I



I. E-M Radiation and interaction with matter (particles)

McHale, Ch 3 / Struve, Ch1 / Bernath, Ch1 / Flygare, Ch11

For molecular spectra we take “hybrid” approach →

treat molecules with QM

Ε-M fields classically – (as waves)

Maxwell’s Eqn describe Classical Ε-M

E, B fields in phase

mutual perpendicular to k

ν(Ηz) → freq = 1/λ [pic]

SI unit: [pic] [pic] or [pic] [pic] ε = μp = 1.0 vacuum

e ~ 1 – 80 (H2O) material dependent

[pic] (no magnetic monopoles) [pic] B → magnetic induction

H → magnetic field

[pic]

[pic] (μ0ε0)-1 = c2 speed of light - related to ε0 electric permitivity

Alternate: (add current) [pic] [pic] current

- charge density x velocity

μ0 - magnetic permeability

1st equation - relates to electro-statics – how fixed fields work

How do these relate to Ε-M? - they couple B, E (3rd, 4th equations)

by substitution can derive wave equations:

[pic] wave equations, 2nd order differential in space, time

Solve with [pic] general form – B, E will vary with this functionality

Can rewrite using a scalar and vector potential: φ, A

[pic] if no charge Δφ = 0

[pic] satisfy [pic]

clearly Ε and B are in phase and orthogonal

Wave equation becomes: [pic] with solution of: A = A0ei(k.r – ωt)

Wave Equations see Flygare: Chap 1

Maxwell eqn: (SI) (e0μ0)-1 = c2 vacuum

1. [pic] [pic] μp = e = 1

2. [pic] [pic] μ0, ε0 cont

3. [pic] J = ρv - current density

4. [pic] recall: [pic] - gradient operator

then: [pic] - divergence, [pic] - curl

and [pic] - LaPlacian operator

Meaning of equations:

1 → (Faraday) time dep [pic]→ induce [pic] (since [pic] cancels [pic])

2 → (Ampere-Oersted) time dependent electric (displacement) field produces [pic]

3 → (Coulomb Law) field relate to change

4 → no magnetic monopoles

Define scalar and vector potentials

Scalar Potential:

φ: static (time independent) 1. [pic]

i.e. fits Max. eqn 1: [pic] – due to cross product, no B field)

note: [pic]

[pic] Poisson’s eqn - scalar potential from charge

(vacuum: LaPlace eqn: [pic])

Vector Potential:

A: time varying consideration 2. [pic]

Subst. into Max. Eqn. #1 [pic]

Constrain variables (f=0) [pic] [pic] [pic]

Scalar [pic] vector: [pic] (eqn. 5)

(but if use the Lorentz convention) [pic]

Wave equation for the scalar potential

Use the definitions with Maxwell #3: [pic]

[pic]

from above take [pic](eqn.5): [pic]

plug in and rearrange (no current): [pic]

Wave equation for the scalar potential -- Now some arithmetic:

Next take eqn (2): [pic]

substitute: [pic]

[pic]

If J = 0 then [pic]

Solution:

let A [pic]

[pic] [pic]

[pic] [pic]

-k2A [pic]

wave vector: k~1/l: k[pic]

note: mp = 1 (non-magnetic medium) [pic] (refractive index- light)

propagation vector [pic] speed of light: [pic]

Poynting vector [pic] in vacuum n = 1.0

Now use equations

[pic] [pic]

[pic] [pic] [pic]

B [pic]

[pic] [pic]

E[pic] Ε ІІ A

E, B are time oscillating (w) and spatial varying (k) fields [pic] Ε-M radiation

Interaction of Radiation and Matter

Turn Back to Molecules

In 542 you learned many problems need approximation → many particle systems

ex: Consider benzene 6 – C’s

6 – H’s

36 + 6 – electrons

Huge dimensionality – relatively small molecules.

Here we will discuss variants of two methods

Perturbation Theory – extend to time dependent

Variation Theory – modify for Hartree-Fock (elect structure calculation)

Time dependent Perturbation Theory – Struve, Ch 1

Levine, Ch 9.9 - 10

Electrostatic fields McHale, Ch 3 - moments

Molecule has changes qε and dipole μ

[pic] en – change of nth particles or electron nucleus

[pic] rn – position of nth particles or electron nucleus

Energy of interaction with electrostatic field

[pic]

where [pic] is the scalar potential at the origin

[pic] is the electric field

[pic] is the quadrupole tensor element ij

Magnetic fields since no magnetic monopoles

[pic]

where [pic] is the magnetic field

[recall [pic] is actual field (magnetic indulation), [pic] is applied field]

Time Independent Perturbation Theory

Recall if [pic] both [pic], let [pic]

[pic]

[pic]

[pic]

Connections to energy depend on [pic],

higher connections → more powers [pic] or more terms from use higher order φ:

[pic]

[pic]

Effect of perturbation is to mix the states with it on, initial state now has some fraction of other states so can say there is some probability

here: [pic]

of [pic]

having the characteristics of [pic]

aside: If [pic] time dependant, can view state as evolving in time [pic] can change its nature

Polarizability – above we have

[pic] [pic]

but this only addresses “permanent” dipole moment of molecule – [pic]applies force to changes, separates them and induces a dipole [pic]

modify [pic]

classical Ε: [pic]

[pic]

in general, α is a tensor – molecular response more complex

Compare this to 2nd order perturbation theory [pic] see terms to power Ε2 are

α[pic] [pic] note eigen slip

Molecules and atoms with biggest electron systems or most loosely bound elections → big α

ex: H = 0.667 Å3, Li = 24.3 Å3, Cs = 59.6

Now light [pic] radiation is an electro-magnetic field

interaction will parallel this Ε = -μ – Ε

but here Ε = Ε(t) and B = B(t)

used to modify the approach

Time dependent fields

Maxwell equations lead to description of Ε-M field

[pic]

[pic]

[pic] and [pic] are in phase, but oriented 90º apart expressed ??? a vector potential[pic]and

the Coulomb Gauge:

[pic] φ = 0 (free space)

then:

(from[pic]) [pic] ([pic] II [pic])

(from[pic], and [pic]) [pic] ([pic])

We can show [Struve, p.11] that effect on Η is:

[pic] [pic] conservative potential

i.e. change interaction inside the molecule

[pic]

expand [pic]

Η0 [pic] remember

0 Coulob operator

[pic]

[pic]

time independent

2nd term in A2 ~ 0 since fields (pert.) small

[pic]

time variation let [pic]

[pic]

To use Time dependent Perturbation Theory McHale, Ch 4 consider

[pic]

time independent w/f [pic]

when [pic] form complete set

when [pic]wave functions must change but – recall expand in complete set

[pic]

now if [pic]is turned off, molecule will be in a time independent state

or cn = 1

cn = 0 n[pic]k

Time dependent Schrödinger Equation goes:

[pic]

multiply left by [pic] and use [pic]to get orthogonal normality condition

(m[pic]n)

[pic]

[pic]

rearrange to: [pic]

now recall initially ck = 1

cn≠k = 0

so can approximate (i.e. for “short” time)

time variation of wave function is in coefficient: (on “weak” perturb)

[pic]

[pic]

where [pic]

operative equation:

integrate to give: [pic]

can do higher orders but they are not normally needed unless very big

perturbations – ex. laser-intense fields

For linear spectroscopy: [pic] [pic]

so substitute

[pic] [pic]

[pic]

[pic]

expand [pic]

1st term: [pic]

assume [pic] (i.e. Ε = Εx, B = By, k = k2)

[pic]

from Struve: [pic]

[pic]

so [pic] [pic]

[pic]

[pic]

Electric dipole transition

[pic] [pic] [pic]

[pic] [pic]

dipole moment [pic]

expression correct

continuing [pic] [pic]

[pic]

Now recall [pic]

so probability that at a time = t

system will be in a state [pic]

conditions [pic][pic]

[pic]

integral if [pic]

[pic][pic]

Draw [pic] get 1, otherwise get 0

now if [pic], delta function not exactly connect [pic] get a very sharp peaked

function center at ω = -ωkm

Also should do this for real part of [pic]and result is sine function

[pic] (no integral)

[pic]

Now one can go beyond [pic]level to include terms from

[pic] prep in y, pull on x

these give rise to [pic]

M1 (Lz) E2

magnetic dipole: [pic] M1

electric quadrupole: [pic] E2

and others could follow

M1 – responsible for EPR, NMR transitions can be important in trans metal spectra

and central to optical activity

E2 – rare but can occur (Electric quadrupole)

Relative sizes: recall expansion [pic]

[pic] visible light λ ~ 5000 Å

infrared light ~ 100,000 Å

r ~ size molecule – medium benzene ~ 5 Å

[pic] visible/benzene

uv a little bigger (factor 5)

in order magnitude smaller

Then recall probability [pic] bigger reduction yet!

So M1 and E2 effects can be neglected for most molecules except nmr, epr, CD

Selection rules – since [pic] for E1

E1 need to have [pic] [pic] is odd [pic] must be even

a) [pic] must be opposite parity (odd, even)

b) polarization will affect transitions of oriented molecules [pic]

[if gas on liquid average one [pic]]

rotation [pic] must have permanent dipole / vibration [pic] elect dipole must change

E2, M1 – similar but

a) [pic] same parity (even, even) (odd, odd)

b) Orientation can affect

Operate E1 – μ-wave, IR, uv-vis absorption (electronic)

M1 – ESR, NMR, CD, weak electronic

E2 – same select rules as Raman, 2 phota but not mechanical

Error in last lecture:

McHale, Chap 4

agree: [pic]

Assume ck = 1, cm = 0, m[pic]k start at t = 0

Probability: [pic]

Note: integral from 0[pic]t because assume that cm = 0 at t = 0 (and before)

small correction (after expansion and electric dipole approximation):

[pic] [pic] Real part eiωt

[pic]

[pic] Substitution: [pic]

[pic]

Now consider probability term, time integral:

[pic]

so there are 2 terms, one dominates in absorbance

where ω = ωmk , emission: ω = - ωmk

Return to probability – square the integral, absorbance: choose ω = ωmk

on ( – ) term dominant

[pic]

[pic]

Δω = ωmk – ω

Plot: [pic]: (ωmk – ω) [pic]

Long time: [pic] Note: [pic]

Transition Rate: [pic]

absorption stimulate

emission

Formal Golden Rule: δ(Δϖ) = 2π δ(Δϖ)

Probability linear in time → longer expose sample to light

the higher probability of a transition

Rate is what we measure experimentally – flex of light

stimulate an absorbance (loss of flex [pic] rate abs)

Uncertainty – lifetime

f(Δω, t) has a width: [pic]

δt → lifetime of state or duration on pulse (especially f-sec)

[pic]

Ch 4.3 Book does nice relationship of density of photon states

and the rate of transition. Development not central

Ch 4.4 Then a detailed discussion of polarizability. We will put

this off until we address scattering. Now focus on dipole

Ch. 3 Frequency dependent polarizability – note complex due to relax

[pic]

here τ is a relaxation of state, [pic] is rate of decay [pic]

express quantum mechanically

[pic]

this picture misses live widths → relaxation → complex function

can insert [pic] into denomination.

Allows quantum mechanic definition of oscillation strength

[pic] [pic]

convenient method of categorizing transition: [pic]

How is this evidenced in matter?

Aside: refraction: speed light in vacuum – C (const)

speed light in material – [pic]

refractive index [pic] = [pic]

most non-magnetic μ ~ μ0 (4π x 10-7 [pic]) ?

[pic] relative permitivity

actually complex – real → dispersion (refraction)

imaging → absorption

since index normally > 1, = 1 vacuum, refraction

– will cause denaturation from path on charge n

– will be greater at an absorbance

Absorbance – attenuation intensity: [pic]

[pic] b = x

γ = 2,3 ε c

[pic] at w ~ wkm

absorbance relates to probability of charge state

McHale,Ch. 3 Polarizability is response of material to electric field

induced dipole moment: [pic]

if model e response to force as Hook Law → harmonic oscillation → [pic]

[pic] [pic]

multiplication: [pic]

see text – time dependent: [pic]

[pic]

Kuernes Kronig: [pic]

[pic]

QM: [pic]

see similar resonance [pic] big α

Now see oscillation strength: [pic]

relative permetivity response of medium to field

[pic] Ε – apparent field / Ε0 – applied

dielectric constant – factor reduction Coulombic force

Apply field induces polarization (P) in medium to oppose it

[pic] Elective susceptibility

[pic] λ = εr - 1

isotropic medium

Parallel polarizability:

if using frequency [pic]

same with refraction [pic]

since: n2(ω) = εr(ω) [pic]

[pic]

plus [pic] into [pic]

[pic]

Now: [pic]

[pic]

so absorb coefficient: [pic]

for solution: [pic] [pic]

relate to dipole expression (Einstein?)

McHale, Ch 6; Struve, Ch 8; Bernath, Ch 1

Einstein relationships are phenormalized expressions of rates of ???

up r12 = N1 B12 ρ(ν) N1 – population lower state

B12 – stimulated rate constant at ν

down r21 = N2 (B21 ρ(ν) + A21) ρ(ν) – energy density

A21 – spontaneous rate

note: only interested in ν = ν12 → resonant frequency

simple kinetics – no light [pic]

[pic]

radiative lifetime: [pic] 1st order decay

if light on a long time system comes to equilibrium

N1 B12 ρ = N2(B21 ρ + A21) and [pic]

solve for [pic] ΔΕ = hν

(relating ρ(ν) to kinetics) [pic]

if let ρ(ν) be a black body light source (also equilibrium)

[pic] (gives denomination term)

[pic]

see that A21 depends strongly on ν3 → probability of spontaneous emission increase

as go to the uv

Two emission processes

Compare rates [pic] high ν → uv – spontaneous dominance

low ν → IR – stimulated dominance

Important → spontaneous (fluorescence) – incoherent

→ stimulated (e.g. laser) – same properties as

incident photoreduction [pic] and polarization [pic]

So how do lasers work in vis-uv (note kT ~ 200 cm-1 – for IR)

non-equilibrium devices → population inversion

must make N2 > N1 (non ???)

Recall Lumbert Law: dI = -γ I dx I = I0 e-γx

positive [pic] absorption [pic] loss of intensity

negative [pic] emission (stimulated) [pic] amplification

relate power/volume to energy density/time: [pic]

assume B12 = B21

relate to einstein: [pic] (correct c for n) non degenerate B12 ρ(ν)

[pic] ρ(ν) → nI(ν)g(ν)/c

Lumbert Law: [pic]

now back to macroscopic: [pic] [pic] complex part

[pic] of induced polarization

[pic] ~ 2n(ω) k(ω)

Relate back to Golden Rule: [pic]

[pic]

rethink Coulent Law:

[pic] assume electronic/no stimulated emission

I in increment dx (cross section: 1-unit) (N2 ~ 0)

Beens Law: -dI = 2.303 a(ν)CIdx I = I0e-acx a(ν) = e(ν)

(C-cone M, a(n) – malar absorption (per cm) molec abs

Combine to get rate: [pic]

([pic] u(ν) energy density [pic] [pic])

To account for bandwidth [pic]

[pic]

but normally ρ(ν) constant over bandwidth – take out and compare to relationship for B12

[pic]

so if meamic spectra, integrate, connect for path and core

can determine B12 experimentally also

[pic]

[pic] in D2

(devices for homework!)

McHale 6.6? Line shapes

Homogenous → “all affect same way” → typical lifetime

[pic] time and freq complementary variable (inverse)

Fourier Transform relate then: [pic] (on correlation function)

Lomentzian: [pic] FWHM [pic]

(very long tails)

This concept for single state transition →

for electronic –vibration (mix) or (rotation-vibration) the

distribution of states (if unresolved) [pic] shape

Bernath, p31 Inhomogenous broadening → collection of molecules has a distribution

Fig 1-22 of resonant freq ν12

Gaussian dist [pic]

(broaden FWHM but tails drop fast)

G = standar deviation: [pic] (often use[pic])

gas phase – velocity distribution means random with r/t detected

shift frequency due to Doppler effect [pic]

[pic] straight on (+) or away (-)

Voigt profile [pic]

convolutes both allows “resolution” homogeneous and in homogeneous (numerical)

-----------------------

Approx 1

1st order

pert

Approx 2

1st order

in field

Approx 3

1st order

in expansion

k

E

B

E

lð

λ

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