lo600
J. Phys. Chem. 1991, 95, 10600-10609
Transition-Metal Ion-Rare Gas Clusters: Bond Strengths and Molecular Parameters for Co+(He/Ne), , Ni+(He/Ne), , and Cr+(He/Ne/Ar) Paul R. Kemper, Ming-Teh Hsu, and Michael T. Bowers* Department of Chemistry, University of California, Santa Barbara, California 93106 (Received: June 3, 1991; In Final Form: July 15, 1991)
Using an equilibrium technique, we have determined enthalpies and entropies for a number of M+ + RG clustering reactions, where the M+ ions used were Cr', Co', and Ni+ and RG was He, Ne, and Ar. Clusters with up to three He ligands and two Ne ligands were examined. The clusters are very weakly bound with bond strengths typically 1-3 kcal/mol for He clusters, 1-2 kcal/mol for Ne clusters, and 6.5 kcal/mol for the Cr+.Ar cluster. The Co+.RG and Ni+.RG bonds were much stronger than the corresponding Cr+-RG bonds. Bond strengths of electronically excited (Ni+)*-He/Neclusters were determined using electronic-state chromatography. Effects on cluster bond strength due to M+ 3do-4s hybridization were observed and confirmed theoretical predictions made by Bauschlicher and co-workers. Finally, it was determined that simple relationships exist between bond strength and frequency as well as bond strength and bond length. These relationships are discussed in relation to the predictions of a simple polarizability theory and are used to obtain accurate entropies for the clustering reactions.
Introduction There has been an increasing interest in transition-metal ion chemistry over the last decade. Recently, a number of papers, both experimentalI4 and t h e o r e t i ~ a l , ~have - ~ discussed binding between metal ions and rare gases (primarily argon). The systems serve as relatively simple examples of transition-metal chemistry and as ideal models for clustering reactions of these metals. Lessen and Brucat have determined photodissociation thresholds for M+.Ar and M+.Kr where M = Cr, V, Co, and Ni.I4 They were limited to Ar and Kr adducts by the low binding energy (typically -12 kcal/mol for Co+ and Ni+ with Ar). They determined the dissociation energy directly, although in some cases the M+ electronic state had to be inferred. They also directly determined the vibrational frequency progression in four cases and, using a simple electrostatic binding model, inferred vibrational frequencies in the other cases. The same model was used to predict bond lengths in the M+.Ar ions, since these could not be measured directly. Prior to their work, there were a number of mobility experiments which indirectly determined binding between several closed-shell ions and the rare gases (see refs 8 and 9 for a summary). Direct measurements, however, were limited to a single spectroscopic study of Hg+.Ar9,Ioalong with estimates of V+ + Ar/Kr/Xe binding" and lower limits for T1+ + Ar/Kr/Xe binding.I2 These systems are well-suited to high-level theoretical calculations. Bauschlicher and co-workers have calculated binding ( I ) Lessen, D.; Brucat, P. J. Chem. Phys. Lett. 1988, 152, 473. (2) Lessen, D.; Brucat, P. J. J. Chem. Phys. 1989, 90, 6296. (3) Lessen, D.; Brucat, P. J. J. Chem. Phys. 1989, 91, 4522. (4) Lessen, D.; Asher, R.L.; Brucat, P. J. Int. J. Mass Spectrom. Ion Proc. 1990, 102, 331. ( 5 ) (a) Bauschlicher, C. W., Jr.; Langhoff, S . R. Int. Rev. Phys. Chem. 1990,9, 149; (b) Chem. Phys. Lett. 1989,158,409. (b) Bauschlicher, C. W., Jr.; Partridge, H.; Langhoff, S. W. J . Chem. Phys. 1989, 91, 4733. (6) Bauschlicher, C. W., Jr.; Partridge, H.; Langhoff, S . R. Chem. Phys. Lett. 1990, 165, 272. (7) Hammond, B. L.; Lester, W. A., Jr.; Braga, M.; Taft, C. A. Phys. Reu. B 1990,41, 10447. (8) Keesee, R. G.; Castleman, A. W., Jr. J. Phys. Chem. Ref: Data 1986, 15, 1011. (9) Huber, K. P.; Hertzberg, G. Molecular Spectra and Molecular Structure IV: Constants of Diatomic Molecules; Van Nostrand, Reinhold: New York, 1979. (10) (a) Linn, S . H.; Brom, J. M., Jr.; Tzeng, W.-B.; Ng, C. Y. J. Chem. Phys. 1985,82, 648. (b) Bridge, N. J. J. Mol. Spectrosc. 1972, 42, 370. (c) Hougen, J. T. J. Mol. Spectrosc. 1972.42, 381. (d) Santaram, C.; Winans, J. G. Can. J. Phys. 1966, 44, 1517. (1 1) Aristov, N.; Armentrout, P. B. J. Phys. Chem. 1986, 90, 5135. (12) (a) Parks, E. K.; Hansen, N. J.; Wexler, S. J. Chem. Phys. 1973,58, 5489. (b) Parks, E. K.; Wagner, A.; Wexler, S. J. Chem. Phys. 1973, 58, 5502. (c) Parks, E. K.; Kuhry, J. G.; Wexler, S. J. Chem. Phys. 1977, 67, 3014.
energies, bond lengths, and vibrational frequencies for the above-mentioned M+.Ar systems as well as closed-shell systems (Cu+.Ar, Cu+.He, Li+.Ar, Na+.Ar, and K+.Ar)? The calculated bond strengths are typically 0.1 eV smaller than the experimental values, while predicted bond lengths are considerably longer: Calculations on double clusters (M+-Ar2)have been done in many cases as Hammond et al.7 have also reported calculations on M+.Ar systems for M+ spanning the second half of the first transition series. Their bond strengths are typically one-half to one-third of the experimental values, however. In the present work we have determined bond strengths for M+.(He/Ne/Ar), complexes using equilibrium measurements. The He and N e complexes are very weakly bound (typically 1-3 kcal/mol). In contrast, the M+-Ar bond strengths are typically 6-12 kcal/mol. By measuring the temperature variation of the equilibrium constants, we were able to determine the clusterin entropy, which in turn allowed a precise determination of AI$ for the clustering reaction. The entropy values together with previous work on vibrational frequencies and bond lengths in the Ar systems allowed us to estimate frequencies and bond lengths in the He/Ne/Ar systems. A unique capability of our experiment is the ability to separate clustering of ground- and excited-state metal ions. One of the important aspects of transition-metal ion chemistry is the large number of low-lying electronic states which are present and which may react very differently.I3 These states are long lived (typically on the order of seconds).14 We have recently developed a technique (electronic-state chr~matography)'~ to separate transition-metal ions with electronic states of different configuration. All these states have valence electrons of either 3d" or 4s3d"' configuration. In the present work this allows determination both of the fraction of ground- and excited-state reactants and of their separate clustering equilibria. Lessen and Brucat have studied M+.Ar clusters for several excited M+ states,24 but unfortunately the identity of the states is unknown. Excited-state clusters have also been investigated theoreti~ally.~ We have also recently completed a study of the mobilities of ground- and excited-state Cr+, Co+, and Ni+ in H e and Ne.16 These measurements allowed an estimation of bonding ~~~
~
(13) Armentrout, P. B. Annu. Rev. Phys. Chem. 1990, 41, 313. (14) All low-lying states in these ions have either 3d" or 4s3d"' electronic
configurations. Radiative transitions are thus parity forbidden and lifetimes are estimated to be on the order of seconds. See: Garstang, R. H.Mon. Nor. R . Astron. Soc. 1%2, 124, 321. The lifetime of the Mn+ 5S first excited state has been measured to be 5.8 & 0.7 s; see: Strobel, F.; Ridge, D. P. J. Phys. Chem. 1989, 93, 3635. (15) (a) Kemper, P. R.; Bowers,M . T. J . Phys. Chem. 1991,95,5134. (b) J . Am. Chem. Sor. 1990, 112, 3231. (16) von Helden, G.; Kemper, P. R.; Hsu,M.-T.; Bowers, M. T. J. Chem. Phys., submitted for publication.
0022-3654/91/2095-10600$02.50/00 1991 American Chemical Society
Transition-Metal Ion-Rare Gas Clusters energies, frequencies, and bond lengths of both ground and excited states. Finally, equilibrium measurements with Ni+ in He and Ne unexpectedly allowed us to determine the rate of collisional deactivation of electronically excited (Ni+)*. Experimental Section The Instrument. A complete description of the instrument has
been given.I7 Briefly, ions are formed by electron impact on volatile metal compounds (Cr02C12,Co(CO),NO, cpCo(CO)2, and Ni(CO)+ The ions are accelerated and mass selected in a double-focusing sector mass spectrometer. The resulting M+ ions are decelerated to 1-3-eV kinetic energy and injected into a high-pressure drift cell. Cell pressures are typically 3-5 Torr of H e or Ne; cell temperatures are variable between 100 and 525 K. Ions are drifted through the cell with a small uniform electric field. Ions which exit the cell are quadrupole mass analyzed and detected. The Arrival Time Distribution. The arrival time distribution (ATD) plots ion intensity as a function of time between injection into the cell and arrival at the detector. In the ATD experiment, ions are pulsed into the cell in 1-10-ps bunches. The input pulse initiates a multichannel scalar scan, and ions are collected as a function of their arrival time at the collector. This time depends mainly on the drift time through the cell, which in turn depends on the drift field across the cell and the ion mobility. The technique of electronic-state chromatography (ESC) exploits the fact that the different transition-metal ion electronic configurations have very different m~bilities.'~ Consequently, use of ESC allows a separation of the M+ ions according to configuration (and often according to state). The different state populations are then determined by the individual ATD peak areas. Arrival time distributions have three applications to the present work. First, the fraction of ground-state reactant (required to calculate the equilibrium constant) can be directly determined via ESC. Second, comparing parent and cluster ATDs allows us to determine if the two are in equilibrium, since two ions in equilibrium must have ATDs which are superimposible (except for an intensity scaling factor). Third, it follows from the above that cluster products of ground- and excited-state metal ions will also be separated by ESC. Thus, we can determine their individual fractions of the total cluster product. The result is a direct measure of ground- and excited-state clustering equilibria. Equilibrium Experiments. In these experiments, pure H e or N e was used with a constant gas density of ~ 1 . X6 lOI7 m/cm3 (5 Torr at 300 K). In the Cr+.Ar experiments, 0.5 Torr of Ar was added to the He. An ATD for the metal ion was collected at a high drift field to determine the fraction of ground and excited states present. The high drift field gives maximum ESC resolution. Mass spectra were then collected at progressively longer residence times until the M+.RG/M+ ratio remained constant. This observation indicated two things: first, that the ions had sufficient time to attain equilibrium; second, that the heating due to the drift field did not measurably influence the equilibrium. (The effect of the field is discussed below.) At these pressures, equilibria were typically achieved in 5250 ps. Arrival time distributions at the longer times could be collected for M+ and M+.RG to determine the fraction of ground- and excited-state products. The equilibrium constant for the ground state of M+ is then calculated from
where M+.RG/M+ is the ratio of cluster and reactant intensities, PRGis the bath gas pressure in Torr, andAMGs+) andf(M+.RGGs) are the fractions of ground-state reactant and cluster present at equilibrium. An analogous relationship is used to determine K p for excited states. Sources of Error. There are four areas of possible error: the M+.RG/M+ intensity ratio, the pressure, the fraction of M+ (17) Kemper, P. R.; Bowers, M. T. J . Am. SOC.Mass Spectrom. 1990, 1 , 197.
The Journal of Physical Chemistry, Vol. 95, No. 26, 1991
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ground state, and the temperature. The cell pressure is measured directly at the cell with a capacitance manometer (0-10 Torr). The unit was recently recalibrated and the absolute error is less than lo4 Torr.'* At these high cell pressures, thermal transpiration does not measurably perturb the pressure measurement.I6 The He and Ne gases were purified in a liquid nitrogen cooled molecular sieve trap and no impurities were observed (other than 5 1 ppm of H2). In the Ar experiments measurable amounts of M+-H20were observed but did not affect the K p . The platinum resistor used to measure the cell temperature has an absolute uncertainty of Cr+.Ne. Again, the Me size probably plays a large role. Bond energies from mobility studies agree fairly well (f20%) with the present values for the M+-Ne systems. Looking further, the M+-Ne clusters are unique in several ways. First, the Co+ and Ni' clusters are less strongly bound than the corresponding H e clusters. This is surprising since the polarizability of Ne (0.396 A3) is about twice that of H e (0.204 A3), In contrast, binding between Cr' and Ne is stronger than with He. The Cr'SNe bond length (2.48 A, from Figure 5) is about 0.23 A longer than that in Cr+.He (2.25 A). The corresponding increases for Co+ and Ni' are much larger (-0.50 A). The change between Cr+.He and Cr+-Ne corresponds closely to the increase of atomic radius between He ( r 0.85 A) and Ne (r N 1.08 A).21 Thus, bonding in Cr+-RG seems to follow the predictions of classic charge-induced-dipole theory, possibly suggesting that hybridization is unimportant in this system, as was indicated by the Cr+.He data. The much larger increase in bond length with Co' and Ni+ may indicate a reduction in the 3 d d s hybridization, which reduced the M+-He bond length. This would be surprising, however, since calculations indicate that such hybridization is present in the M'-Ar clusters.6 Very recent theoretical work on Co' + He/Ne and Cr' + He/Ne is in excellent agreement with the binding energy trends and absolute values presented here.22 A second point of difference between M'aHe and N+.Ne clusters concerns the relative Co+ and Ni+ binding. The Ni+.Ne bond energy (2.37 f 0.1 kcal/mol) appears slightly greater than that of Co+-Ne (2.18 f 0.1 kcal/mol). This is in line with the relative Co+-Ar and Ni+.Ar bond strengths (1 1.71 and 12.68 kcal/mol, respectively). In contrast, binding in the Co+.He and Ni+-He clusters was identical. The entropy values found for the M'.Ne clustering reactions are comparable with those of the M+.He systems. The more (21) Dushman, S. Scientific Foundations of Vacuum Technique; Wiley: New York, 1962. (22) Partridge, H., private communication.
10608 The Journal of Physical Chemistry, Vol. 95, No. 26, 1991 TABLE I V (Ni+)* + Ne Collisional Deactivation Rate Coefficientsa temp, K k, cm3 s-' temp, K k, cm3 s-I 173 0.59 149 1 .o 161 1.5 135 2.0 a Deactivation occurs from the 3d84sl configuration (probably the a4Fand a2F states) to the ground 3d9 configuration (a2D) (see ref 15).
negative ASmm(due to increased RG mass) is compensated mainly by an increase in A&, due to the much lower frequencies in the M+.Ne systems. These are in turn due to both the reduced bond strength and increase in reduced mass relative to the M+.He systems. Finally, it should be noted that mass discrimination factors in the M+-Ne systems were 2.8-4.3, considerably greater than in the M+.He systems and about equal to that found with Cr+.Ar, where the mass difference is twice as great. Remember that the mass discrimination is the only variable used to fit the data in the M+-RG systems (see Data Analysis); consequently it should be well-determined. In order to reduce the mass discrimination to 1.5 in the Co+.Ne system (a factor of 2 reduction), w must be reduced from 140 to -80 cm-I and rminincreased from 2.3 to -2.7 A. These parameters seem too far outside the values derived from Figures 4 and 5 to be reasonable, and we conclude that substantial mass discrimination in these systems is present. Again, this is in keeping with the mass discrimination results noted above (Sources of Error). Clustering of Excited (Ni+)*with Ne. As with (Ni+)* and He, ATDs of Ni+ and Ni+.Ne allowed determination of the relative amounts of ground- and excited-state reactant and product. The peaks were not completely resolved in the Ne bath gas and peak heights were used (instead of areas) for the populations. The known fraction of Ni+ ground state was reproduced well by this method. The (Ni+)*-Ne bond strength was 0.73 f 0.2 kcal/mol, about 1.6 kcal/mol less than that of ground state Ni+. This reduction is again due to the presence of the large 4s orbital in the (Ni+)*.Ne, as noted in the (Ni+)*.He system. The result for (Ni+)*.Ne is in good agreement with the bond strength determined by mobility methods (0.90 kcal/mol).16 Deactivation of (Ni+)*. When K p for the Ni+-Ne system was plotted versus reaction time (see Equilibrium Experiments above) an unusual effect was noted. The equilibrium constant increased quickly from 0 to 150 ps (as with helium buffer gas) but then, instead of remaining constant, continued to increase at a very slow rate even at the longest times we could measure. This continuing increase was not present in the Ni+-He experiments. We believe it is due to collisional deactivation of (Ni+)* followed by rapid equilibration of the resultant ground-state Ni+. This is observed in Ni+-Ne but not Co+ + Ne due to the large amount of excited (Ni+)* (typically 83%) relative to (Co+)* (typically 17%). The derived deactivation rate coefficients are listed in Table IV. The deactivation is inefficient at T 150 K (kdeact cm3/s); however, a temperature dependence is evident. The rate increases by more than a factor of 3 as the temperature drops from 173 to 135 K. The rate below 100 K may be substantial. At 173 K deactivation was barely noticeable (kdcact= 5.9 X cm3/s). Since we did not observe deactivation with Ni+.He, kdeact(He)5 5 X IO-'' cm3/s. M+.Ne2. The ordering of bond strengths in M+.Ne2 reproduces that seen previously: Ni+ Co+ > Cr+. As in the first cluster, the Ni+-Nezbonds appear slightly stronger than those in Co+.Ne2. A surprising result is the weaker binding of the second Ne relative to the first. The average bond strengths are weaker by about 2.5% (Ni'), 4%(Co'), and 18% (Cr+). The M+.Hez clusters had 7% stronger bonds. This effect can be explained if hybridization in these M+.Nez systems is small relative to that in M+.Hez. The ordering of bond strengths (Ni+ 2 Co+ > Cr+) is then largely determined by the M+ size (see M+.He). The decrease in bond strength when the second Ne is added is due to charge delocalization on M+. In M+.Ne, the positive charge (really an electron hole) can be localized near the Ne for a maximum attractive interaction. In M+-Nezthe charge must be shared, reducing the
-
-
-
-
Kemper et al. individual attractions. As noted, this would also help explain observations in the M+.Ne systems. Theoretical investigation of the M + S N ~clusters ~ , ~ would help to resolve this point. Analysis of the clustering entropy proceeded as with M+.He2. The symmetric stretch frequency and bond length were taken from Figures 4 and 5; the asymmetric stretch frequency was set 20% higher than the symmetric stretch; the mass discrimination was fixed at the M+-Ne value; the bend frequency was varied to fit the data. The resulting bend frequencies were extremely low (6-8 cm-I). The data fit is insensitive to the bond length and the higher frequency modes. Thus, calculations in which higher bond frequencies were used (say 20 cm-') required huge mass discrimination factors (>40) to match the experimental data. In support of these surprisingly low frequencies, there is limited data on Ni+.Ne3 which show a positiue AGO versus T slope. This can only be true if A&, is enormous [about +30 cal/(K mol)], which in turn implies extraordinarily low frequency modes in Ni+.Ne3. Cr+.Ar. Due to limits on cell temperature, we were unable to examine the more strongly bound Co+ and Ni+ cases. The binding energy of the Cr+.Ar turned out to be weaker than that of Co+& or Ni+.Ar, and hence we could measure equilibrium constants. From these data we obtain = 6.5 f 0.4 kcal/mol. The larger uncertainty is mainly due to the longer extrapolation to 0 K. This is the only case in the present work where an independent spectroscopic determination is available for comparison. Lessen and Brucat have determined the Cr+.Ar bond strength to be 0.27 eV (6.23 kcal/mol): No uncertainty is quoted for their measurement, but their value is well within our uncertainty limits. Their C o + k and Ni+.Ar bond energies are 11.71 and 12.68 kcal/mol (respectively)2J again showing the relative weakness of the Cr+-RG bond. No vibrational spacings could be determined in the photodissociation of Cr+.Ar. Thus, value of w and rmi,for our analysis were taken from Figures 4 and 5 (see Table 111). In this case, however, the values are especially well-determined by the large amount of other M+Ar data and calculations. The resulting mass discrimination factor (the variable in the data fit) was 3.5 f 0.5. As noted above (Sources of Error) this was also found between the Fe+ and Fe+.C3H8 ions, where the mass difference is very similar to the Cr+.Ar case.
a
Summary and Conclusions (1) By measuring ion-cluster equilibria as a function of temperature, we have determined binding energies for a number of weakly bound transition-metal ion-rare gas clusters: Co+.Hel,2,3r Co+.Nel,z,Ni+.Hel,z,3,Ni+.Nel,2,Cr+.He, Cr+.Nel,2,and Cr+.Ar. Typicaly uncertainties were fO.1 kcal/mol. The bond strength between electronically excited (Ni+)* and Ne was also directly determined. (2) The clusters are weakly bound with binding energies of typically 1-3 kcal/mol. Bond strengths of Ni+ and Co+ clusters are generally similar and are considerably stronger than those in the corresponding Cr+ systems. The Cr+.Ar bond strength was measured to be 6.5 f 0.4 kcal/mol, in agreement with existing spectroscopic data. Bond strengths in the M+.He2 clusters are slightly stronger than those in M+.He. In contrast, adding the second Ne to Co+ and Ni+ leads to a weaker bond than adding the first. The differences were interpreted in terms of 3d-4s orbital hybridization and ion size. The third rare gas neutral is very weakly bound. (3) The bond strength between He and electronically excited Ni+ (4s3d8 valence configuration) was much weaker than with ground-state Ni+ (3d9 configuration). This was interpreted as due to increased repulsion between the 4s electron and He. (4) Critical examination of existing vibrational frequency and bond length data, together with the present temperature-dependent data, allowed estimation of bond lengths and frequencies in the various cluster ions. In the M+.Ar case, these appear to be predictable to within f 5 % . (5) Collisional deactivation of electronically excited Ni+ with Ne was observed. The process is inefficient with a rate coefficient cm3/s. The rate exhibited a large negative temof 1 X
-
10609
J. Phys. Chem. 1991, 95, 10609-10617 perature dependence, however, and deactivation could be important at very low temperatures. Deactivation with He was not measurable. ( 6 ) Reaction entropies were also determined. These data strongly supported the estimated bond lengths and vibrational frequencies obtained (see part 4 above).
In the second clustering reaction, the reactants do have rotational entropy and the change in rotational entropy is given by
Acknowledgment. The support of the National Science Foundation under Grant C H E 88-17201 is gratefully acknowledged.
where I and I1 refer to M+-RG and M+.RG2. Rotations of M+.RG2 are identical to those of an RG2 molecule with RG-RG spacing equal to that in the M+.RG2 (Le., 2rminII). Since Fred11 = RG/2 and u = 2, it is clear from eq A-6 that ASm,is very small in the second clustering reaction. There is a minor correction when RG = Ne due to the second isotope, since CT = 1 in those M+.RGis with different isotopes. The total vibrational entropy for the n vibrational modes is given by
Appendix: Summary of Statistical Mechanics Formulas
The total entropy of a reaction is ( A -1 ) ASo = hsp,ans + U r o t + M v i b for a gas in its standard state ( 1 atm = 760 Torr), the translational entropy [in cal/(K mol)] is given by
c
grans = R In (T5/2M3/2) - 2.314 (A4 where Tis the temperature in kelvin, R is the gas constant, and M is the mass in daltons. For the clustering reactions M+.RG, RG M+*RG,+I
+
where Uj = ovib/T = wj/0.695T. Clustering enthalpies are calculated from
+
= 2.314 - 3/2R In
(pred)- S/zR In
T
T
(A-3)
AHo
where prd is the reduced mass of the reactants in daltons. The rotational entropy of the gas is given by
The AS,,, for M+
+ RG
-
\I
0
('4-8)
The resulting A(AHtranS) = -2.5RT; A(AHrot) = RT/2(nPrdnreact), where n refers to the number of rotational modes (again assuming the rotations are completely active). The vibrational enthalpy is calculated from
where u is the symmetry number, I is the geometric mean of the individual moments of inertia ( I = (IaI&1/3),k is Boltzmann's constant, h is Planck's constant, and n is the number of rotational axes. Since Or,, for these molecules is 6 2 K, use of the integral form for S,, does not introduce significant error. For linear M+.RG and M+.RG2 (4.1230 X lO-*)Tpred r"
= a +AC,dT
Note added in proot The effect of electronic degeneracy is neglected in the above treatment. Such entropy effects are included in the mass discrimination term in our analysis. Recent theoretical work by Partridge and co-workersI6 has determined the cluster electronic states. Thus, the electronic degeneracies can now be calculated exactly. Registry No. Cr', 14067-03-9; Co', 16610-75-6; Ni+, 14903-34-5; He, 7440-59-7; Ne, 7440-01-9; Ar, 7440-37-1.
(A-5)
' I
M+.RG is given directly by eq A-5.
Graph Theoretical Analysis of Water Clusters T. P. Radhakrishnan**tand William C. Herndon Department of Chemistry, The University of Texas at El Paso, El Paso, Texas 79968 (Received: October 12, 1990)
A graph theoretical representation of the hydrogen bond network in a water cluster is developed using the concept of directed graphs. Ab initio STO-3G calculations on a large collection of small linear and branched, open chain and monocyclic water clusters are presented. The invariants of the digraph representations are utilized for a statistical analysis of the ab initio results, and this procedure provides physically meaningful factorizations of the hydrogen bond energies. The usefulness of a graphical analysis of this type is further demonstrated by applying it to (i) the analysis of a previous energy data set (MCY potential/many-body corrections) for highly condensed clusters of water molecules and (ii) the prediction of the cohesive energy of ice. The graph theory parameters based on the many-body calculation results give an ice cohesive energy in good agreement with experiment.
I. Introduction The study of small water clusters is apposite to understanding hydrogen bonding and to elucidating the structure of liquid water and ice. A large amount of work has been published in this area, so we will not attempt an exhaustive listing. A representative 'Present address: School of Chemistry, University of Hyderabad, Hyderabad 500 134, India.
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collection with emphasis on more recent publications is refs 1-30. The review by Vigasin3' is a good survey of the ab initio calcu(1) Shipman, L. L.; Owicki, J. C.; Scheraga, H. A. J . Phys. Chem. 1974, 2055-2060. ( 2 ) Matsuoka, 0.; Clementi, E.; Yoshimine, M. J . Chem. Phys. 1976.64, 1351-1361. (3) Slanina, Z. J. Chem. Phys. 1980, 73, 2519-2521. (4) Newton, M. D.; Kestner, N. R.Chem. Phys. Lert. 1983.94, 198-201.
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0 1991 American Chemical Society
