Multi-Resolution Analysis
Multiresolution Analysis (MRA) brings an effective way to solve Schrödinger equation.
\[ \hat{H}\Psi=E\Psi \tag{1}\]
It is a real space numerical method used to project functions and operators with arbitrary precision given by the following formula.
\[ f^n (x)= \sum_k \sum_l s^n_{kl} ~ \phi^n_{kl}(x) \]{eq-function-projection}
where \(f^n (x)\) is the function resolved at \(n^{th}\) level, \(\phi^n_{kl}(x)\) are the scaling functions and \(s^n_{kl}\) are the scaling coefficients. The scaling or basis functions are derived from Legendre Polynomials. For a detailed overview, book chapter written by Bischoff1 is a good starting point. Assuming the potential energy function in the system of interest is \(\hat{V}=V(r)\) and the rearrangement of Equation 1 will give the following expression
\[\begin{align} \hat{H}\Psi &= E\Psi \\ \Rightarrow (\hat{T} + \hat{V})\Psi &= E\Psi \\ \Rightarrow (\hat{T} - E)\Psi &= -V\Psi \\ \Rightarrow \Psi &= -(\hat{T} - E)^{-1}V\Psi \\ \end{align}\]
The resulting equation above can be algorithmically considered to be a fixed point iteration problem. It can be self-consistently solved with a given good guess of \(\Psi\). At the end, MRA would produce results in complete basis set (CBS) limit being a real space grid based method.
Perpendicular Paramagnetic Bonding
Besides covalent and ionic bonding, there exists a third kind of distinct bonding which occurs primarily in presence of strong magnetic fields. There is a rehybridization of antibonding orbitals in the systems which are non-bonded in ambient atmospheres. For example, singlet He2, it does not have a primary binding interaction according to Molecular Orbital Theory (MOT). But it shows strong primary binding interaction with the increase in \(\mathbf{B}\) values.
Limitations of Gaussian basis functions
In this particular scenario MRA wavelets are much better suited for strong \(\mathbf{B}\) calculations, as it do not require subtle handwork which one needs to do in case of atom centered Gaussian basis functions. The wavefunction in \(\mathbf{B}\) is not gauge-invariant which makes them meaningless entities. To overcome it, one needs to make them gauge-invariant, which in practise done using LONDON orbitals. But MRA does not have these problems as it is a real space grid based approach.