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Quantum Mechanics inChemistry 

Chemistry 110A

Prof. C. William. McCurdy2016 spring quarter

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Syllabus is Posted on SmartSite •  Text: We will use Donald A. McQuarrie and John D.

Simon, Physical Chemistry - A Molecular Approach

(University Science Books, 1997).

•  There is a solution manual available: Caveat emptor ! It

contains errors and a few nonsensical explanations.

• 

Minimum Prerequisites: Chem. 2C, one year of college

 physics, and Math 16C or Math 21C.  Having taken the

entire B.S. in Chemistry mathematics curriculum ( 21A-

D and 22A, 22AL (MatLab), and 22B. ) is better

•  Grading: The distribution of the points in the course

will be:Homework 20%Midterms (2) each 20%Final exam 40%

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Week Date TopicI March 29 Role of quantum mechanics in Chemistry (Chap. 1 is background)

March 31 Chapter 2 Classical Wave Equation,Superposition of solutions – interference

II April 5 Chapter 3 The Schrödinger Eq. and quantum interference,

Particle in a Box, separation of variables April 7 Chapter 4 Principles and Postulates of Quantum Mechanics

III April 12 Chapter 4 Hermitian operators, matrix elementsUncertainty, Ehrenfest’s theorem, commutators

 April 14 Chapter 5 Harmonic Oscillator + Rigid RotorIV April 19 Chapter 5 Harmonic Oscillator + Rigid Rotor

 April 21 Midquarter Exam on “Principles of Quantum Mech”-- in class  V April 26 Chapter 6 Schrödinger equation for Hydrogen

atom and separation of angular from radial motion April 28 Chapter 6 cont Degeneracy, and symmetry in the H atom

VI May 3 Chapter 7 Approximation methods: The variational principleMay 5 Chapter 7 cont.  First order perturbation theory, examples

VII May 10 Chapter 8 Spin, Stern Gerlach experiment, beginning many-electron atomsMay 12 Slater determinants, Hartree-Fock approximation

VIII May 17 Chapter 8 cont. Term Symbols, atomic excited states, Hund’s rules

May 19 Chapter 9 Diatomic Molecules, Born Oppenheimer approxMolecular HamiltoniansIX May 24 Midterm Exam on “Atoms and approximations” – in class  

May 26 Chapter 9 Molecular orbitals, H2+,X May 31 Chapter 9 Hartree-Fock & molecules, Bonding in first row diatomic

June 2 Electronic excited states of molecules

June 8 Final Exam 10:30 am Check date and time in last week of class

Note Date

Note Date

Note Date

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Studying for this course!  In the Chemistry curriculum this is the first course that

asks you to integrate what you learned in othercourses:!  General Chemistry! 

Calculus and Linear Algebra!  Physics

!  There are math and physics review sections in thetext -- use them if you have forgotten any of the skillsthat are assumed in the text.

!  Read the text and do the homework problemsearly. This subject is learned by doing -- not bywatching. 

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Mathematica, MATLAB, Python and theProblem Sets

•  Mathematica, MATLAB or Python will be required for someproblems after week 1.

•  It is highly recommended that you use one of them frequentlyto help check your algebra on homework problems.

• 

Mathematica, MATLAB and Python examples will be postedto help you improve your skills. 

•  Mathematica Tutorial by Professor McCurdy will beannounced for an evening in the week of April 4. Needed

Skills: –  Symbolic integration and differentiation –  Line graphs: f(x), and plotting of surfaces: f(x,y)

 –  Do and If statements, use of arrays

 –  Eigenvalues and eigenvectors of matrices

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Chapter 1: Why Quantum?

The Failures of Classical Mechanics•  Blackbody Radiation – This is important,

and we will look at it now.

• 

You saw the others in FreshmanChemistry: –  Photoelectric Effect –  Quantum interference -- wave properties of

particles –  Hydrogen atom spectrum

We will just briefly review them today

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Black Body Radiation•   A cavity at thermal equilibrium at temperature T•  What spectrum of electromagnetic radiation does it

emit?

• 

 An idealization of how any object characterized bya temperature glows (emits radiation)

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Where does the black body law come from?

Light consists of electromagnetic wavesobeying physics similar to mechanical waves

Classical modes of oscillation of a string. Each has adifferent frequency, but can have any amount of energy.

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Where does the black body law come from?

  Modes of light oscillation in a cavity.Light consists of electromagnetic wavesobeying physics similar to mechanical waves

Light modes have quantized energies with n = 1, 2, 3 … ! -- called “photons”

 E  =  n h" 

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Where does the black body law come from?

8!" 2

c3

 

"   k  B

T 

!

h!  e"h! /k  BT 

1" e"h! /k  BT 

Number of modesof frequency v  Average energyof each mode 

Classical

Quantum

Boltzmanndistribution

Classical Thermodynamics gave the same energy to each modeQuantum Mechanics gives a different mean energy to each mode

8!" 2

c3

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Black Body Radiation Law

• 

The “ultraviolet catastrophe” of classical physics•

 

Spectral radiation density [Energy/Vol/(unit offrequency)] -- Planck distribution law 

" (# )d # = 8$ kT 

# 4  d # 

 

" (# )d # = 8$ hc# 5

1ehc/# kT 

%1d # 

" (# )d # =8$ h

c3

# 3

eh# /kT 

%1d # 

Equivalent distribution in terms of frequency:

Classical mechanical distribution" 

max=

hc

4.96511 kT 

 

 E =  n h" Key concept that leadsto Planck law

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We see it every day

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Distributions (like the Planck distribution)

• 

 A distribution, like the density (a massdistribution), gives the relevant quantity onlywhen integrated over an interval or domain.

 Mass =   ! density   r( )Vol

 !    d 3r

Energy

Volume between " 1  and " 2  =

  ! " ( )" 1

" 2

 !    d " 

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From first problem set

c/4 times Planck distribution gives the distributionof Radiated Power

(Watts/unit area/unit frequency)

So integrating this distribution over all frequencies

gives

! " (T )d "   = c

48# hc3

" 

3

eh" /k  BT 

!1d " 

! " (T )d " 

0

!

 "    = Radiated Power/Area (Watts/m2 )

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Photoelectric Effect--freshman

chemistry version 

•  In chemistry this phenomenon is calledphotoionization and the work function

is called the “ionization potential”

 E kin=

mv2

2= h"  #$ 

! is the "work function"

 E kin= h"  # h" 

0

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Wave properties of particles

The De Broglie relationDiffraction of electrons from a crystalline surface 

" =h

 p

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Hydrogen Spectrum

• 

Line spectra appear for all atoms in absorption andemission.

•  The freshman chemistry story relates thosefrequencies to transitions between discrete energylevels given by the Bohr formula for hydrogen energy

levels. 

h"  = # E = E  f   $ E i

 

1

" =

1

n1

2 #

  1

n2

2

$ 

% & 

' 

( ) 109680cm

#1

Bohr formula

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Freshman chemistry

figures showing old

 pictures of spectradon’t remotely reflect

today’s spectroscopy 

Those were pictures ofspectra taken with 1930s

technology.

We know the energy levels

of atoms to many

significant figures from both experiment and

quantum theory

(From the NIST website) 

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Hydrogen Atom Transitions and Photoionization 

13.6 eV ionization energy

Ejected electron kinetic energy

1

" =

1

n1

2 #

  1

n2

2

$ 

% & 

' 

( ) 109680cm

#1

 

 E n=  "

 R H 

n2

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Correcting a false impression: The hydrogen atom absorbs

all  frequencies corresponding to energies above 13.6 eV --

 NO discrete lines, as there are for lower frequencies 

Photoionization

spectrum (photo

absorption) plottedversus kinetic

energy of ejected

electron

 E kin= h"  #13.6  eV 

Remember thephotoelectric effect –same equation for ejected

electron Energy

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Photoionization of an Atom Gives the

Binding Energies of the Atomic Orbitals

For each type of atom we see

“edges” in the photoionization cross

section (probability) corresponding

to the energies of the atomic orbitals

that are occupied in the atom

X-ray photoionization -- for example

at the Advanced Light Source at the

Lawrence Berkeley Lab or Advanced

Photon Source at Argonne

log-logplot 

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The history of early 20th centuryscience is NOT why we study QM 

• 

Bonding in molecules cannot beunderstood without Quantum Mechanics.

•  No chemical spectroscopy could be

interpreted without Quantum Mechanics.

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IR spectra: vibrational androtational transitions 

Visible/UV spectra - electronic,

vibrational, rotational transitions 

From this spectrum: CO bonddistance in CO2 = 1.16Å,

16O is aBoson (missing rotational lines) 

Electronic transition in I2 visiblespectrum 

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Time-dependent spectroscopy has beeninvented using short pulsed lasers

! 

Molecular Rotation: tens of picoseconds (10

-12

 s)!  Molecular Vibration: tens of femtoseconds(10-15 s)

!  Electronic motion: order of tens to hundreds of

attoseconds (10-18

 s) 

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ModernSpectroscopy

in the time

domain:“femtosecondchemistry”

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Femtosecond spectroscopyto follow a reaction

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Essential Mathematics

• 

Simple concepts of complex variables are essential

• 

Solutions of Schrödinger are frequently intrinsically

complex valued functions

!  Polar representation of a complex number

 z = x+ iy = rei" 

r

r =   z =   x2+  y

2

cos" =  x /r =  x /   x2+  y

2

sin" =  y /r =  y /   x2 +  y2

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Euler Identity and Complex Conjugation

!  Euler identity

!  You’ll use it constantly as the basis of other simple identities

!  Complex conjugation

ei" = cos" + i sin" 

 

ei" +2# i

= ei" 

 

 z = x + iy

 z*= x " iy

 f ( z)* means change the sign of i and

complex conjugate  z wherever it appears in  f ( z)

 

for example :

 f ( z) = 5 z2 + iz+ 2

 f ( z)* = 5   z*( )2

" iz* + 2

 

i = ei"  / 2

 

1

i

= e"i#  / 2

= "i

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Integrals of Complex Functions of a Real Variable 

• 

Same principles as integration of real functions, but keep

track of the complex quantities.

!  Simple example that will be used with the wave functions of the

hydrogen atom

 f (" )* g(" )

0

2# 

 $    d " =   e%i" e2i" 0

2# 

 $    d " =   ei" 0

2# 

 $    d " =ei

" 

i0

2# 

= %i e2# i % e0( ) = %i  1%1( ) = 0 

 f (" ) = ei" 

g(" ) = e2i" 

What is  f (" )*g(" )

0

2# 

 $    d "   ?

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Basic Linear Algebra

• 

Dot products of vectors appear with complex conjugation

in QM

• 

A vector is normalized to one if

•  Two vectors are orthogonal if

u  =

u1

u2

.

.

.

uN 

v  =

v1

v2...

vN 

u · v =

N X

i=1

ui  vi

u∗

· v =

N X

i=1

u∗

i  vi

u∗

·

u   = 1

u∗

· v   = 0

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Eigenvalues and eigenvectors of matrices

• 

A real symmetric (or Hermitian) matrix

• 

has N eigenvectors and eigenvalues

• 

and the eigenvalues are real numbers 

•  the eigenvectors are orthogonal

M =

M 11   M 12   M 13   · · ·   M 1N 

M 21   M 22   M 23   · · ·   M 2N 

.

.

.

.

.

.

.

.

.

.

.

.

M N 1   M N 2   M N 3   · · ·   M NN 

Mvn  = λ

nvn

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Eigenvalues are the roots of an Nth order polynomial

• 

Eigenvalues satisfy the secular equation (seculardeterminant = 0)

• 

So they are the zeroes of an Nth order polynomial, e.g.

for a 2 x 2 they satisfy a quadratic equation

det[M− λI] =  det

M 11 − λ   M 12   · · ·   M 1N 

M 21   M 22 − λ   · · ·   M 2N 

..

.

..

.

..

.

..

.M N 1   M N 2   · · ·   M NN  − λ

= 0

det

M 11 − λ   M 12

M 21   M 22 − λ

 = 0

(M 11 − λ)(M 22 − λ)−M 12M 21  = 0