Infrared band intensities and bond polarities

169 Journal of Molecular Structure, 18 (1973) 169476 Q Elsevier Scientific Publishing Company, Amsterdam

INFRARED

- Printed

in The Netherlands

BAND INTENSLTIES AND BOND POLARITIES

PART 4. THE BOND-BENDING

VIBRATION

OF ACETYLENE

3. GALABOV Department of Chemistry, Unicersity of Sofia, Sofia 26 (Bulgaria) W. J. ORVILLE-THOMAS Department of Chemistry and Applied Chemistry, Uniuersity of Salford, Salford MS 4WT, Lanes. (England) (Received

19 March

1973)

ABSTRACT

Bond moment theory which takes rehybridization effects into account is used to analyze the band intensity of the 7r”vibrational mode of acetylene. A value of pLb(Ct+ H) = 0.82 D is obtained which is in accord with that of ~0.22 D, sense C t+ H, previously obtained for the CH bond in ethylene.

INTRODUCTION

Quantitative information on the ionic character of valence bonds can be obtained from experimental and theoretical studies. On the experimental side molecular dipole moments can be obtained from dielectric and microwave studies, whilst bond moment theory and a normal coordinate analysis yield bond polar parameters’. In this way, the intensity data are used to provide an experimentally based estimate of the electronic charge distribution in the molecule as a whole and of the polar properties of the individual chemical bonds. This experimental approach has been used for a number of linear triatomic molecules’* 3. The results obtained re-emphasized that the usual assumptions of zero-order bond moment theory4 are seriously at fault. Using zero-order- bond moment theory, one puts equal to zero, derivatives of bond dipole moments with respect to coordinates other than that of the mode involving -vibration of the bond concerned. It seems clear, howeyer, that as a particular bond is stretched the polar properties of other neighbouring valence bonds present in the molecule must be changed. The same content& is also clearly t&e -fo; bond-bending vibrations. This type ‘of b&d/bond

170 interaction is taken into account in first-order bond moment theory by the introduction of cross-terms into the bond moment function’* 5. This procedure has the disadvantage of making the analysis more difficult, since the number of unknown bond moment constants is increased_ For example, for linear YXZ moiecules the number of bond moment constants is four using zero-order theory; this number increases to eight when first-order theory is applied. Tn practice, then, an explicit solution is not possible, since the number of unknown bond moment constants exceeds the number of known factors consisting of the molecular dipole moment and the infrared intensities of the three fundamental bands. These considerations emphasize that using first-order theory instead of zero-order bond moment theory is, in general, not particularly profitable unIess the molecule being considered has a high degree of symmetry, in which case the number of uniquely different crossterms is small. It was thought worthwhile, therefore, to seek a modification of zero-order bond moment theory6 which contains fewer and more physically meaningful bond moment parameters than first-order theory. Tn the next section a possible approach developed previously6 is briefly described and applied to the 7~” bond bending vibration of acetylene.

DIPOLE

MOMENTS

FROM

ELECTRONIC

WAVE

FUNCTIONS

If the most general electronic wave function is denoted Cartesian components of the dipole moment are given by

by (jl, then the

where 2, is the charge on nucleus A, whose coordinate is XA and Xi is the timeaveraged x-component of the radius vector of the i’th electron; similar expressions hold for the y and I components. For closed shell molecules, described by single determinantal wavefunctions, eqn. (1) can be rearranged to give

p, = -2eT

f

$*x$;dV+eCZ,X, A

where the first summation extends over the doubly occupied LCAO-MO’s, form is given by

(2) whose

where 4, is an atomic orbital. Equsition (2) can therefore be expressed in terms of integrals over atomic orbitals, +,,

171 where

P PV

=

2CCipCiv

and

X,,

=

f

t#~,x&,dr.

By rearrangement of the summation in eqn. (3) and collection belonging to the same atom one obtains the expression

DEFINITION

of the terms

OF STATIC BOND MOMENT

The first term in eqn. (4) corresponds to the point charge approximation and the second arises from the homopolar momen&. These two dipoIe moment contributions are bond-directed. During the small displacements involved in molecular vibrations it can be assumed that for a particular bond their sum remains effectively constant and can be identified with the “static” bond moment’, &,-

HYBRIDIZATION

MOMENTS

Bond moments arise because of an asymmetry of the overall charge between the two atoms concerned. One of the basic factors giving rise to asymmetry is the use by either or both atoms of hybrid atomic orbitals for (i) bond formation and (ii) for the accommodation of lone pair electrons. In case (i) the contribution can be termed an atomic moment, whilst for case (ii) the term lone pair moment is appropriate. In this paper they will both be referred to as hybridization moments and denoted by ph. The contributions arising from the third term of eqn. (4) are precisely the hybridization moments mentioned above. In general, they are not bond directed but it is possible to calculate the change of direction of ,CQ,during a bending vibration (Fig. 1). Zero-order bond moment theory relies on the assumption that p(tn), the molecuIar

dipole moment,

can be represented

as the vector sum of a unique set

Fig.- 1. Hybridization moments induced during bond bending vibrations.

172 of bond moments,

i.e.,

(5) bonds

rf hybridization moments moment is given by

are taken

into

account

the total

molecular

dipole

Comparison of eqns. (5) and (6) shows clearly that the bond moments obtained from zero-order theory are quite different quantities from those obtained using the definition of “static” bond moment, rub, given in the last section.

REHYBRIDIZATION MOMENTS

It is generally accepted that when valence bond angles change the valence state of the central atom will also tend to alter; i.e. rehybridization occurs at the central atom, so that the valence state corresponds more closely to the new value of the bond angle. Taking acetylene as an example, the CH bond bending vibration (class n,) is infrared active. In the linear equilibrium configuration the carbons are approximately s--hybridized. During the x, vibration the valence state of the carbons changes from C: sp to C: spx where I < x < 2. In a simple fashion it can be imagined that this process leads to the appearance of a third hybrid orbital associated with each carbon atom (Fig. I). It can furthermore be assumed that electronic charge will flow into these hybrid orbitals, leading to the appearance of a hybridization moment, pLh, whose value will clearly vary periodically as the vibration proceeds. This phenomenon occurs because the carbon atoms undergo a “rehybridization” process during vibration - the valence state of the atoms

i I

i I

Fig. 2. pipole moment contributions during bond bending.

173 tending to follow the changes in bond angle. It is thus possible to define a “rehybridization moment” as (&Jar), where y is the angle through which the bond is bent (Fig. 2). This hypothesis of the appearance of an electronic rehybridization moment accompanying certain deformational modes was implicit in the earlier work of Hornig and McKean4 and identified by Coulson and Stephens. Recently its importance was re-emphasized by Steele and Wheatleyg and a quantitative theoretical approach has been developed 6-8 . This approach helps to avoid some of the discrepancies of the zero-order bond moment treatment.

BOND

MOMENT

CONSTANTS

FROM

MOLECULAR

ORBITAL

THEORY

By quantum mechanical calculations the molecular charge distribution can be obtained and this, of course, is directly related to the molecular and to the bond dipole moments. Semiempirical calculations, such as the CNDO method’ ‘, give reasonably good results for molecular dipole moments, as well as dipole moment derivatives with respect to symmetry coordinates”* 12. In previous papers6s7 a more detailed analysis of the dipole moment contributions has been developed and this has enabled individual bond moment values to be obtained. Briefly, the alterations of certain dipole integrals during the bending vibrations leads to the appearance of a rehybridization moment. These dipole integrals arise from the use of different orbitals on the same atom.

BOND

POLAR

PARAMETERS

IN ACETYLENE

For linear molecules in the equilibrium configuration the hybridization moments are necessarily colinear with the bonds and hence the amount contributed by p,, to the overall or effective bond moments cannot be estimated experimentally. For a bond the effective bond moment can be written as

/I(-)

= P(,(AB)+Ph

When a bond bends through an angle y the molecule is non-linear and & need not necessarily follow the bond (Fig. 2). In these circumstances &@B)l@

= ~~,lay

which is finite, since d&,/dy # 0. That is, there is a variation of the effective bond moment during a bond bending vibration contrary to one of the basic assumptions of zero-order bond moment theory. This state of affairs is illustrated in the vectorial diagram Fig. 2. Since the xcomponents for the two CH bonds cancel out, only the z components need to be considered. From Fig. 2 one sees that P: = PLY

sin Y fph,

I

174 Differentiating

with respect to y one obtains for the equilibrium

configuration

(r = 0) (a0y),

= ps(CH) + (aph, ,izy)o

(7)

where (a&Zy)0 is the effective bond moment which governs infrared band intensities. This relation emphasizes that the effective and static bond moments are not iden tical - the difference between them is just the rehybridization moment, (%LZPY)O

-

MODIFICATION

OF THEORY

The treatment outlined above follows the original approach of Jalsovszky and Orville-Thomas6 and a value for p,,(CH) can be obtained from eqn. (7) once values for the effective bond moment and the rehybridization moment are known. In general, however, this theory is difficult to apply and is best modified as indicated below. The use of zero-order bond moment theory leads to expressions of the form a&m)/asi

= f&,

a/.fi/ari)

(i = 1,2 . _ _)

(8)

where Si is a symmetry coordinate, the symbol, f, stands for “a function of” (see eqn. (9)) and pi and ri are the effective bond moment and bond length of the i’th bond. When zero-order theory is modified to include the effect of rehybridization moments, then expression (8) becomes 6&ni)/iYSj -

ap,(m)/asi

=

f (/lbi, apbi/dri)

(9)

where ~Lbiis now the static bond moment of the i’th bond etc. For the infrared bond bending vibration of acetylene eqn. (9) has the form i3&)/aSi

- E$,(m)ii3Si = J&,(CH)

(10)

is the symmetry coordinate where Si = (yI-tyt)f,/2 bending -angles, y (Fig: 2).

DETERMINATION

OF THE

STATIC

A value for p,, (CH)

DIPOLE

MOMENT

FOR

THE

associated with the bond

CH

BONDS

IN ACETYLENE

can be obtained from eqn. (10) by substituting values

for &/as and apJaS. A value of + 14.82 D nm -l has been obtained for &@z)/aSi from the experimentally determined band intensityi3. A value for the rehybridization moment a&aSj was obtained using CNDO/2 theory, following the procedure previously described 6* 7_ The results of the calculation are given in Table 1. In the

175 TABLE

1

THE VARIATION

Y 0”

5” 10”

OF lib IN ACETYLENE

WITH

THE BENDING

ANGLE

‘/ (in

Debyes)

[(la=

jch,”

fP

jih.z(m

+bw

-0.1776 -0.1789 -0.1827

0 -0.0195 -0.0395

0” 6”lO 12”22’

0 -0.0391 -0.0789

-3.2

as,

= Dipole moment contribution on one carbon atom. TABLE

2

VALUES

FOR pb(CH)

a/l ww

as,

IN ACETYLENE

%h (nl)/ =,

(D run-‘)

(D nm- *)

- 14.82 f 14.82

-3.2 -3.2

jcb(cn)

Valrre(D) Sense 0.82 1.27

CcH Ct-tH

course of the calculation the atomic motions were in accord with the symmetry coordinate and were such that Sj remained positive. The data given in Table 1 lead to the two possible values for /l,(CH) given in Table 2. Clearly, the most reasonable value is 0.82 D with the sense &C c+ H). This is in good accord with the value of p(C tt H) = 0.22 D obtained for ethylene, The numerical difference is what one would expect for two CH bonds in which the valence state changes from -sp2 to sp and the sense is expected to be the same for these two molecules_ The value obtained here for the static bond moment can be compared with the range of values from 1 .OO to 1.05 D found for the effective CH bond moment in acetylene’.4, ’ 3- 14. It is seen that the rehybridization moment makes a substantial contribution to the infrared intensity of the CH bond bending mode. From Fig. 2 it is seen that ph,=

=

ph sin

q

where cp is the angle between the vector & and the molecular axis. Values for cpare given in Table 1 and it is seen that cpchanges more rapidly than the bond bending angle y.

REFERENCES 1 L. A. GRIBOV, Intensity Theory for Infrared Spectra of Polyatomic Molecules, Consultants Bureau, New York, 1964. 2 P. R. DAVIES AND W- J. ORVILLE-THOMAS, J. Mol. Structure, 4 (1969) 163. 3 G. A. THOMAS, J. A. LADD AND W. J. ORVILLE-THOMAS, J. Mol. Structure, 4 (1969) 179. 4 D. F. HORNIG AND D. C. MCKEAN, J. Phys. Chem., 59 (1955) 1133.

176 5 G. A. THOMAS, G. JALSOVSZKY,J. A. LADD AND W. J. ORVILLE-THOMAS, J. Mol. Strrrctwe, 8 (1971) 1. 6 G. JAUOVSZKY AND W. J. ORVILLE-THOMAS, Trans. Faraday Sot., 67 (1971) 1894. 7 B. GALABOV AND W. J. ORVILLE-THOMAS,JCS Faraday ZZ, 68 (1972) 1778. 8 C. A. COU~N AND M. J. STEPHEN, Trans. Faraday SOL, 53 (1957) 272. 9 D. STEELEAND W. WHEATLEY, J. Mol. Spectrosc., 32 (1969) 265. IO J. A. POPLE AND D. L. BEVERIDGE, Approximate Molecular Orbital i%eor_v, McGraw-Hill, London, 1970. 11 G. A. SEGAL AND M. L. KLEIN, J. Chenz. Phys., 47 (1967) 4236. 12 G. A. SEGAL, R. BURNS AND W. B. PERSON,J. Chem. Phys., 50 (1969) 3811. 13 D. F. EGGERS JR., I. C. HISATSUNEAND L. VAN ALTEN, J. Phys. C’hern.,59 (1955) 1124. 14 R. L. KELLY, R. RO~LEFSONAND B. S. SCHURIN, J. Chetn. Phys., 19 (1951) 1595.