Polarizability kernel

For larger displacement UA-, the variation of i(r, r ) relative to UK can be computed by using the nonlinear polarizability kernels defined below [26] (see Section 24.4). Forces and nonlocal polarizabilities are thus intimately related. 

One uses a simple CG model of the linear responses (n= 1) of a molecule in a uniform electric field E in order to illustrate the physical meaning of the screened electric field and of the bare and screened polarizabilities. The screened nonlocal CG polarizability is analogous to the exact screened Kohn-Sham response function x (Equation 24.74). Similarly, the bare CG polarizability can be deduced from the nonlocal polarizability kernel xi (Equation 24.4). In DFT, xi and Xs are related to each other through another potential response function (PRF) (Equation 24.36). The latter is represented by a dielectric matrix in the CG model. 

The second derivative with n = 2 and m = 0 (a E/5v(r) 5v(r )), j corresponds to the linear response function, also called the polarizability kernel, and will not be considered in this contribution, as is also the case for higher derivatives [14], Quantities such as %, ti and S are global quantities as they are a property of the system as a whole, whereas position dependent quantities such as f(r) are local quantities. A local version of the softness, s(r), was successfully proposed by Parr and Yang [15] as... 

This quantity can be viewed as a generalization of Fukui s frontier MO concept and plays a key role in linking Frontier MO Theory and the HSAB principle. It can be interpreted either as the sensitivity of a system s chemical potential to an external perturbation at a particular point r, or as the change of the electron density p(r) at each point r when the total number of electrons is changed. The former definition has recently been implemented to evaluate this function,but the derivative of the density with respect to the number of electrons remains by far the most widely-used definition. The second order derivative of the energy with respect to the external potential is the linear response function xCi tO called the polarizability kernel... 

In any chemical reaction, the approaching molecular systems experiences both electron transfer (in some cases, spin polarization) and external potentials changes while the interacting system evolves towards the final state. Behind the perturbative approximation we are here concerned, and within the context of the [A( , Np, v (r), v (r)] representation of spin polarized DFT, the nonlocal descriptors are defined as first (and higher) order derivatives of the electron density of a given spin p (r) with respect to the spin external potentials Vo-(r). In particular, the symmetric linear response (or polarizability) kernels, defining the spin density... 

Figure 6 Singly occupied molecular orbital (SOMO) of a propeller-like trimer radical anion of acetonitrile obtained using density functional theory. The structure was immersed in a polarizable dielectric continuum with the properties of liquid acetonitrile. Several surfaces (on the right) and midplane cuts (on the left) are shown. The SOMO has a diffuse halo that envelops the whole cluster within this halo, there is a more compact kernel that has nodes at the cavity center and on the molecules.

Table 1 shows a gradual improvement in the calculated a along the series vx A < vxc < vxc°P < vxc- The LDA leads, for these relatively hard systems, to a polarizability that is systematically too high, which is related to the excess energies of the occupied orbitals, see below. Note, that the GGA-BP gradient correction of the xc potential produces only a relatively small reduction in the LDA/ALDA average absolute error from 8.8% to 5.6%. The improved SAOP potential reduces the error substantially to 2.9%. Still, further significant improvement is achieved with the accurate xc potential the error from the combination (accurate vxc)/ALDA is only 1.0% (See Table 1). Therefore, the crucial improvement of the TDDFRT results for the molecules considered is achieved with just an alteration of the xc potential, while keeping ALDA for the xc kernel.

TDDFT with an OEP (TDOEP) for excitation energies [109, 110] and frequency-dependent polarizabilities [111] has the same working equations (2-17), (2-21), or (2-22) with (2-60) and (2-61). The corresponding exchange-correlation kernel derived by Gorling [112, 113] is frequency dependent ... 

The ease with which the valence electrons are displaced from their resting position depends to a very large extent on the attraction the atomic kernel of X has for electrons. The greater affinity for electrons it has, the closer it holds them to itself, and the more difficult it is for an approaching ion to displace them. Consequently, polarizability decreases in moving across a row in the periodic table, and the following series of decreasing bond polarizabilities is observed ... 

Table 5. The static and dynamic average polarizability and the second hyperpolarizability of the nitrogen molecule in the aug-cc-pVTZ basis set. The HF, the DFT Potential/Kernel combinations and the CCSD methods were used. Reproduced from [42]...

Recent theoretical work in this respect [60] has shown (via a local approximation to the softness kernel) that the static dipole polarizability is linearly dependent on the global softness. 

For the central ion of a complex the character of the electronic shell is connected with the nature of the atomic orbital. The s, p, d, and f electrons are characterized by an increasing distance to the kernel and therefore can be seen as increasingly polarizable electrons. In the language of Pearson the character of these electrons varies from extremely hard (s-electrons) to extremely soft (f-electrons). The hardness increases with ionization and for different valance states with an increased oxidation state. 

Any evaluation of the polarizability demands the knowledge of the linear response function. The general expression for this function in DFT can be formulated as in Eq. (4.197) with in terms of softness kernel, the local and the global softness, respectively, all related with the introduced local response function L r). However, for atomic systems a very sensitive approximation consists in neglecting all non-local contributions in the linear response function (Putz et al., 2003, 2012b,c) ... 

In Eq. [4.3.5] is the instantaneous part of the solvent response, associated with its electronic polarizability. For simplicity we limit ourselves to the Debye model for dielectric relaxation in which the kernel s in [4.3.5] takes the form... 

Analysis of molecular orbitals obtained by accurate calculations on the dihalides of the heavier Group 2 metals Ca, Sr, and Ba indicate that valence shell d orbitals are involved in the bonding and that the atomic kernels (corresponding to ions) are polarized in the molecules that are angular [10], It would seem that both the polarizable ion and the hybridization model have captured a part of reaUty. 

Varsano et al have shown that a many-body derived XC kernel can explain within TDDFT the optical saturation in molecular chains in terms of excitonic confinement. This has been illustrated for both the static and dynamic polarizabilities of H2 and polyacetylene chains of finite or infinite length. 

The static polarizability of K clusters containing an even number of atoms ranging from 2 to 20 has been calculated at different levels of theory encompassing the ab initio MP2 and CCSD(T) methods as well as using TDDFT with the LDA or SAOP potential in combination with a LDA kernel . Sufficiently large basis sets have been employed. For the small K2 and K4 clusters, MP2, CCSD(T), and TDDFT results are in close agreement. Then, for clusters up to K14, the agreement between TDDFT/SAOP and MP2 is excellent while the TDDFT/LDA values are systematically... 

The polarizability and first hyperpolarizability of two dipolar organoiron(ii) compounds have been evaluated at the TDDFT/BLYP/ZORA level of approximation with LDA kernels in order to rationalize trends in SHG measurements of the guest molecules in polymer matrices. 

Keywords Density response function Polarization propagator Density response kernel Generalized polarizability Polarizability Conceptual density functional theory Random phase approximation... 

The temperature in the canonical BOMD simulation was controlled by a Nos6-Hoover chain thermostat (Hoover 1985 Martyna et al. 1992 Nos6 1984). In order to study the temperature dependency of the sodium cluster polarizabilities the polarizability tensor a was calculated along the recorded trajectories. For this purpose the first 20 ps of each trajectory were discarded and a was then calculated in 100 fs time steps along the remaining 200 ps. Due to the computational demand of the analytical polarizability calculation along the BOMD trajectories we employed the LDA kernel. Thus, the computational level for the calculation of the temperature dependent part of the cluster polarizabilities was VWN/TZVP-FIP/GEN-A2. The temperature dependent mean sodium cluster polarizability was then calculated as ... 

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