Oct. 20, 1956
ENERGY AND INTERNUCLEAR DISTANCE RELATION BETWEEN POTENTIAL
5171
for Cr(C0)a. An error of more than f1 kcal./mole is unlikely in our results, but a considerably larger error could have occurred in measurements made using a Beckman type thermometer. It may be TABLE V recalled, however, that Sharafov and Rezukhina OF PRESENT VALUES WITH THOSE OF SHARAFOV reported the formation of an appreciable amount of COMPARISON AND REZUKHINA~ a peculiar dark chromium oxide which they preAH& sumably regarded as thermodynamically identical Sharafov and Rezukhina This with ordinary Cr203. Compound As reported Corrected" research Acknowledgment.-Thanks are due to the -257.6 -257.08 -249.4 Cr(COh Mallinckrodt Chemical Company for their con-235.6 -234.8 -233.12 Mo(COh tribution to some of the research expenses (A.K.F.). -225.3 -227.3 W(C0)B -219.29 Corrected as described in text by using more recent We are grateful to Dr. E. 0. Brimm of Linde Air heats of formation of metal oxides. Products Co., Tonawanda, N. Y., for samples of indicating a lack of systematic error in either set. the carbonyls, and to Professor G. B. Kistiakowsky The results for Mo(C0)e and W(C0)a agree surpris- for the use of the Mueller bridge. ingly well whereas a considerable discrepancy exists CAMBRIDGE, MASS.
their reported values as described should be comparable to the ones determined in this research. It is seen in Table V that the differences are random,
5
[CONTRIBUTION FROM
THE
DEPARTMENT O F CHEMISTRY, UNIVERSITY O F MARYLAND]
General Relation between Potential Energy and Internuclear Distance. atomic Molecules1
11. Poly-
BY ELLISR. LIPPINCOTT AND RUDOLPH SCHROEDEK RECEIVED APRIL9, 1956 The simple form of a general relation between potential energy and internuclear distance derived from a quantum mechanical model is applied to the bonds of a large number of polyatomic molecules. The relation has the form V = D,[1 exp(-nAR2/2R)], where the parameter n is related to De by the equation De = k,R,/n. Using known values of bond stretching force constants as determined from molecular force models with known bond lengths, dissociation energies of bonds in polyatomic molecules have been calculated more accurately than was hitherto possible. One important feature of the method is that no empirically evaluated parameters are used. The relationship between bond dissociation energy and average bond energy as related to the derived potential function is discussed. For non-polar molecules or for polar molecules containing hydrogen or molecules with atoms of atomic number less than nine the calculated dissociation energies agree with the thermochemical bond energies with about 5% accuracy. A number of other applications of this function will be suggested. With due consideration to its limitations this function should be useful as a tool forelucidatingother problems of bond formation and structure.
Introduction Many attempts have been made to derive from quantum theory or to formulate empirically analytic relationships between potential energy and internuclear distance for diatomic molecules. Except for the simplest systems such as the Hz+ and Hz molecules quantum theory has not given definite relationships, while empirical internuclear potential functions have not correlated satisfactorily such bond properties as dissociation energy, bond lengths, bond stretching force constants and anharmonicity constants. I n no case has i t been possible to apply diatomic internuclear potential functions to the bonds of polyatomic molecules and obtain even a qualitative correlation of bond properties. (1) Presented in part at the 125th Meeting of the American Chemical Society, Kansas City, March, 1954. ( 2 ) P. M. Morse, P h y s . Rev., 84, 57 (1929); Coolidge, James and Vernon, ibid., 64, 726 (1938); H. M. Hulhurt and J. 0. Hirschfelder, J. Chem. P h y s . . 9, 61 (1941); M . L. Huggins, ibid., 8, 473 (1935); 4, 308 (1936); R . Rydberg, Z . Physik, 7 8 , 376 (1932); M. F. Manning and N. Rosen, P h y s . Rev., 44, 953 (1933); G. Poschl and E. Teller, Z . Physik, 88, 143 (1933); E. A. Hylleraas. ibid., 96, 661 (1935): J . W. Linnett, Trans. Faraday Soc., 86, 1123 (1940); 38, 1 (1942); G. B. B. M . Sutherland, J . Chem. Phys., 8, 161 (1940); Proc. Indian Acad. Sci., 8, 341 (1938): A. A. Frost and B. Musulin, THIS JOURNAL, 1 6 , 2045 (1954); J . Chem. P h y s . , P I , 1017 (1954). (3) (a) E. Teller, 2. Physik, 61, 458 (1930); (h) H. M. James and A. S . Coolidge, J . C h e m . Phys., 1 , 823 (1933).
We have recently derived an internuclear potential function from a quantum mechanical model which has found extensive application in quantitatively predicting and correlating the bond prbperties of a large number of diatomic molecule^.^ It will be shown that this internuclear potential function can be used to predict and correlate the bond properties of a number of bonds in polyatomic molecules. One important feature of the method is that no empirically evaluated constants are used, since all necessary parameters have been evaluated from a quantum mechanical modeL4 The form of the function which has been derived previously for diatomic molecules is V = D,[1
- exp(--nL\R2/2R)1
(1)
where Debond dissociation energy referred to the bottom of the potential curve, A R = R - Re, and Re = equilibrium bond length. The derivation of (1) shows that n may be obtained from the relation n = no(l/lo)~'/z(l/l,)~I/~ cm.-'
(2)
where ( I / I ~ )and A ( I / I o ) Bare the ionization potentials of atoms A and B in the bond A-B, relative to those of the corresponding atoms in the same (4) E. R. Lippincott and R . Schroeder, J . Chem. P h y s . , 28, 1131 (1955); E. R . Lippincott, ibid., $3, 603 (1955); E. R . Lippincott, to be published.
5172
ELLISR. LIPPINCOTT AND RUDOLPH SCHROEDER
VOl. 78
The energy for this process should be computed for the state in which both the reactant M and the products R1 and Rz are in the gas phase a t zero pressure and 0°K. This dissociation energy is thus unambiguously defined by the description of the initial electronic state of the molecule and the final specified electronic states of the fragments. However, in using equation 6 to predict the bond dissociation energies of bonds in polyatomic molecules one has the problem of knowing what type of bond stretching force constant to use in the calIZ,,(IL) = 5.34 x 108 C I I L - ~ culations. For diatomic molecules this force connQ(Hrl) = 5.G7 X 1 0 8 cm.-' stant is unambiguously determined by the vibran o ( . l a ) = 6.03 x 1us c111.--1 (3) tional frequency and atomic weights. However, By imposing the conditions for stability that for polyatomic molecules this force constant de(a V-/a& = 0 (,$) pends on the vibrational frequency, atomic weights alld and the type of force field assumed in computing the force constant. In order to test the application ( CPJ'/dR2) e,i = k, (5) the following relation can be derived giving the of equations 1, 2 and 6 for use with polyatornic relation of n to bond dissociation energy, equilib- molecules we will make separate calculations using rium bond length, and bond stretching force con- bond stretching force constants determined from the following three different types of force fields6: stant, k,. (A) a diatomic type force field in which the bond 11, = k,R,/n (6) stretching force constant is obtained from the bond The use of equations I, 2 and 6 has been demon- vibrational frequency considering the molecule to strated previously by correlating and predicting a be diatomic-like. This is obviously a poor aplarge amount of spectroscopic data for diatomic proximation and can only be expected to give rea~nolecules.~In particular, it was shown that the sonable results for bonds of molecules which consist function (1) was much more accurate and easier to of two heavy atoms or of molecules having a light use than the well known Morse potentid function,2 atom attached to a heavier atom. Fortunately and that with it calculations of bond dissociation these two conditions are usually fulfilled for energies and bond anharmonicity constants of di- most molecules where the bond dissociation is atomic molecules are readily made with greater ac- known reasonably well; (B) a simple valence force curacy than was hitherto possible. Relation (2) model (SVF). This model assumes that the powas originally obtained empiri~ally.~The theo- tential energy of vibration of a molecule can be obretical values of no(HA) (5.67 X lo8 cm.-I) and tained by considering only the stretching of bonds n(&) (6.03 X lo8 cm.-') agree remarkably well and the bending of bond angles without any bondwith the values found previously by empirical angle, bond-bond, angle-angle, or other intermethods (5.92 X lo8 cm.-' and 6.32 X lo8 cm.-' action terms. Such force constants are known for re~pectively).~In order to keep our methods for most of the simple polyatomic molecules. (C) A polyatomic molecules independent of any em- modified valence force model (MVF). ;Z number pirically evaluated parameters we will use the of modified valence force models have been used theoretical values of no given in equation 3 in our to describe the vibrational motion of a number of calculations. molecules depenaing on the type of interaction I n the following discussion of the application of terms assumed in the vibrational potential functhe potential function for the prediction and cor- tion. However, there is no general agreement relation of bond properties, we shall group the among molecular spectroscopists as to the best bonds of polyatomic molecules into the following type of modified valence force field. classifications : (a) single bonds, (b) isolated mulThe force constants used in equation 0 should tiple bonds, (c) adjacent multiple bonds, and (d) be ones determined from zero order vibrational bonds for which only the average bond energy is frequencies (corrected for anharmonicity) . There known. is only a limited number of polyatomic molecules where such force constants have been calculated. Application to Polyatomic Molecules The application of the potential function (1) to This anharmonicity correction is about 8% for polyatomic molecules requires consideration of a bonds involving hydrogen and about 3% for bonds number of factors. The process of dissociation of which do not contain hydrogen. If force cona bond in a polyatomic molecule is analogous to stants determined from zero-order frequencies are the dissociation of a diatomic molecule in the sense not available we will approximate a correction in that dissociation leads to two fragments, either our calculations by increasing the force constant radicals, molecules or atoms. The dissociation by 8 or 3%, for A-H or A-B bonds, respectively. Since the quantum mechanical derivation of the energy of a bond A-B in the molecule M may be defined as the endothermicity of the reaction in internuclear potential function (1) implies that which 11 is decomposed into two fragments R1 equation 6 will give dissociation energies i t is irna discussion of a number of molecular force field models, see and Rz fornietl by breaking bond A-B only.6 E.( R6 ). I'or Wilson, J . D . Decius and P. C . Cross, "Molecular Vibrations." row and first column of the periodic table.4 We will assume here that the potential function (1) and relation (2) are valid for bonds in polyatomic molecules, and that the value of no calculated for diatomic molecules can be used with the bonds of polyatomic molecules. The theoretical value of no for the Hz molecule is 5.34 X los ern.-' while for bonds involving multi-electronic atoms no has a value of 6.03 X lo8 cm.-'. If a geometric mean rule is assumed, the no value for bonds of type H-A is 5.67 X lo8 cm.-l, thus
(a) h1. Szwarc aud RI.
C;. ICvans, J . Cheris. P h y s . , 18, GI8 (1050).
hlcCraw--Hill Book Crr , h'ew Yurk, N. Y.. 1!lLj5, Chapter 8.
RELATION BETWEEN POTENTIAL ENERGY AND INTERNUCLEAR DISTANCE
Oct. 20, 1956
3173
TABLE I PREDICTED BONDDISSOCIATION ENERGIES OF SINGLE BONDS IN POLYATOMIC MOLECULES Molecule
HsC-H
II6CZ-H
HhC8-H
n (10
-I,
cm.-
8.22 8.22 8.22
ClaC-H
8.22
HzN-H
9.30
HO-H
HS-H
Hac-CHs
H~C-NHZ H3C-OH HSC-Br
H3C-I
H2N-NH2 02N-XOZ
9.07 8.05 12.GO 14.34 13.99 14.43 13.82 16.25 16.28
ON-c1 ON-Br HO-OH
15.72 16.32 15.32
HS-SH
12.18
k( 10-9, d y n e ~ / c m . ~
Rptd.
Type
Cor.
D, kcal./mole Calcd. Exptl.
4.72 5.04
Error, kcal./ mole
Di-a 5.10 93 101 =k 1 8 SVF 5.44 100 1 MVF 5.394* 99 2 4.82 Di-a 5.21 96 98 2 4.79 SVF 5.17 96 2 MVF 5.35* 99 1 5.49 Di-a 5.93 108 102 6 5.05 SVF 5.45 99 3 .5.065 MVF 5.47 99 3 5.37 Di-a 105 89 st 2 16 5.80 4.80 MVF 5.18 92 3 6.22 Di-a 6.72 101 104 =k 2 3 6.42 SVF 6.93 104 0 MVF 7.171* 108 4 7.47 Di-a 8.07 117 118 rt 0.7 1 7.76 SVF 8.38 122 4 MVF .i 8.428* 123 3.93 Di-a 4.24 97 90 rt 5 7 3.96 SVF 8 4.28 98 MVF 2 4.01* 92 4.36 Di-a 4.49 78 80 & 6 2 5.62 SVF 20 5.79 100 MVF 4.57* 79 1 4.88 Di-a 5.03 73 w80 7 4.86 Di-a 5.01 75 -90 15 2.78 Di-a 54 67 2.86 13 12 2.82 SVF 2.90 55 2.863 MVF 11 2.949 56 2.25 Di-a 2.32 51 53 2 2.23 SVF 3 2.30 50 2.25 MVF 2.32 51 2 3.62 Di-a 3.73 47 60-64 13 1.13 Di-a 1.16 16 13 3 1.47 SVF 1.51 21 8 1.36 MVF 1.40 20 7 1 2.0 SVF 2.06 36 37 1.5 SVF 1.55 29 28 1 1 3.85 Di-a 3.97 54 53 3.84 MVF 3.96 54 1 2.52 Di-a 2.60 62 72-78 10 Av. error (kcal./mole); MVF = 3.5;SVF = 5.3;Di-a =
Error,
%
8.0 1.0 2.0 2.0 2.0 1.0 5.9 2.9 2.9 18.0 3.4 2.9 0.0 3.9 0.8 3.4 4.2 7.8 8.9 2.2 2.5 26.0 1.3 8.8 16.7 19.4 17.9 16.4 3.8 5.7 3.8 21.7 23.1 61.5 53.8 2.7 3.6 1.9 1.9 13.9 8.0
E, kcal./mole
Ref. for D k
98.2 c
d
98.2
E
I d
95.2 h I
i k
92.2
’ rn
109.4
“
e
88
P
a T
80
9
I d
66 79 66.5
P 1
” w z
U w
37
1
