Pole approximation

T ) obtain the Schrodinger equation for the interaction of a molecule with the quanted radiation field, that is, the Schrodinger equation for the (matter + radiation) fr steni. we need the quantum analog of 77MR, the matter-radiation interaction, hi the 1,1 pole approximation HMR depends, according to Eq. (1.51), on the transverse... 

At small k(long-wave approximation) in a onemode pole approximation this dependence has the form 4>(k) = [4tt/(1 — 1/e)] (l + Ak2) where A is the correlation length. This form corresponds to a gradient expansion of the Landau free energy functional... 

Eq. (12) is much smaller than the resonance frequency >eg and the frequency range over which fhe infegrand of Eq. (16) varies significantly, the "pole approximation" [7,8, 32] may by applied so that Eq. (16) reduces to... 

Eq. (18) and the validity of the pole approximation). It demonstrates that in the radiation zone the electromagnetic field energy is concentrated completely in the polarization direction e. Integrating Eq. (28) over all space (thereby neglecting contributions outside the radiation zone) the electric field energy at time t is given by... 

Implementation of an All-Electron GW Approximation Based on the Projector Augmented Wave Method without Plasmon Pole Approximation to Si, SiC, AlAs, InAs, NaH and KH. 

In all these transitions, LSD numbers are substantially in error compared to the HF theory results. On the other hand, results obtained by employing the MLSDSIC functional are highly accnrate. This shows the correctness of the MLSDSIC functional for excited states. We also compare these numbers with those obtained by applying TD-DFT within a single-pole approximation. The A-SCF numbers are better than the TD-DFT nnmbers. 

We again see that the MLSDSIC functional gives transition energies that are very close to the corresponding HF values. On the other hand, single-pole approximation TD-DFT in this case does not improve the LSD numbers by any significant amount. 

Note The last column presents energies obtained using TD-DFT within a single-pole approximation. Numbers are given in atomic units. 

A review of the approximations in any time-depedendent density functional calculation of excitation energies is given. The single-pole approximation for the susceptibility is used to understand errors in popular approximations for the exchange-correlation kernel. A new hybrid of exact exchange and adiabatic local density approximation is proposed and tested on the He and Be atoms. 

T ble 2. Same as Table 1, but within the single pole approximation. 

By neglecting the higher-order terms, a simple manipulation of (4.101) yields the so-called single-pole approximation (SPA) to the excitation energies... 

Table 4.3. Comparison of the excitation energies of neutral helimn, calculated from the exact xc potential [49] by using approximate xc kernels. SPA stands for single pole approximations , while fuU means the solution of (4.107) neglecting continuum states. The exact values are from a non-relativistic variational calculation [53]. The mean absolute deviation and mean percentage errors also include the transitions from the Is until the 9s and 9p states. All energies are in hartrees. Table adapted from [17]...

Several simple methods have evolved for understanding TDDFT results qualitatively. The most basic of these methods is the single-pole approximation (SPA), which includes only one pole of the response function. The easiest way to see the SPA is to truncate Eq. [47] to a 1 x 1 matrix, yielding an (often excellent) approximation to the change in transition frequency away from its KS value ... 

Table 10 Transition Frequencies and Oscillator Strengths (OS) Calculated Using the Double-Pole Approximation (DPA) for the Lowest...

Pole Approximation in Time-Dependent Density Functional Theory. 

One has to consider that in case of FFT, the calculated power is not a continuous function but is only defined at specific frequencies. More sophisticated methods to determine the density power spectrum from FFT, by means of data windowing or pole approximation (maximal entropy method), can be found in Press et al (1988). 

The plasmon dispersion is obtained from the roots of e q,o)) = 0. Cappellini et al. (1993) have generalized their expression (4.2.9) for the static dielectric constant of semi-conductors to include dynamic processes, in a one-pole approximation (Hybertsen and Louie, 1986) ... 

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