Closed-shell molecules reference functions

However, in a large number of closed shell molecules, a single Slater determinant describes the ground state wave function fairly accurately. Even in such cases inclusion of excited state configuration results in substantial lowering of total electronic energy, and this is referred to as nondynamic electron correlation. 

Today we know that the HF method gives a very precise description of the electronic structure for most closed-shell molecules in their ground electronic state. The molecular structure and physical properties can be computed with only small errors. The electron density is well described. The HF wave function is also used as a reference in treatments of electron correlation, such as perturbation theory (MP2), configuration interaction (Cl), coupled-cluster (CC) theory, etc. Many semi-empirical procedures, such as CNDO, INDO, the Pariser-Parr-Pople method for rr-eleetron systems, ete. are based on the HF method. Density functional theory (DFT) can be considered as HF theory that includes a semiempirical estimate of the correlation error. The HF theory is the basie building block in modern quantum chemistry, and the basic entity in HF theory is the moleeular orbital. 

As already noted, all the examples in the preceding section refer to SCF ab initio wave functions, which do not take into account electron correlation. It is well known, however, that the SCF approximation at the Hartree-Fock limit is good enough to give a reliable representation of a one-electron, first-order observable like the electronic potential, at least for closed-shell ground-state systems like those considered here. If we keep to the field of SCF ab initio wave functions, the differences in accuracy between several wave functions for the same molecule depend upon the adequacy of the expansion basis set / employed in the calculations. In the next paragraph, however, we will also treat the case of semiempirical SCF wave functions. 

Cluster expansion representation of a wave-function built from a single determinant reference function [1] has been eminently successful in treating electron correlation effects with high accuracy for closed shell atoms and molecules. The cluster expansion approach provides size-extensive energies and is thus the method of choice for large systems. The two principal modes of cluster expansion developments in Quantum Chemistry have been the use of single reference many-body perturbation theory (SR-MBPT) [2] and the non-perturbative single reference Coupled Cluster (SRCC) theory [3,4]. While the former is computationally economical for the first few orders of the perturbation expansion... 

The next step is to develop a wavefunction. We will restrict our discussion to closed-shell atoms and molecules and to the most common approach that chemists take in solving Schrodinger s equation. That is, the wavefunction j is assumed to be a function of -electron coordinates with the nuclear coordinates frozen and is approximated by n one-electron functions referred to as orbitals. We will refer to these one-electron functions with the symbol x> > or i i, depending on the particular circumstance that we are discussing. But more about that later. We restrict our discussion to atoms for the moment. 

In practice, there is quite a difference between atoms and molecules as far as the representation of the one-particle spectrum is concerned. In atomic physics, one often utilizes a radial-angular representation of the one-electron orbitals in order to allow for the analytic integration over all spin-angular coordinates of the system [35,36]. This so-called angular reduction will be briefly discussed below in Subsection 4.4. However, in order to exploit the symmetry of free atoms the reference state must coincide with a closed-shell determinant o> = < >>, i.e. a reference state which should not depend on the magnetic quantum numbers of the one-electron functions. Unfortunately, the complexity of the perturbation expansions increases very rapidly if the number of electrons in the physical states of interest differ from (the number of electrons in) the reference state. 

The equilibrium geometries of all considered molecules were obtained in the previous study [108] at the all-electron CCSD(T) level in the correlation consistent core-valence triple- basis set (cc-pCVTZ) of Dunning [95] and Woon and Dunning [140]. For the hydrogens contained in the molecules, the cc-pVTZ basis set was used [95]. All the molecules are closed-shell species, therefore the RHF reference wave function was used as the reference for the geometry relaxation. The equilibrium structures are presented in Figures 8, 9, 10, 11 and 12. 

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