Computer-Aided Nano-Engineering - Recently Updated Pages

Acknowledgements

Saddle Point Search

Introduction

Transition Path Search

Student's Tiny World

Kinetic Monte Carlo

Research Topics

Phase Transition Simulation

Cool Links

Tiny Tutorials

Event Calendar

How tiny is nano?

News

2008

Accelerated Molecular Dynamics

Transition State Theory

Multiscale Simulation

Geometric Modeling

Researcher Spotlight

Software Tools

Computer-Aided Nano-Engineering - Recently Updated Pages

Acknowledgements

Saddle Point Search

Introduction

Transition Path Search

Student's Tiny World

Kinetic Monte Carlo

Research Topics

Phase Transition Simulation

Cool Links

Tiny Tutorials

Event Calendar

How tiny is nano?

News

2008

1. Chain-of-States Methods

2. Other Methods

What is Phase Transition?

How Do We Simulate Phase Transition?

2008

2009

February 2008

March 2008

April 2008

May 2008

June 2008

July 2008

August 2008

September 2008

October 2008

November 2008

December 2008

Student of the Week

1. Chain-of-States Methods

2. Other Methods

What is Phase Transition?

How Do We Simulate Phase Transition?

2008

2009

February 2008

Local Search Methods

Global Search Methods

Nudged Elastic Band (NEB)

String Methods

External Links:

Maths Is Fun!

How tiny is nano?

Oct. 27-31, MMM2008, Tallahassee, Florida, USA

News:

Updates:

Send a shout-out!

Nominate a classmate for Student of the Week!

Local Search Methods

Global Search Methods

See Also

Nudged Elastic Band (NEB)

String Methods

External Links:

Maths Is Fun!

How tiny is nano?

Oct. 27-31, MMM2008, Tallahassee, Florida, USA

News:

Updates:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

yan.wang — Sun, 25 Jan 2009 09:09:04 CST

This website is sponsored by the National Science Foundation.

yan.wang — Thu, 07 Aug 2008 15:24:41 CDT

The automated surface walking algorithm is one of the original saddle point search methods. With local quadratic approximations of PES, the search is based on eigenvectors of the Hessian matrix, which is updated iteratively similar to the quasi-Newton technique. It also involves the scaling of one of the active coordinates in order to make the Hessian eigenvalues lie in a required range. In the partitioned rational function optimization method, PES is approximated by rationalized quadratic surfaces, and the search is conducted by partitioning the minimax problem into two separate maximization and minimization subproblems.

References:

Simons, J., Joergensen, P., Taylor, H., and Ozment, J. (1983) Walking on Potential Energy Surfaces. J. Phys. Chem., 87: 2745- 2753
Banerjee, A., Adams, N., Simons, J., and Shepard, R. (1985) Search for Stationary Points on Surfaces. J. Phys. Chem., 89: 52-57

More recently, the ridge method searches the saddle point by traveling down along the ridge with a pair of images. The direction of the connecting line between the two images can be constrained to make sure the pair is kept on the ridge. Similarly, the dimer method searches the saddle point based on a pair of images. However, the small distance between the two images is fixed. Starting from one basin, the dimer moves uphill in the translation step. In the rotation step, it rotates towards the lowest curvature mode of PES by minimizing the dimer energy through the conjugate gradient approach. Later in the improved dimer method, the curvature is calculated differently with reduced numbers of gradient calculations to improve the performance and robustness under numerical noises. In the synchronous transit method, the transition state is initially estimated by minimizing the interpolated inter-atomic distances. Then the saddle point estimate is further refined with the conjugate gradient optimization. In the reduced gradient following method and the reduced potential energy surface model method, stationary points are intersections of zero-gradient curves and surfaces respectively with distinguished coordinates removed. Saddle point search is within the subspace of these zero-gradient curves or surfaces.

References:

Ionova, I.V. and Carter, E.A. (1993) Ridge method for finding saddle points on potential energy surfaces. J. Chem. Phys., 98: 6377-6386
Henkelman, G. and Jónsson, H. (1999) A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J. Chem. Phys., 111(15): 7010-7022
Heyden, A., Bell, A.T., and Keil, F.J. (2005) Efficient methods for finding transition states in chemical reactions: Comparison of improved dimer method and partitioned rational function optimization method. J. Chem. Phys., 123: 224101-224114
Halgren, T.A. and Lipscomb, W.N. (1977) The synchronous-transit method for determining reaction pathways and locating molecular transition states. Chemical Physics Letters, 49(2): 225-232
Govind, N., Petersen, M., Fitzgerald, G., King-Smith, D., and Andzelm, J. (2003) A generalized synchronous transit method for transition state location. Computational Materials Science, 28: 250–258
Quapp, W., Hirsch, M., Imig, O., and Heidrich, D. (1998) Searching for Saddle Points of a Potential Energy Surface by Following a Reduced Gradient. J Comput Chem, 19: 1087-1100
Hirsch, M. and Quapp, W. (2002) Improved RGF Method to Find Saddle Points. J. Comput. Chem., 23: 887-894
Anglada, J.M., Besalú, E., Bofill, J.M., and Crehuet, R. (2001) On the Quadratic Reaction Path Evaluated in a Reduced Potential Energy Surface Model and the Problem to locate Transition States. J. Comput. Chem., 22: 387-406
Rothman, M.J. and Lohr, L.L. (1980) Analysis of an energy minimization method for locating transition states on potential energy hypersurfaces. Chem. Phys. Lett., 70(2): 405-409
Scharfenberg, P. (1982) An improved method for the evaluation of transition states. J Comput Chem, 3: 277-282

Different from local search, the global search methods ensure the saddle point with the maximum energy is located if the search converges to one. The DHS method searches the saddle point by iteratively reducing the distance between reactant and product images while minimizing the energy subject to the equal distance constraint at each step. The Activation-Relaxation Technique searches saddle points in two steps. In the activation step, one image jumps from a local minimum towards a saddle point according to a controlled force. In the relaxation step, it moves from the saddle point to another minimum. Thus it can traverse many saddle points without the knowledge of final product. In the Step and Slide method, the energy levels of two images from the initial and final states are gradually increased. Then the distance between the two is minimized while both images are kept in the same isoenergy surfaces. The interval Newton’s method finds all stationary points by solving the equation of vanishing gradient.

References:

Dewar, M.J.S., Healy, E.F., and Stewart, J.J.P. (1984) Location of Transition States in Reaction Mechanisms. J. Chem. Soc., Faraday Trans., 2(80): 227
Mousseau, N. and Barkema, G.T. (1998) Traveling through potential energy landscapes of disordered materials: The activation-relaxation technique. Physical Review E, 57(2): 2419-2424
Miron, R.A. and Fichthorn, K.A. (2001) The Step and Slide method for finding saddle points on multidimensional potential surfaces. J. Chem. Physics, 115(19): 8742-8747
Lin, Y. and Stadtherr, M.A. (2004) Locating stationary points of Sorbate-Zeolite potential energy surfaces using interval analysis. J. Chem. Phys., 121(20): 10159-10166

yan.wang — Sat, 02 Aug 2008 19:05:31 CDT

Computer-aided nano-engineering is to apply modeling and simulation methods in design of material systems at nano and meso scales, where research streams in mathematics, engineering, materials, chemistry, physics, and computer science merge. Some interesting issues to study include (but not limited to):

Mathematical and computational advancement with applications in nanoscience and nanoengineering
Modeling of geometric structure and physical properties (e.g. electrical, magnetic, mechanical, optical) of organic and inorganic materials
New approaches to understand phenomena and predict behavior of atomic, molecular, and macromolecular structures
Multi-scale multi-physics simulation methods (e.g. ab initio, density functional theory, molecular dynamics, Monte Carlo, finite element) and discrete-to-continuum integration (e.g. quasi-continuum, concurrent coupling, multiresolution, multigrid)
Simulation of fabrication and self-assembly processes in manufacturing and synthesis
Computer based enabling technologies (e.g. visualization, optimization, imaging, prototyping) for engineering of nano structure and morphology
Novel materials and devices by design and their applications

yan.wang — Sat, 02 Aug 2008 18:28:08 CDT

In chain-of-states methods, the transition pathway is represented as a number of intermediate states, which can be looked as snapshots of configurations as the atomic structure transforms from initial to final state along the transition pathway. After the search converges, the discrete states are chained to each other, usually by interpolating between the states, to obtain the transition pathway and the saddle point. They work well in transitions where there may be more than one saddle point. In situations where there may be multiple transition pathways, the methods will converge to the pathway closest to the initial guess for the transition pathway.

The Nudged Elastic Band (NEB) method and its extensions are among the most used and well-developed chain of states methods. NEB is an extension of the plain elastic band (PEB) method, where images, which are points in the configuration space corresponding to intermediate states, are connected by springs. In PEB, images move subject to both the true force due to the gradient of potential energy and the spring force. The relaxed band should converge to the MEP. However, when the spring constant is large enough, the perpendicular components of spring forces (with respect to the direction of path) pull images away from saddle points at the sharp turns of the path. This corner cutting makes PEB overestimate the saddle point energy. On the other hand, when the spring constant is small, the parallel components of true forces make images slide down towards the minima. This sliding down reduces the resolution of the region of interest (ROI) near the saddle point. The NEB method is to solve the problems of corner cutting and sliding down by removing the perpendicular component of spring force and the parallel component of true force in the total force on each image. The only effect of the springs now is to keep images evenly distributed within the path. Nevertheless, when the potential energy changes rapidly, the path has kinks where the parallel component of the energy gradient is much larger than the perpendicular one, because the restoring perpendicular components of forces are weak. Another shortcoming of NEB is that the actual saddle point may not be located by one of the images directly. These issues are solved by some further extensions of the NEB method.

References:

Jonsson, H., Mills, G., and Jacobsen, K., 1998. Classical and Quantum Dynamics in Condensed Phase Simulations. World Scientific, Hackensack, NJ, Chap. 16, pp. 385-404. See also http://www.hi.is/~hj/paperNEBleri.pdf

The improved tangent NEB (IT-NEB) method reduces the chances of getting kinks by better estimating the tangent direction of the path to approximate MEP at each image. Instead of the central finite difference approximation between the image and its two neighbours as in the original NEB, only one neighbor is used for tangent estimation. The climbing image NEB (CI-NEB) method was further developed so that the image with the highest energy actively climbs up to locate the actual saddle point. An alternative to resolve the issue of kinks is the doubly nudged elastic band (DNEB) method where a manipulated perpendicular component of spring force is introduced back into the total force such that MEP can be restored. To locate the actual saddle points, the eigenvector following (EF) optimization approach can be applied to the result of NEB. To increase the resolution of ROI, adaptive spring constants can be applied such as energy-weighted or gradient-weighted. The original NEB method requires that the initial and final states should be known. A free end NEB (FENEB) method was recently proposed to keep one end of the band free so that it can swing at a fixed energy level, thus the number of images can be reduced while still maintaining enough resolution. Similar to the String method, Spline-based interpolation was also used in NEB.

References:

Henkelman, G., and Jonsson, H., 2000. “Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points”, J. Chem. Phys., 113, pp. 9978-9985.
Henkelman, G., Uberuaga, B., and Jonsson, H., 2000. “A climbing image nudged elastic band method for finding saddle points and minimum energy paths”, J. Chem. Phys., 113, pp. 9901-9904.
Trygubenko, S., and Wales, D., 2004. “A doubly nudged elastic band method for finding transition states”, J. Chem. Phys., 120, 2082-2094.
Zhu, T., Li, J., Samanta, A., Kim, H.G., and Suresh,S., 2007. "Interfacial plasticity governs strain rate sensitivity and ductility in nanostructured metals", Proceedings of the National Academy of Sciences of the USA, 104, 3031-3036
Galvan, I., and Field, M., 2008. “Improving the Efficiency of the NEB Reaction Path Finding Algorithm”, J. Comp. Chem., 29, pp. 139-143.

The second group of chain-of-states methods is the String methods. The transition path is represented continuously as Splines. Subject to perpendicular forces the curve evolves and converges to the MEP alternately in two steps. In the evolution step the curve is discretized as a string of points and solved by standard ODE solvers. In the reparameterization step, the points are redistributed along the string based on parameterization constraints and Spline interpolations. Compared to NEB methods where the number of images is fixed during searching, the number of points in the String method can be dynamically modified based on needs. The Growing String method is an extension where the points of strings are initially located at the two ends of reactant and product. Then the string grows by inserting points so as to meet at the saddle point upon convergence. The quadratic String method is formulated based on a multi-objective optimization approach. Based on the local quadratic approximation of the PES, the quasi-Newton technique is applied to search the MEP.

References:

E, W., Ren, W., and Vanden-Eijnden, E., 2002. “String method for the study of rare events”, Phys. Rev. B, 66, pp. 052301(1)-052301(4).
Ren, W., 2003. “Higher order string method for finding minimum energy path”, Comm. Math. Sci., 1, pp. 377-384.
E, W., Ren, W., and Vanden-Eijnden, E., 2007. “A Simplified and improved string method for computing the minimum energy paths in barrier-crossing events”, J. Chem. Phys., 126, pp. 164103(1) -164103(8).
Peters, B., Heyden, A., Bell, A., and Chakraborty, A., 2004. “A growing string method for determining transition states: Comparison to the nudged elastic band and string methods”, J. Chem. Phys., 120, 7877-7886.
Burger, S., and Yang, W., 2006. “Quadratic string method for determining the minimum-energy path based on multiobjective optimization”, J. Chem. Phys., 124, pp. 054109(1)- 054109(12).

In the conjugate peak refinement (CPR) method, saddle points and the approximated MEP are found by searching the maximum of one direction and the minima of all other conjugate directions iteratively, because exactly one eigenvalue of the Hessian matrix at the saddle points is negative. The accelerated Langevin dynamics (ALD) method is a stochastic transition path sampling method. Paths are sampled by solving the modified Langevin equation with negative friction coefficients to accelerate the transition. The MEP is the average of all sampled paths. The Hamilton-Jacobi (HJ) method generates the MEP by solving a Hamilton-Jacobi type equation with special cost functions.

References:

Fischer, S., and Karplus, M., 1992. “Conjugate Peak Refinement: an algorithm for finding reaction paths and accurate transition states with many degrees of freedom”, Chem. Phys. Lett., 194, 252-261.
Chen, L., Ying, C., and Ala-Nissila, T., 2002. “Finding transition paths and rate coefficients through accelerated Langevin dynamics”, Phys. Rev. E, 65, pp. 042101(1)-042101(4).
Dey, B., and Ayers, P., 2006. “A Hamilton–Jacobi type equation for computing minimum potential energy paths”, Mol. Phys., 104, pp. 541–558.

yan.wang — Sat, 02 Aug 2008 18:10:39 CDT

Have fun!

See Also

yan.wang — Sat, 02 Aug 2008 18:00:39 CDT

Kinetic Monte Carlo (KMC) is a time-dependent Monte Carlo simulation approach to simulate structural changes such as reaction and adsorption, which could have the time scale from nanosecond to second. These time scales are usually too long for regular Molecular Dynamics.

kinetic Monte Carlo

yan.wang — Sat, 02 Aug 2008 17:57:56 CDT

Geometric Modeling: how to represent nanoscale structures in computers.
Multiscale Simulation: how to simulate complex systems from the perspectives of multiple length and time scales.
Phase Transition Simulation: how to simulate dynamic structural changes for functional nano-materials design.
[more to add]

yan.wang — Sat, 02 Aug 2008 17:56:08 CDT

A phase transition is a geometric and topological transformation process of materials from one phase to another, each of which has a unique and homogeneous physical property.

In an effort to simulate a reaction or transition, a potential energy surface (PES) that characterizes the process is first generated. Then, a minimum energy path (MEP) is computed which represents the transition pathway in the configuration space. With the knowledge of activation energy, the rate constant that define the speed of the process (the rate of the reaction) then can be calculated using transition state theory (TST). Finally the reaction or transition can be simulated by methods such as kinetic Monte Carlo and accelerated molecular dynamics. The most important step involved in modeling phase transition is the knowledge of the activation energy barrier and rate constant involved in the transition, which depends on searching transition paths and saddle points on PES.

yan.wang — Thu, 31 Jul 2008 12:03:18 CDT

yan.wang — Thu, 31 Jul 2008 11:29:09 CDT

yan.wang — Thu, 31 Jul 2008 11:14:42 CDT

Here is the list of related conferences.

yan.wang — Thu, 31 Jul 2008 11:10:12 CDT

yan.wang — Thu, 31 Jul 2008 10:42:48 CDT

yan.wang — Thu, 31 Jul 2008 10:36:23 CDT

January 2008

Sun	Mon	Tues	Wed	Thurs	Fri	Sat
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15 St. Patrick's Day
16	17	18	19	20	21	22
23	24	25	26	27	28	29
30	31

Sun	Mon	Tues	Wed	Thurs	Fri	Sat
					1	2
3	4	5	6	7	8	9
10	11	12	13	14	15	16
17	18	19	20	21	22	23
24	25	26	27	28	29	30
31

Sun	Mon	Tues	Wed	Thurs	Fri	Sat
			1	2	3	4
5	6	7	8	9	10	11
12	13 Columbus Day	14	15	16	17	18
19	20	21	22	23	24	25
26	27 MMM2008 Tallahassee, Florida, USA	28 MMM2008 Tallahassee, Florida, USA	29 MMM2008 Tallahassee, Florida, USA	30 MMM2008 Tallahassee, Florida, USA	31 Halloween MMM2008 Tallahassee, Florida, USA

Sun	Mon	Tues	Wed	Thurs	Fri	Sat
						1
2	3	4	5	6	7	8
9	10	11 Veteran's Day	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27 Thanksgiving Day	28	29
30

yan.wang — Thu, 31 Jul 2008 09:21:36 CDT

There is no abstract available for this page revision.

yan.wang — Thu, 31 Jul 2008 09:20:31 CDT

There is no abstract available for this page revision.

yan.wang — Thu, 31 Jul 2008 09:03:54 CDT

There is no abstract available for this page revision.

yan.wang — Thu, 31 Jul 2008 09:03:16 CDT

There is no abstract available for this page revision.

yan.wang — Thu, 31 Jul 2008 09:02:16 CDT

This is how a spotlight page could look: Use it "as is," modify it, or delete all the text and start over with your own idea. It's a wiki page, so you can change it any time you want.

Name:
Grade:
Why he/she is being honored:

[Use this section to say "way to go" to this week's student! Add photos, videos, or words of congrats. Or, send them a message by going to their profile page...]

Enter their name here, and describe why they should be recognized.

See also: Announcements | The Syllabus | Assignments | Study Guides

yan.wang — Thu, 31 Jul 2008 09:00:52 CDT

There is no abstract available for this page revision.

1
New Year's Day

21
Martin Luther King Jr. Day