FIG. 2: (Color online) The retardation coefficient \( f _ { 3 } ( a ) \) for the Cs atom as a function of the distance a from a perfect conductor, Au (metal) metal and Si (semiconductor).

FIG. 3: (Color online) The retardation coefficient \( f _ { 3 } ( a ) \) for the Cs atom as a function of the distance a from the differ-ent dielectric surfaces such as \( \mathrm { { S i O _ { 2 } } } \),\( \mathrm { S i N } _ { \mathrm { v } } \), Sapphire (ordinary, extraordinary, birefringent) and YAG.

for the 6 \( S \) state is 399.8 a.u.. This is in very good agree-ment with the value (399 a.u.) obtained by Borschevsky et al. [36] and experimentally measured value 401.0(6) a.u. of Amini et al. [38]. As seen, contributions from \( \alpha _ { c v } \) and \( \alpha _ { t } \) are quite small andj ustify use of a lower method for their evaluation. Our \( \alpha _ { c } \) from RRPA also match with the value of Derevianko et al. [42]. Good agreement between the calculated and experimental results of \( \alpha ( 0 ) \) indicates that our approach can also give similar accura-cies for the estimated dynamic polarizabilities. Next, we compute the atom-surface dispersion \( C _ { 3 } \) coefficients for a perfect conductor by numerically evaluating Eq. (4). Dynamic polarizabilities used for this purpose are plotted in Fig. 1(a). In the same plot, we also show the values obtained by the single oscillator model (SOM) that are used by other works as discussed below. We compare our \( C _ { 3 } \) coefficient for the perfect conductor with the re-sult obtained by Derevianko and co-workers [42] in Table II. Though our \( \alpha ( 0 ) \) value match quite well with Dere-vianko et al. [37], but we find difference in the \( C _ { 3 } \) value. A similar finding was also observed for the other alkali atoms that were studied by us in Ref. [23]. Our analysis shows that the main reason for the discrepancy is because of different numerical integration methods used in both the works. Derevianko et al. use a Gaussian quadrature integration method with 50 point formula while we have used an exponential grid in our calculations as discussed in [23]. In the same table we have given our calcu-lated \( C _ { 3 } \) coefficients for the other material media using their dynamic dielectric constants that are plotted in Fig. 1(b). In our earlier work [23], we have also given these constants for Au, Si, \( \mathrm { { S i O _ { 2 } } } \) and \( \mathrm { S i N } _ { x } \). Hence, we only present the final results without giving the fine details of their evaluations. Compared to these materials we find a different trend of \( \epsilon ( \iota \omega ) \) values for the ordinary, extraor-dinary and birefringent sapphires and also for the YAG surfaces, especially at the low frequency range. We could not find another study to verify directly the validity of this trend, but a recent calculation of \( S ( \omega ) \) in Ref. [55]shows almost a similar trend. This somewhat assures us about accurate determination of the \( \epsilon ( \iota \omega ) \) values for the sapphires and YAG surfaces.

We also give the \( C _ { 3 } \) coefficients for all the considered material media in Table II along with the values known in the literature. As seen, there is another evaluation of \( C _ { 3 } \) for the Au metal reported by Lach et al. [26] They use the SOM model to estimate the dynamic polarizabil-ity values whereas the dynamic dielectric constants are estimated using the Drude model. Nevertheless, we find a reasonable agreement between these results. We, how-ever, did not find any data to compare our results for the dielectric materials and semiconductors. From Table II, it can be seen that the valence correlation is dominant among the core, core-valence and tail correlations, but core contributions are quite significant as compared to their contributions in the evaluation of the polarizabili-ties. Among all the interacting media, the \( C _ { 3 } \) coefficients for the interacting perfect conductor is the highest and it is approximately 40%, 51%, 75%, 64%, 62% and 63% larger than the Au, Si, \( \mathrm { { S i O _ { 2 } } } \),\( \mathrm { S i N _ { x } } \), sapphire and YAG surfaces respectively. The decrease in the interaction co-efficients for the cases of dielectric media might be due to the charge dangling bonds in the materials which ac-counts for the additional interactions in the dielectrics at the shorter separations [56].

Using our described procedure, we also evaluate the re-tarded \( f _ { 3 } \) functions for all the considered materials inter-acting with Cs. In Table III, we compare our results for \( f _ { 3 } \) for the interaction of the Cs atom with the Au surface with the results obtained by Lach et al. [26] at certain separation distances. We also find reasonable agreement between both the results. It can also be observed from this table that at the short separation distances (\( a \rightarrow \) 0),