Primary input to our codes are the Kohn-Sham states and their energy eigenvalues.
Codes are serial and can be easily interfaced with Quantum Espresso 7.5.


Subroutines primarily perform two types of tasks as post-processing applications:

A. Construction of localized orbitals from first principles:

• Template free construction of maximally localized orbitals(MLO) for isolated systems - atoms, molecules and clusters.

• Construction of localized orbitals from Kohn-Sham states for a system following template of localized orbitals of isolated atoms - same as Wannierization in case of perioidic systems.
  Depending on the nature of template and the number of KS bands used, the resultant localized orbitals can be atomic or bonding-orbitals.

• Construction of tight-binding model in the basis of localized(Wannierized) atomic orbitals.

• Computation of Mayer bond order(MBO) in the localized orbital basis and MBO based partitioning of total charge density in real space into components shared and retained by atoms - an unbiased visual proof of covalent interaction of arbitrary strength.

Briefly on the methodologies used in the codes..

Our efforts started in search of tight-binding basis in which we could construct easily transferrable Hamiltonians aimed at inexpensive yet workably accurate computation of properties of the ground and excited states of experimentally realizable nanostructures which are typically made of few 100s to 1000s of atoms - still very large to compute as is explicitly from first priciples.

We started with construction of a suitable localized orbitals from first principles to constitute a minimal basis to maximally represent the electronic structure of a given system.

Why hybridized orbitals?
We found that hybridized atomic orbitals are convenient[1,2] because they can be directed towards directions of coordination, which leads to a sparse Hamiltonian for covalent systems where each nearest neighbour interaction can be represented predominantly by a single off-diagonal element. Locked to the neighbourhood, tight-binding parameters in such a directed basis can be easily transferred across isomorphic systems. Directions of coordinations around atoms often closely follow coordination polyhedra constituted by directions of hybridized orbtials, such as sp², sp³, sp³d², sp³d⁵ etc.

Construction of hybridized orbitals
To construct the hybridized atomic orbitals we took clue from the Foster-Boyz localization scheme which says that the maximally hybridized orbitals are also the maximally separated or localized ones.
We then turned to construction of maximally localized orbitals which turned out to be easily possible for isolated systems through maximal joint diagonalization [2] of the generally non-commuting first moment matrices along X,Y and Z directions in the basis of Kohn-Sham(KS) states.
For an isolated atom the process naturally renders the hybridized orbtials given the choice of KS states[2].

For systems with non-ideal bond-angle, which is predominantly the case, appropriate hybridized orbitals can be constructed with little more of procedures[4].

Wannierization of KS states
Subsequently the hybridized atomic orbitals for individual atoms are placed on atoms in the given systems with appropriate orientation as per the nearest neighbourhood around atoms to constitute a guiding template for Wannierization of the KS states of the given system.


Choice of gauge for Wannierization is obtained in two ways - (1) through Lowdin symmetric orthogonalization[1,2] of representation of the templates within the KS subspace, and (2) through SVD based maxial allignemt of the KS subspace to that of the templates[3].

Tight-binding Hamiltonian
Tight-binding parameters are calculated in the basis of the Wannierized hybrid atomic orbitals constructed with a larger subspace of KS states beyond the occupied manifold.

Self energy corrected TB parameters can be calculated simply by using self-energy corrections to single particle levels within the GW approximation[1,2].

TB parameters can be transferred from smaller systems to larger isomorphic systems through mapping of nearest neighborhoods of atoms. We did see about 90% recovery of self-energy corrected band gap bigger nanodiamonds with self-energy corrected TB parameters mapped from much smaller nano-diamond[1,2].

Bond-order in Wannierized basis
Computation of Mayer bond order(MBO) in the Wannierized orbital basis, and MBO based partitioning[5] of total charge density in real space into charge density components shared and retained by atoms along coordination segments.

The shared charge densities present an unbiased visual proof of covalent interaction of arbitrary strength.



Correlated hybrid Wannier functions from geometric phases
For periodic systems the first moment matrices can be obtained from the matrix generalization of Berry phases incurred by Bloch electrons multiple bands as they evolve through the Brillouin zone.

The eigenvalues of the first moment matrix for a given direction render centres of WFs which have maximum localization in that direction while Bloch like in other linearly independnet directions. These WFs are thus called hermaphrodite or hybrid WFs(hWF).

We then used maximal joint diagonalization of the matrix generalization of Berry phases computed in linearly independent directions in the Brillouin zone to correlate three hWFs localizing simultaneously in the three directions and plotted there centres in real space to render possible centres of electron localization in matter[6].

Up coming project : Template free Wannierization in higher dimension from geometric phases
Then we asked the question : Can we Wannierize Bloch states in more than 1D simultaneously exclusively form geometric phases ? Is a choice of gauge possible without using any template?
We did see above that Wannieraization, is inherently biased by the choice of template. Of course furher numerical localization of WFs, as undertaken in Wannier90, although known not to substantially improve localization, can dilute the bias, but only upto an arbitrary degree unless we are sure to reach the global minima of total spread.