Talk:Molecular Hamiltonian

message: 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)
replies: text: Nope, feel free to edit as you see fit. --HappyCamper 04:33, 10 December 2006 (UTC)
thread_title:  Born-Oppenheimer 

message: The notation used in this article:
: \frac{\partial}{\partial R_i}
is unusual for differentiation with respect to a vector. More common is either \nabla_i
or introduction of components 
: \frac{\partial}{\partial R_{i\alpha}} 
with  \alpha = x,y,z .
Is this notation on purpose, or just by mistake?--P.wormer 10:56, 26 December 2006 (UTC)
replies: text: I have seen that kind of notation before, but I think that we should go with the \nabla 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 Hedegaard 16:08, 26 December 2006 (UTC)
thread_title:  Unusual notation 

message: 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)
replies: text: 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.wormer 10:10, 5 January 2007 (UTC)
replies: text: Wow, really? That's news to me. Okay, we'll not use the arrows and keep what we have already. --HappyCamper 11:00, 9 January 2007 (UTC)
thread_title:  Too many bold letters? 

message: 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)
replies: text: What do you mean by skimming along WP:NOR? --HappyCamper 12:09, 13 January 2007 (UTC)
replies: text: 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.wormer 15:11, 13 January 2007 (UTC)
replies: text: 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. --HappyCamper 15:21, 13 January 2007 (UTC)
thread_title:  Finished overhaul 

message: 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)
replies: text: I have fixed it. Born-Oppenheimer Approximation redirect now to Born-Oppenheimer approximation. --Bduke 11:29, 19 January 2007 (UTC)
thread_title:  Redirects 

message: 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)
replies: text:  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.wormer 13:19, 23 January 2007 (UTC)
thread_title:  Potential energy surface(s) 

message: 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)
replies: text: I agree that "adiabatic approximation" should not link here.  Instead, it should redirect to Adiabatic theorem.  I have made the change.  Cgd8d (talk) 20:54, 8 May 2008 (UTC)
thread_title:  Redirected from Adiabatic Approximation 

message: The bricks and mortar analogy is lame...can we rewrite this? Punctilius (talk) 05:14, 9 September 2008 (UTC)
thread_title:  Rewrite 

message: 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 I_{n,eq} \,\; calculated at the equilibrium configuration.
This is also called the "Harmonic vibrational and rigid-rotor model."
===Vibronic Hamiltonian===
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
:\hat{H}_{M,vibronic}=\hat{T}_{e,vibronic}+\hat{T}_{n,vibronic}+V(x_e,X_n)
with 
:\hat{T}_{e,vibronic}=\frac{-\hbar^2}{2m_e}\sum_{x_e} \hat{\nabla}^2_{x_e}\quad \mathrm{and} \quad \hat{T}_{n,vibronic}=\frac{-\hbar^2}{2}\sum_{X_n} \frac{\hat{\nabla}^2_{X_n}}{M_{X_n}}
with the (x_e,X_n) 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 3N-5 or 3N-6 vibrations if N is the number of nuclei, and whether the molecule is linear or nonlinear.
===Solving the molecular Schrödinger equation===
The molecular Schrödinger equation is given by 
:\hat{H}_M \psi_a(x_e,X_n)=E_a \psi_a(x_e,X_n)
where E_a refers to the energy of the state \psi_a(x_e,X_n). 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 \psi_a(x_e,X_n) to a product of the electronic wavefunction and the nuclear vibration wavefunction. This is given by
:\psi_a(x_e,X_n)=\phi_e(x_e,X_n)\cdot \chi_{e,\nu} (X_n)
where e,\nu 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
:\hat{H}_e \phi_e(x_e,X_n)=E_e(X_n)\phi(x_e,X_n)
where
:\hat{H}_e=\hat{H}_M-\hat{T}_n
with the electronic wavefunctions found the nuclear vibrational coordinates X_n=\{X_1,X_2,...X_{3N-6}\} or X_n=\{X_1,X_2,...X_{3N-5}\} 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 X_n. The electronic wavefunctions defines a complete orthonomal set of functions for each X_n so the molecular wavefunction can be expanded in the basis.
:\psi_a(x_e,X_n)=\sum_{e,\nu} \phi_e(x_e,X_n)\cdot \chi_{e,\nu} (X_n)
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
:(\hat{T}_{n,vibronic}+E_{e'}(X_n))\chi_{e',\nu'}(X_n)=E_{e',\nu'}\chi_{e',\nu'}(X_n)
where the expansion coefficients \chi_{e,\nu}(X_n) describes the vibrational eigenfunctions and the E_{e}(X_n) describe the vibrational potential energy. The eigenvalue, E_{e}(X_n) is often approximated by an harmonic function for simplification.
=== Limitations ===
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
:\langle \phi_e(x_e,X_n)|T_n|\phi_e(x_e,X_n)\rangle
where T_n is the nuclear kinetic energy operator and the electronic wavefunction \phi_e 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)
thread_title: Rotational Hamiltonian

message: 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
:\langle \phi_e(x_e,X_n)|T_n|\phi_e(x_e,X_n)\rangle
where T_n is the nuclear kinetic energy operator and the electronic wavefunction \phi_e 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)
thread_title:  Limitations 

message: 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)
replies: text: Nope, feel free to edit as you see fit. --HappyCamper 04:33, 10 December 2006 (UTC)
thread_title:  Born-Oppenheimer 

message: The notation used in this article:
: \frac{\partial}{\partial R_i}
is unusual for differentiation with respect to a vector. More common is either \nabla_i
or introduction of components 
: \frac{\partial}{\partial R_{i\alpha}} 
with  \alpha = x,y,z .
Is this notation on purpose, or just by mistake?--P.wormer 10:56, 26 December 2006 (UTC)
replies: text: I have seen that kind of notation before, but I think that we should go with the \nabla 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 Hedegaard 16:08, 26 December 2006 (UTC)
thread_title:  Unusual notation