Morse potential

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 D_e 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

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

Here ${\displaystyle r}$ is the distance between the atoms, ${\displaystyle r_{e}}$ is the equilibrium bond distance, ${\displaystyle D_{e}}$ is the well depth (defined relative to the dissociated atoms), and ${\displaystyle a}$ controls the 'width' of the potential (the smaller ${\displaystyle a}$ is, the larger the well). The dissociation energy of the bond can be calculated by subtracting the zero point energy ${\displaystyle E_{0}}$ from the depth of the well. The force constant (stiffness) of the bond can be found by Taylor expansion of ${\displaystyle V'(r)}$ around ${\displaystyle r=r_{e}}$ to the second derivative of the potential energy function, from which it can be shown that the parameter, ${\displaystyle a}$, is

${\displaystyle a={\sqrt {k_{e}/2D_{e}}},}$

where ${\displaystyle k_{e}}$ 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

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

which is usually written as

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

where ${\displaystyle r}$ is now the coordinate perpendicular to the surface. This form approaches zero at infinite ${\displaystyle r}$ and equals ${\displaystyle -D_{e}}$ at its minimum, i.e. ${\displaystyle r=r_{e}}$. 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

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 ${\displaystyle \Psi _{n}(r)}$ and ${\displaystyle E_{n}}$ of the following Schrödinger equation:

${\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+V(r)\right)\Psi _{n}(r)=E_{n}\Psi _{n}(r),}$

it is convenient to introduce the new variables:

${\displaystyle x=ar{\text{; }}x_{e}=ar_{e}{\text{; }}\lambda ={\frac {\sqrt {2mD_{e}}}{a\hbar }}{\text{; }}\varepsilon _{n}={\frac {2m}{a^{2}\hbar ^{2}}}E_{n}={\frac {\lambda ^{2}}{D_{e}}}E_{n}.}$

Then, the Schrödinger equation takes the simple form:

${\displaystyle \left(-{\frac {\partial ^{2}}{\partial x^{2}}}+V(x)\right)\Psi _{n}(x)=\varepsilon _{n}\Psi _{n}(x),}$

${\displaystyle V(x)=\lambda ^{2}\left(1-e^{-\left(x-x_{e}\right)}\right)^{2}.}$

Its eigenvalues (reduced by ${\displaystyle D_{e}}$) and eigenstates can be written as: ^[2]

${\displaystyle \varepsilon _{n}=\lambda ^{2}-\left(\lambda -n-{\frac {1}{2}}\right)^{2}=2\lambda \left(n+{\frac {1}{2}}\right)-\left(n+{\frac {1}{2}}\right)^{2},}$

where

${\displaystyle n=0,1,\ldots ,\lfloor \lambda -{\frac {1}{2}}\rfloor ,}$

with ${\displaystyle \lfloor x\rfloor }$ denoting the largest integer smaller than ${\displaystyle x}$, and

${\displaystyle \Psi _{n}(z)=N_{n}z^{\lambda -n-{\frac {1}{2}}}e^{-{\frac {1}{2}}z}L_{n}^{(2\lambda -2n-1)}(z),}$

where ${\displaystyle z=2\lambda e^{-\left(x-x_{e}\right)}{\text{; }}N_{n}=\left[{\frac {n!\left(2\lambda -2n-1\right)a}{\Gamma (2\lambda -n)}}\right]^{\frac {1}{2}}}$ (which satisfies the normalization condition ${\displaystyle \int \mathrm {d} r\,\Psi _{n}^{*}(r)\Psi _{n}(r)=1}$ ) and ${\displaystyle L_{n}^{(\alpha )}(z)}$ is a generalized Laguerre polynomial:

${\displaystyle L_{n}^{(\alpha )}(z)={\frac {z^{-\alpha }e^{z}}{n!}}{\frac {d^{n}}{dz^{n}}}\left(z^{n+\alpha }e^{-z}\right)={\frac {\Gamma (\alpha +n+1)/\Gamma (\alpha +1)}{n!}}\,_{1}F_{1}(-n,\alpha +1,z).}$

There also exists the following analytical expression for matrix elements of the coordinate operator: ^[3]

${\displaystyle \left\langle \Psi _{m}|x|\Psi _{n}\right\rangle ={\frac {2(-1)^{m-n+1}}{(m-n)(2N-n-m)}}{\sqrt {\frac {(N-n)(N-m)\Gamma (2N-m+1)m!}{\Gamma (2N-n+1)n!}}}.}$

which is valid for ${\displaystyle m>n}$ and ${\displaystyle N=\lambda -1/2}$. The eigenenergies in the initial variables have the form:

${\displaystyle E_{n}=h\nu _{0}(n+1/2)-{\frac {\left[h\nu _{0}(n+1/2)\right]^{2}}{4D_{e}}}}$

where ${\displaystyle n}$ is the vibrational quantum number and ${\displaystyle \nu _{0}}$ has units of frequency. The latter is mathematically related to the particle mass, ${\displaystyle m}$, and the Morse constants via

${\displaystyle \nu _{0}={\frac {a}{2\pi }}{\sqrt {2D_{e}/m}}.}$

Whereas the energy spacing between vibrational levels in the quantum harmonic oscillator is constant at ${\displaystyle h\nu _{0}}$, the energy between adjacent levels decreases with increasing ${\displaystyle v}$ in the Morse oscillator. Mathematically, the spacing of Morse levels is

${\displaystyle E_{n+1}-E_{n}=h\nu _{0}-(n+1)(h\nu _{0})^{2}/2D_{e}.\,}$

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

${\displaystyle n_{m}={\frac {2D_{e}-h\nu _{0}}{h\nu _{0}}}}$ integer part.

This failure is due to the finite number of bound levels in the Morse potential, and some maximum ${\displaystyle n_{m}}$ that remains bound. For energies above ${\displaystyle n_{m}}$, all the possible energy levels are allowed and the equation for ${\displaystyle E_{n}}$ is no longer valid.

Below ${\displaystyle n_{m}}$, ${\displaystyle E_{n}}$ 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

${\displaystyle E_{n}/hc=\omega _{e}(n+1/2)-\omega _{e}\chi _{e}(n+1/2)^{2}\,}$

in which the constants ${\displaystyle \omega _{e}}$ and ${\displaystyle \omega _{e}\chi _{e}}$ can be directly related to the parameters for the Morse potential. Specifically,

${\displaystyle a={\sqrt {\frac {8\pi ^{2}c\ m\ \omega _{e}\chi _{e}}{h}}}}$

and

${\displaystyle D_{e}={\frac {\omega _{e}}{4\chi _{e}}}}$

Note that if ${\displaystyle \omega _{e}}$ and ${\displaystyle \omega _{e}\chi _{e}}$ are given in cm${\displaystyle ^{-1}}$, ${\displaystyle c}$ is in cm/s (not m/s), ${\displaystyle m}$ is in kg, and ${\displaystyle h}$ is in J·s, then ${\displaystyle a}$ will be in m${\displaystyle ^{-1}}$ and ${\displaystyle D_{e}}$ will be in cm${\displaystyle ^{-1}}$.

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

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.

Morse/Long-range potential

Main article: Morse/Long-range potential

An extension of the Morse potential that made the Morse form useful for modern (high-resolution) 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

• Lennard-Jones potential
• Molecular mechanics

References

• 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 (1): 57–64. Bibcode:1929PhRv...34...57M. doi:10.1103/PhysRev.34.57.
• Girifalco, L. A.; Weizer, G. V. (1959). "Application of the Morse Potential Function to cubic metals". Phys. Rev. 114 (3): 687. Bibcode:1959PhRv..114..687G. doi:10.1103/PhysRev.114.687. hdl: 2027/uiug.30112106908442 .
• Shore, Bruce W. (1973). "Comparison of matrix methods applied to the radial Schrödinger eigenvalue equation: The Morse potential". J. Chem. Phys. 59 (12): 6450. Bibcode:1973JChPh..59.6450S. doi:10.1063/1.1680025.
• Keyes, Robert W. (1975). "Bonding and antibonding potentials in group-IV semiconductors". Phys. Rev. Lett. 34 (21): 1334–1337. Bibcode:1975PhRvL..34.1334K. doi:10.1103/PhysRevLett.34.1334.
• 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): 463–466. Bibcode:1967PhRv..157..463L. doi:10.1103/PhysRev.157.463.
• Dong, Shi-Hai; Lemus, R.; Frank, A. (2001). "Ladder operators for the Morse potential". Int. J. Quantum Chem. 86 (5): 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): 2323–2329. Bibcode:2002JChPh.116.2323Z. 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" (PDF). The Journal of Chemical Physics. 88 (7): 4535. Bibcode:1988JChPh..88.4535D. doi:10.1063/1.453761. S2CID   97262147.
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. Bibcode:2005JPhB...38..815D. doi:10.1088/0953-4075/38/7/004. S2CID   119976840.
4. 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. Bibcode:2009JChPh.131t4309L. doi:10.1063/1.3264688. PMID   19947682.
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. Bibcode:2006JChPh.125p4310L. doi:10.1063/1.2354502. PMID   17092076. S2CID   32249407.
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. Bibcode:2007MolPh.105..663L. doi:10.1080/00268970701241656. S2CID   94174485.
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. Bibcode:2007JChPh.126s4313S. doi: 10.1063/1.2734973 . PMID   17523810. S2CID   26105905.
8. 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. Bibcode:2013JPCA..11713373H. doi:10.1021/jp406680r. PMID   24093511. S2CID   23016118.
9. 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" (PDF). Molecular Physics. 109 (3): 435–446. Bibcode:2011MolPh.109..435L. doi:10.1080/00268976.2010.527304. S2CID   97119318. Archived from the original (PDF) on 2019-01-08. Retrieved 2013-11-30.
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. Bibcode:2007JPCA..11112495S. CiteSeerX   10.1.1.584.8808 . doi:10.1021/jp075704a. PMID   18020428.
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. arXiv: 1101.1361 . Bibcode:2011JMoSp.268..199D. doi:10.1016/j.jms.2011.03.030. S2CID   119266866.
12. Gunton, Will; Semczuk, Mariusz; Dattani, Nikesh S.; Madison, Kirk W. (2013). "High-resolution photoassociation spectroscopy of the ⁶Li²A(1¹Σ⁺
    _u) state". Physical Review A. 88 (6): 062510. arXiv: 1309.5870 . Bibcode:2013PhRvA..88f2510G. doi:10.1103/PhysRevA.88.062510. S2CID   119268157.
13. Semczuk, M.; Li, X.; Gunton, W.; Haw, M.; Dattani, N. S.; Witz, J.; Mills, A. K.; Jones, D. J.; Madison, K. W. (2013). "High-resolution photoassociation spectroscopy of the ⁶Li₂ c-state". Phys. Rev. A. 87 (5): 052505. arXiv: 1309.6662 . Bibcode:2013PhRvA..87e2505S. doi:10.1103/PhysRevA.87.052505. S2CID   119263860.
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. Bibcode:2011JChPh.135b4303X. doi:10.1063/1.3606397. PMID   21766938.
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. Bibcode:2010JChPh.132i4105C. doi:10.1063/1.3319739. PMID   20210387.
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. arXiv: 1001.2741 . Bibcode:2010EPJD...57..171S. doi:10.1140/epjd/e2010-00058-y. S2CID   119243162.
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. Bibcode:2010JMoSp.264...83P. doi:10.1016/j.jms.2010.08.007. hdl: 20.500.11850/210096 .
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. Bibcode:2011JChPh.135q4303I. doi:10.1063/1.3652755. PMID   22070298.
19. Steinke, M.; H. Knockel; E. Tiemann (27 April 2012). "X-state of LiNa studied by Fourier-transform spectroscopy". Physical Review A. 85 (4): 042720. Bibcode:2012PhRvA..85d2720S. doi:10.1103/PhysRevA.85.042720.
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. Bibcode:2013JMoSp.283...32Y. doi:10.1016/j.jms.2012.12.006.
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. Bibcode:2013JChPh.138i4303K. doi:10.1063/1.4792725. PMID   23485290.
22. 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. Bibcode:2013JQSRT.130..284L. doi:10.1016/j.jqsrt.2013.07.019.
23. 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. Bibcode:2015JQSRT.151..133C. doi:10.1016/j.jqsrt.2014.08.028.
24. Meshkov, Vladimir V.; Stolyarov, Andrey V.; Heaven, Michael C.; Haugen, Carl; Leroy, Robert J. (2014). "Direct-potential-fit analyses yield improved empirical potentials for the ground XΣg+1 state of Be2". The Journal of Chemical Physics. 140 (6): 064315. Bibcode:2014JChPh.140f4315M. doi:10.1063/1.4864355. PMID   24527923.
25. Dattani, Nikesh S. (2015). "Beryllium monohydride (BeH): Where we are now, after 86 years of spectroscopy". Journal of Molecular Spectroscopy. 311: 76–83. arXiv: 1408.3301 . Bibcode:2015JMoSp.311...76D. doi:10.1016/j.jms.2014.09.005. S2CID   118542048.
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. Bibcode:2015JChPh.142d4305W. doi:10.1063/1.4906086. PMID   25637985. S2CID   2481313.