Morse/Long-range potential

Computational physics
Mechanics Electromagnetics Multiphysics Particle physics Thermodynamics Simulation
Potentials Morse/Long-range potential Lennard-Jones potential Yukawa potential Morse potential
Fluid dynamics Finite difference Finite volume Finite element Boundary element Lattice Boltzmann Riemann solver Dissipative particle dynamics Smoothed particle hydrodynamics Turbulence models
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Particle N-body Particle-in-cell Molecular dynamics
Scientists Godunov Ulam von Neumann Galerkin Lorenz Wilson Alder Richtmyer
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The Morse/Long-range potential (MLR potential) is an

1. the c-state of dilithium (Li₂): 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 the 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]
2. the A-state of Li₂: where Le Roy et al.^[1] constructed an MLR potential which determined the C₃ value for atomic lithium to a higher-precision 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.
3. the a-state 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.^[6]

Historical origins

The MLR potential is based on the classic

Function

The Morse/Long-range potential energy function is of the form

$${\displaystyle V(r)={\mathfrak {D}}_{e}\left(1-{\frac {u(r)}{u(r_{e})}}e^{-\beta (r)y_{p}^{r_{\rm {eq}}}(r)}\right)^{2}}$$

where for large ${\displaystyle r}$,

$${\displaystyle V(r)\simeq {\mathfrak {D}}_{e}-u(r)+{\frac {u(r)^{2}}{4{\mathfrak {D}}_{e}}},}$$

so ${\displaystyle u(r)}$ is defined according to the theoretically correct long-range behavior expected for the interatomic interaction. ${\displaystyle {\mathfrak {D}}_{e}}$ 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:

$${\displaystyle \beta (r)y_{p}^{r_{\rm {ref}}}(r)\simeq \beta _{\infty }=\ln \left({\frac {2{\mathfrak {D}}_{e}}{u(r_{e})}}\right),}$$

where ${\displaystyle r_{e}}$ 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 ${\displaystyle \beta (r)}$ a polynomial that is constrained to become ${\displaystyle \beta _{\infty }}$ at long-range:

$${\displaystyle \beta (r)=\left(1-y_{p}^{r_{\textrm {ref}}}(r)\right)\sum _{i=0}^{N_{\beta }}\beta _{i}y_{q}^{r_{\textrm {ref}}}(r)^{i}+y_{p}^{r_{\textrm {ref}}}(r)\beta _{\infty },}$$

$${\displaystyle y_{n}^{r_{x}}(r)={\frac {r^{n}-r_{x}^{n}}{r^{n}+r_{x}^{n}}},}$$

where n is an integer greater than 1, which value is defined by the model chosen for the long-range potential ${\displaystyle u_{\text{LR}}(r)}$.

It is clear to see that:

$${\displaystyle \lim _{r\to \infty }\beta (r)=\beta _{\infty }.}$$

Applications

The MLR potential has successfully summarized all experimental spectroscopic data (and/or virial data) for a number of diatomic molecules, including: N₂,^[7] Ca₂,^[8] KLi,^[6] MgH,^[9][10][11] several electronic states of Li₂,^{[1][2][15][3][10]} Cs₂,^[16][12] Sr₂,^[17] ArXe,^[10][18] LiCa,^[19] LiNa,^[20] Br₂,^[21] Mg₂,^[22] HF,^[13][14] HCl,^[13][14] HBr,^[13][14] HI,^[13][14] MgD,^[9] Be₂,^[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 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^[26] and KBe.^[27]

See also

• Dilithium
• Morse potential
• Lennard-Jones potential

References