sirius::Stress Class Reference

Stress tensor. More...

#include <stress.hpp>

Public Member Functions
	Stress (Simulation_context &ctx__, Density &density__, Potential &potential__, K_point_set &kset__)
r3::matrix< double >	calc_stress_vloc ()
	Local potential contribution to stress. More...
r3::matrix< double >	stress_vloc () const
r3::matrix< double >	calc_stress_har ()
	Hartree energy contribution to stress. More...
r3::matrix< double >	stress_har () const
r3::matrix< double >	calc_stress_ewald ()
	Ewald energy contribution to stress. More...
r3::matrix< double >	stress_ewald () const
template<typename T >
void	calc_stress_kin_aux ()
	Kinetic energy contribution to stress. More...
r3::matrix< double >	calc_stress_kin ()
r3::matrix< double >	stress_kin () const
r3::matrix< double >	calc_stress_nonloc ()
r3::matrix< double >	stress_nonloc () const
r3::matrix< double >	calc_stress_us ()
	Contribution to the stress tensor from the augmentation operator. More...
auto	stress_us () const
auto	stress_us_nl () const
r3::matrix< double >	calc_stress_xc ()
	XC contribution to stress. More...
auto	stress_xc () const
r3::matrix< double >	calc_stress_core ()
	Non-linear core correction to stress tensor. More...
auto	stress_core () const
r3::matrix< double >	calc_stress_hubbard ()
auto	stress_hubbard () const
r3::matrix< double >	calc_stress_total ()
auto	stress_total () const
void	print_info (std::ostream &out__, int verbosity__) const

Private Member Functions
template<typename T , typename F >
void	calc_stress_nonloc_aux ()
	Non-local contribution to stress. More...

Private Attributes
Simulation_context &	ctx_
Density const &	density_
Potential &	potential_
K_point_set &	kset_
r3::matrix< double >	stress_kin_
r3::matrix< double >	stress_har_
r3::matrix< double >	stress_ewald_
r3::matrix< double >	stress_vloc_
r3::matrix< double >	stress_nonloc_
r3::matrix< double >	stress_us_
r3::matrix< double >	stress_xc_
r3::matrix< double >	stress_core_
r3::matrix< double >	stress_hubbard_
r3::matrix< double >	stress_total_

Detailed Description

Stress tensor.

The following referenceces were particularly useful in the derivation of the stress tensor components:

• Hutter, D. M. A. J. (2012). Ab Initio Molecular Dynamics (pp. 1–580).
• Marx, D., & Hutter, J. (2000). Ab initio molecular dynamics: Theory and implementation. Modern Methods and Algorithms of Quantum Chemistry.
• Knuth, F., Carbogno, C., Atalla, V., & Blum, V. (2015). All-electron formalism for total energy strain derivatives and stress tensor components for numeric atom-centered orbitals. Computer Physics Communications.
• Willand, A., Kvashnin, Y. O., Genovese, L., Vázquez-Mayagoitia, Á., Deb, A. K., Sadeghi, A., et al. (2013). Norm-conserving pseudopotentials with chemical accuracy compared to all-electron calculations. The Journal of Chemical Physics, 138(10), 104109. http://doi.org/10.1103/PhysRevB.50.4327
• Corso, A. D., & Resta, R. (1994). Density-functional theory of macroscopic stress: Gradient-corrected calculations for crystalline Se. Physical Review B.

Stress tensor describes a reaction of crystall to a strain:

\[ \sigma_{\mu \nu} = \frac{1}{\Omega} \frac{\partial E}{\partial \varepsilon_{\mu \nu}} \]

where \( \varepsilon_{\mu \nu} \) is a symmetric strain tensor which describes the infinitesimal deformation of a crystal:

\[ r_{\mu} \rightarrow \sum_{\nu} (\delta_{\mu \nu} + \varepsilon_{\mu \nu}) r_{\nu} \]

The following expressions are helpful:

• strain derivative of a general position vector component:

  \[ \frac{\partial r_{\tau}}{\partial \varepsilon_{\mu \nu}} = \delta_{\tau \mu} r_{\nu} \]

• strain derivative of the lengths of a general position vector:

  \[ \frac{\partial |{\bf r}|}{\partial \varepsilon_{\mu \nu}} = \sum_{\tau} \frac{\partial |{\bf r}|}{\partial r_{\tau}} \frac{\partial r_{\tau}}{\partial \varepsilon_{\mu \nu}} = \sum_{\tau} \frac{ r_{\tau}}{|{\bf r}|} \delta_{\tau \mu} r_{\nu} = \frac{r_{\mu} r_{\nu}}{|{\bf r}|} \]

• strain derivative of the unit cell volume:

  \[ \frac{\partial \Omega}{\partial \varepsilon_{\mu \nu}} = \delta_{\mu \nu} \Omega \]

• strain derivative of the inverse square root of the unit cell volume:

  \[ \frac{\partial}{\partial \varepsilon_{\mu \nu}} \frac{1}{\sqrt{\Omega}} = -\frac{1}{2}\frac{1}{\sqrt{\Omega}} \delta_{\mu \nu} \]

• strain derivative of a reciprocal vector:

  \[ \frac{\partial G_{\tau}}{\partial \varepsilon_{\mu \nu}} = -\delta_{\tau \nu} G_{\mu} \]

• strain derivative of the length of a reciprocal vector:

  \[ \frac{\partial |{\bf G}|}{\partial \varepsilon_{\mu \nu}} = \sum_{\tau} \frac{\partial |{\bf G}|}{\partial G_{\tau}} \frac{\partial G_{\tau}}{\partial \varepsilon_{\mu \nu}} = -\frac{1}{|{\bf G}|}G_{\nu}G_{\mu} \]

  In the derivation of the stress tensor contributions it is important to know which variables are invariant under lattice distortion. This are:
• scalar product of the real-space (in the Bravais lattice frame) and reciprocal (in the reciprocal lattice frame) vectors
• normalized plane-wave coefficients of the wave-functions

  \[ \psi({\bf G}) = \frac{1}{\sqrt{\Omega}} \int e^{-i {\bf G}{\bf r}} \psi({\bf r}) d{\bf r} \]

• unnormalized plane-wave coefficients of the charge density

  \[ \tilde \rho({\bf G}) = \int e^{-i {\bf G}{\bf r}} \sum_{j} |\psi_j({\bf r})|^{2} d{\bf r} \]

Definition at line 96 of file stress.hpp.

Constructor & Destructor Documentation

Stress()

sirius::Stress::Stress ( Simulation_context &  ctx__, Density &  density__, Potential &  potential__, K_point_set &  kset__  )

inline

Definition at line 222 of file stress.hpp.

Member Function Documentation

calc_stress_nonloc_aux()

template<typename T , typename F >

void sirius::Stress::calc_stress_nonloc_aux

private

Non-local contribution to stress.

Energy contribution from the non-local part of pseudopotential:

\[ E^{nl} = \sum_{i{\bf k}} \sum_{\alpha}\sum_{\xi \xi'}P_{\xi}^{\alpha,i{\bf k}} D_{\xi\xi'}^{\alpha}P_{\xi'}^{\alpha,i{\bf k}*} \]

where

\[ P_{\xi}^{\alpha,i{\bf k}} = \langle \psi_{i{\bf k}} | \beta_{\xi}^{\alpha} \rangle = \frac{1}{\Omega} \sum_{\bf G} \langle \psi_{i{\bf k}} | {\bf G+k} \rangle \langle {\bf G+k} | \beta_{\xi}^{\alpha} \rangle = \sum_{\bf G} \psi_i^{*}({\bf G+k}) \beta_{\xi}^{\alpha}({\bf G+k}) \]

where

\[ \beta_{\xi}^{\alpha}({\bf G+k}) = \frac{1}{\sqrt{\Omega}} \int e^{-i({\bf G+k}){\bf r}} \beta_{\xi}^{\alpha}({\bf r}) d{\bf r} = \frac{4\pi}{\sqrt{\Omega}} e^{-i{\bf (G+k) r_{\alpha}}}(-i)^{\ell} R_{\ell m}(\theta_{G+k}, \phi_{G+k}) \int \beta_{\ell}(r) j_{\ell}(|{\bf G+k}|r) r^2 dr \]

Contribution to stress tensor:

\[ \sigma_{\mu \nu}^{nl} = \frac{1}{\Omega} \frac{\partial E^{nl}}{\partial \varepsilon_{\mu \nu}} = \frac{1}{\Omega} \sum_{i{\bf k}} \sum_{\xi \xi'} \Bigg( \frac{\partial P_{\xi}^{\alpha,i{\bf k}}}{\partial \varepsilon_{\mu \nu}} D_{\xi\xi'}^{\alpha}P_{\xi'}^{\alpha,i{\bf k}*} + P_{\xi}^{\alpha,i{\bf k}} D_{\xi\xi'}^{\alpha} \frac{\partial P_{\xi'}^{\alpha,i{\bf k}*}}{\partial \varepsilon_{\mu \nu}} \Bigg) \]

We need to compute strain derivatives of \( P_{\xi}^{\alpha,i{\bf k}} \):

\[ \frac{\partial}{\partial \varepsilon_{\mu \nu}} P_{\xi}^{\alpha,i{\bf k}} = \sum_{\bf G} \psi_i^{*}({\bf G+k}) \frac{\partial}{\partial \varepsilon_{\mu \nu}} \beta_{\xi}^{\alpha}({\bf G+k}) \]

First way to take strain derivative of beta-projectors (here and below \( {\bf G+k} = {\bf q} \)):

\[ \frac{\partial}{\partial \varepsilon_{\mu \nu}} \beta_{\xi}^{\alpha}({\bf q}) = -\frac{4\pi}{2\sqrt{\Omega}} \delta_{\mu \nu} e^{-i{\bf q r_{\alpha}}}(-i)^{\ell} R_{\ell m}(\theta_q, \phi_q) \int \beta_{\ell}(r) j_{\ell}(q r) r^2 dr + \\ \frac{4\pi}{\sqrt{\Omega}} e^{-i{\bf q r_{\alpha}}}(-i)^{\ell} \frac{\partial R_{\ell m}(\theta_q, \phi_q)}{\partial \varepsilon_{\mu \nu}} \int \beta_{\ell}(r) j_{\ell}(q r) r^2 dr + \\ \frac{4\pi}{\sqrt{\Omega}} e^{-i{\bf q r_{\alpha}}}(-i)^{\ell} R_{\ell m}(\theta_q, \phi_q) \int \beta_{\ell}(r) \frac{\partial j_{\ell}(q r)}{\partial \varepsilon_{\mu \nu}} r^2 dr = \\ \frac{4\pi}{\sqrt{\Omega}} e^{-i{\bf q r_{\alpha}}}(-i)^{\ell} \Bigg[ \int \beta_{\ell}(r) j_{\ell}(q r) r^2 dr \Big(\frac{\partial R_{\ell m}(\theta_q, \phi_q)}{\partial \varepsilon_{\mu \nu}} - \frac{1}{2} R_{\ell m}(\theta_q, \phi_q) \delta_{\mu \nu}\Big) + R_{\ell m}(\theta_q, \phi_q) \int \beta_{\ell}(r) \frac{\partial j_{\ell}(q r)}{\partial \varepsilon_{\mu \nu}} r^2 dr \Bigg] \]

Strain derivative of the real spherical harmonics:

\[ \frac{\partial R_{\ell m}(\theta, \phi)}{\partial \varepsilon_{\mu \nu}} = \sum_{\tau} \frac{\partial R_{\ell m}(\theta, \phi)}{\partial q_{\tau}} \frac{\partial q_{\tau}}{\partial \varepsilon_{\mu \nu}} = -q_{\mu} \frac{\partial R_{\ell m}(\theta, \phi)}{\partial q_{\nu}} \]

For the derivatives of spherical harmonics over Cartesian components of vector please refer to the sht::dRlm_dr function.

Strain derivative of spherical Bessel function integral:

\[ \int \beta_{\ell}(r) \frac{\partial j_{\ell}(qr) }{\partial \varepsilon_{\mu \nu}} r^2 dr = \int \beta_{\ell}(r) \frac{\partial j_{\ell}(qr) }{\partial q} \frac{-q_{\mu} q_{\nu}}{q} r^2 dr \]

The second way to compute strain derivative of beta-projectors is trough Gaunt coefficients:

\[ \frac{\partial}{\partial \varepsilon_{\mu \nu}} \beta_{\xi}^{\alpha}({\bf q}) = -\frac{1}{2\sqrt{\Omega}} \delta_{\mu \nu} \int e^{-i{\bf q r}} \beta_{\xi}^{\alpha}({\bf r}) d{\bf r} + \frac{1}{\sqrt{\Omega}} \int i r_{\nu} q_{\mu} e^{-i{\bf q r}} \beta_{\xi}^{\alpha}({\bf r}) d{\bf r} \]

(second term comes from the strain derivative of \( e^{-i{\bf q r}} \)). Remembering that \( {\bf r} \) is proportional to p-like real spherical harmonics, we can rewrite the second part of beta-projector derivative as:

\[ \frac{1}{\sqrt{\Omega}} \int i r_{\nu} q_{\mu} e^{-i{\bf q r}} \beta_{\xi}^{\alpha}({\bf r}) d{\bf r} = \frac{1}{\sqrt{\Omega}} i q_{\mu} \int r \bar R_{1 \nu}(\theta, \phi) 4\pi \sum_{\ell_3 m_3} (-i)^{\ell_3} R_{\ell_3 m_3}(\theta_q, \phi_q) R_{\ell_3 m_3}(\theta, \phi) j_{\ell_3}(q r) \beta_{\ell_2}^{\alpha}(r) R_{\ell_2 m_2}(\theta, \phi) d{\bf r} = \\ \frac{4 \pi}{\sqrt{\Omega}} i q_{\mu} \sum_{\ell_3 m_3} (-i)^{\ell_3} R_{\ell_3 m_3}(\theta_q, \phi_q) \langle \bar R_{1\nu} | R_{\ell_3 m_3} | R_{\ell_2 m_2} \rangle \int j_{\ell_3}(q r) \beta_{\ell_2}^{\alpha}(r) r^3 dr \]

where

\[ \bar R_{1 x}(\theta, \phi) = -2 \sqrt{\frac{\pi}{3}} R_{11}(\theta, \phi) = \sin(\theta) \cos(\phi) \\ \bar R_{1 y}(\theta, \phi) = -2 \sqrt{\frac{\pi}{3}} R_{1-1}(\theta,\phi) = \sin(\theta) \sin(\phi) \\ \bar R_{1 z}(\theta, \phi) = 2 \sqrt{\frac{\pi}{3}} R_{10}(\theta, \phi) = \cos(\theta) \]

Template Parameters

T	One of float, double, complex<float> or complex<double> types for generic or Gamma point case.

Definition at line 37 of file stress.cpp.

calc_stress_vloc()

r3::matrix< double > sirius::Stress::calc_stress_vloc

(

)

Local potential contribution to stress.

Energy contribution from the local part of pseudopotential:

\[ E^{loc} = \int \rho({\bf r}) V^{loc}({\bf r}) d{\bf r} = \frac{1}{\Omega} \sum_{\bf G} \langle \rho | {\bf G} \rangle \langle {\bf G}| V^{loc} \rangle = \frac{1}{\Omega} \sum_{\bf G} \tilde \rho^{*}({\bf G}) \tilde V^{loc}({\bf G}) \]

where

\[ \tilde \rho({\bf G}) = \langle {\bf G} | \rho \rangle = \int e^{-i{\bf Gr}}\rho({\bf r}) d {\bf r} \]

and

\[ \tilde V^{loc}({\bf G}) = \langle {\bf G} | V^{loc} \rangle = \int e^{-i{\bf Gr}}V^{loc}({\bf r}) d {\bf r} \]

Using the expression for \( \tilde V^{loc}({\bf G}) \), the local contribution to the total energy is rewritten as

\[ E^{loc} = \frac{1}{\Omega} \sum_{\bf G} \tilde \rho^{*}({\bf G}) \sum_{\alpha} e^{-{\bf G\tau}_{\alpha}} 4 \pi \int V_{\alpha}^{loc}(r)\frac{\sin(Gr)}{Gr} r^2 dr = \frac{4\pi}{\Omega}\sum_{\bf G}\tilde \rho^{*}({\bf G})\sum_{\alpha} e^{-{\bf G\tau}_{\alpha}} \Bigg( \int \Big(V_{\alpha}(r) r + Z_{\alpha}^p {\rm erf}(r) \Big) \frac{\sin(Gr)}{G} dr - Z_{\alpha}^p \frac{e^{-\frac{G^2}{4}}}{G^2} \Bigg) \]

(see sirius::Potential::generate_local_potential for details).

Contribution to stress tensor:

\[ \sigma_{\mu \nu}^{loc} = \frac{1}{\Omega} \frac{\partial E^{loc}}{\partial \varepsilon_{\mu \nu}} = \frac{1}{\Omega} \frac{-1}{\Omega} \delta_{\mu \nu} \sum_{\bf G}\tilde \rho^{*}({\bf G}) \tilde V^{loc}({\bf G}) + \frac{4\pi}{\Omega^2} \sum_{\bf G}\tilde \rho^{*}({\bf G}) \sum_{\alpha} e^{-{\bf G\tau}_{\alpha}} \Bigg( \int \Big(V_{\alpha}(r) r + Z_{\alpha}^p {\rm erf}(r) \Big) \Big( \frac{r \cos (G r)}{G}-\frac{\sin (G r)}{G^2} \Big) \Big( -\frac{G_{\mu}G_{\nu}}{G} \Big) dr - Z_{\alpha}^p \Big(-\frac{e^{-\frac{G^2}{4}}}{2 G}-\frac{2 e^{-\frac{G^2}{4}}}{G^3} \Big) \Big( -\frac{G_{\mu}G_{\nu}}{G} \Big) \Bigg) = \\ -\delta_{\mu \nu} \sum_{\bf G}\rho^{*}({\bf G}) V^{loc}({\bf G}) + \sum_{\bf G} \rho^{*}({\bf G}) \Delta V^{loc}({\bf G}) G_{\mu}G_{\nu} \]

where \( \Delta V^{loc}({\bf G}) \) is built from the following radial integrals:

\[ \int \Big(V_{\alpha}(r) r + Z_{\alpha}^p {\rm erf}(r) \Big) \Big( \frac{\sin (G r)}{G^3} - \frac{r\cos (G r)}{G^2}\Big) dr - Z_{\alpha}^p \Big( \frac{e^{-\frac{G^2}{4}}}{2 G^2} + \frac{2 e^{-\frac{G^2}{4}}}{G^4}\Big) \]

Definition at line 706 of file stress.cpp.

stress_vloc()

r3::matrix< double > sirius::Stress::stress_vloc ( ) const

inline

Definition at line 279 of file stress.hpp.

calc_stress_har()

r3::matrix< double > sirius::Stress::calc_stress_har

(

)

Hartree energy contribution to stress.

Hartree energy:

\[ E^{H} = \frac{1}{2} \int_{\Omega} \rho({\bf r}) V^{H}({\bf r}) d{\bf r} = \frac{1}{2} \frac{1}{\Omega} \sum_{\bf G} \langle \rho | {\bf G} \rangle \langle {\bf G}| V^{H} \rangle = \frac{2 \pi}{\Omega} \sum_{\bf G} \frac{|\tilde \rho({\bf G})|^2}{G^2} \]

where

\[ \langle {\bf G} | \rho \rangle = \int e^{-i{\bf Gr}}\rho({\bf r}) d {\bf r} = \tilde \rho({\bf G}) \]

and

\[ \langle {\bf G} | V^{H} \rangle = \int e^{-i{\bf Gr}}V^{H}({\bf r}) d {\bf r} = \frac{4\pi}{G^2} \tilde \rho({\bf G}) \]

Hartree energy contribution to stress tensor:

\[ \sigma_{\mu \nu}^{H} = \frac{1}{\Omega} \frac{\partial E^{H}}{\partial \varepsilon_{\mu \nu}} = \frac{1}{\Omega} 2\pi \Big( \big( \frac{\partial}{\partial \varepsilon_{\mu \nu}} \frac{1}{\Omega} \big) \sum_{{\bf G}} \frac{|\tilde \rho({\bf G})|^2}{G^2} + \frac{1}{\Omega} \sum_{{\bf G}} |\tilde \rho({\bf G})|^2 \frac{\partial}{\partial \varepsilon_{\mu \nu}} \frac{1}{G^2} \Big) = \\ \frac{1}{\Omega} 2\pi \Big( -\frac{1}{\Omega} \delta_{\mu \nu} \sum_{{\bf G}} \frac{|\tilde \rho({\bf G})|^2}{G^2} + \frac{1}{\Omega} \sum_{\bf G} |\tilde \rho({\bf G})|^2 \sum_{\tau} \frac{-2 G_{\tau}}{G^4} \frac{\partial G_{\tau}}{\partial \varepsilon_{\mu \nu}} \Big) = \\ 2\pi \sum_{\bf G} \frac{|\rho({\bf G})|^2}{G^2} \Big( -\delta_{\mu \nu} + \frac{2}{G^2} G_{\nu} G_{\mu} \Big) \]

Definition at line 613 of file stress.cpp.

stress_har()

r3::matrix< double > sirius::Stress::stress_har ( ) const

inline

Definition at line 315 of file stress.hpp.

calc_stress_ewald()

r3::matrix< double > sirius::Stress::calc_stress_ewald

(

)

Ewald energy contribution to stress.

Ewald energy:

\[ E^{ion-ion} = \frac{1}{2} \sideset{}{'} \sum_{\alpha \beta {\bf T}} Z_{\alpha} Z_{\beta} \frac{{\rm erfc}(\sqrt{\lambda} |{\bf r}_{\alpha} - {\bf r}_{\beta} + {\bf T}|)}{|{\bf r}_{\alpha} - {\bf r}_{\beta} + {\bf T}|} + \frac{2 \pi}{\Omega} \sum_{{\bf G}} \frac{e^{-\frac{G^2}{4 \lambda}}}{G^2} \Big| \sum_{\alpha} Z_{\alpha} e^{-i{\bf r}_{\alpha}{\bf G}} \Big|^2 - \sum_{\alpha} Z_{\alpha}^2 \sqrt{\frac{\lambda}{\pi}} - \frac{2\pi}{\Omega}\frac{N_{el}^2}{4 \lambda} \]

(check sirius::DFT_ground_state::ewald_energy for details).
Contribution to stress tensor:

\[ \sigma_{\mu \nu}^{ion-ion} = \frac{1}{\Omega} \frac{\partial E^{ion-ion}}{\partial \varepsilon_{\mu \nu}} \]

Derivative of the first part:

\[ \frac{1}{\Omega}\frac{\partial}{\partial \varepsilon_{\mu \nu}} \frac{1}{2} \sideset{}{'} \sum_{\alpha \beta {\bf T}} Z_{\alpha} Z_{\beta} \frac{{\rm erfc}(\sqrt{\lambda} |{\bf r}_{\alpha} - {\bf r}_{\beta} + {\bf T}|)}{|{\bf r}_{\alpha} - {\bf r}_{\beta} + {\bf T}|} = \frac{1}{2\Omega} \sideset{}{'} \sum_{\alpha \beta {\bf T}} Z_{\alpha} Z_{\beta} \Big( -2e^{-\lambda |{\bf r'}|^2} \sqrt{\frac{\lambda}{\pi}} \frac{1}{|{\bf r'}|^2} - {\rm erfc}(\sqrt{\lambda} |{\bf r'}|) \frac{1}{|{\bf r'}|^3} \Big) r'_{\mu} r'_{\nu} \]

where \( {\bf r'} = {\bf r}_{\alpha} - {\bf r}_{\beta} + {\bf T} \).

Derivative of the second part:

\[ \frac{1}{\Omega}\frac{\partial}{\partial \varepsilon_{\mu \nu}} \frac{2\pi}{\Omega} \sum_{{\bf G}} \frac{e^{-\frac{G^2}{4 \lambda}}}{G^2} \Big| \sum_{\alpha} Z_{\alpha} e^{-i{\bf r}_{\alpha}{\bf G}} \Big|^2 = -\frac{2\pi}{\Omega^2} \delta_{\mu \nu} \sum_{{\bf G}} \frac{e^{-\frac{G^2}{4 \lambda}}}{G^2} \Big| \sum_{\alpha} Z_{\alpha} e^{-i{\bf r}_{\alpha}{\bf G}} \Big|^2 + \frac{2\pi}{\Omega^2} \sum_{\bf G} G_{\mu} G_{\nu} \frac{e^{-\frac{G^2}{4 \lambda}}}{G^2} 2 \frac{\frac{G^2}{4\lambda} + 1}{G^2} \Big| \sum_{\alpha} Z_{\alpha} e^{-i{\bf r}_{\alpha}{\bf G}} \Big|^2 \]

Derivative of the fourth part:

\[ -\frac{1}{\Omega}\frac{\partial}{\partial \varepsilon_{\mu \nu}} \frac{2\pi}{\Omega}\frac{N_{el}^2}{4 \lambda} = \frac{2\pi}{\Omega^2}\frac{N_{el}^2}{4 \lambda} \delta_{\mu \nu} \]

Definition at line 490 of file stress.cpp.

stress_ewald()

r3::matrix< double > sirius::Stress::stress_ewald ( ) const

inline

Definition at line 362 of file stress.hpp.

calc_stress_kin_aux()

template<typename T >

void sirius::Stress::calc_stress_kin_aux

Kinetic energy contribution to stress.

Kinetic energy:

\[ E^{kin} = \sum_{{\bf k}} w_{\bf k} \sum_j f_j \frac{1}{2} |{\bf G+k}|^2 |\psi_j({\bf G + k})|^2 \]

Contribution to the stress tensor

\[ \sigma_{\mu \nu}^{kin} = \frac{1}{\Omega} \frac{\partial E^{kin}}{\partial \varepsilon_{\mu \nu}} = \frac{1}{\Omega} \sum_{{\bf k}} w_{\bf k} \sum_j f_j \frac{1}{2} 2 |{\bf G+k}| \Big( -\frac{1}{|{\bf G+k}|} (G+k)_{\mu} (G+k)_{\nu} \Big) |\psi_j({\bf G + k})|^2 =\\ -\frac{1}{\Omega} \sum_{{\bf k}} w_{\bf k} (G+k)_{\mu} (G+k)_{\nu} \sum_j f_j |\psi_j({\bf G + k})|^2 \]

Definition at line 649 of file stress.cpp.

calc_stress_kin()

r3::matrix< double > sirius::Stress::calc_stress_kin

(

)

Definition at line 692 of file stress.cpp.

stress_kin()

r3::matrix< double > sirius::Stress::stress_kin ( ) const

inline

Definition at line 385 of file stress.hpp.

calc_stress_nonloc()

r3::matrix< double > sirius::Stress::calc_stress_nonloc

(

)

Definition at line 758 of file stress.cpp.

stress_nonloc()

r3::matrix< double > sirius::Stress::stress_nonloc ( ) const

inline

Definition at line 392 of file stress.hpp.

calc_stress_us()

r3::matrix< double > sirius::Stress::calc_stress_us

(

)

Contribution to the stress tensor from the augmentation operator.

Total energy in ultrasoft pseudopotential contains this term:

\[ \int V^{eff}({\bf r})\rho^{aug}({\bf r})d{\bf r} = \sum_{\alpha} \sum_{\xi \xi'} n_{\xi \xi'}^{\alpha} \int V^{eff}({\bf r}) Q_{\xi \xi'}^{\alpha}({\bf r}) d{\bf r} \]

The derivatives of beta-projectors (hidden in the desnity matrix expression) are taken into account in sirius::Stress::calc_stress_nonloc. Here we need to compute the remaining contribution from the \( Q_{\xi \xi'}({\bf r}) \) itself. We are interested in the integral:

\[ \int V^{eff}({\bf r}) Q_{\xi \xi'}^{\alpha}({\bf r}) d{\bf r} = \sum_{\bf G} V^{eff}({\bf G}) \tilde Q_{\xi \xi'}^{\alpha}({\bf G}) \]

where

\[ \tilde Q_{\xi \xi'}^{\alpha}({\bf G}) = \int e^{-i{\bf Gr}} Q_{\xi \xi'}^{\alpha}({\bf r}) d{\bf r} = 4\pi \sum_{\ell m} (-i)^{\ell} R_{\ell m}(\hat{\bf G}) \langle R_{\ell_{\xi} m_{\xi}} | R_{\ell m} | R_{\ell_{\xi'} m_{\xi'}} \rangle \int Q_{\ell_{\xi} \ell_{\xi'}}^{\ell}(r) j_{\ell}(Gr) r^2 dr \]

Strain derivative of \( \tilde Q_{\xi \xi'}^{\alpha}({\bf G}) \) is:

\[ \frac{\partial}{\partial \varepsilon_{\mu \nu}} \tilde Q_{\xi \xi'}^{\alpha}({\bf G}) = 4\pi \sum_{\ell m} (-i)^{\ell} \langle R_{\ell_{\xi} m_{\xi}} | R_{\ell m} | R_{\ell_{\xi'} m_{\xi'}} \rangle \Big( \frac{\partial R_{\ell m}(\hat{\bf G})}{\partial \varepsilon_{\mu \nu}} \int Q_{\ell_{\xi} \ell_{\xi'}}^{\ell}(r) j_{\ell}(Gr) r^2 dr + R_{\ell m}(\hat{\bf G}) \int Q_{\ell_{\xi} \ell_{\xi'}}^{\ell}(r) \frac{\partial j_{\ell}(Gr)}{\partial \varepsilon_{\mu \nu}} r^2 dr \Big) \]

For strain derivatives of spherical harmonics and Bessel functions see sirius::Stress::calc_stress_nonloc. We can pull the common multiplier \( -G_{\mu} / G \) from both terms and arrive to the following expression:

\[ \frac{\partial}{\partial \varepsilon_{\mu \nu}} \tilde Q_{\xi \xi'}^{\alpha}({\bf G}) = -\frac{G_{\mu}}{G} 4\pi \sum_{\ell m} (-i)^{\ell} \langle R_{\ell_{\xi} m_{\xi}} | R_{\ell m} | R_{\ell_{\xi'} m_{\xi'}} \rangle \Big( \big(\nabla_{G} R_{\ell m}(\hat{\bf G})\big)_{\nu} \int Q_{\ell_{\xi} \ell_{\xi'}}^{\ell}(r) j_{\ell}(Gr) r^2 dr + R_{\ell m}(\hat{\bf G}) \int Q_{\ell_{\xi} \ell_{\xi'}}^{\ell}(r) \frac{\partial j_{\ell}(Gr)}{\partial G} G_{\nu} r^2 dr \Big) \]

Definition at line 371 of file stress.cpp.

stress_us()

auto sirius::Stress::stress_us ( ) const

inline

Definition at line 441 of file stress.hpp.

stress_us_nl()

auto sirius::Stress::stress_us_nl ( ) const

inline

Definition at line 446 of file stress.hpp.

calc_stress_xc()

r3::matrix< double > sirius::Stress::calc_stress_xc

(

)

XC contribution to stress.

XC contribution has the following expression:

\[ \frac{\partial E_{xc}}{\partial \varepsilon_{\mu \nu}} = \delta_{\mu \nu} \int \Big( \epsilon_{xc}({\bf r}) - v_{xc}({\bf r}) \Big) \rho({\bf r})d{\bf r} - \int \frac{\partial \epsilon_{xc} \big( \rho({\bf r}), \nabla \rho({\bf r})\big) }{\nabla_{\mu} \rho({\bf r})} \nabla_{\nu}\rho({\bf r}) d{\bf r} \]

Definition at line 284 of file stress.cpp.

stress_xc()

auto sirius::Stress::stress_xc ( ) const

inline

Definition at line 461 of file stress.hpp.

calc_stress_core()

r3::matrix< double > sirius::Stress::calc_stress_core

(

)

Non-linear core correction to stress tensor.

Definition at line 217 of file stress.cpp.

stress_core()

auto sirius::Stress::stress_core ( ) const

inline

Definition at line 469 of file stress.hpp.

calc_stress_hubbard()

r3::matrix< double > sirius::Stress::calc_stress_hubbard

(

)

Definition at line 114 of file stress.cpp.

stress_hubbard()

auto sirius::Stress::stress_hubbard ( ) const

inline

Definition at line 476 of file stress.hpp.

calc_stress_total()

r3::matrix< double > sirius::Stress::calc_stress_total

(

)

Definition at line 87 of file stress.cpp.

stress_total()

auto sirius::Stress::stress_total ( ) const

inline

Definition at line 483 of file stress.hpp.

print_info()

void sirius::Stress::print_info	(	std::ostream &	out__,
		int	verbosity__
	)		const

Definition at line 561 of file stress.cpp.

Member Data Documentation

ctx_

Simulation_context& sirius::Stress::ctx_

private

Definition at line 99 of file stress.hpp.

density_

Density const& sirius::Stress::density_

private

Definition at line 101 of file stress.hpp.

potential_

Potential& sirius::Stress::potential_

private

Definition at line 103 of file stress.hpp.

kset_

K_point_set& sirius::Stress::kset_

private

Definition at line 105 of file stress.hpp.

stress_kin_

r3::matrix<double> sirius::Stress::stress_kin_

private

Definition at line 107 of file stress.hpp.

stress_har_

r3::matrix<double> sirius::Stress::stress_har_

private

Definition at line 109 of file stress.hpp.

stress_ewald_

r3::matrix<double> sirius::Stress::stress_ewald_

private

Definition at line 111 of file stress.hpp.

stress_vloc_

r3::matrix<double> sirius::Stress::stress_vloc_

private

Definition at line 113 of file stress.hpp.

stress_nonloc_

r3::matrix<double> sirius::Stress::stress_nonloc_

private

Definition at line 115 of file stress.hpp.

stress_us_

r3::matrix<double> sirius::Stress::stress_us_

private

Definition at line 117 of file stress.hpp.

stress_xc_

r3::matrix<double> sirius::Stress::stress_xc_

private

Definition at line 119 of file stress.hpp.

stress_core_

r3::matrix<double> sirius::Stress::stress_core_

private

Definition at line 121 of file stress.hpp.

stress_hubbard_

r3::matrix<double> sirius::Stress::stress_hubbard_

private

Definition at line 123 of file stress.hpp.

stress_total_

r3::matrix<double> sirius::Stress::stress_total_

private

Definition at line 125 of file stress.hpp.

---

The documentation for this class was generated from the following files:

• stress.hpp
• stress.cpp