Total Energies and the Hartree-Fock Limit

In contemporary theories, a is taken to be and correlation energies are explicitly included in the energy functionals [15]. Sophisticated numerical studies have been performed on uniform electron gases resulting in local density expressions of the form F j.[p(r)] = K [p(r)] -l- F. [p(r)] where represents contributions to the total energy beyond the Hartree-Fock limit [22]. It is also possible to describe the role of spin explicitly by considering the charge density for up and down spins p = p -i- p. This approximation is called the local spin density approximation [15]. 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

Double-2eta basis sets were introduced by Roetti and Clementi to provide greater flexibility in the orbital expansion and to avoid the need to reoptimize the orbital exponents when the basis set is used in a molecular calculation. Double-zeta basis sets contain two functions for every function in a minimum basis set. The accuracy which can be achieved in calculations of total energies using such basis sets is illustrated in Table IV where calculations using double-zeta basis sets are compared with those using minimum basis sets and the Hartree-Fock limit. 

A more useful quantity for comparison with experiment is the heat of formation, which is defined as the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states. The heat of formation can thus be calculated by subtracting the heats of atomisation of the elements and the atomic ionisation energies from the total energy. Unfortunately, ab initio calculations that do not include electron correlation (which we will discuss in Chapter 3) provide uniformly poor estimates of heats of formation w ith errors in bond dissociation energies of 25-40 kcal/mol, even at the Hartree-Fock limit for diatomic molecules. 

As another example, the potential surface of the He-CH4 complex (studied in Ref. lOe) is described. In these calculations a basis set 7s6p3dlflg on the C atom, 4s3pld on the H atoms, and 8s4p2dlf on the He atom, consisting of a total 182 functions, was adopted. The s and p functions were optimised so as to reproduce energies close to the Hartree-Fock limit of the CH4 molecule and He atom, respectively. The exponents of high-order polarisation functions were determined by maximising directly the dispersion contribution. 

When comparing STO and CGTF basis sets of a particular size, it should be taken into account that with STO basis sets the results are only due to exponents of STO s, whereas with the CGTF basis sets also the effects of the number of primitives and their contractions are involved. Furthermore it should be kept in mind that the total energy is a rather insensitive test of the quality of wave functions, so that also some other molecular properties should be considered. Finally, a rigorous comparison should also include calculations going beyond the Hartree-Fock limit. 

With sufficiently large basis set, the Hartree-Fock (HF) method is able to account for 99% of the total energy of the chemical systems. However, the remaining 1% is often very important for describing chemical reaction. The electron correlation energy is responsible for the same. It is defined as the difference between the exact nonrelativistic energy of the system ( 0) and Hartree-Fock energy (E0) obtained in the limit that the basis set approaches completeness [36] ... 

Structure calculation in which the only surviving residue is the relativistic correction to the energy. So, although the total electronic energies fluctuate by 0.1 a.u. in this study, and the results for the largest basis sets are at least 0.005 hartree above the numerical Hartree-Fock limit in all cases, the fluctuation in the relativistic corrections, Er, are significantly less than 10 hartree, which is more than sufficiently accurate for the present study. 

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