Talk:Morse potential

message: For the sake of accuracy it may be worth noting the following fact which is sometimes overlooked.
The Schrodinger equation for the Morse potential V(r) can be solved analytically, obtaining the well-known eigenfunctions psi(r) involving Laguerre associated functions and the closed-form expression for the eigenvalues quoted in the article, imposing as boundary conditions psi(-infty)=psi(+infty) = 0. However, as the meaning of r is that of a bond distance, the physical range where the wavefunctions should live is (0, +infty), not (-infty, +infty).
The eigenfunctions which obey the correct boundary conditions psi(0)=psi(+infty)=0 can be formally written down using expressions involving the confluent hypergeometric function but they are unwieldy to manipulate. Also, there's no closed-formed expression for the eigenvalues in this case.
From the practical point of view, however, for parameters of the potential relevant for molecules V(0) is very large (say, 10 times D_e) and increases exponentially for r<0 so the wavefunctions are vanishingly small for r<0 and the "standard" solutions are exceptionally good (a numerical study of a test-case shows that the relative error on the eigenvalues of the standard expression is of the order of 1.e-30 and hence completely negligible).  —Preceding unsigned comment added by L0rents (talk • contribs) 22:37, 6 July 2009 (UTC)
thread_title:  Adding a note on boundary conditions? 

message: This section needs to be expanded or at least a reference. Where it stands right now is unhelpful
08:36, 1 November 2008 (UTC)
thread_title:  Solving Schrödinger's equation for the Morse oscillator 

message: I am confused with the expression for the eigenvalues. It seems to me that expression for \nu_0 is incorrect and corresponds in fact to \omega_0 ( 2\pi difference). Can anybody check?
195.131.214.163 08:36, 11 September 2007 (UTC)
thread_title:  Spectrum 

message: Can somebody please fix this page so that there is only one symbol for the vibrational quantum number? 'v' (not nu) is the most commonly used vibrational quantum number (just like in the picture... except somebody used 'nu' instead of 'v', which can be confusing as well). Thanks
I am not an expert on this subject, but I think that we are missing -De
in the potential equation, potential of a molecule is negative right?
look at:
http://www.users.csbsju.edu/~frioux/h2-virial/virialh2.htm
and
http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/hmol.html#c1
:: I have changed the vibrational quantum number to  v  (AKA " v ") to match the figure. I have also changed the morse constants to \nu's, which looks quite a bit like "v", and may confuse some people. Oh well.  That gives them the chance to ask "what's \nu (new)?"
:: As for De, I do not favor changing the Morse potential to include such a constant. This article is about the solution to the quantum mechanical system, which is the same, regardless of the (arbitrary) choice of zero for the potential energy. Besides, a Morse potential is also a valid approximation for molecular dissociation on an excited electronic surface, which can produce excited electronic states of the atoms and lead to very different dissociation energies.Zolot 00:55, 24 February 2007 (UTC)
thread_title:  Notations 

message: I feel the section "Vibrational States and Energies" might do well to focus equally on the \omega_e and \chi_e \omega_e formulation for the Morse potential.  Specifically, show how  a and D_e are related to \omega_e and \chi_e \omega_e.  This is not easily findable elsewhere on the web, and it's the dominant formulation in spectrometry. It would be especially helpful for budding chemists that are trying to interpret and compare spectrometric constants  —Preceding unsigned comment added by 134.84.1.67 (talk) 02:32, 21 December 2010 (UTC)
thread_title:  Connection of vibrational frequency to standard fitting parameters 

message: I'm curious whether this potential is somehow derived from actual interaction potentials in some way. Is it just made to be computationally convenient for analytical calculations? If so, when might I choose Morse over Lennard-Jones or some other potential of the same qualitative shape? --Nanite (talk) 08:19, 19 August 2013 (UTC)
replies: text: The references include a link to the abstract of Morse's original 1929 paper (and the paper itself behind a paywall). The abstract says that the potential energy function is assumed to be of a form similar to those required by Heitler and London. This would be a reference to the valence bond theory of Heitler and London, which assumed a certain form for the wavefunction of H2 and other diatomics. From the wavefunction at a given internuclear distance, one can calculate Epot = <ψ|H|ψ> / <ψ|ψ> which I presume leads to a Morse-like potential. Dirac66 (talk) 00:35, 20 August 2013 (UTC)
replies: text: There is no indication of anything except mathematical convenience in Morse's paper.  He says The function which it is proposed to use here is the simple one... This function...in general gives curves of a very similar form to the few potential energy curves which have been calculated theoretically, citing three papers that include Heitler and London's 1927 paper in Zeits. Physik. 150.203.123.132 (talk) 03:14, 12 August 2014 (UTC)
thread_title:  where does this potential come from? 

message: Hmm.  I think that there is a missing factor of a in the normalisation factor for the wave functions.  I'll check when I get around to it.  150.203.123.132 (talk) 03:16, 12 August 2014 (UTC)
replies: text: Yes, I think a factor of a is missing in the normalisation. The cited article doi:10.1063/1.453761 is missing that factor, but it is present for example in doi:10.1103/PhysRev.34.57. JPMalhado (talk) 17:36, 24 July 2020 (UTC)
replies: text: I also think that a factor a1/2 is missing since the wavefunctions as presented are dimensionless. They should have a physical dimension of L-1/2, so that when integrated the corresponding probability amplitudes are dimensionless. I have checked numerically that this correction is fine for the following parameters (for v<2): re = 2.618887 a0, a = 1.096648 a0-1, De = 0.2740735 Ha. 81.223.14.210 (talk) 14:46, 6 June 2023 (UTC)
replies: text: Agreed. The cited article expresses the wavefunction as a function of y = ax (where −ax is the exponent in the Morse potential). Since the authors normalize Ψ(y), the appropriate normalization constant for them has indeed no a, but for our Ψ(x) = Ψ(y/a), the a-term must be included. --Auranofin (talk) 18:37, 1 September 2023 (UTC)
thread_title:  Typo?