The function BlockBandDiagonalize() can be used to reduce the number of basis (spin-)orbitals by making linear combinations of (spin-)orbitals, according to the tight-binding structure (hopping matrix elements) within the (spin-)orbitals. As a simple example to make the idea clear, consider the following 3-by-3 matrix: $$M = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}$$ Now, assuming that $M$ corresponds to a tight-binding Hamiltonian defined on some basis, we can linearly combine the second and third basis orbitals, such that we get a single orbital which mix with the first orbital via the matrix M. Consider the following unitary rotation matrix: $$U = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}$$ Now, transforming the matrix $M$ using the unitary matrix $U$ results in: $$M' = U M U^{T} = \begin{pmatrix} 0 & \frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ In the new representation, the first basis orbital only mixes with the second The basis orbital and not with the third one. The function BlockBandDiagonalize() can accept 3 types of objects as an arguments: Tight-binding object, Operator, or Matrix.

Case 1:

• matrix: hermitian matrix
• blockSize: size of the block (as number) or list of vectors representing the starting states

Case 2:

• operator: hermitian operator
• wave function : single wave function of list of wave functions

Case 3:

• tightbindingObject: tight binding object
• startingBlock : list of atoms with positions, shells and orbitals used as starting block

(Optional) Third argument (in all cases)

• NTri : (integer) maximum number of blocks included (default: $\infty$)

• NOrtho : (integer) maximum number of reorthogonalizations (default: $\infty$)

• ReOrthogonalize: (boolean) use additional Gran-Schmidt orthogonalization after the Löwdin orthogonalization (default: true)

• matrix: block-band diagonal matrix