Talk:Morse potential

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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)Reply

This section needs to be expanded or at least a reference. Where it stands right now is unhelpful 08:36, 1 November 2008 (UTC) —Preceding unsigned comment added by 169.234.139.178 (talk)

I am confused with the expression for the eigenvalues. It seems to me that expression for ${\displaystyle \nu _{0}}$ is incorrect and corresponds in fact to ${\displaystyle \omega _{0}}$ (${\displaystyle 2\pi }$ difference). Can anybody check? 195.131.214.163 08:36, 11 September 2007 (UTC)Reply

Yes, indeed. I added the denominator 2\pi in the expression for \nu_0. The anharmonicity constant is however correct. --Zbyszek.szmek (talk) 22:23, 10 December 2007 (UTC)Reply

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 ${\displaystyle v}$ (AKA " v ") to match the figure. I have also changed the morse constants to ${\displaystyle \nu }$'s, which looks quite a bit like "v", and may confuse some people. Oh well. That gives them the chance to ask "what's ${\displaystyle \nu }$ (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)Reply

I wonder if it would be better to shift the energy of the morse oscillator downwards, so that all the bound states are less than zero? --HappyCamper 17:57, 23 April 2007 (UTC)Reply

I think this suggestion is equivalent to that above, of subtracting a constant (${\displaystyle D_{e}}$) from the equation for the Morse potential. I feel that such a term makes the equation unnecessarily complicated, and if anything obscures physical insight. That is to say, I would expect anyone interested in this article probably already understands that the choice of zero potential energy is arbitrary. Also see my comment about excited electronic states, which (adiabatically) correlate to excited states of the atoms in the Morse approximation. Thus, the only physically relevant quantities are the well depth (${\displaystyle D_{e}}$) and characteristic inverse length (${\displaystyle a}$). Zolot 17:30, 24 April 2007 (UTC)Reply

Many visitors to this page seem to want to re-zero the potential energy function -- which only obscures the physics, as I have already discussed. In response, I have added a brief statement about the zero of a potential energy being arbitrary. Should we also add the discussion, above, about different values being used as the zero? This gets into quite a few ideas that may not be appropriate for a page that's just supposed to be about the Morse oscillator. Zolot 12:15, 17 May 2007 (UTC)Reply

I disagree. I added the alternative formula not because I wanted to re-zero the potential, but because it is a formulation that is also commonly used, and I think it may be helpful to readers who may not immediately recognize the other formulation. Also, it is worth noting that the formula I added is the one originally used by Morse, so it is notable for historical reasons, although I forgot to mention that detail in the article. (Actually, the original formula in the 1929 paper didn't factorize D, so it was something like Dexp(-2ar) - 2Dexp(-ar)). --Itub 14:30, 17 May 2007 (UTC)Reply

Prior to Feb. 2007, the potential on this page was expressed as V(r) = T_o + D_e ( 1-e^{-a(r-r_e)} )^2, algebraically identical to your Dexp(-2ar) - 2Dexp(-ar)), if T_o = -De. Apparently, each of these formulations is commonly used. Which belong on this page? We could list all of them, in which case this page may become another cluttered wiki entry with redundant information. I prefer to list one concise form, and let the reader handle conversion to other formats. Perhaps it should just be emphasized that other formulations are used and are equivalent.

I don't buy the history arguement. Any discussion of calculus (outside of the history of calculus) uses Leibniz's formulation (for it's lucidity), even if Newton's was first chronologically. I don't know if "the history of the Morse potential" is an interesting enough topic to warrant discussion, but make such a section if you like. Zolot 14:35, 18 May 2007 (UTC)Reply

I'm curious, why not simply tack +V(r0) onto the formula? In that form it is generally true for any particular offset specific to the system of interest. —Preceding unsigned comment added by 130.91.197.117 (talk) 19:55, 15 May 2008 (UTC)Reply

Look folks, there is simply no good reason I can see to NOT to include the term +V(r0). It lets the equation be generally true for any applicable system. If anyone wants to further discuss it great, let's discuss it, but please don't just change it back. 130.91.197.117 (talk) 14:21, 3 July 2008 (UTC)Reply

I feel the section "Vibrational States and Energies" might do well to focus equally on the ${\displaystyle \omega _{e}}$ and ${\displaystyle \chi _{e}\omega _{e}}$ formulation for the Morse potential. Specifically, show how a and ${\displaystyle D_{e}}$ are related to ${\displaystyle \omega _{e}}$ and ${\displaystyle \chi _{e}\omega _{e}}$. 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)Reply

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)Reply

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 E_pot = <ψ|H|ψ> / <ψ|ψ> which I presume leads to a Morse-like potential. Dirac66 (talk) 00:35, 20 August 2013 (UTC)Reply

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)Reply

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)Reply

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)Reply

I also think that a factor a^1/2 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): r_e = 2.618887 a₀, a = 1.096648 a₀^-1, De = 0.2740735 Ha. 81.223.14.210 (talk) 14:46, 6 June 2023 (UTC)Reply

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)Reply

I believe there's a problem with the expression of the Morse potential. It is presented as

${\displaystyle V(r)=D_{e}(1-e^{-a(r-r_{e})})^{2}}$

The problem with this expression is that it gives a value for V(r) of 0 at the minimum position r_e, and D_e at infinity. Instead, these should be -D_e at minimum position r_e, and 0 at infinity.

The correct expression for the Morse potential would be

${\displaystyle V(r)=D_{e}(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})})}$ — Preceding unsigned comment added by Jrifkin (talk • contribs) 19:06, 9 March 2025 (UTC)Reply

This article was the subject of a Wiki Education Foundation-supported course assignment, between 26 August 2025 and 17 December 2025. Further details are available on the course page. Student editor(s): Botrma0 (article contribs).

— Assignment last updated by Botrma0 (talk) 02:26, 25 November 2025 (UTC)Reply