Morse/Long-range potential

Owing to the simplicity of the Morse potential (it only has three adjustable parameters), it is not used in modern spectroscopy. The MLR (Morse/Long-range) potential is a modern version of the Morse potential which has the correct theoretical long-range form of the potential naturally built into it. It was first introduced by professor Robert J. Le Roy of University of Waterloo, professor Nikesh S. Dattani of Oxford University and professor John A. Coxon of Dalhousie University in 2009 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 long-range constants. Cases of particular note include:

1. the c-state of Li₂: where the MLR potential was successfully able to bridge a gap of more than 5000 cm⁻¹ in experimental data. 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. The accuracy of these predictions was much better than the most sophisticated ab initio techniques at the time.
2. the A-state of Li₂: where Le Roy et al. 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. 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".
3. the a-state 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.

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 N₂. This primitive form was used on Ca₂, KLi and MgH, before the more modern version was introduced in 2009 by Le Roy, Dattani, and Coxon. A further extension of the MLR potential referred to as the MLR3 potential was introduced in a 2010 study of Cs₂, and this potential has since been used on HF, HCl, HBr and HI.

Function

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

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

where for large $r$,

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

so $u(r)$ 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:

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

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

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

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

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

It is clear to see that:

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

Applications

The MLR potential has successfully summarized all experimental spectroscopic data (and/or virial data) for a number of diatomic molecules, including: N₂, Ca₂, KLi, MgH, several electronic states of Li₂, Cs₂, Sr₂, ArXe, LiCa, LiNa, Br₂, Mg₂, HF, HCl, HBr, HI, MgD, Be₂, BeH, and NaH. 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 inito calculations can more easily be treated described in the long-range). Examples of molecules for which the MLR has been used to represent ab initio points are KLi, KBe.

See also

• Dilithium
• Morse potential
• Lennard-Jones potential