2.1 Computational Methods
49
It is convenient to begin by considering the total energy of a unit cell composed
of N atoms, with indices l. Each atom has three degrees of freedom, denoted as α,
and can describe displacement along the x,y and z axes. DFPT is based on a Taylor
(i.e. derivative) series of small perturbations, u, of the atoms (l) in the direction α
(i.e. x, y and z), such that,
E = E 0 +
l,α
∂ E
∂ u l,α
· u l,α +
1
2
l,α,l ,α
∂
2 E
∂ u l,α ∂ u l ,α
· u l,α .u l ,α + . . .
(2.33)
where E 0 is a constant (referring to the energy of the equilibrium structure) and is
not considered further. At equilibrium geometry, the forces on all atoms are zero, and
hence the first derivative summation disappears. Within the harmonic approximation,
all terms above the second term are ignored. Hence only the second summation
remains. Note that this is akin to Hooke’s classical law, U = 1/2kx
2 . It is convenient
to define the force constant matrix,
l,l
α,α =
∂
2 E
∂ u l,α ∂ u l ,α
(2.34)
It is worth noting here that because phonon calculations are based on Eq. 2.33,
atomic forces must be converged as close to zero as possible such that the above
approximation holds. This requires geometry optimisation of the structure prior to
phonon calculations. Furthermore, as will be discussed below because the energy is
defined by the wavefunction, a well-optimised wavefunction is also required.
Assuming periodic boundary conditions (i.e. that u(R + N a) = u(R) where N a
is an integer translational vector in the crystal), the displacement u of Eqs. 2.33 and
2.34 can be written as a plane waves,
u l,α = q ex p(i(q · R − ω t)
(2.35)
Here, is the polarisation vector, which determines the direction in which ions
move, and q is the associated phonon wave vector. Inserting Eq. 2.35 into the classical
equation of motion gives
D
l,l
α,α (q) q,α,l = ω
2
q,α,l
(2.36)
This is an eigenvalue equation that links the vibrational frequencies, ω to the
dynamical matrix, defined as
D
l,l
α,α (q) =
1
√
M l M l
α
C
l,l
α,α (q) =
1
√
M l M l
α
l,l
α,α e
−i q.r α
(2.37)
where M l is the mass of atom l. That is to say that the dynamical matrix is the
mass-reduced Fourier transform of the real-space force constant matrix.
49
It is convenient to begin by considering the total energy of a unit cell composed
of N atoms, with indices l. Each atom has three degrees of freedom, denoted as α,
and can describe displacement along the x,y and z axes. DFPT is based on a Taylor
(i.e. derivative) series of small perturbations, u, of the atoms (l) in the direction α
(i.e. x, y and z), such that,
E = E 0 +
l,α
∂ E
∂ u l,α
· u l,α +
1
2
l,α,l ,α
∂
2 E
∂ u l,α ∂ u l ,α
· u l,α .u l ,α + . . .
(2.33)
where E 0 is a constant (referring to the energy of the equilibrium structure) and is
not considered further. At equilibrium geometry, the forces on all atoms are zero, and
hence the first derivative summation disappears. Within the harmonic approximation,
all terms above the second term are ignored. Hence only the second summation
remains. Note that this is akin to Hooke’s classical law, U = 1/2kx
2 . It is convenient
to define the force constant matrix,
l,l
α,α =
∂
2 E
∂ u l,α ∂ u l ,α
(2.34)
It is worth noting here that because phonon calculations are based on Eq. 2.33,
atomic forces must be converged as close to zero as possible such that the above
approximation holds. This requires geometry optimisation of the structure prior to
phonon calculations. Furthermore, as will be discussed below because the energy is
defined by the wavefunction, a well-optimised wavefunction is also required.
Assuming periodic boundary conditions (i.e. that u(R + N a) = u(R) where N a
is an integer translational vector in the crystal), the displacement u of Eqs. 2.33 and
2.34 can be written as a plane waves,
u l,α = q ex p(i(q · R − ω t)
(2.35)
Here, is the polarisation vector, which determines the direction in which ions
move, and q is the associated phonon wave vector. Inserting Eq. 2.35 into the classical
equation of motion gives
D
l,l
α,α (q) q,α,l = ω
2
q,α,l
(2.36)
This is an eigenvalue equation that links the vibrational frequencies, ω to the
dynamical matrix, defined as
D
l,l
α,α (q) =
1
√
M l M l
α
C
l,l
α,α (q) =
1
√
M l M l
α
l,l
α,α e
−i q.r α
(2.37)
where M l is the mass of atom l. That is to say that the dynamical matrix is the
mass-reduced Fourier transform of the real-space force constant matrix.