7.2 Energy gradients

Having calculated the total energy, both the density-kernel $K^{\alpha \beta}$ and the expansion coefficients for the localised orbitals $\{ c^{n \ell m}_{(\alpha)} \}$ are varied. Because of the non-orthogonality of the support functions, it is necessary to take note of the tensor properties of the gradients [163], as noted in section 4.6.

7.2.1 Density-kernel derivatives

The total energy depends upon $K^{\alpha \beta}$ both explicitly and through the electronic density $n({\bf r})$. We use the result

$$\frac{\partial K^{ij}}{\partial K^{\alpha \beta}} = \delta_{\alpha}^i \delta_{\beta}^j .$$ (7.20)

7.2.1.1 Kinetic and pseudopotential energies

From equations 7.6, 7.16 and 7.18 we have that

$$E_{\mathrm{kin,ps}} = T_{\mathrm s}^{\mathrm J} + E_{\mathrm... ...NL}} = 2 K^{ij} (T + V_{\mathrm{loc}} + V_{\mathrm {NL}})_{ji}$$ (7.21)

and therefore

$$\frac{\partial E_{\mathrm{kin,ps}}} {\partial K^{\alpha \bet... ... 2 (T + V_{\mathrm{loc}} + V_{\mathrm {NL}})_{\beta \alpha} .$$ (7.22)

7.2.1.2 Hartree and exchange-correlation energies

The sum of the Hartree and exchange-correlation energies, $E_{\mathrm {Hxc}}$ depends only on the density so that

$$\frac{\partial E_{\mathrm{Hxc}}}{\partial K^{\alpha \beta}} ... ...f r})} \frac{\partial n({\bf r})}{\partial K^{\alpha \beta}} .$$ (7.23)

The functional derivative of the Hartree-exchange-correlation energy with respect to the electronic density is simply the sum of the Hartree and exchange-correlation potentials, $V_{\mathrm{Hxc}}({\bf r})$. The electronic density is given in terms of the density-kernel by

$$n({\bf r}) = 2 \phi_i({\bf r}) K^{ij} \phi_j({\bf r})$$ (7.24)

so that we obtain

$$\frac{\partial n({\bf r})}{\partial K^{\alpha \beta}} = 2 \phi_{\alpha}({\bf r}) \phi_{\beta}({\bf r}) .$$ (7.25)

Finally, therefore

$$\frac{\partial E_{\mathrm{Hxc}}}{\partial K^{\alpha \beta}} ... ...r}) \phi_{\alpha}({\bf r}) = 2 V_{\mathrm{Hxc},\beta \alpha} .$$ (7.26)

7.2.1.3 Total energy

Defining the matrix elements of the Kohn-Sham Hamiltonian in the representation of the support functions by

$$H_{\alpha \beta} = T_{\alpha \beta} + V_{\mathrm{Hxc},\alpha... ... V_{\mathrm{loc},\alpha \beta} + V_{\mathrm {NL},\alpha \beta}$$ (7.27)

the derivative of the total energy with respect to the density-kernel is simply

$$\frac{\partial E}{\partial K^{\alpha \beta}} = 2 H_{\beta \alpha} .$$ (7.28)

7.2.2 Support function derivatives

Again we can treat the kinetic and pseudopotential energies together, and the Hartree and exchange-correlation energies together. We use the result that

$$\frac{\partial \phi_i({\bf r})}{\partial \phi_{\alpha}({\bf r'})} = \delta_i^{\alpha} \delta({\bf r}-{\bf r'}) .$$ (7.29)

7.2.2.1 Kinetic and pseudopotential energies

We define the kinetic energy operator ${\hat T} = -{\textstyle{1 \over 2}} \nabla^2$, whose matrix elements are

$$T_{ij} = \int {\mathrm d}{\bf r}~ \phi_i({\bf r}) {\hat T} \phi_j({\bf r}) .$$ (7.30)

Since the operator is Hermitian,

$$\frac{\delta T_{ij}}{\delta \phi_{\alpha}({\bf r})} = \delta... ...\phi_j({\bf r}) + \delta_j^{\alpha} {\hat T} \phi_i({\bf r}) .$$ (7.31)

Therefore

$$\frac{\delta T_{\mathrm s}^{\mathrm J}}{\delta \phi_{\alpha}({\bf r})} = 2 \frac{\delta}{\delta \phi_{\alpha}({\bf r})} \left( K^{ij} T_{ji} \right) = 2 K^{ij} \frac{\delta T_{ji}}{\delta \phi_{\alpha}({\bf r})}$$

$$= 4 K^{\alpha \beta} {\hat T} \phi_{\beta}({\bf r}) .$$ (7.32)

The derivation for the pseudopotential energy is identical with the replacement of ${\hat T}$ by the pseudopotential operator, and so the result for the sum of these energies is just

$$\frac{\delta E_{\mathrm{kin,ps}}}{\delta \phi_{\alpha}({\bf ... ... + {\hat V}_{\mathrm{ps,tot}} \right) \phi_{\beta}({\bf r}) .$$ (7.33)

7.2.2.2 Hartree and exchange-correlation energies

Again this gradient is derived by considering the change in the electronic density.

$$\frac{\partial n({\bf r'})}{\partial \phi_{\alpha}({\bf r})}... ..._i({\bf r'}) \delta({\bf r} - {\bf r'}) K^{i \alpha} \biggr] .$$ (7.34)

Therefore

$$\frac{\delta E_{\mathrm{Hxc}}}{\delta \phi_{\alpha}({\bf r})... ...K^{\alpha \beta} {\hat V}_{\mathrm{Hxc}} \phi_{\beta}({\bf r})$$ (7.35)

7.2.2.3 Total energy

The gradient of the total energy with respect to changes in the support functions is

$$\frac{\delta E}{\delta \phi_{\alpha}({\bf r})} = 4 K^{\alpha \beta}{\hat H} \phi_{\beta}({\bf r})$$ (7.36)

where ${\hat H}$ is the Kohn-Sham Hamiltonian which operates on $\phi_{\beta}({\bf r})$.