Is there a typo in this modified Lennard-Jones potential?

Question

The standard 12-6 Lennard Jones potential is given by

$$U(r_ij) = 4\epsilon\left[ \left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12} - \left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6} \right]$$

where $\epsilon$ and $\sigma$ are the parameters.

I am reading a paper (Pohorille, A.; Jarzynski, C.; Chipot, C. J. Phys. Chem. B 2010, 114, 10235-10253), and my mind is very cloudy right now. In the paper, in equation (65) on page 15 (p. 10249), the authors define a so-called soft-core potential by modifying the original Lennard-Jones potential using parameters $\lambda$ and $\alpha$:

$$U(r_{ij}; \lambda) = 4\epsilon\left(1-\lambda\right)^{n}\left\{ \frac{1}{\left[\alpha\lambda^{2}+\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right]^{2}}-\frac{1}{\alpha\lambda^{2}+\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}}\right\} $$

Then the paper says:

... the original Lennard-Jones potential is fully recovered at $\lambda = 0$.

But I think that if I plug $\lambda = 0$ into the equation above, I get:

$$U(r_ij) = 4\epsilon\left[ \left(\frac{r_{ij}}{\sigma_{ij}}\right)^{12} - \left(\frac{r_{ij}}{\sigma_{ij}}\right)^{6} \right]$$

which is not the original Lennard-Jones potential.

Is there something wrong with my reasoning, or is there a typo in the paper?

Answer 1

You are right, there is most likely a typo in the paper. Presumably they mean $$U(r_{ij}; \lambda) = 4\epsilon\left(1-\lambda\right)^{n}\left\{ \frac{1}{\left[\alpha\lambda^{2}+\left(\frac{r_{ij}}{\sigma_{ij}}\right)^{6}\right]^{2}}-\frac{1}{\alpha\lambda^{2}+\left(\frac{r_{ij}}{\sigma_{ij}}\right)^{6}}\right\}. $$ This is what's given in their reference (Chem. Phys. Lett. 222 no. 6, pp. 529-539 (2010), equation 7, p. 532) except for a reparametrization $\lambda\leftrightarrow(1-\lambda)$.