Hierarchies of Ab Initio Theory

An important characteristic of ab initio computational methodology is the ability to approach the exact description - that is, the focal point [11] - of the molecular electronic structure in a systematic manner. In the standard approach, approximate wavefunctions are constructed as linear combinations of antisymmetrized products (determinants) of one-electron functions, the molecular orbitals (MOs). The quality of the description then depends on the basis of atomic orbitals (AOs) in terms of which the MOs are expanded (the one-electron space), and on how linear combinations of determinants of these MOs are formed (the n-electron space). Within the one- and n-electron spaces, hierarchies exist of increasing flexibility and accuracy. To understand the requirements for accurate calculations of thermochemical data, we shall in this section consider the one- and n-electron hierarchies in some detail [12]. 

A standard method of improving on the Hartree-Fock description is the coupled-cluster approach [12, 13]. In this approach, the wavefunction CC) is written as an exponential of a cluster operator T working on the Hartree-Fock state HF), generating a linear combination of all possible determinants that may be constructed in a given one-electron basis, 

The cluster operator T creates excitations out of the Hartree-Fock determinant and may be written as 

To understand the structure of the coupled-cluster wavefunction, let us Taylor expand the exponential in Eq. (2.1). Sorting the resulting expansion according to the level of excitation, we obtain 

At each excitation level beyond the single-excitation level, a number of terms contribute. For example, double excitations are generated both by means of the double-excitation operator T2 (connected excitations) 

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For the one-electron and //-electron hierarchies of ab initio theory to be useful, it is necessary to carry out a careful and extensive calibration of their performance. Such a calibration is best carried out by comparing, in a statistical manner, calculated values of different properties with experimental measurements. In the present section, we give two examples of such comparisons, for bond lengths and reaction enthalpies. In Figs. 6 and 7, we have plotted the normal distributions of the errors in calculations of bond distances and reaction enthalpies, respectively. The statistics underlying the normal distributions are based on calculations on 19 closed-shell first-row molecules (e.g., HF and C2H4) and 13 reactions involving these molecules (for more details, see Refs. 1 and 17). 

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