LOCATING TRANSITION STATES ON POTENTIAL ENERGY SURFACES BY THE GENTLEST
ASCENT DYNAMICS

Josep Maria Bofill
Departament de Qu´imica Org`anica, Universitat de Barcelona and 
Institut de Qu´imica Te`orica i Computacional, 
Universitat de Barcelona, (IQTCUB), 
Mart´i i Franqu`es, 1, 08028 Barcelona, Spain

Wolfgang Quapp
Mathematisches Institut, Universität Leipzig, 
PF 100920, D-04009 Leipzig, Germany

Marc Caballero
Departament de Química Física, Universitat de Barcelona and 
Institut de Química Teòrica i Computacional, 
Universitat de Barcelona, (IQTCUB), 
Martí i Franquès, 1, 08028 Barcelona, Spain

June  2013

Abstract

The system of ordinary differential equations for the method of the gentlest
ascent dynamics (GAD) has been derived which was previously proposed by 
[W.E and X. Zhou, Nonlinearity 24, 1831 (2011)].  For this purpose we use
diverse projection or reflection operators to a given initial direction.  
By a row of simple examples we explain the two possibilities of a GAD curve: 
it can directly find the transition state of interest by a gentlest ascent, 
or it can go the roundabout way over a turning point and then find the
transition state going downhill along its ridge.  An outlook to generalised
formulas for a search of saddle points of a higher order is added.