C6 parameters

We consider the London’s method:

<math>E=\frac{3}{2}\alpha_A\alpha_B\frac{I_AI_B}{I_A+I_B}\left(\frac{1}{R^ 6}\right)</math>

where α is the static frequency and I is the ionization potential.

then arrived at an approximate expression for the <math>C_6</math> coefficient describing the vdW interaction between two atoms or molecules A and B.

<math>C_6^{AB}=\frac{3}{2}\alpha_A\alpha_B\frac{I_AI_B}{I_A+I_B}</math>

In case of A = B, we obtain

<math>C_6^{A}=\frac{3}{4}\alpha_A^2I_A</math>

To obtain the parameter we only have to evaluate the polarizability for a surface metals. This can be done applying an external electric field (E) and then evaluating the response of the dipole, μ. The latter can be computed from the charge density obtained from DFT simulation (see JCTC, 2014, 10, 5002-5009).

μ=αE

Computational Details

All surface simulations will be represented by different slab thickness and an external electric field perpendicular to the slab will be imposed using the method implement in VASP and then:

1. The electronic charge densities for different values of electric field is computed; the different between this values and charge density without electric field is plotted against the distance from the bottom of the cell.

2. The integration of the curve give us the differential charge, q, we can use this value to compute the surface dipole moment (ρ). We define the position vector like:

<math>d=z_{total}\left(z_{q=0}-\frac{z_{1L}-z_{2L}}{2}\right)</math>

3. We have the dipole, then, We can plot it versus the external electric field. The slope corresponds to the polarizability, in <math>e\AA^2/V)</math>

4. Finally, these polarizabilities can be used to calculated the coefficients, as described below.

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Table 1: Parameters for metals

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                       <math>C_6(Jnm^6/mol)</math>     <math>r_0(\AA)</math>  
                                                    
             Ni          2.6263         1.357         
             Pd 	 5.0242         1.374     
             Pt 	 5.3667	        1.387     
             Cu          2.7404	        1.261  
             Ag          5.4809	        1.438  
             Au          7.3079	        1.450    
             Ru          4.1678	        1.639      
             Pb         18.1555	        1.749

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