Ground-state correlations

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. 

In the 1950s, many basic nuclear properties and phenomena were qualitatively understood in terms of single-particle and/or collective degrees of freedom. A hot topic was the study of collective excitations of nuclei such as giant dipole resonance or shape vibrations, and the state-of-the-art method was the nuclear shell model plus random phase approximation (RPA). With improved experimental precision and theoretical ambitions in the 1960s, the nuclear many-body problem was born. The importance of the ground-state correlations for the transition amplitudes to excited states was recognized. 

The local ground-state correlation potential is defined in RDFT as the functional derivative of Eq.(7) with respect to p. When infinitesimal variation of occupation numbers is allowed, a more practical definition follows from the fact that the unsymmetrical energy formuala used to construct Eq.(7) is itself a Landau functional of the occupation numbers [19]. Correlation energies of Landau quasiparticles, expressed as diagonal elements of a one-electron Hamiltonian matrix, are defined by differentiating with respect to occupation numbers to give... 

We shall now analyze the self-energy 2j(E) of the hole in a way that connects the physical picture of the relaxation process with the more technical pictures based on expansions in terms of one-electron basis orbitals having the full symmetry of the system, involving concepts like monopole relaxation and shake-up, fluctuation and correlation. To this end, let us divide the self-energy -Xj(E) into two parts corresponding to relaxation (R) (Figs. 9e, f) and Fermi sea (ground-state) correlation (C) (Fig. 9g)... 

Although there is reasonable agreement between theory and experiment, there is still about 1 eV shift to be accounted for and the main line carries too much strength compared with the satellites. However, this can possibly be explained by the fact that the 5p (5p 5d 3P, 3D) levels were not included in the calculation and also that ground-state correlations were omitted. 

Clearly, the monopole relaxation and ground-state correlation shifts approximately cancel, and the HF MO eigenvalues should lead to reasonable binding energies for the valence levels provided there were no other important interaction mechanisms. 

For the general case of model (2) the calculation of the singlet ground-state correlation functions can be performed in a similar way. The final result in the thermodynamic limit is [14, 11] ... 

Recently, the response theory formalism was applied for the calculation of singlet-triplet spectra in polyenes and aromatic hydrocarbons [83,133,134,132]. In this section we shall discuss mainly the vertical Tn <— So transitions probability in polyenes. The minimum level of accuracy includes CAS calculations of the ground state correlating all x electrons followed by linear and quadratic multi-configuration response calculations. Because the ethene molecule is the simplest representative of unsaturated hydrocarbons, different types of active spaces and AO basis sets were tested more thoroughly for this molecule in order to study the convergence of the S-T transition probability with respect to the level of accuracy. 

If the valence band is completely filled in a polymer (or in a molecular crystal) we can subdivide the ground-state correlation into a long range and a short range part. For the long range correlation the electronic polaron model33 can be used. It has been applied already to the periodic DNA model polycytosine (polyC),34... 

There are states at 35.63 eV and 37.15 eV that have the 3p momentum distribution (fig. 11.7(b) and (c)). Fig. 11.7(b) includes both the 3p momentum distribution with S35,63(3p) = 0.01 and 0.67 of the cross section calculated with full correlation by Mitroy et al (1984), marked CI(/ = 1). The ground-state correlations cause a small difference in shape. The respective observed spectroscopic factors 0.01 and 0.03 for the two states agree with a number of many-body calculations of the 3p manifold. 

There is a Cl-like eigenvalue equation for the excitation energy, AE, except that the Hamiltonian (H) has been replaced by H and includes the ground state correlation. Again, in practical calculations, the T operator must be truncated to a suitable level of excitation, typically, T = + Ti, which... 

---