Spectrochimica Acts, 1958, vol. 12, pp. 154 to 161. Pergamon Press Ltd., London

A relationship between some bond properties of diatomic molecules and the ionization potentials of their constituent atoms*

BRIAN STEVENS? The James Forrestal Research Center, Princeton University, Princeton, New Jersey, U.S.A.

(Received 13 May 1957)

Abstract-The relationship kr, 2 = 2N (Ia + IB) between the force constant k, equilibrium internuclear distance T, and bond order N of the diatomic molecule AB in its ground state, and the first ionization potentials 1, aud IB of atoms A and B, is examined. For covalent molecules with eight, ten or eleven valence shell electrons the relationship is satisfactory, but for molecules with two, twelve or fourteen electrons in the valence shell the expression must be modified by the introduction of two constants.

For the forty-nine covalent molecules examined the average deviation of calculated from observed bond length is less than B’per cent. Bond lengths predicted for twenty-two molecules are reasonably self-consistent and the derived covalent radii are used to extend the table of values introduced by PAULING [ 141.

Introduction No simple expression relating the force constant Ic, dissociation energy D, order N and equilibrium length re of a chemical bond has yet been proposed which has more than a very limited application. Recently, semi-empirical expressions have been put forward [l, 21 which relate one or more of these bond properties to the electro-negativities of the bonded atoms. These have led to further definitions of the concept introduced by PAULING [3] and MULLIKEN [4] and several tables of electro-negativities now exist [5].

WALSH [6] has pointed out the advantage of relationships between bond constants and atomic ionization potentials which are well defined and which are known with precision; he observed a linear dependence of k for the A-H bond on the second power of the first ionization potential of atom A. For restricted groups of molecules, the force constant [7], equilibrium frequency [8] and dis- sociation energy [9] have also been related to the first ionization potential of one or both of the bonded atoms.

The purpose of this note is to examine the empirical relationship (1) between k and re

kr,2 = 2N(I, + I,) (1)

of the diatomic molecule AB in its ground state, and IA and I,, the energies required to ionize atoms A and B, all expressed in c.g.s. units. N in most cases is equal to the bond order but is more specifically defined to include those mole- cules in which A and B have different group valencies as the average group valency of A and B; thus for CO, NO and 0,, N has the values 3, 2.5 and 2 res- pectively.

* This research was supported by the United States Air Force under Contract No. AF33(038)-23976 monitored by the office of Scientific Research. ASTIA Document No. AD.126489

t Present address: Department of Chemistry, The University, Sheffield 10, England.

154

Diatomic molecules and the ionization potentials of their constituent atoms

Covalent molecules Values of T, calculated from equation (1) are given in Table 1 together with

the experimental bond lengths for seventeen molecules with eight, teh and eleven valence shell electrons*. Apart from the halogen hydrides PO, N, and PN, these molecules are not treated by GORDY [l] and comparison with his expression is not easily made. Use of GORDY’S expression IA leads to calculated bond lengths for the molecules in Table 1 which on the average are some 13 per cent greater

Table 1. Bond lengths calculated from kr, 2 = 2N(I, + IB) for covalent molecules with 8. 10 and 11 valence shell electrons

Molecule

HF HCl HBr HI NO PO

N, p2 PN co SiO GeO SnO PbO cs SiS PbS

k x1W5

(dyn/cm)

9.653 5.160 4.117 3.141

15.94 9.41

22.96 5.556

10.17 19.02

9.242 7.522 5.615 4.557 8.485 4.929 2.991

-

L

L

N

-

I 1 0.917 1.015 1 1.275 1.284 1 1.414 I.407 1 1.604 1.566 2.5 1.151 1.189 2.5 1.449 1.447

3 1.094 1.103 3 1.894 1.948 3 1.491 1.553 3 1.128 1.121 3 1.510 1.504 3 1.651 1.666 3 1.838 1.893 3 1.922 2.106 3 1.534 1.565 3 1.929 1.897 3 2.395 2.390

re (-4 r, (A) Observed , Calculated

-

% Deviation

10.7 0.7 0.5 2,4 3.3 0.2 0.8 2.9 4.2

0.6 0.4 0.9 3.0 9.6 2.0 1.7 0.2

Avg. 2.5

r, (A) t

0.919 1.236 1.397 1.603

1.482 1.082

1.592

-

-

-

% Deviation

0.2 3.2 0.9 0.0

2.3 1.2

6.8

-

Avg. 2.1

t From GORDY'S expression IA.

than the experimental values. Oxygen could be included in this table, the devia- tion of calculated from observed value being O-5 per cent; however, this molecule is grouped with other twelve valence electron molecules in Table 4.

In the case of hydrogen, the alkali metals and their hydrides with two valence shell electrons, re calculated from equation (1) is uniformly larger than the experi- mental value, the linear relationship being expressed by equation (2).

k(0.32 + 1.28 rJ2 = 2iV(1, + 1s) (2)

For the eighteen molecules of this type shown in Table 2, the agreement between the observed bond lengths and those calculated from equation (2) is almost as

* Unless othervise stated, the ionization potentials, force constants and experimental bond lengths are taken from or calculated from the data listed by HERZBERQ [lO,ll].

155

BRIAN STEVENS

Table 2. Bond lengths calculated from k(0.32 + 1.28 T,)~ = 2N(Ia + IB) for covalent molecules with two valence shell electrons

Molecule

J32 Li,

N% K2 Rb2 cs2 LiH

NaH KH RbH CsH LiK LiRb LiCs NaK NaRb NaCs RbCs

-

k x lo-5

(dynlcm)

5.759 0.741 0.711 4.0 0.744 0.4 0.254 2,672 2.630 1.6 2.558 4.3 0.172 3.079 3.167 2.9 3.096 0.6 0.099 3.923 3.890 0.9 3.845 2.0 0.081 4.22t 4.24 0.5 4.22 0.0 0.069 4.50t 4.45 1.1 4.44 1.3 1.026 1.596 1.652 3.5 1.602 0.4 0.779 1.888 1.918 1.6 1.892 0.2 0.560 2.244 2.252 0.4 2.254 0.4 0.514 2.368 2.360 0.8 2.368 0.0 0.467 2.494 2.455 1.6 2.488 0.2 0.148 3.30t 3.33 0.9 3.29 0.3 0.129 3.50.t 3.56 1.7 3.53 0.9 0.109 3.75t 3.83 2.2 3.83 2.2 0.130 3.507 3.53 0.9 3.47 0.9 0.120 3.59t 3.65 1.7 3.59 0.0 0.108 3.76”r 3.79 0.8 3.75 0.3 0.075 4.367 4.34 0.5 4.31 1.1

-

-

r, (4 Observed

r, (-4 Calculated

i

A-

% Deviation

Avg. 1.5

re (A)* % Deviation

_

- Avg. 0.9

* From GORDY'S expressions IB and IC. t Computed from the SCHOMAEER-STEVENSON~L&[~~].

close as that obtained by GORDY [l] who uses a different set of constants for hydrogen and the hydrides.

Lone-pair repulsion [S] may account for the fact that experimental bond lengths for the halogens are larger than those calculated from equation (1). For these molecules with a valence shell of fourteen electrons shown in Table 3, expression (3) is satisfactory.

k(0.63 + 0.50 rJ2 = 2N(I, + 1s) (3)

The experimental bond lengths for those molecules containing two Group VI atoms with twelve valence shell electrons are also larger than the corresponding values calculated from equation (1). In Table 4, the bond lengths calculated from equation (4)

k(0.38 + 0.69 T,)~ = 2N(I, + Is) (4)

are given. GORDY [l] prescribes a bond order of less than 2 for the molecules in this group.

Covalent radicals In Fig. 1 the bond lengths calculated from expression (1) with N = 1 are

plotted against the observed values for some diatomic hydride radicals. A regular trend is apparent and linear relationships appear to exist for the hydrides of elements in the two short periods as well as for those of the elements in a particular

156

Diatomic molecules and t.he ionization potentials of their constituent atoms

group. As observed above, the increase in bond length calculated from equation (1) seems to depend on the number of electrons in the valence shell.

2.5

r, ohs., A

Fig. 1. Plot of T, calculated from expression (1) with N = 1 against observed bond length for some diatomic hydride radicals. The dashed line is drawn through the origin with

unit slope.

Ionic molecules

The combination of highly electropositive and electronegative atoms results in virtually complete electron transfer and the bond constants may be expected to depend on the electron affinity of the electronegative constituent. The deviation of ?e calculated from equation (1) from the observed value is fairly large and irregular for the typical ionic molecules in Table 5; however, the use of electro- negativities in GORDY’S expression does not lead to a significant improvement in the calculated bond length for the molecules chosen.

Table 3. Bond lengths calculated from k(0.63 + 0.50 rJ2 = 2N(I, + Is) for covalent molecules with fourteen valence shell electrons

Molecule j k x 10-5 i r, 6) 1 r, (A) : % (dyn/cm) 1 Observed / Calculated ~ Deviation T, (A) *

% Deviation

I I

Cl, Br, 12 ClF BrF IF BrCl

ICI IBr

3.286 1.989

2.458 2.284

1.721 2.667 4.483 1.628

4.070 1.756

3.621 1.906?

2.755 2.118?

2.383 2.321

2.064 2.439t

1.918 3.6 1.3 2.254 ~ 1.3 1.4 2.682 0.6 2.4 1.686 3.6 14.1 1.776 1.1 9.1 1.880 1.4 3.2 2.136 0.9 4.1 2.286 1.5 0.1 2.460 ~ 0.9 2.5

Avg. 1.7 Avg. 4.2

2.014

2.317 2.730 1.857 1.916 1.967 2.210 2.323 2.500

* From GORDY’S expression IA. 7 Computed from SCHOMAKER-STEVENSON role by COLE and ELVERUM [13].

4 157

BRIAN STEVENS

Table 4. Bond lengths calculated from k(0.38 + 0.69 rJ2 Y 2N(I 1 + I,) for covalent molecules

-

Molecule

02 82 Se2 Te2 so

k x 1W5

(dyn/cm)

11.77 1.207 I.207 0.0 4.958 1.889 I.820 3.6 3.614 2.157 2.144 0.6 2.367 2.59 2.650 2.3 7.929 1.493 1.465 1.9

L -

with twelve valence shell electrons

r, (A) Observed

T, (A) Calculated

% Deviation

Avg. 1.7

Some predicted bond lengths

The agreement between r, calculated from equations (1) and (4) and the experi- mental values shown in Tables 1 and 4 is sufficiently close to justify the prediction of bond lengths for similar molecules shown in Table 6 for which no experimental values are available. For purposes of comparison the appropriate sums of covalent radii are given in the fifth column of Table 6, these being calculated from PAULINO’S values or as otherwise stated in Table 7. The data in Tables 6 and 7 are reasonably self-consistent, but it appears that some revision of PAULING’S double-bond radius for Ge, As and Sn is required if experimental verification of the predicted bond lengths in Table 6 is forthcoming.

Discussion

For covalent diatomic molecules in the ground state, the expression discussed above appears to be as satisfactory as that proposed by GORDY [I] which may be written as

(k - b) T,1’5 = aN(X,X,)0”5 (5)

where X, and X, are the electro-negativities of the bonded atoms and a and b are constants with a = 1.67 = 2°‘76 in most cases. If, as WALSH has suggested [6], I, may be taken as a rough approximation of X,, equations (1) and (5) are of the same form with the exception that the sum and product of electro-negativities are involved respectively; moreover, if it is accepted that k is determined by the energies of the bonding electrons in their atomic orbitals [6] i.e. by the quantity N (1, + Is), the semi-empirical derivation of expression (1) may proceed along the lines by which expression (5) is obtained.

It may be expected that the use of valence ionization potentials [4] would improve the results obtained from expression (1). As these are some 22 per cent higher than the measured ionization potentials for the halogens [S] their use in expression (1) would lead to an increase of 11 per cent in the calculated bond length for the molecules in Table 3. However, the experimental bond lengths in this case are approximately 25 per cent higher than the values calculated from expression (1) using experimental ionization potentials and it is felt that the modification of expression (1) by the introduction of empirical constants leads

15s

Diatomic molecules and the ionization potentials of their constituent atoms

Table 5. Bond lengths calculated from kr, 2 = 2N(I d + IR) for some ionic molecules

N

1

1 1

1

1 1

2

2

2

2 2

% Deviation

rt? (A) re (fv Observed Calculated

k x lo-5

(d&cm)

1.186

1.044

0.938 , 0859

0.825

0.704 7516

3.485

2.843

3.401 3.784

~Rnv’s expressic ,n IA.

% Deviation

6.4

8.8 14.8

4.9 11.0

14.8

5.3 13.2 2.10

20.6 ( 2.23

0.5 1.95

7.8 1.71

2.51 2.71

2.81 3.23

3.18

3.17

1.34

Avg. 9.8

Molecule

X&l

NaBr NaI

KC1

KBr KI

Be0

MgO CaO SrO

BaO

* From

I

2.36t I 2.21 6.4

8.8 3.7

24.7

12.8

3.9

0.7 20.0

27.4

1.6 11.9

Avg. 11.0

2.50j ’ 2.28

271t 2.31

2.67t 2.54

2.82? 2.51

3.057 / 2.61

1.33 , 1.40

1.75 ’ 1.98

1.75 2.11

1.92 1.91

1.94 ~ 1.79

- -

t Reference [15].

Table 6. Bond lengths predicted from kra2 = 2N(I, + IB)

k x 1O-5

(dynlcm) N r, (4

Predicted rll + rB) (A)* I DeJ$ion Molecule

Se0 TeO

As0 SbO

BiO

As2

Sb2

Bi,

AsN

SbN SbBi GeS

SnS

csc SiSe

GeSe SnSe

PbSe SiTe

GeTe

SnTe PbTe

2

2

2.5

2.5 2.5

3 3

3

3 3

3

3 3

3

3 3

I

1.658.t 1.63 1.7 1.8457 1.85 0.3 1.630 1.66 1.8 1.790 1.81 1.2 1.970: 1.93 2.0 2.226 2.22 0.3 2.521 2.52 0.0 2.760: 2.76 0.0 1.743 1.66 5.0 1.842 1.81 1.8 2.640: 2.64 0.0 2.018 2.02 0.1 2.193 2.21 0.8 1.752 1.632 7.4 2.050 2.03 1.0 2.142 2.18 1.8 2.314 2.37 2.4 2.520 2.55 1.2 2.295 2.24 2.0 2.382 2.39 0.4 2.537 2.58 1.7 2.750 2.76 0.4

-

6.444

5.304

7.265 5.560

4.315

4.069 2.612

1.835

7.929 6.566

2.193

4.359 3536

6.583 4.094

3.743

3.064 2.594

3.130 2.902

2.440 2.086

Avg. 1.5 I

lent radii-see text and Table 7. - -

va t Calculated from k(0.38 + 0.69 r,)a = 2N (4 + Is). z Calculated using 7.25 eV for ionization energy of Bi [16].

159

BRIAN STEVENS

Table 7. Covalent radii (A)*

Double-bond 1 Triple-bond 1

Double -bond

Triple-bond

Double-bond Triple-bond

C

0.665

0.602

N

0.60

0.547

-____

0

0.55

0.50

Si Ge Sn

I.07 1.12 1.30

1.00 1.151_ 1.34.t

P As Sb

I.00 1.11 1.31

0.93 1.11s 1.268

S Se Te

0.94 1.08** 1.30**

0.87 1.0377 1.24::

Pb

1.52:

Bi

1.38 4

___-

* Values are those given by PAULING [14] unless otherwise stated. t Obtained from X0 bond length minus 0.5 A. : Obtained from PbS bond length minus 0.87 A. $ One half predicted bond length of X,.

** One half observed bond length of X,. tt Average value of predicted XSe bond length minus r,A (X = C, Si, Ge, Sn, Pb). $$ Average value of predicted XTe bond length minus r,A (X = Si, Ge, Sn, Pb).

to a more useful relationship than one employing corrected values of the related quantities to which some uncertainty is attached [6]. The dependence of these constants on the number of valence shell electrons indicates that the improvement of expression (1) will involve a correction for this factor rather than any other [17].

One disadvantage of expression (1) is its limited application to polyatomic molecules which requires that the ionization potentials of the bonded radicals are known. However, there is evidence that the electro-negativity of an atom also depends on its environment [l] which together with the possibility of inter- action between non-bonded atoms imposes a similar limitation to the use of expression (5), although to a much lesser degree.

References [l] GORDY W. J. Chem. Phys. 1946 14 305.

[2] ARNOLD J. R. J. Chem. Phys. 1956 24 181. [3] PAULING L. J. Amer. Chem. Sot. 1932 54 3570. [4] MULLIKEN R. S. J. Chem. Phys. 1934 2 782.

[5] PRITCHARD H. 0. and SKINNER H. A. Chem. Rev. 1955 55 745; GORDY W. and THOMAS W. J. 0. J. Chem. Phys. 1956 24 439.

[S] WALSH A. II. Proc. Roy. Sot. A 1951 207 13.

[7] TORKINGTON P. Nature, Lond. 1948 161 724.

[8] VARSHNI Y. P. 2. Phys. 1953 135 512.

[9] MITRA S. S. 2. Phys. 1954 137 520. [IO] HERZBERG G. Atomic Spectra and Atomic Structure (2nd Ed.) Dover Publications, New

York 1944. [ll] HERZBERG G. Spectra of Diatomic Molecules (2nd Ed.) Nostrand, New York 1950.

160

Diatomic molecules and the ionization potentials of their constituent atoms

[la] SCHOMAKER V. and STEVENSON D. P. J. Amer. Chem. Sot. 1941 63 37. [13] COLE L. G. and ELVERUM G. W. J. Chem. Phys. 1952 20 1543.

[14] PAULING L. Nature of Chemical Bond (2nd Ed.) Oxford University Press, London 1950. [15] HONIG A., M~NDEL M., STITCH M. L. and TOWNES C. H. Phys. Rev. 1954 96 629. [16] RICE 0. K. Electronic Structure and Chemical Binding (Table 4) McGraw Hill, New

York 1940. [17]. WILLIAMS R. L. J. Phys. Chem. 1956 60 1016.

161