Hartree-Fock approximation potential energy surfaces

There are certain types of system for which this approximate parallelism between the Hartree-Fock and exact potential energy surfaces holds not only in the region of the equilibrium geometry, but over the total surface. In such cases, one can obtain a Hartree-Fock description of the complete chemical change, a description at a level of approximation, which, from the above examples, is clearly adequate for answering many if not most of the questions concerning the static behaviour of the system along the so-called reaction pathway. It is to a discussion of such systems that we now turn. 

However, it should be emphasized that use of these less rigorous methods for the calculation of potential-energy surfaces should be viewed with great caution. Even for a system which dissociates properly in the Hartree-Fock approximation, recent research by Kaufman and co-workers [142] has shown that even the INDO method is not capable of giving an accurate or even realistic surface for Li+ + H2 when compared point by point to Lester s accurate Hartree-Fock surface [117]. 

Fig. 1. Schematic representation of the potential energy surface for the electronic (el) ground state of a molecule existing in two tautomeric forms, A and B. Superscripts exp, HF, CNDO/2, MINDO/3 indicate that energy differences 8 a,b calculated for potential energy surfaces determined either experimentally (exp) or calculated by means of ab initio method in the Hartree-Fock (HF) approximation or by semiempirical methods (CNDO/2, MINDO/3). The symbol eq stands for the geometrical equilibrium of both tautomers, while 2a and Qb indicate nonequilibrium geometries of tautomers A and B, respectively. Note that the theoretical potential surface calculated by sophisticated quantum-mechanical methods ( exact solution of electronic Schrbdinger equation includes electron correlation with geometry optimization) should be the same (or very similar) as that determined experimentally [in this case i>eor) ei

Exact calculations of the potential energy surfaces for complex molecular systems are impossible to carry out from a practical point of view. Such calculations involve the solution of the electronic Schrodinger equation for the system including electron correlation effects and full geometry optimization. However, an estimate of the 8 A lq> value can be obtained in a different way. One can carry out an ab initio calculation in the Hartree-Fock (HF) approximation by using a simple basis set, e.g., (7s, 3p/3s), contracted to a minimal basis set, STO-3G, or 3-21G, etc., with full geom-... 

The extension of the basis can improve wave functions and energies up to the Hartree-Fock limit, that is, a sufficiently extended basis can circumvent the LCAO approximation and lead to the best molecular orbitals for ground states. However, this is still in the realm of the independent-particle approximation 175>, and the use of single Slater-determinant wave functions in the study of potential surfaces implies the assumption that correlation energy remains approximately constant on that part of the surface where reaction pathways develop. In cases when this assumption cannot be accepted, extensive configuration interaction (Cl) must be included. A detailed comparison of SCF and Cl results is available for the potential energy surface for the reaction F + H2-FH+H 47 ). 

Ideally, the best approach would be to be able to solve equation 1 directly to obtain the potential energy surface for the system. The most accurate way of doing this is by using one of the classes of ab initio QM methods that have been developed to solve equation 1 with as few as approximations as possible. Popular ab initio algorithms are Hartree-Fock (HF) molecular orbital (MO) [3] and density functional theory (DFT) methods [4]. The problem with all these techniques is that they are expensive to apply and are generally limited to handling relatively small systems (of a few tens of atoms at the most). As we shall see in section 3.3, recent algorithmic advances have improved this situation somewhat [5], but quicker methods are needed nevertheless. 

Fig. 14.1 Energy relationships for weakly interacting molecules. Solid curves indicate a ground-state potential-energy surface and part of an excited-state surface the broken line being an approximation (e.g. Hartree-Fock) to the former. Typically, the required interaction energy (a) is less than 10 of the correlation error (b), while the correlation and excitation energies (b, c) are of the same order of magnitude (several eV).

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