Dipole Moments and IR Spectra

Suppose we have a one dimensional system of lattice constant $a$ , where we impose PBC over $M$ cells: there are $M$ equally spaced Bloch vectors in the reciprocal cell $[0,2\pi/a)$

$$q_s = {2 \pi \over Ma } s, \enspace \enspace \enspace s = 0, 1, \ldots, M-1 \enspace .$$ (138)

The size of the periodically repeated system is $L = M a$ . The orbitals can be chosen to have the Bloch form

$$\Phi_{q_s,m}(x + \tau) = e^{i q_s \tau} \Phi_{q_s,m}(x) \enspace ,$$ (139)

where $\tau = l a$ is a lattice translation, and m is a band index. There are $N/M$ occupied bands in the Slater determinant wave function, which we write as

$$\mid \Psi \rangle = {\rm A} \prod_{m=1}^{N/M} \prod_{s=0}^{M-1} \Phi_{q_s,m} \enspace ,$$ (140)

where A is the antisymmetrizer. A new set of Bloch orbitals can be defined

$$\tilde \Phi_{q_s,m}(x) = e^{-i {2\pi \over L} x} \Phi_{q_s,m}(x) \enspace .$$ (141)

The position expectation value can now be written as

$$\langle X \rangle = - {L \over 2\pi} \; \enspace \; \langle \Psi \mid \tilde \Psi \rangle \enspace ,$$ (142)

where $\mid \tilde \Psi \rangle$ is the Slater determinant of the $\tilde \Phi$ 's. The overlap among two determinants is equal to the determinant of the overlap matrix of the orbitals:

$$\langle X \rangle = - {L \over 2\pi} \; \enspace \enspace S \enspace ,$$ (143)

where

$$S_{sm,s'm'} = \int_0^L dx \; \Phi^\star_{q_s,m}(x) e^{-i {2\pi \over L} x} \Phi_{q_{s'},m'}(x) \enspace .$$ (144)

Because of the orthogonality of the Bloch functions, the overlap matrix elements vanish except for $q_{s'} = q_s + 2\pi/L$ , that is $s' = s + 1$ . The $N \times N$ determinant can then be factorized into M small determinants

$$S = \prod_{s=0}^{M-1} S(q_s,q_{s+1}) \enspace ,$$ (145)

where for the small overlap matrix the notation

$$S_{m,m'}(q_s,q_{s+1}) = \int_0^L dx \; \Phi^\star_{q_s,m}(x) e^{-i {2\pi \over L} x} \Phi_{q_{s+1},m'}(x) \enspace ,$$ (146)

and $\Phi_{q_{M},m}(x) = \Phi_{q_{0},m}(x)$ is implicitly understood. Finally we get for the electric polarization

$$P_{\rm el} = - {e \over 2\pi} \lim_{L \rightarrow \infty} \enspace \enspace \prod_{s=0}^{M-1} S(q_s,q_{s+1}) \enspace .$$ (147)

The total dipole of a system is calculated as the sum of nuclear and electronic contributions

$$P_{\rm tot} = P_{\rm nuc} + P_{\rm el} \enspace .$$ (148)

Only the total dipole will be independent of the gauge and a proper choice of reference point will be discussed later. First we will give the formulas for the electronic contribution in three dimension and for the case where only the $\Gamma$ point of the supercell Brillouin zone is used

$$P_{\rm el}^{\alpha} = - {2 e \over 2\pi \mid {\bf G}_{\alpha} \mid } \; \enspace \enspace S^{\alpha} \enspace .$$ (149)

The additional factor of 2 comes from the assumed spin degeneracy. The matrix $S$ is defined as

$$S^{\alpha}_{mn} = \langle \Phi_m \mid e^{-i {\bf G}_{\alpha} x} \mid \Phi_n \rangle \enspace .$$ (150)

The index $\alpha = 1, 2, 3$ labels the reciprocal-lattice basis vectors $\{ {\bf G}_{\alpha} \}$ , $P_{\rm el}^{\alpha}$ is the projection of the electronic dipole moment along the direction defined by ${\bf G}_{\alpha}$ , and $\Phi_m$ are the $\Gamma$ point Kohn-Sham orbitals.

The IR absorption coefficient $\alpha(\omega)$ can be calculated from the formula [82]

$$\alpha(\omega) = { 4 \pi \; \omega \; \mbox{tanh} ( \beta \hbar \... ...nfty}^{\infty} dt \; e^{-i \omega t} \langle P(t) \cdot P(0) \rangle \enspace ,$$ (151)

where V is the volume of the supercell, T the temperature, $\beta = 1/k_{\rm B}T$ , $n(\omega)$ is the refractive index, c is the speed of light in vacuum. The angular brackets indicate a statistical average. The correlation function $\langle P(t) \cdot P(0) \rangle$ is calculated classically and quantum effect corrections are taken into account through the factor tanh$( \beta \hbar \omega /2 )$ . More sophisticated treatments of quantum effects are also available. They consist of replacing the classical correlation function with a more involved procedure [83].

As mentioned above only the total dipole moment is independent of the reference point. This can cause some problems during a molecular dynamics simulation. The electronic contribution can only be calculated modulo the supercell size and a unfortunate choice of reference might lead to frequent changes of the dipole moment by amounts of the cell size. Therefore it is most convenient to use a dynamic reference calculated from the nuclear positions and charges. If the reference point is chosen to be the center of charge of the nuclei, the nuclear contribution to the dipole will always be zero

$${\bf Q} = {1 \over \sum_I Z_I} \sum_I Z_I {\bf R}_I \enspace .$$ (152)

$Z_I$ are the nuclear charges and ${\bf R}_I$ the nuclear positions within the supercell. The electronic contribution is then

$$P_{\rm el} = - {2 e \over 2\pi } {\bf h} {\bf d}$$ (153)

$${\bf d} = \tan^{-1}\left[ \mbox{Im} D_{\alpha} / \mbox{Re} D_{\alpha} \right]$$ (154)

$$D_{\alpha} = \exp \left[i ({\bf h}^{T})^{-1} {\bf Q} \right] \enspace S^{\alpha} \enspace ,$$ (155)

where ${\bf h}$ is the matrix defining the supercell.