Density response

This functional can directly be used in the DFPT equations above. Further, for the special case of a magnetic perturbation it can be shown that the response density p(1) analytically vanishes, making the calculation significantly easier. The working equations simplify to ... 

Because of the separation into a time-independent unperturbed wavefunction and a time-dependent perturbation correction, the time derivative on the right-hand side of the time-dependent Kohn-Sham equation will act only on the response orbitals. From this perturbed wavefunction the first-order response density follows as ... 

For MBPT and CC methods, evaluation of the reduced density requires determining a response vector (A) as well as T. This defines a response density p = e Oo)(o (l + A)e. In addition, we want to allow the molecular orbitals to relax. The latter consideration adds another term, p", to the one-particle density. This relaxed density, p = p -I- p", is the critical quantity in CC and MBPT analytical gradient (and property) methods. " For just the one-particle part, we have p(l) = p (l) -I- p" = D(l) which will show up again when we discuss properties. 

Using Ek]s.(2.349) and (2.350) we can express the response density function in the dielectric constant One finds for the bulk ... 

Giese, T. J., and York, D. M. (2005). Improvement of semiempirical response properties with charge-dependent response density, /. Chem. Phys. 123(16), 164108. 

Global minimum of bonding is attained by optimizing the convolution of the applied potential with the response density (Putz 2003, 2008a, 2009a, 2011a)... 

This approximate form can be derived many ways from renormalized second-order perturbation theoryGaussian (linear response) density field theory,or via density functional expansions. Pictorially, the medium-induced potential between a pair of tagged sites is determined by coupling to the surroundings via an effective potential (direct correlation function), which is mediated by density fiuctuations of the condensed phase. The integrated strength of the medium-induced potential is... 

The components sj " (m = 0, 1) are the vector operator components of the total spin. The spin-densities are the response densities with respect to a SOC perturbation. They are calculated from a non-standard set of coupled-perturbed equations analogous to the ones described above for the g-tensor as... 

One-electron properties calculated as energy derivatives are closely related to the response density. Provided that the basis set chosen is independent of the perturbation the corresponding one-electron property is given as the product of the response density matrix Zy with the corresponding property integrals. 

Typical MP correlation effects are reflected by the response density distribution as discussed for the case of the CS molecule. In CS, one of the electron lone pairs of S can be shared between the two atoms thus establishing a semipo-lar bond beside the two normal bonds. This leads to charge transfer from S to C and relatively large partial charges at C and S contrary to what one might expect in view of the similar electronegativities of the two elements This is... 

The same features of p(r) are found at the MP3, MP4(SDQ), MP4(SDTQ), and even higher correlation corrected levels. Qualitatively, there are no differences in the corresponding response densities which means that MP2 already includes the most important correlation corrections. However, the difference density pres,MP2 shown in... 

Figure 13(b) reveals that MP3 correlation corrections reduce MP2 effects, i.e., the MP2 response density is slightly changed back into the direction of the HF electron density distribution. 

The changes in the response density distribution of CS are parallel to calculated changes in other one-electron properties of CS. Most properties oscillate dependent upon the order of PT applied where HF and MP2 results often represent the upper and lower bounds of computed values. Oscillations in calculated one-electron properties are observed in many cases (charges, dipole moments, quadrupole moments, electric field gradients, nuclear quadrupole moments, etc.) and are largely independent of the basis set used. In general, one can draw the following conclusions ... 

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