14 Carbon Based Nanostructures

Table of Content

Module I

5.1 Atom to molecule

5.1.1 Atomic orbitals

5.1.2 Molecule from atomic orbitals

5.1 Atom to Molecule

To study the different properties of carbon based nanostructure, one has to start from carbon atom, then molecule to solid. Quantum mechanically each atom is represented by set of atomic orbital, a mathematical function obtained by solving Schrödinger equation of the atom. However, except hydrogen atom, exact solution is not possible. Determining the energies and wavefunctions of many electrons in a atom or in a solid is truly many body problem. Nevertheless, there is much to be learnt from the exact solution of hydrogen atom. Under appropriate approximation, many electron atoms and many atomic molecules can be built up from the hydrogenic atomic orbital.

5.1.1 Atomic orbitals

Hydrogen is the simplest element. One requires accurate calculations of electronic structure. There are an electron and a positively charged nucleus comprised of a single proton. Hydrogen atom can be treated as particle in a quantum well defined by following electrostatic potential

The electron experiences the attractive potential of the nucleus. The atomic potential is spherically symmetric Coulomb potential. The corresponding Schrodinger equation

As the potential is spherically symmetric, wavefunction ψ(r,θ,f) can be expressed as product of three functions, radial wavefunction R(r), angular wavefunftion f(θ) and azimuthal wavefunction g(f) using separation of variable.

Due to radial symmetry in the problem, the Eq.1 can be expanded in spherical polar co-ordinates and given by

The solution of Eq.2 would be

spherical harmonic function of degree l and order m . Three quantum nos principal quantum no (n), orbital angular momentum quantum no (l) and magnetic quantum no. (m) take the values, n=1,2,3……………, l=0,1,2,3……….(n-1) amd m=-l………..l.  The energy eigenvalues including fine structure correction  are given by

where α is the fine-structure constant and ℓ ± 1/2 is the “total angular momentum” quantum number, ± stands
for spin up and spin down. The probability of finding the electron in a differential volume element dτ is.

Now, the probability distribution function can be given by

For atomic orbital and distribution electron density in atom, radial distribution is sufficient

The atomic orbital is basically mathematical function that describes the wave-like behavior of electron in an atom and can be used to calculate the probability of finding the electron of an atom in any specific region around the atom’s nucleus. The particular form of atomic orbital essentially defines the probability of finding the electron in a particular region inside an atom. A unique set of values of the three quantum numbers n, ℓ, and m define each orbital in an atom. Each such orbital can be occupied by a maximum of two electrons. The names of orbitals, s, p, d and f refer to orbitals with angular momentum quantum number ℓ = 0, 1, 2 and 3 respectively. The radial wavefunctions for n=1,2,3 are

Each of the solutions of Eq.4 . is shown in Figure 5.3 is labeled either s (l=0) or p (l=1) or d (l=2). These letters also describe the angular symmetry of different orbitals.     S   orbital has spherical symmetry and exhibit even symmetry about the origin in every dimension. Other orbitals  exhibit odd symmetry about the origin.

5.1.2 Molecule from atomic orbitals

To determine the  electronic structure of molecules, we have to consider multiple electrons and multiple nuclei. Considerable simplification can be achieved by assuming that the nuclear positions are fixed. Under Born-Oppenheimer approximation the Schrödinger equation is then solved for the electrons in a static potential. Finally the solution is optimized by minimizing total energy out of different arrangements of the nuclei are chosen and the solution is optimized. However, with all simplification the solution of multiatomic molecule is extremely complex and generally solved by numerical techniques. But a good starting point is that the molecular orbital are linear combinations of atomic orbital,

Where Φ is molecular orbital and ψi is atomic orbital. A chemical bond is defined as a filled molecular orbital with lower energy than the constituent atomic orbital stabilizing the molecule.

In carbon based compounds and nanostructures, there are two types of bonds, sigma bond (σ) and pi bond (π). Sigma bond are usually stronger that Pi bonds. Sigma bonds are formed by the overlap of orbital that are pointing directly towards one another. The sigma bonds are the strongest type of covalent chemical bond. Robustness of the sigma bond comes from the fact that if two atoms are connected by a sigma bond, rotating one of the atoms around the bond axis doesn’t break the bond, as schematically shown in Figure 5.4. On the contrary, in case of pi bond, the bond between two atoms gets broken when one atom is rotated around the bond axis, as schematically shown in Figure 5.4. Common forms of sigma bonds are s+s, pz+pz, s+pz and dz2+dz2. The C=C double bond, composed of one sigma and one pi, has a bond energy less than twice that of a C-C single bond. This indicates that the stability added by the pi bond is less than the stability of a sigma bond. The weakness of pi bond can be explained by significantly less overlap between the component p-orbitals due to their parallel orientation, whereas sigma bonds which form bonding orbitals directly between the nuclei of the bonding atoms, result from the greater overlap.