J . Phys. Chem. 1990, 94. 180-185
180
irreversible combination reaction. With these effects present the product ( a m ~ l ) ( A Bcan ) ~ be greater than given in eq 1 through these additional processes. Acknowledgment. Financial support from NSERC is appre-
ciated. We gratefully acknowledge experimental assistance from Z. Wu, J. M. Stadlbauer, and B. W. Ng and technical help by the wSR at TRIUMF. Registry No. Acrylonitrile, 107-13-1 ; toluene, 108-88-3.
Intermolecular Vibrational Frequencies of (HF), and (HCN), Weak Complexes by Electrical Molecular Mechanics Clifford E. Dykstra Department of Chemistry, University of Illinois, Urbana, Illinois 61801 (Received: April 25, 1989)
A model of intermolecular interaction based on simple electrical energetics, and recently formulated for molecular mechanics treatment of clusters, has been extended to find harmonic vibrational frequenciesand transition moments for the intermolecular modes of clusters. Calculation of these values is helpful for estimating zero-point stability corrections and for distinguishing between strong and weak transitions. Application to certain binary complexes shows good agreement with high level ab initio values of the harmonic frequencies. The approach used in the model calculations is presented along with applications to cyclic complexes of HF, with as many as six monomers, and to linear chains of HCN molecules, up to eight in number. This demonstrates the feasibility of the method for studying intermediate-sized clusters, and where comparison values do exist, this offers an assessment of the reliability.
Introduction The potentials, or generally the potential surfaces, for weakly interacting assemblies of atoms and molecules make up an important frontier of contemporary chemical physics. This is a problem area that promises to unite gas- and condensed-phase chemistry in the sense of seeing potentials evolve from that of a pair of molecules to that of a droplet of molecules. The idea that classical electrical interaction, expressed in one form or another, plays a key role in these potentials'-13 offers a fundamental basis for constructing potentials. This notion has recently been carried to the point of a molecular mechanics capabilityI3 for assemblies of weakly interacting species. The interaction of the fixed (permanent moment) charge fields of a pair of molecules dictates the orientational features of the potential surface at very long range,2,3J4J5and this is a well-established idea. It is a powerfully simple picture in that it identifies the optimum long-range orientations from electrical multipole interactions. At closer range, though, it has been more difficult to weigh and compare the various contributions to the interaction, a difficulty recognized very early.I6 Appropriately chosen expansions and representations of the charge fieldssJ0J2J3J7have ( I ) Dykstra, C. E. Acc. Chem. Res. 1988, 21, 355. (2) Buckingham, A . D. Adu. Chem. Phys. 1967, 12, 107. (3) Kollman, P.J . Am. Chem. SOC.1977, 99, 4875. (4) Rendell, A. P. L.; Bacskay, G. B.; Hush, N. S. Chem. Phys. Lett. 1985, 11 7, 400. (5) Buckingham, A. D.; Fowler, P. W. Can. J. Chem. 1985, 63, 2018. (6) Buckingham, A. D.; Fowler, P. W.; Stone, A. J. Int. Reu. Phys. Chem. 1986, 5, 107. (7) Cummins, P. L.; Rendell, A. P. L.; Swanton, D. J.; Bacskay, G.B.; Hush, N. S. Int. Rev. Phys. Chem. 1986, 5 , 139. (8) Liu, S.-Y.: Dykstra, C. E. Chem. Phys. 1986, 107, 343. (9) Spackman. M . A. J. Chem. Phys. 1986, 85, 6579; 1986, 85, 6587. (IO) Dykstra, C. E. J . Phys. Chem. 1987, 91, 6216. ( 1 1 ) Dykstra, C. E.; Liu, S.-Y. In Structure and Dynamics of Weakly Bound Molecular Complexes; Weber, A., Ed.; NATO AS1 Series C; Rediel: Dordrecht, 1987; Vol. 212. (12) Spackman, M. A. J . Phys. Chem. 1987, 91, 3179. (13) Dykstra, C. E. J . A m . Chem. SOC.1989, I l l , 6168. ( 14) Chemical Applications of Atomic and Molecular Electrostatic Potentials; Politzer, p., Truhlar, D. G., Eds.; Plenum: New York, 1981. ( I 5 ) Dykstra, C. E. In Calculation and Characterization of Potential Energy Surfaces; Dunning, T. H., Ed.; JAI Press: Greenwich, CT, 1989. (16) Coulson, C. A. Research 1957, 10, 149. ( I 7) Stone, A. J. Chem. Phys. Lett. 1983,83, 233. Stone, A. J.; Alderton, M . Mol. Phys. 1985, 56, 1047.
helped in using electrical interaction at distances as close as those dictated by van der Waals radii. Rounding out the picture of electrical interaction has been the extensive incorporation of polarization effects.'JOJs Though polarization effects tend to be small much beyond a van der Waals separation distance, they add a noticeable element to the electrical interaction at the van der Waals separation and closer-in. The composite electrostatic interaction (permanent charge fields and polarization) offers a good basis for constructing potential energy surfaces generall~.'~ From analysis of properties of small complexes and comparison with ab initio calculations, we have argued that the primary electronic structure change upon weak interaction is that which arises as poIarization.'JoJ9 Polarization must comprise more than dipole polarization, since even for simple diatomics, dipolequadrupole and quadrupole polarizabilities can be comparable to dipole polarizabilities in their importance.1° The essential notion is that the charge readjustment of a molecule in a weak complex is largely that which arises because of the intrinsic polarizability properties of the molecule and the electrical potential that the molecule experiences as a result of its proximity to another molecule or molecules. The implication of this argument is that, with the incorporation of polarization, electrical interaction becomes a more complete picture. This is especially noticed for properties of a complex such as the dipole moment. If charge polarization is the primary electronic structure change upon weak bond formation, then classical electrical analysis should be quite successful in yielding the dipole moments of complexes. This does seem to be the case,13 and it has been found to hold for even subtler properties of complexes.19 Charge penetration and nonelectrical contributions, such as exchange and dispersion, become more important with decreasing intermolecular separation. In total, these give rise to the wall potentials that only become significant at about van der Waals separation distances. These are sharply repulsive potentials, except for dispersion, which gives an attractive contribution. The notion that the primary electronic structure change upon weak interaction is polarization implies that the wall potentials should be dependent largely on features of the isolated species, their shapes, and so, they should be rather simple to represent. The electrically based, (18) Dykstra, C. E. J. Comput. Chem. 1988, 9, 476. (19) Dykstra, C. E.; Liu, S.-Y.; Malik, D. J. J . Mol. Struct. (Theochem) 1986, 135, 357.
0022-3654,I90 12094-0180$02.50/0 , 0 1990 American Chemical Society I
-
Intermolecular Vibrations of (HF), and (HCN), molecular mechanics for clusters (MMC) model that has been introduced13 assumes a very simple form for these contributions and empirically selects the few parameters required. The net result is a model that relies on parameters and electrical properties associated with constituent species, and it is the nature of the model that the parameters and properties are transferable. This means that parameters used to construct an A-A and a B-B potential may be used just as well to construct an A-B potential. In one sense, other existing ideas for weak interaction potentials aim for the same result but follow an almost opposite direction. For instance, the damped-dispersion Hartree-Fock (HFD)2‘t23 approach is to use ab initio S C F level calculations on a small complex to establish a wall potential. Then, correlation effects, which are primarily associated with dispersion in weak complexes, are built-in with a model function. The model function may employ polarizability values, sometimes as adjustable parameters, for representing the dispersion. Though these ideas head toward the same end result of generating potential energy surfaces for weak complexes, there is a difference relative to electrical interaction notions. That difference amounts to choosing the important elements at long range or the important elements at short range to be the primary concern in developing the potential. No approach succeeds without some incorporation of all the various possible contributions; the question is how to treat each contribution and how then to combine them all. Consequently, one may expect that an approach that focuses on short-range elements will be more suited at short range and vice versa. So, the MMC approach,I3which focuses on long-range elements, is likely to be at its worst for highly repulsive, close-in regions. But, the MMC approach offers much greater applicability, since the only ab initio calculations ever required are those on individual monomers (i.e., the calculations that yield monomer electrical properties). Thus, the electrical view of weak interaction has great practical implications for the general construction of potentials for weakly interacting assemblies, and that is explored for several problems in this report. The extensive incorporation of polarization effects in the MMC scheme means that cooperative or nonpairwise additive elements of the potential will occur. With the notion that polarization is the primary electronic structure change upon weak interaction, other cooperative elements are expected to be much smaller than those associated with polarization. This implies that MMC potentials may be as realistic for small complexes as for large ones (e.g., tens of atoms or molecules) where cooperative effects are more sizable. Calculations on systems that range in size are the way to test this concept, and so the calculations reported here on H F and HCN complexes include the binary members of each series and continue to intermediate-sized clusters. The latest extension of the MMC scheme is the evaluation of harmonic vibrational frequencies. Frequencies test the validity of the shapes of surfaces and in some ways the validity of the surfaces away from equilibrium regions. This new capability, of course, opens up many more applications. Furthermore, a doubly harmonic evaluation of dipole transition moments has also been implemented, and this should aid in the interpretation of vibrational spectra of medium-sized clusters. It is important to realize that the evaluation of transition moments is possible because of the fundamental physical nature of the MMC potentials. Any type of force field approach would only yield energetics, not properties such as the dipole. And, an electrical model withouf polarization would be almost as limited: Any weak-bond stretching mode, even if infrared-active, would have an unchanging dipole (zero transition moment) with the neglect of polarization. Only ab initio electronic structure calculations offer the same capability as MMC. As the results given here show, MMC values seem rather comparable with those of most practical levels of ab initio calculations on small complexes, but of course, the MMC model (20) Hepburn, J.; Scoles, G.; Penco, R. Chem. Phys. Lett. 1976, 36, 451. (21) Ahlrichs, R.; Penco, R.; Scoles, G. Chem. Phys. 1977, 19, 119. (22) Douketis, C.; Scoles, G.; Marchetti, S.; Zen, M.; Thakkar, A. J. J . Chem. Phys. 1982, 76, 3057. (23) Barrow, D. A,; Aziz, R. A. J. Chem. Phys. 1988, 89, 6189.
The Journal of Physical Chemistry, Vol. 94, No. I, 1990 181
is far quicker and can be applied to substantially bigger clusters. It makes many problems tractable that cannot now be pursued by entirely a b initio means. Theoretical Approach
Electrical interaction in a cluster is evaluated with a truncated set of permanent moments, multipole polarizabilities, and hyperpolarizabilities. An efficient computational procedurela that uses Applequist’s polytensor organization of electrical propertiesz4 has been implemented, and it completely solves the mutual polarization equations for electrostatic interaction from an arbitrary multipole expansion. Molecule-centered properties have been used for the small molecules studied to date, but the procedure does not preclude use of properties distributed at various centers throughout a molecule, as Stone has advocated.”J5 The electrical properties used in the calculations have been obtained from a b initio electronic structure calculations.z6 Permanent moments are from well-correlated wave functions, and polarizabilities are large-basis, SCF values obtained with the derivative Hartree-Fock method.27 Vibrational averaging and vibrational polarization effects for the ground vibrational state have been incorporated for the properties of HF. The augmenting (wall) potentials in MMC are atom-atom Lennard-Jones or ‘6-12” potentials:
This means that augmenting the electrical interaction is a Lennard-Jones-type pair potential for every A-B pair of interacting monomers in the system. The sums are over centers located in each molecules, generally the atomic centers. Transferability of the c and d parameters is enforced in their selection. That is, c and d parameters determined for a given molecule are always the same, regardless of the partner species. Values of these parameters and the associated truncation point in the multipole expansion have already been given.I3 The information used most often in finding these parameters is the separation distance for some binary pair. The MMC potentials are constructed for rigid constituent species in most cases, and equilibrium values are used for these structures. Because of this, any coupling with intramolecular vibrations is frozen out. This is a reasonable restriction since there tends to be good separation in the frequencies of intra- and intermolecular modes and, thus, just small coupling. A “laboratory” Cartesian-coordinate system is used to arrange the constituents of a cluster. One set of position coordinates (x,y,z) in this system gives the location of the mass center of a constituent species, while a set of Euler angles (8,8,8,) specifies the orientation relative to the laboratory axes. The convention followed in the MMC program is that relative to its initial orientation, the molecule is considered to have been first rotated by 8, about a local z axis (Le., an axis parallel to the laboratory z axis, but passing through the mass center of the species), then by 8, about a local y axis, (24) Applequist, J. Math. Phys. 1983.24, 736; J . Chem. Phys. 1985,83, 809. (25) Stone, A. J. Chem. Phys. Lett. 1989, 155, 11 1. (26) Dykstra, C. E.; Liu, S.-Y.; Malik, D. J. Adu. Chem. Phys. 1989, 75, 37. (27) Dykstra, C. E.; Jasien, P. G. Chem. Phys. Lett. 1984, 109, 388. (28) Dykstra, C. E.; Malik, D. J. J . Chem. Phys. 1987, 87, 2806. (29) Gaw, J. F.; Yamaguchi, Y.; Vincent, M. A,; Schaefer, H. F.J . Am. Chem. SOC.1984, 106, 3133. (30) Hancock, G. C.; Truhlar, D. G.; Dykstra, C. E . J . Chem. Phys. 1988, 88, 1786. [Another correlated ab initio potential surface for the H F dimer with a much larger set of grid points has also been obtained: Schwenke, D. W.;Truhlar, D. G. J . Chem. Phys. 1988,88,4800. Redmon, M. J.; Binkley, J. S. J. Chem. Phys. 1987, 87, 969.1 (31) Michael, D. W.;Dykstra, C. E.; Lisy, J. M. J. Chem. Phys. 1984.81, 5998. (32) Frisch, M. J.; Del Bene, J. E.; Binkley, J. S.; Schaefer, H. F. J. Chem. Phys. 1986, 84, 2279.
182 The Journal of Physical Chemistry, Vol. 94, No. I , 1990
and then by Ot about the local z axis again. With this set of coordinates, [xy,z,O,O,,O,), for each molecule, the set of geometrical coordinates for a cluster will be an overcomplete specification of the internal structural parameters, but this presents no difficulty or additional computational effort. Forces are evaluated by numerical differentiation. A cluster’s energy is reevaluated for successive small displacements along each geometrical coordinate and pair of coordinates, and from these the second derivatives of the energy with respect to the displacement coordinates are obtained. Tests were carried out to ensure numerical soundness in the choice of the sizes of the small displacements. The angular forces, though, are not obtained merely from Euler angle displacements. Instead, angular forces are obtained for rotations about the local x, y , and z axes of a molecule, the three orthogonal rotations. A difference from simple Euler angle displacements is that a displacement corresponding to rotation by an angle 6 about the x axis is accomplished by the sequence 8, = n/2, Os = 6, and 8, = 1 / 2 . Off-diagonal rotational force constants may be obtained numerically by successive rotations (displacements) about two axes. The harmonic force constants for each of the 6 N degrees of freedom, where N is the number of constituents, are implicitly arranged in a square matrix or force tensor. Then, for each molecule, a six-by-six mass/inertial tensor is constructed in the following way:
I( :!E ; !) M
O
O
0
0
0
0
Iyx
0
0
0
I,,
0
Izy
I,*
M is the total molecular mass, and the lower three-by-three block is the moment of inertia tensor for rotation about the mass center. These six-by-six blocks are implicitly arranged along the diagonal of a 6N-by-6N mass/inertial tensor. Simultaneous diagonalization of this and the force tensor yields the normal coordinates in terms of the original displacement coordinates and yields the squares of the intermolecular vibrational frequencies as the nonzero roots. Transition moments are obtained within the doubly harmonic approximation, which means that the dipole moment is taken to vary linearly with each geometrical coordinate. The slopes of the (laboratory) x, y , and z dipole moment components with respect to each of the 6 N geometrical parameters are obtained by finite differences at the same point at which forces are evaluated. These are arranged in three column vectors corresponding to the three components of the dipole. The vectors are transformed to the normal-coordinate system thereby yielding the slopes of the three components with respect to each of the normal coordinates. The transition moments are obtained from these values and the normal mode frequencies in the usual way. For instance, the doubly harmonic value for the x component of the transition (0 to 1) dipole of the ith normal vibration is
where wi is the normal-mode frequency and qi is the normal coordinate. The total transition moment for the ith mode of vibration is then computed via (di))OI
=
((MP)0l2
+ (/.L$))o12 + (Fp)o12)1/2
(3)
In the MMC approach, the transition moment evaluations require negligible additional computing. Vibrational Frequencies and Transition Moments of HF and HCN Dimers
The first test system is the well-studied H F dimer, and Table I gives the MMC results that were obtained. These are an important set of results because, for H F dimer, there are high-level ab initio calculations to compare with. The comparison values in Table I serve to indicate how similar MMC is to levels of ab
Dykstra TABLE I: Intermolecular Harmonic Vibrational Frequencies (cm-’) of (HFh ab initio f r e d DZPJ DZPJ TZP/ TZ2P/ MMC mode SCF29 CISD29 ACCD’O*” MP2ba32 frea (u),,,, D 156 163 181 0.026 a’ (stretch) 143 156 a’ 193 218 223 231 259 0.567 a” 442 486 430 427 0.480 516 a‘ 529 607 521 582 641 0.308 Representative list of reported calculations. bThe basis was identified as 6-311++G(2d,2p), which is at least of TZ2P size.
initio calculation. It does not entirely reveal the accuracy of the MMC results, though, because there are also errors in each of the ab initio treatments in the Table I values. These errors are primarily in basis set incompleteness and perhaps somewhat in the correlation effects, where included. Also, the ab initio values have been obtained without freezing out the intramolecular vibration, and this could slightly contribute to the differences. When general methods for vibrational analysis of weakly bonded systems become available, better tests for comparing surfaces will be possible. For now, one sees in the comparison in Table I a good agreement between the MMC frequencies and ab initio results, given the range of those values: For each mode, the MMC frequency is within 10% of one of the ab initio values from a correlated level, while the range for those ab initio values is 5-20% of the mean value for a given mode. A value of 154 cm-’ for the weak stretching frequency has been obtained from the measured rotational constant and centrifugal distortion constant,33and this is in line with the a b initio and MMC values, given that this is an indirect experimental evaluation. An important feature of the MMC scheme is that it has a physical basis, electrical interaction, such that it yields predictions of electrical properties of a complex. In vibrational analysis, this means it can yield transition moments. Those calculated for the H F dimer are included in Table I. There is very little available information to compare with these values; however, the ab initio calculations of Michael et aL31 did include an evaluation of the transition moment for the weak bond stretch. This value was for a pseudodiatomic analysis of the stretch (Le., uncoupled to other motions) but was fully anharmonic. The reported value was 0.056 D, but estimating the uncoupled, doubly harmonic transition moment from the data collected in ref 31 gives 0.03 D. The MMC value is 0.026 D. Table I1 lists the harmonic vibrational frequencies and the doubly harmonic transition dipoles of the HCN dimer, along with comparison ab initio values of Anex et. a134and Krofanek et al.35 Similar ab initio results have been obtained by Somasundram et al.36 The agreement with a b initio harmonic frequencies is in line with the results obtained for H F dimer, except that the MMC bending potentials for (HCN)* appear softer than ab initio potentials. The MMC bending frequencies are about two-thirds of the ab initio vdues, though the stretching frequencies are about the same as the ab initio results. The differences for HCN bending between MMC and ab initio treatments may arise from several sources. For one, the MMC frequenices are obtained with intramolecular degrees of freedom being frozen, while the reported a b initio treatments allow for mixing of the intra- and intermolecular motions. The HCN bend frequency is much lower than the stretches, and so this motion could mix with the intermolecular bends, more so than mixing of intra-/intermolecular stretches. Another possible source for the difference may be in the actual 6-1 2 parameters chosen for HCN. They were selected primarily so that the HCN-HCN separation would compare well with the experimental value, and (33) Howard, B. J.; Dyke, T. R.; Klemperer, W. J. Chem. Phys. 1984,81, 5417. (34) Anex, D. S.; Davidson, E. R.; Douketis, C.; Ewing, G. E. J . Phys. Chem. 1988, 92, 2913. (35) Kofranek, M.; Karpfen, A,; Lischka, H. Chem. Phys. 1987, 113, 53. (36) Somasundram, K.; Amos, R. D.; Handy, N. C. Theor. Chim. Acta 1986, 69. 491
Intermolecular Vibrations of (HF), and (HCN), TABLE 11: Intermolecular Harmonic Vibrational Frequencies of Linear (HCN), ab initio MMC
complex (HCN)I stretch bend (HCN)3 stretch bend
WN)4 stretch bend
freqi cm-l b 129 161 56
c 122 158 54
122 107 38
0.053 0.48 0.06
153 85 176 130 63 17
165 92 193 142 67 17
162 92 128 88 49 11
0.0
188 136 73 217 169 136 73 22 11
182 134 73 138 111 79 54 19 5.9
0.01 0.0 0.08 0.58 0.002 0.28 0.10 0.26 0.005
202 162 1 I5 60
193 158 113 61 143 124 100 75 56 24 12 3.7
0.0 0.02 1
200 173 139 98 52 145 131 1 I3 93 72 58 27 16 7.9 2.6
0.005 0.0 0.03 0.0 0.09 0.68 0.003 0.28 0.002 0.21 0.13 0.15 0.007 0.35 0.009
0.07 0.52 0.006 0.29 0.003
The Journal of Physical Chemistry, Vol. 94, No. 1, 1990
TABLE 111: Intermolecular Harmonic Vibrational Frequencies (em-’) of ( W 3
mode e’ (stretch) a’ (stretch) e” (out-of-plane torsion) e’ (in-plane torsion) a” (out-of-plane torsion) a’ (in-plane torsion)
bend
bend
0.0 0.084 0.63 0.003 0.27 0.029 0.22 0.01 1 0.31 0.009
‘Transition dipoles calculated to be less than 0.001 D are listed as 0.0. bThe values from ref 34 are 631G**/MP2 level results for the HCN dimer but are 631G**/SCF results for HCN trimer. CDZ/SCF results from 35.
so the parametrization might turn out to be biased for the end-on parts of the molecule. It will take other information about the HCN-dimer surface to refine the parameters and to determine if the apparent softness in the bending potentials is an artifact. A third source for the difference with ab initio values is lingering basis set effects in those values. Basis set improvements would likely reduce the ab initio bending frequencies; for instance, the trimer bending frequencies go from 193, 142, 67, and 17 cm-I at the DZ/SCF level to 170, 123, 60, and 14 cm-’ upon improvement to a TZP/SCF and further improvement might continue this. For the MMC calculations, electrical properties were from a large, triply polarized basis set calculation. In this one respect, a b initio calculations on H C N dimer34-36are with bases that are not as large and do not have the same flexibility for describing mutual polarization. Where such polarization is relatively important, the use of small bases will produce potentials that are too stiff. For example, a comparison involving basis set
ab initio freq DZP/SCFZ9
freq
( P ) ~ ~D ,”
176 20 1 449 535 664 897
177 240 406 599 629 1016
0.23 0.0 0.0 0.47 0.56 0.0
MMC
‘Transition dipoles calculated to be less than 0.001 D are given as 0.0.
TABLE I V Intermolecular Harmonic Vibrational Frequencies, w (em-’), and Transition Dipoles’ (D) of (HF),, (HF)5, and (HF),
stretches 22 106 230 d 266 284
( H W 5
stretch
183
torsions
501
0.0 0.0
0.0 0.21 0.0 0.0
d 543 655 685
0.0 0.0 0.62
d 862
0.42
1145
0.0
3.3 47 77 85 212 d 261 d 310
0.02 0.04 0.10 0.05 0.03 0.20 0.05
503 536 d 541 610 755 871 d 944
0.22 0.28 0.25 0.44 0.17 0.47 0.35
1110
0.02
d 39 d 52 68 88 219 d 253 d 302 325 509 d 530 d 539 549 d 930
0.09 0.0 0.0 0.15 0.0 0.20 0.0 0.11 0.0 0.53
d 952
0.36 0.53 0.0
956 1046
0.0 0.25
0.0
‘Transition dipole moments calculated to be less than 0.001 D are given as 0.0. bDoubly degenerate modes are preceded by d. For (HF),, convergence to a strictly symmetric equilibrium structure did not result, and so the modes labeled as degenerate actually correspond to pairs of frequencies that were 1.2 cm-’ apart, on average. For the same reason, certain transitions that would be forbidden were there full symmetry show small transition moments. effects with very extended bases, done for the H F dimer,* showed noticeable refinement in the angular potential, as evidenced by a change in the equilibrium angles. And as mentioned, enhancement of a basis to just TZP from DZ yields a noticeable change in the HCN-trimer bending frequencie~?~ a change toward the MMC values. Most likely, all three sources play a role in the difference between ab initio values and MMC values. Improvement that could be made in MMC via reparametrization should await values for which direct comparison is more appropriate (e&, extremely large basis a b initio results), or it should await theoretical tools that make comparison with experimental values much more direct (e.g., evaluation of transition frequencies). But at the current state, the MMC values are realistic and especially good, given the simplicity of their evaluation.
(HF), Vibrations and Vibrations of Linear HCN Chains Table 111 lists results obtained for the cyclic H F trimer. The comparison with ab initio values, from a modest level calculation, is in line with the like comparison for (HF),. Table IV lists the results obtained for the other three cyclic H F complexes. For all systems, the stretching modes are lower frequency than all the bending modes. A very special set of calculations has been carried out on (HF), by Kolenbrander et aL3’ These calculations used electrical in~
(37) Kolenbrander, K.D.; Dykstra, C. E.; Lisy, J. M. J . Chem. Phys. 1988, 88, 5995. (38) Maroncelli, M.; Hopkins, G . A,; Nibler, J. W.; Dyke, T. R. J . Chem. Phys. 1985, 83, 2129. (39) Jucks, K. W.; Miller, R. E. J . Chem. Phys. 1988,88, 2196.
184
The Journal of Physical Chemistry, Vol. 94, NO. 1. 1990
teraction, though no augmenting wall potential, to generate a potential surface for in-plane torsions (three degrees of freedom) and a surface for out-of-plane torsions (three degrees of freedom). By nature of the surface construction, coupling between in-plane and out-of-plane torsions was neglected and coupling of any torsion with stretching motion was frozen out. However, with this reduced problem size (three degrees of freedom in each case), it was tractable to variationally determine a manifold of vibrational-state energies, fully treating the anharmonicity of the surfaces, and from that, transition energies were extracted. The comparison with two measured gas-phase overtone transition frequencies3, proved quite good, with agreement to within about 30 cm-l or about 3%. Taking the fundamental transition frequencies from those calculations to be the best known values for the H F trimer, we can assess the role of anharmonicity by comparison with harmonic frequencies. Of course, a proper comparison requires that same forced separation of the two types of torsions and the freezing of the stretching motion be applied in the harmonic frequency evaluation. When this is done, the calculated harmonic frequencies are found to be greater than the fundamental transition frequencies by about 50 cm-l (1 3%) for the e” mode, 120 cm-l (24%) for the e’ mode, 70 cm-I ( 1 2%) for the a” mode, and 230 cm-l (29%) for the a’ mode. Since the potential surfaces were the same for both vibrational treatments, this comparison indicates that harmonic frequencies can differ significantly (e.g., 30%) from observed frequencies for weak complexes. Of course, this is not surprising; it is just that evaluations of the size of such differences are still uncommon. The results in Table 1V show the “evolution” of stretching and bending frequencies as the number of monomers increases. The frequencies have been arranged in the table so that the most similar values from one complex to the next line up. This reveals certain frequencies to be “characteristic” of all H F complexes. For instance, there are frequencies for each cluster in the vicinity of 240 cm-’. With each added monomer, this frequency decreases slightly, which is expected, in part, for stretching modes as the effective mass becomes greater. Also, the high-frequency, in-plane symmetric torsion of the trimer nicely matches values found for the larger complexes. Across the series, this frequency goes as 1015, 1145, 1 1 10, and 1046 cm-I, and that is a clear trend upon which to anticipate a frequency for a still larger complex. Another way in which these calculations may help unravel the vibrational spectra of some collection of different H F n-mers is with the set of transition moments. Most often, only certain transitions have more than negligible oscillator strengths, according to the values in Table IV. Also, the magnitude of the transition dipoles tends to follow from one complex to the next (e.g., the transitions around 260 cm-I). It is typical that the strongest transitions have more bend character than stretch character. For these, permanent dipoles are the major contributors to the transition moment; large changes in the net dipole come about through the reorientation of dipole vectors in the course of a bend. On the other hand, nonzero transition moments for intermolecular stretches arise from changes in the induced moments, since the permanent moment contributions are unchanging along a pure stretch coordinate. The values given in Table 111 show the results for short chains of HCN monomers. As with the H F complexes, there is an evolution of frequencies as chain length increases, but it is qualitatively different in that it shows a simple spreading pattern. For example, the stretching frequencies start out as just the single value 122 cm-’ for the dimer. Then, for the trimer, where there are two stretching frequencies, we have a range for the values, 92-162 cm-I, which is almost a shift up and a shift down by like amounts from 122. In the tetramer, the range is extended another 20 cm-I, up and down. and then for the pentamer, another 10 cm-l. For the hexamer, the range of the stretching frequencies is extended by less than 10 cm-’, and so convergence to an ‘envelope” of stretching frequencies is becoming apparent at this size. This is also apparent for just the biggest stretching frequency of a (40) Ruoff, R. S.; Emilsson, T.; Klots. T. D.; Chuang, C.; Gutowsky. H. S. J . Chem. Phys. 1988, 89, 138.
Dykstra TABLE V: Stabilities (cm-’) of D. DJn (HF), 1537 (HF), 4983 1661 (HF), 8735 2184 (HF), 11360 2272 (HF), 13949 2325
(HF), from MMC Calculations AD2 D,b DoJn ADOa 3446 3752 2625 2589
782 2858 5250 6899 8472
953 1313 1380 1412
2076 2392 1649 1691
QThe incremental stab es, AD, are the differences between the corresponding stability of an n-mer complex and a complex with n - 1 monomers. *The Do values are De (equilibrium stabilities) less the zero-point energy of only the intermolecular modes, with each zeropoint contribution taken as half the harmonic frequency. complex, where the calculated value for (HCN), is 199.7 cm-’, while it is 204.3 cm-’ for (HCN), and 207.4 cm-’ for (HCN)8. Discussion Perhaps the most essential information to come from a potential surface for a weakly bound cluster consists of the stability and the structural parameters of the cluster. But if these can be determined with a fair measure of reliability, it must mean that the potential is realistically shaped. That in turn means that vibrational information from that surface should be realistic as well. To the extent that harmonic frequencies test the shaping of the surface, small cluster calculations reported here show that the electrically based potentials of the MMC model do display the shaping of the true surfaces-as well as we presently understand those surfaces. The MMC analysis provides two additional types of prediction for the study of weak clusters, harmonic frequencies for intermolecular vibrational modes and doubly harmonic transition dipoles. And naturally, the MMC potentials are also usable in an anharmonic analysis, and with an incomplete form of MMC, this has been done for one problem.37 The harmonic analysis, though, is simple and is rapidly carried out, thereby making it feasible to examine systems with a good number of interacting monomers. Harmonic frequencies in some way characterize or describe a small complex. We may identify trends, as in the two series examined here, or notice distinctions between bending and twisting stiffness. Unfortunately, a set of harmonic frequencies is unlikely to be a very precise road map for finding one’s way through a complex vibrational spectrum of some interesting cluster. Anharmonicity effects and coupling of excited vibrational states are potentially significant in determining what the actual transition frequencies are. Even so, harmonic frequencies provide some guidance, especially when combined with transition dipole information that distinguishes strong and weak transitions. From a practical standpoint, harmonic frequencies are useful but not always worth arduous computation. MMC evaluation is certainly not arduous, yet it is based on intrinsic properties of the constituents and it is generally applicable to weak clusters. Harmonic frequencies do aid in stability determinations, and it is meaningful to use the harmonic frequencies for zero-point energy corrections. As shown by the values in Tables V and VI, the zero-point corrections based on the MMC harmonic frequencies are quite often around 20% of the equilibrium stability. However, as the results for H F dimer indicate, the zero-point corrections may tend to be relatively more sizable for binary clusters than for larger ones. Anharmonicity effects will diminish the zero-point correction from the harmonic value. For H F dimer, Pine and Howard4’ have experimentally determined the dissociation energy and have converted it to an equilibrium energy using an intermolecular anharmonic zero-point correction obtained from the model surface of Barton and Howard.42 Their intermolecular zero-point energy correction is 619 cm-I; the harmonic MMC value is 754 cm-I, or about 22% larger. Taking 30% as an upper estimate for anharmonicity effects means harmonic zero-point values are (41) Pine, A. S.; Howard, B. J . J . Chem. Phys. 1986, 84, 590. (42) Barton, A . E.; Howard, B. J. Faraday Discuss. Chem. SOC.1982, 73, 45.
J. Phys. Chem. 1990, 94, 185-189
existing chain approaches a plateau of about 1700 cm-' and that this is nearly reached at the trimer. In contrast, (HF), and (HF), are seen as standouts in stability for that series. The incremental energetics of adding an HF to an existing cyclic cluster will favor addition to (HF), over addition to other n-mers. The physical understanding that continues to come from efforts such as the MMC approach is the basic role of electrical interaction, that of permanent charge fields and polarization, in weak bonding. It offers a picture with unifying concepts for potentials of binary species and extended clusters. The qualitative understanding has been clear for some time. Consequently, various ways of using point charges to get at electrical components in model potentials are widespread. They represent attempts, sometimes quite successful, at using the understanding quantitatively. But, it is only in recent years that theoretical capabilities for finding subtle molecular electrical properties have led to a much greater sophistication in working out intermolecular electrical interaction. The results given here show quantitative capability for finding vibrational parameters with use of electrically developed potentials. Spackman has also found that harmonic frequencies of binary complexes are well determined given a reasonable incorporation of electrical effects.43 Thus, it is clear that a straightforward basis exists for quantitative evaluations of structure, energetics, and vibrational parameters of weakly bonded clusters.
TABLE VI: Stabilities (cm-I) of Linear (HCN), from MMC Calculations De DJn ADea Dob Dofn ADOa (HCNh
1452
cyclic (HCN)3C 2523 (HCN), (HCN), (HCN)S (HCN)6 (HCNh (HCN)B
3196 5035 6913 8812 10721 12637
841 1065 1259 1383 1469 1532 1580
1452 1071 1744 1839 1878 1899 1909 1916
1246 2208 2792 4434 6114 7818 9527 11245
736 931 1108 1223 1303 1361 1406
185
1246 962 1546 1642 1680 1704 1709 1718
'The incremental stabilities, AD, are the differences between the corresponding stability of an n-mer complex and a complex with n - 1 monomers. bThe Do values are De (equilibrium stabilities) less the zero-point energy of only the intermolecular modes, with each zeropoint contribution taken as half the harmonic frequency. CBothcyclic and linear forms of the HCN trimer have been spectroscopically identified.3840 Ab initio calculations that give stabilities of both cyclic and linear HCN complexes have been presented by Kofranek et al.35 They report that the difference between zero-point-corrected stabilities of the linear and cyclic HCN trimers is 630 cm-I, while the corresponding value obtained from MMC is 584 cm-I.
likely to contribute about a 7% error to the net stability. And actually, it is unlikely that all modes would show such sizable anharmonicities, while those that do are most often the weakest ones, the ones that contribute the least to the zero-point correction. Thus, it is conservative to take the actual error in stabilities from using harmonic frequencies for zero-point corrections as 7% of the De value for most clusters. With zero-point-corrected stabilities (Tables V and VI), one sees that the incremental stability for adding an HCN onto an
Acknowledgment. This work was supported, in part, by the Chemical Physics Program of the National Science Foundation (Grant No. CHE-8721467). Registry No. HF, 7664-39-3; HCN, 74-90-8. (43) Schiller, W. S.; Spackman, M. A. Chem. Phys. Lea. 1988, Z51, 547.
A New Time-Independent Approach to the Study of Atom-Diatom Reactive Collisions: Theory and Application Daniel Neuhauser and Michael Baer* Department of Physics and Applied Mathematics, Soreq Nuclear Research Center, Yaune 70600, Israel (Received: April 25, 1989; In Final Form: July 5, 1989)
This work describes a new approach for the study of atom-diatom reactive collisions. The method is reminiscent of a time-dependent approach presented earlier but is formulated within the time-independent framework. As before, the method makes use of the projection operator formalism to form a coupled system of Schrodinger equations and of optical potentials, to reduce the reactive multiarrangement system to a (single arrangement) nonreactive system. However, in contrast to the time-dependent approach, here the optical potentials impose outgoing boundary conditions,reducing significantlythe numerical effort required to solve the Schrodinger equations. The reactive probabilities are obtained by calculating the fluxes at the exit region of each reactive channel. As an example, the method is applied to the collinear reactive H + H2 system. The extension of this method, to three dimensions like in the time-dependent approach, is straightforward.
I. Introduction A new approach to treat atom-diatom exchange collisions employing the time-dependent wave-packet approach was recently introduced.]+ The numerical procedure was found to be efficient enough to yield the correct collinear'g2 and three-dimen~ional~.~ reactive probabilities for the H + H2 system. The method, as will be seen later in more detail, is based on the idea that it is enough to solve a (slightly extended) nonreactive problem and still obtain ( I ) Neuhauser, (2) Neuhauser, ( 3 ) Neuhauser, 1989, 90, 5882. (4) Neuhauser, publication.
D.; Baer, M . J . Phys. Chem. 1989, 93, 2872. D.; Baer, M. J . Chem. Phys., in press. D.; Baer, M.; Judson, R. S.;Kouri, D. J. J . Chem. Phys.
D.; Baer, M.; Judson, R. S.; Kouri, D. J. Submitted for
0022-3654/90/2094-0185$02.50/0
the correct reactive transition probabilities. This is achieved by substituting short-range negative imaginary potentials at any exit to another arrangement. These potentials were found not to affect the wave function in the reagents' (inelastic) channel, but once in the product channel, they force the wave function to go to zero. The reactive transition probabilities are obtained by calculating the fluxes into each reactive channel along the boundary located just before the imaginary potential region (see Figure 1). This method is still not as efficient as some of the other in treating the fully symmetrical H + H2 system, but (5) Zhang, J. 2. H.; Kouri, D. J.; Haug, K.; Schwenke, D. W.; Shima, Y.; Truhlar, D. G.J . Chem. Phys. 1988,88, 2492. ( 6 ) Baer, M. J . Phys. Chem. 1987, 91, 5846. Baer, M. J . Chem. Phys. 1989, 90, 3043. Baer, M. Phys. Rep. 1989, 178, 99.
0 1990 American Chemical Society
