Download Density Functional Theory

Density Functional Theory
N
LCAO approach: Ψa = ∑cia φi
Ψa = one-electron molecular orbital
i=1
φi = atomic orbitals for N orbitals (Slater or Gaussian)
cia = orbital coefficients -- variational parameters
Variation Theorem: vary cia’s to find the lowest energy:
Hartree-Fock Equations:
f1 Ψa(1)σ(1) = εa Ψa(1)σ(1)
for MO a, b, c, …
εa = one-electron orbital energy
n/2
f1 = h1 + ∑ {2Jj(1) - Kj(1)}
j = all the other filled orbitals
j=1
rA1
for electron 1, 2, 3, …
core Hamiltonian:
––
h2 2
h1 = 2m ∇1 -
rB1
m
∑
k=1
Zke2
4πεork1
k = all nuclei
r12
HF electron-electron repulsion:
Coulomb operator:
Jj(1)
Exchange operator:
Kj(1)
Density Functional Theory:
n/2
f1 = h1 +
∑ {2J (1) - VXC
j (1)}
j
E =ET + EV + EJ + Exc
j=1
kinetic + e-n attraction + Coulomb e-e repulsion +exchange-correlation energy
EXC is taken as a function of the electron density: ρ(r) = 2 ∑|Ψ(r)|2
Uniform Electron Gas (Quantum Monte Carlo Simulations)
Change electron density in one spot
Other electrons respond—motion correlated
.
"Wrap" this correlation around the nuclei
difference density
"Pure" DFT: BP, BLYP, EDF:
"Hybrid" B3LYP:
Colby College
HF Coulomb
HF Coulomb and Exchange (exact)