atomic and molecular orbitals

1.1 Introduction

The most dramatic success in the history of quantum mechanics was the understanding of the details of the spectra of some simple atoms and the understanding of the periodicities which are found in the table of chemical elements.
The study of the behavior of the electron in the hydrogen atom by Shroedinger back in 1926 was considered to be a corner stone in the process of this understanding, since the solution of his equation provided answers for the allowed energy levels of the electron and the probability density of finding the electron of the hydrogen atom in any specific region around the atom's nucleus, in great accordance with Heisenberg's uncertainty principle (1927).

This mathematical function that describes the wave-like behavior of an electron in an atom was named atomic orbital and was first introduced by Robert Mulliken in 1925 [1]. In this page, we will try to explain how the atomic orbitals (AO) are related to the three quantum numbers n, l, m, as well as what happens in the case of homonuclear and heteronuclear molecular bonding.

1.2 The Hydrogen atom

Shroedinger's equation written as

\begin{align} \i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = (-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}))\Psi(\mathbf{r},\,t) \end{align}

can be transformed into

\begin{align} \Psi_{r,\theta,\phi}=R(r)\Theta(\theta)\Phi(\phi) \end{align}

after separation of variables in radial and angular contributions.

The separation leads to three equations for the three spatial variables, and their solutions give rise to three quantum numbers associated with the hydrogen energy levels, the shape and the orientation of the orbitals. Therefore, the equations for the first three quantum numbers are all interrelated:

  • n:The principal quantum number arose in the solution of the radial part of the wave equation as shown below.The Schroedinger wave equation describes energy eigenstates having corresponding real numbers En with a definite total energy which the value of defines. The bound state energies of the electron in the hydrogen atom are given by:
\begin{align} \ E_{n}= \frac {E_1}{n^2} = \frac {-13.6eV}{n^2}, n=1,2,3... \end{align}
  • l:The azimuthial number, denotes the angular momentum of an electron in an orbital and corresponds to the shape of the orbital (i.e. the most probable electron distribution). It can take the values 0, 1, 2, …, n-1, where n is the principal quantum number and the numbers 0, 1, 2, and 3, correspond to the s, p, d, and f orbitals, respectively.
  • ml:The magnetic number, is responsible for the orientation of the orbital around the nucleus and also determines the energy shift of an atomic orbital due to an external magnetic field, hence its name. It can take the values -l, (-l+1),…,0,…,(l-1), l, where l is the azimuthial quantum number.
  • ms: Finally, the fact that no two electrons belonging to the same atom can have an identical set of quantum numbers according to the Pauli exclusion principle introduces the fourth quantum number, named as the spin quantum number, which parameterizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of the electron and can take only the values +1/2,-1/2.

Being a little observant we will notice that Shroedinger's results verify Rydberg calculations held back in 1888 for the allowed energy levels, an aspect which was incorporated in Bohr's conception of the atom in 1913 through the calculation of the smallest possible orbit for the electron, that with the lowest energy, which is found at a distance from the nucleus called the Bohr radius:

\begin{align} \ a_{0} = \frac{4 \pi \epsilon_0 \hbar^2}{m_e e^2} \end{align}

However, it was with Shroedinger's contribution that different pieces of information fall in place and managed to give a more complete perspective of the solution of the problem. Since the orbitals are nothing but the eigenfunctions corresponding to the eigenvalues En, solving explicitly the hydrogen problem will give us, starting from the first state, orbitals named as 1s,2s,2p,3s,3p,4s,… The numbers "1,2,3,4,…" correspond to the energy level and the letter "s,p,d,f" to the shape of the orbital. For the hydrogen case, there will be only one electron sitting on the 1s orbital, but for more complicated atoms the number of electrons increases and so does the amount of orbitals.

Before moving on, let's first solve for the first 5 eigenfunctions to see what information we can obtain:

\begin{align} \Psi_{100}=\Psi_{1s}=A e^\frac{-r}{a_0} \end{align}
\begin{align} \Psi_{200}=\Psi_{2s}=A (2-\frac{r}{a_0}) e^\frac{-r}{2 a_0} \end{align}
\begin{align} \Psi_{210}=\Psi_{2px}=A (\frac{r}{a_0}) e^\frac{-r}{2 a_0} \cos{\theta} \end{align}
\begin{align} \Psi_{21\overline1}=\Psi_{2py}=A (\frac{r}{a_0}) e^\frac{-r}{2 a_0} \cos{\phi} \sin{\theta} \end{align}
\begin{align} \Psi_{211}=\Psi_{2pz}=A (\frac{r}{a_0}) e^\frac{-r}{2 a_0} \sin{\theta} \sin{\phi} \end{align}

The picture shows the shapes and orientation of the atomic orbitals [2]


whereas the energy spectrum for the H atom is shown here


We can easily observe that for the last three wave functions there is contribution of both radial and angular coefficients, whereas for the first and second case the coefficient is only radial. This explains why s orbitals have spherical shapes, while p orbitals have the shape of lobes. Apart from the shape, the equations provide us with information about the energy levels the wave functions correspond to. Thus, we can identify four different orbitals (2s,2px,2py,2pz) the same energy level. Orbitals of this kind are called degenerate and any linear combination of the functions corresponding to a given set of degenerate orbitals serves as an equivalent representation of this set of orbitals [3].

The above mentioned analysis gives rise to the interesting question of how the energy spectrum of the H2+ molecule is affected. As the picture shows, when two H atoms come close and form a bond the 1s, 2s etc levels are doubly degenerate (since there are two atoms) because the electron in the 1s state has the "option" of occupying that level in any of the two atoms. The interaction of the two atoms leads to the split of the energy level into two sub levels, which are nothing but our well known bonding and antibonding orbitals [3].


1.3 From Hydrogen Atom to Molecules and the effect in their properties

1.3.1 The simple case of H2+molecule

After having worked for the hydrogen atom, we can move on to the understanding of molecular orbitals. Similarly to the case of AO, MO is a mathematical function that describes the wave-like behavior of an electron in a molecule.
In chemistry, molecular orbital theory (MO theory) is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule [4].

For example, in the simplest case of H2+ the Hamiltonian will be set

\begin{align} \mathcal{H}=(-\frac{\hbar^2}{2m}\nabla^2 + \frac{e^2}{4 \pi\epsilon_0}(\frac{1}{r_1} +\frac{1}{r_2}) \end{align}

and Shroedinger's equation will be

\begin{align} \mathcal{H} \Psi(r)=E_{electron} \Psi(r) \end{align}

In this theory, each molecule has a set of molecular orbitals, in which it is assumed that the molecular orbital wave function may be written as a simple weighted sum of the n constituent atomic orbitals, according to the following equation:

\begin{align} \Psi(r) = c_1 \phi_1+c_2 \phi_2 \end{align}

This method is called the linear combination of atomic orbitals (LCAO) approximation and is used in computational chemistry. Through several simplifications we can determine numerically the c1,c2 coefficients by substitution of this equation into the Schrodinger equation. Eventually, two eigenfunctions which correspond to two energy levels will be the solution of (11).

\begin{align} \Psi_{bonding}=\frac{1}{\sqrt{2}} (\phi_1+\phi_2) \end{align}
\begin{align} \Psi_{antibonding}=\frac{1}{\sqrt{2}} (\phi_1-\phi_2) \end{align}

So, let's check what is different in this case in comparison to the previous one of the H atom: In the case of H2+ molecule when two H atoms come close to each other to form a bond, meaning to share one electron, the energy level that corresponded to the 1s, say, atomic orbital will split in two energy levels as shown in picture [5].


We call the lower level "bonding state", whereas the upper is known as the "antibonding" one and the wave functions corresponding to each level are named accordingly. The electron shared by the two nuclei will sit on the bonding state, since this is the stable one and the antibonding will be empty.

1.3.2 Expanding to other homonuclear cases and the effect of MO in the properties of molecules

Let's now work for the cases of B2,N2 and O2 and see what kind of information we obtain from this analysis

Molecule Electronic configuration Number of bonding electrons Number of bonds
B2 1s2,2s2,2p1 2 1
N2 1s2,2s2,2p3 6 3
O2 1s2,2s2,2p4 4 2

The triple bond in N2 is apparently the reason why this bond is the strongest among the homonuclear diatomic molecules. Thus, according to the 3-s rule, which states that stronger bonds are shorter in length and related to smaller bonding energies, N2 will have the stiffer and shorter bond among all the three.

Additionally, the electronic configuration provides us with information regarding the magnetic properties of the molecules. Consequently, according to our previous discussion about the bonding and antibonding orbitals, O2 will have two unpaired electrons sitting on two of the three 2p antibonding orbitals, denoting that it has paramagnetic properties, even though it has an even number of valence electrons.


1.3.3 Heteronuclear molecules-The case of hybridization

The case of heteronuclear molecules follows the same principles as the one for the homonuclear molecules.
Here, however there are some new aspects contributing in the description of MO, which we will examine carefully.

Since the atoms that form the molecule are no longer identical, the bond formed will be the result of mixing of the atomic orbitals. This process is known as hybridization and was promoted by chemist Linus Pauling in order to explain the structure of molecules such as methane (CH4) [6]. The number of these new hybrid orbitals must be equal to the numbers of atoms and non-bonded electron pairs surrounding the central atom. The hybridized orbitals are named according to the number and kind of AO from which they are built. This will become more clear in the forthcoming examples:

In the case of methane [7], the three 2p orbitals of the carbon atom are combined with its 2s orbital to form four new orbitals called sp3 hybrid orbitals. The name is simply a mixing of all the orbitals that were blended together to form these new hybrid orbitals. Four hybrid orbitals were required since there are four atoms attached to the central carbon atom. These new orbitals will have an energy slightly above the 2s orbital and below the 2p orbitals as shown in the following illustration [7].


In the boron trifluoride molecule [7], only three groups are arranged around the central boron atom. In this case, the 2s orbital is combined with only two of the 2p orbitals (since we only need three hybrid orbitals for the three groups) forming three hybrid orbitals called sp2 hybrid orbitals. The other p-orbital remains unhybridized and is at right angles to the trigonal planar arrangement of the hybrid orbitals. The trigonal planar arrangement has bond angles of 120o.


Finally let's look at beryllium dichloride [7]. Since only two groups are attached to beryllium, we only will have two hybrid orbitals. In this case, the 2s orbital is combined with only one of the 2p orbitals to yield two sp hybrid orbitals. The two hybrid orbitals will be arranged as far apart as possible from each other with the result being a linear arrangement. The two unhybridized p-orbitals stay in their respective positions (at right angles to each other) and perpendicular to the linear molecule.


In a similar way, hybridization in higher orbitals may occur too, so sp3d and sp3d2 orbitals can be formed.

1.3.4 sigma and pi bonds: How are they formed?

One of the most interesting aspects of hybridization is the fact that it is responsible for the formation
of double and triple bonds in molecules. So, what do we mean when saying that double bonds are formed from one sigma and one pi bond, whereas triple bond is formed from one sigma and two pi bonds?

In order to understand this, we will start from the case of a simple molecule that has a double bond:
Ethene is built from hydrogen atoms (1s1) and carbon atoms (1s2,2s2,2px1,2py1) [8].
The carbon atom doesn't have enough unpaired electrons to form the required number of bonds, so it needs to promote one of the 2s2 pair into the empty 2pz orbital. This is exactly the same as happens whenever carbon forms bonds.Now there's a difference, because each carbon is only joining to three other atoms (another carbon and two hydrogens) rather than four - as, for example, in methane. The carbon re-organises the s orbital and two of the p orbitals to give three new orbitals with exactly the same energy. The other p orbital is left unchanged.


The various atomic orbitals which are pointing towards each other now merge to give molecular orbitals, each containing a bonding pair of electrons. Molecular orbitals made by end-to-end overlap of atomic orbitals are called sigma bonds [8].


This sideways overlap also creates a molecular orbital, but of a different kind. In this one the electrons aren't held on the line between the two nuclei, but above and below the plane of the molecule. A bond formed in this way is called a pi bond [8]. Notice that the p orbitals are so close that they are overlapping sideways.


Similarly, when two pi bonds and one sigma are formed we have a triple bond.

1.4 Conclusion
With this study, I hope I covered some of the fundamental aspects related to the formation of orbitals both in atoms and in molecules and thus, managed to explain to some extent how orbitals affect the formation of bonds and define properties in the molecular level.

1.5 References
1. Mulliken, Robert S., 1967. Nobel Lecture. Science, 157 (3785), pp 13-24.
2. [Online] Available at: [Accessed 2 March 2009].
3. Omar, Ali M., 1975. "Elementary Solid State Physics". Addison-Wesley Publishing Company. Philippines
4. Wikipedia. 2009. Molecular orbitals. [Online]. Available at: [Accessed 2 March 2009].
5. [Online] Available at: [Accessed 2 March 2009].
6. Pauling L., 1931. J. Am. Chem. Soc, 53 (1367), pp 1367–1400
7. Chemistry. 2009. Bonding and Hybridization. [Online]. Available at: [Accessed 2 March 2009].
8. Chemguide. 2009. Covalent bonding-Double bonds. [Online]. Available at: [Accessed 2 March 2009].

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