Renormalized Energy Functional

As we said in the introduction, the only consistent framework for a relativistic many-electron system is QED. By means of the Hartree-Fock limit of this theory, after renormalization, and using gradient techniques, Engel and Dreizler [22] found a complete energy functional where both terms of the two previous sections appear naturally. 

Due to the dependence of / on active nuclear distortions, the Heisenberg exchange interaction, equation (38), summed up over all nearest neighbour centres in a chain, should be added from the start to the energy functional in equation (32). Equation (41) shows, however, that this addition merely leads to the renormalization of the effective force constant Kt in equation (32) by... 

D. Shalloway, Application of the renormalization group to deterministic global minimization of molecular conformation energy functions. Journal of Global Optimization 2 (1992), 281. 

The additional contributions to the renormalized Greens functions resulting from renormalization are usually called counterterms. From Eqs,(227),(230) one extracts as counterterms to the self-energy,... 

These are given in Eqs. (C29) and (C38). The self-energy expressions (C38) and (C40) are calculated perturbatively to second order in the electron-phonon coupling in terms of the zeroth order Green functions (Eq. (55)). The simplest expression for current is obtained by substituting Eqs. (55), (C29) and (C38) in Eq. (51). This zeroth order result can be improved by using the renormalized Green functions obtained from the self-consistent solution of the Dyson equation (44). 

The application of direct variational method to Eq. (3.51) yields, similarly to the previous case of cylindrical particles, the free energy function with renormalized coefficients... 

Finally, the total RHF band energy per cell, band> calculated. Eband defined by the standard atomic expressions but is computed with the renormalized wave functions, including the self-consistent average 5d and 6s conduction band wave functions and occupation numbers. Eband differs from E ai of eq. (3) only by the exclusion of the Hund-rule correction for the 4f electrons discussed in section 2.2. 

Equation (23) expresses the exact density of states as the solution of an integral equation, the kernel of which is the renormalized energy correlation function D( , ). The function D(E, ) has the following simple physical interpretation. For a given , D( , ) shows how the density po(E ) i-s dispersed over the range of energies in the transformation to the exact density of states p( ). The calculation of the exact... 

Fig. 3.5. Renormalized energy levels for cations (magnesium s level full line) and for anions (the highest oxygen p level dashed line) at the surface of thin unsupported MgO films, as a function of their number of layers n, for three surface orientations. Continuous lines are drawn to guide the eye.

In the opposite case of slow flip limit, cojp co, the exponential kernel can be approximated by the delta function, exp( —cUj t ) ii 2S(r)/coj, thus renormalizing the kinetic energy and, consequently, multiplying the particle s effective mass by the factor M = 1 + X The rate constant equals the tunneling probability in the adiabatic barrier I d(Q) with the renormalized mass M, ... 

Once the functions / and g have been established the blocking procedure may be repeated many times to produce a sequence of effective coupling constants K0, Ki, K2 , and fields bo, bx, b2..., together with regular parts of the free energy Gro, Gr, Gr2 The operation of finding the new values of K, b and Gr from the old is renormalization and the set of them all constitute the RG. 

Each of the reconstructions contains many contributions from higher orders of perturbation theory via the 1- and 2-RDMs and thus may be described as highly renormalized. The CSE requires a second-order correction of the 3-RDM functional to generate second-order 2-RDMs and energies, but the ACSE can produce second-order 2-RDMs and third-order energies from only a first-order reconstruction of the 3-RDM. 

That is precisely which is reported say in [123] on example of Pd complexes (and for other systems in Ref. [124]) the TDDFT excitation energies are systematically lower than the experimental ones. In this context it becomes clear that the TDDFT may be quite useful for obtaining the excitation energies in those cases when the ground state is well separated from the lower excited states and can be reasonably represented by a single determinant wave function may be for somehow renormalized quasiparticles interacting according to some effective law, but shall definitely fail when such a (basically the Fermi-liquid) picture is not valid. 

The parameter X has been embedded in the definition of Hp. The wave function from perturbation theory [equation (A.109)] is not normalized and must be renormalized. The energy of a truncated perturbation expansion [equation (A.110)] is not variational, and it may be possible to calculate energies lower than experimental. ... 

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