CHE 210A - Quantum Chemistry Winter 2023 Problem Set 2

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CHE 210A - Quantum Chemistry

Winter 2023

Problem Set 2

1. Born-Oppenheimer approximation

In Lecture 3, we have shown that after applying the Born-Oppenheimer ap- proximation, the nuclear Hamiltonian becomes

Hˆnuc = KˆN + T() + Eelec ( )                              (1)

In this problem, we will investigate why it’s a good approximation to neglect T().

To make the problem simpler, consider the following problem only in 1-dimensional. Given a H molecule, denote the coordinates of the two protons as X1  and        X2  respectively, and denote the coordinate of the electron as x. Suppose the        electron is the state described by the following wavefunction

ψ(x;X1 ,X2 ) = c (e −i + e −i )                        (2)


In the above equation, a0  = 5.29 × 10 − 11m is the Bohr radius and c is the normalization constant such that ⟨ψσ |ψσ ⟩ = 1

(1) Compute the expectation value of the electronic kinetic operator, i.e.

Ke (X1 ,X2 ) = \ ψ ∗ (x;X1 ,X2 )Kˆe ψ(x;X1 ,X2 )dx                  (3)

You can keep physical constants (e.g.  ℏ , a0 , electron mass) in the result. (Hint: make use of ⟨ψσ |ψσ ⟩ = 1 to simplify the calculation.)

(2) Compute the value of T() using the nuclear kinetic operator KˆN , i.e.

T(X1 ,X2 ) = \ ψ ∗ (x;X1 ,X2 )KˆN ψ(x;X1 ,X2 )dx                  (4)

You can keep physical constants in the result.

(3) Based on your results in (1) and (2), explain why T() can be neglected in the nuclear potential, i.e. Vnuc ( ) = T() + Eelec ( ) ≈ Eelec ( )

2. Ladder operator of harmonic oscillators

1) Show that the Hamiltonian of a 1D harmonic oscillator can be written in terms of the ladder operators as

Hˆ = ℏω ( − + − )                                    (5)

2) Given an eigenfunction ψ of the Hamiltonian with eigenvalue E, i.e. Hˆ ψ =

Eψ, show that (− ψ) is also an eigenfunction of Hˆ with eigenvalue E − ℏω ,

Hˆ (− ψ) = (E − ℏω)(− ψ)                                (6)

3) Given two arbitrary wavefunctions f(x) and g(x), show that − satisfies

⟨ − f|g⟩ = ⟨f|+g⟩                                                    (7)

4) Given an eigenfunction of the Hamiltonian with quantum number n , show that −   satisfies

− |n⟩ =^n|n − 1⟩ (8)

5) Use ladder operators and show that the expectation value of the Kinetic operator satisfies

⟨n|Kˆ |n⟩ = En = ℏω (n + )                           (9)

3. Multi-dimensional harmonic oscillator

The Hamiltonian of a 3-dimensional harmonic oscillator is defined as

Hˆ = − − − + (x1 − 2)2 + x 2(2) − x2x3 + 2x3(2)          (10)

Find its energy levels as a function of its quantum numbers. (Hint: Transform the Hamiltonian first into displacement coordinates, then into mass weighted coordinates, and finally into diagonalized coordinates).

4. Franck-Condon factor

Follow the setup of diatomic molecule in the lecture, the normal modes on the ground state qg  and excited state qe are related as qg  = qe +d; the vibrational frequency both on the ground and excited states is ω . The mass along normal mode is m = 1.

(1) Compute the overlap integral ⟨0g |0e ⟩ as a function of ℏ , ω and d. Indicate how the size of the overlap integral changes as d increases.

(2) Find the relation between the two lowering operators and .

(3) Find the recurrence relation of ⟨ng |0e ⟩ .

(4) Suppose ⟨0g |0e ⟩  = 0.01, and use the zero-temperature approximation. Compute the values of Franck-Condon factors ⟨0g |ne ⟩ for 1 ≤ ne  ≤  5, and use these values to draw the stick spectrum for vibronic absorptions 0g  → ne where 0 ≤ ne  ≤ 5. (You can draw by hand.)

(5) Following (4), compute the valuse of Franck-Condon factors ⟨ng |0e ⟩ for 1 ≤  ng   ≤  5, and use these values to draw the stick spectrum for vibronic emissions 0e  → ng  where 0 ≤ ng  ≤ 5. Based on what you’ve drawn, how are the absorption and emission spectrum related to each other?


2023-02-09