Phonons

Theory

A phonon corresponds to small displacements of the ionic positions with respect to their equilibrium positions. The electrons principally follow them, in order to minimize again the energy of the system.

The expansion of the Hamiltonian in powers of the displacement $u^R_\alpha$ of the ion (labeled by its position $R$ ) in the cartesian direction $\alpha = 1, 2, 3$ consists of two parts³:

$$H^{(1)} = H^{(1)}_C + H^{(1)}_{PP}$$ (249)

The contribution $H^{(1)}_C$ comes from the Coulomb term, the electrostatic potential:

$$H^{(1)}_C = u^R_\alpha \frac \partial {\partial R_\alpha} \; \frac{Z_R}{\vert\mathbf{r}- \mathbf{R}\vert}.$$ (250)

The second is due to the pseudopotential which is rigidly attached to the ions and which must be moved simultaneously. In particular, the nonlocal pseudopotential projectors must be taken into account as well:

$$H^{(1)}_{PP} = u^R_\alpha \frac \partial {\partial R_\alpha} \; \left[\sum_i \vert\text{\tt P}^R_i\rangle \langle\text{\tt P}^R_i\vert \right]$$ (251)

$$= u^R_\alpha\sum_i\;\left[ \left[\frac\partial{\partial R_\alpha}\;... ...[\frac\partial{\partial R_\alpha}\; \langle\text{\tt P}^R_i\vert\right] \right]$$ (252)

where $\vert$P$_i^R\rangle$ designates the projectors, whatever type they are. The index $i$ comprises the $l$ and $m$ quantum numbers, for example. The superscript $R$ just says that of course only the projectors of the pseudopotential of the displaced atom at $R$ are considered in this equation.

In CPMD 3.13.2, these projectors are stored in G-space, and only one copy is stored (that is the one for a fictitious ion at the origin, $R=0$ ). The projectors for an ion at its true coordinates is then obtained as

$$\langle \mathbf{G}\vert _i^R\rangle = ^{i\mathbf{G}\cdot\mathbf{R}}\;\langle \mathbf{G}\vert _i^{R=0}\rangle.$$ (253)

This makes the derivative $\frac{\partial}{\partial R_\alpha}$ particularly simple, as only the $i\mathbf{G}$ comes down, and the translation formula (253) remains valid for $\vert$P$_i^R\rangle$ . Thus, there is only an additional nonlocal term appearing which can be treated almost in the same way as the unperturbed pseudopotential projectors.

A perturbative displacement in cartesian coordinates can have components of trivial eigenmodes, that is translations and rotations. They can be written a priori (in mass weighted coordinates in this case) and thus projected out from the Hessian matrix;

$$\ensuremath{\mathbf{t}}_j=\left(\begin{array}{c} \sqrt{m_1}(\ensu... ...nsuremath{\mathbf{e}}_j\times\ensuremath{\mathbf{R}}_N) \\ \end{array} \right)$$ (254)

where j=x,y,z, $\ensuremath{\mathbf{e}}_j$ = unit vectors in the Cartesian directions and $\ensuremath{\mathbf{R}}_k$ = positions of the atoms (referred to the COM). The rotations $\ensuremath{\mathbf{s}}_j$ constructed in this way are not orthogonal; before being used they are orthogonalized with respect to the translations and with each other.

The projection on the internal mode subspace is done this way: first one constructs a projector of the form,

$$\ensuremath{\mathbf{P}}=\sum_j(\ensuremath{\mathbf{t}}_j\cdot\ensuremath{\mathbf{t}}_j^T+\ensuremath{\mathbf{s}}_j\cdot\ensuremath{\mathbf{s}}_j^T)$$ (255)

and then applies the projector to the Hessian matrix,

$$\ensuremath{\mathbf{A}}=(\ensuremath{\mathbf{I}}-\ensuremath{\mat... ...ot\ensuremath{\mathbf{A}}\cdot(\ensuremath{\mathbf{I}}-\ensuremath{\mathbf{P}})$$ (256)

with $\mathbf{I}$ being the unit matrix. The projection is controlled by the keyword DISCARD, vide infra.

Phonon input

The input for the phonon section is particularly simple due to the absence of any special keywords. Only the word PHONON should appear in the &RESP section.

Phonon output

In total analogy to CPMD 3.13.2's VIBRATIONAL ANALYSIS, the displacement of all atoms in all cartesian directions are performed successively. The difference is, of course, that there is no real displacement but a differential one, calculated in perturbation theory. Thus, only one run is necessary per ion/direction⁴.

At the end, the harmonic frequencies are printed like in the VIBRATIONAL ANALYSIS. They should coincide to a few percent.