Dispersion interaction

In chemistry and physics, the dispersion interaction is one of the forces acting between stable molecules. The dispersion interaction operates between molecules and atoms ("monomers") of any kind and is always attractive, that is, it pulls monomers together. Between noble gas atoms it is the only attractive force in operation and hence it is the cause of noble gases undergoing phase transitions, transforming into stable liquids or crystals for low temperature and/or high pressure. Also for non-polar molecules such as methane, benzene, and other hydrocarbons, dispersion gives one of the most important attractive forces between the molecules. It explains, for instance, why benzene under standard temperature and pressure (STP) is a liquid.

In 1930 R. Eisenschitz and F. London gave the first complete explanation of intermolecular forces (also referred to as van der Waals forces). They distinguished different effects and among them the effect that we know now as dispersion interaction. The corresponding dispersion force (minus the gradient of the potential, sometimes referred to as London force) falls of as R⁻⁷, where R is the distance between the monomers. Eisenschitz and London point out that so-called "oscillator strengths" f are an important ingredient of their theoretical description. These f-values entered previously the classical theory of the dispersion of light formulated by Lorentz and Drude. The same f-values also appear in the "old" (Planck-Bohr-Sommerfeld) quantum theory of dispersion formulated by Kramers and Heisenberg. Because of the similarity between the several theories needing f-values, London baptized the corresponding part of the intermolecular interaction the dispersion effect.

Because the Eisenschitz-London work is very complete and accordingly not easily grasped, London gave in the same year a less mathematical description of the dispersion force in which he modeled each monomer of a dimer as a three-dimensional isotropic harmonic oscillator. The oscillators consist of a particle of mass m and charge e and the two oscillating charges interact through the electric Coulomb force. By an approximate quantum mechanical treatment London derived the following form of the interaction potential of the oscillators in their ground (lowest energy) state:

$E=-{\tfrac {3}{4}}h\nu _{0}{\frac {\alpha ^{2}}{R^{6}}},\qquad \qquad \qquad (1)$

an expression which is similar (quadratic in α, depending on R⁻⁶) to the dispersion formula he derived together with Eisenschitz for real atoms and molecules. The gradient (derivative of E with respect to R) is proportional to R⁻⁷; minus the gradient is the attractive force associated with this potential. London stresses that his formula is due to the zero point motion of the free oscillators. The zero point motion is a typical quantum mechanical phenomenon related to Heisenberg's uncertainty principle. Without zero point motion, London's model does not give an attraction between oscillators in their ground state.

It is of some importance to note that London simply modeled the totality of electrons in an atom as one harmonically oscillating dipole, and that he did not attempt to explain why, for instance, two interacting noble gas atoms would become dipolar under their mutual interaction. Representing atomic electrons by oscillators was quite a common procedure before the discovery of quantum mechanics: Lorentz explained the Zeeman effect by it, Planck assumed that a radiating black body consists of oscillators that are quantized, and also in the theory of Drude, Kramers, and Heisenberg for the dispersion of light, oscillating electrons ultimately explain the effects under study.

In the section London forces of the article intermolecular forces the Eisenschitz-London theory, derived by perturbation theory and based on oscillator strengths, is discussed. In this article the simple London model of two interacting oscillating dipoles will be worked out in some detail.

Proof of equation (1)

The London theory leading to equation (1) will be reviewed. The theory is based on two interacting three-dimensional isotropic harmonic oscillators. Isotropy means that the spring (Hooke) constants are equal: k_x = k_y = k_z ≡ k. An isotropic three-dimensional oscillator factorizes into three identical one-dimensional oscillators and the total energy of one oscillator is the sum of the three energies of the three one-dimensional oscillators.

Oscillating dipole

A one-dimensional electric dipole μ₀ consists of two equal charges e of opposite sign a distance x₀ apart, μ₀ = e x₀. In an external electric field E the distance between the charges will change and correspondingly the dipole does too:

$\Delta \mu =e\Delta x=\alpha \;E$

where α is the polarizability of the system. Hooke's law states that there is the spring force kΔx, which balances the force eE by the external field

$k\Delta x=eE\;\Longrightarrow \;k\Delta \mu =e^{2}E\Longrightarrow \;k\alpha E=e^{2}E$

so that Hooke's constant is seen to be inversely proportional to the polarizability

$k={\frac {e^{2}}{\alpha }}.\;$

Quantum mechanical harmonic oscillator

The potential corresponding to the force kx = ½ k x². Let the mass of the oscillating particle be m and the Schrödinger equation becomes,

$\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {1}{2}}kx^{2}\right]\Psi _{n}={\mathcal {E}}_{n}\Psi _{n},$

where ℏ is the reduced Planck constant. It is shown in the article on the quantum harmonic oscillator that

${\mathcal {E}}_{n}=(n+{\tfrac {1}{2}})\hbar \omega _{0}\quad {\hbox{with}}\quad \omega _{0}={\sqrt {\frac {k}{m}}}={\frac {e}{\sqrt {m\alpha }}}.$

Interaction

One can expand the total Coulomb interaction in a series of multipoles. Since the electric dipoles are neutral, the first non-vanishing term in the expansion is the dipole-dipole interaction. If we place the z-axis along the vector R that connects the two dipoles, this interaction takes the form

${\frac {1}{R^{3}}}\left({\boldsymbol {\mu }}_{1}\cdot {\boldsymbol {\mu }}_{2}-3{\frac {({\boldsymbol {\mu }}_{1}\cdot \mathbf {R} )\,({\boldsymbol {\mu }}_{2}\cdot \mathbf {R} )}{R^{2}}}\right)={\frac {e^{2}}{R^{3}}}\left(x_{1}x_{2}+y_{1}y_{2}-2z_{1}z_{2}\right),$

here R = |R|.

Schrödinger equation of dimer

The total 6-dimensional Schrödinger equation becomes

$\left[-{\frac {\hbar ^{2}}{2m}}(\nabla _{1}^{2}+\nabla _{2}^{2})+{\frac {e^{2}}{2\alpha }}(r_{1}^{2}+r_{2}^{2})+{\frac {e^{2}}{R^{3}}}(x_{1}x_{2}+y_{1}y_{2}-2z_{1}z_{2})\right]\Psi ={\mathcal {E}}\psi ,$

where r stands for |r| and ∇ is the nabla operator consisting of derivatives with respect to x, y, and z.

Solution

We make the following substitutions to obtain a 6-dimensional Schrödinger equation that can be factorized into 6 one-dimensional Schrödinger equations of the harmonic oscillator type,

${\begin{aligned}\mathbf {r} _{+}&={\frac {1}{\sqrt {2}}}(\mathbf {r} _{1}+\mathbf {r} _{2})&\qquad \mathbf {r} _{-}&={\frac {1}{\sqrt {2}}}(\mathbf {r} _{1}-\mathbf {r} _{2})\\\mathbf {r} _{1}&={\frac {1}{\sqrt {2}}}(\mathbf {r} _{+}+\mathbf {r} _{-})&\qquad \mathbf {r} _{2}&={\frac {1}{\sqrt {2}}}(\mathbf {r} _{+}-\mathbf {r} _{-})\end{aligned}}$

The Schrödinger equation becomes

$\left[-{\frac {\hbar ^{2}}{2m}}(\nabla _{+}^{2}+\nabla _{-}^{2})+{\frac {e^{2}}{2\alpha }}(r_{+}^{2}+r_{-}^{2})+{\frac {e^{2}}{2R^{3}}}(x_{+}^{2}-x_{-}^{2}+y_{+}^{2}-y_{-}^{2}-2(z_{+}^{2}-z_{-}^{2})\right]\Psi ={\mathcal {E}}\psi .$

It factorizes into the following six equations

${\begin{aligned}\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx_{+}^{2}}}+x_{+}^{2}{\Big (}{\frac {1}{\alpha }}+{\frac {1}{R^{3}}}{\Big )}{\frac {e^{2}}{2}}\right]\psi _{n_{x}^{+}}&={\mathcal {E}}_{x}^{+}\psi _{n_{x}^{+}}\\\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dy_{+}^{2}}}+y_{+}^{2}{\Big (}{\frac {1}{\alpha }}+{\frac {1}{R^{3}}}{\Big )}{\frac {e^{2}}{2}}\right]\psi _{n_{y}^{+}}&={\mathcal {E}}_{y}^{+}\psi _{n_{y}^{+}}\\\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx_{-}^{2}}}+x_{-}^{2}{\Big (}{\frac {1}{\alpha }}-{\frac {1}{R^{3}}}{\Big )}{\frac {e^{2}}{2}}\right]\psi _{n_{x}^{-}}&={\mathcal {E}}_{x}^{-}\psi _{n_{x}^{-}}\\\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dy_{-}^{2}}}+y_{-}^{2}{\Big (}{\frac {1}{\alpha }}-{\frac {1}{R^{3}}}{\Big )}{\frac {e^{2}}{2}}\right]\psi _{n_{y}^{-}}&={\mathcal {E}}_{y}^{-}\psi _{n_{y}^{-}}\\\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dz_{+}^{2}}}+z_{+}^{2}{\Big (}{\frac {1}{\alpha }}-{\frac {2}{R^{3}}}{\Big )}{\frac {e^{2}}{2}}\right]\psi _{n_{z}^{+}}&={\mathcal {E}}_{z}^{+}\psi _{n_{z}^{+}}\\\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dz_{-}^{2}}}+z_{-}^{2}{\Big (}{\frac {1}{\alpha }}+{\frac {2}{R^{3}}}{\Big )}{\frac {e^{2}}{2}}\right]\psi _{n_{z}^{-}}&={\mathcal {E}}_{z}^{-}\psi _{n_{z}^{-}}\\\end{aligned}}$

The total energy is the sum

${\mathcal {E}}={\mathcal {E}}_{x}^{+}+{\mathcal {E}}_{y}^{+}+{\mathcal {E}}_{x}^{-}+{\mathcal {E}}_{y}^{-}+{\mathcal {E}}_{z}^{+}+{\mathcal {E}}_{z}^{-}.$

The 6 equations are are one-dimensional harmonic oscillator equations with the following ω values:

${\begin{aligned}\omega _{x}^{+}=\omega _{0}{\sqrt {1+{\frac {\alpha }{R^{3}}}}}&&\qquad \omega _{x}^{-}=\omega _{0}{\sqrt {1-{\frac {\alpha }{R^{3}}}}}&&\qquad \omega _{y}^{+}=\omega _{0}{\sqrt {1+{\frac {\alpha }{R^{3}}}}}&&\\\omega _{y}^{-}=\omega _{0}{\sqrt {1-{\frac {\alpha }{R^{3}}}}}&&\qquad \omega _{z}^{+}=\omega _{0}{\sqrt {1-{\frac {2\alpha }{R^{3}}}}}&&\qquad \omega _{z}^{-}=\omega _{0}{\sqrt {1+{\frac {2\alpha }{R^{3}}}}},&&\\\end{aligned}}$

where ω₀ = e/(mα)^½ was introduced earlier. The exact energy is

${\mathcal {E}}=(n_{x}^{+}+{\tfrac {1}{2}})\,\omega _{x}^{+}\;+\;(n_{x}^{-}+{\tfrac {1}{2}})\,\omega _{x}^{-}\;+\;(n_{y}^{+}+{\tfrac {1}{2}})\,\omega _{y}^{+}\;+\;(n_{y}^{-}+{\tfrac {1}{2}})\,\omega _{y}^{-}\;+\;(n_{z}^{+}+{\tfrac {1}{2}})\,\omega _{z}^{+}\;+\;(n_{z}^{-}+{\tfrac {1}{2}})\,\omega _{z}^{-},$

where the six n-quantum numbers run independently over 0, 1, 2, ….

Expansion

We now make an expansion of the exact energy that is the analogue of the perturbation expansion and use

${\sqrt {1+x}}=1+{\tfrac {1}{2}}x-{\tfrac {1}{8}}x^{2}+\ldots$

in the expansion of the six ω values. Collecting terms multiplying the same power of R, we get

${\begin{aligned}{\mathcal {E}}\approx \hbar \omega _{0}\;{\Big [}&(n_{x}^{+}+n_{x}^{-}+n_{y}^{+}+n_{y}^{-}+n_{z}^{+}+n_{z}^{-}+3)\\+&\left({\frac {\alpha }{2R^{3}}}\right)(n_{x}^{+}-n_{x}^{-}+n_{y}^{+}-n_{y}^{-}-2n_{z}^{+}+2n_{z}^{-})\\-&\left({\frac {\alpha ^{2}}{8R^{6}}}\right)(n_{x}^{+}+n_{x}^{-}+n_{y}^{+}+n_{y}^{-}+4n_{z}^{+}+4n_{z}^{-}+6){\Big ]}\\\end{aligned}}$

(To be continued)