Frustrated Lewis Pairs - Quantum Chemical

Birgitta Schirmer

Frustrated Lewis Pairs Quantum Chemical Investigation of Structures, Reactions and Mechanisms

- 2013 -

Theoretische Chemie

Frustrated Lewis Pairs Quantum Chemical Investigation of Structures, Reactions and Mechanisms

Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Chemie und Pharmazie der Mathematisch-Naturwissenschaftlichen Fakultät der Westfälischen Wilhelms-Universität M¨ unster

vorgelegt von

Birgitta Katharina Ulrike Schirmer aus Essen

- 2013 -

.

Dekan:

Prof. Dr. Bart-Jan Ravoo

Erster Gutachter:

Prof. Dr. Stefan Grimme

Zweiter Gutachter:

Prof. Dr. Gerhard Erker

Tag der Diputation: Tag der Promotion:

Abstract Frustrated Lewis Pairs (FLPs) consist of a Lewis acid and a Lewis base which are prevented from forming a classical adduct by their large substituents. The electric field between both partners allows to polarise or even split bonds leading to a plethora of possible reactions, as the following scheme shows: H B(C6F5)2

CH2Ph O

Mes2P Mes2P

B(C6F5)2 O

B(C6F5)2

O

*

H

Mes2P

Ph

H

N Ph

H Ph

Ph N C O

CH2

H2 Ph

CH3 N

N

O Mes2P •

R2

B(C6F5)2 O

R1

R Mes2P

R1

R

*

.

.

R

*

H O Ph

Mes2P H

B(C6F5)2

Mes2P

R

B(C6F5)2

NO

CO2

C6F5 C6F5 B

B(C6F5)2 O

P Mes2P

Ph

*

N O

Mes2P

B(C6F5)2

B(C6F5)2 O

O

*

Mes

Mes

*

*

The reactions marked with a * in the above scheme will be presented in this thesis. After a short overview of the quantum chemical methods used in this thesis several studies are presented – some purely theoretical, but most of them cooperations between theoretical and experimental chemistry. Two methodological studies concentrate on the applicability of DFT for the calculation of hydrogen splitting in the electric field generated by an FLP and give a tutorial on how to generate accurate thermochemical results. Spectroscopic properties like IR-shifts are discussed with the help of the example of H2 7

8

Abstract

activation. To the author’s knowledge, this is the first systematic study of the accuracy of DFT for the calculation of H2 -IR-shifts. Further studies on the activation of H2 present a series of FLP-systems which activate H2 already at extremely low temperatures of down to -60◦ C and discuss the relation between their reactivity, stability and basicity. The following chapter concentrates on the activation of other gaseous substrates like the greenhouse gases CO2 and SO2 , which can be turned into highly useful building blocks, as well as the activation of NO with an intramolecular FLP, which leads to a new family of persistent N-oxyl-radicals with a higher reactivity than their famous relative TEMPO. The quantum chemical calculations presented here were able to confirm the structure, explain the high reactivity of the product and also helped to estimate if the reactivity of the radical can be tuned by exchanging the substituents on the bridging C2 -unit of the FLP. A similar reaction behaviour can be observed for the two substrates CO and R-CN when reacted with intramolecular FLPs. Here the presented calculations could explain why t-Bu-isonitriles form a different addition product than n-Bu-isonitriles. The last chapter of this thesis focusses on the activation of carbon-carbon double and triple bonds, which are conjugated with each other or with a carbonyl group. Activation reactions of diynes, enynes and ynones reveal that the 1,4-addition to intramolecular FLPs occurs under kinetic reaction control while for intermolecular FLPs it happens under thermodynamic reaction control. The corresponding 1,2-addition products are rarely found in cases with especially small substituents on the substrates. In this context it was also possible to transfer hydrogen to a C-C-triple bond. The results of the conformational study of the reaction products could later be confirmed by NMR-spectroscopic results. In the last study two non-conjugated C-C triple bonds could be activated via a complex reaction mechanism including the formation of an intermediate FLP. Some of the intermediate structures were not accessible spectroscopically and could only confirmed by the presented calculations.

Publications Parts of this thesis have already been published: 9. M. Sajid, A. Klose, B. Birkmann, L. Liang, B. Schirmer, T. Wiegand, H. Eckert, A. J. Lough, R. Fröhlich, C. G. Daniliuc, S. Grimme, D. W. Stephan, G. Kehr, G. Erker, Reactions of Phosphorus/Boron Frustrated Lewis Pairs with SO2 , Chem. Sci. (2013), 4, 213-219. DOI:10.1039/C2SC21161K. 8. B. Schirmer, S. Grimme, The Quantum Chemistry of FLPs and their Activation of small Molecules - Methodological Aspects. in: Topics in Current Chemistry, (2012). DOI:10.1007/128 2012 389. 7. L. Greb, P. O˜ na-Burgos, B. Schirmer, S. Grimme, D. W. Stephan, J. Paradies, Metal-free Catalytic Olefin Hydrogenation: Low-Temperature H2 Activation by Frustrated Lewis Pairs, Angew. Chem. Int. Ed. (2012), 51(40), 10164-10168. DOI:10.1002/anie.201204007. 6. P. Feldhaus, B. Schirmer, B. Wibbeling, C. G. Daniliuc, R. Fröhlich, S. Grimme, G. Kehr, G. Erker, Frustrated Lewis pair addition to conjugated diynes: Formation of zwitterionic 1,2,3-butatriene derivatives, Dalton Trans., (2012), 41, 9135-9142. DOI:10.1039/C2DT30321C. 5. M. Sajid, A. Stute, A. J. P. Cardenas, B. J. Culotta, J. A. M. Hepperle, T. H. Warren, B. Schirmer, S. Grimme, A. Studer, C. G. Daniliuc, R. Fröhlich, G. Kehr, G. Erker, N,N-Addition of Frustrated Lewis Pairs to Nitric Oxide: An Easy Entry to a Unique Family of Aminoxyl Radicals, J. Am. Chem. Soc. (2012), 134 (24), 10156-10168. DOI:10.1021/ja302652a. 4. I. Peuser, R. Neu, X. Zhao, M. Ullrich, B. Schirmer, G. Kehr, R. Fröhlich, S. Grimme, G. Erker, D.W. Stephan, CO2 and Formate Complexes of Phospine-Borane Frustrated Lewis Pairs, Chem. Eur. J. (2011), S, 9640-9650. DOI:10.1002/chem.201100286. 3. B.-H. Xu, G. Kehr, R. Fröhlich, B. Wibbeling, B. Schirmer, S. Grimme, G. Erker, Reaction of Frustrated Lewis Pairs with Conjugated ynones - Selective Hydrogenation 9

10

Publications of the Carbon-Carbon Triple Bond, Angew. Chem. Int. Ed. (2011), 50, 7183-7186. DOI:10.1002/anie.201101051. 2. B. Schirmer, S. Grimme, Electric field induced activation of H2 – Can DFT do the job? Chem. Commun. (2010), 46, 7942-7944. DOI:10.1039/C0CC02569K. 1. C. M. Mömming, G. Kehr, B. Wibbeling, R. Fröhlich, B. Schirmer, S. Grimme, G. Erker, Formation of Cyclic Allenes and Cumulenes by Cooperative Addition of Frustrated Lewis Pairs to Conjugated Enynes and Diynes, Angew. Chem. Int. Ed. (2010), 49, 2414-2417. DOI:10.1002/anie.200906697.

Two additional manuscripts have been submitted and await acceptance: 2. L. Greb, S. Quack, B. Schirmer, P. O˜ na-Burgos, S. Grimme, J. Paradies, Jan, Electronic effects of triarylphosphines in the metal-free hydrogen activation: A kinetic and computational study, Chem. Sci. (2013), under revision. 1. O. Ekkert, G. González Miera, T. Wiegand, H. Eckert, B. Schirmer, J. L. Petersen, C. G. Daniliuc, R. Fröhlich, S. Grimme, G. Kehr, G. Erker, Remarkable Coordination Behavior of Alkyl Isocyanides Toward Unsaturated Vicinal Frustrated P/B Lewis Pairs, Chem. Sci. (2013), submitted.

Further publications: 2. R. Huenerbein, B. Schirmer, J. Moellmann, S. Grimme, Effects of London dispersion on the isomerization reactions of large organic molecules: a density functional benchmark study, Phys. Chem. Chem. Phys. (2010), 12, 6940-6948. DOI:10.1039/c003951a. 1. F. Neese, T. Schwabe, S. Kossmann, B. Schirmer, S. Grimme, Assessment of Orbital-Optimized, Spin-Component Scaled Second-Order Many-Body Perturbation Theory for Thermochemistry and Kinetics, J. Chem. Theory Comput. (2009), 5, 3060-3073. DOI:10.1021/ct9003299.

Presentations on International Meetings and Conferences 10. ”The Quantum Chemistry of FLPs and their Activation of small Molecules – Methodological Aspects”, 09/2012, 48th Symposium on Theoretical Chemistry, Karlsruhe, Germany (oral presentation). 9. ”The Quantum Chemistry of Frustrated Lewis Pairs – A Methodological Study”, 07/2012, 12th Sostrup Summer School of Quantum Chemistry and Molecular Properties, Ry, Denmark (poster presentation). 8. ”A new persistent N-Oxyl Radical and the Activation of SO2 – Computational Studies on Activation Reactions with FLPs”, 05/2012, M¨ unster Symposium on Cooperative Effects in Chemistry, M¨ unster, Germany (poster presentation). 7. ”New reactions with Frustrated Lewis Pairs – Theoretical studies on the Calculation of IR Spectra, the formation of a new persistent N-oxyl radical and the activation of SO2 ”, 04/2012, Meeting of the International Research Group 1175, University of Zurich, Switzerland (oral presentation). 6. ”Cyclic Allenes and a New Persistent N-Oxyl-Radical – New Activation Reactions with Frustrated Lewis Pairs.”, 10/2011, Modeling of Molecular Properties, Heidelberg, Germany (poster presentation). 5. ”Cyclic Allenes and a New Persistent N-Oxyl-Radical – New Activation Reactions with Frustrated Lewis Pairs.”, 07/2011, Ninth triennial Congress of the World Association of Theoretical and Computational Chemists, Santiago de Compostela, Spain (poster presentation). 4. ”Activation of Conjugated Ynones with Frustrated Lewis Pairs – Selective MetalFree Hydrogenation of the C-C Triple bond”, 05/2011, M¨ unster Symposium on Cooperative Effects in Chemistry, M¨ unster, Germany (poster presentation). 3. ”Activation of Dihydrogen with Frustrated Lewis Pairs”, 02/2011, Meeting of the International Research Group 1175, ETH Zurich, Switzerland (oral presentation). 11

12

Presentations on International Meetings and Conferences 2. ”Quantum Chemical Investigation of Activation Reactions of Frustrated Lewis Pairs”, 09/2010, 46th Symposium on Theoretical Chemistry 2010, M¨ unster, Germany (poster presentation). 1. ”Quantum Chemical Calculations of Frustrated Lewis Pairs”, 03/2010, Meeting of the International Research Group 1175, University of Zurich, Switzerland (oral presentation).

Contents

Abstract

7

Publications

9

Presentations on International Meetings and Conferences

11

1 Motivation

17

2 Theoretical Background

23

2.1

Basic Quantum Mechanics

23

2.2

The Hartree-Fock Approximation

25

2.2.1

27

2.3

2.4

The Correlation Energy

Density Functional Theory (DFT)

29

2.3.1

Kohn-Sham DFT (KS-DFT)

30

2.3.2

Jacob’s Ladder

32

2.3.3

Dispersion-corrected DFT

34

Wave Function Theory

38

2.4.1

Perturbation Theory

38

2.4.2

Other Wave Function-based Methods - CI and CC

41

3 Computational Studies 3.1

45

Methodological Studies

47

3.1.1

47

Electric Field induced Activation of H2 - Can DFT do the Job? 13

14

CONTENTS 3.1.2

Quantum Chemistry of FLPs and their Activation of small Molecules: Methodological Aspects

3.2

Activation of H2 3.2.1

3.2.3 3.3

Activation: A kinetic and computational Study

72

Activation of H2 by a novel Norbornyl-FLP

76

Infrared Spectroscopy

79

3.3.1

Some first Calculations of four small Complexes of a LA/LB with H2 79

3.3.2

Systematic methodological Investigations on the Accuracy of DFT 83

Activation of other gaseous Substrates

89

3.4.1

Activation and Hydrogenation of Carbon Dioxide (CO2 )

89

3.4.2

Activation of SO2 with Frustrated Lewis Pairs

96

3.4.3

N,N-Addition of Frustrated Lewis Pairs to Nitric Oxide (NO): A New Type of Persistent N-Oxyl Radical

3.4.4

Activation of CO by a norbornyl-based intramolecular FLP

3.4.5

Remarkable Coordination Behaviour of alkyl isocyanides (R-CN) towards unsaturated vicinal Frustrated P/B Lewis Pairs

3.5

69

Electronic Effects of Triarylphosphines in the metal-free Hydrogen

Methods for the Calculation of IR Spectra 3.4

69

Metal-free catalytic Olefin Hydrogenations: Low Temperature H2 -Activation by Frustrated Lewis Pairs

3.2.2

53

Activation of C-C Bonds 3.5.1

109 111

111

Activation of a conjugated Diyne with an intermolecular Frustrated Lewis Pair

3.5.3

107

Activation of a conjugated Enyne and Diyne with an intramolecular Frustrated Lewis Pair

3.5.2

99

114

Reaction of Frustrated Lewis Pairs with conjugated Ynones – Selective Hydrogenation of the Carbon-Carbon triple Bond

116

CONTENTS 3.5.4

15 Stepwise Formation of Phospholes from Dialkynylphosphanes and BCF via intermediate FLP formation

120

4 Conclusion

127

References

129

Appendix

141

A List of Abbreviations

141

B List of Figures

145

C List of Tables

149

D Appendix to 3.2.1

155

E Appendix to 3.2.2

159

F Appendix to 3.4.3

165

G Appendix to 3.4.5

167

H Appendix to 3.5.3

173

I

179

Appendix to 3.5.4

J Acknowledgements

187

K Curriculum Vitae

189

16

CONTENTS

1. Motivation The title of this thesis, Frustrated Lewis Pairs – Quantum Chemical Investigation of Structures, Reactions and Mechanisms, may rise several questions among which certainly are: ‘what is a Frustrated Lewis Pair?’ and ‘I know a chemist with labcoat and safety glasses, but what does a quantum chemist do?’ I would like to try and answer these questions first, at the very beginning of this thesis. Now, for the first question: ‘what is a Frustrated Lewis Pair?’ While normal, so-called Brønsted acids and bases exchange protons, Lewis acids and bases (short LA and LB) exchange electrons. In both types, acids and bases like to react with each other giving up their original functionality, the chemist says “they quench”. Lewis acids and bases do so by forming a bond between both partners, the product is called a classical Lewis-adduct. Frustrated Lewis pairs do exactly the same – the LB wants to give its extra electrons, for example from a lone pair, to the LA partner and form a bond. Unfortunately, there are two big problems: the large substituents on the LB and the large substituents on the LA. The steric repulsion between these makes it impossible for the two partners to quench, which leaves them literally frustrated – a Frustrated Lewis Pair (FLP). Scheme 1 visualises the difference between classical and frustrated Lewis pairs. For the LA part there is basically only one choice: B(C6 F5 )3 , short BCF, and variants of it in which one or two substituents are exchanged or altered. Other boranes have proven unreactive in experiments and could not show the typical FLP behaviour. Research is currently trying to find alternatives to BCF, but this has turned out be rather challenging. The Lewis basic partner offers more variety since all that is needed is a possible electron donor like a lone pair: phosphines, 1,2 amines, 3 but also compounds like carbodiphosphoranes 4 and N-heterocyclic carbenes (NHCs) 5 are possible. Even all-carbon FLPs have been proposed. 6 17

18

1 Motivation classical

frustrated

R R R LB R

R' LA R' R'

R

R'

B

P R

R'

R'

F5

P

F5

F B F F

P

B

F5

Scheme 1: Classical and frustrated Lewis acid-base pairs.

The first description of FLP-type reaction behaviour dates back to 1959/1960, when Wittig and Tochtermann described the unexpected formation of phosphoniumborate from 1,2-didehydrobenzene with a triphenylphosphin/triphenylborane mixture 7 and the formation of an unexpected 1,2-addition product of Ph3 C− and BPh3 to butadiene during an anionic polymerisation reaction. 8 Tochtermann called this kind of Lewis pair an “antagonistic pair”.

Wittig 1959: F

BPh3 PPh3

Mg

PPh3

Br

BPh3

Tochtermann 1960: Ph3C Na BPh3

BPh3

Na

Ph3C

Scheme 2: First FLP-type reaction behaviour observed by Wittig and Tochtermann.

19 Only recently, in 2006, Douglas Stephan found a similar reaction behaviour when he observed that the zwitterionic species Mes2 PH(C6 F4 )BH(C6 F5 )2 was able to release H2 at 150◦ C, but not at room temperature (RT). Surprisingly, the back-reaction of this intramolecular FLP with H2 gas proceeded readily already at RT. These remarkable results were published in the renowned journal Science. 9 In the following studies the term (sterically) Frustrated Lewis Pair was used to describe this special type of reactivity. At the same time, Gerhard Erker started working on this aspiring topic. Two of the first FLPs from both groups have become some kind of guinea pigs for new reactions and will appear many times within this thesis: 1,2 F5 F

F

F

F5 P

P

B

B

F

F F F

F F5

F

F

Scheme 3: The intermolecular FLP from the Stephan group (left) and the intramolecular FLP from the Erker group (right).

As one can see, these two FLPs differ in one essential point: while the Stephan system is an intermolecular FLP, the Erker system is an intramolecular FLP which means that the two moieties are part of the same molecule and connected by a bridging unit. The latter one exists in a ‘closed’ form in which LA and LB form a weak connection and in a reactive ‘open’ form where this connection is absent and the distance between the partners is larger. 1 At the same time, the Stephan system is more flexible which allows a different product formation, e.g. in an addition reaction not only the cis but also the trans isomer can be formed. Up to now, a plethora of new FLPs has been developed by several groups all over the world 5,10− 12 and an even larger number of reactions with them has been investigated (for a recent review see reference 13). Scheme 4 shows only a small number of reaction variants for the described intramolecular FLP. All of these reactions have already been investigated with the means of computational chemistry, those carrying an asterisk (*) or very closely related reactions will be discussed in this thesis.

20

1 Motivation H B(C6F5)2

CH2Ph O

Mes2P Mes2P

B(C6F5)2 O

B(C6F5)2

O

*

H

Mes2P

Ph

H

N Ph

H Ph

Ph N C O

CH2

H2 Ph

CH3 N

N

O Mes2P •

R2

B(C6F5)2 O

R1

R Mes2P

R1

R

*

B(C6F5)2

Mes2P

R

B(C6F5)2

R

*

H O Ph

Mes2P H

.

.

NO

CO2

C6F5 C6F5 B

B(C6F5)2 O

P Mes2P

Ph

*

N O

B(C6F5)2

*

Mes2P

B(C6F5)2 O

O

Mes

Mes

*

*

Scheme 4: Some possible reactions with the introduced intramolecular FLP.

Many, but not all, of the presented studies are already published in peer-reviewed journals – the corresponding references can be found in the chapters themselves and in the list of publications at the very beginning of this thesis. For probably all FLPs the first test-case is always the activation of H2 . Interesting aspects are not only if the reaction takes place, but also under which conditions (temperature, H2 pressure, in solution or in gas phase) and if it is reversible. The last of these points usually gains most attention since FLPs were discussed as possible hydrogen storage materials for different applications, e.g. for fuel cells. 14− 16 In synthetic chemistry the bond activation and hydrogen transfer abilities of FLPs are highly interesting because they may replace metal catalysts, which are usually expensive, sometimes toxic and often difficult to separate from the product solution. Therefore, one keyword which is often connected to FLP-chemistry is ‘metal-free activation reactions’, even though some FLPs exist in which the LA and/or LB functions are bound to a metalcontaining backbone like ferrocene or zirconocene. 17− 19

The second question from the beginning was ‘what does a computational chemist do?’ As the term suggests, a computational chemist does not work in a normal laboratory with flasks, solvents or protective gases, but uses a computer for his or her research. Usually

21 the geometry (or guessed geometry) of a molecule is inserted into a quantum chemical program which optimises the geometry, calculates different properties, calculates spectra, looks for conformers or evaluates correction terms to account for solvent effects. Several different techniques and methods are available to go from a structure via solution of the Schrödinger equation to the requested properties. The following chapter, named Theoretical Background, introduces some of these methods briefly. As in the framework of this thesis no new methods have been developed, but the focus is laid on the application of existing methods and the evaluation of their results, the theory part is kept short and limited to those methods which are actually used in the subsequent chapters. For further details and information on other existing methods the reader is referred to the standard textbooks. 20− 22 After this brief methodological introduction, several studies are presented which will show the broad field of applications of FLPs. Two computational studies will discuss the applicability of quantum chemical methods to FLP reactions as well as the “ingredients” which are necessary to obtain accurate and reliable results. Afterwards, the activation of H2 and, in this framework, the accuracy of density functional methods for the calculation of infra red spectra of H2 -complexes will be investigated. To the author’s knowledge, this is the first in-depth study of this topic. The following chapters will present some most interesting reactions, which can turn the greenhouse gases CO2 and SO2 into valuable building blocks or form a new family of persistent N-oxyl radicals, very similar to the well-known TEMPO-radical, from NO gas. A very similar reaction behaviour can be observed in the activation of CO gas and isonitriles in which the size and flexibility of the substrate determines the structure of the reaction product. Thermodynamic or kinetic reaction control – that is the question in the fifth chapter, in which the activation behaviour of carbon-carbon double and triple bonds is investigated for conjugated and non-conjugated examples. With the help of quantum chemistry a complex reaction mechanism will be investigated and experimentally not accessible intermediate structures may be verified. The final chapter of this thesis concludes the obtained results and gives an outlook onto future topics of interest. Additional data to some projects can be found in the corresponding part of the appendix.

22

1 Motivation

2. Theoretical Background

2.1

Basic Quantum Mechanics

While most things of daily life follow the rules of classical mechanics, smaller particles like atoms and electrons follow the rules of quantum mechanics. Among other things they define that each particle can be expressed as a wave (the so-called particle-wave dualism). The basic equation of quantum mechanics is the well-known non-relativistic Schrödinger equation (SE) which resembles the equations of motions from classical mechanics. It describes how the energy E of a system is accessible via the Hamiltonian operator H . This Hamiltonian is nothing else than a prescription how to get the demanded properties out of the system’s wave function ψ which contains all possible information about the system.

H ψ = εψ

(2.1)

The Hamiltonian operator itself can be separated into a part containing the potential energy contributions (V ) and a part containing the kinetic energy contributions (T ). Each of them contains contributions from the electrons (index i and small indices) and from the nuclei (index N and capital indices). 23

24


H

= T +V

(2.2)

= Te + TN +Vee + VeN + VN N n

(2.3)

M

1X 2 X 1 ∇2 ∇i − 2 2 MA i i A n−1 i n X M N X M XX 1 X ZA X ZA ZB + ri − rj − Ri − rA + RA − RB

= −

i

j

i

A

A

(2.4)

B

In this equation MA is the mass of the nuclei, Z is their charge, ∇ is the second derivative of the wave function (WF) and r and R describe the positions of the electrons and the nuclei, respectively (all quantities are given in atomic units). The potential energy terms in this equation are simply the classical Coulomb interaction terms for particles or charges and describe the interactions between two electrons i and j, two nuclei A and B and an electron i and a nucleus A. In this formulation a positive sign is used for repulsive terms, a negative sign for attractive terms. To further simplify this formula an approximation is used, the Born-Oppenheimer approximation (BO). As the nuclei are significantly heavier than the electrons it is assumed that the electrons are able to move a lot faster than the nuclei. This means that if a nucleus is moved its electrons follow instantly. One could also say the nuclei do not move (relative to the electrons), so their kinetic energy should be zero and their repulsion is constant. This assumption makes it possible to separate the Hamiltonian into an electronic part and a static potential for the nuclei-nuclei interaction.

H

= Te + VeN + Vee +VN N | {z } =

He

He ψe = Ee ψe

+ VN N

(2.5) (2.6)

(2.7)

2.2 The Hartree-Fock Approximation

25

Correspondingly, the SE can be reformulated as an electronic SE (eqn. 2.7) – the form which shall from now on be used unless stated differently. For this reason the indices e will be omitted.

2.2

The Hartree-Fock Approximation

Since the exact wave function is usually unknown, another approximation needs to be made. The so-called Slater-determinant (SD) or configuration is an anti-symmetrized product of all one-electron-wave functions. It fulfils the condition that the electrons are interchangeable but that upon such an interchange the wave function needs to change sign. This rule is dictated by the Pauli-principle for fermions. χ1 (1) χ1 (2) . . . χ1 (N ) 1 χ2 (1) χ2 (2) . . . χ2 (N ) SD ψ (1, 2, . . . , N ) = √ .. .. .. N . . . χN (1) χN (2) . . . χN (N )

(2.8)

If only one such SD is used as an approximation to the exact wave function, this procedure is called Hartree-Fock approximation, the corresponding energy is the Hartree-Fock energy. Z

ψ SD H ψ SD dV = E HF

(2.9)

If this procedure is used within the framework of the linear variation principle the resulting energy will always be higher than the exact energy (exact within the BO approximation and if relativistic effects are omitted):

E HF ≥ E0

(2.10)

Still, one problematic term is left – the two-electron potential. It is approximated by the Hartree-Fock-Potential VHF in that way that each electron experiences a mean field

26


created by all other electrons around it. That way the complicated term

1 r12

is evaded.

On the other hand, only the so-called Fermi correlation between electrons of parallel spin is taken into account, the contribution for the Coulomb correlation between electrons of anti-parallel spin is missing. In total, this leads to an uncorrelated wave function. The Hartree-Fock potential itself can be separated into the Coulomb operator Jij and the exchange operator Kij which are defined by the way they work on the orbitals χa and χb : VHF =

XX i

Jij − Kij

(2.11)

j

"Z

# 1 Ji (1) · χa (1) = χb (2) χb (2) dr2 · χa (1) r12 "Z # 1 Ki (1) · χa (1) = χb (2) χa (2) dr2 · χb (1) r12

(2.12) (2.13)

While the Coulomb operator corresponds to the classical repulsive interaction between two electrons, the exchange operator does not have a classical equivalent. It results from the principle of antisymmetry and describes a pseudo-interaction between the electrons. With these approximations it is now possible to define the Hartree-Fock equations

fiHF = hi +

n X

(Jij − Kij )

(2.14)

j

fiHF φi = εi φi

(2.15)

with Fock operator fiHF , molecular orbitals φi , resulting energy εi and index i for the electrons. As the Fock operator itself depends on the orbital wave function via the Hartree-Fock potential these equations can only be solved iteratively. As the true wave function is unknown it is described by a linear combination of atomic orbitals (LCAO-ansatz).

φ=

X i

ci ψ i

(2.16)

2.2 The Hartree-Fock Approximation

27

This ansatz allows to write the Hartree-Fock equations in Roothaan-Hall notation as a matrix equation:

FC = SC

(2.17)

in which F is the Fock matrix, which contains the expectation values of the Fock operator for the different atomic orbitals, eigenvector C contains the LCAO coefficients, energy vector contains the eigenvalues of F, and S is the overlap matrix, which results from the non-orthogonality of the atomic orbitals. This equation can now be solved iteratively by creating a first Fock matrix from the starting orbitals, calculating new orbitals from this Fock matrix and so on until – within certain thresholds – no more change can be observed. This procedure is called self-consistent field (SCF) procedure. The resulting formula for the Hartree-Fock energy can be written as

E HF =

n X

n

εi −

i

εi = hi +

1X (Jij − Kij ) 2 i,j

n X

(Jij − Kij )

(2.18)

(2.19)

i,j

The atomic orbitals for the LCAO ansatz are also called basis functions. They are collected in so-called basis sets of different sizes. The size of a basis set determines the accuracy but also the computational time for a calculation. With a (hypothetical) complete basis set containing infinitely many basis functions it would be possible to obtain the exact energy for a system (within the mentioned approximations). This energy is called the Hartree-Fock limit.

2.2.1

The Correlation Energy

Even at the Hartree-Fock limit the resulting energy will never be the same as the true, exact energy of the system. The difference between them is called the correlation energy.

28


ECorr = Eexakt − EHF

(2.20)

This contribution contains e.g. the part of the electron-electron interaction which is missing in the Hartree-Fock approximation because of the mean field ansatz. It is still one of the main challenges of quantum chemistry to find a way for the exact calculation of this term. Meanwhile, several different approaches and approximations are used some of which will be shortly introduced in the next chapters.

2.3 Density Functional Theory (DFT)

2.3

29

Density Functional Theory (DFT)

Whereas in Hartree-Fock theory it is assumed that all information about a system is stored in its wave function, another approach defines the electron density as the basic property. This approach has the advantage that its results can be directly compared to experimental data – the electron density can be observed, e.g. by X-ray diffraction, while the wave function is not observable. At the same time, all properties of a system can also be determined from the electron density. One example is the total number of electrons in the system which is nothing else than the integral over the electron density; another example are the cusps which specify the position of the nuclei and, via the slope in the cusp region, the atomic number. The connection between electron density and properties like the energy of a system is made by a so-called density functional

Eexact = F exact [ρ (e)]

(2.21)

as could be shown in the first Hohenberg-Kohn theorem form 1964. 23 Unfortunately, this true, exact functional is unknown. Additionally, it could be shown that the energy given by this functional is variational – this means that even with the exact functional the exact energy can never be reached as long as only a trial density is used (second HohenbergKohn theorem). 23 The functional for the electronic energy can be divided into several parts, a kinetic energy functional T [ρ], a functional for the interaction between nuclei and electrons which has basically the same form as described for the HF-method and a functional for the electronelectron interaction, which can again be splitted into a Coulomb part J[ρ] and an exchange part K[ρ].

30


E[ρ] = T [ρ] + Eee [ρ] + EN e [ρ] = T [ρ] + J[ρ] + K[ρ] + EeN [ρ] M Z X ZA (RA )ρ(r) EeN [ρ] = − dr |r − r0 | A Z Z 1 ρ(r)ρ(r0 ) J[ρ] = drdr0 2 |r − r0 |

(2.22) (2.23) (2.24) (2.25)

Two possible formulas for the kinetic energy functional and the exchange functional were suggested by Thomas and Fermi in 1927 and by Slater and Dirac in 1951, respectively:

Z 5 3 √ 3 4 · 9π · ρ 3 (~r) d~r [ρ] = 10 r Z 4 3 3 3 SD · ρ 3 (~r) d~r K [ρ] = − · 4 π

F TUTEG

(2.26) (2.27)

Both formulations are based on the model of a uniform electron gas (UEG) which resembles the situation in metals but is not applicable for chemical problems. One basic drawback of this formulation is that is not able to describe or predict chemical bonds. At the same time it is worth mentioning that the formulations presented up to here are orbital free and in principle exact (for the UEG – to be truly exact also in real systems the exact functional would be necessary).

2.3.1

Kohn-Sham DFT (KS-DFT)

One major reason for the poor performance of the UEG-based formulations is the bad description of the kinetic energy contribution. To solve this problem Kohn and Sham suggested an approach which is very similar to Hartree-Fock theory. In their approach a model system of non-interacting electrons (Vee = 0) but with the same electron density as the real system is assumed and orbitals are reintroduced. Defining the Kohn-Sham operator f KS and the exchange-correlation potential vXC it is possible to formulate the Kohn-Sham equations which, like the Hartree-Fock equations, can be solved iteratively:


fiKS [ρ]φi

= hi [ρ] +

31

X

Jij [ρ] + vXC [ρ] φi = εi φi

(2.28)

j

The KS-operator is an effective one-electron operator like its relative, the Hartree-Fock operator. One major difference between both is the exchange part, which in the KSapproach is replaced by the exchange-correlation potential as functional derivative of the exchange-correlation functional:

vXC [ρ] =

δEXC [ρ] δρ

(2.29)

If a sufficiently good approximation to EXC is found, electron correlation effects as well as up to 99% of the true kinetic energy can be obtained with this KS-DFT approach at about the same computational cost as for a Hartree-Fock calculation. Unfortunately, there is no such thing as a free lunch or, in terms of DFT, as an exact formulation for EXC and it is not even possible to improve its accuracy systematically. The usual procedure is to develop a functional, test it with the help of existing sample sets and compare the results to those of existing functionals – a trial and error approach. Still some improvements are possible which will be described in the next section. One known problem of DFT is a phenomenon called self-interaction error (SIE). This error arises from insufficient cancellation of Coulomb and exchange contributions for the same electron which leads to a contribution for the interaction of this electron with itself. By construction, this error cannot occur in HF-theory and the inclusion of Fock-exchange into the functional is one possible remedy. The long-range description of the potential is also not correct in most density functionals which results in an erroneous description of long-range exchange-correlation effects. In the last years so-called range-separated functionals have been introduced to take care of this effect, but they are not yet applicable to all chemical problems. 24 This is also a general problem in DFT – most functionals were designed for a certain purpose like accurate description of thermodynamics, kinetics, solid state properties or excited state properties. A generally applicable functional which performs equally well in all areas one can imagine has not yet been found. 25

32


Another long-range effect which is badly described by standard DFT is dispersion interactions – special corrections to account for the missing disperion contributioin will be discussed in section 2.3.3.

2.3.2

Jacob’s Ladder

One possibility to sort density functionals is according to their complexity or accuracy. John Perdew illustrated this by citing a passage from the Bible which describes Jacob’s dream:

“ And he dreamed, and behold, there was a ladder set up on the earth, and the top of it reached to heaven. And behold, the angels of God were ascending and descending on it!” (Genesis 28,12).

In Perdew’s picture, the ladder has one step for each rung of density functional theory and leads from ‘Hartree hell’ to the ‘heaven of chemical accuracy’. 26

First Rung - LDA The lowest rung of the ladder is the local density approximation (LDA). As the name suggests, it is based on a local approach saying that the energy density at one point in space is fully described by the electron density at this very point, interactions to other points are not considered. At the same time, the conditions of the UEG are assumed. One possible formulation is the Slater-Dirac formulation already described above. A generalization, the local spin density approximation (LSD or LSDA), uses the separated electron densities of α- and β-spins. Although results obtained by both approximations offer a some improvement over HFresults, this improvement often stems from error cancellation effects. Atomisation energies are usually overestimated and, as mentioned before, chemical bonds cannot always be described sufficiently. 20 At the same time, good results can be obtained for metallic systems like sodium which explains the common use of LDA and LSDA in solid state physics.


33

Second Rung - GGA A significant improvement over LDA is possible if the gradient of the electron density is included in the formulation. That way, it is possible to account for the drastic variations in the densities of molecules. This method is called generalized gradient approximation or GGA and can be generally described as:

GGA [ρ] EXC

Z =

f (ρα (~r), ρβ (~r), ∇ρα (~r) , ∇ρβ (~r) ) d~r

(2.30)

For both exchange and correlation part different formulations and parametrisations are available to combine to different functionals. Common examples are BLYP which contains an exchange functional by Becke 27 and the correlation by Lee, Yang and Parr 28,29 or the PBE functional of Perdew, Burke and Ernzerhof. 30 Third Rung - meta-GGA Seeing the improvement from the inclusion of the first derivative, it is obvious to also try and include the second derivative of the electron density. This is one possible approach for the formulation of a meta-GGA functional. Another approach is to include a term for a more accurate description of the kinetic energy. Both types of functionals are located on the third rung and provide a sufficiently good description of several properties like atomisation energies. TPSS 31 , as one example for this rung of functionals, has also proved to yield rather accurate geometries. For the systems investigated in this thesis it was therefore used in most geometry optimisations. Fourth Rung - hybrid-GGA Hybrid-GGA functionals are also based on GGA functionals but include an additional contribution of ‘exact’ Fock-exchange (based on KS-orbitals) to improve on the description of the exchange contribution. This non-local contribution leads to a significant improvement in the accuracy of molecular energies. Analogous to GGA functionals, a large variety of different exchange and correlation parts exists which can be combined to various hybrid-GGA functionals. A famous and commonly used example is the B3LYP functional, 32,33 a relative to the GGA functional BLYP, which

34


contains three empirical parameters determining the fraction of Fock-exchange, Becke exchange and of the two different correlation types:

B3LY P LSD HF B88 EXC = (1 − a) EX + a EX + b EX + c ECLY P + (1 − c) ECLSDA

(2.31)

∧ a = 0.20; b = 0.72; c = 0.81

Fifth Rung - DHDF While the fourth rung of Jacob’s ladder introduced occupied KS-orbitals, the fifth rung introduces the corresponding virtual KS-orbitals. If this is done by adding a perturbative correction to the correlation part, the resulting functionals are called double hybrid density functionals (DHDFs). One example of this class of functionals is B2PLYP 34,35 which includes 73% LYP-correlation and 27% correlation from a perturbative treatment of second order. In this thesis it is mostly used to calculate accurate electronic energies (a geometry optimisation on this level of theory would not be feasible due to the high computational costs for systems of FLP-size).

B2P LY P B88 HF PT2 EXC = 0.47 EX + 0.53 EX + 0.73 ECLY P + 0.27 EXC

2.3.3

(2.32)

Dispersion-corrected DFT

As already mentioned above, most density functionals show an incorrect behaviour in the asymptotic region of the potential. This especially affects the correct description of longrange interactions like dispersion interaction. Several approaches have been published to remedy this problem (for an overview see the review ref. 36) out of which only the DFTD approach shall be introduced here. Designed to be easily applicable without any large additional computational cost, a dispersion term is calculated which can simply be added to the result of a standard DFT calculation.

E DF T −D = E DF T + E disp.

(2.33)


35

This scheme was developed by Stefan Grimme in 2004 37 and modified in 2006 38 to give what will be called “DFT-D” in the following. A third version called “DFT-D3” was published in early 2010 39 and, slightly later, a modified version which will be referred to as “DFT-D3(BJ)” 40 was presented. Each of these versions shall now be introduced shortly. The DFT-D Correction The correction scheme from 2006 includes a global scaling factor s6 which depends on the functional, the dispersion coefficient for atom pair AB (C6AB ), the interatomic distance DF T −D (RAB ) and a damping function (fdamp ):

1 X C6AB DF T −D D Edisp. = − s6 fdamp (RAB ) 6 2 A6=B RAB

(2.34)

In this formulation a ‘perfect’ C6 -value would have a value of one as it assures the ideal −6 RAB behaviour. The C6AB values were determined semi-empirically from element-specific

C6A values: C6AB

=

q

C6A C6B

(2.35)

DF T −D The above formula also contains a damping function fdamp (RAB ). This damping func-

tion is necessary to avoid near-singularities for small values of R and double counting effects of correlation at medium distances and reads

DF T −D fdamp (RAB ) =

1 1+

e−d(RAB /Rr −1)

(2.36)

with the damping parameter d = 20 and the sum of atomic van der Waals-radii Rr . A more detailed description of this dispersion correction can be found in the original publication from 2006 (reference 38). This dispersion correction, sometimes also called “D2” to distinguish it from the original version from 2004, has been proven rather successful in the chemical community with almost 2000 citations since it was published.

36


The DFT-D3 Correction The basic formula for the newer D3 dispersion correction is in principle very similar to the just described one. It reads

D3 Edisp. =−

1 X X CnAB DF T −D3 sn n fdamp,n (RAB ) 2 A6=B n=6,8 RAB

(2.37)

The first, obvious difference is that now not only multipoles of sixth order, but also of eighth order are included. This is important to account for medium-range correlation effects. A second important difference which may not be visible at first glance is the origin of the dispersion coefficients CnAB which now for every single atom in the molecule depend on its coordination sphere. For most common functionals the scaling factor s6 is set to unity to assure the correct asymptotic behaviour, whereas s8 is fitted for each functional. The damping function used here is a relative to the damping function in DFT-D:

DF T −D3 fdamp,n (RAB ) =

1 AB −1)

1 + e−γ(RAB /sr,n R0

(2.38)

in which R0AB is a cut-off radius for atom pairs AB. Here, sr,6 is fitted to the functional and sr,8 is set to unity. The constant γ determines the steepness of the function for small distances where it goes to zero. Because of this behaviour this damping version is also called zero-damping. 40 In total, two parameters need to be fitted for each functional in DFT-D3. One special case of functionals are the functionals of the Minnesota family, namely M05 and M06 as well as most of their flavours. These functionals are highly parametrised and contain already a certain amount of medium-ranged dispersion interaction. A combination with the presented D3 correction would therefore lead to double-counting effects and even worse energy values. For this reason, the parameters for the Minnesota functionals have been chosen slightly differently by setting the s8 to zero. With this modified version of the zero-damping, it is possible to increase the accuracy also for this group of functionals.


37

The DFT-D3(BJ) Correction The latest version of the D3-dispersion correction is marked by the suffix “-D3(BJ)”. In contains a different, rational damping function as developed by Becke and Johnson 41 which leads to a constant contribution to the dispersion energy for each pair of spatially close atoms. For this variant the formula is slightly different from the two presented above:

D3 =− Edisp.

1X X CnAB sn DF T −D3(BJ) n 0 2 A6=B n=6,8 RAB + fdamp (RAB )n

(2.39)

with the damping function and cut-off radius s DF T −D3(BJ)

fdamp

0 0 + a2 ) = a1 RAB (RAB

∧

0 = RAB

C8AB . C6AB

(2.40)

The parameters a1 and a2 as well as s8 are fitted. For the Minnesota functionals this version of D3 is not available since they proved incompatible during the fitting procedure.

38

2.4 2.4.1


Wave Function Theory Perturbation Theory

In many-body perturbation theory (PT or MBPT) it is assumed that if the solution to a problem is known, it is possible to obtain the solution for a similar problem by adding a (usually small) perturbation to the known problem and evaluating correction terms. This means that the Hamiltonian H for a test system can be obtained by adding a perturbation V to a Hamiltonian H0 of a known and similar reference system. 21,20 H = H0 + λV

(2.41)

H ψ = Wψ

(2.42)

The parameter λ determines the strength of the perturbation and with this parameter it is possible to write energy W and wave function ψ as Taylor expansions in λ. Assorting the expansion terms according to the power of λ gives:

λ0 : H0 ψ0 = W0 ψ0

(2.43)

λ1 : H0 ψ1 + V ψ0 = W0 ψ1 + W1 ψ0

(2.44)

λ2 : H0 ψ2 + V ψ1 = W0 ψ2 + W1 ψ1 + W2 ψ0 n X λn : H0 ψn + V ψn−1 = Wi ψn−i

(2.45) (2.46)

i=0

Which leads to the general formula for the energy:

Wn = hψ0 |V |ψn−1 i

(2.47)

If in this formulation the unperturbed Hamiltonian is selected to be the sum over Fockoperators, this method is called Møller-Plesset perturbation theory, commonly abbreviated as MPn in which n gives the order in which the Taylor expansion has been truncated.

2.4 Wave Function Theory

39

As in this formulation the averaged electron-electron interaction is included twice (compare formulas 2.14 and 2.18), the perturbation must be defined as the difference between the potentials for the test system and the reference system:

V = H − H0

(2.48)

For this reason V is also called the fluctuation potential. Unfortunately, it can no longer be called a small perturbation, but it has the advantage of making this theory size consistent and size extensive. This means that an infinitely far separated dimer of a molecule should posses exactly twice the energy of the monomer and that the energy should be proportional to the system size. Although these two principles appear straight forward they are not valid for all available methods as will be pointed out later. As the zeroth order energy term is only the sum of the MO-energies

E M P 0 = W0 =

X

εi ,

(2.49)

i

the first order correction to the energy is necessary to reach the HF-energy:

E M P 1 = W0 + W1 X = εi + hψ0 |V |ψ0 i

(2.50)

i n

=

X i

n

X 1X εi + (Jij − Kij ) − (Jij − Kij ) 2 i,j i,j

= E HF

The first new contributions to correlation energy are thus included by the second order energy and wave function corrections. If canonical orbitals are used, only double excitations have to be considered as single excitations do not contribute according to Brillouin’s theorem. 21 The second order correction term to the energy thus becomes:

40


W2 =

occ. X virt. X hψ0 |V |ψijab ihψijab |V |ψ0 i i