I Effective molecular Hamiltonian and Hund cases

[BillKet]

I haven't gone through your derivation yet, but yes, there's a way easier method, which is how B&C derive all their matrix elements.

Look at equation 11.3. Its derivation is literally three steps, by invoking only two equations (5.123 first and 5.136 twice, once for $F$ and once for $J$). The whole point of using Wigner symbols is to avoid the Clebsch-Gordan coefficient suffering.

By the way, almost every known case is in the B&C later chapters for you to look up. Every once in a while it's not. It happened to me actually, but I was able to derive what I needed using the process above.

Thanks a lot! This really makes things a lot easier!

I have a few questions about electronic and vibrational energy upon isotopic substitution. For now I am interested in the changes in mass, as I understand that there can also be changes in the size of the nucleus, too, that add to the isotope effects.

We obtain the electronic energy (here I am referring mainly to equation 7.183 in B&C) by solving the electrostatic SE with fixed nuclei. Once we obtain these energies, their value doesn't change anymore, regardless of the order of perturbation theory we go to in the effective Hamiltonian. The energy of the vibrational and spin-rotational will change, but this baseline energy of the electronic state is the same. When getting this energy, as far as I can tell, all we care about is the distance between the electrons and nuclei, as well as their charges. We also care about the electron mass, but not the nuclear one. This means that the electronic energy shouldn't change when doing an isotopic substitution. This is reflected in equation 7.199. However in equation 7.207 we have a dependence on the mass of the nuclei. From the paragraphs before, the main reason for this is the breaking of BO approximation. However, this breaking of BO approximation, and hence the mixing of electronic levels is reflected only in the effective Hamiltonian. As I mentioned above, the electronic energy should always be the same as its zero-th order value. Where does this mass dependence of the electronic energy $Y_{00}$ from equation 7.207 come from?

For vibrational energy, we have equation 7.184. I assume that the $G^{(0)}_{\eta\nu}$ term has the isotopic dependence given by 7.199. Do the corrections in 7.207 come from the other 2 terms: $V^{ad}_{\eta\nu}$ and $V^{spin}_{\eta\nu}$? And if so, is this because these terms can also be expanded as in equation 7.180? For example, from $V^{ad}_{\eta\nu}$ we might get a term of the form $x_{ad}(\nu+1/2)$ so overall the first term in the vibrational expansion becomes $(\omega_{\nu e}+x_{ad})(\nu+1/2)$ which doesn't have the nice expansion in 7.199 anymore but the more complicated one in 7.207? Is this right? Also do you have any recommendations for readings that go into a bit more details about this isotopic substitution effects? Thank you!

 

[amoforum]

I'm much less familiar with vibrational corrections. And as you've probably noticed, it's not the main focus of B&C either. A couple places to start would be:

1. Dunham's original paper: http://jupiter.chem.uoa.gr/thanost/papers/papers4/PR_41(1932)721.pdf
It shows the higher order corrections that are typically ignored in all those $Y_{ij}$ coefficients.

2. In that section B&C refer to Watson's paper: https://doi.org/10.1016/0022-2852(80)90152-6
I don't have access to it, but it seems highly relevant to this discussion.

 

[BillKet]

I'm much less familiar with vibrational corrections. And as you've probably noticed, it's not the main focus of B&C either. A couple places to start would be:

1. Dunham's original paper: http://jupiter.chem.uoa.gr/thanost/papers/papers4/PR_41(1932)721.pdf
It shows the higher order corrections that are typically ignored in all those $Y_{ij}$ coefficients.

2. In that section B&C refer to Watson's paper: https://doi.org/10.1016/0022-2852(80)90152-6
I don't have access to it, but it seems highly relevant to this discussion.

Thanks for the references, they helped a lot. I was wondering if you know of any papers that extended this isotope shift analysis to molecules that are not closed shell. For example the isotope dependence of spin-orbit, spin-rotation or lambda doubling parameters. I see in B&C that they mention that this hasn't been done, but the book was written in 2003 and perhaps someone did the calculations meanwhile.

 

[BillKet]

I looked a bit at some actual molecular systems and I have some questions.

1. In some cases, a given electronic state, say a $^2\Pi$ state is far from other electronic states except for one, which is very close (sometimes even in between the 2 spin-orbit states i.e. $^2\Pi_{1/2}$ and $^2\Pi_{3/2}$) and the rotational energy is very small. Would that be more of a Hund case a or c?

2. I noticed that for some $^2\Pi$ states, some molecules have the electronic energy difference between this state and the other state bigger than the spin-orbit coupling and the rotational energy, which would make them quite confidently a Hund case a. However, the spin orbit coupling is bigger than the vibrational energy splitting of both $^2\Pi_{1/2}$ and $^2\Pi_{3/2}$. How would I do the vibrational averaging in this case? Wouldn't the higher order perturbative corrections to the spin-orbit coupling diverge? Would I need to add the SO Hamiltonian to the zeroth order hamiltonian, together with the electronic energy?

3. In the Hund case c, will my zeroth order Hamiltonian (and I mean how it is usually done in literature) be $H_{SO}$, instead of the electronic one, $H_e$ or do I include both of them $H_e+H_{SO}$? And in this case, if the spin orbit coupling would be hidden in the new effective $V(R)$, how can I extract the spin-orbit constant, won't it be mixed with the electronic energy?

 

[EigenState137]

... One question I have is: is this Hamiltonian (with the centrifugal corrections) correct for any $J$ in a given vibrational level? I have seen in several papers mentioned that this is correct for low values of $J$ and I am not sure why would this not hold for any $J$. I understand that for higher $J$ the best Hund case might change, but why would the Hamiltonian itself change? ...

Greetings,

I am late to this party and forgive me please if I have missed some of the discussion given a rather quick read of a complex topic.

I have not seen any explicit comments regarding Rydberg-Rydberg or Rydberg-valence perturbations (interactions). Such interactions certainly influence observed rotationally resolved spectra, often in very subtle and unexpected ways. Lefebvre-Brion and Field is the most comprehensive discussion of such perturbations of which I am aware.

Just another detail to keep you up at night.ES

 

[Twigg]

I've not heard of these perturbations. Are we talking Rydberg as in electrons that are excited to >>10th electronic state? I knew Rydberg molecules are a thing, but I always assumed that stuff was limited to alkali-alkali dimers.

 

[EigenState137]

I've not heard of these perturbations. Are we talking Rydberg as in electrons that are excited to >>10th electronic state? I knew Rydberg molecules are a thing, but I always assumed that stuff was limited to alkali-alkali dimers.

Greetings,

If you have an unpaired outer electron, for example as in $\textup{NO}$, there is an associated set of Rydberg states corresponding to excitations of that unpaired outer electron. The valence states correspond to excitations of an inner, core electron. Thus doublet states $(S= 1/2)$ would have a set of Rydberg states.

The perturbations occur, for example, when two rotational transitions associated with different electronic states are fortuitously nearly degenerate. A Fortrat diagram, $E= f\left ( J \right )$, will show small discontinuities resulting from mixing of the nearly degenerate rotational states. Figuring out the details can be a challenge!ES

 

[BillKet]

Hello again. So I read more molecular papers meanwhile, including cases where perturbation theory wouldn't work and I want to clarify a few things. I would really appreciate your input @Twigg @amoforum. For simplicity assume we have only 2 electronic states, $\Sigma$ and $\Pi$ and each of them has only 1 vibrational level (this is just to be able to write down full equations). The Hamiltonian (full, not effective) in the electronic space is:

$$
\begin{pmatrix}
a(R) & c(R) \\
c(R) & b(R)
\end{pmatrix}
$$

where, for example $a(R) = <\Sigma |a(R)|\Sigma >$ and it contains stuff like $V_{\Sigma}(R)$, while the off diagonal contains stuff like $<\Sigma |L_-|\Pi >$. If we diagonalize this explicitly, we get, say, for the $\Sigma$ state eigenvalue:

$$\frac{1}{2}[a+b+\sqrt{(a-b)^2+4c^2}]$$

Assuming that $c<<a,b$ we can do a first order Taylor expansion and we get:

$$\frac{1}{2}[a+b+(a-b)\sqrt{1+\frac{4c^2}{(a-b)^2}}] = $$

$$\frac{1}{2}[a+b+(a-b)(1+\frac{2c^2}{(a-b)^2})] = $$

$$\frac{1}{2}[2a+\frac{2c^2}{(a-b)})] = $$

$$a+\frac{c^2}{(a-b)} $$

Here by $c^2$ I actually mean the product of the 2 off diagonal terms i.e. $<\Sigma|c(R)|\Pi><\Pi|c(R)|\Sigma>$This is basically the second order PT correction presented in B&C. So I have a few questions:

1. Is this effective Hamiltonian in practice a diagonalization + Taylor expansion in the electronic space, or does this happened to be true just in the 2x2 case above?

2. I am a bit confused how to proceed in a derivation similar to the one above, if I account for the vibrational states, too. If I continue from the result above, and average over the vibrationally states, I would get, for the $\Sigma$ state:

$$<0_\Sigma|(a(R)+\frac{c(R)^2}{(a(R)-b(R))})|0_\Sigma> = $$

$$<0_\Sigma|a(R)|0_\Sigma>+<0_\Sigma|\frac{c(R)^2}{(a(R)-b(R))}|0_\Sigma> $$

where $|0_\Sigma>$ is the vibrational level of the $\Sigma$ state (again I assume just one vibrational level per electronic state). This would be similar to the situation in B&C for the rotational constant in equation 7.87. However, if I include the vibration averaging before diagonalizing I would have this Hamiltonian:

$$
\begin{pmatrix}
<0_\Sigma|a(R)|0_\Sigma> & <0_\Sigma|c(R)|0_\Pi> \\
<0_\Pi|c(R)|0_\Sigma> & <0_\Pi|b(R)|0_\Pi>
\end{pmatrix}
$$

If I do the diagonalization and Taylor expansion as before, I end up with this:

$$<0_\Sigma|a(R)|0_\Sigma>+\frac{<0_\Sigma|c(R)|0_\Pi><0_\Pi|c(R)|0_\Sigma>}{(<0_\Sigma|a(R)|0_\Sigma>-<0_\Pi|b(R)|0_\Pi>)} $$

But this is not the same as above. For the term $<0_\Sigma|c(R)|0_\Pi><0_\Pi|c(R)|0_\Sigma>$, I can assume that $|0_\Pi><0_\Pi|$ is identity (for many vibrational states that would be a sum over them that would span the whole vibrational manifold of the $\Pi$ state), so I get $<0_\Sigma|c(R)^2|0_\Sigma>$, but in order for the 2 expression to be equal I would need:

$$\frac{<0_\Sigma|c(R)^2|0_\Sigma>}{(<0_\Sigma|a(R)|0_\Sigma>-<0_\Pi|b(R)|0_\Pi>)} =
<0_\Sigma|\frac{c(R)^2}{(a(R)-b(R))}|0_\Sigma>
$$

Which doesn't seem to be true in general (the second one has vibrational states of the $\Pi$ states involved, while the first one doesn't). Again, just to be clear by, for example, $<0_\Sigma|a(R)|0_\Sigma>$
I mean $<0_\Sigma|<\Sigma|a(R)|\Sigma>|0_\Sigma>$ i.e. electronically + vibrational averaging.

What am I doing wrong? Shouldn't the 2 approaches i.e. vibrational averaging before or after the diagonalization + Taylor expansion give exactly the same results?

 

[DrDu]

Thank you for your reply. I will look at the sections you suggested for questions 1. For the second one, I agree that if that $R_\alpha$ is a constant, we can take the electronic integral out of the vibrational integral, but I am not totally sure why can we do this. If we are in the BO approximation, the electronic wavefunction should be a function of $R$, for $R$ not constant, and that electronic integral would be a function of $R$, too. But why would we assume it is constant? I understand the idea behind BO approximation, that the electrons follow the nuclear motion almost instantaneously, but I don't get it here. It is as if the nuclei oscillate so fast that the electrons don't have time to catch up and they just the see the average inter-nuclear distance, which is kinda the opposite of BO approximation. Could you help me a bit understand this assumption that the electronic integral is constant? Thank you!

You should read some day the original Born-Oppenheimer paper.
The point is that the electronic wavefunction changes on a distance $O(1)$, while the nuclear wavefunctions change on a distance $O(\sqrt{m_\mathrm{e}/M_\mathrm{nuc}})$ around the equilibrium distance. So you can expand the electronic matrix elements in a power series in $R-R_0$. The matrix elements of the vibrational functions of $(R-R_0)^n\sim O((\sqrt{m_\mathrm{e}/M_\mathrm{nuc}})^n)$ whence usually all but the term with n=0 are negligible.
I think this expansion of the electronic dipole moment is called Herzberg-Teller coupling.

 

[BillKet]

You should read some day the original Born-Oppenheimer paper.
The point is that the electronic wavefunction changes on a distance $O(1)$, while the nuclear wavefunctions change on a distance $O(\sqrt{m_\mathrm{e}/M_\mathrm{nuc}})$ around the equilibrium distance. So you can expand the electronic matrix elements in a power series in $R-R_0$. The matrix elements of the vibrational functions of $(R-R_0)^n\sim O((\sqrt{m_\mathrm{e}/M_\mathrm{nuc}})^n)$ whence usually all but the term with n=0 are negligible.
I think this expansion of the electronic dipole moment is called Herzberg-Teller coupling.

I am not sure how does this answer my question. I agree with what you said about the perturbative expansion, this is basically what I used in my derivation in the Taylor series. My question was why the 2 methods I used (the 2 different perturbative expansions) don't give the same result. I also think that Herzberg-Teller coupling doesn't apply to diatomic molecules, no?

 

[DrDu]

@amoforum Also the expectation value $<\eta'|T_q^1(d_{el})|\eta>$ is a function of R (the electronic wavefunctions have a dependence on R), so we can't just take them out of the vibrational integral like B&C do in 6.332. What am I missing?

I was trying to answer this question.

 

[Twigg]

I'm also going to invite @EigenState137 to chime in (see questions in post #98), as they seem to know more about diatomic spectroscopy than I do.

Neat trick, @BillKet! As far as question 1, this feature (Taylor series = perturbation theory results) is not unique to the effective Hamiltonian. It's a mathematical fact that Taylor expanding the exact spectrum will give you the same results as perturbation theory of the same order, for any hamiltonian. That's just the way that perturbation theory works (if you want to convince yourself, review the derivation of PT in an undergrad level textbook-- the grad level stuff is too stuffy and notational for a chump like me ). Alternatively, you can just try to directly compute the first and 2nd order perturbation terms of the toy Hamiltonian $\left( \begin{array} aa(R) & c(R) \\ c(R) & d(R) \end{array} \right)$. To sum up, perturbation theory is just a shortcut to the terms in the taylor series when you don't have a closed formula for the spectrum to begin with.

Gimme some time to think about #2. I mostly responded just to get ES137 in on this. To my eyes, Eqn 7.85 looks more like your second expression, since $\eta \neq 0$. Am I missing something?

 

[BillKet]

I'm also going to invite @EigenState137 to chime in (see questions in post #98), as they seem to know more about diatomic spectroscopy than I do.

Neat trick, @BillKet! As far as question 1, this feature (Taylor series = perturbation theory results) is not unique to the effective Hamiltonian. It's a mathematical fact that Taylor expanding the exact spectrum will give you the same results as perturbation theory of the same order, for any hamiltonian. That's just the way that perturbation theory works (if you want to convince yourself, review the derivation of PT in an undergrad level textbook-- the grad level stuff is too stuffy and notational for a chump like me ). Alternatively, you can just try to directly compute the first and 2nd order perturbation terms of the toy Hamiltonian $\left( \begin{array} aa(R) & c(R) \\ c(R) & d(R) \end{array} \right)$. To sum up, perturbation theory is just a shortcut to the terms in the taylor series when you don't have a closed formula for the spectrum to begin with.

Gimme some time to think about #2. I mostly responded just to get ES137 in on this. To my eyes, Eqn 7.85 looks more like your second expression, since $\eta \neq 0$. Am I missing something?

Thank you! I guess what confused me and made me ask the first questions was that the B&C derivation of the effective Hamiltonian is long and he basically gives 2 (or 3) derivations for it, when all that is, is just a Taylor series expansion. I was afraid I was missing something.

For the second questions, I agree with you, 7.85 looks like my second expression i.e. first diagonalize then take the vibrational averaging. However, I am not sure why doing it the other way around i.e. vibrational averaging of the matrix elements and then diagonalization doesn't give the same result (or maybe it does and the 2 expressions are equivalent?). Actually, equation 7.69 in B&C confuses me even more. In that equation he seems to take the vibrational average only of the numerator i.e. $<i|H'|k>$, but not the denominator. So the denominator would still have an R dependence. But in 7.85, he implies that the vibrational averaging should include the denominator, too. So that kinda makes me believe that the 2 approaches are equivalent (or he did a mistake?), but I am not sure why I don't get that in my derivation.

 

[DrDu]

It's a mathematical fact that Taylor expanding the exact spectrum will give you the same results as perturbation theory of the same order, for any hamiltonian. That's just the way that perturbation theory works (if you want to convince yourself, review the derivation of PT in an undergrad level textbook-- the grad level stuff is too stuffy and notational for a chump like me ).

I fear that's not true in general. Most perturbation series in physics are singular perturbation series. Take the Born-Oppenheimer theory expanding the Hamiltonian in the ratio of nuclear to electron mass. The nuclear mass premultiplies the highest derivative (the second derivative with respect to R in diatomic molecules). Hence the zeroth order Hamiltonian would be qualitatively different from the case with finite nuclear mass.

 

[EigenState137]

Greetings,

@Twigg , thank you for your confidence in me. I hope it is not totally misplaced.

I have rather quickly read over this thread and a few questions come immediately to mind that I would like to address to @BillKet . You opted to label this discussion at an "I" level. That surprises me because I would consider it a rather esoteric topic. Thus my questions:

Is this just an exercise in understanding how to construct an Hamiltonian? If so, I think you have already received substantive responses.

Is this part of a research program with which you are associated? If so, what is the research objective? To develop an Hamiltonian to be used for the analysis of experimental spectra?

If the objective is the analysis of experimental spectra, then you need to consider the experimental data in detail. Do you know what the diatomic molecule is? What is the spectroscopic resolution: is it sub-Doppler for example? What angular momenta are relevant? Perhaps most importantly, why approach the analysis of a spectrum by creating an Hamiltonian rather than beginning with one of the numerous Hamiltonians already in the literature? Why begin by attempting to reinvent the wheel?

I would make two additional general comments.

First, keep it as simple as possible. Why on Earth even think about the quagmire that is the Stark Effect (DC and/or AC) unless absolutely necessary? Same for the Zeeman Effect.

Second, if this is indeed part of a formal research program, then I will not intrude on your research. Your research is for you to do in collaboration with your immediate colleagues and your mentor.ES

 

[Twigg]

Take the Born-Oppenheimer theory expanding the Hamiltonian in the ratio of nuclear to electron mass.

Isn't $\frac{m_{nuc}}{m_e} \geq 1000$? How is that a perturbative power series? If it's a lot of work to type up in TeX, can you provide a citation to the original Born-Oppenheimer paper you mentioned?

 

[Twigg]

@EigenState137 Yeah, sorry, you got invited late to a very long thread with a lot of different questions in it. Right now, we're just looking at the questions in Post #98 (post numbers are in the top right corner of each message), which aren't publication worthy (if that's what your concern is).

Here's a link to Post #98, which we are currently responding to.

From what I've seen in this thread, mostly the OP is just working their way through Brown & Carrington and checking in with folks to sanity check or to fill in the myriad gaps in the logic of B&C. Nothing asked here has seemed novel or publication worthy, so I don't think there's a need to worry about intruding on anything. Further, if the OP worked in a group with diatomic experts, I expect they wouldn't need to post questions here.

 

[EigenState137]

Greetings,

From what I've seen in this thread, mostly the OP is just working their way through Brown & Carrington and checking in with folks to sanity check or to fill in the myriad gaps in the logic of B&C

Not your everyday activity even under Covid19 restrictions. I am not concerned about publishing, I am concerned about a potential research student doint their own research. Old school I admit, but so be it.

Regarding Post#98. I fail to understand the concept of averaging over vibrational states.ES

 

[Twigg]

I fail to understand the concept of averaging over vibrational states.

I think they're just trying to evaluate this mixing Hamiltonian in a given vibrational state. I think the jargon was just a little loose. The OP stated they are only considering one vibrational state, as a simplifying assumption.

I am not concerned about publishing, I am concerned about a potential research student doint their own research. Old school I admit, but so be it.

I get what you mean. I'm sympathetic to the OP's cause because I personally find B&C very difficult to learn from. I had the privilege of working with very smart, very knowledgeable people in my research group who helped me along. Just trying to pay it forward. I respect your position on this

 

[EigenState137]

Greetings,

I get what you mean. I'm sympathetic to the OP's cause because I personally find B&C very difficult to learn from. I had the privilege of working with very smart, very knowledgeable people in my research group who helped me along. Just trying to pay it forward. I respect your position on this

As did I. But that was before the days of online forums. I am just being careful not to step on the toes of some PI--for me it is a policy decision.ES

 

[DrDu]

Isn't $\frac{m_{nuc}}{m_e} \geq 1000$? How is that a perturbative power series? If it's a lot of work to type up in TeX, can you provide a citation to the original Born-Oppenheimer paper you mentioned?

See my post No. 100. The perturbation parameter is evidently $m_\mathrm{e}/m_\mathrm{nuc}$, not the other way round. Try author:Born and author:Oppenheimer in Google Scholar which also yields a link to an english translation:
https://www2.ulb.ac.be/cpm/people/bsutclif/bornopn_corr.pdf

 

[BillKet]

@EigenState137 Yeah, sorry, you got invited late to a very long thread with a lot of different questions in it. Right now, we're just looking at the questions in Post #98 (post numbers are in the top right corner of each message), which aren't publication worthy (if that's what your concern is).

Here's a link to Post #98, which we are currently responding to.

From what I've seen in this thread, mostly the OP is just working their way through Brown & Carrington and checking in with folks to sanity check or to fill in the myriad gaps in the logic of B&C. Nothing asked here has seemed novel or publication worthy, so I don't think there's a need to worry about intruding on anything. Further, if the OP worked in a group with diatomic experts, I expect they wouldn't need to post questions here.

@EigenState137, @Twigg is right. For now I just want to understand diatomic molecules better and while reading B&C (and several other papers meanwhile) I came across different ideas that I asked for help here in order to understand them better. Similarly, the Zeeman and Stark shift came in the same context, me trying to understand diatomic molecules from different perspectives. Also as @Twigg said, right now the question in post #98 is what I am curious about, the other questions I had were well answered in this thread.

I am sorry if "vibrational averaging" was misleading (I am kinda using the B&C terms, but I am not sure how general they are). Basically what I want to understand is this: if I build the full electronic and vibrational Hamiltonian (in my case there are only 2 electronic states with only 1 vibrational level each, but the question can be generalized to realistic cases) and then I diagonalize it, would I get the same results as in B&C, where they first diagonalize the electronic Hamiltonian alone, and then they build a vibrational Hamiltonian for each entry on the diagonal of the electronic Hamiltonian. Intuitively I would say the 2 approaches are the equivalent, but the derivation I wrote above for the simplified case doesn't seem to work. Please let me know if the notation in my derivation in post #98 is confusing, I would be happy to clarify what I meant there.

 

[EigenState137]

Greetings,

For now I just want to understand diatomic molecules better and while reading B&C (and several other papers meanwhile) I came across different ideas that I asked for help here in order to understand them better.

Depending on what you mean by "better" you have articulated a potentially very broad and ambitious aspiration. Most professionals in the field spend their careers focused on one aspect of diatomic molecules, even on one single molecule. It is rare to find those with full breadth knowledge of the field.

Thus I will assume for the moment that your objective is to comprehend a Hamiltonian that describes the structure of a diatomic molecule. Are you comfortable with all of the angular momenta in question? Are you conversant with the corresponding Hamiltonian describing the detailed structure of an atom? If not, I urge you to backtrack and cover those prerequisites.

If you want more than that, then you will ultimately need to master the dynamical properties of diatomics, especially their excited states: photo-ionization, photo-dissociation, pre-dissociation, autoionization and interstate perturbations as examples. That is a life-long quest and frankly one that I simply cannot see a non-practicing professional even attempting to pursue.

Please note that I am not attempting to discourage you. I am attempting to make certain that you understand the breadth and complexity of the field.

On a purely pragmatic note, why have you chosen to follow the development as presented by Brown and Carrington? Our colleague #Twigg has stated

but ho boy do I have a lot of traumatizing books! Brown & Carrington tops that list

There are very complete and high-quality discussions of the structure of diatomic molecules available going back to Herzberg and Townes and Schalow. Lefebvre-Brion and Field, and Carrington, Levy and Miller have published Hamiltonians. Many scholarly treatments are available and I urge you to make use of them rather than to focus too tightly on a single treatment.ES

 

[Twigg]

@DrDu Thanks for bearing with me. With my level of journal access, I was only able to find the original in German. I'm rusty but working my way through it. Is there a particular equation or section that you wanted to highlight, or just the whole paper? Edit: I just re-read your post #99 and I think I've got it.

@EigenState137 Yeah, another user convinced BillKet to branch out into Lefebvre-Brion and Field as well as Brown & Carrington. Also, I called B&C "a traumatizing book" as opposed to "a bad book" because "traumatizing" is subjective and I'm possibly the weak link .

Returning to the questions in post 98, I think your second result (taking the expectation value *before* diagonalizing) is the right approach. Here's my logic for this:

When you say $$\langle \Lambda' | H | \Lambda \rangle = \left( \begin{array} aa(R) & c(R) \\ c(R) & b(R) \\ \end{array} \right)$$, what you are really saying is $$\langle \Lambda'; R | H | \Lambda; R \rangle = \left( \begin{array} aa(R) & c(R) \\ c(R) & b(R) \\ \end{array} \right)$$ where $| \Lambda; R \rangle$ is the state where the $\Lambda$ is well-defined but the vibrational number $\eta_{\Lambda}$ is not defined but instead the vibrational wavefunction has collapsed into the position $R$ (i.e., $\langle R | \psi_{vib} \rangle = \delta (R)$). In other words, if you solve for the eigenvalues of your matrix $\langle \Lambda' | H | \Lambda \rangle$, you are really solving for the energies when the vibrational state is concentrated around a position $R$. It might be a good approximation for coherent states on a dissociating potential in the classical limit (maybe?). However, if you want to talk about the spectrum when the molecule is in the well-defined vibrational states $| \eta_\Lambda \rangle$, then you need to take the vibrational expectation values first like you did in your second approach. If you had more than one vibrational state per electronic manifold, then you would have a block matrix for the Hamiltonian. I'm not betting a kidney on this being correct, but perhaps others can correct me if I'm wrong.

Edit: I thought of a better example in which your first approach is valid. Atomic collisions! So long as the change in interatomic potentials over one de Broglie wavelength is small, then you can approximate the atoms as classical particles. There's a name for this approximation and it eludes me.

 

[EigenState137]

Greetings,

However, if you want to talk about the spectrum when the molecule is in the well-defined vibrational states , then you need to take the vibrational expectation values first like you did in your second approach. If you had more than one vibrational state per electronic manifold, then you would have a block matrix for the Hamiltonian.

I believe that is correct, and any real molecule has more than one vibrational state, thus the general treatment is better. Also, I would treat the vibrational states as anharmonic in general.

I thought of a better example in which your first approach is valid. Atomic collisions! So long as the change in interatomic potentials over one de Broglie wavelength is small, then you can approximate the atoms as classical particles. There's a name for this approximation and it eludes me.

Are you thinking of van der Waals broadening?

Yeah, another user convinced BillKet to branch out into Lefebvre-Brion and Field as well as Brown & Carrington.

Levy and Miller should not be overlooked. ``Electron Resonance of Gaseous Diatomic Molecules,'' A. Carrington, D. H. Levy, and T. A. Miller, Adv. Chem. Phys. 18, 149 (1970). http://dx.doi.org/10.1002/9780470143650.ch4 I think that is the right presentation.ES

 

[Twigg]

Are you thinking of van der Waals broadening?

I just found it in Metcalf's book, and what I was thinking of was the Gallagher-Pritchard model (see pg. 18 of this review) for atomic collisions in the presence of a near-resonant laser field. Similar ideas to vdW broadening. It's mainly relevant for collisions of laser-cooled, trapped alkali atoms. It contributes to loss of atoms out of traps and it is relevant to the field of photoassociation/magnetoassociation of alkali dimer molecules at ultracold temperatures (like the fermi-degenerate KRb gas they reported at JILA a few years back).

 

[EigenState137]

Greetings,

More homework! I'll read it over breakfast in the morning.

Sorry for being slow--I was posting a belated introduction to the community.ES

 

[Twigg]

Ah, no need to put a lot of time into it. It's a bit niche haha. I just threw out the link so people would have something to follow if they were interested.

 

[EigenState137]

Ah mon ami, breakfast is for reading!

 

[BillKet]

Greetings,

Depending on what you mean by "better" you have articulated a potentially very broad and ambitious aspiration. Most professionals in the field spend their careers focused on one aspect of diatomic molecules, even on one single molecule. It is rare to find those with full breadth knowledge of the field.

Thus I will assume for the moment that your objective is to comprehend a Hamiltonian that describes the structure of a diatomic molecule. Are you comfortable with all of the angular momenta in question? Are you conversant with the corresponding Hamiltonian describing the detailed structure of an atom? If not, I urge you to backtrack and cover those prerequisites.

If you want more than that, then you will ultimately need to master the dynamical properties of diatomics, especially their excited states: photo-ionization, photo-dissociation, pre-dissociation, autoionization and interstate perturbations as examples. That is a life-long quest and frankly one that I simply cannot see a non-practicing professional even attempting to pursue.

Please note that I am not attempting to discourage you. I am attempting to make certain that you understand the breadth and complexity of the field.

On a purely pragmatic note, why have you chosen to follow the development as presented by Brown and Carrington? Our colleague #Twigg has stated

There are very complete and high-quality discussions of the structure of diatomic molecules available going back to Herzberg and Townes and Schalow. Lefebvre-Brion and Field, and Carrington, Levy and Miller have published Hamiltonians. Many scholarly treatments are available and I urge you to make use of them rather than to focus too tightly on a single treatment.ES

Thank you for your reply. I totally agree with you about the fact that trying to understand *all* (or most of) diatomic molecular physics would be basically impossible. But my questions are quite basics. I guess I just want to understand a basic Hamiltonian better and this vibrational averaging would be an important step before aiming to understand mode advanced topics. Also, I am not using just B&C. @Twigg adn @amoforum directed me towards Lefebvre-Brion and Field and I am also using Herzberg and Demtroder. However, I didn't find an answer to this particular question in their books (or maybe I missed/missunderstood it).

I think I am now more comfortable with the coupling cases. I am not sure exactly what you mean by the atomic case. Do you mean the electronic part of the Hamiltonian only?