Model Predictions of the Dissociation Energies of Homo-nuclear and Heteronuclear Diatomic Molecules of Two Transit ion Metals BY ANDRIES R.MIEDEMA Philips Research Laboratories Eindhoven The Netherlands Received 29th August 1979 It is demonstrated that by taking the solid metal at zero temperature as the reference state the enthalpy of formation of homonuclear diatomic molecules can be obtained as the surface energy of a piece of metal having the size of two atoms. It is suggested that the energy of larger homonuclear metal clusters can be derived similarly. The heat of formation of heteronuclear diatomic molecules relative to the two pure metal dimers is derived by means of a model description for the interaction energy which is very similar to that used for interfacial energies and alloy heats of solution in the condensed phase.1 INTRODUCTION In a series of papers1-’ we have recently demonstrated that the energies of metal- metal combinations can be described in terms of contact energy effects generated at the interface between dissimilar metal atoms. In our energy considerations the reference state we normally take is the pure condensed metal at zero temperature. If atomic cells of metal A are no longer completely surrounded by similar A-cells there is a change of energy which can be related to the change in atomic cell boundary condi- tions. It is assumed that this energy change is proportional to the fraction of the surface area of an atomic cell for which the boundary conditions have become differ- ent.This means that atomic cells are treated as if they were comparable with macroscopic pieces of metal. It follows that the atomic size is characterized by the value of Vg3,where V is the molar volume. The interfacial energy on a microscopic scale which forms the basis of our model description 1-3 of alloy heats of formation contains two terms (here we limit ourselves to combinations of two transition metals). The first term which is negative is introduced as the energy of an electrostatic dipole layer analogous to the dipole layer that would be generated at the interface between two macroscopic pieces of metal. The energy of the dipole layer can be expressed in terms of the difference in work function p of the two metals. The second term which is positive is derived from the difference in electron density nws,that would exist at the interface if the two types of metal atom cells were taken from the respective pure metals without any charge rearrangement.The electron density is required to become continuous across the interface. This is achieved by either6 changing the mixed electron configuration of the metallic atoms (for transition metals an interchange of d-electrons to s-electrons will increase the cell boundary A. R. MIEDEMA electron density) or by a volume change expanding the metal which originally had the higher value of nws and compressing the metal with the lower nws (or both). One would expect that the positive surface energy of a metal-vacuum interface could also be derived from the discontinuity in the electron density between a bulk metal atomic cell and empty space.Indeed it has been demonstrated4 that the surface energy of a solid metal at T =IZ 0 yo is related approximately linearly to nws the proportionality constant being a weakly varying function of the number of valence electrons per atom of the metal. For a metal-vacuum interface the macroscopic atom approach which is similar to that used in the description of alloy heats of formation suggests that the heat of vaporization of a metal is comparable with the surface energy of a piece of metal having the size of a single atom. Indeed such a relation does make sense3ys (see also below). In addition the heats of formation of homonuclear dimers are obtained when this type of approach is used as the difference in surface energy of pieces of metal consisting of a pair of atoms as compared with two single atoms (see below).Also the heat of formation of monovacancies in solid metals can be dealt with quite easily3y4 in terms of the surface energy of a hole of atomic size. A number of situations in which metal-metal and metal-vacuum interfaces play a role are surveyed in fig. 1 which shows the boundary conditions for atomic cells FIG.1.-Metallic adhesion on microscopic and macroscopic scales. The figure is intended to suggest that there are relations between the heat of formation of intermetallic molecules AB (relative to pure dimers) the heat of formation of intermetallic compounds AB (relative to pure metals) the heat of solution of A in B the interfacial energy between two crystals of A and B and the heal of adsorption of B on a substrate of A.in a variety of situations i.e. a heteronuclear AB molecule an ordered AB intermetallic compound the heat of solution of A in liquid B the interfacial energy between two crystals (ignoring the dislocation-determined density deficit) and the heat of adsorp-tion of an atom of B metal on a metallic substrate of A. In this example of A atoms that are clearly smaller than B atoms there is little difference between the heat of formation of the AB compound (per mol AB) and the heat of solution of A in B when the macroscopic atom approach is used. Once DISSOCIATION ENERGIES OF DIATOMIC MOLECULES assumptions are made about the geometry of atomic cells and interfaces the inter- facial energy between two crystals is directly related to the heat of solution.s The adsorption energy of B on A can be expressed in terms of the surface energies of the two metals and once again the heat of solution of A in B.One must make an assumption with regard to the fraction by which the adsorbed atom is in contact with the substrate. It has been found' that a fairly accurate description of experimental heats of adsorption of metals on metallic substrates is obtained by taking this fraction to be 0.35. The energy of the diatomic molecule is derived analogously. Compared with the average of the pure metal dimers the intermetallic molecule presents an interface between dissimilar A and B atoms and a slight increase in the B-vacuum interface (B larger than A).By expressing the interfacial energy in terms of the same para- meters (differences in work function differences in bulk cell boundary electron den- sities) that were used in the condensed phase we have been able to calculate the disso- ciation energies of diatomic molecules of two transition metals. Predicted values are presented in tables l(a)and l(b). TABLE1.-PREDICTEDVALUES OF THE DISSOCIATION ENTHALPIES OF DIATOMIC MOLECULES OF TWO TRANSITION METALS. DiB IN UNITS OF kJ mol-I. The starred values for homonuclear molecules are experimental. (4 Sc Ti V Cr Mn Fe Co Ni Cu 3d metals sc 158" Ti 128 125" V 213 183 238" Cr 207 163 191 151" Mn 139 96 133 94 42" Fe 230 177 185 131 80 100* co 292 232 229 167 123 129 167" Ni 316 262 272 205 163 163 199 229" cu 222 166 188 145 98 125 166 205 196" 4d metals Y 143 114 200 195 128 217 280 295 214 Zr 228 194 267 255 185 276 334 360 260 Nb 266 249 279 243 179 248 295 341 238 Mo 312 274 279 212 159 193 227 265 202 Tc 461 399 363 251 215 205 223 252 244 Ru 479 414 382 263 230 214 228 256 254 Rh 467 397 362 244 213 193 210 237 242 Pd 353 310 270 157 128 103 121 148 164 Ag 231 178 170 111 68 93 139 178 5d metals La 166 131 216 209 145 232 295 300 236 Hf 226 193 256 239 172 256 312 341 244 Ta 282 263 287 248 185 25 1 297 343 243 W 405 363 343 262 212 237 265 301 249 Re 513 450 395 276 242 227 241 268 264 0s 522 458 415 294 260 244 257 284 282 Ir 538 468 411 281 252 225 236 261 272 Pt 488 441 391 261 234 206 215 239 257 Au 369 314 258 159 127 116 144 175 Th 201 160 236 227 159 248 308 333 241 A.R. MIEDEMA 139 TABLE1-continued I___ (b) Y Zr Nb No Tc Ru Rh Pd Ag Y 156" Zr 212 309 Nb 250 333 371 NO 296 370 360 325 Tc 446 512 455 356 330 Ru 463 529 467 362 325 327 Rh 452 514 447 339 302 303 281" Pd 339 404 373 263 211 204 185 104" Ag 223 278 252 198 244 250 239 172 158" La 186 231 266 312 462 479 470 356 253 Hf 210 303 327 352 483 499 483 395 259 Ta 266 350 383 366 456 467 447 373 259 W 388 464 444 385 398 400 376 299 261 Re 496 567 504 395 349 343 317 231 273 0s 505 574 512 405 368 361 336 244 287 Ir 522 588 514 391 337 328 304 210 279 Pt 470 544 502 391 308 298 275 188 275 Au 357 420 373 258 234 231 216 139 Th 223 268 296 335 479 496 484 374 256 La Hf Ta W Re 0s Ir Pt Au La 241 * Hf 230 304 Ta 283 342 395 W 404 443 449 452 Re 511 535 505 436 376 0s 521 543 513 445 386 405 lr 537 555 513 427 352 371 333 Pt 484 516 502 425 328 339 299 278 Au 362 393 376 308 255 269 244 234 220" Th 251 263 312 427 531 540 555 508 392 The present paper is a summary of 3 earlier papers dealing with the heats of forma- tion of homonuclearsa and heteronuclear8b diatomic molecules and the special properties9 of molecules of divalent metals.In addition we shall extend the previous paper on homonuclear dimers to include predictions of the dissociation energies of larger homonuclear clusters.2. HOMONUCLEAR DIATONIC MOLECULES The starting point of the present description is that the difference between the energy of a homonuclear cluster of n atoms and that of n atoms in the bulk metal can be derived as the surface energy of a metallic particle. Defining Do as the zero ternpera- ture dissociation energy of a cluster into isolated free atoms we expect for diatomic homonuclear clusters that 2AHoVap -Do is linearly related to both the zero tempera- ture surface energy of the solid metal yo and the atomic size parameter VZ3(AHovap is the zero temperature heat of vaporization.). The validity of this linear relation is illustrated by means of fig. 2 in which all available experimental information" on 0' 140 DISSOCIATION ENERGIES OF DIATOMIC MOLECULES 800 600 7 z ? Gd" a 400 Ino / 0' X 3-/Pb -4 N 200 0 I I 0 0.4 0.8 1.2 y0V,2'3/ rnJ FIG.2.-Relation between the dissociation enthalpy * into free atoms of metallic dimers Do,the heat of vaporization" of the bulk metal AH!, and the surface energy4 per unit molar surface area ~O?ffn/~.dimers of transition metals and those of monovalent and trivalent non-transition metals has been included. Divalent metals have been omitted; the s2 outer free atom electronic configuration with large s -+p promotion energies leads to exceptional behaviour as will be dis- cussed at the end of this section. Tetravalent metals Si Ge Sn have also been omit- ted; they are considered to be at least partly covalent (i.e.? have more directional bonding than the average metal).Lead is considered to be a borderline case. The slope of the straight line in fig. 2 corresponds to (2AH:, -D0)/70V$3= 0.78 x lo9. It is of interest to compare this result with that derived for the surface energy of N spherical particles of volume np(Vm/N)for np = 2 (N is Avogadro's number np is the number of atoms in the cluster) which equals AHs,,f/~oVm= (36nN)1/3ni/3= 0.65 x lo9. (2.1) The agreement is quite convincing suggesting (1) that dimer interatomic separa- tions and bulk metal volumes are related and (2) that the surface energy remains a meaningful parameter for metal-vacuum interfaces of microscopically small pieces of metal. In fig. 3 we go one step further and investigate to what extent the energy difference between a pair of atoms in a dimer and two separated atoms can be regarded as a difference in surface energy.Again we find an approximate linear relation between Do and yoVz3. However due to the relatively small numerical values of Do [much smaller than (2AH,0ap-DO)],the scatter in the data points for the individual ele- ments is considerably larger than in fig. 2. The slope of the straight line drawn corre- sponds to D0/y0Vk/3= 0.265 x lo9. For spherical particles we would expect this factor to be (36nN)ll3(2 -22/3)= 0.17 x lo9. A. R. MIEDEMA 300 1 I 4 0 Rh 200 I + 0 E 7 Y \ -c ::100 Lio /oIn om Q 0 0.4 1.2 <Vi'3/ rnJ 0.8 FIG.3.-Relation between the dissociation enthalpy Do, of metallic dimers (in units kJ mol-' M2)and the product of the surface energy and molar surface area yoVz3,at zero temperature.Note however that the 3d metals Ti V Cr Mn Fe Co and also Pd that were represented in fig. 2 are omitted from fig. 3. It is suggested that the atomic closed shell d10electron configuration of Pd and magnetic energy contributions in the above 3d metals lead to the irregular behaviour of Do. In the case of a free atom the opti- mum magnetic state makes a larger contribution than in the dimer (or the solid). The reason for accepting that the above 3d metals (and Pd) as free atoms represent an exceptional situation while suggesting that the remaining metals of fig. 3 are the " normal" case is as follows.Combining the linear relations of fig. 2 (for 2AHta -Do and yoVg3)and fig. 3 (for Do and y0VZ3)we find that for the metals included in fig. 3 AHtapalone must also be linearly related to yoYz3. This is seen in fig. 4 (the open circles). The linear relation thus obtained also holds for a large number of 4d and 5d transition metals for which there is no information on Do,but for which surface energies4' and heats of vaporization" are known. The slope of the straight line in fig. 4 corresponds to AH$,,/y0Vm = 0.52 x lo9. This value has to be compared with 0.41 x lo9 the value expected for a pseudo-macroscopic spherical particle of volume VJN. It will be clear that having once accepted that the additional transition metals of fig. 4 represent the normal situation and are thus comparable with the other metals of fig.3 and 4 we can easily derive a predicted value for the missing transition metal dimer dissociation energies. They can be derived from the heat of vaporization the product of surface energy and molar surface parameter or the difference between the two. A suitable average result see ref. 8(a),is included in table 1. The results of fig. 3 and 4 imply that for a large number of metallic elements the dissociation enthalpy of the homonuclear dimer can be related to the heat of vaporiza- tion by Do(M2) = AHt,Ja (2.2) with a = 1.96. This value implies that the effective area of contact between two atoms in a homonuclear dimer equals about one fourth (0.255)of the surface area per atomic I42 DISSOCIATION ENERGIES OF DIATOMIC MOLECULES FIG.4.-Relation between the heat of vaporization of " normal metals " and the surface energy per unit molar surface area YOY:'~.The open circles represent the metals included in fig. 3; the closed circles represent transition metals for which there is no experimental information on Do. cell which seems a reasonable fraction. Relation (2.2) was suggested by Verhaegen et aZ.I2 These authors expected to find the same CI within a family of elements i.e. within a given column of the Periodic Table. It can be concluded from this section that a is an approximate constant for a large fraction of the metallic elements only the 3d metals Ti V Cr Mn Fe Co the divalent metals and Pd being clear exceptions. We emphasize at this point that the relation between 2AH,OaP-Do and y0Yg3is of a more general validity; only divalent metals are exceptions here.The exceptional properties of divalent mstals are illustrated by fig. 5. On the left-hand side we have indicated the position of free atom and dimer for the normal metals M in an energy scale in which the unit of energy is y0Vg3for the metal under consideration and where the solid metal is the zero of energy. Free atoms of the divalent metals are unusually stable as can be observed for instance in the sequence of first ionization potentials of elements as a function of atomic number. The free atom of Cd which in fig. 5 may also represent Hg Zn and Mg has an energy which is lower than that of the normal metallic dimer i.e.the dimer with a mixed electronic configuration comparable with that of the bulk solid. As a conse- quence admixture of 5p electronic wave functions is scarcely possible in the Cd-dimer. The pure s2 configuration of the free atom is retained leading to a van der Waals noble-gas-like molecule with for dimers an unusually large interatomic separation. The alkaline-earth metals Ca Sr and Ba are a borderline case. The free atom is clearly more stable than in the average situation but the free atom is slightly less stable than the " normal " metallic dimer would be. Hence whether we are concerned with a van der Waals type (large interatomic separation) molecule of Ca Sr or Ba or a more ordinary metallic one of mixed s-d configuration the dissociation enthalpy will in any case be small.A. R. MIEDEMA 0.75 0.50 C31?-0.25 0 FIG.5.-Exceptional and normal positions of free atom and dimer in an energy scale in which the unit is 109y0V:’3. The levels for M represent the majority of the transition metals and mono- and tri- valent non-transition metals. For the divalent metals Hg Cd Zn and Mg (represented there by Cd) the free atom energy level lies unusually low clearly below that of a “ normal metallic ” dimer. For Ca (also representing Sr and Ba) the free atom level is near to that of the normal metallic dimer so that metallic and van der Waals type dimers differ little in energy. For Fe (also representing other magnetic 3d-metals and Pd) the free atom energy level lies somewhat too low but the dimer is “ normal ” metallic.We expect the interaction potentials for a pair of alkaline-earth atonis (and to a minor extent also that for Mg,) to be quite unusual being only weakly dependent on interatomic separation over a wide range of distances. For both Mg and Ca there is evidence that the pair potential is indeed of such an unusual In fig. 5 we have included Fe which also represents other irregular 3d-metals. The dimer takes up a normal position in the energy scale but the dissociation en- thalpies are too low. It is of interest to note that the divalent metal Be falls into the Fe class. 3. LARGER HOMONUCLEAR CLUSTERS We have demonstrated that the energy of a cluster of two atoms relative to the bulk metal can be described fairly accurately as the surface energy of a spherical piece of metal of volume 2 VJN the difference being a factorf = 1.2.There is little reason to suppose that dimers are exceptional in this respect. Hence energies of larger clusters can also be derived from surface energies the above fact0r.f being expected to decrease fromf = 1.2tof = 1 with increasing n,. Effects of odd/even npare assumed to be relatively unimportant for larger n values. Intuitively we assumefto decrease asf= 1 + an;+; this assumption immediately gives us the energy of homonuclear clusters relative to the bulk solid in units of 0.41 x 109ni’3y0VZ3(see table 2). The dissociation energy relative to free atoms is obtained by subtracting the cluster surface enthalpy from n,AH&,. For the smaller clusters (dimers trimers) this means that we subtract two large terms so that uncertainties in the solid metal surface energies4 are multiplied.However for the larger clusters the uncertainties are much smaller. As an example we have included calculated values of the dissocia- ion energies (per mole of atoms) for clusters of Cu Pt or A1 in table 2. 1 44 DISSOCIATION ENERGIES OF DIATOMIC MOLECULES TABLE2.-PREDICTED VALUES OF THE HEAT OF VAPORIZATION OF HOMONUCLEAR METALLIC CLUSTERS RELATIVE TO THE BULK METAL AHsurrAND THE DISSOCIATION ENTHALPIES Do OF CLUSTERS OF Cu Pt AND A1 FOR VARIOUS NUMBERS OF ATOMS PER PARTICLE, nP. UNITSOF AHsurf ARE 0.41 x lo9 ni/3y0VA/3,i.e. THE SURFACE ENERGY OF A MACROSCOPIC SPHERICAL PARTICLE. UNITSOF DoARE' kJ mol-' atom-'.3 1.17 108 191 143 4 1.16 131 229 162 6 1.14 160 276 185 8 1.125 178 305 199 10 1.115 191 326 210 20 1.09 223 379 236 50 1.07 255 431 261 100 1.055 272 460 276 200 1.04 286 482 287 500 1.03 300 505 298 1000 1.025 307 517 304 Note the difference from fig. 2 and 3 and table 1 where units of Do are per mol of two atoms. There are few experimental data with which to make comparisons. Wu15 has studied the dissociation enthalpy of Li,. The experimental value of Do is 58 kJ mol-1 atom-' which can be compared with a calculated value of 75 kJ mol-' atom-l. Gingerich et a1.16 have reported on Pb3 and Pb,. The experimental values of Doare 73 and 104 kJ mol-' atom-' to be compared with predictions based on the macro- scopic atom considerations of 76 and 87 kJ mo1-I atom-' respectively.If one accepts the macroscopic atom approach as a correct method of estimating cluster dissociation energies one has a powerful criterion for evaluating theoretical calculations of cluster energies. For instance Anderson l7 has calculated that the binding energy per atom for CuL3clusters is much smaller than that of smaller clusters. (76 kJ mol-' atom-'). Anderson's prediction strongly disagrees with present macroscopic atom considerations. As another example we point to theoretical investigations of the binding energy of hydrogen with metals by means of cluster calculations18 with and without added hydrogen. One should correct calculated binding energies for the increase in the cluster surface energy by adding z2 cm3 per H atom to the cluster molar volume.In principle the atomic volume of small metal particles will be reduced because the surface energy represents an appreciable fraction of the binding energy. However when the volume compression is estimated from surface energy and solid metal bulk modulus K the effect is found to be not very large. For macroscopic spherical particles -AAH~~~~/AH~~~~ = 0.09 x 109n;1/3y0~,/~~,,,. Values of y0Vi'3/KVm are assembled in fig. 6. For transition metals they vary around y0V$3/KVm= 0.5 x corresponding to energy corrections of a few percent even for np = 2. Note that if the change of atomic volume with cluster size plays a role it will be seen first of all for Li clusters. We note that the calculations of table 2 cannot be used to derive dissociation ener- gies of relatively small molecules of the divalent metals Mg Zn Cd Hg.It can be A. R. MIEDEMA 0 li oNa,K .Zr oAs Ti Hf .V oFe Mn .Co oNi snsPb .Ru a Pd Rh Ir .Pt *Au 8 *Os 1 I 1 1 I 1 I 1 I ? t 2 3 4 5 6 7 8 9 10 z FIG.6.-Parameter yo V23/KVmwhich characterizes the relative volume contractions of small particles generated to reduce the total energy of the particle-vacuum interface. K is the bulk modulus yo the surface energy and V the molar volume. For the majority of transition metals y0V,?3/KVmw 0.5 x 10-9. shown’ that Hg particles of up to 20 atoms will be van der Waals type molecules. For Mg Zn Cd this critical number lies around np = 5.Also the more covalently bonded molecules of Si Ge or Sn cannot be expected to be described by the present surface energy considerations. 4. DIATOMIC INTERMETALLIC MOLECULES In a recent papeeb we have suggested a method for calculating dissociation energies of intermetallic molecules. The interactions between two atoms in a molecule are treated in a similar way to those between dissimilar nearest neighbour atoms in solid alloys. There are two differences in the heat of formation of condensed alloys the reference states are the two pure metals. Consequently the heat of formation of intermetallic molecules will be derived relative to the two pure metal dimers. Secondly the contact area between dissimilar atoms in a molecule is only a fraction of the atomic surface area.From the observed relation between the dissociation energy and the heat of evaporation of “normal metals” we derive this fraction as 1/2a=0.255 for atoms of equal size. Where atoms are of different size (V&I3> VZ3),we assume that the contact surface area is determined by the smaller atoms so that there is a positive contribution to the heat of formation of the molecule reflecting an increase in the B-atom-vacuum interface. As a result AH^^^^,^^ = 0.13 x 109~;(v;/3 -~23). (4.2) For AHchem,AB we start out from the corresponding relation for the case of condensed alloys; from the effective contact surface between A and B (0.255 of that of atom A) DISSOCIATION ENERGIES OF DIATOMIC MOLECULES we would expect AHchem,AB to be about one fourth of the heat of solution of A in €3 AHsol,Ain B.This quantity can be calculated from Both terms in relation (4.3) are proportional to the A-B interfacial area per atom A i.e. Vy3. The first term represents the energy of an interfacial electrostatic dipole layer; the average electrostatic shielding length is found in the denominator. The second term is due to the difference in the electron densities at the boundaries of the atomic cells in the two pure metals which must be equalized in the alloy. P and Q are empirical constants. Values for the effective work function q* and cell boundary electron densities nws can be found in previous paper^.^^'^ Previously we have combined the two terms to Analysing experimental data for diatomic we found that AHchem,AB could not be obtained by simply reducing AHso,,Ain by the above factor of 0.255 (accounting for the difference in A-B interface area).Instead the two terms in rela- tion (4.4)must be treated differently Q' is indeed reduced by a factor of z 0.255 but P is only slightly reduced. As a consequence the ionic term in the heat of formation is very much the same for molecules as for an ordered AB compound. Values of P and Q' used in the present calculations of AHche,,,,AB for molecules can be found in ref. [8(b)]. Results for DiBhave been assembled in table 1. Some additional comments are (a)In calculating DiBfor molecules in which one of the metals is Cu Ag or Au we have not used the experimental value of DiAin relation (4.1).It is suggested that the symmetric dimer of a metal with a single s electron and otherwise closed electron shells will have some additional s2 stability. This stability is observed in fig. 3 where data points for the dissociation energies of Li, Na, Cu, Ag, Au are all above the average line. We have introduced " ordinary metallic " Cu, Ag and Au with dis- sociation energies' of 165 140 and 180 kJ mol" respectively (points on the line in fig. 3). The correction is not very substantial. (b) In the present paper we treat the metals at the end of a transition metal series Ni Pd Pt and Au slightly differently from other metals. In the Periodic Table the electronegativity increases as a rule with increasing total number of electrons. There are exceptions there is a steep fall in q* from Ni to Cu from Pd to Ag and from Pt to Au to Hg.Since in the case of metals in a molecule which have an appreciable difference in q* the electron transfer may well become of the order of 1 electron atom" the resistance to charge transfer is higher than average if the electronegative metal is Ni Pd Pt or Au. In order to include this effect in our predictions of DABwe have used the following approximation. The value of P as given previously,' is reduced by 7 13 or 19% if one of the metals in the diatomic molecule is Ni Pd Pt or Au and Ay* lies between 1 and 1.5 1.5 and 2 or 2 and 2.6 V respectively. (c) In fig. 7 the calculated values of DOAB of table 1 are compared with experimental data. On the whole the agreement is quite satisfactory although in the region of small dissociation energies the relative deviations are fairly large.In the deviating cases we are concerned with molecules of Au with a magnetic 3d metal. For molecules such as RhLa IrTh and RhTh for which Aq* is large the dissocia- tion energies range up to very high values. It appears that whereas for homonuclear diatomic molecules the binding energy per atom varies around 1/4 of that of the solid metal this fraction may increase to above 0.4 for the strongly ionic intermetallic A. R. MIEDEMA t 600 RuThe IrTt/' PtTho / / 400 RuYy ePtLu RhV/ RhTi c /# AuLa 'd 0 € /-AuLu 7 Y I AuNi /-\ -.. AuCre. AuCo Ii".Rh :200 AuFewAuM,!?' CuNi 0 0 t .cucrdCuCo AgMn /AuPd I // '/ I 1 I I I 0 0 200 400 600 Doca,c /kJ mol-' FIG.7.-Comparison of experimental and calculated values of the dissociation energy of the diatomic molecules of two transition metals.Experimental values are from Gingerich,lo calculated values from table 1. The straight line corresponds to D",,,, = molecules. Consequently it is possible that molecules consisting of a metal from the left-hand side of the series of transition metals with one from the right-hand side have a higher dissociation energy than the homonuclear dimer of a transition metal in the middle of the series (Mo W). (d) In principle it is also possible to calculate heats of formation of dimers of a transition metal with a polyvalent non-transition metal. In this case the corre-sponding heat of formation of condensed alloys contains a large negative term from d-p hybridization which does not depend much on which transition metal is com- bined with which p-metal partner.It has been foundsb that apparently such a hybrid- ization term also plays a role in molecules. Experimental results mainly on transi- tion metal silicides suggest that the hybridization term like the ionic term is relatively larger in molecules. Since experimental data are restricted mainly to silicides and germanides quantitative predictionssb for d-p intermetallic molecules are less reliable than the predictions for molecules of two transition metals of table 1. 5. CONCLUSIONS We have argued that dissociation energies of metallic clusters can be related to surface energies and interfacial energies such as occur in metallurgy.Predictions of the dissociation energies of diatomic molecules of two transition metals have been tabulated. It is suggested that the binding energy of large homonuclear clusters can be easily derived relative to the bulk metal by comparing it with the surface energy of a metal sphere of the same volume. DISSOCIATION ENERGIES OF DIATOMIC MOLECULES A. R. Miedema F. R. de Boer and R. Boom J. Less-Common Metals 1976,45,237. A. R. Miedema J. Less-Common Metals 1976 46 271. ’A. R. Miedema and P. F. De Chatel Proc. Symp. Theory of Alloy Phase Formation ed. L. Bennett (Amer. Soc. for Metals New Orleans 1979). A. R. Miedema 2.Metallkunde 1978 69 287; 1979 70 345.A. R. Miedema and F. J. A. den Broeder 2.Metallkunde 1979,70,46. A. R. Williams Proc. Symp. Theory of Alloy Phase Formation ed. L. Bennett (Amer. SOC. for Metals New Orleans 1979). ’J. A. Alonso and L. A. Girifalco J. Phys. F 1978 8 2455. A. R. Miedema and K. A. Gingerich J. Phys. B 1979,12,2081 and 2255. A. R. Miedema and J. W. F. Dorleijn Phil. Mag. 1979 submitted. lo K. A. Gingerich Current Topics Muter. Sci. to be published. l1 R. Hultgren P. D. Desai D. T. Hawkins M. Gleiser and K. K. Kelley Selected Values of the Thermodynamic Properties of Metals and Alloys (Amer. Soc. Metals Ohio 1973). l2 G. Verhaegen F. E. Stafford P. Goldfinger and M. Ackermann Trans. Faraday Soc. 1962 58 1926. l3 R. 0.Jones J. Chem. Phys. 1979,71 1300. l4 W. J. Stevens and M.Krauss J. Chem. Phys. 1977 67 1977. l5 C. H. Wu J. Chem. Phys. 1976 65 3181. K. A. Gingerich D. L. Cocke and F. Miller J. Chem. Phys. 1976 64,4027. l7 A. B. Anderson J. Chem. Phys. 1978 68 1744. la A. Lodder personal communication 1979.