Physics in a nutshell

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# Atomic Orbitals

The aim of this chapter is to understand the various bonds acting between atoms and molecules. This is however not possible without some knowledge about the atomic structure. To that end a bit of quantum mechanics is required.

## Quantum Mechanical Description

### Wave Function and Schrödinger Equation

In quantum mechanics the state of a physical system is determined by its wave function $\psi$ which in turn is the solution of the Schrödinger equation for this system.

### Atomic Orbitals and Quantum Numbers

For a single electron in an atom such a particular state is called orbital and specified by three quantum numbers:

• The principal (energy) quantum number $n = 1,2,\dots$
• The angular momentum quantum number $l = 0, 1, \dots, n-1$. For historical reasons the different numbers are denoted by letters, namely $0 \rightarrow \text{s}$, $1 \rightarrow \text{p}$, $2 \rightarrow \text{d}$ and $3 \rightarrow \text{f}$)
• The magnetic quantum number ${m = -l, -(l-1), \dots, l-1, l}$

In addition to these orbital quantum numbers there is another so-called spin quantum number ${s = \pm \frac{1}{2}}$ which can be thought of as an intrinsic angular momentum. This will become relevant later.

#### Probability Density

One can now calculate the probability densitiy of finding the electron in a specific location for a certain orbital. It is the squared value ${\left| \psi (x) \right|^2}$ of the wave function at this particular location. One can visualise the orbital by creating three dimensional figures showing the areas with a high detection probability densitiy.

This is done in the following for s-, p- and d-orbitals. In the notation (1s, 2p, 3d) the numbers represent the principal quantum number $n$ and the letters the angular momentum quantum numbers $l$. The different colors mark different signs of the wave function.

#### 1s-Orbital For each value of $n$ there is only one s-orbital since there is no $m$ degeneracy. The 1s-orbital is lowest in energy and thus the ground state of the electron in a hydrogen atom. All s-orbitals are spherically symmetric.

#### 2p-Orbitals For $n \gt 1$ the electron can be in an excited p-orbital as well. There are three different p-orbitals due to the 3-fold degeneracy of $m$. The orbitals are formed by a double lobe that point in the direction of the coordinate axes.

#### 3d-Orbitals For $n \gt 2$ d-orbitals are possible. There are five different d-orbitals due to the 5-fold degeneracy of m. The first three orbitals are composed of 4 lobes each pointing along the coordinate system diagonals. The fourth orbital looks the same but is orientated along the $x$- and $y$-axes. The fifth orbital looks like the 2p$_z$-orbital enclosed by an additional ring.

## References

  J. Barrett, A. G. Davies, D. Phillips, E. W. Abel, J. Woollins Atomic Structure and Periodicity The Royal Society of Chemistry 2007 (ch. 2.1.2 and 2.1.4)  D. Meschede Gerthsen Physik Springer 2010 (p. 822)  D. Meschede Gerthsen Physik Springer 2010 (p. 701)  J. Barrett, A. G. Davies, D. Phillips, E. W. Abel, J. Woollins Atomic Structure and Periodicity The Royal Society of Chemistry 2007 (ch. 2.1.2)  J. Barrett, A. G. Davies, D. Phillips, E. W. Abel, J. Woollins Atomic Structure and Periodicity The Royal Society of Chemistry 2007 (p. 31 ff.)

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