The Journal of
Physical Chemistry
0 Copyright, 1984, by the American Chemical Society
VOLUME 88, NUMBER 10
MAY 10, 1984
LETTERS An Analytical Model of the Heat of Chemisorption: Some Quantitative Estimates for Atomic Radicals Evgeny Shustorovich Research Laboratories, Eastman Kodak Company, Rochester, New York 14650 (Received: September 27, 1983)
Our analytical LCAO-MO model of the heat of chemisorption Q has been modified to incorporate the self-consistent adjustment of a radical adorbital energy. We present the new analytical expressions for Q and discuss the parametrization procedures. By using the experimental data on Q for H and N on W( 1 lo), we calculated the values of Q for H, N, and 0 on Ir( 111) and Pt( 111) as well as for 0 on W( 1 lo), in remarkable agreement with experiment. Some conceptual conclusions and model perspectives are briefly discussed.
Introduction Recently we have developed an analytical LCAO-MO perturbation model of the heat of chemisorption Q’ which proved to be successful for a qualitative understanding of the observed periodic trends.’s2 Our band-structure model1%* employs the Hiickel-type Hamiltonian and treats Q as the difference in total energies of an adatom A and a metal surface M before and after chemisorption. Since our model’,2is not self-consistent, however, it could not provide for quantitative estimates of Q, ostensibly for chemisorption of electronegative radicals when Q may be very large (several electronvolts) and significant electron transfer may occur. The aim of the present work is to incorporate some self-consistency in our analytical model’ to make it capable of quantitative projections on Q. Accordingly, we will develop a new analytical formalism to treat Q and discuss the appropriate choice of parameters. As examples, we will calculate Q for atomic chemisorption such as H, N, and 0 on W( 1lo), Ir( 11 l ) , and (1) Shustorovich, E. Solid State Commun. 1982, 44, 567. (2) Shustorovich, E.; Baetzold, R. C.; Muetterties, E. L. J . Phys. Chem. 1983.87, 1100.
0022-3654/84/2088-1927$01.50/0
Pt( 111). We will compare our results with experiment and briefly comment on them. As earlier,’ we consider the formation of a monolayer A / M and assume that within each irreducible representation for the relevant unit cell of A/M, an adorbital xA interacts with one effective (symmetry-adapted) metal d orbital xM. The xA band is assumed to be degenerate (of initial energy E ~ ) ,but the xM subband (width W) is divided by the Fermi level EF into the and vacant WvaCparts ( W = WCc + Wac). For occupied WCc the normalized xMorbital density of states n ( e ) , we choose the simplest rectangular type (first introduced by Friede13and now in common use4), namely n ( e ) = 1/ W = constant
(1)
(3) Friedel, J. In “The Physics of Metals”; Ziman, J. M., Ed.; Cambridge University Press: London, 1969. (4) See, for example: (a) Harrison, W. A. “Electronic Structure and the Properties of Solids”; W. H. Freeman: San Francisco, 1980; Chapter 20. (b) Pettifor, D. G . Phys. Rev. Lett. 1979, 42, 896. (c) Heine, V.; Samson, J. H.; Nex, C. M. M. J. Phys. F1981,11,2645. (d) Chan, C. T.; Louie, S. G. Phys. Reu. B: Condens. Matter 1983, 27, 3325.
0 1984 American Chemical Society
1928 The Journal of Physical Chemistry, Vol. 88, No. IO, 1984
TABLE II: Atomic Chemisorotion on bcc W(11OI" . ,
TABLE I: Parameters Used in Calculations of O(EF)' H N 0
13.60 13.40 14.80
1.3 1.950 2.275
12.85 12.04 14.05
'Standard EH values for ZA and {A; UA = 0.58yA0 ( T H O = 0.625{H, = 0.3914{~,0au). See text.
YN,O'
Assuming all the d orbitals (d subbands) to be degenerate,'s2 we have
lowac/w Nd/Nh = v / w v a c
N,j = l o p / w
Nh =
(2)
where Nd ( N h )is the total d-electron (hole) count. Remember also that within our nonself-consistent perturbation approach,'V2 the heat of chemisorption of an atomic radical is, to first order
Q = k[EF - EA
+ 2P2wac//(E~-
EA)]
(3)
where (3 is the usual Hiickel-type (off-diagonal) matrix element between xA and xM,k is the number (degeneracy) of the singly occupied orbitals xA ( k = 1 for H or halogens, k = 2 for 0, k = 3 for N , etc.). Qualitatively, lEFl may be identified with the work function 4 (EF = -4).
Results Consider a singly occupied adatom orbital xi as a set of two (cy and P) spin orbitals I);+; where = (xAa)' is occupied but r&?+ = (xAP)O is vacant. For a free atom A, $A and dA are assumed to be degenerate and of energy eA. Under chemisorption, since 0
N
IA
- EA
(5)
that the electrons will flow until becomes equal to E p Then, from eq 4,the "self-consistent" electron transfer q A (eq 6) results qA
=
(EF
- €A)/UA
(6)
in the energy gain Q(EF) (eq 7), which is the first contribution to the heat of chemisorption Q.
QWF)=
LAfiF
(EF- E ) / ~ A = (EF- E
ad181, Q((3),' Q(EF)$ Qib eV (kcal)/mol atom eV eV/mol eV/mol calcdb exptl H 0.671 0.25 2.70 2.95 (68) 2.95 (68)g N 1.99' 3.96 2.76 6.72 (155) 6.72 (155)* 0 1.62' 1.97 3.24 5.21 (120) 4.5 (104)5.6 (129y 1.20k 1.25 3.24 4.49 (104)
% f f
'The fcc W(110) parameters used: EF = -5.25 eV, W = 10 eV, Nd = 5 (Nh = 5). See text. 'Equation 11 (n = 1). 'Equation 10 (n = 1). dEquation 7. ' A Q = (QeXptl - QWld)/Q,, tl. /Scaled from the experimental data on Q. BTable I1 in ref 2. RTable 5.3 in ref lob. 'Scaled from via Sm/SMo.jReference 16. kTaken equal to the "experimental" value of blro(see Table 111).
TABLE III: Atomic Chemisorption on fcc Ir(ll1)' Q, eV (kcal)/mol adIPI, Q(PL Q(EF), atom eV eV/mol eV/mol calcd exptl H 0.67' 0.23 2.39 2.62 (60) 2.73 (63)' N 1.9gd 3.44 2.43 5.87 (135) 5.49 (127)' 1.82' 3.06 2.43 5.49 (127) 0 1.621 1.73 2.91 4.64 (107) 4.03 (93)' 1.508 1.56 2.91 4.47 (103) 2.91 4.03 (93) 1.20' 1.12
AQ9
% 5 -6 e -15 -10 e
'The fcc Ir(ll1) parameters used: E , = -5.75 eV, W = 10 eV, Nd = 8 (Nh= 2). See text and Table I1 for explanations of the table entries. 'Taken equal to PWH. CTable5.3 in ref lob. dTaken equal to OWN. 'Scaled from the experimental data on Q. /Taken equal to Pw0. gScaled from the "experimental"value of PIrNvia S M N / S M e
TABLE IV: Atomic Chemisorption on fcc Pt(ll1)'
(4)
can assumes the energy destabilization AE(eq 4) to be proportional to the electron transfer q. The coefficient UA is commonly identified with the one-center Coulomb integral yA,which, in turn, relates to the difference between the valence-state ionization potential I, = -eA and electron affinity EAJj6 (eq 5 ) . Suppose UA = YA
Letters
A ) ~ / ~ ~ (7) A
Another contribution to Q (let us denote it as Q(P))originates from mixing of the $A and +A subbands with the XM one. Within the Hiickel-type approximation, once eA' = EF, changes in the +A occupancy do not affect the total energy of the system (since there cannot be occupied levels above EF and vacant levels below EF). Thus, we can start with the fully occupied subband I ): of energy t A and the vacant subband & of energy EF,so that Q(P) will, in principle, include both the donor @ (from qA)and acceptor Q" (from dA)contributions. By using the analytical expressions (5) See, for example: Einstein, T. L.; Hertz, J. A,; Schriffer,
J. R. In
"Theory of Chemisorption"; Smith J. R., Ed.; Springer-Verlag: West Berlin, 1980; p 191. (6) See, for example: (a) Pariser, R. J . Chem. Phys. 1953, 21, 568. (b) Hinze, J.; Jaffe, H. H. J. Am. Chem. Soc. 1962, 84, 540. (c) Pilcher, G.; Skinner, H. A. J . Inorg. Nucl. Chem. 1962, 24, 937. (d) Orloff, M. K.; Sinanoglu, 0. J. Chem. Phys. 1965, 43, 49.
H N 0
0.67' 1.9gd 1.82e 1.621 1.20*
0.20 2.76 2.49 1.43 0.96
2.29 2.32 2.32 2.80 2.80
2.49 (57.4) 2.47 (57)' 5.08 (117) 5.50 (127)' 4.81 (111) 4.23 (97) 3.68 (85)g 3.76 (87)
-1 8 12 -14 -2
'The fcc Pt(ll1) parameters used: EF = -5.93 eV, W = 10 eV, Nd = 9 (Nh = 1). See text and Table I1 for explanations of the table entries. bTaken equal to PWH.CTable5.3 in ref lob. dTaken equal to PwN. 'The "experimental" value of PIrN(see Table 111). /Taken equal to pw0. gcampbell, C. T.; Ertl, G.; Kuipers, H.; Segner, J. Surf. Sci. 1981,107,220. *Taken equal to the "experimental" value of PIrO(see Table 111). for and Q" found earlier' (for the xAsubband of double density of states compared with the $A or ones), we have
Q(P) = h(QD + QA) = P2wac/W(EF-EA)
+ P2[ln (WacPc//3')+ 3 / ] / W (8)
Now assume that the self-consistent effects on Q(P) may be incorporated via some variational coefficients m and n, namely Q ( @ )= l / z ( m Q D+ ne"). Since QA >> QD for all reasonable values of the parameter~,'*~,~ henceforth we will assume Q(P) E 1/2nQA,so that the total heat of chemisorption Q will be
Q = QWF)+ Q(P)
(EF- eAl2/2uA + np2[ln ( ~ a c " u m c c / / p z )
+ 3/]/w(9)
When we turn from a monovalent to a k-valent atomic radical A (each participating orbital xA is treated as eq~ivalent'*~,~), the Q(EF)term remains analytically the samegbut the Q(P) one should (7) Shustorovich, E. J . Phys. Chem., in press. ( 8 ) (a) Varma, C. M.; Wilson, A. J. Phys. Rev. E Condenr. Matter 1980, 22, 3795. (b) Andreoni, W.; Varma, C. M. Ibid. 1981, 23, 437.
The Journal of Physical Chemistry, Vol. 88, No. IO, 1984 1929
Letters be multiplied by k (cf. eq 3), namely Q(P) = W 2 b( W a c p / P 2+) 3/21/W and Q = (EF- fA)2/2uA
+ knP2[h (
(10)
+ yz]/w (11)
~ a c ~ / @ z )
By using eq 11, we can now make quantitative estimates of Q provided we have accurate enough values of the parameters UA, n, 0, and W = Wac F . W e have explored various ways of obtaining these parameters and found the results (by eq 11) to be surprisingly reproducible. The thorough discussion will be given el~ewhere.~ Here we will show only a few examples of how our model works. More specifically, we will calculate Q for chemisorption of H, N, and 0 on bcc W( 1lo), fcc Ir( 11l ) , and fcc Pt( 11 l ) , where the most systematic experimental data are available.1° The results are summarized in Tables I-IV. Table I lists the parameters needed for calculating the Q(EF) term (eq 7). We used the standard extended Huckel (EH) values of ZA = -eA and of the orbital exponents CAS For H, the experimental values of ZH (1 3.60 eV6b)and EH (0.75 e V b ) lead to the “empirical” value of UH =‘12.85 eV. The “theoretical” value of UH may be obtained as UH = Y~ (eq 5) if we use the closed form of yHo.= 0.625fH aull (1 au = 27.21 eV) and some correcting coefficient rA(eq 12) to take care of electron correlation and other
+
VA = YA = TAYA’ (12) effects.6d For CH = 1.3, we find T H = 0.58. We accept the same coefficient T~ = 0.58 for all A’s because for the first-row atoms (where for the 2p orbital, yA0 = 0.3914fA au”), a similar procedure gives T A = 0.59, 0.56, and 0.54 for C, N , and 0, respecti~ely.~ As said above, we will focus here on the 5d transition series. In particular, for W, Ir, and Pt, self-consistent values of W = u.ooC + Wacare 11.95 = 6.05 5.90, 10.07 = 8.06 2.01, and 8.17 = 7.33 + 0.84 eV, respectively,12which in our scheme (cf. eq 2) correspond to Nd = 5.06, 8.00, and 8.97. Thus, henceforth we assume for W, Ir, and Pt the same bandwidth W = 10 eV and the d-band occupancies 5, 8, and 9 (which uniformly correspond to the metal d%I electronic configurations). The procedure was as follows. First, by using the experimental data on EF = -413 and the parameters ZA and UA(Table I), we calculated Q(EF) (eq 7). Then, by using the experimental data on Q for tungsten, available for H and N (Table 11), we estimated Q(P) and then P (eq 9-1 1). The preferred choice of tungsten is clear from the structure of Q(P) (eq lo), which appears to be least sensitive to the values of Nd (Nh)for the middle (5-7 group) metals but most sensitive for the ending (83 group) metals.14 The
+
+
(9) (a) The reviewer suggested to multiply the Q(EF)term in eq 11 by factor k because each orbital xAcan contribute to the overall electron transfer. However, this is just a matter of scaling of the parameter UA(eq 4-7 and 12), and our scheme takes care of this difference. Indeed, we scale UAeither as UA= ZA - E A from empirical values of the ionization potential I , and electron affinity EA or as the approximate expression UA 0.557, (eq 12) where, for the 2p orbitals, the relevant Coulomb integrals Y~ corresponding to one- vs. two-orbital interactions are very close (0.39c vs. 0.35f, respectively”). One can add that even for transition-metal atoms (where all five d orbitals can participate in bondin the use of the same (averaged) parameter UAof the eq 4 type is common% The scaling of UA is discnssed in detail elsewhere.’ (b) See, for example: Pasturel, A,; Hicter, P.; Cyrot-Lackmann, F. Solid State Commun. 1983, 48, 561 and the references therein. This paper also provides further examples of the appropriateness of the rectangular d-band density-of-state (10) (a) See, for instance, Tables 5.3 and 5.4 in ref 10b and Table I1 in ref 2. (b) Ertl, G. In ’The Nature of the Surface Chemical Bond”; Rhodin, T L., Ertl, G., Eds.; North-Holland Publishing Co.: Amsterdam, 1977; Chapter 5. (11) Roothaan, C. C. J. J . Chem. Phys. 1951, 19, 1445. (12) The renormalized atom calculations by Hodges, Watson, and Ehrenreich (cited in Table I of ref 8a). !13) See, for example: Holzl, J.; Schulte, F. R. “Solid Surface Physics”; Springer-Verlag: West Berlin, 1979; Table 4.3. (14) For the same reasons, in the straightforward band-structure calculations on Q,8a the results for N i and Pd could not be obtained directly but only by extrapolating those for the d-band occupancies less than 9.
-
“experimental” values of P for the W-H and W-N bonds were used also for other metal-adsorbate bonds. More specifically, for a given A, the values of PAM across the M series were kept but for a given M, the values of P A M were taken proportional to the relevant overlap integrals SAM,as justified elsewhere.7 The experimental values of Q for N and 0 on Ir( 111) were also used in a similar way to independently scale p)s, which were compared with those obtained from N/W(110). The accuracy of calculations was characterized by the error AQ = (Qexptl - Qcald)/Qexptl.We found7 Qcalcdto be rather insensitive to the choice of the “self-consistent” parameter n (eq 9-1 1) within the range n = 0.5-2. Because the best overall accuracy was found 1 (which nicely corresponds to realistic expectations for for n the “self-consistent” hole count of the acceptor spin orbital +A), we will report here (Tables 11-IV) the representative data for n = 1 only. The appropriateness of the values of the parameters used will be discussed in detail e l ~ e w h e r e . ~
Discussion The essence of our model (cf. also eq 3) is the partitioning of Q into two terms, Q(EF) and Q(P), reflecting, after all, the dual nature of Q originating from both delocalized and localized metal-adsorbate interactions.2 As follows from Tables 11-IV, the values of Q(EF) increase along the series H < N < 0, in accordance with the conventional electronegativity of these atoms. However, the relative contributions of Q(EF) to Q change in a different order: namely, the Q(EF) term is overwhelming for H, major for 0, but minor for N, reflecting a rapid increase of Q(P) along the series H < 0 < N ( k = 1, 2, and 3, respectively). For the M-H bonds, the accuracy of calculations on Q is very high (AQ = 1-5%).15 For the M-N bonds, where the scaling of P is most direct (taken as the “experimental” value for the W-N bond), the accuracy remains high (AQ = 6-8%). Even for M - 0 bonds, when the scaling of is less direct (via the “experimental” PWNcorrected by the overlap integrals S M N / S M o 7 ) , the results are still fairly accurate (AQ I15%). Moreover, if we use the “experimental” value of P for the Ir-0 bond, AQ for the Pt-0 bond proves to be amazingly small (
