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\title{Algorithm for Choosing ABM Parameters}         % Enter your title between curly braces
\author{R. T Pack and G. A. Parker}        % Enter your name between curly braces
\date{March 1993}          % Enter your date or \today between curly braces
\maketitle


The following is for reactions with barriers where few, if any, 
asymptotically closed $\cal E$'s dip down and become locally open in the interaction region.  
For reactions where potential wells cause many asymptotically closed 
levels to become locally open, test calculations will be necessary to 
determine to which asymptote those levels diabatically connect, 
and enough basis functions included in that arrangement channel to 
assure their presence and convergence.

\begin{itemize}
\item
Get the values of $r_e,$  $\omega_e,$ and $\omega_ex_e$ for the 
diatomics of all three arrangements from a spectroscopic tabulation 
such as Huber and Herzberg. 
\item
Determine the largest $\rho$ to be used from the requirement that 
all the long range potentials be negligible there compared to 
scattering energies or accuracies to which resonances are needed. 
\item
Determine accurate values of the vibrational energy levels at this 
large $\rho$.  This can be done with abm using a large unoptimized 
vibrational basis but only the lowest allowed rotational states, 
or it can be done with the Delves Basis or Jacobi Basis parts of APH3D.  
If the last of these is used,
it will give true asymptotic levels which will lie slightly above the
actual levels at any large but finite $\rho$.  
\item
Choose maxvib (noscil) to be at least two quanta larger than the 
largest $v$ expected to be open at {\it any} $\rho$.  
This is necessary to maintain completeness in performing the 
overlaps between sectors.     
\item
Keeping only the lowest allowed rotational state in each arrangement 
and the same large $\rho$, optimize the vibrational energy levels in 
each arrangement by varying the scaling parameters $c_\alpha,$ $ r_x,$ 
and $a_n$ cyclically in that order until all are optimum.  
In doing this, the number of quadrature points, nhermt, must be large 
enough that the energy levels are stable with respect to it to at 
least one more significant figure than the accuracy needed for the 
energy levels.  This process goes quickly if one looks at one 
arrangement at a time with only one basis function in the other 
arrangement channels.  During this, one minimizes the average 
energy (selected using nave1 and nave2) of all the levels 
expected to be open anywhere plus one that will be closed 
everywhere (if possible).  If this results in vibrational 
levels that compare well enough with those determined in 
step 3 (i.e., to about 0.1 meV), one is done with this step.  
If not, he must either increase the size of the basis or decrease the 
number of levels included in the average enegy and then reoptimize.
\item
Choose the jmax in each arrangement so that all the asymptotically open
vibration-rotation states are included.  jmax need usually 
be no larger than this because the high j states are repulsive 
and play little role in the strong interaction region.  nglegn 
needs to be larger than jmax at large $\rho$ and considerably 
larger than that at small $\rho$ to keep the channel quadratures converged.	 
\item
Estimate the smallest $\rho$ needed, $\rho_{min}$.  That may be 
obtained from a few dvr,  fem or preliminary abm calculations 
and WKB arguments to assure that the $\psi (\rho)$ are down to 
about $10^{-5}$ their turning point values. 
\item
For each arrangement $f$, set $\rho$ to the larger of $\rho_{min}$ 
and $s_{mf}$, the scaled equilibrium distance ($r_{xf} r_{ef}/d_f$).  
Check to be sure the quadrature to be used there (alternate or channel) 
gives $\cal E$'s that are solidly converged  w/o quadrature and 
eigenvalues of the primitive overlap matrix that are all positive.  
Minimize the average over all the open (or low-lying) $\cal E$'s 
under variations of $\zeta_f$ and $\delta_f$ to obtain optimum 
values of these parameters.     
\item
If the alternate quadrature is necessary to converge the 
integrals at $\rho_{min}$, choose a reasonable $\rho_{switch}$, 
which is usually the smallest rho at which the channel quadratures 
are readily convergable.  Check to be sure both quadratures 
are converged there.     
\item
The parameters $b_\alpha,$ $ r_\alpha,$ and $d_\alpha$ can be 
used to make $c_\alpha$ depend on $\rho$.  That comes in if 
Old Way = yes is chosen.  That is not usually necessary in the 
current version of the code, and the $d_\alpha$'s can be kept 
equal to zero.
\item
If one of the channels is much smaller than the other(s) and 
has a correspondingly tighter quadrature mesh, setting intwt=2 
for that arrangement will cause overlap integrals between 
basis functions in that channel and those in other channels to be 
done only in the channel with the better quadrature and may allow 
use of the channel quadratures at smaller $\rho$ than otherwise.
\item
abm leaves out basis functions whose eigenvalue of the overlap 
matrix is less than eigmin.  For ordinary double precision 
arithmetic, eigmin should be kept set to 1.e-06.  A message is 
generated if overlaps between two arrangements evaluated by 
quadratures in the two channels differ by more than ovrerr, 
and  1.e-04 is a convenient value for it.
\item
ngood is the number of surface eigenfunctions to be written out.  
One should set nfreq equal to ngood.  If the number of eigenvalues 
generated falls below that, the program will stop. 
\end{itemize}


\par\vfill\eject

The parameters in Table I of the ABM paper or Table I of the big
recent HO2 paper (Prof. Lagana has a copy) are determined from
the masses, spectra of the diatomic molecules, and the highest energy
you plan to use as in the Algorithm writeup.  The only things in
Table I which can vary are the numbers of quadrature points, and
it is important that you check at several rho to make sure your
quadratures are always better converged than your energies.

The parameters $r_x$, $a_n$, and $c_alpha$ are always near unity.
 I start with
them equal to unity and then vary them at the largest rho needed, as
in step 5 of the Algorithm writeup to find their optimum values for
the size basis I am using.  They are scaling parameters.  The quantities
they scale are determined by the program from the input parameters of Table
1, but the scaling parameters are determined totally by optimizing the
energy levels at very large rho.  In our work thus far, the optimum value
for $r_x$ has turned out to be slightly larger than 1, and the optimum values
for $c_alpha$ and $a_n$ have turned out to be slightly smaller than 1.

During the above calculations, I usually take zeta to be 1 and delta to be
0.01.  The calculations at large rho should be rather insensitive to them.
Then, I go to the smallest rho at which the ABM calculations will be done
and optimize the energies by varying zeta and delta.  If your smallest
rho is still larger than the $s_m$ in all of your channels, your results
should not depend much on zeta and delta and a rough optimization will
suffice.  Otherwise, you will have to vary them more carefully.  In all
this, you need to be careful to be sure your quadratures are still well
converged, or the nonlinear least squares minimization (of the average
of the N energies you want to be accurate) will converge to values which
make the quadratures diverge and give spurious energies.


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