Comparing Energy Levels in Atomic and Molecular Systems

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.

• Energy levels
• Atomic systems
• Molecular structures
• Electron transitions
• Confined spaces

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1. 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

2. 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

3. 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

4. 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

5. Switch slide orientation in design

6. 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

7. 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

8. 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