(86d) Accelerated Saddle Point Refinement with Sella

Identification and refinement of first order saddle point (FOSP) structures on the potential energy surface (PES) of chemical systems is a computational bottleneck in the characterization of reaction pathways. For small systems, it may be feasible to evaluate the full Hessian, which determines both the most probable direction of the reaction coordinate and the optimal step size and direction for refinement to the FOSP. However, evaluation of the full Hessian quickly becomes prohibitively expensive as the system size grows. Thus, many FOSP refinement methods designed for use with larger systems (such as those relevant to heterogeneous catalysis) do not evaluate the Hessian in full. Instead, these approaches calculate the lowest-curvature mode of the Hessian using an iterative diagonalization procedure. These procedures approximate Hessian-vector products by performing finite-difference on the gradient vector. Crucially, the full Hessian is no longer available in this approach for the determination of optimal geometry steps during FOSP refinement, which are instead determined using an approximate Hessian. This drastically increases the number of geometry steps required to converge to a FOSP, and may result in a failure to converge all-together.

In a recent publication[1], we describe a FOSP refinement procedure that accelerates convergence to a saddle point using iterative Hessian diagonalization. The Hessian-vector products that have been evaluated during diagonalization are used to construct a highly accurate approximate Hessian. In the full-diagonalization limit, the approximate Hessian constructed in this way becomes exact. The total number of diagonalization steps performed is controlled by a single convergence parameter, γ. Tighter convergence yields a more accurate Hessian – thereby reducing the number of refinement steps needed to locate the FOSP – at the cost of additional diagonalization iterations. The cost of diagonalization iterations can be balanced against the cost of additional geometry refinement steps by carefully tuning the value of γ.

Our method has been implemented in an open source software package, Sella[2]. Sella is written in Python and is compatible with all major operating systems and computer architectures. Sella uses the ASE[3] library to interface with a wide range of software packages for PES evaluations, including electronic structure theory packages and classical force-field packages. In particular, we have used the ASE interface to LAMMPS[4] for several test systems.

We measure the performance of our approach as implemented in Sella on two FOSP refinement benchmarks from optbench.org[5]. On one benchmark, we see an average reduction of over 50% in the number of gradient evaluations (consisting of diagonalization iterations plus geometry refinement steps) required to converge to a FOSP. On a second benchmark, we see an average reduction in the number of gradient evaluations of over 25%. Additionally, our implementation of iterative diagonalization requires an average of 30% fewer iterations to converge to the leftmost eigenvector of the Hessian compared to the best performing codes listed on optbench.org. When an approximate Hessian is available, our Jacobi-Davidson-based diagonalization routine requires on average 40% fewer iterations to converge than non-preconditioned Lanczos. These performance improvements are a direct result of the increased accuracy of the approximate Hessian matrix that our method enables.

[1] Hermes, E.D., Sargsyan, K., Najm, H.N., Zádor, J. J. Chem. Theory Comput. 15, 6536-6549 (2019).

[2] Hermes, E.D. Sella, 2019, DOI: 10.5281/zenodo.3379094.

[3] Larsen, A.H., Mortensen, J.J., Blomqvist, J., Castelli, I.E., Christensen, R., Dułak, M., Friis, J., Groves, M.N., Hammer, B., Hargus, C., Hermes, E.D., Jennings, P.C., Jensen, P.B., Kermode, J.B., Kitchin, J.R., Kolsbjerg, E.L., Kubal, J., Kaasbjerg, K., Lysgaard, S., Maronsson, J.B., Maxson, T., Olsen, T., Pastewka, L., Peterson, Al., Rostgaard, C., Schiøtz, J., Schütt, O., Strange, M., Thygesen, K.S., Vegge, T., Vilhelmsen, L., Walter, M., Zeng, Z., Jacobsen, K.W. J. Phys.: Condens. Matter 29, 273002 (2017).

[4] Plimpton, S. J. Comput. Phys. 117, 1 (1995).

[5] Chill, S.T., Stevenson, J., Ruehle, V., Shang, C., Xiao, P., Farrell, J.D. and Wales, D.J. J. Chem. Theory Comput. 10, 5476 (2014).

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division, as part of the Computational Chemistry Sciences Program (Award Number: 0000232253).

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The views expressed in this article do not necessarily represent the views of the U.S. Department of Energy or the United States Government.