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Hmm...Merge with Electronic Hamiltonian? -- HappyCamper 20:43, 23 September 2006 (UTC)
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.wormer 15:31, 8 December 2006 (UTC)
The notation used in this article:
is unusual for differentiation with respect to a vector. More common is either or introduction of components
with . Is this notation on purpose, or just by mistake?-- P.wormer 10:56, 26 December 2006 (UTC)
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.wormer 14:02, 1 January 2007 (UTC)
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? -- HappyCamper 03:00, 5 January 2007 (UTC)
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.wormer 16:06, 10 January 2007 (UTC)
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.wormer 17:54, 13 January 2007 (UTC)
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. Gershwinrb 08:34, 19 January 2007 (UTC)
The collection of electronic energies for varying nuclear coordinates R forms one potential energy surface that enters one nuclear Schroedinger eq.-- P.wormer 15:00, 22 January 2007 (UTC)
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).
Should this redirect be eliminated? —Preceding unsigned comment added by Kaiserkarl13 ( talk • contribs) 14:25, 30 April 2008 (UTC)
The bricks and mortar analogy is lame...can we rewrite this? Punctilius ( talk) 05:14, 9 September 2008 (UTC)
Commented Out Section
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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:
This is also called the "Harmonic vibrational and rigid-rotor model." Vibronic HamiltonianThis 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. Solving the molecular Schrödinger equationThe molecular Schrödinger equation is given by 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. LimitationsWhen 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.
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I moved the commented out sections from the front page to the box above. -- Kkmurray ( talk) 14:32, 19 December 2010 (UTC)
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.wormer 15:31, 8 December 2006 (UTC)
The notation used in this article:
is unusual for differentiation with respect to a vector. More common is either or introduction of components
with . Is this notation on purpose, or just by mistake?-- P.wormer 10:56, 26 December 2006 (UTC)
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)
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||||||||||||
|
Hmm...Merge with Electronic Hamiltonian? -- HappyCamper 20:43, 23 September 2006 (UTC)
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.wormer 15:31, 8 December 2006 (UTC)
The notation used in this article:
is unusual for differentiation with respect to a vector. More common is either or introduction of components
with . Is this notation on purpose, or just by mistake?-- P.wormer 10:56, 26 December 2006 (UTC)
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.wormer 14:02, 1 January 2007 (UTC)
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? -- HappyCamper 03:00, 5 January 2007 (UTC)
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.wormer 16:06, 10 January 2007 (UTC)
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.wormer 17:54, 13 January 2007 (UTC)
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. Gershwinrb 08:34, 19 January 2007 (UTC)
The collection of electronic energies for varying nuclear coordinates R forms one potential energy surface that enters one nuclear Schroedinger eq.-- P.wormer 15:00, 22 January 2007 (UTC)
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).
Should this redirect be eliminated? —Preceding unsigned comment added by Kaiserkarl13 ( talk • contribs) 14:25, 30 April 2008 (UTC)
The bricks and mortar analogy is lame...can we rewrite this? Punctilius ( talk) 05:14, 9 September 2008 (UTC)
Commented Out Section
|
---|
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:
This is also called the "Harmonic vibrational and rigid-rotor model." Vibronic HamiltonianThis 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. Solving the molecular Schrödinger equationThe molecular Schrödinger equation is given by 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. LimitationsWhen 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 moved the commented out sections from the front page to the box above. -- Kkmurray ( talk) 14:32, 19 December 2010 (UTC)
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.wormer 15:31, 8 December 2006 (UTC)
The notation used in this article:
is unusual for differentiation with respect to a vector. More common is either or introduction of components
with . Is this notation on purpose, or just by mistake?-- P.wormer 10:56, 26 December 2006 (UTC)
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)