The Sigma Program - Lennard-Jones Potential

Experiments have shown that the collision cross section depends fairly strongly on temperature even for a rigid structure like fullerene C60.[4] This phenomenon is due to the nature of the ion-helium interaction, which is attractive at long distances and repulsive at very short distances. The model potential used in the sigma program (in Lennard-Jones mode) to describe the interaction between a specific atom in the polyatomic ion and a buffer gas helium atom (separated by a distance R) is described by the equation below.[2]

V(R) = A [ B(r*/R)n - C(r*/R)6 - D(r*/R)4]

with

A = n E* / [n(3+γ)-12(1+γ)]

B = 12(1+γ)/n

C = 4γ

D = 3(1- γ)

In this (n,6,4) potential, E* and r* are the depth and position of the potential well, respectively, n is the exponent describing the ion-neutral repulsion, and γ is a dimensionless parameter defining the relative contributions of the R-6 and R-4 terms. The
R-4 term, V4(R), accounts for the attraction between the charge q of the particular atom of the polyatomic ion and the dipole it induces in the neutral helium atom of polarizability α. The term is known accurately and is given by the following simple expression.

V4(R) = - (q2α) / (2R4)

The remaining part, Vn,6(R) = V(R) - V4(R), has the form of an (n,6) Lennard-Jones (LJ) potential and can be expressed by using the LJ parameters r and E (LJ well position and depth) as shown below.

Vn,6(R) = (nE)/(n-6) [ (6/n) (r/R)n - (r/R)6 ]

For given parameters n, r, E, and q, the potential V(R) is completely defined, and the parameters γ , r*, and E* can be determined.
The LJ parameters for the He-C interaction were obtained in our group by fitting theoretical cross sections, calculated on the basis of the potential introduced above, to experimental data obtained for fullerene C60.[4] The He-H interaction was obtained by a fit to PEG data.[4] No good model systems are available for O and N, and therefore the He-O and He-N interactions are assumed to be the same as for He-C. The He-alkali ion interactions were obtained from studies of alkali-ion-cationized glycine systems.[5] Additionally we found that atomic radii are smaller for ions with fewer atoms like glycine compared to larger ions like fullerene C60. Therefore, for ions composed of N atoms, the LJ radius r(N) has to be scaled as a function of N to account for the decreasing interaction with decreasing ion size using the following empirical formula.

r(N) = r(60) (0.86882 - 0.99427 N + N 0.99913)

This equation was obtained by a three-parameter fit to hundreds of experimental data points for ions with 11 to 170 atoms.[5] Presently (since 1999) we are using the ion-helium LJ parameters r(60) and E shown in the following table optimized for a (12,6,4) potential. In addition, the table has values for r(20) and r(160) obtained by using the formula for r(N) above.

 Element E(60) (kcal/mol) r(60) (Å) r(20) (Å) r(160) (Å) H 0.340 2.38 2.22 2.57 Li 0.357 2.75 2.56 2.97 C, O, N 0.370 3.02 2.81 3.26 Na 0.364 2.89 2.69 3.12 K 0.374 3.10 2.89 3.35 Rb 0.383 3.30 3.07 3.57 Cs 0.393 3.50 3.26 3.78