4.2 Partial occupation of the Kohn-Sham orbitals

In the Kohn-Sham scheme, the single-particle orbitals $\{ \vert \psi_i \rangle \}$ were either empty or doubly occupied (two spin states). It will prove to be useful if we now generalise to include partial occupation [131] so that each orbital contains $2 f_i$ electrons where $0 \leq f_i \leq 1$. The electronic density is now defined as

$$n({\bf r}) = 2 \sum_i f_i \left\vert \psi_i({\bf r}) \right\vert^2 .$$ (4.6)

Following the constrained search formulation we now define a generalised non-interacting kinetic energy functional $T_{\mathrm s}^{\mathrm J}[n]$ as

$$T_{\mathrm s}^{\mathrm J}[n] = \mathop{\rm min}\limits _{\{f... ...ft( - {\textstyle{1 \over 2}} \nabla^2 \right) \psi_i({\bf r})$$ (4.7)

where the search is over all orthonormal orbitals $\{ \vert \psi_i \rangle \}$ and occupation numbers $\{f_i\}$ which yield the density $n({\bf r})$ (which implies that $2 \sum_i f_i = N$).

Janak's functional is defined as

$$E_V^{\mathrm J}[\{f_i\},\{\vert\psi_i\rangle\}] = 2 \sum_i f... ...\int {\mathrm d}{\bf r}~n({\bf r}) V_{\mathrm{ext}}({\bf r}) .$$ (4.8)

The minimisation is now performed with respect to both the occupation numbers $\{f_i\}$ and the orbitals $\{ \vert \psi_i \rangle \}$.

For a fixed set of occupation numbers, the Euler-Lagrange equations for the variation of the functional with respect to the orbitals again yield Schrödinger-like equations:

$$\left[ -\textstyle{1 \over 2} f_i \nabla^2 + f_i V_{\mathrm{... ... ({\bf r}) \right] \psi_i({\bf r}) = \lambda_i \psi_i({\bf r})$$ (4.9)

in which we can identify $\lambda_i = f_i \varepsilon_i$ to obtain the Kohn-Sham equations

$$\left[ -\textstyle{1 \over 2} \nabla^2 + V_{\mathrm{KS}} ({\bf r}) \right] \psi_i({\bf r}) = \varepsilon_i \psi_i({\bf r}) .$$ (4.10)

Multiplying by $\psi_i^{\ast}({\bf r})$ and integrating gives

$$\int {\mathrm d}{\bf r}~\psi_i^{\ast}({\bf r}) \left( -\text... ...f r}) \right\vert^2 V_{\mathrm{KS}}({\bf r}) = \varepsilon_i .$$ (4.11)

We obtain the dependence of the energy functional on the occupation numbers by varying one of the $\{f_i\}$ while allowing the orbitals to relax (i.e. solving equations 4.6 and 4.10 self-consistently). We define the kinetic energy for orbital $i$, $t_i$, by

$$t_i = \int {\mathrm d}{\bf r}~\psi_i^{\ast}({\bf r}) \left( -\textstyle{1 \over 2} \nabla^2 \right) \psi_i({\bf r})$$ (4.12)

in terms of which the generalised kinetic energy functional $T_{\mathrm s}^{\mathrm J}[n]$ is

$$T_{\mathrm s}^{\mathrm J}[n] = 2 \sum_i f_i~t_i .$$ (4.13)

Then

$$\frac{\partial E_V^{\mathrm J}}{\partial f_i} = 2 t_i + 2 \s... ...ft\vert \psi_j({\bf r}) \right\vert^2}{\partial f_i} \right) .$$ (4.14)

Using equation 4.11 we can rewrite the terms not involving a summation over orbitals:

$$\frac{\partial E_V^{\mathrm J}}{\partial f_i} = 2 \varepsilo... ...ft\vert \psi_j({\bf r}) \right\vert^2}{\partial f_i} \right) .$$ (4.15)

From the definition of $t_j$ (4.12) we obtain

$$\frac{\partial t_j}{\partial f_i} = \int {\mathrm d}{\bf r} ... ...right) \frac{\partial \psi_j({\bf r})}{\partial f_i} \right] .$$ (4.16)

Substituting this result in equation 4.15 we obtain for the second term on the right-hand side

$$2 \sum_j f_j \int {\mathrm d}{\bf r} \left[ \frac{\partial \... ...right) \frac{\partial \psi_j({\bf r})}{\partial f_i} \right] .$$ (4.17)

Now using equation 4.10 we find

$$\frac{\partial E_V^{\mathrm J}}{\partial f_i} = 2 \varepsilo... ... {\mathrm d}{\bf r} \left\vert \psi_j({\bf r}) \right\vert^2 .$$ (4.18)

The second term on the right-hand side vanishes since the orbitals are normalised and so the final result is that

$$\frac{\partial E_V^{\mathrm J}}{\partial f_i} = 2 \varepsilon_i .$$ (4.19)

Variation of the functional subject to the constraint of constant electron number (i.e. unconstrained variation of $E_V^{\mathrm J} - \mu N$) gives

$$\delta [ E_V^{\mathrm J} - \mu N ] = 2 \sum_i (\varepsilon_i - \mu) \delta f_i .$$ (4.20)

This generalised functional is not variational with respect to arbitrary variations in the occupation numbers [132]. Objections have been raised [133] to considering occupation numbers other than zero or one in zero-temperature density-functional theory, but the conclusion is still that at self-consistency, orbitals above the Fermi energy are unoccupied and orbitals below are fully occupied, and we recall that this state of affairs corresponds to an idempotent density-matrix.

If the occupation numbers are allowed to vary in the interval $[0,1]$ we see that the lowest value of the generalised functional is obtained for the correct choice of occupation numbers outlined above. However, if the occupation numbers are allowed to vary outside this interval, this result no longer holds since the energy can be lowered by over-filling ( $f_i \rightarrow \infty$) orbitals below the Fermi level, or negatively filling ( $f_i \rightarrow -\infty$) orbitals above the Fermi level, while still keeping the sum of the occupation numbers correct. Constraining the occupation numbers to avoid these unphysical situations is discussed in section 4.4.