Physics:Morse potential

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The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. Due to its simplicity (only three fitting parameters), it is not used in modern spectroscopy. However, its mathematical form inspired the MLR (Morse/Long-range) potential, which is the most popular potential energy function used for fitting spectroscopic data.

Potential energy function

The Morse potential (blue) and harmonic oscillator potential (green). Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced by ħω, the Morse potential level spacing decreases as the energy approaches the dissociation energy. The dissociation energy De is larger than the true energy required for dissociation D0 due to the zero point energy of the lowest (v = 0) vibrational level.

The Morse potential energy function is of the form

[math]\displaystyle{ V'(r) = D_e ( 1-e^{-a(r-r_e)} )^2 }[/math]

Here [math]\displaystyle{ r }[/math] is the distance between the atoms, [math]\displaystyle{ r_e }[/math] is the equilibrium bond distance, [math]\displaystyle{ D_e }[/math] is the well depth (defined relative to the dissociated atoms), and [math]\displaystyle{ a }[/math] controls the 'width' of the potential (the smaller [math]\displaystyle{ a }[/math] is, the larger the well). The dissociation energy of the bond can be calculated by subtracting the zero point energy [math]\displaystyle{ E_0 }[/math] from the depth of the well. The force constant (stiffness) of the bond can be found by Taylor expansion of [math]\displaystyle{ V'(r) }[/math] around [math]\displaystyle{ r=r_e }[/math] to the second derivative of the potential energy function, from which it can be shown that the parameter, [math]\displaystyle{ a }[/math], is

[math]\displaystyle{ a=\sqrt{k_e/2D_e}, }[/math]

where [math]\displaystyle{ k_e }[/math] is the force constant at the minimum of the well.

Since the zero of potential energy is arbitrary, the equation for the Morse potential can be rewritten any number of ways by adding or subtracting a constant value. When it is used to model the atom-surface interaction, the energy zero can be redefined so that the Morse potential becomes

[math]\displaystyle{ V(r)= V'(r)-D_e = D_e ( 1-e^{-a(r-r_e)} )^2 -D_e }[/math]

which is usually written as

[math]\displaystyle{ V(r) = D_e ( e^{-2a(r-r_e)}-2e^{-a(r-r_e)} ) }[/math]

where [math]\displaystyle{ r }[/math] is now the coordinate perpendicular to the surface. This form approaches zero at infinite [math]\displaystyle{ r }[/math] and equals [math]\displaystyle{ -D_e }[/math] at its minimum, i.e. [math]\displaystyle{ r=r_e }[/math]. It clearly shows that the Morse potential is the combination of a short-range repulsion term (the former) and a long-range attractive term (the latter), analogous to the Lennard-Jones potential.

Vibrational states and energies

Harmonic oscillator (grey) and Morse (black) potentials curves are shown along with their eigenfunctions (respectively green and blue for harmonic oscillator and morse) for the same vibrational levels for nitrogen.

Like the quantum harmonic oscillator, the energies and eigenstates of the Morse potential can be found using operator methods.[1] One approach involves applying the factorization method to the Hamiltonian.

To write the stationary states on the Morse potential, i.e. solutions [math]\displaystyle{ \Psi_n(r) }[/math] and [math]\displaystyle{ E_n }[/math] of the following Schrödinger equation:

[math]\displaystyle{ \left(-\frac{\hbar ^2 }{2 m }\frac{\partial ^2}{\partial r^2}+V(r)\right)\Psi_n(r)=E_n\Psi_n(r), }[/math]

it is convenient to introduce the new variables:

[math]\displaystyle{ x=a r \text{; } x_e=a r_e \text{; } \lambda =\frac{\sqrt{2 m D_e}}{a \hbar } \text{; } \varepsilon _n=\frac{2 m }{a^2\hbar ^2}E_n. }[/math]

Then, the Schrödinger equation takes the simple form:

[math]\displaystyle{ \left(-\frac{\partial ^2}{\partial x^2}+V(x)\right)\Psi _n(x)=\varepsilon _n\Psi _n(x), }[/math]
[math]\displaystyle{ V(x)=\lambda ^2\left(e^{-2\left(x-x_e\right)}-2e^{-\left(x-x_e\right)}\right). }[/math]

Its eigenvalues and eigenstates can be written as:[2]

[math]\displaystyle{ \varepsilon _n=-\left(\lambda -n-\frac{1}{2}\right)^2, }[/math]


[math]\displaystyle{ n=0,1,\ldots,\left[\lambda-\frac{1}{2}\right], }[/math]

with [x] denoting the largest integer smaller than x.

[math]\displaystyle{ \Psi _n(z)=N_nz^{\lambda -n-\frac{1}{2}}e^{-\frac{1}{2}z}L_n^{(2\lambda -2n-1)}(z), }[/math]

where [math]\displaystyle{ z=2\lambda e^{-\left(x-x_e\right)} \text{; } N_n=\left[\frac{n!\left(2\lambda-2n-1\right)}{\Gamma (2\lambda - n)}\right]^{\frac{1}{2}} }[/math] and [math]\displaystyle{ L_n^{(\alpha) }(z) }[/math] is a generalized Laguerre polynomial:

[math]\displaystyle{ L_n^{(\alpha) }(z) = \frac{z^{-\alpha }e^z}{n!} \frac{d^n}{d z^n}\left(z^{n + \alpha } e^{-z}\right)=\frac{\Gamma (\alpha + n + 1)/\Gamma (\alpha +1)}{n!} \, _1F_1(-n,\alpha +1,z), }[/math]

There also exists the following important analytical expression for matrix elements of the coordinate operator (here it is assumed that [math]\displaystyle{ m\gt n }[/math] and [math]\displaystyle{ N=\lambda -\frac{1}{2} }[/math]) [3]

[math]\displaystyle{ \left\langle \Psi _m|x|\Psi _n\right\rangle =\frac{2(-1)^{m-n+1}}{(m-n)(2N-n-m)} \sqrt{\frac{(N-n)(N-m)\Gamma (2N-m+1)m!}{\Gamma (2N-n+1)n!}}. }[/math]

The eigenenergies in the initial variables have form:

[math]\displaystyle{ E_n = h\nu_0 (n+1/2) - \frac{\left[h\nu_0(n+1/2)\right]^2}{4D_e} }[/math]

where [math]\displaystyle{ n }[/math] is the vibrational quantum number, and [math]\displaystyle{ \nu_0 }[/math] has units of frequency, and is mathematically related to the particle mass, [math]\displaystyle{ m }[/math], and the Morse constants via

[math]\displaystyle{ \nu_0 = \frac{a}{2\pi} \sqrt{2D_e/m}. }[/math]

Whereas the energy spacing between vibrational levels in the quantum harmonic oscillator is constant at [math]\displaystyle{ h\nu_0 }[/math], the energy between adjacent levels decreases with increasing [math]\displaystyle{ v }[/math] in the Morse oscillator. Mathematically, the spacing of Morse levels is

[math]\displaystyle{ E_{n+1} - E_n = h\nu_0 - (n+1) (h\nu_0)^2/2D_e.\, }[/math]

This trend matches the anharmonicity found in real molecules. However, this equation fails above some value of [math]\displaystyle{ n_m }[/math] where [math]\displaystyle{ E(n_m+1) - E(n_m) }[/math] is calculated to be zero or negative. Specifically,

[math]\displaystyle{ n_m = \frac{2D_e-h\nu_0}{h\nu_0} }[/math] integer part.

This failure is due to the finite number of bound levels in the Morse potential, and some maximum [math]\displaystyle{ n_m }[/math] that remains bound. For energies above [math]\displaystyle{ n_m }[/math], all the possible energy levels are allowed and the equation for [math]\displaystyle{ E_n }[/math] is no longer valid.

Below [math]\displaystyle{ n_m }[/math], [math]\displaystyle{ E_n }[/math] is a good approximation for the true vibrational structure in non-rotating diatomic molecules. In fact, the real molecular spectra are generally fit to the form1

[math]\displaystyle{ E_n / hc = \omega_e (n+1/2) - \omega_e\chi_e (n+1/2)^2\, }[/math]

in which the constants [math]\displaystyle{ \omega_e }[/math] and [math]\displaystyle{ \omega_e\chi_e }[/math] can be directly related to the parameters for the Morse potential.

As is clear from dimensional analysis, for historical reasons the last equation uses spectroscopic notation in which [math]\displaystyle{ \omega_e }[/math] represents a wavenumber obeying [math]\displaystyle{ E=hc\omega }[/math], and not an angular frequency given by [math]\displaystyle{ E=\hbar\omega }[/math].

Morse/Long-range potential

An important extension of the Morse potential that made the Morse form very useful for modern spectroscopy is the MLR (Morse/Long-range) potential.[4] The MLR potential is used as a standard for representing spectroscopic and/or virial data of diatomic molecules by a potential energy curve. It has been used on N2,[5] Ca2,[6] KLi,[7] MgH,[8][9][10] several electronic states of Li2,[4][11][12][13][9] Cs2,[14][15] Sr2,[16] ArXe,[9][17] LiCa,[18] LiNa,[19] Br2,[20] Mg2,[21] HF,[22][23] HCl,[22][23] HBr,[22][23] HI,[22][23] MgD,[8] Be2,[24] BeH,[25] and NaH.[26] More sophisticated versions are used for polyatomic molecules.

See also


  • 1 CRC Handbook of chemistry and physics, Ed David R. Lide, 87th ed, Section 9, SPECTROSCOPIC CONSTANTS OF DIATOMIC MOLECULES pp. 9–82
  • Morse, P. M. (1929). "Diatomic molecules according to the wave mechanics. II. Vibrational levels". Phys. Rev. 34: pp. 57–64. doi:10.1103/PhysRev.34.57. Bibcode1929PhRv...34...57M. 
  • Girifalco, L. A.; Weizer, G. V. (1959). "Application of the Morse Potential Function to cubic metals". Phys. Rev. 114 (3): p. 687. doi:10.1103/PhysRev.114.687. Bibcode1959PhRv..114..687G. 
  • Shore, Bruce W. (1973). "Comparison of matrix methods applied to the radial Schrödinger eigenvalue equation: The Morse potential". J. Chem. Phys. 59 (12): p. 6450. doi:10.1063/1.1680025. Bibcode1973JChPh..59.6450S. 
  • Keyes, Robert W. (1975). "Bonding and antibonding potentials in group-IV semiconductors". Phys. Rev. Lett. 34 (21): pp. 1334–1337. doi:10.1103/PhysRevLett.34.1334. Bibcode1975PhRvL..34.1334K. 
  • Lincoln, R. C.; Kilowad, K. M.; Ghate, P. B. (1967). "Morse-potential evaluation of second- and third-order elastic constants of some cubic metals". Phys. Rev. 157 (3): pp. 463–466. doi:10.1103/PhysRev.157.463. Bibcode1967PhRv..157..463L. 
  • Dong, Shi-Hai; Lemus, R.; Frank, A. (2001). "Ladder operators for the Morse potential". Int. J. Quantum Chem. 86 (5): pp. 433–439. doi:10.1002/qua.10038. 
  • Zhou, Yaoqi; Karplus, Martin; Ball, Keith D.; Bery, R. Stephen (2002). "The distance fluctuation criterion for melting: Comparison of square-well and Morse Potential models for clusters and homopolymers". J. Chem. Phys. 116 (5): pp. 2323–2329. doi:10.1063/1.1426419. 
  • I.G. Kaplan, in Handbook of Molecular Physics and Quantum Chemistry, Wiley, 2003, p207.
  1. F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific, 2001, Table 4.1
  2. Dahl, J.P.; Springborg, M. (1988). "The Morse Oscillator in Position Space, Momentum Space, and Phase Space". The Journal of Chemical Physics 88 (7): 4535. doi:10.1063/1.453761. Bibcode1988JChPh..88.4535D. 
  3. Lima, Emanuel F de; Hornos, José E M. (2005). "Matrix elements for the Morse potential under an external field". Journal of Physics B 38 (7): 815–825. doi:10.1088/0953-4075/38/7/004. Bibcode2005JPhB...38..815D. 
  4. 4.0 4.1 Le Roy, Robert J.; N. S. Dattani; J. A. Coxon; A. J. Ross; Patrick Crozet; C. Linton (25 November 2009). "Accurate analytic potentials for Li2(X) and Li2(A) from 2 to 90 Angstroms, and the radiative lifetime of Li(2p)". Journal of Chemical Physics 131 (20): 204309. doi:10.1063/1.3264688. PMID 19947682. Bibcode2009JChPh.131t4309L. 
  5. Le Roy, R. J.; Y. Huang; C. Jary (2006). "An accurate analytic potential function for ground-state N2 from a direct-potential-fit analysis of spectroscopic data". Journal of Chemical Physics 125 (16): 164310. doi:10.1063/1.2354502. PMID 17092076. Bibcode2006JChPh.125p4310L. 
  6. Le Roy, Robert J.; R. D. E. Henderson (2007). "A new potential function form incorporating extended long-range behaviour: application to ground-state Ca2". Molecular Physics 105 (5–7): 663–677. doi:10.1080/00268970701241656. Bibcode2007MolPh.105..663L. 
  7. Salami, H.; A. J. Ross; P. Crozet; W. Jastrzebski; P. Kowalczyk; R. J. Le Roy (2007). "A full analytic potential energy curve for the a3Σ+ state of KLi from a limited vibrational data set". Journal of Chemical Physics 126 (19): 194313. doi:10.1063/1.2734973. PMID 17523810. Bibcode2007JChPh.126s4313S. 
  8. 8.0 8.1 Henderson, R. D. E.; A. Shayesteh; J. Tao; C. Haugen; P. F. Bernath; R. J. Le Roy (4 October 2013). "Accurate Analytic Potential and Born–Oppenheimer Breakdown Functions for MgH and MgD from a Direct-Potential-Fit Data Analysis". The Journal of Physical Chemistry A 117 (50): 13373–87. doi:10.1021/jp406680r. PMID 24093511. Bibcode2013JPCA..11713373H. 
  9. 9.0 9.1 9.2 Le Roy, R. J.; C. C. Haugen; J. Tao; H. Li (February 2011). "Long-range damping functions improve the short-range behaviour of 'MLR' potential energy functions". Molecular Physics 109 (3): 435–446. doi:10.1080/00268976.2010.527304. Bibcode2011MolPh.109..435L. 
  10. Shayesteh, A.; R. D. E. Henderson; R. J. Le Roy; P. F. Bernath (2007). "Ground State Potential Energy Curve and Dissociation Energy of MgH". The Journal of Physical Chemistry A 111 (49): 12495–12505. doi:10.1021/jp075704a. PMID 18020428. Bibcode2007JPCA..11112495S. 
  11. Dattani, N. S.; R. J. Le Roy (8 May 2013). "A DPF data analysis yields accurate analytic potentials for Li2(a) and Li2(c) that incorporate 3-state mixing near the c-state asymptote". Journal of Molecular Spectroscopy 268 (1–2): 199–210. doi:10.1016/j.jms.2011.03.030. Bibcode2011JMoSp.268..199.. 
  12. Gunton, Will; Semczuk, Mariusz; Dattani, Nikesh S.; Madison, Kirk W. (2013). "High-resolution photoassociation spectroscopy of the 6Li2 A(11Σ+u) state". Physical Review A 88 (6): 062510. doi:10.1103/PhysRevA.88.062510. Bibcode2013PhRvA..88f2510G. 
  13. Semczuk, M.; Li, X.; Gunton, W.; Haw, M.; Dattani, N. S.; Witz, J.; Mills, A. K.; Jones, D. J. et al. (2013). "High-resolution photoassociation spectroscopy of the 6Li2 c-state". Phys. Rev. A 87: pp. 052505. doi:10.1103/PhysRevA.87.052505. Bibcode2013PhRvA..87e2505S. 
  14. Xie, F.; L. Li; D. Li; V. B. Sovkov; K. V. Minaev; V. S. Ivanov; A. M. Lyyra; S. Magnier (2011). "Joint analysis of the Cs2 a-state and 1 g (33Π1g ) states". Journal of Chemical Physics 135 (2): 02403. doi:10.1063/1.3606397. PMID 21766938. Bibcode2011JChPh.135b4303X. 
  15. Coxon, J. A.; P. G. Hajigeorgiou (2010). "The ground X 1Σ+g electronic state of the cesium dimer: Application of a direct potential fitting procedure". Journal of Chemical Physics 132 (9): 094105. doi:10.1063/1.3319739. PMID 20210387. Bibcode2010JChPh.132i4105C. 
  16. Stein, A.; H. Knockel; E. Tiemann (April 2010). "The 1S+1S asymptote of Sr2 studied by Fourier-transform spectroscopy". The European Physical Journal D 57 (2): 171–177. doi:10.1140/epjd/e2010-00058-y. Bibcode2010EPJD...57..171S. 
  17. Piticco, Lorena; F. Merkt; A. A. Cholewinski; F. R. W. McCourt; R. J. Le Roy (December 2010). "Rovibrational structure and potential energy function of the ground electronic state of ArXe". Journal of Molecular Spectroscopy 264 (2): 83–93. doi:10.1016/j.jms.2010.08.007. Bibcode2010JMoSp.264...83P. 
  18. Ivanova, Milena; A. Stein; A. Pashov; A. V. Stolyarov; H. Knockel; E. Tiemann (2011). "The X2Σ+ state of LiCa studied by Fourier-transform spectroscopy". Journal of Chemical Physics 135 (17): 174303. doi:10.1063/1.3652755. PMID 22070298. Bibcode2011JChPh.135q4303I. 
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  20. Yukiya, T.; N. Nishimiya; Y. Samejima; K. Yamaguchi; M. Suzuki; C. D. Boonec; I. Ozier; R. J. Le Roy (January 2013). "Direct-potential-fit analysis for the system of Br2". Journal of Molecular Spectroscopy 283: 32–43. doi:10.1016/j.jms.2012.12.006. Bibcode2013JMoSp.283...32Y. 
  21. Knockel, H.; S. Ruhmann; E. Tiemann (2013). "The X-state of Mg2 studied by Fourier-transform spectroscopy". Journal of Chemical Physics 138 (9): 094303. doi:10.1063/1.4792725. PMID 23485290. Bibcode2013JChPh.138i4303K. 
  22. 22.0 22.1 22.2 22.3 Li, Gang; I. E. Gordon; P. G. Hajigeorgiou; J. A. Coxon; L. S. Rothman (July 2013). "Reference spectroscopic data for hydrogen halides, Part II:The line lists". Journal of Quantitative Spectroscopy & Radiative Transfer 130: 284–295. doi:10.1016/j.jqsrt.2013.07.019. Bibcode2013JQSRT.130..284L. 
  23. 23.0 23.1 23.2 23.3 Coxon, John A.; Hajigeorgiou, Photos G. (2015). "Improved direct potential fit analyses for the ground electronic states of the hydrogen halides: HF/DF/TF, HCl/DCl/TCl, HBr/DBr/TBr and HI/DI/TI". Journal of Quantitative Spectroscopy and Radiative Transfer 151: 133–154. doi:10.1016/j.jqsrt.2014.08.028. Bibcode2015JQSRT.151..133C. 
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  26. Walji, Sadru-Dean; Sentjens, Katherine M.; Le Roy, Robert J. (2015). "Dissociation energies and potential energy functions for the ground X 1Σ+ and "avoided-crossing" A 1Σ+ states of NaH". The Journal of Chemical Physics 142 (4): 044305. doi:10.1063/1.4906086. PMID 25637985. Bibcode2015JChPh.142d4305W.