On hybridization and bonding in the alkaline earths

On hybridization and bonding in the alkaline earths: Be, Mg, and Ca Charles W. Bauschlicher Jr., Paul S. Bagus, and Brian N. Cox Citation: The Journal of Chemical Physics 77, 4032 (1982); doi: 10.1063/1.444313 View online: http://dx.doi.org/10.1063/1.444313 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/77/8?ver=pdfcov Published by the AIP Publishing

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On hybridization and bonding in the alkaline earths: Be, Mg, andea Charles W. Bauschlicher, Jr. Polyatomic Research Institute. Mountain View. California 94043

Paul S. Bagus IBM Research Laboratory. San Jose. California 95193

Brian N. Co)(') Joint Institute for Advancement of Flight Sciences. The George Washington University. NASA-Langley Research Center. Hampton. Virginia 23665

(Received 26 March 1982; accepted 14 June 1982) We have computed the dissociation energy (D,) for tetrahedral Be., Mg., and Ca. from configuration interaction wave functions: Be. (59.2 kcallmol), Mg. (11.8 kcal/mol), and Ca.(18.3 kcallmol). We are able to relate these De's to the degree of s to p hybridization of the alkaline atom. Using a minimum basis we have computed SCF wave functions for Be l3 and Mg t3 . Using the central atom as a model for a bulk atom, we are able to relate the degree of hybridization in the metal to the heats of sublimation. Combining these calculations and other experimental data on alkaline earths, we can explain many of the similarities and differences between these compounds on the basis of arguments related to degree of hybridization and bond strength.

I. INTRODUCTION One would expect to find many similarities among the alkaline earths Be, Mg, and Ca. The atomic excitation energies from ns2 to nsnp(3po) are reasonably similar; those for Be (21979 cm -1) and Mg (21850 cm -1) are virtually identical and that for Ca (15158 cm -1) is only somewhat smaller. 1 The behavior of simple molecules formed with these alkaline earths is also similar. The two known dimers Be2 and Mg 2 are very weakly bound. 2 The hydrides are all fairly strongly boundS: BeH has a dissociation energy De of 50 kcal/mol; for MgH and CaH, the De are 31 and (39 kcal/mol, respectively. The heats of sublimation of the metals differ somewhat more, although they only range over a factor of - 2: they are 78, 35 and 43 kcal/mol for Be, Mg, and Ca, respectively.· Both Be and Mg are hexagonal close-packed solids and Ca forms a closely related face centered cubic crystal. However, the tetrahedral molecules Be 4 and Mg. have been investigated theoretically and have surprisingly different binding energies. Be4 is known to be strongly bound at the SCF level 5 and even more strongly bound at the correlated level. 6 However, Mg 4 is unbound at the SCF level and only weakly bound at the CI level. 7 Considering the many Similarities, it seems odd to find the binding in Bet and Mg 4 differing by a factor of - 5. In this paper we report on calculations on Be4, Mg 4, Ca 4, Be u , and Mg 13 and discuss the origin of the differences between the bond strengths for Be-Be, Mg-Mg, and Ca-Ca. Our analysis relates the stability of these systems to the degree of s - p hybridization found in the wave functions of the various systems. In the next section, we describe the computations performed on the tetramers (X4 ) and present our realpresent address: Rockwell International Science Center, 1049 Camino dos Rios, Thousand Oaks, Calif. 91360.

suits for these molecules. For Be 4 and Mg 4, we compare with the results of previous calculations which also include the effects of electron correlation. In Sec. III, we compare the bonding in these systems and discuss this bonding; in particular, we relate the X4 bonding to the heats of sublimation found for the metals. For our analysis of the Be and Mg solids, we consider the bonding in SCF wave functions for Be a and Mg 13 clusters. Finally we summarize our conclusions in Sec. IV.

II. COMPUTATIONAL RESULTS

A. Be4 Our initial Be4 basis is a double-zeta (DZ) basis set described as DZII in Ref. 8. This uses the 9s basis of 9 van Duijneveldt and 4p functions optimized for the sp state of Be, 10 and is contracted (9s4p/4s2P). This basis set is similar to that of Dykstra, Schaefer, and Meyer (DSM).l1 We used DSM's computed CI equilibrium bond length of 4.0 bohr l l in our exploratory calculations; in particular those to determine an optimum polarization function exponent. With this optimum d polarization function, we recomputed the equilibrium bond length. The calculations proceeded as follows. We obtained SCF wave functions and USing these orbitals, performed CI calculations including all single and double excitations from the SCF "reference" configuration denoted CI(SD). In all calculations, the 1s orbitals were constrained to be doubly occupied. We have computed the dissociation energy with respect to 4 Be atoms by two methods. The first is as a supermolecule with the Be- Be distances at 40 bohr with and without Davidson's estimate of the quadruple excitations 12 [QE = (1 - cg) x EcorrolatiOn]' When the Davidson estimate was used, we applied it to the energy of Be4 at both the binding distance - 4 bohr and also at the large 40 bohr separated

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TABLE I. Summary of dissociation energies, in kcal/mol, for Be4 calculated with different basis sets and for different wave functions. For the CI wave functions, the dissociation energy is calculated using Davidson's correction for quadruple excitations, the directly calculated CI(SD) values are given in parenthesis. The notation CI (separated atoms) and CI (supermolecule) refer to different methods of obtaining the energy at the separated atom limit (see the text). The Be-Be bond distance, R(Be-Be), is given in bohrs. Dissociation energy

SCF

CI (separated atoms)

CI (super molecule) 32. 35(37.42)

d function

exponents

Calculation

R(Be-Be)

DZ basis set

4.0a

35.91

30.26(15. 91)

TZ basis set

4.0a

35.67

30.33(15.86)

DZP basis set

4.0a

38.88

46.93(30.95)

40.28

59. 15(41.36)

41.59

64.23(45. 85)

0.60

DZP basis set

4.0a

0.35

TZP basis set two d functions

3.92c

0.60 0.20

b

48.99(52.18)

aEquilibrium bond length found by DSM (Ref. 11). boptimized value of the d function exponent. CEquilibrium bond length obtained from the CI(SD) energies, including a correction for quadruple excitations.

atom distance. In the second method, the infinite separation is treated as the sum of 4 atomic calculations. In this case, Davidson's correction was used only for Be. at the binding distance, because each individual Be atom is a full two-electron CI. The results of our calculations are summarized in Table I. A comparison with those of DSM and of Pople and his co-workers (WKPKR)6 is discussed in Sec. II C. With the DZ baSiS, the De values obtained using Davidson's estimate of the quadruple excitations agreed within _ 2 kcal/mol whether we treated the separated system as a supermolecule or as 4 Be atoms. We also agree, within the same limit, to the De obtained by DSM. We tested these two methods for obtaining De with one additional baSis set and again found that they gave very similar results. Therefore, only the second method, separated atom limit, was used in all further work, For SCF wave functions for Be., De is not significantly changed when d polarization functions are included in the 5 basis set. However, WKPKR showed that including d functions in the basiS for correlated wave functions leads 'to a significant increase in De. They used a 6- 31G* baSis set for a Moller Plesset/4, denoted MP4(SDQ), many-body treatment of correlation. With this basiS, they computed the De to be 67.5 kcal/mol. Based on this evidence, we also included d functions in our calculations. The D Z basis set with a single d function added, (9s4p1d/4s2p1d) is denoted DZP. When we used the standard d exponent for Be, 13 of 0.6, we found a _17 kcal/mol increase in De. As with the DZ baSiS, the supermolecule results and the sum of separated atoms agree to within - 2 kcal/mol for this DZP basis. The d exponent was then optimized to yield the lowest CI energy of Be. and the optimum was found to be O. 35. It is interesting to note that the optimum value of the d exponent for the Be. cluster is also the optimum value for the (2 electron CI) Be atom. At the DZP CI level

with the optimized d exponent, we have computed - 59 kcal/mol of binding. With this d exponent, we reoptimized the bond length at the CI level by performing calculations at 0.15 bohr bond length intervals. We obtained an equilibrium bond length of 3. 92 bohrs. This is slightly shorter than the value of 4.0 bohrs obtained without d functions by DSM. In order to obtain basis set limit results, we considered a more extended basis set. For the valence type functions, we constructed a triple zeta (TZ) set by contracting the l1s functions of van Duijneveldt 9 and the 5P functions of Dunning and Hay1s (l1s5p/6s3P). With this TZ basis set, the De is very similar to that obtained with the DZ basis (see Table I). We then constructed a T ZP set by adding two d polarization functions with exponents 0.6 and 0.2 for each Be atom (11 s5p2d /6s3P2d). These exponents were chosen so that they were reasonably separated with one above and one below the optimized d exponent found for the DZP set. With this baSiS, the CI value for the De is 64 kcal/mol or - 5 kcal/mol larger than that found with the DZP set. B. Mg4 and Ca4

The work performed on Mg. and Ca. is similar to that performed on Be.. The basis set used for Mg is McLean and Chandler's14 12s9P augmented Huzinaga set15 and uses their DZ contraction (12s9p/6s4p). The basiS set for Ca is based on the Roos et al. 18 12s6p basis set and adds Wachter's17 two additional p functions to represent the 4p orbital. This is contracted using a segmented contraction with overlapping exponents; (643111/ 5311). The contracted basis set (12s8p/6s4p) is of double zeta quality in the valence region, but only of Single zeta quality in the core. The basiS set is tabulated in Table II along with the atomic total energy. We also used basiS sets where we added one d polarization function per atom. We denote the baSis sets without the d functions by DZ and those with d by DZP. As in Be.,

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Bauschlicher, Bagus, and Cox: Bonding in the alkaline earths

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TABLE II. The Ca basis set. The total energy for the uncontracted basis is - 676.61281 and for the contracted basis is - 676. 536 29 hartrees. Exponents

Contraction coefficients ls

41913.600000 6 177. 050 000 1398. 880000 396.028000 128.244000 44.750300 10.717500 4.350460 1.032220

0.001504 0.011796 0.058557 0.205637 0.449856 0.409969

2s

3s

- O. 042 831 - 0.106620 0.404966 0.693415

4s

48 "

- 0.095486 - O. 507 759 1. 293 081

0.436709 0.063625 0.026995

1.0 1.0 1.0

Exponents

Contraction coefficients

2p 229.840000 53.765100 16.515000 5.580880 1.417140 0.447944 0.075000 0.027000

4s'

0.025593 0.161629 0.451065 0.493315 0.075894

4p'

4p

3p

-0.017815 0.503649 0.591071 1.0 1.0

the clusters are treated as a single SCF configuration and followed by CI(SD). For each atom, only the ns electrons are correlated, and thus the cluster is treated as an eight-electron problem. The infinite separation is treated as the sum of 4 atomic results. Davidson's estimate for the effect of quadruple excitations is used. For the DZP basis sets, the value of the d exponent is the value optimized for the two-electron atomic CI calculations; Mg (a. = O. 2) and Ca (a.= 0.1). For Mg, this is very similar to the a. of 0.14 used by Chiles, Dykstra, and Jordan (CDJ)7 which was optimized for Mg 2 • We limited our study of the clusters to metal-metal bond lengths near the region where we expected to find

the equilibrium separation, In order to locate the equilibrium bond length, we performed calculations at grid intervals of 0.2 bohrs. (CDJ reported a shallow, van der Waals-like minimum for Mg. at -10 bohr bond distance. We have not investigated features like this at all.) The results of the Mg. and Ca. calculations are summarized in Table ITI. With the DZ basis, we find, for both the SCF and CI(SD) level, that Mg. is unbound for bond lengths in the region of 6 bohrs; the same result is found by CDJ. With the DZP basis and with CI(SD) wave functions, Mg. is bound; however, our results for the equilibrium bond length,

TABLE III. Summary of dissociation energies, in kcal/moi. for Mg4 and Ca4' For the calculation of the dissociation energy with CI wave functions, see the caption for Table I. The equilibrium bond length, r e' is given in bohrs. Dissociation energy

Calculation

SCF

DZ P basis set

C~

DZ basis set

C~

DZP basis set

CI (super molecule)

Unbound in the region around 6 bohr

Mg4 DZ basis set M~

CI (separated atoms)

Unbound Unbound

11.75(- 3.12) 11. 86

12.16(9.51)

Unbound in the region around 8 bohr c Unbound Unbound

18.27(4.18) 18.30

"Obtained from the CI(SD) energies without a correction for quadruple excitations. t>obtained including a correction for quadruple excitations. clf a correction for quadruple excitations is used, there is a shallow well for R> 8. 6 bohr. J. Chern. Phys., Vol. 77, No.8, 15 October 1982 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 74.201.16.75 On: Wed, 11 Dec 2013 18:35:14

Bauschlicher, Bagus, and Cox: Bonding in the alkaline earths Y e, and for De are somewhat different from those reported by CDJ. We find Y e =6.1 bohr using the directly computed CI(SO) energies and Y e =6. 02 bohr if we use the Davidson 12 quadruple corrected CI energies; at Y e =6. 02 bohrs, De =11. 9 kcal/mol including Davidson's correction. CDJ find Y e =6. 4 bohrs and De =12 kcal/mol where these values are obtained with the use of a different correction, based on the coupled electron pair approximation (CEPA), 18 for the effect of quadruple excitations.

The results for C~ are much more similar to those for Mg. than to those for Be.. With the DZ basis set, Ca. is unbound (or very slightly bound) at both the SCF and CI levels. With the DZP basiS set, the Ca. SCF wave function is also unbound. The DZP CI(SD) wave function leads to a weakly bound system De =18.3 kcal/ mol (including the quadruple correction) at Y e =7.9 bohr. This is similar to the 12 kcal/mol De of Mg. and much less than the 59 kcal/mol value obtained for Be.. For both Mg. and Ca., adding the correction for higher excitations to the CI(SD) energy for the molecule leads to a decrease in the value of Y e from that obtained based on the uncorrected CI(SD) energies. For Be., the use of this correction leads to an increase in Ye. C. Comparison of this work with previous work

While the major focus of this paper is the relationship between the hybridization and bonding as a function of cluster size, we feel that it is important to compare our results with previous work. Different methods were used and somewhat different results were obtained for Be. and Mg.. However, the comparisons are entirely contained in this section of the paper. It is worth considering the role of basis set superposition error in order to understand the difference between our Be. results and those of WKPKR. Our best SCF value for De, obtained with the TZP baSis set is De =41. 6 kcal/mol. This is 5 kcal/mol less than the value of 46.4 obtained by WKPKR. It is clear that their basis set must have an SCF superposition error. SuperpOSition error results from the fact that the basis is effectively larger in the bonding region than at infinite separation, i. e., atom A uses basis functions from atom B to improve the description of the atomiclike features for atom A. Obviously, a basis set superposition error will lead to too large a value for De. Tavouktsoglu and Huzinaga19 have pOinted out that the procedure used to obtain the exponents for the 6-31G basis set used by WKPKR involves fitting GTO functions to the Slater exponents optimized for minimal basis set atomic STO wave functions. However, the Single STO function does not necessarily have the same shape as the numerical Hartree-Fock orbital. Thus, in this procedure, one is fitting the GTO's to a possibly poor orbital; this may lead to a reasonably large superposition error. We can estimate the magnitude of this error for the WKPKR correlated MP4 value for De by assuming that the error here is the same as in the SCF case. We estimated their SCF superposition error above as - 5 kcal/mol by taking the difference between their SCF De and the SCF value which we obtained with our TZP basis set. Since the WKPKR basis set is a DZP set,

4035

it may be more appropriate to compare with the result we obtained with our DZP basis with the optimized d exponent; De =40. 3 kcal/mol. This leads to a slightly larger superpOSition error of 6 kcal/mol. Adjusting their calculated MP4/SDQ De by 6 kcal/mol leads to a value of 61 kcal/mol, which is in rather close agreement with our Davidson quadruple corrected DZPCID e =59 kcal/mol.

WKPKR intoduced a correction to their MP4/SDQ De value because there is a very significant 2s to 2p neardegeneracy effect in the free Be atom. For this reason, they included a special near-degeneracy correction only for the Be atoms but not for the Be. molecule. This atomic correction reduced the De for Be. from the MP4 calculated value of 67.5 to 56 kcal/mol. If we take into account the - 6 kcal/mol adjustment for the superposition error as discussed above, the resultant De'!:2 50kcal/mol is considerably smaller than our CI value with either the DZP or TZP basis. In order to examine the extent to which the atomic 2s to 2p near degeneracy continues to be present in Be., we carried out an MCSCF calculation on Be 4 USing the D ZP basis. In this MCSCF we correlated the 8 valence electrons by including all single and double excitations from the SCF reference configuration into one a1, one e, and two t2 orbitals. This orbital set was chosen on the basis of the natural orbital (NO) occupations in our CI(SD) calculation. We included in the MCSCF the set of orbitals which had large (greater than 0.01) occupation numbers in the CI(SO). In fact, this set is smaller than would be required to fully account for the 2s to 2p near-degeneracy effect. The atomic Be 2P orbitals lead, in Be., to one a" one e, two t2, and one t1 orbital. However, this limited MCSCF is able to recover 80% of the total CI correlation energy, indicating much of the near degeneracy that is present in the atoms carries into the Be4 complex. Because of this, it is quite likely that the near-degeneracy effect which was the reason for WKPKR to correct the MP4 treatment of the Be atom is also present in Be.. Thus it is likely that the atomic near degeneracy correction which they introduced led to a value of De which was too small. In order to understand the reasons for the differences 'in the Mg. calculations, we carried out calculations which closely parallel those of CDJ. 7 In Table IV, we compare the details of our CI results for Mg4 at 6.0 bohr bond distance with theirs. We also compare calculations for Mg. at effectively infinite separation; we used R =40. 0 bohrs and they used R =22. 0 bohrs. The infinite separation results are used to obtain dissociation energies for Mg. at R = 6. 0 bohrs; we denote this as DE to distinguish it from De. Our basis set is somewhat better than the one they used and, as a consequence, we obtain total energies (both SCF and CI which are - o. 1 hartree lower. However, the smaller differences between the results for R = 6. 0 bohrs and R =00 are more important. Their SCF energy is 75.7 kcal/mol higher at 6 bohr and 78.0 kcal/mol higher at infinite separation. Thus, it appears that they had - 2.3 kcal/mol of SCF superposition error. In this case, we may also examine whether an additional superposition error arises for the CI wave functions. This additional CI superposition

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TABLE IV. A comparison of this work and that of Chiles, Dykstra. and Jordan (CDJ) for Mg,. Total energies of Mg, given in hartree and QE and DE are given in kcal/mol. CDJa

This work 00

Separation

6.0 bohr

b,(kcal/mol)

SCF CI Correlation QE

- 798. 427 601 - 798. 541 294 0.113693 12.22

-798.30329 -798.41351 0.11022 11.8

78.00

SCF CI Correlation QE

- 798.415315 - 798. 556448 0.141133 14.98

-798.29466 - 798.43550 0.14034 20.46

75.70

DE(SCF) DE(CI) DE(CI+QE)

-7.71 9.51 12.27

2.18

0.18

-5.44 13.80b 22.46

aReference 7. ~ote CDJ's minimum is 6.4 bohr and De at that point is 16.3 kcal/mol.

error arises because one atom uses basis functions centered on another atom to improve a basically atomiclike correlation effect. At R =6. 0 bohr, we both obtain essentially the same correlation energy [E(correlation) =E(SCF) - E(CI)] before the quadruple excitation estimates are included; the difference being only 0.2 kcal/ mol. However, at R =00, CDJ obtain an E(correlation) which is 2.2 kcal/mol less than our value. We interpret this as an indication of 2.0 kcal/mol of additional CI superposition error. The SCF and CI superposition errors combined lead to CDJ obtaining a DE which, before correction for quadruple excitations, is 4.3 kcal/mol lower than our value. While we both obtain nearly the same correction for higher (quadruple) excitations at R = 6. 0 bohr, our results are quite different at R =00. ThUS, with these corrections, we obtain an increase in DE of 2.8 kcal/mol while CDJ have an 8.7 kcal/mol increase. Most of this difference probably arises from our use of Davidson's correction12 while CDJ use CEPA2 18 to estimate the higher excitations. Considering the approximate nature of these estimates, the difference does not warrant further comment. The calculations used different grid sizes for the Mg-Mg bond distance; our work was performed with a O. 2 bohr grid while CDJ used a 1. 0 bohr bond distance step. This difference in step size is likely to be the reason' that we obtain a significantly different value of reo

In some respects, our results are similar to the previous work; in particular, they show that Mg, is much less strongly bound than Be, (see summary in Table V). Although one would expect some small changes with an improved calculation, we feel that the large difference between Be, and Mg, is real and is far greater than one would expect from the heats of sublimation of the metals (78 vs 35 kcal/mol) or from the binding energies in the hydrides (50 vs 31 kcal/mol). Since it is unlikely that any bonding can arise in the alkaline earths unless some hybridization takes place, we have investigated the relationship between bonding and hybridization. Jordan and Simons 2o have also noted that the difference in bonding must be related to hybridization and have suggested that Mg is unable to hybridize as well as Be, since the 3p orbital must remain orthogonal to the 2p. This seems unlikely; in particular, consider ScH, 21 where the 3~ state forms a 4s-4p hybrid for the ScH bond and thus the inner orbitalS do not prevent hybridization. The atomic excitation energy from the ns 2 ground state to nsnp(Spo) is about the same

TABLE V. Comparison of Be, Mg, and Ca. Be

Mg

Ca

Dissociation energies in keal/mol

III. DISCUSSION Our goal was to obtain results for Be" Mg" and Ca, at the same level of approximation so that we would be able to identify differences and similarities among the systems. Certainly our calculations with the DZP basis and with CI(SD) wave functions using the Davidson quadruple excitation estimate can be improved. This is shown by the different results for Be, which we obtained using the largest, TZP, basis set and USing the DZP set. It is also shown by the different higher excitation corrections which were obtained for Mg, by us and by CDJ. However, we do feel that this approach does treat all three systems equivalently and that the results are sufficiently accurate to allow meaningful comparisons among them.

De (X_H)"

49.8

30.9

De (X2)b

810 em-!

430 em-!

De(X 4) present work

59.2

11.9

18.3

De (metal)C

78

35

43

< 39. 2

Promotion energies (em-I) 21979

21850

15158

liP population analysis X4

1. 06

0.22

X!, (central atom)

1. 44

0.85

X I3 (average of 12 outer atoms)

1.11

0.40

aReference 3. bReference 2.

0.27

cReference 4. dReference 1.

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Bauschlicher, Bagus, and Cox: Bonding in the alkaline earths for Be and Mg and is somewhat smaller for Ca; the experimental values 1 are 21979 cm -1 for Be, 21850 cm- 1 for Mg, and 15158 cm -1 for Ca. These excitation energies are accurately reproduced by our calculations at the two-electron atomic CI level. For Mg, e. g., we compute an excitation energy of 21 339 cm -1 with an error of only 511 cm -t. These atomic excitation energies suggest that hybridization should be about as energetically difficult for both Be and Mg and less difficult for Ca. We can obtain a useful measure of hybridization from a Mulliken population analysis 22 of the SCF wave functions. For Be 4, Mg4, and Ca4, this shows that each atom has some np population: Be 1. 06 (2P), Mg 0.22 (3P), and Ca 0.27 (4P). It is interesting to note that the ratio of the populations is in reasonable agreement with the ratio of the De's. Thus the expected relationship between hybridization and bonding is observed. However, this observation does not explain the reason for the difference in the extent of hybridization. In order to analyze the relationship between hybridization and heats of sublimation, we computed SCF wave functions for BelS and Mg13 clusters using minimal baSis sets. These clusters are intended to represent the behavior of the bulk metal. They have one central atom surrounded by all 12 neighbors which are present in a hexagonal closed packed metal. The central Be atom has a population of 1. 44 2p electrons and the surrounding atoms have an average of 1. 11 2p electrons. The Mg central atom has 0.85 3p electrons, while the surrounding atoms have an average of 0.40 3p electrons. As with the 4 atom clusters, the heats of sublimation show a good correlation with the extent of hybridization on the central atom (see Table V where these properties are summarized). It is also interesting to note that the ratio of the Be to Mg np population changes from 1. 7 for the central atoms to 2.8 for the edge atoms. It is clear from the relationship between hybridization (as measured by the population analysis) and De, that the promotion of electrons from ns to np is, as expected, a major mechanism responsible for the bonding in both the tetrahedral molecules X4 and the metals. The larger differences between the X4 De's and heats of sublimation of the metals for Be and Mg is a result of the very different ratio of hybridization between a 4 atom cluster and the solid. We conclude that this very large difference arises from the fact that Mg-Mg bond is weaker than the Be-Be bond, presumably because of the more diffuse nature of the 3p orbital relative to the 2p orbitals. In the case of the dimers where there is only one metal-metal bond, the energy gained by forming the single bond is not sufficient to overcome the promotion energy. Thus for both Be 2 and Mg 2, only very weak van der Waals bonds are formed 2 (see Table V). When each atom has three neighbors as in X4, the stronger Be- Be bonds can "pay" the promotion energy and form a very stable cluster. However, for Mg4, the weaker Mg-Mg bonds are unable to fully overcome the promotion energy, even though it is the same as for Be. Thus Mg4 is not as stable as Be4" When one conSiders the metals, we find that 12 neighbors for Be can only cause

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a 50% increase (relative to Be4) in the hybridization of the central Be, since a large amount of hybridization has occurred in Be4' Mg is a different case since 12 weak bonds are able to induce much more promotion of 3s to 3p than 4 weak bonds. Thus the central Mg atom in Mg 13 has a fourfold increase in 3p population over that found in Mg 4 • A second measure of how much more sensitive the more weakly bonding Mg is to the number of neighbors than is the more strongly bonding Be can be made be considering the outer atoms in the Xu clusters. These edge atoms do not have the 12 near neighbors of the central (or of a bulk) atom. In the case of Ben, the edge atoms have, on the average, 23% less 2p population than does the central atom. However, for Mg 13, the edge atoms have 52% less 3p population than the central Mg atom! When these alkaline earths bond with hydrogen, a reasonably strong bond is formed. Since there are no core electrons in hydrogen, it is able to "move" into the np hybrid and maximize the bond strength. The bonding in the hydrides is therefore less sensitive to the diffuse nature of the bonding hybrid orbital of the alkali atom. As can be observed in Table V, the range in De for X-H is smaller than for the X4 molecules or for the bulk metals. On the basis of the more diffuse nature of the 4sp hybrid bonding orbital of Ca over the 3sp hybrid of Mg, one might expect that Ca containing systems would be more weakly bound than Mg containing systems. This is not the case. Ca is more strongly bound, in Cat, Ca metal, and probably in CaH, than is Mg. While we do believe that Ca would form even weaker bonds than Mg, the promotion energy is 30% less; thus there are two competing effects. From the Similarities between Ca and Mg, it appears that these two effects have about canceled. We should also note that Ca has a low-lying 3lJ state with the configuration 4s 1 3d1 • Our d function was chosen as a polarization function for the 4s and 4p region and probably could not accurately describe the 31J state. This additional low-lying state might also account for some of the difference between Ca and Be or Mg. An improved series of calculations, which would properly take this effect into account, might therefore show a larger change for Ca than the other atoms. However, the largest effect in describing these systems is the s-p near degeneracy and the resulting hybridization, which we have been able to describe well. IV. CONCLUSION

While the promotion energy, ns to np, is identical for Be and Mg, the strength of the bonds in analogous Be and Mg compounds varies. When they bond to hydrogen, where a strong bond is formed, the difference is small. When they bond with themselves as in Bet and Mgt, the difference is much greater. This is due to the much weaker Mg-Mg bond, which is attributed to the much larger, more diffuse 35 and 3p bonding orbltals of Mg. The weaker bond leads to a reduction in the degree of hybridization. When many bonds are formed, as in the metal, the many weak Mg-Mg bonds allow substantial

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Bauschlicher, Bagus, and Cox: Bonding in the alkaline earths

hybridization and the difference with Be is smaller for the metal than for the tetrahedral X4 molecules. Be with the much more compact bonding, 2s and 2p, orbitals is able to form stronger bonds and therefore the difference in the degree of hybridization of the Be atom between Be4 and Be metal is much smaller. For Ca, the bond is likely to be still weaker, but the promotion energy is 30% lower. These two effects appear to approximately cancel and Ca is therefore found to be similar to Mg. It is interesting to comment on how well small metal clusters may be able to qualitatively represent the bonding in solids. It is clear, from the De's apd from the population analyses, that while Be 4 may provide a fair description of the metal-metal bond in the solid Mg 4 and Ca4 provide, at best, only a very limited description. We have argued that this arises because of the weaker interatomic bonding for Mg and Ca compared to Be. Given this weaker bonding, we can reasonably expect the larger number of near neighbors in the solid compared to the cluster to have a large and significant effect on the nature of the bond formed. This interpretation is supported by the results for the larger, Ben and Mg13 clusters; in particular, consider the different hybridization found for the central (bulklike) atom and the edge atoms. However, the bonding in the X4 clusters involves a considerably larger extent of hybridization and leads to a much stronger bond than that found for the X2 dimers. Thus, it is clear that while extrapolations from clusters to solids may provide useful information, they should be made cautiously.

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