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Structural Chemistry-2014 -- Atkins Ch. 7, alternate: Hanna, House, Blinder
To understand structure must come to terms with
atoms ↔ make molecules and then with electrons/nuclei ↔ make up atoms!
Study of interaction of small particles Quantum Mechanics
Here “Quantum Chemistry” Molecular scale theory explains: Spectroscopy, Dynamics, Binding, Conformation…
big issue particles ↔ waves: frequency = c/wavelength
Why a special mechanics for small particles? 19
th century “everything worked out”
Newton mechanics – continuous (particle - all energies) Maxwell relations (E & M – waves describe light)
Some problems with this: a. Excite elements in discharge
emission spectra, only at specific wavelength/frequency
1/ = /c = ˜
H-atom: 1/ = R (1/n12 – 1/n2
2)
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Balmer-vis n1=2; Lyman – uv n1=1; Paschen near-IR n1=3 R = 109,677 cm
-1 – why discrete?
frequency
Engel Fig. 12.6 picture of line spectra
b. Black-body radiation
If all equally probable, - probability density would blow
up at short since more short waves in a cavity –Rayleigh-Jeans UV catastrophe
Actual Shape (energy distribution with or )
observed rises with dec. (inc. ) to maximum,
then steep fall off at short high
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Observations:
Stefan-Boltzmann – Energy density ~ T4 (integral of )
M = T4 ~ 5.67 x 10
-8 Wm
-2 K
-4
Wein – distribution maximum depend inversely on T
maxT = constant = 2.9 x 10-3
m K Several people tried to derive it classically – fail
Planck assume oscillators could only have energies
E ~ 0, h, 2h, 3h, … (hc/) – restriction, non-classical h – constant – function to distributions
= c/ frequency Then used Statistical-Thermo ideas to get distribution
energy density from “hole”
()= as function (and T) like Boltzman probability
Test: long – h/kT 0
Mult. top/bot. by e-hc/kT, expand e
-x = 1 – x + ½ x
2 etc.
() ~
... 1 1
... 1hc8
kThc
kThc
5
() ~
hc
kThc8
5 ~ 8kT/4
classical (Wein displacement) result fits long behavior,
i.e. hc/kt is very small (short 8kT/4 ∞,UV catastrophe)
Short (rewrite): () (8hc/5)(e
-hc/kT)/(1- e-hc/kT)
–small exp
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but as increase exponential term builds from zero to 1, denom. collapse – blow up
In between long - short , () must have a maximum
max: (d/d) = 0 = [-5/6 – (hc/kT
5 )(-1/
2)]
maxT = constant = hc/5k -Wein: max fit vs.1/T, slope ~ h Result: a) h = 6.626 x 10
-34 J
.s - note: h is an angular momentum
b) form of () fits observation “perfectly”
Justify assumption E = 0, h, 2h, etc. by fit observation (not “proven”)
Photoelectric Effect – Einstein explain--light impinge on metal,
electrons emitted, accel. in V, grid can stop e- if eVg = KE
K.E. of electrons depend on – long no elect.
th – threshold for appearance
K.E. linear dep. on Intensity not affect E,
just number of electrons (current) light acting like a particle
K.E. = ½ m e2 = h( - Th)
= h –
– work function
no e- escape e
- with KE=h-
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So far: Energy limited to discrete values – Planck h Light (wave) behave particle – Einstein photelectric
key – threshold – frequency energy – intensity (classically E-field)
number of electrons, not energy
Complement: Davisson-Germer e-beam – fixed energy onto (metal)
ordered surface
diffract at specific angles due to periodic structure
analogous to diffract x-rays or light w/ grating
electrons are particles behave as a wave like light (wave behavior) diffract from grating
Combine Louis de Broglie (PhD thesis): = h/p or p = h/
momentum of particle relate to 1/ = /c (or frequency of light relate to its momentum)
↔ p Wave-Particle Duality – Central to Schrödinger QM
– property of small particles / high energies
Catch: a wave has no position so specify x problem
Also: if particle is localized can’t oscillate problem Heisenberg propose this to be fundamental limit
pxx ħ/2 position and momentum complementary variables
Uncertainty principle – cannot know both x,px precisely (works for complementary variables, also E,t )
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So now: have energy restricted (quantized) have light behave like particles particle like light (quantized)
Big Deal – not continuous like classical
So what about those line spectra? Somehow light / frequency limited -- discrete
Atoms (Rutherford) became clear dense nucleus (+) electrons must be outside – postulate in “orbits”?
Oct.7, 2012
Google doodle
Niels Bohr birthday
Bohr (Niels) postulate (we moved back a decade – 1913) a) Energy (light) emit only when change orbits b) Frequency of light jump orbits
E/h = (relate to energy, E – like Einstein) c) “Orbit” → relate electrostatic attraction e
- and p
+
to centripetal force
* d) Angular momentum restricted (nħ): nh/2 n = 0,1,2,. .
Look up derivation – fairly simple
E = T + V = ½ m2 – Ze
2/r Z - atomic number
c) equate coulomb-centripetal forces: -Ze2/r
2 = m(-
2/r)
Newton: force ┴ to motion for change in direction
solve: m2 = Ze
2/r then plug in: E = -Ze
2/2r
d) subst: angular momentum assume: r·p = r·mv = nħ
r = n2ħ
2/Ze
2m where: ħ = h/2m
2= Ze
2/r = n
2ħ
2/mr
2
E = - Z2e
4m/2n
2ħ
2 EH = h = R(1/n1
2 – 1/n2
2) (H-atom)
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For Z=1 Rydberg: R = e4m/2ħ
2 in J (norm. as 109,677 cm
-1)
Fits the Balmer series and Rydberg formula Bohr result explain H-atom discrete line spectra
E~1/n2 E ~ R(1/n1
2 – 1/n2
2)
Also worked for He+, Li
2+ etc. 1-e atoms (change e
4 Z
2e
4)
Depended on assuming angular momentum quantized This is same as assuming – fixed orbits
Recall de Broglie = h/p = h/m
if particle on ring 2r = ncircumference have integer number
wave lengths
plug in: 2r = n(h/m)
mr = nħ matches Bohr’s assumption (10 yrs later)
Again waves particles key With this set-up time was right to generalize
Bohr model great for H but fail for He0
Problem assumption pseudo classical
1926 Schrödinger developed a wave – mechanics
treat particles with wave properties (relate to wave equation of Maxwell but different)
Heisenberg same time did a matrix – mechanics fully consistent but judged more difficult concept
[Dirac later added relativity – source of spin]
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Postulates – simplest approach - wave–quantum mech. idea just like Thermo – postulate set of rules – derive properties – test against observables Catch – initially must just accept, no rationale, then test
little physical picture Atkins or House QM book Postulate 1: State of a system fully described by a
wavefunction: (r1, r2, … t)
> variables (r1, r2 - xyz coord. particle 1, 2, t = time) > Anything not in wave fct. cannot be known in QM
Shorthand – represent state by quant. num. n,ℓ,m,. . .
– or observables a,b,c - helpful for us, not w/f
Alternative: m n vector bracket from Heisenberg
Postulate 2: Observables correspond to operators
summary: constant: c → c concept: mult.
variable: x → x e.g. position
function: (x) → (x) e.g. PE: V(x)
momentum: px → -(ih/2)d/dx = -iħ d/dx
Operators act on w/f– typical: multiplication, derivation
[, ] = [ – ] – commutator, normally expect 0
Constraint is that [x, px] [xpx – pxx] = i ħ
recall: d/dx(y=(yd/dx)+(dy/dx)
here x, px do not commute, since [x, px]
Properties: if (x) = w (x) eigenvalue equation
if operate on (x) and result is w, a constant *(x)
is eigenfunction of with eigenvalue w
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Many possibilities examples: a) let A = d/dx
Aeax
= aeax
a const., = eax
eigenfunction of A
Aeax2
= 2ax eax2
2ax not const., eax2
not eigenfct. of A onstant
Asin ax = -a cos ax sin ax ≠ cos ax not eigenfct of A b) or B = d
2/dx
2
Bsin bx = -b2 sin bx g = sin bx is eigenfunction B
Beax
= a2e
ax g' = e
ax also eigenfunction B
Could also do linear combinations
{n} set of eigenfunction of
if complete set g(x) = 1n
cnn and let n = wn n
then g = cnn = n
cn n = n
cn wn n
This is not an eigenfct relation unless wn = w all wn
g = c'nn cg
= w cnn = wg if all wn equal
degenerate (product of symmetry)
Born (Max) model: Wave function relates to probability For volume element dV, probability of finding the particle
is * dV, so probability between a and b is
integral from a to b: ∫* dV key to solving problems Evaluating Observables
Postulate 3: If system is described by i and i is an eigen-
function of where operator corresponds to observ. a
Then every measurement yields ai, the eigenvalue i = aii
i – set of eigenfunction
ai – corresponding eigenvalue, for operator
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Postulate 4: If is not eigenfunction then the average observed value is “expectation value”
*- complex conjugate ,change i -i
= – integrate over all variables
ex: a) let = eax
eval. px – momentum, operate: px on - iħ (d/dx)(e
ax)= (- iħ a) e
ax
this is eigenvalue equation = - iħ a (const. mom.)
b) let = cos ax - iħ d/dx cos ax = iħ a sin ax not eigenvalue
-
2
-
2x
dxax cos
dxax sinax cos i
a
dx xacos
dx )ax cos( dx
d
i )ax (cos
p
= 0
(numerator odd function, symmetric integral zero)
[meaning: px is or so average cancels out] Schrödinger equation – many ways to get
– here use postulates create an Energy operator “Hamiltonian” – classical operator for total Energy
H = T + V = ½ mv2 + V(x) = p
2/2m + V(x) KE+PE
Quantum Mechanics energy operator H = T + V = (-ħ
2/2m) d
2/dx
2 + V(x) 1-D
H = -ħ2/2m
2 + V(x,y,z) 3-D Hamiltonian operator
2 = d
2/dx
2 + d
2/dy
2 + d
2dz
2 Laplacian =
r = xi + yj + zk = d/dx i+ d/dy j+ d/dz k - vector operator
H = E = [(-ħ2/2m)
2 + V(x,y,z)] = E "gradient”
If is eigenfct of H then eigen values (E)total energy
d *
d *
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Use these eigenvalue properties:
Expand arbitrary wavefcts in set of eigenfct {n} of operator
n = a n {n} complete set of functions
= n
cn n arbitrary function rep.as linear comb. of fn
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so in general for H without time dependence can just use time independent result
Spectroscopy light interaction with matter is time dependent Properties of operators (bit detailed – can skip)
Hermitian operators – have real egienvalues (i.e. can be measured) hence all observables correspond to Hermitian operators (not vice versa)
Hermitian definition: m* n d = { n* m d}*
equivalent to: m* n d = m)* n d
to simplify n m = m* n d
change notation bracket – Dirac notation
Orthonormality mn = nm = n* n d
Hermiticity n m = m R n *
a) Prove: eigenvalues of Hermitian operator are real
n = n n operator left by n
n = n n = n n * = * real
b) Prove: eigenfunction of Hermitian op. are orthogonal
n = n n m = m m
m n = n nm
n m = m nm
c.c. and subtract (recall, c.c --> take neg. of all i)
n m – m n * = (n – m) mn
n = m or mn = 0 – orthogonal
important result - generalize operator form
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Important c) If two operators have simultaneous, arbitrarily precise
eigenvalues in one state, then this is an eigenfunction of both and the operators commute.
i.e. A = a and B = b
AB = Ab = bA = ba = ab = a(B)= B(a) = BA
[A, B] = 0 * prove to yourself the converse –
[A, B] = 0 eigenfunction of both A, B
Relate back to Uncertainty This proves that momentum, position of the same variable can not be measured with arbitrary accuracy because [x, px] = iħ x, px said to be complementary
however since [x,y] = 0 can be measured x,y are independent variables
complementary operators do not commute and involve an uncertainly relationship, such as x,px
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