Covalent radius

The covalent radius, r_cov, is a measure of the size of an atom that forms part of one covalent bond. It is usually measured either in picometres (pm) or angstroms (Å), with 1 Å = 100 pm.

In principle, the sum of the two covalent radii should equal the covalent bond length between two atoms, R(AB) = r(A) + r(B). Moreover, different radii can be introduced for single, double and triple bonds (r₁, r₂ and r₃ below), in a purely operational sense. These relationships are certainly not exact because the size of an atom is not constant but depends on its chemical environment. For heteroatomic A–B bonds, ionic terms may enter. Often the polar covalent bonds are shorter than would be expected on the basis of the sum of covalent radii. Tabulated values of covalent radii are either average or idealized values, which nevertheless show a certain transferability between different situations, which makes them useful.

The bond lengths R(AB) are measured by X-ray diffraction (more rarely, neutron diffraction on molecular crystals). Rotational spectroscopy can also give extremely accurate values of bond lengths. For homonuclear A–A bonds, Linus Pauling took the covalent radius to be half the single-bond length in the element, e.g. R(H–H, in H₂) = 74.14 pm so r_cov(H) = 37.07 pm: in practice, it is usual to obtain an average value from a variety of covalent compounds, although the difference is usually small. Sanderson has published a recent set of non-polar covalent radii for the main-group elements,[1] but the availability of large collections of bond lengths, which are more transferable, from the Cambridge Crystallographic Database[2][3] has rendered covalent radii obsolete in many situations.

Average radii

The values in the table below are based on a statistical analysis of more than 228,000 experimental bond lengths from the Cambridge Structural Database.[4] For carbon, values are given for the different hybridisations of the orbitals.

Covalent radii in pm from analysis of the Cambridge Structural Database, which contains about 1,030,000 crystal structures[4]

H																	He
1																	2
31(5)																	28
Li	Be											B	C	N	O	F	Ne
3	4	Radius (standard deviation) / pm										5	6	7	8	9	10
128(7)	96(3)											84(3)	sp^3 76(1) sp^2 73(2) sp  69(1)	71(1)	66(2)	57(3)	58
Na	Mg											Al	Si	P	S	Cl	Ar
11	12											13	14	15	16	17	18
166(9)	141(7)											121(4)	111(2)	107(3)	105(3)	102(4)	106(10)
K	Ca	Sc	Ti	V	Cr	Mn	Fe	Co	Ni	Cu	Zn	Ga	Ge	As	Se	Br	Kr
19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36
203(12)	176(10)	170(7)	160(8)	153(8)	139(5)	l.s. 139(5) h.s. 161(8)	l.s. 132(3) h.s. 152(6)	l.s. 126(3) h.s. 150(7)	124(4)	132(4)	122(4)	122(3)	120(4)	119(4)	120(4)	120(3)	116(4)
Rb	Sr	Y	Zr	Nb	Mo	Tc	Ru	Rh	Pd	Ag	Cd	In	Sn	Sb	Te	I	Xe
37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54
220(9)	195(10)	190(7)	175(7)	164(6)	154(5)	147(7)	146(7)	142(7)	139(6)	145(5)	144(9)	142(5)	139(4)	139(5)	138(4)	139(3)	140(9)
Cs	Ba		Hf	Ta	W	Re	Os	Ir	Pt	Au	Hg	Tl	Pb	Bi	Po	At	Rn
55	56		72	73	74	75	76	77	78	79	80	81	82	83	84	85	86
244(11)	215(11)		187(8)	170(8)	162(7)	151(7)	144(4)	141(6)	136(5)	136(6)	132(5)	145(7)	146(5)	148(4)	140(4)	150	150
Fr	Ra
87	88
260	221(2)
		La	Ce	Pr	Nd	Pm	Sm	Eu	Gd	Tb	Dy	Ho	Er	Tm	Yb	Lu
		57	58	59	60	61	62	63	64	65	66	67	68	69	70	71
		207(8)	204(9)	203(7)	201(6)	199	198(8)	198(6)	196(6)	194(5)	192(7)	192(7)	189(6)	190(10)	187(8)	175(10)
		Ac	Th	Pa	U	Np	Pu	Am	Cm
		89	90	91	92	93	94	95	96
		215	206(6)	200	196(7)	190(1)	187(1)	180(6)	169(3)

Radii for multiple bonds

A different approach is to make a self-consistent fit for all elements in a smaller set of molecules. This was done separately for single,[5] double,[6] and triple bonds[7] up to superheavy elements. Both experimental and computational data were used. The single-bond results are often similar to those of Cordero et al.[4] When they are different, the coordination numbers used can be different. This is notably the case for most (d and f) transition metals. Normally one expects that r₁ > r₂ > r₃. Deviations may occur for weak multiple bonds, if the differences of the ligand are larger than the differences of R in the data used.

Note that elements up to atomic number 118 (oganesson) have now been experimentally produced and that there are chemical studies on an increasing number of them. The same, self-consistent approach was used to fit tetrahedral covalent radii for 30 elements in 48 crystals with subpicometer accuracy.[8]

Single-,[5] double-,[6] and triple-bond[7] covalent radii, determined using typically
400 experimental or calculated primary distances, R, per set.

H																	He
1																	2
32 - -																	46 - -
Li	Be											B	C	N	O	F	Ne
3	4	Radius / pm:										5	6	7	8	9	10
133 124 -	102 90 85	single-bond double-bond triple-bond										85 78 73	75 67 60	71 60 54	63 57 53	64 59 53	67 96 -
Na	Mg											Al	Si	P	S	Cl	Ar
11	12											13	14	15	16	17	18
155 160 -	139 132 127											126 113 111	116 107 102	111 102 94	103 94 95	99 95 93	96 107 96
K	Ca	Sc	Ti	V	Cr	Mn	Fe	Co	Ni	Cu	Zn	Ga	Ge	As	Se	Br	Kr
19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36
196 193 -	171 147 133	148 116 114	136 117 108	134 112 106	122 111 103	119 105 103	116 109 102	111 103 96	110 101 101	112 115 120	118 120 -	124 117 121	121 111 114	121 114 106	116 107 107	114 109 110	117 121 108
Rb	Sr	Y	Zr	Nb	Mo	Tc	Ru	Rh	Pd	Ag	Cd	In	Sn	Sb	Te	I	Xe
37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54
210 202 -	185 157 139	163 130 124	154 127 121	147 125 116	138 121 113	128 120 110	125 114 103	125 110 106	120 117 112	128 139 137	136 144 -	142 136 146	140 130 132	140 133 127	136 128 121	133 129 125	131 135 122
Cs	Ba	La-Lu	Hf	Ta	W	Re	Os	Ir	Pt	Au	Hg	Tl	Pb	Bi	Po	At	Rn
55	56		72	73	74	75	76	77	78	79	80	81	82	83	84	85	86
232 209 -	196 161 149		152 128 122	146 126 119	137 120 115	131 119 110	129 116 109	122 115 107	123 112 110	124 121 123	133 142 -	144 142 150	144 135 137	151 141 135	145 135 129	147 138 138	142 145 133
Fr	Ra	Ac-Lr	Rf	Db	Sg	Bh	Hs	Mt	Ds	Rg	Cn	Nh	Fl	Mc	Lv	Ts	Og
87	88		104	105	106	107	108	109	110	111	112	113	114	115	116	117	118
223 218 -	201 173 159		157 140 131	149 136 126	143 128 121	141 128 119	134 125 118	129 125 113	128 116 112	121 116 118	122 137 130	136 - -	143 - -	162 - -	175 - -	165 - -	157 - -
			La	Ce	Pr	Nd	Pm	Sm	Eu	Gd	Tb	Dy	Ho	Er	Tm	Yb	Lu
			57	58	59	60	61	62	63	64	65	66	67	68	69	70	71
			180 139 139	163 137 131	176 138 128	174 137	173 135	172 134	168 134	169 135 132	168 135	167 133	166 133	165 133	164 131	170 129	162 131 131
			Ac	Th	Pa	U	Np	Pu	Am	Cm	Bk	Cf	Es	Fm	Md	No	Lr
			89	90	91	92	93	94	95	96	97	98	99	100	101	102	103
			186 153 140	175 143 136	169 138 129	170 134 118	171 136 116	172 135	166 135	166 136	168 139	168 140	165 140	167	173 139	176	161 141 -

See also

• Atomic radii of the elements (data page)

References

1. Sanderson, R. T. (1983). "Electronegativity and Bond Energy". Journal of the American Chemical Society. 105 (8): 2259–2261. doi:10.1021/ja00346a026.
2. Allen, F. H.; Kennard, O.; Watson, D. G.; Brammer, L.; Orpen, A. G.; Taylor, R. (1987). "Table of Bond Lengths Determined by X-Ray and Neutron Diffraction". J. Chem. Soc., Perkin Trans. 2 (12): S1–S19. doi:10.1039/P298700000S1.
3. Orpen, A. Guy; Brammer, Lee; Allen, Frank H.; Kennard, Olga; Watson, David G.; Taylor, Robin (1989). "Supplement. Tables of bond lengths determined by X-ray and neutron diffraction. Part 2. Organometallic compounds and co-ordination complexes of the d- and f-block metals". Journal of the Chemical Society, Dalton Transactions (12): S1. doi:10.1039/DT98900000S1.
4. Beatriz Cordero; Verónica Gómez; Ana E. Platero-Prats; Marc Revés; Jorge Echeverría; Eduard Cremades; Flavia Barragán; Santiago Alvarez (2008). "Covalent radii revisited". Dalton Trans. (21): 2832–2838. doi:10.1039/b801115j. PMID 18478144. S2CID 244110.
5. P. Pyykkö; M. Atsumi (2009). "Molecular Single-Bond Covalent Radii for Elements 1-118". Chemistry: A European Journal. 15 (1): 186–197. doi:10.1002/chem.200800987. PMID 19058281.
6. P. Pyykkö; M. Atsumi (2009). "Molecular Double-Bond Covalent Radii for Elements Li–E112". Chemistry: A European Journal. 15 (46): 12770–12779. doi:10.1002/chem.200901472. PMID 19856342.. Figure 3 of this paper contains all radii of refs. [5-7]. The mean-square deviation of each set is 3 pm.
7. P. Pyykkö; S. Riedel; M. Patzschke (2005). "Triple-Bond Covalent Radii". Chemistry: A European Journal. 11 (12): 3511–3520. doi:10.1002/chem.200401299. PMID 15832398.
8. P. Pyykkö (2012). "Refitted tetrahedral covalent radii for solids". Physical Review B. 85 (2): 024115, 7 p. Bibcode:2012PhRvB..85b4115P. doi:10.1103/PhysRevB.85.024115.