JOYCEJ. KAUFMAN
2648
The equations and discussion of this paper show that experimental investigations of relative substituent effects in HzO and D 2 0 may be of considerable value. I n cases where AG,,t = AG,,t' and A s i n t = ASintr for symmetrical reactions of types (1) and (a), the Hammett p-parameter for a given reaction is the same in both H 2 0 and DzO and differences in equilibrium (or rate) constants can be attributed to Hammett CP values or AHintand AHint' value^.^ It is also apparent
that relations between O-H stretching frequencies, depths of potential curves, and pK values are worth further attention. Acknowledgments. The author thanks Dr. S. D. Hamann and Professors R. H. Stokes and N. V. Riggs for their helpful comments and Professor R. B. Martin for calling his paper to the author's attention, which led to this investigation.
LCAO-MO-SCF Calculations of COO,Systems
by Joyce J. Kaufman Research Institute for Advanced 8tudies (Martin Company)I Baltimore, Maryland (Received April 87,1964)
BIB12
Simple Huckel MO calculations by West predicted a biradical nature for C606-4, due to the degeneracy of molecular orbitals 48 and 4 9 , while experiments showed C606-4 to be diamagnetic. To test the sensitivity of the prediction to M O calculational methods, Pariser-Parr and Pople SCF calculations were performed for C606, c606-', and C60ec4. It is difficult to ascertain whether electron interaction terms alone are sufficient to destroy the calculated degeneracy of 48 and 49 since the evaluation of the elements of the configuration interaction matrix will be critically dependent on having accurate values for the integrals. Evaluating the configuration interaction matrix elements utilizing the integral values for a Pople SCF calculation of neutral C606 or C606-' indicates that electron interaction terms alone do not seem to be sufficient to make the singlet state be lower in energy than the triplet state. The calculational results indicate that for charged species it is not sufficient to use integral values derived from the neutral species, since this leads to physically unmeaningful results for the absolute energies of the species. It is suggested that the splitting of the degeneracy may possibly be due to a type of Jahn-Teller effect-which, however, must necessarily be weak since the electrons involved do not participate strongly in the binding of the molecule.
C,O,-" and recently West, et al., have isolated C G O ~ - * . ~ (1) R. West, H.-Y. Niu, D. L. Powell, and M. V. Evans, J. Am. They reported that- simple Huckel LCAO molecular Chem. See., 8 2 , 6204 orbital calculations of c606-' predict a biradical nature (2) R. West and H.-Y. Niu, ibid., 84, 1324 ( 1 9 6 ~ ) . for C&-* since the two lowest unoccupied nlOleCUlar (3) R. West and D. L. Powell, ibid., 85, 2577 (1963). T h e Journal of Physical Chemistry
2649
LCAO-MO-SCF CALCULATIONS OF C606 SYSTEMS
Table I : RIolecular Orbital Energies Energy level
---
(in 6)-
,----Hilokel Set a
Set b
c606
E7 Es
E*
-0.4396 -0.9025 - 0.9025
Set d
Pople SCF, e.v.------
7 -
Set c
Set d
(neutral)
- 4,4330
0.3820 0 0000 0.0000
-
Pariser-Parr, e.v.-
Set c
-3.3104 -3,3104
- 1.2478 -0.4266 - 0.4266
- 4,5050 - 3.4052 - 3,4052
-0.5537 0,2876 0.2876
4 6852 9 5875 9 5875
0 6737 6 0840 6 0840
4 6851 9 5875 9 5875
14 3014 15 5562 20 0861
10 2613 11 6345 16 6959
14 0104 15 0830 19 7766
c606-2
E7 Es Es
Same as GO6
1 2088 6 5969 6 5969
CaO(,-*(singlet) E7
Same as CsOe
11 1161 12 6724 17 2023
E8 E9
electron correlation effects, neglected in the Hiickel I-A [sets c and d, I = 11.22 e.v., A = 0.69 e.v.1 and the calculations, niay operate to remove the degeneracy. two-center two-electron repulsion integrals are scaled according to the usual Pariser-Parr extrapolation It was of interest to test this assumption by permethod. Pariser-Parr calculations (Table I) using forming comparative LCAO-340 calculations for both sets c and d indicate E7 is bonding and Ea and E9 C606-4in the Framework of the three major techniques in current use for A T 0 calculations on n-systems: the are degenerate and still bonding. Poplr SCF calculations (Table I) using parameter set Huckel method, which neglects interelectronic repuision, the Pariser-Parr method, which starts from thc c indicate ETis bonding and E8 and E9 are degenerate correct many-electron Hamiltonian and simplifies the and still bonding; results using set d show E, to be calculations by a seriies of systematic approximations for bonding but E8 and Eg, while still degenerate in energy, the integrals, and the Pople SCF technique, which apare now indicated to be antibonding. These calculaplies the systematic integral approximations of Pariser tions provide another confirmation of the great care and Parr to the self-consistent equations of R ~ o t h a a n . ~ which must be exercised in inferring absolute bonding or In an SCF calculation there are no subscquent interantibonding properties from the calculated energy actions of configurations arising $om single-electron levels of unoccupicd orbitals since these calculations excitations but oiily those arising from two-electron exare so critically dependent upon the choice of input citations. parameters. (This same conclusion was reached in the author’s research on LCAO-S’IO calculations on Calculations and Results acceptor molecules in charge-transfer Neutral c606. H&kel Calculations. Huckel calcuC606-’. Pariser-Parr and Pople S C F Calculations. lations were performed for neutral e606 with two difThe Pariser-Parr calculations were performed using as ferent sets of so-called standard parameters (set a, Pullstarting >‘IO the Huckel orbitals of c606-’, but retainman5; set b, Streitwieser6). Results (Table I) using ing the same values for the valence state input paraniset a indicated that ET, the energy of the lowest unfilled eters and electron repulsion integrals as for c606. This molecular orbital, is moderately antibonding and that Ea and E9 are degenerate in energy and more antibond(4) R. Daudel, R. Lefebvre, and C. Moser, “Quantum Chemistry: ing, while results of set b indicated E7 to be bonding Methods and Applications,” Interscience Publishers, Inc., New York, with Ea and E, degenerate and nonbonding. N. Y., 1959. (5) B. Pullman and A. Pullman, Rev. M o d . Phys., 32, 428 (1960). Pariser-Parr and Pople S C F Calculations. The (6) A. Streitwieser, “Molecular Orbital Theory,” John Wiley and Pariser-Parr and Pople SCF calculations were each perSons, Inc., New York, N. Y., 1961, p. 135. formed with two different sets of input parameters for (7) J. W. Sidman, J . Chem. Phys., 27, 429 (1957). acore [set c, aceore= 11.22 e.v., aocore = 17.17 e.v.; set (8) J. J. Kaufman and J. R. Hamann, “Some Theoretical Aspects of d, (Sidnian’) aceore= 9.50 e.v., aoCore = 13.00 e.v.1. Charge Transfer Complexes. I. LCAO-MO Calculations of Various Electron Acceptors,” presented before the Division of The one-center two-electron repulsion integrals are Physical Chemistry at the 146th National Meeting of the American calculated from the Pariser-Parr formula (pp pp) == Chemical Society, Denver, Colo., January, 1964.
1
Volume 68, Number 9 September, 1964
JOYCE J. KAUFMAN
2650
artifact of retaining the same parameter values resulted for set c in indicating E7 to be slightly antibonding and levels E8 and E9 to be degenerate and quite antibonding and for set d in indicating E, to be antibonding and E8 and E9 to be degenerate and very antibonding. It is obvious from these results that in order to perform properly any calculations on charged species, one must take into account the variations in atomic valence state ionization potentials and electron repulsion integrals due to the charges. A method which attempts to do this is the VESCF (variable electronegativity self-consistent field) m e t h ~ d . ~ Calculations of the C G Oions ~ by the VESCF niethod are presently being carried out in this laboratory. However, although the absolute magnitudes of E7 and Es (and Eg) will change with improved parameter values, the degeneracy of E8 and E9, which is the crucial question being examined in this article, will remain unchanged. Pariser-Parr and Pople XCF Cahutations. C~OG-~ . As an attempt to probe the behavior of E8 and E9 in the CSOG-~ ion itself, Pariser-Parr and Pople SCF calculations were performed (parameter sets c and d) for the case in which the two electrons were placed into 4s to force a singlet state. Because of the artifact of using the same integrals as for the neutral C606, the arithmetic results must indicate too much antibonding character for all energy levels ,of C606-4; this can be seen in Table I. However, the inathematical and physical implications that E8 and E, are no longer degenerate in this situation emerge clearly. Rather than attempt the corresponding calculation for the triplet state to see if its total energy is lower than that of the singlet, since this will be dependent upon the absolute magnitudes of the integrals over the molecular orbitals, the question of SCF calculations for the ions is being investigated by the alternative VESCF method. The calculated coefficients of the 40's in the MO's for the Hiickel and Pople SCF methods are available from the author and will be issued in an BIAS-TR, which will also include a much more detailed description of the configuration interaction calculation presented in the following section. Configuration Interaction. Electron repulsion introduces a resonance between different configuration wave functions of the same multiplicity and symmetry. A more general wave function niay be taken which is a linear combination of such configurations and the energy minimized by the variational method. From the 1 4 0 dBand + g one has in singlet and triplet states the following four antisymmetric combinations (which, however, do not belong to irreducible representations when $8 and # 9 are degenerate). Xs and XT are singlet and triplet spin functions. The Journal of Physical Chemi8try
$b
9"
=
49(1)48(2)x~
(1/2)1/2[48(1)49(2)
=
+ @9(1)48(2)]xg (2)
('/2>1/214*(1)49(2)-
$t
=
(1)
(3)
$9(1)$8(2)]xT
48 (1 148 (2)X S
(4) One may make the following linear combinations from the various occurring charge distributions which do belong to irreducible representations $a
$0
= ('/~)'/~[48(l)h(2) -/- 49(1)49(2)1x8 =
(5)
('/%)"'[48(1)48(2) - 49(1)49(2)]Xs (6)
lE%d
1c/d
lEZW
$" = ('/2)"'[48(1)49(2)
3EZ,t
1C.t =
+ 49(1)48(2)]xs
('/2)1'2[48(1)49(2) - $9(1)&(2)
]xT
(7) (3)
If one uses for 48 and q5g the virtual orbitals found from the solution of neutral C606or from CaOa-2 egcore =
€score -
J88
'Ai,
EO
= eaCore
=
ea core
Jgg
+ '/zJss + + + + eacore
' / ~ J Q QK89 = 26,c0re 'E~gv
E"
=
~a~~~~
+
eacore
+
JEg
+
2taCor8
'Ezgt
Et
=
€Loore
+ eilcore +
J89
J88
K89
f K E Q (8) =
+ + J89
- &9
=
Ed
+ JSS+ + + = 2eaCore + Jss -
E?
(9)
=
2egcore
'Ezgd
K89
(10)
K89
eaCore
'/2J88
(11) The determinant corresponding to the configurationinduced interaction between $" and $d is K89
'/2J99
+ +
'2etLcore
J89
/[48481$86CJl
-
K89
K89
[4848/4849]- [484914949]
[48$91d'Q49]
+
2esCore
i=o
J88
-
K89
(12)
The equation to be solved for the energies is
+ +
(2taCore
J89
-!- Jss - K89) -
K89)(2EaDore
{ [4848/48+91-
['$849/'$949]]
0
(13)
This evaluation of the energies of the two singlet states after configuration interaction will be critically dependent on having accurate values of the electronic repulsion integrals (which have been corrected for the excess negative charge in the ions). In order for a singlet state to be lower than the triplet state its energy must be
