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48.2 Tunnelling splittings

Instantons used to compute tunnelling splittings are local minima on the ring-polymer surface. They are located using the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method. All publications describing work using this method should quote at least one of these references:

• J. Nocedal. ``Updating quasi-Newton matrices with limited storage''. Math. Comput. 35, 773 (1980).
• D. C. Liu and J. Nocedal. ``On the limited memory BFGS method for large scale optimization''. Math. Program. 45, 503 (1989).

The geometry of the well minimum should be specified with the geometry keyword. The action of the instanton, its fluctuations and the fluctuations about the well minimum are printed to the output file. MOLPRO computes the tunnelling-matrix element which can be used to construct the tunnelling-splitting pattern for a molecular cluster with two or more degenerate wells as described in the latter of the two references. It is necessary to converge the results in the limit that $\beta\rightarrow\infty$ and $\beta/N\rightarrow0$.

References for the ring-polymer instanton method are:

• J. O. Richardson and S. C. Althorpe. ``Ring-polymer instanton method for calculating tunneling splittings.'' J. Chem. Phys. 134, 054109 (2011).
• J. O. Richardson, S. C. Althorpe and D. J. Wales. ``Instanton calculations of tunneling splittings for water dimer and trimer.'' J. Chem. Phys. 135, 124109 (2011).

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