Self-consistent renormalization

Moriya and Kawabata (1973) made significant improvements to the RPA theory of spin excitations above by renormalizing the dynamic susceptibility. Their theory is known as the self-consistent renormalization (SCR) theory. In the SCR theory x q ) is given by... 

SCR self-consistent renormalization xS third-order paramagnetic... 

The other difference refers to the way in which the different groups approximate A. Thus, while VCP focus the attention on a renormalization of the approximated 3-RDM which uses the complementary holes matrix, NY again inspire their procedure on the assumption of an analogy with the Dyson equation. Finally, Mazziotti uses a self-consistent algorithm which may be sumarized as The A-RDM is calculated by means of an algorithm which coincides with the one proposed by NY, expressed in a compact form as ... 

In modest sized systems, we can treat the nondynamic correlation in an active space. For systems with up to 14 orbitals, the complete-active-space self-consistent field (CASSCF) theory provides a very satisfactory description [2, 3]. More recently, the ab initio density matrix renormalization group (DMRG) theory has allowed us to obtain a balanced description of nondynamic correlation for up to 40 active orbitals and more [4-13]. CASSCF and DMRG potential energy... 

Keywords strongly correlated electrons nondynamic correlation density matrix renormalization group post Hartree-Fock methods many-body basis matrix product states complete active space self-consistent field electron correlation... 

Zgid, D., Nooijen, M. The density matrix renormalization group self-consistent field method Orbital optimization with the density matrix renormalization group method in the active space. J. Chem. Phys. 2008, 128, 144116. 

Using the self-consistently obtained solutions of Eq. 19, the calculated chemical shift (Jiso = (To + a(T) is calculated and compared to the experimental data in Fig. 4. Even though the experimentally observed transition is broader than the calculated one, the agreement between theory and experiment is good. As the discontinuity in the lattice-related mode is small at Tc, where Tc corresponds to a = 0, the chemical shift does not show a discontinuity at Tc within numerical accuracy. It is important to note here that the S-shape in the cf T) data is a direct consequence of using the renormalized frequencies as defined in Eq. 19. 

To solve the full problem of finding an approximate ground state to Hamiltonian (13), one is faced to a self-consistent loop which can be proceeded in two steps. First one can get the occupations nia)o from a HWF, and a set of bare levels. Then one obtains a set of configuration parameters, the probabilities of double occupation, di by minimizing (18) with respect to these probabilities. Afterwards the on-site levels are renormalized according to (21) and the next loop starts again for the new effective Hamiltonian He// till convergence is achieved. 

MCT can be best viewed as a synthesis of two formidable theoretical approaches, namely the renormalized kinetic theory [5-9] and the extended hydrodynamic theory [10]. While the former provides the method to treat both the very short and the very long time responses, it often becomes intractable in the intermediate times. This is best seen in the calculation of the velocity time correlation function of a tagged atom or a molecule. The extended hydrodynamic theory provides the simplicity in terms of the wavenumber-dependent hydrodynamic modes. The decay of these modes are expressed in terms of the wavenumber- and frequency-dependent transport coefficients. This hydrodynamic description is often valid from intermediate to long times, although it breaks down both at very short and at very long times, for different reasons. None of these two approaches provides a self-consistent description. The self-consistency enters in the determination of the time correlation functions of the hydrodynamic modes in terms of the... 

In the formulations developed from the renormalized kinetic theory approach, these self-consistencies were avoided either by using values obtained from computer simulation and experiments or by using some exactly known limiting values for the transport coefficient. For example, in the treatment of Mazenko [5-7], and of Mehaffey and Cukier s [8] the transport coefficients are replaced by their Enskog values. In the theory developed by Sjogren and Sjolander [9], the velocity autocorrelation function is required as an input that was obtained from the computer simulated values. This limits the validity of the theories only to certain regimes and for certain systems where the experimental or computer-simulated results are available. 

Cu is no longer in evidence and the corrected AS values range from —3 to —7 eV. This energy arises from extraatomic relaxation, from charge renormalization and from any accumulated errors such as the omission of Self-consistent band calculations, made for all the metals except Zn, provide estimates of the renormalization shifts, — 3 eV for Ti and Cu and —4 to — 5 eV for the other metals (21). These values suggest quite modest contributions to AS from extraatomic relaxation. 

As HJ point out, in practice the above scheme is complicated by the slight overlap of the core densities on different atoms. In their work, the ground-state energy EA> its components EA and and core and valence densities Py(r) and Pc(r), respectively, are determined self-consistently for each constituent atom. The core density is then renormalized to a suitable radius Rc by the addition of a term... 

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