J . Phys. Chem. 1993, 97, 9374-9379
9374
Partially Coupled Electrical Model of Vibrational Frequency Shifts in Weak Atom-Diatomic and Diatomic-Diatomic Complexes Carol A. Parish?and Clifford E. Dykstra’ Department of Chemistry, Indiana University-Purdue University at Indianapolis, 402 North Blackford Street, Indianapolis, Indiana 46202-3272 Received: April 28, 1993’
A simple model has been found to account for the blue and red shifts in the stretching transition frequencies of diatomic molecules that occur because of weak bonding to rare-gas atoms or to other diatomics. The primary element in the model is that of electrical interaction between the interacting species. This affects transition frequencies because the electrical properties change upon vibrational excitation. A second important element is the coupling of the intramolecular stretch to the weak mode vibrations. This comes about because of a change in the shape of the potential surface upon excitation of the diatomic and because of the change in the on-average bond length of the diatomic. Calculations with this model were carried out for a number of complexes, and comparison with 19 available experimental values indicates that shifts are predicted with a mean error of 10 cm-I, excluding one interesting anomalous case, the HF frequency shift in OC-HF. The model is applicable in cases of both blue and red shifts. Introduction Weak intermolecular interaction among closed-shell neutral molecules, including hydrogen-bonding interaction, has been aggressively pursued because the nature of weak molecular attachment is strikinglydifferent from covalent and ionic bonding. It is an interaction with features distinct from those of chemical bonding and yet extremely important in so much of chemical phenomena. Every property and characteristic of weakly bonded complexes is a crucial piece of the puzzle that explains the nature of weak interaction. One ultimate goal of our understanding is predictive capability in the form of a model, and indeed, a number of models for different purposes have been applied to weak complexes (see, for example, refs 1-7). A particular notion underlying our efforts to model weak interaction7 is that the electronic structure change upon weak interaction is primarily that associated with classical charge polarization.8*9Polarization is used in a global sense and meant to include multipole polarization beyond dipole polarization, mutual or back polarization, and hyperpolarization. In practice, the polarization is evaluated through a solution of the classical mutual polarization equations with a property set which for small molecules is truncated7 after including the dipole polarizability, CY, the dipole-quadrupole polarizability, A , the quadrupole polarizability, C, and the dipole hyperpolarizability, B. Describing changes in electronic structure usually calls for a quantum mechanical picture; however, if the electronic structure change is truly that of charge polarization, then a simpler level of physics, classical electrostatics,is appropriate. Under our basic notion, then, we have devised a model of the interaction energetics that avoids quantum mechanical analysis. This has been applied to the structures and stabilities of a large number of small complexes.7JJ-15 An interesting manifestation of weak interaction is the shift that occurs in the vibrational transition frequencies of the constituent molecules of a cluster. Liu and Dykstra used the notion of electrical polarization to develop a model for vibrational frequency shifts of diatomic molecules in binary complexes.16 The essence of the model was that because the various electrical properties of a diatomic vary with the bond distance, r, any electrical influence must lead to a change in the stretching t Purdue Research Foundation Fellow, 1992-1994.
Abstract published in Aduance ACS Abstracts, August 15, 1993.
0022-3654/93/2091-9374%04.00/0
potential. For example, the permanent quadrupole moment of a nearby partner molecule will give rise to a dipole-quadrupole interaction energy that varies with r because the diatomic’s dipole moment is a function of r. And of course, the dipole polarizability, quadrupole moment, and so on, are functions of r, too. A simplification that was taken as part of the model comes from the typically linear variation of a diatomic’s electrical properties with r in the vicinity of the equilibrium. So, in this early model, electrical interaction between a diatomic and its weakly bonded partner was evaluated using the electrical properties of the diatomic at two chosen bond lengths, rl and rz. The interaction energytalues at rl and at rz were used to construct a potential function, linear in r, that was added to the isolated monomer’s stretching potential to account for the effects of weak interaction. This perturbing potential could be used in a low-order perturbative treatment to give thechangein the diatomic’svibrational transition frequency due to weak bonding to the partner.I6 This simple model neglected not only electrical anharmonicity (nonlinear variation of the electrical properties with r) but also coupling with other modes. The latter seems to be a limitation in applying it generally as judged by recent results of Nesbitt on Ar,HF c0mp1exes.l~An attempt to apply the model to a triatomic in a study of (HCN), clusters gave unsatisfactory results;18 however, since this failed to include the electrical effect on anything but the potential of one stretching coordinate, it did not rightly test the basis for the model. We have developed a more complete analysis of vibrational state energetics that includes coupling with weak modes and electrical anharmonicity effects, and we have carried out calculations on a number of weak complexes. Vibrational frequency shifts have been obtained for HF, DF, HCl, Hz,and CO bonded to neon and argon and to other diatomic molecules. The fundamental assumption of this new model is exactly the same as in the earlier model.16 That is, the primary change in electronic structure due to weakbonding is assumed to be electrical polarization. However, this notion is applied more completely and to more cases. The results show agreement which may be characterized as semiquantitative. It is consistently correct in showing ordering of the sizes of the shifts among series of complexes, and it properly gives both red and blue shifts. There are. however. limitations to the model. and one anomalous case. OC-HF. For this complex the model is in error by almost 80 cm-1 for the H F red shift. Possible reasons for this are discussed. 0 1993 American Chemical Society
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Vibrational Frequency Shifts
Model of Vibrational Shifts in Weakly Bonded Diatomics The specific model we examine here uses the molecular mechanics for clusters (MMC) model’ with an imDortant embellishment that allows for partial evaluation of modemode coupling effects and certain anharmonicities: We generate potential surfaces for interacting molecules in specific vibrational states and evaluate the intermolecular vibrational frequencies on those surfaces. In this way, we obtain complete energy differences, those between zero-point levels on different surfaces, which are the frequency shifts. An inherent approximation is an on-average treatment of the rapid intramolecular stretches of the diatomics for coupling with the weak, intermolecular modes. Application to complexes of polyatomic molecules is not precluded; however, the approximations may be less suited, especially in the case of low frequency intramolecular modes. Let us consider the evaluation of frequency shifts in terms of potential surfaces. MMC can be employed to generate an interaction energy (potential) between two or more rigid molecules. The potential is implicitly a function of the intermolecular structural parameters of a cluster, and as in a conventional ab initio electronic structure calculation, one obtains energies at specificallychosen (intermolecular) geometries. The dependence on intramolecular parameters may be treated differently because of the physical basis of MMC. Typically, the intramolecular stretchingvibrationsof diatomic molecules are significantlyhigher in frequency than the intermolecular modes of a weakly bound complex. For instance, the H F monomer stretching frequency of about 3960 cm-1 is much greater than the HF-to-HF stretching frequency of about 180 cm-1 in the dimer. This suggests a BornOppenheimer-likeseparation in thevibrations of weak complexes. The weak or intermolecular modes are sluggish in comparison to the stretching frequencies of diatomics, and so there is a justification for an approximation: The diatomic stretches may be treated in an on-average sense so as to construct effective potentials for the weak modes. Within the context of the MMC calculationalmodel this means using diatomic electricalproperties specific to individualvibrational states. We obtain such electrical properties through ab initio calculations, as discussed in the next section. These effective potentials neglect dynamical monomermonomer coupling. The MMC interaction energy is a sum of the total electrostatic interaction energy of the species in the cluster and an empirical term that is chosen to represent nonelectrical contributions, such as penetration and exchange repulsion. This empirical term is essentially an atom-atom Lennard-Jones potential between atoms in different molecules. It, too, needs to be specificto thevibrational state in order to construct effective potentials. Essentially, we have to account for the small changes in on-average separation distances that occur with changes in vibrational excitation of the diatomic. That is, if excitation from the n = 0 to the n = 1 vibrational state leads to an on-average increase in bond length of 0.01 A, then an interaction calculation for the excited molecule must be performed with the “Lennard-Jones atoms” separated by an extra increment of 0.01 A. With a representation of a diatomic molecule specific to a given vibrational state, application of the MMC model allows us to generate different potential energy surfaces(over intermolecular coordinates) for different levels of excitation in the interacting species. Thus, we achieve an effective potential for the intermolecular interaction, one that reflects the averaged intramolecular vibrational motion of the diatomic(s). The effective potentials that are generated in this way necessarily include coupling between the intramolecular and weak modes; this is evident from the fact that a change in a diatomic’s vibrational state leads to a change in the shape of the potential surface over the intermolecular coordinates. The coupling, though, is with the on-average diatomic stretch; it is not an instantaneous or completeaccountingof the coupling. Naturally, as intermolecular interaction strength increases, the difference between intramo-
The Journal of Physical Chemistry, Vol. 97, No. 37, I993 9375
TABLE I: Calculated Changes in Equilibrium Diatomic Bond Lengths Due to Weak Bonding
clustep HF bond length in OC-HF HF-HCI HC1-HF HF - HF HF
-m
HCI bond length in OC-HCI HCI - HCI HCI - HCl
HF-HCI HCI-HF
6ry(A) (ab initio) 0.005219 0.003119 0.002420
0.005019
n=O
Dartner
&r) (4 n=l n-0-1
Partner
difference
0.0030 0.0023 0.0049 0.0055
0.0029 0.0025 0.0052 0.0060
0.0001 0.0002 0.0003 0.0005
0.0072
0.0078
0.0006
0.0011 0.0024 0.0022 0.0034 0.0052
0.0012 0.0024 0.0023 0.0038 0.0056
0.0001
0.003720
0.0 0.0001 0.0004 0.0004
CO bond length in
OC-HCI -0.0017 -0.0018 -0.OOO1 OC-HF -0.003419 -0.0031 -0.0033 -0.0002 a An underline identifies the monomer in a homonuclear dimer whose vibrational potential was analyzed. lecular vibrational frequencies and the weak mode frequencies will diminish, and then this approximation will be less suited. Neglected in this treatment is the monomer-monomer vibrational coupling, but such neglect appears to be a reasonable approximation in most cases. One measure of the significance of monomer-monomer coupling is the effect on the equilibrium bond length of a diatomic upon vibrational excitation of its partner diatomic. Shown in Table I are calculational results demonstrating that this is a very small effect. These calculations were performed for each A-B complex by using the MMC representation for the partner (B) in its n = 0 or n = 1 state and finding the interaction energy as a function of the bond length of diatomic A. This interaction potential was added to an ab initio stretching potential for isolated A, and then the new equilibrium was obtained. The important result is that though the presence of a partner diatomic affects the bond length, there is little dependence of this effect on the vibrational state of the partner. This is an indication that the stretching potential of monomer A in an A-B complex is essentially the same if B is in its n = 0 or n = 1 state; that is, there is little monomer-monomer coupling. However, in the case of degenerate or quasidegenerate monomer stretches, even a small coupling in the potential might be important, and then this approximation would not be as sufficient. Also shown in Table I are results from correlated ab initio electronic structure studies of the full potential surfaces of OCH F and (HF)*. These give the changes in the monomer bond lengths at the calculated equilibrium structures of the complexes. These equilibrium changes are very similar to those we have calculated for the n = 0 and n = 1 vibrational states from our electrical perturbation analysis. So, from full quantum mechanical treatments, one obtains essentially the same effects on the intramolecular potentials. A report by Dinur19 highlights the fact that the CO bond contracts upon forming OC-HF, and this is a feature reproduced with the simple level of physics of our approach. The developmentof an effectivepotential approach in our model means that for a binary complex of diatomic molecules, A and B, with vibrational quantum numbers n A and n ~ we , obtain a different intermolecular potential energy surface for A(~A=O) and B(~B=O),for A ( n ~ = l )with B(~B=O),for A ( n ~ z 0 )with B ( n ~ = l )and , so on. Each surface has its own minimum energy and equilibrium structure. We shall designate these equilibrium energies as E,,,,, and so the minimum energy of the first surface is Em, while the second surface’s minimum energy is Elo. The difference in the minimum energy of A(na=O) - B ( n p 0 ) and A(n~=l) B(~B=O) is part of the change in the = 0 to n~ =
9376 The Journal of Physical Chemistry, Vol. 97, No. 37, 199’3 1 vibrational excitation energy of the A diatomic, and this is designated AE~JO. Part of the effect of weak modestrong mode (inter- to intramonomer)coupling on the frequency shift may be important for accurate predictions. This is incorporated in our model by adding differences in weak mode zero-point energies to A E ~ J O . Realize that each surface has a slightly different shape. Often, the surfaces with one monomer vibrationally excited are deeper, and so the weak modes tend to be slightly higher in frequency. Thus, we take the actual transition energy from A(n*=O) B(ne=O) to A ( n ~ = l )- B(ne=O) to be the energy difference between the cluster’s overall vibrational ground state, with quantum numbers of zero for the weak modes, and the “ground” state on the potential surface where the A molecule’s stretch (only) has been excited. If the weakmodes and strong modes are completely uncoupled, meaning that the weak mode zero-point energies are unaffected, then this difference is identical with the value AEOOJO. Within our model, we may freely calculate the zero-point energies(ZPE) oftheweakmodesonboththeA(nA=O) -B(ne=O) and of A ( n ~ 3 1 )- B(ne-0) potential surfaces. The resulting energy difference, Av = AEOOJO A(weak mode ZPE), is the value we propose as an approximate model result for the diatomic vibrational frequency shifts in weakly bound clusters. It incorporates (1) the mechanical and electrical anharmonics of the monomers in an on-average sense, (2) the intermolecular effects on electronic structure through electrical polarization, (3) certain vibrational coupling effects by use of effective potentials specific to the high-frequency intramolecular vibrational states of the diatomics,and (4) the changes in the weak mode energies. From this, we arrive at the overall change in transition frequency. Of course, there are limitations that follow from the way we have elected to model the interaction. One is that since the ZPE is based on harmonic frequencies, there is an inherent neglect of anharmoniccoupling of weak modes and intramolecularstretches.
+
Calculational Approach We have previously obtained ab initio potential energy curves for isolated CO,12 HF,16,21 and HC13 with large basis set, wellcorrelated calculations. To obtain a given property for a specific vibrational state calls for values of the property as a function of the diatomic’s separation distance. Only a certain number of these properties are required, and they are the ones used for the MMC representation of a molecule. With MMC, the electrostatic energy is obtained from a complete solution of the mutual nonlinear polarization/hyperpolarizationequations,and we have devised efficient algorithms to carry out this solution with arbitrarily high order of multipoles and hyperpolarization.22The usual truncation, though, is at a rather low order. For the species discussed here, the electrical properties were the dipole and quadrupole moment, the dipole, dipole-quadrupole, and quadrupole polarizabilities, and the dipole hyperpolarizability. For CO, the octupole moment was also part of the property list for the MMC representation.12 These are the properties that must be known as a function of the stretchingcoordinateof the diatomic in order to obtain properties specificto different vibrationalstates. Values have been obtained from high-level ab initio calculations and equilibrium results have been reported elsewhere.23 In prior ab initio calculations, we have obtained sets of CO, HF, HCl, and Hz electrical properties at a number of separation distances. Vibrational state energies for the unperturbed molecules were obtained by numerical solution via the NumerovCooley method24 which allows for a complete incorporation of the effects of anharmonicity in the stretching potential. The electrical properties of the n = 0, n = 1, and n = 2 vibrational states were then obtained from the numerical derivativeNumerovCooley (DNC) method.2s DNC is an analytical method for the evaluation of all orders of energy derivatives (i.e., properties) of one-dimensional oscillators. DNC provides a complete incor-
Parish and Dykstra
TABLE II: Ab Initio Cdudatim of Bond Lmgthening for Isolated Diatomics upon Vibrational Excitation bond lengthening (A) diatomic
HF DF HC1
co Hz
-
n=0 n= 1 ( ~ ) I I- ( d m
0.03 16 0.0227 0.0324 0.0082 0.051 1
n=0
-
(r)22-
n =2 Woo
0.0641 0.0460 0.0658 0.0170 0.1047
ratio ( ( r ) z- ( r ) ~ ) / (WIl - (doe) 2.028 2.026 2.03 1 2.073 2.049
poration of electrical and mechanical anharmonicity in the generation of electrical properties of specific vibrational states. With properties for specific vibrational states as the input data for MMC, potential surfaces for molecules in different vibrational states are generated. The calculated bond lengtheningof the isolated HF, HCl, DF, H2, and CO monomers upon vibrational excitation from the ground state to the first excited state is given in Table 11. These values wereused in the MMC calculationson each complex where one of the partner molecules was vibrationally excited. The incremental changes are the changes in separation positions for thecenters (atoms)of thenonelectricalpart of the MMC potential, which has an atom-atom Lennard-Jones form.’ So, to carry out calculationsfor a diatomic molecule in an excited state as opposed to its ground state, the atomic centers were set to be further apart by the amounts shown in Table 11.
Results and Discussion Table I11 lists the calculated vibrational frequency shifts and the corresponding experimental values. The errors are all less than 35 cm-1 with one exception, the OC-HF complex, an interesting case which we discuss later. Excluding OC-HF, the mean error in the calculated frequency shifts relative to available experimental data (nineteen values) is 10 cm-1. An interesting result from the calculations of the frequency shifts is that the coupling with the intermolecular modes, as measured by the A(ZPE) values, is up to about 25 cm-l. These values are based on the harmonic intermolecular frequencies, and so, they may overestimate this effect, perhaps by 10%or so. It seems less likely, yet not impossible, that they could be an underestimate. Part of the A(ZPE) contribution to the vibrational frequency shift comes about from the change in the potential surface upon monomer excitation. Where the upper surface is deeper and stiffer, implying a red shift in AE00,01, the weak mode frequencies will be higher. So, in the absence of other effects, this will lead to positive A(ZPE) values (shift to the blue). However, there may be an opposing kinetic contribution to A(ZPE). Vibrational excitation of a monomer leads to a longer mean bond length, which in turn leads to a smaller monomer rotational constant. In a rough sense, the weak mode torsions in a complex are hindered monomer rotations, and so a decrease in the rotational constant implies lower torsional frequencies (shift to the red). A blue-shift A(ZPE) arising from stiffening of the intermolecular potential upon monomer excitationeffect goes along with a strong red shift in AEm.01. For most of the complexes, there is a net red shift, and typically for these we obtain an overall blue-shift contribution from A(ZPE). However, in several complexes there is a small overall red-shift contribution from A(ZPE), and this points to there being but little stiffening of the intermolecular potential upon vibrational excitation of the monomer. It is interesting that a blue-shifted AEw.01 usually goes along with a red-shift A(ZPE) contribution. We do not see a direct correlation between the monomer bondlength changes (Table I) and the frequency shifts, although there is a qualitative correspondencethat red shifts go along with bond lengthening and blue shifts with bond contraction. In this qualitative sense, we agree with Dinur,19 but we see both the
Vibrational Frequency Shifts
The Journal of Physical Chemistry, Vol. 97, No. 37, 1993 9377
TABLE IIk Vibrational Frequency Shifts (cm-1)for Diatomics in Weakly Bound Clusters clustefl
,500
hE10.w
A(ZPE) Av(ca1c)
Av(expt)
HF shifts
Ne-HF Ar-HF HrHF NrHF OC-HF HF-HCl HC1-HF HF-HF HF-HF
DF s h i f i y N ~ D F
Ar-DF Hz-DF NrDF OC-HF DF-HCI HCI-DF
-86 -192 -299 -473 -732 -832 -1068 -1537 -1537
-6.3 -17.1 -24.2 -30.5 -47.5 -36.7 -77.2 -87.8 -113.7
4.8 8.9 17.7 6.1 8.0 9.2 18.1 22.4 25.1
-1.5 -8.2 -6.5 -24 -39 -27 -59 -65 -89
-94 -210 -329 -505 -783 -821 -1154
-5.6 -15.2 -21.1 -25.2 -39.2 -26.8 -65.8
3.2 6.5 14.5 4.6 6.3 6.2 14.4
-2.4 -8.7 -6.6 -2 1 -3 3 -2 1 -5 1
1.4 -2.9 -10.8 -5.0 -4.1 3.9 3.6 9.5 11.4
1.1 -5.5 -1 1 -1 1 -14 -15 -14 -2 5 -34
-0.P -9.7273 -1 1.329*’0 -4331-33 -11734*35 -2 136 -9436 -3237.38 -9431.38 -9.0’’ -10.439 -3439
HCI shifts
Ne-HCI HrHCl Ar-HCI NrHCl OC-HCl HC1- HC1 HCl - HCl
-87 -0.3 -163 -2.6 -171 -0.7 -302 -5.8 -443 -9.6 -667 -18.9 -667 -17.7 HF-HF -34.6 -832 HCl-HF -1068 -45.3 CO shifts -87 ‘ -0.01 CO-Ne -158 -0.05 CO-Ar -177 -1.1 CO-Nz OC-HCl -443 13.4 25.6 -732 OC-HF H2 Shifts HtNe -23 -0.17 HFAr -41 -0.33 -2.9 HrNz -59 -15.9 HtHCl -163 -46.3 HrHF -299
0.03 0.02 0.4 -3.8 -7.3
0.0 0.0 -0.7 10 18
0.66 1.1 1.8 8.4 28.4
0.5 0.8 -1.1 -7.5 -18
-643 -2843
1244 2445 -0.02M -1.147
a An underline identifies the monomer whose vibrational shift has been calculated.
bond-length changes and the vibrational frequency shifts simply as two manifestations of the same thing, intermolecular electrical interaction. For shifts that are only about 1 cm-l or so, the qualitative prediction of a red shift versus a blue shift depends on accuracy to a fraction of a wavenumber. Use of harmonic vibrational frequencies for ZPE almost certainly precludes such accuracy. An exaggerated instance of this sort is Ar-HC1. In this case, vibrational excitation of HCl leads to a flattening of the potential surface along the bending coordinate. The surface is so flat and so anharmonic that the slight surface difference in going to the n = 2 vibrational state of HCl puts the minimum slightly away from a linear arrangement. Use of harmonic frequenciesto obtain the ZPE is poor relative to the small size of the frequency shift, and so the model’s greatest error for a rare gas-diatomic complex is found for Ar-HC1. The electrical contributions to the shifts in HF-rare gas complexes are much larger than in HC1-rare gas complexes-more so than the monomer dipole differences between n = 0 and n = 1 states of 0.046 D for H F and 0.032 D for HC1. Pine carried out an analysis for Ar-HF that yielded an electrical contribution to the H F red shift of 2.4 cm-l,48 which is in contrast to the 17.1-cm-’ value we obtained. Pine’s analysis was restricted to the effect of the H F dipole and quadrupole via argon’s dipole polarizability. Truncation in the electrical analysis at the point of using the dipole moment of H F and the dipole polarizability of argon yields 1.1 cm-1 in Pine’s analysis48 and 1.6 cm-l in ours. If we also include the H F quadrupole, we obtain 8.4 cm-1. It turns out that the quadrupole moment of H F also has a large
effect in concert with the quadrupole polarizability of argon, and so it mostly accounts for the 17.1-cm-1 electrical contribution value. The calculated A(ZPE) of 8.9 cm-1 (Table 111) brings the net calculated red shift for Ar-HF to 8.2 cm-I versus the measured value of 9.7.27928The closeness of these values, however, does not imply such accuracy in our evaluation of the electrical contribution. Since A E l ~ and , ~ A(ZPE) contributions tend to be opposed, there may be some cancellation of errors from the two contributions in this case. It may be best to conclude that the electrical contribution to the red shift is within 5 or so of 17 cm-1. A challenge for this model is (HF)2. The model predicts the two monomer shifts to differ by 34 cm-1, whereas the measured shifts differ by 62 cm-l. That is, the model sees the two monomers as being more nearly equivalent than they are. In HC1-HC1, the size of the red shifts and presumably the intermolecular perturbation is smaller, but the same problem appears. The model predicts the vibrational red shifts of the two monomers to be within 1 cm-l whereas the measuredvalues a r e 4 and-28 cm-l.43 The model is limited in these two cases by its lack of an instantaneous treatment of monomer-monomer coupling. The coupling via the potential is small for these and other binary complexes of diatomics; however, the zero-order degeneracy of the vibrational states of the homomolecular complexes means a much greater effect of small couplings. First-order degenerate perturbation theory tells us that the two degenerate states will spread apart in energy from any coupling of the states. The error we seem to have from neglecting this is in having pairs of states that are too close in energy. The model is a better predictor of the mean of the shifts in a homomolecular binary complex. For instance, the calculated and measured mean shifts (see Table 111) are -77 and -63 cm-1, respectively, for HF-HF, and -1 5 and -17 cm-l for HCl-HCl. There have been a number of recent ab initio calculations of the potential energy surface of (HF);! and evaluations of the vibrational frequency ~hifts.~%56 Calculations of Bunker et al.49 yielded a difference in the harmonic frequencies of the monomers of 42.5 cm-l (from w1 = 4178.2 and w;! = 4135.7). With an effective potential to include anharmonic effects, they obtained a difference in the monomer frequencies of 52.0 cm-l (from w1 = 3926.0 and 0 2 = 3874.0). Among the results in the extensive study by Quack and SuhmSo were H F monomer stretching frequencies obtained from one-dimensional cuts through the potential surface of Bunker et al.51 In this particular set of results, the difference in the two monomer stretches was 27 cm-1 when the slices followed the H-F stretching coordinates but 113 cm-l when they followed normal coordinates. These calculations of vibrational frequencies from full ab initio potential surfaces do not correspond to the treatment of our model; however, they suggest that the tendency of our model to make the monomers slightly more equivalent than they are is a consequence of the neglect of coupling, both harmonic and anharmonic, between the monomer stretches. OC-HFis the key problem case in the application of our model. The error in the H F shift is 78 cm-I, whereas the error in the CO shift is only 6 cm-l. In our previous study of weak CO containing complexes,12the surface information on the ground state of OCHF obtained from the MMC model seemed in good accord with spectroscopic data. So then why is the H F frequency shift so poorly predicted by this model? In what way is the model wrong, and is that a general problem? There are three limitations to the model, and we may expect the answer to these questionsto involve one or more of them. The limitations come about from (1) the assumption that intermolecular quantum effects on the potential are relatively unimportant such that electrostatic analysis is suited, (2) the neglect of monomer-monomer vibrational coupling, and (3) the neglect of anharmonic coupling between the weak modes and the monomer stretches. The first of these is the underlying physical notion of our approach, whereas the last two are merely simplifications of the dynamical treatment that we have elected to use. We have attributed the error in our model’s prediction
Parish and Dykstra
9378 The Journal of Physical Chemistry, Vol. 97, No. 37, 1993 TABLE Iv: Calculated Intermolecular Vibrational Frequencies in cm-1 of OC-HF surface nco nWF WI (deg torsion) w2 (stretch) 03 (deg torsion) 0 0 1 0 2
0 1 0 2 0
68 70 66 72 65
108 112 106 117 104
288 292 283 296 278
TABLE V: Relative Effect of Isotopic Substitution on Frequency Shifts HF/DF ratio of cluster Em AE10.00 A(ZPE) Av(ca1c) Av(expt) Ne-H( D) F 0.924 1.132 1.491 0.639 Ar-H(D) F 0.914 1.123 1.374 0.938 HrH(D)F 0.908 1.147 1.224 0.980 NrH(D)F 0.936 1.211 1.312 1.189 1.2936 OC-H( D)F 0.934 1.211 1.284 1.197 HCl-H( D)F 0.926 1.174 1.256 1.151 HF - H(D)F 0.921 1.167 1.202 1.158 HF as the acceptor:
H(D)F-HCl H(D)F- HF
1.014 1.026
1.367 1.417
1.491 1.473
1.313 1.399
70
e
0
HP
-70 -70
0
70
Figure 1. Energy difference contours in cm-1 for the OC-HF complex
as a function of the OC angle, 81, and the HF angle, 02. The equilibrium structure,linear OC-HF, corresponds to81= 82 = 0. Theenergydifference is between the potential energy surface for the complex with both monomers in their ground states and the surface with HF in its first excited state. For each of the two surfamused to generatethis difference plot, the separation distance between the monomers' mass centers was fixed at the calculated equilibrium separation.
for Ar-HCl to the third limitation, and we have attributed the tendency toward equivalencyin homomolecular dimers (e.g., HFH F and HCl-HCl) to the second. It is helpful to try to anticipate the source of the difficulty for OC-HF, but we do not yet have sufficient information available to understand it fully. Two spectroscopicstudies of the OC-HF c o m p l e provide ~ ~ ~ ~ ~ ~ Ab initio calculations of the OC-HF complex offer insight, even though the values can not be immediately dissected into the some interesting ideas. Fraser and Pine reported evidence of physical elements of our model. Curtiss et al.58reported HF/ anharmonic coupling of the H F stretch to CO bending.35 Jucks 4-31G calculations that gave a harmonic frequency red shift of and Miller found lifetime values for excited states that pointed 30 cm-1 for H F and a blue shift of 43 cm-1 for CO. Alberts et to some type of vibrational state mixing, perhaps involving ya al.59 reported MP2/TZ2P calculations showing a 154-cm-l HF near degeneracy" of vibrational states.5' Our calculated weak red shift in the harmonic frequency and a 27-cm-l blue shift for mode vibrational frequencies (Table IV) show torsions at about CO. There is clearly a sizable correlation effect on the frequency 70 and 290 cm-1, and perhaps there is some combination of shifts and significant interplay with basis set effects. As in many excitation quanta to bring about a strong mixing of the pure mode states. Of course, this happens with anharmonic potential weak bonding problems, it is not clear how far one has to go to coupling, which is what Fraser and Pine suggested and which achieve a converged result in a frequency shift. Recent MP2 calculations with effective core potentials and a singly polarized unfortunately is neglected in our model. We only speculate that double-l: valence basislg gave 120 cm-l for the red shift of H F this is the source of the error for this complex. and 29 cm-1 for the blue shift of CO. Given the small basis, this It is particularly interesting to contrast the isoelectronic is quite remarkable in its closeness to the experimental values. complexes, NTHFand OC-HF. They are both linear complexes, The highest level treatment of which we are aware is that of and they are not greatly different in stability. The measured Botschwina,W who used a 100-function basis and a CEPA-1 vibrational frequency shift of H F in N r H F , however, is 43 treatment of electron correlation. He obtained an H F red shift cm-1,31,32 much less than in OC-HF. It is not apparent that there should be a radical difference in the electronic interaction of H F of 117 cm-1 and a CO blue shift of 27 cm-I. Botschwina's vibrational analysis is a diagonalization of an anharmonic with carbon monoxide versus nitrogen. However, there could be a difference in an accidental near degeneracy of vibrational levels vibrational Hamiltonian matrix. The very close comparison of and/or anharmonic coupling of monomer stretches with the weak these high-levelab initio values with the experimentallymeasured modes. This may be testable by isotopic substitution because it values may mean that coupling between the weak modes and the is possible (but not guaranteed) that there would be a sizable H F stretch is truly the key problem for our model in this case. change in effects upon substitution of DF for HF. In N r H F , From application of MMC, we can show that the anharmonicity Nesbitt and co-workers39 find that DF substitution yields a DF in the weak mode torsions is very noticeable; the potential surface red shift of 34 cm-I, which is 77% of the H F red shift in N r H F . along either of the bending coordinates is not parabolic. A more Our calculationson N2-DF (Table 111) givea smaller, but similar, revealing way of illustrating this is the contour plot in Figure 1. This shows the energy difference between the OC(n=O) - HFreduction of 85% and this reduction is mostly associated with the mass effect on the moment of inertia of the HF. A summary of (n=O) complex and the OC(n=O) - HF(n=l) complex as a calculated ratios of shifts is given in Table V. If the DF red shift function of the two torsion angles, 81 for OC and 82 for HF. in OC-DF is also around 80%of the red shift in original isotopomer A final aspect of this workis applicationof our model toovertone (Le., OC-HF), and that would be about 94 cm-1, then there transitions. From ab initio calculations we have obtained values would be no reason to believe there are unusual dynamical for the electrical response properties of the second excited interactions in OC-HF. However, finding a much smaller vibrational states of several diatomics. We have used these in the frequency shift would point toward different vibrational mixing generation of effective potential surfaces corresponding to a and/or anharmonicity effects for OC-DFversus OC-HF. Either monomer being excited with two vibrational quanta. The results result would provide information on the nature of vibrational of these calculations are in Table VI. In the absence of unusual coupling in OC-HF relative to N2-HF, and so we hope vibrational couplings (e.g., near degeneracies), the overtone spectroscopistsare encouraged to find the DF transition frequency transitions are likely to obey a simple relationship that Scoles in OC-DF. and co-workers have discussed recently.61 It is that the ratio of
Vibrational Frequency Shifts
The Journal of Physical Chemistry, Vol. 97, No. 37, 1993 9379
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TABLE VI: Vibrational Frequency Shifts (cm-I) for n = 0 n = 2 Overtone Transitions in Diatomics in Weakly Bound Clusters.
cluster HF shifts Ne-HF Ar-HF HrHF NrHF HF-HC1 HC1-HF HCI shifts Ne-HCl HrHCl Ar-HCI NrHCl OC-HC1 HF-HCI HCI-HF CO shuts CO-Ne CO-Ar CO-Nz OC-HCl OC-HF
EO0
AEz0.m
A(ZPE)
Au(ca1c)
-86 -192 -299 -473 -832 -1068
-13.6 -37.0
10.1 18.8 36.8 12.5 18.6
-3.4 -1 8
-51.1 -63.6 -76.0 -163.0
38.1
-8 7 -163 -171 -302 -443 -832 -1068
-0.76 12.0 -1.9 -11.4 -21.2 -66.9 -97.5
2-78 -23.2 -7.7 -1 1.6 -9.1 21.1 24.2
-87 -158 -177 443 -732
-0.03 -0.10 -2.2 26.8 51.4
0.05 0.09 0.9 -7.8 -14.9
-14 -5 1 -57 -125 2.0 -1 1 -9.6 -23 -30 -46 -7 3 0.02 -0.01 -1.3 19 36
the overtone shift to the fundamental shift should equal the ratio of the bond lengthening in the second excited state to the bond lengthening in the first excited state. Values for the bond length ratios are given in Table 11. Our model, which cannot account for strong coupling of nearly degenerate excited states, etc., shows the expected ratio holds to about 10%. For example, the bond lengthening ratio for H F is 2.028 (Table 11), and the ratio of the overtone to the fundamental frequency in HCl-HF is 2.119. In HF-HCl, it is 2.1 11. The important conclusion of this work is not so much the utility of the model but the physical significance of the model’s success. The physical picture underlying the model is that the intramolecular vibrational potential of a diatomic is altered upon weak bonding to another species via the electrical influence of the other species. This is the same idea advanced in the more limited treatment of vibrational freqyency shifts that was presented several years ago.16 The implications of this statement go beyong the model we have presented here. Our model is one of a simplified, convenienttreatment of dynamics, and consequently,certain cases appear problematic; however, the idea of electrical perturbation of intramolecularvibrational potentialsis not tied to the dynamical simplifications. It is general, and in principle, it can be paired with any scheme for dynamics. That generality is our goal in this study and prior work, an overall basis for weak interaction potentials. From this work, we reach the specificconclusion that weak interaction effects on intramolecular vibrational potentials can be extracted from electrical analysis. If the dependence of the electrical properties on a molecule’s vibrational coordinates is known, if the vibrational potential of the isolated molecule is known, and if the on-average electrical properties of a partner species are known, then an intramolecular potential surface for the molecule weakly bonded to the partner can be generated by electrical interaction analysis. This potential is the sum of the isolated molecule’s potential and the electrical interaction energy as a function of the intramolecular coordinates.
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