An Accurate Analytic H4 Potential Energy Surface

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

Journal of Chemical Physics, 116, 666-689   (2002)

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Abstract: The interaction potential energy surface (PES) of H4 is of great importance for quantum chemistry, as a test case for molecule-molecule interactions. It is also required for a detailed understanding of certain astrophysical processes, namely collisional excitation and dissociation of H2 in molecular clouds, at densities too low to be accessible experimentally. The 6101 ab initio H4 energies reported in 1991 by Boothroyd et al. demonstrated large inaccuracies in analytic H4 surfaces available at that time. Some undesirable features remained in the more accurate H4 surfaces fitted to these energies by Keogh (1992) and by Aguado et al. (1994), due in part to the relatively sparse coverage of the 6-dimensional H4 conformation space afforded by the 6101 ab initio energies. To improve the coverage, 42079 new ab initio H4 energies were calculated, using Buenker's multiple reference (single and) double excitation configuration interaction (MRD-CI) program. Here the lowest excited states were computed as well as the ground state, and energies for the original 6101 conformations were recomputed. The ab initio energies have an estimated rms "random" error of about 0.5 millihartree and a systematic error of about 1 millihartree (0.6 kcal/mol).

A new analytical H4 PES was fitted to these 48180 ab initio energies (and to an additional 13367 points generated at large separations), yielding a significant improvement over previous H4 surfaces. This new PES has an rms error of 1.43 millihartree relative to these 48180 ab initio energies (the fitting procedure used a reduced weight for high energies, yielding a weighted rms error of 1.15 millihartree for these 48180 ab initio energies). For the 39064 ab initio energies that lie below twice the H2 dissociation energy, the new PES has an rms error of 0.95 millihartree. These rms errors are comparable to the estimated error in the ab initio energies themselves. The new PES also fits the van der Waals well to an accuracy of about 5%. For relatively compact conformations (energies higher than the H2 dissociation energy), the conical intersection between the ground state and the first excited state is the largest source of error in the analytic surface. The position of this conical intersection forms a somewhat complicated 3-dimensional hypersurface in the 6-dimensional conformation space of H4. A large portion of the position of the conical intersection has been mapped out, but trying to include the conical intersection explicitly in an analytic surface is beyond the scope of the present paper.

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This paper describes the BMP H4 surface, and the H4 ab initio data on which the surface is based. The paper is available as:

• a PDF file (1.7 Mb), OR
• a gzip-compressed PostScript file (1.2 Mb, expands to 4.3 Mb).
• (note that this paper first appeared as preprint CITA-2001-38)

See also the text file README.TXT (6 kb) for a list of the available Appendices, H4 Fortran programs, and H4 data (which supersede the old 1991 H4 data). These can be downloaded from this website using the links listed below, and are also available from EPAPS, Document No. E-JCPSA6-115-304140, except that the REVISED versions of the BMKP H4 Fortran program are only available below (note that EPAPS is the electronic archive of the AIP).

The Appendices to the above paper:

• as a PDF file (0.43 Mb), OR
• as a gzip'd PostScript file (0.15 Mb, expands to 0.63 Mb).

The H4 Fortran program:

Note that the following files are the revised versions of 24 May 2002 (with improved extrapolation to short distances), and are NOT the ones available from EPAPS:

• the text file h4progREADME.txt (5 kb) describing the BMKP H4 surface Fortran routines and test program,
• the Fortran routines h4bmkp.f (0.22 Mb) for the BMKP H4 surface,
• and the Fortran test program h4bmkp_test.f (52 kb) to check that the surface is being evaluated correctly on your computer.
• Alternatively, the gzip'd tar-file h4prog.tar.gz (67 kb; expands to 0.29 Mb) contains all three of the above files.
• the OLD (original: 6 July 2001) Fortran routines h4bmkp.OLD.f (0.21 Mb) and test program h4bmkp_test.OLD.f (24 kb) are also available; these are the versions contained in the EPAPS archive.

The H4 ab initio data:

• the text file h4ptsREADME.txt (18 kb) describing the two available archives of H4 energies,
• the gzip'd tar-file cih4usen.tar.gz (3 Mb; expands to 8 Mb) containing all the energies used to fit the BMKP H4 surface. Contains 5 files of ground-state ab initio energies, 5 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 cih4ean.tar.gz (17 Mb; expands to 76 Mb) containing all the ab initio H4 energies computed by the above authors. Contains 7 files of ab initio energies, including excited state energies, plus the README file above that describes the contents of this archive.

Spherical Averages of Analytic Surfaces

The rigid-rotor H4 surface of J. Schaefer and W. E. Kohler, Z. Phys. D 13, 217 (1989) is defined only for r1 = r2 = 1.449 bohrs. This SK surface was used to constrain a "reasonable" van der Waals well for our BMKP H4 surface (which however dies away exponentially rather than as 1/R^6 in the intermolecular separation R). For comparison purposes, we present:

• A plain text file v0_h4_SK_BMKP.txt (9 kb) comparing the spherical average of the SK and BMKP surface for r1 = r2 = 1.449 bohrs with R from 2.0 to 9.0 bohrs. (Note that for this case the BMKP surface should be better than the SK surface for R < 4.5 bohrs or so, in which region the aniosotropic terms of the SK surface are significantly less accurate than its spherical average, as discussed in our above paper. The anisotropy is quite large: at any separation R, different orientations of the H2 molecules yield interaction energies differing by a factor of 2 or more.)
• A Fortran program v0ri_h4bmkp.f (31 kb) that can be used with our above BMKP H4 program to compute spherical averages of the BMKP H4 surface for other H2-molecule sizes (i.e., where the SK surface is not defined: note that the BMKP van der Waals well is then less well constrained).
• Files containing the output of this spherical-average program for H2-molecule sizes r1,r2 of 0.9, 1.28, 1.449, 1.618, and 2.0 bohrs:

Old (1991) H4 Ab Initio Data

The above data include and supersede the old H4 ab initio data published by A. I. Boothroyd, J. E. Dove, W. J. Keogh, P. G. Martin, and M. R. Peterson 1991, "Acurate Ab Initio Potential Energy Computations for the H4 System: Tests of Some Analytic Potential Energy Surfaces", J.Chem.Phys, 95, 4331-4342, and AIP document PAPS JCPSA-95-4331-170. The old H4 data are available for comparison purposes: the gzip-compressed data file h4allpts.dat.gz (0.23 Mb, expands to 0.8 Mb) comprises the ab initio energies fitted in this 1991 paper, the gzip-compressed data file h4allpts.full.gz (0.33 Mb, expands to 1.3 Mb) comprises all the old ab initio data, and h4errfix.dat (4 kb) comprises list of the nine erroneous points, with corrected energy values. NOTE that the nine erroneous ab initio energies in this old (1991) dataset are also corrected in the above new (2002) data, and the other old ab initio energies have been very slightly modified (by amounts comparable to or smaller than the quoted uncertainties).