Transition state search
***********************

A transition state (TS) search is very much like a geometry
optimization: the purpose is to find a stationary point on the
potential energy surface, primarily by monitoring the energy
gradients, which should vanish. The difference between a transition
state and a minimum is that at the transition state the Hessian has a
negative eigenvalue: We are at a saddle point, not a minimum, with the
"negative" mode connecting the two basins on the potential energy
surface.

See also:

  * Examples

  * Tutorial Transition state search and characterization of a Ziegler
    Natta Catalyst

  * Tutorial Tips and Tricks for Transition State Searches for Click
    Reactions

A transition state search in AMS is performed by selecting the
corresponding task:

   Task TransitionStateSearch

Due to the similarities between energy minimization and transition
state search, the "TransitionStateSearch" task in AMS is actually
implemented as a special kind of geometry optimization using either
the quasi-Newton or the dimer optimizer. As such most settings and
keywords described on the geometry optimization manual page also apply
to transition state searches.

In a geometry optimization with a quasi-Newton based optimizer the
Hessian is used to make a reasonably sized step in the "downhill"
direction on the potential energy surface, as the goal is simply to
minimize the energy. A transition state search is a bit different: In
the first step a normal mode is picked along which the energy is to be
*maximized*, while it is *minimized* along all other directions.
Normally the mode with the lowest eigenvalue is picked, since we know
that there should be exactly one negative eigenvalue at the TS
geometry, but one can also choose an approximate reaction coordinate,
either by setting the "ReactionCoordinate" or by adding another System
block named "final", similar to how it's done for NEB. If the initial
geometry is sufficiently close to the transition state, i.e. we are
close to the saddle, the lowest mode is normally the correct one to
follow in order to get to the ridge of the saddle. Alternatively a
different mode can also be selected manually.

   TransitionStateSearch
      ModeToFollow integer
      ReactionCoordinate
         Angle string
         Block string
         BlockAtoms integer_list
         Coordinate string
         Dihedral string
         Distance string
      End
   End

"TransitionStateSearch"
   Type:
      Block

   Description:
      Configures some details of the transition state search.

   "ModeToFollow"
      Type:
         Integer

      Default value:
         1

      Description:
         In case of Transition State Search, here you can specify the
         index of the normal mode to follow (1 is the mode with the
         lowest frequency).

   "ReactionCoordinate"
      Type:
         Block

      Description:
         Specify components of the transition state reaction
         coordinate (TSRC) as a linear combination of internal
         coordinates (distances or angles).

      "Angle"
         Type:
            String

         Recurring:
            True

         Description:
            The TSRC contains the valence angle between the given
            atoms. Three atom indices followed by the weight.

      "Block"
         Type:
            String

         Recurring:
            True

         Description:
            Name of the region. Only atoms of the region will be
            included in the TSRC. It is useful when computing the
            reaction coordinate from the initial Hessian, in which
            case only part of the Hessian will be analyzed.

      "BlockAtoms"
         Type:
            Integer List

         Value Range:
            value > 0

         Recurring:
            True

         Description:
            List of atom indices. Only the listed atoms will be
            included in the TSRC. It is useful when computing the
            reaction coordinate from the initial Hessian, in which
            case only part of the Hessian will be analyzed.

      "Coordinate"
         Type:
            String

         Recurring:
            True

         Description:
            The TSRC contains Cartesian displacement of an atom: atom
            index followed by [x|y|z] and the weight.

      "Dihedral"
         Type:
            String

         Recurring:
            True

         Description:
            The TSRC contains the dihedral angle between the given
            atoms. Four atom indices followed by the weight.

      "Distance"
         Type:
            String

         Recurring:
            True

         Description:
            The TSRC contains the distance between the given atoms.
            Two atom indices followed by the weight.

This selection happens only in the first step. Subsequent steps will
attempt to maximize along the mode that resembles most (by overlap)
the previous maximization direction.

Practice shows that a transition state is much harder to find than a
minimum. For a large part this is due to the much stronger
anharmonicities that usually occur near the TS, which threaten to
invalidate the quasi-Newton methods to find the stationary point. For
this reason it is good advice to be more cautious in the optimization
strategy when approaching a transition state:

* We recommend starting the transition state search with an initial
  geometry that is already close to the transition state. One can use
  a potential energy surface scan along something resembling the
  reaction coordinate to get a rough idea where the transition state
  is. This geometry can then be used as an initial geometry for the
  transition state search.

* It is strongly recommended to provide a good initial Hessian for the
  transition state search. Otherwise the first step of the search may
  not be taken in the correct direction and subsequent steps may keep
  steering in the wrong direction. In AMS this is easily possible by
  loading a Hessian from a previous calculation, see the initial
  Hessian section of this manual. A good way to obtain a reasonable
  Hessian is to compute it explicitly with one of the fast engines
  (i.e. at a lower level of theory) and read that Hessian as the
  initial Hessian for the transition state search at a higher level of
  theory.

* When no accurate initial Hessian is available, it may be a good idea
  to specify an approximate normal mode vector using the
  "ReactionCoordinate" input block. A reaction coordinate (RC) is a
  linear combination of distances and/or valence and dihedral angles:

     TransitionStateSearch
       ReactionCoordinate
         Distance  i j     fac
         Distance  i j     fac
         Angle     i j k   fac
         Dihedral  i j k l fac
       End
     End

  Here, "i", "j", "k", and "l" are atom indices, and the "fac" is the
  factor with which the internal coordinate enters the linear
  combination.

  One should be careful when specifying more than one bond or angle as
  the reaction coordinate. For example, suppose atom 2 is located
  between atoms 1 and 3. Then the following ReactionCoordinate block:

     TransitionStateSearch
       ReactionCoordinate
         Distance 1 2  1.0
         Distance 2 3 -1.0
       End
     End

  means that the RC consists of two distances: R(1-2) and R(2-3). The
  positive direction of the RC is defined as an increase of the R(1-2)
  and a decrease of the R(2-3). In other words, this RC corresponds to
  atom 2 moving along the R(1-3) axis.

* As an alternative to "ReactionCoordinate" one can add a second
  System to the input and call it "final". The difference between the
  initial and the "final" geometry will be used as the reaction
  coordinate. For non-periodic systems, the difference is computed in
  the internal coordinates and only coordinates that differ
  significantly (by more than about 0.1 Bohr or radian) are included.
  For this reason, the "final" geometry should not be too close to the
  initial. For periodic systems, the difference in the Cartesian
  coordinates is used. In this case, only coordinates that differ by
  more than 0.5 Bohr are included.

* When the method converges it is usually a good idea to verify that
  the found geometry is indeed a transition state. This can be done by
  performing a frequency analysis and checking whether the Hessian has
  exactly one negative eigenvalue (represented by a mode with a
  negative frequency). Doing so is expensive however. Since we are
  really only interested in the lowest two normal modes calculating
  the full Hessian is however not necessary and one can use the faster
  PES point characterization instead. This uses a Davidson-type
  algorithm to obtain the lowest few normal modes without constructing
  the full Hessian. The user is referred to the PES point
  characterization documentation page for further details.
