AB initio studies of the structure, energetics and vibrational spectra of hydrogen bonded systems

Journal of Molecular Structure (Theochem), 202 (1989) 177-192 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

177

AB INITIO STUDIES OF THE STRUCTURE, ENERGETICS AND VIBRATIONAL SPECTRA OF HYDROGEN BONDED SYSTEMS

STEVE SCHEINER Department of Chemistry and Biochemistry, Southern Illinois University, Carbondale, IL 62901-4409 (U.S.A.) (Received 17 October 1988)

ABSTRACT The accuracy of various levels of theory in calculating the properties of hydrogen bonded systems is assessed. An adequate treatment of both electron correlation and basis set superposition error is generally required for quantitative estimates of the equilibrium intermolecular distance and interaction energy. Polarized double-c basis sets can provide useful information about the perturbations caused by hydrogen bond formation upon the vibrational frequencies and intensities of the individual monomers. The uniqueness of the hydrogen bond is underscored by detailed comparison with a lithium bonded complex. In particular, the calculations provide insights into the origin of the very different behavior of the v, stretch in the two different systems. The cooperativity involved in a number of sequential hydrogen bonds is examined using (HF), and (HCl), as model systems. The internal stretching frequencies exhibit an especially large change on going from dimer to trimer.

INTRODUCTION

A flurry of theoretical activity in the 1960s and 1970s added a great deal to our knowledge of the fundamental character of the hydrogen bond [l-5]. This work dissected the total interaction energy into a number of physically meaningful phenomena such as electrostatic and polarization energies. Insights were provided into the various forces that combine to produce the equilibrium geometry of a given complex. However, computational limitations of the time imposed a cap on the type of information that could be gathered and the level of certainty with which some questions could be answered. For example, small basis sets are known to lead to incorrect estimates of the various contributions to the interaction energy. If not carefully removed, basis set superposition error can further distort the physical picture of the hydrogen bond. The impracticality of including electron correlation in the earlier calculations completely overlooked any role played by dispersion. More recent advances in computer technology and quantum chemical programming have spurred a fresh re-examination of the matter. It is now prac-

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tical to apply fairly large basis sets that provide a well-balancedtreatmentof each component; techniquesof incorporatingcorrelationhave become almost routine. Moreover, gradienttechniques now permit rapid and reliable identification of equilibriumstructures,in contrastto the earlierwork wherecertain assumptionsof symmetrywere commonly imposed and many of the geometrical parameterswere fixed at arbitraryvalues. Accompanying these theoretical advances has been the recent progress in experimentaltechniques for studying hydrogenbonded complexes in the gas phase [ 5-101. Geometries have been determined with unprecedentedaccuracy; spe~roscopic me~uremen~ have provided valuable info~ation about the dynamics of these systems as well. The latter spectra have motivated a number of theoretical calculations which have been greatlyfacilitated by recent additionsto the computer codes. The present paper summarizesthe results of calculationsaimed at an improved understandingof the fundamentalnature of the hydrogenbond. The next sectiondiscussesa particularexample:the resultsfor Hz0 *HCl illustrate some of the more recent findings and the possible pitfalls that may trap an unwaryresearcher.The vibrationalspectrumis evaluatedwithinthe harmonic approximationand the effects of going beyond this approximationdiscussed. The unique character of the hydrogen bond is the subject of a later section which contains a detailed comparison with lithium bonding, with regard to vibrational spectra in addition to geometry and energetics.The final section delvesinto the perturbationof one hydrogenbond by anotherwhen more than one is present in a given system. l

The Hz00 *HClcomplex was studied [ 11]using a doubly polarizedbasis set developed especiallyfor treatmentof molecularinteractions [ 121. Beginning with the standard 6-31G basis set, the scaling factor of each orbital on every atom was optimized at the SCF level within the context of the individualsubunits, namely Hz0 and HCL.This valence set was supplemen~d by a diffuse sp-shell and two sets of d-functions on 0 and Cl in addition to a single set of p-functions on hydrogen; all orbital exponents were optimized, again within the H,O and HCl molecules.The resultingbasis set, designated + VPs( 2d)S [ 121,has been found to produce dramaticreductionsin basis set superposition error,relativeto sets whereorbital exponentsare optimizedwithinthe context of the isolated atoms. Structure and energetics

Various salient featuresof the optimized geometryof H,O* lHCl are listed in Table 1.The first row revealsthat the equilibrium0. *Cl separationis cal-

179 TABLE 1 Structural and energetic features of HzO*. HCI complex’ SCF Uncorrected

R(O..Cl) (A) 8(OClH) (degrees) (Ydegreesb r(HC1) (A) dr(HC1)” (A) - AE, kcal mol-’

3.369 3.2 138.7 1.275 0.008 3.97

“Calculated with +VPS(2d)’ ‘r(HCl),,L,-r(HCl),,,,,,,.

MP2 Uncorrected

Corrected

3.235 3.5 131.3 1.286 0.012 5.42d

3.281 3.4 133.2 1.289 0.015 4.74

basis set. bAngle between HOH bisector and dCalculat.edusing corrected MP2 geometry.

0.*Cl axis.

culated to be 3.369 A at the self-consistent field (SCF) level. This distance shortens by 0.134 A to 3.235 A when correlation is included via the secondorder Meller-Plesset perturbation theory approach (MP2) [ 131. Both of the above values are contaminated by basis set superposition error (BSSE) which tends to artificially shorten intermolecular distances. When the BSSE is corrected by the Boys-Bemardi counterpoise technique [ 141, the MP2 equilibrium 0. *Cl distance stretches a small amount to a final value of 3.281 A, as indicated in the-last column of Table 1, in comparison to an experimental estimate of 3.21 A [ 15 1. The 0.07 A discrepancy conforms to expectations concerning the fundamental distinction between R, and R,, remaining deficiencies of the basis set, and truncation of the perturbation expansion [ 111. As indicated by the next row of Table 1, the hydrogen bond is very nearly linear in H,O* -HCl, with the bridging proton deviating from the 0. *Cl axis by only 3” or so. An interesting aspect of the equilibrium geometry concerns the orientation of the H20 subunit. At the SCF level, the HOH bisector makes an angle of 139” with the O.-Cl axis. As discussed in earlier work [ 16-181, this results from a compromise between various factors such as the dipoledipole and dipole-quadrupole terms of the electrostatic interaction energy. Correlation lowers this angle by only a small amount, with (Yremaining in the 130-140 ’ range. While analysis of the experimental data suggests that c~= 180 ‘, this corresponds to a vibrationally averaged value. The observed average of 180” is due to the flatness of the inversion potential such that the ground vibrational state is near the top of the interconversion barrier (which is less than 0.5 kcal mol-‘) [16]. F (HCl) refers to the equilibrium bond length of the HCl subunit. It may be noted that correlation tends to elongate this bond. The stretch induced in the bond as a result of formation of a complex with H20 is presented as dr (HCl) in Table 1. At the SCF level, the HCl molecule is 0.008 A longer as part of the

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complex than it is as an isolated molecule. This stretch is increasedby correlation to greaterthan 0.01 A. The last row of Table 1 contains the interaction energy, calculated as the differencein energybetween the complex on the one hand and the sum of the two isolated (optimized) monomerson the other. The SCF interactionenergy is about 4.0 kcal mol-‘. This value is nearly free of superpositionerror (the SCF BSSE is 0.09 kcal mol-l) due to the natureof the + VPS(2cQSbasis set [ 121. The correlatedcomplexation energy is significantlyhigher at 5.42 kcal mol-‘. However, removal of the full (SCF+MP2) BSSE reduces the interaction energyto 4.74 kcal mol-‘; 1.00 kcal mol-l, or 21% of this total, is due to correlation effects. This quantity representsthe purely electronic component of the full interaction. Addition of zero-point vibrationalterms reduces the interactionby some 1.4 kcal mol-l, leading to a d.E (0 K) of -3.30 kcal mol-‘. dH (298 K), evaluatedby considerationof translationaland rotational terms, a vibrationalcorrection and ApV, comes to - 3.96 kcal mol-’ [ 111. Several points concerning the above results are worth emphasizing.SCF results,by their very nature, do not include attractive dispersionforces. The correlationcontributionto the interactiondoes includea certainportion of the dispersionenergy (in addition to a number of other terms). A previous work has suggesteda direct link between the H-Cl bond stretchpointed out earlier and the magnitude of the dispersion energy [ 161. The overall effect is to strengthenthe hydrogenbond, also contractingthe distancebetween the two subunitsand increasingthe H-Cl stretchassociatedwith formationof the bond. Restrictingthe calculationsto the SCF levelwouldneglecttheseeffectsentirely. Nonetheless,there are a number of cases in the literaturewhere SCF-level computationshaveproducedhydrogenbond geometriesand energeticsin quite reasonablecoincidence with experiment.This agreementis undoubtedlydue in large measureto the influence of basis set superposition.The artificial attraction associatedwith this phenomenon, if not corrected,can partiallycompensate for the failure to include correlation. However, one certainly cannot expect the cancellation to be quantitativein the generalcase. In H,O- *HCl, for example,uncorrectedSCF calculationsunderestimatethe corrected MP2 interactionenergyby 16%. This caution is particularlycrucial if one is interested in extendedregionsof configurationspace in additionto the equilibrium geometry since BSSE and dispersion have very different functional forms. Moreover, and perhaps most importantfrom a theoreticalperspective,failure to correct the superpositionerror or to includecorrelationis certainto provide a distortedpicture of the fundamentalnature of the hydrogenbond. Vibrational analysis

The frequenciesand intensities of all vibrational modes of the individual subunits H20 and HCl were calculated using the double-harmonicapproxi-

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mation at the SCF level. As is normally the case, the SCF frequencies were found to be somewhat larger than experimental values, in this case by between 4% and 8%. Absolute intensities tend to be rather more difficult to calculate accurately, having much more stringent requirements in terms of basis set flexibility and correlation. In this case, the calculated intensities are all somewhat stronger than experimental measurements although the degree of overestimation is not uniform from one mode to the next. Rather than attempting to reproduce the absolute frequencies and intensities, the calculations are better adapted to a study of perturbations caused by formation of a hydrogen bonded complex. Data of this sort are reported in Table 2 which contains the effects of complexation upon the intramolecular vibrational modes in Hz00 *HCl. Also included for purposes of comparison are corresponding data for the very similar H,O* lHF system, taken from recent calculations of Somasundram et al. [ 191who made use of a doubly polarized triple-c basis set. The first column of Table 2 indicates that the frequencies of the two bond stretching modes of HzO, V~and Y,, are diminished by several wave numbers when this molecule hydrogen bonds to HCl; the bending mode v, is virtually unaffected. These small changes conform to the earlier argon matrix spectrum of Ayers and Pullin [ 201. By far the largest perturbation occurs in the H-Cl stretch which undergoes a red shift of over 100 cm-‘. These changes are duplicated in the H,O. *HF complex except that they are all magnified significantly, consistent with the stronger hydrogen bond in this system. The changes in the vibrational intensities are reported in the last two columns of Table 2 as the ratio between the intensity in the complex and that of the unperturbed monomer. The two bond-stretching modes of H,O are strengthened while a very small intensity drop is observed in the bending mode. These perturbations are characteristic of H,O- lHF as well as H,O* .HCI; the effects are again somewhat magnified in the former complex. The largest change occurs in the H-X stretch which is intensified by a factor of five or six. (The TABLE 2 Effects of complexation upon intramolecular vibrational modes in HzO* .HX (X=Cl,

F)”

b

HzO. .HF Vl

Hz0

vz v3

HX

-4 +1 -3 - 105

-9 0 -10 - 264

‘Data for H,O* *HCl from ref 11 using + VPs(2d)sbasis bin cm-‘.

H,O* . HCI

H20. .HF

1.9 0.9 1.3 6.4

5.5 1.0 1.6 4.6

set; H,O..HF

fromref. 19 using TZ+SP.

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larger enhancement in the weaker H20- - HCl complex appears rather puzzling and warrants further scrutiny). Both the red shift and intensity magnification of the stretching mode of the proton donor are characteristic of hydrogen bonding, having been observed on numerous occasions. It is notable that the calculated frequency shift represents only a fraction of what is measured in low-temperature matrices. There are several possible reasons for this underestimate which include interactions with the matrix, need for electron correlation, and inaccuracies of the harmonic approximation [ 111. It would indeed be useful to quantitatively assess the relative contributions of each of these candidates. Table 3 lists the vibrational frequencies and intensities calculated for the intermolecular modes of the H20**HX complexes. It is important to stress that all frequencies of the stronger H20*-HF complex exceed the analogous values for H20* *HCl. Despite the expected errors arising from the use of the harmonic approximation for these rather weak interactions, the calculated frequencies are in reasonable coincidence with available experimental estimates. The two highest-frequency modes correspond to wags of the proton-donor HX molecule which tend to fairly high intensity. Mode number 4 is associated with the intermolecular stretch while the remaining two modes involve primarily wags of the proton-acceptor HzO. The very low sensitivity of the energy to the wagging of Hz0 suggests the harmonic approximation is a particularly poor one for this mode. Indeed, Legon and Millen [ 211 had previously measured the first overtone frequency to be some four times larger than the fundamental in the analogous H,O..HF system. The H20 wagging motion was probed further by a non-harmonic treatment which elucidates the ground and excited vibrational levels within the computed double-well potential [ 161. The agreement with experiment was quite good for both the fundamental and first overtone frequencies, particularly when a correlated potential was used. TABLE 3 Vibrational frequencies and intensities of H,O* *HX” A(km mol-‘)

v(cm-‘)

1 2 3 4 5

HzO- -HCl

H20--HF

H,O-*HCl

H20--HF

459 351 143 118 94

786 644 234 220 182

77 38 33 3 28

194 226 3 87 155

*Data for H,O- -HCl from ref. 11; H,O- -HF from ref. 19.

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Anharmonicity in this same H20**HF complex was also considered with regard to force constants up through fourth-order in both R (0. *F) and r (HF) [ 161. This approach of coupling the intra- and inter-molecular stretches has more recently been extended to an explanation of the rather anomalous band shape for the ys mode in MezO* -HF [ 221. The frequency and intensity patterns calculated in this manner bear a striking resemblance to the fundamental band of the observed spectrum although the agreement is not as good for the first overtone. COMPARISON OF HYDROGEN BONDING WITH LITHIUM BONDING

The hydrogen bond is frequently thought of as a unique type of interaction. Yet there are other atoms that could conceivably replace hydrogen as a bridge between two different molecules, at least in certain respects. The most obvious possibility for such a substitute is lithium. A systematic comparison of hydrogen bonds with lithium bonds was carried out for complexes composed of NH, on the one hand and HF, LiF, HCl, and LiCl on the other [ 231. Beginning with a 6-31G type of basis set, polarization functions were added to all atoms. In the case of the nitrogen, fluorine and chlorine atoms, two sets of d-functions were used; the bridging proton was also doubly polarized (with p-functions). Structure and energetics The entries in the first row of Table 4 indicate that lithium bonds tend to be considerably longer than hydrogen bonds. That is, the R(N- -F) distance is neatly 1.0 A longer in H3N**HF than in H,N* *LiF; the difference is roughly 0.8 A for R (N- *Cl). This distinction is not surprising since lithium contains a fully occupied core of 1s electrons. One interesting feature concerns the influTABLE 4 Calculated properties of hydrogen bonded and lithium bonded complexes H,N--HF

R(N..X! (A) r(AX) (A!, dr(AX)” (A) -AEb P(D) AP” (D)

HsN- *LiF

H,N--HCl

HsN- -LiCl

SCF

MP2

SCF

MP2

SCF

MP2

SCF

2.728 0.923 0.022 11.8 4.39 0.97

2.693 0.950 0.028 15.1

3.652 1.582 0.018 22.9 8.74 0.91

3.665 1.592 0.014 23.5

3.297 1.293 0.023 9.3 3.90 1.13

3.144 1.317 0.040 11.0

4.118 2.082 0.020 26.7 10.16 1.14

“A = I-KLi; Ar = rC’,,plex - r-,nom.l. bin kcal mol-‘, using geometry optimized at indicated level of theory (SCF or MP2 ) . cpc,,,plex -vector sum of dipoles of isolated monomers.

184

ence of electron correlation upon the calculated properties of the two complexes. Note that the MP2 lengths for both hydrogen bonds are shorter than their SCF counterparts whereas an opposite effect is noted in H3N* *LiF where the MP2 value of R(N* l F) is slightly (0.1 A) longer. Inspection of the second row of Table 4 reveals that in all cases, correlation increases the AX bond length (A= H, Li). However, of greater fundamental interest is the comparison of this bond with its length in the isolated monomer. The third row of the table, labeleddr (AX), contains the stretching of this bond which results from formation of the complex. This stretch is about 0.022 A for both hydrogen bonded complexes at the SCF level but increases substantially when correlation is included. At the MP2 level, the HF bond elongates by 0.028 A upon complexation with NH3 while the stretch of the HCl analogue is 0.040 A. Correlation produces an opposite effect on the lithium bond, diminishing the LiF stretch by 0.004 A. The interaction energies reported in the fourth row of Table 4 are considerably larger for the lithium bonded complexes than for their hydrogen bonded analogues. It is interesting also that whereas in the case of the hydrogen bonds, H,N* *HCl is less tightly bound than is H3N. *HF, the lithium bonds follow an opposite pattern wherein the interaction is stronger in H,N* lLiCl than in H3N* *LiF. Although correlation strengthens both types of interactions to some extent, it clearly plays a more important role in hydrogen bonds. The trends observed in the dipole moments of the various complexes are surprisingly similar to the interaction energies. The lithium bonded systems contain much higher moments than their hydrogen bonded analogues, larger by a factor of perhaps two or more. In the case of lithium bonds, substituting LiF by LiCl raises the moment while an equivalent replacement of HF by HCl leads to a reduction in the moment. The last row of Table 4 reports the enhancement of the dipole moment caused by formation of the complex. Specifically, it lists the difference between the moment calculated for the complex and that obtained by summing together in vector fashion the moments of the individual isolated monomers. The dipole moment enhancement does not seem to indicate any fundamental difference between hydrogen bonds and lithium bonds; one sees only a larger value for complexes containing AC1 as compared to AF. The results concerning the structure and energetics of these two types of bonds may be summarized as follows. Lithium bonds are substantially stronger than their hydrogen bonded counterparts. This greater strength is due in large part to the much higher dipole moments of the LiX monomers, as compared to HX. This higher moment adds to the electrostatic energy which makes up the bulk of the total interaction energy in lithium bonds. It is their largely electrostatic nature which seems to make lithium bonds fairly insensitive to correlation effects. In contrast, hydrogen bonds contain substantial contributions from other components such as dispersion energy. The latter term is

185

introduced by correlation effects and it is largely due to its influence that the hydrogen bond is significantly strengthened by electron correlation, i.e. increased interaction energy, shorter R (N. ax), larger r (HX) stretch. Vibrational spectra Workers in this laboratory have very recently extended the comparison of hydrogen and lithium bonds to the vibrational frequencies and intensities of H,N* *HCl and H,N* lLiCI [24]. These properties were computed using the double-harmonic approximation, within the context of the 6-31G** basis set, augmented by a second set of d-functions on chlorine. The changes induced in the internal vibrations of each monomer by formation of these complexes are reported in Table 5. This study computed the frequencies and intensities at both the SCF and correlated MP2 levels; the relevant data are supplied for each. The stretching frequencies of NH3 are affected very little by complexation with HCl whereas the intensities increase markedly. In contrast, formation of a lithium bond produces notable red shifts of these stretches as well as strong intensifications. The first bending frequency is substantially blue-shifted by hydrogen bonding; this shift is approximately doubled in the lithium bonded complex. Whereas the intensity of this mode is weakened in H,N* l HCl, the opposite effect of greater intensity is observed in H,N. *LiCl. A small red shift TABLE 5 Changes in frequency” and intensityb in subunits caused by complexation” (from ref. 24)

Stretch Stretch (2 ) Bend Bend(B)

SCF MP2 SCF MP2 SCF MP2 SCF MP2

H,N..HCl

HsN**LiCI

u

A

v

A

27 20 15 36 0.9 0.7 1.3 1.6

-13 -10 -26 -26 176 131 -12 -15

173 37 111 60 2.6 1.1 3.7 1.6

73 93

2.5 0.9

0

-3 0 -3 74 71 -7 -14 AC1

SCF MP2

-424 -898

9 50

‘Degeneracy of mode indicated in parentheses. ’ vcomp~sx - G,~,,~,~ (cm-’ 1.b&,mp~J~monomr.

186

occurs in the second bending mode of NH3 in either case, coupled to a small intensity enhancement. It is evident from comparison of successive rows of Table 5 that most of the frequency shifts are not significantly changed when electron correlation is included directly. A major exception to this rule is associated with the HCl stretch: the MP2 frequency shift is approximately twice that calculated at the SCF level. It is probably in the frequency shift of the HCl or LiCl subunits that the most dramatic distinction between hydrogen and lithium bonds appears. In contrast to the very strong red shift of the HCl stretch of at least several hundred wavenumbers, the LiCl stretch is shifted in the opposite direction, i.e. toward higher frequencies, and by a much lesser amount. This contrast is not limited to SCF level calculations but is confirmed by inclusion of electron correlation and agrees with prior experimental observations [ 251. A second major distinction concerns the intensification of this stretch upon formation of a complex. While the HCl stretch strengthens by a factor of perhaps 50 or so, there is little change calculated for the LiCl stretch, again conforming to experimental measurements [ 251. The intensity enhancements of the NH3 modes associated with formation of a hydrogen bond are affected to only a small degree when correlation is included. In the case of the HCl stretch, however, the MP2 intensification is several times larger than SCF calculations would suggest. In contrast to the lack of sensitivity of the intensities of the NH3 vibrations to correlation in H3N. *HCl, the MP2 intensity enhancements are generally a factor of two or so smaller than the SCF values in H3N* -LiCl. The calculated frequencies and intensities for the intermolecular modes are exhibited in Table 6. The higher frequencies obtained for the lithium bond stretch as compared to the hydrogen bond are consistent with the stronger TABLE 6 Frequency (cm-’ ) and intensity (km mol-’ ) of intermolecular modes in HBN- *ACl” (A = H, Li) (from ref. 24) HBN- - HCl

Stretch Bend(B) Bend(a)

SCF MP2 SCF MP2 SCF MP2

H,N- -LiCl

V

A

V

A

173 200 239 276 700 912

10 43 46 44 150 114

248 238 525 499 56 70

24 10 550 208 80 22

*Degeneracy of mode indicated in parentheses; intensities have been premultiplied by this factor.

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nature of the former interaction. Note also that correlation induces a significant increase in the hydrogen bond stretching frequency, just as the MP2 interaction energy is greater than the SCF value; the lithium bond parameters are relatively unaffected by correlation. Another indication of the stronger character of the lithium bond is the larger frequencies for the first bending mode in Table 6, corresponding to a bend of the NH, molecule. The pattern of SCF vs. MP2 frequencies for this bend is also reminiscent of the trends for the hydrogen bond stretch. The final bending motion refers to the HCl or LiCl molecule. The frequencies in the two systems are vastly different, due in large part to the much larger mass of lithium as compared to hydrogen. The intensity of the stretching mode appears to be fairly low for both systems. Correlation has a strong damping effect on all intensities for H3N- -LiCl whereas SCF and MP2 intensities are not drastically different for the hydrogen bonded system (with the exception of the stretch where the SCF value is magnified by a factor of four). Atomic polar tensors were used to provide insights into the reasons for the above observations concerning the intensities [ 241. The strong intensification of the HCl stretch is due in large part to the fact that the hydrogen atom becomes more positively charged as it moves away from the chlorine, due to the reduced ability of the electron density to move along with it once the hydrogen bonded complex has been formed. This effect is magnified by correlation effects, explaining the greater intensification of the HCl stretch at the MP2 level. The situation for the lithium bond is quite different. Considering first the isolated LiCl monomer, this molecule is highly polar with lithium bearing nearly a full positive charge. A stretch of the bond further increases this charge, which translates into a fairly sizeable charge flux term already in the monomer. Complexation with NH3 increases the charge flux, but to only a fraction of the amount observed in HCl. Moreover, some of the electron density extracted from the NH3 winds up on the lithium atom, lowering its positive charge. The latter diminished charge produces a damping effect upon the intensity of the LiCl stretch, counteracting any increase arising from the charge flux; the net result is that complexation leads to no substantial change in intensity. Another fundamental distinction between hydrogen and lithium bonds is associated with the charge distributions occasioned by formation of the complex. These redistributions were explored by way of “spectroscopic” atomic charges which act to mimic experimental quantities such as vibrational intensities [ 241. It was noted that whereas most of the charge extracted from the ammonia in H,N* lHCl was picked up by the chlorine atom, it is the lithium atom in HsN- -LiCl that is the ultimate sink of electron density. A relationship was also discerned between the total charge transferred from NH, to the electron acceptor and the calculated intensity of the intermolecular stretch. This finding conforms to the requirement of a changing molecular dipole moment to lend intensity to this vibration.

188

COOPERATIVITY

The formation of a hydrogen bond produces shifts in electron density in both molecules involved. Taking the XH* lXH interaction as an example, some density is transferred from the proton acceptor molecule to the donor. In addition, an internal polarization within the proton donor further adds to the electron density build-up on the X atom. The increased density on this X atom facilitates a transfer of charge to a third molecule. That is, the proton donor molecule of a dimer is better equipped to act as proton acceptor in another hydrogen bond than is the same molecule if it is not involved in a hydrogen bond [ 41. This cooperative effect leads to the chains of sequential hydrogen bonds that are commonly observed in aggregate phases in addition to a number of other phenomena such as the shorter hydrogen bond lengths in ice as compared to an isolated water dimer. Structure and energetics Cooperativity was examined by comparison of the monomer, dimer and trimer of hydrogen halide molecules [ 261. A polarized double-c basis set ( + VP’) was used; the exponents of this set have been previously optimized within the context of the HX molecules. The salient results for (HF), and (HCl), are summarized in Table 7. The first line of data reveals that formation of the dimer leads to a greater stretch of the H-F bond in the proton donor than in the acceptor (0.004 A as compared to 0.002 A). The R(FF) hydrogen bond length is computed to be 2.819 A at the SCF value and slightly shorter, 2.786 A, at the MP2 level. The hydrogen bond energy is 4.0 kcal mol-‘, with correlation adding only 0.1 kcal mol-l. The geometry of the HF dimer contains a nearly linear hydrogen bond in that the bridging proton lies within 10” of the F- SFaxis. The acceptor molecule is oriented at an angle of about 120” from this axis. In contrast, the three F atoms in (HF), form an equilateral triangle in this trimer with overall C,, symmetry. Each of the HF molecules acts as proton donor and acceptor simultaneously. The three protons lie about 27” from the corresponding F--F axis. As may be seen in Table 7, the H-F bonds are considerably longer in ( HF)3 than in the dimer. In addition, the R (FF ) distances have contracted by some 0.1 A relative to their values in (HF),. The binding energy of (HF), (i.e. relative to three isolated HF molecules) is about 12 kcal mol-‘. In the absence of any cooperativity effects, the binding energy of three hydrogen bonds within the context of the trimer might be expected to approximate three times the energy of a single bond in a dimer. The difference between these two quantities provides some measure of the cooperativity and is reported as Scoopin Table 8 where it may be seen to be quite small for (HF ) 3.

189 TABLE 7 Geometric and energetic aspect@ of (HX),

r(HF)b (A, n

Donor

2 3

0.906

- AE’ (kcal mol-‘)

R(FF) (A) Acceptor

0.904 0.911

SCF

MP2

SCF

MP2

2.819 2.716

2.788 2.695

4.01 11.80

4.10 12.51

- AE’ (kcal mol-‘)

r(HCl)d (A,

r(ClC1) (A)

n

Donor

Acceptor

SCF

MP2

2 3

1.270

1.270

4.119 4.135

3.859

1.271

SCF 0.59 2.92

MP2 1.05 3.62

“Computed with + VP’ basis set. br (HF) = 0.902 A in monomer (SCF ) . ‘All energies are corrected for basis set superposition error; for trimer, AE(HX),=E(HX),-3E(HX). dr(HC1) ~1.269 A in monomer (SCF ) . TABLE 8 Measures of cooperativity (kcal mol-’ ) in trimers of HF and HCl

U-W,

E coopa ERbCb

(HCl)s

SCF

MP2

SCF

MP2

0.23 - 1.66

-0.21 -1.73

-1.15 -0.23

-0.47 -0.22

eometries of monomer, dimer and trimer optimized individually. bThree-body non-additivity same as E_ except that geometries of all fragments are taken from the optimized structure of the trimer.

“E,,=AE(HX),-3[E(HX),-2E(HX)];g

However, there is in fact a significant amount of cooperativity in this trimer which is masked by a second effect. It must be noted that to form a triangular trimer, each hydrogen bond is significantly distorted from its optimized geometry in the dimer, e.g. the 27” deviation of the bridging proton from the F--F axis. This distortion energy competes with and approximately cancels the cooperative effect of three sequential hydrogen bonds, leading to the near zero values of E,, in Table 8. One may circumvent this second complicating factor by defining a three-

190

body non-additivity, Eabo which evaluates all interaction energies within the context of a single geometry, that of the trimer. In this prescription, any distortion energies imposed by the structural constraints of the trimer occur in the dimer as well. The values listed for EabCin Table 8 are negative, indicating that the mutual polarizations of the three monomers add to the overall stability of the trimer. Computations of a similar nature on the HCl analogues suggest a much smaller degree of cooperativity, consistent with the weaker interaction. The binding energy of the dimer is of the order of 1 kcal mol-‘, about half of which is due to correlation. The HCl bond lengths in the monomer, dimer and trimer are all within 0.002 A of one another; R(ClC1) is actually slightly longer in (HCl), than in the dimer. Ecoopis considerably more negative for (HCl), than for ( HF)3. However, this result does not necessarily indicate greater cooperativity, but rather a lesser amount of hydrogen bond distortion energy, as is evident by the rather small three-body energies for (HCl), in the last row of Table 8. It is perhaps important to note finally that the SCF and MP2 values of Eabc are virtually identical, indicating that the essential features of cooperativity are treated adequately at the SCF level. Vibrational spectra

The effects of dimerization and trimerization upon the H-X stretching vibration are displayed in Table 9. Upon formation of the dimer, the frequency of both molecules is decreased, that of the proton donor more than that of the acceptor by a factor between two and three. These red shifts are nearly an order of magnitude larger for the HF dimer as compared to (HCl ),. The intensity of the proton donor stretch is substantially magnified while that of the acceptor shows only a small increase. The H-X stretches are highly coupled in the C,, trimers. The symmetric TABLE 9 Changes in H-X stretching frequency and intensity caused by dimerization and trimerization” X=F n 2 3

Proton donor Proton acceptor Symmetric Antisymmetric”

X=Cl

AUb

A,/A,m,

A vb

&/km,

-84 -36 -309 -167

2.6 1.0 0 3.2

-13 -5 -21 -21

2.0 1.3 0 3.1

“Calculated with + VP’ basis set at SCF level. bv [ (HF) .] - v [ HF] in cm-‘. ‘Doubly degenerate; intensity ratio not multiplied by factor of two.

191 TABLE 10 Frequencies of intermolecular modes* in (HX),

O-W,

Stretch Out-of-plane bend In-plane bend

U-ICI),

n=2

n=3

140 436 520

161 415 410

*Calculated with + VP’basis

222 628 143

n=2

n=3

44 147 233

45 140 182

52 203 336

set at SCF level in cm-‘.

mode involves a simultaneous stretch of all three H-X bonds; its a’ symmetry causes the predicted IR intensity to vanish. Most remarkable about this vibration is its very large red shift. In the case of (HF), the symmetric stretching frequency is more than 300 cm- ’ lower than that of the monomer, representing a shift of almost four times that observed for the proton donor in the dimer. Although not quite as dramatic, the red shift of the antisymmetric stretch in the trimer is likewise considerably larger than in the dimer. It is also notable that the intensity of this mode is enhanced relative to (HF),. The situation for the HCl analogs is quite similar in that the red shifts of the trimer frequencies are considerably greater than the dimer results, as is the intensity of the antisymmetric stretch. The calculated intermolecular frequencies of the dimers and trimers are presented in Table 10. Like the internal H-X stretches, the intermolecular vibrations of the dimer also form symmetric and antisymmetric combinations in the trimer. As may be noted from the first row of the table, the frequencies of both the symmetric and antisymmetric stretching modes of the trimer are larger than those of the dimer, reflecting the cooperativity of the sequential hydrogen bonds. With regard to the bending modes on the other hand, the two frequencies in the trimer bracket the single value for the dimer; i.e. one lower and one higher. ACKNOWLEDGMENTS

I am deeply indebted to my very knowledgeable and talented coworkers. This work was supported by grants from the National Institutes of Health (GM29391, GM36912, and AM01059) and from the National Science Foundation (DMB-8612768). Some of the calculations were performed on the SIU Theoretical Chemistry Computer, funded in part by a grant from the Harris Corporation.

192 REFERENCES 1 2 3 4 5

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

M.D. Joesten and L.J. Schaad, Hydrogen Bonding, Dekker, New York, 1974. P.A. Kollman, in H.F. Schaefer (Ed.), Applications of Electronic Structure Theory, Plenum Press, New York, 1977, pp. 109-152. K. Morokuma, Ace. Chem. Res., 10 (1977) 294. S. Scheiner, in E. Wyn-Jones and J. Gormaliy (Eds.), Aggregation Processes in Solution, Elsevier, Amsterdam, 1983, pp. 462-508. P. Schuster, G. Zundel and C. Sandorfy (Eds.), The Hydrogen Bond - Recent Developments in Theory and Experiments, North-Holland, Amsterdam, 1976. T.R. Dyke, Top. Curr. Chem., 120 (1984) 85. A.C. Legon, Ann. Rev. Phys. Chem., 34 (1983) 275. A.C. Legon and D.J. Millen, Act. Chem. Res., 20 (1987) 39. A. Weber (Ed.), Structure and Dynamics of Weakly Bound Molecular Complexes, D. Reidel, Dordrecht, 1986. R.E. Miller, Science, 240 (1988) 447. 2. Latajka and S. Scheiner, J. Chem. Phys., 87 (1987) 5928. Z. Latajka and S. Schemer, J. Comput. Chem., 8 (1987) 663,674. C. Meher and M.S. Plesset, Phys. Rev., 46 (1934) 618; J.S. Binkley and J.A. Pople, Int. J. Quantum Chem., 9 (1975) 229. S.F. Boys and F. Bemardi, Mol. Phys., 19 (1970) 553. A.C. Legon and L.C. Willoughby, Chem. Phys. Lett., 95 (1983) 449. M.M. Szczesniak, S. Scheiner and Y. BouteilIer, J. Chem. Phys., 81 (1984) 5024. U.C. Singh and P.A. Kohman, J. Chem. Phys., 80 (1984) 353. MM. Sxcxesniak and S. Scheiner, J. Chem. Phys., 83 (1985) 1778. K. Somasundram, R.D. Amos and N.C. Handy, Theor. Chim. Acta, 69 (1986) 491. G.P. Ayers and A.D.E. Pullin, Spectrochim. Acta, Part A, 32 (1976) 1641. AC. Legon and D.J. Millen, Faraday Discuss. Chem. Sot., 73 (1982) 71. Y. Bouteiller, C. Mijoule, M.M. Szcxesniak and S. Scheiner, J. Chem. Phys., 88 (1988) 4861. Z. Latajka and S. Scheiner, J. Chem. Phys., 81 (1984) 4014. M.M. Szcxesniak, Id. Kurnig and S. Scheiner, J. Chem. Phys., 89 (1988) 3131. B.S. Ault and G.C. Pimentel, J. Phys. Chem., 79 (1975) 621. Z. Latajka and S. Scheiner, Chem. Phys., 122 (1988) 413.