Accurate estimate of $C_{5}$ dispersion coefficients of the alkali atoms interacting with different material media

The imaginary part of dynamic electric permittivity $\epsilon_{i}(\omega)$ can be given by

where $n(\omega)$ and $\kappa(\omega)$ are the refractive indices and extinction coefficients of the materials at the real frequencies, respectively. Discrete $n(\omega)$ and $\kappa(\omega)$ values of the considered material media for a wide range of frequencies are tabulated in the Handbook on optical constants of solids by Palik Palik (1998). Using these values, we have extrapolated values of $\epsilon_{i}(\omega)$ for continuous frequencies for a large range. Now using the Kramers-Kronig relation, we can express the real part of dynamic permittivities ($\epsilon_{r}(\iota\omega)$) at imaginary frequencies such that

For the case of semiconductor - Si and dielectrics - SiO₂, YAG, ordinary sapphire (oSap) and extraordinary sapphire (eSap), we have used the optical constants ranging from 0.1 eV to 10000 eV from Handbook of optical constants by Palik Palik (1998). For the case of Au, the $\epsilon$ values at very small energies are very significant, hence in addition to the experimental values from Ref. Palik (1998), we have extrapolated the values of real permittivity at imaginary frequencies using the Drude model for metals as

where $\omega_{p}$ is the plasma frequency and $\gamma$ is the relaxation frequency. We have used $\omega_{p}$=9.0 eV and $\gamma$=0.035 eV as referred in  Łach et al. (2010); Klimchitskaya et al. (2000). For the case of SiN_x, an amorphous dielectric material, we use Tauc–Lorentz model for amorphous materials Perreault et al. (2005) for estimating the electric permittivity at imaginary frequencies, the expression of which is given as follows

where the parameters $g_{0}=0.588$ and $\omega_{0}=0.005$ are the SiN_x’s response functions Perreault et al. (2005).

We have already reported $\alpha_{d}(\iota\omega)$ values of the Li to Cs alkali atoms in our previous works Arora and Sahoo (2014); Kaur et al. (2016). Here, we give $\alpha_{d}(\iota\omega)$ values of the Fr atom and the $\alpha_{q}(\iota\omega)$ values of all the alkali-metal atoms by evaluating them as given in the following procedures.

Total electron correlation contributions to $\alpha_{d}(\iota\omega)$ and $\alpha_{q}(\iota\omega)$ of atomic states of the alkali atoms can be expressed as Kaur et al. (2015)

where $l=d$ corresponds to dipole polarizability and $l=q$ corresponds to quadrupole polarizability. Subscripts $core$, $vc$ and $val$ corresponds to core, valence-core and valence contributions, respectively, to the total polarizability. In the alkali atoms, $\alpha_{l,val}(\iota\omega)$ contributes predominantly followed by $\alpha_{l,core}(\iota\omega)$ and contributions from $\alpha_{l,vc}(\iota\omega)$ are negligibly small. These contributions are estimated in the following way.

To begin with, the electronic configuration of alkali atoms is divided into a closed-core and a valence orbital in order to obtain the mean-field Dirac-Fock (DF) wave function of the respective closed-shell ($|0_{c}\rangle$) using DF method. The mean-field wave functions of the atomic states of the alkali atoms are then defined by appending the respective valence orbital $v$ as

Using these mean-field DF wave functions, we calculated the $vc$ contributions to the dipole and quadrupole polarizability using the following formula

where $J_{v}$ corresponds to total angular momentum of the state. Similarly, the core contributions can be given by

where the first sum for core orbitals is restricted from $a$ to total core orbitals $N_{c}$, second sum is restricted by involving intermediate states $m$ up to allowed bound states $B$ using the respective dipole and quadrupole selection rules, $L=1$ and $\textbf{O}_{1}=\textbf{D}$ is for the dipole operator to give $\alpha_{d}$, $L=2$ and $\textbf{O}_{2}=\textbf{Q}$ is for quadrupole operator to give $\alpha_{q}$ and $\mathcal{E}_{i}$ is the DF energy of the state. We have adopted the random phase approximation (RPA) to evaluate the above expression to account for the core correlations Singh and Sahoo (2014).

The major contributions to the total dipole and quadrupole polarizabilities are provided by the $val$ contributions, hence it is important to calculate the same with accurate methods. We divide the $val$ contributions into two parts - main and tail. The main part corresponds to polarizability contributions by the low-lying dominant transitions responsible for very large polarizability contributions. For the evaluation of the main part of the $val$ contribution of total polarizability, we have employed the AO method to evaluate the accurate wave functions. These wave functions $|\psi_{v}\rangle$, with $v$ denoting the valence orbital, are represented using singles and doubles (SD) approximation of AO method as Safronova et al. (1999)

where $a^{\dagger}$ and $a$ represent the second-quantization creation and annihilation operators, respectively, whereas excitation coefficients are denoted by $\rho$. The subscripts $m,r$ and $a,b$ refer to the virtual and core orbitals, respectively. $\rho_{ma}$ and $\rho_{mv}$ are the single whereas $\rho_{mrab}$ and $\rho_{mrva}$ are the double excitation coefficients. To take into account the important experimental contributions these ab initio wave functions are modified by changing the valence excitation coefficient with modified $\rho_{mv}$ using the scaling procedure such that

After obtaining wave functions of the considered states of alkali-metal atoms using AO method, we determine matrix elements with $k$ as intermediate state using the following expression Blundell et al. (1989)

where $O_{L,vk}$ corresponds to either dipole E1 or quadrupole E2 matrix elements depending on D or Q operators, respectively. Using these matrix elements, the final expression for the main part of the $val$ contribution to either the E1 or E2 polarizability at imaginary frequency is then given as

where the sum now restricted by entailing the intermediate states $m$ after $N_{c}$ and up to $I$. We have considered 10 - 12 E1 and E2 matrix elements for the dominant transitions of considered atoms using the AO method. For precise calculations, we use experimental energies $(E_{i})$ from the National Institute of Science and Technology (NIST) database Kramida et al. (2020). Contributions from the remaining high-lying states are referred as tail part and are evaluated as

where $m>I$ means that states included in the main contribution evaluation are excluded here. Since the tail contributions are much smaller in comparison to the main part, we calculate them using the DF method.

Previously, we had calculated the $C_{3}$ coefficients for various material walls interacting with alkali atoms Arora and Sahoo (2014); Kaur et al. (2016) except for Fr, so we are not repeating the results for other alkali atoms in the present work. Here we provide the $C_{3}$ coefficients only for Fr with a number of material walls. We give the static $\alpha_{d}(0)$ value along with reduced E1 matrix elements of Fr and compare with the other high-precision calculations in Table 1. We have taken the experimental values of E1 matrix elements of the dominant dipole transitions of Fr Simsarian et al. (1998). Other E1 matrix elements are calculated using the method given in Sec. III. Our value is in excellent agreement with value given by Derevianko et al. who used high precision experimental values for E1 matrix elements for the principal transitions and other E1 values by SD method Derevianko et al. (1999). The value given by Safronova et al. is evaluated using SD method which deviates from our value by around 1% Safronova et al. (2007). Though the method opted by us is same as of Refs. Derevianko et al. (1999); Safronova et al. (2007) to calculate the dipole polarizability of Fr but we have also scaled the E1 matrix elements using experimental energies as explained in Sec. III. Recently, Smialkowski et al. used a molecular MOLPRO package to evalaute the dipole polarizability of Fr which is overestimated and diverges by $\sim$3% from our value Śmiałkowski and Tomza (2021). Using reduced E1 matrix elements given in the same table, we have estimated the dynamic $\alpha_{d}(\iota\omega)$ values and used them to estimate different contributions to $C_{3}$ as given in Table 2. The dominant contributor of $C_{3}$ coefficients is the main part followed by core, tail and $vc$. The total value of $C_{3}$ coefficient differs from material to material. Consequently, the various contributions have been added up to provide a final value of $C_{3}$ coefficient. The core is providing 27%-38% of the share of total value of $C_{3}$ which is in accordance with the work by Derevianko et al. Derevianko et al. (1999) where they emphasized the sensitivity of core towards dipole $C_{3}$ values. On the other hand, the tail contribution is $\sim 1\%$ of the total value.

To evaluate the $C_{5}$ coefficients, we require quadrupole polarizability of alkali atoms. In Table 3, we present the static values of quadrupole polarizability $\alpha_{q}(0)$ of the ground states of alkali-metal atoms and compared our resulted values with the available literature. We calculated the static polarizability by putting $\omega=0$ in the Eq. 12. Since E2 matrix elements are required for calculation of dynamic polarizability, therefore we have provided the matrix elements of the dominant E2 transitions in Table 3 for all the alkali atoms. The breakdown of polarizability into the main, tail, core and $vc$ polarizabilities are also presented. The main part of valence polarizability provides the dominant contribution followed by tail and core polarizability. The $vc$ contributions for Li, Na and K are zero due to non-availability of $D$ orbitals in the core of these atoms whereas very insignificant contributions have been encountered for Rb, Cs and Fr. For the final value of total static polarizability, we have added the core polarizability values from RPA. We did not find experimental $\alpha_{q}(0)$ results for any alkali atom to compare our theoretical values with. However, in the same table, we have compared our results with the most recent work by Smialkowski et al. where they calculated the static quadrupole polarizability of alkali atoms using MOLPRO package of ab initio programs Śmiałkowski and Tomza (2021). Our static value of quadrupole polarizability deviates from the values reported by Smialkowski et al. by less than 1% for Li to Cs alkali atoms whereas for Fr, the discrepancy is about 8%. In another work Jiang et al. (1984), Jiang et al. evaluated the dynamic quadrupole polarizability of Na and Cs using oscillator method. We believe that our values are much more reliable than the compared ones due to accurate calculations of the matrix elements evaluated using the AO method.

Using the E2 matrix elements, dynamic quadrupole polarizability $\alpha_{q}(\iota\omega)$ of the alkali atoms over a range of frequencies has been calculated as presented in Fig. 1. Since our static values are accurate, we believe that the dynamic values are also reliable. We find the static RPA and DF values for the core contributions are quite close, so we have estimated the dynamic values of core polarizabilities using the DF method without losing much accuracy. As the frequency increases the polarizability value decreases and reaches a small value beyond $\omega=1$ a.u.. This trend is seen for every atom considered in the present work. Since these dynamic polarizability values can be important for experimental purposes, we have inferred these values at a particular frequency by providing a fitting model. In our previous work Dutt et al. (2020), we gave the fitting formula for dipole polarizability of alkali atoms at imaginary frequencies. Here, we have fitted the quadrupole polarizabilities of all the alkali atoms using the following fitting formula