Build a band from localized orbitals

Take one orthonormal orbital per site in a one-dimensional chain, with onsite energy ε₀ and a nearest-neighbor Hamiltonian matrix element −t, where t is positive. Apply the Hamiltonian to a Bloch sum whose site amplitudes are proportional to exp(ikja). Hopping from the two neighbors contributes opposite phases, whose sum is twice a cosine. This yields a band with its minimum at zero wavevector for the chosen negative off-diagonal convention.

Other authors call the actual off-diagonal element itself t, which changes the displayed sign of the cosine. Neither convention is wrong if the Hamiltonian is stated. Complex hopping requires the reverse hop to be its complex conjugate to maintain Hermiticity. Local changes of orbital phase can redistribute hopping phases, whereas phase accumulated around a closed loop can encode physical magnetic flux. An arbitrary phase assignment cannot be interpreted as a magnetic field without checking gauge-invariant loops.

If each primitive cell contains multiple orbitals, Bloch sums form a small orbital-space Hamiltonian for each wavevector. Fourier-transform hopping over cell displacements, using the same sign convention as the chosen Bloch basis. One can include intracell orbital positions in the phases or leave them in the basis functions; this changes matrix representations and eigenvector phases while leaving properly computed energies invariant. Quantities involving derivatives of eigenvectors require consistent handling of that basis convention.

Atomic orbitals centered on different atoms are not automatically orthogonal. If their overlap is retained, the coefficient vector obeys a generalized eigenvalue equation with overlap matrix S. Its normalization is c†Sc=1, not c†c=1. Simply dropping S while keeping the same matrix elements changes the model. Orthogonalization can be useful, but it transforms the Hamiltonian too and may introduce longer-range matrix elements that were absent in the original nonorthogonal representation.

Longer-range hopping adds higher reciprocal harmonics and can alter extrema, curvature, or Fermi-surface topology even when its energy scale is smaller than the nearest-neighbor term. Orbitals omitted from the basis can also shift effective parameters through virtual mixing. Fitting several energies does not guarantee correct velocities or orbital weights between them. A model should therefore be validated against the observable and energy range for which it will be used, rather than judged only by whether it resembles a textbook band.

The spectral lab provides a Fourier-weight illustration. Reset and set mode count to 1 and coefficient decay to 0; predict highest-mode coefficient 1. Reset and set mode count to 3 and decay to 2; predict highest coefficient 1/9 and a smallest displayed wavelength of 2π/3. This prescribed sine sum uses sin(nx−t)/nᵖ. It is not the cosine hopping dispersion, a Bloch eigensolver, or a calculation of orbital overlap.