quantum mechanical / molecular mechanics study of … · for high-lying singlet states above 7 ev....

Quantum Mechanical / Molecular Mechanics Studyof Electronically Excited states and Assessment of

Methods for Calculating Vertical Excitation Energies

Inaugural-Dissertation

Zur Erlangung des Doktorgradesder Mathematisch-Naturwissenschaftlichen Fakultät

der Heinrich-Heine-Universität Düsseldorf

vorgelegt von

Mario Ramos da Silva Junioraus Recife

Mülheim an der Ruhr/Düsseldorf 2011

Aus dem Institut für Theoretische Chemie und Computerchemieder Heinrich-Heine Universität Düsseldorf

Gedruckt mit der Genehmigung derMathematisch-Naturwissenschaftlichen Fakultät derHeinrich-Heine-Universität Düsseldorf

Referent: Univ.-Prof. Dr. Walter ThielKoreferent: Univ.-Prof. Dr. Christel Marian

Tag der mündlichen Prüfung: 09.05.2011

Hiermit versichere ich, die hier vorgelegte Arbeit eingeständig und ohne unerlaub-te Hilfe angefertigt zu haben. Die Dissertation wurde in der vorgelegten oder inähnlicher Form noch bei keiner Institution eingereicht. Ich habe keine erfolglosenPromotionsversuche unternommen.

Düsseldorf, den 16. Mai 2011

Mario Ramos da Silva Junior

v

The present work is a cumulative dissertationaccording to § 4 Sec. 1 from the Promotionsordnung der Mathematisch-

Naturwissenschaftlichen Fakultät der Heinrich-Heine-Universität Düsseldorfin order of granting the grade “Dr. rer. nat.”

from 13. 10. 2008

This doctoral thesis is based on seven original articles either published or sub-mitted in international peer-reviewed journals. It covers two major topics in thefield of computational photochemistry and photobiology. The first part addres-ses the benchmarking of theoretical methods for computing electronically excitedstates, including the derivation of theoretical best estimates for vertical excitationenergies that should be useful for the validation of existing methods and as refe-rence data for the parametrization of new approximate methods. The second partreports an application of QM/MM methodology to electronically excited biomo-lecules, with an investigation of the photoinduced processes in two artificial flavinderivatives inserted into the YtvA protein.

In the following the articles that form this thesis are listed, together with the tasksI performed in every publication:

M. Schreiber, M. R. Silva-Junior , S. P. A. Sauer, W. Thiel “Benchmarks for elec-tronically excited states: CASPT2, CC2, CCSD and CC3”, J. Chem. Phys., 128,pp. 134110/1–25, 2008.I carried out the extensive literature survey of the benchmark molecules and wasinvolved in the assignment of states. In addition, I produced part of the SupportingInformation.

M. R. Silva-Junior , M. Schreiber, S. P. A. Sauer, W. Thiel “Benchmarks for elec-tronically excited states: Time-dependent Density Functional Theory and Densi-ty Functional Theory Based Multireference Configuration Interaction”, J. Chem.Phys., 129, pp. 104103/1–14, 2008.I performed the TD-DFT calculations together with the assignment of the TD-DFT and DFT/MRCI results. I also carried out the statistical evaluations for theexcitation energies and one-electron properties and provided the literature survey.

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S. P. A. Sauer, M. Schreiber, M. R. Silva-Junior , W. Thiel “Benchmarks for elec-tronically excited states: A comparison of noniterative and iterative triples correc-tions — CCSDR(3) versus CC3”, J. Chem. Theoy. Comput., 5, pp. 555–563, 2009.In this paper I supported the assignment of the states and produced some of thefigures.

M. R. Silva-Junior and W. Thiel “Benchmark of electronically excited states forsemiempirical methods: MNDO, AM1, PM3, OM1, OM2, OM3, INDO/S and IN-DO/S2”, J. Chem. Theor. Comput., 6, pp. 1546-1564, 2010.All calculations and statistical evaluations were done by myself.

M. R. Silva-Junior , S. P. A. Sauer, M. Schreiber and W. Thiel “Basis set effectson coupled cluster benchmarks for electronically excited states: CC3, CCSDR(3)and CC2”, Mol. Phys., 108, pp. 453–465, 2010.The CC2 calculations and the assignment of all coupled-cluster results were mymain tasks in this publication. I also provided the statistical results and their dis-cussion.

M. R. Silva-Junior , S. P. A. Sauer, M. Schreiber and W. Thiel “Benchmarks ofelectronically excited states: Basis set effects on CASPT2 results”, J. Chem. Phys.,133, pp. 174318/1-13, 2010.I performed part of the CASPT2 calculations and all statistical evaluations. I alsocontributed to the discussion of the results.

M. R. Silva-Junior , M. Mansurova, W. Gärtner, W. Thiel “Photophysics of structu-rally modified flavin derivatives in the blue light photoreceptor Ytva: A combinedexperimental and theoretical study”, submitted.I carried out all the theoretical calculations and the analysis of the theoretical re-sults.

In addition, during the period of my doctoral studies, the following article hasbeen published, which is not included in this thesis:

S. Salzmann, M. R. Silva-Junior , W. Thiel, C. M. Marian “Influence of the LOVdomain on low-lying excited states of flavin: A combined quantum-mechanicsmolecular-mechanics investigation”, J. Phys. Chem. B, 113, pp. 15610–15618,2009.The support of the QM/MM system setup and of the geometry optimizations weremy main tasks in this contribution.

vii

Abstract

A systematic and extensive benchmark of theoretical methods for computing elec-tronically excited states is presented in the first part of this dissertation. The cho-sen reference set comprised 28 organic molecules and diverse types of singletand triplet valence excited states. Using the same ground-state geometries andthe medium-sized TZVP basis set, MS-CASPT2 and coupled-cluster CC3 cal-culations were carried out and compared with high-level theoretical work in theliterature. Generally there was excellent agreement between the results from the-se ab initio calculations, especially when the weight of singles excitations washigher than 90% which is usually the case for triplet and low-lying singlet exci-tations. On the basis of these own calculation and high-level literature results, aset of theoretical best estimates (TBE) for vertical excitation energies was deri-ved. Subsequently, basis set effects were analyzed both at the MS-CASPT2 andthe coupled cluster level. Basis set extension from TZVP to aug-cc-pVTZ led to arather small and systematic lowering of the computed vertical excitation energies,which prompted a minor update in the recommended TBE values.

The ab initio benchmark data were used to evaluate the performance of me-thods based on density functional theory (DFT) and of semiempirical methods.Time-dependent density functional theory (TD-DFT) gave different accuraciesfor singlet and triplet states; among the three functionals studied, B3LYP yieldedthe lowest mean absolute deviations, whereas BP86 (BHLYP) tended to unde-restimate (overestimate) the reference excitation energies. The DFT/MRCI me-thod performed uniformly well for singlet and triplet states and was the bestamong the tested DFT-based approaches. Among the semiempirical methods, theorthogonalization-corrected approaches (OM1, OM2, and OM3) were ranked asbest, almost comparable in performance with TD-DFT/B3LYP. The most popularground-state semiempirical methods (AM1, PM3, and MNDO) were found notsuitable for the treatment of excited states, since they systematically gave muchtoo low excitation energies. The spectroscopy-parametrized INDO/S method per-formed well for low-lying singlet excited states, but less so for triplet states andfor high-lying singlet states above 7 eV.

In the second part of this dissertation, the photoinduced processes in a blue-lightphotoreceptor with artificial chromophores were studied using combined quan-tum mechanics / molecular mechanics (QM/MM) methods. This work was mo-tivated by experimental findings that the insertion of the chromophores 1- and5-deaza flavin mononucleotide (1DFMN and 5DFMN) into the binding pocket of

viii

the LOV domain of the YtvA protein leads to unexpected and different behavior:1DFMN-YtvA does not undergo any photoinduced reactions, whereas 5DFMN-YtvA forms a photoadduct in a triplet reaction (similar to the parent FMN chro-mophore) followed by photochemical dark state recovery. The QM/MM calculati-ons reproduced the observed spectral shifts of the protein-incorporated 1DFMN-YtvA and 5DFMN-YtvA reasonably well. For 1DFMN-YtvA, they identified aradiationless deactivation channel along a hydrogen transfer pathway from theactive-site cysteine residue to the flavine that might be responsible for the lack ofphotochemical activity. For 5DFMN-YtvA, they provided a comprehensive me-chanistic scenario that is compatible with the available experimental evidence.According to the calculations, photoexcitation of 5DFMN-YtvA into the brightS1 state is followed by relaxation, intersystem crossing to the triplet manifold,hydrogen transfer on the T1 surface via a moderate barrier to yield a biradical, in-tersystem crossing to the ground state, and recombination in the biradical to forman adduct with a S–C4a covalent bond. Photoexcitation of this adduct triggerscleavage of this bond in the S1 state, and the resulting biradical can return to theground state via another hydrogen transfer in the T1 state. Alternative photoad-ducts with a S–C5 bond are found more difficult to generate and do not offer aneasily accessible path for the photochemically induced dark state recovery.

ix

Zusammenfassung

Im ersten Teil dieser Dissertation wird ein systematischer und umfassender Ver-gleichstest theoretischer Methoden zur Berechnung elektronisch angeregter Zu-stände präsentiert. Zunächst wurde ein Vergleichssatz ausgewählt, bestehend aus28 organischen Molekülen mit verschiedenen Typen angeregter Singulett- undTriplett-Zuständen. Unter Verwendung identischer Molekülgeometrien und desmittelgroßen TZVP Basissatzes wurden MS-CASPT2 und Coupled Cluster CC3Rechnungen durchgeführt und deren Ergebnisse mit denen hoch-genauer ab initioRechnungen aus der Literatur verglichen. Dabei wurde generell eine ausgezeich-nete Übereinstimmung zwischen diesen Resultaten gefunden, insbesondere wenndie Beiträge von Einfachanregungen mehr als 90 % betrugen, was normalerweisebei Triplett-Zuständen und bei niedrig liegenden Singulett-Zuständen der Fall ist.Auf der Grundlage der eigenen Rechnungen und einer Analyse der Literaturdatenwurde ein Satz von besten Referenzwerten für die vertikalen Anregungsenergienerstellt. In der Folge wurden Basissatzeffekte auf die MS-CASPT2 und CoupledCluster Ergebnisse untersucht. Die Erweiterung des Basissatzes von TZVP zurgrößeren aug-cc-pVTZ Basis führte zu einer relativ geringen und systematischenErniedrigung der berechneten vertikalen Anregungsenergien, die in einer Anpas-sung der empfohlenen Referenzwerte berücksichtigt wurde.

Die ab initio Benchmarkdaten wurden benutzt, um die Genauigkeit von Metho-den der Dichtefunktionaltheorie (DFT) und von semiempirischen Methoden fürelektronisch angeregte Zustände zu evaluieren. Die zeitabhängige Dichtefunktio-naltheorie (TD-DFT) lieferte unterschiedlich genaue Ergebnisse für Singulett- undTriplett-Zustände; von den drei getesteten Funktionalen ergab B3LYP die klein-sten mittleren absoluten Abweichungen, während mit BP86 (BHLYP) eher zukleine (zu große) Anregungsenergien erhalten wurden. Die DFT/MRCI Metho-de lieferte einheitlich gute Ergebnisse für Singulett- und Triplett-Zustände, mitden insgesamt kleinsten mittleren Abweichungen. Von den untersuchten semiem-pirischen Methoden schnitten die Methoden mit Orthogonalisierungskorrekturen(OM1, OM2 und OM3) am besten ab, fast vergleichbar in der Qualität mit TD-DFT/B3LYP. Die Standardmethoden AM1, PM3 und MNDO erwiesen sich alsnicht geeignet zur Behandlung angeregter Zustände, weil die berechneten An-regungsenergien systematisch viel zu klein waren. Die speziell für spektrosko-pische Rechnungen parametrisierte INDO/S Methode ergab gute Resultate fürniedrig liegende angeregte Singulett-Zustände, allerdings auch größere Fehler fürTriplett-Zustände und hoch liegende Singulett-Zustände oberhalb von 7 eV.

x

In dem zweiten Teil der Dissertation wurden photoinduzierte Prozesse in einemBlaulicht-Photorezeptor mit künstlichen Chromophoren mit kombinierten quan-tenmechanischen / molekülmechanischen (QM/MM) Verfahren untersucht. DieseArbeiten waren motiviert durch experimentelle Befunde, nach denen der Einbauder Chromophore 1- und 5-Deazaflavin-Mononukleotid (1DFMN und 5DFMN) indie Bindungstasche der LOV-Domäne des YtvA-Proteins zu einem unterschied-lichen und unerwarteten Verhalten nach UV-Bestrahlung führt: Bei 1DFMN beob-achtet man keinerlei photochemische Reaktion, während 5DFMN in einer Triplett-Reaktion ein Photoaddukt bildet (ähnlich wie FMN), welches dann photoche-misch wieder in den Dunkelzustand zurückgeführt werden kann. Die QM/MMRechnungen reproduzierten die beobachteten spektralen Verschiebungen beim Ein-bau der Chromophore in das Protein gut. Beim 1DFMN-YtvA gelang es, einenstrahlungslosen Deaktivierungspfad zu identifizieren, der mit einem Wasserstoff-transfer von einem benachbarten Cystein zum Flavin einhergeht und der für diefehlende photochemische Reaktivität verantwortlich sein könnte. Beim 5DFMN-YtvA konnte aufgrund der QM/MM Rechnungen ein detaillierter Mechanismusvorgeschlagen werden, der mit den derzeit bekannten experimentellen Befundenvereinbar ist. Demzufolge relaxiert 5DFMN-YtvA nach Photoanregung zunächstin dem S1 Zustand, gefolgt von einem Intersystem Crossing (ISC) zum Triplett, ei-nem Wasserstofftransfer in dem T1 Zustand unter Bildung eines Biradikals, einemerneuten ISC zum Grundstand und anschließender Rekombination des Biradikals,wobei ein Addukts mit einer kovalenten S–C4a Bindung entsteht. Die Photoan-regung dieses Addukts führt zum Bruch der S–C4a Bindung im S1 Zustand, dasso gebildete Biradikal kann dann über einen erneuten Wasserstofftransfer im T1Zustand zum Dunkelzustand zurückkehren. Alternative Photoaddukte mit einer S–C5 Bindung sind schwieriger zu erzeugen, und es wurde bei ihnen außerdem keinleicht zugänglicher Reaktionsweg zur photochemischen Regenierung des Dunkel-zustands gefunden.

Contents

1 Introduction 11.1 Reference data for electronically excited states . . . . . . . . . . . 1

1.1.1 CASPT2 and coupled-cluster methods . . . . . . . . . . . 11.1.2 TD-DFT and DFT/MRCI . . . . . . . . . . . . . . . . . . 21.1.3 Semiempirical methods . . . . . . . . . . . . . . . . . . . 4

1.2 QM/MM applications . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Photoreceptors . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Modifying the chromophore . . . . . . . . . . . . . . . . 6

2 Methods 92.1 CASSCF / CASPT2 . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Coupled-cluster methods . . . . . . . . . . . . . . . . . . . . . . 152.3 Time-dependent DFT and DFT/MRCI . . . . . . . . . . . . . . . 162.4 Semiempirical methods . . . . . . . . . . . . . . . . . . . . . . . 202.5 QM/MM methods . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Benchmarking electronically excited states 273.1 Initial considerations . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Coupled-cluster and MS-CASPT2 results . . . . . . . . . . . . . 28

3.2.1 Statistical evaluation . . . . . . . . . . . . . . . . . . . . 373.2.2 Oscillator strengths . . . . . . . . . . . . . . . . . . . . . 39

3.3 Assignment of theoretical best estimates . . . . . . . . . . . . . . 43

4 Assessment of TD-DFT and DFT/MRCI methods 454.1 Initial considerations . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Statistical evaluation . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Assessment of semiempirical methods 555.1 Initial considerations . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Statistical evaluations . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2.1 Vertical excitation energies . . . . . . . . . . . . . . . . . 55

xi

xii CONTENTS

5.2.2 One-electron properties . . . . . . . . . . . . . . . . . . . 57

6 Improvement of best estimates 616.1 Effect of basis set extension . . . . . . . . . . . . . . . . . . . . . 61

6.1.1 Coupled-cluster methods . . . . . . . . . . . . . . . . . . 616.1.2 CASPT2 results . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 Theoretical best estimates . . . . . . . . . . . . . . . . . . . . . . 666.3 Assessment of approximate methods . . . . . . . . . . . . . . . . 68

7 QM/MM study of LOV domains 717.1 Structurally modified flavines . . . . . . . . . . . . . . . . . . . . 717.2 1DFMN-YtvA-LOV . . . . . . . . . . . . . . . . . . . . . . . . 727.3 5DFMN-YtvA-LOV . . . . . . . . . . . . . . . . . . . . . . . . 74

8 Conclusions 81

9 Outlook 85

10 Appendix 117

List of Figures

1.1 Benchmark set of molecules considered in this study. . . . . . . . 31.2 Flavin mononucleotide (FMN, left) and its 1-deaza, 1DFMN (cen-

ter) and 5-deaza, 5DFMN (right) derivatives. The ribityl-5’-phosphatechain is shown for all compounds. . . . . . . . . . . . . . . . . . 7

3.1 Basis set dependence of the 11B1u (filled symbols, top) and the13B1u (open symbols, bottom) state of ethene with different meth-ods: CCSD (squares, blue), CC3 (triangles, red), CASPT2 with2,2 (diamonds, green) and 8,20 (circles, orange) active space. . . . 29

3.2 Correlation plots for all calculated singlet excited states: CC/TZVPvs CASPT2/TZVP vertical excitation energies. . . . . . . . . . . 38

3.3 Correlation plots for all dipole allowed oscillator strengths: CASPT2(literature) vs CASPT2/TZVP, CCSD/TZVP vs CASPT2/TZVPand CCSD/TZVP vs CC2. . . . . . . . . . . . . . . . . . . . . . 42

4.1 Correlation plots for all calculated triplet excited states: TD-DFT(BP86, B3LYP, BHLYP) and DFT/MRCI vs theoretical best esti-mates for vertical excitation energies. . . . . . . . . . . . . . . . 48

4.2 Correlation plots of oscillator strengths for all dipole-allowed ex-cited states using earlier CASPT2, MS-CASPT2/TZVP and CC2/TZVPresults as reference data. . . . . . . . . . . . . . . . . . . . . . . 49

5.1 Correlation plots of vertical excitation energies for all states withreference values (theoretical best estimates) under 6 eV consid-ered in this study. . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Correlation plots of oscillator strengths for all dipole-allowed ex-cited states using MS-CASPT2/TZVP results as reference data. . . 60

6.1 Histograms (in %) of the deviations between aug-cc-pVTZ andTZVP vertical excitation energies (eV) of singlet excited states atthe CC3, CCSDR(3), and CC2 levels. . . . . . . . . . . . . . . . 62

xiii

xiv LIST OF FIGURES

6.2 Histograms (%) of the deviations between CASPT2/aug-cc-pVTZand CASPT2/TZVP vertical excitation energies (eV) of singlet(left) and triplet (right) excited states. . . . . . . . . . . . . . . . 66

7.1 Schematic representation of 1DFMN-YtvA-LOV photophysics (left),with details on the geometry of S1/S0 conical intersection (right).Only atoms included in the QM region are shown. . . . . . . . . . 73

7.2 Schematic overview of 5-DFMN-YtvA-LOV photophysics. Onlyatoms included in the QM region are shown. R denotes the rybityl-5’-phosphate chain. QM(DFT/MRCI)/MM relative energies aregiven in kcal/mol (averaged over three snapshots). . . . . . . . . . 77

7.3 Schematic photocycle of adduct 2, a possible side product fromblue-light irradiation, with possible intermediates. QM(DFT/MRCI)/MMrelative energies are given in kcal/mol (averaged over three snap-shots). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

List of Tables

2.1 DFT/MRCI parameters for BH-LYP functional for singlet andtriplet states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Vertical singlet excitation energies ΔE (eV) of all statisticallyevaluated molecules. . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Vertical triplet excitation energies ΔE (eV) of all statistically eval-uated molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Deviations in excitation energies of singlet excited states of all 28benchmark molecules with respect to CASPT2/TZVP. . . . . . . 38

3.4 Deviation in excitation energies of triplet excited states of 19 bench-mark molecules with respect to CASPT2/TZVP. . . . . . . . . . 39

3.5 Oscillator strengths f for allowed vertical singlet transitions in thebenchmark molecules. . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 TD-DFT and DFT/MRCI deviations of vertical excitation ener-gies (eV) for singlet excited states with respect to ab initio refer-ence data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 TD-DFT and DFT/MRCI deviations of vertical excitation ener-gies (in eV) for triplet excited states with respect to ab initio ref-erence data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 TD-DFT and DFT/MRCI deviations of oscillator strengths (inlength representation) of optically allowed states with respect toab initio reference data. . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 TD-DFT and DFT/MRCI deviations of state dipole moments (inD) with respect to ab initio reference data. . . . . . . . . . . . . . 53

5.1 Deviations of vertical excitation energies (in eV) for singlet stateswith respect to theoretical best estimates. . . . . . . . . . . . . . . 56

5.2 Deviations of vertical excitation energies (in eV) for triplet stateswith respect to theoretical best estimates. . . . . . . . . . . . . . . 58

xv

5.3 Deviations of state dipole moments (in D) for the semiempiricalhamiltonians studied with respect to ab initio reference data. . . . 60

6.1 Deviations in excitation energies (eV) of singlet excited states ob-tained with the aug-cc-pVTZ basis set with respect to TZVP results. 62

6.2 Deviations in the vertical excitation energies (eV) of singlet ex-cited states: CC3/TZVP-based estimates with CC2 and CCSDR(3)basis set corrections vs. CC3/aug-cc-pVTZ results. . . . . . . . . 63

6.3 Deviations between the CC2/aug-cc-pVTZ and CC2/TZVP re-sults for the vertical excitation energies (eV) of singlet and tripletexcited states and for the oscillator strengths and dipole moments(D) of singlet excited states: Statistics for all 28 benchmark molecules.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.4 Deviations between the CASPT2/aug-cc-pVTZ and CASPT2/TZVP

results for vertical excitation energies (eV) of singlet and tripletexcited states and for the oscillator strengths and dipole moments(D) of singlet excited states: Statistics for all 28 benchmark molecules. 67

6.5 Deviations in the theoretical best estimates of vertical excitationenergies (eV): TBE-1 versus TBE-2. . . . . . . . . . . . . . . . . 68

6.6 Deviations of vertical excitation energies (eV) for singlet excitedstates from DFT and semiempirical methods with respect to theo-retical best estimates. . . . . . . . . . . . . . . . . . . . . . . . . 69

6.7 Deviations of vertical excitation energies (eV) for triplet excitedstates from DFT and semiempirical methods with respect to theo-retical best estimates. . . . . . . . . . . . . . . . . . . . . . . . . 70

7.1 Relative energies of 5-deaza-DFMN-YtvA-LOV calculated at theQM(DFT/MRCI)/CHARMM level of theory for some selectedsnapshots. Values in kcal/mol. . . . . . . . . . . . . . . . . . . . 79

Chapter 1

Introduction

1.1 Reference data for electronically excited states

Benchmark sets of molecules with reliable reference data are essential tools for thevalidation of existing computational methods and for the parameterization of im-proved approximate methods. For ground-state molecules, such sets are available,for example the G2 and G3 sets [1–3] which collect a large number of small andmedium-sized molecules with accurate experimental thermochemical data. Thesesets have been used extensively to assess the accuracy of quantum-chemical meth-ods in thermochemistry.[3–5]Corresponding representative benchmarks have been missing for electronicallyexcited states. Traditionally, the primary target of theoretical work in this area hasbeen the calculation of vertical excitation energies and oscillator strengths to pre-dict the electronic spectra, both for small molecules and for medium-sized chro-mophores that are important in organic photochemistry. It is hard and often evenimpossible to obtain reliable and accurate experimental reference data for thesetarget properties, due to various well-known reasons: The observed band max-ima do not exactly match the vertical excitation energies, bands are often foundto overlap, spectral assignments may be uncertain, and spectra may be availableonly in solution and not in the gas phase. These problems are most pronouncedfor larger chromophores which are often most interesting with regard to photo-chemistry.

1.1.1 CASPT2 and coupled-cluster methods

Given this situation, one may turn to high-level ab initio theory and generate therequired reference data computationally. Unfortunately, theoretical calculationswith very high accuracy such as full CI are still restricted to systems containing

1

2 CHAPTER 1. INTRODUCTION

up to 4-6 atoms, [6] and are thus prohibitive for most organic chromophores ofinterest. During the past two decades, CASPT2 (complete-active-space second-order perturbation theory) [7–9] has emerged as the standard ab initio method forcalculating excited-state properties of medium-sized molecules, and it has beenwidely applied, e.g., for the assignment of absorption spectra. There are manyexamples in the literature that illustrate the ability of this multireference methodto properly describe all types of excitations. The introduction of the multi-stateCASPT2 (MS-CASPT2) approach [10] has alleviated the problems with intruderstates and improved the description of states of the same symmetry with strongvalence-Rydberg mixing which had been troublesome in the original CASPT2approach.Alternatively, there are methods based on CC (coupled cluster) theory such asCC2, CCSD, CC3, and CCSDT [11–18] which have been proposed more recentlyand have thus been applied less extensively than CASPT2. Recent developmentsfor CC2 such as the use of the resolution of identity [19] as well as a spin-component scaled version [20] make this methodology suitable for applicationson large chromophores. In the literature, there has been some debate about therelative merits of CASPT2 and CC-based methods, but a balanced assessment hasbeen difficult because of technical differences in the published work. Since thecoupled-cluster methods derive the excited-state properties from linear responsetheory, the valence-Rydberg mixing is intrinsically considered and can thus not bedecoupled.After an extensive survey of the literature, a benchmark set of 28 medium-sizedorganic molecules was defined, and the properties of their valence excited states(with emphasis on vertical excitation energies) were computed using MS-CASPT2,CC2, CCSD, and CC3 at a uniform level (MP2/6-31G* ground-state equilibriumgeometries, TZVP basis set, standard CASPT2 conventions). These benchmarkmolecules cover the most important chromophores found in organic photochem-istry (see Figure 1.1). The focus on valence states assures that the chosen setup isappropriate to obtain a realistic description of the character and the properties ofthe excited states studied. Based on a careful comparison of results presented inthis work and published high-level ab initio results from the literature, theoreticalbest estimates of the excitation energies are derived for the benchmark molecules.

1.1.2 TD-DFT and DFT/MRCIThe application of accurate ab initio methods such as MS-CASPT2 and CC3 isstill limited in practice to rather small molecules, and there is clearly a need forreliable approximate methods that can deal with larger systems. Over the pastdecade, time-dependent density functional theory (TD-DFT) [21] has become oneof the most prominent methods for calculating excited states. [22] A major ad-

1.1. REFERENCE DATA FOR ELECTRONICALLY EXCITED STATES 3

N

NNH

N

NH2

N

NH

NH2

O

NH

NH

O

O

NH

NH

O

O

H3C

O

NH2

OHN

N

H2C O H3C

O

NH2H

O

NH2

Ethene E-Butadiene all-E-Hexatriene all-E-Octatetraene Cyclopentadiene

Benzene Naphthalene Furan Pyrrole Pyridine

Formaldehyde Formamide Acetamide Propanamide

Cytosine UracilThymine Adenine

Cyclopropene Norbornadiene

N

HN

ImidazoleN

N

Pyrazine

N

N

Pyrimidine

N

N

Pyridazine

N N

N

s-Triazine

N

N N

N

s-Tetrazine

O

H3C

H3C

Acetone

OO

p-Benzoquinone

Unsaturated Aliphatic Hydrocarbons

Aromatic Hydrocarbons and Heterocycles

Aldehydes, Ketones and Amides

Nucleobases

Figure 1.1: Benchmark set of molecules considered in this study.

vantage of TD-DFT is its low computational cost, [23] roughly comparable withsingle excitation theories based on the Hartree-Fock (HF) ground state such asconfiguration interaction with singles or the random phase approximation. Sincethe commonly used adiabatic approximation represents the response of a moleculein terms of single excitations, TD-DFT is best suited for excited states that aredominated by single excitations. TD-DFT calculations can be easily standardizedand are thus user-friendly (without the need to define an active space or to selectreference configurations).Evaluations of TD-DFT performance have been reported for various density func-tionals. [24–33] Most of these studies have compared TD-DFT vertical excitationenergies either with experimental results or with published CASPT2 calculations.Generally speaking, these evaluations are less comprehensive and less systematicthan desirable.An alternative DFT-based method for electronically excited states arises from theuse of Kohn-Sham (KS) orbitals in a multi-reference configuration interaction(MRCI) framework. [34] In this ansatz, the major part of dynamic electron cor-relation is captured by the KS-DFT treatment, and static correlation effects areintroduced by the MRCI technique which is also flexible enough to properly de-scribe states with double excitation character. The DFT/MRCI configuration state


functions (CSFs) are built from KS rather than HF orbitals. The method hasbeen tested for a number of organic chromophores [34, 35] and applied to thesimulation of gas phase UV/Vis spectra (see for example [36–45]) and CD spec-tra. [46] Although these previous applications indicate a promising performance,DFT/MRCI has not been used as widely as TD-DFT, and further validation isdesirable.A systematic benchmark of TD-DFT with three functionals (BP86, B3LYP andBHLYP) and DFT/MRCI is presented using the set of accurate ab initio referencedata previously established (see above). The same molecular geometries and basissets are employed as in the previous CASPT2 and coupled-cluster calculationsto allow for a direct comparison between the results. Statistical evaluations andgeneral assessments of performance will be given that highlight the merits andshortcomings of each method.

1.1.3 Semiempirical methodsAnother promising and pragmatic approach for large molecules is provided bysemiempirical molecular orbital (MO) methods such as MNDO, [47] AM1, [48]and PM3 [49] coupled with a configuration interaction (CI) procedure. Thesemethods are based on the NDDO (neglect of diatomic differential overlap) inte-gral approximation and have been parametrized against ground-state properties,in particular heats of formation and geometries. They have been widely appliedin computational studies of ground-state processes (for reviews, see refs. [50–52]). Applications to electronically excited states [53] are rare, however, mainlybecause these standard methods normally underestimate their energies strongly,as result of the integral approximations and the ground-state parametrization. Astraightforward remedy for this shortcoming would be a system-specific reparam-eterization [54] for a given application (see for example ref. [55]) which is how-ever cumbersome in practice and also unsatisfactory from a conceptual point ofview.Among the semiempirical approaches designed to describe excited-state processes,the INDO/S [56–58] method has become the standard procedure. In combinationwith CIS (configuration interaction with single excitations), it has been widelyused for organic molecules, [59–61] transition metals [62, 63], and even lan-thanides. [64] The lack of higher excitations in the INDO/S CIS treatment restrictsapplications to states dominated by single excitations. Another limitation is thefocus on vertical processes: by its design, INDO/S targets spectroscopy ratherthan photochemistry, and it is thus not made for the exploration of excited-statepotential energy surfaces (PES).Semiempirical methods using orthogonalization corrections (OM1 [65, 66], OM2[67, 68] and OM3 [69]) offer a better description of virtual orbitals and their

1.2. QM/MM APPLICATIONS 5

energies. [65] For ground state-properties, they are superior in performance tostandard NDDO methods like AM1 and PM3. [5] Despite the fact that they havebeen parametrized with regard to ground-state reference data, they also provideencouraging results in case studies on vertical excitation energies, [68, 70, 71]conical intersections, [72] and semiclassical excited-state dynamics. [73] How-ever, a thorough benchmark on the performance of semiempirical methods forelectronically excited states has been missing. Such a systematic evaluation is re-ported here for the standard NDDO-based semiempirical methods (MNDO, AM1,PM3), the commonly used INDO-based approaches (INDO/S, INDO/S2), andthe orthogonalization-corrected methods (OM1, OM2, OM3). Reference data areagain taken from the previous benchmark work outlined in the Section 1.1.1.The discussion addresses some specific troublesome cases, followed by statisticalevaluations and an outlook with comments that may guide future reparametriza-tions.

1.2 QM/MM applications on electronicallyexcited biomolecules

For very large molecular systems with hundreds of atoms, all the methods outlinedabove become prohibitively expensive. An alternative that has been used progres-sively is the combined Quantum Mechanics / Molecular Mechanics (QM/MM)approach. The system is divided into an inner region, which is treated with aquantum-chemical method as accurately as needed (e.g., ab initio or DFT), andan outer region, which is described by a less accurate classical force field.Recently, Senn and Thiel [74, 75] reviewed the QM/MM methodology and sur-veyed QM/MM applications for many diverse types of biomolecular systems. Thegreat majority (above 90%) of these studies address processes that occur on theground-state potential energy surface (PES). QM/MM investigations of excited-state processes are still relatively rare, although they have become more feasiblein recent years due to the development of analytical excited-state gradients in sev-eral QM codes.

1.2.1 Photoreceptors

Photoreceptors are a special class of biomolecules that undergo excited-state pro-cesses which control biological activity and mediate diverse physiological re-sponses to light irradiation. This class includes xanthopsins, phytochromes, andcryptochromes, BLUF (Blue-Light Using Flavin) proteins and LOV (light-oxygen-


voltage) sensors. [76, 77] The function of these photoreceptors requires the pres-ence of a chromophore, namely 4-hydroxycinnamic acid (in xanthopsins), tetrapyrrolderivatives (in phytochromes), flavin adenine dinucleotide FAD (in cryptochromes,BLUF), and flavin mononucleotide FMN (LOV domains).The phototropins were the first LOV proteins identified in plants. They normallycontain two LOV domains that are bound to a serine/histidine kinase. They con-trol a variety of biological processes such as phototropism, chloroplast reloca-tion and stomatal opening. [78, 79] Based on sequence homology to these plantphototropins, a prokaryotic YtvA protein was found in the bacterium Bacillussubtilis, which contains a single LOV domain, attached to a STAS domain. [80]When this protein is activated by blue-light irradiation, it mediates environmentalstress, [81, 82] in a manner similar to phototropins. However, there is still limitedknowledge about this signalling mechanism which has been the subject of activeresearch in the last decade (for a review see [83]). In order to understand these sys-tems at a molecular level, two main strategies have been followed experimentally:modifications of the protein through site-directed mutagenesis and, more recently,modifications of the chromophore in recombinant systems. Here the focus is onthe latter strategy.

1.2.2 Modifying the chromophoreLOV sensors undergo specific sequences of reactions after blue-light irradiation.In isolated LOV domains (LOV447) irradiation triggers a photocycle, with theformation of a blue-shifted FMN-cysteine C4a-thiol adduct (LOV390) [84] thatslowly reverts to LOV447 in the dark. [76] There is experimental [85, 86] and the-oretical [38, 87–89] evidence that a reactive flavin triplet state plays a fundamentalrole in the adduct formation, at an intermediate step of the mechanism. Dark staterecovery occurs thermally, but rather slowly, with kinetics up to hours in bacterialLOV proteins. [90]Recently experimental methods have been developed to incorporate modified co-factors into photoreceptors [91] and thus to help in the design of recombinantproteins with specially tailored chomophores inside the binding pocket. Particu-larly, two modified flavin derivatives have raised the interest of experimentalistsdue to their exotic photochemistry compared with the parent FMN compound,namely 1- and 5-deaza-flavin mononucleotide (1DFMN and 5DFMN, see Fig-ure 1.2). They have been recently investigated in aqueous solution using spec-troscopic measurements and theoretical calculations. [92, 93] 1DFMN shows astrong bathochromic shift of its first absorption band, and there is no evidenceof a triplet state being formed. By contrast, 5DFMN exhibits a significant hyp-sochromic shift, with formation of triplet states detected by transient absorptionspectroscopy and confirmed by the theoretical calculations. An obvious question

1.2. QM/MM APPLICATIONS 7

is whether the photophysical behavior of such molecules changes when insertedinto a LOV domain.QM/MM calculations of these two modified cofactors incorporated into the LOVdomain of the YtvA protein will be presented and discussed. On this basis acomprehensive mechanism of the photophysics will be outlined for each systemand compared with recent experimental findings.

N

N

NH

N

O

O

O

OH

OHHO

PO OH

OH

N

N

NH

O

O

O

OH

OHHO

PO OH

OH

FMN 1-deaza-FMN

N

NH

N

O

O

O

OH

OHHO

PO OH

OH

5-deaza-FMN

1 2

344a

10a

5

109a

5a

9

67

81

5I II III

CS1

Figure 1.2: Flavin mononucleotide (FMN, left) and its 1-deaza, 1DFMN (center)and 5-deaza, 5DFMN (right) derivatives. The ribityl-5’-phosphate chain is shownfor all compounds.

Chapter 2

Methods

This section presents a brief overview of the methods employed in this study.For more detailed descriptions the reader is advised to consult either the originalpublications or book references where the entire formalism is derived. This sec-tion is divided into five major parts covering the following methods: (a) CASSCF(complete-active-space self-consistent field) and CASPT2 (multiconfigurationalsecond-order perturbation theory); (b) methods derived from coupled-cluster the-ory (CC2, CCSD, CCSDR(3), and CC3); (c) methods based on density functionaltheory (TD-DFT and DFT/MRCI); (d) semiempirical methods, with emphasis onorthogonalization-corrected approaches (OM1, OM2, and OM3); and (e) com-bined quantum mechanics / molecular mechanics (QM/MM).

2.1 CASSCF / CASPT2The description of these methods is derived from ref. [9], with detailed formal-ism found in the references [94, 95]. CASSCF is a variant of the general MC-SCF (multiconfigurational self-consistent field) approach. A multiconfigurationalwavefunction can be written as follows:

Ψ =M∑I

cIΦI = c1Φ1 + c2Φ2 + · · ·+ cMΦM , (2.1)

where cI is the coefficient associated with configuration ΦI and the summationextends over all possible M configurations ΦI , each of which is normally writtenas a linear combination of Slater determinants. This expansion includes only thoseconfigurations which have the same spatial and spin symmetry as the state ofinterest. Such configurations are called configuration state functions (CSFs). Thefinal wavefunction is obtained by optimizing both the CI coefficients cI and themolecular orbitals (MOs) used to generate the electronic configurations ΦI . The

9

10 CHAPTER 2. METHODS

expression for the energy of a multiconfigurational wavefunction (Eq. 2.1) isgiven by:

E =

〈∑I

cIΦI |Ĥ|∑j

cjΦj

〉

=∑tu

∑IJ

cIcJdIJtu htu +

1

2

∑tuvw

∑IJ

cIcJDIJtuvw (tu|vw)

=∑tu

dtuhtu +1

2

∑tuvw

Dtuvw (tu|vw) ,

(2.2)

where Ĥ is the electronic Hamiltonian operator (within the Born-Oppenheimerapproximation), htu and (tu|vw) are one- and two-electron integrals, respectively,and dtu and Dtuvw are the one- and two-electron density matrices, respectively.Usually, the electronic energy is expressed as a function of the variational param-eters of the wavefunction, namely the CI and MO coefficients. In order to optimizethe wavefunction one needs to find the stationary energy point (Eq. 2.2), basedon the variation of the coefficients cI and of the molecular orbitals φt, with theconstraints: ∑

I

c2I = 1, (2.3)

〈φt|φu〉 = δtu, (2.4)which ensure wavefunction normalization (Eq. 2.3) and the orthonormality of theMOs (Eq. 2.4). Two common approaches to optimize the wavefunction are: (a)the use of Lagrange multipliers together with a Taylor expansion of the energy,iterated using the Newton-Raphson procedure, or (b) the application of a super-CIapproach. These methodologies are explained in detail in the references given atthe beginning of this section. Within the MCSCF formalism, many diverse typesof wavefunction can be constructed. In the most popular CASSCF variant, theorbital set (occupied and virtual) is divided into four subsets:

1. Core: Internal orbitals that are assumed to remain unaltered during excita-tion/deexcitation of valence electrons;

2. Inactive: Orbitals that do not participate in these processes, but should stillbe reoptimized to improve the description of the system;

3. Active: Orbitals that are directly involved in the processes mentioned initem 1 and thus must be reoptimized;

2.1. CASSCF / CASPT2 11

4. Virtual (or secondary): Orbitals that are not be occupied and remain un-changed.

With these four subsets defined, the configuration generation with the CASSCFmethod is as follows: The core orbitals are always doubly occupied and main-tained in their original form (e. g., as obtained from an initial standard SCF cal-culation - this is frequently called the frozen-core approach). The inactive orbitalsare also always doubly occupied, but are reoptimized during the CASSCF cal-culation. The active orbitals are optimized both with regard to their form andtheir occupation number, normally starting from initial occupations of zero, oneor two. The virtual (secondary) orbitals remain empty. After choosing the orbitalsthat will be part of each subset, all possible CSFs within the active space are gen-erated (full CI, FCI). The CASSCF wavefunction is the FCI solution built fromthese configurations.One major challenge in CASSCF applications is exactly how to choose these or-bital subsets. Unfortunately there is not a straightforward and unique way to de-fine the active space for a given system. Through the years, however, a consensusapproach has developed, with construction of a standard active space as start-ing point. For organic molecules, for example, the whole π-system (π and π∗

orbitals) is usually considered the minimal active space that should be selectedwhen π → π∗ transitions are studied. The size of the active space is normally themain limiting factor in CASSCF calculations. The number of CAS configurationsis given by the Weyl-Robinson equation [96]:

NCAS =2S + 1

n+ 1

(n+ 1

N/2− S)(

n+ 1

N/2 + S + 1

), (2.5)

where n is the number of active orbitals, N is the number of active electrons, andS is the total spin. In practice, the number of orbitals in the active space is usuallystill limited to 12-15, with a similar or smaller number of electrons. In CASSCFcalculations of excited states, one often encounters a “root flipping”: during op-timization of the wavefunction for a given excited state, the ground-state energygrows gradually and, at a certain point, becomes lower than the energy of the ex-cited state in question. A remedy for this problem is the use of a common set oforbitals for all states during the optimization process, through the so-called state-averaged CASSCF (SA-CASSCF) method. [97, 98] The mean energy value is de-fined as weighted sum over the energies of each state considered in the averaging,and the density matrix for a single state is likewise substituted by mean densitymatrices (see Equation 2.2). One thus obtains a mean set of MOs (state-averagedMOs) and a CASSCF wavefunction for every state included. This method hasmany advantages over the single-state approach: one computes several states ina single job, the wavefunction generally converges faster, and the orthogonality


between the states is ensured which facilitates the calculation of transition prop-erties.The CASSCF method has the intrinsic flexibility to properly describe diverse elec-tronic states of different character. However, it recovers only part of the corre-lation energy arising from the so-called static (non-dynamical) correlation, andtherefore one needs to also account for the remaining dynamical correlation whenaiming at accurate results. One popular method for this purpose is CASPT2 [7–9]which applies second-order perturbation theory to the multiconfigurational zero-order CASSCF wavefunctions, similar in spirit to the second-order Møller-Plesset(MP2) treatment on top of a single-reference Hartree-Fock wavefunction. Thezero-order Hamiltonian for CASPT2 can be written in different ways, the com-monly chosen analogue to Møller-Plesset perturbation theory has the form:

Ĥ0 = P̂0FP̂0 + P̂SDFP̂SD + P̂XFP̂X , (2.6)

with the CASSCF wavefunction as eigenfunction (|Φ0〉). The configuration spaceis divided into different subspaces: 0 (the reference configuration space), SD (allCSFs singly and doubly excited with respect to the CAS space, i. e., the interact-ing configuration space VSD), and X (the space of the remaining configurationsthat do not interact with the reference function). The symbols P̂i represent thecorresponding projection operators, and F denotes the generalized Fock operator.The first-order CASPT2 wavefunction is composed of CAS wavefunctions thatbelong to VSD:

|Ψ1〉 =M∑j=1

Cj|Ψj〉, |Ψj〉 ∈ VSD, (2.7)

|Ψ1〉 =∑p,q,r,s

Cpqrs|pqrs〉 =∑pqrs

EpqErs|Ψ0〉, (2.8)

In the expressions above, Epq denotes an excitation operator, defined as the prod-uct of the creation operator a†p and the annihilation operator aq. The indexesp, q, r, s are general indexes for the orbitals. More specifically, there are three cat-egories: i, j, k, . . . , refer to the inactive orbitals; t, u, v, . . . to the active orbitals;and a, b, c, . . . to the secondary orbitals. VSD represents the first-order interactionspace, and {Cj = 1, . . . ,M} is the solution of the linear equations:

M∑i=1

Cj〈Ψi|F̂ − E0|Ψj〉 = −〈Ψi|H|Ψ0〉. (2.9)

The configurations belonging to the interaction space VSD can be classified intoeight categories, according to the number of electrons created in the virtual space:

2.1. CASSCF / CASPT2 13

• Internal (no electron in the virtual space):EtiEuv|Ψ0〉EtiEuj|Ψ0〉

• Semi-internal (one electron in the virtual space):EatEuv|Ψ0〉EaiEtu|Ψ0〉 and EtiEau|Ψ0〉EtiEaj|Ψ0〉

• External (two electrons in the virtual space):EatEbu|Ψ0〉EaiEbt|Ψ0〉EaiEbj|Ψ0〉

The energy, correct to second order, is given by:

E2 = 〈Ψ0|H|Ψ〉. (2.10)and the normalized wavefunction, correct to first order, is:

|Ψ〉 = √ω|Ψ0〉+√1− ω|Ψ1〉, (2.11)

where ω is the weight of the CAS reference wavefunction and |Ψ1〉 is the first-order normalized wavefunction. This completes the outline of the basic CASPT2method.The extensive use of the CASPT2 method over the years revealed several short-comings that arise from the use of a multiconfigurational wavefunction and a trun-cated active space. Examples of artifacts are the appearance of so-called intruderstates and the strong mixing of valence and Rydberg states. Such problems are ex-plained in more detail in the following, together with strategies to overcome them.Because CASPT2 is a perturbative method, its proper use requires the perturbationto be small. Therefore, it is essential to include into the reference wavefunction Ψ0all configurations which have high coefficients. In other words, when the weightof the reference wavefunction (ω) is small in a CASPT2 calculation, the activespace should be expanded.


When several states of a given symmetry are calculated, it is important that all ofthem have similar values of ω to ensure a balanced treatment of the correlation ef-fects. If any state has a very low value of ω, this indicates problems with “intruderstates”, which belong to the first-order interaction space VSD.There are two ways to treat the problems with an intruder state. If this state inter-acts strongly with the reference wavefunction, the contribution to the energy fromthis state can not be ignored. One remedy is to increase the active space, in or-der to include the orbitals which cause the appearance of intruder states (throughsmall energy denominators in the perturbation treatment). When the intruder stateinteracts only weakly with the other states included in the reference wavefunction(large coefficients, but small energy contributions), the level-shift technique [99]can be applied to remove the intruder state. In this technique, the HamiltonianH0 is substituted by a shifted Hamiltonian (Ĥ0 + �P̂I), where � is a small positivenumber and P̂I is a projector operator to the first-order interaction space. Whenusing this technique, one must apply an a posteriori correction to the second-orderenergy which is now dependent on the parameter �. This is done by testing dif-ferent values of � and comparing the corresponding CASPT2 energies with andwithout the shift (which will be different when an intruder state is present andequal otherwise).Another relevant issue to be considered is the form of the generalized Fock oper-ator. The diagonal matrix elements of the one-electron generalized Fock operatorare related either to ionization potentials (occupied orbitals) or electron affinities(empty orbitals). For orbitals with an occupation number of one, the correspond-ing diagonal Fock matrix element is the average of the ionization potential andthe electron affinity. [100] This implies that the energy denominators in the ex-pression for second-order energy correction can become very small in such cases,causing a systematic underestimation of computed excitation energies. Ghigo et.al.[100] proposed a new definition for the zero-order Hamiltonian, called IPEAshift, which depends on an average shift parameter �. After some calibration withatomic data, they proposed an � value of 0.25 a.u., which has now become thedefault in CASPT2 calculations.Finally, there are cases when the SA-CASSCF wavefunction is not a good startingpoint for the CASPT2 calculation, for instance, when states of the same symme-try and of different character (valence, Rydberg) lie close in energy and contributeto similar extent to the final wavefunction. Dissociation paths near avoid cross-ing regions also suffer from this deficiency. Such problems have largely beenovercome in the multistate CASPT2 (MS-CASPT2) method [10] which is nowan established standard procedure in CASPT2 calculations. It is based on theconstruction of an effective Hamiltonian whose diagonal elements correspond tothe CASPT2 energies, whereas the off-diagonal elements represent the couplingterms between the relevant roots up to second order of the dynamic correlation

2.2. COUPLED-CLUSTER METHODS 15

energy. This effective Hamiltonian is diagonalized first before performing theperturbation treatment.

2.2 Coupled-cluster methodsThis section provides a brief general description of coupled-cluster (CC) theory,highlighting the methods used in this study, namely CC2, CCSD, CCSDR(3), andCC3. Details of the methodology are found in the original publications [12, 13,101–103], with derivation here following the recent review [104]. The coupled-cluster ansatz is given by:

|ΨCC〉 = eT̂ |ΨHF 〉 (2.12)where |ΨHF 〉 denotes the Hartree-Fock reference wavefunction and T̂ is the so-called cluster operator:

T̂ = T̂1 + T̂2 + T̂3 + · · ·+ T̂n =n∑

i=1

∑μi

tμiτμi . (2.13)

In this section, the following notation is used. Electrons are promoted from occu-pied orbitals i, j, k, . . . to unoccupied orbitals a, b, c, . . .. Excitation operators insecond quantization are written as a†aai for single excitations, a

†aa

†baiaj for double

excitations, and so on. These operators can be denoted by τμi , for i-tuple exci-tations. Thus the index i corresponds the level of excitation and μ stands for aparticular index set (ai, aibj, . . . etc). Referring back to equation 2.13, the num-ber of electrons in the system is n and tμi refers to the cluster amplitudes. Whenthe CC wavefunction ansatz is introduced into the time-independent Schrödingerequation and pre-multiplied with e−T̂ one obtains:

e−T̂HeT̂ |ΨHF 〉 = ECC |ΨHF 〉, (2.14)The expression for the CC energy is derived by projecting on the Hartree-Fockreference space:

ECC = 〈ΨHF |HeT̂ |ΨHF 〉 (2.15)since the e−T̂ operator drops out due to the condition: τνj |ΨHF 〉 = 0.

eμi = 〈μi|e−T̂HeT̂ |ΨHF 〉 = 0 (2.16)These are the central equations of coupled cluster theory. Truncations of the clus-ter operator T̂ formally define a particular method: T = T1 (CCS), T1 + T2(CCSD), T1 + T2 + T3 (CCSDT), etc. Starting from this hierarchy of methods,


one can derive approximate approaches that include higher excitations via a per-turbative framework, and thus reduce the scaling of the computational effort withrespect to the number of orbitals, without significant loss of accuracy. The mostpopular such methods for treating excited states by CC theory are CC2 [12, 15]and CC3 [13, 14, 18], which can be characterized as follows.CC2 is an approximation to CCSD where the singles equations are treated in thesame manner as CCSD, whereas the doubles equations for the nonlinear ground-state amplitudes as well as for the linear response are approximated by the first-order terms only. The singles excitation operator (T1) is treated as zeroth order,since the response of the singles excitations to an external one-electron perturba-tion, such as electromagnetic radiation, is of zeroth order. The equations for theground-state doubles amplitudes are almost identical to expressions of the MP2method, apart from the fact that the two-electron integrals are transformed with T1.Concerning excited electronic states, singles dominated excitations are correctedthrough second order, whereas pure double excitations are only correct to zerothorder. CC3 is an approximation to full CCSDT in a similar manner as CC2 is anapproximation to CCSD. The triples equations in CC3 are approximated by thesecond-order terms following the condition that the singles excitations are treatedagain as zeroth order, whereas the CCSDT singles and doubles equations are keptunchanged. Singles (doubles) dominated excitations are now correct to third (sec-ond) order. CCSDR(3) is based on a pseudoperturbation theory expansion of theCC3 eigenvalue problem and thus includes essentially only a non-iterative triplescorrection which is added to the CCSD excitation energy, scaling formally also asN7 with the number of orbitals N .

2.3 Time-dependent DFT and DFT/MRCIThis section starts with a brief overview of time-dependent density functionaltheory (TD-DFT). The derivation outlined here follows the reference [105]. For adetailed description and derivation of the main equations, see [22, 106–108].Let us consider an N -electron system that is initially found at time t = t0 inits ground state |Φ0〉. At times t > t0, a time-dependent perturbation λÛ(t) isapplied. The electric dipole part (μ) of this perturbation, when considering amonochromatic radiation field, is given by the expression:

λÛ(t) = −μEe−iωt (2.17)oscillating with frequency ω at a (constant) field strength E. The linear responseof an observable is defined as the first variation of the expectation value withrespect to the scalar variable λ. Considering the Fock operator F̂ , written as afunctional of the first order density matrix γ, we have:

2.3. TIME-DEPENDENT DFT AND DFT/MRCI 17

F̂ [γ](t) = ĥ[γ](t) + νH[γ](t)− νxc[γ](t) + λÛ(t), (2.18)where h is the one-electron part of the Hamiltonian, the index H denotes the two-electron Coulomb part, and vxc represents the exchange-correlation potential. Thelinear response of the mean-field density matrix then has the form:

γ(1)(t) =d

dλγ(t)|λ=0 (2.19)

By a Fourier transformation to the frequency domain and after expansion into thecomplete set of Kohn-Sham one-electron eigenstates, one obtains the followingequation:

γ(1)(ω) =∑i

∑a

(uai(ω)

ω − (εa − εi) |i〉〈a|)−

(u∗ai(ω)

ω + (εa − εi) |a〉〈i|), (2.20)

where i and a are occupied and virtual orbitals, respectively; ε denotes their en-ergies and uia the matrix elements of the perturbation in the basis of the orbitals.After some algebraic manipulations and insertion of equation 2.20 into 2.18, oneobtains a non-hermitian eigenvalue problem. This should be solved with the con-straint ω = ΔE. It has the form:(

A BB A

)(X

Y

)= ΔE

(1 00 −1

)(X

Y

)(2.21)

with X and Y as the solution vectors of the single particle-hole (excitations, |i〉〈a|)and hole-particle (deexcitations, |a〉〈i|) amplitudes, respectively; A and B are re-lated to the Hamiltonian matrices between the single (de)excitations that are con-stituted of one-particle eigenvalues and the Coulomb and exchange integrals overmolecular orbitals. The matrix elements can be expressed as:

Aiaσ,jbτ =δστδijδab (�aσ − �iτ ) + (iσaσ|jτbτ )− δστcHF (iσjσ|aτbτ ) + (1− cHF) (iσaσ|fστ |jτbτ )

(2.22)

Biaσ,jbτ =(iσaσ|bτjτ )− δστcHF (iσbσ|aτjτ )+ (1− cHF) (iσaσ|fστ |bτjτ )

(2.23)

where σ and τ are the spin variables. The parameter cHF denotes the amount ofHartree-Fock exchange in hybrid functionals. The last terms in equations 2.22 and2.23 are defined by:


(iσaσ|fστ |jτbτ ) =∫∫

i(r1)a(r1)fxcστ (r1, r2)j(r2)b(r2)dr1dr2. (2.24)

The time-dependent exchange-correlation kernel fxcστ , when derived from the time-independent ground-state functional, defines the so-called adiabatic approxima-tion, having the form:

fxcστ (r1, r2) =δ2Exc

δρσ(r1)δρτ (r2)(2.25)

This approximation should work best when ω → 0, i. e., for energetically low-lying excited states. This concludes the description of fundamental aspects of theTD-DFT methodology.The TD-DFT calculations reported in this work were performed with the TUR-BOMOLE program (version 5.7.1). [109] Three functionals were investigated:the gradient corrected BP86 functional, [110, 111] the hybrid Becke3-Lee-Yang-Parr (B3LYP) functional [110, 112, 113] with 20 % of HF exchange, and theBecke-half-and-half-Lee-Yang-Parr (BHLYP) functional [112, 114] with 50 % ofHF exchange. In the BP86 case, the resolution of identity (RI) method was em-ployed to evaluate the two-electron integrals, making use of auxiliary basis setsfrom the TURBOMOLE library. [115, 116] Calculations of vertical excitationenergies [23, 117–120] were done in the adiabatic approximation. [23, 121] Forevery state symmetry, the energies, dipole moments and oscillator strengths werecomputed at least for the five lowest states. [120, 122–124]Another DFT-based method for describing electronically excited states is the com-bination of DFT and configuration interaction (CI). Grimme [125] initially pro-posed such a combination with a CIS treatment involving only single excitations.This was later extended and generalized to a multireference singles and doublesCI treatment (DFT/MRCI) [34, 126] which is briefly outlined in the following.The general idea behind the DFT/MRCI method is to capture dynamical electroncorrelation by standard DFT and the nondynamical effects by a rather small CIexpansion. Considering the fact that the electronic Hamiltonian in a CI methodcan be written in terms of a one-electron Fock operator and the SCF energy ofa particular parent configuration, any matrix element can be formulated in termsof its difference from the SCF result. [125] Denoting the occupation number ofthe ith MO in the parent configuration by ni, the Fock operator, the Hartree-Fock(HF) energy, and the Hamiltonian are given by:

F̂HFij = ĥij +∑k

nk [(ij|kk)− 1/2 (ik|jk)] (2.26)

EHF =∑i

niF̂HFii −

1

2

∑ij

ninj [(ii|jj)− 1/2 (ij|ij)] (2.27)

2.3. TIME-DEPENDENT DFT AND DFT/MRCI 19

Ĥ =∑

ij FHFij Ê

ji −

∑ijk [(ij|kk)− 1/2 (ik|jk)]nkÊji

+1

2

∑ijkl (ij|kl)

[Êji Ê

lk − δjkÊli

], (2.28)

where the two-electron integrals are written in Mulliken notation and Êlk is thesingle-particle (de)excitation operator. Defining the occupation number vector vfor the space part and the spin-specific coupling pattern w, the Hamiltonian matrixelements between two CSFs 〈wv|Ĥ|w′v′〉 can be divided into four classes: [34,126]

Ia Same space part, same spin-coupling (diagonal);

IIa Same space part, different spin-coupling (off-diagonal);

Ib Space part differs by one-electron excitation (off-diagonal);

IIb Space part differs by two-electron excitation (off-diagonal).

The key approximation of the DFT/MRCI method is to introduce empirical cor-rections of the exact Kohn-Sham matrix elements and scaled two-electron inte-grals, to avoid double counting of correlation effects and to be able to tune theresults. The diagonal elements of the effective DFT/CI Hamiltonian ĤDFT havethe form:

〈wv|ĤDFT − EDFT |wv〉 = 〈wv|Ĥ − EHF |wv〉 −nexc∑c

(FHFcc − FKScc

)+

nexc∑a

(FHFaa − FKSaa

)+

1

nexc

nexc∑a

nexc∑c

pJ (aa|cc)− p [No] (ac|ac)(2.29)

where FKSij is a matrix element of the Kohn-Sham Fock operator of the parent stateand EDFT is the pure DFT energy of the parent determinant |wv〉. The indices cand a refer to creation and annihilation operators, and nexc is the excitation levelwith respect to the space part of the parent determinant. p [No] has different formsfor singlet and triplet states: 1p [N0] = p [0] + N10α and

3p [N0] = N30α. The

off-diagonal elements are obtained by an appropiate scaling of the exact matrixelements:

〈wv|ĤDFT |w′v′〉 = 〈wv|Ĥ|w′v′〉p1e−p2ΔE4ww′ (2.30)where p1 and p2 are scaling factors and ΔEww′ is the energy difference betweenthe diagonal elements of the two CSFs. This damping is necessary to avoid anoverestimate of electron correlation effects, which are already partly included inthe diagonal term. The parameters pJ , p [0] and α are obtained by a least-squaresfit of calculated excitation energies against experimental data of selected reference


Table 2.1: DFT/MRCI parameters for BH-LYP functional for singlet and tripletstates.

Parameter

Multiplicity p1 p2 pJ p [0] α

singlet 0.619 3.27 0.510 0.595 0.106triplet 0.619 3.27 0.493 – 0.056

molecules, using BHLYP as DFT functional. The final DFT/MRCI parameters areshown in Table 2.1.In the DFT/MRCI calculations reported in this work, two-electron integrals wereevaluated using the RI approximation [115, 127] with the optimized RI-MP2 basissets from the TURBOMOLE library. Calculations were carried out using the mrciprogram. [34] The adopted energy threshold value of 1.0 hartree always yieldedconverged results for the excitation energies. Initial reference configurations weregenerated by promoting up to two (active) electrons out of ten electrons in tenactive orbitals. The corresponding number of reference configurations varied be-tween 14 for cyclopentadiene and 244 for s-tetrazine. The oscillator strengthswere calculated in the dipole length representation. This standard DFT/MRCIprocedure was applied to all benchmark molecules.

2.4 Semiempirical methods

This section reviews the basic formalism of semiempirical quantum-chemicalmethods. Detailed derivations are available in several reviews. [50, 58, 128–130]Semiempirical methods are formulated in the framework of Hartree-Fock molec-ular orbital (MO) theory. They employ a minimal value basis set and make use ofrather drastic integral approximations to speed up the calculations. The most im-portant interactions are retained, and the corresponding integrals are representedby parametric functions that are parametrized against experimental data. The MOsψi are expressed as a linear combination of atomic orbitals φi, with expansion co-efficients and energies obtained from the solution of the secular equations 2.32:

ψi =∑μ

cμiφμ, (2.31)

0 =∑ν

(Fμν − Sμνεi) cνi, (2.32)

2.4. SEMIEMPIRICAL METHODS 21

where Sμν is an the overlap integral. The Fock matrix elements Fμν are definedas:

Fμν = Hμν +∑λ

∑σ

Pλσ

[(μν, λσ)− 1

2(μλ, νσ)

]. (2.33)

Hμν and Pλσ are elements of the one-electron core Hamiltonian and the densitymatrix, respectively, and (μλ, νσ) is a two-electron integral. The total energy Etotof a molecule is the sum of its electronic energy Eel and the nuclear repulsionVNN :

Eel =1

2

∑μ

∑ν

Pμν (Hμν + Fμν) , (2.34)

Etot = Eel + VNN . (2.35)

The chosen integral approximations characterize a semiempirical method. Usu-ally most of the integrals are neglected, whereas many of the remaining ones areparametrized against experimental data or derived from analytical expressions(see below). Most semiempirical methods employ the zero-differential-overlap(ZDO) approximation:

μ (1) ν (1) dτ (1) = δμνμ (1) ν (1) dτ (1) . (2.36)

As a consequence, all three-center and four-center two-electron integrals vanish,i.e. the computation of these integrals scales with N2. For the calculation of theFock matrix, only the one- and two-center terms remain for the one- and two-electron integrals. Historically, further approximations in this equation have beenintroduced, defining three classes of methods. The first scheme is the completeneglect of differential overlap (CNDO) [131, 132] which assumes:

μA (1) νB (1) dτ (1) = δμνsA (1) sA (1) dτ (1) , (2.37)

where the subscripts assign an atomic orbital (AO, μ or ν) to an atom (A or B)and sA denotes a spherically symmetric s AO at atom A. This implies that all non-vanishing two-electron integrals for a given pair A − B are identical in CNDO.An improved scheme is the so-called intermediate neglect of differential overlap(INDO) [133] where all integrals from CNDO are considered, plus all one-centertwo-electron integrals. A modified version of INDO combined with a singles CItreatment (CIS) was parametrized by Ridley and Zerner (INDO/S) [56, 57] forcalculating electronic spectra, particularly vertical excitation energies at a givenground-state geometry. The best ZDO-based integral approximation is the neglectof diatomic differential overlap (NDDO) which requires


μA (1) νB (1) dτ (1) = δABμA (1) νB (1) dτ (1) (2.38)

during integral evaluation. This NDDO scheme retains the heterogeneity of thetwo-center two-electron integrals and thus improves on the INDO or CNDO mod-els that use a common value for all these integrals.The most widely used semiempirical methods are based on the NDDO scheme,e.g., MNDO, [47] AM1, [48] and PM3. [49] They use of a minimal valence basisset which contributes to their computational efficiency. In the following the mainequations of the MNDO model are presented, with differences between AM1 andPM3 methods outlined thereafter:

0 =∑ν

(Fμν − δμνεi) cνi, (2.39)

FμAνA =HμAνA +∑λA

∑σA

PλAσA

×[(μAνA|λAσA)− 1

2(μAλA|νAσA)

]+∑B

∑λB

∑σB

PλBσB(μAνA|λBσB),

(2.40)

FμAνB = HμAνB − 12∑λA

∑σB

PλAσB(μAλA|νBσB), (2.41)

Etot = Eel +∑ ∑

A

2.4. SEMIEMPIRICAL METHODS 23

MNDO differs from the other two methods (AM1 and PM3) mainly in the treat-ment of core-core repulsion term:

EcoreAB = ZAZB(sAsA|sBsB)[1 + e−αaRAB + e−αbRAB

], (2.43)

The extra exponential terms (with parameters αa(b)) are introduced to account forPauli repulsion and to better model repulsive interactions at smaller interatomicdistances. The core-core repulsion terms are modified in AM1 and PM3 by addingfurther Gaussian correction terms:

EAB =ZAZB(sAsA|sBsB)[1 + e−αaRAB + e−αbRAB

]+

ZAZBRAB

4∑i=1

{ai(A) exp

[−bi(A)(RAB − ci(A))2]+ai(B) exp

[−bi(B)(RAB − ci(B))2]} ,(2.44)

with further adjustable parameters ai(A), bi(A), and ci(A). PM3 differs fromAM1 and MNDO also in that it treats the one-center, two-electron integrals aspure parameters, instead of deriving them from atomic spectroscopy.A new generation of semiempirical methods arises from the explicit inclusionof orthogonalization corrections in the semiempirical NDDO-MO scheme. Allone-electron integrals are strongly affected by these orthogonalization corrections.Conceptually, the inclusion of these corrections in the Fock matrix is expected toimprove the description of virtual orbitals which are shifted to higher energies. InOM2 (Orthogonalization Model 2) the Fock matrix elements are given by:

Fμμ = Hμμ +A∑ν

Pνν

[(μμ|νν)s − 1

2(μν|μν)s

] ∑B �=A

B∑λ

B∑σ

Pλσ(μμ|λσ)s,

(2.45)

Fμν = Hμν +1

2Pμν [3(μν|μν)s − (μμ|νν)s]

∑B �=A

B∑λ

B∑σ

Pλσ(μν|λσ)s, (2.46)

Fμλ = Hμλ − 12

A∑ν

B∑σ

Pνσ(μν|λσ)s. (2.47)

The matrix elements of the core-electron Hamiltonian are:


Hμμ = Uμμ +∑B

[ZB(μμ|ss)s + Vμμ,B(PI) + Vμμ,B(ORT ) + Vμμ,B(ECP )] ,(2.48)

Hμν =∑B

[ZB(μν|ss)s + Vμν,B(PI) + Vμν,B(ORT ) + Vμν,B(ECP )] , (2.49)

Hμλ = βμλ +∑C

Vμλ,C(ORT ). (2.50)

Explicit formulas for the terms in equations 2.45–2.50 are listed below:

(μν|λσ)s = fKO(μν|λσ)a, (2.51)

fKO =(ss|ss)s(ss|ss)a , (2.52)

βμλ = ±12(βAμ + β

Bλ )

√RABe

−(αAμ+αBλ )R2AB, (2.53)

Vμν,B(PI) =[V aμν,B − ZB(μν|ss)a

]fKO, (2.54)

Vμν,B(ORT ) =− 12FA1

B∑λ

(Sμλβλν + βμλSλν)

+1

8FA2

B∑λ

SμλSλν(Hlocμμ +H

locνν − 2H locλλ ),

(2.55)

Vμλ,C(ORT ) =− 12GAB1

C∑ρ

(Sμρβρλ + βμρSρλ)

+1

8GAB2

C∑ρ

SμρSρλ(Hlocμμ +H

locλλ − 2H locρρ ),

(2.56)

GABx =1

2(GAx +G

Bx ), x = 1, 2 (2.57)

Vμν,B(ECP ) = −(SμαGαν +GμαSαν − SμαSανFαα), (2.58)

2.5. QM/MM METHODS 25

Gμα =1

2(βAμ + β

Bα )

√RABe

−(αAμ+αBα )R2AB. (2.59)

The subscripts s and a on the integrals refer to their semiempirical [134, 135]and analytical forms, and their ratio defines the Klopmann-Ohno factor (fKO).The parameters FAx and G

ABx control the amount of correction that is taken into

account for the resonance integrals.There are two other two semiempirical methods with such orthogonalization cor-rections (OM1 and OM3) which can be considered as variants of OM2. The OM1method [65, 66] does not contain the extra term that takes into account interactionsof non-neighbor atoms (Vμλ,C(ORT )); in addition, it makes use of ab initio pseu-dopotentials in the term Vμν,B(ECP ) that are then scaled by the Klopmman-Ohnofactor. The OM3 method [69] neglects some small contributions to the correctionsVμν,B(ORT ) and Vμλ,C(ORT ), by setting FA2 and G

AB2 equal to zero.

In the semiempirical calculations on electronically excited states reported in thiswork, a standard closed-shell SCF treatment was followed by a CI calculationto determine the excited-state properties. For six of the investigated methods(MNDO, AM1, PM3, OM1, OM2 and OM3), MRCI (multireference configu-ration interaction) calculations were carried out using the GUGA (graphical uni-tary group approach) implementation in our semiempirical code [136]; these cal-culations utilized a standard active space and all configurations up to an exci-tation level of four (CISDTQ). In addition, INDO/S and INDO/S2 (an oxygen-reparametrized version of INDO/S [137]) computations were performed using theZINDO-MN [138] program and a standard CIS treatment.

2.5 QM/MM methods

This final section presents a brief introduction to quantum mechanics / molecularmechanics (QM/MM) methods. Comprehensive reviews on the methodology anddetailed surveys of applications of biological relevance are available in the recentliterature. [74, 75]The central concept of the QM/MM scheme is to divide a large system into aninner region, treated quantum-mechanically (called QM region or I), and an outerregion, which is described by a force field (called MM region or O). The total en-ergy of the system is not simply the sum of the energies from the inner and outerregions, since these regions interact strongly. The boundary between the subsys-tems must be treated carefully, especially when it crosses covalent bonds. Thesimplest and most popular strategy is the link-atom approach, where an additionalhydrogen atom is placed in the covalent bond being cut. This atom is included in


the QM treatment, while the covalent bond between the QM and MM regions isdescribed at the MM level.There are two types of QM/MM energy expression in the literature, i.e., a sub-tractive scheme and an additive scheme. The former can be regarded as a MMapproach, in which a certain region has been cut out and replaced by a higher-level treatment. One major shortcoming of this approach is the need to have MMparameters for the whole system, which are often cumbersome to obtain for elec-tronically demanding QM regions. By contrast, in the additive scheme, the MMcalculation is performed only for the outer MM region which alleviates the needfor extra MM parameters. The energy expression of the additive scheme is givenby:

EQM/MM = EMM(O) + EQM(I+ L) + EQM/MM(I,O). (2.60)

The coupling term EQM−MM(I,O) collects all the interaction terms between theinner and outer region and includes the electrostatic, van der Waals, and bondedinteractions between the atoms in the QM and MM regions:

EQM/MM(I,O) = EelQM/MM + E

vdWQM/MM + E

bQM/MM. (2.61)

The last two terms of equation 2.61 are handled purely at the MM level. Thefirst term determines the type of embedding used in the calculation (mechanicalor electrostatic or polarized embedding). The commonly applied electrostatic em-bedding takes into account the polarization of the QM region due to MM pointcharges. The QM Hamiltonian has the following form:

ĤelQM/MM = −electrons∑

i

∑M∈O

qM|ri −RM | +

∑α∈I+L

∑M∈O

qMZα|Rα −RM | (2.62)

where qm denotes the MM point charges and Zα is the nuclear charge of the QMatoms. The index i runs over all electrons, M over the point charges, and α overthe QM nuclei.

Chapter 3

Benchmarking electronically excitedstates

3.1 Initial considerations

In this section, high-level ab initio results from CASPT2 and coupled-cluster cal-culations are presented. The choice of the medium-sized basis set TZVP is ex-plained, followed by a brief discussion of general trends observed in the results.Together with an extensive survey of the literature of high-level ab initio valuesfor the molecules contained in the benchmark set, guidelines for the definition oftheoretical best estimates are established. The developed procedure is exempli-fied for four molecules: Ethene, naphtalene, furan, and adenine. Statistical resultsfor singlet and triplet states are outlined and briefly discussed. For one-electronproperties, such as oscillator strengths, results are listed and commented. Finallybasis-set effects on CASPT2 and coupled cluster methods are evaluated, improv-ing the definition of theoretical best estimates to some extent.In the MS-CASPT2 calculations for π → π∗ states, the active space comprisedall π and π∗ orbitals; for n (σ)→ π∗ states, the relevant n (σ) orbitals were alsoincluded. This active space is denoted as (n,m), with n electrons in m orbitals.It is the smallest reasonable choice one can take. Additional orbitals were added,when more flexibility in the active space was needed or when one had to avoid in-truder states. Multistate CASPT2 energies as well as one-electron properties suchas oscillator strengths and excited-state dipole moments were computed with theprogram MOLCAS (version 6.4). [139] All linear response coupled-cluster calcu-lations were performed with the Dalton 2.0 [140] program. The same geometriesand basis set were employed as for CASPT2 calculations.

27

28CHAPTER 3. BENCHMARKING ELECTRONICALLY EXCITED STATES

3.2 Coupled-cluster and MS-CASPT2 resultsA central issue for comparison of vertical excitation energies obtained by differentmethods consists of intrinsic details and common features of how the calculationswere set up. The use of the same ground-state equilibrium geometries (MP2/6-31G*) and medium-sized basis set (TZVP) contributes mostly for this consistency.Moreover, for CASPT2, other parameters such as the active space as well as theuse of a single-state or multi-state treatment play a role for the final excitationenergy. However, it is important to analyse how the quality of the results dependsof the setup employed.In order to address the basis set dependence, CASPT2, CCSD, and CC3 resultsare obtained with the TZVP basis and 12 different correlation-consistent basissets [141, 142] (up to augmented polarized quintuple-zeta) for the π → π∗ tran-sitions in ethene, illustrated in Figure 3.1. The triplet state (13B1u, lower panel)is a well-characterized low-lying state, dominated by a single excitation. Conse-quently, variations of less than 0.1 eV are found, with TZVP results sufficientlyaccurate for CCSD and CC3 methods. In contrast, the high-lying singlet state(11B1u, top panel), which is known to be rather diffuse and susceptible to valence-Rydberg mixing, shows a large basis set dependence: An extension of basis inthe cc-pVXZ series (X=D,T,Q,Z) significantly lowers the excitation energy by ca.0.6 eV, and the TZVP result is much too high (by more than 0.4 eV). Augmentedbasis sets with added diffuse functions are surely most appropriate for this singletstate, confirmed by the fast convergence observed in the aug-cc-pVXZ series. Inthis context, another methodological aspect is worth to be mentioned (see Fig-ure 3.1): The single-state (SS) CASPT2 treatment with a (2,2) active space, evenusing extended basis sets such as in the aug-cc-pVXZ series, overestimates theexcitation energy to the 11B1u state by more than 0.5 eV compared to the CC3method or a multistate (MS) CASPT2 treatment [143] with a (8,20) active space.Previous results [144] have already pointed out the tendency of this procedure toexaggerate valence-Rydberg mixing, which is corrected by MS-CASPT2. [143]In the following, the singlet and triplet excitation energies of the benchmarkset are addressed, collected in the Tables 3.1 and 3.2, respectively. These ta-bles contain previous CASPT2 results from the Roos group, [143–161] the cur-rent MS-CASPT2, CC2, CCSD, CCSDR(3) (only for singlet states) and CC3 re-sults with the TZVP basis, and a set of theoretical best estimates (TBE) basedon the current calculations together with a survey of literature data (references[13, 15, 16, 28, 37, 41, 43, 141, 143–277]).Looking at the computational results in the first five data columns of Tables 3.1and 3.2, general trends are observed: (a) The excitation energies from the cur-rent CASPT2 calculations are often about 0.2 eV higher than the previously pub-lished CASPT2 values, consistent with the inclusion of the IPEA shift [100]

3.2. COUPLED-CLUSTER AND MS-CASPT2 RESULTS 29

cc-pVDZ

TZVPcc-pVTZ

cc-pVQZ

cc-pV5Z

aug-cc-pVDZ

aug-cc-pVTZ

aug-cc-pVQZ

aug-cc-pV5Z

d-aug-cc-pVDZ

d-aug-cc-pVTZ

d-aug-cc-pVQZ

d-aug-cc-pV5Z

4.4

4.6

7.8

8.0

8.2

8.4

8.6

8.8

13B1u

11B1u

Exc

itatio

n en

ergi

es in

eV

Figure 3.1: Basis set dependence of the 11B1u (filled symbols, top) and the 13B1u(open symbols, bottom) state of ethene with different methods: CCSD (squares,blue), CC3 (triangles, red), CASPT2 with 2,2 (diamonds, green) and 8,20 (circles,orange) active space.

in the present calculations. (b) The deviations between the current and previ-ous CASPT2 results tend to be larger for high-lying excited states. This is ex-pected since the current calculations use the TZVP basis without diffuse functionswhereas the previous CASPT2 studies normally included diffuse functions in theirbasis to take into account valence-Rydberg mixing effects. The current use of theMS-CASPT2 approach [143] is expected to reduce the effects of valence-Rydbergmixing, but the lack of diffuse functions in the TZVP basis will still cause prob-lems for high-lying valence states. (c) Among the coupled cluster results, thesinglet excitation energies from the most reliable coupled cluster method (CC3)are normally lower than those from CC2 and always lower than those from CCSDusing the TZVP basis set. CCSDR(3), on the other hand, yields results virtuallyidentical to CC3. (d) For excited states that are dominated by single excitations(single excitation weight - T1 - above 90%, see Table 3.1), the current CASPT2and CC3 results are generally very close to each other (within 0.1 eV). Such ex-cellent agreement is found particularly often for triplet states.

Table 3.1: Vertical singlet excitation energies ΔE (eV) of all statistically evalu-ated molecules.

Molecule State CASPT2a CASPT2b CC2c CCSDc(%T1)d CCSDR(3)e CC3c(%T1)

d TBEd

Ethene 11B1u (π → π∗) 7.98 8.62 8.40 8.51 (97.2) 8.36 8.37 (96.9) 7.8E-Butadiene 11Bu (π → π∗) 6.23 6.47 6.49 6.72 (95.5) 6.56 6.58 (93.7) 6.18

21Ag (π → π∗) 6.27 6.63 7.63 7.42 (85.8) 6.95 6.77 (72.8) 6.55all-E-Hexatriene 11Bu (π → π∗) 5.01 5.31 5.41 5.72 (95.0) 5.56 5.58 (92.6) 5.10

21Ag (π → π∗) 5.20 5.42 6.67 6.61 (84.7) 6.04 5.72 (65.8) 5.09all-E-Octatetraene 21Ag (π → π∗) 4.38 4.64 5.87 5.99 (85.4) 5.40 4.97 (62.9) 4.47

11Bu (π → π∗) 4.42 4.70 4.72 5.07 (94.7) 4.93 4.94 (91.9) 4.66Cyclopropene 11B1 (σ → π∗) 6.36 6.76 6.96 6.96 (94.5) 6.89 6.90 (93.0) 6.76

Continued on Next Page. . .

30CHAPTER 3. BENCHMARKING ELECTRONICALLY EXCITED STATES

Table 3.1 – Continued

Molecule State CASPT2a CASPT2b CC2c CCSDc(%T1)d CCSDR(3)c CC3c(%T1)

d TBEf

11B2 (π → π∗) 7.45 7.06 7.17 7.24 (96.3) 7.10 7.10 (95.5) 7.06Cyclopentadiene 11B2 (π → π∗) 5.27 5.51 5.69 5.87 (95.7) 5.72 5.73 (94.3) 5.55

21A1 (π → π∗) 6.31 6.31 7.05 7.05 (89.4) 6.76 6.61 (79.3) 6.3131A1 (π → π∗) 7.89 8.52 8.86 8.95 (95.8) 8.72 8.69 (93.1)

Norbornadiene 11A2 (π → π∗) 5.28 5.34 5.57 5.80 (95.3) 5.65 5.64 (93.4) 5.3411B2 (π → π∗) 6.20 6.11 6.37 6.69 (94.5) 6.51 6.49 (91.9) 6.1121B2 (π → π∗) 6.48 7.32 7.65 7.87 (95.5) 7.65 7.64 (93.8)21A2 (π → π∗) 7.36 7.44 7.66 7.87 (95.0) 7.73 7.71 (93.0)

Benzene 11B2u (π → π∗) 4.84 5.05 5.27 5.19 (90.5) 5.12 5.07 (85.8) 5.0811B1u (π → π∗) 6.30 6.45 6.68 6.74 (95.6) 6.70 6.68 (93.6) 6.5411E1u (π → π∗) 7.03 7.07 7.44 7.65 (94.5) 7.45 7.45 (92.2) 7.1321E2g (π → π∗) 7.90 8.21 9.03 9.21 (84.9) 8.71 8.43 (65.6) 8.41

Naphthalene 11B3u (π → π∗) 4.03 4.24 4.45 4.41 (90.5) 4.34 4.27 (85.2) 4.2411B2u (π → π∗) 4.56 4.77 4.96 5.21 (94.3) 5.08 5.03 (90.6) 4.7721Ag (π → π∗) 5.39 5.90 6.22 6.23 (90.1) 6.09 5.98 (82.2) 5.9011B1g (π → π∗) 5.53 6.00 6.21 6.53 (91.4) 6.26 6.07 (79.6) 6.0021B3u (π → π∗) 5.54 6.07 6.25 6.55 (93.9) 6.35 6.33 (90.7) 6.0721B1g (π → π∗) 5.87 6.48 6.82 6.97 (93.8) 6.81 6.79 (91.3) 6.4821B2u (π → π∗) 5.93 6.33 6.57 6.77 (93.8) 6.60 6.57 (90.5) 6.3331Ag (π → π∗) 6.04 6.71 7.34 7.77 (88.4) 7.29 6.90 (70.0) 6.7131B2u (π → π∗) 7.16 8.18 8.46 8.77 (93.5) 8.53 8.44 (87.9)31B3u (π → π∗) 7.18 7.76 8.85 9.03 (84.1) 8.50 8.12 (58.7)

Furan 11B2 (π → π∗) 6.04 6.43 6.75 6.80 (94.9) 6.64 6.60 (92.9) 6.3221A1 (π → π∗) 6.16 6.52 6.87 6.89 (90.8) 6.71 6.62 (84.9) 6.5731A1 (π → π∗) 7.66 8.22 8.78 8.83 (94.2) 8.57 8.53 (90.7) 8.13

Pyrrole 21A1 (π → π∗) 5.92 6.31 6.61 6.61 (91.2) 6.47 6.40 (86.0) 6.3711B2 (π → π∗) 6.00 6.33 6.88 6.87 (94.2) 6.74 6.71 (91.6) 6.5731A1 (π → π∗) 7.46 8.17 8.44 8.44 (93.7) 8.20 8.17 (90.2) 7.91

Imidazole 11A′′ (n → π∗) 6.52 6.81 6.86 7.01 (92.4) 6.87 6.82 (87.6) 6.8121A′ (π → π∗) 6.72 6.19 6.73 6.80 (92.0) 6.64 6.58 (87.2) 6.1931A′ (π → π∗) 7.15 6.93 7.28 7.27 (93.1) 7.15 7.10 (89.8) 6.9321A′′ (n → π∗) 7.56 7.91 8.00 8.15 (93.3) 7.98 7.93 (89.4)41A′ (π → π∗) 8.51 8.15 8.62 8.70 (92.7) 8.49 8.45 (88.8)

Pyridine 11B2 (π → π∗) 4.84 5.02 5.32 5.27 (90.6) 5.20 5.15 (85.9) 4.8511B1 (n → π∗) 4.91 5.14 5.12 5.25 (92.8) 5.12 5.05 (88.1) 4.5911A2 (n → π∗) 5.17 5.47 5.39 5.73 (92.4) 5.55 5.50 (87.7) 5.1121A1 (π → π∗) 6.42 6.39 6.88 6.94 (95.3) 6.88 6.85 (92.8) 6.2631A1 (π → π∗) 7.23 7.46 7.72 7.94 (94.2

quantum mechanical / molecular mechanics study of … · for high-lying singlet states above 7 ev....

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