Download Quantum Chemistry - Winona State University

Introductory Quantum Mechanics/Chemistry
Historical
Bohr's Atom
Wave vs. Particle
De Brogile's Hypothesis
Quantum Tools
Heisenberg
Operator Algebra
Postulates
Applications
Spring 2014
Translational
Vibrational
Rotational
Spectroscopy (NMR)
Genealogy of Quantum Mechanics
Classical Mechanics
(Newton)
High
Low
Velocity
Mass
Relativity
Wave Theory of Light
(Huygens)
Maxwell’s
EM Theory
Quantum Theory
Quantum Electrodyamics
Electricity and Magnetism
(Faraday, Ampere, et al.)
Energy and Matter
Size of Matter
Particle Property
Wave Property
Large – macroscopic
Mainly
Unobservable
Intermediate – electron Some
Some
Small – photon
Mainly
Few
E = m c2
The Wave Nature of Light
c   
E  h 
The speed of light is constant (2:30 into video)
Continuous
Spectrum
Line Spectra and the Bohr Model
Line Spectra
Bohr Model
• Colors from excited gases arise because electrons move between energy states in
the atom. (Electronic Transition)
Class Data
Line Spectra and the Bohr Model
Line Spectra
Line Spectra - Bohr Model

En   2.178 10
18
 Z2
J
 n2






Z  charge ( H  1 )
Ei f  E f  Ei  h
Ei  f
1 1
hc
18
 h    2.178  10 J  2  2 

 n f ni 
Line Spectra
Mathcad Calns
Fall 12
Bohr Model Experiment
1. Calculate the expected electronic transitions ( λ in nm ) in the Balmer
Series. [from higher n’s to n=2]
2. Calculate the series limit transition in the Balmer Series.
3. Using the Spectroscope, record observable lines (color and λ) for the
Hydrogen discharge tube and compare them to your Bohr Model
calculations.
4. Record observable lines for Oxygen and Water. [Calculate possible
lines in the visible region for the Oxygen atom using an appropriate
Z.]
5. Compare the observed lines for the three discharge tubes.
6. Using the following sample table (spreadsheet) as a guide, discuss
your results.
Tube
Color
Observed λ
Bohr Model λ ( ni →nf )
Hydrogen
…..
Class Data
Bohr Model Calculations for the H-atom
The Balmer Series: higher-n's down to nf=2
8
c  3.00 10 m s
1
n i  3
 34
h  6.626 10
n f  2
 19
E  3.025  10
 
h c
E
J s
 J 
 18
E  2.178 10
1
 nf2



2
ni

1
9
nm  10
J
7
  6.571  10
m
m
  657.1 nm
Class Data
Line Spectra and the Bohr Model
•
•
•
•
Limitations of the Bohr Model
Can only explain the line spectrum of hydrogen
adequately.
Can only work for (at least) one electron atoms.
Cannot explain multi-lines with each color.
Cannot explain relative intensities.
• Electrons are not completely described as small particles.
• Electrons can have both wave and particle properties.
The Wave Behavior of Matter
The Uncertainty Principle
• Wavelength of Matter:
h

mv
• Heisenberg’s Uncertainty Principle: on the mass scale
of atomic particles, we cannot determine exactly the
position, direction of motion, and speed simultaneously.
• If x is the uncertainty in position and mv is the
uncertainty in momentum, then
h
x· (mv) 
4
Mathcad Calns
The Heisenberg Uncertainty Principle
Comparison between microscopic (electron) and macroscopic (SR)
BBT QM
Micro: Determine the uncertainty in finding the el ectron in an atom with a 1% uncertainty in
determination of its speed.
Macro: Determine the uncertainty in finding student along a 100 m track with a 1% uncertainty
in determination of his speed.
h
x ( m v ) 
4
h  6.626 10
4
h
x
 34
h
x m ( v ) 
v
4 m v
v  1%
 34
xe 
6.626 10

  kg   1 107 
4  
 9.1 10
 31
 10
xe  5.8  10
%xe
xe
de
 100
 J s
1
100
 m s
1
m
xe
%xe 
 11
5 10
100
3
%xe  1.2  10
m
 34
6.626 10
xSR 
 J s
4  [ ( 90)  kg ]  ( 0.1)  m s
 36
xSR  5.9  10
m
%xSR 
1
xSR
100m
100
 36
%xSR  5.9  10
 J s
Energy and Matter
Size of Matter
Particle Property
Wave Property
Large – macroscopic
Mainly
Unobservable
Intermediate – electron Some
Some
Small – photon
Mainly
Few
E = m c2
Quantum Mechanics and Atomic Orbitals
• Schrödinger proposed an equation that contains both
wave and particle terms.
^
H  E 
• Solving the equation leads to wave functions.
Quantum Chemistry
• Bohr orbits replaced by Wavefunctions
• Three Formulations
– Differential Equation Approach by Schrödinger
– Matrix Approach by Heisenberg
– Operator/Linear Vector-Space approach by Dirac
(Use Scaled-down version here)
Operator Algebra
• Operator Equation
• Algebraic rules:
–
–
–
–
•
•
•
•
Equality
Addition/Subtraction (linear operators)
Multiplication (order of operation)
Division (reverse operation)
Commutators
Eigenvalue Equation
Compound Operators
Ladder Operators
Operator Equation
^
 f  g
Algebraic Rules
(i) Equality
(ii) Addition(/Subtraction) [Linear Operators follow Distributive Law]
Algebraic Rules
(iii) Multiplication [Order of Operation]
(iv) Division
Commutators
^
^
^
^
^
^
^
C  [ ,  ]       
BBT-V
Commutators – S10
^
^
^
^
^
^
^
C  [ ,  ]       
Eigenvalue Equation
^
  f ( x)  a  f ( x)
Eigenvalue Equation – S10
^
  f ( x)  a  f ( x)
Compound Operators
^
^2
h  x d
2
2
2
2
 (cartesian )  2  2  2
x
y
z
2
1  2 
1


1
2
  2  r   2

 sin  


r r
r r  sin  
 r 2  sin 2   2
2
Ladder Operators
For commutator s of the form :
[ ˆ , ˆ ]  k  ˆ
where :
ˆ is a " generator" for the eigenfunct ions of ˆ .
( that is : ˆ is a Ladder Operator. )
Sequential examples :
1) Define :
ĥ  x̂ 2  dˆ 2 & Â  x̂  dˆ
Then :
  raising operator for ĥ
B̂  lowering operator for ĥ
&
B̂  x̂  dˆ
Ladder Operators: Seq. Eg. Cont/…
2)
Show that :
[ ĥ , Â ]  2  Â
3)
Show that :
[ ĥ , B̂ ]  - 2  B̂
4)
(a)
(b)
(c)
Is f(x)  e
-
x2
2
an eigfcn. of ĥ ? [Ans : Yes, e.v.  1]
Show that   f(x)  (x - d̂)  e
x2
2
 2x  e
x2
2
Show that the new function, g(x)  2x  e
x2
2
,
is an eignfcn. of ĥ with e.v.  2  1  3 .
(d)
-
x2
2
Show that B̂  g(x)  B̂  (2x  e ) reproduces
the original function, f(x) (within a constant); that is,
the operators  & B̂ are raising & lowering operators
for the eigfcn' s of ĥ .
Ladder Operators
Ladder Operators (S11)
Postulates of Quantum Theory
• The state of a system is defined by a function (usually
denoted  and called the wavefunction or state
function) that contains all the information that can be
known about the system.
• Every physical observable is represented by a linear
operator called the “Hermitian” operator.
• Measurement of a physical observable will give a result
that is one of the eigenvalues of the corresponding
operator for that observable.
TBBT: QM-joke
Postulate I
 (x, y, z)  stationary state
Often,  is complex :   Z(a, b)
where : a  Real(Z) & b  Imag(Z)
For example :
Z  a  ib in which
Complex Conjugate  Z  a  ib
i  complex number unit vecto r
*
1
i  1 ; i  i ; i  1 ;  i .
i
2
3
4
Z  Z  Z  a 2  b2
*
2
Postulate I …cont…
    Probabilit y Distributi on Function
*
Criteria of 
 must be single - valued over all space
 must be finite & continuous over all space
  must have a finite integral over all space
2
Normalizat ion of 
*

    d  1
all space
(1D - cartesian) : d  dx
(3D - cartesian) : d  dx  dy  dz
(3D - spherical) : d  r 2  dr  sin   d  d
Postulate I …cont…
Probabilit y of finding particle within a certain region of space (1D - x only) :
Prob( x1

xx )

2
x2
x1


 dx
2
 dx
2
x2
  
   dx
if normalized
x1
2
Postulate II
Variable
CM
QM
Position
x,y,z
xˆ , yˆ , zˆ
Momentum
Potential Energy
mv
V
pˆ x  i
  

x i x
pˆ y  i
  

y i y
pˆ z  i
  

z i z
Vˆ ( x, y, z )  V ( x, y, z )
Postulate II…cont…
Variable
Kinetic Energy
Total Energy
CM
QM
1 2 p2
T  mv 
2
2m
T+V
ˆ  Hamiltonia n

2
2
2
2
2







Tˆ  
2
 2  2  2
2m  x y z 
2m
 
2

ˆ 

 2  V ( x, y , z )
2m
Starting Point in all Quantum
Mechanical Problems.
Heisenberg’s Uncertainty Principle
Variable A : ˆ  i  ai  i
Variable B : ˆ   i  bi   i
[ˆ , ˆ ]  0
A fundamental incompatibility exists in the measurement of
physical variables that are represented by non-commuting
operators:
“A measurement of one causes an uncertainty in the other.”
1
A  B   [ˆ , ˆ ] 
2
 [ˆ , ˆ ]   * (ˆˆ  ˆˆ )  d
Heisenberg’s Uncertainty Principle
1
A  B   [ˆ , ˆ ] 
2
1
p x  x   [ pˆ x , xˆ ] 
2

    ˆ
[ pˆ x , xˆ ]  
, x   [d , xˆ ] 
i
 i x  i
1
p x  x   
2
h

2
Particle in a Box (1D) – 1 – S14
∞
∞
V
V=0
V=∞
V=∞
0
x
a
Figure 11.6 Potential energy for the particle in a box. The
potential (V) is zero for some finite region (0<x<a) and
infinite elsewhere.
Particle in a Box (1D) – 2 – S14
Particle in a Box (1D) – 3 – S14
Particle in a Box (1D) – 4 – S14
Wolfram Integrator
Particle in a Box (1D) – 5 – S14
Particle in a Box (1D) – 1 – S13
∞
∞
V
V=0
V=∞
V=∞
0
x
a
Figure 11.6 Potential energy for the particle in a box. The
potential (V) is zero for some finite region (0<x<a) and
infinite elsewhere.
Particle in a Box (1D) – 2 – S13
Particle in a Box (1D) – 3 – S13
Particle in a Box (1D) – 4 – S13
Particle in a Box (1D) – 5 – S13
Particle in a Box (1D) - Interpretations
●
MC plots of Wavefunctions
n 
Excel plots
●
2
 n   x 
 sin 

a
 a 
Plots of Squares of Wavefunctions
a
●
Check Normalizations
2

  dx  1
0
●
How fast is the particle moving? Comparison of
macroscopic versus microscopic particles.
Calculate v(min) of an electron in a 20-Angstrom box.
Calculate v(min) of a 1 g mass in a 1 cm-box
n2  h2
En 
8  m  a2
Particle in a Box (1D) - Applications
Particle in a Box (3D) – 1 – S14
z
Explained using S13 slides
a
0
a
y
a
x
Figure 11.8 The cubic box. For the three-dimensional particle in a box, the
potential is zero inside a cube and infinite elsewhere. This could represent
the situation of a particle inside a container with perfectly rigid,
impenetrable walls.
Particle in a Box (3D) – 2 – S14
Particle in a Box (3D) – 3 – S14
Particle in a Box (3D) – 1 – S13
z
a
0
a
y
a
x
Figure 11.8 The cubic box. For the three-dimensional particle in a box, the
potential is zero inside a cube and infinite elsewhere. This could represent
the situation of a particle inside a container with perfectly rigid,
impenetrable walls.
Particle in a Box (3D) – 2 – S13
Particle in a Box (3D) – 3 – S13
Particle in a Box (3D) - Solutions
3
2
 ny    y 
 nx    x 
2
 nz    z 


   x  y  z     sin 
 sin 
  sin 


a
a
 a 
 a 



h2
2
2
2
E  Ex  E y  Ez 

n

n

n
x
y
z
2
8ma

Particle in a Box (3D) -Degeneracies
Energy*
3
6
9
11
12
14
17
38
54
g
1
3
3
3
1
6
3
9
12
States
(1,1,1)
(2,1,1) (1,2,1) (1,1,2)
(2,2,1) (2,1,2) (1,2,2)
(3,1,1) (1,3,1) (1,1,3)
(2,2,2)
(3,2,1) (3,1,2) (2,3,1) (2,1,3) (1,2,3) (1,3,2)
(3,2,2) etc
(5,3,2) etc; (6,1,1,) etc
(5,5,2) etc; (6,3,3) etc; (7,2,1) etc
*Energy given in units of h2/8ma2
Particle in a 2D (non-Symmetric) Plane - Solutions
y
L
V0 :
0xL & 0 yW
W
x
2 D
 ny    y 
2
2
 nx    x 

  x  y 

sin 
  sin 
L W
 L 
 W 
2
2

ny 
nx
h
 Ex  E y 
 2  2 
8m  L W 
2
E2 D
MCad
Particle in a 2D (non-symmetric) Plane
L  20
 ( x y ) 

W  10
2
L

2
W
n x  10
 nx  x   n y  y 
  sin

 L   W 
 sin 
n y  2
The Harmonic Oscillator – Model for Vibrations
Use: V=½kx2 ; reduced mass; Ladder Operators.
Applications include: Heat Cap’s, Blackbody radiation.
Eigenfunct ions : v  Av  [ H v ( y )]  e
y2

2
v  quantum number
H v  Hermite Polynomial s
1

Eigenvalue s : Ev   v    h  vo
2

where : vo 
1
k

2 
reduced mass :  
m1  m2
m1  m2
Mathcad
The Rigid Rotor – Model for Rotations
Use: m &  ; spherical polar coord’s; Ang. Mom. Operators.
App’s: Structural Spectr; Rotat. Spectr; NMR, ESR, Mössbauer.
Eigenfunct ions : combinatio ns of sin n and cos n
  (  1)  h 2
Eigenvalue s : E 
8 2  I
where :   integral quantum numbers
Moment of Inertia : I    R 2
R  radius of rotation of system
Mathcad
Quantum Numbers of Wavefuntions
Quantum #
Symbol
Values
Description
Principle
n
1,2,3,4,…
Size & Energy of orbital
Azimuthal

0,1,2,…(n-1)
for each n
Shape of orbital
Magnetic
m
-…,0,…+ 
for each 
Relative orientation of orbitals within same

Spin
ms
+1/2 or –1/2
Spin up or Spin down
Azimuthal Quantum #
Name of Orbital(CD)
0
s (sharp)
1
p (principal)
2
d (diffuse)
3
f (fundamental)
4
g
Quantum Mechanics and Atomic Orbitals
Orbitals and Quantum Numbers
Figure 6.27
MO-Mcad’s
Spectroscopy: Quantum Interpretations
Measurements of quantized E-levels
involving discrete transitions.

E
hc
Selection Rules – Depends on allowed QM changes in quantum
number between pairs of stationary states ( Ψ’s ).
Golden Rules of Transition s
I x   i  x   j  d
*
I y   i  y   j  d
*
I z   i  z   j  d
*
If any of above integrals is non-zero, then transition allowed.
Intensity  Ix2 + Iy2 + Iz2
If all integrals are zero => Forbidden Transition
Introductory Quantum Mechanics
Ei  f  h 
1 1
hc
  2.178  10 18 J  2  2 

 n f ni 
Bohr's Atom
1
2
p x  x   

Wave vs. Particle
h
2
Heisenberg
2
ˆ     2  V ( x, y , z )

2m
E

hc
E = m c2
Historical
De Brogile's Hypothesis
^
Quantum Tools
Operator Algebra
Applications
H  E 
Postulates
2
 n   x 
n 
 sin 

a
 a 
Translational
Vibrational
Rotational
Spectroscopy (NMR)
a
2

  dx  1
0
n2  h2
En 
8  m  a2