Quantum Chemistry and Schrödinger’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 proven to be computationally intractable and analytically unsolvable once it goes beyond the H atom. So, it is necessary to use relevant approximations to solve the Equation 1. Some of the ways to do it is using basis set expansion using atom-centred gaussians, Plane Wave methods, Finite Element methods or numerical tensor based methods.
Multi-Resolution Analysis
One of the ways that my research is interested in, is to solve Equation 1 is using Multiresolution Analysis (MRA). MRA is a real space numerical method to project functions and operators with arbitrary precision.
\[ F(x)= \sum_{n,l} \sum_{k=0}^{k_{\mathrm{max}} - 1} \phi^n_{k,l}(x) s^n_{k,l} \]
where \(\phi^n_{k,l}(x)\) is the function under consideration and \(s^n_{k,l}\) are the scaling coefficients obtained from Legendre Polynomials. For details one can refer the book chapter from Bischoff, F. A.
How can one use MRA to solve the Equation 1? It is pretty straightforward in the language of MRA. Let us assume the potential energy function in the system of interest is \(\hat{V}=V(r)\) and now we will rearrange the Equation 1 as following
\[\begin{align} \hat{H}\Psi &= E\Psi \\ \Rightarrow (\hat{T} + \hat{V})\Psi &= E\Psi \\ \Rightarrow (\hat{T} - E)\Psi &= -V\Psi \\ \Rightarrow \Psi &= -(\hat{T} - E)^{-1}V\Psi \\ \end{align}\]
Point to be noted here is that \(V(r)\) can be any form of potential - particle in a box, simple harmonic oscillator, hydrogen atom, coulomb and exchange from Hartree-Fock method etc. Hence, using variational principle1 now we have an equation that is algorithmically a fixed point iteration. It can be self-consistently solved with a given guess of \(\Psi\) in a complete basis set limit. In that regard, it is a basis set incompleteness error free technique.