Download Lecture 3 - Engineering

Lecture 3
Summary of Lecture 2
• The three types of interactions that optical methods exploit to yield
biomedical information about cells and tissues are:
– Scattering
• Elastic scattering (scattered frequency same as incident)
– Multiple scattering ----Diffuse reflectance spectroscopy, Diffuse optical
tomography
– Single scattering ----Light scattering spectroscopy, Microscopy, Optical
Coherence Tomography
• Inelastic scattering (scattered frequency shifted with respect to incident)
– Raman spectroscopy
– Absorption
• Radationless relaxation ---- Diffuse reflectance spectroscopy, Diffuse optical
tomography
– fluorescence
• To understand the wavelength/energy dependent nature of these
processes, we need to understand how light interacts with matter
Summary of Lecture 2
• The basic unit of matter is the
– atom
• Atoms consist of
– a nucleus surrounded by electron(s)
• It is impossible to know exactly both the location
and velocity of a particle at the same time
• Describe the probability of finding a particle
within a given space in terms of a
– wave function, y
Summary of Lecture 2
• The wavefunction of an electron is also called an
– orbital
• We draw orbitals to represent the space within
which we have 90% probability of finding an
electron
• To find the wavefunction(s) representing the
electronic state(s) of an atom we need to solve
– the Schrödinger equation

H  E
Summary of Lecture 2
• The particle confined in a one-dimensional box of length
a, represents a simple case, with well-defined
wavefunctions and corresponding energy levels
2
nx
y n ( x) 
sin
a
a
2
2
nh
En 
2
8ma
• n can be any positive integer, 1,2,3…, and represents
the number of nodes (places where the wavefunction is
zero)
• Only discrete energy levels are available to the particle
in a box----energy is quantized
Atomic orbitals: Hydrogen atom
• The Schrödinger equation can be formulated and solved for a
hydrogen atom, consisting of a negatively charged electron moving
around a positively charged nucleus (i.e. electron has potential
energy due to nuclear attraction ,
)
e2
V 
40 r 2
y n,l ,m r , ,    Rn,l r lm  ,  ;
Rnl describes how wave function varies with distance of electron from
nucleus
Ylm describes the angular dependence of the wave function
Subscripts n, l and m are
the quantum numbers of hydrogen
Quantum numbers
• Principal quantum number, n
–Has integral values of 1,2,3…… and is related to size and energy of the
orbital
• Angular quantum number, l
–Can have values of 0 to n-1 for each value of n and relates to the angular
momentum of the electron in an orbital; it defines the shape of the orbital
• Magnetic quantum number, ml
–Can have integral values between l and - l, including zero and relates to
the orientation in space of the angular momentum.
• Electron spin quantum number, ms
–This quantum number only has two values: ½ and –½ and relates to spin
orientation
Rules for filling electronic states
Pauli exclusion principle
No two electrons can have the same set of quantum
numbers: n, l, ml and ms
Aufbau principle
Electrons fill in the orbitals of successively increasing
energy, starting with the lowest energy orbital
Hund’s rule
For a given shell (example, n=2), the electron
occupies each subshell one at a time before pairing
up
Example: Br (35 electrons)
• Electronic configuration:
1s22s22p63s23p64s23d104p5
Molecular Orbitals
1.
2.
3.
Introduction to molecular orbitals
Bonding vs. antibonding orbitals
s (sigma) and  (pi) bonds
Introduction to molecular orbitals
•
–
•
•
Molecular orbitals (chemical bonds) originate from
the overlap of occupied atomic orbitals
Only the valence electrons of atomic orbitals contribute significantly
to molecular orbitals
–
–
Oxygen 1s22s22p4
Has 6 valence electrons
–
–
Xenon : 1s22s22p63s23p64s23d104p65s24d105p6
has 8 valence electrons
Each molecular orbital can hold two electrons; spins must be opposite
Bonding vs. anti-bonding orbitals
•
•
Bonding molecular orbital: lower in energy than the atomic orbitals of
which it is made
Antibonding molecular orbital: higher in energy than the atomic
orbitals of which it is made
Antibonding character indicated by asterisk.
Molecular orbitals
s Bonds
•
Involve s orbitals and p orbitals
•
Overlap of two atomic orbitals along the line joining nuclei of bonded
atoms
•
Charge distribution is localized along bond axis
•
Electrons in s bonds are tightly bound; lots of energy required to
vacate molecular orbitals
Sigma (s) bonds
bonding
Anti-bonding
Molecular orbitals
 Bonds
•
Involves p or d orbitals
•
Overlap of two atomic orbitals at right angles to the line joining the
nuclei of bonded atoms
•
Charge distribution is above and below plane containing  bond
•
Less tightly bound
Pi () bonds
bonding
Anti-bonding
Bond
order: 3
Bond
order=3
(# of e- in bonding orb)-(# of e- in anti-bonding orb)
2
Bond order: 2
Bond order: 2
Molecular orbitals
• Molecular orbitals (chemical bonds) originate from the overlap of
occupied atomic orbitals
• Bonding molecular orbitals
– are lower in energy than corresponding atomic orbitals (stabilizes the
molecule)
• Anti-bonding orbitals
– are higher in energy than corresponding atomic orbitals and destabilizes
the molecule
 s bonds
– involve overlapping s and p orbitals along the line joining the nuclei of
the bond-forming atoms
  bonds
– involve p and d orbitals overlapping above and below the line joining the
nuclei of the bond-forming atoms
Molecular orbitals
• Only the valence electrons of atomic orbitals contribute
significantly to molecular orbitals
– Oxygen has 6 valence electrons: 1s22s22p4
– Xenon has 8 valence electrons:
1s22s22p63s23p64s23d104p65s24d105p6
• Each molecular orbital can hold two electrons; spins
must be opposite
• Number of available molecular orbitals equals the sum of
original atomic orbitals
Polyatomic molecules: hybridization
• Valence bond theory cannot explain the bonding or the
structure of polyatomic molecules
• Carbon, for example, has in its ground state only two
unpaired electrons in two 2p orbitals.
• Thus, carbon should form only two bonds.
• However, carbon almost always forms four bonds
Hybridization: sp3 orbitals
• To explain how carbon forms its four bonds, we assume that one of
the 2s electrons is “promoted” or “excited” to one of the unoccupied
2p orbitals
• With the 2s orbital half empty and the 2p orbitals all having electrons
with parallel spin, the orbitals merge to form four equal energy
orbitals, arranged in a tetrahedral geometry (bond angle-109.5º)
Methane: sp3 orbitals
• Why sp3 orbitals?
– Each sp3 orbital has a large lobe, which
makes it easier to overlap with another orbital,
such as that of a hydrogen atom
sp hybrid formation
• When a 2s and a 2p bond combine they need to form two bonds,
equivalent in shape and energy
• Think of the resulting orbitals as the shapes that arise either when
you add the 2s and the 2p orbitals or when you subtract the 2p from
the 2s orbital
sp hybrid orbitals: triple bonds
• A carbon atom with 2 sp orbitals still has one electron in each one of
the two p bonds
• What happens when two such atoms form a molecular bond?
acetylene
sp2 orbitals: double bonds
• What happens when a 2s orbital mixes with two 2p orbitals? How
many orbitals are formed and at what orientation?
•Three sp2 hybrid orbitals form, arranged on a plane at 120 º from each other
sp2 hybrid orbitals: double bonds
• Does a carbon atom with 3 sp2 orbitals still have any other electrons?
What happens when two such atoms form a molecular bond?
One s sp2 bond and one  bond are formed
Origin of UV-Visible spectra:
conjugated bonds
• Conjugated organic molecules consist of alternating
single and multiple bonds between chains of carbon
atoms
1,3-butadiene
H2C=CH-CH=CH2
• Carbon and hydrogen atoms are bonded so that each
carbon atom is left with an unused electron in a 2p
orbital, with the 2p bonds parallel to each other
Conjugated bonds
•
The four 2p orbitals can combine to form these  orbitals, arranged according to
energy, with the lowest energy  orbital at the bottom.
•
•
Can you think of a set of wavefunctions that may describe what is going on?
These are similar to the wavefunctions we got for a particle in the box, with the
length of the box corresponding to the length of the carbon chain
Conjugated bonds and particle in a
box
•
The four 2p orbitals can combine to form these  orbitals, arranged according to
energy, with the lowest energy  orbital at the bottom.
4*
4 2p AO
3* Lowest Unfilled Orbital
2 Highest Filled Orbital
1
•
•
How will the electrons be distributed?
Each of the orbitals can accommodate two electrons. Since there are 4
electrons, the two lower orbitals will be occupied
Conjugated bonds and particle in a
box
•
What will be the energy required for an electron to be excited from such a
bonding to an anti-bonding  orbital?
4*
3* Lowest Unfilled Orbital
4 2p AO
2 Highest Filled Orbital
1
2

h
2
2
E  ELUO  EHFO 
nLUO  nHFO
2
8mL

•What can provide the energy for this transition?
•The energy for this transition can be provided by a photon with energy
E=hv=hc/l
Origin of UV-visible spectra
• For UV-Visible spectroscopy relevant electronic
transitions involve n→* and  →* transitions
l
Compound
nm)
transition with lowest energy
CH4
122
ss* (C-H)
CH3CH3
130
ss* (C-C)
CH3OH
183
n-s* (C-O)
CH3SH
235
n-s* (C-S)
CH3NH2
210
n-s* (C-N)
CH3Cl
173
n-s* (C-Cl)
CH3I
258
n-s* (C-I)
CH2=CH2
165
* (C=C)
CH3COCH3
187
* (C=O)
273
n-* (C=O)
CH3CSCH3
460
n-* (C=S)
CH3N=NCH3
347
n-* (N=N)
Some chromophores of interest
Beta carotene
Energy Levels
Definition
Energy levels are characteristic states of a molecule
Ground state is state of lowest energy
States of higher energy are called excited states
Energy Levels
Classification of Energies
Can you think of some types of energies associated with a molecule?
A molecule can be thought of as having several distinct reservoirs of energy
Emolecule = Etranslation (motion of the molecule’s center of mass through space )
+ Eelectron spin (orientation of nuclear spin in a magnetic field)
+ Enuclear spin (orientation of electron spin in a magnetic field)
+ Erotation (rotation of the molecule about its center of mass)
+ Evibration (vibration of the molecule’s constituent atoms )
+ Eelectronic (electronic transitions between available energy states)
The energy associated with each of these are quantized
Energy Levels
Energy Levels
Energy
Energy Level Separation
(J)
Translation
Very small
Spin
10-32
Rotation
10-28
Vibration
10-25
Electronic
10-19
Energy Levels
EM spectrum
Energy Levels
EM Radiation Energy Levels
Radio
Frequency
Microwave
Spin
Orientation
Rotational
IR
Vibrational
UV- VIS
Electronic
U
Energy Levels
Electronic energy levels
•
Electronic energy levels of molecules are described by molecular
orbitals
•
When an electron undergoes an electronic transition, it is transferred
from one molecular orbital to another
•
UV-VIS absorption / fluorescence spectroscopy involves electronic
energy transitions
Energy Levels
Atomic energy levels
•
Each energy level in the system corresponds to the potential energy
between the positive and negative charges
•
The potential energy results from the force between particles, i.e., the
nucleus and electron (Coulombic force)
-e F
+
+e
r (radius)
Energy Levels
Atomic energy levels
•
The electronic energy level is constant for each energy level, when the
distance between the electron and nucleus is constant
Energy Levels
Molecular energy levels
How does energy level change with inter-nuclear distance? Example,
1s orbital of Hydrogen.
R = infinity
R = 0*
Sigma bond
*chemically unfeasible limit when nuclei fuse together
Energy Levels
Molecular energy levels
•
Energy of a pair of atoms as a function of distance between them is given
by the Morse curve, where R2 is the equilibrium bond distance
•
Stretching or compressing
the bond gives an increase
in the energy
•
Morse curve can be
approximated by a simple
Hooke’s law function
V=.5*k(R-R2)2+.5*k’(R-R2)3 +.5*k’’(R-R2) 4