We present a simple procedure to obtain the maximally localized Wannier function of isolated bands in one-dimensional crystals with or without inversion symmetry. First, we discuss the generality of dealing with real Wannier functions. Next, we use a transfer-matrix technique to obtain nonoptimal Bloch functions which are analytic in the wave number. This produces two classes of real Wannier functions. Then, the minimization of the variance of the Wannier functions is performed, by using the antiderivative of the Berry connection. In the case of centrosymmetric crystals, this procedure leads to the Wannier-Kohn functions. The asymptotic behavior of the Wannier functions is also analyzed. The maximally localized Wannier functions show the expected exponential and power-law decays. Instead, nonoptimal Wannier functions may show reduced exponential and anisotropic power-law decays. The theory is illustrated with numerical calculations of Wannier functions for conduction electrons in semiconductor superlattices.

Generalized Wannier functions of a couple of bands in a one-dimensional crystal are investigated. A lower bound for the global minimum of the total spread is obtained. Assumption of such a value being the minimum leads to a first-order differential equation for the transformation matrix. Simple analytical solutions leading to real generalized Wannier functions are presented for consecutive bands in a crystal with inversion symmetry. Results are displayed for a particle in a diatomic Kronig-Penney potential. For the lowest couple of bands, calculated single-band Wannier functions resemble orbitals of a diatomic molecule, whereas generalized Wannier functions seem like orthogonalized atomic orbitals. The latter functions are neither symmetric nor antisymmetric and display increased exponential decay. For the next two pairs of bands, Wannier functions retain their centers and symmetries, and exponential decay does not increase. Results are also shown to be in agreement with solutions of the eigenvalue problem of the band-projected position operator.

The maximally localized magnetic and electric Wannier functions of a one-dimensional photonic crystal with inversion symmetry are investigated. The calculated Wannier functions are real and either symmetric or anti-symmetric about an inversion centre of the crystal. The magnetic and electric Wannier functions of each band are centred at the same point, but they have opposite inversion symmetries. Interestingly, for the first band, they show different kinds of asymptotic behaviour. In turn, for higher bands, both types of Wannier functions decay in the same way. When dealing with localized electromagnetic modes in a perturbed one-dimensional crystal, the knowledge of these properties should help to build an appropriate basis set of Wannier functions.

It is shown that the standard sp n hybrid orbitals are orthogonal orbitals that minimize the total quadratic spread. This is done in a concise way that may improve the understanding of hybrid orbitals. The fact that maximally localized Wannier functions of crystalline materials may resemble hybrid orbitals is discussed.

Exponentially localized Wannier functions of cumulene are calculated from the Bloch functions obtained through a tight‐binding approach. Numerical results and discussions are given for the π and σ bands. In the latter case, the single‐band Wannier functions are similar to the orbitals of a diatomic molecule, while the two‐band Wannier functions resemble hybrid atomic orbitals. Contour plot of an sp‐like Wannier function of cumulene.