J. Phys. Chem. 1985, 89, 3324-3325
3324
A Num6rkal Test on the Equivalence of Intramolecular Potential Expanslons in Normal and Valence Dkplacement Coordinates for H20 Barbel Maessen, Max Wolfsberg,* Department of Chemistry, University of California, Irvine, California 9271 7
and Lawrence B. Harding Technical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (Receioed: January 30, 1985)
The present study presents a numerical test on the equivalence of potential expansions in normal and valence displacement coordinate space by calculating the lower vibrational states in a variational procedure. The calculations are performed for H 2 0 and they indicate that a fourth-order Taylor series expansion in valence coordinates is not well represented by an expansion in normal codrdinates which is truncated at fourth order.
1. Introduction
The usual analysis of vibrational-rotational spectra within the framework of perturbation theory' is related to a Taylor series expansion of the potential energy in normal-mode coordinates. Such a potential expansion is dependent on the isotopic constitution of a molecule; thus in the analysis of spectral dtita on isotopic variants of a molecule, it has been usual to relate the potential expansion in normal coordinates (Q expansion) to an isotope independent Taylor series expansion in valence displacement coordinates ( R expansion) such as stretches and bending coordinates. The latter coordinate system is also usually used in the mnstruction of Taylor series potential expansions from a b initio electronic structure calculations. Thus, analytical methods have been developed for converting an R expansion into a Q expansion? This transformation is, however, quite cumbersome, sufficiently so that one of the major virtues of the widely used method of Whitehead and Handy3 for carrying out variational calculations of rotational-vibrational energy levels is that it avoids the potedtial expantion transformation. Recently, Harding and Ermler4 have constructed a computer algorithm called SURVIB which first constructs an R expansion by fitting B grid of ab initio electronic structure energies, then numerically fits a expansion to this R expansion, and then uses the parameters o this Q expansion to calculate the perturbation theory parameters for evaluating vibrational s p t r a l data. A Q expansion obtained with the SURVIB algorithm has also been used by Romanowski, Bowman, and Harding5 in a variational calculation on the vibrational spectrum of formaldehyde. The usual form of the potential function in the R-expansion form is a Taylor series containing terms up through fourth order (pexpansion). When converting this expansion into the Q form, it is usual again to carry terms through fourth order in Q (Q" expansion). One notes here that the transformation from Q to R is not a linear one. The question arises how well does a expansion approximate the R4 expansion. This point is explored here for the example of H20. Ab initio data from the work of Bartlett, Shavitt, and Purvis (BSP)6 are used to determine the R4 potential expansion. This analytical potential expansion is then reexpanded to obtain four separate
7
(1) H. H. Niclsen, Reo. Mod.Phys., 23,90 (1951).
(2) M.A. Pariseau, I. Suzuki, and J. Overend, J. Chem. Phys., 42, 2335 (1965). (3) R. J. Whitehead and N. C. Handy, J . Mol. Spectrosc., 55, 356 (1975); 59, 459 (1976). (4) L. B. Harding and W. C. Ermler, J. Comput. Chem., 6, 13 (1985). (5) H. Romanowski, J. M. Bowman, and L. 8. Harding, J. Chem. Phys., 82, 4155 (1985). (6) R.J. Bartlett, I. Shavitt, and G. D. Purvis 111, J. Chem. Phys., 71, 281 (1979).
0022-3654/85/2089-3324$01.50/0
normal-mode potential expansions: Qz,Q", @, Q",which contain terms up to second, fourth, sixth, and eight order, respectively. The fit of the a b initio data to the R4 force field and the reexpansion of this force field to the Q force fields is done here by the numerical procedures of the SURVIB program. The equivalence of the five different potential expansions is tested. This test is perfbrmed by variational calculations of the lower vibrational energies. Details of the different steps of the calculations are outlined in section 11, and the results are presented and discussed in section 111.
XI. Details of the Calculations 1 . Ab Initio Data Set and Fit to Valence Displacement Potential. It should be emphasized that the concern of the present communication is not the ab initio potential surface itself. Neither the quality of the ab initio energies nor the best selection of the a b initio grid are explored here. The actual ab initio points used for water are taken from Table 11, column D-MBPT (a)of the work of BSP.6 The 36 ab initio points in that reference are given for a grid formed by R,,Rz = 1.811096 f 0.06 bohr and a = 104.4492 f 3'. The p analytical fit obtained here is quite similar to that given by BSP. 2. Reexpansion of Fourth-Order Force Field in Valence Coordinates to Force Fields in Normal Coordinates. The transformation of the R4 force field to an expansion in normal coordinates is carried out by the numerical methods of the SURVIB program. This program was described r e ~ e n t l y .The ~ transformation is done by the following steps: (1) The procedure of Gwinn* is used to carry abt a normal-mode analysis. The second-derivative matrix used for this step is derived numerically for the valence coordinate potelltial. (2) A grid of small displacements in normal coordinates is created and the corresponding energies are calculated from the initial internal coordinate potential. (3) The resulting energy grid in normal coordinates is then fit in a linear least-square procedure to a Taylor series polynomial up to a certain order. In this latter transformation, the grid points are chosen very close to the potential minimum; the maximum distortions in the valence coordinates correspond to about 1% of the respective values at the potential minimum with corresponding potential energy changes up to about 0.5 X lo4 au. It is found necessary to employ quadruple precision in the fitting procedure to generate reasonable fits for the and Q" expansions. Note that the grid used in this step is much smaller in extent than the grid of the ab initio electronic structure data. The closeness of the grid points to the (7) See, for instance, G. J. Sexton and N. C. Handy, Mof.Phys., 51, 1321 (1984). (8) W. D.Gwinn, J. Chem. Phys., 55, 477 (1971).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3325
Equivalence of Intramolecular Potential Expansions TABLE I: Vibrational Energiesa (cm-') in H20 Calculated from Different Potential Expansionss6 assignment R' 02 cr, Q6 es 4677.9 4677.9 4731.2 4672.4 4677.8 GST *2 2v2 "1
v3
3v2 v1 + Y2 v2
+ v3
4v2
v , + 2v2 2v2 + Yj 2Vl v1 + "3 2v3 max error
rms error
1634.1 3233.4 3723.7 3820.2 4796.6 5347.8 5438.9 6321.3 6936.7 7024.6 7427.9 7513.4 7616.9
1697.1 3394.0 3845.9 3967.9 5090.6 5543.2 5691.3 6787.0 7240.2 7414.2 7691.8 7814.0 7935.9
1622.8 3210.7 3702.7 3801.7 4748.7 5277.4 5473.2 6201.3 6827.7 7014.9 7372.7 7557.3 7520.5
1634.3 3234.1 3723.5 3819.9 4798.8 5348.3 5438.8 6328.6 6939.1 7026.2 7426.8 7511.6 7615.6
1634.1 3233.4 3723.7 3820.2 4796.6 5347.8 5438.8 6321.3 6936.6 7024.6 7427.9 7513.3 7616.9
465.7 264.6
114.0 59.2
7.3 2.3
0.1 0.1
"Absolute energies are given for the ground state (GST); energies of other states are given with respect to the corresponding ground state. *Column R' refers to internal coordinate expansion up through fourth order. Columns Q', @, @, and @ refer to normal coordinate expansions up through second, fourth, sixth, and eighth order, respectively. potential minimum means that the coefficients of the Taylor expansion approximate the appropriate derivatives at the origin of the expansion. Use of the methods of ref 2 has confirmed this statement for the third and fourth derivatives here. 3. Calculation of the Vibrational Energies. The equivalence of the different potential expansions can be judged by comparing potential energies for an extended energy range or by calculating some specific properties. This communication chooses to consider the lower vibrational energies of the H 2 0 molecule. The vibrational energies are calculated in a variational procedure, with use of a computer code based on the method of Whitehead and Handy.3 This method uses the Watson Hamiltonian in normal coordinates and has been described earlier.3*9 Only the evaluation of the potential energy contributions to the matrix elements will be described here. The integrals over the potential energy are calculated numerically by Gauss-Hermite quadrature. Thus, potential energy points with distortions in Q space are used which are 35-70 times larger than those employed in the R Q fitting procedure; admittedly, these very large distortions correspond to small weights. The potential energy of the R4 potential expansion for a specific point in Q space is calculated by use of the transformation coefficients of the normal-mode analysis. The variational calculations are based on 159 A I and 118 B1 basis functions, which are products of harmonic normal mode oscillator functions; the Gauss-Hermite integration procedure uses the integration point set9 (Mk) = (12,20,12).
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(9) B. Maessen and M. Wolfsberg, J . Chem. Phys., 80, 4651 (1984).
111. Discussion of the Results The lower vibrational energies calculated for the five different potential expansions are reported in Table I. The concern here is not a comparison with experimental data, but rather with the consideration of the error introduced by the cutoff of the Q potential expansions. Comparison of the vibrational energies calculated with the different Q expansions shows that the deviations from the values obtained with the potential expression decrease drastically when one goes from second-, to fourth-, to sixth-, to eighth-order expansion. The @ potential yields agreement with the R4 results to within a maximum error of 0.1 cm-I; this result is very gratifying and attests the convergence of the SURVIB procedure. The present calculations indicate that for H 2 0 a fourth-order force field in valence displacement coordinates needs to be expanded in the SURVIB procedure up to sixth-order terms in normal-mode coordinates, if one is concerned about an agreement of the order of 1 cm-I in the lower vibrational energies. Thus, some doubt is cast on quantitative meaning in perturbation theory analyses of the assumption of the equivalence of R4and Q' expansions. The authors are, of course, aware of the work of Amat, Nielsen, and Tarrago'O in which the formal vibrational perturbation theory is carried out with use of higher than fourth-order terms in Q. These considerations again raise a concern which has already been raised by the studies1IJ2which have compared R4 Taylor expansions with Simon-Parr-Finlan expansions;" namely, a concern about the adequacy of Taylor series expansions for obtaining reasonable accuracies (1 cm-I) in the calculation of even the lower vibrational energy levels of polyatomic molecules with vibrational amplitudes comparable to those of H20. In calculations for D20, similar to those for H 2 0 reported here, the differences between the R4 and Q' force fields for the states v2 and v1 respectively are 4.2 and 9.1 cm-I; for H 2 0 the corresponding numbers obtainable from Table I are 11.3 and 21.0 cm-I.
Acknowledgment. This work was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, US. Department of Energy, under contract No. DE-AT03-76ER1088 (University of California) and W-3 1-109-ENG-38 (Argonne National Laboratory). B.M. gratefully acknowledged support by the Alexander von Humboldt Foundation as a Feodor Lynen Fellow. The authors also acknowledge discussions with Professor J. M. Bowman and Dr. H. Romanowski as well as the help of Ms. Annette Stumpf in obtaining derivatives by the methods of ref 2. Registry No. H 2 0 , 7732-18-5. (10) G . Amat, H. H. Nielsen, and G . Tarrago, "Rotation-Vibration of Polyatomic Molecules", Marcel Dekker, New York, 1971. (1 1) G. D. Carney, L. A. Curtiss, and S . R. Langhoff, J. Mol. Spectrosc., 61, 371 (1976). (12) B. Maessen and M. Wolfsberg, J . Phys. Chem., 88, 6420 (1984). (13) G . Simons, R. G . Parr, and J. M. Finlan, J . Chem. Phys., 59, 3229 (1973); G . Simons, J . Chem. Phys., 61, 369 (1974).
