (ω1, ω2), D

]pq+

1

2([δκT (ω1),

[δκT (ω2), D

]]+[δκT (ω2),

[δκT (ω1), D

]])pq

(3.36)

24

3.6. THE APPLEQUIST EQUATIONS

By taking rC to be the center of molecular charge, the second-orderchange in the molecular dipole moment equates to summing up theatom and bond contributions to the density

δ2〈−~rC〉(ω) =∑AB

∑l∈Am∈B

−(~rAB)lmδ2Dlm(ω1, ω2)

+∑AB

∆2QAB(ω1, ω2)(~RA − ~RB)

where ~rAB is the electronic coordinate with respect to the bondmidpoint between atoms A and B. ∆2QAB(ω) is the atomic charge-transfer matrix∑

B

∆2QAB(ω1, ω2) = δ2QA(ω1, ω2) (3.37)

where

δ2QA(ω1, ω2) = −∑l∈A

δ2Dll(ω1, ω2) (3.38)

are the second-order local response with respect to the external field.The localized hyperpolarizability can then be written as

βAB(ω1, ω2) = −∑l∈Am∈B

~rlmδ2Dlm(ω1, ω2) + ∆2QAB(ω1, ω2)(~RA − ~RB)

(3.39)

3.6 The Applequist Equations

The first order Applequist procedure is presented, followed by thesecond-order extension.

3.6.1 First order

In the point-dipole model, each particle pi ∈ [p1,p2, . . . ,pN ], willinduce its dipole-moment due to the external field for the entire

25

CHAPTER 3. THEORY

collection of N particles, and simultaneously the dipole field of allother induced dipoles. For particle i the induced dipole moment tothe first order is

pi = p0i +αi · Ei (3.40)

For external fields much longer than the size of the system it can beassumed to be uniform for all positions ri.

Ei = Fi +∑j 6=i

Tij · pj (3.41)

where Tij is the dipole field coupling tensor.

The above leads to a system of equations, that when solved, givesthe relay matrix

Rij = (δij1−αiTij)−1 ·αj (3.42)

Which couples the change in the local dipole moment at site i withrespect to the external field change at site j

Rij =δpiδFj

(3.43)

Per definition, the polarizability tensor is the change of the totaldipole moment with respect to an external field component, whichmeans that summing up the relay matrix gives the total linear re-sponse

αm =∑ij

Rij (3.44)

3.6.2 Second order

By including the local hyperpolarizability tensor βi, the local dipolemoments can be written

pi = p0i +αi · Ei +

1

2βi : EiEi (3.45)

26

3.6. THE APPLEQUIST EQUATIONS

taking the differential

δpi = αi · δEi + βi : EiδEi (3.46)

leads to a system of linear equations analogous to the first-orderscenario, where the relay tensor is modified with a tilde ( )

Rij = (1δij − αi ·Tij)−1 · αj (3.47)

where α is the induced local polarizability due to the external staticfield

αi = αi + βi · E0i (3.48)

giving a new total linear polarizability due to the local hyperpolar-izabilities

For the total hyperpolarizability, we differentiate the point-dipolesto the second order

δ2pi = αi · δ2Ei + βi : δEiδEi (3.49)

and look for the terms which are linear with respect to the externalfield δ2F = 0

δ2Ei =∑j 6=i

Tij · δ2pj (3.50)

which sets up the equations for the second-order dipole shifts

(δij1− αiTij) · δ2pj = βi : δEiδEi (3.51)

The first order shifts in the local fields are obtained from the first-order response

δEi = (1 +∑jk

Tij · Rjk) · δF (3.52)

inserting Eq. (3.52) into Eq. (3.51) gives the final hyperpolarizability

βm =∑i

δ2piδF2

=∑ij

Rij · βj : (1 +∑kl

Tjk · Rkl)(1 +∑kl

Tjk · Rkl)(3.53)

27

CHAPTER 3. THEORY

3.7 Tholes damping model

A non-neutral charge distribution may be considered a point chargeat large distances. At very short range, the point approximation ofa charge distribution is not physical, and thus the potential causedby an ideal monopole will not be accurate.

The method proposed by Thole was to consider charges being distri-butions rather than idealized points. This lead to the introductionof a parametric damping coefficient, which would only depend on theinter-atomic distances and the isotropic polarizabilities of two point-dipoles. Thole tested various charge distributions of atoms, whichyielded various screening factors, and the most successful model ob-tained was the exponential model

u =rij

(αisoi α

isoj )

1/6(3.54)

with the Gaussian exponential damping with an arbitrary dampingparameter a

ρ(u) =a3

8πe−au (3.55)

the damping factors for the potential (fV ), electric-field (fE), andthe gradient of the field (fT ) are

v = au (3.56)

fV = 1−(

1

2v + 1

)e−v (3.57)

fE = fV −(

1

2v2 +

1

2v

)e−v (3.58)

fT = fE −1

6v3e−v (3.59)

These damping factors come in the expression of the dyadic tensor,and the static field created by all the point monopoles and dipoleswhich gives the induced polarizability in Eq (3.48).

28

3.8. NON-LINEAR OPTICS

3.8 Non-linear optics

In the molecular frame, the total dipole moment of a molecule in-teracting with an external field is Taylor expanded as

µind = µ0 +α ·E +1

2!β·E2 +

1

3!γ ·E3 + . . . (3.60)

where α is the molecular polarizability, and β, γ are the first, andsecond hyperpolarizabilities, respectively.

When comparing the theoretical α, β, and γ properties to exper-iments, there is a need to relate the molecular frame to the bulkproperties in the experimental frame.

In the bulk phase, one models the light-matter interaction usingMaxwell’s-equations, where the equivalent to Eq. (3.60) is the in-duced polarization density. The induced polarization density in amaterial can be expanded in powers of the applied external field,

Pind

εo= χ(1) · E + χ(2) · E2 + χ(3) · E3 + . . . (3.61)

where χ(n) is the electric susceptibility of order n, ε0 is the permit-tivity of vacuum, and P ind is the average induced dipole momentper volume

P ind =

⟨pind

⟩V

(3.62)

The terms n > 1 will manifest in phenomena which depend on theexternal field intensity to quadratic, cubic e.t.c. order. For weaklasers fields, these terms become negligible, with only the first termcontributing to the induced polarization density. The terms χ(n)

in Eq. (3.61) are furthermore frequency dependent. For the linearterm we can write

χ(1) = χ(1)(ω) (3.63)

For the higher order terms n > 1, a mixing of the external fieldswill result in different macroscopic phenomena. The notation todescribe wave-mixing phenomena is

χ(n) = χ(n)(−ωs;ω1, ω2, . . . ωn) (3.64)

29

CHAPTER 3. THEORY

where ωs is the total sum of individual frequencies

ωs = ωi (3.65)

which can individually be positive or negative, depending on thephenomena. For SHG, the outgoing photon of frequency ωs willthus be a sum of 2 photons with frequency ω

χSHG = χ(2)(−2ω;ω, ω) (3.66)

Hyper-Rayleigh scattering is an other phenomenon which could bedescribed by the second susceptibility as noted in Eq. (3.66).

30

Chapter 4

Applications

The main applications of the work in this thesis relates to propertiesof chemical systems. In this chapter, different aspects of the appli-cations are presented, alongside summaries of the attached papers.

4.1 The Green Fluorescent Protein

The green fluorescent protein, GFP, was first extracted from thejellyfish Aequorea victoria 44, and has become an important proteinin bio-imaging technology. The usage of GFP has found variousapplications such as Ca2+-probes45, As-probes46, intra-cellular pHprobes47,48, and even for use in visualization of cancer49.

The main interesting component of GFP is the photo-active site lo-cated in the middle of its barrel-like structure. It is a chromophoreconsisting of 3 amino acids which have internally converted to asingle residue. A visual representation of the protein with the chro-mophore is shown in Figure 4.1.

Due to its light-absorbing and light-emitting properties, the greenfluorescent protein (GFP) has also become an interesting systemfor computational challenges and investigations. As different muta-tional species of the protein will yield vastly different fluorescent col-ors, it is important to include the environment of the chromophorein the computational modeling of the protein in order to fully sim-

31

CHAPTER 4. APPLICATIONS

Figure 4.1: The Green Fluorescent Protein

ulate the complex nature of the light-absorbing properties of thechromophore.

Among the first applications of the polarizable embedding scheme3

for QM/MM, was the calculation of the OPA of enhanced GFP50,and how the absorption properties where affected by the inclusionof polarizable environments. It was found that the gas-phase val-ues of the OPA were red-shifted when the polarizable embeddingwas included. Further work on other GFP mutations gave the samefindings51, while also incorporating the molecular dynamics to fullysimulate the conformational dependence of the absorption proper-ties. The force-fields for these calculations were derived using theoriginal LoProp approach implemented in the MOLCAS52,53 pack-age.

A phenomenon of mutated GFP models denoted as dual-emission(deGFP)54,55, was observed where the OPA, and fluorescence, wasstrongly pH-dependent. For two of the forms, deGFP1/deGFP4,the absorption spectra recorded a decrease in the 400/400 nm ab-sorption peak, with the simultaneous increase in the 504/509 nmpeak when going from pH=6.96 to pH=9.02. The stability of theseprotein mutations were found to be very high in a broad pH-range.

32

4.2. FREQUENCY DEPENDENT FORCE-FIELDS

In Paper I, we choose the pdb structure 1JBY of the deGFP1 pro-tein54, and use MD to sample the configuration space dynamics atfinite temperature, and calculate the full absorption spectrum forthe different protonation states of the chromophore. Our implemen-tation of LoProp, as described in Section 3.5, was used in order toconstruct the polarizable atomic force-field via a modified MFCCprocedure, see supporting information for Paper I.

By using various levels of the force-field inclusion in the MM region,we compare the effect of the external MM perturbation to the OPAof the various chromophore states, as to elucidate each hydrogenatedstates contribution to the full experimental spectrum.

Although similar applications to the GFP protein have been per-formed51, here we study the direct pH-dependence of the OPA-spectrum observed experimentally, and whether it could be ex-plained by the possible protonation states of the chromophore.

We found that the gas-phase values would predict the zwitter-ionicstate going to the neutral state as being the mechanism for thechange in the absorption spectrum w.r.t. the change in pH. However,by inclusion of molecular dynamics, charge-only, and charge pluspolarization with the protein only, the trend reversed and pointedto the neutral form going to the anionic for high pH to be the mainmechanism. This is illustrated in Figure 4.2.

Moreover, it was found that the water transport around the differentchromophore protonation states would be different for each proto-nation state, which could also contribute to the significant change inabsorption spectrum between the different force-field models tested.

4.2 Frequency dependent force-fields

The water molecule, arguably being the most vital molecule to life56

and engineering, has been extensively studied throughout the years.Despite this, many fundamental properties of water are not wellunderstood. The unique hydrogen-bonding network of interactingwater molecules can have many different patterns of bonding, whichcan for instance lead to various different water phases in different

33


Figure 4.2: (a) Absorption spectra from dynamical sampling usingno force field, (b) super-molecular calculations for the chromophoreand water molecules included up to 10A from the chromophore cen-ter of mass, (c) using ff03 force field with only charges and (d, e,f) using our polarizable force field with the MM-2A, MM-2B andMM-2 method, respectively.

34

4.2. FREQUENCY DEPENDENT FORCE-FIELDS

0.000 0.043 0.091 0.114

Frequency ω used for the MM force fields

190

200

210

185.0

187.5

192.5

195.0

197.5

202.5

205.0

207.5α

( ω) i

nat

omic

units[A.U.]

QMMM mean α(ω) as a functionof the frequency dependent MM region

QMMMfrequency

0.000

0.043

0.091

0.114

Figure 4.3: The average polarizability of the TRP residue at dif-ferent frequencies. The x-axis corresponds to MM-properties calcu-lated at the corresponding frequencies.

thermodynamical conditions57,58.

Being a very well studied system, alongside all of its importancein the modeling of biological systems, the water molecule is oftenused as starting ground for benchmarking new methods and force-fields. We chose to study the frequency dispersion of the molecularpolarizability for clusters of water molecules.

In Paper II, we use the water model system to elucidate the variouscontributions to its dynamical polarizability, both in the pure semi-classical Applequist picture and quantum mechanically. We alsoinvestigate the frequency-dependent polarizability using QM/MM,and how the frequency dependency in the MM region affects theproperties in the QM area of a biological system.

We show that the inclusion of frequency-dependent properties in theMM-region, for biological system, can affect the dynamical polariz-ability by around 5%, seen in Figure 4.3.

35


4.3 Hyperpolarizability of water

Our second interest in water comes from experiments measuringthe EFISHG signal, and specifically the anomalously varying signalwith respect to liquid and gas-phase. The first experimental valuefor the first hyperpolarizability of liquid phase water was measuredby Levine et al.59 and estimated to β|| (−2ω;ω, ω) = 8.3±1.2 ·10−32

esu. Later, both Ward et al.60 and Kaatz et al.61 observed a negativevalue of β|| (−2ω;ω, ω) = −9.5±0.8·10−32 esu and = −8.3±0.8·10−32

esu, respectively, for water in gas phase.

This elusive behavior of the sign change of β|| would later be con-firmed for small water clusters using, at that time, high level oftheory calculations62,63. We choose to apply the first calculations towater as a starting point for LoProp hyperpolarizabilities and thequadratic Applequist formalism for this reason.

In Paper III, we implement the quadratic Applequist formalism fromSection 3.5.2, using the localized hyperpolarizabilities, and test iton the water system. After finding the largest source of errors inthe model applied to the water system, we perform calculations ofthe larger 500 water containing cluster. We found that the possi-ble source for the β|| sign change could be the lack of long-rangeinteractions. This is illustrated in Figure 4.4.

4.4 Rayleigh-scattering of aerosol par-

ticles

The light-matter interactions that occur in the atmosphere has awide impact on the earths climate. First, the ionosphere absorbsthe ultra-high energy photons in the x-ray region. The sunlightin the high UV-region that reaches the upper troposphere is thenmostly absorbed by the photo-chemical ozone reaction. The rest ofthe electromagnetic radiation, which is in the low UV-region andlower in frequency will continue until it hits a light-absorbing or-ganism or some surface of the earth. For the light not absorbed byany particles, a lot of it is reflected back into space, and the rest

36

4.4. RAYLEIGH-SCATTERING OF AEROSOL PARTICLES

Figure 4.4: Property convergence with respect to how long-rangeinteractions are included for each point-dipole.

refracted. The refraction of light is described by different theories,depending on the size of the particles involved.

For small particles, Rayleigh-scattering theory is used. This theorypredicts that light with short wavelengths will be scattered morethan light with longer wavelengths. Since blue light is scatteredmore than red light, the sky will have a blue tone at day, and a redtone at dusk.

In Paper IV, the Rayleigh scattering is calculated for water clusterscontaining adsorbed cis-pinonic acid molecules on its surface. TheApplequist equations are applied to LoProp dynamic polarizabilitiesof water clusters up to 1000 waters, as a function of a growingdroplet. For a full cluster of 1000 water molecules, the scatteringis calculated as a function of adsorbed cis-pinonic acid molecules.The contribution to the scattering intensity for the water systemonly, at different external fields, can be seen in Figure 4.5.

The main conclusion was that the isotropic component of α gives themost dominant contribution to the scattering intensity for pure wa-ter clusters, and that weakly adsorbed surfactant molecules will giverise to less scattering, compared to strongly adsorbed ones. This isdue to the lack of anisotropicity in the calculated polarizability of

37


Figure 4.5: The relative increase in scattering intensity from theApplequist equations for a growing droplet with different amountsof water molecules in the cluster.

the water/cis-pinonic acid system.

4.5 Hyperpolarizability of Collagen

The collagen protein exists in many forms, and its main purposeis to act as a stabilizing network in the body, holding up muscletissue, bones, and cells. The protein structure of collagen can con-sist of different compositions, but the general structure, depictedin Figure 4.6, is the triple-helix, formed by a repeating trimer unitconsisting of the amino acids glycine, proline, and hydroxy-proline.Different types of collagen also have other acids included in one ofthe three chains. Bulky groups cause less tighter packing of thetriple-helix, leading to various levels of tensile strength, while thesmaller glycine and proline groups lead to a tighter packing of thetriple-helix chains.

The quadratic response properties of collagen is important as theconcentration of collagen in various places in the human body can

38

4.5. HYPERPOLARIZABILITY OF COLLAGEN

serve as a probe for medicinal diagnostics, and specifically, the di-agnosis of heart-disease64,65.

The collagen has been shown to exhibit a large β(−2ω;ω, ω) re-sponse in SHG experiments66. The explanation to this large re-sponse is the collective directional stacking of the peptide bonds,which builds up the large non-linear response. This was elucidatedwith the help of a bottom-up approach67, and later also with calcu-lations using the ONIOM method68.

In Paper V, we apply the developed theory to the calculation of theHRS intensity of collagen.

In modeling the hyper-Rayleigh scattering intensity,

βHRS =√< β2

XXZ > + < β2ZZZ > (4.1)

we use the rat-tail69 collagen, for which structural X-ray data isavailable. The trimer sequence is the repeating (PPG) unit. Thecrystal structure provides three chains, with ten repeating trimerunits in each. By using the MFCC procedure, we calculate theLoProp β for each individual amino acid. Then the inter- and intra-chain hyperpolarizability is obtained from the quadratic Applequistequations to give the total collagen β.

To avoid over-polarization between atoms and bonds in close prox-imity between neighbor residues we furthermore implement Tholesdamping equations with the standard damping parameter27,28.

Figure 4.7 shows how the static hyper-Rayleigh scattering intensitygrows sigmoidally with the inclusion of more and more residues fromthe N-terminal to the full crystal structure. For the most natural de-composition of properties into atoms, and using the Thole damping,we obtain a scattering close to the experimental value for all quan-tum chemical methods. However, the depolarization ratio for thefull collagen model is slightly inconsistent with the measurementsperformed by Tuer et al66 at 7.12, 5.13, 6.30, for the TDHF, TD-B3LYP, and TD-CAMB3LYP, respectively, with the experimentalvalue being 8.4.

39


Figure 4.6: Collagen triple-helix visualized with large sticks repre-senting bonds.

Figure 4.7: The total static scattering intensity βHRS for a grow-ing collagen triple helix. For 29 residues, the rat-tail collagen isobtained.

40

4.6. C6 COEFFICIENTS FROM COMPLEX POLARIZABILITIES

4.6 C6 coefficients from complex polar-

izabilities

The forces which govern intermolecular attraction and repulsion areoften partitioned into different terms. This is convenient as approx-imating many-body interactions is quite beneficial when computingthe forces acting on atoms in large systems such as proteins.

The attractive forces are of different origin. At close intermolecu-lar distances, the electrostatics govern the energetics, which givesrise to ionic and to some degree hydrogen bonding. The groundstate multipole moment of a molecule will also lead to dipole-dipoleinteractions in systems such as water.

The second attractive force occurs due to the polarizability of moleculesand atoms, which gives induced dipoles which attract each other.This is called the induction, or polarization, energy. The inductionenergy will depend on the ground state static polarizability of amolecule.

The dispersion comes in when considering larger distances, or atomsand molecules which have zero ground state multipole moment. Thedispersion is responsible for the bonding of noble-gases at low tem-peratures. The dispersion attraction, also commonly denoted asvan der Waals attraction, can be computed by the infinite integralof the polarizabilities of two molecules over all imaginary frequen-cies. Whereas the electrostatic and polarization attraction wears offat long distances, the dispersion attraction is always non-zero, andcan be quite large if many particles are considered, in systems suchas proteins. Thus, even if the individual two-body dispersion termsare small, they add up and their accuracy and parametrization areimportant to consider in force-fields.

In Paper VI, we compute the LoProp polarizabilities from the com-plex polarization propagator code implemented in Dalton, and com-pare both the anisotropic and isotropic contributions to the net dis-persion energy between different dimer systems of molecules. Asexperimental measurements of this energy is extremely difficult, wecompare our results to the high-level SAPT2(HF) / SAPT(DFT)method, which serves as a good reference method.

41


We find that the anisotropic formula for the LoProp dipole-dipolepolarizability is in good agreement with the reference SAPT2(HF)method for the H2 and N2 dimer systems overall. For the ben-zene dimer system, all models underestimate the dispersion of theT-shaped configuration of the molecules. The anisotropic LoPropdispersion removes this descrepancy to some degree. One way totest the validity of the presented models would be to incorporatethe quadrupolar-quadrupolar and dipole-octupolar tensor operatorsin the CPP formulation and calculate the LoProp contributions forthis benzene configuration.

4.7 Calculations of X-ray photoelectron

spectra of polymers

As a final example of applications of our locally decomposed forcefields we carried out quantum mechanical molecular mechanical(QM/MM) calculations of X-ray photoelectron spectra (XPS), wherethe LoProp force fields, together with the MFCC capping procedure,was used to construct the MM part.

Here the the X-ray induced core-ionization occurs in a specific atom,which together with its nearest surrounding is described by quan-tum chemistry electronic structure theory (Hartree-Fock or DensityFunctional Theory), and where the outer environment is describedby atoms equipped with the LoProp force-fields, that is local chargesand polarizabilities. We believe that this approach is the most gen-eral to date for studying XPS spectra of polymers as it includes shortas well as long range effects, and, in view of the original XPS mod-els, both initial and final state effects. The application on polymerfibers is also a new aspect for QM/MM on XPS spectra — previousapplications have concerned other aggregates like liquid solutionsand surface adsorbates.

The polymethyl methacrylate (PMMA) polymer was chosen as demon-stration example, partly because XPS spectra of this polymer hasbeen used in practical applications such as in e.g. artificial eye lenses,in which case it is desirable to predict a pure XPS spectrum in or-

42

4.7. CALCULATIONS OF X-RAY PHOTOELECTRON SPECTRAOF POLYMERS

Figure 4.8: Model of PMMA from different views.

der to identify unwanted contamination. A goal of the study was tostudy how well the shape of the polymer PMMA spectrum can bereproduced using a long oligomer sequence as an approximation forthe structure, and how this shape is changed with oligomer length.An illustration of the model can be seen in Figure 4.8.

The study assumed high-energy X-ray photon energies, for whichcore-electron ionization cross sections are equal for each element,meaning that the full spectrum is determined totally by the chemialshifts of the binding energy of each core site. It was found that pureQM calculations (DeltaSCF method) of the monomer can have somepredictive power, but that the long range interactions must be ac-counted for the full polymer XPS shifts, which range approximatelyto an eV towards lower binding energy.

The polymer shifts vary somewhat with respect to the position of theatom in the monomer and show a monotonous decrease with respectto the number of monomer units, making it easy to extrapolate asmall remaining part to infinity. For the tested PMMA polymerthe order of the shifts are maintained and even the relative sizeof the differential shifts are kept intact in most cases. The resultsindicate that good first estimates of the polymer spectrum can beobtained by few monomer units. Another useful result obtainedis that the polymeric effect to a large degree is present already atthe frozen orbital approximation since the coupling of the internalrelaxation to the environment was found to be very small (one ortwo tenths of an eV). Although we investigated just one polymer, itstill indicates that substantial computational simplification thereby

43


can be reached.

We used two quantum descriptions, with one or three monomer unitstreated quantum mechnially. It was clear that the former couldgenerate some anomalies in the spectra, which are traced to thecloseness of some QM ionized atoms to the MM interface. The threeQM monomer results reduced, or even removed, these anomalies,giving monotonous trends of BE changes with respect to the numberof monomer units in each case. The results of our study supportsthe utility of the quantum classical QM/MM method for predictingX-ray photoelectron spectra of polymeric systems.

44

Chapter 5

Conclusions and Outlook

The introduction spanned general topics of the work presented inthe thesis. Sought after applications in QM/MM, force field de-velopment, and semi-classical light-matter interaction models werepresented.

In the Theory part, the standard solutions to the Schrodinger equa-tion was presented in the BO-approximation though the use of Slaterdeterminants, resulting in Hartree-Fock theory. The limitations ofHF and a brief discussion of Density Functional Theory was pre-sented. This was followed by response theory, and its implementa-tion for the light-matter interactions of atoms and molecules. Thesewell-known topics were presented mostly in a broad fashion.

Equations to evaluate the perturbed density from the interaction ofa molecule with an external field was presented. Using this theoreti-cal basis, the LoProp method, which is at the core of the thesis, waspresented in its original form, and with the extension for the gen-eral frequency-dependent framework. A summary of the Applequistequations were then presented.

Lastly, a summary of the applications to models of real systemsusing the underlying theories was presented.

All in all, this thesis work was a concoction of established theo-ries from quantum mechanics mixed with molecular mechanics andmethod development of a new semi-classical formalism of the point-dipole model which uses ab initio properties. The work presents

45

CHAPTER 5. CONCLUSIONS AND OUTLOOK

some immediate practical applications for similar systems as thosepresented in the thesis. These include the isotropically averagedpolarizability in the case of the linear response properties, and thehyper-Rayleigh averaged scattering intensity in the case of the hy-perpolarizability.

For the future, the theory presents applications in the evaluationof dispersion coefficients C6 with the aid of complex LoProp polar-izabilities of molecules. This could be important for the accurateparametrization of the dispersion energy in MD force-fields.

The implementation of the frequency-dependent hyperpolarizabilitycan also be constructed, where the dynamic contribution to the totalβ(−2ω;ω, ω) can be calculated.

The applications in this thesis has focused on models representingreal systems. We ultimately do this to learn more about the worlds,be it environmental concerns in the atmosphere, early heart-diseasediagnostics, or academic insight into force-field parametrizations.Quantum mechanics provides us with the tools to learn about fun-damental phenomena in nature. As the accessible computing powerhas reached a point where it is incredibly difficult to scale larger,and full reference calculations using the most accurate quantummechanical methods are still not reachable beyond the smallest ofsystems, I think it is crucial to explore QM/MM and semi-classicalideas such as those presented in this thesis.

It is important to look at the gap between the classical and quan-tum world, and for this reason I believe that this work, or perhapsderivatives of it, can find useful applications in the future.

46

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