Bond strength-bond length relationships for some metal-oxygen bonds

] in,lrt' nl,, ,+ (tlt,li

1978 \ oi J,II. pp 275 285

Pergamon Press

Printed in (]real Britain

BOND STRENGTH-BOND LENGTH RELATIONSHIPS FOR SOME METAL-OXYGEN BONDS+ STANLEY SIEGEl, Chemical Engineering Division, Argonne National Laboratory, 97(}0 South Cass Avenue, Argonne, IL 6(}439, U.S,A. (Received 4 April 1977)

Abstract--Examination of crystal structure data for uranyl salts has led to the development of the relation D = l)ll ~ KI1 - ( I/s )) where D is the bond length for bond strength s, D(1) is the bond length corresponding to unit bond strength, and K is a constant which can be obtained from bond length data. The expression is analogous to the Pauling logarithmic relationship D = D(I)- 2 k log s. It can be shown, however, that a more general form of the bond length-bond strength expression is D= A+ KIs. This equation has been applied to U+~-(), B-t}, V~-(} and Mo++'-O bonds. Ihe range of application is limited, however, to s-values near unity and greater. A similal expression D = ,4' + K~/s can be used to obtain D-s information for ionic bonds. For these cases, bond distance> and bond strengths are also limited to a restricted range of values. The quantities A, A', K/s and K']s appear to be radii with the s-dependent terms representing perhaps a polarization quality. The statement that these may be radii is not proven, but is implied because of numerical similarity to known radii values. An empirical method is suggested for computing all bond strengths over the extended range of bond distances.

INTRODUCTION The form of the relationship between bond length and bond strength has been the subject of investigations by numerous authors[I-7]. These studies have centered on the introduction of expressions with adjustable parameters and methods for computing bond distances. However, by examination of the extensive structural information on U +'+-O bonding for which bond strength data have been published[7, 8], it is possible to obtain a simple expression applicable to any structure containing U+%O bonds within the range of bond strengths from .s= I to s - 2 . This relationship can be obtained by determining the manner in which average bond distances within a coordination complex change when the coordination number (with an average bond strength s) is reduced to a lower coordination number (and with a different average bond strength s'). The results appear to be general and have been applied to B-O, V+s-O and Mo+~-O bonds. For these cases, the applicable range of bond strengths is also limited to values of s near unity up to larger values. It is found that the bond strength-bond length relationship is usefui for Na-O, K-O, Ca-O and -]'his work performed under the auspices of the United States Energ} Research and Development Administration.

Mg-O bonds but is limited to s-values corresponding to a restricted range of bond distances. DEVELOPMENT OF THE D.s RELATIONSHIP FROM STRUCTURES CONTAINING U++-O BONDS

The U++-O bonds within a coordination complex show substantial variations in bond lengths. However, for a specific coordination number, CN, the sum of the individual bond distances, ZD~, is approximately constant. and average D- and s-values will show some correspondance with known quantities obtained from structures[8]. This is demonstrated in ]able I for HzU3Om[9], which exhibits CN values of 6 and 7, H.P, (high pressure) UO3[10] with CN 7, and MgUOaII1] and CuUO4[12] with CN values of 6. The quantities listed as s(curve) are bond strength values taken from the Zachariasen data[8] for the specific average distance Y D J C N , and may be compared with bond strengths computed from s - v/CN where r is the valence[13]. If one considers only those structures for which the oxygen coordination about a uranium atom is 6, slight discrepancies are found between the ideal bond strength and those values deduced from the D--s curve. Furthermore, the average bond distance for this coordination is about 2.1 A, whereas for s - I. it should

Table I. 5'Di and average s H2U3Olo

tI.P.

UO3

It2U3010

MgUO4

CuUO 4

CN

7

7

6

6

6

?;I)i

15.31

15.39

12.64

12.56

12.5B

ZDi/( N

2.187

2.199

2.107

2.09~

2.097

s ( c u >,,~ ) *

O. 82

0.80

0.95

0.98

0.9 7

s = v/CN

0.857

0.857

1.00

1.OO

i .00

*Zacha~iasen

rep~rted

parc, d from this in d e d u c i n g

individual

information,

a bond

D a[~d s values.

Use of a curve will

stren!~th or a bond

length.

A curve

i~ usually

fetid to slight

pre-

error~;

276

S. SIEGEL

be 2.083 A according to the D-s curve. (See Table 5(a) for D-s data.) It will be shown that these differences are not too significant. The problem is to relate changes in average bond distance within a coordination to changes in coordination, with the knowledge that Y~Diis a reasonably reliable quantity. Thus, in H2U30]o , CN 7, ~D~ is 15.31 A, while for MgUO4, CN 6, ED~ is 12.56 A. A question arises as to whether each individual bond distance Di is capable of making a "contribution" to the total change in bond length when the coordination number is reduced from, say, 7 to 6. This is indeed the case and the "contribution" can he obtained from the following considerations by reference to Table 2 for H2U3OIo. In the first row, the sums of all the bond distances for the seven uranium-oxygen bonds are tabulated. Numerical values for the experimentally determined bond distances, D~, are given in the second row. The difference between ED~ and D~ (row 3) is the new total bond distance for CN 6. The fourth row shows the average distance for the developed 6-coordinated configuration. Clearly, each of these average distances is not correct, for in an ideal sixfold coordination with each bond characterized by bond strength unity, the Zachariasen data require that D = 2.083 A. Hence, each bond requires a "correction," k, to make it ideal, as shown in the last row, If the k, corrections are plotted against the

corresponding Di values, the curve of Fig. 1 is obtained. Use of H.P. UO3, also with CN 7, but with different bond lengths, leads to the same curve. For either H2U3Olo or H.P. UO3, Ek~- 0.74 A. Reduction from CN 8 to CN 7 will lead to a similar curve, but it is displaced slightly. In the following, s and CN will be used interchangeably by way of the expression s = v/CN. This usage leads to fictitious coordination numbers. Thus, according to the Zachariasen D--s data, a bond of about 1.97 A has an s value near 1.2. Hence, CN =6/1.2=5 so that in such a fivefold complex the length of each uraniumoxygen bond is 1.97 A and the bond strength for each bond is 1.2. With this definition, the process of coordination reduction can be continued in the manner described in Table 2. The results are presented in Table 3, using Y,DI values of MgUO4, CN 6. In this table, (EDI-Di)/5 gives the average distance in a fictitious 5-coordinated complex. Each average distance is high compared with the expected bond length of 1.97 A so that [(£D~ - D~)/5]- 1.97 is the ki correction required. The various corrections are compared with k, values taken from Fig. 1. It is found that the procedure can be reversed; that is, the correction k~ appropriate to the experimental D~ can be taken from Fig. 1 and used to deduce the bond distance, For example, in Table 3, (£D~-1.92)/5 = 2.128A. The k~ value for 1.92A is 0.157, leading to a

i

0.20 0.15

k/

0.10 0.05

0

1.80

2.00

2.20

D/

2.4.0

2.60

2.80

Fig. I. Variationof corrections k, with bond distances Di. +. H.P. UO~;0, H2U;O, . Table 2. Change in EDi with change in CN for H2U3Oi0 ~D i

15.31

Di Difference

15.31

15.31

15.31

15.31

15.31

15.31

1.74

1.78

2.48

2.33

2.27

2.31

2.40

13.57

13.53

12.83

12.98

13.04

13.00

12.91

Ave. = diff/6

2.262

2.255

2.138

2.163

2.173

2.167

2.152

Ave. - 2.083

0.179

0.172

0.0~5

0.080

0.090

0.084

0.069

Table 3. Reduction of CN 6 to CN 5 with ED~ for MgUO4 ~D i Di Difference

12.56

12.56

12.56

1.92x2

2.16x2

2.20x2

10.64

10.40

10.36

Ave. = diff/5

2.128

2.080

2.072

Ave. - 1.97 A

0.158

0.11

0.]02

k

0.157

0.116

0.i]0

(Fig. i) i

Bond strength-bond length relationships value of 2.128-0.157 1.97A for CN5 (s = 1.2). This process can be continued to derive bond distances for CN4 (s = 1.5, D = 1.83 A) and CN3 (s =2, D = 1.70A) by subtracting D~ as many times as required to reach the correct coordination number and applying k~ the same number of times. Uranium-oxygen bond distances are not known with high accuracy, and it is therefore necessat}' to average the data from several structures when carrying out this process. In ]able 2, each of the seven uranium-oxygen bonds is associated with a specific kf which can be applied to give the proper distance in a structure of lower coordination number and which contains bonds of equal length. However, use of the Zachariasen D-s data leads to a v e o simple procedure for obtaining the same results. For example, in H y ~ O . , with CN 7, ]~k i 0,729 A and the axerage value of k is 0.729/7 or 0.104A per bond for ~ unit change in coordination number (in this case from ('N 7 m CN 6). This average value of k is the same as 2.187 2.083 0.104A, w h e r e D = 2 . 1 8 7 A i s t h e a v e r a g e t I-() distance for the seven bonds in H,_U3()t,, and D : 2.083 A again is the Zachariasen bond distance for .s I {CN 61. Thus, k:,,, is just the difference between the average bond distance in the sevenfold structure and the distance for s - 1. For tt.P. UO3, with an average dislance of 2.199A, k., is 2.199-2.083=0.116A. Reference to Fig. 1. which was based on data for both H : U d ) , . and H.P. U O , shows that k~ ~ 0.112 A for the average of 2,187 A and 2.199 A. The average bond strength for CN 7 is 6/7 and the U-() distance associated with this value is 2.16A according to the Zachariasen D-s curve. The latter value is much lower than the average distances of 2.187 A and 2,199A given above. The k~ values thus reflect corrections which apply to individual distances within actual structures. It appears, nevertheless, that k~,,, values, appropriate to the published bond strength-bond length curve, can be obtained directly from the D-s data, as show n belo,a.

277

these values will apply to individual bond distances, 1), in structures of CN 6. Thus, the assignment of ks corrections to the experimental bond distances in MgU(h, CuUO4 and H2U3Om (CN 6) leads to Eki of 0.774, 0.766 and 0.758 A, respectively. The average of these values is 0.766 A and the k~v per bond is 0.766/6 or 0.128 A. To the extent that A D and k~,~are the same. the above information suggests the use of k,,~ ~ A D ~ 0 . 1 2 7 A for D - 2.083 A (s = 1, CN 61 in deriving bond distances. The results are shown in Table 4. In general, for s > 1, D may be computed from D = D(I) -0.127 (6--CN), where D(I) is the bond distance tor s - 1. If CN is replaced by 6/s, the expression becomes

(')

D=D(I)

K I- s

with

K=0.762{0.127>6).

:

~ I.O 1.2 15 2.0

"('N '" 6 ~ 4 3

D 2.083 1.975 1.83 1.70

AD

0.108 0.142 0.130

l'he fact that A D (~k~,,I is not constant is noted and is discussed belo,a. ]'he average value of AD in the tabulation above is 0,127 A, which is essentially the k~ value of i1.13 A taken from Fig. I for D - 2 . 0 8 3 A ( s - 1 ) . The k~ corrections of Fig. 1 were, of course, derived for a coordination number of 7, but, to a close approximation, • lnitialb presented in term,, of radii. :;Ihcse unpubli,,hed ne'a ~alues for bond distances and bond stlcngths v,ere kindl) supplied to me by Prof, W. H. Zachari~I~CI]

This may be compared with the Pauling equation[ 141 D - D(1)- 2 k log sl. Although the term (1- 1/s) is an approximation to log s, it is explicit in the manner by which it was obtained. Examination of Table 4 shows that a more general form of the D-CN expression is D = A+0.127CN, indicating a linear relationship with CN. If, in fact, the Zachariasen data are plotted with D as a function of CN, an approximate linear relationship is obtained. In this plot, D extrapolates to A - 1.32 A for CN = 0. A refined value for A is obtained from Table 4, leading to D : : 1.321+0.127CN, or, in terms of s, D-1.321+0.762/s. This expression is, of course, D = [ D ( I ) - K ] + K/s. The agreement between observed distances and those cornpuled from this formula is shown in Table 5(at. A refined set of D-s values$ for uranium-oxygen bonds leads to the equation D : 1.3517+0.768/s. ]'able 5(b) shows a comparison between the new distance assignments and bond lengths calculated on the basis of the modified A and K terms. These expressions do not account for experimental bond distances with bond strengths less than one. The value of the A term is close to that of the ILl" covalent radius, while K/s approximates that of a single bond oxygert covalent radius with s-dependency. However, the accepted single bond radius for U '~ is 1.42 All5]. It should be recognized that oxygen positions in uranyl salts, as determined by X-rays, are not known with high accuracy, and uncertainties in A and k will arise because of the bond distance errors. For B-O bonds, k is 0.ll; for V-O bonds it is 0.126 with an uncertainty of 0.015: and, for Mo-O bonds, the value is 0.108 with an uncertainty of 0.01. If the value 0.11 is chosen for uranium-oxygen bonds, then t9 1.42 + 0.66/s, an expression that leads to reasonable U-O distances. The uncertainties in AD are apparent in a comparison of Table 5{a) with Table 5(bl.

Table 4. Variation of D based on D(I) and k, of 0.127 A CN

Dca I

6

2,083

5

2.083-I

4

2.083

3

2.083-3

× 2

I) f o r

0.12;

61(N

(Xachar%~sen)

s

=' 6 / n

n

2.083

1.96

A

s

-

6/3

[) =

I .97

A

6/4

D

l .83

:\

6/3

1) -

].70

h

x 0.]2!

-

] .83

A

s

x

:

1.70

A

s

0.12[

s ~

=

A

278

S. SIEGEL Table 5. Calculation of D for U-O bonds (a) D = 1.321 + 0.762/s s

DZach"

Deal.

s

DZach"

Dca I

2.0

1.70

1.70

2.0

1.735

1.736

1.9

1.71

1.72

1.9

1.752

1.756

1.8

1.73

1.74

1.8

1.775

1.778

1.7

1.76

1.77

1.7

1.802

1.803

1.6

I. 79

1.80

1.6

1.834

1.832

1.5

1.83

1.83

1.5

1.870

1.864

1.4

1.88

1.87

1.4

1.908

1.900

1.3

1.92

1.91

1.3

1.948

1.942

1.2

1.97

1.96

1.2

1.990

1.992

i.i

2.03

2.01

i.i

2.035

2.050

1.0

2.08

2.08

1.0

2.083

2.120

B-O BONDS

Zachariasen[16] has also reported D-s data for boronoxygen bonds. His tabulated mean B-O bond lengths for triangular and tetrahedral configurations lead to grand mean values of 1.368 A for CN 3 and 1.475 A for CN 4. Assuming again the applicability of the expression D = A+k(CN), the data give D ~ _ o - A + 0 . 1 0 7 C N , from which A - 1 . 0 4 A . Slight refinement leads to DB_o = 1.034+0.11 CN, or 1.034+0.33/s on an s-basis. Table 6 gives a comparison between the reported D values and those calculated from Da_o above. Failure to account for observed bond distances with s values less than 0.7 is apparent; however, the DB_o expression appears to be sufficiently accurate to permit bond strength calculations for both tetrahedral and triangular configurations. In the NaBO2 structure[17], B-O distances for the triangular bond show large variations. Table 7 gives experimental and computed bond strength values for these bonds. Each sodium atom is bonded to seven oxygen atoms as follows: Na-101 2.461A

Na-20i 2.607A

Na-20j

Na-20, 2.482

2.474

(b) D = 1.3517 + 0,768/s

In the following, all primed quantities, A' and k', will refer to those M-O bonds for which the reference distance is the sum of the ionic radii. The average Na-7(O) distance is 2.5124A. It is assumed that k'v can be obtained from the difference between this average distance and the ionic radii sum of 2.44 A (CN 6). This gives k'av= 0.0724 A from which A' = 2.5124 - 7 × 0.0724 or 2.006 A and DNa_ o = 2.006 + 0.0724/s. If a "coordination reduction" from 7 to 6 is carried out in a manner similar to that described in Table 2 for H2U30,o, it is found that the specific k'i corresponding to 2.44 A is 0.0845 A. This leads to DNa-O = 1.933 + 0.0845/s. It is assumed here that ki-Di data obtained from CN 7 apply to CN 6. Although the differences in the A' and k' coefficients for both expressions are small, the DNa_o equation with A ' = 2.006 A yields bond strengths which agree slightly better with the reported values. This comparison is shown in Table 8. For the examples given here and others which follow, it must be emphasized that A, A', k and k' values, when changed, even substantially, can lead to approximately the same bond strengths as long as the reference distance is unchanged. This relationship arises because the shape

Table 6. Experimental and calculated B-O bond lengths s

DeEp,

Dcal.

s

Dexp.

Deal.

1.4

1.274

1.270

0.9

1.409

1.407

1.3

1.287

1.288

0.8

1.453

1.447

1.2

1.304

1.309

0.7

1.497

1.505

i.I

1.330

1.334

0.6

1.541

--

1.0

1.365

1.364

0.5

1,585

--

Table 7. Experimental and calculated bond strengths for B-O bonds in NaBO2 Bond B-101

Sex p . 1.280 A

B-2011 1.433

Sea I .

1.32 2 x 0.84

1.68 Es 3.00

1.341 2 x 0.827

1.654 Es 3.00

Bond qrength-bond length relationships

?_79

Table g. Bond strengths for Na-O bonds in NaBO, Bond

Sexp,

Scal.

Na-lO I

2,461 A

I x 0.16

0.16

i x 0.159

0.159

Na-201

2,474

2 x O.l~)

0.32

2 x 0.!547

0.309

Na-201

2.607

2 x 0.I0

0.20

2 x 0.:205

0.241

Na-2Oii

2.482

2 x 0.16

0.32

2 x O.

52

Es 1.00

of the D - s curve derived from the general expression .4 + K/s is not altered too much in the vicinity of the distances of interest. It is therefore of little value to attempt to refine these terms much beyond those values obtained from the structural information. Similar calculations may be made for CaB2Q[18]. The B-O oxygen distances are given in Table 9 along with experimental and calculated bond strengths, The calcium atom is bonded to eight oxygen atoms at the following distances:

Q.3[14 gs

1.0]

Table 10. Experimental and calculated bond strengths ior t'a () bonds in CaB:O4 Bond

Sexp.

'; c a i .

Ca-20 r

2 x 0.37

2 x O.LqO

Ca-20

2 x 0.133

'2 x 0 , ; ! ' ,

2 x 0,0~

2 x r).](~'~

2 x 0.22!

2 x 0.-

t

Ca-20] Ca-20

I

}:s 2 . 0 0

Ca-2(h

2.347 A Ca-201

2.727A

Ca-201

2.399

2.549.

Ca-20.

The average bond distance is 2.5055 A. Hence, 2 kL = 2.5055-2.40 A or (I.106, where 2.40 A is the ionic radii sum. Bond strengths can then be computed from Dc~.o =2.082+0.053CN. or Dc~o =2.082+0.106/s. However, if, as in the case of the Na-O bonds, the k'~ values are obtained, the specific k'~ for 2.40 A is 0.0703 A. The expression is then Dc,,_o = 1.978 + 0.141/s. The first formulation based on k,;, leads to better agreement with reported bond strengths. Experimental bond strengths and those computed from Dc,~_o=2.082+0.106/s are shown in Table 10. The major discrepancy is in the calculated bond strength for the long Ca-(h bond of 2.727 A. Although the calculated bond strength sum is high, the accuracy is ]able 9. Bond strengths for B-O bonds in CaBe(h D

S

~; " ct] .

exp.

g-O {

] . 32i" ,a

1.12

1 . ] 3{)

55 0

I . ¢'~'

0.95

) <~4r

11

}~--[? [ ! ,

] . 40!

0.2) ~ Ls 2 . 9 9

) . F,c]q

K-201

3.034A

K-20m

2.819A

K-20v

2.856

K-20vi

2.903,

Fable I 1. Experimental and computed bond strengths for CAB2()4

i

0,3}

0

1

It

~- 0.33 + 0.08

0.22

; ,). 33 + 0 . i 6 1

(O.2~)

Es 1.00

Ca (0.40

1 .12

(1

0.95. ~)

2~; 2 . 2 q

sufficient to warrant comparison with the overall repoaed bond strength sums within the structure. Fhe results are shown in Table l l. This table is presented in the same format used by the authors to display their bond strengths and sums. Quantities shown in parentheses are computed. The agreement is reasonably good. Bond strength sums for B-O bonds in several structures are presented in Table 12 using the general expression DB_O= t.034 + 0.33/s. The equation D = A + KIs accounts well for s-values greater than 0.7 for B-O bonds in the various structures reported here. However, it is not certain that the relationship is always applicable for those M-O bonds which are considered ionic. Examination of several structures shows that the expression fails for the longer bond distances and possibly under conditions in which the average bond length within, say, an eightfold M-() coordination, is not much greater than that within a sixfold coordination (corresponding to the ionic radii sum). The DM o expression works well, for example, for the K-8(O) bonds in KB~Os. 4HeO[201. The bond distances are as follows:

Es 2 . ~ i 7

0

:7

(0.94

(l.12)

+ 0.92

2.99

"~ 0 . 9 0 )

(2.97)

1.90

2.09

3.99

(2.02)

(2.07)

(4.09)

Es

280

S. SIEGEL Table 12. 52s for several structures containing B-O bonds Compound

Ref.

CAB204 III

~s B-3(O)

19

2.96, 3.05

2.95, 3.01, 3.00, 2.97

KB508'4H20

20

3.04, 3.06

2.98

HBO 2 y

21

HBO 2 B

16

2.98, 2.99

3.04

K2[B508(OII)]'2H20

22

2.95, 2.97, 2.96

2.95, 2.95

K20"2B203

23

2.96, 2.95, 2.98, 2.95

3.00, 2.98, 3.01, 2.98

3.02

The average bond length is 2.903 A and the ionic radii sum is 2.79 A. Hence, 2 k'v is 0.113 A, leading to DK_O= 2.451+0.0565/S. Use of the specific k'i for 2.79A gives DK_O = 2.338 + 0.0753/S. The first expression leads to the bond strengths and bond strength sum shown below. The second equation based on k'i for 2.79 A gives a slightly higher bond strength sum. Scal.

K-20,

Is B-4(O)

0.0969×2

Scal.

K-20m 0.1535x2

K-20v 0.1395x2 K-20v, Es = 1.03

0.1250x2

The reader can readily determine that the above rules do not always apply and that substantial changes in the coefficients A' and k' will still lead to very reasonable bond strength sums. Proof that the individual values are useful can be established only by using such data to account for appropriate bond strength sums for other O-M or M-O bonds within a structure. V+s-O BONDS

Sauerbrei et a1.[24] in a discussion of the structures of Co3V208 and Ni3VzO8 state that a grand average for a V-O bond in tetrahedral coordination is 1.721 A. Average bond distances for six- and fivefold coordination vary considerably; however, approximate average bond lengths for these coordinations are 1.95 and 1.83 A, respectively. The value of k is roughly 0.11 A (1.83- 1.72) and a plot of D with CN leads to the approximate expression for Dv_o of 1.28 + 0.11 CN. Improved values for A and k could be obtained by trial-and-error refinement, but a simpler procedure is based on the following. An average value of k was obtained from structures exhibiting coordination 6 and for which the average V-O bond length within a coordination varied from 1.943 to 1.994A. The grand average bond length was 1.963A, so that the average k is then (1.9631.721)/2 or 0.121 A. The spread in k is -+0.015 A. This variation is not an error; it simply represents the range of values to be expected in actual structures. A value of 0.126 A was finally used leading to Dv_o = A + 0.126 CN. The average V-O bond distance for tetrahedrally coordinated vanadium is 1.721 A as noted, so that Dv_o = 1.217 + 0.126 CN, or 1.217 + 0.63/s. The spread in k and therefore K is high; nevertheless, the Dv_o expression satisfactorily accounts for bond strength sums in the V-O bonds. The expression is not useful for bond strengths near s < I and, for this reason, only structures exhibiting tetrahedral coordination are considered. Bond strength sums for V--4(O) bonds in several structures will

be given. However, Es on all atoms will be calculated for Mg3(VO4)2 [25]. The V-O distances and computed bond strengths for Mg3(VO4)2 are shown in Table 13. Mg, and Mg2 are each bonded to six oxygen atoms so that k' cannot be evaluated. If k' is taken as 0.055, the average value for Ca-O and K-O, Es is 2.12 for Mgt and 2.18 for Mg2. Improvement is obtained by use of k' = 0.0724 derived for Na-O bonds. This value leads to the following bond strength values and sums, based on DMg_o= 1.67+0.145/S, as given in Table 14. The average of the k'i values for Na-O, Ca-O and K-O bonds is 0.07669A. If this average is taken as the approximate specific k'i value for the Mg + O ionic radii sum of 2.11A, the DMg_o expression is 1.65+0.153/s. Bond strength sums computed from these coefficients are somewhat higher than those given in Table 14. Compatability between the DM~_o and Dv_o expressions is established if the bond strength sums for oxygen are reasonable, as shown in Table 15. Analogous results are obtained for the isomorphous Co3VzO8 and Ni3V208 structures [24]. Bond strength sums for V-4(O) bonds in several other structures are presented in Table 16. In Ca3(VO4)21291, the V--4(O) distances are shorter than the average value of 1.721 A which was used to Table 13. Calculated bond strengths for V-4(O) bonds in Mg3(V04)2 Bond

D

V-O 1

1.716 A

V-O 2 V-203

Scal. 1.263

1.809

1.064

2 x 1.695

2 x 1.318 Es 4.96

Table 14. Mg-O bond strengths in Mg3(V04)2 Bond

D

Mgl-202

2 x 2.034 A

Mg2-403

2 x 2.130

Scal. 0.7967

(2 x 0.3983)

1.261

(4 x 0.3152)

Is 2.06 Mg2-201

2 x 2.022

0.8239

(2 x 0.4119)

Mg2-202

2 x 2.135

0.6237

(2 x 0.3118)

Mg2-203

2 x 2.118

0.6473

(2 x 0.3237)

ES 2.09

281

Bond slrength-bond length relationships

expression D = D{I) - 2 k log s by considering O-V distances in V20~ and applied the equation to V-O bonds. Bond D Seal, In this structure D ( O l - V ) - I . 5 8 5 A , D(()3 2V] 1.78A, and D ( O z - 3 V ) - 1 . 9 3 A. (These distances are O]-2Mg 2 2.022 A 0.8239 taken from the later work on Ve(h by Bachmann ~'t (/ V ] . 716 1.263 a/.[31].) The O2-3V distance is an average of one short ] and two longer bond lengths and will not be used here. Es 2.09 Using the distances corresponding to one- and two-fold O2-2M82 2,135 0. 6237 coordination, the change in D with coordination is 0.195A. A plot of D with CN gives an intercept of O2-MgI 2,034 0. 3984 1.40A; hence, I)o v - 1.40+0.39/s. Evans[32] used the O2-V i , .°,09 1. 064 Pauling logarithmic expression with 2 k = 0.78 as derived Zs 2.09 by Bystr6m and Wilhelmi[30] to obtain Xs values f¢¢ several vanadates. Evan's values and those compuled 03-Mf, ] : . I 30 0. 3152 from D,,._o and Do v are shown in Table 17. O3-Mg 2 2. t 18 0,3237 The failure of the Dv (, expression to uccoui]t for bond strengths related to the longer ~-O bonds is O3-V ] ,695 1.3180 evident. The differences in results obtained for tl',e Es 1.96 various D-s equations in the table stem not only from their inherent approximations, but also from the chok:e of a particular reference distance. With regard to lhe Table 16. Bond strength sums for tetrahedrally coordinated Dv_o and Do v expressions, the different A and k values vanadium which lead to similar bond strengths are again indications that the shapes of the D-s curves in the vicinity of the Ref, gs V-4 (0) s-values of interest are not too different. Clearly, in order for the Dv_o and Do v expressions to yield the O,15V2010 26 5 . 0 3 , 5.01 same bond strengths, the coefficient would have to be the
Table 17. Bond strength sums for several ~anadates Compound

Dobs.

KVO3

1.66 A

4

s(I)v. O=].217~,? (%,'~1

5(b<,_,r::] ./~ ]4), ~i ':

!.50

1,422

]. 500

.65

1.53

i.&55

]. ~,(0

2 x 1.8]

2 x 1.00

2 :,: ] . 0 6 2

Es 5.02

Es 5,00

]

N

s ( D = l . 8 1 - 0 . 7 8 l o g s)

V)

].67

1.45

1,39]

] .65

1.53

1.455

2 x !.80

2 x 1.02

2 x ],0,~]

Is 5.02 KVO 3 "H90~

Xs 5,O1

x .9; Ys 4 . t!~ ~..44'.-

2 >" .9/~ )s

4.9%

i. 45

] . 39]

1.444

1.65

],53

] .455

1 • 560

] .97

0,62

.837

2 x 1.93

2 x 0,70

] .67

Es 5.00

2 x

,88'~

zs 5.45

.6S'~ 2 x

.7V~

Fs 5 . ] 6

282

S. SIEGEL Table 18. Bond strengths obtained from different D-s relationships

Vl-O 1

1.744 A

1.195

1.133

1.14

VI-O 2

1.658

1.429

1.511

1.48

Vl-O 3

1.721

1.250

1.215

1.22

VI-O 4

1,764

1.152

1.071

Zs 5.03

gs 4.93

1.08 Zs 4.93

V2-O 5

1.650

1.455

1.560

1.52

V2-O 6

1,780

1.119

1.026

1.03

V2-O 7

1,728

1.233

1.]89

1.20

V2-O 8

1,742

1.200

1.140

1.15

Zs 5.01

Zs 4.92

to D M o _ o ~ 1.32 + 0.11, assuming as before that the relationship is linear. However, better average values of A and k were obtained from average bond distances in many 6- and 4-coordinated configurations. For CN 6, the average D chosen was 1.974 A, while for CN 4, the value was 1.765 A. In Gd2(MoO4)3133], the bond distances within the three tetrahedral configurations are quite uniform, with an average value of 1.756 A. However, because the average Mg-4(O) distance in many structures is slightly higher, the larger value of 1.765 A will be used. This leads to 2kav=l.974--l.765A or kav=0.105A with a variation of about 0.01 A. The final value used in the calculations was 0.108A, leading to DM,~o = 1.333+ 0.648/S. The equation is limited to bond distances for which s is greater than one, so that only tetrahedral configurations can be considered. Bond strength sums for several structures exhibiting tetrahedral coordination are shown in Table 19. No attempt has been made to evaluate bond strengths for other M-O bonds within these structures. COMMENTS The expression D = A + K / s is an approximation and its general reliability and applicability have yet to be established. Much more data will be required in order to evaluate the coefficients A and K; however, it is noted that the value of A approximates that of a single bond covalent radius for the specific cation. The A' values are slightly higher than known radii. If these values are reduced by the amount K', the

Zs 4.90

difference approximates that of a metal radius for coordination 12. This is shown below. Metal in M-O bond Na Ca K Mg

A'-K' 1.93 (1.93) 1.98 (1.98) 2.39 (2.34) 1.53 (1.65)

The quantities in parentheses above are A' terms which are obtained when specific k~ values are assigned to the sum of the ionic radii for the particular bond. Bond strength sums can also be derived with the oxygen radius of 1.40 A as the A' term. However, use of this radius cannot be completely justified. It is emphasized again that, in many cases, A, A', k and k' can be altered substantially without producing major changes in bond strength sums. Hence, the above coeffcients may not be known with high accuracy and the similarities to radii may be fortuitous. Empirical relationships between D and s for bond strengths less than unity can be established for both U-O and B-O bonds utilizing the published bond strength data. For example, the expression DB o = 1.365 + 0.44(1 - s) accounts for bond strengths between 1 and 0.5 for boron-oxygen bonds. A more general form of the relationship which appears to apply over the extended range of s-values is based on D = a - ~ s ~, where a, 13 and x are constants to be evaluated. This expression has not been developed from structural information, but it is useful for computing bond strengths from known distances.

Table 19. Mo--4(O)bond strength sums Compound

R .... (CN 12)[40] 1.90 1.97 2.35 1.60

Ref.

Zs Mo-4(O)

CuMoO 4

34

5.89, 6 . 0 2 , 6.02

La2(Mo04) 3

35

5.96, 5.94, 5.96, 5.91, 5.81

ZnMoO 4

36

5.97

~MnSloO4

37

6.06, 6.17

Gd2(MoO4) 3

33

6.12, 6.12, 6.19

Cu3Ho209

38

5.89

6.26

NaCo2.31(Mo04)2

39

6.03

5.89

5.96, 6.15

283

Bond strength-bond length relationships The expression D = A + ( B / s ) was obtained by considering the behavior of the D - s relationship in the region near CN - 0. The function begins to deviate substantially near s = 1 (CN 6) and for smaller s-values. This deviation can be suppressed by arbitrarily changing the functional dependence on s. One way to carry this out is to expand the Pauling logarithmic expression for the extended range of s-values. This leads to D = D(ll 2 k ( s I) using only one term of the logarithmic expansion. Replacing 2 k with the K value of 0.768 for U-O bonds gives D = 2.85 - 0.768 s. This expression, of course, will not work. However, if s is replaced with sL rough agreement between calculated and observed distances can be obtained for x ~ 1/2. The value 2.85 A also corresponds to D .... of 2.75 A e,,lablished b~, Donnay and AIImann[1] as the maximum cation-anion distance for U - O bonding. It is derived by extrapolating the effective ionic radii of Shannon and Prewitt[41] to s - 0 and adding the value 1.45 A for lhe oxygen ionic radius. (Curiously, the distance D = 2 . 8 3 A is found for k = 0 (Fig. I) and is the value for which the contribution to the coordination change is zero.) These values suggest that a D - s relationship might be based on D = 2 . 8 5 - 1 . 4 5 f ( C N ) . Assuming that f(CN) is C N ' and utilizing the known D - s data for uranium-oxygen bonds, it is found that D ~ 2.85 0.75 s"4 but with a Large variation in the exponent. The 1,45 A radius has disappeared to become 0.75 A when the distances are given in terms of bond strengths. On the assumption that poor agreement between D ~ , and D ..... arose because of uncertainties in the numerical values of the coefficients, these values were entered as input data for a least squares refinement on the uraniumoxygen bond strength and bond length data. The final paranteters were a = 3.21,/3 = 1.13, and x = 0.398, leading to excellent agreement with the observed distances. The values of the coefficients at the 95% confidence inter~al are c r - 3 . 0 2 to L40, /3 =0.94 to 1.33, and x 0.33 to 0.47. ()n the other hand, the correlation matrix of the parameters shows such strong interaction between any pair of coefficients as to cast some uncertainty on the refined values. Thus, the correct coefficients and the .~-dependency may not yet have been established. Nevertfiele,,s, the expression O - o~-/3s ~ appears to be very u~,eful in deriving s-values for metal-oxygen bonds. The coefficients would, however, have to be obtained from bond distance data. For those who may wish 1o consider the uraniumoxygen data only, "fable ~,(b) is extended below s 1.0 0.9 11.8 1t.7

D 2.1/83 2.131 2.181 2.233

s 0.6 0,5 0.4 0.33

D 2.287 2.346 2.420 2.481.

No:c added m proo/. In the expression D [DIll+/',2] Ks aho~c, lhe ill',( term, ~, agrees well with Dm,,xllI. To obtain the .~' dependency, assume D - a K -~-AN, where &%' is the Zachariasen coordination correction. - 0.274 In (CN/61[15] If this correclion is phtced o n an s ba,,i,,, K + AN is approximalely /3s'

with /3 and x readily evaluated. The exponem will depend t~n valence: x ~- 11.36for U +6. ~ 0.53 for B ~a, etc. Values of a tor Om,xI, /3 and x. so obtained, lead only to crude agreement with known s. However, refinement of these initial quantities gives reasonable D-s relationships for all s (or D). REFERENCES

1. 2. 3. 4. 5.

G. Donnay and R. Allmann, Am. Min. 55, 1003 (19711). W. H. Bauer, Trans. Am. Crysr Assoc. 6, 129 !197111. Yu. A. Pyatenko, Soy, Physics-Crysr 17, 677 119731. 1. D. Brown and R. D. Shannon, Acta Cryst. A29, 266 (1973). G. Donnay and J. D. H. Donnay, Acta Crysr B29, 1417 (19731. 6. G. Ferraris and M. Gatti, Acta Crysr B29, 2006 (19731. 7. W. H. Zachariasen, Heavy Element Properties (Edited by W. Mtiller and H. Blank), p. 91, North-HoLland. Amsterdam 11976). 8. W. H. Zachariasen and H. A. Plettinger, Acta ('rysZ. 12. ~26 ( 19591, 9. S. Siegel, A. Viste, H. Hoekstra and B. Fani. Acta Crvsr B28, 117 (19721, 10. S. Siegel, H Hoekstra and E. Sherry, Acta Crv,sr 20, 292 (19661. 11. W. H. Zachariasen, Acta Crysr 7, 788 11954). 12. S. Siege( and W. H. Hoekstra, Acta Cryst. B24, 967 11968). 13. L. Pauling, J. Am. Chem. Sot'. 51, 1010 11929). 14. L. Pauling, J. Am, Chem. Soc. 69, 542 11947). 15. W. H. Zachariasen, The Actinide Elements. p. 769. McGrawHill, New York (Edited by G. T. Seaborg and J. J. Katz) 11954). 16. W. H. Zachariasen, A~ta Crysr 16, 385 11963). 17. M. Marezio, H. A. Plettinger and W. H. Zachariasen, Acta Cryst. 16, 594 11963). 18. M. Marezio, H. A. Plettinger and W. H. Zachariasen, 4,eta ('o'sr 16, 390 11963). 19. M. Marezio, J. P. Remeika and P. D. Dernier, 4eta Crysr B25. 955 11969). 20. W. H. Zachariasen and H. A. Plettinger, Acta Crvst. 16, 376 (1963). 21. W. H. Zachariasen, Acta Crysr 16, 380 11963) 22. M. Marezio, Acta Cryst. B25, 1787 11969). 23. J. Drogh-Moe, Acta Cryst. B28, 3089 (1972). 24. E. E. Sauerbrei, R. Faggiani and C. Cairo, Acta ('rvsr B29, 2304 11973). 25. N. Krishnamachari and C. Cairo, Can. J. ('hem. 49, 1629 11971). 26. R. D. Shannon and C. Cairo, Acta Crysr B29, 1338 119771). 27. R. Gopal anti C. Cairo, Can, J. Chem. 51, 1004 (1973). 28. E. E. Sauerbrei, R. Faggiani and C. Cah'o, Acta ('rvst. 1130. 2907 (1974). 29. R. Gopal and C. CaNo, Zeir f. Krisr 137, 67 (1973i. 311. A. Bystr6m and K. A. Wilhelm), Acta Chem. Stand, 5, 1003 i19511. 31. H. G. Bachmann, F. R. Ahmed and W. H. Barne~,. Zeir f. Krist. 115, ll0 (1961). 32. H. T. Evans, Jr., Zeir f. Krisr 114, 257 (19601. 33. W. Jeitschko, Acta Cry'st. B28, 60 (1972). 34. S, C. Abrahams, J. L. Bernslein and P. B. Jamieson. J. Chem. Phys. 48, 2619 (19681, 35. W. Jeitschko, Acta Crysr B29. 2074 11973). 36. S. C. Abrahams. J. Chem. Phys. 46, 2052 11967). 37. S. C Abrahams and J. M. Reddy. J. Chem. Phys. 43, 2!533 11965). 38. 1,. Kihlborg and R. Nonestam, Acta Crysr B28. 31197t1972i. 39. J. A. lbers and G. W. Smith, Acta Crysr 17, 19(I 11964). 40. L, Pauling. The Nature of the Chemical Bond. Cornell University Press, Ithaca i19601. 41. R. D. Shannon and C. Y. Prewitt. Acta Crvsr B25. 925 11969).