introduction to computational chemistry -...

Introduction to Computational

Chemistry Lehrstuhl für Theoretische Chemie

- Winter term 2007/2008 -

Organisation:

Frank Neese, Thomas Bredow, Frank Wennmohs

Table of Contents Foreword

1

Study Plan

2

1 INTRODUCTION – FUNDAMENTALS AND GOALS OF COMPUTATIONAL CHEMISTRY 5

1.1 WHY COMPUTATIONAL CHEMISTRY? 5

1.2 MODELS VERSUS CALCULATIONS (OPTIONAL READING) 7

1.3 FUNDAMENTALS OF THEORETICAL CHEMISTRY 8

1.3.1 STATES AND TOTAL ENERGIES 8

1.3.2 THE BORN-‐OPPENHEIMER HAMILTONIAN AND POTENTIAL ENERGY SURFACES 9

1.3.3 THEORETICAL METHODS 11

1.4 SLATER DETERMINANTS AND MOLECULAR ORBITALS 13

1.5 BASIS SETS 16

1.6 MODEL CHEMISTRIES 19

Foreword 1

Foreword 2007

The present course is a resdesign of the course we have first organized in 2006. Several changes were necessary: (a) since Barabara Kirchner left to create her own chair of theoretical chemistry at the university of Leipzig we can no longer support the first principles molecular dynamics part of the course. (b) we got “banned by Gaussian” and consequently we had to delete all aspects of the course that had to do with the Gaussian program. (c) we have revised several experiments according to the feedback we have received from the students. (d) we have created two new computer experiments: relativistic effects in quantum chemistry and simulation of biomolecular structures.(e) we have fixed the errors that came to our attention. In the next version we hope to add experiments concerned with mixed-‐valency, redox potentials, metal-‐metal bonding and Jahn-‐Teller systems to the coordination chemistry section. We hope that these changes make the course more attractive for students and – as usual – we appreciate any positive or negative feedback on this material.

Frank Neese Thomas Bredow

Frank Wennmohs Taras Petrenko

Bonn, December 2007 Foreword 2006

The present course has been designed ‚from scratch’ in the fall of 2006. Its main goal is to provide experimental chemists, who have not been exposed in depth to computational chemistry, with a working knowledge that allows them to tackle their own research problems with the now widely available techniques of computational chemistry. Over the pas two decades computational methods have evolved into routine techniques that can be productively applied by every chemist in perfect analogy to, e.g. the use of an NMR spectrometer – it is not necessary to understand all of the complicated physics that goes into the design of the instrument (quantum chemistry program), but it is essential to understand the possibilities and limitations of the measurements (theoretical methods). It is also necessary to be critical about the results of the experiments (calculations) and to detect when they are useless. Since we take this analogy very serious we have deliberately called the problems to be solved in this course ‘Computer Experiments’. Large parts of the course should be suitable for all students after the second year of their studies and may also be attractive for Ph.D. students who did not have an in-‐depth training in theoretical and computational chemistry. In addition, we have also added some more advanced material for theoretically oriented students who have already completed four courses in theoretical and quantum chemistry as they are taught at the university of Bonn (TC I+II and QC I+II). In preparing the material we have tried to find a reasonable balance between theoretical concepts and practical applications. We have essentially avoided lengthy mathematical treatments since it is our main goal to make this material accessible. Although these lecture notes are not devoid of equations they do not play an essential role in this course. Rather, the equations serve to clarify the points that we whish to bring across and they are, with very few exceptions, of an elementary nature. In preparing the lecture notes, we felt that at least a minimal amount of background information is necessary in order to appreciate the contents of the calculations and this background is provided as an introduction to each computer experiment. We have tried

to be as clear and as concise as possible and hope that we have come as close as possible to this goal. However, inevitably each newly designed course will involve some errors or misleading descriptions. We appreciate the feedback of students, teachers and teaching assistants in this respect in order to improve on the contents and presentation of the course in the coming years. After having defined the desirable contents of an introductory course in computational chemistry, we have quickly realized that it will not be possible for the students to do all of the computer experiments within the time frame that we envision (basically six weeks full time). We have therefore divided the course into three parts as explained below. We have enjoyed working on this course and putting it together was a real ‘joint venture’ of essentially all theoretical chemists that presently work at the university of Bonn. We hope that we have succeeded in showing the students that computational chemistry is a powerful branch of chemistry with many exciting applications – and also – that it’s fun to do! Bonn, October 2006,

Frank Neese Thomas Bredow Barbara Kirchner Frank Wennmohs Dmitry Ganyushin

Roman Gupta Simone Kossmann

Taras Petrenko Werner Reckien

Christian Spickermann Yang Su

Jens Thar Shengfa Ye

Study Plan 3

Study Plan This course contains a significant amount of material. In order to carefully work through all of the material will probably require several months of work. In order to keep the workload manageable we have structured this course into three parts:

1. Essential Computer Experiments

2. Advanced and Special Subjects

3. Research Related Subjects

In part I we have collected what we feel is the “canonical” contents of an introductory course in computational chemistry. This material is elementary and does not require special skills or prior knowledge. It should be suitable for students after their second year in the chemistry curriculum. In part II we have collected a number of subjects that we found interesting and that relate to somewhat more specialized techniques that belong to the toolbox of theoretically oriented chemist. We feel that experiments 8 and 9 relate to students with some background knowledge in spectroscopy. The elementary parts of these experiments should be appropriate for all students that manage to work through part I while some subjects in these experiments are slightly more demanding. Experiments 12 -‐ 14 are of a slightly more advanced nature that are perhaps tackled by students in their third and fourth year after they have heard the lectures on advanced quantum chemistry. Experiments 7, 10 and 11 are of an elementary nature and should be approachable by all students. In part III, we have tried to hint at research areas in applied computational chemistry that are of particular interest to us. They deal with computational coordination chemistry (AG Neese), computational solid state chemistry (AG Bredow) and biomolecular simulations (AG Wennmohs). The intention of these experiments is to provide a feeling of how one could tackle problems of actual chemical relevance with computational chemistry. In order to successfully complete the course we require the following:

1. All students should do part I completely. These are simple and fast experiments that should be straightforward to perform once an elementary familiarity with the computational environment has been achieved.

2. Of part II, two experiments should be chosen. However, experiments 8, 9 and 13 are more time-‐consuming than the other experiments. If these experiments are chosen, it is sufficient to either complete it together with one of the smaller experiments (7,10,11,14) or to complete about half of two of the larger experiments.

3. Of part III one subject must be chosen. While experiments 16 and 17 should be done completely, experiment 15 is more time consuming and it is sufficient to complete a part of it as will be explained in detail in chapter Error! Reference source not found..

1 Introduction – Fundamentals and Goals of Computational Chemistry

1.1 Why Computational Chemistry? Chemistry is an experimental science. Insights concerning the behaviour of matter,

the properties of molecules and their interaction with the environment are studied

using a broad variety of experimental techniques. New molecules are being

synthesized using a mixture of rational strategies and trial and error procedures. At

all stages of a chemical investigation the same basic questions arise:

1. How do I analyze the results of a chemical experiment or a physical measurement?

2. Could I possibly exclude several possibilities from consideration by predicting the

outcome of the experiment beforehand?

3. What are the underlying principles that govern the behaviour of classes of

substances?

These basic questions pose the framework for the discipline of theoretical

chemistry. Question number (1) is concerned with the correct interpretation of

physical measurements with respect to the behaviour of the molecular system

under investigation. This could possibly be an NMR or IR spectrum, the

measurement of a pKa value, a kinetic measurement or a thermochemical

measurement. In very many cases, the analysis of the experiment immediately rules

out several possibilities but leaves other alternatives open –Theory can help to

discern between alternatives if it is able to reliably predict the outcome of the

experiment for all possible alternatives.

Question number (2) is concerned with predictive power – if I have a model or a

theory that realibly tells me what is possible and what is not possible I may save a

lot of time in the laboratory chasing compounds that cannot exist or running

reactions that can never give the desired result -‐ A truly predictive theory can thus

save a lot of trial and error time and guide the experimental work.

Question number (3) is concerned with the question of qualitative insight –

chemists very often want to understand the behaviour of classes of substances, they

want to categorize a large body of experimental evidence in terms of the properties

of functional groups or structural motives. Thus, a good theory should not only

provide results that pertain to individual molecules but should provide a framework

and language in which the results obtained for related substances can be phrased.

Fortunately, we are in possession of a good theory that – at least to some extent –

satisfies all of the criteria mentioned above. This theory is quantum mechanics.

Quantum mechanics applied to chemistry defines the field of quantum chemistry –

its technical aspect is just theoretical physics while the objects studied are mostly of

a chemical nature. Quantum chemistry itself is just one subdiscipline of theoretical

chemistry which contains many other branches which are not directly related to

quantum mechanics. Examples are: simulation of the dynamics of chemical

networks, molecular dynamics and mechanics based on classical force-‐fields or

pattern recognition techniques to name only a few.

1.2 Models versus Calculations (optional reading) Closely related to the desire for qualitative understanding is the nature of

developing models of reality. A model aims at providing a conceptual framework in

which a restricted part of reality can be understood and discussed on the basis of a

minimum number of preferably simple rules. Typical examples for models that have

deeply invaded chemical thinking are the Hückel model of aromatic molecules and

the ligand field theory of inorganic chemistry. Both models, if taken literally, are

physically absurd and produce nonsensical numbers. Yet, after the introduction of a

minimum number of parameters (resonance integrals, ligand field strengths, Racah

parameters) they provide a tremendous amount of insight into the behaviour of

very large classes of compounds.

Thus, models are deliberately wrong if one defines rigorous first principles physics

as the ultimate target. Yet, in a good model, the parameters have a clean connection

to those first principles. In this sense, the Hückel theory and ligand field theory are

relatively mediocre models (neither the ligand-‐field strength nor the resonance

integrals have any rigorous theoretical definition) while the model of the spin-‐

Hamiltonian used to interpret NMR and ESR spectra is a very good model since the

This course is concerned with quantum chemistry in its practical

realization – computational chemistry. It is intended to show with an

absolute minimum of theoretical background how can approach chemical

problems with the aid of computers. It is not a substitute for the study of

considerable body of theory that goes into the development of the

computer program that are used throughout the course. It is highly

recommended to the student to study the theory more closely as a deeper

understanding of the underlying theory will ultimately also enable you to

do better calculations and will enhance your ability to judge the strengths

and weaknesses of your calculations. A warning – despite the usefulness

and sophistication of the theoretical approaches: Don’t blindly believe

everything the computer tells you! Be critical, seek feedback from

experiment.

parameters occurring in it (chemical shifts, spin-‐spin couplings, g-‐factors, hyperfine

coupling constants) can be rigorously related to first physical principles.

On the opposite extreme are high-‐accuracy calculations that can presently be

performed with sophisticated computer programs and large-‐scale computational

facilities. The outcome of such calculations are highly accurate numbers that refer to

individual molecules in particular electronic (vibrational, rotational, …) states. One

could, and indeed some researches do, argue, that the only thing that is of interest

are physical observables such as the energy change during a chemical reaction of

spectroscopic transition. Thus, each individual molecule is a completely separate

physical object that is to be calculated to high-‐precision and the only thing of

interest is its energy. The usefulness of a given theoretical method is then judged in

a statistical means by recording its average and maximum error in application to a

large collection of molecules. Using this philosophy one basically discards the great

deal of chemical regularity that is close to the heart of the vast majority of

experimentally working chemists.

Both, the model philosophy and the rigorous philosophy have much to recommend

themselves. It is undisputable that models are of great importance for the past,

present and future of chemistry. Equally important is the development of theories

that make ever more precise predictions in ever shorter turnaround times. Yet, in

our opinion, it is important to at least try on a qualitative level to try to “find” the

chemical concepts inside the complicated computations. We will try to at least hint

at such possibilities.

1.3 Fundamentals of Theoretical Chemistry

1.3.1 States and Total Energies

Quantum mechanics is centered around the Schrödinger equation which exists in

time.-‐dependent and time-‐independent form. For a given molecular system with N-‐

electrons and M-‐nuclei, the time-‐independent Schrödinger equation has an infinite

number of solutions that define the states of definite energy that the system can

adopt. Associated with each state is a:

- Total energy EI ( I = 0,1,...) and a

- N-‐electron wavefunction !

Ix

1,x

2,...,x

N( ) where xi denotes the space and

spin-‐coordinates of a single electron. The square of the many electron

wavefunction evaluated at the point x ! x1,x

2,...,x

N is the probability

density associated with the probability to find the N-‐electrons in an

infinitesimal volume element dx around point x .

Transitions between such states are observed in spectroscopic experiments.

Dynamics of evolving systems can be studied by solving the time-‐dependent

Schrödinger equation. In this course we are mainly concerned with molecules in

stationary states (however, see chapter Error! Reference source not found. which

deals with molecular dynamics and chapter Error! Reference source not found.

which deals with chemical kinetics) and in particular in their lowest energy state E0

which is called the electronic ground state (however, excited electronic states will

be studied in chapters Error! Reference source not found., Error! Reference

source not found., Error! Reference source not found. and Error! Reference

source not found.).

1.3.2 The Born-‐Oppenheimer Hamiltonian and Potential Energy Surfaces

In addition to the total energy E and the many-‐electron wavefunction Ψ , the

Schrödinger equation contains the Hamilton-‐operator (HO) H which corresponds

to the total energy of the system. Fortunately, for the vast majority of chemical

applications, the HO contains only a few terms, namely those that describe the

kinetic energy of the particles (electrons, nuclei) and their electrostatic (Coulombic)

interactions which is inversely proportional to the interparticle distance. All other

terms that contribute to the total energy are (usually) much smaller and can be well

treated using the methods of perturbation theory. Despite this deceptive simplicity,

the Schrödinger equation for a molecule is still far too complex to be solved either

analytically or numerically and therefore approximations are necessary. The

refinement of such approximations represents a major research goal of quantum

chemistry.

The first approximation that is central to quantum chemistry is the so-‐called Born-‐

Oppenheimer (BO) approximation. The BO approximation allows one to treat the

motions of electrons and nuclei independently. It rests on the fact that nuclei are

much heavier than electrons1 and consequently move much more slowly than

electrons. Thus, the “fast” electrons are always “in equilibrium” with the “slow”

nuclei. As a consequence, the nuclei can be assumed to be “at rest” from the point of

view of the electronic system and the Schrödinger equation needs to “only” be

solved for the electrons at fixed nuclear positions. As a consequence, the total

energies and many electron wavefunctions become parametric functions of the

nuclear coordinates R1,R

2,...,R

M: E

IR

1,R

2,...,R

M( )! EI

R( ) and !

Ix | R( ) . The

function E

IR( ) is called a „potential energy surface“2 and it can be studied by

calculating the energy E

IR( ) at different points R .3 Unfortunately, the variation of

the energy with respect to R can only be visualized graphically for at most two

coordinates which represents two-‐dimensional cuts through, in general, very

complicated energy landscapes.

Particular points on a given potential energy surface deserve special attention.

These are so called stationary points R and they are characterized by a vanishing

first derivative:

!EI

R( )!R

A

= 0 (for all A)

( 1) Such points can be of different nature. They can either be minima, maxima, or saddle

points of varying order. The nature of a stationary point can be determined by

calculating the so-‐called Hessian matrix:

1 The ratio of the proton mass to the electron mass is ≈1822. 2 Note that this energy only represents a “potential energy” from the point of view of the nuclei but in fact also contains contributions from the kinetic energy of the electrons. 3 Once the function E(R) is known, it can be used to calculate the nuclear wavefunctions.

H

AB=!2

EI

R( )!R

A!R

B

( 2)

6 eigenvalues (or 5 eigenvalues for linear molecules) of the Hessian matrix are

precisely zero and correspond to nuclear motions that describe the molecular

translations and rotations. For minima, all remaining eigenvalues of the Hessian

matrix must be positive. A single negative eigenvalue corresponds to a “transition

state” in a chemical reaction. Higher order saddle points are not of chemical interest.

Note, that for a given electronic state, say the ground state, the function E

0R( ) may

have many minima which correspond to different isomers of a molecular system in

the chemical sense. Each stationary point is associated with a different total energy

and the lowest of such energies represents the most stable of the possible isomers.

1.3.3 Theoretical Methods

Unfortunately, we cannot even solve the BO problem exactly (except for 1-‐

dimensional 2 -‐electron systems) – the task is still far too complex. Over the

decades, the community of theoretical chemists and physicists has intensely studied

this problem and has developed many different approximations to the problem.

Presently, the most popular of these methods may be broadly categorized as

follows:

• Wavefunction based methods: These methods try to compute an-‐accurate-‐as-‐

possible many-‐electron wavefunction which then automatically leads to an

accurate total energy. These treatments almost invariably built upon the

Hartree-‐Fock (HF) method, which is a variant of a mean-‐field theory. Due to its

mean-‐field nature, the HF method is also called ‘independent particle’ model.

The HF approach is probably the simplest possible model that takes proper

account of the required antisymmetry of the N-‐electron wavefunction (e.g. it

satisfies the Pauli principle). The errors that remain in the HF method are called

“correlation errors” and they are reduced as much as possible by using various

“correlated ab initio” methods. Impressive progress has been made along these

lines and there are now many close-‐to-‐exact solutions available. The large-‐scale

use of such methods in chemistry is presently still prevented by the high-‐

computational cost of these methods which scale as O(N5) and O(N7) with

molecular size where N is a measure of the system size (e.g. the number of

electrons or nuclei) and ‘O’ means that the leading term of the computational

effort requires a time that is proportional to the indicated power of N.

• Density Functional Theory. Many electron wavefunctions contain much more

information than is strictly necessary since the BO operator does not contain

more than two-‐body interactions. In fact the only “difficult” term is the electron-‐

electron interaction. Thus, in principle, the problem is fully solved if one would

know the electron-‐pair distribution function ! x

1,x

2( ) (this function is closely

related to the probability of finding a pair of electrons at please x1 and x2

respectively). Unfortunately, it has been found to be impossible so far to

calculate this function without the detour of the many-‐electron wavefunction

itself. However, there is a famous theorem that was awarded with the Nobel

price for chemistry in 1998 which essentially states that one does not even need

to know the pair distribution function – the knowledge of the electron density

! x( ) is „in principle“ enough to deduce the exact total energy of the electronic

ground state (Hohenberg-‐Kohn theorem). Unfortunately, nobody is in

possession of the exact recipe that describes how to deduce the exact energy

from the electron density alone. Nevertheless, over the years, many highly

intelligent guesses to this “universal functional” have been made and each of

these constitute a different “density functional”. The practical realization of DFT

Due to a very good price/performance ratio DFT is presently the methodology of

choice for most chemical applications. It can be applied to molecules with 100-‐

200 atoms using standard computational hardware (PC’s).

is the so-‐called Kohn-‐Sham method which is essentially at the same level of

complexity as Hartree-‐Fock theory but almost invariably provides better results.

• Semiempirical methods. Such methods are designed as „cheap“ substitutes for

either DFT or Hartree-‐Fock based method. In contrast to the latter they contain

many adjustable parameters that are used to compensate for the drastic neglect

of certain integrals that are time-‐consuming to calculate in the more precise

methods. Semiempirical methods can be used for much larger molecular systems

than either DFT or ab initio methods. However, these methods are also much less

reliable and are only available for certain ranges of elements.

1.4 Slater Determinants and Molecular Orbitals The simplest reasonable Ansatz that one can make for the many electron

wavefunction ! x | R( ) and that satisfies the fundamental physical requirements for

a valid N-‐electron wavefunction is the form of a single so-‐called Slater-‐determinant.

In this course we will essentially not be concerned with the various methods of

quantum chemistry. Instead we will focus on very few successful standard

methods (The Hartree-‐Fock method, the simplest correlated ab initio method

(MP2) as well as the so-‐called B3LYP DFT method) and use them to address

various chemical problems. However, one should always be aware, that every

quantum chemical method only provides an approximate solution which can fail

in some instances. Again: nothing substitutes for seeking careful feedback

from experiments in judging the quality and reliability of a given quantum

chemical calculation!

! x | R( ) =1

N !

!1

x1( ) !

2x

1( ) ! !N

x1( )

!1

x2( ) !

2x

2( ) ! !N

x2( )

" " # "!

1x

N( ) !2x

N( ) ! !N

xN( )

( 3)

The quantities (lower-‐case ψ’s) appearing inside the determinant only depend on a

single set of electronic coordinates. Such single-‐electron wavefunctions are called

orbitals and the integral over all space of their square has to exist in order for them

to be physically acceptable. Loosely speaking, orbitals describe the “motion” of

individual electrons. The Slater determinant is nothing but an antisymmetrized (in

the electron coordinates) product of such single-‐particle wavefunctions. It is often

abbreviated as: ! x | R( ) = !

1!

2...!

N. As explained in great detail in the lectures on

theoretical chemistry such a single N-‐electron Slater determinant describes the

uncorrelated motion of N-‐electrons. Since !

ix( )

2 describes the probability

distribution for a single electron, it is not surprising that the total electron density

described by a single Slater determinant is simply:

!(r) = "i

r,s( )2ds!

i=1

N

"

= "i# r( )

2

i=1

N#

" + "i$ r( )

2

i=1

N$

"= !# r( )+ !$ r( )

( 4)

The integral in the first line is taken over the spin degree of freedom only. Since

electrons carry either a “spin-‐up” or a “spin-‐down” intrinsic angular momentum, the

second and third lines follow which describe the decomposition of the total electron

density into the separate densities of the spin-‐up and spin-‐down electrons. In this

way it becomes obvious how each orbital contributes to the total electron density. In

chemistry, “orbitals” are used to explain many observations. However, since

electrons interact with (actually repel) each other, such a single-‐determinant

wavefunction, is a drastic, however, useful, simplification and can not provide an

accurate description of an actual N-‐electron state via one-‐electron functions.

We will not go into any detail about how the orbitals are actually determined in

quantum chemical program packages. It suffices to say that orbitals are used in

essentially three different contexts:

• Minimization of the energy of a single Slater-‐determinant with respect to the

shapes of the orbitals yields a set of “optimum” occupied orbitals of a given

system. Such orbitals are called “Hartree-‐Fock” orbitals. The Slater determinant

composed of the occupied Hartree-‐Fock orbitals form the total “Hartree-‐Fock

wavefunction”. It is a simple 0th-‐order approximation to a genuine N-‐electron

eigenfunction of the BO operator.

• In so-‐called “post-‐Hartree-‐Fock” methods (such as MP2), “excited” determinants

are formed by replacing 1, 2, 3… occupied orbitals of the Hartree-‐Fock

determinant with empty orbitals. The “correlated wavefunction” is then

determined as a linear or nonlinear combination of the Hartree-‐Fock

determinant and the set of “excited” determinants. One can think of the

admixture of excited determinants as describing the effect of electrons

“jumping” briefly out of their occupied orbitals into virtual orbitals in order to

minimize their repulsion with the other electrons in the system. Note that the

“excited” determinants do not describe genuine excited states of the molecule

but are merely used as “building blocks” for the improvement of the HF

wavefunction.

• In density functional theory a Slater determinant is used in a completely

different context – it just provides the decomposition of the total electron

density that is necessary to apply the so-‐called Kohn-‐Sham scheme. Thus, the

exact system and the so-‐called ‘non-‐interacting’ reference system just share the

same electron density which is (formally) used to calculate the exact ground

state energy. In practice we don’t know how to go from the exact electron

density to the exact energy and therefore all practical DFT is approximate.

No matter to what end we use orbitals, they are just helping us in building up

molecular wavefunctions or densities but they are not fundamental objects of theory

– in fact, the entire discipline of quantum chemistry can be carried to high-‐precision

without any recourse to orbitals whatsoever. This necessarily means that individual

orbitals do not have a fundamental physical or chemical reality or relevance.

Nevertheless, orbital pictures are so prevalent in chemistry that we will not refrain

from using them in this course too. We, however, want to point out that the

theoretical justification of the orbital based pictures is – at the very least – not

unambiguous and a much more careful study of this subject is highly recommended

to the interested student. Having said that – we enjoy as much as you probably do to

look at orbitals of some kind and use them to think about the chemistry that

interests us.

1.5 Basis Sets In practice, we can – again unfortunately – not determine the shapes of the orbitals

(whichever we are talking about) directly. Instead, we have to use an approximate

decomposition of the orbitals in terms of a finite set of, say L, known functions ({ϕ}).

!

ix( )! c

µi"

µx( )

µ=1

L

"

( 5)

This set of basis functions {ϕ} constitutes the so-‐called “basis set” for a given

calculation. Only in the (unreachable) limit that the set is complete in the

mathematical sense can the orbital be exactly expanded in the basis set. For any

incomplete set, the results of our calculations will depend on the nature and size of

the basis set that we use. In general, larger basis sets (if properly designed) are

more accurate than smaller basis sets but will also lead to longer computation times.

Thus, there always is a “cost-‐versus-‐accuracy” trade-‐off to be made in choosing a

specific basis set for a given computational task.

The basis set problem has been thoroughly studied by quantum chemists. As a

consequence, there are many different “standard basis sets” available. Each basis set

has a different name and a different range of elements that is supported. Broadly

speaking, basis sets can be categorized by their construction rules. Below we give an

elementary introduction:

From the qualitative discussions that are commonly found in textbooks one must

get the impression that there is a carbon 2s orbital and three carbon 2p orbitals that

entirely determine the chemical, spectroscopic, ... behaviour of any carbon atom in

any molecule. From this point of view it is rather confusing to be exposed to the

subject of basis sets in quantum chemical calculations.

The suggestion that “in principle” not more than a single 2s and three 2p orbitals on

a carbon center are necessary to describe it in a molecular environment remains

valid even in quantum chemistry. However, the important point is that the atomic

orbitals get distorted from the shapes they have in the neutral atom upon entering

the molecule. Furthermore, the shape of the orbitals also changes for each state of

the neutral carbon atom and they also change from the neutral atom to the cation or

the anion. In molecules a combination of all of these effects need to be taken into

account -‐ the state of a carbon atom in a molecule maybe a mixture of several atomic

states and the carbon center might have a partial charge. The symmetry around a

given carbon center in a molecule is less then spherical and therefore there will be

an additional distortion of the shape of the basic orbitals that is due to this

symmetry lowering.

Basically there are two major types of distortions that are taken into account in the

basis set design. The first is a radial distortion characterized by the expectation

value <r> of a given atomic orbital. Atomic orbitals will tend to contract the more

they are involved in bonding and the more positive charge the center that they are

attached to carries. Importantly, the contraction or expansion can be different for

different MOs. Thus for a σ-‐bonding p-‐AO the distortion is generally different

compared to a π-‐bonding MO which in turn is different from a nonbonding π-‐MO, an

antibonding σ*-‐ or an antibonding π*-‐MO. The second type of distortion is an

angular distortion. Depending on the local symmetry and bonding interaction the

orbital is involved in for example the two lobes of a ‘p-‐orbital’ may become

inequivalent or an s-‐orbital is polarized to deviate from spherical symmetry.

In general, chemical core orbitals (e.g. the carbon 1s orbital) distort very little since

they are held very tightly by the nuclear charge. Generally it is sufficient to

represent such orbitals by a single basis function.

The radial distortion is simulated in actual calculations by supplying several types of

the “principal” atomic orbitals but with different spatial extent. Thus, if there is a

single set of 2s-‐type orbitals and 2p type on a carbon one speaks of a single-‐zeta

basis set, two sets of 2s,p functions constitute a double-‐zeta basis set, three sets a

triple-‐zeta basis set etc. In most cases, triple-‐zeta basis sets allows for an adequate

modelling of the radial distortion effects that arise in molecules. Double-‐zeta is,

however, a minimum requirement. Calculations with single-‐zeta basis sets are

completely unreliable and should not be done. Higher accuracy can be reached with

quadruple-‐zeta or pentuple-‐zeta basis sets but such calculations become very time

consuming. In calculations with anions, the charge density will tend to become very

diffuse. In such a case it is advisable to add a set of particularly diffuse basis

functions to the basis set in order to describe such additional expansions of the

electron cloud.

The angular distortion effects are modelled by including in the basis sets orbitals

with higher angular momentum quantum numbers – these are called polarization

functions since they describe an angular distortion of the atomic orbitals which is

brought about by the anisotropic molecular environment. Thus, for carbon atoms,

one includes d-‐functions and perhaps f-‐functions in the basis sets. These basis

functions are by no means representations of the atomic 3d and 4f orbitals. On the

contrary, they have a similar spatial extent as the 2s and 2p functions and mainly

help these orbitals (through small admixtures) to depart from their “pure” s-‐ or p-‐

shapes. In general, a single-‐set of polarization functions on non-‐hydrogen atoms is a

minimum requirement for a reasonable calculation (single-‐set of polarization

functions) but at least a single set of polarizing p-‐functions should also be included

on the hydrogens. Two sets of polarization functions provide better results and

three sets (2d and 1f on non hydrogens and 2p and 1d on hydrogens) provide good

results. However, such calculations already become rather time consuming.

All of the basis sets mentioned here were developed by the Karlsruhe quantum

chemistry group and are particularly efficient. Other basis sets can be found on the

internet (try for example http://www.emsl.pnl.gov/forms/basisform.html).

We have only scratched the very surface of the basis set problem and only mention

in passing that (essentially for practical reasons), the individual basis functions are

almost universally taken as a linear combination of so-‐called ‘Gaussian’ functions

(that decay as exp(-‐αr2)). Such functions show behaviour at short and long distances

r from the nucleus that is incorrect if compared to the known limiting behaviour of

exact atomic Hartree-‐Fock orbitals. However, Gaussians have important advantages

that can be exploited very efficiently in sophisticated quantum chemical program

packages such as the ORCA program used in this course.

1.6 Model Chemistries The task of computational chemistry can now be defined more clearly:

a) Choose an initial set of atomic coordinates that represent your system of

interest

b) Choose a theoretical method

c) Choose a basis set

d) Carry out the calculation of molecular structure, properties, energies,

dynamics, etc.

The combination of a given theoretical method with a given basis set defines a “model chemistry”. This combination will determine the achievable accuracy in a given study. Accurate model chemistries take up significant (or even prohibitive)

In this course we will use just a small number of standard basis sets. The so-‐called

SVP basis set is a double-‐zeta basis set with one-‐set of polarization functions on all

atoms. The more accurate TZVP basis set is a triple-‐zeta basis set with one-‐set of

polarization functions and the TZVPP basis set is of triple-‐zeta quality with three

sets of polarization functions and results that are fairly good (meaning close enough

to the basis set limit in order to properly reflect the theoretical method used and not

the limitations of the basis set).

computer resources while less accurate model chemistries yield necessarily predictions of lower reliability. The choice of model chemistry is therefore an important aspect of a theoretical study and is dictated by experience and available computational resources. Model chemistries of unknown accuracy need to be calibrated before they should be applied in chemical application studies. The process of calibration consists of testing the predictions of the model chemistry for a range of molecular systems and the properties of interest. A proper calibration yields a solid confidence level (or error interval) for the predictions of the model chemistry with respect to the chosen property.

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