Valence

Question 1

What is the Hamiltonian for a moleuclar system?

Answer 1

Ĥ = T̂_e + T̂_n + V̂_en + V̂_nn + V̂_en

Where: T̂ - KE terms

V̂ - PE term from Coulomb interaction

V̂_en is attractive, V̂_nn & V̂_en is repulsive

Question 2

What is the T̂_e operator?

Answer 2

Kinetic energy term for the electrons

T̂_e = (-ħ/2m_e)*∇²

Question 3

What is the form of the V̂_ee operator?

Answer 3

PE term for attraction between e-

V̂_ee = +e²/4πε₀r_ij

Question 4

What is the general form of the Schrodinger equation for a molecule?

Answer 4

Ĥψ(r,R) = Eψ(r,R)

Where r and R are the position vectors (wrt to centre of mass) of the e- and nuclei respectively

Question 5

Why can locations in the Schrodinger equation for H₂ be given wrt the centre of mass?

Answer 5

Constant velocity when there are no external forces

So factor it out

Question 6

What is the sign of a PE operator mean?

Answer 6

-ve is an attractive force

+ve is a repulsive force

Question 7

What are you stating if atomic units are being used?

Answer 7

e = 1

Question 8

What is the form of the T̂_n operator for H₂ in atomic units?

Answer 8

T̂_n = (-1/2μ)*∇_R²

Question 9

What is the form of the V̂_en for H₂ in atomic units?

Answer 9

V̂_en = - 1/r_1A - 1/r_1B - 1/r_2A - 1/r_2B

Attractive

Question 10

What is the form of the V̂_nn for H₂ in atomic units?

Answer 10

V̂_nn = 1/R

Can write as atomic units

Question 11

What is the form of the V̂_ee opreator for H₂ in atomic units?

Answer 11

V̂_ee = 1/r₁₂ = 1/|r₁-r_{<strong>2</strong>}|

Where r₁₂ = separation between e- which is difficult to determine

Question 12

What does the Born-Oppenheimer assume?

Answer 12

Adiabatic separation: Assumes nuclei are stationary on the timescale of electronic motion

ψ(r,R) = χ_n(R)ψ_e(r|R)

where χ_n(R) = wavefn describing motion of nuclei

ψ_e(r|R) = electronic wavefn (depends parametrically on coordinates of nuclei)

Question 13

Why does the electronic wavefn depend parametrically on coordinates of nuclei?

(in the Born-Oppenheimer)

Answer 13

As the nuclear position changes (bond length changes) then so does the position of the electrons

Question 14

How is an electronic Hamiltonian defined in the Born-Oppenheimer approx?

Answer 14

Ĥ_e = Ĥ - T̂_n

Where T̂_n is the KE of nuclei

Question 15

Whsat is the solution of the electronic Schrodinger equation at fixed R?

Answer 15

Ĥ_eψ_e(r|R) = E_e(R)ψ_e(r|R)

Gives electronic states a bit like atomic orbitals

Where E_e(R) are electronic energy levels which depend on position of nuclei R

Question 16

What do electronic states with energy E_e(R) represent?

Answer 16

PE function (curves) in the form V(R) that a nuclei experience at a given separation R

Includes the Coulomb replusion between fixed nuclei V̂_nn

Question 17

How are the PE curves whihc nuclei experience found?

Answer 17

Performing calculations fo the electronic energy states at many values of R

Question 18

How can the states with energy E_e(R) be used to solve the Schrodinger eqn?

Answer 18

Ĥ_nχ_n(R) = [T̂_n + E_e(R)]χ_n(R) = E_nχ_n(R)

Where E_n are energy levels from vibrational and rotational motion of nuclei in a specific electronic state

Question 19

How are each electronic states described?

Answer 19

Described by a different PE curve, and own rotation-vibration states

obtained from solution of Schrodinger for nuclear motion: Ĥ_nχ_n(R) = [T̂_n + E_e(R)]χ_n(R) = E_nχ_n(R)

Question 20

How does the BO approx make the schrodinger equation solvable?

Answer 20

Ĥψ(r,R) = Eψ(r,R) , where E = E_n

Ĥψ(r,R) = (Ĥ_e + T̂_n)χ_n(R)ψ_e(r|R) = Ĥ_eχ_n(R)ψ_e(r|R) + T̂_nχ_n(R)ψ_e(r|R)

Ĥ_e term, doesnt operate on χ_n(R): χ_n(R)Ĥ_eψ_e(r|R) = χ_n(R)E_e(R)ψ_e(r|R) = χ_n(R)V(R)ψ_e(r|R)

T̂_n more complicated: T̂_nχ_n(R)ψ_e(r|R) = χ_n(R)T̂_nψ_e(r|R) + ψ_e(r|R)T̂_nχ_n(R), can then neglect a term to give T̂_nχ_n(R)ψ_e(r|R) = ψ_e(r|R)T̂_nχ_n(R)

Ĥψ(r,R) = ψ_e(r|R)[T̂_n + V(R)]χ_n = Eψ_e(r|R)χ_n​

Question 21

Why is the small r bold in Ĥψ(r,R) = Eψ(r,R) ?

Answer 21

As there is potentially more than 1 electron

Question 22

Why can the χ_n(R)T̂_nψ_e(r|R) term be ignored in the BO approx?

T̂_nχ_n(R)ψ_e(r|R) = χ_n(R)T̂_nψ_e(r|R) + ψ_e(r|R)T̂_nχ_n(R)

Answer 22

Small is reasonable providing ψ_e(r|R) is varying slowly in R (nuclear positions)

Question 23

What approx can be made with T̂_e in terms of T̂_n?

Answer 23

T̂_eψ_e(r|R) ≈ μ/m_eT̂_nψ_e(r|R)

As the T̂_e term is significantly greater as heavier