Drugs and Light. A computational approach to predict phototoxicity

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Drugs and Light

A computational approach to predict phototoxicity

Neus Aguilera i Porta

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Theoretical Chemistry and Computational Modelling

Drugs and Light

A computational approach to predict phototoxicity

Neus Aguilera-Porta

�. Reviewer Rachel Crespo-Otero

School of Biological and Chemical Sciences Queen Mary University of London

�. Reviewer Daniel Escudero

Department of Chemistry Katholieke Universiteit Leuven

Supervisors

Inés Corral Pérez , Giovanni Grannuci

January, ��

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Neus Aguilera-Porta Drugs and Light

A computational approach to predict phototoxicity

Theoretical Chemistry and Computational Modelling, January, ��

Reviewers: Rachel Crespo-Otero and Daniel Escudero Supervisors: Inés Corral Pérez , Giovanni Grannuci

Universidad Autónoma de Madrid Facultad de Ciencias

Departmento de Química Francisco Tomás y Valiente �

�� Madrid, Spain

GlaxoSmithKline Research and Innovation Computational Toxicology Park Road

SG�� DP Ware, England

Universitá di Pisa Facoltá di Scienze

Dipartimento di Chimica e Chimica Industriale Via Giuseppe Moruzzi, ��

�� Pisa, Italy

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Acknowledgement

In these next lines, I would like to take the chance to thank the academic supervisors (Inés Corral and Giovanni Granucci), the industrial supervisor (Jordi Muñoz-Muriedas) and the Jury Committee members (Mar Reguero, Cristina Trujillo, Daniel Escudero, Pedro Braña Coto y Lara Martínez-Fernández) for their guidance and valuable feedback during this project.

Moreover, I would like to thank the support given by the European Unions Horizon �� research and innovation programme under the Marie Curie Skodowska-Curie grant agreement (N��).

I would also like to thank Rachel Crespo-Otero and Daniel Escudero for being the external reviewers during my thesis defence.

Special acknowledgement is due to all the friends I made during these last three years. Thank you for your help and support.

Moltes Gràcies, Muchas gracias,

Grazie Mille, Thank you very much!

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Abstract

Drugs exposure, either to visible (Vis) or ultraviolet (UV) light, along the pharmaceutical chain is inevitable; from their manufacture until their dispensation or even after administration. Thus, it is crucial to investigate their photophysics and the photoreactive paths activated upon photon absorption for predicting either an eventual loss of potency of the active pharmaceutical products or a possible production of photochemical reactive species.

Photosafety recommendations are outlined in the International Conference on Harmonization (ICH)S�� guidance. TheICHS�� guidance (and the associatedICHM� guidance) suggests the characterization of the UV visible absorption spectrum as the initial assessment because it can obviate any further photosafety evaluation.

First attempts to predict photostability and phototoxicity of drugs in-silico, were based on the calculation of the HOMO-LUMO energy gap.

However, the predictive power of this indicator is limited for certain classes of drugs, such as for instance,Non-Steroidal Anti-Inflammatory Drugs (NSAIDs)and therefore they are sometimes complemented with indicators of light absorption intensity, i.e. the molar extinction coe�cient.

As a first approach to predict photostability, we have modelled the absorption spectra of a selected group ofNSAIDs, such as aspirin (in gas phase as well as in solvent) and ibuprofen, indomethacin, carprofen and suprofen (in gas phase).

Multistate second order perturbation theory on state average complete active space self-consistent field wavefunctions MS-CASPT�//SA-CASSCF and time dependent density functional theory (TD-DFT) were the computational protocols for this purpose.

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simulations, performed with the surface-hopping algorithm incorporating spin orbit coupling.

Higher chances of producing phototoxic species are expected the longer the drug remains excited, so the keys behind photostability of a drug is the accessibility of S0, although reactive processes may also take place from a hot ground state.

Therefore, our goal with these results is to generate a model that would allow us to predict photostability with the information of the di�erent deactivation mechanisms and new indicators and at the same time, improve the assessment of the photophysical properties of drugs.

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Resumen

La exposición de los medicamentos, ya sea a la luz visible (Vis) o ultravioleta (UV), a lo largo de la cadena de producción farmacéutica es inevitable. Desde su fabricación hasta su dispensación o incluso después de su administración. Por lo tanto, es crucial investigar su fotofísica y las rutas fotorreactivas activas en la absorción de fotones para predecir una pérdida eventual de potencia de los productos farmacéuticos activos o una posible producción de especies fotoquímicas reactivas.

Las recomendaciones de seguridad de la fotoseguridad se describen en la guíaICH S��. La guíaICHS�� (y la guíaICHM� asociada) sugiere la caracterización del espectro de absorción visible UV como evaluación inicial porque puede obviar cualquier otra evaluación de fotoestabilidad.

Los primeros intentos de predecir la fotoestabilidad y la fototoxicidad de los medicamentos in silico se basaron en el cálculo de la brecha de energía entre HOMO y LUMO. Sin embargo, el poder de predicción de este indicador es limitado para ciertas clases de medicamentos, como por ejemplo, NSAIDs y, por lo tanto, a veces se complementan con indicadores de la intensidad de absorción de la luz, es decir, el coeficiente de extinción molar.

Como primer acercamiento a predecir la fotoestabilidad, hemos modelado los espectros de absorción de un grupo seleccionado de NSAIDs, como la aspirina (en la fase gaseosa, así como en el disolvente) y ibuprofeno, indometacin, carprofeno y suprofeno (en fase gas).

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activo completo MS-CASPT� // SA-CASSCF así como la teoría del funcional de la densidad dependiente del tiempo (TD-DFT).

Exploramos el mecanismo de desactivación fotofísica más probable de las moléculas excitadas con simulaciones de dinámica semi-clásicas, realizadas con el algoritmo de salto de superficie que incorpora acoplamiento de órbita de espín.

Se esperan mayores posibilidades de producir especies fototóxicas cuanto más tiempo permanece excitada la droga, por lo que las claves están en su desactivación. La fotoestabilidad de un medicamento es la accesibilidad que tiene para volver a S0, aunque los procesos reactivos también pueden tener lugar desde un estado fundamental caliente.

Por lo tanto, nuestro objetivo con estos resultados es generar un modelo que nos permita predecir la fotoestabilidad con la información de los distintos mecanismos de desactivación y nuevos indicadores a su vez, permitiéndonos mejorar la evaluación de las propiedades fotofísicas de los fármacos.

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Introduction

�.� Motivation

An Active Pharmaceutical Ingredient (API) can be sensitive to the action of light - from formulation, manufacturing, storage to administration - leading to di�erent adverse events which are likely to reduce its potency or modify its structure as well as its properties and biological e�ects.

"The photostability of a drug is defined as the response of the drug (or drug product) to the exposure to solar radiation in the solid, semisolid or liquid state that leads to physical or chemical change" [�].

Figure �.�: Drug photoinstability e�ects adapted from [�]

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Some consequences of drug photoinstability are illustrated in figure

�.�, where we can schematically see that drug photodegradation leads to the loss of potency of the product [�].

There are also light-induced side e�ects: as reacting with oxygen species or with endogenous substances, which may lead to some unexpected biological effects. For these main reasons, the pharmaceutical industry raised awareness about drug-light interaction to be assessed before anAPIis ready to be commercialised.

�.� Photostability Studies

Human exposure to a drug can be systemically or topically, but in order for an eruption to occur, the drug or its metabolites must be present in the skin at the time of the exposure to the radiation. Additionally, the drug and/or its metabolites must be able to absorb either visible or UV radiation.

Elucidating photochemical and photophysical information of a system will make possible to assess its shelf life, as well as, the e�cacy of the stabilizing agents to overcome the light exposure e�ects [�][�] .

The objective of drug-light interaction evaluation is the intrinsic knowledge of the photostability properties of the compounds. This process involves to define photostability characteristics; physical and chemical changes; photodegradation pathways and mechanisms.

However, adverse drug reporting database largely underreport the incidence of these reactions, particularly for drugs that have been on the market for some time and are already known photosensitizers [�].

In Fig.�.�, the drug product absorbs a photon and reaches its excited state. The extra energy acquired by the drug can be released photochemically; as shown in the two processes that follow: transfering energy to the system; to the cell or to an oxygen molecule present; or generating a free radical, which can further interact withReactive Oxigen Species (ROS); or photobinding to DNA, as well as, to proteins.

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�.�. Photochemical Evaluation

DRUG PRODUCT

EXCITED STATE hν DRUG

ENERGY TRANSFER

FREE RADICAL GENERATION

CELLULAR MEMBRANE

SINGLET OXYGEN

SUPEROXIDE

PHOTOBINDING

OXIDATION

CELLULAR LIPID/PROTEIN

DRUG

DNA

PROTEINS

PHOTOIRRITATION

PHOTOINSTABILITY

PHOTOGENOCITY

PHOTOALLERGY

O2

Figure �.�: Representation of possible pathways for phototoxic responses induced by photosensitive drugs adapted from [�]

So in this case, our quest is understanding light e�ect on the drug, and to be able to predict the key parameters that describe a photostable compound, i.e. a molecule that, upon photoexcitation, photophysically deactivate to the ground state and do not undergo any photochemical transformation.

The knowledge about their photochemical and photophysical behaviour can provide guidance for handling, packaging and labelling of the di�erentAPI, in addition to the production of saver drugs.

�.� Photochemical Evaluation

Once chemicals are newly synthesised, it is needed to examine their photochemical properties to rationalise their phototoxic potential.

For this purpose UV spectral analysis provides information of the photoexcitability of the chemicals but the true photochemistry or reactivity leading to phototoxic events is unclear because, as seen in figure�.�, many processes can occur.

The complexity of the phenomena of excitation has generated some debate around the ontology [�] and the usefulness of photostability data to assess photosafety evaluation [�].

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�.� Pharmaceutical Industry

The historical chronology of events for phototoxicity drug-testing protocols in pharmaceutical industry [��] started for the first time in the mid-��s, the need to establish a regulatory photosafety testing framework both in Europe and United States was disclosed.

In ��; the Organisation for Economic Cooperation and Development (OECD) drafted the first proposal of a guideline, Acute Dermal Photoirritation Screening Test (TGP��)[��] which defined in general terms the use of the rabbit or guinea pig as preferred species for in vivo phototoxicity evaluations.

However, after the comment period, no further work was done with this draft guideline.

In ��, the European Medicine Agency (EMA)issued the Note for Guidance on Photosafety Testing; followed in �� by the United States Food and Drug Administration (FDA)Guidance for Industry, Photosafety Testing. These documents stablished a formal framework under which a photosafety program for pharmaceuticals should be conducted.

Both guidances indicate that photosafety testing is warranted for chemicals when the spectrophotometric absorption is between �� and

�� nm, and the test material is topically applied to the skin, or reaches the eyes or skin following systemic administration.

Under this framework, industry along with regulatory groups began to generate data and assess it in order to determine how well the Guidance-recommended approaches were addressing photosafety concerns.

These reviews soon revealed shortcomings in the approaches that both of these documents presented, and raised concerns about the utility of the recommended approaches to testing and the data generated.

In ��, The Drug Information Association (DIA) Workshop on Photosafety Evaluation of Drugs, brought key stakeholders from industry, regulatory bodies and academia together to discuss the current state of photosafety testing. Basic photochemistry and physics related to preclinical and clinical evaluation of phototoxicology, regulatory photosafety assessment and risk assessment were reviewed.

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�.�. Pharmaceutical Industry

��

OECD��

��EMA

��FDA

��DIA

Concept Paper��

��ICH

Step ��

Step � - ��

Figure �.�: Guidance chronology

The organizers recognized that only one validated assay for phototoxicity (the �T� PT NRU In Vitro �T� NRU Phototoxicity Test assay) existed and while not formally validated, the UV-Local Lymph Node Assay was recognised as validated in distinct test facilities.

The lack of validated assays for other in vitro or in vivo photosafety testing was recognised, and indeed, the lack of real standards on how to perform those assays, the varying endpoints used and ways to interpret the data generated, and the overall lack of clarity and consistency in the performance of the assays. This lack of clarity was compounded by the lack of consistency between the two guidances regarding testing strategies (e.g., the utility of the photocarcinogenesis assay).

In January ��, the EMA adopted a Concept paper on the Need for Revision of the Note for Guidance on Photosafety Testing [��]. The problem statement recognised the substantial shortcomings of the current guideline recommendations and recognised that new data and developments in the field allowed for better-designed guidances.

The recognised deficiencies included:

�. The criteria used were non≠specific and caused the testing of many pharmaceuticals that may not have been necessary

�. The parallel approach to testing was recognised as not allowing testing to be stopped, leading, in theory, to testing for photoallergy and / or photogenotoxicity, even after a negative response in the in vitro phototoxicity test was obtained.

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Oversensitivity (the �T� assay) and “pseudoe�ects” (photogenotoxicity), were recognized as serious problems. The timing of testing in the drug development cycle was not addressed.

In ��, ICH formally issued the Final Concept Paper for S��:

Photosafety Evaluation of Pharmaceuticals, a complete revision of the guidance that was required and endorsed by theICHSteering Committee.

This document formalized, with a Statement of the Perceived Problem, many of the concerns that had arisen in the years since the issuance of the EMA andFDAguidance documents and recognized in theEMAConcept Paper. The issues identified were:

• Criteria of light absorbance and skin exposure to initiate phototoxicity testing should be defined;

• Criteria of tissue levels achieved and/or retained in the skin and eye should be defined;

• A consensus on the triggers for photosafety testing should be developed;

• The value of several in vitro phototoxicity and photoclastogenicity test should be evaluated after examining their correlation with clinical data;

• The value of photogenotoxicity testing should be evaluated.

EMA issued “Questions and Answers on the Note for Guidance on Photosafety Testing”, in parallel withICHe�orts.

In ��, the “Step � ≠ document” was released for public comment and in �� the final “Step � - document” [��] was published and currently is the guidance in use.

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�.�. In-Silico Methodologies

�.� In-Silico Methodologies

This section addresses the contribution of in silico methods (performed on a computer or via computer simulations) to toxicology, as a computational toxicology approach.

It is believed that the interplay in between in vivo ≠in vitro and in silico is critical, but the di�culty resides in the heterogeneous approaches that computations can bring to toxicology.

Non-testing data can be generated in three main groups

�. Grouping approaches, which include read-across and chemical category formation;

�. Both Structure - Activity Relationship (SAR) and Quantitative Structure - Activity Relationship (QSAR)and

�. Expert systems

These very heterogeneous approaches of computer in toxicology have to rationalise the photostability parameters in order to reduce and improve in vitro and in vivo analysis. The exploration of many systems will lead to improved data analysis procedures, as well as, imaging analysis technologies. The prediction models are expected to be able to correctly translate the behaviour of a system from in vitro to in vivo.

Advantages of these methods [��]

Higher throughput Less expensive

Less time consuming

Constant optimisation possible

Higher reproducibility if the same model is used Low synthesis requirements

Have potential to reduce the use of animals Limitations [��] [��]

Quality and transparency of training set experimental data Transparency of the program

(clear understanding of what is being modelled) Descriptors sometimes confusing

Applicability domain sometimes not clear

Absorption, Distribution, Metabolism, and Excretion (ADME) features, specially metabolism not taken into account

Carcinogenicity prediction unapplicable on non-genotoxic compounds

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In silico tools have a bright future in toxicology. They add the objectivity and the tools to appraise our toolbox. They help to combine various approaches in more intelligent ways than a battery of tests [��]

More specifically in photo-toxicity there are only � assessments currently used [��][��]:

�. Deductive Estimation of Risk from Existing Knowledge (DEREK) structure-based photosafety prediction [��] [��]

�. Homo-Lumo Gap (HLG)[��] [��]

�. QSARmodel Structure-based photosafety prediction (see�.�.�)

�.� Computational Background

During years, phototoxicity and photosensibility have been studied using in vivo models. However, the concern of light exposure associated to the modern lifestyle is increasing the number of compounds to be tested in�T� - Neutral Red Uptake (�T�-NRU)in vitro, but we need to take into account that this analysis needs resources: both in terms of cost and time.

Computational methods o�er the advantatge to be cost-e�ective and to provide rapid estimation of biological activities. Therefore, they can reduce the number of chemical systems to be tested in vitro. Several European projects aim at developing integrated test strategies expected to significantly increase the use of non-testing information for regulatory decision-making.

The use of existing knowledge, categories based on mechanistic rationale and/or structural similarity [��], read-across and mathematical models based on structure-activity relationships promotes the use of non-testing information for many diverse toxicological endpoints.

Phototoxic potential was initially predicted with a QSAR model by Giesy [��] onpolycyclic aromatic hydrocarbons (PAH)by investigating structural factors that could be contributing to their toxicity.

Among the first descriptors used were the lowest singlet excitation energy, as well as, the one from the lowest triplet of the excited molecule and the phosphorescence lifetime. Due to the triplet persistency, it was postulated that this excited form of the molecule would be able to react with other species, i.e nearby oxygen (ground state) and create and

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�.�. Computational Background

excited singlet oxygen which would react with biomolecules to produce a toxic e�ect.

�.�.� Quantitative Structure-Activity Relationship

The main objective of aQSARis a direct comparison in between several active compounds and inactive with similar structures and determine the odds and resemblance among them in order to explain their activity, for example in phototoxicity. The general application has been to relate biological activity to the presence of physicochemical features.

QSAR analysis builds knowledge on prediction and design of biologically active compounds. The actual definition is based on Hansch and Fujita [��] which correlates biological activity with physicochemical properties.

Recent research are evolving QSAR analysis in three main blocks:

the development of new descriptors, the search for new methods able to calculate the best correlation and the so-called "reverse-problem", meaning, the development of data basis that will allow to obtain the compound that would better suit the scientist requirements and at the same time accomplish the equation from the analysis.

Setubal principles [��] [��] have been formulated, which state that (Q)SAR should:

�. be associated with a defined endpoint of regulatory importance

�. take the form of an unambiguous algorithm

�. ideally, have a mechanistic basis

�. be accompanied by a definition of domain of applicability

�. be associated with a measure of goodness-of- t, robustness and predictivity

�. be assessed in terms of their predictive power by using data not used in the development of the model.

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�.�.� Non-Steroidal Anti-Inflammatory Drugs

The research in this project will be focused onNSAIDsbecause there is a major failure on predicting their phototoxicity by the actual techniques.

The studiedNSAIDsare aspirin, ibuprofen, carprofen, suprofen, naproxen and indomethacin, from which one can find the molecular structure in Fig.�.�

NSAIDs are a diverse group of compounds that are mainly used to inhibit the physiological responses of pain, inflammation and fever by inhibiting the enzymesCyclooxygenase (COX)� and � - isoforms of each other.NSAIDsare among the most frequently used classes of medications, which, as of ��, represented a market worth around $� billion among prescribed medications in the in the seven major economies (the US, Japan, France, Germany, Italy, Spain and the UK) [��].

They are the second most prescribed class of drugs in Europe today, after antibiotics. While antibiotics are only sold in pharmacies as prescription drugs, NSAIDs can be bought either with a prescription or Over-the-Counter (OTC)in pharmacies and retail shops. Two generations of NSAIDs are available depending on the binding selectivity to COX isoforms. They have side-e�ects that could be counteract by a third generation with a tailored ratio of binding to theCOX-� orCOX-�.

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�.�. Computational Background

O

O O

OH

Aspirin

OH O

N H Cl

Carprofen

OH O O

Naproxen

O OH

Ibuprofen

OH O O

S

Suprofen

OH

O

N O

O Cl

Indomethacin

Figure �.�: Molecular structures for the studied NSAIDs

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Photochemistry

�.� Light absorption phenomena

The interaction of UV-Visible light with a molecule promotes electronic excitation. In many cases, this process can be represented, to a good approximation, as a promotion of an electron from an occupied to a virtual orbital: for example, from theHighest Occupied Molecular Orbital (HOMO)to the Lowest Unoccupied Molecular Orbital (LUMO).

The minimum energy needed to change the state of this molecule follows the Bohr equation, which corresponds to the energy di�erence between the two electronic states.

A0 h‹

≠≠æ¹A (�.�)

h‹ = Ef≠ Ei (�.�)

where Ef is the energy of the final excited state and Ei is the ground state energy.

The radiative transition probability between two states Âⁿand Â^mdepend on theTransition Dipole Moment (TDM)between them:

TDM =

⁄ _Œ

0 nµ _md· (�.�)

and is in particular proportional to the square of TDM. A related

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Chapter �. Photochemistry

dimensionless quantity id the oscillator strength f

f = 2

3 E|TDM|² (�.�)

Allowed transitions have f larger than, say, �.��. But not all the transitions among electronic states are equally permitted. These will be discussed later in this section.

�.�.� The Franck-Condon Principle

The Franck-Condon (FC) principle concerns radiative transitions between electronic states. It assumes that when an electronic transition happens the nuclear geometry remains stationary. This is a non radiative process. For clarity, I would not talk about that in this subsection. This principle relies on theBorn-Oppenheimer (BO) approximation.

The statements of theFCprinciples are (and apply also to radiationless processes);

• The transitions which imply small nuclear geometry and momentum changes are favoured

• The energy must be conserved

S0 S1

Energy

Internuclear Distance FRANCK-CONDON

Figure �.�: FCprinciple representation of di�erent transitions from theGround State (GS)and its associated absorption spectra.

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�.�.� Absorption Spectra

A part from the FC principle there are other complementary rules that govern the obtention of the absorption spectra of every molecule.

�. The spin conservation rule dictates that during an electronic transition, total spin angular momentum must be conserved S = 0.

Then, the only allowed transitions are the ones with the same function (singlet-singlet or triplet-triplet).

�. Emission can usually be observed only from the lowest excited state of any multiplicity This is a consequence of Kasha’s rule [��]

�. El-Sayed’s Rules: In the case of intersystem crossing transitions in between states with the same excited configuration will have the same spatial wavefunction. Belonging to the totally symmetric irreducible representation but the spin-operator which is not totally symmetric will make its value to be zero. So the next selection rules can be written:

1(n, fi^ú)¡³ (fi, fi^ú) & ³(fi, fi^ú)¡¹ (fi, fi^ú) allowed processes

1(n, fi^ú)=³ (n, fi^ú) & ¹(fi, fi^ú)=³(fi, fi^ú) forbidden processes

�.�.� Light Spectral Regions

Before we start, to set frame the energies that are used to excited the drugs in this project.

Since the majority of photochemical reactions occur due to solar light which includes theultraviolet-A (UVA),ultraviolet-B (UVB)andvisible (Vis) range, sinceultraviolet-C (UVC)cannot reach the Earth surface thanks to the ozone layer.

• UVC: �� - �� nm.

• UVB: �� - �� nm.

• UVA: �� - �� nm.

• Vis: �� - �� nm.

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Chapter �. Photochemistry

�.� Deactivation mechanisms

Once the system is electronically excited by the absorption of a photon the excess energy of this electron lying in a higher-energy states must dissipate to reach equilibrium again.

But this deactivation could undergo two di�erent type of processes:

�. Photophysical: which involve in the deactivation through the states of the original molecule back to the ground state

�. Photochemically: which would involve the use of the photon energy to form other species, for example, isomers.

The characterization of the photochemical properties of drug substances and drug formulations is a part of the formulation work and cannot be ignored but the scope of this project is to understand their decay to the ground state and in order to explore a higher number of compounds we focused on photophysical deactivation mechanisms.

�.�.� Photophysical Deactivation mechanisms

In figure�.�we can see the Jablonski diagram [��]. As a simplification and for schematic purposes the electronic states are represented by horizontal lines at di�erent heights in relation to the ground state to indicate the relative energy of the states, Singlet State (S) and Triplet State (T). The vibrational states are in grey.

S1

T1

S2

VR

ISC VR

ISC IC

IC

S0

VR

Absorption Fluorescence Phosphorescence

Figure �.�: Jablonski diagram on the photophysical deactivation mechanisms

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The di�erent procesess depicted are :

• Light absorption: promotion to a higher level of energy A0 h‹

≠≠æ¹A (�.�)

• Fluorescence: transition between states of the same multiplicity with the release of a photon (time scale: 10^≠10≠ 10^≠8 s )

1Aæ A0+ h‹ (�.�)

• Phosphorescence: transition between states of di�erent multiplicity with the release of a photon (time scale: 10^≠6≠ 10^≠1 s )

3Aæ A0+ h‹ (�.�)

• InterSystem Crossing: non-radiative transition between two states of di�erent spin multiplicity. For example:

1Aæ³A (�.�)

• Internal Conversion: non-radiative transition between two states of the same spin multiplicity. For example:

1A2æ¹A1 (�.�)

• Vibronic Relaxation Conversion: non-radiative transition between two states of the same spin multiplicity.

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Chapter �

Computational Methodology

This chapter o�ers an overview of the theoretical background of the methods employed in this project.

�.� Schrödinger equation

The Schrödinger Equation arose from the failure of classical mechanics to describe the motion of systems at atomic scale or smaller. A central concept in this theory is the duality waveparticle of matter and radiation.

H = E ˆ (�.�)

Being the wavefunction of the system and ˆH the non-relativistic Hamiltonian operator of a molecule (Total Energy), which includes the sum of kinetic ˆT and Coulomb potential ˆV energy terms for electrons and nuclei:

H = ˆˆ T + ˆV =≠ 1

2}²m + ˆV (�.�)

The Schrödinger equation �.�for a many-particle system consisting of n electrons with mass m^e and N nuclei with mass mⁱ that move in the three dimensions is:

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H =ˆ C

≠ }² 2me

ÿn i=1

Ò²i

¸ ˚˙ ˝

Tˆe

≠}² 2

ÿN I =1

Ò² mI

¸ ˚˙ ˝

TˆN

≠ ÿn

i

ÿN I

ZIe² 4ﬁÁ0rIi

¸ ˚˙ ˝

VˆeN

+ ÿn i<j

e² 4ﬁÁ0rij

¸ ˚˙ ˝

Vˆee

+ ÿN I <J

ZIZJe² 4ﬁÁ0rIJ

D

¸ ˚˙ ˝

Vˆ_NN

(�.�) where the ˆTeand ˆTNterms represent the kinetic energy of the electrons and nuclei (Laplacian operator Ò² = _ˆx^ˆ²2 + _ˆy^ˆ²2 +_ˆz^ˆ²2).

The ˆVeN term: electron-nucleus attraction energy operator, the ˆV^ee term: electron-electron repulsion energy operator and the ˆVNN term the nucleus-nucleus repulsion energy operator.

However, analytical solutions for the Schrödinger equation can only be obtained for hydrogenic atoms, consisting of bare nuclei of charge Z^e and one electron. Several approximations have been introduced to solve it for other atoms and molecules.

H ( ˛ˆ x1, ˛x2, ..., ˛xN; t) = i}ˆ

ˆt ( ˛x1, ˛x2, ..., ˛xN; t) (�.�)

�.� Born-Oppenheimer approximation

The BOapproximation assumes that the electronic and the nuclear motion in molecules can be separated because nuclei are much heavier than electrons. So, we can consider the electrons in a molecule to be moving in the field of fixed nuclei. This allows the non-relativistic Schrödinger equation to be solved for the electronic and the nuclear part separately. This approximation is in general very reliable for theGSbut less reliable for theExcited State (ES).

Equation �.�can be written including electronic (r) and nuclear (R) coordinates as follows

HˆTot Tot(r, R) = ETot Tot(r, R) (�.�) By this approximation in equation�.�what remains is the electronic Hamiltonian -or Hamiltonian describing the motion of n electrons in the field of M point charges- since the kinetic energy of the nuclei can be neglected and the repulsion between nuclei can be considered constant.

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�.�. Born-Oppenheimer approximation

HˆTot= ˆTe+ ˆTN + ˆVee+ ˆVNN + ˆVNe (�.�)

Hˆe (r, R) = C

≠ }² 2me

ÿn i=1

Ò²i ≠ ÿn

i

ÿN I

ZIe² 4ﬁÁ0rIi+

ÿn i<j

e² 4ﬁÁ0rij

D

(r, R) (�.�)

The Schrödinger equation is written more compactly as

H = ˆˆ TN(R) + ˆTe(r) + ˆVeN(r, R) + ˆVNN(R) + ˆVee(r) (�.�) where R is the set of nuclear coordinates and r is the set of electronic coordinates. If spin-orbit e�ects are important, they can be added through a spin-orbit operator ˆHSO

The total energy provides a potential for nuclear motion. This function constitutes a potential energy surface, this theBOapproximation move on a potential energy surface obtained by solving the electronic problem, excluding the description of the vibration, rotation and translation of the molecule (nuclear problem)

Hˆel= ˆTe(r) + ˆVeN(r, R) + ˆVNN(R) + ˆVee(r) (�.�)

Unfortunately, the ˆVeN(r , R) term prevents separating ˆH into nuclear and electronic parts, which would allow to write the molecular wavefunction as a product of nuclear and electronic terms, (r, R) = (r )‰(R). Now we introduce the Born-Oppenheimer approximation, by which we conclude that this nuclear and electronic separation is approximately correct. The term V^eN (r, R) is large and cannot be neglected; however, we can make the R dependence parametric, so that the total wavefunction is given as (r; R)‰(R). We can fix R, the nuclear configuration, at some value Ra, and solve for the electronic wavefunction (r; Ra), which depends only parametrically on R. If we do this for a range of R, we obtain the potential energy curve along which the nuclei move. Initially, T^N(R) can be neglected since T^N is smaller than T^e. This approximation is in general very reliable for theGSbut less reliable for theES.

First, the equation involving the electrons movement is solved, keeping fixed the nuclei coordinates. The electronic energy is used to build the potential su�ered by the nuclei, find the nuclei energy, and therefore the total final energy.

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From this point, the BO approximation is used to solve the non-relativistic Schrödinger Equation.

�.� Pauli principle

In the electrostatic approximation, the electronic Hamiltonian depends only on the spatial coordinates of the electrons. However, we have to introduce the spin coordinate to properly take into account the symmetry properties of the wavefunction with respect to permutation of identical particles: "A many-electron wavefunction must be antisymmetric with respect to the interchange of the coordinate x (both space and spin) of any two electrons " [��]

The simplest n-electron wavefunction that fulfils this requirement of antisymmetry is anAntysimmetrised Spinorbital Product (ASP)expressed in terms of a set of spinorbitals ( which are assumed to be orthonormal among them).

(x1, ... , xi, ... , xj, ... , xn) =≠ (x1, ... , xi, ... , xj, ... , xn) (�.��)

(x1, ... , xn) = 1 Ôn!

-- -- -- -

Ï₁(x1) ... Ï1(xn) ... ... ...

Ï_n(x1) ... Ïn(xn) -- -- -- -

= 1

Ôn!|Ï1(x1) ... Ïn(xn)| (�.��)

This requirement is also called the antisymmetry principle A spinorbital is a one-electron wavefunction and is the product of a one-electron spatial wavefunction (or orbital) and a one-electron spin function, ‡Ï1(x1) = Ï1(r1)‡1(s1). The spin function can be –(ms= +¹/2)or

—(ms=≠¹/2)

�.� Hartree-Fock theory

The HartreeFock (HF) method is an approximation to solve the electronic Schrödinger equation applying the Variational Principle after invoking theBOapproximation.

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�.�. Hartree-Fock theory

This principle consist in choosing an initial wavefunction dependent on one or several parameters for which the expected value of the energy is the lowest possible.

E =⁺ 0(x1, ... , xn)|ˆHe| 0(x1, ... , xn)^,Ø E0 (�.��) where E⁰ is the varying the spin-orbitals, ⁰; of the full electronic Hamiltonian ˆHe.

If we recall equation �.�; the two-electron contributions are the Coulomb operator, that takes into account the Coulomb repulsion between electrons

ˆJjf(1) =

5⁄ ‰^ú_j(2)‰j(2) r12 d·2

6

f(1) (�.��)

and the exchange operator takes into account quantum corrections to the Coulomb repulsion due to spin correlation which has no classical analogue because it comes from the non-classical antisymmetry principle

Kˆjf(1) =

5⁄ ‰^ú_j(2)f(2) r12 d·2

6

‰_j(1) (�.��)

The two-electron term can be rewritten as the di�erence of the total Coulomb and exchange operators.

E₍₂₎ = 1 2

ÿ

ij

È‰i|ˆJj≠ ˆKj|‰iÍ (�.��)

The final Hartree-Fock equations are written after applying the variational method to optimise the wavefunction [��]

Fˆ|‰jÍ = ‘|‰jÍ (�.��) where ‰j are theHForbitals, ‘j are the orbital energies and ˆF is the Fock operator:

F = hˆ core+^ÿ

j

(ˆJj≠ ˆKj) (�.��)

Being h^core the core-Hamiltonian operator (one-electron operator including kinetic energy and interaction with the nuclei), the second

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term being an e�ective one-electron potential operator called the HF potential, ‹^HF. This way, theHFenergy expression is

E =^ÿ

i

‘i≠1 2

ÿ

ij

È‰i|ˆJj≠ ˆKj|‰iÍ (�.��)

The first term includes all orbital energies but it causes a double counting since it includes all interactions of a particular electron with all other electrons, is then the second term that eliminates this problem.

In order to solve theHFone needs to apply aSelf Consistent Field (SCF) process to obtain the Fock operator: a set of guess spinorbitals is used to calculate the Fock operator, then theHFequations are solved to obatin a new set of improved spinorbitals and a revised Fock operator and so on.

This iterative process is repeated until the convergence criterion is met.

The outcomes of a HF calculation for an n-electron system are an infinite number of optimised spinorbitals and corresponding energies.

These spinorbitals arranged in order of increasing energy, and the n lowest are the occupied and are used to build the Fock operator and the HF wavefunction. The remaining ones are virtual orbitals.

TheHFequations can be solved numerically for atoms and diatomic systems but are too complicated to solve polyatomic systems. Atomic orbitals of many-electron atoms can be used as a starting point for the description of molecular orbitals of many-electron molecules Introducing a basis set transforms the HFequations into the Roothaan equations, denoting the atomic orbital basis functions as „(r), we have the expansion

i(r) =^ÿ

j

cij„j(r) (�.��)

This defines the molecular orbitals using a Linear Combination of Atomic Orbitals (LCAO), leading to

F(r)^ÿ

j

cij„_j(r) = ‘i

ÿ

j

cij„_j(r) (�.��)

or even a simpler representation as matrices

FC = ÁSC (�.��)

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where Á is a diagonal matrix of the orbital energies Ái. This is like an eigenvalue equation except for the overlap matrix S. One performs a transformation of basis to go to an orthogonal basis to make S vanish.

Then it is just a matter of solving an eigenvalue equation. Since F depends on it is own solution (through the orbitals), the process must be done iteratively.

This process can be accelerated by several algorithm which allow faster convergence [��], [��], [��], [��]

�.�.� Hartree-Fock limitations

In the Hartree-Fock theory the electron correlation is not included explicitly since the electron-electron repulsion is treated as each electron moves in an averaged potential generated by the remaining electrons.

Therefore, the Coulomb electron correlation is not taken into account and the expectation value of the energy from Hartree-Fock calculations is overestimated. Although, one could decrease the energy with the use of larger basis set, there is also a computational limit which leads to the Hartree-Fock limit.

The electron correlation is defined as the di�erence between the exact non-relativistic energy of the electronic Schrödinger Equation and the Hartree-Fock limit. (see equation�.��)

Ecorr = Eexact≠ EHF (�.��)

Electron correlation is classified as : dynamical or non dynamical. The dynamical correlation is related to electron movement. Since electrons are all correlated and the movement of an electron can a�ect the others individually, then, a strict mean-field picture would not be su�cient to describe it accurately.

The mono-configurational nature of the HF method is unable to describe excited states. We should apply more than one determinant to recover the nondynamical correlation. These methods are called MulticonfigurationalSCF(MCSCF) methods. Di�erent approaches such as Møller-Plesset (MP) perturbation theory, Configuration Interaction and Coupled-Cluster Theory try to recover the dynamical correlation.

Although, the correlation energy is a relatively small part of the total energy of a system, it can be very crucial to correctly describe photochemical problems.

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�.�.� Multiconfiguration and Multireference Approaches

�.�.� Complete Active Space Self-Consistent Field

One of theMultiConfiguration SelfConsistent Field (MCSCF)methods used in this project is the Complete Active Space SelfConsistent Field (CASSCF) approach. It is based on the optimisation of the expansion coe�cients as well as the molecular spinorbitals to obtain the multiconfigurational wavefunction.

In CASSCF the orbitals are divided in subsets which will give them a di�erent weight in the wavefunction. In the primary space or active orbitals space one can have occupation between � and �. In CASSCF, all the possible configurations, corresponding to di�erent occupations of the active orbitals, are considered. The secondary space is the one belonging to the core orbitals or inactive orbitals with an occupation of �. The third space is the one containing the virtual orbitals with an occupation of �.

The notation for this calculations is CASSCF(N,n) where N electrons are distributed in n orbitals. In practice, anFull Configuration Interaction (FCI) restricted to the active space is performed, considering all the possible determinants resulting from the di�erent available distributions of electrons in the active space. These determinants are included in a MCSCFcalculation in order to recover the non-dynamical correlation.

The number of configurations in the Configuration Interaction (CI) expansion is given by the Weyl formula as a function of the number of active orbitals, electrons and total spin.

NCAS= 2S + 1 n + 1

A n + 1 N/2≠ S

BA n + 1 N/2 + S + 1

B

(�.��)

The limiting factor in CASSCFcalculations is size of the active space due to its dependence on n. However, larger active spaces might be necessary for some systems, in this cases, one can useRestricted Active Space SCF (RASSCF)method to repartitioning the previously selected active space. In theRASSCFmethod the original active space is RAS�

where the fullCIis performed, RAS� is a subspace of the inactive space where a maximum number of holes is allowed per configuration, and RAS� a subsection of the virtual spaces where a maximum number of electrons are allowed, as you can see in figure�.�.

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Active Orbitals (RAS 2) Virtual Orbitals RAS 3

RAS 1 Inactive Orbitals

RASSCF CASSCF

Active Orbitals (Full CI) Virtual Orbitals

Inactive Orbitals

Figure �.�: Representation of a CASSCF space and a RASSCF space The RASSCF with its advance features is able to recover some dynamical correlation. Nevertheless, RASSCFone needs to remember that it is not a complete active orbital space which hinders the convergency by introducing orbital rotation among the three subspaces.

The orbital convergence is performed using super- CI method and a quasi Newton updated can also be used to improve convergence.

In some cases one might face convergency issues or even root flipping.

A solution to this problems is to average the energy of all the states involved

Eaverage = N^≠1^ÿ

i,N

Ei (�.��)

In State Average Complete Active Space Self - Consistent Field (SA-CASSCF)the orbitals are optimized for the number of interest states.

This type of calculations are very interesting when treating systems which states lie close in energy, i.e in regions of the Potential Energy Surface (PES)as avoided crossings or conical intersections. (SECTION???)

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Molecular Dynamics

The main goal of the molecular dynamics study of the excited state deactivation mechanism of chromophores is to get insight on the actual relaxation pathways, the time scales and quantum yields of the di�erent photophysical processes contributing to the decay.

Semiclassical nonadiabatic dynamics simulations based on independent trajectories are proven in many cases to yield quantitative or semiquantitative results for medium / large sized molecular system such as those considered in the present study. Moreover, they allow to treat all the nuclear degrees of freedom, which is especially important in the absence of mechanistic indications on the dynamics.

The purpose of our molecular dynamics is to depict the evolution the population of di�erent electronic excited states following the electronic excitation of the system as a function of time, and also unravel the photophysical behaviour of theNSAIDsdrugs.

�.� Semiempirical Methods

Semiempirical (SE)methods are an alternative to ab initio methods for the treatment of molecules of large size, and/or when a great number of calculations has to be performed, such as in on the fly molecular dynamics simulations. SEmethods use the sameSCF or post-SCF procedures as the ab initio ones, but the atomic integrals are either obtained from

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Chapter �. Molecular Dynamics

simple formulas, or replaced by empirical parameters (hence the name

"semiempirical"), or neglected.

ASEmethod is defined by the following specifications:

�. Valence electrons. In a SE calculation only the valence electrons are explicitly taken into account. Core electrons are treated as a part of the atomic nuclei. Therefore, only basis functions for the valence electrons are needed.

�. The integrals. Three and four centers integrals are neglected. The others are evaluated in a way which depend on the SE method considered, or replaced by empirical parameters.

�. The basis functions. Atomic basis functions are assumed to be orthonormal. These methods use minimal basis sets.

Three levels of integral approximations are used in semiempirical methods: complete neglect of diferential overlap (CNDO), intermediate neglect of diferential overlap (INDO) and neglect of diatomic diferential overlap (NDDO). The modified neglect of diatomic overlap (MNDO) [��]

model is based on the NDDO approximation and it employs a minimal basis of real atomic orbitals for the valence electrons.

For a closed shell system the molecular orbitals are obtained as eigenvectors of the Fock matrix. Here, where ‘ is the diagonal matrix collecting the molecular orbital energies (notice that the overlap matrix do not appears as it coincides with the unit matrix).

FC = C‘ (�.�)

The total energy, Etotis obtained adding the SCF energy EHF and the core repulsion E^coreAB

Etot=¹1 2

ÿ

µ‹

Pµ‹(hµ‹+ Fµ‹)²

¸ ˚˙ ˝

EHF

+^ÿ

A<B

E^core_AB (�.�)

The MNDO model includes the following interactions and parameters:

�. One-center one-electron integrals hµA‹A = Uµ”_µ_A_‹_A≠ ^ÿ

B”=A

ZB(µA‹_A|sBsB) (�.�)

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where Uµis an empirical parameter representing the energy of the AO ‰^µin the atom A. The second term is an approximation of the electrostatic core-electron attraction.

�. Two-center one-electron integrals

hµ‹ = 1

2Sµ‹(—µ+ —‹) (�.�) where S^µ‹ is the overlap between ‰^µand ‰^‹ atomic orbitals and —^µ is an empirical parameter for a given atom.

�. One-center two-electron repulsion integrals (µA‹_A|⁄A‡_A) are represented by five empirical parameters based on s and p orbitals.

�. Two-center two-electron repulsion integrals (µA‹_A|⁄B‡_B) are computed considering a semiempirical parameter called atomic orbital exponent, which depends on the atom and the orbital.

�. Two-center core-core repulsions are E^coreAB = E^coul_AB + E^e↵_AB

E^coul_AB = ZAZB(sAsA|sBsB) (�.�) and E^e↵ABcontains four atomic empirical parameters and represents Pauli exchange repulsion.

The Austin Method � (AM�) [��], which is the SE method used throught this work for electronic structure calculations used in the Molecular Dynamic (MD) simulations, was developed to overcome core-core repulsion problems of NDDO model. Gaussian functions centered at internuclear points are added to obtain a better description of these repulsions. The MNDO model and its standard implementations (AM�, PM�, PM�) have been parametrised with respect to the ground state properties and emptyingHFwavefunctions. Then those methods are able to reproduce ground state properties and their equilibrium geometries.

�.�.� Floating Occupation Number

In the algorithm used in the dynamics simulations, the electronic states are obtained with the FOMO-CI scheme, where a CI calculation (usually of CAS-CI type) is performed using molecular orbitals obtained from an SCF with floating occupation numbers [��] .

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Chapter �. Molecular Dynamics

In particular, the occupation number of each orbital is obtained as the integral of Gaussian function (along the energy axis ‘) centered at the corresponding Fock eigenvalue. For the i-th orbital with ‘i energy

fi(‘) = Ô2 ÔﬁÊe^≠

(‘≠‘i)2

(2Ê2) (�.�)

where Ê is an arbitrary orbital energy width which determines the population distribution around ‘i.

The occupation numbers are computed at eachSCFiteration for each MO.

Oi=

⁄ _‘_F

≠Œfi(‘)d‘ (�.�)

The Fermi level (‘F) is obtained imposing that the sum of the occupation numbers is equal to the total number of electrons.

N =^ÿ

i

⁄ ‘F

≠Œfi(‘)d‘ (�.�)

The floating occupation numbers are assigned to the orbitals of the selected CI space. In this method the SCF energy depends on the Gaussian width Ê, as you can see from Fig. �.�. The advantage of the FOMO-CI approach is that the occupation numbers adopt smoothly to changes in the nuclear geometries ensuring a balanced treatment of degenerate orbitals. The low lying orbitals ‘^F≠ ‘i ∫ Ê will have an occupation of �, while for higher energy virtual orbitals Oi= 0

Active Orbitals (Full CI) Virtual Orbitals

Inactive Orbitals

numberFO Gaussian width

=

FOMO CI SCHEME

Figure �.�: Floating occupation number representation.

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�.� Semiclassical Dynamics

�.�.� Electronic Trajectories

Semiclassical dynamics means that the nuclear motion is treated with the Newton equations (classical trajectories) while the electrons are treated as quantum mechanically and the time dependent Schrödinger equation (TDSE) must be solved.

Let Q(t) be the nuclear trajectory, which is propagated according to the classical equations of motion. The electronic hamiltonian as function of the nuclear coordinates ˆHel(Q) as well as its eigenstates ÂK(q:Q) and eigenvalues E^K(Q) are then implicitly time-dependent:

Hˆel(Q(t))|Â^K(Q(t))Í = E^K(Q(t))|Â^K(Q(t))Í (�.�) In the semiclassical approach, the electronic motion is ruled by a time-dependent schroedinger equation (TDSE) for the electrons only, which is, in atomic units (~=�):

id

dt| (t)Í = ˆHel| (t)Í (�.��) Here (t) is the time-dependent electronic wavefunction which will be expanded as a linear combination of the adiabatic states,

| (t)Í =^ÿ

L

AL(t)e^≠i“^L^(t)|ÂL(Q(t))Í (�.��) where

“_L(t) =

⁄ _t

0 EL(Q(t^Õ))dt^Õ (�.��) The probability to be in state L at time t is PL(t) =|AL(t)|². The time derivative of (t)Í is

ˆ

ˆt| el(t)Í =^ÿ

L

e^≠i“^L^(t)^Ë( ˙AL≠ i“LAL)| LÍ + AL

ÿ

r

ˆ _L ˆQ_rÍ ˙Qr

È (�.��)

Multiplying by È ^K| and substituting into the TimeDependent Schrödinger Equation (TDSE)

Drugs and Light. A computational approach to predict phototoxicity

Drugs and Light

Neus Aguilera i Porta

Drugs and Light

Neus Aguilera-Porta

�. Reviewer Rachel Crespo-Otero

�. Reviewer Daniel Escudero

Supervisors

Acknowledgement

Abstract

Resumen

Contents

Chapter �

Introduction

�.� Motivation

�.� Photostability Studies

�.� Photochemical Evaluation

�.� Pharmaceutical Industry

�.� In-Silico Methodologies

�.� Computational Background

Photochemistry

�.� Light absorption phenomena

�.� Deactivation mechanisms

Chapter �

Computational Methodology

�.� Schrödinger equation

�.� Born-Oppenheimer approximation

�.� Pauli principle

�.� Hartree-Fock theory

Molecular Dynamics

�.� Semiempirical Methods

�.� Semiclassical Dynamics