About Huang-Rhys parameter and electron-phonon coupling

[HAYAO]

Hello all, I am confused about Huang-Rhys parameter and electron-phonon coupling.

I read a chapter of a book The biophysics of photosynthesis, called "Electron–Phonon and Exciton–Phonon Coupling in Light Harvesting, Insights from Line-Narrowing Spectroscopies".

I am confused about the following part (Line 27 of Page 55):

The Huang–Rhys factor is associated with the displacement of the equilibrium positions of the nuclei upon a photoexcitation of the chromophore as illustrated in Fig. 2.2. Therefore, S (Huang-Rhys factor) is a measure for the strength of the linear electron–phonon coupling and characterizes the average number of phonons accompanying a particular
electronic transition.

If I understand correctly, Huang-Rhys parameter in the regime of Born-Oppenheimer approximation and Franck-Condon approximation, basically shows the equilibrium configuration displacement between initial and final state of a transition. Since Born-Oppenheimer approximation ignores the non-adiabatic part of the Hamiltonian, under this approximation, electron-phonon coupling is ignored. If electron-phonon coupling is to be considered, then it needs to be accounted for by considering the non-adiabatic part of the Hamiltonian in which there are several techniques available. And it is precisely these method that we can actually figure out how strong the electron-phonon coupling is.

So then how can the Huang-Rhys parameter in the regime of Born-Oppenheimer approx. shows the strength of electron-phonon coupling? Am I fundamentally missing something here?

 

[DrDu]

I think this refers to the diabatic reppresentation rather than to the BO-adiabatic one.

 

[TeethWhitener]

It’s still Born-Oppenheimer. This site:
https://ocw.mit.edu/courses/chemist...pring-2009/lecture-notes/MIT5_74s09_lec08.pdf
gives a good overview.

A lot of concepts are getting jumbled up here when precision is required. BO says that the electronic and nuclear Schrodinger equations are separable: the total wave function can be represented as a product of electronic and vibrational, with the electronic depending only parametrically on the nuclear coordinates:
$$\Psi (\mathbf{r,R})= \psi_e (\mathbf{r;R}) \psi_N (\mathbf{R})$$
The absorption coefficient is going to be proportional to the overlap of the excited state wavefunction with the dipole operator acting on the ground state:
$$A \propto \langle \Psi_E | \mu | \Psi_G \rangle$$
A further approximation (Condon) states that the dipole operator only acts on electronic states. In other words, there is no nuclear displacement due to the action of the dipole operator (vertical transition). However, in many real systems, the nuclear coordinates of the potential energy surface of the excited electronic state are displaced from those of the ground state potential energy surface. The greater this displacement, the smaller the integral representing the absorption coefficient will be (see link for detailed calculation for HO PES’s). The Huang-Rhys parameter is simply a measure of this “misalignment” between PES’s.

Edit: in terms of “electron-phonon coupling,” the Huang-Rhys parameter says that the vibrational states on the excited PES with nuclear coordinates most closely matching the ground state wavefunction will have a proportionally larger absorption coefficient than vibrational states further away.

 

[HAYAO]

Thank you DrDu and TeethWhitener.

Like TeethWhitener said, the above chapter discusses Huang-Rhys parameter under BO approximation, and that I thought Huang-Rhys parameter is indeed the "misalignment". At least that is how I understood.

However, this parameter doesn't seem to be linked with electron-phonon coupling. That is, I thought that the term "electron-phonon coupling" refers to the "mixture" (for lack of better words, someone please tell me the correct term) of electronic and vibrational wavefunctions, so that it cannot be separated into product like in the BO-approximation. That is, there is this electron-phonon coupling Hamiltonian that contributes to the off-diagonal matrix elements that mixes vibrational wavefunctions into the electronic wavefunctions. In another words, nonadiabatic. I believe this is the source of internal conversion. (Of course this is difficult to calculate. So one can try perturbation approach with Herzberg-Teller expansion.)

I can understand that the overlap between vibrational wavefunction are important in electronic transition. I can also understand well that Franck-Condon coefficients can be (at zero temperature to be precise) given by equation 6.38 of the link TeethWhitener provided. However, Huang-Rhys parameter only shows the displacement, not the electron-phonon coupling itself. In fact, the Franck-Condon factor has nothing to do with electron-phonon coupling. It is derived from the BO approximation, which ignores electron-phonon coupling. Therefore, it only has to do with vibrational wavefunction overlap, not the degree of mixture of vibrational and electronic wavefunction.I think I am starting to figure that both the vibrational wavefunction overlap AND mixture of vibrational and electronic wavefuction are confusingly termed as "electron-phonon coupling". Could this ambiguity possibly be the confusion I am having here?

 

[TeethWhitener]

I think I am starting to figure that both the vibrational wavefunction overlap AND mixture of vibrational and electronic wavefuction are confusingly termed as "electron-phonon coupling". Could this ambiguity possibly be the confusion I am having here?

Yes it might be better if you let us know what you mean precisely when you say electron-phonon coupling. Like I said, absorption is given by:
$$A\propto \langle \Psi’| \mu | \Psi\rangle$$
BO approximation let's us rewrite this as:
$$A \propto \langle \psi_e’ | \langle \psi_N’|\mu |\psi_N \rangle | \psi_e\rangle$$
And Condon let's us write:
$$A \propto \langle\psi_N’|\psi_N\rangle\langle \psi_e’| \mu |\psi_e\rangle$$

As for “off-diagonal,” I’m not sure what exactly you mean, but I will point out that even if you had a diagonal Hamiltonian, the dipole operator mixes eigenfunctions. (This is the whole reason light can cause transitions in the semi-classical approximation.)

 

[HAYAO]

Yes it might be better if you let us know what you mean precisely when you say electron-phonon coupling. Like I said, absorption is given by:
$$A\propto \langle \Psi’| \mu | \Psi\rangle$$
BO approximation let's us rewrite this as:
$$A \propto \langle \psi_e’ | \langle \psi_N’|\mu |\psi_N \rangle | \psi_e\rangle$$
And Condon let's us write:
$$A \propto \langle\psi_N’|\psi_N\rangle\langle \psi_e’| \mu |\psi_e\rangle$$

As for “off-diagonal,” I’m not sure what exactly you mean, but I will point out that even if you had a diagonal Hamiltonian, the dipole operator mixes eigenfunctions. (This is the whole reason light can cause transitions in the semi-classical approximation.)

Yes, I know very much about those concepts and equations. I have no problem with it whatsoever. I am not quite talking about the dipole operator (or actually the Hamiltonian of the interaction of the molecule with the electromagnetic field), but the electron-phonon Hamiltonian.

The chapter essentially says that Huang-Rhys parameter IS the electron-phonon coupling parameter. However, electron-phonon coupling arises from electron-phonon interaction Hamiltonian, which has a off-diagonal matrix element that becomes hard to handle in most practical cases. For example, if you have two electronic states that have close energies (for example conical intersection or avoided intersections), then the off-diagonal matrix element is somewhat too large to be ignored. In such cases, you can start off from adiabatic case and use diabatic transformation. This has nothing to do with Huang-Rhys parameter, which only has a well-described definition in the realm of BO approximation. Huang-Rhys parameter does not provide how large the off-diagonal matrix elements of the electron-phonon coupling Hamiltonian are.

So then either 1) the term "electron-phonon coupling" is ambiguous that could refer to the Huang-Rhys parameter or the extent of electron-phonon coupling strength determined by electron-phonon coupling Hamiltonian, depending on the context of the discussion, OR 2) there is some fundamental mistake that I am making that we haven't figured out yet.

 

[TeethWhitener]

I am not quite talking about the dipole operator (or actually the Hamiltonian of the interaction of the molecule with the electromagnetic field), but the electron-phonon Hamiltonian.

But the dipole operator is required to couple the electronic and vibrational modes. Keep in mind that the vibrational eigenfunctions for the ground electronic state are not, in general, vibrational eigenfunctions for the excited electronic state. This is why the Franck-Condon factor $\langle\psi_N’|\psi_N\rangle$ (where the prime denotes that the wavefunction is on a different PES) is non-zero for electronic transitions.

1) the term "electron-phonon coupling" is ambiguous that could refer to the Huang-Rhys parameter or the extent of electron-phonon coupling strength determined by electron-phonon coupling Hamiltonian, depending on the context of the discussion

I think maybe you're using "electron phonon coupling" to refer to the correction to the time independent Schrodinger equation. So the BO Schrodinger equation is:
$$(H_{el} + H_{ph})\psi_{el}\psi_{ph} =(E_{el} + E_{ph})\psi_{el}\psi_{ph}$$
and when you add a $H_{el-ph}$ correction term to this, you get off-diagonal terms such that the equation is no longer separable into nuclear and electronic coordinates. In which case, you're right, BO does not hold. Whereas the Huang-Rhys parameter has to do with the time-dependent Schrodinger equation where the Born-Oppenheimer approximation does hold:
$$i\hbar\frac{\partial\psi_{el}(\mathbf{r;R},t)\psi_{ph}(\mathbf{R},t)}{\partial t} = (H_0 + V(t))\psi_{el}(\mathbf{r;R},t)\psi_{ph}(\mathbf{R},t)$$
where $V(t)$ is the perturbing radiation field $V(t) \propto e^{i\omega t} \mu$. The $V(t)$ interaction is the only term which mixes up the electronic and vibrational wavefunctions. Without it there is no electron-phonon coupling in this picture.

 

[DrDu]

As I already mentioned, the derivation of the Huang Rhys factor is based on a diabatic or crude adiabatic approximation. This means that the Hamiltonian is expressed in a basis of electronic states which do not depend on position of the nuclei but all refer to the same equilibrium position of the ground state. In this approximation, there won't be any nuclear momentum dependent non-adiabatic couplings but there will be couplings linear in the nuclear displacement. Namely, if the electronic states are $|i\rangle$, $|j\rangle$ ..., then the operator for the nuclear motin becomes $T_n +\sum_{ij} (e_i \delta_{ij} +((\partial /\partial R)\langle i| H_\mathrm{el}| j\rangle ) (R-R_0) +C_{ij} (R-R_0)^2+\ldots$. The part linear in $R-R_0$ describes the coupling of the electronic and nuclear degrees of freedom. Even if the potential energy part of the hamiltonian is diagonalised to obtain the usuall BO-hamiltonian, this coupling will determine the shift of the equilibrium postions of the excited states relative to the ground state.

 

[HAYAO]

Due to the lack of ability on my side, I have a hard time understanding.

I think maybe you're using "electron phonon coupling" to refer to the correction to the time independent Schrodinger equation. So the BO Schrodinger equation is:
$$(H_{el} + H_{ph})\psi_{el}\psi_{ph} =(E_{el} + E_{ph})\psi_{el}\psi_{ph}$$
and when you add a $H_{el-ph}$ correction term to this, you get off-diagonal terms such that the equation is no longer separable into nuclear and electronic coordinates. In which case, you're right, BO does not hold. Whereas the Huang-Rhys parameter has to do with the time-dependent Schrodinger equation where the Born-Oppenheimer approximation does hold:
$$i\hbar\frac{\partial\psi_{el}(\mathbf{r;R},t)\psi_{ph}(\mathbf{R},t)}{\partial t} = (H_0 + V(t))\psi_{el}(\mathbf{r;R},t)\psi_{ph}(\mathbf{R},t)$$
where $V(t)$ is the perturbing radiation field $V(t) \propto e^{i\omega t} \mu$. The $V(t)$ interaction is the only term which mixes up the electronic and vibrational wavefunctions. Without it there is no electron-phonon coupling in this picture.

But the dipole operator is required to couple the electronic and vibrational modes. Keep in mind that the vibrational eigenfunctions for the ground electronic state are not, in general, vibrational eigenfunctions for the excited electronic state. This is why the Franck-Condon factor $\langle\psi_N’|\psi_N\rangle$ (where the prime denotes that the wavefunction is on a different PES) is non-zero for electronic transitions.

I am getting confused now. I am starting to get the feeling that we aren't talking about the same thing. Tell me if I am understanding this right.

The Hamiltonian of the time-independent Schrodinger equation for an entire molecule is:
$H(\textbf{r},\textbf{R}) = T(\textbf{r},\textbf{R}) + V(\textbf{r},\textbf{R})$　　　　　　　　　　　　　　　　[Eq.1]
where
$T(\textbf{r},\textbf{R}) = -\sum _{I}\frac{\hbar}{2M_{I}}\nabla_{I}^{2} -\sum _{i}\frac{\hbar}{2m_{e}}\nabla_{i}^{2}$　　　　　　　　　　　　[Eq.2]
$V(\textbf{r},\textbf{R})=\frac{e^{2}}{4\pi \varepsilon _{0}}\left ( -\sum _{I>I'}\frac{Z_{I}Z_{I'}}{R_{I,I'}}-\sum _{I,i}\frac{Z_{I}}{r_{I,i}}+\frac{1}{2}\sum _{i>i'}\frac{1}{r_{i,i'}} \right )$　　　[Eq.3]
I believe that in Born-Oppenheimer approximation, you take out the nuclear kinetic energy operator (first term of the right side of [Eq.2]) for the moment for calculating the electronic part of the wavefunction under the assumption that it does not contribute to the electronic part. And then we come back after solving the Schrodinger equation for the electronic part, and solve the Schrodinger equation for the nuclear motion.

BO approximation assumes that the off-diagonal matrix element of the kinetic energy operator of nuclei is zero, and that only the diagonal part survives (Born-Huang approx.), which is further approximated to contain terms that only depend on the nuclear positions and not electronic. It is precisely this assumption that the off-diagonal matrix element is zero that the electronic and vibronic wavefunctions remain separated and that the entire wavefunction can then be expressed as the product of the two. However, these off-diagonal elements are not truly negligible especially when two PES closes in energy, and it contributes to the nonradiative decay process. The energy gap law is based on this framework and thus so is Kasha's rule. So I was considering this as the fundamental electron-phonon coupling. Now I am starting to understand that this was an incorrect nomenclature, so that is my fault. I would like to know how this type of interaction is actually called.Now, I believe what you are talking about in your latter half of the post is about taking into account the perturbing radiation field (operator of $V(t)$ that you mentioned) in the time-dependent Schrodinger equation under the assumption that BO and FC is a good approximation, which mixes electronic and vibrational wavefunctions. So it seems to me like I was confused about "electron-phonon coupling" at two different stages of the problem, one being the fundamental interaction in the ABSENCE of a radiation field (which is what I described right above), and the other being the interaction in the PRESENCE of a radiation field (which is what you are describing).Now, getting back to Huang-Rhys parameter, I don't think this parameter represents how well the radiation field couples electronic and vibrational wavefunctions, or in short, "electron-phonon coupling". Yes, as far as I can see from the original paper by Huang and Rhys, that the derivation assumes radiation field, but the derivation of the so-called Huang-Rhys parameter [Eq. 4.21] (actually it's more like a definition than a derivation), comes after all that radiation field has been accounted for and now only shows the equilibrium displacement between two PESs. It doesn't represent "electron-phonon coupling" itself. It just shows the degree of displacement. In fact, the FC factor also depends on which vibrational quantum number we are looking at, and not just the Huang-Rhys parameter. And yet, so many people call this parameter "electron-phonon coupling parameter", and because I not a confident man, I felt that I might be fundamentally wrong about something that need someone to point it out. Then, DrDu said:

As I already mentioned, the derivation of the Huang Rhys factor is based on a diabatic or crude adiabatic approximation. This means that the Hamiltonian is expressed in a basis of electronic states which do not depend on position of the nuclei but all refer to the same equilibrium position of the ground state. In this approximation, there won't be any nuclear momentum dependent non-adiabatic couplings but there will be couplings linear in the nuclear displacement. Namely, if the electronic states are $|i\rangle$, $|j\rangle$ ..., then the operator for the nuclear motin becomes $T_n +\sum_{ij} (e_i \delta_{ij} +((\partial /\partial R)\langle i| H_\mathrm{el}| j\rangle ) (R-R_0) +C_{ij} (R-R_0)^2+\ldots$. The part linear in $R-R_0$ describes the coupling of the electronic and nuclear degrees of freedom. Even if the potential energy part of the hamiltonian is diagonalised to obtain the usuall BO-hamiltonian, this coupling will determine the shift of the equilibrium postions of the excited states relative to the ground state.

So then let's assume that BO approximation is valid. I believe that the second term and beyond are ignored in BO approximation. So we only have the first term, which is the kinetric energy matrix element purely of the nuclei. Am I wrong?

Also, isn't diabatic and crude adiabatic approximation two opposite approximation? As far as I can see from Huang and Rhys's original paper, they derive equations including derivation (or rather the definition) of Huang-Rhys parameter (Eq 4.21), under the Franck-Condon principle (which in turn assumes BO approximation). I don't know what paper followed after this paper and any reformulation or rederivation of Huang-Rhys parameter, but purely from this standpoint, I don't think diabatic approximations are used.

 

[TeethWhitener]

I am starting to get the feeling that we aren't talking about the same thing

I think you might be right.

The Hamiltonian of the time-independent Schrodinger equation for an entire molecule is:
H(r,R)=T(r,R)+V(r,R)H(r,R)=T(r,R)+V(r,R)H(\textbf{r},\textbf{R}) = T(\textbf{r},\textbf{R}) + V(\textbf{r},\textbf{R})　　　　　　　　　　　　　　　　[Eq.1]
where
T(r,R)=−∑Iℏ2MI∇2I−∑iℏ2me∇2iT(r,R)=−∑Iℏ2MI∇I2−∑iℏ2me∇i2T(\textbf{r},\textbf{R}) = -\sum _{I}\frac{\hbar}{2M_{I}}\nabla_{I}^{2} -\sum _{i}\frac{\hbar}{2m_{e}}\nabla_{i}^{2}　　　　　　　　　　　　[Eq.2]
V(r,R)=e24πε0(−∑I>I′ZIZI′RI,I′−∑I,iZIrI,i+12∑i>i′1ri,i′)V(r,R)=e24πε0(−∑I>I′ZIZI′RI,I′−∑I,iZIrI,i+12∑i>i′1ri,i′)V(\textbf{r},\textbf{R})=\frac{e^{2}}{4\pi \varepsilon _{0}}\left ( -\sum _{I>I'}\frac{Z_{I}Z_{I'}}{R_{I,I'}}-\sum _{I,i}\frac{Z_{I}}{r_{I,i}}+\frac{1}{2}\sum _{i>i'}\frac{1}{r_{i,i'}} \right )　　　[Eq.3]
I believe that in Born-Oppenheimer approximation, you take out the nuclear kinetic energy operator (first term of the right side of [Eq.2]) for the moment for calculating the electronic part of the wavefunction under the assumption that it does not contribute to the electronic part. And then we come back after solving the Schrodinger equation for the electronic part, and solve the Schrodinger equation for the nuclear motion.

BO approximation assumes that the off-diagonal matrix element of the kinetic energy operator of nuclei is zero, and that only the diagonal part survives (Born-Huang approx.), which is further approximated to contain terms that only depend on the nuclear positions and not electronic. It is precisely this assumption that the off-diagonal matrix element is zero that the electronic and vibronic wavefunctions remain separated and that the entire wavefunction can then be expressed as the product of the two. However, these off-diagonal elements are not truly negligible especially when two PES closes in energy, and it contributes to the nonradiative decay process. The energy gap law is based on this framework and thus so is Kasha's rule. So I was considering this as the fundamental electron-phonon coupling.

This is fine.

Now I am starting to understand that this was an incorrect nomenclature, so that is my fault. I would like to know how this type of interaction is actually called.

I'm not sure this is true. It might be that "electron-phonon coupling" is a fairly vague term with different meanings across different fields. Maybe it's more precise to say static versus dynamic electron-phonon coupling. I'm more familiar with molecular quantum mechanics than solid state, so I would refer to these as "vibronic transitions," where the electronic motion is coupled with vibrational motion. Same idea though.

Now, I believe what you are talking about in your latter half of the post is about taking into account the perturbing radiation field (operator of V(t)V(t)V(t) that you mentioned) in the time-dependent Schrodinger equation under the assumption that BO and FC is a good approximation, which mixes electronic and vibrational wavefunctions. So it seems to me like I was confused about "electron-phonon coupling" at two different stages of the problem, one being the fundamental interaction in the ABSENCE of a radiation field (which is what I described right above), and the other being the interaction in the PRESENCE of a radiation field (which is what you are describing).

Yes.

Now, getting back to Huang-Rhys parameter, I don't think this parameter represents how well the radiation field couples electronic and vibrational wavefunctions, or in short, "electron-phonon coupling". Yes, as far as I can see from the original paper by Huang and Rhys, that the derivation assumes radiation field, but the derivation of the so-called Huang-Rhys parameter [Eq. 4.21] (actually it's more like a definition than a derivation), comes after all that radiation field has been accounted for and now only shows the equilibrium displacement between two PESs. It doesn't represent "electron-phonon coupling" itself. It just shows the degree of displacement. In fact, the FC factor also depends on which vibrational quantum number we are looking at, and not just the Huang-Rhys parameter. And yet, so many people call this parameter "electron-phonon coupling parameter", and because I not a confident man, I felt that I might be fundamentally wrong about something that need someone to point it out.

I haven't read the entire paper very closely, but just glancing at it, they seem to be talking about the FC factors in terms of "modified normal coordinates," where the effect of the electric field is incorporated into the coordinates. (eqs. 3.6, 3.8)

In terms of off-diagonal matrix elements, let's consider two PES's separated by energy E that are identical with zero nuclear displacement (zero Huang-Rhys parameter). Let's assume Born-Oppenheimer ($\Psi = \psi_N\psi_e$ and $H = T_N+H_e$ which is diagonal). The entire hamiltonian is diagonal, but we can also think of it as "block diagonal", where each block is a single electronic state with its set of vibrational states. When we apply the perturbation (oscillating E field), we couple electronic states, but if the HR parameter is zero, then $\langle v'|v\rangle = \delta_{v'v}$ and the off-diagonal blocks connecting electronic states are "diagonal" in their vibrational states. Turning the HR parameter on (i.e., displacing the equilibrium position) dials in the off-diagonal-off-diagonal (for lack of a better term) states and couples vibrational modes of different electronic states. I don't know if thinking about it that way helps at all.

 

[HAYAO]

I haven't read the entire paper very closely, but just glancing at it, they seem to be talking about the FC factors in terms of "modified normal coordinates," where the effect of the electric field is incorporated into the coordinates. (eqs. 3.6, 3.8)

Yes, indeed.

In terms of off-diagonal matrix elements, let's consider two PES's separated by energy E that are identical with zero nuclear displacement (zero Huang-Rhys parameter). Let's assume Born-Oppenheimer ($\Psi = \psi_N\psi_e$ and $H = T_N+H_e$ which is diagonal). The entire hamiltonian is diagonal, but we can also think of it as "block diagonal", where each block is a single electronic state with its set of vibrational states. When we apply the perturbation (oscillating E field), we couple electronic states, but if the HR parameter is zero, then $\langle v'|v\rangle = \delta_{v'v}$ and the off-diagonal blocks connecting electronic states are "diagonal" in their vibrational states. Turning the HR parameter on (i.e., displacing the equilibrium position) dials in the off-diagonal-off-diagonal (for lack of a better term) states and couples vibrational modes of different electronic states. I don't know if thinking about it that way helps at all.

Ooooh, now I'm starting to understand.

So then the electron-phonon coupling is the RESULT of the displacement of equilibrium position under a radiation field? Then the two "parameters" are indeed linked. I might have to re-study the papers more rigorously, but I think I'm getting there.

Thank you!

 

[TeethWhitener]

So then the electron-phonon coupling is the RESULT of the displacement of equilibrium position under a radiation field?

Just to be clear, the displacement exists regardless of whether there’s a radiation field or not. The radiation field attempts to couple various states and the displacement determines the strength of that coupling.

 

[HAYAO]

Just to be clear, the displacement exists regardless of whether there’s a radiation field or not. The radiation field attempts to couple various states and the displacement determines the strength of that coupling.

Oh yes, of course. I phrased it wrong on the above post.

 

[HAYAO]

I'm sorry for the follow-up questions.

So what we've have so far is that :
1) Huang-Rhys parameter shows the displacement of two PES equilibrium position (assuming BO approx.) that indicates indirectly the degree of radiation field-induced electron-phonon coupling (off-diagonal matrix elements of block off-diagonal elements of the Hamiltonian has an value, depending on how displaced the PES equilibrium positions are).
2) "Vibrational relaxation" of excited state is something else; it is caused by intrinsic off-diagonal matrix element (nonadiabatic property where BO approx. fails) that exist even without radiation field.So then that leads to my second question. If we have a material under constant radiation field and a pulse radiation field, would the two property differ at this level of discussion? That is, would the phonon sideband be smaller for pulse excitation than constant excitation? Would the vibrational relaxation rate constant differ between pulse excitation and constant excitation?

 

[TeethWhitener]

"Vibrational relaxation" of excited state is something else; it is caused by intrinsic off-diagonal matrix element (nonadiabatic property where BO approx. fails) that exist even without radiation field.

Vibrational relaxation in real systems tends to be mediated by a coupling to a bath. So, for example, an elecronically excited molecule in solution will vibrationally relax primarily through interactions with solvent molecules. I'm less familiar with these processes in solids, but I suppose you could characterize this process as nonadiabatic (off-diagonal), with the excited vibrations relaxing to the ground state via couplings through low-energy acoustic phonons (essentially a thermalization process: transfer of energy away from the excitation as heat).

That is, would the phonon sideband be smaller for pulse excitation than constant excitation? Would the vibrational relaxation rate constant differ between pulse excitation and constant excitation?

Under continuous radiation, you get a constant population distribution of the vibrational levels (determined by the Franck-Condon factors). So vibrational relaxation is in equilibrium with excitation. But I'm not sure how or if the nature of the radiation affects the vibrational relaxation rate constant. The rate constant is pretty much set in stone by the intrinsic coupling between excited-state vibrations (the on-diagonal off-diagonal, to abuse the unfortunate terminology that we've introduced), whereas the radiation affects the coupling between ground-state vibrations and excited-state vibrations (the off-diagonal off-diagonal).

Caveat: I freely allow that there's something I might be missing here. I welcome any comments from folks more familiar with these processes than me (especially in solids).

 

[HAYAO]

I thank you very much for your time in answering my questions.

Vibrational relaxation in real systems tends to be mediated by a coupling to a bath. So, for example, an elecronically excited molecule in solution will vibrationally relax primarily through interactions with solvent molecules. I'm less familiar with these processes in solids, but I suppose you could characterize this process as nonadiabatic (off-diagonal), with the excited vibrations relaxing to the ground state via couplings through low-energy acoustic phonons (essentially a thermalization process: transfer of energy away from the excitation as heat).

You are absolutely right as far as I know. I guess you can consider this as an "internal" or "external" effects, the same way we consider reorganization energy of both for molecules in solution. However, for localized luminescent centers in a solid, the host material itself can be considered as a "solvent" as well. At this point, the term "external" and "internal" is ambiguous.

That being said, I am primarily thinking about lanthanides, where the excitation is very localized. Huang-Rhys parameter of lanthanide 4f-states can be in the order of 10^-2 which is quite small. In such case, I assume that the contribution from nonadiabatic type relaxation is also quite small. Indeed, nonradiative relaxation rate of lanthanides (for example Eu(III) ⁵D₀→⁷F_J transition) is very small and can be in the order of 10¹ - 10² s^-1. Nonetheless, it still nonradiatively relax to the ground state hence why I had to focus on the adiabatic type relaxation. In this sense, I should've provided this paper by W. Siebrand first. He explains the nonradiative relaxation as an adiabatic process by Franck-Condon factor in which the motivation is that nonadiabatic process cannot quantitatively nor completely explain the phenomenon. He also further explain that it is not the displacement between PES (Huang-Rhys parameter) that governs the nonradiative relaxation, but shifts in frequency between ground and excited state vibrational modes.

EDIT: However, this paper by Miyakawa and Dexter essentially says that nonadiabatic process is the main process of vibrational relaxation for lanthanides as well, so in this sense we have two different ideas.

Side note: Confusingly, I work on lanthanide complexes, which has the property of both a molecule in solvent AND solid.

Under continuous radiation, you get a constant population distribution of the vibrational levels (determined by the Franck-Condon factors). So vibrational relaxation is in equilibrium with excitation. But I'm not sure how or if the nature of the radiation affects the vibrational relaxation rate constant. The rate constant is pretty much set in stone by the intrinsic coupling between excited-state vibrations (the on-diagonal off-diagonal, to abuse the unfortunate terminology that we've introduced), whereas the radiation affects the coupling between ground-state vibrations and excited-state vibrations (the off-diagonal off-diagonal).

Caveat: I freely allow that there's something I might be missing here. I welcome any comments from folks more familiar with these processes than me (especially in solids).

Yes, I was exactly confused for the reasons you provided here. Like you said, the rate constant is supposedly dominated by the on-diagonal block off-diagonal elements and that is what I have always thought. But it didn't particularly make sense to me that lanthanides, despite their small Huang-Rhys parameter, still show notable vibrational relaxation. The intrinsic coupling is prominent near intersection of two PES but because lanthanide have low Huang-Rhys parameter, the intersection must be quite high in energy between that of ground and excited state. This is especially true for lanthanide complexes, which contain higher energy vibrational modes compared to solids. Indeed, vibrational relaxation rate is experimentally larger for lanthanide complexes than for lanthanide-doped solids, generally speaking (of course, I am omitting the case for extra processes such as accidental low-energy intersection of PESs or energy transfer to other states).In summary, we have few facts at hand:
1) Vibrational relaxation occur due to nonadiabatic intrinsic coupling (on-diagonal block off-diagonal matrix element has a non-negligible value).
2) Even systems where there is very small Huang-Rhys parameter can still relax to ground state by vibrational relaxation.
3) There can also be vibrational relaxation by adiabatic-type coupling (by electromagnetic field, or off-diagonal block off-diagonal matrix element has a value) provided by W. Siebrand.

Then this raises the question(s):
1) Does that mean constant excitation and pulse excitation will provide different rate constant? Different emission spectrum? (Experimentally, no. I've tried time-resolved fluorescent spectroscopy myself for organic molecules and they show pretty much the same emission spectra.)
2) So then something must be missing here, so what logical mistake am I making here?

 

[DrDu]

Sorry that I could not participate more in increasing the confusion :-), but I was on vacation. I had a look at the paper by Miyakawa and Dexter and I think they wrote down more precisely what I could write on my smartphone. Namely the quality V defined in 2.12 is what I would call the electron phonon coupling. It's electronically diagonal elements are responsible for the shift of the equilibrium position in 2.11 while it's electronically non-diagonal elements determine the strength of the nonadiabatic coupling 2.17. I think you should have a look at the original paper by Born and Oppenheimer and on the appendices by Born and Huang to see how the crude adiabatic and adiabatic picture are linked.

Furthermore I think that the mechanism proposed by Siebrand is also via non-adiabatic couplings, only that it involves the quadratic shifts in Q which Miyakawa and Dexter neglect. These papers are quite old and I suppose that Siebrands mechanism had been disproven to be effective in the case of lanthanides when M&D published their paper. However, it is important in other systems.
The relaxation rate depends on both the strength of the nonadiabatic couplings and on the shifts of the equilibrium positions. In many cases, the modes which lead to a strong nonadiabatic interaction are not the same ones as the modes which show a strong shift of the equilibrium position in the excited states or a strong shift in frequency in the excited state. The modes are often classified according to which of these effects is strongest for them.
Anyhow none of the relaxation mechanisms involves the exciting electromagnetic field. So the relaxation rate is independent on the excitation mechanism.

Edit: You may also like to read how exactly diabatic electronic states are defined https://en.wikipedia.org/wiki/Diabatic
and how this is similar to the crude adiabatic approximation

 

[HAYAO]

First, DrDu, I thank you sincerely for your help.

Sorry that I could not participate more in increasing the confusion :-), but I was on vacation. I had a look at the paper by Miyakawa and Dexter and I think they wrote down more precisely what I could write on my smartphone. Namely the quality V defined in 2.12 is what I would call the electron phonon coupling. It's electronically diagonal elements are responsible for the shift of the equilibrium position in 2.11 while it's electronically non-diagonal elements determine the strength of the nonadiabatic coupling 2.17. I think you should have a look at the original paper by Born and Oppenheimer and on the appendices by Born and Huang to see how the crude adiabatic and adiabatic picture are linked.

I already understood that the off-diagonal elements resulting from the kinetic energy operator is the source of nonadiabatic process (namely, vibrational relaxation) before this thread started. But I admit I don't fully understand the paper by Miyakawa and Dexter.

I am familiar with crude adiabatic, Born-Oppenheimer, and Born-Huang approximations, especially when all of this needs to be considered at hand. But just in case, tell me if I am wrong.
Born-Huang approximations: neglects off-diagonal matrix element of nuclei kinetic energy operator.
Born-Oppenheimer approximations: Born-Huang approx. plus neglect one of the two terms of diagonal matrix elements, namely, the one consisting of electronic wavefunctions with nuclei momentum operator squared.
Crude adiabatic approximation: Born-Huang approx. plus fix the nuclear configuration of the electronic wavefunction to equilibrium position.

However, many of the references seems to have their own definition as to what they each mean. For example, Herzberg calls all of the approximations mentioned above as Born-Oppenheimer, while Jortner do not mention any term for Born-Oppenheimer and Crude adiabatic approximation and name, and considers Born-Huang as the Born-Oppenheimer adiabatic. Very confusing. And unfortunately, I am not educated well in the area of quantum chemistry and am almost fully self-taught. Thus, I easily get confused when different formulations are used and different names are used for the adiabatic approximations.

So then how about the case for Miyakawa and Dexter? I interpreted it as Born-Huang approximation along with Teller-Herzberg expansion at the equilibrium position for deriving Equation (2.13) through (2.15). So I didn't think $V$ employed in (2.12) is electron-phonon coupling. But they did not employ those approximation for Equation (2.8) to getting (2.16). The equation (2.16) is the nonadiabatic part, therefore the off-diagonal part. Perhaps I am wrong with this.

Also, Born and Oppenheimer's original paper is German if I remember correctly. I cannot read German.

Furthermore I think that the mechanism proposed by Siebrand is also via non-adiabatic couplings, only that it involves the quadratic shifts in Q which Miyakawa and Dexter neglect. These papers are quite old and I suppose that Siebrands mechanism had been disproven to be effective in the case of lanthanides when M&D published their paper. However, it is important in other systems.
The relaxation rate depends on both the strength of the nonadiabatic couplings and on the shifts of the equilibrium positions. In many cases, the modes which lead to a strong nonadiabatic interaction are not the same ones as the modes which show a strong shift of the equilibrium position in the excited states or a strong shift in frequency in the excited state. The modes are often classified according to which of these effects is strongest for them.

So then we have a disagreement between TeethWhitener and you, and I probably have to start from scratch again. So then electromagnetic field do nothing about coupling vibrational and electronic wavefunctions?

Also, I thought that Siebrand worked from Born-Huang approximation, not nonadiabatic.

Edit: You may also like to read how exactly diabatic electronic states are defined https://en.wikipedia.org/wiki/Diabatic
and how this is similar to the crude adiabatic approximation

Yes, I already understand this. I also already read this page too.I am sorry for the post getting this long.

 

[DrDu]

Yes, terminology is confusing and not always consistent.
I could only read the abstract of the article by Siebrand, but nonradiative transitions are always non-adiabatic, and him talking of vibronic coupling does not seem to contradict.

I think one of the big differences between quantum chemistry and solid state physics is that the former almost always start form Born-Oppenheimer or Born-Huang, while the latter ones always start from the crude adiabatic approximation. The reason is that in solid state it is of tremendous advantage to take into account the periodicity of the lattice, which only prevails at the equilibrium position of the nuclei. As a phonon only exists in solid state systems, terms like electron-phonon coupling always point to the use of the crude Born Oppenheimer approximation. In the papers you cited, the ordinary BO approximation is derived from the crude one by successive diagonalisation. This is similar to what Born and Oppenheimer did in their article, which is also available in english translation:

http://elib.bsu.by/bitstream/123456789/154381/1/1927-084 AP Born & Oppenheimer - On the Quantum Theory of Molecules.pdf

PS: I don't see any point where I were in disagreement with Teethwhitener.

 

[TeethWhitener]

There can also be vibrational relaxation by adiabatic-type coupling (by electromagnetic field, or off-diagonal block off-diagonal matrix element has a value) provided by W. Siebrand.

I got deluged with work so I might not be much help in the next few days/weeks, but as far as I can tell, the model Siebrand uses is nonadiabatic. Equation 1 explicitly couples off-diagonal states with the nuclear kinetic energy operator. Siebrand even remarks on page 442:

There is a formal similarity between the above equations and the equations appropriate to vibrationally induced radiative transitions. The transformation to the optical case requires replacement of the operator $\mathcal{T}_N$ by the dipole operator $\mathcal{P}$ and the addition of a photon $h\nu$ to (10)

So the nuclear kinetic energy operator takes the place of the dipole operator in determining the off-diagonal coupling.

1) Does that mean constant excitation and pulse excitation will provide different rate constant? Different emission spectrum?

As @DrDu said,

Anyhow none of the relaxation mechanisms involves the exciting electromagnetic field. So the relaxation rate is independent on the excitation mechanism.

I will try to go through the Miyakawa paper if I get a moment.

PS: I don't see any point where I were in disagreement with Teethwhitener.

I also don't see any disagreement.

 

[HAYAO]

Okay, so then I will have to reinterpret what you guys said to mean that, the nonadiabatic process is somehow described through any form of adiabatic approximation. That doesn't seem to make sense to me. Whether you use Born-Huang, Born-Oppenheimer, or crude adiabatic, you are still neglecting the off-diagonal matrix element of the nuclei kinetic energy operator. It is exactly this off-diagonal matrix element that causes vibrational relaxation. So by neglecting the off-diagonal matrix element by any form of adiabatic approximation, it would be impossible that vibrational relaxation can be accounted for.

The paper by Miyakawa and Dexter sounds good to me. This is precisely because they account for the nonadiabatic part of the Hamiltonian by coming back to it after accounting for the adiabatic part. I don't see any problem with this.

The problem is what Siebrand says. For example, Teethwhitener talks about Equation (1), but Siebrand says nothing about exactly how much he is going beyond Born-Oppenheimer approximation. That is, it could still be Born-Huang approximation, or could be nothing at all. In either case, however, Equation (1) only talks about the "nonzero matrix element", but does not say anything about whether that is diagonal or off-diagonal part. In Born-Huang approximation, you are neglecting off-diagonal part resulting from nuclei kinetic energy operator while still accounting for the diagonal part. If no approximations are used, then you account for both diagonal and off-diagonal elements. But that cannot be solely judged based on Equation (1) (nor (3)) or what Siebrand says in page 442. But if what you guys say is true, then this quite simplifies the problem I was having.
I had to interpret this (as my main advisor did in his Ph.D. thesis) as meaning that there is a vibrational relaxation through adiabatic means. I asked my advisor directly as well, and I didn't want to oppose or deny his entire Ph.D. thesis. This is why I was trying to comprehend Huang-Rhys parameter and its supposedly means of coupling vibrational and electronic wavefunctions, which I thought would somehow explain vibrational relaxation (and not phonon sideband).

It seems, however, at this point that this interpretation is indeed wrong, which will make the story much more simpler (my advisor was wrong!).

 

[TeethWhitener]

but Siebrand says nothing about exactly how much he is going beyond Born-Oppenheimer approximation.

He expands the kinetic energy matrix elements about the equilibrium nuclear positions and neglects all terms higher than linear. I don’t have access to the paper at this exact moment, but I remember that specifically.

 

[HAYAO]

He expands the kinetic energy matrix elements about the equilibrium nuclear positions and neglects all terms higher than linear. I don’t have access to the paper at this exact moment, but I remember that specifically.

Yes, I know that. But which kinetic energy matrix elements? The diagonal or the off-diagonal? Because it seems like he's expanding on the second term of the diagonal matrix elements, or maybe it's not?

 

[DrDu]

Okay, so then I will have to reinterpret what you guys said to mean that, the nonadiabatic process is somehow described through any form of adiabatic approximation.

I think this is a misunderstanding. At least what I am trying to say is that the nonadiabatic processes can be described using different adiabatic representations, not approximations. With a representation I mean using a product basis for the electronic and nuclear Hilbert space. As these products of electronic and nuclear wavefunctions form a complete set, it is clear that they span the whole Hilbert space. The adiabatic approximation in turn consists in approximating the eigenstates of the full hamiltonian with only a single product function.

 

[HAYAO]

I think this is a misunderstanding. At least what I am trying to say is that the nonadiabatic processes can be described using different adiabatic representations, not approximations. With a representation I mean using a product basis for the electronic and nuclear Hilbert space. As these products of electronic and nuclear wavefunctions form a complete set, it is clear that they span the whole Hilbert space. The adiabatic approximation in turn consists in approximating the eigenstates of the full hamiltonian with only a single product function.

Well, adiabatic representation basically means adiabatic basis, right? So are you are talking about adiabatic-to-diabatic transformation when describing nonadiabatic processes? Then it seems reasonable. I think this is what Miyakawa and Dexter essentially do in their paper (EDIT: and what I suspected Siebrand of not doing), although it is hard for me to follow exactly what kind of calculations were done to achieve some of the equations.

 

[DrDu]

Well, adiabatic representation basically means adiabatic basis, right? So are you are talking about adiabatic-to-diabatic transformation when describing nonadiabatic processes? Then it seems reasonable. I think this is what Miyakawa and Dexter essentially do in their paper (EDIT: and what I suspected Siebrand of not doing), although it is hard for me to follow exactly what kind of calculations were done to achieve some of the equations.

Not quite. I just mean that you can express the non-adiabatic coupling matrix elements in an adiabatic product basis.

 

[HAYAO]

Not quite. I just mean that you can express the non-adiabatic coupling matrix elements in an adiabatic product basis.

I see. Thank you!

 

[TeethWhitener]

Yes, I know that. But which kinetic energy matrix elements? The diagonal or the off-diagonal? Because it seems like he's expanding on the second term of the diagonal matrix elements, or maybe it's not?

From the sentence before equation 1:

In the Born-Oppenheimer approximation this (meaning $\phi_B\Lambda_B$) is an exact eigenstate. But if we go beyond this approximation then there is a nonzero matrix element of the form
$$H_{AB}=\langle \phi_A\Lambda_A| \mathcal{T}_N|\phi_B\Lambda_B\rangle$$
where $\mathcal{T}_N$ is the nuclear kinetic-energy operator.

Here $\phi_B$ is the electronic state and $\Lambda_B$ is the vibrational state. So the off-diagonal terms of the nuclear kinetic energy operator are clearly nonzero in this treatment.

EDIT: The hand-wavy overall idea here is that the lowest vibrational state of the excited electronic state may be very close in energy to some high vibrational state of the ground electronic state. These states, being so close in energy, should mix, but they can't do so under the BO approximation. Siebrand adds the term back into couple them (and does so to first order in the nuclear coordinates).

 

[HAYAO]

From the sentence before equation 1:

Here $\phi_B$ is the electronic state and $\Lambda_B$ is the vibrational state. So the off-diagonal terms of the nuclear kinetic energy operator are clearly nonzero in this treatment.

Now that you mentioned it, you are absolutely right. I overlooked the fact that the subscript designates a specific state of the system, not a general state. I'm sorry.

EDIT: The hand-wavy overall idea here is that the lowest vibrational state of the excited electronic state may be very close in energy to some high vibrational state of the ground electronic state. These states, being so close in energy, should mix, but they can't do so under the BO approximation. Siebrand adds the term back into couple them (and does so to first order in the nuclear coordinates).

I see. Thank you.It's been a long discussion for lack of understanding on my side. I apologize for taking this long to understand. Now I think I am starting to get it, and things are much more clear now.

Thank you, TeethWhitener and DrDu for your great help.

HAYAO