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Stable conformations of a magnesium ion-icosahedral borane complex
CHEMICAL PHYSICS LETTERS
Volume 130, number 3
STABLE CONFORMATIONS
OF A MAGNESIUM
ION-ICOSAHEDRAL
3 October I986
BORANE
COMPLEX
C.L. BECKEL and I.A. HOWARD Department of Physrcs and Astronomy, University of New Mexrco, 800 Yale Boulevard, NE, Albuquerque, NM87131,
USA
Received 16 June 1986; in final form 14 July 1986
The stable positions of a magnesium ion relative to an icosahedral borane ion, B,2H,2, with the complex Mg+B,,H,2 neutral are investigated. Configurations in which the Mg is inside the boron cage have been included. Using a self-consistent molecularorbital calculational technique, it is established that the center of the boron cage is a stable energy minimum position, probably the global minimum, for the Mg ion.
1. Introduction The icosahedral borane, B , 2H, *, is well known and has been thoroughly studied, both experimentally and theoretically. Longuet-Higgins and Roberts [ 1 ] proposed, as has been confirmed by Lipscomb and others [ 21, that B , *HI 2 would exist in its dianionic form, B,,H:,. It exists in this form, for instance, in potassium borane crystals, in which two K atoms donate two electrons to the borane [ 21. Muetterties et al. [ 31 have observed the IR and Raman spectra of B 12H:s and have interpreted these in terms of regular icosahedral (I,, point group) structure. There also exist carboranes, C2BI,,HIZ, in which the two “extra” C valence electrons serve to till the inner icosahedral bonding orbitals [ 45 1. It is known that several elements (besides carbon) can go substitutionally into the B, 2 cage (e.g. Be, P) , and that other atoms can replace H in bonding to the cage [ 51. To our knowledge, however, complexes of B,*H, 2 plus a divalent atom have not been theoretically investigated (although icosahedral solid borides such as Mg,B,_, ar known to exist [ 61); a divalent metal atom could doubly ionize to supply thetwo extra bonding electrons required by the boron icosahedron. We have used a self-consistent field (SCF) molecular-orbital computational technique to find the orbitals, one-electron eigenvalues, Mulliken charges and total SCF energies of a complex consisting of a single Mg atom plus B , *HI 2. The calculations run over a variety of configurations, including those 254
in which the Mg is inside the B12 cage. Magnesium was chosen because its single and double ionization energies are relatively small; as Mg+ or Mg2+ its ionic radius permits it to lit sterically inside the boron icosahedron [ 71; and its radius is not so small that it could readily leak through an icosahedral face. It is of interest to consider such complexes both on a molecular level and as indicators of possible stable configurations in the solid borides based on linked icosahedral B,* units. Although none of these icosahedral borides is known to contain heteroatoms internal to the icosahedra, such a configuration exists as a possibility. In addition, a number of compounds (including at least one of the “heavy fermion” superconductors) have a structure which can partially be interpreted in terms of atoms trapped in icosahedral structures, e.g. NaZn,,, ThBe,,, andUBe,, [8,9].
2. Methods and models We have made use of the PRDDO (partial retention of diatomic differential overlap) technique [ lo] in examining the neutral Mg+B12Hi2 clusters. This is a self-consistent quantum-chemical molecularorbital calculational method developed by the Lipscomb group. It has been applied widely to calculations involving boranes and carboranes, as well as numerous other molecular structures. It goes beyond such methods as CNDO and MNDO in retaining and approximating some three-center inte0 009-2614/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 130, number
3
CHEMICAL
PHYSICS
LETTERS
3 October
1986
WalS. Molecular orbitals are expressed as a linear combination of Slater-type atomic orbitals. In our calculations we have retained the minimum basis Is, 2s and 2p orbitals for boron, 1s for H, and have used a greater than minimum basis for Mg. In two separate sets of calculations, the Mg basis included 1s through 3p and 1s through 3d basis functions. The only coordinate varied in the calculations was the distance of the Mg ion from the center of the B ,zH, z cage. Two different directions of approach of the Mg to the borane cage were also considered. Since we are interested in the Mg+B12HL2 cluster partially as an indicator of what might happen in an icosahedral boride solid, the geometry we have used for the B,2HIZ cage is not precisely that expected in the borane molecular ion, where all B-B bond lengths are x 1.77 A and all B-H lengths are z 1.19 A. Instead, we have used a somewhat longer B-B bond length, as one finds in the icosahedral solids. We have also slightly broken the icosahedral symmetry to Djd, as is found in the solids. As a result, there are two inequivalent groups of B atoms: one group of six on the top and bottom triangular faces of the cage, and another group of six which comprises the equatorial puckered hexagon of the cage. The bond lengths are those we had determined in an earlier calculation [ 111 to correspond to a neutral BIZH,2 cluster in a simulated crystalline environment in which all the neighboring atoms (here, the H atoms) are chargeneutral. They are: B-B (triangular faces): 1.888 A; B-B (equatorial hexagon): 1.847 A; B-B (from triangle to hexagon): 1.870 A; B-H (all): 1.09 A. The distance from the cage center to a B atom in the equatorial hexagon is thus 1.757 A, while the distance from the center to a B in a top or bottom triangular face is 1.798 A. Self-consistent calculations were performed as the Mg atom was moved along a path from the center of the BIZH12 group to a point 10 8, from the center (i.e. outside the B,,H,, cage). This was done either by moving the Mg through the center of one of the triangular faces of the B, 2 cage or by moving it through the center of one of the edges of the cage (see fig. 1) . Results are qualitatively the same in either case.
Fig. 1. Pathways, indicated by the dashed lines, along which the Mg ion is moved relative to the borane molecular ion. Hydrogens on the borane are omitted here for clarity. Inequivalent B( I ) and B(2) type atoms are labelled.
3. Results and discussion 3. I. Total energies We first consider the set of calculations in which there are 1s through 3p basis functions on Mg. If we plot the total SCF energy of the cluster, relative to its total energy when the Mg is at the center, as we move the Mg from the center through a triangular face to its 10 A position, we produce fig. 2. Stable energy minima are at the center and at z 3.5 A from the center (outside the cage); the minimum at 3.5 A is more stable by x 7.6 eV. Between 0.5 8, and x 2 8, there is a region, left blank in the figure, in which we have large inconsistencies in our calculation of total energies due to effects of the size of the basis set. In this region the Mg atom is in or near the plane of the 16 12 6 k E&CV)
I
; 2 -4-6-
Fig.2. Total Mg+B,,H,zcluster energy as the Mg ion is moved along a path from the center of the borane through a triangular face to a point 10 8, from the center. Energy is relative to the total energy when the Mg is at the center of the borane. Energies calculated are for the case of a Is through 3p basis set on the Mg ion.
255
CHEMICAL PHYSICS LETTERS
Volume 130, number 3 24
i
20 16 ETOT(*v)
i
12 6 4 lY_Lc Oo
DISTANCE 2
4
6
6
IO
CENTER
;KlM A
Fig.3. Total cluster energy, relative to energy when the Mg ion is at the center ofthe borane, as the Mg ion is moved from the center to a point 10 A from the center of the borane. The Mg basis in this case includes Is through 3d orbitals.
boron atoms. The (atomically) unfilled Mg 3p orbitals then essentially act as “extra” B basis functions; that is, in the absence of B basis functions beyond the 2p level, electrons associated with the B atoms in the plane flow into the Mg orbitals to take advantage of the additional flexibility in spatial distribution offered. As a result, the total SCF energies calculated in this region are very low. To some extent, this effect will be present for all configurations; however, it will be much more severe as the Mg nears the plane of the borons. Therefore, we have left blank the region in our plots of total energy versus Mg distance which corresponds to a Mg position near the B plane. To accurately calculate the total energy in all spatial configurations would require the use of extended basis sets on all B atoms as well as on the Mg atom. When the Mg basis set is increased to include 3d orbitals, movement of the Mg from the center through a triangular face to a position 10 8, from the center produces the variation in total energy shown in fig. 3. Qualitatively, the features of the plot are similar to those in fig. 2: one stable minimum at the center of the cage, and another outside, at z 3.0 A from the center. In this case, however, the minimum at the center is 4.7 eV below the one at 3.0 A. The addition of the extended and highly directional d orbitals has made the central minimum much more stable relative to the farther one. While we do not show a comparable plot for movement of the Mg through an edge of the cage, features are qualitatively similar: total energy rising sharply as the Mg is displaced from the center, falling outside the cage boundary, and rising somewhat and leveling off as the distance increases to 10 A. Since basis set limitations prevent us from obtaining an accurate comparison of the total energies near 256
3 October 1986
the B face and away from it, we cannot estimate from our calculations the activation barrier for the Mg to pass through the BIZHi2 cage. What we have shown is that the configuration of the cluster with the Mg at the center is stable, with a total energy comparable to and probably lower than the total energy when Mg is external to the B , ZHI2 cage. 3.2. Charge distributions and orbital occupancies It is of interest to consider the variation of charge distribution within the cluster as configuration - i.e. position of the Mg - and basis set are varied. In this section we consider, for several positions of the Mg atom, the distribution of charge among the atomic basis functions. There can be, of course, significant differences in charge distribution according to whether the Mg 3d functions are omitted or included. We first consider the situation when the Mg atom is 10 8, from the center of the borane. At this distance, the Mg 3d functions play no role; charge distribution among the atomic basis functions is unaffected by their presence. Two electrons are transferred from Mg to the borane. While 3d orbitals are unimportant, the highly directional Mg 3p orbitals contain mO.3 electrons. The 3s occupancy, however, is small, only ~0.03 electrons. Lower-energy basis functions are, of course, almost fully occupied: the 2p orbitals contain 5.69, the 2s orbital 1.97, and the Mg 1s orbital, 2.0 electrons. Since the Mg 3d functions are unimportant at this distance from the borane, there is no difference in the total’energy of the cluster in their presence or absence. As the Mg is moved in to 5 A from the center, the 3d functions remain unimportant, containing less than 0.01 electrons in all; the energy lowering from 10 to 5 A is Coulombic. When the Mg is at the center of the borane, however, the presence of the 3d orbitals in very important. The total energy of the cluster is 14.5 eV lower when the 3d are present than when they are absent. If we look at the components of the total energy, we find that upon addition of the Mg 3d orbitals, the total kinetic energy declines by 16.8 eV, the electron-nuclear attraction (i.e. the bonding energy) is lower by.29.2 eV, and the electron-electron repulsion rises by 3 1.5 eV. The Mulliken charge on the Mg is informative in that it becomes much more positive
Volume 130, number 3
CHEMICALPHYSICS LETTERS
when the 3d functions are removed from the basis set; the net charge is + 0.52 (el in the presence of 3d and + 1.23 (e( in their absence. Thus, electronic charge tends to flow into the 3d orbitals when they are available. The consequent gain in bonding energy is cancelled by the rise in electron-electron repulsion, and the deciding factor in making the presence of the 3d orbitals energetically favorable is the lowering of kinetic energy that takes place in their presence due to electron delocalization. When the 3d orbitals are present, and the Mg is at the center of the cluster, we find that, by comparison with the case when they are absent, the H atoms carry 0.46 Iel less negative charge altogether. The Mg 2s through 3p orbitals lose x 0.07 Iel , and the B 1s, 2s and 2p orbitals lose x 0.26 Ie I of negative charge. Together with minor contributions from 1s orbitals, this charge goes into the 3d orbitals, which then carry - 0.79 Ie I. The Mg has a net positive Mulliken charge of + 0.52 Ie 1, as noted above. The gap between the highest occupied and lowest unoccupied cluster molecular orbitals shows a dependence on configuration that is qualitatively the same for the two different Mg basis sets. The gap when Mg is at the center of the cluster is 16.9 eV when only 3p Mg basis functions are included, and 16.2 eV when 3d functions are added. Gaps decrease when the Mg is in the vicinity of the B or H atoms of the borane, and increase again to 11.3 eV at 10 A from the center, for both of the Mg basis sets. The gap, both when the Mg is in the center and when it is at 10 A, is bounded by energy levels corresponding to borane molecular orbitals; the (most filled) Mg atomic orbitals participate in lower-lying molecular orbitals. It is of some interest to look at the overlaps between the Mg 3d orbitals and the B orbitals when the Mg is at the center of the borane. The total overlap population between the Mg and the B atoms is almost twice as large for the equatorial B atoms as for the B atoms in the polar triangles (this is reasonable since the equatorial B atoms are closer to the center of the borane). There is, of course, very little overlap between any B 1s and Mg 3d. Although there is significant overlap with B 2s orbitals on every B atom, the principal overlaps are between the B 2p and the 3d, and are not therefore centered on the atoms themselves but are in the triangular faces of the borane.
3 October 1986
4. Summary We have investigated the stability of a complex consisting of a Mg atom plus a borane, BIZH12. Two Mg basis sets, and a variety of different distances of the Mg from the center of the borane, were used. The Mg atom was allowed to be either inside or outside the borane cage. We find two stable minima for the Mg atom, one outside the borane cage at a range of % 3.0-3.5 8, from the center, and the other at the center of the cage. If we allow 1s through 3p basis functions on the Mg atom, the external minimum is more stable, whereas the internal minimum is more stable if 3d functions are added. In either case, the Mg atom, being divalent, supplies the two “extra” electrons needed to fill the bonding orbitals of the borane. The present calculations are approximate in several ways. Though we varied the basis set on the Mg atom somewhat, we have restricted boron to a minimal basis; greater flexibility in the B wavefunctions would ensure more accurate calculation of the total energies. Also, no variation of the borane geometry has been allowed. We have shown [ 111 that a Djd borane, upon gaining two electrons, tends to contract (although the exact form of the distortion has not been established). We may expect, then, that there will be a contraction of the borane when the Mg is external to it, and a consequent lowering of the total energy, although probably only by several tenths of an eV. Some relaxation of the geometry will also occur when the Mg is inside the borane, although it may be an expansion. There are a number of ions which, in their doubly ionized state, will “fit” inside the borane cage. That is, their effective radius will be approximately equal to = 0.9 the boron radius. A list of suitable ions under these conditions encompasses more than 20 species, includingGe2+,Nb2+,Fe2+,andZn2+ [7].Itispossible that some ions otherwise Suitable will be too small to have a significant stable energy minimum at the center, and if placed inside the borane will be thermally activated over the barrier at its boundaries. Be2 + , for instance, may be such an ion. It seems, however, that there are many ions which codld be accommodated inside the borane cage, and quite possibly in the B,, units which make up many of the solid box-ides. Whether such compounds can be synthesized and characterized is a question of 257
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CHEMICAL PHYSICS LETTERS
considerable interest in light of the growing research on the borides for thermoelectric, and possibly semiconductor, applications.
Acknowledgement The present research was supported by the Jet Propulsion Laboratory under Contract No. 956970, and by the Center for High Technology Materials, UNM. We would like to thank Professor W.N. Lipscomb of Harvard University, and Dr. D. Emin of Sandia National Laboratories, for valuable discussions, and Professors Riley Schaeffer and J.V. Ortiz of UNM for helpful discussions and for supplying the PRDDO codes used in this work.
References [ I ] H.C. Longuet-Higgins and M. de V. Roberts, Proc. Roy. Sot. A230 (1955) 1IO.
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[2] W.N. Lipscomb, Boron ‘hydrides (Benjamin, New York, 1963). [ 31 E.L. Muetterties, R.E. Merritield, H.C. Miller, W.H. Knoth Jr. and J.R. Downing, J. Am. Chem. Sot. 84 (1962) 506. [4] E.L. Muetterties, Boron hydride chemistry (Academic Press, New York, 1975). [5] R.N. Grimes, Carboranes (Academic Press, New York, 1970). [6] A. Guette, M. Barrett, R. Naslain, P. Hagenmuller, L.-E. Tergenius and T. Lundstrom, J. Less-Common Metals 82 (1981) 325. [ 71 R.C. Weast, ed., Handbook of chemistry and physics, 5 1st Ed. (The Chemical Rubber Co., Cleveland, 1970). [ 81 D.P Shoemaker, R.E. Marsh, F.J. Ewing and L. Pauling, Acta Cryst. 5 (1952) 637. [ 91 A.I. Goldman, S.M. Shapiro, D.E. Cox, J.L. Smith and Z. Fisk, Phys. Rev. B32 (1985) 6042. [IO] T.A. Halgren and W.N. Lipscomb, J. Chem. Phys. 58 (1973) 1569; D.S. Marynick and W.N. Lipscomb, Proc. Natl. Acad. Sci. US 79(1982) 1341. [ 111 LA. Howard, CL. Beckel and D. Emin, in: Boron-rich solids, eds. D. Emin, T. Aselage, C.L. Beckel, LA. Howard and C. Wood (American Institute of Physics, New York, 1986) p. 240.
Volume 130, number 3
STABLE CONFORMATIONS
OF A MAGNESIUM
ION-ICOSAHEDRAL
3 October I986
BORANE
COMPLEX
C.L. BECKEL and I.A. HOWARD Department of Physrcs and Astronomy, University of New Mexrco, 800 Yale Boulevard, NE, Albuquerque, NM87131,
USA
Received 16 June 1986; in final form 14 July 1986
The stable positions of a magnesium ion relative to an icosahedral borane ion, B,2H,2, with the complex Mg+B,,H,2 neutral are investigated. Configurations in which the Mg is inside the boron cage have been included. Using a self-consistent molecularorbital calculational technique, it is established that the center of the boron cage is a stable energy minimum position, probably the global minimum, for the Mg ion.
1. Introduction The icosahedral borane, B , 2H, *, is well known and has been thoroughly studied, both experimentally and theoretically. Longuet-Higgins and Roberts [ 1 ] proposed, as has been confirmed by Lipscomb and others [ 21, that B , *HI 2 would exist in its dianionic form, B,,H:,. It exists in this form, for instance, in potassium borane crystals, in which two K atoms donate two electrons to the borane [ 21. Muetterties et al. [ 31 have observed the IR and Raman spectra of B 12H:s and have interpreted these in terms of regular icosahedral (I,, point group) structure. There also exist carboranes, C2BI,,HIZ, in which the two “extra” C valence electrons serve to till the inner icosahedral bonding orbitals [ 45 1. It is known that several elements (besides carbon) can go substitutionally into the B, 2 cage (e.g. Be, P) , and that other atoms can replace H in bonding to the cage [ 51. To our knowledge, however, complexes of B,*H, 2 plus a divalent atom have not been theoretically investigated (although icosahedral solid borides such as Mg,B,_, ar known to exist [ 61); a divalent metal atom could doubly ionize to supply thetwo extra bonding electrons required by the boron icosahedron. We have used a self-consistent field (SCF) molecular-orbital computational technique to find the orbitals, one-electron eigenvalues, Mulliken charges and total SCF energies of a complex consisting of a single Mg atom plus B , *HI 2. The calculations run over a variety of configurations, including those 254
in which the Mg is inside the B12 cage. Magnesium was chosen because its single and double ionization energies are relatively small; as Mg+ or Mg2+ its ionic radius permits it to lit sterically inside the boron icosahedron [ 71; and its radius is not so small that it could readily leak through an icosahedral face. It is of interest to consider such complexes both on a molecular level and as indicators of possible stable configurations in the solid borides based on linked icosahedral B,* units. Although none of these icosahedral borides is known to contain heteroatoms internal to the icosahedra, such a configuration exists as a possibility. In addition, a number of compounds (including at least one of the “heavy fermion” superconductors) have a structure which can partially be interpreted in terms of atoms trapped in icosahedral structures, e.g. NaZn,,, ThBe,,, andUBe,, [8,9].
2. Methods and models We have made use of the PRDDO (partial retention of diatomic differential overlap) technique [ lo] in examining the neutral Mg+B12Hi2 clusters. This is a self-consistent quantum-chemical molecularorbital calculational method developed by the Lipscomb group. It has been applied widely to calculations involving boranes and carboranes, as well as numerous other molecular structures. It goes beyond such methods as CNDO and MNDO in retaining and approximating some three-center inte0 009-2614/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 130, number
3
CHEMICAL
PHYSICS
LETTERS
3 October
1986
WalS. Molecular orbitals are expressed as a linear combination of Slater-type atomic orbitals. In our calculations we have retained the minimum basis Is, 2s and 2p orbitals for boron, 1s for H, and have used a greater than minimum basis for Mg. In two separate sets of calculations, the Mg basis included 1s through 3p and 1s through 3d basis functions. The only coordinate varied in the calculations was the distance of the Mg ion from the center of the B ,zH, z cage. Two different directions of approach of the Mg to the borane cage were also considered. Since we are interested in the Mg+B12HL2 cluster partially as an indicator of what might happen in an icosahedral boride solid, the geometry we have used for the B,2HIZ cage is not precisely that expected in the borane molecular ion, where all B-B bond lengths are x 1.77 A and all B-H lengths are z 1.19 A. Instead, we have used a somewhat longer B-B bond length, as one finds in the icosahedral solids. We have also slightly broken the icosahedral symmetry to Djd, as is found in the solids. As a result, there are two inequivalent groups of B atoms: one group of six on the top and bottom triangular faces of the cage, and another group of six which comprises the equatorial puckered hexagon of the cage. The bond lengths are those we had determined in an earlier calculation [ 111 to correspond to a neutral BIZH,2 cluster in a simulated crystalline environment in which all the neighboring atoms (here, the H atoms) are chargeneutral. They are: B-B (triangular faces): 1.888 A; B-B (equatorial hexagon): 1.847 A; B-B (from triangle to hexagon): 1.870 A; B-H (all): 1.09 A. The distance from the cage center to a B atom in the equatorial hexagon is thus 1.757 A, while the distance from the center to a B in a top or bottom triangular face is 1.798 A. Self-consistent calculations were performed as the Mg atom was moved along a path from the center of the BIZH12 group to a point 10 8, from the center (i.e. outside the B,,H,, cage). This was done either by moving the Mg through the center of one of the triangular faces of the B, 2 cage or by moving it through the center of one of the edges of the cage (see fig. 1) . Results are qualitatively the same in either case.
Fig. 1. Pathways, indicated by the dashed lines, along which the Mg ion is moved relative to the borane molecular ion. Hydrogens on the borane are omitted here for clarity. Inequivalent B( I ) and B(2) type atoms are labelled.
3. Results and discussion 3. I. Total energies We first consider the set of calculations in which there are 1s through 3p basis functions on Mg. If we plot the total SCF energy of the cluster, relative to its total energy when the Mg is at the center, as we move the Mg from the center through a triangular face to its 10 A position, we produce fig. 2. Stable energy minima are at the center and at z 3.5 A from the center (outside the cage); the minimum at 3.5 A is more stable by x 7.6 eV. Between 0.5 8, and x 2 8, there is a region, left blank in the figure, in which we have large inconsistencies in our calculation of total energies due to effects of the size of the basis set. In this region the Mg atom is in or near the plane of the 16 12 6 k E&CV)
I
; 2 -4-6-
Fig.2. Total Mg+B,,H,zcluster energy as the Mg ion is moved along a path from the center of the borane through a triangular face to a point 10 8, from the center. Energy is relative to the total energy when the Mg is at the center of the borane. Energies calculated are for the case of a Is through 3p basis set on the Mg ion.
255
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Volume 130, number 3 24
i
20 16 ETOT(*v)
i
12 6 4 lY_Lc Oo
DISTANCE 2
4
6
6
IO
CENTER
;KlM A
Fig.3. Total cluster energy, relative to energy when the Mg ion is at the center ofthe borane, as the Mg ion is moved from the center to a point 10 A from the center of the borane. The Mg basis in this case includes Is through 3d orbitals.
boron atoms. The (atomically) unfilled Mg 3p orbitals then essentially act as “extra” B basis functions; that is, in the absence of B basis functions beyond the 2p level, electrons associated with the B atoms in the plane flow into the Mg orbitals to take advantage of the additional flexibility in spatial distribution offered. As a result, the total SCF energies calculated in this region are very low. To some extent, this effect will be present for all configurations; however, it will be much more severe as the Mg nears the plane of the borons. Therefore, we have left blank the region in our plots of total energy versus Mg distance which corresponds to a Mg position near the B plane. To accurately calculate the total energy in all spatial configurations would require the use of extended basis sets on all B atoms as well as on the Mg atom. When the Mg basis set is increased to include 3d orbitals, movement of the Mg from the center through a triangular face to a position 10 8, from the center produces the variation in total energy shown in fig. 3. Qualitatively, the features of the plot are similar to those in fig. 2: one stable minimum at the center of the cage, and another outside, at z 3.0 A from the center. In this case, however, the minimum at the center is 4.7 eV below the one at 3.0 A. The addition of the extended and highly directional d orbitals has made the central minimum much more stable relative to the farther one. While we do not show a comparable plot for movement of the Mg through an edge of the cage, features are qualitatively similar: total energy rising sharply as the Mg is displaced from the center, falling outside the cage boundary, and rising somewhat and leveling off as the distance increases to 10 A. Since basis set limitations prevent us from obtaining an accurate comparison of the total energies near 256
3 October 1986
the B face and away from it, we cannot estimate from our calculations the activation barrier for the Mg to pass through the BIZHi2 cage. What we have shown is that the configuration of the cluster with the Mg at the center is stable, with a total energy comparable to and probably lower than the total energy when Mg is external to the B , ZHI2 cage. 3.2. Charge distributions and orbital occupancies It is of interest to consider the variation of charge distribution within the cluster as configuration - i.e. position of the Mg - and basis set are varied. In this section we consider, for several positions of the Mg atom, the distribution of charge among the atomic basis functions. There can be, of course, significant differences in charge distribution according to whether the Mg 3d functions are omitted or included. We first consider the situation when the Mg atom is 10 8, from the center of the borane. At this distance, the Mg 3d functions play no role; charge distribution among the atomic basis functions is unaffected by their presence. Two electrons are transferred from Mg to the borane. While 3d orbitals are unimportant, the highly directional Mg 3p orbitals contain mO.3 electrons. The 3s occupancy, however, is small, only ~0.03 electrons. Lower-energy basis functions are, of course, almost fully occupied: the 2p orbitals contain 5.69, the 2s orbital 1.97, and the Mg 1s orbital, 2.0 electrons. Since the Mg 3d functions are unimportant at this distance from the borane, there is no difference in the total’energy of the cluster in their presence or absence. As the Mg is moved in to 5 A from the center, the 3d functions remain unimportant, containing less than 0.01 electrons in all; the energy lowering from 10 to 5 A is Coulombic. When the Mg is at the center of the borane, however, the presence of the 3d orbitals in very important. The total energy of the cluster is 14.5 eV lower when the 3d are present than when they are absent. If we look at the components of the total energy, we find that upon addition of the Mg 3d orbitals, the total kinetic energy declines by 16.8 eV, the electron-nuclear attraction (i.e. the bonding energy) is lower by.29.2 eV, and the electron-electron repulsion rises by 3 1.5 eV. The Mulliken charge on the Mg is informative in that it becomes much more positive
Volume 130, number 3
CHEMICALPHYSICS LETTERS
when the 3d functions are removed from the basis set; the net charge is + 0.52 (el in the presence of 3d and + 1.23 (e( in their absence. Thus, electronic charge tends to flow into the 3d orbitals when they are available. The consequent gain in bonding energy is cancelled by the rise in electron-electron repulsion, and the deciding factor in making the presence of the 3d orbitals energetically favorable is the lowering of kinetic energy that takes place in their presence due to electron delocalization. When the 3d orbitals are present, and the Mg is at the center of the cluster, we find that, by comparison with the case when they are absent, the H atoms carry 0.46 Iel less negative charge altogether. The Mg 2s through 3p orbitals lose x 0.07 Iel , and the B 1s, 2s and 2p orbitals lose x 0.26 Ie I of negative charge. Together with minor contributions from 1s orbitals, this charge goes into the 3d orbitals, which then carry - 0.79 Ie I. The Mg has a net positive Mulliken charge of + 0.52 Ie 1, as noted above. The gap between the highest occupied and lowest unoccupied cluster molecular orbitals shows a dependence on configuration that is qualitatively the same for the two different Mg basis sets. The gap when Mg is at the center of the cluster is 16.9 eV when only 3p Mg basis functions are included, and 16.2 eV when 3d functions are added. Gaps decrease when the Mg is in the vicinity of the B or H atoms of the borane, and increase again to 11.3 eV at 10 A from the center, for both of the Mg basis sets. The gap, both when the Mg is in the center and when it is at 10 A, is bounded by energy levels corresponding to borane molecular orbitals; the (most filled) Mg atomic orbitals participate in lower-lying molecular orbitals. It is of some interest to look at the overlaps between the Mg 3d orbitals and the B orbitals when the Mg is at the center of the borane. The total overlap population between the Mg and the B atoms is almost twice as large for the equatorial B atoms as for the B atoms in the polar triangles (this is reasonable since the equatorial B atoms are closer to the center of the borane). There is, of course, very little overlap between any B 1s and Mg 3d. Although there is significant overlap with B 2s orbitals on every B atom, the principal overlaps are between the B 2p and the 3d, and are not therefore centered on the atoms themselves but are in the triangular faces of the borane.
3 October 1986
4. Summary We have investigated the stability of a complex consisting of a Mg atom plus a borane, BIZH12. Two Mg basis sets, and a variety of different distances of the Mg from the center of the borane, were used. The Mg atom was allowed to be either inside or outside the borane cage. We find two stable minima for the Mg atom, one outside the borane cage at a range of % 3.0-3.5 8, from the center, and the other at the center of the cage. If we allow 1s through 3p basis functions on the Mg atom, the external minimum is more stable, whereas the internal minimum is more stable if 3d functions are added. In either case, the Mg atom, being divalent, supplies the two “extra” electrons needed to fill the bonding orbitals of the borane. The present calculations are approximate in several ways. Though we varied the basis set on the Mg atom somewhat, we have restricted boron to a minimal basis; greater flexibility in the B wavefunctions would ensure more accurate calculation of the total energies. Also, no variation of the borane geometry has been allowed. We have shown [ 111 that a Djd borane, upon gaining two electrons, tends to contract (although the exact form of the distortion has not been established). We may expect, then, that there will be a contraction of the borane when the Mg is external to it, and a consequent lowering of the total energy, although probably only by several tenths of an eV. Some relaxation of the geometry will also occur when the Mg is inside the borane, although it may be an expansion. There are a number of ions which, in their doubly ionized state, will “fit” inside the borane cage. That is, their effective radius will be approximately equal to = 0.9 the boron radius. A list of suitable ions under these conditions encompasses more than 20 species, includingGe2+,Nb2+,Fe2+,andZn2+ [7].Itispossible that some ions otherwise Suitable will be too small to have a significant stable energy minimum at the center, and if placed inside the borane will be thermally activated over the barrier at its boundaries. Be2 + , for instance, may be such an ion. It seems, however, that there are many ions which codld be accommodated inside the borane cage, and quite possibly in the B,, units which make up many of the solid box-ides. Whether such compounds can be synthesized and characterized is a question of 257
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considerable interest in light of the growing research on the borides for thermoelectric, and possibly semiconductor, applications.
Acknowledgement The present research was supported by the Jet Propulsion Laboratory under Contract No. 956970, and by the Center for High Technology Materials, UNM. We would like to thank Professor W.N. Lipscomb of Harvard University, and Dr. D. Emin of Sandia National Laboratories, for valuable discussions, and Professors Riley Schaeffer and J.V. Ortiz of UNM for helpful discussions and for supplying the PRDDO codes used in this work.
References [ I ] H.C. Longuet-Higgins and M. de V. Roberts, Proc. Roy. Sot. A230 (1955) 1IO.
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[2] W.N. Lipscomb, Boron ‘hydrides (Benjamin, New York, 1963). [ 31 E.L. Muetterties, R.E. Merritield, H.C. Miller, W.H. Knoth Jr. and J.R. Downing, J. Am. Chem. Sot. 84 (1962) 506. [4] E.L. Muetterties, Boron hydride chemistry (Academic Press, New York, 1975). [5] R.N. Grimes, Carboranes (Academic Press, New York, 1970). [6] A. Guette, M. Barrett, R. Naslain, P. Hagenmuller, L.-E. Tergenius and T. Lundstrom, J. Less-Common Metals 82 (1981) 325. [ 71 R.C. Weast, ed., Handbook of chemistry and physics, 5 1st Ed. (The Chemical Rubber Co., Cleveland, 1970). [ 81 D.P Shoemaker, R.E. Marsh, F.J. Ewing and L. Pauling, Acta Cryst. 5 (1952) 637. [ 91 A.I. Goldman, S.M. Shapiro, D.E. Cox, J.L. Smith and Z. Fisk, Phys. Rev. B32 (1985) 6042. [IO] T.A. Halgren and W.N. Lipscomb, J. Chem. Phys. 58 (1973) 1569; D.S. Marynick and W.N. Lipscomb, Proc. Natl. Acad. Sci. US 79(1982) 1341. [ 111 LA. Howard, CL. Beckel and D. Emin, in: Boron-rich solids, eds. D. Emin, T. Aselage, C.L. Beckel, LA. Howard and C. Wood (American Institute of Physics, New York, 1986) p. 240.