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## Metric tensor formulation of strain in density‐functional perturbation theory, Phys (2005)

Citations: | 2 - 0 self |

### Citations

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and references therein.
- Baroni, Gironcoli, et al.
- 2001
(Show Context)
Citation Context ...tion of occupancy number with eigenvalue and solve this problem. Eq. (10) must be modified in this case for states in a band of energies around the Fermi energy Fε , 2,16 and the first-order wave functions in this band can be expressed in a form reminiscent of ordinary finite-temperature perturbation theory. While first-order variations of Fε vanish for perturbations with finite wave vector, the fires-order Fermi energy (1)Fε and its contributions to (1)αψ and hence (1)n must be included for zero-wave-vector perturbations including strain.4 An expression for (1)Fε is given in Eq. (79) of Ref. [4], but we prefer a simple alternative expression, (1) (1) (0) (0) (0) (0)F F F F( ) ( ) ,F f fα α α α α ε ε ε ε ε ε′ ′= − −∑ ∑ (12) where F ( )f ε′ is the derivative of the Fermi function, (0) αε and (0) Fε are the zero-order eigenvalues and Fermi energy, and the first-order eigenvalues are given by (1) (0) (1) (0) .Hα α αε ψ ψ= (13) We note that the energy dependence of Ff ′ confines the contributions in the sums in Eq. (12) to states within the band discussed above. Since the self-consistent contributions to (1)H depend on (1)Fε , it must be converged in the iterative process of solving the S... |

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and references therein.
- Resta
- 1994
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Citation Context ...n interactions, which are an important contribution to the total energy of a solid, but lie outside the framework of DFPT since they don’t involve the electrons. The second derivatives of the 7 ion- ion energy must be added to the terms in Eq. (15) to obtain the elastic and internalstrain tensors. A. Canonical transformation formulation The application of homogeneous strain to a crystal lattice simply moves the positions of the atoms and hence changes the DFT external potential,1 cell cell ( ) ( ) ( ) [ ( ) ( ) ] ,ext extV V V V= → = ⋅ ⋅∑∑ ∑∑?t t? R t R t r r - t - R r r - 1 + ? t - 1 + ? R (17) where t denotes the positions of atoms within a unit cell, R is the set of lattice vectors, and ? is the Cauchy infinitesimal strain tensor.18 From the point of view of the infinite lattice the difference, ext extV V− ? , can never be a small perturbation. Within a single unit cell, of course, an infinitesimal strain will produce an infinitesimal change in potential. However, it also changes the boundary conditions, so the perturbed wave functions cannot be expanded in a basis of the unperturbed wave functions, and DFPT is not applicable. One solution to this problem was proposed by Baroni et... |

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An Introduction to Continuum Mechanics (Kluwer Academic Publishers,
- Smith
- 1993
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Citation Context ...ext extV V− ? , can never be a small perturbation. Within a single unit cell, of course, an infinitesimal strain will produce an infinitesimal change in potential. However, it also changes the boundary conditions, so the perturbed wave functions cannot be expanded in a basis of the unperturbed wave functions, and DFPT is not applicable. One solution to this problem was proposed by Baroni et al.10 They introduced a fictitious strained self-consistent Hamiltonian obtained from the unstrained Hamiltonian through a scale transformation, 1( , ) (1 ) ,(1 ) .SCF SCFH H − ∇ = + ⋅ + ⋅∇ ? r ? r ?% (18) Eigenfunctions of SCFH ?% obey the same boundary conditions as those of the actual strained Hamiltonian SCFH ? . The spectrum of SCFH ?% is identical to that of the unstrained Hamiltonian since the two are related by a unitary transformation, and the wave functions and charge density n?% of SCFH ?% are generated by simple transformations of the corresponding unstrained quantities. The energy difference between the fictitious and unstrained systems is easily computed. The strategy is to then compute the energy difference between the system described by SCFH ?% and that described by the real st... |

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The dependence is more complex in the case of gradient-corrected functionals. See
- Corso, Resta
- 1994
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Citation Context ... δ β βγ α δ δ α γδ αβ αδ β γ γ β βδ α γ γ α δ δ η η δ δ ∂ ϒ ϒ ≡ = + + + ∂ ∂ + + + + (28) where we have introduced the notation of parenthesized Cartesian superscripts to denote strain derivatives. It can be verified that these formulas are invariant under interchange of ( , )α β or ( , )γ δ index pairs. This is a manifestation of the fact that antisymmetric components of ? correspond to rotations rather than strains, under which the metric tensors are invariant. The strain derivative of the unit-cell volume Ω is sufficiently simple so as not to warrant additional notation, .αβ αβ δ η ∂Ω = Ω ∂ (29) The extension to second derivatives is obvious. Finally, it is easily shown from Eq. (19) that 2 ,π⋅ = ⋅K X K X% % (30) so dot products between real and reciprocal vectors do not involve the metric tensors and are strain independent. 10 We note that DFPT yields second derivatives of the energy per unit cell. This has the consequence that the naturally defined “elastic tensor” as calculated in DFPT, 2 * , 1 ,el E Cαβγδ αβ γδη η ∂ ≡ Ω ∂ ∂ (31) is not equal to the conventional elastic tensor *, 1 ,el E C Cγδαβγδ αβγδ αβ γδ αβ αβ γδ σ δ σ η η η ∂ ∂∂ ≡ = = − ∂ ∂ Ω ∂ (32) where γδσ is the stress te... |