Morse/Long-range potential

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. Le Roy and colleagues for a study on N₂.[7] This primitive form was used on Ca₂,[8] KLi[6] and MgH,[9][10][11] before the more modern version was introduced in 2009.[1] A further extension of the MLR potential referred to as the MLR3 potential was introduced in a 2010 study of Cs₂,[12] and this potential has since been used on HF,[13][14] HCl,[13][14] HBr[13][14] and HI.[13][14]

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.

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:

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

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]

• Dilithium
• Morse potential
• Lennard-Jones potential