J. Phys. Chem. 1994, 98, 968-971
968
Relationship among Force Constants Implied by the Principle of Bond-Order Conservation in Chemisorbed Systems Harrell Sellers Department of Chemistry, South Dakota State University, Brookings, South Dakota 57007, and National Center for Supercomputing Applications, Beckman Institute for Advanced Science and Technology, University of Illinois, Urbana, Illinois 61801 Received: September 7 , 1993’
We consider the implications of the bond-order conservation Morse potential (BOC-MP) model for some vibrational force constants and molecular structural parameters of polyatomic adsorbates on surfaces. W e show that a form of the Pauling-Badger rule results from the principle of bond order conservation and, more importantly, that relationships among vibrational force constants arise from the conservation of bond order. W e give as examples the chemisorption of SCHj and SH on Pd( 1 1 1 ) . Introduction Researchers have long sought relationships between the chemical idea of bond order and bond lengths and vibrational force constants.1-2 In the 1930s Badger developed an empirical relationship that became known as Badger’s rule that expresses a relationship between quadratic stretching force constants and theequilibrium bond distance.2 It was later noticed that quadratic and higher order stretching force constants seemed to be roughly an exponential function of bond distance,3 and a modified form of the Badger rule was put forward:’ (-l)kFk = exp{-(re - aij)/bi,}
The i a n d j subscripts refer to bonds between atoms in rows i and j o f the periodic table.3 In theoriginal formulation the exponential was in reference to the base ten rather than e; however, the use of e rather than 10 just shifts the values of the parameters. The k value in eq 1 is the order of the force constant; k = 2,3,4. The parameters ai, and bU in this modified version of Badger’s rule were refined by Herschbach and Laurie by fitting them to empirical force constant data.3 In the 1940s, as part of his model of resonating covalent bonds in metallic systems, Pauling put forward an empirical relationship that describes the dependence of internuclear distance on bond order in metallic crystals:’ R ( n ) = R( 1 ) - a ln{n)
where n is the Pauling bond order value and a is a constant. Solving eq 2 for n gives the Pauling expression for the bond order in terms of the equilibrium distances:
n = exp{-(R(n) - R(l ) ) / a }
(3) Since Pauling’s relationship allowed a mathematical definition of bond order in terms of the internuclear distance and Badger empirically connected the vibrational force constants to the bond distances, it was natural to combine the two empirical relationships into a single equation that gives the vibrational stretching force constants directly in terms of the bond orders.4 A given ai, in q 1 was interpreted as a standard bond length3 and for a given family of related molecules the force constants can be expressed in terms of one another through eqs 1-3 giving what we call the Pauling-Badger rule:4 F(n) = nF(1) (4) The bond order conservation Morse potential model (BOC-MP)S*6 ~
~~
0
Abstract published in Advance ACS Absrracrs, January 1, 1994.
has been used very successfully by Shustorovich and co-workers to describe strong and weak chemisorpti~n,~?~ intrinsic and diffusion activation barriers,6v7and catalyzed chemical reactionsM Although the model is elegant in its simplicity, the prediction of enthalpies of chemisorption and activation barriers has been strickingly accurate. Chemisorption energies are usually predicted to within 1-3 kcal/mol by this mode1.5-6 It is very difficult for much more complex a b initio calculations to achieve this level of absolute accuracy. Only the best and most computationally expensive ab initio calculations ever approach this accuracy level. In cases in which our a b initio results have disagreed with the predictions of the BOC-MP model we have experienced that we just get closer to the BOC-MP result upon improvement of our ab initio calculation. Researchers in the 1960s employed a constraint of constant total bond order in gas-phase reaction kinetics modeling to obtain the activation energy of hydrogen-transfer reactions.4 The bond energy-bond order (BEBO) methodof Johnston4employs a bond order conservation constraint, but the relationship between bond order and bond energy is obtained from an empirical rule. When force constants are needed in the BEBO method, the Pauling rule is used. Sims and Lewis,4who employ a method similar to BEBO, note that their computed results for other types of reactions are in better agreement with experimental quantities when an assumption of constant bond order is introduced. What we do herein that to our knowledge has not been done before is to consider what relationships among the vibrational force constants result from the principle of bond order conservation. We do not introduce any empirical relationships. The BOC-MP model we start with employs parameters that can be obtained from a b initio calculations or experimental measurements, but it does not make use of any empirical relationships between energy and bond order, etc. The fact that the BOC-MP model has been so successful at predicting accurately the chemisorption energies of diatomic and polyatomic adsorbates is very important to the present work. We take this high accuracy level (the reader is directed to Shustorovich’s convincing works” and our own efforts to test thevalidity of the principle of bond order conservation in condensed systemsgJO as evidence that the model assumptions are close to physical reality. It would be surprising if a model that gives such accurate chemisorption energies would simultaneously fail to give reasonable force constants. Shustorovich has reportedS structural parameters and vibrational frequencies in terms of the BOC-MP parameters for atomic adsorbates. Herein we consider what relationships among the force constants are implied by the principle of bond order conservation in chemisorbed systems
0022-3654/94/2098-0968%04.50/0 0 1994 American Chemical Society
Principle of Bond-Order Conservation
The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 969
involving polyatomic adsorbates. This is fundamentally different from what is done in the BEBO method in several respects. As mentioned above, rather than use the force constant relationships required by the principle of bond order conservation, the BEBO method as practiced4simply adopts the Badger rule (or PaulingBadger rule when used in conjunction with the Pauling bond order). In the BEBO method the goal is not usually to obtain equilibrium quantities as it is herein. Also there is an already large and growing body of evidence that indicates that the principle of bond order conservation is obeyed to a high degree in systems chemisorbed on metal surfaces.5-”J This fact explains, as we would judge from earlier works,&1° that the BEBO method is not as successful as the BOC-MP method of Shustorovich.
that the AB bond order changes in a nonrelaxed situation, as just mentioned, due to the changes in the a! and & Morse parameters of Z A B of eq 7 as the adsorbate is moved in this constrained fashion. From this point of view, moving off the RABtrajectory for the relaxed system effectively puts the AB species on a different energy surface described by a different set of Morse parameters. For the relaxed RABtrajectory (or reaction coordinate) the mapping that we employ to associate bond order with the bond distance is eq 3 with Morse parameters for the free AB molecule and Morse parameters for the chemisorbed A atom as per Shustorovich.s~6 Substitution of eq 5 into eq 6 gives the BOCM P energy expression for the chemisorption energy of the AB molecule:
Theory
(8) v = -QA( 1 - ZA,Z) - DAB(~ZAB- ZA;) Equation 8 gives the expression for the energy curve for the “relaxed system”. Solving for the AB bond order a t the minimum in the potential yields
The main tenet of the BOC-MP modeP6 is that the sum of the bond orders in a chemisorbed system is a constant along a path of motion: ciZi = constant. The constant is usually taken to be unity.s.6 In the case that a diatomic adsorbate, AB, is chemisorbed on a surface, the BOC-MP model considers that the sum of the bond orders, ZA, ZAB,and Z B is constant (here Z B is the bond order between the B atom (or group of atoms) and the surface, and ZAB is the bond order associated with the AB adsorbate bond). In the case that the chemisorption of AB species is with the A atom toward the surface having the B atom directed away from the surface, then the B-surface bond order can be negle~ted.~ The A-surface bond order plus the AB bond order is conserved and normalized to unity:
The QA and DAB are Morse potential dissociation energy parameters and Zi is the bond order (defined below). If one would consider the bond orders to be independent of one another and therefore the two oscillator coordinates to be independent (i.e., neglecting eq 5 or considering the sum of the bond orders to be nonnormalized) then the force constant that couples the two Morse oscillators would be zero. This is a statement that, without eq 5 , eq 6 is just a sum of two noninteracting Morse potentials. The important point is that the effect of the bond order conservation condition, eq 5, is to tell the otherwise independent Morse oscillators how to respond to (or couple with) each other. In the BOC-MP model care must be taken when associating the bond order with the bond length. In a relaxed system the bond order is taken to be the Pauling bond order:
Z = exp(-(R - R , ) / a ]
(7) where Ro and a! are the Morse potential parameters that describe the A-surface potential and the free AB molecule. By “relaxed system” we mean that for a fixed A-surface bond order, Z A = expi-(RA - RA,o)/cYAI,the AB bond order and AB distance are fixed by eqs 5 and 7. Johnston4describes this path as the reaction coordinate. However, there is no a priori criterion that fixes the mapping of bond order onto bond distance in all situations. We can, for example, hold the AB bond distance fixed and move the adsorbate around over the surface and the AB bond order should not be considered to be constant just because the AB distance is not changing. One could consider a normalization scheme that would always maintain the sum of the bond orders at unity by dividing each of the bond orders as defined by eq 7 by the sum of the bond orders. This would introduce a normalization factor that is a function of the system geometry. We could also consider
(9) ZAB= DAB/@AB+ Q A ) This is a thermodynamic definition of bond order a t equilibrium. Here We have used dV/dRAB = aV/dZABdZAB/aRAB. Since dZAB/dRAB can never be zero, the extremum is defined as the vanishing of aV/aZAB. This procedure can also be applied to the RAgeometry parameter, but the group bond order must be used rather than the individual A-Mi bond orders.s.6 Shustorovich has shown that the Q parameters we have used in eqs 8 and 9 are the experimental binding energies only for strongly bound systems (for example, SH or SCH3 or many if not most radicals). One could derive similar parameters for weakly bound systems, but, for example, the QAwould not be the experimental atomic binding energy for atom A. In weakly bound systems QA needs to be a function of the binding site, which is almost always different for a closed-shell molecule than for the isolated A atom. For more strongly bound systems, such as many radicals, the adsorbate and the isolated atom usually prefer similar binding situations. Putting eq 9 equal to eq 7 gives RAB RAB,O- AB WZABI RAB = RAB,,- ~
A W B AB/(DAB + QA))
RAB= RAB,,- [ ~ D A B / ~1’2A B ~ ~] ( D A B / ( D+AQB A ) ) (10) where we have used ~ A =B [2DAB/3AB11/2for a Morse potential. The harmonic force constant, AB, is that for the free AB species. Equations 9 and 10 specify that the bond order and bond length shift that one can expect when AB chemisorbs. Equation 10 is the empirical Pauling rule’ for the dependence of bond length on bond order, n: R, = Ro - c ln(n). Here we use the symbol Z to represent bond order while Pauling used n. That we get back Pauling’s rule is trivial, since the definition of bond order that we use is from Pauling who essentially solved the first of eqs 10 for ZAB. The second and third equations of eqs 10 make the connection to the thermodynamic quantities Qt and DAB. By considering the implications of the principle of bond order conservation on force constants we can derive relationships between the value of the force constant and the binding energies and the bond order:
a2v/aR2AB= ~
( Q A-tD
(11) AB)~~AB/~’AB
or
a2v/aRZAB = 3ABZAB
970
Sellers
The Journal of Physical Chemistry, Vol. 98, No. 3, 1994
and 3 A B is the harmonic force constant for thefree AB molecule. In going from eq 11 to eq 12, we have used eq 9 and ( Y ~ A B= 2 D A B / S A B . The first of eqs 12 is the combined (empirical) Pauling-Badger rule,4 eq 4, for the variation of the harmonic stretching force constant with bond order. It is important to keep in mind here that the above expression for the diagonal stretching force constant corresponds to the AB stretching force constant along the reaction coordinateor along the “relaxed RAB” trajectory and has a different definition than the usual spectroscopic force constants as will be seen below. Hence, simply adopting the Pauling-Badger rule for the force constants, as in the BEBO m e t h ~ dwould , ~ be tantamount to moving away from the definition of spectroscopic force constants. (However, the use of empirical relationships4 tends to wash out this effect.) The thing that the BOC condition adds here is a cause and effect relationship. The cause of the change in theforce constant is the interaction between the two Morse oscillators which is determined by the principle of bond order conservation and, this interaction between the Morse oscillators is such that the Pauling-Badger rule results as the prescription for the change in the harmonic stretching force constant. We have stated that the BOC condition is responsible for coupling the oscillators in the BOC-MP model. It is straight forward to show that the BOC condition fixes the ratio of the diagonal stretching force constants and thevalue of the interaction force constant between the two oscillators. The usual Taylor series force field expansion of the potential energy around the equilibrium geometry of the coupled oscillator system is given by
a very small displacement in R1 causes the following displacement in R2 in the relaxed system (or along the reaction coordinate):
6 4 = -(F12/F22)8R1
(14)
From the BOC condition (eq 5):
6 Z l = -6Z2 = (dZl/dRl)GRI = -(dZ,/dR,)dR,
(15)
Equations 14, 15, 9, and 7 require
Here we have defined oscillator 1 as the atomic A-surface oscillator (AS) and oscillator 2 is the AB bond. Equation 16 could be written as F12/F22 = ( D 1 3 ~ / D 2 3 2 )with 1 / 2 QA= D I , AS = gl, and so on. In the case that QA and DABare of similar magnitude and the force constants for the free oscillators are of similar size, as in the case of SCH3chemisorption on most metals (in this case the 3~ is the atomic sulfur-surface stretching force constant), one can expect the coupling constant F12 to be roughly the size of the F11 and F22. The force constant matrix for the chemisorbed system cannot be considered to be diagonally dominant. As mentioned above, eq 12 gives the force constant for the AB stretching coordinate along the relaxed trajectory or “reaction coordinate” which allows the A-surface distance to respond to changes in the AB distance. On the other hand the usual spectroscopic force constants are defined as partial derivatives of the energy holding the other vibrational degrees of freedom constant as is the case in eq 13. So the Pauling-Badger rule as implied by the principle of bond order conservation does not strictly apply to the usual definition of spectroscopic force constants. However, we can derive a similar expression to the PaulingBadger rule that contains a correction term. Using eq 14, it is
straightforward to show that (17) = FAB - FI?/FAS where FABand FM correspond to F22 and F11 in eq 13 and 3 A B is the AB stretching force constant in the free AB species. Equation 17 has the form of a “modified” or “corrected” PaulingBadger rule with the last term on the right-hand side being the correction. In the nomenclature of eq 13 the above equation can be written as zl3ll = F11 - F12~/F22, also Z2 sc22 = F22 - F1z2/ F11. These two relationships leads to zAB3AB
F ~ ~ / F 2 2 =3 ~ ~ z ~ / 3 2 2 z 2
= 3~~Di/322D2
or
3lIZllF11 = 322z2/F22 (18) In the first of eqs 18 DI and D2 are the dissociation energies of the oscillators (QA and DAB). Equations 16 and 17 imply that F12 = (F11F22)l/~. Therefore, when this relationship holds (i.e., when the principle of bond order conservation is valid), the interaction constant will be larger than one of the diagonal constants. Equations 12 and 16-1 8 are the primary result of this work. Equation 12 shows that the Pauling-Badger rule gives the force constant along the relaxed energy curve or reaction coordinate. Equations 16-1 8 relate the spectroscopic force constants for the adsorbatesurface system to those of the free specie and the thermodynamic parameters (enthalpies of binding). Substitutiotl of eqs 12 and 16 into eq 17 yields the expression of the harmonic force constant for the stretching of the AB bond in the chemisorbed state in terms of thermodynamic quantities: FAB = DAB3AB/(DAB + Q A ) + QA3ASFAS/(DAB3AB) (19) Again, eq 19 has the form of a modified Pauling-Badger rule (with DAB/(DAB + QA) being the Pauling bond order) with the second term on the right-hand side being the correction. It is difficult to find experimental data with which one can compare the force constant (or frequency) predictions of this work. Much of the available experimental frequency data contain coverage effects, even when frequency shifts as a function of coverage are not observed (in this case it is likely that the adsorbates are grouping together in islands on the surface as they chemisorb which creates a situation of locally high coverage). Also, different researchers report experimental determinations of the same frequencies for surface adsorbed species under similar conditions that can differ by 30-50 cm-1.11,12 Comparison to ab initio data is not ideal either since cluster truncation effects and other systematic errors can be present. However, some comparison with experimental data or high quality ab initio result is called for. We have computed the force constants for the free SCH3 and SH radicals and the Pd( 11l ) S C H 3and Pd( 11l ) S H condensed systems at the Hartree-Fock + MBPT2 level of theory.” The methods employed herein are the same as in ref 13. For comparison purposes we evaluate the ratio of the diagonal force constants forthePd( 11l)SCH3system, FlllF22, that weobtained from ab initio calculation^,^^ which is also predicted by eq 18 above. According to eq 18 for the Pd( 11l ) S C H 3 system, this ratio should be equal to Qs3s/Ds&c, where Qs and 3 s are the atomic binding energy of sulfur to Pd( 11 1) and the force constant for the vibration of the atom against the surface, respectively, and D ~ and c 3 s are ~ the S-CH3 bond enthalpy and the S-C stretching force constant for the radical. With DSC and 3 s c being 70 kcal/mol and 3.418 m d y n / k respectively, and QSand 3s being 85 kcal/mol and 2.19 mdyn/A, re~pectively,’~ the S C This is the prediction for the quantity Q S ~ S / D S Cis~ 0.78. ratio of the diagonal force constants from this work based on the BOC-MP model. The ratio of our a b initio force constants, Fll/ F22,from ref 13 is 0.80 to compare with 0.78 above. For the Pd( 11l ) S H system13 the ratio of the a b initio force constants
Principle of Bond-Order Conservation is 0.55. The prediction of eq 18 above for this system is 0.52(we obtain 4.385 mdyn/A for the SH radical stretching force constant). There may still be some cluster effects remaining in our previous cal~ulations;*~ however, we consider that this is very good agreement between the BOC-MP model and the ab initib force constant ratios. ‘Onsidering Once again the Of surface adsorption Of an adsorbate: in the case that the adsorbate is bound parallel to the surface the BOC-MP energy expression is the sum of three Morse potentials. The equilibrium bond orders are given by
The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 971
stretching force constant along the relaxed energy curve. We have stated the connection between bond order and thermodynamic quantities. We have given a comparison to ab initio results on the Pd(ll11-SCHq and P d .( l l 1,1 S H systems. . ,
Acknowledgment. The author acknowledges the National Science Foundation (Grant OSR-9108773)and theSouth Dakota State Future Fund for partial of this work. 1 thank Dr. EvgenyShustorovichformany helpfuldiscussionsand Mr. Pierre Dewey La Fontaine, Jr. for his many contributions. References and Notes
Conclusion We have shown that the principle of bond order conservation gives rise to relationships among vibrational force constants. We also showed that the BOC-MP model gives rise to the Badger rule for the variation of harmonic stretching force constant with respect to bond order. The proper interpretation of the PaulingBadger rule in this framework is that it gives the diagonal
(1) Pauling, L. J . Am. Chem. SOC.1947, 69, 542. (2) Badger, R. M. J . Chem. Phys. 1934,2,128; 1935,3,710; Phys. Rev. 1935, 48, 284. Waser, J.; Pauling, L. J. Chem. Phys. 1950, 18, 618. (3) Herschbach, D. R.; Laurie, V. W. J. Chem. Phys. 1961, 35, 458. (4) Sims, L. B.; Lewis, D. E. In Isotopes in Organic Chemistry; Buncel, E., Lee, C. C., Eds.;Elsevier: New York, 1984; Vol. 6, Chapter 4. Johnston, H. S. Gas Phase Reaction Rare Theory; Ronald Press: New York, 1966. ( 5 ) Shustorovich, E. Surf. Sci. Rep. 1986, 6, 1. 16) Shustorovich. E. Adv. Card. 1990. 37. 101. (7) Shustorovich; E. Surf.Sci. 1992, 279,’355. ( 8 ) Shustorovich, E.; Bell, A. T. Surf.Sci. 1993, 289, 127. (9) Sellers, H. L. J. Chem. Phys. 1993, 98, 627. (10) Sellers, H.L. J . Chem. Phys. 1993, 99, 650. (1 1) Timbrell, P.Y.; Gellman, A. J.; Lambert, R. M.; Willis, R. F. Surf. Sci. 1988, 206, 339. (12) Marchon, B. Surf. Sci. 1985, 162, 382. (13) Sellers, H. L. Surf.Sci. 1992, 264, 177. (14) Handbook of Chemistry and Physics, 13d ed.; CRC Press: Boca Raton, FL, 1992.
