Theoretical Chemistry for Chemists

Inhaltsverzeichnis

0. Preface
0.1. Short Description, target audience, required Preknowledge
0.2. Introduction to the used Pathways of Argumentation and examples for the need of understanding more about Theoretical Chemistry

1. Overview about Theoretical-Methods

2. Basis set
2.1. Introduction
2.2. Pictorial-Imagination of Hilbert-Space
2.3. Orthogonality and relative behavior between different Solutions of the Schrödinger-Equation
2.4. First look at real Basis-Functions and appearing Integrals
2.5. The Schrödinger-Equation for the Hydrogen-Atom with a Gaussian-Ansatz
2.6. Ground-Principles of modern Basis-Set
2.6.1. Contracted-Gaussians
2.6.2. Basic Terms of modern Basis-Set

3. Post-Hartree-Fock-Methods
3.1. The Hartree-Product
3.2. Introduction to Slater-Determinants and Properties of Functionals
3.2.1. Construction of Slater-Determinants from Spin-Orbitals
3.2.2. Functionals
3.3. Hartree-Fock-Theory
3.3.1. Introduction, deepening of our Comprehension of Hilbert-Space and Eigenvalue-Equations
3.3.2. Derivation of the Hartree-Fock-Equations
3.3.3. The Hartree-Fock-Equations
3.4. Application of the Hartree-Fock-Theory
3.4.1. Roothaan-Hall-Equations
3.4.2. Open-Shell-Calculations, Configuration-Interaction, Full-CI and Correlation-Energy
3.4.3. Coupled-Cluster-Theory and Introduction to Application of Perturbation-Theory

5. Fundamental Principles of Density-Functional-Theory and Variations of DFT-Methods

6. Other Theories
6.1. Semi-Empirical-Methods
6.2. Quantum-Theory and Basis-Set in Solid-State-Physics (and esoteric☺)
6.3. How to find the best theoretical method, necessary for my desired Answer and how to chose the Optimal Functional for a DFT-Calculation with my System-Properties?

7. The most important Conclusion from modern Theoretical-Chemistry

8. Important Calculation-Types
8.1. Interaction-Energy, Potential-Energy-Surfaces and Counterpoise-Correction
8.2. Geometrical Optimizations vs. Varying Coefficients for Energy-Minims in other Calculations, Definition of the different Variations
8.3. Density-Calculations and other possible Calculations following the Geometry-Optimization
8.4. Spectra-Calculation; Pre-Calculations and Simplification using Group-Theory of Symmetry-Operators and Symmetry-Elements, Symmetry-Adapted-Methods

9. Molecular Dynamics/Monte-Carlo-Simulations and Future-Perspectives of Theoretical Chemistry

10. Literature

11. Acknowledgements

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 additional 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-Theor y (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.

I’m giving an imagination and an overview deep enough to understand coherences, 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 introduce 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 catalyzing 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-dimensional, Hilbert-Space, Basis-Set (maybe without understanding more than that it is something used in some way at theoretical calculations), Force-Field-Calculations/Methods, Gaussian-functions, e-functions, the Schrödinger-Equation, the Nabla-Operator, Coordinate-Transformation, M atrix, 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, websites 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 reference 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 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 concentrated 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.

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 capable 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 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 Potential-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 analytics. Even if the more exact methods find a similar geometry as energy-minimum like the force-field-calculation. First if you stated your geometry with quantum-mechanical calculations and tried the most exact available option you can really make a scientific statement about mechanisms happening in your system, even in solution before crystallization.

Quantum Physics can't be understood without mathematics and how chemistry is, out a philosophical point of view, a part of physics, theoretical chemistry is a part of quantum physics and quantum-physics are based on theories using higher mathematics at a really complex level. But there are good news for all readers who nevertheless want to understand quantum chemistry:

All the used mathematics are, reduced to there ground, just known operations like: Differential and integral calculus, vector algebra, some matrix calculation but on a basically view we are just adding rows and columns all the time and in the end not we but the computer is calculating. These computers are also the reason why nobody has to be scared about complex vector space. Our computer does handle it. Just if you want to develop new methods and become a theoretical chemist you have work with complex-vector-functions in multidimensional cases.

Out of those “simple” basic operations mathematics can construct complex buildings and so more names are needed to categorize different types of such complex mathematical buildings. Something like we are building the variation of a functional sounds impressive but reduced to the basic operation it's a differentiation and that’s a topic every student knows. Really understanding these buildings is a little bit tricky and not everybody's taste but that's not necessary for knowing why a Coupled Cluster Calculation needs a million of complicated Integrals for every tested geometry and therefore much computational time/cost and why other calculations are much faster.

Most authors, introducing quantum chemistry, start presenting some Equations and bring slowly more mathematical detail. I'll try a little bit different way, but don't be scared ;)

The main target of this book is to develop a pictorial imagination of the modern quantum chemical description of molecular systems and to introduce available methods deep enough to understand the content of the calculation so that the reader becomes an imagination of the computational cost. Mathematical details will be introduced directly when they are needed but just at a level, every graduated student of chemistry is definitely able to understand.

First time I saw the complete definition of a Gaussian basic function, I was on my bachelor-thesis. Even in lectures for advanced students we just studied more simple cases. The bigger part of all chemistry-students has not even once seen such a simple case of a used basis function in context of theoretical chemistry. Because Theoretical chemistry II is normally followed by two or three students. But these functions are nothing to fear and so the first mathematically defined gaussian-basis-function you’ll see in this book will be the complete one.

We know now all types of such functions. Even at scholarship we had to work with them until vomiting. And the best is that we don’t have to handle them. Computer does it for us. In this book we’ll work with some much more simple examples to develop our understanding and the reader needs just to watch. But every chemistry-student should have learned enough mathematics to imagine how much more calculation steps are necessary if a used function is more complex then a simple e-function, because he already was pained several times with such calculation in lessons or lectures of mathematics.

After reading my mental outpourings the reader should definitely be able talk with theoretical chemists, he should be able to know which interpretation is possible from the result of a calculation and after reading some user-articles, he can understand now at a sufficient level he should be able to decide which method and which time exposure is necessary to answer his questions.

Why should every graduated chemist be able to do this?

For me the best example is bioactive-supramolecular-chemistry. Complex tailor-made architectures are created, with moving parts and so on. An architect who is building a bridge over a river makes calculations too, otherwise he'll lose much time and material until somebody can walk over that bridge.

Thanks for reading, Tobias Grömke.

1. Overview about Theoretical-Methods

Every Chemist who has ever constructed a chemical structure using a Computer and using the geometry-optimization of the program has used force-field-methods. These methods are very fast and because of that very useful in all fields of theoretical chemistry.

Imagine you have painted something like the following picture for the report of your organic-chemistry-internship:

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Fig.1 Five connected C-Atoms as a first painting- step with a chemical-structure-editor

Then you press a button and get an optimized structure:

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Fig.2 Optimized structure

What made that possible was a forcefield-calculation.

In force-field-methods the atoms are figured as charged balls in space, connected by springs and geometry is calculated with methods of classical mechanics. Different force-field-methods use different springs, mathematically represented, as different functions describing potentials. This can be simple ones like quadratic functions or more exact mathematical descriptions like Morse-potentials and so on. Hybridization-geometries and possible binding-angles are factorized as geometrical boundaries.

The following picture will give an imagination:

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Fig.3 balls in space connected with springs

You know it‘s possible to bend a valence-bond but just in a limited way. There is a point were the bond is breaking. Enzymes work that way to there guests. If the guest enters the enzyme, parts of the enzyme are moving caused by the guest who is entering and those movements are going on until the bond is breaking. Other movements place other atoms near enough that new bonds can develop. Molecular Mechanics have been used to calculate such processes and with the received geometrical data small movies have been produced, showing enzyme-guest-interaction. I’ve already seen this at scientific presentations. That was very impressive. The geometrical-boundary-parameters have to ensure that only possible binding-angles appear as results of our calculations. And when using forcefield-methods as basis for small movies about processes like enzyme-guest-interaction the boundaries have to be exact enough the bonds brake in the right moment. A long time it was said forcefield based molecular-dynamic-simulations aren’t able to simulate Bond-breaking. But many techniques have been developed to factorize these properties in the force-field-calculations and that was really necessary, because Bimolecular-systems are much to big for quantum-chemistry. For those problems it makes sense. The force-field-parameters are based on results of Quantum-chemistry. But it don’t makes sense to do only such calculations for smaller systems. You won’t get any new information about molecular interactions because if you see for example hydrogen-bonds like shown at the next figure, an example that will be used and described later in that book, you just know that these bonds are possible, but not one of the bonds is stated using a physical theory which is able to describe hydrogen bonds at a quantum-chemical level.

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Fig.20 symmetrical conformation with π-sytems and possible hydrogen-bridges

The information that hydrogen-bonds are possible you already can get using a much more simple method with no computational cost :

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Fig. 4 Sketch of the system from our example described in chapter 4

For not making anybody angry, I directly want to accent that force-field methods can do much more than a chemist with pencil and paper, but especially in answering questions from problems happening in solution or gaseous-phase they aren’t able to give reliable answers.

If we look in a molecular system we have electrons with a very small mass moving very fast around cores with a relatively big mass moving relatively slow to the electrons. Cores have quantum-behavior too but it‘s Ok to neglect that and to handle them as classical particles to get a useful insight to core-geometries. It should be noted that in Cristal structures: ionic effects, packing effects and lattice-energy-stabilization are playing dominant roles and so force-field methods already give very interesting results compared with experimental data.

Solid-state-physicist developed special force-field-techniques to make these calculations more accurate.

But one fact we can’t change is that electrons are to fast and can‘t be figured with classical mechanics. A bulk of higher electron-density has big influence to molecular geometry. Think for the space for a π-system of a benzene molecule!

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Fig.4 π-system of benzene from struktura benzena se zato prikazujestrukturnim formulama sa naizmeničnimprostim i dvostrukim vezama

We know that an energy-surface is a mathematical expression for the energy of a molecular system depending on its geometry. The intricacies of an energy-surface can just be calculated if we calculate electrons too. These intricacies are caused by bonds: σ-bonds, π-bonds, hydrogen-bonds, Donor-Acceptor-bonds, π-back-bonds, etc. and because of that fucking-fast electrons we need quantum-mechanical-computational-chemistry to calculate those intricacies of the potential-energy-surface. These are Semi-Empirical-methods like Extended-Hückel-theorie, Ab-Initio-Methods like e.g. Density-Functional-Theory, Hartree-Fock-Theorie, Post-Hartree-Fock-methods like e.g. Møller-Plesset-Perturbation-theory, Coupled-Cluster-theory and Correlation-Interaction.

Force-field-methods are good to get a first insight in core-geometry. The reason is that core-interaction is a dominating part of the molecular energy and so a calculation with reduced core-charge is the first logical step in a quantum-chemical-calculation. For small systems we can use the complete force-field-geometry as start-geometry for quantum-chemical-calculations. If we remember the example at the preface, the first mentioned minimum geometry, which wasn’t trustable enough, maybe was the result of many calculation steps. The atoms are moved stepwise following an algorithm until the minimum-energy-Geometry is found. For the pre-calculation we just need single-step calculations like shown in Figure 1 and 2. We construct a geometry near to that structure we think it is an energy-minimum at our potential-energy-surface and make a forcefield-pre-calculation step to have a first correction of our constructed structure. This is called molecular modeling. Some nice free software for this step is available for all operating systems. I really can suggest to come in contact with linux-systems. Many interesting software is easy available and special software is often based on linux/Unix-systems.

If we have big systems, like enzyme-substrate-interaction we can cut out interesting parts out of lower level calculations and calculate them with more accurate Ab-Initio-methods, but for such calculations additional geometrical boundaries are necessary to avoid geometries which are impossible because of the other atoms we have cut out.

Very exact but very slow (1 year for one calculation is no problem) are the Post-Hartree-Fock-methods.

Much faster are Density-Functional-Theory-calculations but one should be careful with the interpretation. Many interaction forces, like dispersion-interaction can't yet be figured out well enough with DFT. We just get a first geometrical insight to a system with quantum-mechanical electrons and fixed cores. Some properties of electrons are neglected, so we have two spin-densities in our theoretical view but no direct coulomb interaction between single electrons.

Add-Ons like dispersion-correction and Hartree-Fock-DFT-Hybrid-methods can increase accuracy of DFT-Calculations but there you have a new problem. You need to find a functional which is especially good in figuring out your expected special geometrical opportunity. Some theoreticians think that it’s possible find a functional for everything but there are reasons to believe that a calculation with that potentially existing functional wouldn’t be faster than Hartree-Fock-methods. Today, for a substantial statement about your system Post-Hartree-Fock-methods are obligatory.

Based on that methods moving systems can be calculated. This is a topic of molecular dynamics or Monte-Carlo-Simulations. Most time these methods are based at forcefield-calculations, because other methods are to slow. But also quantum-mechanical approaches are possible and have been used.

Now some short pause and some recommended tip^☺ before we slowly start to get more and more mathematical:

To develop your concentration, and the level of necessary concentration will develop exponential until we reached chapter 3.3.3..

I can recommend some simple meditation technique. When I was young I got some meditation-tape^* and in the introduction-text a guy was mentioned who traveled a lot to visit some famous Guru and to ask him for advice to increase the quality of his meditation-technique.

The Guru only answered:

“Sit down and and count your breaths. Beginning from 1 to 10^☺☺, if you get wrong start from the beginning^** ”

2. Basis set

2.1. Introduction

Every chemist has heard about orbitals. Atomic orbitals, molecular orbitals, bonding-orbitals, anti-bonding orbitals, … . Every chemist has some Ideas about these orbitals but not about how these orbitals are calculated and just a very small part of chemists knows the definition of an orbital, which is no more than a mathematical definition and which don't makes any statements about single-electron-behavior. Every chemist who read articles involving theoretical calculations has heard about basis set. There a are abbreviations like 6-31G* or Aug-cc-PVQZ or something like this.

In this chapter we'll first develop a pictorial imagination of Hilbert-space and try extend our understanding of the concepts dimension and space. The next part may evoke aversion in a bigger part of the readers, because we'll solve the Schrödinger-equation for the hydrogen-atom. I promise that this will be the only really mathematical part in this book. It is necessary to understand the big amount of available basis sets and the introduced conventions in this sub chapter are indispensable to read the following chapters.

2.2. Pictorial-Imagination of Hilbert-Space

For theoretical calculations on quantum-chemical level we need to understand the Hilbert-space.

First we imagine a normal three-dimensional coordinate system. Then, in our mind, we add a few more axes, some hundred or how everybody wants. The only important thing is that the amount of axes, in our mind, is a big number. To understand that that picture of a n-dimensional Vector-space we are constructing we need to state that the angle between any of these axes is 90°, even if it don't looks so. That doesn't matter!

In mathematics we can state and define everything we want. If mathematics are used to describe physical phenomena we just need to explain why we do that - and we will.

The following figure will help you developing this picture in your mind. Because some readers could have problems with the imagination of all those 90°-angles between more than 3 axes, but this is really simple ;)

See it at the following picture….

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Fig.5 Illustration: “Axes of an n-dimensional coordinate-system”

Now we take one axis and think for a three-dimensional space instead of that axis. In that Space some function is defined. For example some Gaussian-function. In the next step we image such a coordinate-system with another different function for all axes, for example some Gaussian-function multiplied with some Polynomials like . One coordinate-system with one different function instead of every axis!

To have better pictures of those functions, we imagine some orbital. A p-orbital is the visualization of one solutions of the Schrödinger-equation for the Hydrogen-atom which was used to calculate the space where probability is e.g. 95% to find an electron in that p-state. Gaussian functions multiplied with polynomials can create similar pictures and more. Not all basis sets are containing just Gaussian functions multiplied with polynomials, but more about later.

We try now, in our mind, to see two or more functions at same time. No problem if that don't really works. Maybe in our mind the picture of two of those hundreds of axes and the two chosen functions are trying to overlap. That's enough.

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Fig.2 illustrates a subset from the basis set for the methane-molecule and will help to completely clarify all confusions respective the dimension-concept.

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Fig.6 subset, from the basis set, of an ansatz to solve the Schrödinger-equation for CH4

This Schrödinger-equation can be solved with an approximation procedure, like mentioned in the following chapters.

Ф1,2,3 (x,y,z,σ) are described by single electron-functions provided by the C-Atom at the origin and the H-Atom placed somewhere at the z-axis. These functions are depending on three spatial-dimensions and one spin-dimension visualized by different colors.

In this special case it’s necessary to say that not all functions are orthogonal. Only Ф1 and Ф3 are orthogonal. This is because of the different core-coordinates emerging in our basis function, see also equation (1). Because of the different core coordinates, these functions are solutions to different atomic Schrödinger-equations. Otherwise the C- and H-orbitals can’t overlap to build the molecular orbitals they should build.

If we want to construct equations for polyatomic systems, just orbitals localized at the same atom need to be orthogonal. That really easy to understand. Bonds are happening between different atoms. And so we don’t want to have interactions between orbitals of the same atom. OK, we have such interaction, if we think of hybrid-orbitals like sp3-hybridization of a carbon-atom, but to make our calculation able to calculate these effects, we can put more basic functions in our calculation.

It's really important to note that these pictures don't show the complete function. Mathematical our picture is a just a mapping from functions which are defined over entire Space, transformed to a 3-dimensional representation which is represented in as a 2 dimensional mapping on the 2 dimensional surface of page 14 of this book. These pictures just show the the space where the probability to find all electrons in the same volume-element is e.g. 95%. The s- and p-orbitals contributed by the carbon-atom and the s-orbital from the hydrogen atom can be mixed to a new geometrical electron-density-object.

With the mentioned approximation techniques we can calculate the optimal atomic coordinates at energy-minima. That object is emerging particular in the solution of the complete CH4-Schrödinger-equation and it is what every chemist calls: a binding-sigma-orbital, constructed out of a sp3-hybrid-orbital provided by the carbon atom and an s-Orbital provided by the z-hydrogen-atom, but it has nothing to do with the mathematical definition of an orbital.

An orbital is mathematically defined as a single-electron function. Chemists normally use the term orbital in a further context. I will often do this mistake, too in the following chapters, because this is not a book for theoretical chemists, it is for chemists who need to use theoretical chemistry in there practical work. But for the theoretical chemists I’ll repeat the fact several times at the right moment.

2.3. Orthogonality and relative behavior between different Solutions of the Schrödinger-Equation

If we multiply scalar (0,0,1) and (1,0,0), two basis-vectors of the three-dimensional vector-space, we get 0 because the angle between those two basis-vectors is 90° another. That's the mathematical way two say that. The Hilbert-space is another kind of space, where no angles are defined and so we use the word “orthogonal”. In cartesian, spatial space all orthogonal vector-pairs give angles of 90°.

In Hilbert-space we say “orthogonal” not with a simple vector-multiplication we say it with multiplication of the two functions (axis of our n-dimensional vector-space) and integration over entire space ( for example, this way the vector product of Hilbert-space is defined). These integrations give zero () for all pairs of “axis-functions”/ orthogonal basis vectors, because the “angles” between our axes are “90°”. Remember - we stated that at the beginning and could we state that because the Schrödinger-equation is an Eigenwert-equation and solutions of Eigenwert^* -equations behave like vectors. For more details we have to become really mathematical – maybe later ;)

In Hilbert-space, these functions are possible solutions of an equation. Such an equation can be the Schrödinger-equation for example. Linear combinations of these functions are in that case solutions of the Schrödinger-equation, too. A linear combination makes something like interference between two wave-functions. That gives a new function which is a solution of the Schrödinger equation, too. Linear Combination of Atomic Orbitals. LCAO. These Solutions can be molecular orbitals with a new shape compared to the start-orbitals contributed by some basis-set constructed with atomic orbitals.

What is that vectorial behavior of our solutions to the Schrödinger-equation?

- They have a scalar-product
- some of those solutions behave like basis vectors (“scalar-product” = 0)
- linear-combinations (vector1+vector2=different vector) are part of the same space, what in case of the Schrödinger-equation means: combining orbitals leads to other orbitals which are solution to the Schrödinger-equation, too That's everything the reader needs to know about n-dimensional Hilbert-spaces.

It is really important to know that the definition of an orbital is completely mathematical. An orbital is an one-electron-function. This function is the description of one possible electron-condition and it can be combined with other functions, using quantum mechanical operators like the Hamilton-Operator from the Schrödinger-equation. Nevertheless we can keep using our mental visualizations of orbitals. They are very useful. As we have seen at the remarks of Fig. 2, the concepts of hybridization-orbitals and binding orbitals describe important facts. The hybridization means that two possible one-electron-functions from the Carbon-Atom are involved in creating that new geometrical construct of electron-density we call sigma-bond.

Theoretical Chemistry constructs molecular orbitals or similar buildings and variates coefficients of those functions to search e.g. for energy-minima at a potential-energy-surface or to calculate interaction-energy between molecules or to calculate particular electron-densities, in a way that will be explained later in more detail.

As bigger the basis set, as bigger the amount of possible functions figuring an electron. Thus excited states can be used in theoretical explanations. One molecular-orbital for example can be build out of one ground-state LUMO and one exited state HOMO.

2.4. First look at real Basis-Functions and appearing Integrals

I promised some real working basis-function. Following there's some cartesian Standard-Gaussian-Type-function:

We have some core-coordinates Ax,y,z , we have some exponents ax,y,z defining orbital-type, we have a scale factorand we have electron-coordinates. In the exponent of the Gaussian function we see the distance between electron and core. What we don't have to do is to calculate integrals with various combinations of these functions, but we can imagine that many steps are included to solve such an integral. And how many steps are included for millions of integrals.

DFT-Calculations don't need so much of these Integrals like Post-Hartree-Fock-methods but they can also need weeks. For the really exact methods months of calculation are normal.

One example for an integral emerging in real theoretical calculations is the so-called Slater-Type-Geminal, defined with equation (2).

With the new Cores B, C, D, a length scale factorand cartesian Standard-Gaussian-Type-functions like shown before. Geminals are used to make our quantum-mechanical electrons move correlated. It's an integration over six spatial dimensions.

It should be mentioned that basis-set-super-position-error means there where not enough possible functions available and so our calculation shows reality with a bigger error. A last important remark to this chapter should be that solutions are in real vector space but the mathematics - quantum chemical software is doing happens in complex vector space. To calm the reader, all examples will use as much as possible simple real functions.

To finish with some pictures from our Hilbert-Space: Orbitals – e.g. like (1) – at the same atom are orthogonal and don't overlap. Possible electron-conditions at the same atom are provided by the basis set. These functions are combined with complicated many-step calculations and give other functional-systems which are solution to our equations too. In these functional-systems we search for local or global energy-minima.^*

2.5. The Schrödinger-Equation for the Hydrogen-Atom with a Gaussian-Ansatz

Before we go on watching our basis set in a more detailed way we need to make a small excursion. We'll solve the Schrödinger-equation with a Gaussian-function as trial function. If we watch again our Standard-Gaussian-Type-basis-function (1) we see that this a Gaussian-function multiplied with some Polynomials. If we remember our first (an for the most chemists the only) lecture in theoretical chemistry, the solution for the Schrödinger-equation for the was an e-Function combined with some polynomials. Those e-functions describe better what's happening in in electronic systems but e-functions make the mathematics more complicated.

A simple example: if we think about an s-orbital with the middle at the origin of some coordinate-system. We watch just an e-function in two dimensions, it's easy to understand why we need to use equation (3):

If we think about this in three dimensions with more complicated orbitals and combinations of them it's easy to imagine how much more complicated calculation-steps are necessary with e-functions as basis-functions many-body-systems. More about later.

Two things might be new for the reader. The first thing is atomic units. The good thing about these units is, that we don't need units. We can calculate our informations out coordinates, values of integrals, …. The next good thing is. We don't need to calculate these informations, computer does For the interested reader, a complete derivation of atomic coordinates can be found in “Attila Szabo/Neil.S.Ostlund Modern Quantum-Chemistry” a cheap, but very good book, every quantum-chemist and every chemist who wants to have some mathematical understanding of the roots of his science should have.

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