Intermolecular potential surfaces from electron gas methods. II. Angle and distance dependence of the A’ and A” Ar–NO(X^2II) interactions
Angle dependent intermolecular potential energy surfaces for the two states (2A’ and 2A”) that arise from the interaction of ground (X^2II) state NO with Ar are calculated using the electron gas model to obtain the short range interactions. The average and difference of the two interaction energies are fit to analytic forms convenient for use in scattering calculations and joined smoothly onto the long range van der Waals potential previously determined. The results, which appear to be of useful accuracy, and the applicability of the electron gas model to such open shell–closed shell interactions are discussed.
Intermolecular potential surfaces from electron gas methods. II. Angle and distance dependence of the A' and A" Ar-NO{X2rr) interactions* Glen C. Nielsont and Gregory A. Parkert Department of Chemistry, Brigham Young University. Provo. Utah 84602 Russell T Pack Theoretical Division, University of California Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545 (Received 22 September 1976) Angle dependent intermolecular potential energy surfaces for the two states eA' and 2A ) that arise from the interaction of ground (X 2 ") state NO with Ar are calculated using the electron gas model to obtain the short range interactions. The average and difference of the two interaction energies are fit to analytic forms convenient for use in scattering calculations and joined smoothly onto the long range van der Waals potential previously determined. The results, which appear to be of useful accuracy, and the applicability of the electron gas model to such open shell-closed shell interactions are discussed. II. CALCULATIONS FIG. 1. Coordinate system used in the present work. The NO orbitals are measured relative to the unprimed axes. A. Electron gas potentials The method of calculating the electron gas potential was given in detail in paper 1. Herein we will only discuss differences between the present procedure and that one. Ar B r I Z A that plane. This is achieved merely by multiplying the rest of the Hartree-Fock Tr orbital by an appropriately normalized sincp or coscp, respectively, to give the proper cp dependence and symmetry. All calculations and results in this paper are nonrelativistic and use a spin-free Hamiltonian. However, intermolecular potentials which incorporate an approximate accounting of spin orbit effects, such as the 121. 1 cm-l spin orbit splitting 8 between the 2nl / 2 and 2n3 /2 states of NO, can easily be generated from the results of this paper by solving a 2 x 2 secular equation as described in detail in an earlier paper. 9 I. INTRODUCTION In paper I of this series 1 we reviewed and discussed applications of the electron gas (EG) model-as developed by Gaydaenko and Nikulin,2 and Gordon and Kim,3 and modified by Rae,4 and Cohen and Pack5-for calculation of interactions between atoms and molecules when both partners have closed shell electronic structure. We also applied the method to the He-C and Ar-C02 interactions. In this paper we use the method to calculate the interaction energy between a closed shell atom Ar and an open shell molecule NO. Closed shell-open shell interactions of atoms have been treated with the EG model by Clugston and Gardon6 who used it to study the noble gas halides. They compared their results with the ab initio calculations of Dunning and Hay 7 for KrF and found that the EG results were good for the n state in which the empty F orbital is perpendicular to the Kr but poor for the state in which the empty F orbital points at the Kr atom. We believe that this is because the EG model cannot account for the charge transfer and mixing 7 of covalent and ionic character that occurs in this state. In applying the model to the Ar-NO system, we are on much safer ground. NO has a much lower electronegativity than F and we expect negligible ionic character or configuration mixing; all the ground state interactions should be of the simple non-bonding type and be adequately described by the EGmodel. The electronic structure of ground (Xzn) state NO is essentially that of closed shell Nz with one additional electron in an antibonding 1T orbital. In the Ar-NO system, which has Cs symmetry, the degenerate n state is split into two states: an A I state (herein called +) in which the wavefunction is symmetric under reflection in the triatomic plane and an A" state (herein called -) asymmetric about that plane. Using the coordinates of Fig. 1, it is clear that the A' (+) state is obtained by putting the extra electron on the NO in a Try orbital lying in the triatomic plane while theA"(-) is obtained by putting it in a 1fx orbital perpendicular to 1396 The Journal of Chemical Physics, Vol. 66, No.4, 15 February 1977 Copyright 1977 American Institute of Physics Nielson, Parker, and Pack: Intermolecular potential surfaces 1397 are easily recovered from our fits to Va and Va, and it turns out that Va and Va are themselves useful quantities in the theory of scattering of atoms by TI-state mole- B. Fitting of EG potentials The difference between the two interaction energies V.(r, 0) and Vjr, 0) resulting from the calculations described above turned out to be much smaller than (ca. 10% of) either one. To treat this difference accurately in fitting V. and V_ to analytic forms, we chose to fit the average, (7) (6) (8) Vja =L v a (r) p (cose). n=2 where i =a or d. In this fitting, the EG results at r =9ao were not used because they were less stable and accurate than the rest of the data. In addition, It was found that at r = 3a0, Va was negative at several of the angles and somewhat irregular. Whether this is a real physical effect, an artifact of the EG model, or just due to the fact that Va is so much smaller than Va at small r as to be in the noise of the quadrature was not clear, and some data at r =3 was omitted in doing the fitting. This close-in region of the potential energy surface is and and the coefficients determined by Gaussian quadrature as in Paper I. This is equivalent to a least squares fit using optimized points and weights. However,as we showed elsewhere, 9 Va is best expanded in associated Legendre polynomials P: with m =2: Vja =L V a (r)Pn(cos8), j =SCF or COR, (5) n=O Now the P form a complete set but are not orthogonal under ordinary Gauss-Legendre quadrature. Hence the v a here had to be determined by ordinary linear least squares fitting. Thus we are not assured as good a fit to V:a as to Va' but the fit is probably as good as justified by the size and accuracy of the original Va' As in paper I, the V l (r) were fit to the analytic forms: v CFI(r) =A l exp(A 2r +A 3 r 2), Vi =V SCF' + VCORI' i =a or d, (4) and each of these was fit separately. V. was first expanded in Legendre polynomials, cules. L3 In addition, as we showed in determining van der Waals coefficients for this system,9 the natural Legendre polynomial expansions of Va and Va are different as detailed below. Both these interaction energies are given by the EG model as the sum of SCF and correlation (COR) contributions: (3) (2) Va=t(V.+VJ, (1) and half the difference, Va =t (V. - VJ, instead. The original potentials of the two states: The wavefunctions used were the Hartree-Fock functions of Green10 for NO and of Clementi 11 for Ar. The energies of the A'(+) and A"(-) states were calculated by storing the charge density due to the extra 11 orbital separately from the rest of the NO charge density and putting the pieces together with the different rp dependences discussed in the previous section. The 3-dimensional numerical quadrature L2 was carried out using Gauss-Legendre quadratures similar to those in I; 32 points were used in the