Electronic System and Molecular Vibrations

Study Notes

Molecular Structure and Bond Lengths

Hexatriene structure: CH2=CH-CH=CH-CH=CH2.
Bond lengths:
- C=C bonds measure 134 pm (3 bonds, total = 402 pm).
- C-C bonds measure 154 pm (2 bonds, total = 308 pm).
- Average bond length calculated as 154 pm.
Effective length 'a' for hexatriene is 864 pm.

Electronic Energy Levels

Energy formula: E = n²h² / 8ma², where 'n' is the quantum number, 'h' is Planck's constant, 'm' is mass, and 'a' is the box length.
Ground state (n=1): ψ1(2) = (h² / 8m * 864²) x 2.
Energy levels for n=2 to n=7 are computed similarly using the formula.

HOMO and LUMO

Two-electron system distinguishes between HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital).
HOMO energy level: En = n²h² / 8ma².
LUMO energy level (n+1) has a corresponding energy calculation: En+1 = (n+1)²h² / 8ma².
The energy difference ΔE between LUMO and HOMO is determined as ΔE = (2n+1)h² / 8ma².

Energy Calculations

For CH2=CH-CH=CH2, 'a' is 578 pm; ΔE between LUMO and HOMO is 9.02 x 10⁻¹⁹ J.
Frequency and wavenumber calculated as ΔE/hc ≈ 4.61 x 10⁴.

Molecular Vibrational States

ψ₁(2) and ψ₂(2) derived from vibrational analysis of CH₂=CH-CH-CH₂.
Integrals solve for wave functions within defined conditions.

Boundary Conditions for Wave Function

Two primary conditions for wave function Ψ:
- At x = 0, Ψ = 0 implies B = 0.
- At Ψ = 0 and x = a, this leads to sine conditions where 0 = sin(k).

Energy Drift and Differences

Defined energy levels for n states:
- E1 (n=1), E2 (n=2), and E3 (n=3) follow the formula E = n²h² / 8ma².
Energy difference ΔE between consecutive levels computed as ΔE = E2 - E1 and ΔE = E3 - E2.

Particle in a Box Model

Characterization of quantum states where potential energy outside the box is infinite while inside it is zero.
Wave functions defined within the box using Ψ = √(2/a)sin(nπ/a x).

3D Schrödinger Equation

General form of the 3D equation incorporates second derivatives along each axis and considers potential energy (V).

Atomic Orbitals and Energy Transition

Atomic orbital defined in terms of 3D space surrounding the nucleus.
Energy required to promote electrons from HOMO to LUMO calculated for linear hexatriene systems.

Summary of Energy Levels

Energy levels and wave function characteristics calculated for n from 0 upwards; concludes with HOMO at n = 1/2 and LUMO at n = 3/2.
The wave function Ψ is normalized to the total probability of finding an electron within the defined region.

Electron Configuration in Conjugated Systems

Conjugate molecules possess overlapping p-orbitals across three or more atoms, affecting their electronic properties.
Provided calculations focus on the structural geometries inherent in conjugated molecules emphasizing atomic characteristics.

Description

Explore the concepts of electronic systems, focusing on HOMO and LUMO energies, as well as the calculation of molecular vibrations. This quiz covers important formulas and calculations relevant to quantum chemistry. Dive into topics like ΔE calculations and frequency determination for organic compounds.