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Atomic-number dependence of relativistic effects on chemical bonding using the non-relativistic and relativistic discrete-variational Xα methods
Volume 2 17, number 1,2
CHEMICAL PHYSICS LETTERS
7 January 1994
Atomic-number dependence of relativistic effects on chemical bonding using the non-relativistic and relativistic discrete-variational Xcu methods Jun Onoe,
Rika Sekine,
Kazuo
Takeuchi
The Insitute ofphysical and Chemical Research (RIKEN), 2-I Hirosawa, Wako, Saitama 351-01, Japan
Hirohide
Nakamatsu,
Takeshi
Mukoyama
Institute for Chemical Reseach, Kyoto University, Gokasho, Uji, Kyoto 61 I, Japan
and Hirohiko
Adachi
Department of Metallurgy, Faculty of Engineering, Kyoto University, Yoshidahonmachi, Sakyo-ku, Kyoto 606-I 1, Japan
Received 22 July 1993; in final form 8 October 1993
We have performed molecular orbital calculations to study the relativistic effects on chemical bonding in hexafluoride molecules such as SF,, SeFs, MoF,, TeF,, WFs, and UF6, using the nonrelatiyistic and relativistic discrete-variational XLYmethods. The atomionumber dependence of the relativistic effects was examined by means of bond overlap population analysis. It is found that the relativistic effects become remarkably important for the molecules containing the elements with atomic number larger than 50.
1. Introduction It is past 60 years since Dirac developed the quantum mechanics combined with relativity [ 11. He predicted that the relativistic quantum mechanics was of no importance in consideration of the electronic structures and chemical bonding in atoms and molecules. However, a rapid development of computer technology has led to the opposite results that relativistic effects are very important in heavy atoms and molecules [ 2-8 1. There are many reports on the relativistic effects on chemical bonding by means of bonding energy analysis with various kinds of molecular orbital (MO) calculations for diatomic molecules until now [ 5-7 1. The comparison of bonding energies between nonrelativistic and relativistic calculations is useful to examine whether the relativity changes the bond length or not [ 6,7 1. However, it is difficult for this 0009-2614/94/$07.00 SSDI
analysis to relate the variations in both orbital contraction and expansion directly to the chemical bonding, though the relativistic contraction of bond length is attributed to the orbital contraction for simple diatomics such as AuH [ 9-23 1. On the other hand, bond overlap populations between valence orbitals obtained by the Mulliken population analysis [ 241 are a good indicator for the strength of covalent bonding and further can provide a detailed contribution of the relativistic changes in orbitals to the chemical bonding, because they directly take account of the effects of variations in both bond length and atomic orbitals on bond strength. This analysis has been successfully applied to the study of the relativistic effects on chemical bonding for polyatomic molecules such as UF6 [25,26]. The aim of the present work is to examine the atomic-number dependence of the relativistic effects on the chemical bonding by comparison of bond
0 1994 Elsevier Science B.V. All rights reserved.
OOOOS-2614(93)E1343-F
61
Volume 217, number 1,2
CHEMICAL PHYSICS LETTERS
overlap populations between the nonrelativistic discrete-variational Hartree-Fock-Slater (DV-HFS) and relativistic DV Dirac-Slater (DV-DS) calculations for hexafluoride molecules such as SF6, SeF6, MoF6, TeF6, WF6, and UF6. Since the DV-DS method can take into account fully the relativistic effects in the Slater exchange potential approximation, it is useful to study the relativistic effects on the chemical bonding [ 27-29 1.
2. Computational method The computational details of the DV-HFS and DVDS methods have been described elsewhere [ 30-331. The molecular wavefunctions were expressed as linear combinations of atomic orbitals obtained by numerically solving the Dirac-Slater or Hartree-FockSlater equations. The atomic potentials for generating the basis functions were derived from the spherical average of the molecular charge density around the nuclei. Thus the atomic orbitals as basis functions were automatically optimized for the molecule. The molecular geometries of the hexafluoride molecules are assumed to be of Oh symmetry with the bond lengths taken from experiments (S-F= 1.561 &Se-F=1.690A,Mo-F=1.820A,Te-F=1.815& W-F= 1.832 A, U-F= 1.999 A) [34,35]. Since the spin function is included in the Dirac-Slater oneelectron equation, the Oh symmetry reduces on the 0: double group in the DV-DS. Symmetry orbitals corresponding to irreducible representations of the 0: symmetry were constructed from the atomic orbitals by using the standard projection operator method [ 36 1, Both relativistic and nonrelativistic calculations were performed with the Slater exchange parameter (Y of 0.7 for all atoms and with 6000 DV sample points, which provided a precision of less than 0.1 eV for valence MO eigenvalues. The basis functions were used up to the 3d orbital for sulfur (S), the 4d orbital for selenium (Se), the 5p orbital for molybdenum (MO), the 5d orbital for tellurium (Te), the 6p orbital for tungsten (W), the 7p orbital for uranium, and the 2p orbital for fluorine (F), respectively. Bond overlap populations were estimated by means of the Mulliken population analysis [ 241. Since the populations depend on the choice 62
7 January 1994
of basis sets, we used the same basis set in both nonrelativistic and relativistic calculations for each molecule. The self-consistent charge method [ 371 was used to approximate the self-consistent field for molecules. The present calculations were carried out selfconsistently until the difference in orbital populations between the initial and final states of the iteration was less than 0.0 1.
3. Results and discussion Fig. 1 shows the bond overlap populations of the hexafluoride molecules XF6 (X=S, Se, MO, Te, W, and U) obtained with the nonrelativistic DV-HFS and relativistic DV-DS calculations as a function of atomic number (Z) of the X element. We have recently shown that the relativistic effects are negligibly small in the valence levels of light molecules such as SF6 and CF4 [ 38 1, though the effects become important in studying the photoelectron spectrum of core levels such as 2p orbitals of sulfur and carbon atoms [ 391. For the valence electrons, the bond overlap population of SF6 obtained with the nonrelativistic calculation is considered to be equal to that obtained with the relativistic one. In fact, fig. 1 indicates that the bond overlap population of 3.86 for the nonrelativistic case agrees with that of 3.85 for the relativistic case. Therefore, the magnitude of the relativistic effects on chemical bonding in the hexafluoride molecules can be discussed with the
0
20
40
60
80
100
Atomic number of X element Fig. 1. Z dependence of bond overlap populations obtained by the nonrelativistic DV-HFS and relativistic DV-DS methods for hexafluorides XF6 (X=S, Se, MO, Te, W, U).
Volume 2 17, number 1,2
7 January 1994
CHEMICAL PHYSICS LETTERS
difference in bond overlap populations between the relativistic and nonrelativistic calculations. It is found from fig. 1 that the magnitude becomes larger with increasing Z. It is interesting to note that the relativistic effects on the X-F chemical bonding remarkably appear in the hexafluoride molecules containing an element with Z larger then 50 (Te, W, and U). Hence the nonrelativistic calculation is inadequate to describe the chemical bonding in molecules containing such heavy elements. The present results of the hexafluorides for the relativistic effects on chemical bonding are essentially applicable to other molecules and solids. Namely, fig. 1 suggests that the relativistic MO calculations should be used to study the bond nature in molecules containing elements with Z larger than 50. As an example, the bond overlap population of UFs for the nonrelativistic calculation is only half of that for the relativistic one. The details of discussion on the role of the relativistic effects on the chemical bonding of UF6 have been described elsewhere [ 25 1. Furthermore, the chalcogen hexafluorides, SeFs and TeF,, show that the bond overlap populations for the relativistic calculation is smaller than those for nonrelativistic ones as shown in fig. 1. For the Pbz molecule, where the relativistic expansion of the Pb-Pb bond length was reported by means of the bonding energy analysis with a relativistic ab initio calculation [ 40 1, a similar trend in bond overlap population has been found ( +0.69 for nonrelativistic and +0.49 for relativistic). This fact indicates that the relativistic expansion of bond lengths occurs in the chalcogen hexafluorides. On the other hand, MoF,, WF6, and UFs show the relativistic contraction of their bond distances, because the AuH has a similar trend ( + 0.53 for nonrelativistic and + 0.63 for relativistic). The details of discussion on the relativistic contraction and expansion of bond length have been described elsewhere [g-23,25,41 I. Pyykko et al. [ 5,421 have systematically examined hydride molecules containing elements with 2 up to 80 with their bonding energies obtained by the Dirac-Fock one-centered approximation and shown that the relativistic contraction of their bond distances roughly increases with Z*. Fig. 2 shows the plot of the difference APx_r in the bond overlap populations between the relativistic (Px& rel. ) ) and
1.0
0.8.
0.2 _
~
LL
"'<.:FB
z u
MoFE,,...'
I/ :' ,...I /'
Y
‘ySeFs '... -0.2-
-0.41 0
“h. 1
.I
2000
4000
6000
8000
10000
z2 Fig. 2. Plot of the difference AP,., in bond overlap populations between the nonrelativistic and relativistic calculations for the hexafluorides as a function of 2’.
nonrelativistic (Px+( nonrel. ) ) calculations for the hexafluorides as a function of Z*. One can see that the AZ’,_, shows Z* dependence for the hexafluorides except UFB, and agrees well with Z* dependence of the relativistic contraction of the bond distances for the hydrides reported by Pyykkij et al. Consequently, the Z dependence of the relativistic effects on chemical bonding in terms of bond overlap populations is equivalent to that of the relativistic effects on the bond distances with analysis of bonding energy. As shown in fig. 2, Z* dependence is not valid for UF6. Since the other actinide hexafluorides such as NpFs and PuF6 have the large ux_r values as much as that for UF6, relativistic effects for Sf-electron system cannot be expressed by the Z* dependence. Although Pyykko et al. have not shown the reason for the Z* dependence of bond distances for the hydrides, we have explained the Z dependence of hpx_, for the hexafluorides including UF6 by the simple model based on Z dependence of the relativistic effects on the 1s orbital for a hydrogenic atom in the present analysis [ 4 11. 63
Volume 217, number 1,2
CHEMICAL PHYSICS LETTERS
4. Summary Z dependence of the relativistic effects on chemical bonding has been examined by comparison of bond overlap populations between the nonrelativistic and relativistic calculations for hexafluorides. It is found that the relativistic effects become remarkably important for the chemical bonding in molecules containing elements with Z larger than 50. In addition, it is found that chalcogen hexafluorides show the relativistic expansion of their bond distances similar to Pbz, while the other hexafluorides indicate the relativistic contraction of their bond lengths in a manner similar to AuH diatomic molecule. The difference in bond overlap populations between nonrelativistic and relativistic calculations can be expressed as a function of Z2 except for UF6 actinide hexafluoride. This result agrees with the 2’ dependence of relativistic contraction of bond length for hydrides containing elements with Z up to 80 by bonding energy analysis.
References [ 1] P.A.M. Dirac, Proc. Roy. Sot. London 123 A ( 1929) 7 14. [2] P. Pyykkii, Advan. Quantum Chem. 11 (1978) 353. [ 31 P. Pyykko, Relativsitic theory of atoms and molecules (a bibliography 1916-1985) (Springer, Berlin, 1986). (41 K, Balasubramanian and KS. Pitzer, Advan. Chem. Phys. 67 (1987) 287. [.5]P. Pyykk8, Chem. Rev. 88 (1988) 563, and references therein. [6] K. Balasubramanian, Chem. Rev. 89 (1989) 1801. [7] K. Balasubramanian, Chem. Rev. 90 ( 1990) 93. [8] P. Pepper and B.E. Bursten, Chem. Rev. 91 (1991) 719. [9] K. Balasubramanian, J. Phys. Chem. 93 (1989) 6585. [ lo] P. Pyykkij and J.P. Desclaux, Chem. Phys. Letters 42 ( 1976) 545. [ 111 P. Pyykkd and J.P. Desclaux, Chem. Phys. 34 (1978) 261. [ 121 J.G. Snijders and P. Pyykkii, Chem. Phys. Letters 75 (1980) 5. [ 131 T. Ziegler, J.G. Snijders and E.J. Baerends, Chem. Phys. Letters 75 (1980) 1.
64
7 January 1994
[ 141 T. Ziebler, J.G. Snijders and E.J. Baerends, J. Chem. Phys. 74 (1981) 1271. [ 151 W.H.E. Schwarz, Physica Scripta 36 ( 1987) 403. [ 161 W. Kutzelniggand W.H. E. Schwarz, Phys. Rev. A 26 (1982) 2361. [ 171 W.H.E. Schwarz, S.Y. Chu and F. Mark, Mol. Phys. 50 (1983) 603. [ 181 P.A. Christiansen, K. Balasubramanian and K.S. Pitzer, J. Chem. Phys. 76 (1982) 5087. [ 191 L.R. Kahn, Intern. J. Quantum Chem. 25 ( 1984) 149. [20] P.A. Christiansen and W.C. Ermler, Mol. Phys. 55 (1985) 1109. [21] E.R. Davidson, Y. Ishikawa and G.L. Malli, Chem. Phys. Letters 84 ( 198 1) 226. [22] N.C. Pyper and P. Marketos, Mol. Phys. 42 (1981) 1059, 1073. [23] W.C. Ermler, R.B. Ross and P.A. Christiansen, Advan. Quantum Chem. 19 ( 1988) 139. [24] R.S. Mulliken, J. Chem. Phys. 23 (1955) 1833,1841,2338, 2343. [25] J. Onoe, K. Takeuchi, H. Nakamatsu, T. Mukoyama, R. Sekine and H. Adachi, J. Chem. Phys., in press. [26] J. Onoe, K. Takeuchi, H. Nakamatsu, T. Mukoyama, R. Sekine and H. Adachi, Chem. Phys. Letters 196 ( 1992) 636. [27] V. Pershina, W.-D. Sepp, B. Fricke and A. Rosen, J. Chem. Phys. 96 (1992) 8367. [28] V. Pershina, W.-D. Sepp, T. Bastug, B. Fricke and G.V. Ionova, J. Chem. Phys. 97 (1992) 1123. [29] V. Pershina, W.-D. Sepp, B. Fricke, D. Kolb, M. Schldel and G.V. Ionova, J. Chem. Phys. 97 ( 1992) 1116. [30] F.W. Averill and D.E. Ellis, J. Chem. Phys. 59 (1973) 6413. [ 3 1 ] H. Adachi, M. Tsukada and C. Satoko, J. Phys. Sot. Japan 45 (1978) 875. [ 321 A. Rosen and D.E. Ellis, J. Chem. Phys. 62 (1975) 3039. [ 331 H. Adachi, A. Rosen and D.E. Ellis, Mol. Phys. 33 ( 1977) 199. [ 341 Handbook of chemistry, The Chemical Society of Japan (Maruzen, Tokyo, 1984) p. II-649 (in Japanese). [35] H.M. Seip, Acta Chem. Stand. 19 (1965) 1955. [36] J. Meyer, W.D. Sepp, B. Fricke and A. Rosen, Computer Phys. Commun. 54 (1989) 55. [ 371 A. Rosen, D.E. Ellis, H. Adachi and F.W. Averill, J. Chem. Phys. 65 (1976) 3629. [38] H. Nakamatsu, H. Adachi and T. Mukoyama, Bull. Inst. Chem. Res. Kyoto Univ. 68 ( 199 1) 304. [39] H. Adachi, H. Nakamatsu and T. Mukoyama, Advan. Xray Chem. Anal. Japan 23 ( 1992) 19 (in Japanese). [40] K. Balasubramanian and K.S. Pitzer, J. Chem. Phys. 78 (1983) 321. [ 411 J. Onoe, in preparation. [42] P. Pyykkli, J. Chem. Sot. Faraday Trans. II 75 (1979) 1256.
CHEMICAL PHYSICS LETTERS
7 January 1994
Atomic-number dependence of relativistic effects on chemical bonding using the non-relativistic and relativistic discrete-variational Xcu methods Jun Onoe,
Rika Sekine,
Kazuo
Takeuchi
The Insitute ofphysical and Chemical Research (RIKEN), 2-I Hirosawa, Wako, Saitama 351-01, Japan
Hirohide
Nakamatsu,
Takeshi
Mukoyama
Institute for Chemical Reseach, Kyoto University, Gokasho, Uji, Kyoto 61 I, Japan
and Hirohiko
Adachi
Department of Metallurgy, Faculty of Engineering, Kyoto University, Yoshidahonmachi, Sakyo-ku, Kyoto 606-I 1, Japan
Received 22 July 1993; in final form 8 October 1993
We have performed molecular orbital calculations to study the relativistic effects on chemical bonding in hexafluoride molecules such as SF,, SeFs, MoF,, TeF,, WFs, and UF6, using the nonrelatiyistic and relativistic discrete-variational XLYmethods. The atomionumber dependence of the relativistic effects was examined by means of bond overlap population analysis. It is found that the relativistic effects become remarkably important for the molecules containing the elements with atomic number larger than 50.
1. Introduction It is past 60 years since Dirac developed the quantum mechanics combined with relativity [ 11. He predicted that the relativistic quantum mechanics was of no importance in consideration of the electronic structures and chemical bonding in atoms and molecules. However, a rapid development of computer technology has led to the opposite results that relativistic effects are very important in heavy atoms and molecules [ 2-8 1. There are many reports on the relativistic effects on chemical bonding by means of bonding energy analysis with various kinds of molecular orbital (MO) calculations for diatomic molecules until now [ 5-7 1. The comparison of bonding energies between nonrelativistic and relativistic calculations is useful to examine whether the relativity changes the bond length or not [ 6,7 1. However, it is difficult for this 0009-2614/94/$07.00 SSDI
analysis to relate the variations in both orbital contraction and expansion directly to the chemical bonding, though the relativistic contraction of bond length is attributed to the orbital contraction for simple diatomics such as AuH [ 9-23 1. On the other hand, bond overlap populations between valence orbitals obtained by the Mulliken population analysis [ 241 are a good indicator for the strength of covalent bonding and further can provide a detailed contribution of the relativistic changes in orbitals to the chemical bonding, because they directly take account of the effects of variations in both bond length and atomic orbitals on bond strength. This analysis has been successfully applied to the study of the relativistic effects on chemical bonding for polyatomic molecules such as UF6 [25,26]. The aim of the present work is to examine the atomic-number dependence of the relativistic effects on the chemical bonding by comparison of bond
0 1994 Elsevier Science B.V. All rights reserved.
OOOOS-2614(93)E1343-F
61
Volume 217, number 1,2
CHEMICAL PHYSICS LETTERS
overlap populations between the nonrelativistic discrete-variational Hartree-Fock-Slater (DV-HFS) and relativistic DV Dirac-Slater (DV-DS) calculations for hexafluoride molecules such as SF6, SeF6, MoF6, TeF6, WF6, and UF6. Since the DV-DS method can take into account fully the relativistic effects in the Slater exchange potential approximation, it is useful to study the relativistic effects on the chemical bonding [ 27-29 1.
2. Computational method The computational details of the DV-HFS and DVDS methods have been described elsewhere [ 30-331. The molecular wavefunctions were expressed as linear combinations of atomic orbitals obtained by numerically solving the Dirac-Slater or Hartree-FockSlater equations. The atomic potentials for generating the basis functions were derived from the spherical average of the molecular charge density around the nuclei. Thus the atomic orbitals as basis functions were automatically optimized for the molecule. The molecular geometries of the hexafluoride molecules are assumed to be of Oh symmetry with the bond lengths taken from experiments (S-F= 1.561 &Se-F=1.690A,Mo-F=1.820A,Te-F=1.815& W-F= 1.832 A, U-F= 1.999 A) [34,35]. Since the spin function is included in the Dirac-Slater oneelectron equation, the Oh symmetry reduces on the 0: double group in the DV-DS. Symmetry orbitals corresponding to irreducible representations of the 0: symmetry were constructed from the atomic orbitals by using the standard projection operator method [ 36 1, Both relativistic and nonrelativistic calculations were performed with the Slater exchange parameter (Y of 0.7 for all atoms and with 6000 DV sample points, which provided a precision of less than 0.1 eV for valence MO eigenvalues. The basis functions were used up to the 3d orbital for sulfur (S), the 4d orbital for selenium (Se), the 5p orbital for molybdenum (MO), the 5d orbital for tellurium (Te), the 6p orbital for tungsten (W), the 7p orbital for uranium, and the 2p orbital for fluorine (F), respectively. Bond overlap populations were estimated by means of the Mulliken population analysis [ 241. Since the populations depend on the choice 62
7 January 1994
of basis sets, we used the same basis set in both nonrelativistic and relativistic calculations for each molecule. The self-consistent charge method [ 371 was used to approximate the self-consistent field for molecules. The present calculations were carried out selfconsistently until the difference in orbital populations between the initial and final states of the iteration was less than 0.0 1.
3. Results and discussion Fig. 1 shows the bond overlap populations of the hexafluoride molecules XF6 (X=S, Se, MO, Te, W, and U) obtained with the nonrelativistic DV-HFS and relativistic DV-DS calculations as a function of atomic number (Z) of the X element. We have recently shown that the relativistic effects are negligibly small in the valence levels of light molecules such as SF6 and CF4 [ 38 1, though the effects become important in studying the photoelectron spectrum of core levels such as 2p orbitals of sulfur and carbon atoms [ 391. For the valence electrons, the bond overlap population of SF6 obtained with the nonrelativistic calculation is considered to be equal to that obtained with the relativistic one. In fact, fig. 1 indicates that the bond overlap population of 3.86 for the nonrelativistic case agrees with that of 3.85 for the relativistic case. Therefore, the magnitude of the relativistic effects on chemical bonding in the hexafluoride molecules can be discussed with the
0
20
40
60
80
100
Atomic number of X element Fig. 1. Z dependence of bond overlap populations obtained by the nonrelativistic DV-HFS and relativistic DV-DS methods for hexafluorides XF6 (X=S, Se, MO, Te, W, U).
Volume 2 17, number 1,2
7 January 1994
CHEMICAL PHYSICS LETTERS
difference in bond overlap populations between the relativistic and nonrelativistic calculations. It is found from fig. 1 that the magnitude becomes larger with increasing Z. It is interesting to note that the relativistic effects on the X-F chemical bonding remarkably appear in the hexafluoride molecules containing an element with Z larger then 50 (Te, W, and U). Hence the nonrelativistic calculation is inadequate to describe the chemical bonding in molecules containing such heavy elements. The present results of the hexafluorides for the relativistic effects on chemical bonding are essentially applicable to other molecules and solids. Namely, fig. 1 suggests that the relativistic MO calculations should be used to study the bond nature in molecules containing elements with Z larger than 50. As an example, the bond overlap population of UFs for the nonrelativistic calculation is only half of that for the relativistic one. The details of discussion on the role of the relativistic effects on the chemical bonding of UF6 have been described elsewhere [ 25 1. Furthermore, the chalcogen hexafluorides, SeFs and TeF,, show that the bond overlap populations for the relativistic calculation is smaller than those for nonrelativistic ones as shown in fig. 1. For the Pbz molecule, where the relativistic expansion of the Pb-Pb bond length was reported by means of the bonding energy analysis with a relativistic ab initio calculation [ 40 1, a similar trend in bond overlap population has been found ( +0.69 for nonrelativistic and +0.49 for relativistic). This fact indicates that the relativistic expansion of bond lengths occurs in the chalcogen hexafluorides. On the other hand, MoF,, WF6, and UFs show the relativistic contraction of their bond distances, because the AuH has a similar trend ( + 0.53 for nonrelativistic and + 0.63 for relativistic). The details of discussion on the relativistic contraction and expansion of bond length have been described elsewhere [g-23,25,41 I. Pyykko et al. [ 5,421 have systematically examined hydride molecules containing elements with 2 up to 80 with their bonding energies obtained by the Dirac-Fock one-centered approximation and shown that the relativistic contraction of their bond distances roughly increases with Z*. Fig. 2 shows the plot of the difference APx_r in the bond overlap populations between the relativistic (Px& rel. ) ) and
1.0
0.8.
0.2 _
~
LL
"'<.:FB
z u
MoFE,,...'
I/ :' ,...I /'
Y
‘ySeFs '... -0.2-
-0.41 0
“h. 1
.I
2000
4000
6000
8000
10000
z2 Fig. 2. Plot of the difference AP,., in bond overlap populations between the nonrelativistic and relativistic calculations for the hexafluorides as a function of 2’.
nonrelativistic (Px+( nonrel. ) ) calculations for the hexafluorides as a function of Z*. One can see that the AZ’,_, shows Z* dependence for the hexafluorides except UFB, and agrees well with Z* dependence of the relativistic contraction of the bond distances for the hydrides reported by Pyykkij et al. Consequently, the Z dependence of the relativistic effects on chemical bonding in terms of bond overlap populations is equivalent to that of the relativistic effects on the bond distances with analysis of bonding energy. As shown in fig. 2, Z* dependence is not valid for UF6. Since the other actinide hexafluorides such as NpFs and PuF6 have the large ux_r values as much as that for UF6, relativistic effects for Sf-electron system cannot be expressed by the Z* dependence. Although Pyykko et al. have not shown the reason for the Z* dependence of bond distances for the hydrides, we have explained the Z dependence of hpx_, for the hexafluorides including UF6 by the simple model based on Z dependence of the relativistic effects on the 1s orbital for a hydrogenic atom in the present analysis [ 4 11. 63
Volume 217, number 1,2
CHEMICAL PHYSICS LETTERS
4. Summary Z dependence of the relativistic effects on chemical bonding has been examined by comparison of bond overlap populations between the nonrelativistic and relativistic calculations for hexafluorides. It is found that the relativistic effects become remarkably important for the chemical bonding in molecules containing elements with Z larger than 50. In addition, it is found that chalcogen hexafluorides show the relativistic expansion of their bond distances similar to Pbz, while the other hexafluorides indicate the relativistic contraction of their bond lengths in a manner similar to AuH diatomic molecule. The difference in bond overlap populations between nonrelativistic and relativistic calculations can be expressed as a function of Z2 except for UF6 actinide hexafluoride. This result agrees with the 2’ dependence of relativistic contraction of bond length for hydrides containing elements with Z up to 80 by bonding energy analysis.
References [ 1] P.A.M. Dirac, Proc. Roy. Sot. London 123 A ( 1929) 7 14. [2] P. Pyykkii, Advan. Quantum Chem. 11 (1978) 353. [ 31 P. Pyykko, Relativsitic theory of atoms and molecules (a bibliography 1916-1985) (Springer, Berlin, 1986). (41 K, Balasubramanian and KS. Pitzer, Advan. Chem. Phys. 67 (1987) 287. [.5]P. Pyykk8, Chem. Rev. 88 (1988) 563, and references therein. [6] K. Balasubramanian, Chem. Rev. 89 (1989) 1801. [7] K. Balasubramanian, Chem. Rev. 90 ( 1990) 93. [8] P. Pepper and B.E. Bursten, Chem. Rev. 91 (1991) 719. [9] K. Balasubramanian, J. Phys. Chem. 93 (1989) 6585. [ lo] P. Pyykkij and J.P. Desclaux, Chem. Phys. Letters 42 ( 1976) 545. [ 111 P. Pyykkd and J.P. Desclaux, Chem. Phys. 34 (1978) 261. [ 121 J.G. Snijders and P. Pyykkii, Chem. Phys. Letters 75 (1980) 5. [ 131 T. Ziegler, J.G. Snijders and E.J. Baerends, Chem. Phys. Letters 75 (1980) 1.
64
7 January 1994
[ 141 T. Ziebler, J.G. Snijders and E.J. Baerends, J. Chem. Phys. 74 (1981) 1271. [ 151 W.H.E. Schwarz, Physica Scripta 36 ( 1987) 403. [ 161 W. Kutzelniggand W.H. E. Schwarz, Phys. Rev. A 26 (1982) 2361. [ 171 W.H.E. Schwarz, S.Y. Chu and F. Mark, Mol. Phys. 50 (1983) 603. [ 181 P.A. Christiansen, K. Balasubramanian and K.S. Pitzer, J. Chem. Phys. 76 (1982) 5087. [ 191 L.R. Kahn, Intern. J. Quantum Chem. 25 ( 1984) 149. [20] P.A. Christiansen and W.C. Ermler, Mol. Phys. 55 (1985) 1109. [21] E.R. Davidson, Y. Ishikawa and G.L. Malli, Chem. Phys. Letters 84 ( 198 1) 226. [22] N.C. Pyper and P. Marketos, Mol. Phys. 42 (1981) 1059, 1073. [23] W.C. Ermler, R.B. Ross and P.A. Christiansen, Advan. Quantum Chem. 19 ( 1988) 139. [24] R.S. Mulliken, J. Chem. Phys. 23 (1955) 1833,1841,2338, 2343. [25] J. Onoe, K. Takeuchi, H. Nakamatsu, T. Mukoyama, R. Sekine and H. Adachi, J. Chem. Phys., in press. [26] J. Onoe, K. Takeuchi, H. Nakamatsu, T. Mukoyama, R. Sekine and H. Adachi, Chem. Phys. Letters 196 ( 1992) 636. [27] V. Pershina, W.-D. Sepp, B. Fricke and A. Rosen, J. Chem. Phys. 96 (1992) 8367. [28] V. Pershina, W.-D. Sepp, T. Bastug, B. Fricke and G.V. Ionova, J. Chem. Phys. 97 (1992) 1123. [29] V. Pershina, W.-D. Sepp, B. Fricke, D. Kolb, M. Schldel and G.V. Ionova, J. Chem. Phys. 97 ( 1992) 1116. [30] F.W. Averill and D.E. Ellis, J. Chem. Phys. 59 (1973) 6413. [ 3 1 ] H. Adachi, M. Tsukada and C. Satoko, J. Phys. Sot. Japan 45 (1978) 875. [ 321 A. Rosen and D.E. Ellis, J. Chem. Phys. 62 (1975) 3039. [ 331 H. Adachi, A. Rosen and D.E. Ellis, Mol. Phys. 33 ( 1977) 199. [ 341 Handbook of chemistry, The Chemical Society of Japan (Maruzen, Tokyo, 1984) p. II-649 (in Japanese). [35] H.M. Seip, Acta Chem. Stand. 19 (1965) 1955. [36] J. Meyer, W.D. Sepp, B. Fricke and A. Rosen, Computer Phys. Commun. 54 (1989) 55. [ 371 A. Rosen, D.E. Ellis, H. Adachi and F.W. Averill, J. Chem. Phys. 65 (1976) 3629. [38] H. Nakamatsu, H. Adachi and T. Mukoyama, Bull. Inst. Chem. Res. Kyoto Univ. 68 ( 199 1) 304. [39] H. Adachi, H. Nakamatsu and T. Mukoyama, Advan. Xray Chem. Anal. Japan 23 ( 1992) 19 (in Japanese). [40] K. Balasubramanian and K.S. Pitzer, J. Chem. Phys. 78 (1983) 321. [ 411 J. Onoe, in preparation. [42] P. Pyykkli, J. Chem. Sot. Faraday Trans. II 75 (1979) 1256.