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{{Computational physics}}
The '''Morse potential''', named after physicist [[Philip M. Morse]], is a convenient model for the [[potential energy]] of a [[diatomic molecule]]. It is a better approximation for the [[oscillation|vibrational]] structure of the molecule than the QHO ([[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 band|overtone]] and [[Hot band#Combination bands|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|Morse/Long-range]]) potential, first developed by Professor [[Robert J. LeRoy|Robert J. Le Roy]] of [[University of Waterloo]], which is currently the most popular potential energy function used for fitting spectroscopic data.

==Potential energy function==
[[Image:Morse-potential.png|400px|thumb|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''<sub>e</sub> is larger than the true energy required for dissociation ''D''<sub>0</sub> due to the zero point energy of the lowest (''v'' = 0) vibrational level.]]
The Morse potential energy function is of the form

:<math>V(r) = D_e ( 1-e^{-a(r-r_e)} )^2</math>

Here <math>r</math> is the distance between the atoms, <math>r_e</math> is the equilibrium bond distance, <math>D_e</math> is the well depth (defined relative to the dissociated atoms), and <math>a</math> controls the 'width' of the potential (the smaller <math>a</math> is, the larger the well). The [[Bond dissociation energy|dissociation energy]] of the bond can be calculated by subtracting the [[zero point energy]] <math>E(0)</math> from the depth of the well. The [[force constant]] of the bond can be found by Taylor expansion of <math>V(r)</math> around <math>r=r_e</math> to the second [[derivative]] of the potential energy function, from which it can be shown that the parameter, <math>a</math>, is

:<math>a=\sqrt{k_e/2D_e}</math>,

where <math>k_e</math> is the force constant at the minimum of the well.

Since the [[Potential energy#Relation between potential energy and force|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>V(r) = D_e (( 1-e^{-a(r-r_e)} )^2 -1)</math>
which is usually written as

:<math>V(r) = D_e ( e^{-2a(r-r_e)}-2e^{-a(r-r_e)} )</math>

where <math>r</math> is now the coordinate perpendicular to the surface. This form approaches zero at infinite <math>r</math> and equals <math>-D_e</math> at its minimum, i.e. <math>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==

Like the [[quantum harmonic oscillator]], the energies and eigenstates of the Morse potential can be found using operator methods.<ref>F. Cooper, A. Khare, U. Sukhatme, ''Supersymmetry in Quantum Mechanics'', World Scientific, 2001, Table 4.1</ref>
One approach involves applying the [[factorization method]] to the Hamiltonian.

To write the [[stationary state]]s on the Morse potential, i.e. solutions <math>\Psi(v)</math> and <math>E(v)</math>  of the following [[Schrödinger equation]]:

:<math>\left(-\frac{\hbar ^2 }{2 m }\frac{\partial ^2}{\partial r^2}+V(r)\right)\Psi(v)=E(v)\Psi(v),</math>

it is convenient to introduce the new variables:

:<math>x=a r

\text{;  }

x_e=a  r_e

\text{;  }

\lambda =\frac{\sqrt{2 m  D_e}}{a \hbar }

\text{;  }

\varepsilon _v=\frac{2 m }{a^2\hbar ^2}E(v).
</math>

Then, the [[Schrödinger equation]] takes the simple form:

:<math>
\left(-\frac{\partial ^2}{\partial x^2}+V(x)\right)\Psi _n(x)=\varepsilon _n\Psi _n(x),
</math>
:<math>
V(x)=\lambda ^2\left(e^{-2\left(x-x_e\right)}-2e^{-\left(x-x_e\right)}\right).
</math>
Its [[eigenvalue]]s and [[eigenstate]]s can be written as:
:<math>
\varepsilon _n=-\left(\lambda -n-\frac{1}{2}\right)^2
</math>
:<math>
\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>
z=2\lambda  e^{-\left(x-x_e\right)}
\text{;  }
N_n=n!\left[\frac{a\left(2\lambda-2n-1\right)}{\Gamma (n+1)\Gamma (2\lambda -n)}\right]^{\frac{1}{2}}
</math>  and <math>L_n^{\alpha }(z)</math> is a [[Laguerre polynomial]]:
:<math>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 + 2)/\Gamma (\alpha +2)}{\Gamma (n+1)} \, _1F_1(-n,\alpha +1,z),
</math>

There also exists the following important analytical expression for [[Matrix (mathematics)|matrix]] elements of the coordinate operator (here it is assumed that <math>m>n</math> and <math>N=\lambda -\frac{1}{2}</math>)
<ref name="Another">E. F. Lima and J. E. M. Hornos, "Matrix Elements for the Morse Potential Under an External Field", J. Phys. B: At. Mol. Opt. Phys. 38, pp. 815-825 (2005)</ref>
:<math>
\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>E(v) = h\nu_0 (v+1/2) - \frac{\left[h\nu_0(v+1/2)\right]^2}{4D_e}</math>

where <math>v</math> is the vibrational quantum number, and <math>\nu_0</math> has units of frequency, and is mathematically related to the particle mass, <math>m</math>, and the Morse constants via

:<math>\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>h\nu_0</math>, the energy between adjacent levels decreases with increasing <math>v</math> in the Morse oscillator. Mathematically, the spacing of Morse levels is

:<math>E(v+1) - E(v) = h\nu_0 - (v+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>v_m</math> where <math>E(v_m+1) - E(v_m)</math> is calculated to be zero or negative. Specifically,

:<math>v_m = \frac{2D_e-h\nu_0}{h\nu_0}</math>.

This failure is due to the ''finite'' number of bound levels in the Morse potential, and some maximum <math>v_m</math> that remains bound. For energies above <math>v_m</math>, all the possible energy levels are allowed and the equation for <math>E(v)</math> is no longer valid.

Below <math>v_m</math>, <math>E(v)</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 form<sup>1</sup>

:<math> E_v / hc = \omega_e (v+1/2) - \omega_e\chi_e (v+1/2)^2\,</math>

in which the constants <math>\omega_e</math> and <math>\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>\omega_e</math> represents a [[wavenumber]] obeying  <math>E=hc\omega</math>, and not an [[angular frequency]] given by <math>E=\hbar\omega</math>.

== Morse/Long-range potential ==

An important extension of the Morse potential that has made the Morse form very useful for modern spectroscopy is the MLR ([[Morse/Long-range potential|Morse/Long-range]]) potential, first developed by Professor [[Robert J. LeRoy|Robert J. Le Roy]] of [[University of Waterloo]]. 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 N<sub>2</sub>,<ref name=LeRoy(N2)>{{cite journal|last=Le Roy|first=R. J.|coauthors=Y. Huang, C. Jary|title=An accurate analytic potential function for ground-state N<sub>2</sub> from a direct-potential-fit analysis of spectroscopic data|journal=Journal of Chemical Physics|year=2006|volume=125|page=164310|doi=10.1063/1.2354502|url=http://scitation.aip.org/content/aip/journal/jcp/125/16/10.1063/1.2354502}}</ref> Ca<sub>2</sub>,<ref name=LeRoy(Ca2)>{{cite journal|last=Le Roy|first=Robert J.|coauthors=R. D. E. Henderson|title=A new potential function form incorporating extended long-range behaviour: application to ground-state Ca<sub>2</sub>|journal=Molecular Physics|year=2007|volume=105|page=663|doi=10.1080/00268970701241656|url=http://www.tandfonline.com/doi/abs/10.1080/00268970701241656#.UpoYFcRQKSo}}</ref> KLi,<ref name=Salami(KLi)>{{cite journal|last=Salami|first=H.|coauthors=A. J. Ross, P. Crozet, W. Jastrzebski, P. Kowalczyk, R. J. Le Roy|title=A full analytic potential energy curve for the a<sup>3</sub>Σ<sup>+</sup> state of KLi from a limited vibrational data set|journal=Journal of Chemical Physics|year=2007|volume=126|page=194313|doi=10.1063/1.2734973|url=http://scitation.aip.org/content/aip/journal/jcp/126/19/10.1063/1.2734973}}</ref> MgH,<ref name=Henderson(MgH,MgD) /><ref name=LeRoy(damping) /><ref name=Shayesteh(MgH)>{{cite journal|last=Shayesteh|first=A.|coauthors=R. D. E. Henderson, R. J. Le Roy, P. F. Bernath|title=Ground State Potential Energy Curve and Dissociation Energy of MgH|journal=The Journal of Physical Chemistry A|year=2007|volume=111|page=12495|doi=10.1021/jp075704a|url=http://pubs.acs.org/doi/abs/10.1021/jp075704a}}</ref> several electronic states of Li<sub>2</sub>,<ref name=LeRoy(A-X)>{{cite journal|last=Le Roy|first=Robert J.|coauthors=N. S. Dattani, J. A. Coxon, A. J. Ross, Patrick Crozet, C. Linton|title=Accurate analytic potentials for Li<sub>2</sub>(X) and Li<sub>2</sub>(A) from 2 to 90 Angstroms, and the radiative lifetime of Li(2p)|journal=Journal of Chemical Physics|date=25 November 2009|volume=131|page=204309|doi=10.1063/1.3264688|url=http://scitation.aip.org/content/aip/journal/jcp/131/20/10.1063/1.3264688}}</ref><ref name=Dattani(c-a)>{{cite journal|last=Dattani|first=N. S.|coauthors=R. J. Le Roy|title=A DPF data analysis yields accurate analytic potentials for Li<sub>2</sub>(a) and Li<sub>2</sub>(c) that incorporate 3-state mixing near the c-state asymptote|journal=Journal of Molecular Spectroscopy (Special Issue)|date=8 May 2013|volume=268|pages=199–210|doi=10.1016/j.jms.2011.03.030|url=http://www.sciencedirect.com/science/article/pii/S0022285211000853#}}</ref><ref name=Gunton(A-X)>W. Gunton, M. Semczuk, N. S. Dattani, K. W. Madison, ''High resolution photoassociation spectroscopy of the <sup>6</sup>Li<sub>2</sub> A-state'', http://arxiv.org/abs/1309.5870</ref><ref name=Semczuk(c-a)>{{cite news|first1=M.|last1= Semczuk|first2=X.|last2= Li|first3=W.|last3= Gunton|first4=M.|last4= Haw|first5=N. S.|last5= Dattani|first6=J.|last6= Witz|first7=A. K.|last7= Mills|first8=D. J.|last8= Jones|first9=K. W.|last9= Madison|title=High-resolution photoassociation spectroscopy of the <sup>6</sup>Li<sub>2</sub> c-state|journal=Phys. Rev. A|year=2013|volume=87|pages=052505|doi=10.1103/PhysRevA.87.052505|url=http://pra.aps.org/abstract/PRA/v87/i5/e052505}}</ref><ref name=Gunton(A-X)>W. Gunton, M. Semczuk, N. S. Dattani, K. W. Madison, ''High resolution photoassociation spectroscopy of the <sup>6</sup>Li<sub>2</sub>A-state'', http://arxiv.org/abs/1309.5870</ref><ref name=LeRoy(damping)>{{cite journal|last=Le Roy|first=R. J.|coauthors=C. C. Haugen, J. Tao, H. Li|title=Long-range damping functions improve the short-range behaviour of ‘MLR’ potential energy functions|journal=Molecular Physics|date=February 2011|volume=109|page=435|url=http://scienide2.uwaterloo.ca/~rleroy/Pubn/11MolP_damping.pdf}}</ref> Cs<sub>2</sub>,<ref name=Li(Cs2)>{{cite journal|last=Xie|first=F.|coauthors=L. Li, D. Li, V. B. Sovkov, K. V. Minaev, V. S. Ivanov, A. M. Lyyra, S. Magnier|title=Joint analysis of the Cs<sub>2</sub> a-state and 1 g (33Π1g ) states|journal=Journal of Chemical Physics|year=2011|volume=135|page=02403|doi=10.1063/1.3606397|url=http://scitation.aip.org/content/aip/journal/jcp/135/2/10.1063/1.3606397}}</ref><ref name=Hajigeorgiou(Cs2)>{{cite journal|last=Coxon|first=J. A.|coauthors=P. G. Hajigeorgiou|title=The ground X <sup>1</sup>Σ<sup>+</sup><sub>g</sub> electronic state of the cesium dimer: Application of a direct potential fitting procedure|journal=Journal of Chemical Physics|year=2010|volume=132|page=094105|doi=10.1063/1.3319739|url=http://scitation.aip.org/content/aip/journal/jcp/132/9/10.1063/1.3319739}}</ref> Sr<sub>2</sub>,<ref name=Knockel(Sr2)>{{cite journal|last=Stein|first=A.|coauthors=H. Knockel, E. Tiemann|title=The 1S+1S asymptote of Sr<sub>2</sub> studied by Fourier-transform spectroscopy|journal=The European Physical Journal D|date=April 2010|volume=57|issue=2|pages=171–177|doi=10.1140/epjd/e2010-00058-y|url=http://link.springer.com/article/10.1140%2Fepjd%2Fe2010-00058-y}}</ref> ArXe,<ref name=LeRoy(damping) /><ref name=Piticco(ArXe)>{{cite journal|last=Piticco|first=Lorena|coauthors=F. Merkt, A. A. Cholewinski, F. R. W. McCourt, R. J. Le Roy|title=Rovibrational structure and potential energy function of the ground electronic state of ArXe|journal=Journal of Molecular Spectroscopy|date=December 2010|volume=264|page=83|url=http://www.sciencedirect.com/science/article/pii/S002228521000189X|doi=10.1016/j.jms.2010.08.007}}</ref> LiCa,<ref name=Ivanova(LiCa)>{{cite journal|last=Ivanova|first=Milena|coauthors=A. Stein, A. Pashov, A. V. Stolyarov, H. Knockel, E. Tiemann|title=The X<sup>2</sup>Σ<sup>+</sup> state of LiCa studied by Fourier-transform spectroscopy|journal=Journal of Chemical Physics|year=2011|volume=135|page=174303|doi=10.1063/1.3652755|url=http://scitation.aip.org/content/aip/journal/jcp/135/17/10.1063/1.3652755}}</ref> LiNa,<ref name=Steinke(LiNa)>{{cite journal|last=Steinke|first=M.|coauthors=H. Knockel, E. Tiemann|title=X-state of LiNa studied by Fourier-transform spectroscopy|journal=Physical Review A|date=27 April 2012|volume=85|page=042720|doi=10.1103/PhysRevA.85.042720|url=http://pra.aps.org/abstract/PRA/v85/i4/e042720}}</ref> Br<sub>2</sub>,<ref name=Yukiya(Br2)>{{cite journal|last=Yukiya|first=T.|coauthors=N. Nishimiya, Y. Samejima, K. Yamaguchi, M. Suzuki, C. D. Boonec, I. Ozier, R. J. Le Roy|title=Direct-potential-fit analysis for the system of Br<sub>2</sub>|journal=Journal of Molecular Spectroscopy|date=January 2013|volume=283|pages=32–43|url=http://www.sciencedirect.com/science/article/pii/S0022285212002354|doi=10.1016/j.jms.2012.12.006}}</ref> Mg<sub>2</sub>,<ref name=Knockel(Mg2)>{{cite journal|last=Knockel|first=H.|coauthors=S. Ruhmann, E. Tiemann|title=The X-state of Mg2 studied by Fourier-transform spectroscopy|journal=Journal of Chemical Physics|year=2013|volume=138|page=094303|doi=10.1063/1.4792725|url=http://scitation.aip.org/content/aip/journal/jcp/138/9/10.1063/1.4792725}}</ref> HF,<ref name="Li(HF, HCl, HBr, HI)">{{cite journal|last=Li|first=Gang|coauthors=I. E. Gordon, P. G. Hajigeorgiou, J. A. Coxon, L. S. Rothman|title=Reference spectroscopic data for hydrogen halides, Part II:The line lists|journal=Journal of Quantitative Spectroscopy & Radiative Transfer|date=July 2013|volume=130|pages=284–295|url=http://www.sciencedirect.com/science/article/pii/S0022407313003026|doi=10.1016/j.jqsrt.2013.07.019}}</ref> HCl,<ref name="Li(HF, HCl, HBr, HI)" /> HBr,<ref name="Li(HF, HCl, HBr, HI)" /> HI,<ref name="Li(HF, HCl, HBr, HI)" /> and MgD.<ref name=Henderson(MgH,MgD)>{{cite journal|last=Henderson|first=R. D. E.|coauthors=A. Shayesteh, J. Tao, C. Haugen, P. F. Bernath, R. J. Le Roy|title=Accurate Analytic Potential and Born–Oppenheimer Breakdown Functions for MgH and MgD from a Direct-Potential-Fit Data Analysis|journal=The Journal of Physical Chemistry A|date=4 October 2013|doi=10.1021/jp406680r|url=http://pubs.acs.org/doi/abs/10.1021/jp406680r}}</ref> More sophisticated versions are used for polyatomic molecules.

==See also==
*[[Morse/Long-range potential]]
*[[Lennard-Jones potential]]
*[[Molecular mechanics]]

==References==
*<sup>1</sup> CRC Handbook of chemistry and physics, Ed David R. Lide, 87th ed, Section 9, ''SPECTROSCOPIC CONSTANTS OF DIATOMIC MOLECULES'' p.&nbsp;9-82
* {{cite news
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}}
* {{cite news
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* {{cite news
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* {{cite news
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* {{cite news
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|first3=P. B.
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* {{cite news
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* {{ cite news
|first1=Yaoqi
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|first2=Martin
|last2=Karplus
|first3=Keith D.
|last3=Ball
|first4=R. Stephen
|last4=Bery
|title=The distance fluctuation criterion for melting: Comparison of square-well and Morse Potential models for clusters and homopolymers
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* I.G. Kaplan, in Handbook of Molecular Physics and Quantum Chemistry, Wiley, 2003, p207.

{{Reflist|2}}

{{DEFAULTSORT:Morse Potential}}
[[Category:Chemical bonding]]
[[Category:Quantum chemistry]]