Morse/Longrange potential
Computational physics 

Numerical analysis · Simulation 
Particle

Owing to the simplicity of the Morse potential (it only has three adjustable parameters), it is not used in modern spectroscopy. The MLR (Morse/Longrange) potential is a modern version of the Morse potential which has the correct theoretical longrange form of the potential naturally built into it.^{[1]} It was first introduced by professor Robert J. Le Roy of University of Waterloo, professor Nike Dattani of Oxford University and professor John A. Coxon of Dalhousie University in 2009^{[1]} Since then it has been an important tool for spectroscopists to represent experimental data, verify measurements, and make predictions. It is particularly renowned for its extrapolation capability when data for certain regions of the potential are missing, its ability to predict energies with accuracy often better than the most sophisticated ab initio techniques, and its ability to determine precise empirical values for physical parameters such as the dissociation energy, equilibrium bond length, and longrange constants. Cases of particular note include:
 the cstate of Li_{2}: where the MLR potential was successfully able to bridge a gap of more than 5000 cm^{−1} in experimental data.^{[2]} Two years later it was found that Dattani's MLR potential was able to successfully predict the energies in the middle of this gap, correctly within about 1 cm^{−1}.^{[3]} The accuracy of these predictions was much better than the most sophisticated ab initio techniques at the time.^{[4]}
 the Astate of Li_{2}: where Le Roy et al.^{[1]} constructed an MLR potential which determined the C_{3} value for atomic lithium to a higherprecision than any previously measured atomic oscillator strength, by an order of magnitude.^{[5]} This lithium oscillator strength is related to the radiative lifetime of atomic lithium and is used as a benchmark for atomic clocks and measurements of fundamental constants. It has been said that this work by Le Roy et al. was a "landmark in diatomic spectral analysis".^{[5]}
 the astate of KLi: where an analytic global (MLR) potential was successfully built despite there only being a small amount of data near the top of the potential.^{[6]}
Historical origins
The MLR potential is based on the classic Morse potential which was first introduced in 1929 by Philip M. Morse. A primitive version of the MLR potential was first introduced in 2006 by professor Robert J. Le Roy and colleagues for a study on N_{2}.^{[7]} This primitive form was used on Ca_{2},^{[8]} KLi^{[6]} and MgH,^{[9]}^{[10]}^{[11]} before the more modern version was introduced in 2009 by Le Roy, Dattani, and Coxon.^{[1]} A further extension of the MLR potential referred to as the MLR3 potential was introduced in a 2010 study of Cs_{2},^{[12]} and this potential has since been used on HF,^{[13]}^{[14]} HCl,^{[13]}^{[14]} HBr^{[13]}^{[14]} and HI.^{[13]}^{[14]}
Function
The Morse/Longrange potential energy function is of the form
where for large ,
 ,
so is defined according to the theoretically correct longrange behavior expected for the interatomic interaction.
This longrange form of the MLR model is guaranteed because the argument of the exponent is defined to have longrange behavior:
 ,
where is the equilibrium bond length.
There are a few ways in which this longrange behavior can be achieved, the most common is to make a polynomial that is constrained to become at longrange:
 ,
 ,
where n is an integer greater than 1, which value is defined by the model chosen for the longrange potential .
It is clear to see that:
 .
Applications
The MLR potential has successfully summarized all experimental spectroscopic data (and/or virial data) for a number of diatomic molecules, including: N_{2},^{[7]} Ca_{2},^{[8]} KLi,^{[6]} MgH,^{[9]}^{[10]}^{[11]} several electronic states of Li_{2},^{[1]}^{[2]}^{[15]}^{[3]}^{[10]} Cs_{2},^{[16]}^{[12]} Sr_{2},^{[17]} ArXe,^{[10]}^{[18]} LiCa,^{[19]} LiNa,^{[20]} Br_{2},^{[21]} Mg_{2},^{[22]} HF,^{[13]}^{[14]} HCl,^{[13]}^{[14]} HBr,^{[13]}^{[14]} HI,^{[13]}^{[14]} MgD,^{[9]} Be_{2},^{[23]} BeH,^{[24]} and NaH.^{[25]} More sophisticated versions are used for polyatomic molecules.
It has also become customary to fit ab initio points to the MLR potential, to achieve a fully analytic ab initio potential and to take advantage of the MLR's ability to incorporate the correct theoretically known short and longrange behavior into the potential (the latter usually being of higher accuracy than the molecular ab initio points themselves because it is based on atomic ab initio calculations rather than molecular ones, and because features like spinorbit coupling which are difficult to incorporate into molecular ab inito calculations can more easily be treated described in the longrange). Examples of molecules for which the MLR has been used to represent ab initio points are KLi,^{[26]} KBe.^{[27]}
See also
References
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Le Roy, Robert J.; N. S. Dattani; J. A. Coxon; A. J. Ross; Patrick Crozet; C. Linton (2009). "Accurate analytic potentials for Li_{2}(X) and Li_{2}(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.
 ^ ^{a} ^{b} Dattani, N. S.; R. J. Le Roy (8 May 2013). "A DPF data analysis yields accurate analytic potentials for Li_{2}(a) and Li_{2}(c) that incorporate 3state mixing near the cstate asymptote". Journal of Molecular Spectroscopy (Special Issue). 268: 199–210. Bibcode:2011JMoSp.268..199.. doi:10.1016/j.jms.2011.03.030.
 ^ ^{a} ^{b} Semczuk, M.; Li, X.; Gunton, W.; Haw, M.; Dattani, N. S.; Witz, J.; Mills, A. K.; Jones, D. J.; Madison, K. W. (2013). "Highresolution photoassociation spectroscopy of the ^{6}Li_{2} 1^{3}Σ^{+} state". Phys. Rev. A. 87. p. 052505. arXiv:1309.6662. Bibcode:2013PhRvA..87e2505S. doi:10.1103/PhysRevA.87.052505.
 ^ Halls, M. S.; H. B. Schlegal; M. J. DeWitt; G. F. W. Drake (18 May 2001). "Ab initio calculation of the astate interaction potential and vibrational levels of 7Li2" (PDF). Chemical Physics Letters. 339 (5–6): 427–432. Bibcode:2001CPL...339..427H. doi:10.1016/s00092614(01)004031.
 ^ ^{a} ^{b} LY. Tang; ZC. Yan; TY. Shi; J. Mitroy (30 November 2011). "Thirdorder perturbation theory for van der Waals interaction coefficients". Physical Review A. 84 (5): 052502. Bibcode:2011PhRvA..84e2502T. doi:10.1103/PhysRevA.84.052502.
 ^ ^{a} ^{b} ^{c} Salami, H.; A. J. Ross; P. Crozet; W. Jastrzebski; P. Kowalczyk; R. J. Le Roy (2007). "A full analytic potential energy curve for the a^{3}Σ^{+} state of KLi from a limited vibrational data set". Journal of Chemical Physics. 126 (19): 194313. Bibcode:2007JChPh.126s4313S. doi:10.1063/1.2734973.
 ^ ^{a} ^{b} Le Roy, R. J.; Y. Huang; C. Jary (2006). "An accurate analytic potential function for groundstate N_{2} from a directpotentialfit analysis of spectroscopic data". Journal of Chemical Physics. 125 (16): 164310. Bibcode:2006JChPh.125p4310L. doi:10.1063/1.2354502.
 ^ ^{a} ^{b} Le Roy, Robert J.; R. D. E. Henderson (2007). "A new potential function form incorporating extended longrange behaviour: application to groundstate Ca_{2}". Molecular Physics. 105 (5–7): 663–677. doi:10.1080/00268970701241656.
 ^ ^{a} ^{b} ^{c} 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 DirectPotentialFit Data Analysis". The Journal of Physical Chemistry A. 117 (50): 131028105904004. Bibcode:2013JPCA..11713373H. doi:10.1021/jp406680r.
 ^ ^{a} ^{b} ^{c} ^{d} Le Roy, R. J.; C. C. Haugen; J. Tao; H. Li (February 2011). "Longrange damping functions improve the shortrange behaviour of 'MLR' potential energy functions" (PDF). Molecular Physics. 109 (3): 435–446. doi:10.1080/00268976.2010.527304.
 ^ ^{a} ^{b} 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. doi:10.1021/jp075704a. PMID 18020428.
 ^ ^{a} ^{b} Coxon, J. A.; P. G. Hajigeorgiou (2010). "The ground X ^{1}Σ^{+}_{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.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} Li, Gang; I. E. Gordon; P. G. Hajigeorgiou; J. A. Coxon; L. S. Rothman (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.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} 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. Bibcode:2015JQSRT.151..133C. doi:10.1016/j.jqsrt.2014.08.028.
 ^ Gunton, Will; Semczuk, Mariusz; Dattani, Nikesh S.; Madison, Kirk W. (2013). "High resolution photoassociation spectroscopy of the ^{6}Li_{2} A(1^{1}Σ_{u}^{+}) state". Physical Review A. 88 (6). arXiv:1309.5870. Bibcode:2013PhRvA..88f2510G. doi:10.1103/PhysRevA.88.062510.
 ^ 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 Cs_{2} astate and 1_{g}(3^{3}Π1_{1g}) states". Journal of Chemical Physics. 135 (2): 02403. Bibcode:2011JChPh.135b4303X. doi:10.1063/1.3606397.
 ^ Stein, A.; H. Knockel; E. Tiemann (April 2010). "The 1S+1S asymptote of Sr2 studied by Fouriertransform spectroscopy". The European Physical Journal D. 57 (2): 171–177. arXiv:1001.2741. Bibcode:2010EPJD...57..171S. doi:10.1140/epjd/e201000058y.
 ^ 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.
 ^ Ivanova, Milena; A. Stein; A. Pashov; A. V. Stolyarov; H. Knockel; E. Tiemann (2011). "The X^{2}Σ^{+} state of LiCa studied by Fouriertransform spectroscopy". Journal of Chemical Physics. 135 (17): 174303. Bibcode:2011JChPh.135q4303I. doi:10.1063/1.3652755.
 ^ Steinke, M.; H. Knockel; E. Tiemann (27 April 2012). "Xstate of LiNa studied by Fouriertransform spectroscopy". Physical Review A. 85 (4): 042720. Bibcode:2012PhRvA..85d2720S. doi:10.1103/PhysRevA.85.042720.
 ^ Yukiya, T.; N. Nishimiya; Y. Samejima; K. Yamaguchi; M. Suzuki; C. D. Boonec; I. Ozier; R. J. Le Roy (January 2013). "Directpotentialfit analysis for the system of Br_{2}". Journal of Molecular Spectroscopy. 283: 32–43. Bibcode:2013JMoSp.283...32Y. doi:10.1016/j.jms.2012.12.006.
 ^ Knockel, H.; S. Ruhmann; E. Tiemann (2013). "The Xstate of Mg2 studied by Fouriertransform spectroscopy". Journal of Chemical Physics. 138 (9): 094303. Bibcode:2013JChPh.138i4303K. doi:10.1063/1.4792725.

^ "Directpotentialfit analyses yield improved empirical potentials for the ground X^{1}Σ_{g}^{+} state of Be_{2}". Bibcode:2014JChPh.140f4315M. doi:10.1063/1.4864355. Missing or empty
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(help)  ^ 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.

^ "Dissociation energies and potential energy functions for the ground X 1Σ+ and "avoidedcrossing" A ^{1}Σ^{+} states of NaH". Bibcode:2015JChPh.142d4305W. doi:10.1063/1.4906086. Missing or empty
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(help) 
^ "The effect of innershell electrons on the ground and lowlying excited states of KLi: Ab initio study with allelectron basis sets". Bibcode:2013JQSRT.129....8X. doi:10.1016/j.jqsrt.2013.05.025. Missing or empty
url=
(help) 
^ "An ab initio study of the ground and lowlying excited states of KBe with the effect of innershell electrons". Bibcode:2013JChPh.139g4305X. doi:10.1063/1.4818452. Missing or empty
url=
(help)