Research
Quantum Chemistry and Schroedinger’s Equation
Materials having one or more dimensions especially below 20 nm shows observable changes in optical, electronic, chemical and other properties than the material in bulk. These materials are prefixed with the word quantum1. The word quantum is associated because of the behaviour of the particles is not anymore understood by classical mechanics. Quantum mechanics is the answer for them. Quantum mechanics and Schrödinger equation are synonymous. The time-indepedent non-relativistic form of the equation looks like following:
\[ \hat{H}\Psi = E \Psi \tag{1}\]
Apparently it might look very straightforward equation, but trust me it is not analytically solvable for a two electron systems/molecules. So, it is necessary to use relevant approximations to solve the Equation 1. Some of the famous ways to do it is using Hartree-Fock approximation, Density Functional Theory, Plane Wave methods, Finite Element methods etc.
Multi-Resolution Analysis (MRA)
One of the modern ways to solve it is by MRA. It is a bit straightforward how one can use MRA to solve the partial differential equation at Equation 1. Let us assume the potential energy function in the system of interest is \(\hat{V}=V(r)\) and now we will split the hamiltonian into the follwing parts
\[\begin{align} \hat{H}\Psi &= E\Psi \\ \Rightarrow (\hat{T} + \hat{V})\Psi &= E\Psi \\ \Rightarrow (\hat{T} - E)\Psi &= -V\Psi \\ \Rightarrow \Psi &= -(\hat{T} - E)^{-1}V\Psi \\ \end{align}\]
Hence now we have an equation that can be self-consistently solved with a given guess of \(\Psi\). And from the [cite=alpert et al] we know that \((\hat{T}-E)^{-1}\) can be approximated to a bound state helmholtz operator. For more details oe can go through the book chapter written by Florian A. Bischoff2, my PhD supervisor.
Molecules in Strong Magnetic Field
There are some atoms that theoretically and practically do not form bonds in earthly environment. One of the example that we know is helium dimer, i.e. He2. If we consider Molecular Orbital Theory we will see that He2 does not exist. For a mathematical explanation here one can say
MO theory says a non-zero number generated by the difference of number of electrons in bonding and non-bonding MOs would assure there is a covalent bonding in the molecular system. For helium dimer this would be
\[ \mathrm{bond ~ order} = N_{bonding} - N_{anti-bonding} = 2-2 =0 \]
which suggests that there is no bonding.