Computational Quantum Chemistry: Molecular Structure and Properties In Silico

by Joseph J W McDouall

Overview

Computational Quantum Chemistry presents computational electronic structure theory as practised in terms of ab initio waveform methods and density functional approaches. Getting a full grasp of the field can often prove difficult, since essential topics fall outside of the scope of conventional chemistry education. This professional reference book provides a comprehensive introduction to the field. Postgraduate students and experienced researchers alike will appreciate Joseph McDouall's engaging writing style. The book is divided into five chapters, each providing a major aspect of the field. Electronic structure methods, the computation of molecular properties, methods for analysing the output from computations and the importance of relativistic effects on molecular properties are also discussed. Links to the websites of widely used software packages are provided so that the reader can gain first hand experience of using the techniques described in the book.

Product Details

ISBN-13:	9781849736084
Publisher:	RSC
Publication date:	06/03/2013
Series:	Theoretical and Computational Chemistry Series , #5
Pages:	292
Product dimensions:	6.30(w) x 9.40(h) x 0.90(d)

About the Author

Walter Thiel studied chemistry at the University of Marburg (West Germany) from 1966 to 1971, where he subsequently obtained his doctorate with A. Schweig in 1973. After a post-doctoral stint at the University of Texas at Austin with M. J. S. Dewar (1973–1975), he obtained his habilitation from the University of Marburg in 1981. He was appointed Professor of Theoretical Chemistry at the University of Wuppertal (West Germany) in 1983 and Professor of Chemistry at the University of Zurich (Switzerland) in 1992. In 1987 he was a visiting professor at the University of California at Berkeley. Since 1999, he is a director at the Max Planck Institute for Coal Research in Mülheim an der Ruhr (Germany) and an honorary professor at the neighbouring University of Düsseldorf (Germany) since 2001.

Read an Excerpt

Computational Quantum Chemistry

Molecular Structure and Properties in Silico

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By Joseph J W McDouall

The Royal Society of Chemistry

Copyright © 2013 Joseph J W McDouall
All rights reserved.
ISBN: 978-1-84973-608-4

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CHAPTER 1

Computational Quantum Chemistry

1.1 What Does Computational Quantum Chemistry Offer?

Computational quantum chemistry has been in development for almost nine decades. Its progress has been intimately linked to developments in computing hardware and technology. Today computational quantum chemistry provides a complementary way of investigating a wide range of chemistry. In particular it provides reliable information on molecular structures, molecular properties, reactions mechanisms and energetics. Detailed mechanistic questions can be addressed using the techniques of computational quantum chemistry. An advantage over traditional experimental techniques is that it provides a route to the study of chemical questions which may be experimentally difficult, or expensive, or dangerous. The purpose is always to answer a chemical question and in that sense computational quantum chemistry is the complement to experiment, either approach on its own is much less convincing. This complementarity of techniques is very familiar to chemists. For example, to determine a molecular structure a range of spectroscopies must be used and each provides a component of the overall picture. Now to these spectroscopies are added quantum chemical techniques that can provide further information.

Computational quantum chemistry is an elegant conjunction of chemistry, physics, mathematics and computer science. Chemistry defines the question. Physics defines the laws that are obeyed by the chemical system. Mathematics formulates a numerical representation of the problem. Computer science solves the mathematical model, yielding numbers that encapsulate physical significance. For example, does a particular alkylation reaction proceed more efficiently with the alkyl chloride or the corresponding iodide? To answer this question, at the simplest level, we could compute the geometries of the transition structures and reactants, from which we would obtain the activation energies and so determine which reaction should be more efficient. The insight gained from such numerical answers can lead to further questions. This often results in an iterative refinement of questions, answers and models, see Figure 1.1. By such a process our understanding of a chemical question deepens.

The historical development of quantum chemistry can be categorised into a number of eras. The earliest, first age of quantum chemistry, was characterised by computational results of a qualitative nature. These did much to help develop understanding of potential energy surfaces, geometries of molecules at equilibrium, reactive transition structures, and molecular orbital concepts. These insights were able to explain the physical origins of experimentally measured properties. The second age of quantum chemistry came about through the development of computer technology and accompanying developments in numerical algorithms. This enabled much more elaborate computations to be performed. In this second era, semi-quantitative agreement with experiment was already obtained for some measured quantities. Despite this improved accuracy, quantum chemical techniques were still not able to displace experimental measurements, but had become sufficiently reliable that they could be applied to situations in which experiments were not yet feasible. For example, the study of very short-lived molecular species, or the study of the properties of postulated molecules that had yet to be synthesised. The third age of quantum chemistry is best summarised by Graham Richards in his influential article of 1979:"The work represents perhaps a near perfect instance of theory being in harmony with experiment, each aspect vital to the other and the combination much more than the sum of the separate parts". Experimental measurements cannot be interpreted or understood in the absence of a reliable theoretical framework. The studies referred to by Richards showed the computational work to be an equal partner to the experiments. There have also been cases where computational studies have preceded experiments through the correct predictions of measured quantities, which have subsequently been confirmed by experiment. Since the dawn of this new age of quantum chemistry, alluded to above, rapid developments have continued and their success has made computational quantum chemistry an essential component of many modern chemical investigations.

Historically, computational quantum chemistry was restricted to the realm of specialists who had access to high performance computing facilities, a good knowledge of software construction and numerical methods, as well as a good understanding of the underlying quantum mechanical models. There is still a strong need for this type of specialist who can push the subject forward by developing new methods, or providing very efficient computer implementations of established techniques. However, the standard models that we shall discuss in this book have been developed to the point that serious molecular questions can be tackled by any good scientist, not just the computational specialist. Today elaborate quantum chemical calculations can be carried out using fast desktop machines, and readily available software, by non-specialists. The same care and rigour must be applied to the design and execution of such calculations as would be applied to the design of any scientific investigation. A poorly thought out study, whether computational or experimental, cannot produce useful results.

1.2 The Model: Quantum Mechanics

An interesting experiment, which the author has carried out on numerous occasions, is to ask a room full of 200 undergraduate chemistry students: What is chemistry? Invariably, one obtains a fascinating range of answers. Many will tell you that chemistry is about "making things", for example materials, medicines, or fuels. Others may tell you that chemistry is about understanding the physical processes that govern chemical properties, for example the rate of reaction between two molecules, or the colour of a molecule. All these answers, and many others beside, are equally valid. Yet the overriding answer is: Chemistry is a game that electrons play! In a sense this answer encapsulates all the other answers, since everything chemical is under the control of the electrons that participate in the chemical process. There are no chemical phenomena that cannot be traced back to the behaviour of electrons.

So chemistry is about electrons. To understand chemistry we need to understand the behaviour of electrons. We are familiar with electrons being negatively charged particles with mass. Additionally we know from the experiments of Davisson and Germer in 1925, involving the diffraction of electrons by a crystal, that electrons can behave as waves. This wave-particle duality is quantified in the de Broglie relation

λ = h/p (1.1)

where λ is the wavelength associated with a particle of mass, m, moving with velocity, v. The linear momentum is, p = mv, and h is Planck's constant. For slow-moving macroscopic objects the wavelength given by eqn (1.1) is undetectably small. However electrons confined within atoms and molecules are very light and fast-moving with comparatively large de Broglie wavelengths. This is the realm of quantum mechanics and the correct description of quantum mechanical particles, such as the electron, is provided by the Schrödinger equation. The electronic structure and properties of any molecule, in any of its available stationary states may be determined, in principle, by solution of Schrödinger's (time-independent) equation.

HΨA = EA ΨA (1.2)

In eqn (1.2), A labels the state of interest. For example, the ground state or the first electronically excited state. To begin we shall concern ourselves with the ground state only and suppress the state label. At the simplest level we want to find the energy, E, and the wavefunction, ψ, based on the hamiltonian operator, H, for the molecular system of interest. The Schrödinger equation can be solved exactly only for one-electron systems. Hence much of the apparatus of computational quantum chemistry is concerned with finding increasingly accurate approximations to the Schrödinger equation for many-electron molecular systems. As we shall see in due course, the accuracy of the approximations is intimately related to the computational cost of the underlying numerical algorithms.

The first chemical application of the Schrödinger equation was undertaken by Heitler and London in 1927. In their landmark paper they calculated the potential energy curve of the hydrogen molecule. Today we are able to perform calculations on much larger systems, perhaps including up to 1000 atoms, and the methods we use are very different from those used by Heitler and London. Developments in computational quantum chemistry have been closely allied to developments in computational hardware as well as algorithmic developments (Figure 1.1). This endeavour shows no sign of abating and the demand for computational studies to complement experimental work grows continually. This is easily understood since, as we have asserted, chemistry is about the behaviour of electrons. The Schrödinger equation furnishes us, in principle, with all information about the behaviour of electrons in molecules and in turn, all information about chemistry. As we have stated already, approximations are key and it emerges that there is in practice no "best" method in computational quantum chemistry. Studies on real chemical problems always involve a trade-off between accuracy and computational cost. For certain methods we can make formal statements about their relative merits as approximations to the Schrödinger equation. However if such methods are too computationally demanding to be applicable to a problem of interest then describing them as "better" is, at best, vague.

1.2.1 The Schrödinger Equation and the Born-Oppenheimer Approximation

Before proceeding to some details, it is useful to briefly describe a very powerful notational expedience, introduced by Paul Dirac in 1939, which we shall use throughout this book. We shall write the many equations and integrals that appear using Dirac notation. For example, consider how we can obtain the energy, E, from the Schrödinger equation as shown below (the conventional notation will be shown on the left hand side and the equivalent in Dirac notation on the right).

HΨ = EΨ [equivalent] H|Ψ> = H|Ψ> (1.3)

Now pre-multiply by ψ(the complex conjugate of Ψ) and integrate over all variables, call them τ,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

Now rearrange to obtain E:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

A quantity denoted in |> is termed a "ket" and here represents the wavefunction. <| is called a "bra" and represents the complex conjugate of |>. For real quantities, <| and |> are the same. When an operator is pre- and post-multiplied by a bra and ket, integration is assumed implicitly. Denoting a general operator as C, we form

e.g. <Ψ|H|Ψ> (1.6)

Accordingly this notation is often referred to as "bracket" notation. It is very widely used. There are many other subtle features, related to the description of vector spaces, which are implicit in the Dirac notation, but they will not concern us here. In the rest of this text we shall use Dirac notation and conventional notations as suits the discussion.

It turns out that despite the simple form in which the Schrödinger equation can be written, its solutions are far from simple to obtain. In fact the Schrödinger equation can only be solved for one-electron systems. To deal with more complex atoms and molecules we must introduce a number of approximations. There are three key ideas which we shall adopt. To motivate the first of these, let us look in more detail at the quantities that enter the Schrödinger equation. The model of the atom that we shall use consists of a set of protons positioned at the atomic nucleus and surrounded by a number of electrons. The number of protons is given by the atomic number, Z, which tells us the number of protons carrying a unit positive charge, e, in the atomic nucleus. For neutral atoms, Z also gives the number of electrons surrounding the nucleus, each with unit negative charge, -e. In the absence of electric or magnetic fields, the hamiltonian operator, H, then includes terms which specify the kinetic and potential energies of the electrons and nuclei. H includes (i) the kinetic energy of motion for electrons and nuclei; (ii) the potential energy of attraction between electrons and nuclei; (iii) the potential energy of repulsion between electrons and similarly the potential energy of repulsion between nuclei. These terms have the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

The quantities that enter eqns (1.7) – (1.9) are:

h: Planck's constant divided by 2ITLπITL,

me: the rest mass of the electron

MA: the mass of nucleus A

e: the charge on the proton

ε0: the permittivity of free-space

[nabla]2i and [nabla]2A: are the kinetic energy operators for electron i and nucleus A, respectively. [nabla]2 is known as the "laplacian operator" and has the general form (in cartesian coordinates) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

riA, rij and RAB: are the distance vector between electron i and nucleus A; the distance vector between electron i and electron j; the distance vector between nucleus A and nucleus B, respectively. For example, if A and B have cartesian coordinates,(xA,yA,zA) and (xB,yB,zB), respectively, the magnitude of the distance vector between A and B can be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. All other distances are defined similarly.

Before proceeding any further we can simplify the forms of eqns (1.7) – (1.9) by introducing atomic units (au). In this system of units a number of fundamental constants take the value of unity, hence

e = 1

me = 1

h = 1

4πε0 = 1

Table 1.1 shows these, and other key quantities assigned a value of 1 au, with their SI equivalents. A more complete list of quantities is given in Appendix 1A.

Using these definitions we can write the kinetic energy terms in au as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

where [bar.M]A is the ratio of the mass of nucleus A to the mass of the electron, [bar.M]A = MA/me. Similarly the potential energy of attraction between electrons and nuclei may be written in au as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

and the repulsive potential energy terms as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

The full molecular hamiltonian now becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

The ratio of the mass of the proton to that of the electron is mp/me [approximately equals] 1836. Therefore even the lightest nucleus, hydrogen, with its single proton is three orders of magnitude heavier than the electron. This large disparity in mass means that the relatively light electron will move much more quickly than the nucleus to which it is attached. This implies a difference in the time scales governing the motion of electrons and nuclei. The electrons will be able to execute several periods of their motion before the nuclei have moved to any significant degree. This means that we can quantise the motion of the electrons for a fixed position of the nuclei. If the nuclei are fixed then their kinetic energy is zero. Hence we can remove the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from the molecular hamiltonian in eqn (1.13), to yield the simpler "electronic" hamiltonian

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

Since the kinetic energy of the nuclei are assumed to be zero, the potential energy of repulsion between nuclei assumes a constant value (for a given position of the nuclei). This is termed the "nuclear repulsion" energy and is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)

(Continues...)

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Excerpted from Computational Quantum Chemistry by Joseph J W McDouall. Copyright © 2013 Joseph J W McDouall. Excerpted by permission of The Royal Society of Chemistry.
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Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Computational Quantum Chemistry;
Computational Electronic Structure Theory: The Computation of Molecular Properties: Understanding Molecular Wavefunctions, Orbitals and Densities: Relativistic Effects and Electronic Structure Theory: Subject Index;

Editorial Reviews

In the past few decades, computational resources have become more powerful every year and in addition methodology development has led to much more effi- cient techniques through parallelization of the calculations and the advent of den- sity functional theory. These reasons make it possible for computational quantum chemists to work on relatively large chem- ical systems with a total number of atoms well over 100 nowadays. As a result of this, interest in applications of computational quantum chemistry has considerably widened and opened up research opportu- nities in novel areas. In particular, applica- tions of “realistic” quantum chemical sys- tems have become possible and as such it is starting to become common practise in fields of, e.g., bioinorganic chemistry and biochemistry, to do experimental studies side-by-side with computational model- ing. This means that computational quan- tum chemistry does not operate in virtual worlds and settings anymore on small atomic systems, but can address major chemical problems. These combined experimental/computational studies gen- erally give a broader perspective of a chemical problem and look into it from different angles and perspectives than stand-alone experimental studies. Thus, the computational studies give important additional information alongside exper- iment and assist in the interpretation of the experimental data. Furthermore, with computational quantum chemistry short- lived catalytic intermediates and their reactivity patterns can be investigated, which coupled to experimental work can explain product distributions and reac- tion rates. In addition, the computationalwork can make predictions that encourage future experimental studies. This symbio- sis of experiment and theory has led to a large field of research, where theoreticians and experimentalists work together. As a result of that it is not uncommon any- more that PhD students and postdoctoral researchers do a combination of experi- ment and computation for a single mul- tidisciplinary project. However, although many experimentally based groups are starting to use computational chemistry methods, almost at a routine basis, nowa- days there are some serious caveats with the methods and techniques and often these computational studies cannot be done through “black-box”-procedures but require expert supervision. Although there is an increased popularity of computational quantum chemistry mainly through the use of computational quantum chemistry meth- ods by experimentalists, this does not mean these methods and techniques are routinely done with little or no prior knowledge of the theories and back- grounds. To highlight the difficulties in doing computational quantum chem- istry research on experimentally relevant chemical systems, McDouall has written a monograph on the chemical procedures and techniques behind the computational chemistry software packages and the many pitfalls the user should be aware of. The book, therefore, tries to address questions for beginners in doing computational chemistry research, including: 1. What does computational quantum chemistry offer? 2. Where do you start? 3. How do you select a theoretical model? 4. What useful output do I generate and how do I relate this to my experiment? The book is subdivided into five chap- ters covering the basics of computational quantum chemistry, electronic structure methods, computation of molecular prop- erties, molecular orbitals, spin densities and relativistic effects. These are the key methods and techniques necessary for computational quantum chemistry in col- laboration with experiment and a descrip- tion of the essential components of the output that can be linked to experiment. Each chapter has a logic set-up that first gives a layman’s explanation of the reasons and the background of the basic theories with clear figures. Of course, a quantum chemistry book cannot be complete with- out equations and there are quite a lot of those in this book. However, these are well explained andMcDouall puts them in a broad context and clearly defines their variables and uses. As such I do not feel that the equations scare off the reader here, but are illustrative of the background. There are plenty of examples in the text that explain the theories in better detail. The book is very well written and is aimed at starters in the field of compu- tational quantum chemistry, such as new Master and PhD students. This book, how- ever, is not like “normal” quantum chem- istry books, where the reader gets drowned in very difficult to understand equations that require a high level of Mathematics knowledge. Instead, the author has cho- sen a selective set of equations and explains in detail what the equation means in chemical terms, what you can do with it and how you can solve the equations. As such, it makes the book highly read- able even for starters in the field that not necessarily have a thorough previous background in computational quantum chemistry. It may even be worthwhile for experimentalists who collaborate with computational quantum chemist to read this book and get understanding of what computational quantum chemistry can offer. What I found particularly useful was the section on converting quantum chem- ical energies into free energies through thermodynamic state functions and thereby gives experimentally measurable variables. The book is illustrated with a large number of drawings and figures that highlight what is explained in the text. In summary, the book on “Computational Quantum Chemistry” by McDouall is highly recommended literature for anyone working in the field, collaborating with computational chemists or interested in moving into the field of computational quantum chemistry. The work is very accessi- ble to the lay-reader and should help and assist with getting started in the field. Received:12August2013;accepted:13August2013; published online:03September2013. Citation:deVisserSP(2013)Gettingstartedincom- putationalquantumchemistry.Front.Chem. 1:14. doi: 10.3389/fchem.2013.00014 This articlewassubmittedtoTheoreticaland ComputationalChemistry,asectionofthejournal FrontiersinChemistry. Copyright©2013deVisser.Thisisanopen-access articledistributedunderthetermsoftheCreative CommonsAttributionLicense(CCBY).Theuse,dis- tributionorreproductioninotherforumsisper- mitted, providedtheoriginalauthor(s)orlicensor arecreditedandthattheoriginalpublicationin this journaliscited,inaccordancewithaccepted academicpractice.Nouse,distributionorrepro- ductionispermittedwhichdoesnotcomplywith theseterms.

doi: 10.3389/fchem.2013.00014 - Sam P de. Visser