J . Phys. Chem. 1987, 91, 4245-4250
4245
Toward a Better Understanding of the Atom Superposition and Electron Delocalization Molecular Orbital Theory and a Systematic Test: Diatomic Oxides of the First Transition-Metal Series, Bonding and Trends Alfred B. Anderson,* Robin W. Grimes,+ and Sung Y. Hong Department of Chemistry, Case Western Reserve University, Cleveland, Ohio 441 06 (Received: January 27, 1987)
Advancement in the understanding of the physical basis of the atom superposition and electron delocalization molecular orbital (ASED-MO) theory is presented. This is followed by the results of a systematic study of the parameter dependence of calculated bond lengths, force constants, and bond energies of first-row transition-metal diatomic oxides. It is shown that, when metal VSIP are varied in 0.1-eV steps to produce the ionicity predicted from the difference between the oxygen and metal electronegativities,and oxygen VSIP are decreased by 1.5 eV, a 0.3 au decrease in oxygen valence Slater orbital exponents is a good average choice. Oxygen exponents decreased by 0.2 and 0.4 are nearly as good. When this simple parametrization scheme is used, calculated R,and k, values are generally closer to experiment than those from recent ab initio Hartree-Fock and configuration-interaction calculations. Variations in calculated and experimental dissociation energies are discussed in terms of variations in covalent stabilization, covalent destabilization, charge-transfer stabilization, and two-body repulsion energies.
Introduction The atom superposition and electron delocalization molecular orbital (ASED-MO) theory] has been applied in numerous diverse studies to predict structures, reaction mechanisms, and vibrational and electronic properties. The theory uses for input data valence state Slater orbital exponentsZand ionization potentials3 (VSIP) for the constituent atoms. These parameters are sometimes altered, particularly in treatments of ionic heteronuclear molecules, to ensure reasonably accurate calculations of ionicities and bond lengths in diatomic fragment molecules. In this way electronic charge self-consistency is taken into account and the same parameters are used in studying larger molecules. The parameter selection procedure used in the past is straightforward. For example, suppose one desires to study atom relaxations on surfaces of zinc oxide, ZnO, and the effect of CO adsorption on the surface atom relaxations, as in a recent study.4 Parameters for Zn and 0 are determined by analysis of diatomic ZnO. The Zn VSIP are increased and the 0 VSIP are decreased by equal amounts in 0.5-eV steps until the calculated charge transfer, from the Mulliken approximation for Slater orbitals, is close to 0.59 as predicted from the difference in electronegativity of zinc and oxygen. The oxygen valence s and p exponents are reduced simultaneously in 0.1 au steps until the calculations yield a reasonable diatomic bond length. In this case shifts of 2.5 eV and 0.2 au were used, yielding Zn0.8704,87and a bond length, Re,of 1.69 A. The diatomic bond length is not known, but this value lies in the range of known values for other oxides. Standard parameters were used for CO with the result that the CO bond energy, De, force constant, k,, bond length, and charge transfer were close to experimental values and the CO adsorption energy, effect on surface lattice relaxation, and vibrational frequency shift also agreed with experiment. In such a study the concepts of molecular orbital theory are used to interpret the results of the calculations and thereby to explain the experimental facts. When experimental data are unavailable, as for the bond length of CO adsorbed on zinc oxide, then the calculated results are predictions subject to future confirmation or correction. Although the above parameter selection procedure has been used many times, we have not before published a systematic study of the sensitivity of calculated Re, k,, and De to variations in exponent and VSIP shifts. This is the purpose of the present study. We have chosen 10 transition-metal oxide diatoms from the first ‘Present address: Department of Chemistry, University of Keele, Keele, , Staffs, ST5 SBG, England.
0022-3654/87/2091-4245$01.50/0
transition series, ScO through ZnO. This study defines and explains periodic trends and demonstrates the accuracy of the ASED-MO theory when compared to state-of-the-art ab initio calculations and to experiment.
ASED-MO Theory The electronic charge density function of a molecule or solid can be partitioned into components in any number of arbitrary ways. The ASED-MO theory is based on partitioning the densty function into free atom components and the rest. The atomic components are spherically symmetric in field-free space and are centered on the nuclei. They follow the nuclei “perfectly”. The remainder changes shape depending on geometry. With respect to the nuclei it is “nonperfectly following”. The theory is most easily depicted for diatomic molecules; its generalization to polyatomics is straightforward. For a diatomic molecule a-b, the perfectly following atomic densities, pa and Pb, and the nonperfectly following density, pnpf,are shown schematically in Figure 1. Some approximations to p n p fbased on self-consistent field calculations may be seen in other work.5 Placing the coordinate origin on nucleus a, the molecular density is given by p(r,
Rb)
= pa(r) + Pb(Rb-r)
+ Pnpf(r,Rb)
(1)
where r and R are electron and nuclear coordinates, respectively. The force on nucleus b is the sum of two components F(Rb)
= F(Rb*pa,za)
+ F(Rb,Pnpf)
(2)
which are evaluated according to electrostatics. The interaction energy is the integral of the force times -1 = ER(Rb)
+ Enpf(Rb)
(3)
where the subscript R indicates repulsive and ER(Rb) = -Zb[ ZaJRa- Rb1-l - l p a ( r ) l R b - rl-’ dr] (4) (1) Anderson, A. B. J Chem. Phys. 1975, 62, 1187. (2) (a) Richardson, J. W.; Nieuwpoort, W. C.; Powell, R. R.; Edgell, W. F. J . Chem. Phys. 1%2,36, 1057. (b) Clementi, E.; Raimondi, D. L. J . Chem. Phys. 1963, 38, 2686. (c) Basch, H.; Gray, H. B. Theor. Chim. Acra 1966, 4 , 367. (3) (a) Lotz, W. J . Opt. Sot. A m . 1970,60, 206. (b) Moore, C. E. Aromic Energy Leuels; NBS Circ. No. 467; National Bureau of Standards, U. S. Government Printing Office: Washington, DC, 1958. (4) Anderson, A. B.; Nichols, J. A. J . A m . Chem. SOC.1986, 108, 1385. (5) Bader, R. F. W.; Henneken, W. H.; Cade, P. E. J . Chem. Phys. 1967, 46, 3342; 1967, 47, 3381; 1969, 50, 5313.
0 1987 American Chemical Society
4246
The Journal of Physical Chemistry, Vol. 91, No. 16. 1987
Anderson et al. orbital occupation numbers (0, 1, or 2), ciab are calculated molecular orbital energies, and t: and e? are atomic orbital energies (-VSIP in practice). The Hamiltonian used so far in the ASED-MO calculations has the form' = -(VSIP):
Figure 1. Schematic of charge density components. Dashed contours represent depletion. Dots are the positions of the nuclei.
p
+ H,bb)Sijab exp(4.13R)
Hilab = 1.125(HiP
where Si, = (i,ak,b) and R is the internuclear distance. Thus the bond energy of a diatomic molecule is given in ASED-MO theory as
E = EER(a,b) 4- AEMo a>b
'Enpf Figure 2. Diatomic potential energy curve and energy components (eq 3 of the text) as functions of internuclear distance R.
Figure 2 shows schematically the energy components and their sum. ER is repulsive because the nuclear repulsion energy is greater than the attractive energy between nucleus b and pa. Enpr is attractive because of the concentration of charge in the internuclear region due to bonding. There is no net density in pnpC: SPnpf(r) dr = 0. If pnpf is assumed to behave like rigid point charges during molecular vibrations, then6
4TZbPa(Rb)
At the equilibrium internuclear value for Rb,Re, eq 6 yields the bond stretching force constant k,: ke
=
4TZbpa(Re)
(11)
For polyatomic molecules one sums over the two-body repulsion energies to get the total E R and adds AEMO:
R
=
(9)
(13)
Experience shows that, when the AEMo approximation is used to Enpf,superior results are usually obtained when ER is calculated by using the charge density of the more electronegative atom of the pair in eq 4. An understanding of this AEMo approximation to can be gained by looking at the one-electron Fock operator, F( l ) , for a closed-shell mole2ule. Let us suppose it can be written as a sum of atomjc parts, V,, which include electron repulsions on atom a, and Vab,which includes off-center electron repulsion: 1
+ xva a + axpab >b
= --VI2 2
Now we estimate the Fock matrix elements. Diagonal elements, neglecting all off-center contributions, become
where $ia is an eigenfunction and E: its eigenvalue. Off-diagonal elements are zero and the other off-diagonal elements are approximated as follows:
(7 1
Higher derivatives of eq 6 yield formulas for higher order force constants.' As shown in ref 7 and elsewhere,*predictions of force constants are usually more accurate when the density of the more electronegative atom is used in eq 6. Equation 7 has suggested a number of empirical relationships which depend on the exponential decay of the atomic electron density function away from the nucleus.' Except for providing the theoretical basis for the charge density formalism for force constants, eq 6 and 7, eq 3 is of no other obvious utility. This is because it is impossible to evaluate eq 5 in practice, though eq 4 is trivial. However, Enqris an energy due to electron delocalization and is roughly the difference between atomic and molecular orbital energies. Enpfis successfully approximated in this way
Dropping the sums in the latter
This is essentially a statement of approximations leading to the extended Hiickel matrix elements, Molecular orbital energies, el, are the solutions of the Fock Hamiltonian and the total electronic energy is
EM0 = x n l t l a b- Vee
(18)
I
where the sum is over the atomic or molecular orbitals, n, are the ~
~~~
( 6 ) Anderson, A. B. J . Chem. Phys. 1974, 60, 2417. (7) Anderson, A. B.; Parr, R. G.Theor. Chim. Acta 1972, 26, 301. (8) Anderson, A. B. Thesis, The Johns Hopkins University, 1970. (9) (a) Anderson, A. B.; Parr, R. G. Chem. Phys. Lett. 1971,10, 293. (b) Anderson, A. B. J . Mol. Spectrosc. 1972, 44, 411. (c) Anderson, A . B . J . Chem. Phys. 1975, 63, 4430.
where V, is the total electronic repulsion energy. In arriving at eq 15 and 16, all off-center intermolecular contributions to Vee were neglected, but on-center intramolecular contributions are included so long as appropriate values for the E, are employed. Therefore the total molecular orbital energy must be written as EMo = ~ n r c l a bCVeea l
a
(19)
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
Physical Basis of the ASED-MO Theory TABLE I: Parameters and Ionization Potentials, IP (ev), from Ref 3a 4s 4P
