J. Phys. Chem. 1986,90, 3097-3103
3097
was not included in the fit. This result is somewhat anomalous and presently cannot be explained. In general, the observed transferability of the four bond moments among the members of the fluoroozonide series has potential usefulness for predicting the dipole moment for new members of the series.
Industrie. All a b initio calculations were carried out a t the Rechnenzentrum der Universitiit Koln. Registry No. t-F,Oz, 54892-65-8; t-F20z-d2,102307-21-1; r-F20z-d,, 102307-22-2; t-F20z-L80p180p, 102342-10-9; t-F20z-1803, 102307-23-3; r-F20z-W!, 102307-24-4; DFE, 1630-77-9.
Acknowledgment. This work was supported by Grants CHE8005471 and CHE-8303615 from the National Science Foundation. Support a t the Universitat Koln was provided by the Deutsche Forschungsgemeinschaft and the Fond der Chemischen
Supplementary Material Available: Tables Sl-S6 listing transition frequencies for the ground vibrational states of the normal and isotopic species (10 pages). Ordering information is given on any current masthead page.
A Theory of Vibrational Transition Frequency Shifts Due to Hydrogen Bonding Shi-yi Liu and Clifford E. Dykstra* Department of Chemistry, University of Illinois, Urbana, Illinois 61801 (Received: January 3, 1986)
Changes in vibrational transition frequencies, usually red shifts, are characteristic of the effects of hydrogen bonding. A theory based on intermolecular electrical interaction has been developed and tested to explain these shifts entirely from monomer properties. Dipole and quadrupole polarizabilitiesand permanent electrical moments through the octupole give rise to important changes in a molecule’s stretching potential when it interacts with the electrical field of another molecule. This interaction potential tends to be almost linear with the stretching coordinate, and so a change in the vibrational transition frequency arises largely because of the intrinsic anharmonicity of the original, or unperturbed stretching potential. Knowledge of the interaction potential, which we show can be obtained from a simple electrical model, and knowledge of the vibrational anharmonicity parameters are then sufficient to adequately account for vibrational shifts. Furthermore, because the interaction potential function prescribes a new equilibrium bond length, the usual correlation between bond length changes and vibrational frequency changes is obtained as a more quantitative relation involving the intrinsic anharmonicity and the slope of the interaction potential. Calculations on a number of hydrogen-bonded complexes of hydrogen fluoride show very good agreement with experiment.
Introduction One of the significant advances in contemporary chemical physics has been the capability to study weakly bound complexes in virtually as much detail as the molecules that make up such complexes. From there being all too infrequent papers on the subject a decade and a half ago, there is now a wealth of information on structures, stabilities, vibrational frequencies, and the dynamics of van der Waals and hydrogen-bonded complexes, including many dimers, some trimers, and even large aggregates. Techniques utilizing molecular beam electric resonance,I4 pulsed beam microwave ~pectroscopy,~-~ vibrational predi~sociation,8*~ matrix isolation,lOJ1long-path-length absorption,12-15and laserinduced fluorescence16 have all played a role in this direct attack (1) Dyke, T. R.; Howard, B. J.; Klemperer, W. J. Chem. Phys. 1972,56, 2442. (2) Novick, S. E.; Davies, P.; Harris, S. J.; Klemperer, W. J. Chem. Phys. 1973, 59, 2273. (3) Dyke, T. R.; Muenter, J. S. J . Chem. Phys. 1974,60, 2929. (4) Klemperer, W. J . Mol. Struct. 1980, 59, 161. (5) Balle, T. J.; Campbell, E. J.; Keenan, M.R.; Flygare, W. H. J. Chem. Phys. 1979, 71, 2723. 1980, 72,922. (6) Balle, T. J.; Flygare, W. H. Reu. Sci. Instrum. 1981, 52, 3 3 . (7) Campbell, E. J.; Read, W. G.; Shea, J. A. Chem. Phys. Lett. 1983,94, 69. (8) Lisy, J. M.; Tramer, A.; Vernon, M. F.;Lee, Y . T. J. Chem. Phys. 1981, 75, 4733. (9) Cassasa, M.P.; Western, C. M.; Cellii, F. G.; Brinza, D. E.; Janda, K. C J. Chem. Phys. 1983, 79, 3227. (10) Johnson, G. L.; Andrews, L. J . A m . Chem. SOC.1982, 104, 3043. (11) Andrews, L. J . Phys. Chem. 1984, 88, 2940. (12) Thomas, R. K. Proc. R . SOC.London, Ser. A 1971, A325, 133. (13) Pine, A. S.; Lafferty, W. J. J . Chem. Phys. 1983, 78, 2154. (14) Pine, A. S.; Lafferty, W. J.; Howard, B. J. J. Chem. Phys. 1984.81, 2939. (15) Kyr6, E. K.; Shoja-Chaghervand, P.; McMillan, K.; Eliades, M.; Danzeiser, D.; Bevan, J. W. J. Chem. Phys. 1983, 79, 78. (16) Levy, D. H. Adu. Chem. Phys. 1980, 31, 197.
on a problem that is fundamental in molecular science. What is particularly exciting is the prospect of this new information being used to aid the refinement of hydrogen-bonding interaction potentials in molecular dynamics simulations of biomolecules or large molecular clusters that serve as models of condensed-phase systems. Beginning with Coulson’s partitioning of the hydrogen bond strength in the water dimer,” hydrogen bonding has been thought of as arising from a number of competing effects. These include electrostatic interaction of the permanent electrical moments of interacting species, polarization of charge, dispersion, charge transfer, and exchange effects. Morokuma developed some of the first calculational means for partitioning energies obtained from a b initio electronic structure wave functions18-20 among such effects. Partitioning methods were also formulated by Kollman,21-24and one recurring conclusion of such analyses is that electrostatic interactions have a significant role in hydrogen bonding. The possible range of hydrogen bond strengths is unquestionably large. At one end are charged complexes such as F H F where the hydrogen bond strength is 39 kca125*26 and the proton is symmetrically located between the two bonding partners. This type of bonding, though, is easily understood in terms of usual chemCoulson, C. A. Reseorch 1957, 10, 149. Kitaura, K.; Morokuma, K. Znt. J. Quontum Chem. 1976, 10, 325. Umeyama, H.; Morokuma, K. J . Am. SOC.1977, 99, 1316. Morokuma, K. Acc. Chem. Res. 1977, 10, 294. Kollman, P. A.; Allen, L. C. Chem. Rev. 1972, 72, 283. Kollman, P. A. J . A m . Chem. SOC.1977, 99, 4875. Kollman, P. A. Ed.; In Applications ofBIectronic Structure Theory, Schaefer, H. F., Plenum: New York, 1977. (24) Singh, V. C.; Kollman, P. A. J . Chem. Phys. 1984,80, 353. (25) Harrel, S. A.; McDaniel, D. H. J. Am. SOC.1964, 86, 4497. (26) Larson, J. W.; McMahon, T. B. J. Am. Chem. SOC.1982,104, 5848. (17) (18) (19) (20) (21) (22) (23)
0022-3654/86/2090-3097$01.50/00 1986 American Chemical Society
3098 The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 ical-bonding principle^.^^ On the weak end of the scale are complexes such as Ar-HCl which has a bond strength of about 0.5 k ~ a l . ~ *The - ~ complexes ~ between these extreme, where the hydrogen bond strengths are 1-10 kcal, are representative of the hydrogen-bonding interactions that are biologically important, for these are the bond strengths typical of water-protein interactions which Coulson was already considering in his energy partitioning of hydrogen bonds.I7 Molecular dynamics simulations afford a computational strategy for understanding the dynamics of proteins, other biomolecules, and liquids, and it is here that accuracy in representing hydrogen-bonding interactions is most valuable. The evaluation of interatomic forces is a computationally slow step in molecular dynamics and so elaborate forms of interaction potentials present computational limitations. If a simple basis can be found for the interaction, and a simple mathematical representation arises from it, then there is a strategic advantage over empirically sorting through various adjustments and corrections. For example, Stillinger’s polarization mode131-33was a compact and successful way of generating hydrogen-bonding potentials, and its first feature was the simple Coulomb’s law interaction of partial molecular charges. Point charge interactions have also been employed by J o r g e n ~ e nand ~ ~by others in constructing hydrogen-bonding potentials for simulations of numerous liquids. Another example is Kollman’s d e ~ e l o p m e n of t ~ ~potentials giving especially good correspondence with measured gas-phase ion-hydration enthalpies because of how polarization of charge is incorporated. Three important aspects of hydrogen bonding are the relative orientations of bonded monomers, the total bond strength, and the perturbation of the constituent molecules’ internal potentials. These are all features associated with the hydrogen-bonded complex’s global potential energy surface (PES) and may be analyzed or identified from cuts, slices, or segments of the PES. These three aspects are addressed in experimental investigation of the complexes that seek to determine the structure of the complex, the binding strength of the complex, and the changes in the vibrational spectra of a constituent species upon forming a complex. On the theoretical side, orientational features in hydrogen bonding are well-known to depend heavily on the permanent moment electrical interaction^,^^^^^-^* while the binding strengths can involve competing effects. The focus in this report is that aspect of a hydrogen-bonding interaction that leads to changes in the vibrational potential of a constituent molecule.
Effects of Vibrational Potential Changes Upon formation of a weak complex, the vibrational transition frequency for a monomer stretch tends to change largely because of how the potential function along the stretching coordinate has been influenced. Coupling of vibrational modes because of complex formation is a lesser effect if the complex is indeed weak. In analyzing the (HF), complex, Michael et al.39 roughly determined that coupling of the monomer stretches to torsional vibrations affected the transition frequencies by -6 cm-I. The very different frequencies of the low-energy modes in a complex (27) Pimentel, G . C. J . Chem. Phys. 1951, 29, 446. Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond; Freeman: San Francisco, 1960. (28) Novick, S. E.; Janda, K. C.; Holmgren, S. L.; Waldman, M.; Klemperer, W. J . Chem. Phys. 1976, 65, 1 1 14. (29) Keenan, M. R.; Campbell, E. J.; Balle, T. J.; Buxton, L. W.; Minton, T. K.; Soper, P. D.; Flygare, W. H. J. Chem. Phys. 1980, 72, 3070. (30) Hutson, J. M.; Howard, B. J. Mol. Phys. 1981, 43, 493. (31) Rahman, A.; Stillinger, F. H. J . Chem. Phys. 1971, 55, 3336. (32) Stillinger, F. H.; Rahman, A. J . Chem. Phys. 1974, 60, 1545. (33) Lemberg, H. L.; Stillinger, F. H. J. Chem. Phys. 1975, 62, 1677. (34) Jorgensen, W.; Chandrasekhar,J.; Madura, J.; Impey, R.; Klein, M. J . Chem. Phys. 1983, 79, 196. (35) Lybrand, T. P.; Kollman, P. A. J. Chem. Phys. 1985, 83, 2923. (36) See, for example: Tomasi, J. in Molecular Inreracrions, Vol. 3. Ratajczak, H., Orville-Thomas,W. J., Eds.; Wiley-Interscience: New York, 1982. (37) Kollman, P. Acc. Chem. Res. 1977, 20, 365. (38) Buckingham, A.; Fowler, P. J . Chem. Phys. 1983, 79, 6426. (39) Michael, D. W.; Dykstra, C. E.: Lisy, J. M. J. Chem. Phys. 1984,81, 5998.
Liu and Dykstra TABLE I: Vibrational Potential Parameters for Hvdrogen Fluoride calculated 3977.1 7786.9 11435.7 3809.8 3648.8 4156.8 91.9 1.61 0.92087 0.00841 0.02562 0.04334 0.93628 0.96776 1.00024 1.03354 0.065732 0.093803 0.1 15939 -0.007236
experiiieital 3961.4 7750.8c 1 1 372.ge 4138.52d 90.06d 0.98d O.9168Oc 0.00879‘ 0.02668~ 0.045 1 5c
Calculated values obtained from fitting the first four vibrational state energies to C(u) = ( u + 1/2)w, -. (u ‘ / 2 ) 2 w g e (u 1 / 2 ) 3 ~ c y c . R, is [ ( r - 2 ) n n ] - ’less / 2 re. It is the vibrationally caused bond elongation as determined from the rotational constant, which is proportional to r-2. Reference 44. Reference 45.
+
+ +
work against coupling to the relatively high-energy monomer stretches, while coupling of two monomers’ vibrations even of like frequency is small since it is indirect. One of the simplest ways that a stretching potential for a molecule such as hydrogen fluoride may change when it forms a hydrogen-bonded complex is through adding a linear potential, p t ( r ) ,to the stretching potential, V(r),of the isolated molecule. Among other things, this corresponds to the molecule’s dipole moment function being linear in the stretching coordinate, what S a n d ~ r f y ~designates ~,~’ “electrically harmonic,” and interacting with a fixed field. We show later that, for realistic systems that are polarizable and possess higher moments of their charge distributions, the interaction is often very nearly linear. The addition of a linear potential to a harmonic potential shifts the potential minimum but does not alter the force constant. Thus, there would be no change in the vibrational level spacing. For an anharmonic potential, however, a linear vnt(r)shifts the location of the minimum so that the curvature at that new minium is no longer the same. This “mechanical anharmonicity” can shift vibrational transition frequencies. This results even at low order perturbation theory. In a basis of the vibrational energy levels of an isolated diatomic, the Hamiltonian matrix elements from a perturbing interaction that happens to be linear in the stretching coordinate, r, are given by where n and m are vibrational quantum numbers, s is the slope of the perturbing interaction potential (Le., pi@) = sr) and ( r ) , = (nlrlm). To first order the energy difference between states will be changed by,
= S((r)n+l,n+l -
(%,A
(2b)
For a harmonic oscillator (r),,“ = re and so only because of anharmonicity does eq 2b correspond to a frequency shift. The size of any shift is simply proportional to the slope of the interaction potential at first order. To use eq 2b, we have calculated a very accurate a b initio potential energy curve for HF. A large basis set and extensive incorporation of electron correlation effects were used to generate (40) DiPaolo, T.; Bourderon, C.; Sandorfy, C. Con. J . Chem. 1972, 50, 3161. (41) Sandorfy, C. Top. Curr. Chem. 1984, 20, 41.
The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 3099
H-Bonding Vibrational Frequency Shifts
TABLE 11: Comparison of Numerical and Perturbation Theory HF Vibrational Shifts with Selected Linear Interaction Potentials A,?%&,shifts, cm-' AL?l-2 shifts, cm-' interaction potential, slope, exact cm-'/A (numerical)
250 500 1000 2000 4000 8000
7.2 15.1 31.0 62.9 127.8 261.8
" The approximation is eq
.first order with first-order pertbn theory 7.9 15.7 31.5 63.0 125.9 25 1.8
second-order pertbn theory 7.9 15.7 31.4 62.5 124.3 245.2
approx. matrix elements" 8.2 16.3 32.7 65.3 130.6 261.2
second-order pertbn theory
8.1 16.2 32.4 64.5 128.1 252.5
first order with approx. matrix elements" 8.2 16.3 32.7 65.3 130.6 261.2
3a.
the potential curve, and complete details are given later. A Numerov-Cooley a n a l y ~ i s was ~ z ~carried ~ out to numerically give the exact vibrational wave functions for this potential. These wave functions were used to calculate ( r ) , (9), ( 1 / 9 ) , etc., and a summary of results is given in Table I. Where experimental values are available, the comparison is extremely good. With the results in Table I, the vibrational interaction Hamiltonian matrix, H', of eq 1 can be constructed for any chosen interaction slope, s. Vibrational energy levels may then be found easily with first-order or second-order perturbation theory. Or, the energy levels can be solved for variationally by diagonalizing the complete Hamiltonian matrix. Or, the energy levels can be obtained exactly by the numerical solution of the vibrational wave function using the Numerov-Cooley whereby a linear potential of slope s is added to the isolated molecule, a b initio potential. Results with this last procedure are given in Table I1 for a range of interaction potentials. The agreement with the first-order shifts obtained from eq 2b is quite good. Also given in Table I1 are the shifts correct through second-order perturbation theory. The second-order changes tend to be small, and to a good extent first-order perturbation theory appears to be well-suited. Therefore, eq 2b is a simple relation between the frequency shift and the interaction potential requiring only certain information related to the molecule's intrinsic anharmonicity. A number of means can be used to obtain the ( r ) n nvalues needed in eq 2b. First, though, it is easy to see the following are reasonable approximations: (r),,"c a + bn (3a) (r)n,n+l c
c + dn
( r ) , , , = 0 for m # n, n f 1
(3b) (3c)
The approximations of eq 3 reduce the task of finding the matrix representation of r to just finding a, b, c, and d. With the values in Table I, for instance, a least-squares fit gives a = 0.93559 A, b = 0.03266 A, c = 0.0681 1 A, and d = 0.02302 A. Using eq 3a with eq 2b indicates that b is the constant needed to find frequency shifts to first order. Detailed infrared spectra of the monomer can yield the value of b. Rotational fine structure can give (r-2)wand ( f 2 ) " , and then b may be approximated as bc
exact first-order (numerical) pertbn theory 7.3 8.1 15.5 16.2 32.5 31.8 64.8 65.0 131.9 129.9 270.8 259.8
2[(r-2)ll-'/2
- (r-2),,,,-'/2]
(4)
The experimental values in Table I give b = 0.03578 which is within 10% of the a b initio value. Another way to extract b from spectroscopic data is to use the vibrationally averaged dipole moments for two vibrational states, probably the lowest two. If the dipole moment function, p(r), is linear in r, i.e. if p ( r ) = f + gr, then b = ((c1)11-
(cc)oo)/g
(5)
In Table 111, values are given for both experimental and ab initio (p),,'s. With g = 1.530 D/A, which is from the a b initio dipole moment curve, the experimental ( p ) values yield b = 0.03007 A. Thus, either approach provides a means of relating readily (42)Cmley, J. W.Math. Comput. 1961, 25, 363. (43)Cashion, J. K. J. Chem. Phys. 1963, 39, 1872.
TABLE 111: Use of Dipole Moment Matrix Elements To Estimate Stretching Coordinate. r . Matrix Elements element X operator matrix elements' between vibrational X = u, X = u. X = r. exDtl X = r. . . states: uI,u2 ab initio exptl usingb p i g ab initio 0.023' 0.015 0.01541 (0,O)-X, 0.02573 0.0644 0.065 73 (091) 0.10229 0.0985d (02) -0.008 51 -0.0127' -0.00830 -0.00724 0.069' 0.045 0.04689 (l,l)-X0.07893 0.1380d 0.0902 0.093 80 ( 172) 0.148 25 "All dipole operator matrix elements are in debyes. r-coordinate matrix elements are in angstroms. bThe experimental values of (p),,to the left have been divided by g = 1.530 D/A. cReference 46. 'Reference 47.
available spectroscopic data of the monomers to the frequency shift. In the next section, we use an electrical interaction picture to relate the interaction slope, s, to monomer properties as well, and thereby produce a means of explaining vibrational frequency shifts in hydrogen-bonded complexes entirely in terms of constituent monomer properties.
The Electrical Vibrational Interaction Potential The vibration interaction potential, P t ( r ) ,arising from electrical effects depends on the changes with r in a monomer's electrical properties. If the properties had no r dependence, P t ( r ) would be a constant function, the interaction slope, s, would be zero, and there would be no shift. Though obvious, this notion is important because the constant part of p t ( r ) is a component of the hydrogen bond strength. Consequently, correlations of vibrational shifts with hydrogen bond strengths may not turn out to be simple. The electrical properties of a molecule may change sharply around but typically the change is quite linear. Dipole moment functions have been found that are very nearly linear, and this is certainly the case for HF49350and many hyd r i d e ~ . ~ ~Interaction ,~' with the field of a nearby dipole will then yield a linear vnt(r). The same would hold for other moments and for all induced moments, provided the associated polarizabilities are linearly dependent on r. One way a quadratic term in P t ( r ) could arise is from the interaction of two induced moments, but most often this is small. Consequently, one expects a nearly linear P t ( r ) and, in fact, that is just about what is found. Figure 1 shows the calculated vnt(r) functions in the vicinity of the equilibrium. Some curvature is seen well away from the equilibrium, in regions where the electrical properties begin to be nonlinear in r, but these are also regions where there is a diminished probability density in the low-energy vibrational wave functions. (44)Guelachvili, G. Opt. Commun. 1976, 19, 150. (45)Talley, R. M.; Kaylar, H. M.; Nielsen, A. H.Phys. Reu. 1950, 77, 529. (46)Gough, T.E.;Miller, R. E.; Scoles, G. Faraday Discuss. 1981, 71, 77. (47)Sileo, R. N.; Cool, T. A. J . Chem. Phys. 1976, 65, 117. (48)Malik, D. J.; Dykstra, C. E. J . Chem. Phys. 1985, 83, 6307. (49)Jasien, P.G.; Dykstra, C. E. Int. J. Quantum Chem. Symp. 1983, 17, 289. (50) Werner, H.-J.; Rosmus, P. J. Chem. Phys. 1980, 73, 2319 (51) Meyer, W.; Rosmus, P. J. Chem. Phys. 1975, 63, 2356.
3100 The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 I
100 ’
50
=-!!
-
0’
‘0 E
3-HF N2-HF
.-c
c
t -50 f > -100
’
OC- HF IH F - E
-150
I
\HCN-HF
0,88
0.90
0.92 rnF
cH 1
I
0.94
Figure 1. Interaction potentials, p t ( r ) ,from electrical interactions. Also shown is the ACCD-calculated stretching potential for an unperturbed HF molecule.
Were pnt(r)treated as a linear function arising from just the permanent electrical moments, the interaction could be computed from derivatives of the moments, e.g., dp/ddre. However, induced effects on the polarization of charge have been included to infinite order, or to convergence in the iterative treatment of intermolecular polarization. Thus, the interaction slope, s, is most directly obtained from the complete electrical interaction at two near equilibrium geometries. The values in Table IV list the properties used a t the two geometries chosen. B u ~ k i n g h a m ~ has * - ~provided ~ the theoretical foundation for the understanding of electrical intermolecular forces and complete formulas are available for evaluating permanent and induced moment interactions to all order^.^^-^^ The recent papers by A p p l e q ~ i s t ~provide ~ ~ ’ ~ a most effective organization of the problem, which we follow in part here. The usual CartesianSs multipole moment tensors of a monomer, either A or B for the A-B dimer, are arranged in canonical orderS9as an infinite-order vector:
MtA = (piA),piA),piA),Q$$),...) (6) In like _order,the linear array T is found from the geometrical vector R A B that connects the A and B centers. (Since A and B represent molecules, what are meant by their center positions are the positions of the points about which their multipole moments have been evaluated.) T = ( t ( O ) , ti1),lit), ri1), t(*), ...) (7) t(n) = (-1)n t,(O)
VnRAB-l
= RAB-l
@a) (8b)
t,U) = -xRAB-3
(8c) The electric potential at A due to B is the polytensor dot prodU C ~ , @~=~T * MA. ~ ~ From , ~ ~@, the field PI),field gradient, Pz), and so forth at A due to B are straightforwardly evaluated as
.
(52) Buckingham, A. D. Q. R . Chem. Soc., London, 1959, 13, 183. (53) Buckingham, A. D. Adu. Chem. Phys. 1967, I2, 107. (54) Buckingham, A. D. In Intermolecular Interactionsfrom Diatomics
to Biopolymers. Pullman, B.,Ed.; Wiley: New York, 1978. (55) Stogryn, D. E.; Stogryn, A. P. Mol. Phys. 1966, II, 371. (56) Applequist, J. Chem. Phys. 1984, 85, 279. (57) Applequist, J. J. Chem. Phys. 1985, 83, 809. (58) Logan, D. E. Mol. Phys. 1981, 44, 1271. (59) Applequist, J. J . Math. Phys. 1983, 24, 736.
Liu and Dykstra TABLE I V HF Monomer Electrical Properties at Two Near-Equilibrium Structures property R W F= 0.9068 Ab R H F= 0.9268 Ab -1 ,8000 -1.8306 -0.7082 -0.7202 -5.5291 -5.7371 -4.3 122 -4.3 5 54 9.2114 8.2236 0.761 1 0.7423 -2.6723 -2.6284 -4.3754 -4.3925 2.2908 2.3728 3.6986 4.0877 -0.1393 -0.1503 0.5955 0.6405 -8.0920 -8.5955 -0.9338 -0.9292 -5.4470 -5.4955 -2.5468 -2.6176 -2.1876 -2.1982 -1.6298 -1.6487 -63.136 -67.496 -13.127 -13.430 -13.303 -13.508 -58.541 -59.426 -20.13 1 -20.375 -22.569 -23.318 -19.205 -19.526 -1.7 164 -1.8185 0.2129 0.2325
‘Obtained from ACCD correlated wave functions. center of mass.
Evaluated at
derivatives, and these become components of FA(B). The polarizabilities can be arranged in a polytensor alsoS7and the induced moments at A due to B may be written compactly as
M i d = PA.FA(B)
(9)
The total moments on the monomers are then MA
= Mf’
MB = Mio)
+ PA*FA(B)
( 1 Oa)
+ PB’FB(A)
(lob)
Since the Fs depend on the Ms, eq 10 is a set of coupled equations that need to be solved to yield values for the unknown total moments. In the interactions we have calculated, certain hyperpolarizabilities have been incorporated and so the coupled equations are nonlinear, e.g. MA =
Mf) + P A . F A ( B )
+ F+A(B)*HA*FA(B)
( 1 1)
where H is the hyperpolarizability in polytensor form. This potentially cumbersome set of equations has been solved iteratively by successive relaxation. Convergence in the interaction energy is usually achieved in 3-5 iterations. The interaction energy is evaluated with a field of total moments. In a more explicit form, eq 9 for the dipole moment elements is ptdA
=
a$y(B)
+ 1/2 A & F p + yzp&F;(B)FI;(B) + y*sg,,,F y B ) F ’ p + ... (12)
This means the dipole induced at A arises from (i) the field due to B (FA@)) via the dipole polarizability, (ii) the field gradient (F”A(B)) via the dipole quadrupole polarizability, and so on. Regardless of the formalism, intermolecular interactions have been calculated with all properties in Table IV treated fully. The resulting vnt(r)functions turn out to be linear and, as discussed in the next section, they depend most strongly on the lowest order moments and polarizabilities. That means that measured or calculated monomer properties can be employed to generate a suitable pnt(r). An important point in multipole moment interactions is the choice of a center since only the first nonvanishing moment is
H-Bonding Vibrational Frequency Shifts
The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 3101
TABLE V Vibrational Frequency Shifts (AEw1,cm-') in Hydrogen Fluoride Complexes electrical electrical model: contributing effects model permanent permanent total' dipole quadrupole complex exptl ref prediction dipoles quadrupoles electrostatic polarizability polarizability HF-HF (free) NZ-HF HF-HF (bonded) OC-HF HCCH-HF HCN-HF
32 43 93 118 168 251
13 61 13 61 61 12
35 34 96 90 187 160
24 0 26 1 0 43
18 17 40 23 76 12
27 23 68 71 107 116
total' induced
3 4 7 5 27 17
7 8 16 7 31 24
8 11 28 19 81 44
'See text for discussion of contributions beyond those specifically listed. invariant to the choice of the center. Center of mass is a possibility, but that would introduce at least a small, but nonphysical isotope dependence in the interaction potential with a truncated set of moments. We have tested a few choices of centers and believe that the results obtained are not strongly affected by center choices in the vicinity of the center of mass, or really the molecule's "center". One approach that has no arbitrariness and no mass dependence is to separate the electrons and nuclei into negative and positive charge distributions, each with their own moments. The evaluation centers for each distribution are the respective charge centers. This procedure was used to calculate the total interactions of permanent and induced moments. The use of separate moment expansions for the negative and positive charge distributions corresponds implicitly to a higher-order moment expansion about any one point. For example, just the zeroeth moments of the separate charge distributions correspond to the first moment (dipole) of the composite distribution. Hexadecapoles were evaluated and their effects on shifts were small, amounting to only 2 cm-l in HCN-HF. Thus, our results indicate a convergence in the moment expansion for the vibrational shifts when analyzed in terms of a linear potential.
vnt
Prediction and Interpretation of Vibrational Frequency Shifts Six HF stretching frequency shifts have been computed for the complexes (HF),, OC-HF, NN-HF, HCCH-HF, and HCNHF. For each complex, the monomer perturbing the HF was treated as a polarizable charge distribution possessing permanent moments up to the octupole, a dipole polarizability and hyperpolarizability, dipole-quadrupole and quadrupole-quadrupole polarizabilities, and dipole-dipole-quadrupole hyperpolarizability. The HF monomer was also treated as a polarizable charge distribution with the same list of tensor properties. The HF and the perturbing monomer were placed at equilibrium mass center separations and orientations. The choice of these structures and the dependence of the results on the structural parameters is discussed later. The classical, infinite-order electrical interaction was then computed by using first one set of H F properties in Table IV and then the other set. From P t ( r = 0.9068) and vnt(r= 0.9268) so calculated, the interaction slope, s, was obtained for each complex. Then, first-order perturbation theory was used to relate s to the vibrational frequency shift by using eq 2. Table V lists the calculated frequency shifts. The agreement with spectroscopic determinations of the shifts is very good, which is remarkable given the calculational and conceptual simplicity of our model. To understand how the shifts arise, Table V decomposes the calculated shifts into contributing factors. This is possible to do, in part, because at first order perturbation theory additive contributions to vnt(r)yield additive contributions to the shifts. For induced or polarization effects, the breakdown in Table V has been accomplished by repeating the infinite-order electrical interaction analysis but with selected polarizabilities. Interactions of permanent dipole and quadrupole moments are always important, but the octupoles also contribute. We have also tested the effects of hexadecapoles, and as already mentioned they are unimportant. The permanent moments alone, though, do not satisfactorily account for the shifts, since the induced effects play an important role. Correlations of shifts with dipole plarizabilities or induced dipoles,60 for instance, have not proven too successful except
50
I
I
150
250
1
Observed HF Frequency Shift (cm-')
Figure 2. Plot of observed HF frequency shifts against electrical model predicted shifts. The dashed line is the line of perfect correspondence. The shaded region represents the range of deviation from this line. perhaps for a series of three quite similar complexes.61 We see here that it is the wole nature of the electrical effects that correlate with the shift. Figure 2 contrasts the electrical model of the vibrational frequency shifts with observed values. It shows a good correspondence but with the model in error by undervaluing more than overvaluing. Part of this is due to the neglect of torsional vibrational coupling and other couplings which, if included, would tend to increase the vibrational shift. For the systems where the perturbation of the HF vibrations are strongest, the greatest effects of coupling are expected, and so the range of error tends to increase with the frequency shift. Additionally, the use of a linear interaction potential becomes less appropriate with increasing intermolecular interaction, just as the whole electrical model does. Nonetheless, the range of values in Figure 2 is not too far removed from the line of perfect correspondence. On the other hand, no such correspondence is achieved with the electrostatic, permanent moment interaction alone, as Figure 3 demonstrates. The electrical picture involves point multipole moments of each monomer interacting. The vibrational frequency shifts are therefore dependent on the choice of the multipole moment centers, which is simply determined from the geometry of the complex. Used to obtain the values in Table V were dimer (60) Nelson, D.D.;Fraser, G. T.; Klemperer, W. J . Chem. Phys. 1985, 82,4483. (61)Kolenbrander, K. D.;Lisy, J. M.,submitted to J . Chem. Phys. (62) Legon, A. C.; Soper, P. D.; Keenan, M. R.; Minton, T. K.; Balle, T. J.; Flygare, W. . J . Chem. Phys. 1980, 73, 583. (63)Soper, P.D.;Legon, A. C.; Read, W. G.; Flygare, W. H. J . Chem. Phys. 1982, 76, 292. (64) Legon, A.C.; Millen, D. J. Rogers, S.C. Proc. R. SOC.London, Ser. A 1980, 370, 213. (65) Read, W.G.;Flygare, W. H. J . Chem. Phys. 1982, 76, 2238. (66) Read, W.G.;Campbell, E. J. J . Chem. Phys. 1983, 78, 6515.
3102
Liu and Dykstra
The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 I
I
TABLE VII: Correlation Effects on Electrostatic Interactions and Vibrational Shifts
electrostatic (permanent moment") contribution to freq shifts, cm-' (HF), NN-HF (HF),
OC-HF HCCH-HF
0
0
0
e
HCN-HF
I 150
50
I
250
Observed H F Frequency Shift (cm-1)
Figure 3. Plot of observed H F frequency shifts against shifts predicted
with only electrostatic interaction. The open circles are from truncating the electrostatic interaction to dipole and quadrupole moment interactions. The closed circles include octupoles and hexadecapoles in the interactions. TABLE VI: Structural Parameters of the Complexes
measd vibrationally av Darameters'
equilibrium parameters (adiusted av values)'
RN-F = 3.082b RC-F = 3.062' RN-F = 2.796d Rill+ = 3.121'
3.030 3.004 2.765 3.100 RFF = 2.1675' 0, = 120.08O 9 2 = -6.42'
"Bond lengths are in angstroms. bReference 63. CReference66. dReference 64. eReference 65. 'Reference 39. that correspond to vibrationally averaged separations, then adjusted to equilibrium values on the basis of a pseudo-diatomic analysis of the intermolecular ~ t r e t c h i n g . Table ~ ~ VI gives these structural parameters. For the HF dimer, vibrationally averaged intermolecular separation values from an ab initio potential surface39agree very well with spectroscopic and the a b initio equilibrium structure39was used. The strongest dependence of the shift on the intermolecular separation distance was in the T-shaped HCCH-HF complex where a 0.01-A change in the separation produces about a 4-cm-' change in the shift. For the other complexes, there was less sensitivity. The dependence of the shifts on the intermolecular separation represents a coupling between the intermolecular stretches and other vibrations, and again, neglecting that coupling is part of the cause of the discrepancy between the model and observed values. Buckingham and Fowler38reported the success of just dipole and quadrupole interactions in explaining orientations for a series of dimers. Baiochi, Reiher, and Klemperer took an opposing view,O ' arguing in part that success of a moment interaction picture cannot be claimed unless the moments are well determined. To be certain that our moments are well determined in our analysis of vibrational shifts, we have used moments obtained from well-correlated wave functions, the details of which are given later. At this level, the agreement with experimental values where (67) Benzel, M. A.; Dykstra, C. E. J . Chem. Phys. 1983, 78, 4052. (68) Howard, B. J.; Dyke, T. R.; Klemperer, W. J . Chem. Phys. 1984,81, 5417. (69) Gutowsky, H. S . ; Chuang, C.; Keen, J. D.; Klots, T. D.; Emilsson, T. J . Chem. Phys. 1985,83, 2070. (70) Baiocchi, F. A,; Reiher, W.; Klemperer, W. J . Chem. Phys. 1983, 79, 6428.
ACCD
SCF
shifts, cm-l
27 23 68 71 107 116
36 24 84 68 136 114
-9 -1 -16 3 -29 2
0
0
comDlex
complex HF-HF (free) N,-HF HF-HF (bonded) OC-HF HCCH-HF HCN-HF
Includes through octupole moments.
0
0
N>-HF OC-HF HCN-HF HCCH-HF HF-HF
net correlation effect on freq
TABLE VIII: Predicted Overtone Shifts and Bond Length Changes' complex AE,,, cm-' AEIW2, cm-' Are, A HF-HF (free) 65 33 0.0021 0.0028 NZ-HF 87 44 0.0062 HF-HF (bonded) 189 96 OC-HF 240 122 0.0079 HCCH-HF 341 173 0.0114 0.0174 HCN-HF 509 259
Shifts were calculated with second-order perturbation theory using the linear interaction potential yielding the experimentally measured hEbl (see Table V). possible is to a few percent.49 The error is likely to increase in going from dipole to quadrupole to octupole moments because of basis set effects, but a countervailing effect is the diminishing importance in the frequency shifts of the higher moments. A comparison of the electrostatic interaction effects from SCF and correlated moments is given in Table VII. Quantitative values for the vibrational shifts do require good, correlated moments, but SCF values at least offer a reasonable picture.
-
Further Vibrational Effects of Hydrogen Bonding
-
Table VI11 gives frequency shifts for the v = 0 1 transition and also for the u = 1 2 transition. To within 5%, the calculated u = 1 2 shifts are the same as the u = 0 1 shifts. This is the consequence of using a linear v n t ( r )and the intrinsic anharmonicity of the unperturbed potential being well represented by eq 3. There do not appear to be experimental results yet available to verify this prediction for the u = 1 2 frequencies. Overtone transition ( 0 = 0 2 ) spectra would provide very important information on the interaction potentials. Another effect of vnt(r)is to change the r, of the monomer in the complex. Again, this is very dependent on the intrinsic anharmonicity of the unperturbed stretching potential. Predicted values for the changes in re's are in Table VIII. The changes in re values will tend to correlate well with the vibrational shifts because both depend on the anharmonicity and the interaction slope.
-
-
-
Computation Approach
The HF Potential Energy Curve. The stretching potential curve of isolated HF was obtained as a set of 16 points ranging from
0.7368 to 1.2168 A. The molecular electronic energy was calculated at each point with extensive incorporation of electron correlation at the ACCD (approximate double substitution coupled cluster) level. The coupled cluster a p p r o a ~ h ~ is ~ -a' ~size-extensive72.73 procedure that brings in higher-order electron correlation (71) Cizek, J. J . Chem. Phys. 1966,45,4256. Ado. Chem. Phys. 1969, 14, 25. (72) Pople, J. A,; Krishnan, R.; Schlegel, H. B.; Binkley, J. S . Int. J . Quantum Chem. 1978, 14, 545. (73) Bartlett, R. J. Annu. Reo. Phys. Chem. 1981, 32, 359. (74) Chiles, R. A,; Dykstra, C. E. J . Chem. Phys. 1981, 7 4 , 4544. (75) Bartlett, R. J.; Paldus, J.; Dykstra, C. E. In Aduanced Theories and Computational Approaches to the Electronic Structure of Molecules, Dykstra. C. E. Ed.; Reidel; Dordrecht, Holland, 1984.
H-Bonding Vibrational Frequency Shifts
The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 3103 can be quicdy evaluated, and D H F employs the Nee, Parr, and Bartlett derivations3 fot. this purpose. Basis set effects can be critical in polarizability evaluation^.^^^^^^^ Large basis sets were used for the polarizability D H F calculations, and basis set specifications are given in Table IX. These basis sets have been tested48~82~s8~s9 and have been found to yield polarizabilities to around 10% and better, and hyperpolarizabilities that are within 20-30% typically of basis set limits. Correlation effects on moments were accounted for by obtaining the moments from correlated wave functions, as described above. Correlation effects in the polarizabilities and hyperpolarizabilities were not included, but for a number of covalent molecules the refinement in S C F determined dipole polarizabilities from including correlation effects is 10%-15%.9~84~w2 Test calculations were performed where polarizabilities were scaled to be around 15% larger or smaller. The effects on the vibrational shifts arising from induced moments were found to change by a corresponding factor, which means the shifts were changed only slightly.
TABLE Ix: Basis Sets for DHF Evaluation of Molecular Polarizabilities and Hywrpohrizabilities
atomic center
Ha
valence set (~s/~s)c
basis augmentation: exponents of uncontracted functions S P d 0.06
0.008 Hb
(6s/3s)‘
0.06
C
(IOs6p/6~3p)~+’
0.05
N
(10s6p/6~3p)~+’
0.06
F
(IOs6p/6~3p)~*” 0.06
0.9 0.12 0.015 0.9 0.1 0.03 0.005 0.04 0.006 0.06
0.008 0.001
0.15
-
0.9 0.13 0.02 0.9 0.13 0.02 0.9 0.13 0.02
OHydrogen basis for HF. bHydrogen basis for HCN and HCCH. ‘Reference 93. dReference 94.
effects. With double substitutions in the exponential CC generator, quadruple, hextuple and further order substitutions enter the correlated wave function with expansion coefficients that are products of the expansion coefficients of the doubly substituted configurations. ACCD takes advantage of a near cancellation of small Hamiltonian matrix terms7678 and thereby achieves especially good computational efficiency. The extensive incorporation of correlation effects at the ACCD level has been shown to give very accurate potential surfaces for vibrational analysis,39*79.80 and the results in Table I reinforce that. The u = 0 1 transition frequency calculated agrees with the measured value47 to better than 0.4%. The basis set used was a (12s7p3d/8s5p3d) contraction on fluorine and a (8s3p/6s3p) contraction on hydr~gen.~~ The poteiltial curve points were least-squares fitted to an eighth-orer polynomial in the bond length. This polynomial was used in the N u m e r o v - C o o l ~ y ~vibrational ~-~~ analysis. The numerical integration for the vibrational analysis was with a step size of less than O.OOO1 A. The exact vibrational analyses carried out with the addition of p t ( r ) were done by adding Vntat each of the 16 potential curve points and refitting to an eighth-order polynomial. Monomer Electrical Properties. Electrical. polarizabilities and hyperpolarizabilities were obtained analytically with the derivative HartreeFock (DHF) approach.81v82 D H F is a uniform procedure for solvihg the derivative Hartree-Fock equations order by order. The derivatives of the Hamiltonian operator that are needed are just the moment component operators. From the nth derivative of the wave function, the 2n 1 order derivatives of the energy
-
+
(76) Chiles, R.A.; Dykstra, C. E. Chem. Phys. Lett. 1981, 80, 69. (77) Jankowski, K.; Paldus, J. Int. J . Quantum Chem. 1980, 28, 1243. (78) Paldus, J.; Cizek, J.; Takahashi, M. Phys. Rev. A 1984, 30, 2193. (79) Bachrach, S.M.; Chiles, R.A.; Dykstra, C. E. J . Chem. Phys. 1981, 75, 2270. (80) Dykstra, C. E.; Secrest, D. J . Chem. Phys. 1981, 75, 3967. (81) Dykstra, C. E.; Jasien, P. G. Chem. Phys. Lett. 1984, 109, 388. (82) Dykstra, C. E. J . Chem. Phys. 1985, 82, 4120.
Conclusions The changes in the vibrational frequency of H F when complexed with a variety of other monomers are primarily a consequence of (i) the mutual, linearly varying part of the electrical interactions of the species in the complex, and (ii) the intrinsic anharmonicity of the HF potential curve. The electrical interaction is easily determined from monomer electrical properties, and so it is possible to directly relate the vibrational frequency shifts to monomer properties. The physical picture is that of a vibrational potential affected by the static and polarization interaction with proximate molecular charge fields. Of course, the polarization of charge can be viewed in alternate ways, such as a transfer of charge or some kind of orbital mixing. The value of the electrical picture is that it uses calculated or measured properties of the constituent species, not the complexes. This is expected to provide a basis for understanding frequency shifts in larger clusters. Acknowledgment. The support of the National Science Foundation’s Chemical Physics Program through Grant CHE84- 19496 is gratefully acknowledged. Discussions with several colleagues were truly motivating, and the earlier work of former graduate students a t Illinois was also important in this development. For this, C.E.D. thanks Drs. M. A. Benzel, L. W. Buxton, and W. G. Read, and Professors L. Andrews, W. H. Flygare, H. S. Gutowsky, J. M. Lisy, and D. J, Malik. The ACCD and D H F calculations were carried out on VAX 11/780 and FPS-164 computers in the School of Chemical Sciences. Registry NO. HF, 7664-39-3; N2,7727-37-9; CO, 630-08-0; HCCH, 74-86-2; HCN, 74-90-8. (83) Nee, T.-S.; Parr, R.G.; Bartlett, P. J. J. Chem. Phys. 1976,64,2216. (84) Werner, H.-J.; Meyer, W. Mol. Phys. 1976, 31, 855. Sadlej, A. J. Chem. Phys. Lett. 1982,86, (85) Karlstrijm, G.; Roos, B. 0.; 374. (86) Christiansen,P. A.; McCullough, E. A. Chem. Phys. Lett. 1978, 55, 439. (87) Bishop, D. M.; Marwulis, G. J . Chem. Phys. 1985,82, 2380. ( 8 8 ) Liu, S.-Y.; Dykstra, C. E. Chem. Phys. Lett. 1985, 119, 407. (89) Dykstra, C. E.; Liu, S.-Y.;Malik, D. J. J. Mol. Srruct. THEOCHEM 1982, 374. (90) Amos,R. D. Chem. Phys. Lett. 1980, 70, 613. 1982, 88, 89. Mol. Phys. 1980, 39, 1. (91) Bartlett, R. J.; Purvis, G. D. Phys. Reu. A 1979, 20, 1313. (92) Morrison, M. A,; Hay, P. J. J . Chem. Phys. 1979, 70, 4034. (93) Huzinaga, S.J . Chem. Phys. 1965, 42, 1293. (94) Dunning, T. H. J . Chem. Phys. 1971,55, 716.
