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Some considerations on the force constants of bond-bond interaction in simple triatomic molecules
Spectrochimica Acta. Vol. 34A. pp. 147 to 152. Pergamon Press 1978. Printed in Great Britain
Some considerations on the force constants of bond-bond interaction in simple triatomic molecules KEN OHWADA Division of Chemistry, Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki-ken, Japan
(Received 7 October 1976) Abstract--The force constants of bond-bond interaction in simple triatomic molecules have been discussed on the basis of a simple molecular orbital theory. It has been shown that the force constant of highly delocalized-bond molecule tends to take a large positive value, while that of highly localizedbond molecule a negative value. It is therefore suggested that the sign of the force constant is a good measure of the electron localization or delocalization in a molecule.
INTRODUCTION
As is well known, the force constants in a molecule represent the magnitude of the forces acting between the atoms. Therefore, the force constant will be closely related to the nature of the chemical bond, i.e. bond energy, bond distance, bond order, electron delocalization, resonance energy, polarizability etc. In the case of diatomic molecules, the relation of the force constants K and the bond distances Re has been systematically investigated by BAthER[I,2], and it has been clarified that there is a simple relationship between them (K-1/3 = ao(R e _ bo); ao' bo: empirical constants). P I ~ T E L et al. [3] have also shown that the force constants in such molecules as B2, C2, N2, 02, F2 and Ne 2 can be mutually related in a simple manner to the bond orders, which are obtained theoretically from the molecular orbital theory, as well as the bond energies and bond distances. Similarly, in the case of polyatomic molecules, it has been found that there are some qualitative correlations between the force constants and the bond energies, bond orders, and bond distances. COULSON and LONGUET-HIGGINS[4] have pointed out from a detailed analysis of the conjugated hydrocarbons as well as the other organic compounds containing n-electrons that the stretching force constant of a bond is a function of the bond order and the polarizability. In addition, LONGUET-HIGGINS and S^LEM[5,6] have been establishing a method by which the resonance energy can be determined from the force constants of symmetric and asymmetric stretching in the conjugated system. On the other hand, it has been shown by SHIMANOUCHIand NAKAGAWA[7-9] that the stretching force constants in some-metal amine complexes give quantitative information on the nature of coordination bonds which have relatively high covalency. The force constants are, in a sense, a measure of electron delocalization. For example, such molecules
as ~ - - O and R--CO2 have large positive values of the force constants of bond-bond interaction between two C - - O bonds. The positive value in the molecule indicates that it is easy to lengthen one bond and to shorten the other. This phenomenon will be connected undoubtedly to the electron delocalization or quantum-mechanical resonance. In the present study, we have paid attention to the force constant of bond-bond interaction in simple triatomic molecules, and discussed in detail the correlation of the force constant with the electron localization or deiocalization in the molecules.
THEORY
Formulation of the force constants In writing the complete Hamiltonian of a molecule, we must take into account four terms: (1) the kinetic energy of electrons, (2) the repulsion energy between electrons, (3) the attractive energy between electron and nucleus, and (4) the repulsion energy between nuclei [10, 11]. In the case of a diatomic molecule, the Hamiltonian Ht may formally be written down as follows;
ZA Z8 e 2 H , = He~oo + -
-
(1)
RAB
where Hdee is the Hamiltonian associated with only the electrons and the second term denotes the Coulomb repulsion energy between the nuclei A and B at a distance of RAn. Therefore, the energy function
E= f ~,*Ut~,de/f ~,*~,d~
(2)
may be expressed as the sum of two parts, one associating with the electrons, the other with the nuclear repulsion. The latter term becomes
f ~l* ZAZae2 /f RA-----~ ~' de 147
~b*d/de -
ZAZBe2 RA~
(3)
148
KEN OHWADA
In the case of a polyatomic molecule, it may also be allowed quantum-mechanically that, to a first approximation, the molecular energy is divided into two parts as follows; E = E,u~l + E~l~
(4)
where E , ~ is the classical potential energy arising from the nuclear positive charges and E,~, is the electronic energy associated with the Hamiltonian H~e,. On the basis of above consideration, let us assume that the potential energy change 3// of a molecule may be approximately expressed as the sum of two parts, one (E,) arising from the nuclear repulsion't, the other (E~) from the valence electrons which participate in the bond formation;
Let us assume further that the nuclear repulsiox. energy is the sum of independent contributions from the bonds in a molecule. On the other hand, the electron binding energy, as we know from the molecular orbital theory, is a function of the resonance or exchange integrals of all the bonds, E~ = ¢(fl~, f12..... fl..... ),
(0~,)R, = 0 (for all n);
(7)
that is,
(8)
\dR.Ja. + \Off. dR.fa.
where R. is the equilibrium bond length. When further energy differentials are evaluated at the equilibrium configuration, we obtain the force constants as follows;
K = \OR2jR" = \dR2jr~" + \dE~ c~fl.J~. + (\dR./
(11)
This form is justified by our knowledge that ft, must vanish for infinite R., and that its decrease, as R. increases, must be roughly proportional to the overlap integral [hence to e x p ( - R . / a ) ] of the bonded atomic orbitals. Differentiating equation (11), we obtain
\
where B~, is the value of resonance integral at equilibrium position. When we now return to equation (10) expressing the b o n d - b o n d interaction force constant in question, it may be concluded that its sign depends only on that of (c~2Eb/Oflnt3flm).
(6)
where the resonance integral ft., from its definition: ft. = ~b,//~bs dz (r :# s), must be some function fl.(R.) of the bond length R,. H is the effective Hamiltonian operator for each valence electron and q~'s are the atomic orbitals. Now let us write the condition that each bond is in equilibrium, so that no forces are acting on the nuclei. The conditions for static equilibrium are
to.
ft, = - - B e x p ( - - R J a ) , ( B , a > 0).
(5)
fiE = fiE, + 3E b.
\
In the foregoing interprekation, we have assumed nothing at all about the analytic form of fl.(R.). Here, we shall for the first time introduce the general function ft., in which the relation between ft. and R. is assumed to be an exponential one:
fl./Jt~.
, (9)
~\dRd \dR./\a#.~#~/J~.R~' where K and k are the bond stretching and b o n d bond interaction force constants. t This is also regarded as including the repulsions between the localized electrons on atoms A and B.
Relation between the (a2 EJOflnt3flm) term and molecular orbital coefficients In order to evaluate the sign of the (cgEgjc3fln63flm), we have need of a lengthy discussion. To do this, it is convenient to start with Hiickel theory of molecular orbital [10-12]. According to the theory, the following secular equations can simply be assumed;
(H,,-Ebj)C~+~'fl,,C~=O(r=
1,2 . . . . . n). (13)
$
Here, H,, and fl,, are the matrix elements of the effective Hamiltonian H in the system of n distinct atomic orbitals ~b,, and defined by
H. = f ~* He, d~, g,
dz.
C,j and Ebi are the numerical coefficients and the electron binding energies of the jth molecular orbitals ~,j. The prime after the summation sign denotes that the term with s = r is omitted from the sum. Also, in equation (13), overlap between atomic orbitals is neglected and each atomic orbital is supposed to be normalized. If we choose a purely real form for every atomic wave function q~,, which is always permitted except when dealing with certain magnetic properties, the H,,, fl,s and C,~ are real also. The condition for consistency of the n equations (13) is that the secular determinant shall vanish. In general this determinant has n roots Eb~ (j = 1, 2..... n). Each of these is substituted back in turn into secular equations (13) to give the corresponding sets of coefficients C~j,
149
Interaction in simple triatomic molecules
C2 (7.. . , C,, or rather their ratios. Their absolute magnitudes~ are fixed by the orthonormalization ; Cij + Cij + “. + Cij = 1, C C,jC,, = 0. (14) I Further, we shall discuss only molecules in which all the electrons are paired, so that each occupied molecular orbital contains exactly two electrons. The zero of energy can be chosen in such a way that Ebj is negative for each of the occupied molecular orbitals and positive for each of the unoccupied molecular orbitals. In the ground state of the molecule the occupied molecular orbital will be measured from 1 to m, and the unoccupied from m + 1 to n. The total electronic binding energy Eb is therefore the sum of the electron energies in the occupied molecular orbitals, 2E,.
(1%
j=l
In a given molecule, the coulomb integrals H, in secular equations (13) will depend upon the density of electrons near atom r. But it will be supposed that in fact H,, depends only on the nature of atom r. The number of electrons on atom r is generally defined as qr=iIljCzj
where nj is the number of electrons in the jth molecular orbital. The coefficients C,j are determined from the secular equations (13). The coefficients may also be used for calculating the binding electron densities Pr, of the bonds. If r and s are neighbouring atoms, an electron in the jth molecular orbital, which is assumed to be real and not complex, makes a contribution C,jC, to the bond rs. So the total binding electron density is defined by Pr, = t njC,jC,j. j
1 C,!jaH,, + C C2CrjCsj6Brs. (18)
tij =
(lg),
= CFj, &j)
E
bj
_
E
ICI, (24) bk
and J
.
all other integrals of this type being zero. So
where c;j
=
i
‘_ ~crk~/%s,
crk(crj:,
k=l
bk
This leads to a change in binding electron density of bond rs given by In
{(Crj
+
Cij)
(csj
+
Cj)
-
crjcsji
j=l
= 2C,jC,j.
(19,20)
t
j=,
(2)
=
($j
t21’
and
s AA.34/2-c
JljH’$kdr
since
,=1
Crjcsj
c
c’ ’ t=i
This may be written
I=1
j=l
+
over t;e occupied molecular
qr= 2 5 cfj=2
P,s = 2 f
*i
which equals
BP,, = 2 C
r
Summation 0;(19,20) orbitals gives
+
(17)
If r and s are not neighbours, PIs is taken to be zero. Both these quantities q, and P,, can be related through the coefficients C,j and C, to the electron energy variations with changes in H,, and /.?,, Now, it follows from equations (13) that for any small changes in H,, fi,# and Crj,
(2)
$j
(16’
i
we obtain from iquation
which correspond to those of equations (9) and (10). An expression for (aP,J~?fi,~) can be obtained by application of ordinary first-order perturbation theory to the secular equations (13). Notice what happens to the coefficients, binding electron densities when the resonance integral /?r, is increased by a small amount S/I,, By first-order perturbation theory the perturbed wave functions can be expanded in the form n
Eb = i
6Ebj =
where E, is given by equation (15). Continuing with the argument, the differentials of P,, with 8. and/or /3,. can be expressed as;
-
,i,
(ff) =;(z),
(22’
,=“+I
(c&k + C&k)' E, - Ebt (26)
150
KEy OHWADA
Similarly, for aP,,/afl,., we obtain
Molecular o~bital
0P,, i -= 2j~" O#,. ~ i:.,+,
~/: =°'¢m- b'¢02+0'q5,,3
(C,jC,k + C,jC,D(CuC.k + C.jC,k) X
1
V,~ :~¢)~,
Ebj -- Ebk
= _1( ~2Eb "~
2
Energy level
(27)
\a#,,a#,d
We can consequently evaluate the force constant of bond-bond interaction of equation (10) under consideration of the molecular orbital coefficients as well as the difference of electron energies. DISCUSSION
The simple triatomic molecules offer an interesting opportunity to test the theory as previously interpreted. We here discuss only two extreme cases of highly delocalized- and localized-bond molecules.
Delocalized-bond molecules For simplicity, let us consider the case of triatomic molecules such as A1 - B2 - A3 and A1 - A2 - A3 types, and suppose the bonds are due to overlapping of a-type atomic orbitals, ~A,, ~n2 (~b,d, and ~A~ as shown in Fig. 1. According to the delocalized-bond description [3, 13, 14], we may form three molecular orbitals ej by linear combination of the three atomic orbitals ~AI, ~B2, and SA,;
- ~1 %
= O@A +, b@s~+ a(hA3
0
E3
©
E~:o
@
El
~6A,: qSA3:2p. [F] @B2:ls F.H] Fig. 2. Molecular orbitals and energy level of hydrogen bifluoride ion. lide ions, and rare-gas bifluoride (XeF2)[-3, 13, 15]. Of course, the n-bonding of carbon dioxide (CO2), carbon disulphide (CS2), etc. is in the category of the three-center bonding. Of these molecules, HF~ is most typical example representing the delocalization of electrons. An approximate diagram of the energy level of HF~ is shown in Fig. 2, wherein only the ¢1 and ~/'2 orbitals are occupied with two electrons, respectively and, however, the latter is entirely confined to the fluorine atoms and represents an essentially non-bonding orbital. In the case of other molecules, similar diagrams of the energy level will be expected. When the molecular orbitals (28) for these molecules are applied to equation (27) derived in the previous section, we immediately obtain
~1 = a¢4, + bCB~ + aCa3, 1
(
1
_ 2(a'b- ab')2 b 12
¢] = a'(aa, -- b'$.~ + a'Oa3,
E2 _ E3,
(30)
where qJ~ denotes the bonding, @2 the non-bonding and q,* the anti-bonding molecular orbitals. The conditions of orthonormalization are 2a 2 + b 2 = 1 , 2 a ' + b ' 2 = t , 2 a a ' - b b ' = 0 .
(29)
These molecular orbitals are in general found as creating a three-centre bond due to the delocalization of electrons over the entire molecule. As examples of the three-centre bonding, we will take such molecules as bifluoride ion (HF2), trihaA,
0
B2(Aa)
0
{i)IEI-E21~IE2-E31
A~
0 I.neo,)
(ii) IE,-E21:IEe-E31
(Bent)
A,
A~
Atomic ~A, (~2(~Aa) ~A~ orbital Fig. 1. Arrangement of atomic orbitals in linear and bent symmetric triatomic molecules.
(Jill IE,-Eal)) IEa-Esl Fig. 3. Energy level diagrams of three extreme cases of three-centre four-electron bonding.
Interaction in simple triatomic molecules where El, E 2 , and E a are the energies associated with the molecular orbitals, ~01, ~2, and q/~'. In order to obtain the value of equation (30), we must solve the secular determinant of third order for each molecule. However, as far as we are concerned with only the sign of equation (30), it will be unnecessary to solve the secular determinant. Along such a point of view let us consider three extreme cases of (i) tIE1 - E21 '~ [E2 - - E3 l, (ii) IE a - E 21 = I E2 - E3 l, and (iii) [E 1 - E 21 >> I E2 - Ea I, for which the energy level diagrams are schematically shown in Fig. 3. For the first case (i), to which the weak bonding molecules belong, the value of equation (30) is of course approximated to zero since the first term is nearly equal to the second one: 2(a'b - ab') 2
b '2
E t - E3
E2 - E3"
The second case (ii) leads to positive value of - b ' 2 / ( E 2 -- E3) since E2 - Ea < 0 and a'b - ab' = O. For a better understanding of this situation, suppose the molecular orbitals with the coefficients of a=a'= 1/2 and b = b ' = l/x/2. Although the last case (iii), to which the strong bonding molecules belong, is somewhat complicated, it may be expected to have relatively large positive value from the consideration of some popular molecular orbitals. It is found from the above discussion that for the case of delocalized-bond molecule such as H F 2 , equation (30) tends to take positive value. Therefore, we can conclude from equation (10) that the force constant of bond-bond interaction in the delocalizedbond molecules is also positive value. In Table l, we show the force constant of bond-bond interaction together with the other constants in some molecules [16-21]. In calculating the force constants, the following potential function V has been assumed; 2V = K {(Arl) 2 + (Ar2) 2} + 2k(Arx) (At2)
Table 1. Force constants of deloealized-bond molecules Molecule
Force constant, md/A k H
K
HF~ DF~ XeF2 CO2 CS2 O3 SO2
2.31 2.30 2.84 15.50 7.50 5.70 10.01
1.71 1.74 0.13 1.30 0.60 1.52 0.024
0.22 0.22 0.20 0.57 0.23 1.28 0.79
h 0* 0* 0* 0* 0* 0.33 0.19
* The force constant of bond-angle interaction was approximated to zero. Applying the method of linear combination of atomic orbital to H 2 0 , we would employ two of the oxygen p-orbitals, say the Px and the py atomic orbitalst, in combination with the hydrogen ls-orbitals (Fig. 4). Two a-bonding and two antibonding molecular orbitals are thus formed, which we might designate in un-normalized form; ~, = ~ba + ~b2, ¢,, = ~b3 + q~,,
~¢' =
4',
-
4~2,
¢,~
=
4~3 -
(32)
~,,
where ~ and ¢* denote the bonding and anti-bonding molecular orbitals. Practically speaking, these orbitals represent those of a completely localized molecule which will not actually exist. Therefore, when applying these molecular orbitals to equation (27), it is necessary for practical purposes that the bicentric localized orbitals are formally transformed to the polycentric delocalized orbitals in which the bicentric localizability remains. COOLSON[10] has discussed in detail this transformation which promises to yield,
~ o - ,,)
= a'q~x + b'~P2 - - b ' ( ~ 3 - - a'~p,,,
(33)
~#J-,,) = a'~ba - b'q~2 - b'~b3 + a'q~,l., ~#J+,,) = aq~a - b~b2 + b~b3 - a~b4. The orthonormal conditions are
+ r02H(Aq5)2 + 2 r o h ( A r l ) ( A q b ) + 2r0h(Ar2) (Aq~),
151
(31)
where K is the stretching, k the bond-bond interaction, H the bending, and h the bond-angle interaction force constants.
2a 2 + 2 b 2 = 1,2a '2 + 2 b ' 2 = 1 , 2 a a ' - 2 b b ' = O .
Applying these molecular orbitals (33) to equation (27), we obtain, dPr~
1(
632Eb
t In order to explain the directed valence, we must use hybrid orbitals in formulating the molecular orbitals~ However, the use of such hybrid orbitals should not give any change in the discussion below.
2(a'b - ab') 2 E1 - E3
Localized-bond molecules
In terms of molecular orbital theory, a localizedbond leads us directly to localized molecular orbitals. In other words, even in very complicated molecules we may still, to a good approximation, consider the bonding molecular orbitals to be bicentric and similar to those existing in diatomic molecules. If we take H 2 0 molecule as example, we would picture two separate O H bonding molecular orbitals relatively independent of each other.
(34)
+
2(ab' - a'b) 2 E2 - E,,
4,,
(35)
¢, qS : q54 : Is[HI : 2px[O] q53 : 2py[O]
Fig. 4. Arrangement of atomic orbitals in water molecule.
152
KEN OHWADA
Table 2. Force constants of locafized-bond molecules Molecule H20 D20 H2S D2S H2Se D2Se HgCI 2 HgBr2 KrF2
K 8.45 8.45 4.03 4.20 3.12 3.18 2.40 2.14 2.46
Force constant, md/A k H -0.11 -0.11 -0.11 -0.22 -0.12 -0.13 -0.26 -0.17 -0.20
0.82 0.82 0.48 0.48 0.34 0.32 0.04 0.05 0.21
h 0.18 0.18 0* 0* 0* 0* 0* 0* 0*
* The force constant of bond-angle interaction was approximated to zero.
where E1 and E2 denote the energies of bonding Sa + tt~ and ~/(I -- If) orbitals, and Ea and E4 the energies of anti-bonding ~bff_ .) and ~'a + ll) orbitals. Equation (35) leads clearly to negative value which is expected to be not SO large. Because in the right hand side of equation (35), the denominators of the energy difference of both the terms are negative (El - E3 < 0, E 2 - E4 < 0) and both the numerators associated with the molecular orbital coefficients are positive (2(a'b - ab') 2 > 0, 2(ab' - a'b) 2 > 0). Of course, for a completely localized molecule, this leads to zero. It is therefore concluded from equation (10) that the force constant of bond-bond interaction in such a highly localized molecule as H 2 0 tends to take negative value. In fact, it has been found on the basis of a general potential function (31) for H 2 0 that the force constant of bond-bond interaction k is negative value as shown in Table 2, wherein the force constants of H 2 0 as well as some other molecules[18, 19] regarded as the localized-bond molecules are given. In Table 2, a striking molecule is krypton difluoride, KrF2. It has been pointed out that the bonding in KrF2 can be described by the delocalized molecular orbitals as creating a three-centre four-electron bond [3,1. These molecular orbitals are quite the ~ m e as those of xenon-difluoride, XeF2 [15,1 as listed in Table 1. According to the present theory, these three-centre molecular orbitals should yield positive value of the force constant of bond-bond interaction. However, CLAASSENet al. [21] have shown from the vibrational study that the interaction force constant in KrF2 is of negative value ( - 0 . 2 md//~) on the contrary to that in XeF2. This difference between KrF2 and XeF2 has been discussed in detail by COULSON[22-1, who has suggested that the negative sign of KrF2 may be due to the greater weight of a no-bond structure in a valence-bond description. In order to explain this difference by the molecular
orbital theory, it is convenient to start from a point of view, based on the localized molecular orbitals rather than the three-center delocalized oneg For obtaining the localized molecular orbitals for KrF2, let us assume the sp diagonal hybrid orbital of krypton atom as follows; 1 Ol = - - ( S + p x
),
1 O2=--(S-px
).
Such a hybridization can be achieved by an excitation, 4s24p6--* 4s24pS5s, which requires only a little less energy than the 4s24p6-*4s24pS4d. If we use these hybrid orbitals, the same description of the localized-bond as in the case of H 2 0 may be made. It is therefore inferred from such a description that the force constant of b o n d - b o n d interaction in KrF2 should take negative value. REFERENCES [1] R. M. BADGER,J. Chem. Phys. 2, 128 (19341 [2] R. M. BADGER,J. Chem. Phys. 3, 710 (1935). [3] G. C. P1MENTALand R. D. SPRA~II.EY,Chemical Bonding Clarified Through Quantum Mechanics. HoldenDay, New York (1969). [4] C. A. COULSONand H. C. LONGUET-HIGGINS,Proc. Roy. Soc. A193, 456 (19481 [5] H. C. LONGUET-HIC~INSand L. SALEM, Proc. Roy. Soc. A251, 172 (1959). [6] L. SALEM, The Molecular Orbital Theory of Conjugated Systems, 3rd edn. W. A. Benjamin, Reading, Massachusetts (19721 [7] T. SHIMANOUCm and I. NAKAGAWA, Spectrochim. Acta lg, 89 (1962). [8] I. NAKAGAWA and T. SHIMANOUCHL Spectrochim. Acta 22, 759 (1966). [9] I. NAKAG^WA,Coord. Chem. Rev. 4, 423 (1969). [10] C. A. COULSON, Valence, 2nd edn. Clarendon Press, Oxford (1961). [11] J. N. MURRELL,S. F. A. KETTLE and J. M. TEDDER, Valence Theory, John Wiley, New York (1965l [12] A. STREITWmSER,JR., Molecular Orbital Theory for Organic Chemists. John Wiley, New York (1961). [13] G. C. PIMENTAL,J. Chem. Phys. 19, 446 (19511 [14] F. A. COTTON,Chemical Applications of Group Theory, 2nd edn. Wiley-Interscience, New York (1964). [15] C. A. COULSON,J. Chem. Soc. 1442 (19641 [16] R. NEWMANand R. M. BADGER,J. Chem. Phys. 19, 1207 (1951). [17] P. DAWSON,M. M. HARGREAVEand G. R. WILKINSON, Spectrochira. Acta 31A, 1055 (1975). [18] G. HERZBERG,Infrared and Raman Spectra of Polyatomic Molecules. Van Nostrand, Princeton, New Jersey (19451 [19] S. MIZUSHIMAand T. SHIMANOUCl-n,Infrared Absorption and Raman Effect. Kyoritu, Tokyo (1958). [20] H. SmBERT, Anwendunoen der Schwinoun#sspekroskopie in der Anorganischen Chemie. Springer-Verlag, Berlin (1966l 1-21] H. H. CLAASSEr~,G. L. GOODMAr~,J. G. MALM and F. SCnREINER,J. Chem. Phys. 42, 1229 (1965). [22] C. A. COULSON,J. Chent Phys. 44, 468 (1966).
Some considerations on the force constants of bond-bond interaction in simple triatomic molecules KEN OHWADA Division of Chemistry, Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki-ken, Japan
(Received 7 October 1976) Abstract--The force constants of bond-bond interaction in simple triatomic molecules have been discussed on the basis of a simple molecular orbital theory. It has been shown that the force constant of highly delocalized-bond molecule tends to take a large positive value, while that of highly localizedbond molecule a negative value. It is therefore suggested that the sign of the force constant is a good measure of the electron localization or delocalization in a molecule.
INTRODUCTION
As is well known, the force constants in a molecule represent the magnitude of the forces acting between the atoms. Therefore, the force constant will be closely related to the nature of the chemical bond, i.e. bond energy, bond distance, bond order, electron delocalization, resonance energy, polarizability etc. In the case of diatomic molecules, the relation of the force constants K and the bond distances Re has been systematically investigated by BAthER[I,2], and it has been clarified that there is a simple relationship between them (K-1/3 = ao(R e _ bo); ao' bo: empirical constants). P I ~ T E L et al. [3] have also shown that the force constants in such molecules as B2, C2, N2, 02, F2 and Ne 2 can be mutually related in a simple manner to the bond orders, which are obtained theoretically from the molecular orbital theory, as well as the bond energies and bond distances. Similarly, in the case of polyatomic molecules, it has been found that there are some qualitative correlations between the force constants and the bond energies, bond orders, and bond distances. COULSON and LONGUET-HIGGINS[4] have pointed out from a detailed analysis of the conjugated hydrocarbons as well as the other organic compounds containing n-electrons that the stretching force constant of a bond is a function of the bond order and the polarizability. In addition, LONGUET-HIGGINS and S^LEM[5,6] have been establishing a method by which the resonance energy can be determined from the force constants of symmetric and asymmetric stretching in the conjugated system. On the other hand, it has been shown by SHIMANOUCHIand NAKAGAWA[7-9] that the stretching force constants in some-metal amine complexes give quantitative information on the nature of coordination bonds which have relatively high covalency. The force constants are, in a sense, a measure of electron delocalization. For example, such molecules
as ~ - - O and R--CO2 have large positive values of the force constants of bond-bond interaction between two C - - O bonds. The positive value in the molecule indicates that it is easy to lengthen one bond and to shorten the other. This phenomenon will be connected undoubtedly to the electron delocalization or quantum-mechanical resonance. In the present study, we have paid attention to the force constant of bond-bond interaction in simple triatomic molecules, and discussed in detail the correlation of the force constant with the electron localization or deiocalization in the molecules.
THEORY
Formulation of the force constants In writing the complete Hamiltonian of a molecule, we must take into account four terms: (1) the kinetic energy of electrons, (2) the repulsion energy between electrons, (3) the attractive energy between electron and nucleus, and (4) the repulsion energy between nuclei [10, 11]. In the case of a diatomic molecule, the Hamiltonian Ht may formally be written down as follows;
ZA Z8 e 2 H , = He~oo + -
-
(1)
RAB
where Hdee is the Hamiltonian associated with only the electrons and the second term denotes the Coulomb repulsion energy between the nuclei A and B at a distance of RAn. Therefore, the energy function
E= f ~,*Ut~,de/f ~,*~,d~
(2)
may be expressed as the sum of two parts, one associating with the electrons, the other with the nuclear repulsion. The latter term becomes
f ~l* ZAZae2 /f RA-----~ ~' de 147
~b*d/de -
ZAZBe2 RA~
(3)
148
KEN OHWADA
In the case of a polyatomic molecule, it may also be allowed quantum-mechanically that, to a first approximation, the molecular energy is divided into two parts as follows; E = E,u~l + E~l~
(4)
where E , ~ is the classical potential energy arising from the nuclear positive charges and E,~, is the electronic energy associated with the Hamiltonian H~e,. On the basis of above consideration, let us assume that the potential energy change 3// of a molecule may be approximately expressed as the sum of two parts, one (E,) arising from the nuclear repulsion't, the other (E~) from the valence electrons which participate in the bond formation;
Let us assume further that the nuclear repulsiox. energy is the sum of independent contributions from the bonds in a molecule. On the other hand, the electron binding energy, as we know from the molecular orbital theory, is a function of the resonance or exchange integrals of all the bonds, E~ = ¢(fl~, f12..... fl..... ),
(0~,)R, = 0 (for all n);
(7)
that is,
(8)
\dR.Ja. + \Off. dR.fa.
where R. is the equilibrium bond length. When further energy differentials are evaluated at the equilibrium configuration, we obtain the force constants as follows;
K = \OR2jR" = \dR2jr~" + \dE~ c~fl.J~. + (\dR./
(11)
This form is justified by our knowledge that ft, must vanish for infinite R., and that its decrease, as R. increases, must be roughly proportional to the overlap integral [hence to e x p ( - R . / a ) ] of the bonded atomic orbitals. Differentiating equation (11), we obtain
\
where B~, is the value of resonance integral at equilibrium position. When we now return to equation (10) expressing the b o n d - b o n d interaction force constant in question, it may be concluded that its sign depends only on that of (c~2Eb/Oflnt3flm).
(6)
where the resonance integral ft., from its definition: ft. = ~b,//~bs dz (r :# s), must be some function fl.(R.) of the bond length R,. H is the effective Hamiltonian operator for each valence electron and q~'s are the atomic orbitals. Now let us write the condition that each bond is in equilibrium, so that no forces are acting on the nuclei. The conditions for static equilibrium are
to.
ft, = - - B e x p ( - - R J a ) , ( B , a > 0).
(5)
fiE = fiE, + 3E b.
\
In the foregoing interprekation, we have assumed nothing at all about the analytic form of fl.(R.). Here, we shall for the first time introduce the general function ft., in which the relation between ft. and R. is assumed to be an exponential one:
fl./Jt~.
, (9)
~\dRd \dR./\a#.~#~/J~.R~' where K and k are the bond stretching and b o n d bond interaction force constants. t This is also regarded as including the repulsions between the localized electrons on atoms A and B.
Relation between the (a2 EJOflnt3flm) term and molecular orbital coefficients In order to evaluate the sign of the (cgEgjc3fln63flm), we have need of a lengthy discussion. To do this, it is convenient to start with Hiickel theory of molecular orbital [10-12]. According to the theory, the following secular equations can simply be assumed;
(H,,-Ebj)C~+~'fl,,C~=O(r=
1,2 . . . . . n). (13)
$
Here, H,, and fl,, are the matrix elements of the effective Hamiltonian H in the system of n distinct atomic orbitals ~b,, and defined by
H. = f ~* He, d~, g,
dz.
C,j and Ebi are the numerical coefficients and the electron binding energies of the jth molecular orbitals ~,j. The prime after the summation sign denotes that the term with s = r is omitted from the sum. Also, in equation (13), overlap between atomic orbitals is neglected and each atomic orbital is supposed to be normalized. If we choose a purely real form for every atomic wave function q~,, which is always permitted except when dealing with certain magnetic properties, the H,,, fl,s and C,~ are real also. The condition for consistency of the n equations (13) is that the secular determinant shall vanish. In general this determinant has n roots Eb~ (j = 1, 2..... n). Each of these is substituted back in turn into secular equations (13) to give the corresponding sets of coefficients C~j,
149
Interaction in simple triatomic molecules
C2 (7.. . , C,, or rather their ratios. Their absolute magnitudes~ are fixed by the orthonormalization ; Cij + Cij + “. + Cij = 1, C C,jC,, = 0. (14) I Further, we shall discuss only molecules in which all the electrons are paired, so that each occupied molecular orbital contains exactly two electrons. The zero of energy can be chosen in such a way that Ebj is negative for each of the occupied molecular orbitals and positive for each of the unoccupied molecular orbitals. In the ground state of the molecule the occupied molecular orbital will be measured from 1 to m, and the unoccupied from m + 1 to n. The total electronic binding energy Eb is therefore the sum of the electron energies in the occupied molecular orbitals, 2E,.
(1%
j=l
In a given molecule, the coulomb integrals H, in secular equations (13) will depend upon the density of electrons near atom r. But it will be supposed that in fact H,, depends only on the nature of atom r. The number of electrons on atom r is generally defined as qr=iIljCzj
where nj is the number of electrons in the jth molecular orbital. The coefficients C,j are determined from the secular equations (13). The coefficients may also be used for calculating the binding electron densities Pr, of the bonds. If r and s are neighbouring atoms, an electron in the jth molecular orbital, which is assumed to be real and not complex, makes a contribution C,jC, to the bond rs. So the total binding electron density is defined by Pr, = t njC,jC,j. j
1 C,!jaH,, + C C2CrjCsj6Brs. (18)
tij =
(lg),
= CFj, &j)
E
bj
_
E
ICI, (24) bk
and J
.
all other integrals of this type being zero. So
where c;j
=
i
‘_ ~crk~/%s,
crk(crj:,
k=l
bk
This leads to a change in binding electron density of bond rs given by In
{(Crj
+
Cij)
(csj
+
Cj)
-
crjcsji
j=l
= 2C,jC,j.
(19,20)
t
j=,
(2)
=
($j
t21’
and
s AA.34/2-c
JljH’$kdr
since
,=1
Crjcsj
c
c’ ’ t=i
This may be written
I=1
j=l
+
over t;e occupied molecular
qr= 2 5 cfj=2
P,s = 2 f
*i
which equals
BP,, = 2 C
r
Summation 0;(19,20) orbitals gives
+
(17)
If r and s are not neighbours, PIs is taken to be zero. Both these quantities q, and P,, can be related through the coefficients C,j and C, to the electron energy variations with changes in H,, and /.?,, Now, it follows from equations (13) that for any small changes in H,, fi,# and Crj,
(2)
$j
(16’
i
we obtain from iquation
which correspond to those of equations (9) and (10). An expression for (aP,J~?fi,~) can be obtained by application of ordinary first-order perturbation theory to the secular equations (13). Notice what happens to the coefficients, binding electron densities when the resonance integral /?r, is increased by a small amount S/I,, By first-order perturbation theory the perturbed wave functions can be expanded in the form n
Eb = i
6Ebj =
where E, is given by equation (15). Continuing with the argument, the differentials of P,, with 8. and/or /3,. can be expressed as;
-
,i,
(ff) =;(z),
(22’
,=“+I
(c&k + C&k)' E, - Ebt (26)
150
KEy OHWADA
Similarly, for aP,,/afl,., we obtain
Molecular o~bital
0P,, i -= 2j~" O#,. ~ i:.,+,
~/: =°'¢m- b'¢02+0'q5,,3
(C,jC,k + C,jC,D(CuC.k + C.jC,k) X
1
V,~ :~¢)~,
Ebj -- Ebk
= _1( ~2Eb "~
2
Energy level
(27)
\a#,,a#,d
We can consequently evaluate the force constant of bond-bond interaction of equation (10) under consideration of the molecular orbital coefficients as well as the difference of electron energies. DISCUSSION
The simple triatomic molecules offer an interesting opportunity to test the theory as previously interpreted. We here discuss only two extreme cases of highly delocalized- and localized-bond molecules.
Delocalized-bond molecules For simplicity, let us consider the case of triatomic molecules such as A1 - B2 - A3 and A1 - A2 - A3 types, and suppose the bonds are due to overlapping of a-type atomic orbitals, ~A,, ~n2 (~b,d, and ~A~ as shown in Fig. 1. According to the delocalized-bond description [3, 13, 14], we may form three molecular orbitals ej by linear combination of the three atomic orbitals ~AI, ~B2, and SA,;
- ~1 %
= O@A +, b@s~+ a(hA3
0
E3
©
E~:o
@
El
~6A,: qSA3:2p. [F] @B2:ls F.H] Fig. 2. Molecular orbitals and energy level of hydrogen bifluoride ion. lide ions, and rare-gas bifluoride (XeF2)[-3, 13, 15]. Of course, the n-bonding of carbon dioxide (CO2), carbon disulphide (CS2), etc. is in the category of the three-center bonding. Of these molecules, HF~ is most typical example representing the delocalization of electrons. An approximate diagram of the energy level of HF~ is shown in Fig. 2, wherein only the ¢1 and ~/'2 orbitals are occupied with two electrons, respectively and, however, the latter is entirely confined to the fluorine atoms and represents an essentially non-bonding orbital. In the case of other molecules, similar diagrams of the energy level will be expected. When the molecular orbitals (28) for these molecules are applied to equation (27) derived in the previous section, we immediately obtain
~1 = a¢4, + bCB~ + aCa3, 1
(
1
_ 2(a'b- ab')2 b 12
¢] = a'(aa, -- b'$.~ + a'Oa3,
E2 _ E3,
(30)
where qJ~ denotes the bonding, @2 the non-bonding and q,* the anti-bonding molecular orbitals. The conditions of orthonormalization are 2a 2 + b 2 = 1 , 2 a ' + b ' 2 = t , 2 a a ' - b b ' = 0 .
(29)
These molecular orbitals are in general found as creating a three-centre bond due to the delocalization of electrons over the entire molecule. As examples of the three-centre bonding, we will take such molecules as bifluoride ion (HF2), trihaA,
0
B2(Aa)
0
{i)IEI-E21~IE2-E31
A~
0 I.neo,)
(ii) IE,-E21:IEe-E31
(Bent)
A,
A~
Atomic ~A, (~2(~Aa) ~A~ orbital Fig. 1. Arrangement of atomic orbitals in linear and bent symmetric triatomic molecules.
(Jill IE,-Eal)) IEa-Esl Fig. 3. Energy level diagrams of three extreme cases of three-centre four-electron bonding.
Interaction in simple triatomic molecules where El, E 2 , and E a are the energies associated with the molecular orbitals, ~01, ~2, and q/~'. In order to obtain the value of equation (30), we must solve the secular determinant of third order for each molecule. However, as far as we are concerned with only the sign of equation (30), it will be unnecessary to solve the secular determinant. Along such a point of view let us consider three extreme cases of (i) tIE1 - E21 '~ [E2 - - E3 l, (ii) IE a - E 21 = I E2 - E3 l, and (iii) [E 1 - E 21 >> I E2 - Ea I, for which the energy level diagrams are schematically shown in Fig. 3. For the first case (i), to which the weak bonding molecules belong, the value of equation (30) is of course approximated to zero since the first term is nearly equal to the second one: 2(a'b - ab') 2
b '2
E t - E3
E2 - E3"
The second case (ii) leads to positive value of - b ' 2 / ( E 2 -- E3) since E2 - Ea < 0 and a'b - ab' = O. For a better understanding of this situation, suppose the molecular orbitals with the coefficients of a=a'= 1/2 and b = b ' = l/x/2. Although the last case (iii), to which the strong bonding molecules belong, is somewhat complicated, it may be expected to have relatively large positive value from the consideration of some popular molecular orbitals. It is found from the above discussion that for the case of delocalized-bond molecule such as H F 2 , equation (30) tends to take positive value. Therefore, we can conclude from equation (10) that the force constant of bond-bond interaction in the delocalizedbond molecules is also positive value. In Table l, we show the force constant of bond-bond interaction together with the other constants in some molecules [16-21]. In calculating the force constants, the following potential function V has been assumed; 2V = K {(Arl) 2 + (Ar2) 2} + 2k(Arx) (At2)
Table 1. Force constants of deloealized-bond molecules Molecule
Force constant, md/A k H
K
HF~ DF~ XeF2 CO2 CS2 O3 SO2
2.31 2.30 2.84 15.50 7.50 5.70 10.01
1.71 1.74 0.13 1.30 0.60 1.52 0.024
0.22 0.22 0.20 0.57 0.23 1.28 0.79
h 0* 0* 0* 0* 0* 0.33 0.19
* The force constant of bond-angle interaction was approximated to zero. Applying the method of linear combination of atomic orbital to H 2 0 , we would employ two of the oxygen p-orbitals, say the Px and the py atomic orbitalst, in combination with the hydrogen ls-orbitals (Fig. 4). Two a-bonding and two antibonding molecular orbitals are thus formed, which we might designate in un-normalized form; ~, = ~ba + ~b2, ¢,, = ~b3 + q~,,
~¢' =
4',
-
4~2,
¢,~
=
4~3 -
(32)
~,,
where ~ and ¢* denote the bonding and anti-bonding molecular orbitals. Practically speaking, these orbitals represent those of a completely localized molecule which will not actually exist. Therefore, when applying these molecular orbitals to equation (27), it is necessary for practical purposes that the bicentric localized orbitals are formally transformed to the polycentric delocalized orbitals in which the bicentric localizability remains. COOLSON[10] has discussed in detail this transformation which promises to yield,
~ o - ,,)
= a'q~x + b'~P2 - - b ' ( ~ 3 - - a'~p,,,
(33)
~#J-,,) = a'~ba - b'q~2 - b'~b3 + a'q~,l., ~#J+,,) = aq~a - b~b2 + b~b3 - a~b4. The orthonormal conditions are
+ r02H(Aq5)2 + 2 r o h ( A r l ) ( A q b ) + 2r0h(Ar2) (Aq~),
151
(31)
where K is the stretching, k the bond-bond interaction, H the bending, and h the bond-angle interaction force constants.
2a 2 + 2 b 2 = 1,2a '2 + 2 b ' 2 = 1 , 2 a a ' - 2 b b ' = O .
Applying these molecular orbitals (33) to equation (27), we obtain, dPr~
1(
632Eb
t In order to explain the directed valence, we must use hybrid orbitals in formulating the molecular orbitals~ However, the use of such hybrid orbitals should not give any change in the discussion below.
2(a'b - ab') 2 E1 - E3
Localized-bond molecules
In terms of molecular orbital theory, a localizedbond leads us directly to localized molecular orbitals. In other words, even in very complicated molecules we may still, to a good approximation, consider the bonding molecular orbitals to be bicentric and similar to those existing in diatomic molecules. If we take H 2 0 molecule as example, we would picture two separate O H bonding molecular orbitals relatively independent of each other.
(34)
+
2(ab' - a'b) 2 E2 - E,,
4,,
(35)
¢, qS : q54 : Is[HI : 2px[O] q53 : 2py[O]
Fig. 4. Arrangement of atomic orbitals in water molecule.
152
KEN OHWADA
Table 2. Force constants of locafized-bond molecules Molecule H20 D20 H2S D2S H2Se D2Se HgCI 2 HgBr2 KrF2
K 8.45 8.45 4.03 4.20 3.12 3.18 2.40 2.14 2.46
Force constant, md/A k H -0.11 -0.11 -0.11 -0.22 -0.12 -0.13 -0.26 -0.17 -0.20
0.82 0.82 0.48 0.48 0.34 0.32 0.04 0.05 0.21
h 0.18 0.18 0* 0* 0* 0* 0* 0* 0*
* The force constant of bond-angle interaction was approximated to zero.
where E1 and E2 denote the energies of bonding Sa + tt~ and ~/(I -- If) orbitals, and Ea and E4 the energies of anti-bonding ~bff_ .) and ~'a + ll) orbitals. Equation (35) leads clearly to negative value which is expected to be not SO large. Because in the right hand side of equation (35), the denominators of the energy difference of both the terms are negative (El - E3 < 0, E 2 - E4 < 0) and both the numerators associated with the molecular orbital coefficients are positive (2(a'b - ab') 2 > 0, 2(ab' - a'b) 2 > 0). Of course, for a completely localized molecule, this leads to zero. It is therefore concluded from equation (10) that the force constant of bond-bond interaction in such a highly localized molecule as H 2 0 tends to take negative value. In fact, it has been found on the basis of a general potential function (31) for H 2 0 that the force constant of bond-bond interaction k is negative value as shown in Table 2, wherein the force constants of H 2 0 as well as some other molecules[18, 19] regarded as the localized-bond molecules are given. In Table 2, a striking molecule is krypton difluoride, KrF2. It has been pointed out that the bonding in KrF2 can be described by the delocalized molecular orbitals as creating a three-centre four-electron bond [3,1. These molecular orbitals are quite the ~ m e as those of xenon-difluoride, XeF2 [15,1 as listed in Table 1. According to the present theory, these three-centre molecular orbitals should yield positive value of the force constant of bond-bond interaction. However, CLAASSENet al. [21] have shown from the vibrational study that the interaction force constant in KrF2 is of negative value ( - 0 . 2 md//~) on the contrary to that in XeF2. This difference between KrF2 and XeF2 has been discussed in detail by COULSON[22-1, who has suggested that the negative sign of KrF2 may be due to the greater weight of a no-bond structure in a valence-bond description. In order to explain this difference by the molecular
orbital theory, it is convenient to start from a point of view, based on the localized molecular orbitals rather than the three-center delocalized oneg For obtaining the localized molecular orbitals for KrF2, let us assume the sp diagonal hybrid orbital of krypton atom as follows; 1 Ol = - - ( S + p x
),
1 O2=--(S-px
).
Such a hybridization can be achieved by an excitation, 4s24p6--* 4s24pS5s, which requires only a little less energy than the 4s24p6-*4s24pS4d. If we use these hybrid orbitals, the same description of the localized-bond as in the case of H 2 0 may be made. It is therefore inferred from such a description that the force constant of b o n d - b o n d interaction in KrF2 should take negative value. REFERENCES [1] R. M. BADGER,J. Chem. Phys. 2, 128 (19341 [2] R. M. BADGER,J. Chem. Phys. 3, 710 (1935). [3] G. C. P1MENTALand R. D. SPRA~II.EY,Chemical Bonding Clarified Through Quantum Mechanics. HoldenDay, New York (1969). [4] C. A. COULSONand H. C. LONGUET-HIGGINS,Proc. Roy. Soc. A193, 456 (19481 [5] H. C. LONGUET-HIC~INSand L. SALEM, Proc. Roy. Soc. A251, 172 (1959). [6] L. SALEM, The Molecular Orbital Theory of Conjugated Systems, 3rd edn. W. A. Benjamin, Reading, Massachusetts (19721 [7] T. SHIMANOUCm and I. NAKAGAWA, Spectrochim. Acta lg, 89 (1962). [8] I. NAKAGAWA and T. SHIMANOUCHL Spectrochim. Acta 22, 759 (1966). [9] I. NAKAG^WA,Coord. Chem. Rev. 4, 423 (1969). [10] C. A. COULSON, Valence, 2nd edn. Clarendon Press, Oxford (1961). [11] J. N. MURRELL,S. F. A. KETTLE and J. M. TEDDER, Valence Theory, John Wiley, New York (1965l [12] A. STREITWmSER,JR., Molecular Orbital Theory for Organic Chemists. John Wiley, New York (1961). [13] G. C. PIMENTAL,J. Chem. Phys. 19, 446 (19511 [14] F. A. COTTON,Chemical Applications of Group Theory, 2nd edn. Wiley-Interscience, New York (1964). [15] C. A. COULSON,J. Chem. Soc. 1442 (19641 [16] R. NEWMANand R. M. BADGER,J. Chem. Phys. 19, 1207 (1951). [17] P. DAWSON,M. M. HARGREAVEand G. R. WILKINSON, Spectrochira. Acta 31A, 1055 (1975). [18] G. HERZBERG,Infrared and Raman Spectra of Polyatomic Molecules. Van Nostrand, Princeton, New Jersey (19451 [19] S. MIZUSHIMAand T. SHIMANOUCl-n,Infrared Absorption and Raman Effect. Kyoritu, Tokyo (1958). [20] H. SmBERT, Anwendunoen der Schwinoun#sspekroskopie in der Anorganischen Chemie. Springer-Verlag, Berlin (1966l 1-21] H. H. CLAASSEr~,G. L. GOODMAr~,J. G. MALM and F. SCnREINER,J. Chem. Phys. 42, 1229 (1965). [22] C. A. COULSON,J. Chent Phys. 44, 468 (1966).