Quantum Chemistry

122 Pages·2017·2.33 MB·English

by  Ajit J Thakkar

Quantum Chemistry
A concise introduction for students of physics, chemistry,
biochemistry and materials science
Ajit J Thakkar
Department of Chemistry, University of New Brunswick, Fredericton, Canada
Morgan & Claypool Publishers
CopyrightªMorgan&ClaypoolPublishers2014
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ISBN 978-1-627-05416-4(ebook)
ISBN 978-1-627-05417-1(print)
DOI 10.1088/978-1-627-05416-4
Version:20140601
IOPConcisePhysics
ISSN2053-2571(online)
ISSN2054-7307(print)
AMorgan&ClaypoolpublicationaspartofIOPConcisePhysics
Morgan&ClaypoolPublishers,40OakDrive,SanRafael,CA,94903,USA
Contents
Preface viii
1 Molecular symmetry 1-1
1.1 Symmetry operations and elements 1-1
1.1.1 Rotations around axes 1-1
1.1.2 Reflections through symmetry planes 1-4
1.1.3 Inversion through a center of symmetry 1-5
1.1.4 Improper rotations around improper axes 1-6
1.2 Classification of molecular symmetry 1-6
1.3 Implications of symmetry 1-9
Problems 1-10
2 Basic quantum mechanics 2-1
2.1 Wave functions specify a system’s state 2-1
2.2 Operators represent observables 2-2
2.2.1 Operators 2-2
2.2.2 Quantum chemical operators 2-4
2.3 Schrödinger’s equation 2-5
2.4 Measured and average values 2-6
Problems 2-7
3 Translation and vibration 3-1
3.1 A particle in a wire 3-1
3.1.1 Solving the Schrödinger equation 3-2
3.1.2 The energies are quantized 3-3
3.1.3 Understanding and using the wave functions 3-4
3.2 A harmonic oscillator 3-5
3.2.1 Molecular vibrations 3-7
Problems 3-8
4 Symmetry and degeneracy 4-1
4.1 A particle in a rectangular plate 4-1
4.2 Symmetry leads to degeneracy 4-2
4.3 Probabilities in degenerate states 4-4
v
QuantumChemistry
4.4 Are degenerate wave functions unique? 4-6
4.5 Symmetry of wave functions 4-7
Problems 4-8
5 Rotational motion 5-1
5.1 A particle on a ring 5-1
5.2 A particle on a sphere 5-4
5.2.1 Rotational wave functions 5-6
5.3 The rigid rotor model 5-7
Problems 5-8
6 The hydrogen atom 6-1
6.1 The Born–Oppenheimer approximation 6-1
6.2 The electronic Hamiltonian 6-2
6.3 The hydrogen atom 6-3
6.3.1 Energy levels 6-4
6.3.2 Orbitals 6-5
6.3.3 Electron density and orbital size 6-6
6.3.4 Spin angular momentum 6-8
6.4 Hydrogen-like ions 6-9
Problems 6-9
+
7 A one-electron molecule: H 7-1
2
7.1 The LCAO model 7-2
7.2 LCAO potential energy curves 7-3
7.3 The variation method 7-5
7.4 Beyond the LCAO model 7-6
7.5 Force constant and dissociation energy 7-7
7.6 Excited states 7-9
Problems 7-10
8 Many-electron systems 8-1
8.1 The helium atom 8-1
8.2 Spin and the Pauli postulate 8-2
8.3 Electron densities 8-4
8.4 The Hartree–Fock model 8-4
8.4.1 Matrix formulation 8-6
vi
QuantumChemistry
8.5 Atoms 8-7
8.6 Diatomic molecules 8-9
8.7 The Kohn–Sham model 8-11
Problems 8-13
9 Qualitative MO theory 9-1
9.1 The Hu€ckel model 9-1
9.2 Cumulenes 9-3
9.3 Annulenes 9-4
9.4 Other planar conjugated hydrocarbons 9-6
9.5 Charges, bond orders, and reactivity 9-7
9.6 The Hu€ckel model is not quantitative 9-9
Problems 9-10
10 Computational chemistry 10-1
10.1 Computations are now routine 10-1
10.2 So many choices to be made 10-2
10.2.1 Selection of a basis set 10-2
10.2.2 Selecting a functional 10-4
10.2.3 Heavy atoms and relativistic effects 10-4
10.2.4 Accounting for a solvent 10-5
10.3 Practical calculations 10-5
Further study 10-7
Appendices
A Reference material A-1
Matrices and determinants A-1
Miscellaneous A-2
Table of integrals A-3
Conversion factors A-4
Constants and Greek letters A-4
Equation list A-5
B Problem hints and solutions B-1
vii
Preface
All chemists and many biochemists, materials scientists, engineers, and physicists
routinely use spectroscopic measurements and electronic structure computations to
assist and guide their work. This book is designed to help the non-specialist user of
these tools achieve a basic understanding of the underlying concepts of quantum
chemistry. The emphasis is on explaining ideas rather than on the enumeration of
facts and/or the presentation of procedural details. The book can be used to teach
introductory quantum chemistry to second-orthird-year undergraduateseither as a
stand-alone one-semester course or as part of a physical chemistry or materials
sciencecourse.Researchersinrelatedfieldscanusethebookasaquickintroduction
or refresher.
The foundation is laid in the first two chapters which deal with molecular sym-
metry and the postulates of quantum mechanics, respectively. Symmetry is woven
through the narrative of the next three chapters dealing with simple models of
translational, rotational, and vibrational motion that underlie molecular spectros-
copyandstatisticalthermodynamics.Thenexttwochaptersdealwiththeelectronic
structure of the hydrogen atom and hydrogen molecule ion, respectively. Having
beenarmedwithabasicknowledgeoftheseprototypicalsystems,thereaderisready
to learn, in the next chapter, the fundamental ideas used to deal with the com-
plexitiesofmany-electronatomsandmolecules. Thesesomewhatabstractideasare
illustrated with the venerable Hu€ckel model of planar hydrocarbons in the penul-
timate chapter. The book concludes with an explanation of the bare minimum of
technical choices that must be made to do meaningful electronic structure compu-
tations using quantum chemistry software packages.
Iurgereaderswhomaybeafraidoftacklingquantumchemistrytorelax.Rumors
aboutitsmathematical contentand difficulty arehighly exaggerated.Comfortwith
introductory calculus helps but an open mind and some effort are much more
important.Youtoocanacquireaworkingknowledgeofappliedquantumchemistry
justlikethevastmajorityofstudentswhohavestudiedit.Sometipsforstudyingthe
material are listed below.
1. The material in later chapters depends on earlier ones. There are extensive
back references throughout to help you see the connections.
2. Solving problems helps you learn. Make a serious attempt to do the end-of-
chapter problems before you look at the solutions.
3. You do need to learn basic facts and terminology in addition to the ideas.
4. Study small amounts frequently. Complex ideas take time to sink in.
This book grew from the quantum chemistry course that I have taught at the
University of New Brunswick since 1985. During the first few years of teaching it,
IwasunabletofindatextbookthattreatedallthetopicswhichItaughtinawayI
liked. So in the fall of 1994, I wrote a set of ‘bare bones’ notes after each lecture
anddistributedthemduringthenextone.Theencouragingandpositiveresponseof
the students kept me going to the end of the course. Having arrived at a first draft
viii
QuantumChemistry
in this manner, the bare bones were expanded over the next few years. Since then,
this book has been revised over and over again using both explicit and implicit
feedback from students who have taken my course; it has been designed with
theirverbalandnon-verbalresponsestomylectures,questions,problems,andtests
in mind.
Fredericton Ajit J Thakkar
31 March 2014
ix
IOP Concise Physics
Quantum Chemistry
A concise introduction for students of physics, chemistry, biochemistry and materials science
Ajit J Thakkar
Chapter 1
Molecular symmetry
1.1 Symmetry operations and elements
Symmetry is all around us. Most people find symmetry aesthetically pleasing.
Molecular symmetry imposes constraints on molecular properties1. A symmetry
operationisanactionthatleavesanobjectlookingthesameafterithasbeencarried
out.Asymmetryelementisapoint,straightline,orplane(flatsurface)withrespectto
whichasymmetryoperationiscarriedout.Thecenterofmassmustremainunmoved
by any symmetry operation and therefore lies on all symmetry elements. When dis-
cussingmolecularsymmetry,wenormallyuseaCartesiancoordinatesystemwiththe
originatthecenterofmass.Therearefivetypesofsymmetryoperation.Theidentity
operation E does nothing and is included only to make a connection between sym-
metry operations and group theory. The other four symmetry operations—rotations
C , reflections σ, inversion i, and improper rotations S —are described next.
n n
1.1.1 Rotations around axes
AsymmetryaxisC ,ofordern,isastraightlineaboutwhich(1/n)thofafull‘turn’
n
(a rotation by an angle of 360 /n) brings a molecule into a configuration indistin-
!
guishable from the original one. A C axis must pass through the center of mass.
n
A C axis corresponds to a 360 rotation and so it is the same as the identity oper-
1 !
ation: C E. A C axis has a 360 /2 180 rotation associated with it. In H O,
1 2 ! ! 2
¼ ¼
C C
2 2
O O
H H
H H
1AsEugeneWignersaid,symmetryprovides‘astructureandcoherencetothelawsofnaturejustasthelawsof
natureprovideastructureandcoherencetoasetofevents’.
doi:10.1088/978-1-627-05416-4ch1 1-1 ªMorgan&ClaypoolPublishers2014
QuantumChemistry
the line bisecting the HOH angle is a C axis; rotation about this axis by 180 just
2 !
interchanges the two hydrogen nuclei. If the z axis is a C axis, then its action on a
2
nucleus is to move it from its original position (x, y, z) to ( x, y, z). Thus
# #
x x
#
C z y y : 1:1
2ð Þ2 3¼2# 3 ð Þ
z z
6 7 6 7
4 5 4 5
AC axisgeneratesonlyoneuniquesymmetryoperationbecausetwo180 rotations
2 !
bring an object back to its original configuration; that is, C C C2 E. Each of
theobjectsA,BandCinfigure1.1hasexactlyoneC axis.T2h2e¼C a2xi¼sisalongthe
2 2
xaxisinobjectA,alongtheyaxisinobjectB,andalongthezaxisinobjectC.The
moresymmetricalobjectDinfigure1.1hasthreeC axes,onealongeachofthex,y
2
and z axes.
A square has a C axis of symmetry as illustrated in figure 1.2. Performing two
4
successive C or 360 /4 90 rotations has the same effect as a single C or 180
4 ! ! 2 !
¼
rotation;insymbols,C2 C .HenceforeveryC axisthereisalwaysacollinearC
4 ¼ 2 4 2
axis.Moreover,C4 C2 E andsoaC axisgeneratesonlytwouniquesymmetry
4 ¼ 2 ¼ 4
operations, C and C3. A clockwise C3 rotation is the same as a counter clockwise
4 4 4
y y
A B
x x
y y
C D
x x
Figure1.1. CanyouseetheC axesinobjectsA,B,C,andD?
2
1-2
QuantumChemistry
a b d a
C
4
d c c b
C
C 2 C
4 4
b c c d
C
4
a d b a
Figure1.2. C symmetryinasquare.
4
C rotation.Weadopttheconventionthatallrotationsareclockwise.AC axiscan
4 4
be found, for example, along each S–F bond in the octahedral molecule SF , and
6
along the axial I–F bond in the square pyramidal IF molecule. (Tip: nuclei not on
5
C occur in sets of n equivalent ones.)
n
In NH , the line passing through the nitrogen nucleus and the center of the
3
triangle formed by the hydrogen nuclei is a C axis; rotation by 360 /3 120 per-
3 ! !
¼
mutes the H nuclei a b, b c, c a . Methane has a C axis along each
3
ð ! ! ! Þ
C–H bond.
H
N
C
H H
H H
H H
AC axisgeneratestwouniquesymmetryoperations,C andC2.BenzenehasaC
3 3 3 6
axis perpendicular to the ring and passing through its center. A C axis generates
6
only two unique symmetry operations, C and C5, because a C and a C axis are
6 6 3 2
always coincident with it, and C2 C , C3 C , C4 C2, and C6 E. In
6 ¼ 3 6 ¼ 2 6 ¼ 3 6 ¼
O C O, the molecular axis is a C axis because rotation by any angle, however
¼ ¼ 1
small, about this axis leaves the nuclei unmoved.
1-3