J . Phys. Chem. 1990, 94, 5625-5630 straightforward to determine the expansion coefficientsas functions of RAB. Further analysis is required to introduce intramolecular
5625
where p is an overall scale factor for the particular plot and F(4) is the integral given below.
vibration. This will be carried out in the notation of rko and rk introduced in section 2.
Acknowledgment. In writing this paper the author has attempted to emulate John Pople by clearly expressing a new point of view. To the extent that he has succeeded, he is indebted to Walter Stockmayer and Donald Hornig for igniting his interest in kinematics, George DeTitta and Xian Fa Zou for rekindling the flame, Gregory Ezra and Brian Sutcliffe for expert criticism, and James McIver for introducing him to symbolic manipulation programs and for providing software and hardware supported by NSF Grant No. CHE8803072. This research was supported by the National Institutes of Health under Grant No. DK19856.
Appendix Distances in the radial direction in Figure 5 , and also Figures 7-10, are scaled by a factor of g(4)-'I2. An orientation of the body which is rotated by an angle 4 from the reference orientation is represented in Figure 5 by a point on a circle of radius p F ( 4 ) ,
This integral diverges logarithmically at 4 = *2x, so we introduce a new variable y . Y 4/(2x), bl < 1 (67) Removing the logarithmic singularity yields a function with a rapidly convergent Taylor series expansion.
The contribution from the infinite series is at mmt half that from the logarithmic term in (68), so numerical stability and accuracy is achieved with only a few terms. Rational expressions for the ck coefficients were obtained, using the symbolic manipulation program M A C S Y M A , ~and ~ converted to the numerical values reported in Table 11.
MP2-R12 Calculations on the Relative Stability of Carbocations Wim Klopper and Werner Kutzelnigg* Lehrstuhl fiir Theoretische Chemie, Ruhr-Universitat Bochum, 0-4630 Bochum, FRG (Received: November 28, 1989)
The MP2-RI2 method (second-order Maller-Plesset perturbation theory with linear r12-dependentterms) is briefly presented, and applications to the ions CHS+,C2H3+,C2H5+,and C3H7+are described. With the MP2-RI2 method one comes rather close to the basis set limit of MP2 theory, although the MP2-RI2 method does not provide strict upper bounds. The basis sets used here are a compromise between what one ought to use for perfectly reliable calculations and what one can afford. The results for CH5+do not change much with respect to conventional calculations. Only the C, structure becomes energetically much closer to the C, equilibrium structure (differenceonly -0.2 kcal/mol) than in the previous most sophisticatedcalculations. For C2H3' and C2H!+ the energy differences between classical and nonclassical structures are larger by -2 kcal/mol as compared to conventional MP2 calculations. We predict that the nonclassical ions C2H3+and C2H5+are more stable by 4-5 and 7-8 kcal/mol, respectively, than the classical ones. In the case of the isopropyl cation, C3H7+,it is confirmed that a "chiral" C, structure has the lowest energy.
1. Introduction Current quantum chemical ab initio methods have two main sources of error: (1) the truncation of the one-electron basis set and (2) the use of an approximation to "full CI" in the given basis. Other approximations usually made, like the Born-Oppenheimer separation or the neglect of relativistic effects, are critical only in special situations. A way to overcome the basis-truncation problem, or more precisely the slow convergence of the energy or of properties with increasing basis set to the basis limit, has recently been found in the use of linear r12-dependentterms in the wave function. It has, e.&, been shown that for a two-electron atom the energy increments to second order of perturbation theory in a partial wave but that with the expansion go conventionally as - ( I ansatz
+
with 4 the unperturbed wave function and x expanded in partial waves, the 1 dependence of the partial wave increments is - ( l + i/2)4.' In separating off the linear r I 2term, which takes care of Kato's correlation cusp condition? and treating it exactly, a
spectacular increase of the speed of convergence is achieved. In order to take advantage of this same idea in more complex systems, it is recommended to use a well-defined and sufficiently simple level-Le., approximation to full C I - o f the treatment of electron correlation and to go to higher levels only when it has been established that on t,he chosen level sufficiently fast convergence to the basis set limit k guaranteed. As this level we have chosen Maller-Plesset perturbation theory to second order (MP2), Le., Rayleigh-Schradinger-perturbation theory with the Hartree-Fock operator as zeroth-order Hamiltonian.'~~ This approach has been recommended by Pople et al.5 as a first nontrivial step in a hierarchy of levels for the treatment of electron correlation. The combination of the MP2 formalism with an ansatz of type (1) which takes care of linear r12-dependentterms will be referred to as MP2-Rl2. This method will be shortly described in section 2. A more detailed presentation will be given elsewhere? Some applications of the MP2-Rl2 method to atoms (Be, Ne, Ar, Ca) and molecules (HF, H20,CH,) were published,'$ and a systematic (2) Kato, T. Commun. Pure Appl. Math. 1957, 10, 151. (3) Maller, C.; Plesset, M. S.Phys. Reu. 1934, 46, 618. (4) Claverie, P.; Diner, S.;Malrieu, J. P. Inr. J . Quantum Chem. 1967, 1. 715.
(1) Kutzelnigg, W. Theor. Chim. Acra 1985, 68, 445.
0022-3654/90/2094-5625$02.50/0
(5) Binldey, J. S.;Pople, J. A. Inr. J . Quanrum Chem .1975, 9, 229. (6) Kutzelnigg, W.; Klopper, W. To be published.
0 1990 American Chemical Society
5626 The Journal of Physical Chemistry, Vol. 94, No. 14, 1990
Klopper and Kutzelnigg
-
study is in p r e p a r a t i ~ n . ~The . ~ ~general experience has been that the computed MP2-Rl2 correlation energies differ by 1% or less from the estimated basis set limits in those cases where such estimates are available. The MP2-R12 method requires that the basis sets are not too small, because certain completeness relations must be satisfied: especially that sufficiently high 1 values are present. For Ne the basis must for example include at least one f function, while for CHI a few d functions are sufficient. The purpose of the present paper is to apply the MP2-Rl2 method to some problems of structural chemistry, namely, to classical vs nonclassical carbonium ions, a problem to the solution of which some time ago both J. A. Pop1e1l.l2and the present senior have contributed. These applications will be described in section 3, where we start with various possible structures of CHS+(section 3.1) before we consider the classical and nonclassical structures of C2H3+(section 3.2) and C2HS+(section 3.3) and finally the problem that arose rather recently16as to the equilibrium structure of the isopropyl cation, C3H7+(section 3.4). Concluding remarks will be given in section 4. 2. Tbe MP2-Rl2 Metbod We start from the Hylleraas functional for the second-order energy (with 4 the zeroth-order and the first-order wave function).
+
3(+) = 2 Re
P(1)+ F(2) - ti - #~rl2[ijl = f/,[W)
+ W ) r121[ijl , - f/z[K(1) + W ) , r121[ijl
= -Y2(712/r12)(V1 - V,)[iJl - r12-1[ijl f/Z[KU)+ W ) , r121[ijl (9) where T is the kinetic energy operator and K the exchange operator. The Coulomb operator is local and commutes with rI2. For the further evaluation of (4) it is recommended to orthogonalize uij - wu to all pairs constructable in the given basis, i.e., to define uij = I1 - Q(1) Q(2)NUij - wi,) = 1/2cij(l - P(1)[1 - Q(2)I - p(2)[1 - Q(1)I - Q(1) Q(2)lr12[ijl (10) Q(l) = Clvp(l)) (vp(2)1
S(tC.1 =
? fluij)
i
