Title: ETH_Sem4 / Advanced_Kinetics / Exercises

Q: 1. Starting from \(\hat \Delta_{mn} = \delta_{mn} \hat T_N - \bra{\varphi_m} \hat T_N \ket{\varphi_n}\), derive the following expression\[\hat \Delta_{mn} = \frac{1}{2M} \bigg[2 \hat F_{mn} \cdot \hat \nabla + \hat G_{mn} \bigg]\]where\[\hat F_{mn} = \braket{\varphi_m|\nabla \varphi_n}\]
\[\hat G_{mn} = \braket{\varphi_m|\nabla^2 \varphi_n}\][...]
2. Prove that \(\hat F\) is anti-Hermitian (Hint: consider \(\nabla(\braket{\varphi_m|\varphi_n})\).
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3. What is the physical effect that non-zero values of \(\hat \Delta_{mn}\) have on the dynamics?
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A: \[\hat \Delta_{mn} = \delta_{mn} \hat T_N - \bra{\varphi_m} \hat T_N \ket{\varphi_n}\]\[= -\delta_{mn} \sum_i \frac{\nabla^2_{N,i} }{2M_i} + \bra{\varphi_m}\sum_i{\frac{\nabla^2_{N,i} }{2M_i} } \ket{\varphi_n}\]\[= \frac{1}{2M} \bigg[\braket{\varphi_m|\varphi_n}(-\nabla^2_{N, i}) + \hat G_{mn} \bigg]\]\[= \frac{1}{2M} \bigg[2 \braket{\varphi_m \nabla_{N, i} \varphi_n} \cdot \hat \nabla + \hat G_{mn} \bigg]\]
\[ = \frac{1}{2M} \bigg[2 \hat F_{mn} \cdot \hat \nabla + \hat G_{mn} \bigg]\]; \[0 = \nabla (\braket{\varphi_m | \varphi_n})\]\[= \braket{\nabla \varphi_m | \varphi_n} + \braket{\varphi_m | \nabla \varphi_n}\]\[= \braket{\varphi_n | \nabla \varphi_m}^{\ast} + \braket{\varphi_m | \nabla \varphi_n}\]\[= \hat F^{\ast}_{nm} + \hat F_{mn}\]; The physical effect is that there is some force of interaction between the particles, deteremined by their kinetic energies.

Breakdown of Born-Oppenheimer Approximation
1. Apply the \(\nabla\) operator (differentiation w.r.t. \(\vec R\)) to both sides of the electronic Schrödinger equation: \(\hat H_e(\vec R) \ket{\varphi_n (\vec R)} = V_n(\vec R) \ket{\varphi_n(\vec R)}\)

2. Rearrange this equation to find an expression for \(\hat F_{mn} = \braket{\varphi_m | \nabla \varphi_n}\) in terms of the expectation value \(\bra{\varphi_m}(\nabla \hat H_e) \ket{\varphi_n}\)

3. Unter what circumstances can \(\hat F_{mn}\) not be neglected?
\(\hat F_{mn}\) cannot be neglected for \(n = m\) or if \(V_n = V_m\).

Q: Conical Intersections
Consider the diabatic potential matrix
\[U = \bigg(\matrix{U_{11}(\vec R) & U_{12}(\vec R) \\ U_{21}(\vec R) & U_{22}(\vec R)}\bigg), \text{ with } U_{12}(\vec R) = U_{21}^{\ast}(\vec R)\]
1. Diagonalize this matrix to find the adiabatic potential-energy surfaces, \(V_{\pm} (\vec R)\)
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2. Under what circumstances can the two adiabatic states be equal?
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3. In the simplest 2D case \(U_{11} = ax, U_{22} = -ax\) and \(U_{12} = U_{21} = by\). Describe/sketch the shape of the adiabatic surfaces.
3. What is the eigenvector corresponding to the lower surface and how does this change on moving in a circle around the conical intersection?
A: Convert to polar coordinates \(ax = r \cos \theta\) and \(by = r \sin \theta\):\[U = \bigg(\matrix{r \cos \theta & r \sin \theta \\ r \sin \theta & -r \cos \theta}\bigg)\]Eigenvalues are \(\pm r\); The lower (Born-Oppenheimer) state has the eigenvector \(\bigg(\matrix{\sin \frac{\theta}{2} \\ -\cos \frac{\theta}{2} }\bigg)\). The total electronic wavefunction \(\phi = c_1 \tilde \varphi_1 + c_2 \tilde \varphi_2\), thus changes sign when \(\theta\) goes from \(0\) to \(2 \pi\).

Q: Time-dependent Hamiltonian
Consider the following ansatz for the time-dependent wavefunction
\[\Psi(t) \stackrel{?}{=} e^{-i \int_0^t H(t') dt'/\hbar} \Psi(0)\]1. Write down series expansion for the matrix exponential.

2. Show why this expression cannot be valid as it violates causality.
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3. When would it be OK?
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4. Can you suggest a way to fix it?
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A: Because from the second (double) integral on we could have \(t'' \gt t'\) implying the future could affect the past, which contradicts the causality.; If the hamiltonians commute with eachother there is no problem (e.g. for time independent hamiltonians).; We can use the time order operator:
.

Hamiltonian in the interation picture
Given the time-dependent Hamiltonian \(\hat H(t) = \hat H_0 + \hat H_{\lambda}(t)\), in the interaction
picture the wavefunction is \(\ket{\Psi_I (t)} = \hat U_I(t, t_0) \ket{\Psi}\) where
\[\hat U_I(t, t_0) = \hat U_0(t,t_0)^{\dagger}\hat U(t,t_0)\]1. Rearrange for \(\hat U(t,t_0)\) and substitute this into the TDSE (with the total Hamiltonian \(\hat H(t)\)) to prove the following:
\[i \hbar \dede{\hat U_I (t,t_0)}{t} = \hat H_I(t) \hat U_I(t,t_0)\]
2. Hence find an expression for \(\hat H_I(t)\).

3. Show that the above TDSE is solved by:

Time-independent spectrum
1. Expand the dipole-dipole correlation function \(C_{\mu \mu}(t) = \tr [\hat \rho_{eq} \hat \mu \hat \mu(t)]\) in the basis of
vibrational eigenstates.

2. Perform the Fourier cosine transform and interpret the expression you get in terms of transition dipole moments, as introduced in previous courses.

Q: Rabi oscillations
where \(\omega_{ab} = \omega_b - \omega_a\)
This can be solved to give the amplitude of the excited state as
where \(\Omega = \sqrt{\Delta \omega^2 + \mu^2 \varepsilon^2/\hbar^2}\) and \(\Delta \omega = \omega - (\omega_b - \omega_a)\).
1. Sketch the population of the excited state against time for various values of \(\Delta \omega\).
Hint: consider the short-time limit carefully. What is the behaviour with \(t\)? How is
this affected by changes to \(\Delta \omega\)?
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2. Design an experimental setup to achieve almost 100% population conversion.
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3. Draw a graph.
A: Population is given by \(|b(t)|^2 = \bigg(\frac{\mu \varepsilon}{\hbar \Omega} \sin(\Omega t / 2) \bigg)^2\).This means we get a sinus-type function where larger \(|\Delta \omega|\) leads to larger \(\Omega\) which leads to smaller amplitude and higher frequency.; To achieve almost 100% population conversion we need to get \(\Delta \omega \approx 0\), meaning \(\omega \approx \omega_b - \omega_a\) being the energy difference between the two states.

Q: Converting between position and momentum representations
Position state in momentum representation is \(\braket{p | x} = [...]\).
Momentum state in position representation is \(\braket{x | p} = [...]\).
1. Make a sketch of a position and momentum state both in the position representation.

2. Explain why this is in agreement with the Heisenberg uncertainty principle?
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3. Show that the normalization is correct such that \(\braket{x'|x''} = \delta(x'-x'').\) Use the fact that \(\int e^{i\omega t} dt = 2 \pi \delta(\omega)\).
A: (2\pi \hbar)^{-1/2} \exp(-ipx/\hbar); (2\pi\hbar)^{-1/2} \exp(+ipx/\hbar); This is in agreement with the Heisenberg uncertainty principle because as x is localized, p is completely delocalized, and if p were localized, x would be completely delocalized.

Q: Gaussian wavepackets
A typical wavepacket looks like:
\(\braket{x | \psi(t_0)} = [...]\)
1. Compute \(\bra{\psi(t_0)} \hat x \ket{\psi(t_0)}\) and \(\bra{\psi(t_0)} \hat p \ket{\psi(t_0)}\).

2. What is the meaning of \(a\) and \(k_0\)?

3. Is the wavepacket an eigenstate of the free-particle Hamiltonian? Evaluate \(\bra{x}\hat H \ket{\psi(t_0)}\) to find out.
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A: \exp(-\gamma(x-a)^2/2 + ik_0(x-a)+c); No!

Trotter splitting
1. Take Taylor expansions (in \(\varepsilon\)) of Eq. 2 and Eq. 3 and confirm that the error is of order \(\varepsilon^2\).

2. Do the same for and confirm that the error is of order \(\varepsilon^3\).

Q: Fermi’s golden rule
Consider the time-dependent amplitude of state \(\ket{n}:\)

1. Write the formula for time-dependent population \(P_{m\to n}(t) = |b_n(t)|^2\).

2. Show that it is equal to \(P_{m \to n} (t) = \frac{\Delta^2_{nm} }{\hbar^2} \frac{4 \sin^2(\Omega_{nm} t/2)}{\Omega_{nm}^2}\).

3. Plot the resulting function against \(\Omega_{nm}\) for short and long times.

4. So, in the long-time limit, what transitions are allowed?
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A: In the long time limit we get a dirac-\(\delta\) function \(\implies\) Only transitions with \(\Omega_{nm} = 0 \implies E_n = E_m\) are allowed.

Floquet theory
1. What are typical expressions for \(\hat H^{(0)}, \hat H^{(1)}, \hat H^{(-1)}, \hat H^{(2)}, \dots\)?

2. Write Eq. (1) in matrix form and interpret the dynamics which will result.

Q: Preparation of an excited electronic state
1. Insert the Gaussian pulse into the time-dependent pertubtation theory expression
above.

2. Perform the integral over \(t'\), assuming that \(t_0 \ll t_p \ll t\) and the pulse width is much smaller that the total experimental time. Hint: \(\int \exp(-a x^2 / 2 + bx) dx = \sqrt{2\pi/a} \exp(b^2/2a)\).

3. Consider which vibration states \(\ket{\chi_{en} }\) will be excited by a pulse (consider short and long pulse limits).

4. Is this in agreement with what you know from the uncertainty principle?
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A: Yes, we have a time-energy uncertainty (balance between knowing when it happened and what energy).

Flux operator
The general definition for the flux operator is
1. Show that for our standard Hamiltonian in Cartesian coordinates and \(P = \theta(\hat x - s)\):

Hint: first expand the operator expression \(\hat p \theta(\hat x -s)\)

Reactive scattering theory
Use \(\bra{p_a} \hat F \ket{p_b} = \frac{1}{2\pi\hbar} \frac{p_1}{m}\) if \(p_a = p_b\) but 0 if \(p_a = -p_b\).
1. Evaluate the flux \(\bra{\psi_b} \hat F \ket{\psi_b}\) at a value of \(s\) in the product asymptotic region where \(\ket{\psi_b} \sim T(E) \ket{p}\).

2. Evaluate the flux \(\bra{\psi_b} \hat F \ket{\psi_b}\) at a value of \(s\) in the reactant asymptotic region where \(\ket{\psi_b} \sim \ket p + R(E) \ket{-p}\).

3. Flux must be equal everywhere, so what relationship can you deduce between \(R(E)\)
and \(T (E)\)? How can we interpret these values?

1D classical rate
1. What is the functional form of \(P_{cl}(E)\) for classical mechanics?

2. Sketch this and also how you think \(P (E)\) looks for quantum mechanics.

3. Use
to derive a formula for the classical rate for a one-dimensional system.

4. Compare with the result of TST.

5. Draw an Arrhenius plot for the classical rate that you have derived.

6. Add a sketch of how you think the quantum rate will behave.

Q: 1D quantum rate: Eckart barrier
1. Sketch \(P(E)\) for a fat barrier (large \(a\)).
2. Sketch \(P(E)\) for a thin barrier (small \(a\)).

3. For a given potential-energy surface, is tunnelling favoured for heavy or light masses?
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A: Tunnelling is favoured for light masses because \(\alpha \propto \sqrt m\) and \(W(E) \propto \alpha\) and \(P(E)\) decreases with larger \(W\).

Q: Diffusion
1. Use linear response theory to rewrite \(D = -\frac{\abk{B(t)}_{\lambda} }{\beta \lambda}\) as \(D = - \abk{\Delta x(0) \dot x(t)}\) in 1D.

2. Justify the two alternative expressions:

3. Justify that in 3D they are written as:
which is Einstein’s famous result connecting microscopic dynamics to macroscopic observables.
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A: To go to 3D just divide by 3 and replace scalars by vectors.

Linear Response Theory
1. Expand \(\abk{B(t)}_{\lambda}\) to first order in \(\lambda\).

2. Hence show that

Rate theory
1. Use the detailed-balance condition \(\frac{\abk B}{\abk A} = \frac{k_+}{k_-}\) and \(A+B = 1\) to show that

2. Use the linear response \(\abk{\Delta B(t)}_{\lambda} = \abk{\Delta A(0) \Delta B(t)}\) and rearrange to get

3. Finally, make the approximation (valid for slow reactions) that \(\abk{\Delta A(0) \Delta B(t)} = \abk{A(0) B(t)} - \abk A \abk B \approx - \abk A \abk B\) to obtain

Q: Recrossing
Consider the three types of trajectory, each starting from
the dividing surface.
1. What contribution does each make to the short-time limit?
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2. What contribution does each make to the long-time limit?
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3. Sketch the form of \(C_{fs}(t)\).

4. Are there any other types of trajectory which contribute?
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A: In the short time limit (1) and (2) look equal and seem to convert fully to product, whereas (3) goes only to the left and goes back to reactant.; In the long time limit, (1) and (3) go to product and (2) to reactant.; There could also be multiple numbers of recrossings, but the probability for that decreases vastly.

Q: Quantum correlation function
For a free particle, \(H = \hat p^2/2m\), it is possible to find an analytic expression for the flux-flux correlation function:\[C_{ff}(t) = \frac{1}{2\pi \beta \hbar} \frac{(\hbar \beta/2)^2}{(t^2+(\hbar\beta/2)^2)^{3/2} }\]
1. Plot \(C_{ff}(t), C_{fs}(t) = \int_0^t C_{ff}(t') dt'\) and \(C_{ss}(t) = \int_0^t C_{fs} (t') dt'\).
 Sketch the equivalent functions from classical mechanics.
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3. What would be the prediction of a quantum-mechanical TST?
4. The integral of this function is \(\int_0^{\infty} C_{ff}(t) = 1/(2\pi\beta\hbar)\). What is therefore the relation
between the classical and quantum rate in this case?
A: Quantum smooths out the correlation function, classical is discontinuous.

Q: System-bath model
1. Confirm that this solves the equations of motion and has the correct initial conditions.

2. Integrate by parts to show that it can be written equivalently as

3. Substitute into \(\dot p = -V'(x) + \sum_j c_j \bigg(q_j - \frac{c_j}{\omega^2_j} x \bigg)\) to show that \(\dot p = -V'(x) - \int_0^t \frac{p(t')}{m} K(t-t') dt' + R(t)\) and hence find expressions for \(K(t-t') \) and \(R(t)\) and interpret them. Hint: think about friction forces.
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A: \(K(t-t')\) is called the memory kernel, because it depends on what has been happening in the past.
\(R(t)\) is called the random forse, because the coefficients \(c_j\) are chose by a Boltzmann distribution.

Q: Langevin dynamics
1. Assume the bath is in thermal equilibrium (for a given value of \(x(0)\)). What is the distribution of the coordinates and momenta of the bath modes?

2. Show that:
Hint: You can either explicitly compute each of the Gaussian integrals, or use symmetry and the equipartition theorem to justify them.
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3. Thus find an expression for \(\abk{R(t)}\) and \(\abk{R(t)R(t')}\).
A: No correlation between \(p_j(0)\) and \(q_j(0)\) \(\implies\)The expectation value is \(0\). The other expectation values are easily found computing the gaussian integrals.

Q: Kramers theory
1. Sketch a contour plot for the 2D PES for the three values of coupling:
\(\qquad c=0\), small, but non-zero \(c\), large \(c\)
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2. Which will have the most recrossing in the \(x\)-coordinate?
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3. Which will have the fastest rate, and why?
4. Write down the Hessian matrix, \(H\), and evaluate \(\det H\).
5. Find a formula for the eigenvalues \(\lambda_0\) and \(\lambda_1\).
A: For \(c=0\) we have a harmonic oscillator PES. For \(c\) small it's slightly shifted and for \(c \) large there is only a narrow valley:; There will be more recrossing if \(c\) gets larger.

Electron-transfer theory
1. Find an expression for \(Z_0 = \tr \bigg[ \exp(-\beta \hat H) \ket 0 \bra 0 \bigg]\) to lowest order in \(\Delta\).

2. Find an expression for \(C_{ff}(t) = \tr \bigg[ \exp(-\beta \hat H) \hat F \exp(i \hat H t / \hbar) \hat F \exp(-i \hat H t /\hbar) \bigg]\) to lowest order in \(\Delta\).

Electron-transfer theory in the golden-rule limit
1. Expand the trace using the eigenstate basis, defined such that \(\hat H_0 \ket{\psi_0^{\mu} } = E_0^{\mu} \ket{\psi_0^{\mu} }\) and \(\hat H_1 \ket{\psi_1^{\nu} } = E_1^{\nu} \ket{\psi_1^{\nu} }\).
Hint: Insert complete sets of states \(1 = \sum_{\mu} \ket{\psi_0^{\mu} } \bra{\psi_0^{\mu} }\) and \(1 = \sum_{\nu} \ket{\psi_1^{\nu} } \bra{\psi_1^{\nu} }\) and show that this is equivalent to the Fermi’s golden-rule formula derived in a previous lecture.

Marcus theory
1. Find a formula for the activation energy \(V^{\ddagger} = V_0(x^{\ddagger})\) where \(x^{\ddagger} = -\varepsilon / 2M\Omega^2\xi\).

2. Show that it can be written in terms of the reorganization energy \(\Lambda = 2M \Omega^2 \xi^2\) as \(V^{\ddagger} = (\Lambda - \varepsilon)^2/4\Lambda\).

3. Describe how the rate depends on \(\varepsilon\) for a fixed value of \(\Lambda\).

4. Make a plot of \(\log k_{MT}\) against \(\varepsilon\) and annotate it.

5. Explain the behaviour by sketching the potential-energy surfaces in different cases.

Q: Landau–Zener–(Stuckelberg) theory
1. Assume that the particle moves at a constant velocity. Express the transition probability in terms of this velocity, \(v\), and the forces \(F_0\) and \(F_1\).

2. Plot the transition probability as a function of \(\Delta\). Find approximate expressions in the weak coupling and strong coupling limits.

3. What is the adiabatic transition probability? Does this behave as you would expect from what we learned in previous lectures?
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A: The adiabatic transition is almost guaranteed not to happen (probability approx. 0). On the contrary to that the diabatic transition is almost guaranteed to happen. This is in agreement with what was found in prev. lectures.

Q: Density of states for a harmonic oscillator
1. Show that the density of states of a classical one-dimensional harmonic oscillator is \(\rho_{cl}(E) = \frac{1}{\hbar\omega}\).

2. Plot the densities of states, \(\rho(E)\), and number of states, \(N(E) = \int_0^E \rho(E') dE'\), for both the quantum and classical H.O.

3. Use the convolution formula to find the density of states of 2 independent classical harmonic oscillators.

4. Hence show that the density of states of \(f\) independent classical harmonic oscillators is:\[\rho(E) = \frac{E^{f-1} }{(f-1)! \prod_{j=1}^f \hbar \omega_j}\][...]
5. Find a very rough estimate for the density of states for a single methane molecule (neglecting translations/rotations) at \(E = 10000\)/cm and \(E = 45000\)/cm.
A: This follows directly from using the formula above and recursion.

Q: RRKM rate theory
where
1. Typically one approximates the reactant and perpendicular modes at the transition state by quantum harmonic oscillators. Write an expression for \(N^{\ddagger} (E)\) in this case.
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2. Sketch how \(N^{\ddagger}(E)\) and the RRKM rate depend on energy. Consider the case of a small molecule and a large molecule.

3, How would this change if everything was classical?

4. How would this change if tunnelling were allwed?
A: \(E_n = \hbar\omega(n+1/2)\)

Classical harmonic oscillators
Using \[\int_0^{\infty} x^{f-1} e^{-x} dx = (f-1)!\]1. Compute the partition function by evaluating the integral:\[Q_r(\beta) = \int_0^{\infty} \rho_r(E) e^{-\beta E} dE\] where \[\rho_r(E) = \frac{E^{f-1} }{(f-1)! \prod^f_{j=1} \hbar\omega_j}\]
2. Evaluate the thermal equilibrium probability distribution\[P_{eq} (E) = \rho_r(E) \frac{e^{-\beta E} }{Q_r(\beta)}\]
3. Plot this function against \(E\) for the case of high-temperature and low-temperature.

Q: Connection to classical mechanics
1. For the case that \(\mathscr L = \frac 1 2 m \dot x ^2 - V(x)\), show that the Euler-Lagrange equation is equivalent to Newtonian mechanics:\[\dd{}{t} \bigg(\dede{\mathscr L}{\dot x} \bigg) = \dede{\mathscr L}{x}\][...]
2. The discretized action is given by\[S_N(x_0,x_1, \dots, x_{N-1},x_N) = \sum_{n=0}^{N-1} \frac{m(x_{n+1}-x_n)^2}{2\varepsilon} - \frac{\varepsilon}{2} V(x_0) - \sum_{n=1}^{N-1} \varepsilon V(x_n) - \frac{\varepsilon}{2} V(x_N)\]Show that paths for which the action is stationary w.r.t. \(x_1, \dots, x_{N-1}\) obey Newtonian mechanics in the \(\varepsilon \to 0\) limit.
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A: Because all the paths following \(F = ma\) are constructively interfering, we see the world as if all was according to classical mechanics.; Which works the same way for all \(i \in \set{1,\dots,N-1}\)

Q: Classical limit
1. For the Hamiltonian \(H = \frac{p^2}{2m} + V(x)\), write an expression for \(\rho(x_0, x_0)\) in terms of \(\beta\) for the simplest (approximate) case \(N=1\).

2. Hence, write an expression for the partition function and an equilibrium average, \(\abk{\hat A}\), within this approximation.

3. Show that these are equivalent to the classical results discussed in the first lecture.
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A: These are equivalent to the classical result from the beginning of the course:

Cross-over temperature
In this exercise, you will estimate the temperature below which tunnelling becomes a dominant effect on the rate.
1. Write the Hessian for the ring polymer when it is collapsed at the barrier top, i.e. \(x_n = x^{\ddagger}\).

By analogy with the Hückel formula for a cyclic alkene (\(E_k = \alpha + 2 \bar \beta \cos \frac{2k\pi}{N}\) for \(k = -N/2, \dots, +N/2\), where \(\alpha = 2m/\beta_N^2\hbar^2 - m \omega_b^2\) and \(\bar \beta = - m/\beta_N^2\hbar^2\)) we find that the lowest eigenvalues of the ring-polymer Hessian are:
where \(\omega_b\) is the imaginary frequency of the potential \(V(x)\) at the barrier top \(x = x^{\ddagger}\).
2. Take the limit \(N\to\infty\) and determine the temperature at which the number of negative eigenvalues changes from one to three.

3. For what range of \(\omega_b\) is tunnelling important at room temperature?