Born-Oppenheimer approximation.pptx

Born-Oppenheimer approximation,Slaters rule

1. Born-Oppenheimer approximation & Slater determinant By Gokila.N

2. The Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated . This involves the following assumptions : The electronic wave function depends upon the nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed .

3. Electrons are much lighter than the nuclei . Hence electrons in a molecule would much more faster than the nuclei The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the speedy electrons

4. Wave function ψ of a molecule is the product of the electronic wave function and the nuclear wave function . ψ = ψ e ψ N Hamiltonian operator of a molecule Ĥ = T N + T e + V eN + V ee + V NN ------------------ (1) Ĥe = T e + V eN + V NN ------------------ (2) T e + V eN + V NN = Hamiltonian for electron motion V NN = Hamiltonian for nu-nu repulsion ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

5. Ĥ = Ĥe + V NN ---------------- (3) Consider hydrogen molecule ^

6. Operator for the kinetic energy of the electron T = - ½ - ½ Operator for the potential energy due to the electron nuclear attraction V eN = Operator for the potential energy due to the electron-electron repulsion V ee =   ^ ^ ^

7. Operator for the potential energy due to the nu-nu repulsion V NN = Hamiltonian of the molecule Ĥ = + + Ĥ = Ĥ e +   ^

8. Slater determinant In  quantum mechanics, a  Slater determinant  is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons

9. The Slater determinant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital, where denotes the position and spin of a single electron. A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function that is zero

10. For Helium atom = =1/     ψ (1,2) =1 /   - ψ (1,2)

11. For Lithium atom =1/     ψ (1,2,3)