Comparing Energy Levels in Atomic and Molecular Systems
E=0
*Energy in units of h
2
/8m
e
L
2
~
1.5*10
-18
J
n=1
n=2
n=3
n=4
n=5
E*
25
16
9
4
1
Energy Manifold: 1 electron in ground state
in box with L=2*10
-10
m (0.2 nm)
m
e
~1*10
-30
kg
E=0
*Energy in units of h
2
/8m
e
L
2
=1.8*10
-19
J
n=1
n=2
n=3
n=4
n=5
E*
25
16
9
4
1
Energy Manifold assuming butadiene
e
-
are in box the length of butadiene (p. 83)
L=5.78*10
-10
m
m
e
~10
-30
g
Assume
4
electrons
resonate freely
E=hf=
5
Comparison of calculated 1D
particle in box magnitudes with real
atomic scale systems
1 electron in 2*10
-10
m box
E(n=1)
1.5*10
-18
J
1 electron in 2 *10
-10
m
diameter orbit around H
atom
2.2*10
-18
J
E
for n=2
n=3 jump in
model for butadiene
(L=5.78*10
-10
m)
5*1.8*10
-19
=9.0*10
-19
J
Observed HOMO-
LUMO
transition
energy
for butadiene
9.1*10
-19
J
System considered
P(x)dx
= 2/L sin
2
(n
x/L) dx
(x) =
2/L sin (n
x/L)
Key Particle-in-box results
1)Normalized Wave function
2) Eigen energy ,
E
E
=n
2
h
2
/8m
e
L
2
3) Probability density,
P(x)
Switch slide orientation in
design
1D particle-in-a-box P(x) vs. n: at high n…the system converges to
continuum
P(x) for 1D box at
n=100,L=1
2D particle-on-a ring predictions for simple porphyrin (26
electrons* running free
(See also:
Supplement 2:
The 2-D particle in-a-box applied to a real
molecule)
L
L
* Assumes lone pair and inner
electrons are part of
delocalization
L = 1*10
-9
m
HOMO=Highest Occupied Molecular Orbit
LUMO = Lowest Unoccupied Molecular Orbit
E
predicted
= k*[(3
2
+4
2
)-(4
2
+2
2
)] = k[25-20] =5k =
1.52*10
4
cm
-1
E =k(n
x
2
+n
y
2
)
= 3.0375*10
3
(n
x
2
+n
y
2
) cm
-1
E
observed
=
=1.70*10
4
cm
-1
After solving 2D problem: (see supplement 2)
Comparing energy levels of electrons in different scenarios, including 1D particle in a box, electrons around atoms, and within molecular structures like butadiene and porphyrin. The calculated and observed energy transitions provide insights into the behavior of particles within confined spaces.
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Presentation Transcript
Energy Manifold: 1 electron in ground state in box with L=2*10-10 m (0.2 nm) E* 25 n=5 16 n=4 n=3 9 4 1 n=2 me ~1*10-30 kg n=1 E=0 *Energy in units of h2 /8me L2~1.5*10-18J
Energy Manifold assuming butadiene e- are in box the length of butadiene (p. 83) E* Assume 4 electrons resonate freely 25 n=5 C C C C C C C C L=5.78*10-10 m 16 n=4 n=3 9 E=hf=5 n=2 me ~10-30 g 4 n=1 1 E=0 *Energy in units of h2 /8meL2=1.8*10-19 J
Comparison of calculated 1D particle in box magnitudes with real atomic scale systems E(n=1) 1.5*10-18 J System considered 1 electron in 2*10-10 m box 1 electron in 2 *10-10 m diameter orbit around H atom 2.2*10-18 J E for n=2 n=3 jump in model for butadiene (L=5.78*10-10 m) 5*1.8*10-19 =9.0*10-19 J Observed HOMO- LUMO transition energy for butadiene 9.1*10-19 J
Key Particle-in-box results 1)Normalized Wave function (x) = 2/L sin (n x/L) 2) Eigen energy , E E=n2 h2 /8meL2 3) Probability density, P(x) P(x)dx = 2/L sin2 (n x/L) dx
Switch slide orientation in design
1D particle-in-a-box P(x) vs. n: at high nthe system converges to continuum P(x) for 1D box at n=4, L=1 P(x) for 1D box at n=1, L=1 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 P(x) P(x) 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x P(x) for 1D box at n=100,L=1 P(x) for 1D box at n=2, L=1 2 1.8 1.6 1.4 1.2 P(x) 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x
2D particle-on-a ring predictions for simple porphyrin (26 electrons* running free (See also: Supplement 2: The 2-D particle in-a-box applied to a real molecule) L L N NH H N N L = 1*10-9 m * Assumes lone pair and inner electrons are part of delocalization
After solving 2D problem: (see supplement 2) E =k(nx2 +ny2) = 3.0375*103 (nx2 +ny 2) cm-1 HOMO=Highest Occupied Molecular Orbit LUMO = Lowest Unoccupied Molecular Orbit Epredicted= k*[(32+42)-(42+22)] = k[25-20] =5k = 1.52*104 cm-1 Eobserved = =1.70*104 cm-1
