Random Phase Approximation RPA

In the next section, an attempt will be made to evaluate further details on the composition fluctuations as related to the experimental scattering function, /(q). 

The calculations will be based on the result (see Eq. (7.55)) that the pair correlation function is expressed in terms of linear response theory. The calculations wUl further be made using a mean-field approximation principally where the excluded volume effects, the density constrain and the interactions between chains are taken into account as perturbations, expressed in terms of potential energies. This calculation is called the random phase approximation (RPA). 

Initially, consider the case where polymers A and B are placed on the lattice at random, without any excluded volume effects or interaction energies. In this case, there is by definition no correlation in the positioning of polymers segments A and B, and the correlation term d4 / (R)d p (R + r)) obviously equals zero. The correlations (d y (R)d /) (R-Fr)) and (3(/)b(R)(3(/)b(R + r)) between A —A and B-B 

Equations (7.67)-(7.69) form a set of simultaneous equations for the unknowns d0A(R). 0b(R)- and V(R). To solve these equations, we will use the Fourier transform of d0(R) and V(R). In setting the formula Eq. (7.67) for the concentration fluctuation expressed in terms of the spatial correlation function, the Fourier transformed d0 (R) acquires the form 

After some mathematical rewritings, using the expression for the unperturbed structure factor of noninteracting polymer chains 

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RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

The simplest polarization propagator corresponds to choosing an HF reference and including only the h2 operator, known as the Random Phase Approximation (RPA). For the static case oj = 0) the resulting equations are identical to those obtained from a Time-Dependent Hartree-Fock (TDHF) analysis or Coupled Hartree-Fock approach, discussed in Section 10.5. 

Saue and Jensen used linear response theory within the random phase approximation (RPA) at the Dirac level to obtain static and dynamic dipole polarizabilities for Cu2, Ag2 and Au2 [167]. The isotropic static dipole polarizability shows a similar anomaly compared with atomic gold, that is, Saue and Jensen obtained (nonrelativ-istic values in parentheses) 14.2 for Cu2 (15.1 A ), 17.3 A for Ag2 (20.5 A ), and 12.1 A for Au2 (20.2 A ). They also pointed out that relativistic and nonrelativistic dispersion curves do not resemble one another for Auz [167]. We briefly mention that Au2 is metastable at 5 eV with respect to 2 Au with a barrier to dissociation of 0.3 eV [168, 169]. 

Using the random phase approximation (RPA), the coherent scattering intensity Icoh(Q, t) of a polymer blend/solvent or a diblock copolymer/solvent system can... 

In the random phase approximation (RPA), that is, when the dephasing is rapid in the timescale under consideration, we have... 

Eqs. (9),(11) the condition eq. (12) in the random phase approximation (RPA) formalism leads to a self-consistency relation between the symmetry potential and the Landau parameter F [26] ... 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

Diblock copolymers represent an important and interesting class of polymeric materials, and are being studied at present by quite a large number of research groups. Most of the scientific interest has been devoted to static properties and to the identification of the relevant parameters controlhng thermodynamic properties and thus morphologies [257-260]. All these studies have allowed for improvements to the random phase approximation (RPA) theory first developed by Leibler [261]. In particular, the role of the concentration fluctuations, which occur and accompany the order-disorder transition, is studied [262,263]. 

The vertical IPs of CO deserve special attention because carbon monoxide is a reference compound for the application of photoelectron spectroscopy (PES) to the study of adsorption of gases on metallic surfaces. Hence, the IP of free CO is well-known and has been very accurately measured [62]. A number of very efficient theoretical methods specially devoted to the calculation of ionization energies can be found in the literature. Most of these are related to the so-called random phase approximation (RPA) [63]. The most common formulations result in the equation-of-motion coupled-cluster (EOM-CC) equations [59] and the one-particle Green s function equations [64,65] or similar formalisms [65,66]. These are powerful ways of dealing with IP calculations because the ionization energies are directly obtained as roots of the equations, and the repolarization or relaxation of the MOs upon ionization is implicitly taken into account [59]. In the present work we remain close to the Cl procedures so that a separate calculation is required for each state of the cation and of the ground state of the neutral to obtain the IP values. 

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 LR fimction within the Random Phase Approximation (RPA) for a Hartree gas (HG) without XC is given by... 

Currently the time dependent DFT methods are becoming popular among the workers in the area of molecular modelling of TMCs. A comprehensive review of this area is recently given by renown workers in this field [116]. From this review one can clearly see [117] that the equations used for the density evolution in time are formally equivalent to those known in the time dependent Hartree-Fock (TDHF) theory [118-120] or in its equivalent - the random phase approximation (RPA) both well known for more than three quarters of a century (more recent references can be found in [36,121,122]). This allows to use the analysis performed for one of these equivalent theories to understand the features of others. 

The frequency-dependent polarizability, and therefore p(1> diverges when is equal to an excitation energy. Casida (54) made use of this property to derive a TDDFT equation for the excitation energy to an excited state J that is essentially the DFT form of the random phase approximation (RPA) equation ... 

Here q is a wavevector (eqn 1.6), ip(q) is the Fourier transform of />(r), and S(q) is the structure factor (Fourier transform of the two-point correlation function). The cubic term, ft, is zero for a symmetric system and otherwise may be chosen to be positive. The quartic term, y, is then positive to ensure stability. For block copolymers, these coefficients may be expressed in terms of vertex functions calculated in the random phase approximation (RPA) by Leibler (1980). The structure factor is given by... 

To date, there has been a very limited effort devoted to developing theory for ionic block copolymers. Gonziilez-Mozuelos and Olvera da la Cruz (1994) studied diblock copolymers with oppositely charged chains in the melt state and in concentrated solutions using the random phase approximation (RPA) (de Gennes 1970). However, this work has not been extended to dilute solutions. 

The theory of Broseta and Fredrickson (1990) was primarily for blends containing AB random copolymer. However, they also found a region of validity of this approach for block copolymers at high temperatures. Using the random phase approximation (RPA), the structure factor associated with fluctuations of the total A monomer volume fraction was found to be (Broseta and Fredrickson 1990)... 

The microscopic electronic structure of the buckyonion can be derived using an effective one-electron model where the screening effects are treated within the random phase approximation (RPA). The particular spherical geometry of these... 

The absorption spectrum is proportional to the imaginary part of the macroscopic dielectric function. Adopting the same level of approximation that we have introduced to obtain GW quasiparticle energies, i.e. neglecting the vertex correction by putting T = 55, we get the so called random phase approximation (RPA) for the dielectric matrix. Within this approximation, neglecting local field effects, the response to a longitudinal field, for q 0, is ... 

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