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Structural Chemistry 参参: << 结结结结 >> 参参参参参参参参参参参 <<Physical Chemistry>> P.W. Atkins << 结结结结结结 >> 参参参参参 , 参参参参参参参 结结结结结结参参 参参参参参参参参参 结结结结结结结结结结结 结结结 参参 QQ 参 : 8675318 ( 参参参参参参参参参参 ) 参参参参参 : 参参 参参 Tel: 2181600 (mobile/office) email: [email protected] 参参参 236 参 http://pcss.xmu.edu.c n/users/xlu/ 参参参参 [email protected] 参参参 [email protected] 参参参参 :

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课程教学组 : 吕鑫 教授 Tel: 2181600 (mobile/office) email: [email protected] 化学楼 236 室 http://pcss.xmu.edu.cn/users/xlu/ 教辅: 王雪 [email protected] 张嘉伟 [email protected] 课程主页 : http://pcss.xmu.edu.cn/users/xlu/group/courses/structurechem/index.html. 参考书 : - PowerPoint PPT Presentation

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Page 1: Structural Chemistry

Structural Chemistry参考书 :

<< 结构化学 >> 厦门大学化学系物构组编

<<Physical Chemistry>> P.W. Atkins

<< 结构化学基础 >> 周公度编著 , 北京大学出版社

《化学键的本质》鲍林

上海科学技术出版社

《结构化学习题解析》周公度 段连运课程 QQ 群 : 8675318 ( 加入时请注明个人学号 )

课程教学组 : 吕鑫 教授 Tel: 2181600 (mobile/office) email: [email protected] 化学楼 236 室 http://pcss.xmu.edu.cn/users/xlu/

教辅:王雪 [email protected] 张嘉伟 [email protected] 课程主页 :http://pcss.xmu.edu.cn/users/xlu/group/courses/structurechem/index.html

Page 2: Structural Chemistry

两点说明:

• 作业要求 (交作业时间、理科作业规范)

• 考勤、考试方式与成绩 (期中 / 期末 / 随堂考试,期平成绩等)

Page 3: Structural Chemistry

What is ChemistryWhat is Chemistry

The branch of natural science that deals with composition, structure, properties of substances and the changes they undergo.

Page 4: Structural Chemistry

Types of substancesTypes of substances

Nano materials

Bulk materials

Geometric Structure

Size makes the difference

Electronic Structure

AtomsMolecules

ClustersCongeries

Page 5: Structural Chemistry

Structure determines propertiesProperties reflect structures

Structure vs. PropertiesStructure vs. Properties

Page 6: Structural Chemistry

Structural ChemistryStructural Chemistry

Inorganic ChemistryOrganic ChemistryCatalysisElectrochemistryBio-chemistryetc.

Material ScienceSurface ScienceLife ScienceEnergy ScienceEnvironmental Scienceetc.

Page 7: Structural Chemistry

Role of Structural Chemistry Role of Structural Chemistry in Surface Science in Surface Science

Page 8: Structural Chemistry

fcc(100) fcc(111)

fcc(775) fcc(10 8 7)

Surface structures of Pt single crystal

Low-index surface :NCS

LIS < NCB.

High-index surface :•Abundant edge sites.•NCes < NCS

LIS < NCB

•Lower NC ~ higher reactivity.

Page 9: Structural Chemistry

(111)

(775)

(100)

(10 8 7)

Different surfaces do different chemistry.

Structure-sensitive Catalysis!

Page 10: Structural Chemistry

418

1

25

Surface Structure vs. Catalytic Activity

N2 + 3H2 2NH3

Fe single crystal ,20atm/700K

Another example of Structure-sensitive Catalysis

Page 11: Structural Chemistry

Role of Structural Chemistry Role of Structural Chemistry in Material Science in Material Science

Page 12: Structural Chemistry

Graphite & Diamond StructuresDiamond: Insulator or wide bandgap

semiconductor: Graphite: Planar structure: sp2 bonding 2d metal (in plane)

Other Carbon allotropes“Buckyballs” (C60, C70 etc) “Buckytubes” (nanotubes),

other fullerenes

C Crystal StructuresC Crystal Structures

Structure makes the difference

Page 13: Structural Chemistry

Zheng LS ( 郑兰荪 ), et al.

Capturing the labile fullerene[50] as C50Cl10 

SCIENCE 304 (5671): 699-699 APR 30 2004

The pentagon-pentagon fusions in pristine C50-D5h are sterically strained and highly reactive. Perchlorination of these active sites stabilizes the labile C50-D5h.

Page 14: Structural Chemistry

Nature Materials, 2008, 7, 790.

Page 15: Structural Chemistry

Chlorofullerenes featuring triple

sequentially fused pentagons

Xie, S.Y., Lu, X., Zheng, L.S. et al

Nature Chemistry, in

press. #540C54Cl8 (a), #864C56Cl12 (b),

#4,169C66Cl6 (c), #4,169C66Cl10 (d).

Page 16: Structural Chemistry

Endohedral Metallofullerene:Sc4@C82 (C3v) vs. Sc4C2@C80 (Ih)

• (Sc2+)4@C828-

Shinohara, H. Rep. Prog. Phys. 2000, 63, 843.

Proposed QM-predicted

Sc4@C82

E = 28.8 kcal/mol

E = 0.0 kcal/mol

C26-@(Sc3+)4@C80

6--Ih

A Russian-Doll endofullerene

X.Lu, J. Phys. Chem. B. 2006, 110, 11098; Lu & Wang, J. Am. Chem. Soc. 2009, 131, 16646.Highlighted by C&EN and Nat. Chem.

Page 17: Structural Chemistry

Role of Structural Chemistry Role of Structural Chemistry in Life Science in Life Science

Page 18: Structural Chemistry

What do proteins do ?Proteins are the basis of how biology gets things

done.

• As enzymes, they are the driving force behind all of the biochemical reactions which makes biology work.

• As structural elements, they are the main constituents of our bones, muscles, hair, skin and blood vessels.

• As antibodies, they recognize invading elements and allow the immune system to get rid of the unwanted invaders.

Page 19: Structural Chemistry

What are proteins made of ?

• Proteins are necklaces of amino acids, i.e. long chain molecules.

Page 20: Structural Chemistry

Definition of Structural Chemistry• It is a subject to study the microscopic

structures of matters at the atomic/molecular level using Chemical Bond Theory.

• Chemical bonds structures properties.

Page 21: Structural Chemistry

Objective of Structural ChemistryObjective of Structural Chemistry

1) Determining the structure of a known substance

2) Understanding the structure-property relationship

3) Predicting a substance with specific structure and property

Page 22: Structural Chemistry

Chapter 1 basics of quantum mechanics 4Chapter 2 Atomic structure 4Chapter 3 Symmetry 3Chapter 4 Diatomic molecules 3 Chapter 5/6 Polyatomic structures 5Chapter 7 Basics of Crystallography 4 Chapter 8 Metals and Alloys 1Chapter 9 Ionic compounds 3Special talk

OutlineOutline

Page 23: Structural Chemistry

Chapter 1 The basic knowledge of quantum mechanics

1.1 The origin of quantum mechanics--- The failures of classical physics

Black-body radiation, Photoelectric effect, Atomic and molecular spectra

• Classical physics: (prior to 1900) Newtonian classical mechanics

Maxell’s theory of electromagnetic waves

Thermodynamics and statistical physics

Page 24: Structural Chemistry

1.1.1 Black-body radiation

Device for the experimentation of black-body radiation.

Page 25: Structural Chemistry

It can not be explained by classical thermodynamics and statistical mechanics.

Black-Body Radiation

Classical Solution:

Rayleigh-Jeans Law

(high energy, Low T)

Wien Approximation

(long wave length)Wave length / m

A number of experiments revealed the temperature-dependence of max and independence on the substance made of the black-body device.

Page 26: Structural Chemistry

The dawn of quantum mechanics

Page 27: Structural Chemistry
Page 28: Structural Chemistry

1.1.2 The photoelectric effect

Page 29: Structural Chemistry

The photoelectric effect

Page 30: Structural Chemistry

The Photoelectric Effect

1. The kinetic energy of the ejected electrons depends exclusively and linearly on the frequency of the light. 2. There is a particular threshold frequency for each metal. 3. The increase of the intensity of the light results in the increase of the number of photoelectrons.

Page 31: Structural Chemistry

Classical physics: The energy of light wave should be directly proportional to intensity and not be affected by frequency.

Page 32: Structural Chemistry

Explaining the Photoelectric Effect• Albert Einstein

– Proposed a corpuscular theory of light (photons) in 1905.

– won the Nobel prize in 1921

1. Light is consisted of a stream of photons. The energy of a photon is proportional to its frequency.

= h h = Planck’s constant

2. A photon has energy as well as mass. Mass-energy relationship: = mc2 m= hc2

3. A photon has a definite momentum. p=mc= hc=h/

4. The intensity of light depends on the photon density.

Page 33: Structural Chemistry

Therefore, the photon’s energy is the sum of the photoelectron’s kinetic energy and the binding energy of the electron in metal.

• Ephoton = E binding + E Kinetic energy

• h=W + Ek

Explaining the Photoelectric Effect

Page 34: Structural Chemistry

Example I: Calculation Energy from Frequency

Problem: What is the energy of a photon of electromagnetic radiation emitted by an FM radio station at 97.3 x 108 cycles/sec?What is the energy of a gamma ray emitted by Cs137 if it has a frequencyof 1.60 x 1020/s?

Ephoton =h = (6.626 x 10 -34Js)(9.73 x 109/s) = 6.447098 x 10 -24J

Ephoton = 6.45 x 10 - 24 J

Egamma ray =h = ( 6.626 x 10-34Js )( 1.60 x 1020/s ) = 1.06 x 10 -13J

Egamma ray = 1.06 x 10 - 13J

Solution:

Plan: Use the relationship between energy and frequency to obtain the energy of the electromagnetic radiation (E = h).

Page 35: Structural Chemistry

Example II: Calculation of Energy from Wavelength

Problem: What is the photon energy of electromagnetic radiationthat is used in microwave ovens for cooking, if the wavelength of theradiation is 122 mm ?

wavelength = 122 mm = 1.22 x 10 -1m

frequency = = = 2.46 x 1010 /s3.00 x108 m/s1.22 x 10 -1m

cwavelength

Energy = E = h = (6.626 x 10 -34Js)(2.46 x 1010/s) = 1.63 x 10 - 23 J

Plan: Convert the wavelength into meters, then the frequency can becalculated using the relationship;wavelength x frequency = c (where cis the speed of light), then using E=h to calculate the energy.Solution:

Page 36: Structural Chemistry

Example III: Photoelectric Effect

• The energy to remove an electron from potassium metal is 3.7 x 10 -19J. Will photons of frequencies of 4.3 x 1014/s (red light) and 7.5 x 1014 /s (blue light) trigger the photoelectric effect?

• E red = h = (6.626 x10 - 34Js)(4.3 x1014 /s)

E red = 2.8 x 10 - 19 J

• E blue = h = (6.626 x10 - 34Js)(7.5x1014 /s)

E blue = 5.0 x 10 - 19 J

Page 37: Structural Chemistry

• The binding energy of potassium is = 3.7 x 10 - 19 J

• The red light will not have enough energy to knock an electron out of the potassium, but the blue light will eject an electron !

• E Total = E Binding Energy + EKinetic Energy of Electron

• E Electron = ETotal - E Binding Energy

• E Electron = 5.0 x 10 - 19J - 3.7 x 10 - 19 J

= 1.3 x 10 - 19Joules

Page 38: Structural Chemistry

1.1.3 Atomic and molecular spectra

Page 39: Structural Chemistry

The Line Spectra of Several Elements

An atom can emit lights of discrete/specific frequencies upon electric/photo-stimulation.

Page 40: Structural Chemistry

• First proposed by Rutherford in 1911.

•The electrons are like planets of the solar system --- orbit the nucleus (the Sun).

• Light of energy E given off when electrons change orbits of different energies.

Why do the electrons not fall into the nucleus?

Why are they in discrete energies?

Planetary model:

Based on classical physics, the electrons would be attracted by the nucleus and eventually fall into the nucleus by continuously emitting energy/light!!

Page 41: Structural Chemistry

Bohr’s atomic model • Niels Bohr, a Danish physicist, combined the Plank’s

quanta idea, Einstein’s photon theory and Rutherford’s Planetary model, and first introduced the idea of electronic energy level into atomic model.

• Quantum Theory of Energy.• The energy levels in atoms can be

pictured as orbits in which electrons travel at definite distances from the nucleus.

• These he called “quantized energy levels”, also known as principal energy levels.

n=1n=2n=3n=4

n : principal quantum number

Page 42: Structural Chemistry

The electron in H atom can be promoted to higher energy levels by photons or electricity.

Page 43: Structural Chemistry

The Energy States of the Hydrogen Atom

Bohr derived the energy for a system consisting of a nucleus plus a single electron eg.

He predicted a set of quantized energy levels given by :

- R is called the Rydberg constant (2.18 x 10-18 J)- n is a quantum number- Z is the nuclear charge

n 1,2,3.. .En RZ2

n2

H He Li2

Page 44: Structural Chemistry

Problem: Find the energy change when an electron changes from then=4 level to the n=2 level in the hydrogen atom? What is the wavelengthof this photon?

Plan: Use the Rydberg equation to calculate the energy change, then calculate the wavelength using the relationship of the speed of light.Solution:

Ephoton = -2.18 x10 -18J -1n1

2

1n2

2

Ephoton = -2.18 x 10 -18J - = 4.09 x 10 -19J14 2

12 2

= = h c E

(6.626 x 10 -34Js)( 3.00 x 108 m/s)

4.09 x 10 -19J= 4.87 x 10 -7 m = 487 nm

Page 45: Structural Chemistry

1.1.1 Black-Body Radiation

Planck’s quanta idea E= nh

Lesson 1 Summary

1.1 The failures of classical physics

1.1.2 The photoelectric effect

A corpuscular theory of light (photons) = h h = Planck’s constantp=h/

1.1.3 Atomic and molecular spectra Planetary model: orbits of electrons around the nucleusBohr’s atomic model: quantized energy levels of orbits

En RZ2

n2

Page 46: Structural Chemistry

1.2 The characteristic of the motion of microscopic particles

Page 47: Structural Chemistry

1.2.1 The wave-particle duality of microscopic particles

• In classical physics, waves and particles behave differently and can be described by rather different theories.

• As mentioned above, Einstein’s Corpuscular Theory of Lights (photons) to explain the photoelectric effect for the first time introduced the wave-particle duality of photon.

• In 1924 de Broglie suggested that microscopic particles such as electron and proton might also have wave properties in addition to their partical properties.

Page 48: Structural Chemistry

De Broglie considered that the wave-particle relationship in light is also applicable to particles of matter, i.e.

•The latter equation clearly reflects the wave-particle duality by involving both a particle property (momentum) and a wave property (wavelength)!

•The wavelength of a particle could be determined by

= h/p = h/mv (v: velocity, m: mass)

E=h

p=h/

h = Planck’s constant,

p = particle momentum,

= de Broglie wavelength

Page 49: Structural Chemistry

de Broglie Wavelength

Example: Calculate the de Broglie wavelength of an electron with speed 3.00 x 106 m/s.

electron mass = 9.11 x 10 -31 kg

velocity = 3.00 x 106 m/s

= =

h mv

6.626 x 10 - 34Js( 9.11 x 10 - 31kg )( 1.00 x 106 m/s )

Wavelength = 2.42 x 10 -10 m = 0.242 nm

J = kg m2

s2hence

Page 50: Structural Chemistry

The moving speed of an electron is determined by the potential difference of the electric field (V)

eVm 2v2

1

If the unit of V is volt, then the wavelength is:

)(10226.1

1

10602.110110.92

10626.6

2v/

9

1931

34

mV

V

Vme

hmh

1eV=1.602x10-19 J

Page 51: Structural Chemistry

The de Broglie Wavelengths of Several particles

Particles Mass (g) Speed (m/s) (m)

Slow electron 9 x 10 - 28 1.0 7 x 10 - 4

Fast electron 9 x 10 - 28 5.9 x 106 1 x 10 -10

Alpha particle 6.6 x 10 - 24 1.5 x 107 7 x 10 -15

One-gram mass 1.0 0.01 7 x 10 - 29

Baseball 142 25.0 2 x 10 - 34

Earth 6.0 x 1027 3.0 x 104 4 x 10 - 63

Page 52: Structural Chemistry

Different Behaviors of Waves and Particles

Page 53: Structural Chemistry

Si Crystal

Electron beam (50eV)

The diffraction of electrons

STM image of Si(111) 7x7 surface

Page 54: Structural Chemistry

Spatial image of the confined electron states of a quantum corral. The corral was built by arranging 48 Fe atoms on the Cu(111) surface by means of the STM tip. Rep. Prog. Phys. 59(1996) 1737

Electron as waves

Page 55: Structural Chemistry

• Wave (i.e., light)

- can be wave-like (diffraction)

- can be particle-like (p=h/)• Particles

- can be wave-like ( =h/p)

- can be particle-like (classical)

The wave-particle duality

•A wave of microscopic particles is a probability wave, neither like the macroscopic mechanical wave nor like the normal electromagnetic wave!

•It reflects the statistic probability of particle presenting.

Page 56: Structural Chemistry

For photon:

p=mc,

E=h=h (c/) = pc = mc2

The differences between photon and microscopic particles

For particles:

p=mv,

E=(1/2) mv2 = p2/2m=pv/2

E=h

p = h/

= u / … u is not the moving speed of particles.

p2/2m =(1/2) mv2

pv

Question: what is the exact expression of u?

Page 57: Structural Chemistry

1.2.2 The uncertainty principle

•In classical Physics, the position and momentum of a macroscopic particle (a body) can be certainly determined at a give time.

•This is not the case for a microscopic particle.

•In the diffraction experiments to reveal the wave-particle duality of electrons, the observed wave pattern is just a statistic distribution of electron motion. The exact position and momentum of an electron at a given time remains uncertain.

Page 58: Structural Chemistry

The experiments of electron beam diffraction revealed that the narrow the slit is, the larger is the central area of the diffraction pattern.

Page 59: Structural Chemistry

hpx

hpxD

hD

hDp

ppp

pp

DDOAOC

APOP

xo

x

//

)0(sin

sin

/2

1/

2

1/sin

2

1

Include higher order,

y

psin

A

O

C

P

A

O

P

A

O

P

2

1

4or

hpx

A quantitative version

electrons. ofy uncertaintposition

slit). theof(width 2OADx

e OA

P

Q

C

x

e OA

P

Q

C

x

e OA

P

Q

C

x

Page 60: Structural Chemistry

Example

The speed of an electron is measured to be 1000 m/s to an accuracy of 0.001%. Find the uncertainty in the position of this electron.

The momentum is

p = mv = (9.11 x 10-31 kg) (1 x 103 m/s)

= 9.11 x 10-28 kg.m/s

p = p x 0.001% = 9.11 x 10-33 kg m/s

x = h / p = 6.626 x 10-34 / (9.11 x 10-33 )

= 7.27 x 10-2 (m)

Page 61: Structural Chemistry

Example

The speed of a bullet of mass of 0.01 kg is measured to be 1000 m/s to an accuracy of 0.001%. Find the uncertainty in the position of this bullet.

The momentum is

p = mv = (0.01 kg) (1 x 103 m/s) = 10 kg.m/s

p = p x 0.001% = 1 x 10-4 kg m/s

x = h / p = 6.626 x 10-34 / (1 x 10-4 )

= 6.626 x 10-30 (m)

Page 62: Structural Chemistry

Example

The average time that an electron exists in an excited state is 10-8 s. What is the minimum uncertainty in energy of that state?

tE

eV 100.66

eV 106.1

1006.1 J 10 x 1.06

s 10Js/ 10 x 1.06 /

7

19

26 26-

-8-34min

tE

Page 63: Structural Chemistry

Measurement

•Classical: the error in the measurement depends on the precision of the apparatus, could be arbitrarily small.

•Quantum: it is physically impossible to measure simultaneously the exact position and the exact velocity of a particle.

Page 64: Structural Chemistry

CLASSICAL vs QUANTUM MECHANICS

Macroscopic matter - Matter is particulate, energy varies continuously. The motion of a group of particles can be predicted knowing their positions, their velocities and the forces acting between them.

Microscopic particles - microscopic particles such as electrons exhibit a wave-particle “duality”, showing both particle-like and wave-like characteristics. The energy level is discrete. …

The description of the behavior of electrons in atoms requires a completely new “quantum theory”.

Page 65: Structural Chemistry

Quantum mechanical description of Electron

• Quantum mechanics is basically statistical in nature.

• Quantum mechanics does not say that an electron is distributed over a large region of space as a wave is distributed.

• Rather it is the probability patterns used to describe the electron’s motion that behave like waves.

Page 66: Structural Chemistry

1.3 The basic assumptions (postulates) of quantum

mechanics

Page 67: Structural Chemistry

Postulate 1. The state of a system is described by a wave function of the coordinates and the time.

Page 68: Structural Chemistry

In CM (classical mechanics), the state of a system of N particlesis specified totally by giving 3N spatial coordinates (Xi, Yi, Zi)and 3N velocity coordinates (Vxi, Vyi, Vzi).

In QM, the wave function takes the form (r, t) that depends onthe coordinates of the particles and on the time.

))(/2exp[( EtxphiA x

The wavefunction for a single particle of 1-D motion is:

For example:

)/(2exp[ vtxiA The wavefunction of plane monochromatic light:

Page 69: Structural Chemistry

A wave function must satisfy 3 mathematical conditions:

1. Single-value; 2. Continuous; 3. Quadratically integrable.

a) The product of wave function (r,t) and its complex conjugate (r,t)* represents the probability distribution function of the system. (Physical meaning of wave function!)

b) The wave function (r,t) must be continuous in space. Otherwise its second derivative would not be attainable.

c) The wave function of a system must be quadratically integrable so as to evaluate the statistical average values of its properties.

Page 70: Structural Chemistry

dxdydztrtr ),(),(* The probability that the particle lies in the volume element dxdydz, located at r, at time t.

The probability

1 ),(),(*

dxdydztrtr

To be generally normalized

0 ),(),(

dxdydztrtr ji

Wave functions of different states for a given system must be generally orthogonal

Page 71: Structural Chemistry

Postulate 2. Each observable mechanical quantity of a microscopic system is associated respectively with a linear Hermitian operator.

To find this operator, write down the classical-mechanical expression for the observable in terms of Cartesian coordinates and corresponding linear-momentum, and then replace each coordinate x by the operator, and each momentum component px by the operator –iћ/x.

Page 72: Structural Chemistry

An operator is a rule that transforms a given function into another function. E.g. d/dx, sin, log

C)BA()CB(A

Operators obey the associative law of multiplication:

d/dxD 5)( 3 xxf23 3)'5()(ˆ xxxfD

f(x)Bf(x)A)f(x)BA(

f(x)Bf(x)A)f(x)BA(

f(x)]B[Af(x)BA

Page 73: Structural Chemistry

• A linear operator means

• A Hermitian operator means

*111

*1 )ψA(ψψAψ *

122*1 )ψA(ψψAψ

• A Hermitian operator ensures that the eigenvalue of the operator is a real number

2121 ψAψA)ψ(ψA

ψAcψA c

Page 74: Structural Chemistry

Eigenfunctions and Eigenvalues

Suppose that the effect of operating on some function f(x) with the operator  is simply to multiply f(x) by a certain constant k. We then say that f(x) is an eigenfunction of  with eigenvalue k.

kf(x)f(x)A

2x2x 2e(d/dx)e

Eigen is a German word meaning characteristic.

Page 75: Structural Chemistry

To every physical observable there corresponds a linear Hermitian operator. To find this operator, write down the classical-mechanical expression for the observable in terms of Cartesian coordinates and corresponding linear-momentum components, and then replace each coordinate x by the operator x. and each momentum component px by the operator

iћ/x.

Mechanical quantities and their Operators

Page 76: Structural Chemistry

Position x

Momentum (x) px

Angular Momentum (z) Mz=xpy-ypx

Kinetic Energy T=p2/2m

Potential Energy V

Total Energy E =T+V

Some Mechanical quantities and their Operators

Mechanical quantities Methematical Operator

22

2

2

2

2

2

2

2

2

2

m8-)

z

y

x(

m8-T

hh

V )z

y

x

(m8

- H2

2

2

2

2

2

2

2

h

xx2-px

iih

)x

-y

(2

-Mz

yxih

xx

VV

Hamiltonian

Page 77: Structural Chemistry

If a system is in a state described by a normalized wave function , then the average value of the observable A corresponding to operator  is given by –

dAa

ˆ *

The average value of the physical observable

Page 78: Structural Chemistry

n

nnn

nnn

nn

a

da

da

dAa

*

*

ˆ*

Thus the only value we measure is the value an.

If the wave function is an eigenfunction of Â, with eigenvalue an,

then a measurement of the observable corresponding to Â

will give the value an with certainty.

n

nn

a

dAa

2

22

ˆ*

0

-

-

22

222

nn

a

aa

aa

Page 79: Structural Chemistry

• When two operators are commutable, their corresponding mechanical quantities can be measured simultaneously.

Commuted operators ( 对易算符 )

0FG-GF]G ,F[

Poisson bracket

Page 80: Structural Chemistry

Postulate 3: The wave-function of a system evolves in time according to the time-dependent Schrödinger equation

Page 81: Structural Chemistry

Assumption 3: The wave-function of a system evolves in time according to the time-dependent Schrödinger equation -

ttzyxH

i ),,,(ˆ

In general the Hamiltonian H is not a function of t, so we canapply the method of separation of variables.

f(t) z)y,(x, ),,,( tzyx

t

zyxtzyx-iE

e ),,( ),,,(

Edt

tdf

f(t)

H

)(1i

z)y,(x,

z)y,(x,ˆ

dt

tdff(t)H

)(z)y,(x,i z)y,(x,ˆ

z)y,(x,z)y,(x,ˆ EH

Edt

tdf

f(t)

)(1i

Page 82: Structural Chemistry

Time-independent Schrödinger’s Equation

V )z

y

x

(m8

- H2

2

2

2

2

2

2

2

h

VTH ˆˆˆ

xx2-Px

iih

m

PT

2

ˆˆ

2

2

04

ZeV

r

e.g. H atom

EH ˆ Eigenvalue equation

2z

2y

2x

2 PPPP

m

pmvT

22

1 22

Page 83: Structural Chemistry

Spherical polar coordinates

2

2

2222

22

sin

1)(sin

sin

1)(

1

rrr

rrr

The Schrödinger equation EH ˆ

e-n2

e2

2

V m8

- H h

2

2

22

2

2

2

2

2 12

rrrryxa

2

2

22

2

2

2 1

ayx

Ema 2

2

2

2

2 ,3,2,1,

8

)(1

),(1

22

22

21

21

nma

nhEE

nCosnSin

Page 84: Structural Chemistry

The Schrödinger’s Equation is an eigenequation.

aψψ A

I. The eigenvalue of a Hermitian operator is a real number.

****A ψaψ

ψψaψψ **A

**** )A( ψψaψψ *aa

Proof:

Quantum mechanical operators have to have real eigenvalues

In any measurement of the observable associated with the operator A, the only values that will ever be observed are the eigenvalues a, which satisfy the eigenvalue equation.

Page 85: Structural Chemistry

0 * ijji d

II. The eigenfunctions of Hermitian operators are orthogonal

Page 86: Structural Chemistry

II. The eigenfunctions of Hermitian operators are orthogonal

Consider these two eigen equations

mmm ψaψ A

nnn ψaψ A

Multiply the left of the 1st eqn by m* and integrate, then take the complex conjugate of eqn 2, multiply by n and integrate

a ˆ *n

* ddA nmnm

a ˆ **m

** ddA mnmn

Page 87: Structural Chemistry

There are 2 cases, n = m, or n m

0**)a - (a

**ˆ ˆ*

mn

d

dAdA

nm

mnnm

0 **)a - (a mn dnm

Subtracting these two equations gives -

If n = m, the integral = 1, by normalization, so an = an*

Page 88: Structural Chemistry

If n m, and the system is nondegenerate (i.e.different eigenfunctions do not have the same eigenvalues, an am ), then

0 *)a - (a mn dnm

0 * dnm

The eigenfunctions of Hermitian operators are orthogonal

0 * ijji d

Page 89: Structural Chemistry

Example:

)2(32

1)(

0

2/

30

20

a

re

aH ar

s

oar

s ea

H /

30

1

1)(

0 02 / / 2 2

1 2 3 0 0 000

1(2 ) sin

4 2r a r a

s s

rd e e r drd d

aa

dra

rre

aa

r

0

0

22

3

30

)2(24

40

0

0

32

3

0

22

3

30

00[2

1dr

a

redrre

aa

r

a

r

0 0

32 ]81

16

27

16[

2

1dyyedyye yy

ya

r

02

3

1 16 16[ (3) (4)]27 812

0]!381

16!2

27

16[

2

1

Page 90: Structural Chemistry

Postulate 4 : Superposition Principle ( 态叠加原理 ) If 1, 2,… n are the possible states of a microscopic system (a complete set), then the linear combination of these states is also a possible state of the system.

i

iinn ccccc 332211

The coefficient ci reflects the contribution of wavefunction i to .

Page 91: Structural Chemistry

Postulate 5 : Pauli’s principle( 泡利不相容原理 ).

Every atomic or molecular orbital can only contain a maximum of two electrons with opposite spins.

Page 92: Structural Chemistry

Energy level diagram for He. Electron configuration: 1s2

N

S

Hparamagnetic – one (more) unpaired electrons

N

S

Hediamagnetic – all paired electrons

Ene

rgy

1

2

3

0 1 2

n

l

Page 93: Structural Chemistry

ms = spin magnetic electron spin

ms = ±½ (-½ = ) (+½ = )

The complete wavefunction for the description of electronic motion should include a spin parameter in addition to its spatial coordinates.

•Two electrons in the same orbital must have opposite spins.

•Electron spin is a purely quantum mechanical concept.

Pauli exclusion principle:

Each electron must have a unique set of quantum numbers.

Page 94: Structural Chemistry

),(),,( sl msmln

The complete wavefunction for the description of electronic motion should include a spin parameter in addition to its spatial coordinates.

Fermions

•Particles that do obey the Pauli Exclusion Principle.

Bosons

•Particles that do not obey the Pauli Exclusion Principle

+ symmetry(Bosons)

- antisymmetry(Fermions))()(

)()(

)(

)(

1,22,1

2

1,2

2

2,1

2,1

qqqq

qqqq

qq

Heatomelectrontwofor

Page 95: Structural Chemistry

1.4 Solution of free particle in a box – a simple application of Quantum Mechanics

Page 96: Structural Chemistry

1.4.1 The free particle in a one dimensional box

V(x)= V(x)=

V(x)= 0

x=0 x=l

I

II

III

1. The Schrödinger’s Equation and its solution

V xm8

- H2

2

2

2

d

dh

In area I, III:

EVxm

h

2

2

2

2

8

08 2

2

2

2

mV

h

x

VEVVVh

m

x

)(08

2

2

2

2

1D box

(meaning no particle in area I and III).

Page 97: Structural Chemistry

II: V=0

Boundary condition and continuous condition: (0)=0, (l)=0

Hence, (0) =Acos0+Bsin0 = A + 0 = 0 => A =0

=Bsinx (B0)

(l) =Bsinx =Bsin l=0, Thus, l=n, =n/l (n=1,2,…)

0

8

2

2

2

2

2

2

xd

d

h

mEset

xBxA

eBeA xixi

sincos

or ''

xm8

- V xm8

- H2

2

2

2

2

2

2

2

d

dh

d

dh

H E

E xm8

-2

2

2

2

d

dh0

82

2

2

2

E

h

m

xd

d

Page 98: Structural Chemistry

...)3,2,1(8

8

2

22

2

22

2

22

nml

hnE

l

n

h

mE

xl

nB

sin

Normalization of wave-function:

xxdxx 2sin4

1

2

1sin 2

xl

n

l

sin2

lB

21

2

1)(

2

1 20

2 ln

l

l

nBx

n

l

l

nB l

1sin0

22 dxxl

nB

l

1 2

0 dx

l

Page 99: Structural Chemistry

2. The properties of the solutions

a. The particle can exist in many states

b. quantization energy

c. The existence of zero-point energy. minimum energy (h2/8ml2)

d. There is no trajectory but only probability distribution

e. The presence of nodes

2

2

1 8ml

hE

l

x

l

sin2

1

2

2

2 8

4

ml

hE l

x

l

2sin

22

2

2

3 8

9

ml

hE

l

x

l

3sin

23

n=1

n=2

n=3

Page 100: Structural Chemistry

Ene

rgy

E2

E3

E4

E1

n=2

• In the ground state (n=1), the highest probability of the particle occurs at the location l/2.

• In the first excited state (n=2), the highest probability of the particle occurs at the locations l/4 and 3l/4, the lowest probability at the location l/2.

nodenode

Page 101: Structural Chemistry

Discussion:

i. Normalization and orthogonality

0sinsin2

)()(00

ll

mn dxl

xm

l

xn

ldxxx

ii. Average value

2sinsin

20

ldx

l

xnx

l

xn

lx

l

3sinsin

2 2

0

22 ldx

l

xnx

l

xn

lx

l

0ˆ0

*

0

* l

nn

l

nnx dxdx

didxPp

2

222

2

22

2

22

22

44sin

2

4

ˆ

l

hncp

l

hn

l

xn

ll

hn

cdx

dicP

Page 102: Structural Chemistry

iii. Uncertainty

3222 l

xxx

l

nhppp

222

34232

nh

l

nhlpx

2

)(1

hpx

stategroundnwhen

Page 103: Structural Chemistry

a. Obtain the potential energy functions followed by deriving the Hamiltonian operator and Schrödinger equation.

b. Solve the Schrödinger equation. (obtain n and En)

c. Study the characteristics of the distributions of n (e.g., boundary conditions)

d. Deduce the values of the various physical quantities of each corresponding state.

The general steps in the quantum mechanical treatment:

Page 104: Structural Chemistry

3. Quantum leaks --- tunneling

V

x0 l

I II III),0(

8

)0(8

2

2

2

2

2

2

2

2

lxxExm

h

and

lxEVxm

h

The probability of penetration is given by

lEVmeVEVEP

)(22

)]/(1)[/(4

When E<V

• In classical mechanics, all particles with E > V can pass through the barrier of potential V, whereas the particles with E<V can not pass through the potential .

• This is not the case in quantum mechanics.

P decreases exponentially with increasing V and its width AWA m!!

Page 105: Structural Chemistry
Page 106: Structural Chemistry

Tunneling

CLASSICAL MECHANICS

QUANTUM MECHANICS

Page 107: Structural Chemistry

Tunneling in the “real world”

• Tunneling is widely exploited:

- for the operation of many microelectronic devices (tunneling diodes, flash memory, …)

- for advanced analytical techniques (scanning tunneling microscope, STM)

• Responsible for radioactivity (e.g. alpha particles)

Page 108: Structural Chemistry

Tip

Piezo-Tube

STM System

Mode: Constant Current mode, Constant height mode

zEmKeP

)(22

C28H588

Free electrons of metals can tunnel between the surfaces of two metals of atomic distances (~ 1 nm) driven by a suitable bias voltage.

Page 109: Structural Chemistry

1.4.2 The free particle in a three dimensional box

Page 110: Structural Chemistry

Let = (x, y, z)= X (x) Y (y) Z (z) (separation of variables)Substituting into 3-D Schrödinger equation 

Out of the box, V(x, y, z) = ∞ In the box, V(x, y, z) =0

Particle in a 3-D box of dimensions a, b, c

zz

by

ax

0

0

0

E

m

h 2

2

2

8

Ezyxm

h

)(8 2

2

2

2

2

2

2

2

Page 111: Structural Chemistry

EXYZXYZzyxm

h

Ezyxm

h

)(8

)(8

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

(separation of variables)

XEXxm

hx

2

2

2

2

8YEY

ym

hy

2

2

2

2

8ZEZ

zm

hz

2

2

2

2

8

E=Ex+Ey+Ez

Let Ez = E – (Ex+Ey)

EXYZz

ZXY

y

YXZ

x

XYZ

m

h

)(8 2

2

2

2

2

2

2

2

xEzZ

Z

yY

Y

m

hE

xX

X

m

h

)(88 2

2

2

2

2

2

2

2

2

2

Page 112: Structural Chemistry

c

zn

b

yn

a

xn

abcXYZ zyx sinsinsin

8

)(8 2

2

2

2

2

22

c

n

b

n

a

n

m

hEEEE zyx

zyx

b

yn

byY ysin

2)(

a

xn

axX xsin

2)(

c

zn

czZ zsin

2)(

The solution is:

c

zn

b

yn

a

xn

abcXYZ zyx sinsinsin

8

)(8 2

2

2

2

2

22

c

n

b

n

a

n

m

hEEEE zyx

zyx

2

22

8=

a

n

m

hE x

x

2

22

8=

b

n

m

hE

y

y

2

22

8=

c

n

m

hE z

z

( n= 1, 2, ……)

Page 113: Structural Chemistry

Multiply degenerate energy level when the box is cubic

(a = b = c)

)(8

)(8

2222

2

2

2

2

2

2

22

zyxzyx

zyx nnnma

h

c

n

b

n

a

n

m

hEEEE

The ground state: nx=ny=nz=1 2

2

8

3

ma

hE

The first excited state: ni=nj=1, nk=2

The wave-functions are called degenerate (triply degenerate)

2

2

8

6

ma

hE

112

121

211

E

Page 114: Structural Chemistry

m-dimensional box (m=1-3)• Wave functions

)1,...( sin2

,1

i mil

xn

l i

ii

i

m

ii

)1,...( 8

;2

2

1

miml

hnEEE

i

ii

m

ii

• Energy

Page 115: Structural Chemistry

1.4.3 Simple applications of a one-dimensional potential box model

Page 116: Structural Chemistry

Example 1: The delocalization effect of 1,3-butadiene

Four electron form two localized bonds

Four electron form a 44

delocalized bonds

l l l

E1

C CC C

E4/9

E1/9

C CC C

E=2×2 ×h2/8ml2=4E1E=2×h2/8m(3l )2+ 2×22 ×

h2/8m(3l )2 =(10/9)E1

Page 117: Structural Chemistry

Example 2: The adsorption spectrum of cyanines

R2N-(CH=CH-)m-CH=NR2+

..

Total electrons: 2m+4

In the ground state, these electrons occupy m+2 molecular orbitals

The adsorption spectrum correspond to excitation of electrons from the highest occupied (m+2) orbital to the lowest unoccupied (m+3) orbital.

)52(8

])2()3[(8 2

222

2

2

mlm

hmm

lm

hE

ee

The general formula of the cyanine dye:

E

n=1n=2

n=m+2n=m+3

Page 118: Structural Chemistry

Table 1. The absorption spectrum of the cyanine dye

R2N-(CH=CH-)mCH=NR2+

..

m max (calc) / nm max (expt) /nm

1 311.6 309.0

2 412.8 409.0

3 514.6 511.0

)52(8

])2()3[(8 2

222

mlm

hmm

lm

h

h

Ev

ee

)( 565248 pmml )(52

3.3

)52(

8 22

pmm

l

mh

clme

Page 119: Structural Chemistry

More Examples

1. Are eim and cos(m) eigenfunctions of operator id/d ? If so, please determine the eigenvalue.

imimim meimiee

d

di

So eim is an eigenfunction of operator id/d with an eigenvalue of –m.

mcmimmmimd

di cossin)sin(cos

So cosm is not an eigenfunction of operator id/d.

Page 120: Structural Chemistry

2. For the -conjugate molecule CH2CHCHCHCHCHCHCH2, its UV-vis spectrum shows the first long-wavelength absorption at 460 nm. Please estimate the length of its carbon chain using the 1D box model.

)8...3,2,1(8

2

22

nml

hnEn

hc

ml

h

ml

hEEE

2

2

2

222

45 8

9

8

)45(

1-4 are occupied and 5-8 are unoccupied in its ground state. The energy of the first excitation is

Ans: It has 8 p-electrons forming a delocalized 88 bond.

Energies of -MOs given by 1D box model are

pmmc

h

mhc

hl 1120

8

9

8

9 2

Page 121: Structural Chemistry

3. Does the following function represent a state of particle in a 1-dimensional box?

a

x

aa

x

ax

2sin

23sin

22)(

If yes, does it have a certain value of energy and what is the energy of this state? If not, determine the average energy.

Ans: Two wave functions of a particle in a 1-d box of length a are

a

x

ax

a

x

ax

2sin

2 )( ;sin

2)( 21

)(3)(2)( 21 xxx

So (x) represents a possible state of a 1d-box particle, according to the superposition principle.

)()(3)(2

)(ˆ3)(ˆ2)](3)(2[ˆ)(ˆ

2211

2121

xcxExE

xHxHxxHxH

So (x) is not an eigenfunction of Hamiltonian and has not a certain energy.

Page 122: Structural Chemistry

To evaluate <E>, (x) must be normalized.

)()(' xcx Suppose , then

1132

sin2

3sin2

2

)()()('

2

0

2

2

0

22

0

2

0

2

cdxa

x

aa

x

ac

dxxcdxxcdxx

a

aaa

13

12 c

2

2

2

22

0

2

2

2

2

2

0

13

55

2sin

23sin

22ˆ2

sin2

3sin2

2

8ˆ ,)('ˆ)'*(

ma

h

ma

hc

dxa

x

aa

x

aH

a

x

aa

x

ac

dx

d

m

hHdxxHxE

a

a

Page 123: Structural Chemistry

What is Quantum Mechanics?

QM is the theory of the behavior of very small objects (e.g. molecules, atoms, nuclei, elementary particles, quantum fields, etc.)

One of the essential differences between classical and quantum mechanics is that physical variables that can take on continuous values in classical mechanics (e.g. energy, angular momentum) can only take on discrete (or quantized) values in quantum mechanics (e.g. the energy levels of electrons in atoms, or the spins of elementary particles, etc).