Ground state approximate methods

A more sophisticated ground-state approximate energy functional can be constructed using the frequency-dependent response function of linear response TDDFT. We now introduce the basic formula and then discuss some of the systems this method is being used to study. 

While it is not essential to the method, frozen Gaussians have been used in all applications to date, that is, the width is kept fixed in the equation for the phase evolution. The widths of the Gaussian functions are then a further parameter to be chosen, although it appears that the method is relatively insensitive to the choice. One possibility is to use the width taken from the harmonic approximation to the ground-state potential surface [221]. 

The simplest molecular orbital method to use, and the one involving the most drastic approximations and assumptions, is the Huckel method. One str ength of the Huckel method is that it provides a semiquantitative theoretical treatment of ground-state energies, bond orders, electron densities, and free valences that appeals to the pictorial sense of molecular structure and reactive affinity that most chemists use in their everyday work. Although one rarely sees Huckel calculations in the resear ch literature anymore, they introduce the reader to many of the concepts and much of the nomenclature used in more rigorous molecular orbital calculations. 

The complexity of molecular systems precludes exact solution for the properties of their orbitals, including their energy levels, except in the very simplest cases. We can, however, approximate the energies of molecular orbitals by the variational method that finds their least upper bounds in the ground state as Eq. (6-16)... 

In a molecule with electrons in n orbitals, such as formaldehyde, ethylene, buta-1,3-diene and benzene, if we are concerned only with the ground state, or excited states obtained by electron promotion within 7i-type MOs, an approximate MO method due to Hiickel may be useM. 

A number of MO calculations has been carried out, and these have had mixed success in predicting chemical reactivity or spectroscopic parameters such as NMR chemical shifts and coupling constants. Most early calculations did not take into account the contribution of the sulfur 3d-orbitals to the ground state, and this accounts for some of the discrepancies between calculations and experimental observations. Of the MO methods used, CNDO/2 and CNDO/S have been most successful the INDO approximation cannot be used because of the presence of the sulfur atom. 

Furthermore, LandS s theory only represents a first-order approximation, and the L and S quantum numbers only behave as good quantum numbers when spin-orbit coupling is neglected. It is interesting to note that the most modem method for establishing the atomic ground state and its configuration is neither chemical nor spectroscopic in the usual sense of the word but makes use of atomic beam techniques (38). 

The Brueckner-reference method discussed in Section 5.2 and the cc-pvqz basis set without g functions were applied to the vertical ionization energies of ozone [27]. Errors in the results of Table IV lie between 0.07 and 0.17 eV pole strengths (P) displayed beside the ionization energies are approximately equal to 0.9. Examination of cluster amplitudes amd elements of U vectors for each ionization energy reveals the reasons for the success of the present calculations. The cluster operator amplitude for the double excitation to 2bj from la is approximately 0.19. For each final state, the most important operator pertains to an occupied spin-orbital in the reference determinant, but there are significant coefficients for 2h-p operators. For the A2 case, a balanced description of ground state correlation requires inclusion of a 2p-h operator as well. The 2bi orbital s creation or annihilation operator is present in each of the 2h-p and 2p-h operators listed in Table IV. Pole strengths are approximately equal to the square of the principal h operator coefiScient and contributions by other h operators are relatively small. 

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