Tight binding

There are already many descriptions of tight binding available in the literature[12,13,14,15,16,17], which will not be repeated here. For now we note the following which will be referred to frequently in the forthcoming sections. The cohesive energy (U_coh) is given by:

 

U_coh = U_band+U_rep-U_atoms (1)

where U_band is the band energy, the calculation of which is the focus of this paper, U_rep is a repulsive energy, generally given as a sum of pair potentials, and U_atoms is the total one electron energy of the free atoms.

The band energy can be expressed in a variety of ways, each of which will be considered below. In terms of the eigenvalues ( $\epsilon^{(n)}$) of the Hamiltonian matrix it is given by:

$$U_{band} = 2\sum_n f((\epsilon^{(n)}-\mu)/k_BT)\epsilon^{(n)},$$ (2)

where $f(x)=1/(1+\exp(x))$ is the Fermi function, $\mu$ is the chemical potential for the electrons, k_B is Boltzmann's constant, and T is the temperature of the electrons. Note that below we will often use x to stand for $(E-\mu)/k_BT$. The eigenvector corresponding to the n^theigenstate is (using Dirac's notation):

$$\mid\psi^{(n)}\rangle=\sum_{i\alpha}C^{(n)}_{i\alpha}\mid i\alpha\rangle,$$ (3)

where i is an atomic site index, $\alpha$ is an atomic orbital index, and $\mid i\alpha\rangle$ is an atomic orbital. The eigenvalues and vectors are found by solving Schrödinger's equation:

$$\sum_{j\beta}H_{i\alpha,j\beta}C^{(n)}_{j\beta}=\epsilon^{(n)}C^{(n)}_{i\alpha},$$ (4)

where $H_{i\alpha,j\beta}$ is the $(i\alpha,j\beta)$ element of the tight binding Hamiltonian matrix. We define the density of states by:

$$n_{total}(E) = \sum_n\delta(E-\epsilon^{(n)}).$$ (5)

Thus the band energy can be rewritten as:

$$U_{band} = 2\int{\rm d}E f(x)En_{total}(E).$$ (6)

A third way to write the band energy is in terms of the density matrix which is defined by:

$$\rho_{i\alpha,j\beta} = \sum_n C^{(n)}_{i\alpha}f((\epsilon^{(n)}-\mu)/k_BT) C^{(n)}_{j\beta}.$$ (7)

In operator notation, the density matrix is given by:

$$\hat\rho = \sum_n\mid\psi^{(n)}\rangle f((\epsilon^{(n)}-\mu)/k_BT) \langle\psi^{(n)}\mid.$$ (8)

The band energy is given by:

$$U_{band}=2\sum_{i\alpha,j\beta}\rho_{j\beta,i\alpha}H_{i\alpha,j\beta},$$ (9)

This last form has the advantage that it allows us to write the band energy contribution to the atomic forces in a very compact way:

$$\vec F_{band}^{(k)}=-2\sum_{i\alpha,j\beta}\rho_{j\beta,i\alpha} {{\partial H_{i\alpha,j\beta}}\over{\partial\vec r_k}}.$$ (10)

Note that this expression for the force assumes that we have included the electron entropy term, since the electrons have a finite temperature (see Ref.[18]). The contribution of the entropy to the free energy is:

$$U_{ent} = -2k_BT\int{\rm d}E \sigma(x)n_{total}(E),$$ (11)

where $\sigma(x)$ is the entropy density given by $\sigma(x)=-[f(x)\log f(x) +(1-f(x))\log(1-f(x))]$.