Physical Properties of Molecular Materials

Physical properties of molecular materials

Vibrational states

Introduction

H(β,ε)ψ_q(ε,R) = E_qψ_q(ε,R) a particular possible state of the system β = {r₁, r₂, r₃......r_3N} electrons ε = {R₁, R₂, R₃, ......R_M} nuclei

H(ε, β|Ψ^o(ε,R)) = T_e(ε) + T_N(R) + V(ε,R)

T_e = - ħ² ∑∇²_i ∇_i = Σ ∇²∇²_i∇: Σ ∈ i ∇

T_N = - ħ² ∑_i∇²_i ∇: Σ_i ∈ i

V̂ = V_ee + V_en + V_nn

V_en = - e° ∑_i Z_i e²_t |R_i - R_j|

V_nn = αΔT_{eoΣZi |Ri - Rj|}

Born-Oppenheimer approximation

The electrons state can be described within a quasi-static field generated

Ĥ (ε, β) = V(ε,β)

E = total energy of the molecular quantum state, including molecular and electronic contribution.

Ĥ (β, ε) ψ_n = E ψ_e Φ_n

Ĥ_e Φ_n T̂_n + V_nn = V_n + V_nn electron wave function, as a parametric function with respect to the nucleus of fraction of the nuclei β and nuclear wave function Φ

Ĥ(ε,β) Ψ^o(ε|R) H_e(β) Ψ⁰ Schrodinger equation for the nuclei

Hellmann-Feynman theorem

The nuclei are to be held fixed in position, no point charges and the force required to be applied

Physical properties of molecular materials

Vibrational states

Introduction

H(ξ,ℰ)Ψ_σ(ξ,ℰ) = E_σΨ_σ(ξ,ℰ) is associated with the full set of quantum numbers that define a particular possible state of the system ξ = {ξ₁,ξ₂,...,ξ_s} electrons ℰ = {ℰ₁,ℰ₂,...,ℰ_μ} nuclei

H(ξ,ℰ) = T_e(ξ) + T_n(ℰ) + V(ξ,ℰ)

T̂_e = -Σ_i ∇²_i, ∇²_i = ∂², ∂², ∂² - Σ_i ∇²_i:

T̂_n = - Σ_t 1/2M_t ∇²_t = -Σ_t ∇²_t:

V̂ = V_ee + V_en + V_nn

V_ee = e² Σ_i>j |r_i-r_j|^-1

V_en = -e² Σ_i Z_t |r_i-R_t|^-1

V_nn = Σ_t>u Z_t Z_u |R_t-R_u|^-1

They are two-body operators but electron-nuclear attraction term can be cast in the form of a one-body operator when it is considered from the point of view of the electrons

Born-Oppenheimer approximation

The electronic state can be described within a quasistatic field generated by almost immobile nuclei (nuclei are less freely particol.)

H(ξ,ℰ)Φ_e(ξ,R) = ε Φ_e(ξ,R) (T̂_n + V_nn) Φ_n = ε Φ_n

H_e(R)Ψ_e(ξ|R) = E_e(R)Ψ_e(ξ|R)

(T̂_n + V_nn) Φ_n = ε Φ_n Adiabatic approx invalid

γ = Ψ_e(R) electronic Hamiltonian (dependent upon R) b + ε ψ_e(ξ|R) = E_e(R)Ψ_e(ξ|R) Schroedinger eq. for the nuclei: H_d(R)Ψ_eΨ_e = E_e(R)Ψ_e

Hellmann-Feynman theorem

The nuclei are to be held fixed in position, no point charges and the force required to be applied to these is to be calculated work done in displacing the nuclei through d2f_i= -∂E_iδE = δS[(q_i∣h₁q_i)]f_i= -δ[(E_iγ(1R₁)] δE(∑ = ∫ρ(q_i∣he(R_i)∣η)]q