A Refined H3 Potential Energy Surface

Arnold I. Boothroyd (CITA), William J. Keogh, Peter G. Martin (CITA), and Michael R. Peterson (Dept. of Computing and Networking, U. of Toronto)

Journal of Chemical Physics, 104, 7139-7152   (1996)

[Download below]

Abstract: In evaluating some low temperature (less than 1000 K) thermal rate coefficients for inelastic rotational excitation of H2 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 H3 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 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 ab initio energies with an overall rms error of 0.27 millihartrees (i.e., 0.17 kcal/mole) and a maximum absolute deviation of 6.2 millihartrees (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 millihartrees and the maximum absolute deviation is 1.7 millihartrees. The refined surface is compared critically to four previous surfaces, including the DMBE surface of Varandas et al. (1987), 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 ab initio energies for 1809 conformations with corresponding predictions from the DMBE surface.

Note: A new H3 surface has been presented by Y.-S. Wu, A. Kuppermann, and J. B. Anderson, Phys. Chem. Chem. Phys., 1, 929-937 (1999), which should be an order of magnitude more accurate than the BKMP2 surface described here (at least for energies up to the H2 dissociation energy: at higher energies, or in the van der Waals well, their accuracy is comparable to that of the BKMP2 surface). They confirm our BKMP2 error estimates, finding that the BKMP2 surface has a typical error of a few tenths of a millihartree, with an error of less than 0.1 millihartree near the saddle point.

More recently, a yet-more-accurate H3 surface has been presented by S. L. Mielke, B. C. Garrett, and K. A. Peterson, J. Chem. Phys., 116, 4142-4161 (2002) [Fortran routines for their surface are available from EPAPS document No. E-JCPSA6-116-305206]. Their surface has an overall estimated rms error of only 10 microhartrees (0.0065 kcal/mole) for energies below about twice the H2 dissociation energy [i.e., fitting errors slightly smaller than the estimated errors of about 16 microhartrees (0.01 kcal/mole) in the ab initio data that they fit], with an estimated error in the van der Waals well of about 2 microhartrees. They consider error estimates at some length. They indicate that the BKMP2 error estimates for the BKMP2 ab initio data are reasonable (the CI error being perhaps slightly larger than estimated, but the basis correction error slightly smaller). They note that, for the BKMP2 analytic surface, the collinear van der Waals well is too deep by about 20 percent (17 microhartrees: 0.011 kcal/mole), but at angles from 30 to 90 degrees they find that the BKMP2 van der Waals well depth is off by only a few percent (less than 4 microhartrees). They also consider corrections to the Born-Oppenheimer approximation, that have not been included yet in H3 surfaces. They point out that the relativistic corrections should be fairly small for H3, but that Born-Oppenheimer diagonal correction (BODC) and nonadiabatic corrections can be significant, with the BODC term being the largest correction at low energy [they estimate that the BODC correction to the H3 barrier height is about 0.25 millihartree (0.154 kcal/mole), comparable to earlier estimates of 0.2 kcal/mole].

Download:

This paper describes the BMKP2 H3 surface and the H3 ab initio data on which the surface is based. The paper is available as:

• a PDF file (0.39 Mb, including figures), OR
• a gzip-compressed tar archive (0.13 Mb, expands to 0.45 Mb) that includes plain TeX text (90 kb: 19 pages) and 14 PostScript figures.

Note that the H3 paper of Mielke et al. (2002) compares the BKMP2 analytic surface and ab initio data to their more accurate results.

H3 Fortran program:

NOTE that versions of the H3 subroutines obtained prior to 29 October 1999 may be less accurate, since some of the required constants were not made explicitly double precision until then. (This only affects some compilers, but can cause the test program to yield incorrect output.)

• the text file h3progREADME.txt (4 kb) describing the BKMP2 H3 surface Fortran routines and test program,
• the Fortran routines bkmp2.f (63 kb) for the BKMP2 analytic H3 surface,
• the Fortran test program bkmp2test.f (8 kb) to check that the surface is being evaluated correctly on your computer,
• and the text file bkmp2test.out (3 kb) containing the correct output from the test program. Note that it is O.K. if the test numerical derivatives disagree in the last few decimal places, but the rest of the output should be identical.
• Alternatively, the gzip'd tar-file h3prog.tar.gz (21 kb; expands to 80 kb) contains all four of the above files.

H3 ab initio data:

• the text file h3ptsREADME.txt (18 kb) describing the two available archives of H3 energies,
• the gzip'd tar-file cih3usen.tar.gz (0.32 Mb; expands to 1.4 Mb) containing all the energies used to fit the BKMP2 H3 surface. Contains 5 files of ground-state ab initio energies, 2 files of non-ab-initio energies that were generated to constrain the surface at short and long range, and the README file above that describes the contents of this archive.
• the gzip'd tar-file cih3ean.tar.gz (1.0 Mb; expands to 5 Mb) containing all the ab initio H3 energies computed by the above authors. Contains 5 files of ab initio energies, including some excited state energies, plus the README file above that describes the contents of this archive.

Legendre coefficients for analytic surfaces:

For comparison purposes, plain text files containing tables of Legendre coefficients for H2 + H can be downloaded for the BKMP2 analytic H3 surface and five previous analytic H3 surfaces. For each surface, separate files are given with V0 (the spherical average), V2, and V4; in each file, five separate hydrogen molecule sizes are included (0.9, 1.28, 1.449, 1.618, and 2.0 bohrs), with H - H2 separations R in the range 1.6 bohrs < R < 20 bohrs. (The six surfaces are discussed in our above paper.)

Surface	V0	V2	V4
BKMP2	v0_h3_BKMP2.txt	v2_h3_BKMP2.txt	v4_h3_BKMP2.txt
BKMP (see below)	v0_h3_BKMP1.txt	v2_h3_BKMP1.txt	v4_h3_BKMP1.txt
DMBE	v0_h3_DMBE.txt	v2_h3_DMBE.txt	v4_h3_DMBE.txt
Aguado 11th order	v0_h3_Agu11.txt	v2_h3_Agu11.txt	v4_h3_Agu11.txt
Aguado 7th order	v0_h3_Agua7.txt	v2_h3_Agua7.txt	v4_h3_Agua7.txt
LSTH	v0_h3_LSTH.txt	v2_h3_LSTH.txt	v4_h3_LSTH.txt

Those interested in Legendre coefficients for H2-molecule sizes other than the five listed above can download:

• vlam_h3.f (12 kb), the H-H2 Legendre coefficient integration program;
• bkmp1.f (51 kb), our old BKMP H3 surface (see also below), which is also required;
• dmbe.f (23 kb), h3aguado.f (50 kb), and lsth2.f (16 kb): the other H3 surfaces (which are also required);
• note that the BKMP2 H3 surface (download above) is of course needed as well.

Note that the H3 paper of Mielke et al. (2002) compares Legendre coefficients of their accurate H3 surface with those of the BKMP2, BKMP, DMBE, and LSTH surfaces.

Old (BKMP) H3 surface of 1991:

The above surface (and data) supersede the old (Fortran) BKMP potential energy surface for H3 h3bkmpsurf.f (55 kb) (includes a short driver program) or bkmp1.f (51 kb) (H3 surface subroutines only) [note that this version has not been updated to have explicitly double precision constants]: from A. I. Boothroyd, W. J. Keogh, P. G. Martin, and M. R. Peterson, "An Improved H3 Potential Energy Surface", J. Chem. Phys., 95, 4343-4359 (1991).