View source for Morse/Long-range potential

{{short description|Model of the potential energy of a diatomic molecule}}
{{Computational physics}}

The '''Morse/Long-range potential''' ('''MLR potential''') is an [[interatomic potential|interatomic interaction model]] for the [[potential energy]] of a [[diatomic molecule]]. Due to the simplicity of the regular [[Morse potential]] (it only has three adjustable parameters), it is very limited in its applicability in modern [[spectroscopy]]. The MLR potential is a modern version of the Morse potential which has the correct theoretical long-range form of the potential naturally built into it.<ref name=LeRoy(A-X)>{{cite journal|last=Le Roy|first=Robert J.|author2=N. S. Dattani |author3=J. A. Coxon |author4=A. J. Ross |author5=Patrick Crozet |author6=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=2009|volume=131|issue=20|page=204309| doi=10.1063/1.3264688| pmid=19947682|bibcode=2009JChPh.131t4309L}}</ref>  It has been an important tool for spectroscopists to represent experimental data, verify measurements, and make predictions. It is useful 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]], [[bond length|equilibrium bond length]], and long-range constants. Cases of particular note include:
# the c-state{{huh|date=September 2024}} of [[dilithium]] ({{chem2|Li2}}): where the MLR potential was successfully able to bridge a gap of more than 5000&nbsp;cm<sup>−1</sup> in experimental data.<ref name=Dattani(c-a)>{{cite journal|last=Dattani|first=N. S.|author2=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 |date=8 May 2011|volume=268 | issue=1–2| pages=199–210|doi=10.1016/j.jms.2011.03.030|bibcode=2011JMoSp.268..199D|arxiv=1101.1361|s2cid=119266866 }}</ref> Two years later it was found that the MLR potential was able to successfully predict the energies in the middle of this gap, correctly within about 1&nbsp;cm<sup>−1</sup>.<ref name=Semczuk(c-a)>{{cite journal|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> 1<sup>3</sup>Σ<sup>+</sup> state|journal=Phys. Rev. A|year=2013|volume=87|issue= 5|page=052505|doi=10.1103/PhysRevA.87.052505 |arxiv=1309.6662 |bibcode=2013PhRvA..87e2505S|s2cid= 119263860}}</ref> The accuracy of these predictions was much better than the most sophisticated ''[[ab initio]]'' techniques at the time.<ref name=Halls(Li2,a-state)>{{cite journal|last=Halls|first=M. S.| author2=H. B. Schlegal |author3=M. J. DeWitt |author4=G. F. W. Drake |title=Ab initio calculation of the a-state interaction potential and vibrational levels of <sup>7</sup>Li<sub>2</sub>|journal=Chemical Physics Letters|date=18 May 2001|volume=339 |issue=5–6|pages=427–432|url=http://www.chem.wayne.edu/schlegel/Pub_folder/236.pdf|doi=10.1016/s0009-2614(01)00403-1 |bibcode=2001CPL...339..427H}}</ref>
# the A-state{{huh|date=September 2024}} of Li<sub>2</sub>: where [[Robert J. LeRoy|Le Roy]] et al.<ref name=LeRoy(A-X) /> constructed an MLR potential which determined the C<sub>3</sub> value for atomic lithium to a higher-precision than any previously measured atomic oscillator strength, by an order of magnitude.<ref name=Tang(C9,C11)>{{cite journal|author= L-Y. Tang|author2=Z-C. Yan|author3= T-Y. Shi|author4=J. Mitroy|title=Third-order perturbation theory for van der Waals interaction coefficients|journal=Physical Review A|date=30 November 2011|volume=84|issue=5|page=052502|doi=10.1103/PhysRevA.84.052502|bibcode=2011PhRvA..84e2502T}}</ref> 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. 
# the a-state{{huh|date=September 2024}} of KLi: where the MLR was used to build an analytic global potential successfully despite there only being a small amount of levels observed near the top of the potential.<ref name=Salami(KLi)>{{cite journal|last=Salami|first=H.|author2=A. J. Ross |author3=P. Crozet |author4=W. Jastrzebski |author5=P. Kowalczyk |author6=R. J. Le Roy |title=A full analytic potential energy curve for the a<sup>3</sup>Σ<sup>+</sup> state of KLi from a limited vibrational data set|journal=Journal of Chemical Physics|year=2007|volume=126|issue=19|page=194313|doi=10.1063/1.2734973|pmid=17523810|bibcode=2007JChPh.126s4313S|doi-access=free}}</ref>

== 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 [[Robert J. LeRoy|Robert J. Le Roy]] and colleagues for a study on [[molecular nitrogen|N<sub>2</sub>]].<ref name=LeRoy(N2)>{{cite journal|last=Le Roy|first=R. J.|author2=Y. Huang |author3=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|issue=16|page=164310 |doi=10.1063/1.2354502 |pmid=17092076|bibcode=2006JChPh.125p4310L}}</ref> This primitive form was used on [[calcium|Ca<sub>2</sub>]],<ref name=LeRoy(Ca2)>{{cite journal|last=Le Roy|first=Robert J.|author2=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|issue=5–7|pages=663–677|doi=10.1080/00268970701241656|bibcode=2007MolPh.105..663L|s2cid=94174485 }}</ref> KLi<ref name=Salami(KLi)/> and [[Magnesium monohydride|MgH]],<ref name=Henderson(MgH,MgD) /><ref name=LeRoy(damping) /><ref name=Shayesteh(MgH)>{{cite journal|last=Shayesteh|first=A.|author2=R. D. E. Henderson |author3=R. J. Le Roy |author4=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|issue=49|pages=12495–12505|doi=10.1021/jp075704a |pmid=18020428 |bibcode=2007JPCA..11112495S|citeseerx=10.1.1.584.8808}}</ref> before the more modern version was introduced in 2009<!-- [removal of names because I added their names elsewhere, other editor can add back if they feel it's warranted] by Le Roy, Dattani, and Coxon -->.<ref name=LeRoy(A-X) /> A further extension of the MLR potential referred to as the MLR3 potential was introduced in a 2010 study of [[cesium|Cs<sub>2</sub>]],<ref name=Hajigeorgiou(Cs2) /> and this potential has since been used on [[hydrogen fluoride|HF]],<ref name="Li(HF, HCl, HBr, HI)">{{cite journal|last=Li|first=Gang|author2=I. E. Gordon |author3=P. G. Hajigeorgiou |author4=J. A. Coxon |author5=L. S. Rothman |title=Reference spectroscopic data for hydrogen halides, Part II: The line lists |journal=Journal of Quantitative Spectroscopy & Radiative Transfer|date=2013|volume=130|pages=284–295 |doi=10.1016/j.jqsrt.2013.07.019 |bibcode=2013JQSRT.130..284L}}</ref><ref name="sciencedirect.com"/> [[hydrogen chloride|HCl]],<ref name="Li(HF, HCl, HBr, HI)" /><ref name="sciencedirect.com"/> [[hydrogen bromide|HBr]]<ref name="Li(HF, HCl, HBr, HI)" /><ref name="sciencedirect.com"/> and [[hydrogen iodide|HI]].<ref name="Li(HF, HCl, HBr, HI)" /><ref name="sciencedirect.com"/>

== Function ==

The Morse/Long-range potential energy function is of the form
<math display="block">V(r) = \mathfrak{D}_e \left( 1- \frac{u(r)}{u(r_e)} e^{-\beta(r) y_p^{r_{\rm{eq}}}(r)} \right)^2</math>
where for large <math>r</math>,
<math display="block">V(r) \simeq \mathfrak{D}_e - u(r) + \frac{u(r)^2}{4\mathfrak{D}_e},</math>
so <math> u(r) </math> is defined according to the theoretically correct long-range behavior expected for the interatomic interaction. <math>\mathfrak{D}_e</math> is the depth of the potential at equilibrium.

This long-range form of the MLR model is guaranteed because the argument of the exponent is defined to have long-range behavior:
<math display="block">\beta(r) y_p^{r_{\rm{ref}}}(r) \simeq \beta_\infty = \ln\left(\frac{2\mathfrak{D}_e}{u(r_e)}\right),</math>
where <math>r_e</math> is the equilibrium bond length.

There are a few ways in which this long-range behavior can be achieved, the most common is to make <math>\beta(r)</math> a polynomial that is constrained to become <math>\beta_\infty </math> at long-range:
<math display="block">\beta(r) = \left(1-y_p^{r_{\textrm{ref}}}(r)\right)\sum_{i=0}^{N_{\beta}} \beta_i y_q^{r_{\textrm{ref}}}(r)^i + y_p^{r_{\textrm{ref}}}(r)\beta_\infty,</math>
<math display="block">y_n^{r_x}(r) = \frac{r^n-r_x^n}{r^n+r_x^n},</math>
where ''n'' is an integer greater than 1, which value is defined by the model chosen for the long-range potential <math>u_\text{LR}(r)</math>.

It is clear to see that:
<math display="block">\lim_{r\to \infty}\beta(r) = \beta_\infty.</math>

==Applications==

The MLR potential has successfully summarized all experimental spectroscopic data (and/or [[virial coefficient|virial data]]) for a number of diatomic molecules, including: N<sub>2</sub>,<ref name="LeRoy(N2)"/> Ca<sub>2</sub>,<ref name="LeRoy(Ca2)"/> KLi,<ref name="Salami(KLi)"/> MgH,<ref name=Henderson(MgH,MgD) /><ref name=LeRoy(damping) /><ref name="Shayesteh(MgH)"/> several electronic states of Li<sub>2</sub>,<ref name=LeRoy(A-X)/><ref name="Dattani(c-a)"/><ref name=Gunton(A-X)>{{cite journal| title=High resolution photoassociation spectroscopy of the <sup>6</sup>Li<sub>2</sub> A(1<sup>1</sup>Σ<sub>u</sub><sup>+</sup>) state|journal=Physical Review A|volume=88|issue=6|pages=062510|arxiv=1309.5870|doi=10.1103/PhysRevA.88.062510 |year=2013 |last1=Gunton|first1=Will|last2=Semczuk|first2=Mariusz|last3=Dattani|first3=Nikesh S.|last4=Madison|first4=Kirk W.| bibcode=2013PhRvA..88f2510G|s2cid=119268157 }}</ref><ref name=Semczuk(c-a)/><ref name=LeRoy(damping)>{{cite journal|last=Le Roy|first=R. J.| author2=C. C. Haugen |author3=J. Tao |author4=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|issue=3|pages=435–446 |url=http://scienide2.uwaterloo.ca/~rleroy/Pubn/11MolP_damping.pdf|doi=10.1080/00268976.2010.527304|bibcode=2011MolPh.109..435L|s2cid=97119318 }}</ref> Cs<sub>2</sub>,<ref name=Li(Cs2)>{{cite journal|last=Xie|first=F.|author2=L. Li |author3=D. Li |author4=V. B. Sovkov |author5=K. V. Minaev |author6=V. S. Ivanov |author7=A. M. Lyyra |author8=S. Magnier |title=Joint analysis of the Cs<sub>2</sub> a-state and 1<sub>g</sub>(3<sup>3</sup>Π1<sub>1g</sub>) states|journal=Journal of Chemical Physics|year=2011|volume=135|issue=2| page=02403| doi=10.1063/1.3606397|pmid=21766938|bibcode=2011JChPh.135b4303X}}</ref><ref name=Hajigeorgiou(Cs2)>{{cite journal|last=Coxon|first=J. A.|author2=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| issue=9| page=094105|doi=10.1063/1.3319739|pmid=20210387|bibcode=2010JChPh.132i4105C}}</ref> Sr<sub>2</sub>,<ref name=Knockel(Sr2)>{{cite journal|last=Stein|first=A.|author2=H. Knockel |author3=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|arxiv=1001.2741|bibcode=2010EPJD...57..171S|s2cid=119243162 }}</ref> ArXe,<ref name=LeRoy(damping) /><ref name=Piticco(ArXe)>{{cite journal|last=Piticco|first=Lorena|author2=F. Merkt |author3=A. A. Cholewinski |author4=F. R. W. McCourt |author5=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|issue=2| pages=83–93| doi=10.1016/j.jms.2010.08.007|bibcode=2010JMoSp.264...83P|hdl=20.500.11850/210096|hdl-access=free}}</ref> LiCa,<ref name=Ivanova(LiCa)>{{cite journal| last=Ivanova| first=Milena| author2=A. Stein |author3=A. Pashov |author4=A. V. Stolyarov |author5=H. Knockel |author6=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|issue=17|page=174303|doi=10.1063/1.3652755|pmid=22070298|bibcode=2011JChPh.135q4303I}}</ref> LiNa,<ref name=Steinke(LiNa)>{{cite journal|last=Steinke|first=M.|author2=H. Knockel |author3=E. Tiemann |title=X-state of LiNa studied by Fourier-transform spectroscopy|journal=Physical Review A|date=27 April 2012|volume=85|issue=4|page=042720| doi=10.1103/PhysRevA.85.042720| bibcode=2012PhRvA..85d2720S}}</ref> Br<sub>2</sub>,<ref name=Yukiya(Br2)>{{cite journal| last=Yukiya|first=T.|author2=N. Nishimiya |author3=Y. Samejima |author4=K. Yamaguchi |author5=M. Suzuki |author6=C. D. Boonec |author7=I. Ozier |author8=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| doi=10.1016/j.jms.2012.12.006|bibcode=2013JMoSp.283...32Y}}</ref> Mg<sub>2</sub>,<ref name=Knockel(Mg2)>{{cite journal|last=Knockel|first=H.|author2=S. Ruhmann |author3=E. Tiemann |title=The X-state of Mg2 studied by Fourier-transform spectroscopy|journal=Journal of Chemical Physics|year=2013| volume=138| issue=9|page=094303|doi=10.1063/1.4792725|pmid=23485290|bibcode=2013JChPh.138i4303K}}</ref> HF,<ref name="Li(HF, HCl, HBr, HI)"/><ref name="sciencedirect.com">{{cite journal|doi=10.1016/j.jqsrt.2014.08.028 |title=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=Journal of Quantitative Spectroscopy and Radiative Transfer |volume=151 |pages=133–154 |year=2015 |last1=Coxon |first1=John A. |last2=Hajigeorgiou |first2=Photos G. |bibcode=2015JQSRT.151..133C }}</ref> HCl,<ref name="Li(HF, HCl, HBr, HI)" /><ref name="sciencedirect.com"/> HBr,<ref name="Li(HF, HCl, HBr, HI)" /><ref name="sciencedirect.com"/> HI,<ref name="Li(HF, HCl, HBr, HI)" /><ref name="sciencedirect.com"/> MgD,<ref name=Henderson(MgH,MgD)>{{cite journal|last=Henderson|first=R. D. E.|author2=A. Shayesteh |author3=J. Tao |author4=C. Haugen |author5=P. F. Bernath |author6=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|volume=117|issue=50|pages=13373–87|date=4 October 2013|doi=10.1021/jp406680r|pmid=24093511 | bibcode = 2013JPCA..11713373H}}</ref> Be<sub>2</sub>,<ref>{{cite journal|doi=10.1063/1.4864355 |pmid=24527923 |title=Direct-potential-fit analyses yield improved empirical potentials for the ground X<sup>1</sup>Σ<sub>g</sub><sup>+</sup> state of Be<sub>2</sub> |journal=The Journal of Chemical Physics |volume=140 |issue=6 |pages=064315 |bibcode=2014JChPh.140f4315M |year=2014 |last1=Meshkov |first1=Vladimir V. |last2=Stolyarov |first2=Andrey V. |last3=Heaven |first3=Michael C. |last4=Haugen |first4=Carl |last5=Leroy |first5=Robert J. }}</ref> BeH,<ref>{{cite journal | doi=10.1016/j.jms.2014.09.005 | volume=311 | title=Beryllium monohydride (BeH): Where we are now, after 86 years of spectroscopy | journal=Journal of Molecular Spectroscopy | pages=76–83| year=2015 | last1=Dattani | first1=Nikesh S. | arxiv=1408.3301 | bibcode=2015JMoSp.311...76D | s2cid=118542048 }}</ref> and NaH.<ref>{{cite journal|doi=10.1063/1.4906086 |pmid=25637985 |title=Dissociation energies and potential energy functions for the ground X 1Σ+ and "avoided-crossing" A <sup>1</sup>Σ<sup>+</sup> states of NaH |journal=The Journal of Chemical Physics |volume=142 |issue=4 |pages=044305 |bibcode=2015JChPh.142d4305W |year=2015 |last1=Walji |first1=Sadru-Dean |last2=Sentjens |first2=Katherine M. |last3=Le Roy |first3=Robert J. }}</ref> 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 long-range 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 spin-orbit coupling which are difficult to incorporate into molecular ''ab initio'' calculations can more easily be treated in the long-range). MLR has been used to represent ''ab initio'' points for KLi<ref>{{cite journal|title=The effect of inner-shell electrons on the ground and low-lying excited states of KLi: ''Ab initio study'' with all-electron basis sets |journal=Journal of Quantitative Spectroscopy and Radiative Transfer|volume=129|pages=8–14|doi=10.1016/j.jqsrt.2013.05.025 |bibcode=2013JQSRT.129....8X |year=2013|last1=Xiao|first1=Ke-La|last2=Yang|first2=Chuan-Lu|last3=Wang|first3=Mei-Shan| last4=Ma|first4=Xiao-Guang|last5=Liu|first5=Wen-Wang}}</ref> and KBe.<ref>{{cite journal|title=An ''ab initio'' study of the ground and low-lying excited states of KBe with the effect of inner-shell electrons |journal=The Journal of Chemical Physics| volume=139|issue=7|pages=074305|doi=10.1063/1.4818452|pmid=23968090| bibcode=2013JChPh.139g4305X| year=2013| last1=Xiao| first1=Ke-La|last2=Yang|first2=Chuan-Lu|last3=Wang|first3=Mei-Shan| last4=Ma|first4=Xiao-Guang| last5=Liu|first5=Wen-Wang}}</ref>

==See also==
*[[Dilithium]]
*[[Morse potential]]
*[[Lennard-Jones potential]]

==References==
{{reflist|colwidth=30em}}

{{DEFAULTSORT:Morse Long-range potential}}
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