J . Phys. Chem. 1985,89,2581-2585
species that are bracketed and denoted with a dagger superscript may or may not be surface stable. Competing dissociation and desorption pathways are available for both N O and N 2 0 in this mechanism. The relative importance of each step is dependent on the nature of the oxygen overlayer with steps 2 and 9 dominating clean W ( 100) chemistry. Dimer formation is predicated on the inhibition of NO dissociation via the presence of surface oxygen. Nitric oxide forms a centrosymmetric dimer in the solid state as established by x-ray d i f f r a c t i ~ n . More ~ ~ recently, dimeric N O has been studied by using X-ray and ultraviolet photoemission in condensed multilayer^^^ and in N O adsorbed on Ag(ll1) at 20 K." Previously reported infrared studies of nitric oxide adsorbed on supported molybdena catalysts also evidence dimer f ~ r m a t i o n . ~Formation ' of dimeric nitric oxide on P t ( l l 1 ) has also been previously proposed on the basis of vibrational although further investigation supported a model with monomeric N O with more than one adsorption (38) Dulmage, W. J.; Meyers, E. A.; Lipscomb, W. N. Acta Crystallogr. 1953, 5, 760. (39) Tonner, B. P.; Kao, C. M.; Plummer, E. W.; Caves, T. C.;Messmer, R. P.; Salaneck, W. R. Phys. Rev. Lett. 1983, 51, 1378. (40) (a) Behm, R. J.; Brundle, C. R. J . Vac. Sci. Technol., A 1984, 2, 1020. (b) Nelin, C . J.; Bagus, P. S.; Behm, J.; Brundle, C. R. Chem. Phys. Let. 1984, 105, 58. (41) Perl, J. B. J . Phys. Chem. 1982, 86, 1615. (42) Ibach, H.; Lehwald, S. Surf.Sci. 1978, 76, 1. (43) Gland, J. L.; Sexton, B. A. Surf. Sci. 1980, 94, 355. (44) Severson, M. W.; Overend, J. J . Chem. Phys. 1982, 76, 1584. (45) The atomic nitrogen on the -(2X1)-0 which results in NZWdesorption above 1000 K may result from either direct NO or N 2 0 dissociation on the
surface. (46) The extent of dissociation to atomic nitrogen of N2, N20, and NO
were dependent on their initial coverage. Quantitative measurements of the dissociation probability could not be made due to the nonlinearity in the heating rate required to attain temperatures sufficient to desorb N, formed via atom recombination.
2581
The nitric oxide dimer formed may not be surface stable, but may react rapidly to form adsorbed N 2 0 . The similarity of the thermal desorption data for NzO produced from N O vs. N,O adsorption suggests that N 2 0 is produced from NO prior to desorption. This assertion is also consistent with the observation of production of adsorbed N z O on W( 110) following high N O exposures at 100 K.21 The nitric oxide dimer may be stable at lower temperatures or on the highest coverage oxygen adlayer, W( 100)-p(2X2)-0. The observation of molecular N O desorption at 175 K on the -p(2X2)-0 surface suggests that nitric oxide dimer may be present on this surface at low temperature. Further spectroscopic investigation is necessary to identify the surface stable intermediates. Conclusion
Key features of the chemistry of N O and NzO on W( 100) and oxygen-pretreated W ( 100) surfaces have been identified. The presence of surface oxygen inhibits the dissociation of NO, NzO, and N,. As a result of the inhibition of N O dissociation, nitric oxide reacts to form N z O via a nondissociative pathway on the oxygen-pretreated surfaces. The production of N 2 0 is postulated to proceed via an N O dimer which reacts to form adsorbed nitrous oxide. We submit that the inhibition of nitric oxide dissociation is critical in the proposed dimer formation, as opposed to a drastic change in the N O bonding mode. The inhibition of dissociation would allow for the natural tendency of N O to form a dimeric species. Comparison of this work with previously reported studies of NO and N 2 0 adsorption on W(110) suggests a relative insensitivity to surface crystallography. Furthermore, qualitatively similar behavior is associated with all oxygen-pretreated surfaces studied.
Acknowledgment. We thank and acknowledge Prof. T. N. Rhodin and the Cornel1 Materials Research Laboratory for supplying the W(100) crystals used in these experiments. This work was supported in part by the Harvard Materials Research Laboratories (NSF DMR 80-20247), the donors of the Petroleum Research Fund, administered by the American Chemical Society, and the Research Corporation (Cottrell Research Grant No. 9787). C.M.F. acknowledges the receipt of an IBM Faculty Development Award, 1983-85. Registry No. NO, 10102-43-9; N1, 7727-37-9; N,O, 10024-97-2; Olr 7782-44-7; W, 7440-33-7.
In-Crystal Polarizability of 02P. W. Fowler University Chemical Laboratory, Cambridge, CB2 1 E W England
and P. A. Madden* Physical Chemistry Laboratory, Oxford, OX1 3QZ England (Received: December 19, 1984)
An ab initio value for the polarizability of the 02-ion in the MgO crystal has been found by coupled Hartree-Fock theory with Mder-Plesset correlation corrections. The calculation includes the effects of the Madelung potential and nearest-neighbor overlap on the oxide polarizability. We find a = 1.83 A3, which is within 10% of an experimental value. The effects of the Madelung potential in CaO, SrO, and BaO on the 0,- polarizability are also calculated. The results of the calculation are used to appraise the treatment of the crystalline environment in other theoretical schemes.
1. Introduction The dielectric permittivity of an ionic crystal at optical fiequencies (em) is related to the "in-crystal" values of the ionic polarizabilities via the Clausius-Mossotti formula (€- - l ) / ( e m + 2) = (3Vt,)4Ca' (1) i
The sum contains the in-crystal polarizability ( a i )of each ion in the unit cell, which is of volume V. Tessman et al.' used this relationship and the assumption that the in-crystal polarizability was a constant, independent of the particular ionic environment (1) Tessman, J. R.; Kahn, A. H.; Shockley, W. Phys. Rev. 1953, 92, 890.
0022-3654/85/2089-2581.%01.50/0 0 1985 American Chemical Society
2582
Fowler and Madden
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985
in order to deduce ionic polarizabilities empirically. For alkali halides this analysis was quite successful; a consistent set of ionic polarizabilities was found to describe a large amount of data. For 02-in the chalcongenides the constant polarizability assumption broke down'; values ranging from 0.5 to 3.2 A3 were needed. Other empirical ana lyse^^^^ have subsequently shown a variation in halide ion polarizabilities in the alkali halides, but it is much weaker than found for 02-.For cations the environment is found to have very little influence on the polarizability, in-crystal values agree well with estimates of gas-phase polarizabilities. The situation has been analyzed by M a h a n 4 He has shown that the crystalline environment greatly reduces the anionic polarizability from its gas-phase value by two physically distinct mechanisms which act by confining the electron density. The electrostatic interaction of the electrons with the ionic lattice (regarded as point charges) may be represented as a potential well whose width is twice the lattice spacing (the Madelung potential). In addition, for closed-shell systems, overlap interactions with the electron clouds of neighboring ions lead to "overlap compression". In Mahan's work these two effects were incorporated into density functional calculations of the polarizability; the overlap effects were represented by a one-electron pseudopotential well which was somewhat narrower (by about the counterion crystal radius) than the Madelung potential well. Only the spherical parts of the Madelung and overlap potentials were included. Mahan's calculations accounted for the large decrease in anionic polarizability between the gas and crystal and identified the reasons for the sensitivity of the in-crystal anion polarizability to the ionic environment. The calculated cation polarizabilities were remarkably insensitive to the crystalline environment. Recently, we have described5-' the results of a series of large-scale ab-initio electronic structure calculations on the free and in-crystal polarizabilities of the fluorides, chlorides, and hydrides of lithium and sodium. The present paper contains the results of a similar set of calculations on the polarizabilities of the oxide ion and the divalent cations of magnesium and calcium. The calculations use the coupled Hartree-Fock (CHF) method with second-order Maller-Plesset estimates of the correlation polarizability.s Very large basis sets have been used so that the C H F polarizabilities of the free ions are extremely close to the Hartree-Fock limits. The Maller-Plesset method gives two estimates of the correlation contribution to the polarizability which will be denoted MPE and MPD as they arise from the energy and dipole series, respectively. The two series converge at the same limit but do not coincide at finite orders of perturbation t h e ~ r y . ~ The closeness or otherwise of MPD and MPE estimates is one indicator of the convergence of the MP series. Although convergence is poor for the free anions'0,'' it is much beetter for the in-crystal halide ion^.^^^ The influences of the Madelung potential and the overlap compression are separately determined in our calculations by performing two series of calculations: "XTAL" calculations in which the ionic environment is represented by a lattice of point charges and "CLUS" calculations in which the electrons and basis functions of the first shell of neighbors are also included. The CLUS polarizabilities of the alkali, halide, and hydride ions were in very good agreement with experiment. The objective of our work has been to provide a reliable basis for attributing the environmental effects on the in-crystal polarizability to different physical factors; for example, to Madelung vs. overlap compression or to the influence of compression (of either kind) on correlation vs. Hartree-Fock polarizability contributions. The underlying motivation is to understand the role (2) Wilson, J. N.; Curtiss, R. M. J . Phys. Chem. 1970, 74, 187. (3) Coker, H. J. Phys. Chem. 1976.80, 2078, 2084. (4) Mahan, G . D. Solid Stare tonics 1980, 1 , 29. (5) Fowler, P. W.; Madden, P. A. Mol. Phys. 1983, 49, 913. (6) Fowler, P. W.; Madden, P. A. Phys. Reu. B 1984, 829, 1035. (7) Fowler, P. W.; Madden, P. A. Phys. Rev. E, accepted for publication. (8) We made use of the HONDO 1984,62, 2161-2167. M.; Rys, J.; King, H. F. QCPE 1981, 13, 403) as modified by R. D. Amos at Cambridge. (9) Amos, R. D. Chem. Phys. Lerr 1982.88, 89. (IO) Diercksen, G . H. F.; Sadlej, A. J. Mol. Phys. 1982, 47, 33. (1 1) Diercksen, G. H. F.; Sadlej, A. J. Chem. Phys. Lert. 1981, 84, 390.
TABLE I: Polarizabilities of Mg2+ and Ca2* in Atomic Units "CHF
Mg2+
FREE XTAL
Ca2+
FREE XTAL
0.4697 0.4700 3.2532 3.2588
"MFQ
~ M P E
0.0157 0.0158
0.0165 0.0 166
-0.0776 -0.0775
-0.0430 -0.0427
of such factors so that they may be correctly incorporated into calculations of the dispersive contribution to the cohesive energy of crystals and into models of the fluctuating polarizability which is responsible for light scattering in condensed ionic systems (see ref 7 ) . Recently, an account of a semiempirical scheme for calculating in-crystal polarizabilities in which the Madelung and short-range effects are separately estimated (and correlation effects neglected) has been given by Pearson et a1.I2 The extension of our calculations to the oxide ion raises new issues because the empirically observed' variability of the 02polarizability suggests a sensitivity to environmental effects which is much greater than for the halide ions considered previously, and so raises the spectre of covalency and possible charge transfer. By understanding the oxide ion polarizabilities we may improve the analysis of the permittivity data for the chalcogenide series and thereby increase the reliability of empirical estimates of the polarizabilities of the ions which form this class of compounds. Our calculations may be expected to be considerably more realistic than Mahan's as we avoid the a priori introduction of a psuedopotential to treat overlap effects as well as the usual approximations of density functional theory, such as the use of the uniform electron gas expressions for the correlation energy. Although our results do not agree quantitatively with Mahan's (in particular in the way in which the environmental effect is partitioned among the contributing factors) our results strongly support the validity of the central concepts introduced in his work. Firstly, the interpretation of our calculations on the fluctuating polarizabilities in distorted lattices' supports the representation of overlap effects by a pseudopotential well. The polarizability responds in an almost identical way to compressions of the electron density induced by changes in the overlap interactions and in the (genuinely one-electron) Madelung potential. As the calculation on an isolated F6Li' cluster discussed in section 7 of ref 6 also shows, the overlap compression dominates the Madelung effect when both are acting; that is, the well formed by the overlap pseudopotential is narrower than the Madelung well. Secondly, we find that only the spherical part of the potentials has a significant influence on the polarizabilities of the halide ions. We can illustrate the relative unimportance of the anisotropic part of the potential by calculting the C H F polarizability of F in two models of the electrostatic environment in LiF. These are (i) the cubic point charge lattice of the XTAL calculations and (ii) the full spherical part of the Madelung potential modeled by nine concentric icosahedra of charges. Icosahedra are used for convenience but their potential differs from spherical only in V, and higher terms, where V, is the coefficient of rank 1 in an expansion of the potential in Legendre polynomials. In particular the lowest-order anisotropic terms of the cubic lattice (V,) are missing. The polarizabilities of F- calculated at the C H F level in the basis of ref 5 are 7.32 aO3for (i) and 7.24 ao3 for (ii) compared with 10.65 aO3for the free ion. The influence of the anisotropic potential becomes significant for ions in which d orbitals are occupied in the ground state; this has been demonstrated for Ag+.I3 Pearson et a1.I2 have suggested that the effects of the spherical part of the electrostatic potential may be simulated with a Watson ~ p h e r e . ' ~We may evaluate this suggestion by considering a third calculation in which only a single icosahedron is included at a radius equal to the lattice constant for LiF and the charges chosen to give the Madelung well depth. For this case the F polarizability is 8.54 aO3,showing that the Watson sphere ( 1 2) Pearson, E. W.; Jackson, M. D.; Gordon, R. G. J . Phys. Chem. 1984, 88, 119. ( 1 3) Fowler, P. W.; Pyper, N. C. Proc. R . Soc. London, Ser. A 1985,398, 377. (14) Watson, R. E. Phys. Rev. 1958, I l l , 1108.
In-Crystal Polarizability of 02TABLE II: Polarizability and Size of 02-in the Electrostatic Crystal Fields of the Alkaline Earth Oxides“
environment FREE XTAL XTAL XTAL XTAL
R
BaO
2.162 2.573 CaO 2.401 MgO 2.104
SrO
~ C H F WPD ~ M P E (r2) a, 392.4 3.874 41.711 23.034 24.559 2.991 25.43 34.461 15.124 16.090 2.841 22.46 29.118 10.397 11.065 2.715 19.99 21.177 5.093 5.475 2.501 16.15
” R is the interionic distance in 8, and other symboles are defined in the text. mP is the estimate of the oxide polarizability in the model of Pearson et al.;I2 it should be compared with aCHF, treatment recovers only 63% of the electrostatically induced compression in LiF. The technical aspects of our calculations have been discussed e l ~ e w h e r e .The ~ ~ ~basis sets used in the present work are described in an Appendix. W e begin by giving the results of the XTAL calculations, which contain the effect of the Madelung potential. W e then describe the use of a CLUS calculation on MgO, to evaluate the overlap compression. We then compare our results with experimental data and comment on their bearing on the issues raised above.
2. Crystal Field Effects on Polarizabilities Table I gives results for FREE and XTAL Mg2+and Ca2+ions. The XTAL environments are those appropriate to MgO and CaO, respectively. The C H F results for the free ions may be compared with numerical C H F resultsIs of cu(Mg2+) = 0.4701 aO3and a(Ca2+) = 3.261 aO3which are larger than the present results by only 0.1% and 0.25%, respectively. The correlation polarizabilities of the cations are small. The negative sign of the correlation polarizability of Ca2+ is perhaps surprising but, like that found for K+,6 may be ascribed to an increase in the importance of angular correlation at the expense of radial correlation for the larger, doubly charged cation. A similar rationalization has been applied to correlation polarizabilities of the atoms.16 The electrostatic crystal field has a very small effect on the cationic polarizabilities and as even this small increase will be opposed by overlap compression we may take the FREE ion results as values for the in-crystal polarizabilities of these cations. Polarizabilities of the FREE and XTAL oxide anion calculated with the present basis are shown in Table 11. XTAL environments appropriate to all the alkaline earth oxides crystallizing in the rock salt structure were used, taking lattice parameters from ref 17. The effect of the crystal field on ionic size is shown by the column in Table 11. This quantity is the mean square radius headed (9) per electron (at the S C F level) and, as a ground-state property, is less environment sensitive than the polarizability. Although the free 0” ion is not a stable species the closed-shell S C F procedure has a solution for this system. The constraints of double occupation and identical radial functions for all orbitals in a p shell are sufficient to produce this bogus solution.ls A similar, spurious solution exists for H-. Unlike H-, however, 02does not become stable even at the correlated level. The C H F polarizability corresponding to this 02-S C F wave function is enormous (see Table 11) and indeed is kept finite only by incompleteness of the basis. When a finite field is switched on the reduction in symmetry allows the ion to dissociate by loss of an electron pair and so no constrained solution is found-the S C F procedure does not converge and therefore we have no values in Table I1 for the correlation polarizability of the free ion. More physically meaningful are the results for 02-in a lattice of point charges. In the electrostatic potential well the ion becomes (15) McEachran, R. P.; Stauffer, A. 0.; Greita, S. J . Phys. R. 1979, 12, 3119. (16) Reinsch, E. A.; Meyer, W. Phys. Rev. A 1978, 18, 1793. (17) Goodenough, R. D.; Stenger, V. A. In “Comprehensive Inorganic Chemistry”; Pergamon Press: New York, 1973; Vol. 1. (18) Cook, D. B. J . Chem. Soc. Chem. Commun. 1980, 622.
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985 2583 bound and its size and polarizability fall accordingly (see Table 11). As found for F,C1-, and H-,s,6the correlation polarizability shrinks faster than the C H F polarizability. Our calculated XTAL polarizabilities at the C H F level may be directly compared with the values for the in-crystal oxide ion polarizabilities “uncorrected for short-range effects” obtained in the work of Pearson et a1.12 The latter values are also appropriate to the polarizability of the anion in the Madelung potential at the S C F level. They are estimated from the moments of the ground-state wave function in an S C F calculation on 02-in a Watson sphere potential (via a scaled version of an approximate expression due to B~ckingham’~) and are shown in the sixth column of Table 11. The model gives values which are systematically lower than our ab initio values and much less dependent upon the lattice parameter. We have discussed the shortcomings of the Watson sphere model above and in ref 6 commented on the relationship between the XTAL polarizability and the second moment of the electron density.
3. Effect of Overlap on a(02.) A CLUS calculation of the type described in ref 5 was performed for a cluster of a single 02-ion and 6 Mg2+ions in a point charge lattice. At the S C F level a finite field calculation gives, for the total polarizability of the cluster acLUs= 11.228 a: We may interpret this as a sum of term@ = a(o2-) + 6a(Mg2+) + “DID
WLUS
+ aBSSE(02-) + aBSSE(cage) (3)
a(02-) is the polarizability of an oxide ion in the crystalline environment and is the quantity of interest here. a(Mg2+)is the polarizability of a free Mg2+ion and for the minimal [2slp] basis is exactly zero. a D I D is the enhancement of the cluster polarizability by dipole-induced-dipole interactions and, since a(MgZ+) is the basis set superposition error in the oxide polarizability due to the virtual orbitals on the Mg2+ neighbors and, since the cage basis is minimal, vanishes. The remianing term, aBssa(cage), is the superposition enhancement of the cage polarizability by the virtual orbitals on the 0 center. From the Hartree-Fock limiting polarizability of Mg2+ we expect
0
< aBssE(cage) < 2.8 aO3
(4)
The conventional counterpoise estimate of this term is found by calculating a for the cage in the full cluster basis with the 0 nucleus replaced by a charge of 2-. Because all functions on 0 are included this is an overestimate. The value found is 0.330 ao3and the bounds on a ( 0 2 - )are, from (2) and (3) 10.899 < a(02-)< 11.228 a:
(5)
Another estimate of BSSE may be obtained by calculating the cage polarizability in the basis extended by just s or p or d functions of 0. The conventional BSSE of 0.330 aO3splits approximately into 0.028 aO3(s), 0.106 a: (p), and 0.182 aO3(d). The s and p contributions are expected to be dominated by the occupied orbitals of 02-; from the d contribution a reasonable estimate of aBssE(cage) is 0.2 aO3. The correlation polarizability of the 02-in MgO is likely to be significant, even though reduced by overlap compression from the XTAL value of around 5 a: (Table 11). To make a calculation of the correlation polarizability feasible the following scheme was adopted. The correlation polarizability was calculated by Maller-Plesset perturbation theory using a frozen core approximation for the 60 Mg2+ electrons and the 1s orbital of 02-.Thus the lowest 31 molecular orbitals of the CLUS SCF calculation were frozen and the 8 valence electrons of 02-allowed to excite into the 65 virtual ~~
(19) Buckingham, R. A. Proc. R . SOC.London, Ser. A . 1937, 160, 1 1 3 .
2584 The Journal of Physical Chemistry, Vol. 89, No. 12, 1985
Fowler and Madden
:.p
orbitals. A C H F value was also calculated in the frozen core approximation and we obtain aCHF = 10.957 aO3 CYM~D =
1.292 ao3
(7a)
CYMPE =
1.438 aO3
(7b)
The C H F polarizability (6) is approximately free of superposition error because the Mg-dominated molecular orbitals were frozen and indeed the C H F polarizability (6) falls in the range of (5), corresponding to aessE(cage) of 0.271 aO3. The result of this calculation is an a b initio value for the polarizability of 02-in MgO found by adding the mean of MPD and MPE to the C H F value ( 6 ) a(02-)= 12.32 aO3= 1.83 A3
(8)
Thus a proper treatment of overlap compression of 02-brings a(02-)down by a factor of two from the value calculated in the purely electrostatic crystal field of MgO (see Table 11) and, as we will see in section 5 , into agreement with experiment. Our CLUS model calculation includes the overlap compression by the charge clouds of nearest neighbors but treats second neighbors as point charges. Because 02-is a large ion the 02--02interactions may have a significant effect on the polarizability. An indication of the expected size of this effect is given by the for a gas-phase (02-)(Mg2+)6cluster following. The C H F acLUs is 1.1% greater than for the same cluster in the lattice. A similar calculation on (F)(Li+)66 showed only 0.1% change. If the presence of second neighbors as point charges produces a compression of 1%, then as extended charge distributions they would be expected to produce a further change of similar magnitude. As correlation polarizability is more sensitive than the C H F polarizability a total reduction of some few percent would be expected if second-nearest neighbors could be included. The model of Pearson et al.I2 gives a reduction of 3.6% in MgO due to next nearest neighbor N-. effects at the S C F level.
4. Comparison with Experiment and Conclusions An experimental value of the polarizability per ion pair in MgO may be obtained from the refractive index of MgO (extrapolated to infinite wavelength)20via the Clausius-Mossotti relationship. Taking the lattice parameter from X-ray data2’ the polarizability per MgO pair is CY,
= 11.831 aO3
Subtracting the theoretical Mg2+ polarizability yields
a(Oz- in MgO) = 11.345 ao3
(expt)
compared with ( ~ ( 0 in~ MgO) = 12.32 a:
1
(6)
(calcd)
so that our calculation overestimates the polarizability by 9%. A discrepancy of this order might be produced by our neglect of 02--02-overlap compression and of higher terms in the Mdler-Plesset series. The “experimental” number itself may be subject to some error caused by breakdown of the ClausiusMossotti relationship. One estimate22is that in MgO the finite size of the ions leads to an error of
