Variational principle

As mentioned in section 4.4.3, Kohn [135] has suggested the use of a penalty functional to impose the idempotency condition, and has proved a variational principle based upon it. We consider trial density-matrices $\rho({\bf r},{\bf r'})$ expressed in diagonal form with real orthonormal extended orbitals $\{ \varphi_i({\bf r}) \}$ and occupation numbers $\{f_i\}$:

$${\rho}({\bf r},{\bf r'}) = \sum_i f_i~ {\varphi}_i({\bf r}) {\varphi}_i({\bf r'}) .$$ (6.1)

The functional ${\cal Q}[{\rho};\mu,\alpha]$ is then formed:

$${\cal Q}[{\rho};\mu,\alpha] \equiv E_{\mathrm{NI}}[{\rho}^2] - \mu N[{\rho}^2] + \alpha {\cal P}[{\rho}]$$ (6.2)

in which

$$E_{\mathrm{NI}}[{\rho}^2] \equiv 2 \int {\mathrm d}{\bf r'} \left\{ \left[ -{\textstyle{1 \over 2}... ...\bf r'}) V_{\mathrm{KS}}({\bf r'}) \right\} = 2 \sum_i f_i^2~ {\varepsilon}_i ,$$ (6.3)

$$N[{\rho}^2] \equiv 2 \int {\mathrm d}{\bf r}~ \rho^2({\bf r},{\bf r}) = 2 \sum_i f_i^2 ,$$ (6.4)

$${\cal P}[{\rho}] \equiv \left[ \int {\mathrm d}{\bf r}~ \left( {\rho}^2(1 - {\rho})^2 \ri... ...r}) \right]^{1 \over 2} = \left[ \sum_i f_i^2 (1 - f_i)^2 \right]^{1 \over 2} ,$$ (6.5)

and where $\mu$ is the chemical potential and $\alpha$ is a positive real parameter.

Kohn proves the following variational principle: that for some $\alpha > \alpha_{\mathrm c}$, the minimum value of ${\cal Q}[{\rho};\mu,\alpha]$ is obtained for the idempotent ground-state density-matrix $\rho_0$ and that the minimum value is the ground-state grand potential i.e.

$$\mathop{\rm min}\limits _{\rho} {\cal Q}[{\rho};\mu,\alpha] ... ...m_{i,\varepsilon^{(0)}_i \leq \mu} (\varepsilon^{(0)}_i - \mu)$$ (6.6)

in which the $\{ \varepsilon^{(0)}_i \}$ are the exact eigenvalues of the self-consistent Hamiltonian, generated by the ground-state density-matrix $\rho_0$.

The critical value of $\alpha$, denoted $\alpha_{\mathrm c}$, is given by

$$\alpha_{\mathrm c} = \mathop{\rm max}\limits _{{\cal P}'} \l... ...athrm d} \Omega({\cal P}')} {{\mathrm d}{\cal P}'} \right\vert$$ (6.7)

in which $\Omega({\cal P}')$ is the conditional minimum defined by

$$\Omega({\cal P}') = \mathop{\rm min}\limits _{{\cal P}[{\rho... ...}'} \left( E_{\mathrm{NI}}[{\rho}^2] - \mu N[{\rho}^2] \right)$$ (6.8)

i.e. the minimum grand potential for all trial density-matrices which give a penalty functional value of ${\cal P}'$. Clearly

$$\alpha_{\mathrm c} \geq \left\vert \frac{{\mathrm d} \Omega(... ..._i \leq \mu} (\varepsilon^{(0)}_i - \mu)^2 \right]^{1 \over 2}$$ (6.9)

although this is only a lower bound on $\alpha_{\mathrm c}$.

Figure 6.1: Behaviour of Kohn's penalty functional ${\cal P}[\rho]$ when a single occupation number $f_i$ is varied and all others are zero or unity.

=1mm

Kohn's variational principle is based on the non-interacting energy $E_{\mathrm{NI}}[\rho^2]$. We now present a simple modification of this functional based upon self-consistent variation of the interacting energy. Consider the functional

$${\tilde {\cal Q}}[{\rho};\mu,\alpha] = E[{\rho}] - \mu N[{\rho}] + \alpha {\cal P}[{\rho}]$$ (6.10)

in which $E[{\rho}]$ is the interacting energy, and ${\rho}$ is a positive semi-definite trial density-matrix. A given set of occupation numbers $\{f_i\}$ fixes the value of the penalty functional ${\cal P}[{\rho}]$ and variation of ${\tilde {\cal Q}}[{\rho};\mu,\alpha]$ with respect to the orbitals $\{ { \varphi}_i({\bf r}) \}$ at fixed occupation numbers and subject to the orthonormality constraint yields Kohn-Sham-like equations. Self-consistent variation of the occupation numbers $\{f_i\}$ (i.e. allowing the orbitals to relax, as in section 4.2) yields

$$\frac{\partial {\tilde {\cal Q}}[{\rho};\mu,\alpha]}{\partia... ... + \frac{\alpha}{{\cal P}[{\rho}]} f_i (1 - f_i) (1 - 2 f_i) .$$ (6.11)

In the case of idempotent density-matrices, for which ${\cal P}[{\rho}]=0$, we obtain the special cases

$$\left.\frac{\partial {\tilde Q}[{\rho};\mu,\alpha]} {\partia... ..._i=(0^{\pm},1^{\pm})} = 2 ({\varepsilon}_i - \mu) \pm \alpha .$$ (6.12)

For this functional the critical value of $\alpha$, again denoted $\alpha_{\mathrm c}$, is given by

$$\alpha_{\mathrm c} = \mathop{\rm max}\limits _{\rho} \left\vert 2({\varepsilon}_i - \mu) \right\vert$$ (6.13)

where the maximum is strictly over those density-matrices searched during the minimisation. For $\alpha > \alpha_{\mathrm c}$ the total functional ${\tilde {\cal Q}}[{\rho};\mu,\alpha]$ takes its minimum value when $f_i = 0,1$ for ${\varepsilon}_i > \mu$ and ${\varepsilon}_i < \mu$ respectively. In particular, for the ground-state density-matrix $\rho_0$, the functional is strictly increasing with respect to all variations in occupation numbers. The discontinuity in the occupation number derivative of the penalty functional at idempotency is required because of the non-variational behaviour of the total energy with respect to these variations (section 4.2). The behaviour of the penalty functional for unconstrained occupation number variation is plotted in figure 6.1, and in figure 6.2 the total functional is sketched schematically for several representative values of the parameter $\alpha$. This demonstrates how the minimising density-matrix is idempotent only for $\alpha \geq \alpha_{\mathrm c}$.

Figure 6.2: Schematic illustration of Kohn's variational principle: behaviour of the total energy (black) and total functional (red) for representative values of $\alpha$.