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,

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 EH

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

• 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

a

xn

axn

sin2

)( 2

22

8ma

hnEn

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 , )

Rnl describes how wave function varies with distance of electron from nucleus

Ylm describes the angular dependence of the wave functionSubscripts n, l and m are

the quantum numbers of hydrogen

;,,, ,,, lmlnmln rRr

20

2

4 r

eV

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 principleNo 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 ruleFor 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. Introduction to molecular orbitals

2. Bonding vs. antibonding orbitals

3. (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 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 bonds are tightly bound; lots of energy required to vacate molecular orbitals

Sigma () bondsbonding

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=3Bond 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

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 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.

• 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

4 2p AO

1

2

3

4

Highest Filled Orbital

Lowest Unfilled Orbital

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 2p AO

1

2

3

4

Highest Filled Orbital

Lowest Unfilled Orbital

222

2

8 HFOLUOHFOLUO nnmL

hEEE

•What can provide the energy for this transition?•The energy for this transition can be provided by a photon with energy E=hv=hc/

Origin of UV-visible spectra

• For UV-Visible spectroscopy relevant electronic transitions involve n→and →transitions

Compoundnm) transition with lowest energy

CH4 122 * (C-H)

CH3CH3 130 * (C-C)

CH3OH 183 n-* (C-O)

CH3SH 235 n-* (C-S)

CH3NH2 210 n-* (C-N)

CH3Cl 173 n-* (C-Cl)

CH3I 258 n-* (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 LevelsDefinition

Energy levels are characteristic states of a molecule

Ground state is state of lowest energy

States of higher energy are called excited states

Energy LevelsClassification 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

RadioFrequency

SpinOrientation

Microwave Rotational

IR Vibrational

UUV- VIS Electronic

Energy LevelsElectronic 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 LevelsAtomic 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

r (radius)+e

F

Energy LevelsAtomic energy levels• The electronic energy level is constant for each energy level, when the

distance between the electron and nucleus is constant

Energy LevelsMolecular energy levels

How does energy level change with inter-nuclear distance? Example, 1s orbital of Hydrogen.

R = 0* R = infinity

Sigma bond

*chemically unfeasible limit when nuclei fuse together

Energy LevelsMolecular 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 compressingthe bond gives an increase in the energy

• Morse curve can be approximated by a simpleHooke’s law function

V=.5*k(R-R2)2+.5*k’(R-R2)3 +.5*k’’(R-R2) 4