Physics in a nutshell

$ \renewcommand{\D}[2][]{\,\text{d}^{#1} {#2}} $ $\DeclareMathOperator{\Tr}{Tr}$

Interactions due to Orbital Overlap

What happens when two or more atoms (both in the ground state) approach each other? At some point the occupied orbitals start to overlap which can give rise to qualitatively very different effects. In this article we will have a closer look at interatomic repulsion, covalent bonds and metallic bonds.

Interatomic Repulsion

First consider the case where both orbitals involved are filled by the maximum of two electrons. When the orbitals start to overlap spatially, there are actually more than two electrons (since they are indistinguishable) in each orbitals which is however forbidden by the Pauli principle Therefore the excess electrons have to move to another orbital Since both atoms were initally in the ground state, the relocated electron will have a higher potential energy afterwards.

Thus, such a transition is energetically unfavourable and gives rise to a strong repulsion between the atoms.[1] [2]

Covalent Bond

Now we consider the case where the orbitals involved are both only partially filled:

When two such orbitals overlap, the total energy of the system changes due to a quantum mechanical exchange interaction (some kind of interference). This change in energy can hardly be understood from a classical point of view. It follows from the solution of the Schrödinger equation for the composite system. This cannot be done analytically in general but there are some approximate solutions for very simple cases:

The LCAO Method

Maybe you have attended a course on quantum mechanics before and discussed the $\text{H}_2^+$ molecule (two protons and one electron). One can solve the Schrödinger equation for both nuclei separately and approximate the actual solution of the composite system as a linear combination of the original solutions. This approach is known as LCAO (Linear combination of atomic orbitals). I will possibly add the corresponding calculations later.

Qualitative Outcomes

One finds that there are two solutions possible, one being higher (bonding) and one being lower (anti-bonding) in energy compared to the combined energy of the separated atoms. The important qualitative results are the facts that

  • atoms can bind due to orbital overlap (form covalent bonds)
  • the change in energy (and thus the strength of a possible bond) increases with increasing overlap. Since most of the outer orbitals are oriented along certain directions (e.g. the 2p-orbital) the covalent bond exhibits a strong directional dependence!
  • usually the involved electrons are located mainly between the atoms

[3] [4] [5] [6]

Hybridisation

Until now we neglected any form of deformation of the atomic orbitals. However, when two atoms interact as they come closer to each other, this affects the shape of the atomic orbitals as well. It is even possible that different orbitals merge and form completely new ones. This process is called hybridisation. The principle is really simple: An electron from a filled orbital is raised to an empty one. This costs some energy but the atom is now able to form more covalent bonds than before since it has more partially filled orbitals. These bonds can lead to a decrease in the total energy that eventually overcompensates the initial expense.[7] [8] [9]

sp-Hybrid Orbitals

A simple hybrid orbital is the sp-hybrid orbital which arises from merging a s- and a p-orbital. To find the concrete shape of sp-hybrid orbitals one uses the following approach: Consider the s- and p$_x$-orbitals and their respective wave functions $\Psi_s$ and $\Psi_{p_x}$. Now the plan is to construct two orbitals \begin{align} \Psi_1^{sp} = c_1 \cdot \Psi_s + c_2 \cdot \Psi_{p_x} \\[2ex] \Psi_2^{sp} = c_3 \cdot \Psi_s + c_4 \cdot \Psi_{p_x} \end{align} as linear combinations of the original ones with $c_i \in \mathbb{R}$ being coefficients that can be obtained by applying the normalisation and orthogonality conditions (which must be fulfilled by any wave function): \begin{align} \int \left| \Psi_i \right|^2 \D V = 1 \\[1ex] \int \Psi_i^* \, \Psi_k \D V = \delta_{ik} \end{align} One receives $c_1 = c_2 = c_3 = \frac{1}{\sqrt{2}}$ and $c_4 = - \frac{1}{\sqrt{2}}$ which produces the sp-hybrid orbitals depicted below.[10]

Formation of the sp-hybrid orbitals as linear combinations of a s- and a p- orbital
The sp-hybrid orbitals are formed as linear combinations of the original ones. The choice of the coefficients is not arbitrary but restricted by the normalisation- and orthogonality conditions. Note that the different colours represent different signs of the wave function. Thus both amplification and cancellation occur!

sp3-Hybridisation of Carbon

Now let us discuss the sp3-hybrid orbitals (which are of high relevance for solid state physics) as another example: As we have seen before, the electron configuration of carbon is 1s$^2$ 2s$^2$ 2p$_x^1$ 2p$_y^1$ (the superscript indicates the number of electrons in that particular orbital). Thus, actually there are only two partially filled orbitals and carbon could form only two bonds.

But what can happen instead is that one of the electrons in the 2s$^2$-orbital gets raised to the empty 2p$_z$-orbital. This may initially entail an extra energy expense. But now there is a total of four partially filled orbitals which will merge and form four new completely equivalent orbitals which are a linear combination of the original ones. Thereby the carbon atom becomes able to develop four bonds instead of two. This can cause a significant gain in energy that compensates the initial expense by far. But this depends on the binding partner!

This kind of hybridisation is very common for instance for carbon (diamond), silicon and germanium and is of high relevance for solid state physics.[11] [12] Covalent bonds of sp3-hybrid orbitals can be very strong due to the perfect overlap of the orbital2.

Shape of the sp3-hybrid orbitals
Shape of the sp3-hybrid orbitals: They form four equivalent lobes which point along the diagonals of a cube with the nucleus being located in the center of the cube. The angles between the lobes are $109.47\deg$.[13]

Metallic Bond

Elements of the first main groups in the periodic table have only few electrons in the outermost shell. In comparison to the inner electrons of the atom these electrons are located considerably further away from the nucleus and therefore only loosely bound.

Furthermore, these electrons occupy atomic orbitals which are quite spacious. In a solid the corresponding wavefunctions can still have a significant value at the positions of the surrounding atoms. Therefore the electrons are not only shared between two but between a large number of atoms!

Besides that metals are characterised by the fact that they posses few valence electrons that are significantly looser bonded than the inner ones because on average they are much further away from the nucleus and most of the positive charge is screened by the inner electrons. Thus it does not take much energy to detach these valence electrons from the atom.

Hence the outer electrons can be regarded as free - especially when an external source of energy is provided as in the case of en electric field (a voltage).

A bond characterised by this phenomenology are referred to as metallic bonds.[14] [15]

References

[1] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 3.2.2)
[2] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (pp. 65-66)
[3] H. Ibach, H. Lüth Solid-State Physics Springer 2009 (ch. 1.2)
[4] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 3.4)
[5] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (pp. 76-77)
[6] J. S. Blakemore Solid-State Physics Cambridge University Press 2004 (pp. 8-9)
[7] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 3.4.3.1)
[8] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (p. 77)
[9] S. Hunklinger Festkörperphysik De Gruyter 2014 (ch. 2.4.3)
[10] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 3.4.3.2)
[11] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 3.4.3.3)
[12] H. Ibach, H. Lüth Solid-State Physics Springer 2009 (ch. 1.2)
[13] J. S. Blakemore Solid-State Physics Cambridge University Press 2004 (pp. 8-9)
[14] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (pp. 135-136)
[15] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (pp. 78-79)

Your browser does not support all features of this website! more