J. Phys. G e m . 1995, 99, 3114-3119
3114
Electronic Structure and Bonding in CaC2 E. Ruiz Departament de Quimica Inorghnica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
P. Alemany" Departament de Quimica Fisica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain Received: September 29, 1994; In Final Form: November 30, 1994@
The electronic structure of CaC2 has been studied by means of periodic Hartree-Fock calculations. Analysis of the band structure and the electronic density reveals a highly ionic character for this compound. The undistorted CaC2 structure, with the dicarbide units aligned along the c axis, is calculated to be the most stable one. According to our results, rotation of these groups at low temperatures should be severely hindered.
Introduction A large number of binary and ternary metal carbides have been known for a long time.' One way of classifying the binary phases is to group them by the extent of C-C bonding: those with short carbon-carbon contacts in one class and the rest in the other. The simplest structure with a short C-C bond is usually referred to as the CaC2 type or CaC2 (I). This structure is adopted by most alkaline-earth carbides and a large number of rare-earth carbides.' The crystal structure is formed by Ca2+ and discrete Cz2- ions arranged in a tetragonally distorted NaCl lattice in which the C2 dimers arrange along the c axis (Figure 1). The structure, as determined from a powder neutron diffraction ~ t u d y , belongs ~,~ to the body-centered tetragonal system with space group I4lmmm (No. 139) and unit cell parameters a = 3.89 8, and c = 6.38 A. The C-C bond distance has been determined to be 1.191 A, which is comparable to that found in acetylene (1.205 A). Due to the difficulty of obtaining pure crystals of adequate size, no refinement of the crystal structure by single-crystal X-ray diffraction has been reported. The validity of this simple structure has been questioned recently. 13C-NMRexperiments on CaCz indicate a marked deviation of carbon atoms from axial ~ y m m e t r y , ~ . ~ which would only be compatible with a structure in which the Cz dimers are not aligned parallel to the c axis. Three further structural modifications for CaC2 which differ in the relative orientations of the C2 units have been reported. CaCz (11) is a low-temperature triclinic modification6 which is only stable in presence of cyanamide ions (CN2*-). In this structure half of the carbide ions are parallel to two of the body diagonals in the cubic NaC1-type subcell, while the other half of the carbide ions are parallel to two of the face diagonals of the cubic subcell. Monoclinic CaC2 (111)7 is very similar to the low-temperature modification of sodium cyanide in which the carbide units are tilted with respect to the c axis. A high temperature form, CaC, (IV),has also been described.6 Rotational disorder of the carbide units yields an average cubic unit cell for this structure. The electronic structure and bonding of several metal carbides have been analyzed using semiempirical band-structure calculations.8-16 Long et al." used the tight-binding extended Hiickel method to study the bonding relations in CaC2 , as well as the possibility of distortions leading to structures in which the Cz2- units are not aligned with the c axis. These authors found that although the tetragonal structure of CaCz (I) is the @
Abstract published in Advance ACS Abstracts, February 1, 1995.
0022-365419512099-3114$09.00/0
C
Figure 1. Crystal structure of tetragonal CaC2.
most stable one, their calculated barrier for rotation of the C2 dimer away from the c axis is extremely small, allowing a significant population of rotated geometries at room temperature. In this contribution, the electronic structure of tetragonal CaC, is studied with the aid of the ab-initio periodic Hartree-Fock method. The calculations presented in this paper were largely motivated by the aim of obtaining a reliable first principles potential for the rotation of the C2 units in calcium carbide.
Methodology The calculations reported in the present work were performed using the CRYSTAL-92 program,18 which provides self consistent solutions to the Hartree-Fock-Roothan equations subject to periodic boundary conditions. Details of the mathematical formulation of this method have been previously described19 and will be omitted here. CRYSTAL-92 uses linear combinations of Gaussian functions to construct a localized atomic basis from which Bloch orbitals are built up. As in the molecular Hartree-Fock calculations, the results can be quite sensitive to the choice of the basis set. Previous work using this methodology has shown that the standard basis sets used in molecular calculations must be modified for their use on periodic systems.19 Parameters for the Gaussian functions used in our work have been obtained from previous calculations on molecular systems by reoptimization of the exponents of the most diffuse functions using the experimental crystal structure. The basis set adopted for the present calculations is reported in Table 1. Computational parameters controlling the truncation of both the Coulomb and exchange infinite series have been chosen to
0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. IO, 1995 3115
Electronic Structure and Bonding in CaC2
TABLE 1: Orbital Exponents (bohr2) and Coefficients for Contracted Basis Sets Used in Calculations on Crystalline CaC2" Ca C coefficient coefficient exponent S P exponent S P 57400.000 0.000 763 3048.000 0.001 826 19130.000 0.001 567 456.400 0.014 060 6377.000 0.007 682 103.700 0.068 760 2126.000 0.028 090 29.230 0.230400 708.600 0.100 900 9.349 0.468 500 236.200 0.299 300 3.189 0.362 800 78.730 0.507 800 26.240 0.205 400 8.748 0.514 900 3.665 -0.395 900 0.236 500 2.916 0.537 400 0.771 1.216OOO 0.860 600 708.600 0.003151 0.185 1.000 OOO 1.000 000 236.200 0.015340 78.730 0.083870 26.240 0.311000 8.748 0.553700 2.916 0.234500 0.972 1.000 000 1.000000 0.299 1.000 000 1.000000 d
d
0.365 1.000000 0.645 1.000000 a The basis set used for Ca is obtained by a 8-21-1 contraction for the s functions and a 6-1-1 contraction for the p functions. For the carbon atom both the s and p functions are obtained from a 6-2-1 contraction. d-type polarization functions are added to both atom types.
0.80
0.40
? d
0.00
-0.40
-0.80
-1.20
r
X
P
r
N
I&
total dC
I
I I
1
I
TABLE 2: Structural Properties Calculated for CaC2; Experimental Data2 Are Provided for Comparison calc exp
3.972 3.890
6.444 6.380
0.4038 0.4020
1.622 1.640
1.239 1.191
14.65 17.36
give a "good" level of accuracy defined in the sense of Pisani, Dovesi, and Roetti.lg
2
Figure 2. Electronic band structure of CaC2.
X10
sc dCa pCa X 1 0 sCa
-1.20 -0.80
-0.40
0.00
0.40
X10
0.80
Energy [a.u.]
Results
Figure 3. Atomic orbital projections of the density of states (DOS) obtained for CaC2. The atomic orbital contributions have been evaluated using a Mulliken partition scheme.
Crystal Structure. The CaC2 (I) structure is tetragonal, space group 14/rnmm (No. 139), with Ca atoms located at 2a positions and C atoms at 4e positions. A total of three parameters (u, c, and the internal coordinate z for the carbon atoms) are needed to fully describe the structure. Table 2 shows the results obtained from optimization of these three parameters together with their experimentally determined values.2 A good agreement between theory and experiment is observed. The internal coordinate z has been obtained by evaluating the total energy as a function of z , keeping u and c at their optimum values. This curve was fitted to a third-order polynomial with the minimum indicated in Table 2. The optimized C-C distance in the crystal is only slightly larger than the value of 1.205 A found for the triple bond in acetylene.' The binding energy is obtained from the difference between the energies calculated for the crystal and for the sum of isolated neutral atoms. In the calculation of the energy of an isolated calcium atom, the basis set used in the crystal calculations (Table l), essentially designed to describe an ionic situation, has been supplemented with an additional sp shell (a = 0.1 bohr-*) in order to provide more variational freedom to account for the tails of the atomic wave function. Agreement with the experimental binding energy is poor. The Hartree-Fock
method is known to underestimate significantly this value due to the neglect of electron correlation. Band Structure and Density of States. The calculated band structure of CaC2 is presented in Figure 2. Calcium carbide is calculated to be a wide gap insulator with a band gap of approximately 10.9 eV. The overestimation of this quantity is a well-known failure of the Hartree-Fock method. Although CaC2 is known to be a colorless insulator (indicating a band gap of at least 3 eV), no experimental determination of this property has been reported to the best of our knowledge. Except for the band gap, our calculated band structure agrees well both in topology and in its main features with the band structure obtained by Long et al.17 As can be appreciated from the different projections of the CaC2 density of states (Figure 3), the valence band is divided into two subbands. The lowest one is very narrow and composed mainly of carbon 2s orbitals. Approximately 11 eV higher in energy one finds the upper subband, mainly formed by carbon 2p orbitals. The conduction band, lying 10.9 eV above the top of the valence band, is also composed basically of carbon 2p orbitals. These results are in good agreement with the detailed analysis of the electronic structure given by Long et al.17 Participation of calcium states is negligible both in the valence and in the conduction band.
3116 J. Phys. Chem., Vol. 99, No. IO, 1995
4
b
Ruiz and Alemany
I Ca
Figure 4. Electronic charge density maps calculated for CaC2. Total electron density (a) on the [llO] plane and (b) on the [001] plane. Electronic density difference between the crystal and the corresponding spherical atomic density superposition arrays (c) on the [110] plane and (d) on the [001] plane. Values corresponding to neighboring isodensity lines differ by 0.005 e/bohrJ. The full and broken curves in the difference maps indicate increase and decrease of electron density, respectively. TABLE 3: Mulliken Population Data for the Optimized Structure of CaCf total 1s 2s 2P 3s,3p 3d Ca 18.142 2.000 2.047 6.010 7.997 0.088 _________~
C
total 6.929
1s 1.997
______~
2s
1.5830
2Pm 2PY 0.9700
~~
2Pz
1.3640
3d 0.044
All occupations are in electrons. The z direction corresponds to the crystallographic c direction, to which C-C bonds in the structure are parallel.
As can be seen from the projections, the participation of d-type polarization functions to the valence and conduction bands is negligible, while their participation in the states above the conduction band is more important, especially for calcium atoms. Electron Density Maps. The calculated electron density maps for two different planes are shown in Figure 4 a,b. Figure 4c,d shows the difference between the crystalline charge density and the superposition of the spherical atomic densities for the same crystallographic planes. Complementary information is supplied in Table 3, where Mulliken population data are reported. As noted in earlier work, when dealing with periodic systems these values should be used in an even more qualitative way than for molecule^.^^ Figure 4 shows an almost perfect spherical charge density around the calcium atoms, indicating a strong ionic character for this compound. The charge density around the C2 dimers indicates strong covalent bonding between both carbon atoms. The difference map in Figure 4c shows an important electron density buildup in the direction of the C-C bonds, both in the bonding region and between the carbon and calcium atoms. On the other side, on formation of the C2 dimers, electron density
is removed from the carbon p orbitals perpendicular to the bond. These results are in good agreement with the qualitative orbital diagram given by Long et al.” that assigns nonbonding (5 character to the states forming the top of the valence band and C-C n antibonding character to the conduction band. Bulk Modulus and C-C Stretching Frequency. The total crystal energy has been evaluated for nine different volumes, keeping the c/a ratio fixed at the value of 1.622 found for the optimized structure. From a third-order least-squares polynomial fit to these data, one can derive the bulk modulus and its pressure derivative. The values obtained for these two properties will be an upper limit for them since no reoptimization of the cla ratio and the z value is performed for each volume. This approximation is reasonable because no important modifications of the crystal structure are expected for small deviations of the optimized crystal structure. The calculated values for these two quantities are B = 63.1 GPa and B’ = 5.86. Unfortunately, to our knowledge no experimental determination of these values has been reported, probably due to the difficulty of obtaining good quality single crystals. The equilibrium volume calculated from this curve is of 101.66 A3. An interesting property is the stretching frequency for the C22- units present in CaCz. Calculations of this quantity for isolated Cz2- ions have been reported by Py~kko.~O The values obtained by this author using the 6-31G* basis set are 1998 cm-’ at the HF level and 1716 cm-’ at the MP2 level. The optimized distances are 1.263 A at the HF level and 1.299 8,at the MP2 level. This author also gives an experimental value of 1831 cm-’ for the frequency obtained for BaC2 crystals which are isostructural to CaC2. To our knowledge, this value has not been reported for calcium carbide. Our aim is to evaluate the effect of the surrounding environment on the equilibrium
J. Phys. Chem., Vol. 99, No. IO, 1995 3117
Electronic Structure and Bonding in CaC2
TABLE 4: Harmonic Vibration Frequencies (in crn-l) and Bonding Distances (in A) for Cz2- Calculated for the Four Models Described in the Text‘ CZ2Cz2- PC Ca&21°+ PC CaC2
+
V(m v(MP2)
d(HF) d(MP2) a
1988 1689 1.263 1.303
1960 1674 1.269 1.307
+
2044 1974 1.242 1.274
2182 1.239
The same basis set (see Table 1) is adopted in all four cases.
distance and the stretching frequency of the C22- units. For this purpose we perform separated calculations for four different models: (a) the isolated Cz2- ion, (b) the Cz2- ion surrounded by a 512 point charge array simulating the Madelung potential of the solid, (c) the Cz2- ion surrounded by the six nearest Ca2+ ions and a 506 point charge array, and (d) the total crystal. The same basis set has been used in all four cases. Calculations for the nonperiodic cluster models have been performed using the GAUSSIAN-92 program.21 To obtain the C-C stretching frequency in the CaC2 crystal, the total energy of the solid has been evaluated as a function of the displacement of the carbon atoms from their equilibrium positions (change in the internal parameter z ) maintaining the symmetry of the crystal. The energy has been evaluated for 11 different C-C distances and the harmonic vibration frequency obtained from the second derivative of a second-order polynomial least-squares fit of these data. The calculated frequencies are indicated in Table 4. The differences in the C-C distances calculated at the HF level for all four models are small, indicating that the surrounding environment has almost no effect on this property. Inclusion of the surrounding Ca2+ ions in the cluster model yields almost the same value as the calculation for the full crystal. This is no longer the case for the C-C stretching frequency. The calculated value is almost 200 cm-’ higher in the crystal than for the isolated dimer. Inclusion of a point charge array around the ion has almost no effect on this frequency. An increase in the stretching frequency is observed when the first coordination sphere is included in the cluster, showing that charge transfer between the Cz2- units and its nearest neighbors is causing an important change on this property. The frequency obtained in this case is still 140 cm-’ smaller than the calculated frequency for the whole crystal. Experience has shown that vibration frequencies obtained by the Hartree-Fock method with basis sets of similar quality as those employed in our work are an average of 12.6% higher than the experimental ones.22 If this trend is suposed to hold in the solid state, an estimated C-C stretching frequency of 1907 cm-’ is obtained for calcium carbide, which is similar to the experimental value found for BaC2 (1831 cm-’). Rotation of the Dicarbide Units. Taking a closer look at the coordination of the Cz2- units, one can see that they are located in an elongated octahedral environment (see la). There are two different types of calcium atoms around each carbon atom: four equatorial atoms at a distance of 2.876 A and two axial calcium atoms at a distance of 2.602 A. The rotation of the C22- unit within the pseudooctahedral environment can be described using the two angles shown in lb. Departure from the regular structure can be gauged by the deviation of 0 from 0”. When 4 = 0”, increasing 0 moves the carbon atoms toward opposite faces of the octahedron, while for 4 = 45” the carbon atoms are moved toward opposite edges of the octahedron. Long et al.” found that, at the extended Huckel level, the undistorted structure is more stable than either of the two distorted ones. Both distortions showed maxima in the total energy near 0 = 35”, with energy barriers of 0.09 and 0.23 eV.
la
lb
These authors also point to the fact that these maxima do not coincide with the points at which the distance between the carbon atoms and the axial calcium atoms is equal to the distance between the carbon atoms and the equatorial calcium atoms (0 = 23” and 0 = 18”, respectively). In these two distortions all dicarbide units are tilted in the same direction, making them incompatible with the neutron diffraction results. A model that includes rotated C22- dimers and is still consistent with the neutron diffraction data needs to consider some sort of disorder in the orientation of the dicarbide units that would statistically average to an axial C22- dimer distribution in the crystal structure. The simplest distortion, shown in 2, results from the
n
A
2
pairing of two neighboring dicarbide units. Extended Huckel results” give also for this case the undistorted structure as the most stable one, although little energy is needed to rotate the G2-dimers out of the c direction. Long et al. conclude that even if the undistorted structure is found to be the most stable one, if one dicarbide unit is provided with enough energy to rotate, then the distortion can, to some extent, propagate through the solid. Since the neglect of ionic interactions in the extended Huckel method could lead to significantly unrealistic results for a highly ionic compound as CaC2, we have calculated the rotation potential for the C2 group in the “frozen” crystal structure using the Hartree-Fock method. The strong ionic character found for this compound suggests that the electronic correlation effects, not included in the Hartree-Fock approach, will be not important in the energy changes due to the rotation of the anions. Figure 5 shows the total energy for the rotation of the dicarbide units plotted for four different values of 4. In all cases, the undistorted structure is the most stable one. The existence of metastable structures for CaC2 with tilted C22- units in which the Ca subnet is slightly different leads to the conclusion that reoptimizating the position of the Ca atoms in our models could leave to more stable structures for large tilting angles. The steep energy change for small tilting angles (for which no significant relaxation of the Ca subnet is expected) indicates however that our “frozen” structure model should be applicable in this case. The potential energy for tilting of the Cz units is almost isotropic in the range 0” < 0 < 20”. For larger deviations from the c
3118 J. Phys. Chem., Vol. 99, No. IO, 1995 2.0
Ruiz and Alemany
1
1
1.5
1.0
0.5
0
10
20
0.0 0
10
20
30
40
e
50
60
70
80
90
(0)
Figure 5. Relative energy for the rotation of the Cz dimers out of the c axis.
axis, the rotation with q5 = 0" (movement of the C atoms toward the faces of the octahedron) is energetically favored while the rotation with 4 = 45" (movement of the C atoms into the edges of the octahedron) represents the most unfavorable case. This trend is inverted for very large deviations of the dicarbide unit from the c axis (8 > SO"). The extended Huckel calculations of Long et al." agree qualitatively with these results, although some important differences should be noticed. The most important one refers to the magnitude of the energy needed to move the dicarbide units away from their alignement with the c axis. The energy necessary to rotate the C2 group by 35" is calculated to be 0.09 eV (for q5 = 0") and 0.23 eV (for 4 = 45") at the extended Huckel level while our calculations give 0.56 and 0.71 eV for these quantities. These values are significantly larger than the thermal energy kT, which at room temperature is approximately of 0.025 eV, indicating that only small displacements of the dicarbide units should be expected. The other important discrepancy between the Hartree-Fock and the extended Huckel results is the shape of the energy curves. The maximum observed in the extended Huckel calculations near 8 = 35" does not appear in the Hartree-Fock results, indicating that the undistorted structure is the only minimum in the energy surface. We have also analyzed the pairing distorsion proposed by Long et al.17 by using a supercell model. A schematic view of this rotation is shown in 2. We have limited our study to the case when both neighboring units move the same amount away from the c axis. The energy curve for this case, shown in Figure 6, is very similar to that obtained for the rotation of all C2 units in the same direction with 4 = 0", showing that the rotation of each C2 unit seems not to be affected significantly by the rotation of the neighboring ones. These results agree qualitatively with those obtained at the extended Huckel 1 e ~ e l . l ~
Concluding Remarks The electronic structure of CaC2 has been studied by means of periodic Hartree-Fock calculations. These reveal a highly ionic character for this compound. Both the calculated C-C distance and stretching frequency are close to the values
30
40
50
60
70
80
90
8 ("1 Figure 6. Relative energy for the pairing distortion of two neighboring CZ dimers. calculated for an isolated Cz2- ion. The undistorted CaC2 structure, with the dicarbide units aligned along the c axis, is calculated to be the most stable one. Our calculations indicate that at low temperatures rotation of these groups is strongly limited. These results do not agree with the spectral broadening observed in 13C-Nh4R4,5that seems to indicate a marked deviation of the CZ units from axial symmetry. The model proposed by Long et al. l7 of a time-averaged alignment of the dicarbide groups along the c axis with a significant population of rotated geometries seems not to be compatible with the potential energy calculated for the C2 rotation at the HartreeFock level. Although the tetragonal form of calcium carbide is probably the stable room temperature modification, the existence of at least three more metastable phases which differ mainly in the orientation of the dicarbide units may be at the origin of the contradiction between our calculations and the I3CNh4R experiments. Probably the solution of this problem will have to wait until good quality, well-characterized crystals of tetragonal CaC2 can be grown.
Acknowledgment. The authors are grateful to C. Pisani, R. Dovesi, and C. Roetti for providing them with a copy of the CRYSTAL-92 code and to S. Alvarez for many helpful discussions. The computing resources at the Centre de Supercomputaci6 de Catalunya (CESCA) were generously made available through a CESCA grant. Financial support to this work was provided by DGYCIT through Grant PB92-0655. References and Notes (1) Wells, A. F. Structural Inorganic Chemistry, 5th ed.; Clarendon Press: Oxford, 1984. (2) Atoji, M.; Medrud, R. C. J. Chem. Phys. 1959, 31, 332. (3) Atoji, M. J. Chem. Phys. 1961, 35, 1950. (4) Duncan, T. M. J. J. Chem. Phys. 1989, 28, 2663. (5) Wrackmeyer, B.; Horchler, K.; Sebald, A,; Merwin, L. H.; Ross, C . Angew. Chem., Int. Ed. Engl. 1990, 29, 807. (6) Vannerberg, N.-G. Acta Chem. Scand. 1961, 16, 1212. (7) Vannerberg, N.-G. Acra Chem. Scand. 1961, 15, 769. (8) Wijeyesekera, S. D.; Hoffmann, R. Organometallics 1984, 3, 949. (9) Bullet, D. W. Inorg. Chem. 1985, 24, 3319. (10) Satpathy, S.; Anderson, 0. K. Znorg. Chem. 1985, 24, 2604. (11) Miller, G. J.; Burdett, J.; Schwarz, C.; Simon, A. Znorg. Chem. 1986, 25, 4437. (12) Hoffmann, R.; Li, J.; Wheeler, R. A. J. Am. Chem. SOC. 1987, 109, 6600. (13) Lee, S.; Jeitscko, W.; Hoffmann, R.-D.Znorg. Chem. 1989, 28, 4094.
Electronic Structure and Bonding in CaC2 (14) Jeitschko, W.; Gerss, M. H.; Hoffmann, R.-D.; Lee, S. J. Less Common Met. 1989,15,397. (15) Li, J.; Hoffmann, R. Chem. Mater. 1989, 1, 83. (16) ~ ~ fR.; Mi ~H,-J.~Z, A ~ ~A&, ~, ~(-hem. , ~1992, ~607,57. 71
I I.
(17) Long, J , F,; Hoffmann, R.; M ~ H,-J.~ ~ Chern. ~ 1992, , 31, 1734. (18) CRYSTAL-92: Dovesi, R.; Saunders, V. R.; Roetti, C. University of Torino (Italy) and Landsbury Laboratory (U.K.), 1992. (19) Pisani, C.; Dovesi, R.; Roetti, C. Hartree-Fock Ab-Znitio Trutment of Crystulline Solids; Springer Verlag: Berlin, 1988.
J. Phys. Chem., Vol. 99, No. IO, 1995 3119 (20) Pyykko, P. Mol. Phys. 1989,67, 871. (21) GAUSSIAN-92: Fnsch, M. J.; Trucks, G. W.; Head-Gordon, M.; P. M. W., G.;Wong, M. W.; Foresman, J. B.; Johnson, B.G.; Schlegel, H. B.; Robb, M. A.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.;Binkley, J. s.;Gonzalez, c.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A., Gaussian Inc., Pittsburgh, PA, 1992. (22) Hout, R. F.; Levi, B. A,; Hehre, W. J. J. Comput. Chem. 1982,3, 234. JP9426177
