Transcript B - Piazza

CHAPTER 10
Molecules
Why do molecules form?
Molecular bonds
Rotations
Vibrations
Johannes Diderik van der Waals
(1837 – 1923)
Spectra
Complex planar molecules
“Life ... is a relationship between molecules.”
Linus Pauling
Prof. Rick Trebino, Georgia Tech, www.frog.gatech.edu
Molecules are combinations of atoms.
When more than one atom is involved, the potential and the wave
function are functions of way more than one position (a position
vector for each nucleus and electron):
V  V (r1 , r2 ,..., rN ; r1, r2,..., rM )
  (r1 , r2 ,..., rN ; r1, r2,..., rM , t )
Electrons’ positions
Solving the Schrodinger Equation in
this case is even harder than for
multi-electron atoms.
Serious approximation methods are
required.
This is called Chemistry!
Nuclei positions
10.1: Molecular Bonding and Spectra
Nucleus
The only force
that binds atoms
together in
molecules is the
Coulomb force.
Nucleus
E 0
Electron cloud
Electron cloud
But aren’t most atoms electrically neutral? Yes!
Indeed, there is no attraction between spherically symmetrical
molecules—the positive and negative charges both behave like
point sources and so their fields cancel out perfectly!
So how do molecules form?
Why Molecules Form
Most atoms are not spherically
symmetrical.
For example, these two “atoms”
attract each other:
Atom #1
Atom #2
+
+
-
This is because the distance between opposite charges is less
than that between charges of the same sign.
The combination of attractive and repulsive forces creates a
stable molecular structure.
Force is related to the potential energy surface, F = −dV/dr, where
r is the position.
Charge is distributed very unevenly in
most atoms.
The probability density for the hydrogen atom for three different
electron states.
Closed shells of electrons are very
stable.
Atoms with closed shells (noble gases) don’t form molecules.
Ionization energy (eV)
Noble gases (difficult to
remove an electron)
Atomic number (Z)
Atoms with closed shells
(noble gases) also have the
smallest atomic radii.
Atomic radius (nm)
Atoms like closed
electron shells.
Atomic number (Z)
But add an extra electron, and it’s weakly bound and far away.
An extra electron or two outside a
closed shell are very easy to liberate.
Atoms with one or two (or even more) extra electrons will
give them up to another atom that requires one or two to
close a shell.
Atoms with extra
electrons are said to
be electropositive.
Those in need of
electrons are
electronegative.
Ionic Bonds
An electropositive atom gives up
an electron to an
electronegative one.
Example: Sodium (1s22s22p63s1)
readily gives up its 3s electron
to become Na+, while chlorine
(1s22s22p63s23p5) easily gains
an electron to become Cl−.
Covalent Bonds
Two electronegative atoms
share one or more electrons.
Example: Diatomic molecules
formed by the combination of
two identical electronegative
atoms tend to be covalent.
Larger molecules are formed
with covalent bonds.
Diamond
Metallic Bonds
In metals, in which electrons are very weakly bound, valence
electrons are essentially free and may be shared by a number of
atoms. The Drude model for a metal: a free-electron gas!
Molecular Potential
Energy Curve
The potential depends on the
charge distributions of the atoms
involved, but there is always an
equilibrium separation
between two atoms in a
molecule.
The energy required to separate
the two atoms completely is the
binding energy, roughly equal
to the depth of the potential well.
Vibrations are excited thermally, that is, by collisions with other
molecules, or by light, creating superpositions of ground plus an
excited state(s).
Molecular Potential
An approximation of the force felt by one
atom in the vicinity of another atom is:
A B
V n- m
r
r
where A and B are positive constants.
Because of the complicated shielding effects of the various electron
shells, n and m are not equal to 1. One example is the Lennard-Jones
potential in which n = 12 and m = 6.
The shape of the
curve depends on
the parameters
A, B, n, and m.
Vibrational Motion: A Simple Harmonic
Oscillator
The Schrödinger Equation can be
separated into equations for the positions
of the electrons and those of the nuclei.
The simple harmonic oscillator accurately
describes the nuclear positions of a
diatomic molecule, as well as more
complex molecules.
Vibrational States
The energy levels are
those of a quantummechanical oscillator.
Evibr  (n  1 2) 
n is called the
vibrational quantum
number. Don’t
confuse it for n, the
principal quantum
number of the
electronic state.
Vibrational-transition
selection rule:
Dn = ±1
The only spectral line is  !
However, deviations from a perfect
parabolic potential allow other
transitions (~2, ~3, …), called
overtones, but they’re much weaker.
Vibrational Frequencies for Various Bonds
Different bonds have different vibrational frequencies (which are
also affected by other nearby atoms).
← Higher energy (frequency)
Wavenumber (cm-1)
Notice that bonds containing Hydrogen vibrate faster because H is
lighter.
Water’s Vibrations
Rotational States
Consider diatomic molecules.
A diatomic molecule may be thought of as two atoms held together
with a massless, rigid rod (rigid rotator model).
In a purely rotational system, the kinetic energy is expressed in
terms of the angular momentum L and rotational inertia I.
Erot
L2

2I
Rotational States
L is quantized.
L  (  1) 
where ℓ can be any integer.
The energy levels are
Erot 
2
(  1)
2I
Erot varies only as a function of the
quantum number ℓ.
= ħ2/I
Rotational transition energies
And there is a selection
rule that Dℓ = ±1.
Erot 
2
(  1)
2I
Transitions from ℓ +1 to ℓ :
Emitted photons have energies at regular intervals:
E ph 
2
2I

(
 1)(  2) - (  1) 
2
 2  3  2 2I
2
2
-   (  1)
I
Vibration and
Rotation
Combined
Note the difference
in lengths (DE) for
larger values of ℓ.
E  Erot  Evib 
2
(  1) 
1
n  
2I
2

DE increases
linearly with ℓ.
Most transitions are
forbidden by the
selection rules that
require Dℓ = ±1 and
Dn = ±1.
Note the
similarity in
lengths (DE)
for small
values of ℓ.
Vibration and Rotation Combined
The emission (and absorption) spectrum spacing varies with ℓ.
The higher the starting energy level, the greater the photon energy.
Vibrational energies are greater than rotational energies. For a
diatomic molecule, this energy difference results in band structure.
The line strengths depend on the populations of the states and the
vibrational selection rules, however.
Weaker overtones
Dn = 0
Dℓ = -1 Dℓ = 1
Dn = 1
Dn = 2
Energy or Frequency →
Dn = 3
Vibrational/Rotational Spectrum
In the absorption spectrum of HCl, the spacing between the peaks
can be used to compute the rotational inertia I. The missing peak
in the center corresponds to the forbidden Dℓ = 0 transition.
ℓi- ℓf = -1
ℓi- ℓf = 1
ni- nf = 1
Frequencies in Atoms and Molecules
Electrons vibrate in their motion around nuclei
High frequency: ~1014 - 1017 cycles per second.
Nuclei in molecules vibrate
with respect to each other
Intermediate frequency:
~1011 - 1013 cycles per second.
Nuclei in molecules rotate
Low frequency: ~109 - 1010 cycles per second.
Including Electronic Energy Levels
A typical large molecule’s
energy levels:
E = Eelectonic + Evibrational + Erotational
2nd excited
electronic state
Energy
1st excited
electronic state
Lowest vibrational and
rotational level of this
electronic “manifold.”
Excited vibrational and
rotational level
Transition
Ground
electronic state
There are many other
complications, such as
spin-orbit coupling,
nuclear spin, etc.,
which split levels.
As a result, molecules generally have very complex spectra.
Studying Vibrations and Rotations
Infrared spectroscopy allows the study of vibrational and
rotational transitions and states.
But it’s often difficult to generate and detect the required IR light.
It’s easier to work in the visible or near-IR.
Input light
DE
Output light
Raman scattering:
If a photon of energy greater than DE
is absorbed by a molecule, another
photon with ±DE additional energy
may be emitted.
The selection rules become:
Δn = 0, ±2 and Δℓ = 0, ±2
Modeling Very Complex Molecules
Sometimes more complex is
actually easier!
Many large organic (carbonbased) molecules are planar, and
the most weakly bound electron
is essentially free to move along
the perimeter. We call this model
the Perimeter Free-Electron
Orbital model.
plus
inner
electrons
This is just a particle in a one-dimensional box! The states are just
sine waves. The only difference is that x = L is the same as x = 0.
So y doesn’t have to be zero at the boundary, and there is another
state, the lowest-energy state, which is a constant:
y 0 ( x)  1/ L
Auroras
Intensity
Typical Aurora Emission Spectrum
Species Present in the Atmosphere
Constituents Contributing to Auroras
+
+
O
N
O
+
2
H
2
N
O2