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Pauli principle causes same spin electrons to repel; a parameterization

Consider localized electrons in a solid. One way to interpret the Pauli principle's effect is to imagine that the electrons have finite extent and are prevented from intersecting each other, like hard spheres. Compressing the solid causes the electrons to squeeze together and shrink, increasing their kinetic energy. This increase in kinetic energy manifests itself in a force resisting compression of the solid. This repulsive force is the dominant interaction between neutral molecules at short range -- it is the basis of the steric effect in chemistry, it prevents stars from collapsing, and prevents the reader from falling through the earth.

Electrons do not have finite extent, of course, and even same spin electrons can interpenetrate each other. A more rigorous way to understand the origin of Pauli repulsion, outlined by Wilson and Goddard [22], is to compare the kinetic energy of an antisymmetrized product wavefunction with that of a Hartree product. In a Hartree product, the kinetic energy of the wavefunction is the sum of the orbital kinetic energies. In an antisymmetrized product, the kinetic energy is the sum of orthogonalized orbital kinetic energies; this mathematical simplification is the reason theorists often work in a basis of orthogonal molecular orbitals. Hence we can approximate the Pauli energy as the kinetic energy difference between orthogonalized versus non-orthogonalized orbitals.

Figure 4.2: Pauli repulsion comes from the kinetic energy increase upon making orbitals orthogonal to each other.

As Figure 4.2 shows, when two same spin electrons intersect in space, their orthogonalized orbitals take on larger slopes to keep their overlap zero. The increase in slope causes an increase in kinetic energy, which causes a large portion of the Pauli repulsion.

In deriving our Pauli potential, we make two assumptions: (1) we can approximate the Pauli energy as a sum over pairs of electrons, and (2) we can assume that the Pauli energy between pairs of electrons is dominated by the kinetic energy change upon forming an antisymmetric wavefunction. This neglects two effects: first, the mutual exclusion of more than two electrons at a time, which may become important when the electron density is high; and second, the fact that electrons, once orthogonalized, may have different electrostatic interactions with each other and with nuclei, which may become important for electrons near nuclei.

Kinetic energy difference-based Pauli potentials have been obtained and used by Boal and Glosli [14], who considered the case of same size nucleons; and by Klakow [16], who considered the more general case of Pauli repulsion between different size electrons. The form of the potentials bear some resemblance to earlier Pauli potentials [9,12] that decay as $ e^{-a x^{n}}$, where $ x$ is the distance between electron centers and $ a$ and $ n$ are arbitrary parameters.

Consider the Slater and Hartree wavefunctions for two same spin electrons:

$\displaystyle \Psi_{\mathrm{Slater}}$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2 - 2 S^{2}}} (\phi_{1}(r_{1})\,\phi_{2}(r_{2}) - \phi_{2}(r_{1})\phi_{1}(r_{2}))$  
$\displaystyle \Psi_{\mathrm{Hartree}}$ $\displaystyle =$ $\displaystyle \phi_{1}(r_{1})\,\phi_{2}(r_{2})$  

where the factor containing $ S = \int \phi_{1} \phi_{j} \, dV$ ensures that the wavefunction is normalized. Then we estimate the Pauli energy between wavefunctions $ \phi_{1}$ and $ \phi_{2}$ as
$\displaystyle E_{u}$ $\displaystyle =$ $\displaystyle \braket{\Psi_{\mathrm{Slater}}\vert-\frac{1}{2} \nabla{^2}\vert\P...
...Psi_{\mathrm{Hartree}}\vert-\frac{1}{2} \nabla{^2}\vert\Psi_{\mathrm{Hartree}}}$ (4.3)
  $\displaystyle =$ $\displaystyle \frac{S^{2}}{1-S^{2}} \left(t_{11} + t_{22} - \frac{2 t_{12}}{S}\right)$ (4.4)

where $ t_{ij} = \braket{\psi_{i}\vert-\frac{1}{2}\nabla^{2}\vert\psi_{j}}$ (detailed derivation given in Appendix A).

Klakow used $ E(\uparrow \uparrow) = E_{u}$ and $ E(\uparrow \downarrow) = 0$; to get our expression, we make use of the reference valence-bond wavefunction

$\displaystyle \Psi_{\mathrm{VB}} = \frac{1}{\sqrt{2 + 2 S^{2}}} (\phi_{1}(r_{1})\,\phi_{2}(r_{2}) + \phi_{2}(r_{1})\phi_{1}(r_{2})).$ (4.5)

Then we compute
$\displaystyle E_{g}$ $\displaystyle =$ $\displaystyle \braket{\Psi_{\mathrm{VB}}\vert-\frac{1}{2} \nabla{^2}\vert\Psi_{...
...Psi_{\mathrm{Hartree}}\vert-\frac{1}{2} \nabla{^2}\vert\Psi_{\mathrm{Hartree}}}$ (4.6)
  $\displaystyle =$ $\displaystyle \frac{S^{2}}{1+S^{2}} \left(t_{11} + t_{22} - \frac{2 t_{12}}{S}\right)$ (4.7)

which is a kind of a correlation energy. We mix $ E_{g}$ and $ E_{u}$ together, and scale the orbital exponents and distance between orbitals by a set of fixed and universal parameters: $ \alpha = \alpha_{actual} / 0.9$, $ r = r_{actual}* 1.125$. Finally we calculate the functions:
$\displaystyle E (\uparrow \uparrow)$ $\displaystyle =$ $\displaystyle E_{u} - (1 - \rho) E_{g}$  
$\displaystyle E (\uparrow \downarrow)$ $\displaystyle =$ $\displaystyle - \rho E_{g}.$  

The universal parameter $ \rho$ and the scaling factors were adjusted to produce correct geometries for a range of test structures. Figure 4.3 shows that the effect of the $ E_{g}$ term is to make the Pauli potential between both opposite and same spin electrons more repulsive; this reduces the known tendency for floating orbitals to coalesce into each other and become linearly dependent.

Figure 4.3: Comparison of Pauli repulsion and electrostatic repulsion between two wavefunctions with $ s=1$.

next up previous contents
Next: Validation against ground state Up: General theory of the Previous: Bonding comes from balancing   Contents
Julius 2008-04-29