Editing Physisorption (section)

==Modeling by quantum-mechanical oscillator==

The [[van der Waals force|van der Waals]] binding energy can be analyzed by another simple physical picture: modeling the motion of an electron around its nucleus by a three-dimensional simple [[harmonic oscillator]] with a potential energy ''V<sub>a</sub>'':{{Clarify|reason=Provide a geometric image representing the equitation|date=August 2021}}

:<math>V_a = \frac{m_e}{2}{\omega^2}(x^2+y^2+z^2),</math>

where ''m<sub>e</sub>'' and ''&omega;'' are the mass and vibrational frequency of the electron, respectively.

As this atom approaches the surface of a metal and forms adsorption, this potential energy ''V<sub>a</sub>'' will be modified due to the image charges by additional potential terms which are quadratic in the displacements:

:<math>V_a = \frac{m_e}{2}{\omega^2}(x^2+y^2+z^2)-{e^2\over 16\pi\varepsilon_0 Z^3}\left(\frac{x^2+y^2}{2}+z^2\right)+\ldots</math>  (from the Taylor expansion above.)

Assuming

:<math> m_e \omega^2\gg{e^2\over 16\pi\varepsilon_0 Z^3},</math>

the potential is well approximated as

:<math>V_a \sim \frac{m_e}{2}{\omega_1^2}(x^2+y^2)+\frac{m_e}{2}{\omega_2^2}z^2</math>,

where

:<math>
\begin{align}
\omega_1 &= \omega - {e^2\over 32\pi\varepsilon_0 m_e\omega Z^3},\\
\omega_2 &= \omega - {e^2\over 16\pi\varepsilon_0 m_e\omega Z^3}.
\end{align}
</math>

If one assumes that the electron is in the ground state, then the van der Waals binding energy is essentially the change of the zero-point energy:

:<math>V_v = \frac{\hbar}{2}(2\omega_1+\omega_2-3\omega)= - {\hbar e^2\over 16\pi\varepsilon_0 m_e\omega Z^3}.</math>

This expression also shows the nature of the ''Z''<sup>&minus;3</sup> dependence of the van der Waals interaction.

Furthermore, by introducing the atomic [[polarizability]],

:<math> \alpha= \frac {e^2} {m_e\omega^2},</math>

the van der Waals potential can be further simplified:

:<math>V_v = - {\hbar \alpha \omega\over 16\pi\varepsilon_0 Z^3}= -\frac{C_v}{Z^3},</math>

where

:<math>C_v = {\hbar \alpha \omega\over 16\pi\varepsilon_0},</math>

is the van der Waals constant which is related to the atomic polarizability.

Also, by expressing the fourth-order correction in the Taylor expansion above as (''aC<sub>v</sub>Z''<sub>0</sub>)&nbsp;/&nbsp;(Z<sup>4</sup>), where ''a'' is some constant, we can define ''Z''<sub>0</sub> as the position of the ''dynamical image plane'' and obtain
{| class="wikitable floatright"
|+Table 1. The van der Waals constant ''C<sub>v</sub>'' and the position of the dynamical image plane ''Z''<sub>0</sub> for various rare gases atoms adsorbed on noble metal surfaces obtained by the jellium model. Note that ''C<sub>v</sub>'' is in eV/Å<sup>3</sup> and ''Z''<sub>0</sub> in Å.
! rowspan="2" |
! colspan="2" | He
!colspan="2" | Ne
!colspan="2" | Ar
!colspan="2" | Kr
!colspan="2" | Xe
|-
!''C<sub>v</sub>''
!''Z''<sub>0</sub>
!''C<sub>v</sub>''
!''Z''<sub>0</sub>
!''C<sub>v</sub>''
!''Z''<sub>0</sub>
!''C<sub>v</sub>''
!''Z''<sub>0</sub>
!''C<sub>v</sub>''
!''Z''<sub>0</sub>
|-
!Cu
|0.225
|0.22
|0.452
|0.21
|1.501
|0.26
|2.11
|0.27
|3.085
|0.29
|-
!Ag
|0.249
|0.2
|0.502
|0.19
|1.623
|0.24
|2.263
|0.25
|3.277
|0.27
|-
!Au
|0.274
|0.16
|0.554
|0.15
|1.768
|0.19
|2.455
|0.2
|3.533
|0.22
|}
:<math>V_v = - \frac{C_v}{(Z-Z_0)^3}+O\left(\frac{1}{Z^5}\right).</math>

The origin of ''Z''<sub>0</sub> comes from the spilling of the electron wavefunction out of the surface.  As a result, the position of the image plane representing the reference for the space coordinate is different from the substrate surface itself and modified by ''Z''<sub>0</sub>.

Table 1 shows the [[jellium]] model calculation for van der Waals constant ''C<sub>v</sub>'' and dynamical image plane ''Z''<sub>0</sub> of rare gas atoms on various metal surfaces.  The increasing of ''C<sub>v</sub>'' from He to Xe for all metal substrates is caused by the larger atomic [[polarizability]] of the heavier rare gas atoms.  For the position of the dynamical image plane, it decreases with increasing dielectric function and is typically on the order of 0.2&nbsp;Å.