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Relation between bond force constants and integrals of orbital overlap
Spectrochimica Acta, 1996, Vol. 22, lap. 201 to 204. Pergamon Press Ltd. Printed in Northern Ireland
Relation between bond force constants and integrals o! orbital overlap MARISA SCROCCO Institute of General Chemistry, University of Rome (Received 17 May 1965)
Abstract--Starting from the definition of a stretching force constant, K = ~,~--~IR. [d~E] the following relation between K and the overlap integrals was obtained: K = S(1 + S)~ \ a R / A R , This relationship has been tested on some series of simple tetrahedral molecules and of different symmetry. I~ X series of previous notes [1] a t t e m p t s were made to correlate the values of experimental stretching force constants with SI~T~.R'S effective nuclear change (Zee.) of bond orbitals of atoms interested in the bond. I t has been possible to observe some linear relationships between the force constant K and the distance of the radial density maxima of atoms bond orbitals. I f we consider also the principal quantum numbers of bond orbitals, it is possible to include all types of simple bonds in a general linear relationship such as:
(where /~, = ge~. (x)/n,)
in which the value of the slope m differs according to the type of the overlapping orbitals: s - s ; s-p; p - p ; s p L s ; sp~-p; spa-s; spa-~p; etc. I n this work a correlation was sought between the bond force constants and the overlap integrals calculated b y MIrLTXK~'S tables [2]. The force constant K which binds two atoms in the molecule is defined b y : IdlE\
where E is the stretching bond energy as a function of interatomic distance R and R, is the equilibrium distance. I t is possible to develop this relation giving to E the value:
fl E ---- ~ + 1 +---~
(2)
where the first term (~) contains the Coulomb energy, and the second term the exchange energy. (fl is the resonance integral [3] and S is the overlap integral.) [1] M. SCROCCO,R/c. sci. 84 (IIA), 573, 579, 587, 595 (1964). [2] R. S. Y ~ . T . ~ N , C. A. Rr~C'H~, D. OaLO~T and H. 0 ~ o ~ , J. Chem. Phys. 17, 1248 (1949). [3] R. S. Ml~.zx~N, J. Chim. Phys. 46, 497 (1949). 201
202
M. Sc~occo
Since the difference between the bonding and antibonding MO energies, in the LCAO approximation is given actually by the term +fl/1 q- S it is clear that in E the term q-fl/1 q- S is substantially responsible for the energies of the bond. The Coulomb part of energy contained in the term ~, changes less with the distance than the part represented by the second term. For this reason we consider, as a first approximation, ~ as independent of/~ and, following the ~IULLIKEN'S approximation [4]: - - 2 fl = A S I
(3)
(where A is a constant and I the ionization potential) we consider the resonance integral ~ proportional to the overlap integral S. Introducing (3) into (4) we have:
E= 0E OR
and, since: we have:
K =
ASI
2(1+S)
(4)
0E 0S OS OR
[(1 A1 1 q- S) a \ 3 ! JR=R,
(5)
I n this relationship we have two values which are difficult to evaluate, namely: the ionization potential I and the constant A. However, according to Mrr,TXKm¢, we can introduce the dissociation energy as: D~ = [ASI/(1 -4- S)]R=R.
(6)
The relationship (5) is then reduced to: K =
[S(1 + S) ~ laS] l =R, \aR/JR
(7)
at the point R,. In practice, however, the dissociation energy of polyatomic molecules is not always directly measurable and it is necessary to use some approximations. In this work, in order to calculate the values of Do t h a t are experimentally unknown, we have used the approximation proposed b y L~PI~COTT et al. [5]. These authors showed, in a series of papers, the good agreement between experimental and calculated D (error 3-5%). The force constants used are corrected for anharmonicity. This anharmonieity correction is (from L~PI~COTT [5]) about 8% for bonds involving hydrogen and 3 % for bonds which do not contain hydrogen. The relationship (7), applied to the series of tetrahedric molecules of Table 1, gives a good linear relationship as we can see from Fig. 1. The value of the overlap integral S has been evaluated using the normalized tetrahedric hybrid: s = 1/2(s +
and the OS/OR has been calculated between an interval AR = 0.4 a.u. [4] R. S. M~r~a-KE~r,J. Phys. Chem. 56, 295 (1952). [5] E. R. Ln,Pr~rco~T,J. Ghem. Phys. ~ , 603 (1955); ibid. 241, 1678 (1957).
Relation between bond force constants and integrals of orbital overlap
203
Table 1. Hydrogenated molecules Molecule
(~)
(10s dyn/cm)
(e.v.)
a'R
BHa-OH4 A1H4Sill4 CH3CI SiHsC1 Call6 C~H4 NH s PH a H20 HaS O==CH~
1.25 1.09 1.62 1.479 1.11 1.483 1.105 1"07 1.008 1.42 0'958 1"34 1.06
2.99 5.44 2.05 3.07 5.74 3.09 5-21 5"48 6"93 3.34 8"38 4"28 4-86
3.32* 4.37 3.46* 3.51 4.84 3"94* 4"24 4-37 4-50 3"31 5.12 3"9 3-91"
3"46 x lO-~ 12.18 2.41 5-56 14.2 6.07 12.27 12"27 19-94 6"35 24.4 9.71 10-8
Re
0: tetr. tetr. tetr. tetr. tetr. tetr. tetr. 120° 107-4° 93° 104-45° 93"4° 125.8°
(1) The D, are calculated from the experimental Do * These values are calculated by the Ln'Pr~COTT'Smethod [6] 10
'
'
'
I
~
I
. . . .
l
,
[
. . . .
1 . . . .
K
~
0
I
5
,
,
r
~
10
~
,
,
]
,
1
15
I
,
I
r
i
20 "
I
25
f s,fl
Fig. 1. Hydrogenated molecules: O - - D , experimental values: O - - D , calculated values. The hypotheses supporting these relations are: (a) t h a t t h e m o l e c u l e is p e r f e c t l y t e t r a h e d r a l (b) t h a t t h e b o n d o r b i t a l s o f t h e a t o m s i n t e r e s t e d t o e a c h b o n d a r e p e r f e c t l y a l i g n e d . W e h a v e b e e n s t u d y i n g t h e p o s s i b i l i t y o f e m p l o y i n g t h e l i n e a r r e l a t i o n (7), f o u n d f o r t e t r a h e d r a l m o l e c u l e s , also f o r m o l e c u l e s o f d i f f e r e n t s y m m e t r y b y a c a l c u l a t i o n of the a p p r o p r i a t e overlap i n t e g r a l on the basis of h y b r i d m i x i n g k n o w n from t h e e x p e r i m e n t a l a n g l e s o f e q u i l i b r i u m a n d u s i n g t h e c o n d i t i o n (b) o f p e r f e c t a l i g n m e n t . I n this k i n d of p r o b l e m , the molecules w i t h u n e x c h a n g e d lone pairs are of p a r t i c u l a r interest because they can easily change the mixing of their bond orbitals with the l o n e p a i r . W e will r e t u r n t o t h i s p r o b l e m i n s t u d y i n g t h e a b s o l u t e i n f r a r e d i n t e n s i t y of b a n d s of some simple molecules in solution.
[6] E. R. LIPPINCOTTa n d R. SCHROEDER,J . chem. Phys. 23, 1131 (1955); J . A m . Chem. Soc. 78, 5171 (1956). [7] C. E. I~EI~IS]K and J. W. LI~CNEa'r, Trans. Farad. Soc. 50, 657 (1954); ibid. 47, 1033 (1951). [8] C. A. CouLsoN, Valence, Oxford Claredon Press (1952).
204
M. S c R o c c o T a b l e 2. T e t r a h e d r a l c h l o r u r a t e d molecules
~6 Molecule
(/~)
CC14 HCCI a CtIaC1 A1Cla-SiC14 HSiC1 a PCI4 +
1.766 1.767 1.78 2.13 2.01 2.02 1-98
K~
D~
(10 s d y n / c m )
D~
(e.v.)
4.51 3-48 3.56 2-61 3.86 3.1 4-37
\aR/ JR, 13.4 × 10-3 12-2 15.0 3.75 8.30 6.74 16.87
3.42 2.76 ( *~ 3.42 3.46 (*~ 3.98 3.17 ( *~ 4.03 ( *~
(*) C a l c u l a t e d f r o m LIPPINCOTT [6],
10
,
K
5
""-5"-
5
0
10
15
l
2O
0e f S'fl LsO+s)2kb R/J%
, [
Fig. 2. T e t r a h e d r a l c h l o r u r a t e d molecules: E]--D~ e x p e r i m e n t a l v a l u e s : O - - D e c a l c u l a t e d values.
In this work we consider the molecules: NH3, PHa, H~O, H2S, C~H4 and O~CH~. The mixing coefficient 2 of s-type and p-type orbitals of hybrid bond has been calculated from the experimental angle of equilibrium using the orthogonality condition: 1 +2 3cos~j=0 valid for equivalent hybrids. The related data are tabulated in the second part of Table 1 and the values are shown in the plot of Fig. 1. To prove the validity of these approximations in other different types of bonds, a series of tetrahedral molecules was studied in which there are bonds of X--CI type. These results are also satisfactory if we consider that for a better approximation it would be necessary to account for the polarity of the bonds. The related data are given in Table 2 and in the Fig. 2.
Acknowl,e~gemenb---We w i s h of t h i s w o r k .
t o t h a n k t h e Consiglio N a z i o n a l e delle R i c e r c h e for f i n a n c i a l s u p p o r t
Relation between bond force constants and integrals o! orbital overlap MARISA SCROCCO Institute of General Chemistry, University of Rome (Received 17 May 1965)
Abstract--Starting from the definition of a stretching force constant, K = ~,~--~IR. [d~E] the following relation between K and the overlap integrals was obtained: K = S(1 + S)~ \ a R / A R , This relationship has been tested on some series of simple tetrahedral molecules and of different symmetry. I~ X series of previous notes [1] a t t e m p t s were made to correlate the values of experimental stretching force constants with SI~T~.R'S effective nuclear change (Zee.) of bond orbitals of atoms interested in the bond. I t has been possible to observe some linear relationships between the force constant K and the distance of the radial density maxima of atoms bond orbitals. I f we consider also the principal quantum numbers of bond orbitals, it is possible to include all types of simple bonds in a general linear relationship such as:
(where /~, = ge~. (x)/n,)
in which the value of the slope m differs according to the type of the overlapping orbitals: s - s ; s-p; p - p ; s p L s ; sp~-p; spa-s; spa-~p; etc. I n this work a correlation was sought between the bond force constants and the overlap integrals calculated b y MIrLTXK~'S tables [2]. The force constant K which binds two atoms in the molecule is defined b y : IdlE\
where E is the stretching bond energy as a function of interatomic distance R and R, is the equilibrium distance. I t is possible to develop this relation giving to E the value:
fl E ---- ~ + 1 +---~
(2)
where the first term (~) contains the Coulomb energy, and the second term the exchange energy. (fl is the resonance integral [3] and S is the overlap integral.) [1] M. SCROCCO,R/c. sci. 84 (IIA), 573, 579, 587, 595 (1964). [2] R. S. Y ~ . T . ~ N , C. A. Rr~C'H~, D. OaLO~T and H. 0 ~ o ~ , J. Chem. Phys. 17, 1248 (1949). [3] R. S. Ml~.zx~N, J. Chim. Phys. 46, 497 (1949). 201
202
M. Sc~occo
Since the difference between the bonding and antibonding MO energies, in the LCAO approximation is given actually by the term +fl/1 q- S it is clear that in E the term q-fl/1 q- S is substantially responsible for the energies of the bond. The Coulomb part of energy contained in the term ~, changes less with the distance than the part represented by the second term. For this reason we consider, as a first approximation, ~ as independent of/~ and, following the ~IULLIKEN'S approximation [4]: - - 2 fl = A S I
(3)
(where A is a constant and I the ionization potential) we consider the resonance integral ~ proportional to the overlap integral S. Introducing (3) into (4) we have:
E= 0E OR
and, since: we have:
K =
ASI
2(1+S)
(4)
0E 0S OS OR
[(1 A1 1 q- S) a \ 3 ! JR=R,
(5)
I n this relationship we have two values which are difficult to evaluate, namely: the ionization potential I and the constant A. However, according to Mrr,TXKm¢, we can introduce the dissociation energy as: D~ = [ASI/(1 -4- S)]R=R.
(6)
The relationship (5) is then reduced to: K =
[S(1 + S) ~ laS] l =R, \aR/JR
(7)
at the point R,. In practice, however, the dissociation energy of polyatomic molecules is not always directly measurable and it is necessary to use some approximations. In this work, in order to calculate the values of Do t h a t are experimentally unknown, we have used the approximation proposed b y L~PI~COTT et al. [5]. These authors showed, in a series of papers, the good agreement between experimental and calculated D (error 3-5%). The force constants used are corrected for anharmonicity. This anharmonieity correction is (from L~PI~COTT [5]) about 8% for bonds involving hydrogen and 3 % for bonds which do not contain hydrogen. The relationship (7), applied to the series of tetrahedric molecules of Table 1, gives a good linear relationship as we can see from Fig. 1. The value of the overlap integral S has been evaluated using the normalized tetrahedric hybrid: s = 1/2(s +
and the OS/OR has been calculated between an interval AR = 0.4 a.u. [4] R. S. M~r~a-KE~r,J. Phys. Chem. 56, 295 (1952). [5] E. R. Ln,Pr~rco~T,J. Ghem. Phys. ~ , 603 (1955); ibid. 241, 1678 (1957).
Relation between bond force constants and integrals of orbital overlap
203
Table 1. Hydrogenated molecules Molecule
(~)
(10s dyn/cm)
(e.v.)
a'R
BHa-OH4 A1H4Sill4 CH3CI SiHsC1 Call6 C~H4 NH s PH a H20 HaS O==CH~
1.25 1.09 1.62 1.479 1.11 1.483 1.105 1"07 1.008 1.42 0'958 1"34 1.06
2.99 5.44 2.05 3.07 5.74 3.09 5-21 5"48 6"93 3.34 8"38 4"28 4-86
3.32* 4.37 3.46* 3.51 4.84 3"94* 4"24 4-37 4-50 3"31 5.12 3"9 3-91"
3"46 x lO-~ 12.18 2.41 5-56 14.2 6.07 12.27 12"27 19-94 6"35 24.4 9.71 10-8
Re
0: tetr. tetr. tetr. tetr. tetr. tetr. tetr. 120° 107-4° 93° 104-45° 93"4° 125.8°
(1) The D, are calculated from the experimental Do * These values are calculated by the Ln'Pr~COTT'Smethod [6] 10
'
'
'
I
~
I
. . . .
l
,
[
. . . .
1 . . . .
K
~
0
I
5
,
,
r
~
10
~
,
,
]
,
1
15
I
,
I
r
i
20 "
I
25
f s,fl
Fig. 1. Hydrogenated molecules: O - - D , experimental values: O - - D , calculated values. The hypotheses supporting these relations are: (a) t h a t t h e m o l e c u l e is p e r f e c t l y t e t r a h e d r a l (b) t h a t t h e b o n d o r b i t a l s o f t h e a t o m s i n t e r e s t e d t o e a c h b o n d a r e p e r f e c t l y a l i g n e d . W e h a v e b e e n s t u d y i n g t h e p o s s i b i l i t y o f e m p l o y i n g t h e l i n e a r r e l a t i o n (7), f o u n d f o r t e t r a h e d r a l m o l e c u l e s , also f o r m o l e c u l e s o f d i f f e r e n t s y m m e t r y b y a c a l c u l a t i o n of the a p p r o p r i a t e overlap i n t e g r a l on the basis of h y b r i d m i x i n g k n o w n from t h e e x p e r i m e n t a l a n g l e s o f e q u i l i b r i u m a n d u s i n g t h e c o n d i t i o n (b) o f p e r f e c t a l i g n m e n t . I n this k i n d of p r o b l e m , the molecules w i t h u n e x c h a n g e d lone pairs are of p a r t i c u l a r interest because they can easily change the mixing of their bond orbitals with the l o n e p a i r . W e will r e t u r n t o t h i s p r o b l e m i n s t u d y i n g t h e a b s o l u t e i n f r a r e d i n t e n s i t y of b a n d s of some simple molecules in solution.
[6] E. R. LIPPINCOTTa n d R. SCHROEDER,J . chem. Phys. 23, 1131 (1955); J . A m . Chem. Soc. 78, 5171 (1956). [7] C. E. I~EI~IS]K and J. W. LI~CNEa'r, Trans. Farad. Soc. 50, 657 (1954); ibid. 47, 1033 (1951). [8] C. A. CouLsoN, Valence, Oxford Claredon Press (1952).
204
M. S c R o c c o T a b l e 2. T e t r a h e d r a l c h l o r u r a t e d molecules
~6 Molecule
(/~)
CC14 HCCI a CtIaC1 A1Cla-SiC14 HSiC1 a PCI4 +
1.766 1.767 1.78 2.13 2.01 2.02 1-98
K~
D~
(10 s d y n / c m )
D~
(e.v.)
4.51 3-48 3.56 2-61 3.86 3.1 4-37
\aR/ JR, 13.4 × 10-3 12-2 15.0 3.75 8.30 6.74 16.87
3.42 2.76 ( *~ 3.42 3.46 (*~ 3.98 3.17 ( *~ 4.03 ( *~
(*) C a l c u l a t e d f r o m LIPPINCOTT [6],
10
,
K
5
""-5"-
5
0
10
15
l
2O
0e f S'fl LsO+s)2kb R/J%
, [
Fig. 2. T e t r a h e d r a l c h l o r u r a t e d molecules: E]--D~ e x p e r i m e n t a l v a l u e s : O - - D e c a l c u l a t e d values.
In this work we consider the molecules: NH3, PHa, H~O, H2S, C~H4 and O~CH~. The mixing coefficient 2 of s-type and p-type orbitals of hybrid bond has been calculated from the experimental angle of equilibrium using the orthogonality condition: 1 +2 3cos~j=0 valid for equivalent hybrids. The related data are tabulated in the second part of Table 1 and the values are shown in the plot of Fig. 1. To prove the validity of these approximations in other different types of bonds, a series of tetrahedral molecules was studied in which there are bonds of X--CI type. These results are also satisfactory if we consider that for a better approximation it would be necessary to account for the polarity of the bonds. The related data are given in Table 2 and in the Fig. 2.
Acknowl,e~gemenb---We w i s h of t h i s w o r k .
t o t h a n k t h e Consiglio N a z i o n a l e delle R i c e r c h e for f i n a n c i a l s u p p o r t