Theoretical Chemistry for Chemists

Natural Science

Tobias Grömke

Everything the 21th Century Chemist Should Know about Theoretical Chemistry

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This book at GRIN: https://www.grin.com/document/373462 Tobias Grömke

Theoretical Chemistry for Chemists

Everything the 21th Century Chemist Should Know about Theoretical Chemistry

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Visit us on the internet: http://www.grin.com/ http://www.facebook.com/grincom http://www.twitter.com/grin_com Inhaltsverzeichnis 0. PREFACE...... 4

0.1. SHORT DESCRIPTION, TARGET AUDIENCE, REQUIRED PREKNOWLEDGE...... 4

0.2. INTRODUCTION TO THE USED PATHWAYS OF ARGUMENTATION AND EXAMPLES FOR THE

NEED OF UNDERSTANDING MORE ABOUT THEORETICAL CHEMISTRY...... 8

1. OVERVIEW ABOUT THEORETICAL-METHODS...... 13

2. BASIS SET...... 20

2.1. INTRODUCTION...... 20

2.2. PICTORIAL-IMAGINATION OF HILBERT-SPACE...... 20

2.3. ORTHOGONALITY AND RELATIVE BEHAVIOR BETWEEN DIFFERENT SOLUTIONS OF THE

SCHRÖDINGER-EQUATION...... 24

2.4. FIRST LOOK AT REAL BASIS-FUNCTIONS AND APPEARING INTEGRALS...... 26

2.5. THE SCHRÖDINGER-EQUATION FOR THE HYDROGEN-ATOM WITH A GAUSSIAN-ANSATZ..28

2.6. GROUND-PRINCIPLES OF MODERN BASIS-SET...... 34

2.6.1. CONTRACTED-GAUSSIANS...... 34

2.6.2. BASIC TERMS OF MODERN BASIS-SET...... 40

3. POST-HARTREE-FOCK-METHODS...... 43

3.1. THE HARTREE-PRODUCT...... 43

3.2. INTRODUCTION TO SLATER-DETERMINANTS AND PROPERTIES OF FUNCTIONALS...... 52

3.2.1. CONSTRUCTION OF SLATER-DETERMINANTS FROM SPIN-ORBITALS...... 52

3.2.2. FUNCTIONALS...... 53

3.3. HARTREE-FOCK-THEORY...... 54

3.3.1. INTRODUCTION, DEEPENING OF OUR COMPREHENSION OF HILBERT-SPACE AND

EIGENVALUE-EQUATIONS...... 54

3.3.2. DERIVATION OF THE HARTREE-FOCK-EQUATIONS...... 58

3.3.3. THE HARTREE-FOCK-EQUATIONS...... 63

3.4. APPLICATION OF THE HARTREE-FOCK-THEORY...... 69

3.4.1. ROOTHAAN-HALL-EQUATIONS...... 69

3.4.2. OPEN-SHELL-CALCULATIONS, CONFIGURATION-INTERACTION, FULL-CI AND

1 CORRELATION-ENERGY...... 70

3.4.3. COUPLED-CLUSTER-THEORY AND INTRODUCTION TO APPLICATION OF PERTURBATION-

THEORY...... 74

5. FUNDAMENTAL PRINCIPLES OF DENSITY-FUNCTIONAL-THEORY AND VARIATIONS OF DFT-

METHODS...... 94

6. OTHER THEORIES...... 97

6.1. SEMI-EMPIRICAL-METHODS...... 97

6.2. QUANTUM-THEORY AND BASIS-SET IN SOLID-STATE-PHYSICS (AND ESOTERIC☺)...... 99

6.3. HOW TO FIND THE BEST THEORETICAL METHOD,...... 103

NECESSARY FOR MY DESIRED ANSWER AND HOW TO CHOSE THE OPTIMAL FUNCTIONAL

FOR A DFT-CALCULATION WITH MY SYSTEM-PROPERTIES?...... 103

7. THE MOST IMPORTANT CONCLUSION FROM MODERN...... 107

THEORETICAL-CHEMISTRY...... 107

8. IMPORTANT CALCULATION-TYPES...... 108

8.1. INTERACTION-ENERGY, POTENTIAL-ENERGY-SURFACES AND COUNTERPOISE-CORRECTION ...... 108

8.2. GEOMETRICAL OPTIMIZATIONS VS. VARYING COEFFICIENTS FOR ENERGY-MINIMS IN

OTHER CALCULATIONS,...... 110

DEFINITION OF THE DIFFERENT VARIATIONS...... 110

8.3. DENSITY-CALCULATIONS AND OTHER POSSIBLE CALCULATIONS FOLLOWING THE

GEOMETRY-OPTIMIZATION...... 112

8.4. SPECTRA-CALCULATION; PRE-CALCULATIONS AND SIMPLIFICATION USING GROUP-

THEORY OF SYMMETRY-OPERATORS AND SYMMETRY-ELEMENTS, SYMMETRY-ADAPTED-

METHODS...... 113

9. MOLECULAR DYNAMICS/MONTE-CARLO-SIMULATIONS AND FUTURE-PERSPECTIVES OF

THEORETICAL CHEMISTRY...... 115

10. LITERATURE...... 116

11. ACKNOWLEDGEMENTS...... 125

2 Theoretical Chemistry for Chemists

Everything the 21’th-Century-Chemist definitely should know about Theoretical-Chemistry.

by B.Sc. Tobias Grömke

3 0. Preface 0.1. Short Description, target audience, required Preknowledge

This small book is condemned to graduate students of Chemistry – from practical path-ways of chemistry, (like organic-synthesis for instance) who already watched some, or like it’s most time the case, just one theoretical lecture(s) during there studies, but (like it’s also most time the case) with some successfully absolved test at the end of that lecture, most time far in the past ;) – who want to understand more while reading articles of theoretical chemistry and who simply want to understand all those abbreviations in papers of chemistry concerning with used methods theoretical chemistry or who wand to use theoretical calculations as addi- tional help for there work. I think it can also be a first orientation for bachelor students who think about deepening in theoretical chemistry, but who aren’t sure if they are able to.

I’ll give some overview about Post-Hartree-Fock-Methods (Coupled-Cluster (incl. Example for Application of Perturbation-Theory), Full-CI, explicitly correlated methods) Density-Functional-Theory (Basic Equations, reason of lower computational cost, important Types of Functionals (LSD-Functionals, GGA-Functionals, Hybrid-Functionals), Important points in searching the right method), Force-Field-methods (Basic Theory, Basic Equations, practical tips as tool in quantum-chemical Calculations), theoretical Solid-State Physics (differences to quantum chemical equations, special behavior of solid-state-systems, atomic groups with single-particle-behavior – like phonons) Role of special Techniques (Perturbation Theory, Group Theory) and it shows the connections of those techniques to molecular dynamics.

Because of the not linear introduction of terms and concepts, used and derived from quantum physics and sometimes very special details of higher mathematics, other readers could become confused, already very early in the text.

4 I’m giving an imagination and an overview deep enough to understand coher- ences, differences, borders and benefits of all the different available high-level- ab-initio-calculations – which sometimes need months until giving first results, even in big calculation-clusters, using hundreds of CPUs – it’s important to intro- duce some mathematics. Every necessary mathematical detail will be presented, the way for solving the Schrödinger-Equation with a gaussian function in atomic units will be shown, while introducing quantum-chemical notation (bracket-style), and before Hartree-Fock-Theory will be derived from Slater- Determinants, I try to commit you my pictorial view of Hilbert-space and basis- set of quantum chemical calculations. After my Prof. and mentor said me he thinks that peoples who have new theoretical ideas for Theory, always have pictures in there mind, I think the pictures in my mind really can help other chemists. And try to give you an overview about influences of basis set to quantum chemical calculations and new insights from using bigger basis sets for quantum-chemical-calculations. The mathematics are really necessary, otherwise the differences between the methods can’t be understood, but it’s enough for that book If you have seen what is done and can approximately imagine how much the calculation steps become more if you go to more exact theory.

I’ll show why theoretical chemistry is so important for the future-developing of chemistry. If we look which complex structures nature is already using for catalyz- ing reactions we’ll easily understand why there’s still much we need to understand for that and why theoreticians and chemists synthesis-focused areas should work more together and why PhD-students of such different parts of our beloved science should have a look at the possibilities of theoretical chemistry from the beginning an of there scientific work.

I think my book can only work if the reader (just) has heard the terms: n-dimen- sional, Hilbert-Space, Basis-Set (maybe without understanding more than that it

5 is something used in some way at theoretical calculations), Force-Field-Calcula- tions/Methods, Gaussian-functions, e-functions, the Schrödinger-Equation, the Nabla-Operator, Coordinate-Transformation, Matrix, Complex Numbers, Atomic Units*, Bracket- or Operator-notation in Quantum-Chemistry.** As already mentioned, he can have heard about all those terms very far in the past and without understanding everything and also without understanding very much. But that reader should need more knowledge about theoretical chemistry in some way and because he don’t wants to search on his own in different books, articles, web- sites and with using secondary and tertiary literature if he was confronted by something he absolutely wasn’t able to understand and what don’t just wanted to believe.

This book should definitely has to be read successively and not be used as refer- ence book. If you first have loan that book, from your local library, you can later buy the ebook-version, with some active some hyperlinks, to some helpful literature-sources. The moment will come, were you aren’t remembering everything you read here.

I promise, If you read this book over maximum two or three (because of to much work) weeks every day minimum a little bit and let you time to thinks about a part you absolutely can’t understand. I got the tip that even professors need sometimes more than one day for just one page at a complicated book, so it was Ok for me to need more time. You really shouldn’t need days for a page, if you spend an half hour every day or something like that.

For not completely bore you until you reached the end of the book, beside a story is hidden, about circumstances the made me writing and the wish to have the last word in a discussion during some studentic-helper-job, which also strongly drove

*In atomic units, e.g., the energy is given in Hartree. 1 Hartree is 4,3598x10-18 Js **If there where just one or two terms he absolutely can’t remember, he can google now and than he still will have success with that book

6 me to finish that book. I think in that case, the result is sanctifying that driving force. In this discussion, I had always a good reasons to be sure that I’m going the right way with my work and arguments, because my Professor and mentor all the time found a moment to answer my questions and so I was all the time able to have a competent assessment of my ideas and knowledge.

Who am I? what’s my qualification? I always was reading stuff about that matter and I always was interested. In periods between regular work and studies I con- centrated my self-studies of higher mathematics. Later I started deepening in theoretical chemistry with very good results and by the way, I was restless going addicted to higher mathematics and theoretical theories describing the quantum- level of matter, light and so on. During my self studies I had all the time single- appointments with my Quantum-Professor, PhD-Students in Quantum-Chemistry, PhDs and Dr. of mathematics, to get answers to my questions and to proof if my new understanding is right. T To understand mathematics deeper and deeper, and in following to have the ability to read literature for theoreticians about theoretical methods with a satisfying yield of information, gave me self esteem because over the years I was staying nerved by fact that I never understood more. Other chemists told me that they feel the same and are nerved by the same fact and so I hope you’ll feel the same while understanding enough to work with the methods presented in my book.

I directly talk open about possible fears everybody might have an sometimes I make jokes for loosening up. I have my own style.

7 0.2. Introduction to the used Pathways of Argumentation and examples for the need of understanding more about Theoretical Chemistry

Today in the most scientific articles and presentations theoretical calculations are included and experience shows that especially after finishing master studies there's an information-deficit about theoretical chemistry among a huge part of practical working chemists.

The following remarks want to help finding entrance to theoretical chemistry. The main task is to mediate the extent of available methods and to explain what these methods can do on which accuracy and what not. Without any insight in the underlying theory of the used methods of theoretical chemistry there's danger to take informations from a calculation which can’t be taken from that calculation, because desired properties are neglected in the underlying theory.

An example: You have the results of a force-field-calculation, which is relatively fast available, for a complex of two organic compounds with π-systems, donor- acceptor-positions for hydrogen-bridges to nitrogen- or oxygen-atoms on both compounds and from the received geometrical data you conclude on which positions in your complex you have π-π-interactions and hydrogen-bonds stabilizing your system. You have chosen a theoretical method cause your real system can’t be directly observed because your system works on a surface like color from a printer on a paper. Your color-complex-system works in an organic solution but not on the paper. Your problem is now that your forcefield-calculation isn’t capa- ble to directly describe hydrogen bonds or π-π-interactions, because it uses classical mechanics and don’t calculates electrons with quantum-mechanical behavior. It only can simulate the development basing on parameters from other calculations. For very big Systems like biological systems, these methods are often the only possibility, for smaller systems they can’t be used to state something or

8 answer a new question, because it’s just saying that something is possible, that it is maybe possible you have already known before. Like all geometrical calculations this method searches for a low energy-geometry and for solid structures the received geometries normally don’t distinguish very much from more exact quantum-chemical calculations, only atom-atom-distances differ a little bit but the ground-geometry is the same. But in solution or at gaseous state there is big amount of other stable low-energy-geometries and they only can be found with more exact quantum-chemical methods. To calculate the energies of these or other possible geometries you need to calculate the influence of electronic structures. Electrons are moving to fast to handle them with classical mechanics. Every possible geometrical structure of a molecular system has another energy. Energy-minima, depending on geometrical orientation need to be able to calculate these structures.

Let’s try to think about two molecules in a coordinate system, one is fixed at the origin. The other molecule is placed at different positions on a spherical surface relative to our fixed molecule, rotated around all possible molecular axes and for a range of distances, to everyone of these different geometrical orientations the energy for your molecular-system is calculated. The results of all those calculations together are values of the so called Potential-energy-surface of your system, depending on various geometrical parameters. Local minima on that Poten- tial-energy-surface are caused by all the possible intermolecular interactions which only can be calculated with quantum-chemical methods. And all those geometries really appear in our systems, depending on the conditions, like aggregate state, solvating molecules and so on. Possible gaseous-phase-energy-minima of the Potential-energy-surface and they can be stabilized by solvating molecules and reach a lower energy-state as that one of our force-field-geometry. So for our investigated system we need more accurate calculations to really answer the questions which can’t be answered using available instrumental analyt-