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They describe the same things, and a merge with the Born-Oppenheimer_approximation and expandning the merged article to include parts about the vibronic and rotational Hamiltonian would make the articles much more usefull--Martin Hedegaard 22.19, 24 September 2006 (UTC)
I agree. Actually, these articles on Wikipedia are a mess right now, and really need some major cleanup. Maybe in a few days we can merge? --HappyCamper18:29, 24 September 2006 (UTC)[reply]
Sure, there is some information at Wikipedia:Merging and moving pages which you might find useful. I agree with your approach too. Is everything on this page in Electronic Hamiltonian now? If so, we can blank this page and make it into a redirect. I also see you have a sandbox set up. I can merge that with the existing electronic Hamiltonian article once its done if you like. --HappyCamper03:53, 26 September 2006 (UTC)[reply]
Yes, I noticed that :-) In that case, we can simply move the sandbox contents over to the main article. Don't cut and paste it though, because it's important that we keep the edit histories which accompany the articles. --HappyCamper16:23, 26 September 2006 (UTC)[reply]
I would like a breif review of the sandbox before moving it, I am not a native english speaker, and its a long time since I wrote anything english. If its okay just go ahead and move the content. The Electronic Hamiltonian should just be a redirect to this page, the new article covers exactly the same things as the old article-- Martin Hedegaard16:45, 26 September 2006 (UTC)[reply]
I would like to separate off the BO lemma again. I have seen the criticisms, especially
by the anonymous from the University of Stockholm (130.237.179.166). I understand what (s)he is saying and I can write an article that will not offend him/her too much. Anybody has any objections?
--P.wormer15:31, 8 December 2006 (UTC)[reply]
I have seen that kind of notation before, but I think that we should go with the notation because its more consistent with the rest of the article, and maybe give a link to a page like Nabla_in_cylindrical_and_spherical_coordinates. The part you are reffering to is one of the last parts of the orginal article thats why the notation is inconsistent, anyway its because of any special reason. Martin Hedegaard16:08, 26 December 2006 (UTC)[reply]
I was not very satisfied with this lemma, so I give it a new try. I started today, but will continue.
Question: why is the article by Handy et al. included in the reference list? In my view it is just one of the many research papers written about different terms in the Hamiltonian. This one is about a specific computational method for the BO diagonal correction and continues similar work by others, e.g. by David Yarkoni. Several of the review papers by Brian Sutcliffe would be more appropriate for the reference list, I would think.--P.wormer14:02, 1 January 2007 (UTC)[reply]
Its there because it was a part of the old aticles, when I rewrote it the first time and I just let i be, if you have better references or some good review papers then go ahead and fill it in. I actually thought the same thing about the paper from Handy et al., but it was not wrong as far as i could see, and referes to the last part of the article, that I dident write.
I wonder if it would be better to replace our bold letters with vector notation instead? The presentation comes across as a bit heavy, I think. Thoughts? --HappyCamper03:00, 5 January 2007 (UTC)[reply]
You mean with arrows on top? I heard that an SI committee advised sans serif letters for matrices and vectors, would that be an idea? --P.wormer10:10, 5 January 2007 (UTC)[reply]
Basically I finished the overhaul of the article. I thought I knew this stuff, but writing it I'm amazed how many holes there are in this theory. So I had to skim along WP:NOR. See also Talk:GF_method. I commented out the last part of the original text, so if somebody feels that (some of) it must be restituted, please uncomment it. --P.wormer16:06, 10 January 2007 (UTC)[reply]
Well, some sentences reasonings are not copied straight from a book, but thought up by me and I hope I didn't make mistakes. I'm still thinking and reading. --P.wormer15:11, 13 January 2007 (UTC)[reply]
Maybe we need something like the introduction in H. Köppel, W. Domcke, and L. Cederbaum, Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation, Adv. Chem. Phys., vol. 57, 1984. I don't have this with me right now, but I vaguely remember there is a nice concise way of summarizing things there. --HappyCamper15:21, 13 January 2007 (UTC)[reply]
As far as I remember those guys write about conical intersections and such things, that is break-down of Born-Oppenheimer. So, their work is relevant for the diabatic and Born-Oppenheimer articles. If you see any room for improvements in those articles, please edit them in.
For the present article Molecular Hamiltonian my problems have to do with (i) nuclear mass polarization: why do many authors (Wilson&Decius&Cross and Papousek&Aliev and Louck) not discuss them? They should get them by transforming to the nuclear center of mass. (ii) Internal versus external coordinates. Wilson&Decius&Cross define linearized valence coordinates without mass weighting. These are internal coordinates. In two other chapters they define internal coordinates with mass weighting (via the Eckart conditions). The Watson nuclear motion Hamiltonian is defined with Eckart conditions. How do Wilson's linearized coordinates fit into this? If anybody knows the answers to this, or knows literature that contains the answers, or sees that I made error(s) here (s)he should not hesitate to edit. --P.wormer17:54, 13 January 2007 (UTC)[reply]
Just thought I'd point out that Born-Oppenheimer Approximation (note the capital A) redirects here, even though it has its own article. My guess is that it got lost in the recent overhaul. I'd fix it myself, but I really don't know where to start. Gershwinrb08:34, 19 January 2007 (UTC)[reply]
The collection of electronic energies for varying nuclear coordinates R forms one
potential energy surface that enters one nuclear Schroedinger eq.--P.wormer15:00, 22 January 2007 (UTC)[reply]
Vb: I spent lots of time and thoughts on this article, so don't be rash and pigheaded. Please read the Born-Oppenheimer article. Within the BO approximation one goes from the Coulomb (electronic) Hamiltonian to the nuclear motion Hamiltonian with the use of ONE potential energy surface. Of course, the Coulomb Hamiltonian has more (often infinitely many) excited states and energies, but the whole idea is that one considers only ONE state with ONE energy (almost always this is the ground state and its energy) for MANY nuclear constellations. Please, if you don't understand this point, leave this article alone. Thank you. --P.wormer13:19, 23 January 2007 (UTC)[reply]
I am sorry but I spent enough time computing sets of potential surfaces to know that the BO idea applies to many coupled potential surfaces. That nowadays most people are most concerned with the groundstate is clear to me. However this article must be clear on this point! User:Vb09:58, 1 February 2007 (UTC)[reply]
Searching for "Adiabatic Approximation" redirects to this article. This article does NOT describe the adiabatic approximation, however (it merely describes the entity that results when this approximation is made for certain physical systems).
Pure rotational spectra are very hard to achieve experimentally, but they can be described by further separation of the vibrational and electronic motions. This requires two things:
Assume that the nuclei only make small oscillations from equilibrium configuration so the vibrational potential can be considered harmonic;
Approximate the inertia tensor with the inertia tensor calculated at the equilibrium configuration.
This is also called the "Harmonic vibrational and rigid-rotor model."
This is the most prevalent form of the molecular Hamiltonian because the vibrations are essentially independent of the surroundings. Hence, vibrational transitions are easily observed. Since the rotational transitions are almost never observed, a good approximation to the molecular Hamiltonian would be obtained by keeping only the part of HM that describes the electronic and vibrational parts. This is called the vibronic Hamiltonian, a portmanteau of "vibrational" and "electronic". The vibronic Hamiltonian is given by
with
with the being internal electronic and nuclear vibration coordinates. The use of the internal coordinates is used since the coulomb interaction only depends on the relative distance between the charged particles. Since the rotational and translational motions are now separated there will be either or vibrations if is the number of nuclei, and whether the molecule is linear or nonlinear.
where refers to the energy of the state . To solve the Schrödinger equation it is needed to decouple the motion of the nuclei and electrons. This is done by approximating the molecular wavefunction to a product of the electronic wavefunction and the nuclear vibration wavefunction. This is given by
where is the electronic and nuclear vibration quantum number. This formulation is termed an adiabatic wavefunction.
There are two main cases used in molecular physics, a dynamic and a static type. The dynamic type the electronic wavefunctions are assumed to follow the vibrations of the nuclei. The static case uses a static reference configuration to calculate the electronic wavefunctions, this is also called the crude adiabatic approximation.
In the dynamic approximation the electronic wavefunction is defined as the solution to the electronic Schrödinger equation
where
with the electronic wavefunctions found the nuclear vibrational coordinates or can be treated as parameters and the solution of the electronic Schrödinger equation then define the dependence of the electronic wavefunction and eigenvalues on the set of nuclear vibration coordinates . The electronic wavefunctions defines a complete orthonomal set of functions for each so the molecular wavefunction can be expanded in the basis.
using this result in the most used vibronic case, and inserting in the electronic Schrödinger equation and neglecting electronic coupling gives a new eigenvalue equation given by
where the expansion coefficients describes the vibrational eigenfunctions and the describe the vibrational potential energy. The eigenvalue, is often approximated by an harmonic function for simplification.
When the assumptions required for the adiabatic Born-Oppenheimer approximation do not hold, the approximation is said to "break down". Other approaches are needed to properly describe the system which is beyond the Born-Oppenheimer approximation.
The explicit consideration of the coupling of electronic and nuclear (vibrational) movement is known as electron-phonon coupling in extended systems such as solid state systems. In non-extended systems such as complex isolated molecules, it is known as vibronic coupling which is important in the case of avoided crossings or conical intersections.
The so-called 'diagonal Born-Oppenheimer correction' (DBOC) can be obtained as
where is the nuclear kinetic energy operator and the electronic wavefunction is parametrically (not explicitly) dependent on the nuclear coordinates.
I would like to separate off the BO lemma again. I have seen the criticisms, especially
by the anonymous from the University of Stockholm (130.237.179.166). I understand what (s)he is saying and I can write an article that will not offend him/her too much. Anybody has any objections?
--P.wormer15:31, 8 December 2006 (UTC)[reply]
I have seen that kind of notation before, but I think that we should go with the notation because its more consistent with the rest of the article, and maybe give a link to a page like Nabla_in_cylindrical_and_spherical_coordinates. The part you are reffering to is one of the last parts of the orginal article thats why the notation is inconsistent, anyway its because of any special reason. Martin Hedegaard16:08, 26 December 2006 (UTC)[reply]
The section describing the separation of the COM motion from the internal motion contained several errors. In particular, the process is not "more cumbersome" quantum mechanically (QM) than it is classically. The mass polarization term that results from separating the 3 COM coordinates from the internal motion appears in both the classical and QM Hamiltonian. Furthermore, it is completely unnecessary to introduce a generalized reduced mass tensor, when the process is done by eliminating the Nth particle from the internal Hamiltonian. There were also several factors of two missing from the Hamiltonian. I'm editing the page to try to correct these errors. 99.11.197.75 (talk) 20:12, 19 February 2012 (UTC)[reply]
Dear 99.11.197.75, in the expression for H′ (first equation in the section The Schrödinger equation of the Coulomb Hamiltonian) you removed the ti from the ∇ and replaced it by i. Further you replaced μi by mi. This suggests that you believe that the sums over i and j are still over nuclei and electrons separately. However, the new coordinates ti are linear combinations of electronic and nuclear coordinates (as is clearly stated in the text in a sentence that you, oddly enough, did not fiddle with). So, in your opinion, what mass mi is associated with given ti ?