NCSA Nanotechnology Initiative
Testing transition state searching algorithms for use with plane-wave DFT codes
Information on activation barriers is of central importance for estimating rates,
providing an understanding of mechanisms and evaluating possible
rate-enhancing modifications. Transition state optimization methods employed
in non-periodic ab initio codes (such as Gaussian 98,
Jaguar, and Spartan generally
use analytic first derivatives together with analytical or good numerical
estimates of the Hessian. Most theoretical work on periodic
systems (surfaces and bulk) is currently carried out using plane-wave methods for which
accurate evaluation of the Hessian is difficult due to the use of grids.
Two promising non-Hessian based transition state optimization
algorithms are the Ridge algorithm1,2 (from Carter's
group at UCLA) and the Nudged Elastic Band (NEB) algorithm 3,4 (from
at U. of Washington).
We are testing these algorithms as implemented in plane wave
codes for locating transition states for various surface
rearrangement processes. Below we summarize the main features
of the NEB algorithm, which has been used in most of our studies
The NEB algorithm, which is an improvement of the previously used
Elastic Band algorithm, only requires evaluation of the potential energy
and its first derivatives. It converges to the minimum energy path
(MEP) in a well-defined limit. The transition state corresponds to
the highest energy point along the MEP.
The NEB algorithm is highly parallelizable
since it involves multiple constrained minimizations that can be carried
out simultaneously and that do not require much interprocessor communication.
The computational effort grows slowly with system size.
Outline of the Elastic Band Method
Each image may be minimized on a separate processor, making the algorithm
ideal for parallel processing.
- Let R0 and RP denote the coordinates of
the reactant and product, respectively. (These are 3N
component vectors where N is the number of atoms.
- A set of intermediate points or "images" of the system is generated
by interpolation between the reactant and product in coordination
space. These images define a discrete representation of a path
connecting reactant and product.
- A constrained minimization of these images is done so that they
descend in energy to the MEP without collapsing to either the
reactant or the product.
- This is accomplished by constructing an object function F
that includes a spring force between neighboring images.
- The spring force causes "corner cutting" and thus the algorithm
does not converge to the exact MEP.
- The density of the images is lowest near the highest points
of the path and so resolution is worst in the region of the saddle
Outline of the Nudged Elastic Band Method
The NEB algorithm addresses the shortcomings of the
simple elastic band methods by setting the
spring force in directions perpendicular to the
the path equal to zero (this solves the "corner
cutting" problem) and by setting the true
forces along the path to zero (this prevents
the images from sliding towards the endpoints
and improves the density of images near the
The unit vector
is tangent to the path at the point t.
The unit vectors
are perpendicular to the path at t. The forces acting on
the images are redefined so as to:
- Zero out the perpendicular component of the net spring force.
- Zero out the parallel component of the true interaction potential.
Minimize F with respect to 3N(P-1) coordinates.
What the Jordan Group is Doing With NEB
Initial tests of NEB interfaced with the VASP
plane-wave DFT code are encouraging.
The figure below depicts the results for a NEB optimization of the reaction path for an H atom
hopping between the Si atoms of an Si-Si dimer on the Si(100)-2x1 surface. The calculated barrier - 1.50eV -
is identical to that found by minimization with symmetry imposed.
We have recently employed the NEB algorithm to map out the MEP for 3x1 ---> 2x1
rearrangements on the Si(100) surface.
The reaction pathway for the rearrangements is very complex.
This research has been accepted for publication in Surface Science.5
In the future, we plan to:
- Develop a web-based implementation of NEB in
conjunction with various model potentials.
- Incorporate NEB into a MM/QM model for
electrons interacting with water clusters.
1I. V. Ionova and E. A. Carter,
J. Chem. Phys., 98, 6377 (1993).
2I. V. Ionova and E. A. Carter,
"DIIS-Induced Acceleration of the Ridge Method for Finding Transition States,"
J. Chem. Phys., 102, 1251 (1995).
3G. Mills, H. Jónsson and G. K. Schenter,
Surface Science, 324, 305 (1995).
4H. Jónsson, G. Mills, K. W. Jacobsen,
`Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions',
in `Classical and Quantum Dynamics in Condensed Phase Simulations',
ed. B. J. Berne, G. Ciccotti and D. F. Coker (World Scientific, 1998), page 385.
5T.-C. Shen, J.A. Steckel, and K.D. Jordan,
"Electron-Stimulated Bond Rearrangements on the H/Si(100)-3x1 surface", Surf. Sci.,
446, 211-218 (2000).