# Parallels between Tight-binding wavefunctions and Bloch states

I was reading Robert Knox's Theory of Excitons and I came across a certain statement that I have trouble reconciling, both mathematically and conceptually. The context concerns the electronic structure description by a Hamiltonian that includes terms akin to the Kohn-Sham equation. I'm familiar with this but then the author discusses the approximations to the wavefunction itself - apart from the well-known Hartree-Fock solutions. The two approximations are:

a) A tight-binding-esque wavefunction wherein one writes it down as an atomic function $$\psi_{lr}$$ where l is a set of atomic quantum numbers and R is one of the lattice sites where $$\psi_{lr}$$ is centered

b) A bloch state $$\psi_{nk}$$ where n is a band index and k runs over the entire Brillouin zone

I am familiar with both models - With the tight-binding model, you have localized orbitals whereas with the bloch state, it is distributed uniformly over the crystal. The author then says that

In a crystal consisting of atoms in closed shells, the ground state is identical in the two schemes. This holds in the formal sense: if from a set of localized states, one constructs Bloch states, $$\psi_{lk} = N^{-1/2} \sum_{R}e^{ik.R} \psi_{lR}(r)$$

The author goes on to say that the determinantal functions (basically a many-particle wavefunction written as a Hartree product) of the two are identical:

$$\psi_{lk_{1}\alpha}(r_1)\psi_{lk_{1}\beta}(r_2)..\psi_{lk_{N}\beta}(r_2N)$$

$$\psi_{lR_{1}\alpha}(r_1)\psi_{lR_{1}\beta}(r_2)..\psi_{lR_{N}\beta}(r_2N)$$

Here $$\alpha$$ and $$\beta$$ (I assume) run over spins -1/2 and +1/2, and the system has 2N valence electrons.

The justification , directly quoted from the author is: Once these two equations are written as determinants, and noting the elements form matrices related by simple multiplication by a unitary matrix ($$U_{pq} = N^{-1/2}e^{ik_{p}.R_{q}}$$) whose determinant is unity.

I have trouble understanding how both these schemes are the same in this context and would be grateful if anyone could break it down for me.

• The two wave functions are related by a unitary transformation. Commented Jan 12, 2021 at 10:27
• So basically, rotating one basis set yields the other? @my2cts and hence they are equivalent, is the reasoning? Commented Jan 12, 2021 at 18:29

Bloch functions are the eigenstates of an electron in a spatially periodic potential, such as that of a crystal lattice: $$\psi(\mathbf{r}) = e^{i\mathbf{k}\mathbf{r}}u(\mathbf{r}).$$ That the eigenstates in a crystal have the form of the Bloch states can be shown by quite general symmetry arguments. However, thsi leaves undefined the periodic part of the Bloch states - $$u(\mathbf{r})$$. There exist approximate methods for calculating these function, one of which is the tight-binding approximation.