Solid State Flashcards
Fig(chapter.page.figonpage)
Position operator
x̂ = x
Momentum operator
p̂ₓ = -iħ d/dx
Angular momentum operator
L̂ₓ = ŷp̂ᶻ - ẑp̂ᵧ, etc.
In spherical,
L̂ᶻ = −iħ ∂/∂φ
Hamiltonian
Ĥ = T̂ + V̂ (kinetic + potential)
= p̂ₓ²/2m + V̂(x̂, t)
= -ħ²/2m d²/dx² + V(x, t) (1D)
Wavefunction (r,t)
Ψ(r, t) = Ψ(r) exp(-iEt/ħ)
TISE
Ĥ Ψ(r) = E Ψ(r)
Expectation value
⟨Â⟩ = ∫ Ψ∗ÂΨ dr / ∫ |Ψ|² dr
= ∫ Ψ∗ÂΨ dr (if Ψ normalised)
Hamiltonian for hydrogen-like atoms
Ĥ = -ħ²/2mₑ ∇² - Ze²/4πε₀r
where Z is number of protons (2nd term is Coulomb potential)
Solved by
Eₙ,ₗ,ₘₗ = Eₙ
= −Z²Eᴿ/n²
where Eᴿ is the Rydberg energy
Eᴿ = e²/8πε₀a₀ = 13.6 eV
and a₀ is the Bohr radius
a₀ = 4πε₀ħ²/mₑe² = 0.53 Å
Split hydrogen wavefunction into its constituent parts
Ψₙ,ₗ,ₘₗ,ₘₛ = ψₙ,ₗ,ₘₗ(r, θ, φ)χₘₛ
ψₙ,ₗ,ₘₗ(r, θ, φ) = Rₙ,ₗ(r)Yₗ,ₘₗ(θ, φ):
* Rₙ,ₗ(r) = const. × (Zᵣ/a₀)ˡ exp(−Zᵣ/na₀) Σ(0→n−l−1) (−1)ᵏ cₖ (Zr/a₀)ᵏ
* Yₗ,ₘₗ(θ, φ) = Pₗ,ₘₗ(θ)Φₘₗ(φ), spherical harmonic functions
* Pₗ,ₘₗ(θ) = associated Legendre polynomials
* Φₘₗ(φ) = exp(imφ)
χₘₛ denotes spin, up or down. (χ1/2 = χ↑, χ−1/2 = χ↓)
Allowed values:
n = 1, 2, …
l = 0, 1, 2, … , n−1
mₗ = −l, −l + 1, … , l − 1, l
mₛ = ±1/2
Degeneracy of an energy level
2n²
n = principle quantum number
2 is spin factor
Wavefunction of helium
2 electrons
Ψ(r₁, r₂) = 1/√2’ ψ₁ₛ(r₁)ψ₁ₛ(r₂)(↑↓ − ↓↑)
Hund’s rules
- Maximize S (fill all mₗ ‘slots’ with aligned spins before any with anti-aligned)
- Maximize L (fill slots from high mₗ to low, with L = Σmₗ)
- If orbital is less than half full, minimize J = |L − S|.
- If orbital is more than half full, maximize J = L + S.
Charge density from wavefunction
Charge density ∝ |Ψ|²
Born-Oppenheimer approximation
Nuclear mass much greater than electron mass, treat electron motion separately. In practice, consider electron’s (quantum mechanical) motion within fixed configuration of nuclei.
Potential for BO approx. of H₂⁺
V = −e²/4πε₀rₐ − e²/4πε₀rᵇ + e²/4πε₀r
where rₐ = |rₑ−a|, rᵇ = |rₑ−b|
There is strong attraction at protons, rₐ → 0 and rᵇ → 0.
see fig(1.12.1) and fig(1.13.1)
What is a LCAO?
Linear combination of atomic orbitals (LCAO). Applies when nuclei are well separated. Since we want to find the ground state, choose the combination of the lowest energy states of the constituents.
A molecular orbital (MO, wave function for one electron in a molecule) is constructed by a LCAO using
ψᵢ(rₑ) = Σₘ cₘᵢ ψₘ(rₘ)
where ψₘ are atomic orbitals located at aₘ with rₘ = rₑ − aₘ, and cₘ are mixing coefficients for lowest energy value. May have many such combinations, as labeled by index i.
Graph wavefunctions for H₂⁺ using LCAO
see fig.(1.13.2) and fig(1.14.1)
Two possibilities, 1s orbitals in phase or out of phase.
In phase, 1s orbitals add, out of phase they subtract so halfway probability is zero. Graphs are normalised.
Bonding in H₂⁺
ψ⁺⁻(rₑ) = N⁺⁻[ψₐ(rₐ) ⁺⁻ ψᵇ(rᵇ)]
Electron distribution:
|ψ⁺⁻|²
“Even” state ψ⁺:
Electron has high probability in area between the two protons, binds the protons together due to Coulomb attraction. Wavefunction ψ⁺ is a bonding orbital.
“Odd” state ψ⁻:
Electron has low probability in area between two protons, will not bind the nuclei.
Wavefunction ψ⁻ is an anti-bonding orbital.
Hamiltonian and energy expectation in H₂⁺
Ĥ = −ħ²/mₑ ∇ₑ² − e²/4πε₀rₐ − e²/4πε₀rᵇ + e²/4πε₀r
Expectation values in states ψ⁺⁻:
E⁺⁻ = ∫ (ψ⁺⁻)∗Ĥψ⁺⁻drₑ / ∫ |ψ⁺⁻|²drₑ
= H±Hₐᵇ/1±S
where
H = Hₐₐ = Hᵇᵇ = ∫ ψₐ∗Ĥψₐdrₑ = ∫ ψᵇ∗Ĥψᵇdrₑ
Hₐᵇ = ∫ ψₐ∗Ĥψᵇdrₑ = ∫ ψᵇ∗Ĥψₐdrₑ
S = ∫ ψₐ∗ψᵇdrₑ
Plot expectation energy as a function of internuclear separation in H₂⁺
E⁺⁻(r) = E₁ₛ + e²/4πε₀r + C(r)±K(r) / 1±S(r)
S(r) ∝ exp(−r/a₀)
K(r) ∝ −exp(−r/a₀)
C(r) ∝ −1/r
Bonding state σᵍ₁ₛ, in phase, E⁺/Eᵍ
Anti-bonding state σ∗ᵤ₁ₛ, antiphase, E⁻/Eᵘ
Eᵍ is lower and has a minimum at r₀, which is referred to as bond length.
Eᵘ is higher without a minimum.
See fig(1.16.1)
What is dissociation energy?
Energy required to separate two bonded atoms.
Difference between bonding orbital energy and individual atomic orbital energy.
For H₂⁺ and H₂:
D = E₁ₛ − Eᵍ for one electron (H₂⁺)
D = 2E₁ₛ − Eᵍ for two electrons (H₂)
Splitting MO energy levels
Energy levels in an MO can be split into bonding (lower energy) and antibonding (higher energy). Electrons will fill these levels as expeced.
MOs up to 2p
1s and 2s split into 1sσ, 1sσ∗, 2sσ, 2sσ∗.
2p splits into 1pσ, 1pπ, 1pπ∗ 1pσ∗.
π can hold 4 electrons, formed from perpendicular 2pₓ- and 2pᵧ-orbitals.
2sσ∗ is close to 2pᶻσ (and they have the same symmetry), so hybridization/linear combination of these two orbitals occurs. The upper hybrid orbital, 2pσ’, will then be above the π-orbtial.
Bond order
A measure of the bond strength of a molecule
bond order = ½(no. bonding e⁻ − no. antibonding e⁻)