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%  contact
%  Peter G. Martin                   pgmartin@cita.utoronto.ca
%  CITA, University of Toronto       (416) 978-6840    
%  Toronto, Ontario M5S 3H8          Fax      -3921
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\def\S{Sec.}

%
%**end of header
%

\noindent
{\bf A Refined H$_3$ Potential Energy Surface }

{\narrower\par
\smallskip\noindent
Arnold I. Boothroyd\footnote{$^a$}{Present address: Department of
Mathematics, Monash University, Clayton, Victoria 3168, Australia}

\noindent
{\it Canadian Institute for Theoretical Astrophysics, University of
Toronto, Toronto, Ontario, M5S 3H8, Canada}

\smallskip\noindent
William J. Keogh\footnote{$^b$}{Present address: Corel Corporation, 1600
Carling Avenue, Ottawa, Ontario, K1Z 8R7, Canada}

\noindent
{\it Department of Chemistry, University of Toronto, Toronto, Ontario,
M5S 3H6, Canada}
  
\smallskip\noindent
Peter G. Martin

\noindent
{\it Canadian Institute for Theoretical Astrophysics, University of
Toronto, Toronto, Ontario, M5S 3H8, Canada}

\smallskip\noindent
Michael R. Peterson

\noindent
{\it Department of Chemistry, University of Toronto, Toronto, Ontario,
M5S 3H6, Canada}

\bigskip\bigskip

%%%%%%%% ABSTRACT %%%%%%

\noindent
In evaluating some low temperature ($T < 1000$~K) thermal rate
coefficients for inelastic rotational excitation of H$_2$ by H atoms,
Sun and Dalgarno have found a marked sensitivity to the potential energy
surface adopted for the calculations.  We have investigated the origin
of the discrepancies between previous H$_3$ potential energy surfaces
and have developed a refined surface which addresses these concerns.
New quasiclassical trajectory calculations of cross sections for low
energy rotational excitation are reported.
% 
The refined surface is based on 8701 {\it ab~initio\/} energies, most
newly computed for this purpose.  It has the same functional form as our
earlier (BKMP) surface, but since the fit of the parameters is more
fully constrained than for any previous surface it is a more accurate
representation.  The refined surface matches the {\it ab~initio\/}
energies with an overall rms error of 0.27~$\millih$
(i.e.,~0.17~kcal/mole) and a maximum absolute deviation of 6.2~$\millih$
(for a very compact high energy equilateral triangle conformation).  For
``noncompact'' conformations (no interatomic distance smaller than
1.15~bohr), the rms error is 0.18~$\millih$ and the maximum absolute
deviation is 1.7~$\millih$.
%
The refined surface is compared critically to four previous surfaces,
including the DMBE surface of Varandas \etal, in several respects:
Legendre expansion coefficients; the interaction region for low energy
rotational excitation; near the collinear saddle point; near conical
intersections of the ground and first excited state surfaces; the van
der Waals well; and compact geometries.
%
We have also compared new first excited state {\it ab~initio\/} energies
for 1809 conformations with corresponding predictions from the DMBE
surface.

\vskip 0.1 pt}

\bigskip

\leftline{\bf I. INTRODUCTION }
\nobreak\medskip\nobreak

In this paper we present a refined H$_3$ potential energy surface which
addresses an important issue raised recently by Sun and Dalgarno~[1].
They carried out quantum mechanical calculations of the rotational
excitation (ground vibrational state) of H$_2$ by H atoms, using the
``DMBE'' H$_3$~potential energy surface of Varandas \etal~[2].
Quite surprisingly, the low temperature ($T < 1000$~K) thermal rate
coefficients for the transitions $0 \rightarrow 2$ [i.e., H + H$_2(0,0)
\rightarrow$ H + H$_2(0,2)$] and $1 \rightarrow 3$ were significantly
higher than those found earlier by Green and Truhlar~[3] and Mandy and
Martin~[4] based on the ``LSTH'' surface developed by Truhlar and
Horowitz~[5].  Sun and Dalgarno concluded ``that the large discrepancies
arise primarily from the different potential energy surfaces employed in
the two calculations.'' As supporting evidence they cited unpublished
quasiclassical trajectory (QCT) calculations of state-to-state cross
sections by S.\ Lepp and by two of us (Keogh and Martin; see \S~V) on
the two potential energy surfaces, which reveal the systematic
differences underlying those in the rate coefficients.

Because of the fundamental significance of this H + H$_2$ system in
theoretical chemistry, and because of important applications to the
rotational excitation and hence cooling efficiency of hydrogen-rich gas
(e.g., in astrophysics of the interstellar medium), it is important to
understand in what ways the potential energy surfaces are different and
which is to be preferred for these rotational excitation (and other)
calculations.  Sun and Dalgarno comment: ``It is claimed by Varandas
\etal\ that their potential is superior to the LSTH potential at
intermediate and long-range separations of H and H$_2$'', from which
they concluded that the DMBE surface should be preferable. However, what
is meant by ``intermediate and long-range'' is critical to this
qualitative assessment.  The range of separation $\rhhtwo$ (distance of
H atom from center of H$_2$) most influential for this low energy
rotational excitation is 3 -- $5\>\bohr$ (in this paper distances are in
bohrs).  One could call this range intermediate, it being well inside
the longer-range van der Waals region and the very long range asymptotic
region.  Varandas \etal\ do argue that the DMBE surface might be better
than the LSTH surface at these intermediate distances (about
$3.5\>\bohr$ is cited).  However, neither the LSTH surface nor the DMBE
surface is well constrained there, this being an interpolation region
between {\it ab initio\/} data at smaller separations and
experimentally-constrained descriptions of the van der Waals region.

Our interest in resolving this question stems in part from our
development of the ``BKMP'' H$_3$ potential energy surface~[6]. This
extended the LSTH surface in several ways based on a more extensive grid
of {\it ab initio\/} interaction energies. Although the BKMP surface
incorporates new data extending to $\rhhtwo = 4\>\bohr$, it too is least
well constrained in the critical range of $\rhhtwo$. The BKMP surface
tends to be closer to the LSTH than to the DMBE surface (see Ref.~6);
%[6]
comparative calculations of QCT rate coefficients at 1000~K by Lepp,
Buch, and Dalgarno [7] bear this out.

Comparison of the Legendre expansion coefficients of the potential
energy surfaces (see Ref.~6 and \S~IV~B)
%[6]
shows that the DMBE, LSTH, and BKMP surfaces have important differences
in this critical unconstrained range of $\rhhtwo$, which of course
explains the different derived cross sections and rate coefficients.
More recent {\it ab initio\/} calculations by Partridge \etal~[8]
including $\rhhtwo > 4\>\bohr$ begin to address this problem, giving
coefficients closer to those from the LSTH and BKMP surfaces than to
those from the DMBE surface.  Comparison of the first expansion
coefficient (spherically averaged interaction energy, $V_0$) shows the
DMBE surface to be too repulsive.  More important for the current issue
of rotational excitation is the second coefficient (first anisotropic
term, $V_2$~[9,10]).  For the DMBE surface uniquely this coefficient
displays a suspicious dip at $\rhhtwo \sim 4\>\bohr$ for an H$_2$
molecule size near $1.4\>\bohr$ (see Ref.~6, Table VIII;
%[6]
Partridge \etal~[8] Table IV and Fig.~6); as discussed below
(\S~IV~B), this feature is unphysical.

Initially we carried out new {\it ab initio\/} calculations for 540
conformations on a preliminary grid spanning this unconstrained region
of $\rhhtwo$, confirming that there were indeed problems with the DMBE
surface in the interpolation region.  We found further support for this
view in the {\it ab initio\/} calculations in Partridge \etal~[8].
Therefore, as the basis for a new analytic surface that
would provide a correct description of this region, we have performed
further {\it ab initio\/} calculations for an extensive grid of 6548
H$_3$ conformations, also including good coverage in the 3 --
$5.5\>\bohr$ region (\S~II~A).

There are other issues to be addressed.  As mentioned, such
``intermediate'' separations are significantly less than the extent of
the van der Waals well ($\rhhtwo \gtapprox 6\>\bohr$). Partridge
\etal~[8] have pointed out that the H$_3$ surfaces could also be
improved in the region of the van der Waals well ($\sim$6 --
$10\>\bohr$; see e.g., their Figure 2). Their results are incorporated
in our new analytic surface as well (see \S~II~C and \S~IV~C).

Mielke \etal~[11] have presented low temperature ($T < 1000$~K)
thermal quantal rate coefficients for the reaction $\rm D + H_2
\rightarrow H + HD$.  For the LSTH and DMBE surfaces there is good
agreement with the experimental data. While the BKMP results agree near
1000~K, they are systematically larger for lower temperatures,
differing from the values for the other surfaces by a factor 2 -- 2.3 at
167~K\hbox{}.  This behavior at low temperature is presumably related in
part to the value of the classical barrier height for this reaction,
which was constrained to be lower in the BKMP surface (9.54~kcal/mol)
than in the LSTH and DMBE surfaces (9.80 and 9.65~kcal/mol). As
described in \S~II~D, our new surface is updated to match the best
current estimate of the classical barrier height, 9.61~kcal/mol.  As
pointed out by Garrett \etal~[12], these rate coefficients also
depend on the width of the barrier and more global characteristics of
the surface which should be well constrained by our large grid of {\it
ab initio\/} calculations (Secs.~II~A and D).

The lowest two potential energy surfaces for H$_3$ exhibit a seam of
conical intersections for $D_{3h}$ conformations [13] (i.e.,
equilateral triangles), at least for ``normal'' sizes (see \S~IV~G).
Mielke \etal\ comment that the DMBE surface ``is the only surface
with the correct behavior in the region of the conical intersection''
(see also Ref.~8).  
%[8]
Certainly it represents the only attempt at an analytic continuation
onto the upper surface and calculation of the associated nonadiabatic
coupling.  It is also the case that in developing the BKMP surface~[6]
we did not incorporate the {\it ab initio\/} data reported by Varandas
\etal\ for 18~nonsymmetric conformations in the vicinity of
$D_{3h}$ symmetry, since there was insufficient information to bring the
energies to a common consistent basis.  However, this does not
necessarily mean that the BKMP (or LSTH) surface has an incorrect
behavior; it has simply not been constrained or modeled so explicitly.
We have in fact compared the shape of the BKMP and DMBE surfaces on cuts
based on the $s$ and~$q$ symmetry coordinates of Varandas \etal,
and the differences are not substantial (\S~IV~E).  This is because
conformations which apply appropriate constraints do arise in our
regular grid of conformations.  Nevertheless, to be assured of a good
treatment of this part of the surface we have made further {\it ab
initio\/} calculations (\S~II~E) to constrain the fit.

Johnson~[14] pointed out that the London equation, which is used as a
component of the three surfaces, can lead to unphysical discontinuities
in the derivative of the potential for certain high energy isosceles
triangle geometries.  This is not addressed in the LSTH or DMBE
surfaces, which thus exhibit this problem, but is corrected here as in
the BKMP surface.  A ``compact'' geometry is defined to be one in which
any of the three distances between the nuclei is less than the
distance~$\rr$ below which the Johnson correction begins to modify the
H$_2$~triplet curve: as with the BKMP surface, $\rr = 1.15\>\bohr$.

Aguado and Paniagua~[15] have proposed a new way of fitting {\it ab
initio\/} data for triatomic molecules.  Using a FORTRAN code supplied
to us by Aguado~[16] we provide an assessment of their 71-parameter
surface.  While this ``AP'' surface has a low rms deviation with respect
to the {\it ab initio\/} BKMP data to which it was fitted, it turns out
to have several deficiencies in its shape (\S~IV).

The surface fitting here follows that in Ref.~6
%[6]
for the BKMP surface and is summarized in \S~III; the new surface is
dubbed ``BKMP2''. The weighted rms deviation with respect to 8701 {\it
ab initio\/} energies used is only $0.27\>\millih$.  In \S~IV we discuss
the various desirable properties which the BKMP2 surface
embodies. \S~IV~B focusses on how the new surface differs in shape
relative to the previous surfaces, particularly in the critical
intermediate region $\rhhtwo \sim 3$ -- $5\>\bohr$; the deviations are
largest for the DMBE surface.  New QCT calculations of cross sections
for rotational excitation are given in
\S~V.  These show the systematic differences arising from the various
surfaces; again, the DMBE values are the most discrepant.  QCT cross
sections should be adequate for computing thermal rate coefficients down
to about 600~K [4, 17], but for lower temperatures such as treated by
Sun and Dalgarno [1] new quantum mechanical calculations will be
required.

\bigskip
\leftline{\bf II. ENERGIES CONSTRAINING THE SURFACE}
\nobreak\medskip\nobreak

The conformations and energies which we used to fit our new analytic
BKMP2 H$_3$~surface are described in this section.  There are a total of
7172 different conformations for which {\it ab initio \/} data are
available. Some of these have energies computed using two or more basis
sets, for a total of 8701 {\it ab initio \/} energies; in addition 125
{\it ab initio \/} data from Ref.~8
%[8]
for van der Waals conformations were also used.  Tables of these data
are available in electronic form from the authors~[18].  For some very
small and large separations, this set of energies is supplemented to
ensure good behavior in those parts of the surface.

Unless otherwise specified, atomic units are used in this paper, i.e.,
distances are in bohrs~($\bohr$) and energies are in hartrees~($\har$),
millihartrees~($\millih$), or microhartrees~($\microh$).  Recall that
$1\>\bohr = 0.529177\>$\AA, while $1\>\millih = 0.0272114\>$eV${} =
0.62751\>$kcal/mole.  Unless otherwise specified, energies are expressed
relative to the energy of isolated hydrogen atoms, i.e.,~zero energy
corresponds to separated ${\rm H + H + H}$.  Relative to this, an
isolated H~atom plus~H$_2$ at the equilibrium separation of
$1.401\>\bohr$ ({\it not\/} a ground state H$_2$~molecule) has an energy
of~$-0.174496\>\har$ (or~$-109.498\>$kcal/mole),
according to the accurate analytic H$_2$~curve presented by
Schwenke~[19] which we used throughout this paper to obtain accurate
H$_2$~energies.

\medskip
\leftline{\bf A. Conformations for new {\it ab initio\/} computations}
\nobreak

Figure~1 in the BKMP paper (Ref.~6)
%[6] 
illustrates the coordinates used to specify our grids of
H$_3$~conformations, which can be described by the shortest interatomic
distance~$r_1$, the next-shortest distance~$r_2$, and the (exterior)
angle between them~$\theta_{12}$ (which we shall simply denote $\theta$
in this paper; $\theta = 0^{\circ}$ is collinear).  Alternatively, one
may specify the shortest distance~$r_1$, the distance~$\rhhtwo$ of the
third atom with respect to the {\it midpoint\/} of~$r_1$, and the
angle~$\chi$ between $r_1$ and $\rhhtwo$ ($\chi = 0^{\circ}$ is
collinear).

In our preliminary exploration of the interpolation region, we adopted
the ($9s3p1d$)/[$4s3p1d$] Gaussian basis set used previously~[20, 6, 21,
22].  The grid was specified as follows: $r_1$ was chosen from $\{$0.8,
1.0, 1.2, 1.4, 1.6, 1.8, 2.1, 2.35, 2.6$\>\bohr\}$, $\rhhtwo$ was
chosen from $\{$3.0, 3.5, 4.0, 4.5, 5.0, 5.5$\>\bohr\}$, and $\chi$ ran from
$0^\circ$ to $90^\circ$ inclusive in increments of $10^\circ$ (note that
this grid included a few points with $r_2 < r_1$).  This preliminary
grid comprised 540 conformations.

For our more comprehensive grid, our ($9s3p2d$)/[$4s3p2d$] basis
set~[22] was used; the only difference from the [$4s3p1d$] basis set was
that the single $d$~function (with exponent~$1.0\>\bohr^{-2}$) is
replaced by two $d$~functions, with exponents 1.76
and~$0.62\>\bohr^{-2}$ (optimized for H$_2$ at equilibrium separation).
The grid was specified as follows: $r_1$ and $r_2$ were chosen from
$\{$0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8,
1.9, 2.0, 2.15, 2.3, 2.45, 2.6, 2.8, 3.0, 3.2, 3.4, 3.7, 4.0, 4.3, 4.6,
5.0, 5.5$\>\bohr\}$, such that $r_1 \le r_2 \le r_3$; $\thot$ ran from
$0^\circ$ to~$90^\circ$ inclusive in increments of~$10^\circ$, and
continued from $\thot = 90^\circ$ through $\chi = 90^\circ$ inclusive in
one, two, or three equally-spaced increments that did not
exceed~$10^\circ$ (note that for $r_1 \le r_2$, $\chi = 90^\circ$
corresponds to $\thot \le 120^\circ$).  This grid contained 5083 {\it
new\/} conformations, as well as 32~conformations from the preliminary
grid, and a number of conformations from the old grid of Ref.~6.
%[6].

To investigate the bending behavior away from the collinear saddle point
(\S~II~D) we computed energies for 50 new conformations with $r_1 = r_2
= 1.757 \bohr$ for a range of $\thot$.  In the vicinity of $D_{3h}$
symmetry (\S~II~E) we computed energies for the 18~conformations of
Varandas \etal~[2] and added 138~similar {\it new\/}
nearly-equilateral conformations.  We also computed energies for 77~{\it
new\/} conformations in between grid-points for $0.7 \>\bohr \le r_1 \le
1.0 \>\bohr$ (where there were few conformations from older grids) to
test (and constrain) the inter-grid-point behavior of the analytic
surfaces (\S~IV~G).

Our comprehensive grid comprised 6548 conformations.  Compared to the
grid used for the BKMP surface, it offers the following improvements:
new coverage in the ``interpolation region'' from 4 to~$5.5\>\bohr$;
better coverage at all $r_1$ and~$r_2$, particularly in the ranges
0.6~to~$1\>\bohr$ and 3 to~$4\>\bohr$; and better angular coverage in
$\thot$ for all separations.

For comparative tests, we computed new {\it ab initio\/} energies for
{\it all\/} the conformations from the old grid and those available to
us from other authors.  This comprised our 403~old conformations~[6],
137~conformations from Liu~[23], 156~conformations from Siegbahn and
Liu~[20] (six of which duplicate conformations from Liu),
31~conformations from Blomberg and Liu~[24] (six of which duplicate
conformations from Liu), and 50~conformations from Bauschlicher
\etal~[25] (one of which duplicates a conformation from our old
grid~[6]).  Besides the 57~conformations from Partridge \etal~[8]
which were contained in one of our above grids, we also computed
energies for the remaining 330~of their conformations that had $\rhhtwo
\le 5.5$ (their grid had comprised 503~conformations in all). An update
for some of these H$_3$ energies was supplied by Partridge~[26].

For further checking, we also carried out computations using the
[$4s3p1d$] basis for all the conformations of other authors described
above, plus the added nearly-equilateral conformations.  Along with our
energies from Ref.~6, 
%[6]
this yielded 1233 conformations with [$4s3p1d$]
energies, all of which could be compared with [$4s3p2d$] energies from
the comprehensive grid, and 710 of which could be compared to energies
obtained by other authors.

\medskip
\leftline{\bf B. {\it Ab initio\/} computations and analysis}
\nobreak

The {\it ab initio\/} computations and analysis of errors follow the
methods described fully in our previous papers~[6, 21, 22]. Here we give
a brief summary and update.  Energies were obtained using a workstation
version of Buenker's \hbox{MRD-CI} program~[27].  For most
conformations, molecular orbitals were obtained from an open shell
SCF~calculation; where the open shell~SCF was slow to converge, mixed
shell~SCF (1~closed and 1~open shell) or closed shell~SCF was
automatically used instead to obtain the molecular orbitals.  In almost
all cases a second CI~energy was also obtained using molecular
orbitals from one of the other two SCF~types.

A fairly extensive set of reference configurations was used: the minimum
size of~$\Crsq$, the sum of reference configuration~$C^2$ values, was~0.976,
and the {\it average\/} size was~0.986.  A configuration selection threshold
of $T = 0.4\>\microh$ was used: the extrapolation to zero threshold
comprised a shift of order~$0.4\>\millih$, with an estimated rms
uncertainty of~$0.02\>\millih$ (this is expected to be ``random'' error,
in that systematic effects are probably on a smaller scale than the grid
spacing of the conformations).  As a check, some conformations were also
computed with $T = 2\>\microh$ (yielding a shift of order~$1\>\millih$,
with an estimated rms extrapolation uncertainty of~$0.07\>\millih$).

A very small semi-empirical Davidson-type correction to full~CI was
made, similar to that described in Ref.~21, 
%[21]
of $\Delta E_{\hbox{\eightrm full-CI}} = \lambda_{\rm DC:H3} \Delta
E_{\rm DC} = \lambda_{\rm DC:H3} (1-\Crsq) \Delta E_{sd} / \Crsq$.  The
values of $\lambda_{\rm DC:H3}$ were chosen to minimize the difference
between results using different SCF-types for molecular orbitals; this
resulted in $\lambda_{\rm DC:H3}^{\rm(open)} = 0.01$, $\lambda_{\rm
DC:H3}^{\rm(mixed)} = 0.19$, and $\lambda_{\rm DC:H3}^{\rm(closed)} =
0.22$ (cf.\ old $\lambda_{\rm DC:H3}$ values of 0.16, 0.3, and 0.5,
respectively, used in Refs.~6 and 21).  
%[6, 21]
This correction is negligible for open shell cases, and of
order~$0.1\>\millih$ for mixed and closed shell cases, but the estimated
uncertainty for all three SCF-types is of order~$0.1\>\millih$ (largely
``systematic'' error, with a somewhat smaller ``random'' component).
Comparing CI~energies obtained using molecular orbitals from different
SCF~calculations yielded: $\rm rms_{open-mixed} = 0.059\>\millih$
(3891~conformations), $\rm rms_{open-closed} = 0.065\>\millih$
(1301~conformations), and $\rm rms_{mixed-closed} = 0.056\>\millih$
(658~conformations).  These differences are somewhat less than the
estimated errors; note, however, that with the old $\lambda_{\rm DC:H3}$
values these rms differences would have been 0.064, 0.148,
and~$0.151\>\millih$, respectively, with the closed shell cases having a
systematic shift averaging~$0.14\>\millih$ with respect to the open and
mixed shell cases.

An independent estimate of the combined uncertainty from the
extrapolation plus full~CI correction may be obtained by comparing our
[$4s3p1d$] basis results with the 186 energies computed previously~[20,
23, 24] with this basis set; the rms difference between the earlier
energies and ours is $0.099\>\millih$, or $0.078\>\millih$ if one leaves
out the three energies in Ref.~20 
%[20]
that lie $0.2\>\millih$ above even our {\it truncated-CI\/} energies.
This suggests that an estimated uncertainty of $0.1\>\millih$ in the
extrapolation plus full~CI correction is reasonable.

Because of the finite basis set, there is a small basis correction which
averaged about~$1.1\>\millih$ for the [$4s3p2d$] basis set, and
$1.8\>\millih$ for the less complete [$4s3p1d$] basis set; in both
cases, the basis correction is larger for compact conformations and
smaller for noncompact conformations.  As before~[6, 21, 22], this
correction was based on the London formula.  Note that the BKMP version
(see Ref.~6)
%[6]
of the triplet curve (without the Johnson correction) was used in the
computation of the London H$_3$~basis correction.  Our London-based
correction has been tested further by Partridge \etal~[8] who
carried out calculations with much larger basis sets, mostly
($11s5p3d1f$)/[$6s5p3d1f$], and concluded that ``the results confirm the
accuracy of the BKMP potential and their scheme for correcting the H$_3$
calculations using the error in the H$_2$ potential.''  Partridge~[26]
has supplied us with the appropriate errors in the H$_2$ potential for
the larger basis sets.

Partridge \etal~[8] suggested that the correction scheme might be
refined further by introducing a slight dependence on~$\thot$ or $\chi$.
Note, for example, that the London basis correction is independent of
angle for $r_1 = r_2$, where one might expect variation by a factor of
order~1.5 from consideration of the change in the geometry.  To gauge
how to implement this we have computed (\S~II~A), with both of our
(smaller) basis sets, energies for the 387 of their conformations with
$\rhhtwo \le 5.5\>\bohr$. For the angular dependence it appears,
empirically, that a relatively simple function~$f_{\rm
basis}(\chi,r_1,r_2)$ multiplying the original correction~$\Delta E_{\rm
basis}^{\rm London}$ of Ref.~[21]
%[21]
should suffice, namely
%
$$f_{\rm basis}(\chi,r_1,r_2) = 1 - f_1 \, e^{-(r_2/r_1-1)^2} \cos^2\chi
 + f_2 \, e^{-f_3(r_2/r_1-1)^2 - f_4(r_1-1.4)} \sin^2\chi \> ,
 \eqno(1)$$
%
where $r_1 \le r_2 \le r_3$, $f_1$ and~$f_2$ are parameters that differ
for different basis sets and $f_3$ and~$f_4$ are simple constants.  The
exponential factors are largely to make~$f_{\rm basis}(\chi,r_1,r_2)$ go
to unity for conformations approaching $\rm H_2 + H$, where the London
basis correction approaches the correct value (by construction).  The
$f_1$~term is used to make a small shift to obtain a collinear saddle
point energy that agrees with the Monte Carlo classical barrier height
of $9.613 \pm 0.006$~kcal/mole obtained by Diedrich~[28] (see also
Ref.~29
%[29]
and \S~II~D).  To do this for Partridge's [$6s5p3d1f$] basis, Liu's
Slater basis, our [$4s3p2d$] basis, and our [$4s3p1d$] basis (the latter
is the same as that of Refs.~20 and 24)
%[20, 24]
one needs $f_1 = 0.182$, 0.09053, 0.073, and 0.06747, respectively.
Note that the resulting shift for linear symmetric geometries
is~$\ltapprox 0.1\>\millih$, smaller than our earlier estimate~[6]
of~$0.15\>\millih$ for the uncertainty in the basis correction for such
geometries; this estimate still seems reasonable.

The $f_2$~term is used to provide additional flexibility in the angular
dependence.  Differencing energies obtained with [$4s3p1d$] and
[$4s3p2d$] basis sets shows no evidence of a systematic trend as a
function of angle; however, differencing energies obtained with
[$4s3p2d$] and [$6s5p3d1f$] basis sets {\it does\/} show a significant
angular effect (note that the size of the H$_2$ basis correction differs
by a factor of~$\sim 1.6$ for both of these pairs of bases).  Since this
angular effect depends on something other than the pure size of the
basis set or basis correction, the most conservative assumption is that
Partridge's [$6s5p3d1f$] basis set requires no additional angular
dependence beyond that introduced by setting $f_1 = 0.182$, i.e., that
$f_2 = 0$ for the [$6s5p3d1f$] basis set.  In this case, one obtains
$f_2 = 0.6$ and~0.35, respectively, for our [$4s3p2d$] and [$4s3p1d$]
basis sets; values of $f_3 = 1.5$ and $f_4 = 0.8$ give the best
agreement.  The resulting shift reduces the rms difference between
Partridge's [$6s5p3d1f$] energies and our [$4s3p2d$] energies from
0.37 to~$0.14\>\millih$ (387~conformations).  The shift is comparable to
our earlier estimate of~$0.3\>\millih$ for the uncertainty in the basis
correction for nonlinear geometries; furthermore, an angular
contribution to the basis correction error amounting to even half the
H$_2$ basis correction would still be only~$0.4\>\millih$, and so our
earlier estimate again seems reasonable.

For conformations that are very close to being equilateral triangles
(i.e., near $D_{3h}$ symmetry),
a systematic difference remains between Partridge's energies and ours;
it can be of order $0.5\>\millih$ for $r_1 = r_2 \ltapprox 2\>\bohr$
and $\chi = 90^\circ$, but vanishes for $\thot \ltapprox
90^\circ$ or $r_2/r_1 \gtapprox 1.3$.  This remaining difference
cannot be characterized well enough to attempt a correction; but it
points out the difficulty of estimating the (systematic) error in the
basis correction, which for equilateral triangles might be of
order~$1\>\millih$.

In summary, our {\it ab initio\/} energies~$E_{\hbox{\eightrm final}}$
have a total ``random'' error of about $0.1\>\millih$ (from
\hbox{MRD-CI} errors), plus a systematic error of about~$0.4\>\millih$
(mostly from basis correction errors; smaller for linear conformations,
larger for equilateral triangles).  The errors are somewhat larger for
the high-energy compact conformations, and smaller for conformations
tending towards~${\rm H_2 + H}$.

\medskip
\leftline{\bf C. The van der Waals region}
\nobreak

The Partridge \etal~[8] {\it ab initio\/} computations also
improve the van der Waals region, filling in a region at large
$\rhhtwo$ from 5.5 to 8~--~$10\>\bohr$ that had been left out in previous
theoretical studies.  Although this region is not at the root of the
discrepancies in inelastic rotational excitation, it seemed worthwhile
to refine the BKMP surface here as well.  In the fitting we have made
use of the {\it ab initio\/} energy values for all 503 conformations of
Ref.~8, 
%[8]
of which 116 have $6\>\bohr \le \rhhtwo \le 10\>\bohr$.

Partridge \etal~[8] showed that their {\it ab initio\/} data in
the well region and beyond were accurately fit by a modified
Tang--Toennies (MTT) surface [this gives energies relative to
H$_2(r_1)$~+~H for fixed $r_1$].  We adopted this MTT representation;
the four parameters describing the dependence on $\rhhtwo$ and $\chi$
were found as a function of $r_1$ by nonlinear least squares fitting to
their {\it ab initio\/} energies for $\rhhtwo \ge 5.5\>\bohr$, the
accuracy of these fits being of order~$1\>\microh$.  Following their
recommendation, the well was deepened slightly for each~$r_1$ (from 66
to $75\>\microh$ at $r_1 = 1.449\>\bohr$) by adjusting the two
parameters describing the spherically symmetric radial dependence.  By
construction, for $\rhhtwo > 10\>\bohr$ this representation extrapolates
to the asymptotic long range dispersion interaction energies~[8].

This MTT form is not incorporated explicitly in our surface. Instead,
energies were computed from it (plus Schwenke's~[19] H$_2$ potential) to
be used as constraints in fitting the new surface (the BKMP surface used
energies from the Gengenbach \etal\ formula~[30]).
The grid was as follows: $r_1$ was chosen from $\{$0.7, 0.8, 1.0, 1.2,
1.449, 1.7, 2.0, 2.45, 3.0, 3.4, 4.0$\>\bohr\}$, and $\rhhtwo$ was
chosen from $\{$5.5, 5.75, 6.0, 6.25, 6.5, 6.75, 7.0, 7.25, 7.5, 7.75,
8.0, 8.5, 9.0, 10.0, 12.0, 15.0$\>\bohr\}$, subject to the constraint
that $r_2 \ge 4.5\>\bohr$; the angle~$\chi$ ran from $0^\circ$
to~$90^\circ$ inclusive, generally in increments of~$15^\circ$ (for $r_1
= 0.7$, 1.7, 2.45, and~$3.4\>\bohr$, the $\chi$~increment
was~$30^\circ$, and values of $\rhhtwo \ge 10.0\>\bohr$ were omitted).
This van der Waals grid comprised 968~conformations (80~of which
coincide with conformations computed in Ref.~8).
%[8]

During the fitting procedure described in \S~III, deviations of the fit
for these conformations were given high weights as follows.  
%
A set of basic weights for $r_1$ near equilibrium ($1.4\>\bohr \le r_1 <
1.5\>\bohr$) were assigned according to $\rhhtwo$: 10~for Partridge
\etal\ conformations with $5.25 \>\bohr < \rhhtwo \le 5.6\>\bohr$;
20~for MMT-derived points with $5.5\>\bohr \le \rhhtwo < 5.6\>\bohr$;
40~for $5.6\>\bohr \le \rhhtwo < 5.9\>\bohr$; 80~for $5.9\>\bohr \le
\rhhtwo < 8.9\>\bohr$; and 360~for the ``distant'' conformations with
$8.9 \>\bohr \le \rhhtwo$.
%
These were then scaled down on the basis of $r_1$: by 0.5 for
$0.9\>\bohr \le r_1 < 1.4\>\bohr$ and $1.5\>\bohr \le r_1 < 2.1\>\bohr$;
by 0.25 for $2.1\>\bohr \le r_1 < 3.2\>\bohr$; and by 0.1 for $r_1 <
0.9\>\bohr$ and $r_1 \ge 3.2 \>\bohr$.
%
Some of the conformations generated by the MTT representation had $r_1$
rather far from equilibrium and so the weight was reduced from the above
by a factor 2 for $1.499 \le r_2/r_1 < 1.7499$ and by 4 for $r_2/r_1 <
1.499$. But in no case was the weight allowed to sink below unity.

\medskip
\leftline{\bf D. Collinear saddle point/classical barrier height}
\nobreak

On the collinear cut the adopted H$_3$ saddle point occurs at $r_1 = r_2
= r_s = 1.7570\>\bohr$ (linear symmetric conformation) and its energy is
$E_{saddle} = -0.1591760\>\har$ (relative to~${\rm H + H + H}$).  Using
the Schwenke equilibrium~H$_2$ energy, this matches the latest Monte
Carlo result (Ref.~28)
%[28]
on the classical barrier height, namely, $9.613 \pm 0.006$~kcal/mol.
This saddle point is incorporated in our surface implicitly through
specification of the effective H$_2$ triplet curve.

Various {\it ab initio\/} estimates of the barrier height have been
discussed recently by Partridge \etal~[8]; these depend on the
basis correction.  Smooth agreement of our surface with the Monte Carlo
result at the saddle point is achieved by fitting {\it ab initio\/} data
which have been corrected using a slight reduction of the London basis
correction for geometries that are not too far from linear symmetric
(see discussion above in~\S~II~B)\hbox{}.

In order to tie down the asymmetric stretch and bending quadratic force
constants at the collinear saddle point, two conformations with high
weight (1000) were added as for the BKMP surface. We also computed
energies for 50 new conformations with $r_1 = r_2 = 1.757 \bohr$ for a
range of $\thot$ to study the angular dependence away from this saddle
point (e.g., the quartic bending force constant; \S~IV~D).

\medskip
\leftline{\bf E. Near $D_{3h}$ symmetry}
\nobreak

Varandas \etal\ report {\it ab initio\/} data for 18~nonsymmetric
conformations explicitly in the vicinity of $D_{3h}$ symmetry.  These
are specified using $s$ and~$q$ symmetry coordinates which are useful in
connection with analytic continuation to the upper electronic surface;
$q$~is the rms~$r_i$, while $s$~is a measure of the deviation from
$D_{3h}$~symmetry ($s = 1$ corresponds to the collinear limit). There is
a third angular coordinate giving the direction of the deviation; only
two choices of this angle were used in their conformations corresponding
to two cases: (a) scalene with
$r_1=q(1-s\cos30^\circ)^{1/2}$, $r_2 = q$, and
$r_3=q(1+s\cos30^\circ)^{1/2}$ (an atom is moved in a direction which is
initially, for small~$s$, parallel to the opposite side, $r_2$, of the
triangle)
%
and (b) isosceles with
$r_1=r_2=q(1-s/2)^{1/2}$ and $r_3=q(1+s)^{1/2}$ (an atom is moved in a
direction perpendicular to and toward the opposite side, $r_3$, of the
triangle).
%
As part of the comprehensive grid (\S~II~A) we computed in the same
manner energies for 156 conformations with $q = \{0.8$, 0.9, 1.0, 1.2,
1.4, 1.6, 1.8, 2.0, 2.3, 2.5, 2.8, 3.2, $3.7 \bohr\}$, $s = \{0.05$,
0.1, 0.2, $0.3\}$.  These include the 18~conformations of Varandas
\etal, which have $q = 1.6$, 2.0, and~$2.5\>\bohr$.  We also added a
third case: (c) isosceles with
$r_1=q(1-s)^{1/2}$ and $r_2=r_3=q(1+s/2)^{1/2}$
%
which is equivalent (geometrically -- the smallest separation is
denoted~$r_1$) to using negative values of~$s$ in case~(b); case~(a) is of
course unaffected by changing the sign of~$s$.

The behavior of the surface in the vicinity of $D_{3h}$ symmetry is also
constrained through many other conformations in the grids described in
\S~II~A (conformations with $\chi = 90^\circ$ including equilateral
triangles, and others with $\thot = 90^\circ$ through $\chi =
90^\circ$ in the appropriate range of $r_2/r_1$).

\medskip
\leftline{\bf F. Extremely compact conformations}
\nobreak

To ensure that {\it extrapolation\/} of our surface to very short
distances is well behaved we added 175~{\it extremely\/} compact
conformations (distances down to~$0.32\>\bohr$) with energies
estimated from the (non-Johnson-corrected) London equation.  The
conformations were generated for $0^\circ \le \chi \le 90^\circ$, at
$15^\circ$~intervals, for $r_1 = 0.5$, 0.4, and~$0.32\>\bohr$ with~$r_2$
taking the values $\{$0.32, 0.4 0.5, 0.6, 0.8, 1.0, 1.2, 1.4,
$1.75\>\bohr\}$ (subject to the constraint~$r_2 \ge r_1$), and also for
$r_1 = r_2 = 0.6\>\bohr$ (for overlapping angles, energies for the
latter agreed with the {\it ab initio\/} energies within $\ltapprox
10$\%).

\bigskip
\leftline{\bf III. THE FITTING PROCEDURE}
\nobreak\medskip\nobreak

Since the BKMP surface~[6] is already a flexible and accurate
representation we used this parametrized functional form as the basis
for the fit to the new set of H$_3$ energies.  Despite the vastly
increased number of constraining energies and conformation coverage, the
same number of parameters was used; the parameters were simply
reoptimized. For the main steps in the fitting procedure we followed
closely the same strategy as for the BKMP surface; we used the same
optimization methods and we adopted the same criteria to assess the
surface fit.

%T Table I
The overall rms error of the BKMP2 surface (as well as those of the
LSTH, DMBE, and BKMP surfaces) with respect to the {\it ab~initio\/}
data is given in Table~I, along with the rms errors for various subsets
of the {\it ab~initio\/} energies, corresponding to various subsections
of the surface.
%
The fitted surface has continuous first and second derivatives
everywhere (the cusp at conical intersections with the first excited
state surface is only rounded off by a negligible, and removable,
amount, namely~$1\>\microh$; see \S~IV~E). Other specific qualities of
the BKMP2 surface are discussed in \S~IV.


\medskip
\leftline{\bf A. The Triplet Equation and the Johnson Correction}
\nobreak

In the first step we optimized the effective H$_2$~triplet curve.  The
form used for~$r \ge \rr$ (no Johnson correction) was actually fitted to
71 {\it ab initio\/} energies for linear symmetric conformations,
including 8 for compact ones.
As in all subsequent stages of the fit, deviations for energies greater
than $0.2\>\har$ were given a reduced weight of $0.2/E$.
In addition, energies for 14 ``distant'' linear asymmetric van der Waals
conformations ($\rhhtwo = 12 \>\bohr$ and $15 \>\bohr$) for 7 values of
$r_1$ ($0.8 \>\bohr$ to $4.0 \>\bohr$) were included in order to force
the triplet curve to decay such that the London energy is not too large
in the region of the van der Waals well.  As described in \S~II~C, the
deviations for these energies were assigned high weights producing a fit
better than $2\>\microh$.
%
Of the triplet parameters only $\gamma_3$, $\gamma_4$, $\gamma_5$,
and~$\gamma_6$ have to be optimized. As in the BKMP and LSTH surfaces,
$\gamma_1$ and~$\gamma_2$ are obtained self-consistently by ensuring
that the surface reproduce the collinear saddle point accurately.  All
linear symmetric geometries were fit very well with this modified form
of the triplet potential (see Table~I).
%
The correct symmetric force constant~$k_s = 0.1068\> \har\, \bohr^{-2}$ is
obtained (for a summary of force constants see Table~VI in Ref.~6).
%[6] 

The Johnson correction coefficients $\gamma_7$, $\gamma_8$,
and~$\gamma_9$ were calculated analytically such that the triplet curve
is continuous (with continuous first and second derivatives) at the point
$\rr = 1.15\>\bohr$ where the Johnson correction takes over.

\medskip
\leftline{\bf B. The Asymmetric Correction}
\nobreak

The second step was optimization of the asymmetric
correction~$V_{asym}$. The optimization of the seven parameters was
carried out using 850 energies for linear conformations, including all
674 {\it ab~initio\/} values for noncompact geometries.
%
A further~67 {\it ab~initio\/} energies for barely compact ($1.01\>\bohr
\le r_1 < \rr$) linear conformations were included so that the
$V_{asym}$~correction term would have the correct shape through~$\rr$.
%
The deviations for the 70 Partridge \etal\ energies among these were
assigned higher weights (the same scheme also applies in later parts of
the fit).  For their 22 conformations in the van der Waals range
($\rhhtwo > 5.25 \>\bohr$) the weights are as described in \S~II~C.  For
smaller $\rhhtwo$ the scheme is essentially the same except that the
basic $\rhhtwo$-based weights at the outset are ramped down to 7 for
$4.75\>\bohr < \rhhtwo \le 5.25\>\bohr$; 4 for $4.25\>\bohr < \rhhtwo
\le 4.75\>\bohr$; 2 for $3.75\>\bohr < \rhhtwo
\le 4.25\>\bohr$.  No weight in this range is less than 1.0. There is no
special weight for $\rhhtwo \le 3.75\>\bohr$.
%
In addition, 109 energies for noncompact linear van der Waals
conformations from the MTT representation were used to ensure a
reasonable van der Waals well in the linear direction for a range of
$r_1$.
%
The correct asymmetric force constant~$k_a = -0.0579\> \har\,
\bohr^{-2}$ is obtained due to the constraining conformation with
weight~1000.

\medskip
\leftline{\bf C. Two Compact Corrections}
\nobreak

The third step was the optimization of the terms $C_{all}$
and~$C_{asym}$, correction terms which have an effect only on compact
geometries (which are affected by the Johnson correction).  The
parameters in both terms (16 in total) were optimized simultaneously
using 356 energies for compact linear conformations.  This included 281
{\it ab initio\/} energies from our grid and 5 from Partridge \etal.
Using the MTT representation, 45 van der Waals conformations ensure that
the van der Waals well has a stable behavior for these
\hbox{nonequilibrium} H$_2$~separations.  Some 25 extremely compact
conformations (\S~II~F) all had $E>0.2\> \har$ and weights $0.007
(0.2/E)$, sufficient to ensure that modest extrapolation of our surface
remains well behaved (\S~IV~G).

\medskip
\leftline{\bf D. The Bending Terms}
\nobreak

The fourth step was the optimization of terms to reduce the residuals
for bent geometries, $V_{bend}^\sa$~and~$V_{bend}^\sb$.  All of the
bending terms (including the compact ones) contain an exponential decay
factor to ensure the correct asymptotic behavior; the same values as for
the BKMP surface were used.

The parameters in $V_{bend}^\sa$ and~$V_{bend}^\sb$ (32 in total) were
optimized simultaneously using 6402 energies for bent conformations,
including all 5453 {\it ab~initio\/} energies for noncompact bent
geometries (91 of the latter are for van der Waals conformations from
Ref.~8)
%[8]
A further~366 {\it ab~initio\/} energies for barely compact bent
conformations were included to obtain the correct shape through~$\rr$.
%
The 390 Partridge \etal\ energies among the above were assigned higher
weights by the above scheme where appropriate.
%
Using energies from the MTT representation, 583 noncompact bent van der
Waals conformations were used to ensure a reasonable van der Waals well
for bent geometries for a range of $r_1$.
%
The correct bending force constant~$k_b = 0.0210\> \har\, \bohr^{-2}$ is
obtained due to the constraining conformation with weight~1000.

\medskip
\leftline{\bf E. The Compact Bending Terms}
\nobreak

The final step was to optimize the term~$C_{bend}$, a correction term
for compact bent geometries.  The 60 coefficients were optimized using
2731 energies for compact bent conformations, including 2350 {\it
ab~initio\/} values (12 of these are for van der Waals
conformations from Ref.~8)
%[8]
%
plus 231 bent compact van der Waals conformations using the MTT
representation.  As mentioned in~\S~II~F, we also added 150~{\it
extremely\/} compact bent conformations of low weight.

\bigskip
\leftline{\bf IV. DISCUSSION}
\nobreak\medskip\nobreak

A FORTRAN computer subprogram which evaluates this surface and its
derivatives with respect to the three internuclear distances is
available from the authors~[18]; it is in a form that readily replaces
the routines supplied to evaluate the LSTH~[31] or BKMP~[6] surface. The
BKMP2 surface (of the BKMP form) is simple enough that it can be used
for quantum mechanical and QCT scattering calculations. Quantum
mechanical calculations using the BKMP surface have been carried out by
Chang and Brown [32] 
and QCT calculations have been reported by Keogh \etal~[33] and
Lepp, Buch, and Dalgarno~[7].  As a benchmark for quasiclassical
trajectory studies, a BKMP2 (or BKMP) evaluation of the interaction
potential and derivatives takes typically 140~$\mu$s on a DEC 3000
workstation (Alpha 150 MHz processor).

\medskip
\leftline{\bf A. Low rms for {\it ab initio\/} energies and other global
comparisons}
\nobreak

Table~I summarizes some comparisons of the various surfaces with the
{\it ab initio\/} data. 

The quality of the fit for the BKMP2 surface is very good.  It fits the
entire set of 8701 {\it ab initio\/} energies with an rms error
of~$0.27\>\millih$ (the maximum deviation being~$6.2 \>\millih$ for a
very compact equilateral triangle with $r_1 = 0.7\>\bohr$).  For 6077
energies for noncompact conformations the rms error is~$0.18\>\millih$
(the maximum deviation being only~$1.67\>\millih$).  These rms
accuracies are comparable to the estimated ``random'' error
of~$0.1\>\millih$ for our newly-computed {\it ab~initio\/} energies and
are actually slightly smaller than the estimated systematic errors in
the {\it ab~initio\/} energies ($0.4\>\millih$ mostly from basis
correction errors; \S~II~B).

At low energies the new BKMP2 surface is quite similar to the BKMP
surface, but it is based on a grid with more extensive coverage and
better resolution; for example, the rms difference between the BKMP2 and
BKMP surfaces is~$0.26\>\millih$ for the noncompact conformations used
for the {\it ab~initio\/} calculations. At higher energies we have used
many more {\it ab~initio\/} data than were used in the BKMP fit, and so
the new BKMP2 representation should be even more accurate there; note
for example the much lower rms error for compact bent conformations.

The BKMP surface fits their version (slightly different basis
correction) of the energies for their 766~independent conformations with
an rms error of~$0.25\>\millih$ (the maximum deviation
being~$1.93\>\millih$).  The BKMP fit to the new energies for this
``BKMP subset'' is slightly worse, having an rms error of
$0.36\>\millih$.  For the same energies, the BKMP2 fit has an rms error
of $0.30\>\millih$, slightly larger that the original $0.25\>\millih$
value in Ref.~6
%[6]
because now the same number of parameters is being used to fit a much
more extensive set of energies.  The BKMP surface fits our entire new
set of energies with an rms error of~$3.76\>\millih$ (maximum
deviation~$262\>\millih$ for an equilateral triangle with $r_1 =
0.6\>\bohr$); comparison with the new value $0.27\>\millih$ for BKMP2
illustrates the overall benefit of the reoptimization of the parameters.

Not shown in Table~I are values for the AP surface; for non-compact
conformations, it has an rms error of~$3.4\>\millih$ (with maximum
deviation~$17\>\millih$), several times worse than any of the other
surfaces.  The overall rms error is~$6.2\>\millih$, with a maximum
deviation of~$46\>\millih$ for an equilateral triangle with $r_1 =
0.6\>\bohr$.  As shown below, this surface also lacks many of the
desirable features seen in the other surfaces.

\medskip
\leftline{\bf B. Legendre coefficients}
\nobreak

We have computed Legendre coefficients $V_{2n}$ for the various surfaces
for several values of $r_1$. These are compared in Figure~1 for the case
$r_1= 1.449\>\bohr$ which is representative.  Also plotted are the
coefficients computed by Partridge \etal\ directly from their {\it ab
initio\/} data, which have less extensive angular coverage than our
comprehensive grid.  

%F Figure 1: Legendre coefficients, several panels

Our fit surface generates a spherical average $V_0$ (Fig.~1a) which is
in good agreement with the other surfaces, the results of Partridge et
al.\ and the best available experimental data discussed therein.  The
DMBE surface stands out somewhat as being more repulsive in the
intermediate (or interpolation) range near $\rhhtwo = 4\>\bohr$.  The AP
surface is the most discrepant (as for all $V_{2n}$), having poor
extrapolation characteristics beyond where it is constrained by the BKMP
data.

For inelastic rotational excitation we are interested in the radial
dependence of the first anisotropic coefficient $V_2$ (Fig.~1b) for the
intermediate range $3\>\bohr < \rhhtwo < 5.5\>\bohr$.  Our {\it
ab~initio\/} energies (see~\S~II~A) provide better radial coverage than
was available to the LSTH, DMBE, and even BKMP fits for this range and
also superior angular coverage; thus the BKMP2 surface should be the
most accurate in the interaction region.  Note how the added coverage
has changed the BKMP2 coefficient slightly relative to the BKMP
coefficient; both are quite close to that from the LSTH surface.  The
DMBE surface produces a dip near $\rhhtwo = 4\>\bohr$; this incorrect
behavior accounts for the discrepant cross sections and rate
coefficients (see also \S~V).  The AP surface has the most discrepant
representation of the minimum and again extrapolates poorly.
There are slight discrepancies of the BKMP2 result compared to Partridge
\etal, but only at the $0.2\>\millih$ level comparable to the random
and systematic errors of the fit (see also Fig.~2).

Calculations of $V_4$, $V_6$, and $V_8$ (the first two shown in Figs.~1c
and 1d respectively) show the continued good behavior of the
coefficients from the BKMP2 fit. Other surfaces are somewhat less
precise, as for the lower order coefficients.

%F Figure 2: cuts to show basic asymmetry; 2 panels

A complementary way of examining how surfaces differ in their basic
asymmetry is to plot cuts of the interaction potential for various fixed
$r_1$ and $\rhhtwo$ as a function of $\chi$.  We have done this for many
combinations; three representative ones are shown in Figure~2a--c for
$r_1 = 1.449\>\bohr$ and $\rhhtwo = 3$, 4, and $5\>\bohr$.  The relative
bulge in the DMBE potential near $\rhhtwo = 4\>\bohr$ at $\chi= 90\deg$
(up by $\sim 1.5~\millih$) is reflected in the anomalous dip in $V_2$.

These plots also show the {\it ab initio\/} energies used in the BKMP2
fit, including the Partridge \etal\ data.  The BKMP2 surface appears
to have low level angular structure (which changes with $\rhhtwo$) somewhat
larger than warranted by just these data. However, recall that the
surface is required to fit simultaneously energies on a grid comprising
many more separations and angles. Therefore, some of the structure just
arises from the decaying tails of functions required for more compact
geometries.  These oscillations are constrained by data at many more
angles than shown in Figure~2, but only at the level of the rms of the
fit, $\sim 0.1\>\millih$ in this region of conformation space ($\rhhtwo
\simeq 4\>\bohr$). Given this and the accuracy of the {\it ab initio\/}
data, we tend to discount the details of the high order coefficients at
levels below $0.1\>\millih$. Even the derived BKMP2 $V_2$ coefficient is
slightly different than directly computed by Partridge \etal\ from
{\it ab initio\/} data for $\rhhtwo > 4\>\bohr$ (Fig.~1b), which suggests
caution in applications at very low energy (or temperature).

\medskip
\leftline{\bf C. The van der Waals well}
\nobreak

%F Figure 3: cuts to show basic van der Waals wells; 3 panels

The BKMP2 functional form for the surface has sufficient flexibility
that it is not necessary to join it explicitly to another description
(like MTT) for larger separations ($5.5\>\bohr < \rhhtwo < 10\>\bohr$).
The BKMP surface fit the constraining data from Gengenbach \etal~[30]
well, but these data have been superseded by the present MTT description
which also has a much more securely-based dependence on $r_1$.  The
BKMP2 fits resulted in a van der Waals well that agrees fairly closely
in position, depth, and shape with the data underlying the MTT
description.  Figure~3 illustrates the angular dependence of the well
for $r_1 = 1.449\>\bohr$ for different analytic H$_3$~surfaces.  Also
shown are the {\it ab initio\/} data from Partridge \etal\ and the
constraining data generated from our MTT formula.  The BKMP2 well
behaves smoothly as a function of angle and also of the size of the
H$_2$~molecule: we have examined the well for $r_1$ as small
$0.6\>\bohr$.  The BKMP2 Legendre coefficients in the van der Waals
region are correspondingly well behaved.  For $r_1 = 1.449\>\bohr$ the
minimum of $V_0$ is $75\>\microh$ occurring at $\rhhtwo = 6.58\>\bohr$,
close to the values recommended by Partridge \etal

The BKMP2 analytic form of the surface (like the BKMP and the LSTH
surfaces) approaches the London surface exponentially at very large
distances ($\rhhtwo > 10\>\bohr$), unlike the DMBE surface which closely
mimics the asymptotic form of the interaction; however, at these
distances the interaction energy is only a few~$\microh$.  The BKMP2
surface also dissociates to give Schwenke's accurate H$_2$~potential
curve.

\medskip
\leftline{\bf D. Saddle point properties}
\nobreak

The collinear saddle point occurs at $r_1 = 1.757\>\bohr$.  The
classical barrier height of our fitted surface is $15.32\>\millih$
(i.e.,~$9.613\>$kcal/mole) in good agreement with the best recent
determinations (\S~II~C).  As mentioned in \S~III, all three {\it
quadratic} force constants for our surface agree closely with Liu and
Siegbahn's~ values and those for the LSTH, DMBE, and BKMP surfaces.

%F Figure 4 bending direction

The reactivity of the system can also depend on the angular dependence
away from this collinear saddle point.  In their discussion of
transition state properties and the possible relevance of the zero-point
corrected barrier height (which correction depends on various force
constants), Garrett \etal\ [12]  used an expansion for fixed
nearest-neighbor distances:
%
$$
V(\thot) = V_{saddle} + {1\over 2} F \thot^2 + {1\over 24} A
\thot^4
+ \, \dots \eqno(2)
$$
%
Adopting the framework of this expansion, Figure~4 shows
$2[V(\thot) - V_{saddle}]/ \thot^2 $ versus $\thot^2$
for $r_1 = r_2 = 1.757\>\bohr$ ($1.755\>\bohr$ for DMBE).
The curves therefore have intercept $F$, which is the quadratic bending
force constant (as expressed in $\millih$~radian$^{-2}$).  For the BKMP2
surface, as for the LSTH and BKMP surfaces, this is determined by a
highly weighted constraining energy (solid square); the DMBE and AP
surfaces also have similar bending force constants.

The slope of the curves in Figure~4 is approximately $A/12$, where $A$
is the quartic bending force constant ($\millih$~radian$^{-4}$) as
defined by Garrett {\etal}  Clearly there is curvature to each surface,
pointing to the need for additional higher-order expansion coefficients
beyond $\thot \simeq 0.1$~radian.  Because of this curvature,
evaluation of the quartic bending constant could depend on the range of
$\thot^2$ over which the (average) slope was measured.  To be
precise in calculating $A$ as one of many expansion coefficients, one
can either restrict the range to very small $\thot^2$ and fit
coefficients only up to $A$ or fit a suitably higher-order polynomial
over a more extended range. The values of $A$ calculated consistently in
this way for the BKMP2, BKMP, LSTH, and DMBE surfaces are 75.6, 78.9,
16.0, and $28.7\>\millih$~radian$^{-4}$, respectively.  We cannot see
how to reconcile these with the values reported for the latter two
surfaces in Table I of Garrett \etal

Away from the collinear geometry, the surfaces have slightly different
curvature as illustrated in Figure~4.  The detailed shape of the surface
here is determined by the functional form of the analytic
representation, whose parameters are in turn determined by energies for
a wide variety of conformations not chosen specifically to isolate this
region near the collinear saddle point. The BKMP2 (and BKMP) surface has
the most angular flexibility and so shows some higher-order
structure. Note that the BKMP2 surface is very close to the BKMP
surface, despite many more constraining energies. In particular, we did
use the {\it ab initio\/} data shown in Figure~4 for the new fit, but with
obviously little effect since the BKMP fit was already so good, well
within the errors (largely systematic) expected for bent geometries
(\S~II~B). The deviation is $0.3\>\millih$ at $\thot \sim 1$~radian and
actually decreases at smaller $\thot$ (the deviations appear amplified
in Figure~4 because of division by $\thot^2/2$).

In particular the differences in the quartic bending force constant are
not being determined by localized {\it ab initio\/} data.  Nor is it
possible to use data of this level of accuracy to choose between the
surfaces in this region.  This is probably not a serious issue.  While a
difference in the zero-point corrected barrier height, which is the
transition state theory activation energy at 0~K in the absence of
tunneling, could influence the relative exchange reaction rate
coefficients according to the usual Boltzmann factor, Garrett \etal~[12]
also emphasize the importance of tunneling contributions for low
temperature thermal rate coefficients. Low energy tunneling
probabilities are sensitive to the global characteristics of the
surface, not just the barrier height, including fairly large-distance
interactions ($\rhhtwo \sim 4\>\bohr$). From our ongoing discussion of
both the increased constraints and the overall accuracy of the BKMP2
surface, we suggest that it should produce the most reliable results.

\medskip
\leftline{\bf E. Near $D_{3h}$ symmetry}
\nobreak

As discussed in \S~II~E, we have computed many additional points to
explore the nature of the $D_{3h}$-symmetry conical intersection (which
for ``normal'' sizes occurs between the ground and first excited state
surfaces; see \S~IV~F).  The minimum energy conical intersection is also
the saddle point for $\thot = 120^\circ$.  On the BKMP2 surface it
occurs at
$r_1 = 1.9711\>\bohr$ with energy $74.33\>\millih$, or
$62.85\>$kcal/mole relative to the equilibrium~H$_2$ energy; the
corresponding values for the DMBE surface are $1.973\>\bohr$ and
$63.36\>$kcal/mole.

%F Figure 5 conical intersection cuts 

Figure~5 shows two cuts of the surfaces near this intersection,
specified using the Varandas \etal\ symmetry coordinates $q$ and~$s$
(see \S~II~E). Both have $q = 2.0\>\bohr$ rather than $1.9711\>\bohr$ in
order to display the {\it ab initio\/} data.  The two cuts are for cases
(a) and (b)+(c) of \S~II~E\hbox{} (scalene and isosceles, respectively).
We have examined such plots for many different values of $q$ (not
shown). Except for the AP surface, which does not even have a cusp in
its formulation and which for other values of $q$ often does not pass
through the {\it ab initio\/} energy for the equilateral triangle
geometry, the surfaces all fit the {\it ab initio\/} energies fairly
well. This is true even of the LSTH and BKMP surfaces, whose shapes were
not constrained by the data shown in Figure~5 (except for the
equilateral triangles); this agreement is possible because of a
sufficiently flexible and apt functional form (not the case for the AP
surface) and because conformations which apply appropriate constraints
do arise in a regular grid of conformations even if not explicitly
generated in symmetry coordinates.

Quantitatively the BKMP2 surface, with constraints including these new
{\it ab initio\/} data, provides the most accurate fit.  For noncompact
geometries (i.e., with shortest distance $r_1 \ge 1.15\>\bohr$) near the
conical intersection ($\vert s \vert \le 0.3$) the rms errors for 325
{\it ab initio\/} energies are $0.29\>\millih$ for BKMP2,
$0.62\>\millih$ for BKMP, $0.82\>\millih$ for DMBE, $1.2\>\millih$ for
LSTH, and $9.4\>\millih$ for AP.  For 54 compact {\it ab initio\/}
energies for $q \ge 0.85\>\bohr$ (see \S~IV~G for the reason for this
cut) and $\vert s \vert \le 0.3$, the rms errors are $0.91\>\millih$ for
BKMP2, $2.4\>\millih$ for BKMP, $5.9\>\millih$ for DMBE, $7.9\>\millih$
for LSTH, and $20\>\millih$ for AP.  For the very high-energy points
with $q < 0.85\>\bohr$, only the BKMP2 surface remains accurate
(\S~IV~G.)

\medskip
\leftline{\bf F. The DMBE first excited state surface }
\nobreak

Porter, Stevens, and Karplus~[13] showed that for $D_{3h}$ symmetry
(equilateral triangles) the doubly-degenerate $E^\prime$ state of H$_3$
corresponds to a conical intersection between two surfaces.
For H$_3$ systems of ``normal'' size the two states intersecting at
$D_{3h}$ symmetry (of symmetry $A_1$ and $B_2$ for isosceles triangles
and otherwise $A^\prime$ for scalene triangles) define the ground and
the first excited potential energy surfaces.

The DMBE surface is intended to be a consistent analytic representation
of the two lowest potential energy surfaces for H$_3$, explicitly making
an analytic continuation based on this conical intersection for $D_{3h}$
geometries.  The comparison in Figure~5 (for $q = 2\>\bohr$) with the
{\it ab initio\/} energies for the first excited state that we have
computed shows that the DMBE representation is indeed quite good near
$D_{3h}$ symmetry.  However, away from the conical intersection it
becomes less accurate, as might be expected because the first excited
state surface was not constrained by these (or any) {\it ab initio\/}
energies. As $q$ decreases, the predicted surface is not as asymmetric
in $s$ (for isosceles triangles) as the {\it ab initio\/} data indicate.

Having computed {\it ab initio\/} excited state energies for 1809 of our
conformations
(these excited state energies are also available from the authors~[18]),
we can examine more globally the accuracy of the DMBE
excited surface prediction; the overall rms error for these energies is
$19\>\millih$.  Near the conical intersection, for $\vert s \vert \le
0.3$ where the surface is best, there is an rms error of $1.1\>\millih$
for 71 noncompact points and $1.9\>\millih$ for 20 compact points with
$q \ge 0.85\>\bohr$.  With no restriction on $s$ the overall rms error
is $7.9\>\millih$ for all 1246 noncompact points and $30\>\millih$ for
531 compact points with $q \ge 0.85\>\bohr$.

\medskip
\leftline{\bf G. Compact conformations}
\nobreak

Except for our studies, the high energy interaction region has received
little attention.  For compact conformations ($r_1 < 1.15\>\bohr$), the
BKMP2 surface should be fairly accurate up to at least the energy of our
highest {\it ab~initio\/} points, of order $1\>\har$ above the energy
of~${\rm H + H + H}$.  The grid coverage (\S~II~A) is more extensive
than for the BKMP surface.  Energies up to~$200\>\millih$ above the
energy of~${\rm H + H + H}$ are given full weight in the fit.  Our
weighting scheme gives lower weight to the deviations for conformations
with very high {\it ab~initio\/} energies, but only by a factor of six
at the very most, thus ensuring that the absolute error at high energy
is acceptable.  Energies above $\sim 1\>\har$ require extrapolation,
and, of course, any fitted surface is best used for interpolation rather
than extrapolation; however, we did ensure (\S~II~F) that our
H$_3$~surface does not differ too greatly from the non-Johnson-corrected
London equation, even when extrapolated in to very short distances where
we did not have {\it ab~initio\/} energies.  For $r_1 = r_2$ the surface
is monotonically increasing for all $\thot$ up to energies $\sim
3\>\har$, down to distances slightly shorter than~$0.35\>\bohr$.  Below
this, for sufficiently bent geometries, the analytic surface turns over
and drops steeply.

Our multiple-root MRDCI calculations near and including $D_{3h}$
symmetry show some interesting changes as the geometry becomes more
compact.  For $q = 0.9\>\bohr$, the energy of the doubly-degenerate
$E^\prime$ root is $341\>\millih$ (all energies relative to $\rm H + H +
H$).  The third root at $D_{3h}$ symmetry is single, with state symmetry
$A_1^{\prime}$; away from $D_{3h}$, the symmetry becomes $A_1$ for
isosceles triangles and otherwise $A^{\prime}$ for scalene
triangles.  The energy of the $A_1^{\prime}$ root is
$347\>\millih$.  In Varandas symmetry coordinates ($s$ and their angle
$3 \theta$) the lowest three surfaces defined by these states are all
fairly convex (``bowl''-shaped) for such compact geometries.  As $q$
decreases the upper surface associated with $A_1^{\prime}$ rises
in energy, but the two formerly lower surfaces associated with
$E^\prime$ rise even more quickly.  The $A_1^{\prime}$ and
$E^\prime$ roots at $D_{3h}$ symmetry coincide at $q = 0.844\>\bohr$
(energy $438\>\millih$).

For even smaller $q$ the $A_1^{\prime}$ root is lower than the
$E^\prime$ root.  For example, by $q = 0.8$ the energy of the
$A_1^{\prime}$ root is $525\>\millih$, clearly separated from the
energy $530\>\millih$ for the doubly-degenerate $E^\prime$ root.  In
fact the surface associated with $A_1^{\prime}$ remains the ground
state for most $s$ and $3 \theta$, except for an isolated intersection
with the first excited state for a squashed isosceles triangle geometry
at $s \approx 0.225$ (i.e., $r_1 = r_2 \approx 0.754\>\bohr$, $r_3
\approx 0.885\>\bohr$), having an energy $\sim 549\>\millih$
($24\>\millih$ up from the bottom of the ``bowl'').  The conical
intersection between the surfaces through $E^\prime$ at $D_{3h}$
symmetry is now between the first and second excited state surfaces.

We can conclude that the DMBE first excited state surface should not be
extrapolated to such compact geometries. The first reason is technical,
since the representation depends explicitly on the existence of the
$D_{3h}$-symmetry conical intersection with the ground state surface;
the second reason is practical, since the rms error with respect to our
{\it ab~initio\/} energies increases to $30\>\millih$ for 531 compact
points with $q > 0.85\>\bohr$, and $50\>\millih$ for 32 compact points
with $q < 0.85\>\bohr$.  (For near-$D_{3h}$ points with $\vert s \vert
\le 0.3$, corresponding rms errors are $1.9\>\millih$ for 20 points with
$q > 0.85\>\bohr$, and $64\>\millih$ for 15 points with $q <
0.85\>\bohr$.)  Of course, the DMBE ground state representation is also
becoming less accurate when extrapolated to compact geometries, having
an rms error of $5.6\>\millih$ for 2118 compact points with $q >
0.85\>\bohr$, and $40\>\millih$ for 51 compact points with $q <
0.85\>\bohr$.  

Even our more accurate BKMP2 surface, which provides a much better fit
to the small-$q$ (compact) {\it ab initio\/} data, still exhibits a
slight cusp at small-$q$ $D_{3h}$ geometries (rather than, for example,
shifting it to the isosceles-triangle geometry where the actual
intersection takes place when $q = 0.8\>\bohr$).  This cusp at $D_{3h}$
symmetry arises from the London term which appears explicitly in the
analytic surface, as it does for the LSTH and DMBE surfaces. For compact
geometries this London term is increasingly overwhelmed by the three
compact correction terms (see Ref.~6) in the BKMP2 surface,
%[6]
but a $D_{3h}$ cusp, now relatively insignificant, must still appear.
The size of both the spurious $D_{3h}$ cusp and features like the
missing isosceles-triangle cusp for small $q$ are of order a few
$\millih$, like the error in the fit, so that while the BKMP2 surface is
qualitatively incorrect, it is quantitatively still acceptably close.

\bigskip

\leftline{\bf V. CROSS SECTIONS FOR INELASTIC ROTATIONAL EXCITATION}
\nobreak\medskip\nobreak

In this section we report on quasiclassical trajectory calculations
carried out to assess low energy state-to-state cross sections for
rotational excitation (nonreactive rotationally inelastic scattering).
The method of calculation generally follows that described by Mandy and
Martin~[4, 17] and references therein. We use a revised three-body
trajectory code written to incorporate several different optional
surfaces and to make use of a state-of-the-art Runge-Kutta variable step
integrator~[33].  Cross sections are computed for transitions in the
downward direction [e.g., for H + H$_2(0,2) \rightarrow$ H +
H$_2(0,0)$]; the upward cross sections are determined from these using
detailed balance~[34].  With the QCT method, cross sections involving
nonreactive excitation and exchange are distinguished. For the low
energy transitions of interest here, only the former are important for
parity-conserving transitions; these are the values reported.

Results for H + H$_2(0,0) \rightarrow$ H + H$_2(0,2)$ are shown in
Figure~6.  This example is sufficient to contrast the results from
different surfaces. Like the original DMBE quantum cross sections~[1],
the QCT cross sections for the DMBE surface remain substantial to a much
lower relative kinetic energy than encountered (in QCT calculations) for
other surfaces (LSTH, BKMP, and now BKMP2).

The relatively high DMBE cross sections at low energy are of course what
cause the discrepantly high rotational rate coefficients at low
temperature. The origin of the high cross sections is the abnormally
large asymmetry in the DMBE surface through the intermediate
(interpolation) region $\rhhtwo \sim 4\>\bohr$ (\S~IV~B; Figs.~1 and 2).
We have been able to verify this explicitly with our QCT calculations as
follows.  First the trajectories (specified by particular initial
conditions) which produce the low energy excess cross section for the
DMBE surface were isolated.  Then the time development of these
trajectories was examined in detail.  The H atom approaches the \h2\
molecule and is strongly deflected in an interaction region which is
seen to be $\rhhtwo \simeq 4\>\bohr$, leaving behind a rotating
molecule. For the same initial conditions, but for the other potential
energy surfaces, there is little deflection or induced rotation.

The smooth curves in Figure~6 are from nonlinear least squares fits of
the data to the Le~Roy class~II excitation function~[35].  The thermal
rate coefficient for rotational excitation can be obtained in analytic
form by the appropriate integral of this excitation function over the
Maxwell-Boltzmann distribution~[35]. For the LSTH surface we verified
that these rate coefficients are within a few percent of those
previously calculated~[4, 17] for temperatures up to 2000~K
(coefficients for higher temperatures would require an extrapolation of
the new cross section data to higher energies than computed).  Our rate
coefficients also compare well with the QCT results by Lepp, Buch, and
Dalgarno~[7] for the BKMP, DMBE, and LSTH surfaces.  For the DMBE
surface the QCT rate coefficients also agree well with the quantum
values above 600~K.

In summary, we have shown that these rate coefficients for nonreactive
rotational excitation using the DMBE surface are indeed too large;
quantitatively this is by a factor 10 at 1000~K and growing toward lower
temperatures (more than a factor 30 discrepancy at 600~K). We have
identified the reason for this excess. Since new BKMP2 surface is the
most directly constrained by {\it ab initio\/} calculations for
conformations in the relevant interaction region and shows less
extraneous structure there (previous surfaces relied on interpolation),
it should be the most reliable for these calculations.  The rate
coefficients for temperatures 600, 800, 1000, 1400, and 2000~K are
0.076, 0.32, 0.76, 2.0, and $4.0 \times 10^{-11}$~\cms, respectively, up
to 30\% lower than the LSTH results at the lower temperatures.

\bigskip
\leftline{ACKNOWLEDGEMENTS}
\nobreak\medskip\nobreak

We wish to thank the following: 
R.~J.~Buenker and his associate P.~Funke for supplying us with a
workstation version of their \hbox{MRD-CI} program; 
H.~Partridge for an electronic version of the {\it ab initio\/} data
reported in Ref.~8,
%[8]
for making new calculations to correct a few of those entries, and for
evaluating the error in the H$_2$ potential using the larger basis sets;
D.~L.~Diedrich for his latest evaluation of the saddle point energy;
and A.~Aguado for the FORTRAN code to evaluate the AP surface.
This work was supported by grants from the Natural Sciences and
Engineering Research Council of Canada (A.~I.~B., P.~G.~M.).

%%%%%
\bigskip\bigskip
%%%%%\vfill\eject

\leftline{\bf REFERENCES}
\nobreak\medskip\nobreak

\bref [1] Y.\ Sun and A.\ Dalgarno\ref{94}{Astrophys.~J.}{427}{1053}

\bref [2] A.~J.~C.~Varandas, F.~B.~Brown, C.~A.~Mead, D.~G.~Truhlar, and
N.~C.~Blais\ref{87}{J.~Chem.\ Phys.}{86}{6258} 

\bref [3] S.\ Green and  D.~G.\
Truhlar{\ref{79}{Astrophys.~J. (Lett.)}{231}{L101}  

\bref [4]  M.~E.~Mandy and P.~G.~Martin\ref{93}{Astrophys.~J. Supp.}{86}{99}

\bref [5] D.~G.~Truhlar and C.~J.~Horowitz, \refnp{78}{J.~Chem.\
Phys.}{68}{2466}; \refnp{79}{J.~Chem.\ Phys.\ (Errata)}{71}{1514} 

\bref [6] A.~I.~Boothroyd, W.~J.~Keogh, P.~G.~Martin, and
M.~R.~Peterson\ref{91}{J.~Chem.\ Phys.}{95}{4343} 

\bref [7] S.\ Lepp, V.\ Buch, and
A.\ Dalgarno\ref{95}{Astrophys.~J. Supp.}{98}{345}  

\bref [8] H.\ Partridge, C.~W.~Bauschlicher, J.~R.\ Stallcop, and E.\
Levin\ref{93}{J.~Chem.\ Phys.}{699}{5951}

\bref [9] J.~R.~Stallcop\ref{74}{J.~Chem.\ Phys.}{61}{5085}

\bref [10] D.~G.~Truhlar, and R.~E.~Wyatt\ref{77}{Adv.~Chem.\ Phys.}{36}{141}

\bref [11] S.~L.\ Mielke, G.\ C.\ Lynch, D.~G.~Truhlar, and
D.~W.~Schwenke\ref{94}{J.~Phys.\ Chem.}{98}{8000}

\bref [12] B.~C.\ Garrett, D.~G.~Truhlar, A.~J.~C.~Varandas, and N.~C.\
Blais\ref{86}{Int.\ J.~Chem.\ Kinet.}{18}{1065}

\bref [13] R.~N.\ Porter, R.~M.\ Stevens, and M.\
Karplus\ref{68}{J.~Chem.\ Phys.}{49}{5163}

\bref [14] B.~R.~Johnson\ref{81}{J.~Chem.\ Phys.}{74}{754}

\bref [15] A.\ Aguado and M.\ Paniagua\ref{92}{J.~Chem.\ Phys.}{96}{1265}

\bref [16] A.\ Aguado, private communication (1992)

\bref [17] P.~G.\ Martin and M.~E.\
Mandy\ref{95}{Astrophys.~J. (Lett.)}{455}{L89}

\bref [18] Electronic files for the data and a program to evaluate the
BKMP2 surface are available via anonymous ftp (currently
ftp.cita.utoronto.ca, file cita/pgmartin/h3pes/readme); for up-to-date
instructions contact the authors by e-mail at the following addresses:
boothroy@vulcan.maths.monash.edu.au; pgmartin@cita.utoronto.ca.

\bref [19] D.~W.~Schwenke\ref{88}{J.~Chem.\ Phys.}{89}{2076}

\bref [20] P.~Siegbahn and B.~Liu\ref{78}{J.~Chem.\ Phys.}{68}{2457}

\bref [21] A.~I.~Boothroyd, J.~E.~Dove, W.~J.~Keogh, P.~G.~Martin, and
  M.~R.~Peterson\ref{91}{J.~Chem.\ Phys.}{95}{4331}

\bref [22] See AIP document no.~PAPS~JCPSA-95-4343-163  for a total
  of 163~pages comprising appendices to Ref.~[21]
%[21]
 (30~pages) and tables of
  H$_3$~energies (15~pages) and H$_4$~energies (118~pages).  Order by PAPS
  number and journal reference~(Ref.\ 21)
%[21]
  from American Institute of Physics, Physics Auxiliary Publication Service,
  335~East 45th~Street, New York, NY~10017.  The price
  is~\${\hbox to 1 cm{\hrulefill}} for each microfiche (98~pages), or
  \${\hbox to 1 cm{\hrulefill}} for photocopies up to 30~pages and
  \${\hbox to 1 cm{\hrulefill}} for each additional page over 30~pages.
  Airmail additional.  Make checks payable to the American Institute of
  Physics. 

\bref [23] B.~Liu\ref{73}{J.~Chem.\ Phys.}{58}{1925}

\bref [24]  M.~R.~A.~Blomberg and B.~Liu\ref{85}{J.~Chem.\ Phys.\
(Notes)}{82}{1050} 

\bref [25]  C.~W.~Bauschlicher, S.~R.~Langhoff, and
H.~Partridge\ref{90}{Chem.\ Phys.\ Lett.}{170}{345} 

\bref [26] H.~Partridge, private communication: H$_2$ energies (1992)
and H$_3$ energies (1994).

\bref [27] R.~J.~Buenker and  P.\ Funke, private communication (1992).

\bref [28] D.~L.\ Diedrich, private communication (1992).

\bref [29] D.~L.\ Diedrich and J.~B.\ Anderson\ref{92}{Science}{258}{186}

\bref [30] R.~Gengenbach, Ch.~Hahn, and J.~P.~Toennies\ref{75}{J.~Chem.\
Phys.}{62}{3620} 

\bref [31] AIP document PAPS~\hbox{JCPSA-68-2466-18}: see Ref.~24 of
Truhlar and Horowitz (Ref.~5).
%[5]

\bref [32] J.\ Chang and N.~J.\ Brown\ref{95}{J.~Chem.\
Phys.}{103}{4097}.

\bref [33]  W.~J.~Keogh, A.~I.~Boothroyd, P.~G.~Martin, S.~L.\ Mielke,
D.~G.~Truhlar, and D.~W.~Schwenke\ref{92}{Chem.\ Phys.\
Letters}{195}{144} 

\bref [34] M.~E.~Mandy and P.~G.~Martin\ref{91}{J.~Phys.\ Chem.}{95}{8726}

\bref [35] R.~J.\ LeRoy\ref{69}{J.~Phys.\ Chem.}{73}{4338}

\vfill\eject

\noindent
%comment: caption
{\bf Table I}. Accuracy of the analytic H$_3$ potential energy surfaces:
rms errors and maximum deviations for various subsets of the 8701 {\it
ab~initio\/} data (125 energies for van der Waals conformations from
Ref.~8 are not included).  ``Compact'' geometries are those having at
least one distance less than~$1.15\>\bohr$.  Note that different authors
used slightly different H$_3$~basis corrections to the {\it ab~initio\/}
energies.
\bigskip

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%=====(
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{\tabskip=0 em
\halign to \hsize{ 
%comment: column template definition; overrule it with \omit
\hfil #\tabskip=0.01 em plus 30 em& \hfil $#$&
\hfil $#$\tabskip=0.01 em plus 20 em& \hfil $#$\tabskip=.01 em plus 30 em&
\hfil $#$\tabskip=0.01 em plus 20 em& \hfil $#$\tabskip=.01 em plus 30 em&
\hfil $#$\tabskip=0.01 em plus 20 em& \hfil $#$\tabskip=.01 em plus 30 em&
\hfil $#$\tabskip=0.01 em plus 20 em& \hfil $#$\tabskip=.01 em plus 10 em\cr
%comment: first overlines
\noalign{\medskip\hrule\smallskip\hrule\medskip}
%
%comment: titles
 & & \multispan2\hfil BKMP2\hfil&
     \multispan2\hfil BKMP\hfil&
     \multispan2\hfil LSTH\hfil&
     \multispan2\hfil DMBE\hfil\cr
\noalign{\vskip -2pt}
 & & \multispan2\hfil surface\hfil&
     \multispan2\hfil surface\hfil&
     \multispan2\hfil surface\hfil&
     \multispan2\hfil surface\hfil\cr
%
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%=====[
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 & & \multispan2\hrulefill&
     \multispan2\hrulefill&
     \multispan2\hrulefill&
     \multispan2\hrulefill\cr
\noalign{\vskip -2pt}
 & & & \omit\hfil Max\hfil&
     & \omit\hfil Max\hfil&
     & \omit\hfil Max\hfil&
     & \omit\hfil Max\hfil\cr
\noalign{\vskip -2pt}
 Description& & \omit\hfil rms\hfil& \omit\hfil dev\rpa\hfil& 
                \omit\hfil rms\hfil& \omit\hfil dev\rpa\hfil&
                \omit\hfil rms\hfil& \omit\hfil dev\rpa\hfil&
                \omit\hfil rms\hfil& \omit\hfil dev\rpa\hfil\cr
\noalign{\vskip -2pt}
 of subset\ \ & N_{\rm data}&  (\millih)& (\millih)&
                              (\millih)& (\millih)&
                              (\millih)& (\millih)& 
                              (\millih)& (\millih)\cr
\noalign{\medskip\hrule\medskip}
%
%comment: data lines; can separate with \noalign{\medskip}
%                    Npts         BKMP2             BKMP        
%         LSTH                    DMBE
 All H$_3$ energies& 8701&    0.27&   6.22&    3.76&     261.62&
    10.87&     \m169.35&     4.34&     \m168.93\cr %1
    BKMP subset\rpb&  766&    0.30& \m2.23&    0.36\rpc&   2.98&
    10.54&      \m98.98&     3.53&      \m47.93\cr %2
    LSTH subset\rpb&  283&    0.22& \m1.78&    0.24&       0.97&
     0.60\rpd&     2.40&     0.66&         4.07\cr %3
    DMBE subset\rpb&  308&    0.23& \m1.78&    0.23&       0.97&
     0.60&         2.40&     0.66\rpe&     4.07\cr %4
\noalign{\medskip}
            Compact& 2624&    0.42&   6.22&    6.82&     261.62&
    19.76&     \m169.35&     7.84&     \m168.93\cr %5
         Noncompact& 6077&    0.18& \m1.67&    0.28&       2.58&
     0.71&         4.06&     0.69&         2.72\cr %6
\noalign{\medskip}
             Linear& 1001&    0.15& \m1.26&    0.22&     \m2.82&
    10.35&     \m169.35&     1.58&         9.01\cr %7
   Linear symmetric&   74&    0.18& \m1.26&    0.38&     \m2.82&
    20.33&     \m169.35&     0.84&       \m3.98\cr %8
  Linear asymmetric&  927&    0.14& \m0.54&    0.20&     \m1.24&
     9.09&     \m116.38&     1.63&         9.01\cr %9
     Linear compact&  286&    0.19& \m1.26&    0.30&     \m2.82&
    19.36&     \m169.35&     2.91&         9.01\cr %10
  Linear noncompact&  715&    0.13& \m0.37&    0.18&     \m0.47&
     0.22&         0.52&     0.35&       \m1.55\cr %11
\noalign{\medskip}
               Bent& 7700&    0.29&   6.22&    3.99&     261.62&
    10.93&     \m165.88&     4.58&     \m168.93\cr %12
       Bent compact& 2338&    0.44&   6.22&    7.23&     261.62&
    19.81&     \m165.88&     8.24&     \m168.93\cr %13
    Bent noncompact& 5362&    0.19& \m1.67&    0.29&       2.58&
     0.76&         4.06&     0.72&         2.72\cr %14
%
%comment: last underline
\noalign{\medskip\hrule\smallskip\hrule}
%
} %comment: end of hsize
} %comment: end of halign

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%=====[
\medskip
%=====]

%comment: various footnotes

\noindent\hang ${}^a$ Largest deviation---analytic surface energy minus
{\it ab initio\/} energy. 

\noindent\hang ${}^b$ The numbers of energies for these sets
are slightly lower than those in the BKMP paper~(Ref.~6),
%[6]
due to elimination of duplicate geometries here and, for the LSTH and
DMBE subsets, to leaving out three Siegbahn and Liu energies (Refs.~20
and 23)
%[20, 23]
that are inaccurate by about $0.5\>\millih$. Differences in rms values
with respect to the BKMP values for these subsets are due largely to the
different basis correction (\S~II~B).

\noindent\hang ${}^c$ Boothroyd \etal~(Ref.~6) 
%[6]
fitted the BKMP surface to {\it their\/} version of these data with an
rms error of~$0.25\>\millih$, and a maximum deviation
of~$1.93\>\millih$.

\noindent\hang ${}^d$ Truhlar and Horowitz~(Ref.~5) 
%[5]
fitted the LSTH surface to {\it their\/} version of these data with an
rms error of~$0.27\>\millih$, and a maximum deviation
of~$0.88\>\millih$.

\noindent\hang ${}^e$ Varandas \etal~(Ref.~2) 
%[2]
fitted the DMBE surface to {\it their\/} version of these data with an
rms error of~$0.39\>\millih$.

\vfill\eject

\leftline{\bf FIGURE CAPTIONS}
\nobreak\medskip\nobreak

Fig.~1. Legendre expansion coefficients for the representative case
$r_1= 1.449\>\bohr$ evaluated for various surfaces: BKMP2 (solid), BKMP
(dashed), LSTH (dotted), DMBE (dashed--dotted), and AP (dashed--triple
dotted).  $\rhhtwo$ spans the intermediate range inside the van der
Waals well.  Open circles present values from Partridge \etal~[8]
at certain $\rhhtwo$.  Note the change in energy range for the
successive panels.  (a)~Spherical average $V_0$. (b)~First anisotropic
term, $V_2$, important for rotational excitation. (c)~$V_4$. (d)~$V_6$.

Fig.~2. Cuts of the various potential energy surfaces as a function of
$\chi$ for fixed $r_1= 1.449\>\bohr$ and selected $\rhhtwo$. Interaction
energy plotted is relative to $\rhhtwo = \infty$ for this $r_1$ (cf.\
Fig.~1a).  Line types as in Fig.~1.  {\it Ab initio\/} data are from our
calculations (squares) and from Partridge \etal\ (open circles).
Note the change in energy range for the successive panels. (a)~$\rhhtwo =
3\>\bohr$. (b)~$\rhhtwo = 4\>\bohr$.  (c)~$\rhhtwo = 5\>\bohr$.

Fig.~3. Profile of the van der Waals well as a function of $\rhhtwo$ for
fixed $r_1= 1.449\>\bohr$ and selected values of $\chi$.  Interaction
energy plotted is relative to $\rhhtwo = \infty$ for this $r_1$.  Line
types as in Fig.~1.  {\it Ab initio\/} data are from our calculations
(squares) and from Partridge \etal\ (open circles).  Bullets
are from our MTT representation.  (a)~$\chi = 0^\circ$.  (b)~$\chi =
45^\circ$. (c)~$\chi = 90^\circ$.

Fig.~4. Angular dependence of the potential with fixed $r_1 = r_2$ for
bent geometries relative to the energy for the collinear saddle point.
The representation plotted is motivated by the expansion in equation 2.
Line types as in Fig.~1.  The solid square on the BKMP2 (and BKMP and
LSTH) curve is the high weight conformation used to constrain the
quadratic force constant. {\it Ab initio\/} results are shown as open
squares.

Fig.~5. Cuts of the potential energy surfaces near the conical
intersection at fixed $q=2\>\bohr$ as a function of symmetry coordinate
$s$ ($s = 0$ is the equilateral triangle, whereas $s = 1$ is the
collinear limit).  Line types as in Fig.~1. Note that all surfaces but
AP have a cusp.  {\it Ab initio\/} results for the ground electronic state
are shown as open squares.  DMBE analytic extension onto the upper
surface is shown for comparison with our {\it ab initio\/} results for the
first excited state (bullets).  See \S~II~E for definition of cases
(a)~scalene and (b)~isosceles (squashed for positive~$s$).

Fig.~6. QCT cross sections for rotational excitation (without exchange)
of \h2\ by H atoms for the transition (0,0)~$\rightarrow$~(0,2) as a
function of relative translational energy.  Symbols are circle, square,
cross, and diamond for the BKMP2, BKMP, LSTH, and DMBE surfaces,
respectively.  Smooth curves (with line types as in Fig.~1) of the
Le~Roy class~II form have been fitted to the data.  The cross sections
from the DMBE surface are too high at low energies.

\vfill
\bye
