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CHEM 543 – Spring 2007
Point of course – move from abstract QM to something we measure → spectra
QM – potential constrains motions of particles quantizes energy levels
Spectra – transitions between levels by absorbing, emitting light (photons)
Types of motion correspond to different = h
different spectroscopies
rotational ↔ wave/far IR
vibrational ↔ IR (or equivalent shifts/Raman or other techniques)
electronic value ↔ vis-UV-vac UV etc.…
core ↔ x-ray
Need to know levels → QM
Need to know probability of transition → spectroscopy
Many particle systems lead to complex QM (approx.)
Symmetry can simplify/reduce size
Group Theory provides a machinery to use symmetry in QM
SYLLABUS – discuss all structure/requirements
Text - Molecular Spectroscopy by Jeanne McHale → guide
Necessary to read outside – Topics list notes chapters in Auxiliary books
Web page – → link via Dept → course work → 543
Quantum Mechanics Review
Describe motion and state of particle or group particles
with a wavefunction (r1, r2, - - - )
can be represented as a vector
Interrogate it as to property with an operation:
Energy operation is Hamiltonian
sum of kinetic and potential
for set of particle summed over i
V(r1, r2, - - - )(V = V(r) only) – same as classical
Shroedinger: – n – index of solu represent one of possible values
IF n eigen fct
Quantized energy → normally a result from V(ri) restrict motion for transition
Spectroscopy
free particle V(r) = 0/const → no quant of translation (continuum)
If not an eigen function
(is an expectation value observable for absorp) =
Uncertainty:conjugate variables
one indeterminant to some degree
similarlynote completely parallel
McHale: this is QM relates to commutator
define uncertainty as deviance (std. dev):
brackets → expectation values
if commute, no uncert
This leads to zero point energies → i.e. particles can not
beat rest on p = 0 and n = const violate uncertainty
Symmetry ↔ degeneracy
Particle in 3-D box
if cubic: degen
if lose this then levels lose degeneracy
True in general, this is an example → symmetryultimatesourcedegeneracy
but need multidimensional problem
atoms → spherical symmetry → express as ang. mom.
molecules → variable symmetry → about a central point
categorize with “pointgroups” – will be subj of 543
other properties wave fct:
– node structure allows i + j to be orthogonal
– nodes closer together (more of them) higher
– smaller boxes constraint (general) – more energy separation (constraint)
larger go to continuum (classical)
Rotating molecules – no potential Energy but ang. mom. restricted
commutation simultaneous eigen values
diatomics: I = R2
representation:
commutation: etc i,j,k x,y,z (any x,y,z)
simultaneous eigen fct
These are
properties m = -j, -j+1 …, +j
needed j = 0, 1, 2 …
Ladder operator
raise and lower eigen value
- spherical harmonics (if want 3-D rep of w/f)
but commutator relationship give all we need
Rigid Rotation describe diatomic → no moment initiate G axis
extreme prolate In = 0
Polyatomics described the sliver but now rotation
different depending on axes → principle axes important
depend on symmetry → degeneracy
Spherical top: Ix = Iy = Iz → CH4, SF6, Carbone, C60, …
Symmetrical top: Ix = Iy = Iz → CH3Cl, benzene, NH3
prolate Iz < Ix, Iyoblate Iz > Ix, Iy
CH3CNC6H6
Harmonic oscillation – model internal motion nuclei (vibration)
→ constraint quantize linear motion (Hooks law F = -kx restoring force)
1-D (diatomic)reduce variable
r = r1 – r2 = q
Solu:
solu are Hermite polynomial – messy – orthog
Alternate representation – raising/lowering operator
0 = 20
then
from [q,p] = i [a, a+] = 1add + subtract
0 = 1-[a,a2] follows from (fixed?)
Now N = a+anumber operator
H
Now What is effect of a?
a – lower
a+ – raise lower operator
Similarly
Approximation Methods
Perturbation Theory idea is know exact solu to a simple problem, add something to it and ask how solu shift → if “something” small → perturbator
1st order connection (single state involved):
2nd order connection (multiple states):
uses 1st order w/f:
states become mixed
in general:
Degenerate states any mix of degen is ok
To get “exact” solu → full perturbation treatment
solu:
done by degenralizing matrix of coeff.
Solutions are 2-fold
Energies are eigen values/solu to det solve polymers
Wavefunctions are vector {Ci} where data by secular eqn
(subst back in and get system of n equations)
Variation Method
often for giving system particles it is relatively easy to express ,
but not easy to solve it
if pick a trial fct then variation says expect ???
0 – lowest energy state
This gives a guide on how to improve a w/f
can add corrections – if goes down then improving
Chemists use this to make MO → LCAO – MO
add AO’s and as improve … [actually add components to AO’s … ]
e.g. = c1f1 + c2f2 + … cnfn
put this in equation →
minimize by setting
yields a set of equations (secular eqn)
det solu for i (n solu)
lowest are 10 by variation
values of ci by solving simultaneous equations
E-M Radiation and interaction with matter (particles)
McHale, Ch 3 / Struve, Ch1 / Bernath, Ch1 / Flyger, Ch11
For molecular spectra we take “hybrid” approach →
treat molecules with QM
-M fields classically – (as waves)
Maxwell’s Eqn describe Classical -M
in phase
mutual pert to
(2) → freq =
SI unit: or = p = 1.0 vacuum
E ~ 1 – 80 (H2O) material
(no magnetic monopoles)B → mag induction
H → mag field
(add current term)(00)-1 = c2 speed light related to 0 electric permitting
relativity and 0 magnetic permeability
1st to relate to electric statistics – how fixed fields work
How do these relate to -M see they couple B, (3rd, 4th) by substitution can derive wave equations:
wave equations, 2nd order differential in space, time
Solve with general form
Connor to rewrite as scalar and vector potential: , A
if no change = 0
satisfy
Wave eigen becomes: A = A0ei(kr – t)
??? or B in phase and orthogonal
Wave Equations:
see Flygare: Chap 1
Maxwell eqn: (SI)(00)-1 = c2vacuum
1. p = = 1
2. 0, 0 cont
3. current density
4. recall: grad
divergence, code
LaPlacian operator
1 → (Faraday) time dep → induce (since ??? )
2 → (Ampere-Oerstat) time dependent electric (displacement) field produces
3 → (Coulomb Law) field relate to change
4 → no magnetic monopoles
Define scalar and vector potentials
:static (time independent)1.
fits eqn 1: – due to cross prod)
note:
(vacuum: LaPlace: Poisson’s eqn
:time varying consideration2.
into #1
Constrain variables
Scalar vector:(5) (Lorentz convention)
Use these with Maxwell: (3)
from above (5):
plug in and rearrange:
wave equation for the scalar potential
Next take eqn (2):
substrate:
If J = 0 then
let A
-k2A
k
note: = 1 (unmagnetic medium) (refraction index)
propagation vectorspeed of light
Poyting vectorin vacuumn = 1.0
Now use equations
B
ІІ A
time oscillating (w) and spatial varying (k) fields -M radiation