2298
JOHN F. OLSENAND LOUISBURNELLE SO
-X M
As indicated in Figure 5 there is a strong attachment of about one HBr per amide group which does not easily desorb. This result, similar to that found for HC1 on Nylon 6,6 by Reyersori and P e t e r ~ o ncould , ~ be a result of the formation of ionic groups HO
I
CH2-
\
/
/
\
C=N+
0
lOOG
-CH2
0
0.2
0.4
0.6
0.8
I. 0
Figure 6. Water sorption at room temperature: sample B, undrawn yarn; sample H-B, undrawn yarn after HBr sorption; mmple D, hot-drawn 5 : l yarn; sample H-D, hot-drawn yarn after HBr sorption.
Br-
H
on the polymer chain. This in turn should greatly modify the sorption properties of the polymer. To demonstrate this the sorption of water was carried out on the HBr modified polymer as shown in Figure 6. The modification of the polymer greatly enhances the water sorption. The HBr modified sample B contained 0.78 HBr/amide and sample D contained 1.01 HBr/ amide so the increase in water sorption capacity increases with the amount of HBr in the polymer. If we consider the increase in water sorption at PIPo = 0.25, we find that about 0.8 HaO is associated with each ionic grouping introduced into the polymer.
The Electronic Structure of Carbon Suboxidel by John F. Olsen and Louis Burnelle Department of Chemistry, New York University, NEWYork, New York
lOOOS
(Received November 18, 1968)
The CNDOIP and the extended Huckel molecular orbital (XHMO) methods have been applied to linear and bent configurations of C302. The CNDOIP method was found to give a satisfactory description of the electronic charge distribution in the molecule. It indicates the presence of a large negative charge on the central carbon atom, with a corresponding reduction in the CC ?r-bond population. These features provide an explanation for the low frequency of the bending vibration about the central carbon atom. The expressions obtained for the highest occupied and lowest unoccupied molecular orbitals are found to explain the dissociations which C302 undergoes in the mass spectrum and in photochemical reactions. The variation of the XHMO orbital energies with the apex angle also provides some insight into the low resistance to bending in this molecule, and the resulting orbital energy diagram can be utilized to make predictions concerning the structure of the ions and of some low-lying excited states of the molecule. The CND0/2 orbital energies appear not to lend themselves t o such a discussion.
Introduction Carbon suboxide ( G O z ) is one of the lesser known oxides of carbon, and what we do know about it has only come about in recent years. In the past few years, it has been the object of numerous infrared and Raman,2-4 photo~hemical,5-~ electron diffraction,* and C3 which has ultraviolet absorption s t u d i e ~ . ~ - ~Like l been observed in the spectra of comets,12C302presents some interest for the astrophysicist. It has been menThe Journal of Physical Chemistry
tioned as a possible constituent in the atmosphere of Venus. l 3 (1) Work supported by the U.S. Army Research Office (Durham). (2) F. A. Miller and W. G. Fateley, Spectrochim. Acta, 20, 253
(1964). (3) F.A. Miller, D. H. Lemmon, and R. E. Witkowski, ibid., 21, 1709 (1965). (4) W.H.Smith and G. E. Leroi, J . Chem. Phys., 45, 1767 (1966). (6) K. D. Bayes, J . Amer. Chem. Soc., 85, 1730 (1963),and references cited therein.
THEELECTRONIC STRUCTURE OF CARBON SUBOXIDE
2299
Carbon suboxide possesses the C3 chain, :C=C=C :, and for such a multiple-bonded structure one would expect a significant barrier to bending about the central carbon atom. On the contrary, the bending about this central atom is found to be remarkably easy. The bending frequency for both C3and C302is only 63 cm-l, which is indeed very sma11.2-6 C302 has been found to produce CzO+ as a fragment ion in the mass ~ p e c t r u m ,that ' ~ is
extends over the occupied valence molecular orbitals only. A convenient way of describing the CKD0/2 method is to start from the Hartree-Fock approximation.22 I n order to make a comparison with eq 5 possible, we can express the total energy in the Hartree-Fock theory in the following manner occ
ET = 2
E %
c 3 0 2
-e, c20'
f
co
(1)
occ occ Ei
-
C C @Jar - Kij) i j
C
+
A B(#A)
Similarly, CzO (probably 38 ground state) has been suggested as an intermediate species in the photolysis of carbon suboxide, V ~ X . ~
c&*
czo f co
(3)
where C3O2*is an electronically excited G O z molecule. The absorption coefficients of c302 have been measured from 2000 to 1050 photoelectrically.1° The 1650-1050-A region is dominated by a ns + r Rydberg series converging to the first ionization potential at 10.6 eV. The main purpose of this study was to examine whether a relatively simple quantum mechanical model could satisfactorily account for the various physical and chemical properties just described.
Computational Methods Two computational methods have been employed in this study: on the one hand, the extended Huckel molecular orbital (XHMO) method as used extensively by H ~ f f m a n n , 'on ~ the other, the CNDO/2 model of Pople, Santry, and Segal.16-18 Both are total valence electron models. In the XHMO approach, the off-diagonal Hamiltonian matrix elements were approximated by the algorism due to Cusachs,lgthat is Hij =
'/2(Hii
+ H,,)(2 -
/Si1I)&j
(4)
where the Hti's are the diagonal matrix elements, which are taken as the negative of the valence state ionization potentials,20and the Sdl'sare the elements of the overlap matrix. Care was taken to ensure the invariance with respect to orientation of the p orbitals in the molecular coordinate system.21 It is useful to recall that XHMO is basically an independent electron model (where the orbital energies are the eigenvalues of an effective one-electron Hamiltonian), according to which the total energy is simply given, for a closed shell molecule, as twice the sum of the orbital energies i
where the
et
are the orbital energies and the summation
ZAZBRAB-~ (6)
where ZA is the nuclear charge on atom A, and RABis the internuclear distance between atoms A and B. In the Hartree-Fock method all the electrons are considered, so that the summations in the first two terms of eq 6 are over all the occupied molecular orbitals, including those associated with the inner shells. The JIj and K d jare the Coulomb and exchange integrals, respectively, these being over the occupied molecular orbitals i and j . The orbital energies E i are the roots of the secular equation IFpv- ESpvl = 0
(7)
where the indices p and v refer to atomic orbitals. The F matrix is a representation of the Hartree-Fock operator. I n the CND0/2 procedure the energy is still given by eq 6; however, the inner-shell electrons are not treated, so that ZA and ZB now represent core charges (nuclear charge - 2 in our case). The first two terms are now summed over the valence orbitals only. Furthermore, in the construction of the F matrix, a series (6) L. J. Stief and V. J. De Carlo, J . Chem. Phys., 43, 2552 (1965). (7) (a) R. T . Baker, J. A. Kerr, and A . F. Trotman-Dickenson, J . Chem. SOC., A , 975 (1966); (b) R. Muller and A. Wolf, J . Amer. Chem. Soc., 84, 3214 (1962); H. Chang, A. Lautzenheiser, and A. Wolf, Tetrahedron Lett., 6295 (1966). (8) R. L. Livingston and C. N. R. Rao, J . Amer. Chem. Soc., 81, 285 (1959). (9) H . W. Thompson and N. Healey, Proc. R o y . Soc. (London), A157, 331 (1936). (10) H . H. Kim and J. L. Roebber, J. Chem. Phys., 44, 1709 (1966); J. L. Roebber, J. C. Larrabee, and R. E. Huffman, ibid., 46, 4594 (1967). (11) 8. Bell, T. S. Varadarajan, A. D. Walsh, P. A. Warsop, J. Lee, and L. Sutcliffe, J . M o l . Spectrosc., 21, 42 (1966). (12) K. S. Pitaer and E. Clementi, J . Amer. Chem. SOC.,81, 4477 (1959), and references cited therein. (13) W. M. Sinton and J. Strong, Astrophys. J., 131, 470 (1960). (14) T. J. Hirt and J. P. Wightman, J . Phys. Chem., 66, 1756 (1962). (15) R. Hoffmann, J . Chem. Phgs., 39, 1397 (1963). (16) J. A. Pople, D. P. Santry, and G. A. Segal, ibid., 43, 9129 (1965). (17) J. A. Pople and G. A. Segal, ibid., 43, 5136 (1965). (18) J. A. Pople and G. A. Segal, ibid., 44; 3289 (1966). (19) L. Cusachs, ibid., 43, S157 (1965). (20) J. Hinze and H. Jaff6, J . Amer. Chem. SOC.,84, 540 (1962). (21) M.D . Newton, J. Chem. Phys., 45, 2716 (1966); L. Cusachs, ibid., 45, 2717 (1966). (22) C. C. J. Roothaan, Rev. M o d . Phys., 23, 69 (1951).
Volume 79, Number 7 July 1969
2300 of approximations is introduced, particularly the complete neglect of differential overlap (CNDO), whereby the overlap distribution t$# (1) (1) between any two different atomic orbitals t$@ and t$v is neglected in all electron-repulsion integrals. The overlap matrix is correspondingly reduced to the unit matrix. The calculations of the remaining one- and two-center Coulomb integrals and the core integrals are described in ref l7and 18. A comparison of eq 5 and 6 shows the essential difference between the two methods (XHh90 and CSD0/2) used in the present study. It might be argued that the independent electron model is an approximation to the Hartree-Fock method, based on the mutual cancellation of the second and third terms of eq 6, and one might think that the great success encountered by the XHMO method in predicting molecular geometries is precisely due to the approximate cancellation of these two terms. However, this argument is not quite correct, since the difference between the nuclear repulsion energy and the sum over the Coulomb and exchange integrals is actually sizable. Insight into this problem has been gained recently by Allen and his collaborator^.^^ Analyzing results from ab initio calculations, they first reasoned that the important quantity relating to molecular shape is the sign of the derivative bET/d8, at every angle 8. I n the Hartree-Fock method, then, the sum of the valence orbital energies will predict the same geometry as the correct expression (that is, eq 6) if, at every angle, the two derivatives, b/d0(2,s,) and dET/bB have the same sign. I n other words, the condition of the Hartree-Fock orbital energies to give a quantitative prediction of the equilibrium bond angle is
JOHN F. OLSENAND LOUISBURNELLE not been subjected to a similar study. It is, of course, an interesting question to know whether the approximate form of the CND0/2 Hamiltonian permits expression (8) to hold in this case also. If the inequality (8) would hold, then one advantage of the CND0/2 method would be to enable one to discuss the geometry of polyatomic molecules in terms of a simple model. I n both models employed, the MO’s were expressed in terms of a basis set of 2s and 2p Slater-type orbitals. The orbital exponents were calculated from Slater’s rules.27 The original parameters of Pople, et u Z . , ’ ~ were employed in the CND0/2 model. The electron diffraction8 results were employed for the geometry of C302, that is, 1.28 A for the CC bond and 1.16 A for the GO bond.
Results and Discussion Leaving aside the inner shells, the ground-state electronic configuration predicted for Ca02 by these calculations is XHMO:
(iau)2
(iu,)2
(ZQ
( 2 4 2 (3Qg)2( i P U ) 4
(3 CND0/2:
(zP,)
(iP,)4 4;
Q,+
( 1 ~ ~(la,)’ ) ’ ( 2 ~ ~( )2’~ ~() ’1 ~( 3~~ )~ )~’ ( i P , ) 4 ( 3 4 2 (2P,)4; 1zg+
From their measurement of the photoelectron spectrum, Baker and Turner’s suggest the ground state of C302may be written as (SUA2
(sau)’
(pa,)’ (pall)’ (P.U)’ (Pall)( (PPA4 (PTLJ4
There are several differences between this ordering of the molecular orbitals and the one resulting from /Val \ our calculations. I n particular, we find a orbitals sandwiched between the three a orbitals. However, as stated by Baker and Turner, the relative order of the deeper P levels and the highest u levels cannot be Peyerimhoff, Buenker, and Allenz4have shown that this deduced unambiguously from the photQelectron specrelation holds for a large number of triatomic molecules. trum. It was found that it no longer holds for species conBetween the computational models themselves there taining atoms of widely differing electronegativities is seen to be only one discrepancy, this being the raising (e.y., Li’O). of the 3a, orbital energy by the CND0/2 method. The fact that eq 8 is satisfied in a very large number We prefer to think the XHniIO results are more correct, of cases explains the success of the simple one-electron since it is known that the CNDO/2 method leads to model used by Walsh25 in constructing his correlation “strong” intermixing of P and u orbitals, usually overthe a-type orbitals. This is presumably due diagrams, and the analysis by Peyerimhoff, et ~ 1 . ) ~stabilizing ~ suggests that the orbital binding energies of the Walsh the use of the same resonance integral (p) for both T and diagrams are nothing more than the orbital energies. I n other words, they represent ionization potentials. (23) L. Allen, “Quantum Theory of Atoms, Molecules, and the As far as the XHMO results are concerned, Allen and Solid State,” P. 0 . Lowdin, Ed., Academic Press, Inc., New York, N. Y., 1966, pp 39-80. Russel126have shown that angle estimates based on the (24) S. D. Peyerimhoff, R. J. Buenker, and L. Allen, J . Chem. Phys., semiempirical term &es parallel those obtained from the 45,734 (1966). ab initio ZiVa1ei. It thus appears that a relation similar (25) A. D. Walsh, J. Chem. Soc., 2325 (1953). to (8) is also satisfied by the XHMO orbital energies, for (26) L. Allen and J. D. Russell, J. Chem. Phys., 46, 1029 (1967). covalent molecules at least. To our knowledge, the (27) J. C. Slater, Phys. Rev.,36, 57 (1930). (28) C . Baker and D. W. Turner, Chem. Commun., 401 (1968). orbital energies produced by the CNDO/2 method have The Journal of Physical Chemistry
2301
THEELECTRONIC STRUCTURE OF CARBON SUBOXIDE Table I : .Ir-Eigenvalues and Eigenvectors of Linear CSOZ symmetry----------1 "E -Mehtod---------
lau
-----
XHMO
Atom
CNDO/2 Eigenvalue, eV--18.93 -20.08
-24.93
0.46 0.46 0.46 0.43 0.43
-0.15 0.15 -0.66 0.66 Symmetry
7
2 "E
XHMO Atom
-9.14
Method CNDO/Z XHMO Eigenvalue, eV 2.68 -4.41
c1 c2 c3 02 03
-0.72 0.72 0.32 -0.32
-0.60 0.60 0.38 -0.38
1
I
-0.38 0.38 -0.60 0.60
---
-13.02
-12.42
0.67 0.35 0.35 -0.23 -0.23
0.65 0.16 0.16 -0.52 -0.52
3 *"
CNDO/2
8.30
0.87 -0.70 -0.70 0.24 0.24
u orbitals in CKD0/2 theory." The effect is exemplified in ethylene, where the method predicts the highest occupied molecular orbital (HOMO) to be u in character (unpublished observation). The P orbitals and their corresponding energies as obtained by these calculations are recorded in Table I. I n this table, the central carbon is labeled C1 and the outer carbons are labeled C2 and C3 with the oxygens attached to these carbons labeled 0 2 and 03, respectively. We have recorded only one component of the degenerate P orbitals (that is, P, or P J . If, assuming the validity of Koopmans' theorem,2g we equate the negative of the orbital energy of the HOMO to the molecular ionization potential, the following results are obtained: 13.0 eV and 12.4 eV by XHMO and CNDO/2, respectively. These calculated values are seen to be in only fair agreement with the experimental ionization potential of 10.6 eV.10$2*The discrepancy may be partly imputed to the fact that we have calculated the vertical ionization potential, whereas the experimental quantity indicated above is the adiabatic value. It is possible that upon ionization there is a substantial change in the molecular dimensions, giving rise to a relatively higher value of the vertical ionization potential. It was mentioned in the Introduction that C802 fragments both photochemically and mass spectrometrically to produce the species C20 (either neutral or positively charged). The expressions which our calculations yield for the molecular orbitals, especially for the HOMO and LUMO (lowest unoccupied), are in agreement with such energetic decompositions. The 2 ~ , orbital (HOMO) is seen to be CC bonding and CO antibonding. The Z n , orbital, on the other hand, is seen to be CC nonbonding and CO antibonding. Thus, the overall
7
CNDO/2
--__.
0.03 0.15 0.15 0.65 0.65
c2 c3 02 03
XHMO
XHMO
CNDO/2
-18.94
c1
~
2 *u
0.61 -0.52 -0.52 0.23 0.23
Table 11: Atomic Orbital Population in Linear CSO~" Method Atom
Orbits1
XHMO
CNDO/2
c1
2s 2P# 2P, 2P"
1.13 0.80 1.19 1.19
0.95 0.88 1.27 1.27
c2
2s 2P, 2PZ 2Pk.
0.94 0.50 0.53 0.53
1.08 0.94 0.76 0.76
0
2s 2P, 2P, 2P,
1.88
1.71 1.87 1.87
1.71 1.36 1.61 1.61
The z axis is oriented along the molecular axis, while the x and y axes are perpendicular to it.
effect of promoting an electron out of the 2n, MO (which is presumably what occurs in these energetic decompositions) is to weaken the CC bonds. This weakening suggests an explanation for the observed fragmentation into CO and CzO. Table I1 records the atomic orbital populations of linear carbon suboxide. When an orthogonal basis is used the atomic population for an orbital t$p is defined as
N , = xn(i)cifi2 z
(9)
where n(i) is the number of electrons in molecular orbital i and Ci, is the LCAO coefficient of the atomic (29) T. Koopmans, Phuaica, 1, 104 (1984).
Volume 75,Number 7 July 1969
JOHNF. OLSENAND LOUISBURNELLE
2302 orbital 6, in A 4 0 i. In the XHR4O method, which deals with a nonorthogonal basis, the corresponding quantity is the gross atomic orbital population defined as
NP = Cn(i)C;JCip a
+ CCiVSrrV) lJ
(10)
ZP
where the orbitals are located on atoms different from the one containing 4,. The resulting atomic charges are recorded in Table 111. It is a well known fact that the coefficients obtained from XHMO methods exaggerate the charge separation.15 This is immediately apparent for the exterior CO bond, whose atoms in the XHMO method have a charge larger than one whole electron. The atomic charges produced by the CNDO/2 method are seen to be much more realistic.
that is, these are the off-diagonal elements of the firstorder density matrix. We have also computed a total bond population between any two bonded at'oms k and 1, which can be defined as
where the index p is associated with all the atomic orbitals on atom k and v is similarly associated with all the atomic orbitals on atom 1. Equation 12 is essenTable IV : Bond Population of Linear C@Z _____---__M&od----------XHMO------. T Total
Bond
c-c
Atom
XHMO
CND0/2
c1
-0.31 1.49 -1.33
-0.36 0.46 -0.28
c2
0
I n a comparative calculation on Cs (labeled as the Co fragment in C3Oz) we obtained the following charge densities
c1 c2
XHMO
CNDO/P
Clementi
0.08 -0.04
-0.16
-0.28
0.08
1.41 0.73
Total
0.62 0.69
1.76 1.93
tially the sum of the squares of the bond orders between the bonded atoms k and LS1 It was necessary to use the square due to the constraint imposed on some bondorder terms by the choice of the coordinate axes." The index defined by eq 12, as pointed out by WibergJ31 seems to be closely related to the bond character (the index reflects the number of bonds between atoms k and 1, corrected for the ionic character of each bond). I n the XHMO method the quantity involved is the total overlap population, defined as occ
0.14
Also included is the result of an ab initio calculation by Clementi.30 It is immediately apparent that the signs of the CND0/2 charges are identical with the nonempirical charges, and also that the magnitudes of the charges agree satisfactorily. On the contrary, the extended Huclcel model gives the atoms an opposite polarization. (Oddly enough, the charge separation is very small by the XHMO method for this molecule. C3 is apparently an exception to the general observation that the XHMO method exaggerates the atomic charges.) This is certainly a good illustration of the superiority of the CND0/2 method over the extended Hiickel method for the calculation of charge distributions. Our CNDO/2 calculations indicate that the effect of adding two exterior oxygen atoms to the Ca fragment results in more negative charge flowing onto the central carbon atom in CaOz, with the outer carbons becoming more positive. This has important consequences as we shall see. The results of the population analysis are made complete by the bond populations which are recorded in Table IV. In the CNDO/2 model the orbital bond population ( p # v ) is defined as
The Journal of Physical Chemistry
0.11
T
--.
Method
7
0.28
c-0
Table I11 : Atomic Charges in Linear CaOs
7
P--CNDO/!~---
Nki
=
k
1
2Cn(i) i
CzukCi)Sfiykl
(13)
P l J
The bond populations may of course be broken down into A and v components. The a populations indicated in the table refer to only one component ( r Zor T,). The bond populations are seen to vary considerably depending upon the method employed just as was found with the atomic orbital populations. Thus the XHMO model predicts a CC bond much stronger than the CO bond, whereas the CND0/2 method places greater electron density between the carbon and the oxygen. The discussion of the population analysis will center mainly on the CNDO/2 results since they appear to offer the most reasonable picture of CaOz. First, it is seen that the =-bond populations for the CC bonds are slightly smaller than those for the CO bonds; the total bond populations are seen to parallel the r-bond populations. These CC a and total bond populations are much smaller than the corresponding populations in ethylene (where Rcc was also set at 1.28 A), where a a(30) E. Clementi and A. D. MoLean, J. Chem. Phys., 36, 45 (1963). (31) K. Wiberg, Tetrahedron, 24, 1083 (1968). Alternatively, overlap integrals could have been used instead of squaring the bond order terms. However, the use of eq 12 is more consistent with the CNDO/2 theory.
THEELECTRONIC STRUCTURE OF CARBON SUBOXIDE
2303
bond population of 1-00is obtained along with the total bond population of 2.08. This reduction in the CC bond population is presumably related to the large negative charge on the central carbon atom in CsOz. Indeed, the results of these electron population analyses are consistent with the following resonance forms contributing to the ground state of CaOz.
-
+
O~C--C=C=O C
+ -
o=c-c=c=o E
- +
o-c=c=c=o G
-8.0
+
-
O=C=C=C=O A
-4.0
O=C=C-C~O B
-
% s P
f
c t
-
O=C=C-C=O
e
< -12.0
D
0
+ O=CdJ=C-O F + + O~C-C-C~O H
16.0
IngInu
3b, lo., 4 a , I
It is to be noted that the resonance forms with a negative charge on the central carbon atom, which must be present to a nonnegligible extent, involve a lowering of the CC a-bond population. This factor undoubtedly contributes to the decrease of the bending force constant with respect to a pure double-bonded structure. Another way - of looking a t the question is to consider that the -C= group is isoelectronic with -N= and that consequently it should have a bent configuration. (It is known that molecules with the -N= group, like HNO and FNOz are bent in their ground state.32) The structures with a negative charge on the central carbon atom, from this point of view, provide a negative contribution to the bending force constant.33 Thus, the wave functions obtained by the CNDO/Z method satisfactorily explain the experimentally observed low bending frequency of C302,namely 63 cm-l, a t least qualitatively. The wave functions obtained from the XHMO method are seen to be less satisfactory in that respect. I n addition to the linear configuration, we have also carried out calculations a t various apex angles between 180 and 90" for C302. Both XHMO and CNDO/% correctly predict the linear form (I&+) as being the most stable. Further insight into the low resistance to bending is gained by looking at the variation of the XHMO orbital energies as a function of the apex angle. Such a plot is presented in Figure 1, where the four lowest and four highest lying molecular orbitals have not been included. Figure 1 may be thought of as a Walsh-typeZ6diagram for ( 2 3 0 2 specifically and for AaBz molecules in general. The interesting feature of this diagram is that the electrons in the highest filled orbital in C302(27,) tend to impose a bent shape on the molecule. Among the other occupied orbitals, 4b2(3u,) operates in the opposite direction and its effect barely outweighs the tendency of
Ib.
-20.0 90'
I
3Q.
20.
135"
180"
CCC angle.
Figure 1. Plot of t h e XHMO orbital energies us. apex angle
for GOz.
the 2n, electrons to keep the system bent. The low bending frequency in this molecule is thus the result of these two opposing tendencies. This interpretation has been proposed in qualitative terms by Bell, et al." The values of the bending force constant resulting from our energy calculations are considerably larger than the experimental value. This is not too surprising in view of the above discussion. One cannot hope to compute quantitatively the variation of the energy with the angle, particularly with the type of wave functions used here, and the discussion of the bending in the molecule must for the time being remain largely qualitative. The discussion can be extended easily to the two ions C302+and c&-, with the help of Figure 1. Both ions should be linear, with a bending force constant larger than in the neutral molecule; the constant is predicted to be the largest in the anion, because the Zaz level varies more steeply than 2bl with the angle. A look at Figure 1 reveals that C3Os should be linear in all states corresponding to the configuration (2nU)3 (27,)'. An increase of the bending frequency should be observed in these states, as appears from the shape of the two curves coalescing into the 2a, level. A similar situation is encountered in CB,where in the first excited state ('Tzu) the bending frequency has been founda4to (32) F.W.Dalby, Can. J. Phys., 36, 1336 (1958); D. W.Magnuson, J . Chem. Phys., 19, 1071 (1951). (33) K.8. Pitzer and 8.J. Striokler, ibid., 41,730 (1964). (34) L. Gausset, G.Herzberg, A. Lagerqvist, and B. Roaen, Astrophys. J., 142, 45 (1965).
Volume 73, Number 7 July 1969
2304
WALTERF. VOGLAND ROBERT C. AHLERT
increase from 63 to 308 cm-l. On the other hand, the )’ give rise to bent configuration ( 2 ~ ” )( ~3 ~ ~ should states, in view of the fact that both components of 3ru show a sizable stabilization upon bending. I n particular, the 6al orbital has a steep slope, and the XHMO wave functions thus predict a strongly bent B1 state of cso2. Finally, it is to be noted that while the total energy of the CNDO/2 method correctly predicts the linear geometry for CaO2 the quantity 2Ziei(CND0/2) does not predict the correct molecular geometry of CsO2. Some results for this quantity us. angle are e 2Cei
100’ -643.22
120° -638.9
135’
180’
-637.10
-635.19
i
where the energies are in electron volts. These results reveal that contrary to the XHMO and the HartreeFock one-electron energies, the CNDOIB orbital energies cannot be used to predict molecular geometries, for this molecule at least. Similar results have also been found by us for CJVZ, CsH4, NOa, and COa-. It
seems that the approximate form of the CNDO/2 Hamiltonian does not permit inequality (8) to hold. More calculations are needed to establish this point, however. I n conclusion, one may say that the two methods used in the present work have provided some insight into the electronic structure of carbon suboxide. The computed wave functions have enabled us to explain a series of properties of the compound, in particular its low resistance to bending, which we have found to be linked with a buildup of negative charge on the central carbon atom. I n general, the CNDO/Z wave functions give a more accurate picture of the charge distribution than the XHMO orbitals, but the XHMO one-electron energy diagram has the advantage of enabling one to predict readily the geometry of the molecule in. its excited and ionized states.
Acknowledgment. A grant of computer time from the New York University AEC Computing Center is gratefully acknowledged. We also express our thanks to a referee for his constructive criticism of our manuscript.
Optimization of Force Constante for Methane and Argon by Walter F. Vogl and Robert C. Ahlert College of Engineering, Rutgers-The
State UniveTsity, New Brunswick, New Jersey
(Received November 18, 1868)
A systematic study of the generation of intermolecular force constants from experimental data is reported. Thermodynamic data and transport property data are compared as source information for the best value search. Thermodynamic data lead to a unique choice of force constants, while transport property data lead to multimodality and insensitivity in the optimization procedure.
Introduction Constants for the empirical equations of state and parameters for intermolecular potentials, used in conjunction with cluster integrals and virial theory, are calculated from experimental data. One or more thermodynamic or transport properties are selected and the variables are manipulated until predictions are in “best” agreement with experimental observations. I n general, such searches are heuristic; few efforts have been made to employ the formalism of modern optimization theory. Limits on the application of generalized search techniques have been dictated by the complexity of calculations, particularly for virial theory, and uncertainty regarding the modality of the problem. The Journal of Physical Chemktry
This study was undertaken to examine several questions : 1. Are thermodynamic or transport property spaces multimodal? 2. How sensitive are search techniques in the regions of local and global extremums? 3. How do modality and sensitivity vary with the choice of experimental data? Virial theory and the highly simplified LennardJones potential, i e . , 6-12, were selected as the basis of state relations. This choice was predicated on the frequency with which (‘best values” have been reported for the variables of this system. Excellent approximations to the cluster integrals exist and permit more efficient use of computer time than is possible when direct numerical integration is required. Methane and argon were selected as vehicles for this study because
