pekka pyykkö department of chemistry ... -...

Covalent Radii

Pekka PyykköDepartment of Chemistry

University of [email protected]

December 2009

Winter School in Theoretical Chemistry, Helsinki, December 2009

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IONIC RADII

1. Introduction. We are looking for effective radii for two ions, A and B, sothat the interionic distance

R(A− B) = rA + rB. (1)

These ionic radii can be regarded as purely operational, as done by Bragg [1],or be related to physical properties, as done by Wasastjerna [3].A first question is the primary standard chosen. Shannon and Prewitt [5,6]

give two sets, based on the values for oxide and fluoride, respectively, at theirCN = 6. Historically, some values are:


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Reference rO2−/pm rF−/pm CommentsBragg [1] 130 135Landé [2]Wasastjerna [3] 132 133Goldschmidt [4] ” ”Shannon [5,6] 140 133 ’IR’

126 119 ’CR’As the early pioneers between 1920 and 1927, Pauling [7] quotes Bragg,Landé, Wasastjerna, Goldschmidt and himself.1. W. L. Bragg, Phil. Mag. 40 (1920) 169. 2. A. Landé, Z. Physik 1 (1920) 191.3. J.A. Wasastjerna, Comm. Phys.-Math., Soc. Sci. Fenn., Vol. 1, No. 38 (1923) 1-25.4. V. M. Goldschmidt, Skrifter Norske Vid. Ak. 1 (1926).5. R.D. Shannon and C.T. Prewitt, Acta Cryst. B, 25 (1969) 925.6. R.D. Shannon, Acta Cryst. A, 32 (1976) 751.7. L. Pauling, The Nature of the Chemical Bond, 3rd Ed., Cornell U. P., Ithaca, NY (1967), p.

224 and Ch. 13.


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COVALENT RADII

2. Molecules. We still wish to use eq. (1). Now Pauling [8] proposed asprimary standards half of the X-X distance in the halogen molecules X2,X=Cl,Br,I. The results are 99, 114 and 133 pm, respectively. These Paulinghalogen radii are still acceptable and exactly agree with Pyykkö-Atsumi(2009). The other Pauling radii are rA = 1

2(A-A), too, for instance for carbon.Bond-order: We take, operationally and by definition,σ2 systems as single bonds,σ2π2 systems as double bonds andσ2π4 systems as triple bonds.The calculated bond-orders can be what they want to be. The purpose is tosimply predict bond lengths.


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Fluorine is special. Schomaker and Stevenson [9] pointed out that Pauling’sfluorine radius of 64 pm refers to an excited state of F2! (The Gale-Monkbands [10]). The current experimental F2 Re is 141.193 pm, yielding an rF of71 pm. Nevertheless the shorter Pauling radius works better.The experimental F2 radius may be ’too long’, due to strong multiconfigurationcharacter (3σ2

g → 3σ2u MCSCF, 3σg1πg → 3σu2πu excitations) [11]. HF: F-F

133 pm [12].The current, Pyykkö-Atsumi (2009) rF = 64 pm.

8. L. Pauling, M. L. Huggins, Z. Krist. A 87 (1934) 205 and ’NCB’, 2nd Ed., Ch. 5.9. V. Schomaker, D. P. Stevenson, J. Am. Chem. Soc. 63 (1941) 37.10. Ap. J. 69 (1929) 77.11. L. G. M. Pettersson, P. E. M. Siegbahn, O. Gropen, Mol. Phys. 48 (1983) 871; M. R. A.

Blomberg, P. E. M. Siegbahn, Chem. Phys. Lett. 81 (1981) 4.12. A. C. Wahl, J. Chem. Phys. 41 (1964) 2600.


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How to handle polar cases? Schomaker and Stevenson [9] suggested theelectronegativity-corrected formula

R(A− B) = rA + rB − c | xA − xB |, (2)

where the x are electronegativities (see Pauling [7, p. 229]). Originally c = 9pm [9]. Not used in current systems of radii.


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3. My Year-1978 attempt. Suppose that we accept the Pauling halogen radiiand derive the radius for other elements, notably d transition metals, with agiven oxidation state and coordination number, from

rA = R(A−X)− rX. (3)

Such radii were derived in [13-14] using all the halogens, X=F-I. The resultswere quoted in [15].As a third step, for an A-H bond for instance, an effective hydrogen radius

becomesrH = R(A−H)− rA. (4)

The old ’hydridic’ value in these papers for electropositive A became rH = 58pm. Similarly, a ’carbidic’ rC = 97 pm.Problem: These radii are not coherent with the M-M distances.

13. P. Pyykkö, J. Chem. Soc., Faraday Trans. II 75 (1979) 1256.14. P. Pyykkö, J. P. Desclaux, Chem. Phys. 34 (1978) 261.15. P. Pyykkö, Chem. Rev. 88 (1988) 563, Table VI.


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4. A halogen-based radius for fluorine. Take now the Pauling radii for theother halogens Cl-I as a standard and see what would they give for rF:

Molecule F-X/pm rF/pmFCl 162.83 63.83FBr 175.89 61.89FI 190.98 57.98

Av. 61.23 ≈ 61 pm.


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Halogen-based covalent radii: Argon5. HArF and all that. The first neutral argon compound with clear covalentbonds to the argon was HArF, reported in [16]. It should be seen as’hydridoargon fluoride’, (HAr)+F−. The calculated H-Ar = 133 pm and Ar-F =197 pm [16].In a ’perspective’ article on rare-gas compounds [17], Pyykkö derived a

covalent radius of 98 pm for argon:

Molecule Ar-X/pm rX/pm rAr/pmArH+ 128.0 30 98ArF+ 162.1 64 98

Av. 98


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An extrapolated rAr of 97 pm is quoted by Huheey [18]. By combining thepotential energy curves of Ar+2 with X2 and X−2 data, Chen and Wentworth [19]deduce a ’virtual covalent’ rAr of 98 pm (Pyykkö-Atsumi 2009: 96 pm.)

16. L. Khriachtchev, M. Pettersson, N. Runeberg, J. Lundell, M. Räsänen, Nature 406 (2000)874.17. P. Pyykkö, Science 290 (2000) 64.18. J. E. Huheey, E. A. Keiter, R. L. Keiter, Inorganic Chemistry, 4th Ed., Harper Collins(1993), p. 292; L. C. Allen, J. E. Huheey, J. Inorg. Nucl. Chem. 42 (1980) 1523.19. E. C. M. Chen, J. G. Dojahn, W. E. Wentworth, J. Phys. Chem. A 101 (1997) 3088.


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METHYL-BASED COVALENT RADII

6. Suresh and Koga [20] chose as the standard radius, instead of thehalogens, the methyl carbon, rC = 77 pm, half of the distance in ethane. Theybase their treatment on B3LYP calculations on CH3-EHn test systems, usingpseudopotentials if necessary. Results are reported for main-group elementsand d transition elements, up to bismuth, Z = 83. This approach takes care ofthe relativistic bond-length contraction.


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Earlier, experimental, carbon-based radii were given by Alcock [21]. They arelargely based on the Cambridge Crystallographic Database study by Allen etal. [22]. For another set, using a corrected formula, see Batsanov [23].New radii from reference C, N, O values by Cordero et al.[20] usingcrystallographic data.

The largest deviations from experimental A-B distances in Table 3 of [20]occur for As-F, As-O, B-B, Cl-Si, F-S, I-I. Some other X2 would not be good,either. In other words, there now are problems with large electronegativitydifferences, and with homonuclear bonds between electronegative elements.The former come out too long, the latter too long for X=Cl, Br and I.

20. C. H. Suresh, N. Koga, J. Phys. Chem. A 105 (2001) 5940.21. N. W. Alcock, Bonding and Structure, Ellis Horwood, Chichester, (1990), pp. 129, 315.22. F. H. Allen, O. Kennard, D. G. Watson, L. Brammer, A. G. Orpen, R. Taylor, J. Chem.Soc., Perkin Trans. II (1987) S1.23. S. S. Batsanov, Russian J. Inorg. Chem. 43 (1998) 437.20. B. Cordero, V. Gómez, A. E. Platero-Prats, M. Revés, J. Echeverria, E. Cremades, F.Barragán, S. Alvarez, Dalton (2008) 2832.


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PYYKKÖ-ATSUMI 2009: A NEW APPROACH

TM–main-group bonds are not single! Omit M(nd)-halide bonds from thedata set! These bonds have some npπ(X)→ n′dπ(M) back donation and musthence be classified as partial multiple bonds.All remaining single bonds can be fitted to a coherent system for Z = 1-118.Recall: A self-consistent fitting. No special elements.For the interalkali dimers an extremely high accuracy resulted, better than anycurrent quantum chemistry.P. Pyykkö, M. Atsumi, Chem. Eur. J. 15 (2009) 186-197.


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MM’ ALKALI-METAL DIMERS

P. Pyykkö, M. Atsumi, Chem. Eur. J. 15 (2009) 186-197. δ=0.84 pm.


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Our r1(Cs) is 232.2 pm (narrow), 232 pm (broad). Earlier value 235 pm (Sanderson 1967)misprinted in the Wiley-VCH table as 253 pm.


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FURTHER QUESTIONS8. Gold versus silver. Our present monovalent, monocoordinate radii for Agand Au are 128 and 124 pm, respectively. Gold is smaller than silver by 4 pm!There are many experimental data for the monovalent, dicoordinate caseshowing a similar trend:S-M-S3− in crystals have experimentally 237 and 230 pm, respectively (see[24]). From a number of experimental cases, CN=2 radii of 68 and 63 pm arederived, corresponding to an oxide radius of 138 pm.In [25], the effective radii are 146 and 137 pm for Ag(I) and Au(I), respectively.In [26], a decrease of about 10 pm is found for 14 compounds.In [27] the decrease is 5.1 pm for M(S-Ad)−2 . In [29], for M(CN)−2 , the M-C are205.2 and 198.4 pm, respectively.

24. M. S. Liao, W. H. E. Schwarz, J. Alloys Compounds 246 (1997) 2. Earlier paper in ActaCryst. B 50 (1994) 9.25. U. M. Tripathi, A. Bauer, H. Schmidbaur, J. Chem. Soc., Dalton Trans. (1997) 2865.


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26. R. E. Bachman, D. F. Andretta, Inorg. Chem. 37 (1998) 5657. These are based onM-phosphine, not M-X.27. K. Fujisawa, S. Imai, Y. Moro-oka, Chem. Lett. (1998) 167.28. E. J. Fernandez et al., Inorg. Chem. 37 (1998) 6002.29. D. B. Leznoff et al., Inorg. Chem. 40 (2001) 6026.


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FURTHER QUESTIONS9. H2. Batsanov [30] argues that this H-H bond of 74 pm is anomalously long,because there is no core. Now r1(H) = 32 pm.10. Lanthanides and actinides. Many open problems. Note the specialposition of Eu(II), Yb(II). Is the actinide contraction larger or smaller than thelanthanide contraction? For the r2 set, Ln=CH+

2 CASPT2 data produced byRoos and Pyykkö [31].11. HF. For the two irregular elements H and F put together, the HF bondlength is 91.7 pm. The VCH radii give 30+64 = 94 pm. Pyykkö-Atsumi (2009)96 pm. Nothing strange here.12. Relation to atomic shells? It is not obvious that this would work.Occasional attempts have been made to use various < rn >.13. Beyond triple bonds, Nagarajan and Morse [32]: TM diatomics withbond-order 5-6.

30. S. S. Batsanov, Struct. Chem. 9 (1998) 65.31. B. O. Roos, P. Pyykkö, Chem. Eur. J. (2009) DOI 10.1002/chem.200902310.32. R. Nagarajan, M. D. Morse, J. Chem. Phys. 127 (2007) 164305.


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Early predictions for trans-An covalent radii

Gr. Syst. Metals ∆R/pm Ref.4 MH4 E104 (Rf), Hf 194-191 = 3 [5]6 MH6 E106 (Sg), W 191-185 = 6 [6]14 MH4 E114, Pb 178.2-179.6 = -1.4 [1,4]

[1] J.P. Desclaux, P. Pyykkö, CPL 29(1974)534.[4] P. Pyykkö, J.P. Desclaux, Nature 266(1977)336.[5] P. Pyykkö, J.P. Desclaux, CPL 50(1977)503.[6] P. Pyykkö, J.P. Desclaux, CP 34(1978)261.


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TRIPLE-BOND COVALENT RADII• RAB = rA + rB.• Pauling 1945: 7 elements, Pauling 1960: 4 elements (C, Si, P,

S).• Now [1]: 81 elements (Be-E112). For 324 data points, standard

deviation 3.2 pm.

[1] P. Pyykkö, S. Riedel, M. Patzschke, Chem. Eur. J., 11 (2005) 3511-3520.


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METHOD

Penalty function over homonuclear and heteronuclear pairs,

∑k

(∆k)2 =Khomo∑

i

(Rii − 2ri)2 +Khetero∑

ij

(Rij − ri − rj)2. (1)

Putting its derivatives with respect to ri equal to zero, one obtains an iterativealgorithm. ’Manual’ starting values were used.

r(N+1)i =

14Khomo +Khetero

Khomo∑i

2Rii +Khetero∑

j

(Rij − r(N)j )

. (2)

Here K is the number of data points in each set and N the iteration. Noconvergence difficulties or signs for multiple minima were noticed.


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The final triple-bond data set had 324 points, used for fitting 81 radii. Thestandard deviation was 3.2 pm

ε =

[(

K∑tot

∆2)/Ktot

]12

Ktot = Khomo +Khetero (3)


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TRIPLE-BOND COVALENT RADII


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Keith-Haring-style Gesamtdiagrams


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6d versus 5d metals: Transactinide break

Group Species R(5d) R(6d) ∆(6d-5d)/pm4 MO 172.3 181.0 8.75 MN 168.3 175.7 7.46 MC 172.1 179.1 7.0

Cl4M≡O 168.9 176.1 7.27 MO− 165.3 172.2 6.98 Cl4M≡O 166.8 173.5 6.79 MC− 165.1 173.2 8.1

MN 159.1 166.6 7.510 MC 167.5 171.8 4.311 MC+ 181.0 176.4 -4.612 MC2+ - 190.8 -


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Some multiply-bonded systemsSpecies Year Ref. NotesTh≡O 1988 1 ThO is a heavy CO, σ2π4.O≡U≡O2+ 1992,1994 2 σ2

gπ4gσ

2uπ

4u.

Au≡C+ 1998, 2000 3 First triple bond to gold.PtSi, PtTh 2003 4 σ2π4.BaPta 2005 5 R=257 pm,

∑r 259 pm.

1 C.L. Marian et al., J. Mol. Str. (Theochem) 169 (1988) 339.2 R. G. Denning, Struct. Bonding 79 (1992) 215; P. Pyykkö, J. Li, N. Runeberg, J. Phys.

Chem. 98 (1994) 4809.3 M. Barysz, P. Pyykkö, Chem. Phys. Lett. 285 (1998) 398.4 M. Barysz, P. Pyykkö, Chem. Phys. Lett. 368 (2003) 538.5 P. Pyykkö, S. Riedel, M. Patzschke, Chem. Eur. J. subm..

aPlatinides, Pt2− first suggested by C. H. L. Goodman, J. Phys. Chem. Solids 6 (1958)305. Reiterated by Pyykkö (2002). Cs2Pt synthesized by A. Karpov, J. Nuss, U. Wedig, M.Jansen Angew. Chem. Int. Ed. 42 (2003) 4818.


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Do van der Waals radii exist?

1) Personal opinion of the author: NO! See e. g. C.R. 97 (1997) 597, Fig. 5.One rather has a series of minima, with log(V (Re)) versus Re being a more orless linear function (ibidem, Fig. 36.)2) The noble gases may form an exception. The mixed dimers obey the sums.See ibid., Table 2. The same sums work for the solid rare gases. SeeRuneberg and Pyykkö, IJQC, 66 (1998) 131.3) Anisotropy: For Group 16-17 X, the vdW radii appear flattened along theA-X axis; see Pyykkö (1997), p. 601. First observed by Pauling.4) Recent summary: S. S. Batsanov, Inorg. Mater. 37 (2001) 1031;Experimental Foundations of Structural Chemistry, Moscow U P (2008).


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THE END


pekka pyykkö department of chemistry ... -...

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