3618
J. Phys. Chem. 1981, 85, 3618-3628
-
empty, increasing the 3d,, character of the 3a, 3b2 orbital by bending the molecule seems to maximize its stabilization. The combination of these two effects seems to be required to allow a stable bent structure. We conclude that the introduction of p-type polarization functions in the basis sets of the metal and the hydrogens results in a much longer metal-hydrogen bond. It also results in a reduction of the electron transfer from the metal to the hydrogens. The addition of these polarization functions appears to have little effect on the qualitative
pattern of molecular-orbital energy variation with HMH angle or on the preference for a bent over a linear equilibrium geometry in certain configurations. However, the introduction of the p-type polarization functions does reduce the relative stability of the bent over the linear geometry as measured by the total energy. Finally, our model for indicating preference for a bent over a linear equilibrium geometry now requires not only an empty metal 3dy, orbital but also the simultaneous occupancy of the metal 3d,z orbital.
Normalized Kinetic Field Potentials for Atom-Diatom Reactions. Three-Dimensional Surfaces from the Relaxed Bond Energy-Bond Order Model‘ Roman F. Nalewajskl** and Roman Pastewskla Department of Theoretical Chemistry, Institute of Chemistry, Jagiellonlan University, 30-060 Cracow, Poland (Receivd: April 7, 108 1)
-
A new empirical potential energy surface, WT(R,8),is constructed for the atom-diatom reactions, A + BC AB + C, via the virial theorem from the rotated Morse function (RMF) representation of the electronic kinetic energy surface, T(R,B);R = (RABJI~c), 0 = ABC, and the RMF swing point is at the united atom. The T-RMF surface is subject to the T-normalization: Jo”[T(sR,O)- T(-)] ds = V,(R,O), where V, is the nuclear repulsion energy and stands for the separated atoms limit. This allows one to determine the WT surface solely from the coordinates and the energy profile of the zero-uirial path, ZVP(B), WT(R,B)= wT[zvP(e)]. Realistic approximationsto the ZVP(B)’s are generated from the relaxed bond energy-bond order (BEBO/R) model devised for the purpose, in which the collinear BEBO configurations are relaxed after bending, due to increased AC repulsion, against the bonding forces of the AB and BC “diatomics”, estimated from the harmonic approximation. The &constrained WT[BEBO/R]surfaces for the H + H2 and F + H2reactions were generated to test usefulness of the T-normalization in controlling the surface shape during reaction. It is shown that the WT[BEBO/R]surface for the H3 system closely approximates the SCF CI surface of Siegbahn and Liu. For the FH2 system and large 0, the W, surface exhibits more coupled reactants-to-productspassage than does the LEPS surface.
-
Introduction The methods of generating the potential energy surfaces (PES’s) for simple, mostly triatomic reactive molecular collisions have undergone rapid progress in the last few years. Recent reviews of such methods can be found in ref 4-9. In addition to standard global techniques of constructing the PES’s, e.g., the London-Eyring-Polanyi-Sat0 (LEPS) or diatomics-in-molecules (DIM) functional expressions, various local methods of fitting a limited number of ab initio data have been developed. The most efficient among the local interpolating functions are perhaps those which combine the rotated Morse function (RMF) approach of Wall and Porterlowith the use of the (1) This work was supported by the Institute of Low Temperatures and Structural Research, Polish Academy of Sciences, Wrocyaw. (2)T o whom correspondence should be addressed. (3)Department of Physical Chemistry, Institute of Inorganic Chemistry and Technology, Technical University of Gdafmk, 80-952Gdaiisk, Poland. (4)G. G. Balint-Kurti, Adu. Chem. Phys., 30,137 (1975). (5)H.F. Schaefer, Annu. Reu. Phys. Chem., 27, 261 (1976). (6)P. J. Kuntz in “Dynamics of Molecular Collisions”, part B, W. H. Miller, Ed., Plenum Press, New York, 1976,Chapter 2. (7)Faraday Discuss.Chem. SOC.,62 (1977). (8)D. G. Truhlar and R. E. Wyatt, Adu. Chem. Phys., 36,141(1977). (9)J. N. L. Connor, Comput. Phys. Commun., 17, 117 (1979). (10)F.T.Wall and R. N. Porter, J. Chem. Phys., 36,3256(1962);39, 3112 (1963). 0022-3654/81/2085-3618$01.25/0
cubic spline functions.l’ Such RMF cubic spline techniques9J2-18were shown to generate smooth, realistic surfaces representing rather accurate fits of the ab initio data. One of us (R.F.N.)lg introduced an empirical PES for collinear triatomics based on rotating a Morse curve about a swing point at the united atom to generate the electronic kinetic energy surface (T) and then, via the virial theorem, the PES (WT). By enforcing the so called T-normalizationm along each uniform scaling cut through the T surface, it was possible to determine the rotation-angle dependence of the PES cut in an a priori manner from known nuclear (11)J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, “The Theory of Splines and Their Applications”, Academic Press, New York, 1967. (12)J. N. L. Connor, W. Jakubetz, and J. Manz, Mol. Phys., 29,347 (1975). (13) J. M. Bowman and A. Kuppermann, Chem. Phys. Lett., 34,523 (1975). (14)S. K. Gray and J. S. Wright, J. Chem. Phys., 66, 2867 (1977). (15)S. K. Gray, J. S. Wright, and X. Chapuisat, Chem. Phys. Lett., 48,155 (1977). (16)J. S. Wright and S. K. Gray, J. Chem. Phys., 69,67 (1978). (17)W. Jakubetz, Chem. Phys., 35, 141 (1978). (18)J. S.Wright and S.-K. Shih, J. Chem. Phys., 73, 5204 (1980). (19)R.F.Nalewajski, Int. J. Quantum Chem., Symp., 14,483(1980). (20)(a) R. F. Nalewajski, Chem. Phys. Lett., 54, 502 (1978); (b) J. Phys. Chem., 82,1439(1978);(c) Int. J. Quantum Chem., Symp., 12,87 (1978).
0 1981 American Chemical Society
The Journal of Physical Chemistry, Vol. 85, No. 24, 1981 3819
Potential Energy Surfaces of Atom-Diatom Reactions
repulsion energy along the zero-uirial path (ZVP)% which is very close for collinear reactions to the steepest descent minimum energy path (MEP). The only data required to generate the PES were the coordinates and energy profile of the ZVP, WT = wT[zvP]. Subsequent extensive, comparativestudy of this novel RMF approach (T-RMF)21 has shown that the resulting WT surfaces for H3 and FHH, although lacking flexibility of other methods (including adjustable parameters), compare quite favorably with the H3 ab initio collinear surface22and the LEPSZ3collinear surface for FHH, respectively. This good agreement provides strong support for the idea that the main qualitative features of potential surfaces can be obtained from the ZVP data alone, by employing the virial theorem and its implication (the T-normalization)in construction of the PES through the T-RMF representation of the electronic kinetic energy component of the Born-Oppenheimer potential energy surface. By using the bond energy-bond order (BEB0)24*26 data to approximate the ZVP path and its profile, relatively accurate collinear WT surfaces for H3 and FHHlep21have been obtained from the T-RMF model. Such WTIBEBO] empirical surfaces represent therefore a direct extension of the BEBO path characteristics toward generating the full PES, similarly to the empirical approaches of Kafri and Berry26and Agmon and Le~ine.~’ The purpose of this paper is to generalize the WT[BEBO] method for the A BC triatomic reactive systems, to include bent geometries, and to show the efficiency of the T-normalization constraint in controlling the shape of general three-dimensional(3D)energy hypersurfaces for such systems. The paths and energy profiles necessary to represent the ZVP input data for bent geometries have been generated from the relaxed BEBO (BEBO/R) model. Within the BEBO/R model the starting, collinear BEBO geometry is modified after bending by allowing the AB and BC internuclear distances to relax against the bonding forces in these diatomic fragments, as a result of an increase in the AC repulsion in the bent structure. The reactions considered are the H + H-H and F + H-H exchange reactions.
>I
D aJ C
w
+
W,[ZVP(B)] Potential Surfaces from the T-RMF Model Consider a constrained triatomic reactive system
with the bond angle 0 held fixed. Construct orthogonal R1 and R2 axes (Figure 1) and choose the united atom configuration, R (R1,R2)= 0, as a swing point about which the Morse function will be rotated. Unlike the standard RMF approaches, the RMF is used within the T-RMF methodlQto represent the electronic kinetic energy surface, T(R,B), from which the potential surface, WT(R,B), is generated by the virial theorem. Following our previous (21)R. F. Nalewajski and R. Pastewski, Int. J. Quantum Chem., Symp., in press. (22)B. Liu, J . Chem. Phys., 58, 1925 (1973). (23)J. T.Muckerman, J. Chem. Phys., 54, 1155 (1970). (24)(a) H.S. Johnston, Adu. Chem. Phya., 3, 131 (1960); (b) H.S. Johnston and C. A. Parr, J. Am. Chem. Soc., 85,2544(1963);( c ) H.S. Johnston, “Gas Phase Reaction Rate Theory”, Ronald Press, New York, 1966. (26)L. Pauliig, J. Am. Chem. Soc., 69,542 (1947). (26)0.Kafri and M. J. Berry, Faraday Discuss. Chem. Soc., 62,127 (1977). (27)N.Agmon and R. D. Levine, J. Chem. Phys., 71, 3034 (1979).
Flgure 1. T-RMF method of rotating (around the united atom axis) a Morse function to construct the electronic kinetic energy surface, TdR), and, via the virial theorem, the model potential energy surface, W,.(R), for the atom-diatom chemical reactions.
notation,21let cp be the RMF rotation angle (Figure 1). Each cut through the T surface along the direction of the uniform scaling path (USP), R, = sR, represented by a single Morse function, is “normalized” to satisfy the following condition:20 Lm[T(sR,B)- T(m)] ds = V,(R,B)
(1)
where s is a scale factor, Vnn(R,B)is the nuclear repulsion energy for the nuclear configuration (R,B), and m refers to the A + B + C dissociation limit. This condition ensures that the WT surface will exhibit the correct, nuclear-repulsion-like behavior at short internuclear distances. Condition 1makes it possible to determine the parameters a, b, and c of the corresponding potential surface cut, defined by a current value of cp
from the coordinates, R(r,B), and energy, W(r,B), of the fixed B potential surface minimum along R,(cp) (point on the ZvP);20c r is the value of the reaction progress variable, e.g., Pauling’sE bond order of the BEBO method,24which corresponds to a current value of the rotation angle cp. To
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The Journal of Physical Chemistry, Vol. 85, No. 24, 1981
a class of triatomic reactions. Within the BEBO model AC interaction is purely repulsive: VAC(RS)= ~ / ~ D J ~ x P [ - ~ -P R3J.I ( R ~ + 2 exp[-P(R3 - R311 (7)
Figure 2. Schematic illustration of the BEBO/R approach used to generate the constrained (with angle 8 held fixed) reaction coordinate R(r,8) for bent trlatomic reactive systems from the collinear BEBO geometry, R(r); r is the Pauling bond order. The noncollinear geometry is obtained via the relaxation (due to the increased A-C repulsion in the bent system) of the bent BEBO geometry against the bonding forces in the AB and BC diatomic fragments.
determine the T-RMF parameters, one first solves the following numerical equation: exp(z)[(z + 1) exp(z) - 22 - 11 + zVnn[R(r,8),8]/W(r,8)= 0
z
>0
(3)
where W(r,8) is relative to the A + B + C separated atoms limit. The parameters, for a given bond angle 8, are c(cp,e) = z/R(r,B)
(4)
a(cp,@= 2vnn[R(r,e),6lz(ex~[zb(cp,B)c(cp,8)1 -
4 exp[b(cp,@~ ( c p , @ I l - ~(6) where R(r,8) = [Rl(r,8)2+ R2(r,8)2]1/2 To generate the WT surface, one requires only the ZVP data for a given angle 8, WT(R,t9)= wT[zvP(8)]. One may either use the ab initio ZVP data (for the ZVP-following algorithm see ref 20c and 21) or, alternatively, develop a reliable empirical model to generate realistic approximations to the ZVP(6)’s and their energy profiles. This approach is being followed in the present work. Our empirical ZVP(8) data are constructed from a simple modification of the collinear BEBO model, which we call the relaxed BEBO (BEBO/R) model, taking into account the bent geometries as well. Relaxed Bond Energy-Bond Order Model and empirical model treatBoth the ab ments6J6i26~34 have shown that both the coordinates and energy of the constrained MEP’s (with 8 held fixed) are changed as 6 increases. For those reactions which prefer the collinear pathway, the saddle-point height increases and the MEP is shifted to larger internuclear distances for increasing 8. Since we intend to base our model surfaces on the BEBO empiricism, they can be applied only to such (28) H. Conroy and B. L. Bruner, J. Chem. Phys., 47, 921 (1967). (29)I. Shavitt, R.M. Stevens, F. L. Minn, and M. Karplus, J. Chem.
DBeand Rse are respectively the AC diatom dissociation energy and the equilibrium internuclear distance, and 0 is the Morse potential parameter (=[k,/(2D,)]1/2 where k, is the harmonic force constant] for diatom AC. After bending the collinear BEBO structure, R(r) (Figure 2) Rl(r) = R1, - a In (1- r) (8) R2(r) = R2, - a In r
(9)
where a = 0.26 A, increased AC repulsion is the dominant contribution to the increase in the total energy along the constrained (with 6 held fixed) reaction coordinate.% Such increase in the repulsion between atoms A and C should cause their displacements along the AB and BC directions, respectively, against the current bonding forces in these diatomic fragments, to new positions (Figure 2): R(r,8) = R(r) x(r,8) (10)
+
Let kl(r) and k,(r) denote the force constants of these “diatomics” in the ABC reactive system, corresponding to a given value of the bond order r. We assume that the displacements x(r,B) can be determined by simply finding a minimum of the function E[x(r,B)I = f/z[k1~1~(r,6) + k2xZ2(r,@1 + V~dR,(r,@l- VAC[R~(~)I (11) where R3(r) = [R12(r)+ R?(r)
+ 2R1(r) R2(r)cos 8]1/2 (12) R3(r,8) = [RI2(r,8)+ R?(r,8) + 2Rl(r,8) R2(r,8)cos el1/’ (13)
The solution of the equation
is discussed in the Appendix. Once the relaxed BEBO geometry (eq 10) is determined from eq 14, the corresponding total energy, W(r,6) (relative to the A + B + C limit), is calculated from the standard BEBO expression: W(r,6) = -Dle(l - r)P1 - D2,rP2 + VAC[R3(r,8)](15) where pi’s are the familiar parameters defined within the BEBO method.24 The BEBO/R model described above requires, however, a knowledge of the diatomic fragment force constants, k[s, as functions of the bond order r. The choice of the k = k(r) relation is the subject of the next section. Finally, it should be noticed that physical intuitions upon which the BEBO/R model is based are similar to those used by Kafri and Berry26 and by Garrett and T r ~ h l a in r ~their ~ extensions of the collinear BEBO model to include bent structures of the reactive triatomic systems. Force Constant-Bond Order Relations for Diatomic Fragments The simplest possible lz = k(r) relation is the one assumed by Garrett and T r ~ h l a rin~ their ~ noncollinear BEBO model. Their modified Badger’s rule for changes (35) B. C. Garrett and D. G. Truhlar, J. Am. Chem. SOC.,101,4534 (1979).
The Journal of Physical Chemistty, Vol. 85, No. 24, 198 1 3621
Potential Energy Surfaces of Atom-Diatom Reactions
in the energy corresponding to a small bending of the collinear MEP configuration implicitly assumes that ki(ri) = rikie
b
1
-rT(H-H) 0.8
0.6
0
0.2
0.4
0.4
0.2
0
(16)
where ri is the bond order of the ith diatomic fragment (rl = 1 - r, r2 = r). Although this relation is in general agreement with what one would intuitively expect, its simple linear form requires at least some verification for both consistency and reliability, since the connection between the bond order r and the force constant is by no means clear in the BEBO model. Bond order r represents some measure of the amount of “chemical bonding”. It can be related to a force constant via empirical correlations between various molecular properties of isolated diatomics, such as k,, Re,and De, by assuming a validity of such relations beyond the equilibrium case, i.e., for the diatomic fragments of the reactive system. Following the BEBO intuitive assumptions, one may interpret the first two terms of expression 15 for the BEBO profile as representing current dissociation energies of “diatomics” AB and BC, respectively, during reaction Di(ri) = Dierp
(17)
Using the F r o s t - M ~ s u l i nempirical ~~ relation
ke(R, - RJ2/De = constant
(18)
where Ri is a constant related to anharmonic force constants, one finds the following k = k(r) relation:
where Ri(ri) is given by eq 8 and 9. Another class of k, = k,(R,) relations which one could In use to relate ki to ri is the Rc3-type c~rrelation.~’-~~ its most simple and reliable form40 it can be written as ki, = f i R i ~ ~
(20)
where f i is a simple function of the constituent atoms “coordinates” in the periodic table. This relation leads to ki(ri) = kie[Rje/Ri(ri)I3
(relation 2)
(21)
A similar relation, including empirical parameters though, results from Badgers’s4I rules
kie(Rie- d J 3 = constant
(22)
The resulting ki = ki(ri) relation is ki(ri) = kie{(Rie- di)/[Ri(ri)- di])3
(relation 3)
(23)
As a final correlation we consider the exponential relation.42i43We use the more universal version of Anderson and Parr43
kie = F exp(-j-Ri,)
(24)
(36) A. A. Frost and B. Musulin, J. Am. Chem. SOC.,76,2045 (1954). (37) R. G. Parr and R. F. Borkman, J. Chem. Phys., 49,1055 (1968); 60, 58 (1969). (38) A. B. Anderson and R. G. Parr, Theor. Chim. Acta, 26,301 (1972). 99, 4869 (1977). (39) R. G. Pearson, J. Am. Chem. SOC., (40) R. F. Nalewajski, J. Phys. Chem., 83, 2677 (1979). (41) R. M. Badger, J. Chem. Phys., 2, 128 (1934); 3, 710 (1935). (42) D. H. Herschbach and V. Laurie, J. Chem. Phys., 35,458 (1961). (43) A. B. Anderson and R. G. Parr, Chem. Phys. Lett., 10,293 (1971). (44) C.F. Bender, S. V. O’Neil, P. K. Pearson, and H. F. Schaefer, Science, 176, 1412 (1972).
I
0.6
0.8 r(H-F) --LI
1
Figure 3. Harmonic force constant, k , of the H-H and H-F diatomic fragments as a function of Pauling’s bond order, r , as suggested by various k, = k,(R,) empirical relations (Re is the equilibrium bond length and k , is the corresponding value of k ; see text): (1) the Frost-Musulin relation; (2) the RL3-type relations; (3) Badger’s rules; (4) the Anderson-Parr version of the exponential Herschbach-Laurie relation; the same key is used in Figure 4.
where, for the whole periodic table, F = 1.72 and l= 0.63 in atomic units. Equation 24 gives ki(ri) = kie exp(-l[Ri(r;) - Riel) (relation 4) (25) One could also consider the Ander~on-Parr~~ exponential relation including an explicit dependence on the nuclear charges 2, and Zy of the diatom XY ki, = 4.rrCZxZy exp(-[Rie) (26) where, for the whole periodic table, C = 0.12 and [ = 2.0 in atomic units. This relation also leads to eq 25, but with much larger { (=& By including the explicit form of the Ri(ri)for the collinear BEBO trajectory (eq 8 and 9) Ri(ri) = Ri, - a In ri (27) one obtains the approximate linear dependence for ki(ri): ki(ri) = kierpt = kieri (28) since a[ = 0.98. One may conclude, therefore, that the Garrett-Truhlar assumption is supported by the second Anderson-Parr relation (eq 26), derived from the Poisson equation treatment38of diatomic force constants. All of the above correlations assume the approximate validity of the k, = ke(R,,De) correlations also for the diatomic fragments of the reactive molecular system, involved in the “bond breaking-bond forming” process leading to an exchange reaction. Some numerical verification of this assumption will be presented in the next section, where the predicted BEBO/R paths and profiles
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The Journal of Physlcal Chemlstty, Vol. 85, No. 24, 1981 I
I
0
l
'
'
'
'
0.5
Nalewajskl and Pastewski
r-
~
'
"
'
I
't
YKk- Y
5
2 1
1
f
5 LL
1
1
1'5
R1/8-
2
Flgure 5. Constrained reaction coordinates(lor fixed 0 = O', 30°, 60°, 90°, and 120') for the ground state of H, resulting from the BEBO/R approach and various estimations of the diatomic fragment forceconstant Variations, based on (a) the Frost-Musulln relation, (b) the RL3-type relation and the Badger rules (indistinguishable wlthin the figure scale), and (c) the Anderson-Parr exponential relation. The Corresponding ab Initio saddle-point positions, obtained by uslng the data of ref 31, are also shown for comparison.
and the opposite is true when relation 4 is used to determine force-constant variations during reaction.
+
+
H H2and F H2Constrained Paths and Energy Profiles from the BEBO/R Model In order to test general reliability of the proposed BEBO/R model, we have applied it to the H + H-H H-H + H and H + H-F H-H + F reactions. The results, for 0 = Oo, 30°,60', 90°, and 120°, are summarized in Tables I and I1 (saddle-point data) as well as in Figures 4-6 (paths and their profiles). It follows from Tables I and I1 that the BEBO/R constrained saddle-point geometries, predicted from all of the considered force constant-bond order relations, are very close to each other, despite the differences observed in the plots of Figure 3. Predictions from relations 2 and 3 and eq 16 are almost identical. This is due to relatively small differences in the corresponding plots shown in Figure 3. As seen in Tables I and 11,as well as in Figure 5, predicted positions of the saddle point compare quite favorably with the ab initio (H,) and LEPS (FHH) data. Close inspection of the collinear paths and profiles (Figure 4) shows that the path and the saddle-point location are slightly shifted outward because of "bond relaxation. At the same time barrier profiles are slightly lowered. Similar trends have been observed in the extended BEBO model of Kafri and Berry.26 As 0 increases, the BEBO/R paths shift out to larger internuclear distances. Figure 5 shows that the best overall agreement with the SCF CI saddle-point positions has been obtained from the R-3-type relation and Badger's rules (Figure 5b). In the next section the BEBO/R path
-
Flgure 4. Collinear reaction coordinates for the (a) H-H-H and (b) F-H-H systems, and their energy profiles, as determined by the ab inlb calculatlons (for H, from ref 22), the standard BEBO model, and the BEBO/R model using various estimations of the diatomic fragment forcaconstant varlations; for key see Figure 3. Double designations represent curves which are practically indistingulshable within the scale of the figure.
are compared with the predictions from ab initio SCF CI calculations. Predicted force-constant variations for the H-H and F-H "bonds", resulting from relations 1-4, are shown in Figure 3. One can see that the plots correspondingto the R,-, and Badger's relations are quite close to the linear behavior shown also in the figure. According to relation 1"bonds" are much weaker compared to the linear plot,
-
Potential Energy Surfaces of Atom-Diatom Reactions
The Journal of Physical Chemistry, Vol. 85, No. 24, 1981 3623
Flgure 6. BEBOlR constralned reaction coordinates (for fixed 6 = O’, ?IO0, 60°, QO’, and 120’) and their energy profiles for the ground state of the H-H-F, as determined by using different estimations of the diatomic force-constant variations during reaction. The deslgnatlons (a-c) are the same as those for Figure 5. The broken-line curves of part b refer to the Badger-rules version.
and profile, based on relation 2, will be used as the ZVP input data in generating the model WT potential energy surfaces. In accordance with the ab initio32and LEPS23results for the FH2 reactive system, all versions of the BEBO/R model predict a significant decrease in the FH distance and a simultaneous increase in the HH distance (Table I1 and Figure 6) at the saddle point as the approach angle goes from 0’ to 120’. Again, relation 1leads to the largest changes in the BEBO/R paths as 0 increases (the most flexible “bonds” during reaction) while relation 4 gives comparatively strong “bonds” as reaction progresses. Note also the quite strong effect which the choice of the ki(ri) relation has on the shape of the BEBO/R energy profile (Figure 6). The BEBO/R energy profiles for H3 underestimate the real increase in the barrier height as 0 increases, as represented by rather accurate SCF CI results of Siegbahn
and Liu. However, the agreement between the LEPS and BEBO/R barrier heights for FH2 and various angles or approach (Table 11) appears to be quite satisfactory.
&ConstrainedWTIBEBO/R]Potential Energy H2and F H2 Surfaces for H The &constrained WT potential energy surfaces, generated from the BEBO/R-ZVP data, are presented in Figure 7 (H + H2 system) and Figure 9 (F + H2 system). In Figure 8 we show a comparison of the saddle-point uniform scaling cuts through the H3 WTIBEBO/R] and the ab initio SCF CI surfaces. For FH2 we compare our model surfaces with the corresponding 6-constrained LEPS surfaces that were obtained by using M u ~ k e r m a n ’ pas~~ rameters. In Table I11 we address the question of the accuracy of the 3D WTIBEBO/R] surface for HS. We give the standard deviation analysis of the WT model surface points from
+
+
3824
The Journal of Physical Chernistty, Vol. 85, No. 24, 198 1
Nalewajski and Pastewskl
R(H-H) 1A.U
/a u
E 14
16
Flgure 7. Summary of the model WTIBEBO] and W,[BEBO/R] fixed-angle potential energy surfaces for H3: (a) ab inltlo (from ref 22) collinear surface; (b) WTIBEBO] collinear surface; (c) the difference surface of a and b; the WTIBEBO/R] surfaces for (d) 6 = 30°, (e) e = 60°, (f) 6 = 90°, and (g) 6 = 120'. The corresponding BEBO/R paths and energy profiles have been generated by using the RL3-type k = k ( R ) relation to estimate the force-constant variations of diatomic fragments during reaction. The contour values are in units of kcai/moi.
Potential Energy Surfaces of Atom-Diatom Reactions
The Journal of Physical Chemistty, Vol. 85, No. 24, 1987 3825
TABLE I: Comparison of Predicted Saddle-Point Geometries ( A ) and Energies (kcal/mol, Relative t o H c H,) for Different Angles ( e ) of Approach of H to H,
120
ki(ri) 87
deg 0
30
method SCFCI BEBO BEBO/R
SCFCI BEBO/R
relation“
1 2 3 4 eq 16 1
2 3 4 eq 16 60
90
120
SCFCI BEBO/R
SCFCI BEBO/R
SCFCI BEBO/R
1 2 3 4 eq 16 1 2 3 4 eq 16 1
2 3 4 eq 16
R
-
(H%) energy 0.93 0.92 0.97 0.95 0.95 0.94 0.95 0.94 0.97 0.95 0.95 0.94 0.95 0.95 0.98 0.96 0.96 0.95 0.96 0.99 1.02 0.99 0.99 0.97 0.99 1.05 1.08 1.03 1.03 1.00 1.03
9.8 9.9 9.3 9.5 9.5 9.6 9.5 11.6 10.1 10.3 10.3 10.5 10.3 16.9 13.2 13.6 13.6 13.9 13.6 30.2 21.4 22.4 22.4 23.1 22.4 62.8 45.3 48.3 48.2 50.3 48.2
I
I
I
I
ref 22 24 this work
31 this work
31 this work
i 4
31 this work W, [ BEBOIR]
31 this work
See text.
the corresponding SCF CI surface points of Liu and Siegbahn22.31as a function of the maximum WT energy sampled. For comparison we also include the corresponding standard deviations of the somewhat related empirical surface of Kafri and Berry.26 As we see from Table 111, the results for H3compare quite satisfactorily with the ab initio surface, exhibiting similar standard deviations as does the BEBO-type empirical surface of Kafri and Berry. The deviation increases with increasing 8. This is obviously due to increasing errors in the BEBO/R energy profiles which we used to approximate the ZVP(8) input data of the wT[zvP(e)] procedure. Nevertheless, we feel that the H3 results (Figure 3) prove our point, which is that, using the virial theorem and the T-RMF model, one can generate quite realistic 3D potential energy surfaces for triatomic systems from the ZVP data and the T-normalization condition. One notes that this constraint produces increasing “blind”-valley features for large values of 8, similar to what has also been observed on other surfaces for this sytem, e.g., ref 26 and 34,
The difference surface contour diagram (Figure 7c) shows that the collinear H~,WT surface represents a rather good approximationto the ab initio surface at the repulsive wall region. This is a direct consequence of the T-normalization. One can expect this capability of the wT[zvP] surface, to reproduce with high accuracy this part of the surface, to be a general property of the T-RMF model, independent of the system to which it is applied, provided, of course, that the accurate ZVP data are used.lg Figure 9 shows that, in general, the main features of the FH~,WTmodel surfaces agree well with those resulting from the LEPS expression. The reactants-to-products
-WT[
1 “ -
BEBO]
I
1
I
2
3
4
i RIA.U.
5
Flgure 8. Symmetric stretching potentials of H3 for fixed angles 8 = Oo, 60°, 90°, and 120’. The ab initio SCF CI results are from ref 31. The WTIBEBO/R] results have been obtained by using the R,-3-type k = k ( R ) relation to estimate the force-constant variations of diatomic fragments during reaction.
channel, however, is somewhat narrower on the WT maps, especially in the F + H2 region. This relative narrowness is probably a result of the insufficient flexibility of the rotated Morse function when representing the uniform scaling cuts through the electronic kinetic energy surface. Note also that the much emphasized “bIind”-valley feature of the 6’ = 90’ LEPS surface (Figure 9e) contrasts with a much smoother passage between the two valleys on the corresponding W, surface (Figure 90. Taking into account the relative reliability of the WT surfaces in this repulsive region of the surface, we conclude that this “blind”-valley characteristic of the 8 = 90’ LEPS surface for FH, is probably an artifact of the model involved.
Discussion The main purpose of this paper was to test the usefulness of the T-normalization procedure in controlling a shape of the 3D potential energy surface for atom-diatom reactions, generated via the virial theorem from the TRMF surface. We consider results obtained for H3 to be very satisfactory since quite good approximations to the accurate ab initio surface have been generated in this way, even when using a rather crude BEBO/R approximation to the exact ZVP(t9) data. We conclude, therefore, that the idea of controlling shapes of the surface valleys through the T-normalization, in accordance with varying nuclear repulsion, is basically sound although the Morse function may not be the best possible analytical representation of
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The Journal of Physical Chemistty, Vol. 85, No. 24, 1981
Naiewajskl and Pastewski
-MEP -*-ZVP
- _"_._
BEBO/R
Flgure 9. Comparison between the LEPS and the W,[BEBO/R] potential energy surfaces of H-H-F for fixed angles (a, b) 8 = 30°, (c, d) 8 = 60°, and (e, 9 8 = 90'. The BEBO/R results correspond to the Re4-type k = k(R) estimate of the force-constant variations of diatomic fragments. The LEPS surfaces have been generated by using the Muckerman parameters (ref 23). The contour values are in units of kcal/moi. The paths shown are the LEPS minimum energy path (MEP), the zero-virial path (ZVP), and the BEBO/R path. The corresponding positions of the saddle point are marked by the circle (LEPS) and cross (BEBO/R) symbols.
Potential Energy Surfaces of Atom-Diatom Reactions
TABLE 11: Comparison of Predicted Saddle-Point Geometries ( A ) and Energies (kcal/mol, Relative to F-H t H) for Different Angles ( e ) of Approach of F to H, energy/ ki(ri) exo0, rela- R, - Rsp- thermdeg method tiona (H-k) (H-F) icityb refC 0 SCFCI 0.74 1.54 36.1/ 44 34.4 SCF CI 0.81 1.37 26.1/ 32 20.4 LEPS 0.76 1.60 29.7 23 0.77 1.54 30.8 BEBO 24 this work BEBO/R 1 0.78 1.67 30.2 0.77 1.57 30.7 2 3 0.77 1.59 30.6 4 0.77 1.56 30.7 eq 16 0.77 1.59 30.6 30 SCFCI 0.83 1.32 27.7/ 32 20.4 LEPS 0.76 1.52 30.2 23 BEBO/R 1 0.79 1.62 30.6 this work 2 0.78 1.55 31.2 3 0.78 1.54 31.1 0.78 1.50 31.3 4 eq 16 0.78 1.54 31.1 60 LEPS 0.81 1.32 33.3 23 BEBO/R 1 0.81 1.50 32.7 this work 0.81 1.44 33.7 2 0.81 1.44 33.5 3 4 0.81 1.40 33.9 eq 16 0.81 1.44 33.5 0.99 1.24 37.9/ 32 90 SCFCI 20.4 0.91 1.24 47.6 23 LEPS BEBO/R 1 0.91 1.34 40.9 this work 2 0.88 1.30 43.0 0.88 1.31 42.5 3 4 0.87 1.27 43.7 eq 16 0.88 1.30 42.6 120 LEPS 1.01 1.32 65.2 23 BEBO/R 1 1.08 1.30 75.1 this work 2 1.03 1.23 81.6 3 1.04 1.25 79.8 4 1.00 1.20 84.5 eq 16 1.03 1.24 80.6 a See text. To facilitate a graphical comparison, the MuckermanZ3value of the reaction exothermicity (28.8 kcal/mol) has been assumed in all of the LEPS and BEBO/ R calculations. The LEPS results for e # 0 have been obtained by using the same parameters as those for the collinear case.
the electronic kinetic energy cut along the uniform scaling direction. It is obvious that there is room for improvement in two directions. First, one may generate more accurate ZVP(8) coordinates and profiles by actually following the ZVP on the explored &constrained surface. This should be especially important for large values of 8 for which ZVP deviates significantly from the minimum energy path and its BEBO/R approximation. The second direction is to use a more flexible rotated function representation. For example, within the WT[BEBO/R] model one could use a four-parameter function, exhibiting the same general shape as does the Morse function, fitted to reproduce also the current curvature of the uniform scaling cut, which results from the corresponding values of the force constants of diatomic fragments, ki(r). We believe that the wT[zvP(e)] model surfaces can be useful in quickly predicting the main features of the triatomic potential energy surfaces from a few cubic spline representations of the ZVP(8) data, obtained from a limited number of points determined from the ZVP-following
The Journal of Physlcal Chemlstry, Vol. 85,No. 24, 1981 3827
TABLE 111: Standard Deviations (SD’s)‘ of the WTIBEBO/R] (Relation 2)Surface Points for H,, from the SCF CI Collinear (LiuZ2)and Noncollinear (Siegbahn-Liu”) Surface Points as a Function of the WT Maximum Energy Sampledb’ maximum energy sampled, kcal/mol all e. dee 20 30 40 PointsC 2.20 3.47 1.81 0 SD 1.33 nb 102 120 137 91 30 SD 2.14 2.17 3.24 5.13 41 29 34 n 25 60 SD 2.70 3.11 3.19 429 n 26 33 39 50 90 SD 1.41 6.19 6.42 6.52 n 4 21 27 33 120 SD 16.28 7 0 0 n 0 total: WT SD 1.80 2.93 3.34 5.08 [ BEBO/R] n 146 185 220 268 Kafri and SD 1.52 2.20 2.82 Berry“ n is the number of sample points. a In kcal/mol. All SCF CI points have been compared.
algorithm. Finally, an application of our model surface to the classical trajectory studies would require a use of the cubic spline representation of the ZVP(0) input data and the parameters of the rotated function, in order to facilitate efficient repeated calculations of the potential energy surface and its gradient. Acknowledgment. R.F.N. is pleased to express here his gratitude to Professor James E. Boggs and Professor Per-Olov Lowdin for their kind invitations to attend the 1981 Texas Conference on Theoretical Approaches to Chemical Dynamics and the 1981 Sanibel Symposium, respectively, where this work has been presented. R.F.N. also thanks Professor Robert G. Parr and the National Institutes of Health for enabling him to spend 3 months at the University of North Carolina, Chapel Hill, NC, where this text was completed.
Appendix Solution of Eq 14. In order to solve eq 14 to find x(r,e), and, hence, the current point on the BEBO/R path, R(r,e) (eq lo), we take partial derivatives
where E(xl,x2)is defined by eq 11. The explicit forms of eq A1 are M ) [ R 3 ( r )+ x~lxi= Y2QePB(r)(Ri(r) + xi + [Rj(r) + x j ] COS e) e x p ( - ~ x ~x) [B(r)exp(-Px3) + 11 i, j = 1, 2; i # j (A2) where (see eq 12 and 13) x 3 = R3[Rl(r)+xl,R2(r)+x2,81- R3[Rl(r),R2(r),el (A3) B(r) = exp(-P[R~(r)- hell (A4) By dividing eq A2 (i = 1, 2) one obtains the equation kl(r)xl - Rl(r) + x1 + [Rz(r)+ x 2 ] cos e -(A5) k2(r)x2 R2(r)+ x 2 + [Rl(r)+ xl] cos 8 which can be solved to give xi = x ; ( x j ) (i # j ) . One obtains then the quadratic equation ki(r) cos 8 xi2 + (ki(r){[Rj(r) + x j ] + Ri(r)cos 8)k,(r)xj)xi- kj(r)xjRi(r)+ [Rj(r)+ x j ] cos 8) = 0 (A6)
J. Phys. Chem. 1981, 85, 3628-3635
3628
For small xi ( x i = xl for R1 IR2, xi = x2 for R2 IR J the solution is xi(xj) = [-ki(r)[Rj(r) + xi + Ri(r) cos e] + kj(r)xj + ((ki(r)[Rj(r)+ xj + Rl(r) cos e] - kj(r)xjI2- 4ki(r) x kj(r)xj(Ri(r) + [Rj(r) + xj cos e]) cos e)1/2]/(2ki(r)cos 8 ) (A71 One finds the solution of eq A1 by numerically solving eq A2 for xi, with xj calculated from eq A7, i.e.
d E[xi,xj(xi)]/dxi = 0
(A8)
Since xi is small compared to Ri(r), one may simplify eq A5 to
Hence
Thus, one may alternatively find approximate solutions to eq 14 by solving numerically the sum of eq A2 (i = 1, 2) for xi, with x calculated from eq A10. Numerical testa of both ways of!solving eq 14 show that, for H + H2 and F + H2reactive systems (0 = O”, 30°,60°,90°, and 120°), the approximate relation, eq A10, gives practically identical solutions (up to three decimal figures).
Adsorption of Hydrocarbons on Silica-Supported Water Surfaces Gllles M. Dorrls and Derek G. Gray” Pulp and Paper Research Institute of Canada and Depatfment of Chemlstty, McGlll Unlvers#y,TMontreal, Quebec H3A 2A7, Canada (Received: March 24, 798 7; In Final Form: June 30, 798 1)
A gas-chromatographic technique was used to measure the adsorption from the vapor phase of a series of n-alkanes, at zero and finite coverages on the surface of water-coated silica having a uniform large pore size. The presence of one to two water monolayers preadsorbed on the glass was sufficient to reduce considerably the London force field of the glass. At water loadings greater than 3.8% by weight, the effect of the solid on the adsorption of n-alkanes appears to vanish, and the surface properties reach a steady state, as revealed by adsorption isotherms and by the differential heats of adsorption at zero coverage. Contrary to some results in the literature, the heats of adsorption of n-alkanes are found to be smaller than the heats of vaporization. Thus, the model postulating water surface restructuring by hydrocarbon molecules is not supported by the GC data. Thermodynamic functions agree well with some other reported data, but the surface excesses and surface pressures for the n-alkanes are larger in the present study. The surface of the water-coated glass has a London component of the surface free energy of 23 mN m-l, compared to 22 mN m-l for bulk water.
Introduction The adsorption of hydrocarbons has been measured by gas chromatography at zero and finite coverages on the surface of water-swollen cellulose’ and film.2 The surface properties did not approach those of pure water at the highest accessible water contents, probably because cellulose absorbs water to form a water-swollen gel. Because the properties of water at the liquid-vapor interface are of great interest in various contexts, our previous studies are here extended to a simpler system. The adsorption of hydrocarbon vapors is measured on a nonswelling macroporous glass/water system, which at high water contents should provide a surface approaching that of pure water. The adsorption of a number of hydrocarbon vapors on water surfaces has been studied by measuring the change in surface tensions with partial pressure of organic vap o r ~ . ~ - ’ Adsorbates ~ such as n-alkanes, branched alkane~,*~**~’ and aromatic hydrocarbon^^^^ invariably yield type-I11 isotherms, indicating that water acts as a lowenergy surface toward nonpolar vapors.13 This conclusion is also consistent with the high interfacial tensions and small works of adhesion a t the liquid hydrocarbon-water interfaces. According to Fowkes,14J5the work of adhesion at a hydrocarbon-water interface is governed mostly by London force interactions so that the active part of the 3420 University Street. 0022-3654/81/2085-3628$01.25/0
surface tension of water (Le., its London component) is 22 mN m-’ at 20 “C. Adamson et al.16-20have questioned this apparently (1)Dorris, G.M.; Gray, D. G. J. Chem. SOC.,Faraday Trans. 1 1981, 77,713,725. (2)Katz, S.; Gray, D. G. J. Colloid Interface Sci. 1981,82,339. (3)Hayes, E. I.; Dean, R. B. J. Phys. Chem. 1953,57,80. (4)Cutting, C. L.;Jones, D. C. J. Chem. SOC.1955,4067. (5)Jones, D. C.; Ottewill, R. H. J. Chem. SOC.1955,4076. (6)Jones, D. C.; Ottewill, R. H.; Chater, A. P. J. h o c . Int. Congr. Surf. Act., 2nd, 1957 1957,1, 199. (7)Blank, M.; Ottewill, R. H. J. Phys. Chem. 1964,68, 2206. (8) Hauxwell, F.; Ottewill, R. H. J. Colloid Interface Sci. 1968,28,514. (9)Hauxwell, F.; Ottewill, R. H. J. Colloid Interface Sci. 1970,34,473. (10)Massoudi, R.;King, A. D., Jr. J. Phys. Chem. 1974, 78, 2262. (11)Massoudi, R.; King, A. D., Jr. In “Colloid and Interface Science”; Kerker, M., Ed.; Academic Press: New York, 1976; Vol. 3,p 331. (12)Jho, C.; Nealon, D.; Shogbola, S.; King, A. D., Jr. J. Colloid Interface Sci. 1978,65,141. (13)Vidal-Madjar, C.; Guiochon, G.; Karger, B. L. J. Phys. Chem. 1976,80,394. (14)Fowkes, F. M. Ind. Eng. Chem. 1964,56,40. (15)Fowkes, F. M. In “Chemistry and Physics of Interfaces 11”;Ross, S., Ed.; ACS Publications: Washington, DC, 1971;p 153. (16)Adamson, A. W.; Dormant, L. M.; Orem, M. J.Colloid Interface Sci. 1967,25,206. (17)Orem, M. W.; Adamson, A. W. J. Colloid Interface Sci. 1969,31, 278.
(18)Adamson, A. W.; Shirley, F. P.; Kunichika, K. T. J. Colloid I n terface Sci. 1970,34,461. (19)Adamson, A. W.; Orem, M. W. In “Progress in Surface and Membrane Science”; Cadenhead, D. A., Danielli, J. F., Rosenberg, M. D., Eds.; Academic Press: New York, 1974;Vol. 8, p 285.
0 1981 American Chemical Society
