from the precisely measured lifetimes of the \( 6 p \, ^ { 2 } P _ { 1 / 2 } \) and \( 6 p ~ \AA ^ { 2 } \)\( ^ 2 P _ { 3 / 2 } \) states [35]. The other important E1 matrix elements are obtained using the RCC method in the sin-gles and doubles approximation (CCSD method) as de-scribed in [45, 46]. Contributions to \( \alpha _ { c } \) are determined using a relativistic random-phase approximation (RRPA) as described in [40]. It has been demonstrated that the RRPA method can give rise to very reliable results for the atomic systems having inert gas configurations [47].Smaller contributions from \( \alpha _ { c v } \) and from the high-lying excited states (tail contribution \( \alpha _ { t } \)) that are omitted in the above sum-over-states approach are estimated in the Dirac-Hartree-Fock (DHF) approximation.

It is not easy to get the dynamic electric permittivity of any real materials, so the convenient way of determin-ing these constants for simple metals such as gold is to use the Drude-Lorentz model as was done in Ref. [26]. In pursuance of obtaining more realistic values of these con-stants for different materials, we prefer to use the known real and imaginary parts of the refractive indices of a ma-terial at few real values of frequency \( \omega \). The imaginary parts of the dielectric permittivities of the materials can then be obtained using the relation

\[ I m \left[ \epsilon ( \omega ) \right] = 2 n ( \omega ) \kappa ( \omega ) , \eqno ( 9 ) \]

where \( n ( \omega ) \) and \( \kappa ( \omega ) \) are the respective real and imag-inary parts of the refractive index of a material at fre-quency \( \omega \). We use the optical data from the handbook by Palik [48] for the frequencies ranging from 0.1 eV to 10000 eV for Au metal to calculate \( I m [ \epsilon ( \omega ) ] \). Thereafter, the required real values of the dielectric constants at the imaginary frequencies \( ( \epsilon ( \iota \omega ) \)) are obtained by using the Kramers-Kronig formula. The available data, however, does not cover the whole frequency range to carry out the integration of Eq. (4). Thus, we extrapolate these val-ues for the lower frequencies to increase the domain over which the integrations are to be performed [49, 50]. For the frequencies below 0.1 eV, the classified values from[48] are extrapolated using the free electron Drude model in which the dielectric permittivity along the imaginary axis is represented as

\[ \epsilon ( \iota \omega ) = 1 - \frac { \omega _ { p } { } ^ { 2 } } { \omega ( \omega + \iota \gamma ) } , \eqno ( 1 0 ) \]

where \( \omega _ { p } = ( 2 \pi c / \lambda _ { p } ) \) is the plasma frequency and \( \gamma \) is the relaxation frequency. The optical data values for \( \omega _ { p } \) and \( \gamma \) available from various sources differ slightly, but we use these values as \( \omega _ { p } = 9 . 0 \) eV and \( \gamma \) = 0.035 eV as out-lined in [26, 49–51]. In case of Si (semiconductor),\( \mathrm { { S i O _ { 2 } } } \) (dielectric), Sapphire and YAG, the complex frequency-dependent dielectric permittivity are quoted for a wide range of energies in the handbook of Palik. Therefore, we use all these values for carrying out the integration and do not extrapolate any data. On the otherhand, experi-mental data of \( n ( \omega ) \) and \( k ( \omega ) \) for \( \mathrm { S i N _ { x } } \) are not available at all and we use Tauc-Lorentz model [33, 52] for esti-mating dielectric constants of this material.

TABLE III: Comparison of the calculated values of damping function \( f _ { 3 } ( a ) \) defined in Eq.(4) for different separation dis-tances a for the interaction of Cs atom with the gold surface.

<table><tr><td rowspan="2">a 
(a. u.)</td><td colspan="2">f3(a)</td></tr><tr><td>This work</td><td>Ref. [26]</td></tr><tr><td>11×10</td><td>1.00</td><td>0.99780</td></tr><tr><td>12×10</td><td>1.00</td><td>0.99408</td></tr><tr><td>15×10</td><td>0.98</td><td>0.97903</td></tr><tr><td>21×10</td><td>0.95</td><td>0.95169</td></tr><tr><td>22×10</td><td>0.89</td><td>0.90210</td></tr><tr><td>25×10</td><td>0.78</td><td>0.79521</td></tr><tr><td>31×10</td><td>0.66</td><td>0.68309</td></tr><tr><td>32×10</td><td>0.53</td><td>0.54485</td></tr><tr><td>35×10</td><td>0.33</td><td>0.33918</td></tr></table>

TABLE IV: Comparison of \( C _ { 3 } \) coefficients ratios of Li, Na, K and Rb atoms with the Cs atom for the perfect conductor, metal, semiconductor and dielectric surfaces.

<table><tr><td></td><td>iC \( C _ { 3 } ^ { \mathrm { L } } \) /Cs3 </td><td>\( \gamma _ { 2 } ^ { \mathrm { N } } \) aC/Cs3 </td><td>C \( \gamma _ { 2 } ^ { \mathrm { F } } \) /Cs3 </td><td>Rb
C s \( \gamma _ { 3 } ^ { 0 } \) 3/</td></tr><tr><td>Perfect conductor</td><td>0.313</td><td>0.393</td><td>0.638</td><td>0.773</td></tr><tr><td>Metal: Au</td><td>0.416</td><td>0.464</td><td>0.715</td><td>0.811</td></tr><tr><td rowspan="2">Semiconductor: Si 
Dielectric:</td><td rowspan="2">0.430</td><td rowspan="2">0.477</td><td rowspan="2">0.726</td><td rowspan="2">0.818</td></tr><tr><td>SiO2</td><td>0.409</td><td>0.458</td><td>0.708</td><td>0.807</td></tr><tr><td>SiNx</td><td>0.418</td><td>0.464</td><td>0.709</td><td>0.801</td></tr></table>

Again, the interactions between the ground states of atoms with an anisotropic surface have been studied be-fore [53, 54]. These studies demonstrate that for a uni-form birefringent dielectric surface, with the symmetry axis normal to the interface, the interaction potential de-scribed Eq. (3) is still applicable if \( \epsilon \) is replaced by an effective quantity \( \bar { \epsilon } \) defined as

\[ \bar { \epsilon } ( \iota \omega ) = \left[ \epsilon _ { | | } ( \iota \omega ) \epsilon _ { \bot } ( \iota \omega ) \right] ^ { \frac { 1 } { 2 } } , \eqno ( 1 1 ) \]

where \( \epsilon _ { | | } \) and \( \epsilon _ { \perp } \) are the respective dielectric permittiv-ities for the electric fields parallel and perpendicular to the interface between the atom and the dielectric surface.

# III. RESULTS AND DISCUSSION

The evaluation of \( C _ { 3 } \) coefficients requires the precise estimation of dynamic polarizabilities of the Cs atom. Table I presents the scalar polarizabilities of the Cs atom in its ground state along with the E1 matrix elements for different transitions that are used to estimate \( \alpha _ { v } \) and other contributions to the polarizabilities. The E1 ma-trix elements for the \( 6 S _ { 1 / 2 } { - } 6 P _ { 1 / 2 , 3 / 2 } \) transitions are taken from the experimentally measured data given in [35].Our calculated value of the scalar polarizability \( ( \alpha ( 0 ) ) \)

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),