dm_propagation_oct_m Module Reference

Data Types
type	dmp_t
	Density matrix dissipation. More...

Functions/Subroutines
subroutine, public	dm_propagation_init_run (adiabatic_st, namespace, space, gr, st, hm, mc)
	Initialise the adiabatic states prior to running TB propagation. More...
subroutine	dmp_init (this, namespace, st)
	Initialise an instance of density matrix dissipation. More...
subroutine, public	dm_propagation_run (dmp, namespace, space, gr, ions, st, mc, hm, ks, iter, dt, ext_partners, update_energy)
	Apply dissipation to a TD run via the Linblad formalism. More...
subroutine	orthogonality_check_ks (ik, st, gr)
	Check orthonality and electron number of TDKS wavefunctions We constructed the density matrix from the TDKS wavefunctions, as. More...
subroutine	construct_residuals (adiabatic_st, ik, st, gr, conj_psi_phi, resd)
	Construct the residual basis from the TDKS wavefunctions. More...
subroutine	construct_density_matrix (ik, st, conj_psi_phi, rho_mat)
	Construct the full density matrix in the adiabatic and residual basis. More...
subroutine	population_in_adiabatic (ik, st, rho_mat, pop)
	Calculate population in adiabatic basis. More...
subroutine	update_st (dmp, ik, gr, resd, st, rho_mat)
	Diagonalize the density matrix and update the wavefunction and occupation. More...
subroutine	update_wfc_occ (adiabatic_st, ik, st, gr, resd, rho_mat, occ)
	Transform the wavefunctions into real-space basis The wavefunctions in real-space basis are given by: More...
subroutine	update_wfc_occ_procrustes (adiabatic_st, ik, st, gr, resd, rho_mat, occ)
	Make TDKS wavefunctions continuous by maximizing their overlap. More...
subroutine	dissipation (dmp, bst, st, ik, dt, nn)
	Apply dissipation to the density matrix This subroutine applies the dissipation operator to the density matrix using Taylor expansion of the matrix exponential upto order 4. More...

Function/Subroutine Documentation

dm_propagation_init_run()

subroutine, public dm_propagation_oct_m::dm_propagation_init_run	(	type(states_elec_t), intent(inout)	adiabatic_st,
		type(namespace_t), intent(in)	namespace,
		type(electron_space_t), intent(in)	space,
		type(grid_t), intent(in)	gr,
		type(states_elec_t), intent(in)	st,
		type(hamiltonian_elec_t), intent(in)	hm,
		type(multicomm_t), intent(in)	mc
	)

Initialise the adiabatic states prior to running TB propagation.

Definition at line 190 of file dm_propagation.F90.

dmp_init()

subroutine dm_propagation_oct_m::dmp_init ( class(dmp_t), intent(out)  this, type(namespace_t), intent(in)  namespace, type(states_elec_t), intent(in)  st  )

private

Initialise an instance of density matrix dissipation.

Definition at line 220 of file dm_propagation.F90.

dm_propagation_run()

subroutine, public dm_propagation_oct_m::dm_propagation_run	(	type(dmp_t), intent(inout)	dmp,
		type(namespace_t), intent(in)	namespace,
		type(electron_space_t), intent(in)	space,
		type(grid_t), intent(in)	gr,
		type(ions_t), intent(in)	ions,
		type(states_elec_t), intent(inout)	st,
		type(multicomm_t), intent(in)	mc,
		type(hamiltonian_elec_t), intent(inout)	hm,
		type(v_ks_t), intent(inout)	ks,
		integer, intent(in)	iter,
		real(real64), intent(in)	dt,
		type(partner_list_t), intent(in)	ext_partners,
		logical, intent(in), optional	update_energy
	)

Apply dissipation to a TD run via the Linblad formalism.

Parameters

[in,out]	dmp	Density matrix propagation instance
[in]	space	Spin dimension
[in]	gr	Underlying mesh
[in]	ions	Geometry
[in,out]	st	Time-dependent wfcs
[in]	mc	MPI communicator
[in,out]	hm	Hamiltonian, updated at each time step
[in,out]	ks	Kohn-Sham potential
[in]	iter	Number of iterations
[in]	dt	Time step
[in]	ext_partners	External interaction partners
[in]	update_energy	Whether update energy when solve adabatic wfcs

Definition at line 313 of file dm_propagation.F90.

orthogonality_check_ks()

subroutine dm_propagation_oct_m::orthogonality_check_ks ( integer, intent(in)  ik, type(states_elec_t), intent(in)  st, type(grid_t), intent(in)  gr  )

private

Check orthonality and electron number of TDKS wavefunctions We constructed the density matrix from the TDKS wavefunctions, as.

\[ \rho_{ij} = \langle \psi_i | \left( \sum_l f_l | \psi_l \rangle \langle \psi_l | \right) | \psi_j \rangle \]

where \( f_l \) is the occupation of the state \(l\) at \(k\)-point.

Parameters

[in]	ik	K-point index
[in]	st	Time-dependent wfcs
[in]	gr	Underlying mesh

Definition at line 412 of file dm_propagation.F90.

construct_residuals()

subroutine dm_propagation_oct_m::construct_residuals ( type(states_elec_t), intent(in)  adiabatic_st, integer, intent(in)  ik, type(states_elec_t), intent(in)  st, type(grid_t), intent(in)  gr, complex(real64), dimension(:, :), intent(out)  conj_psi_phi, complex(real64), dimension(:, :, :), intent(out)  resd  )

private

Construct the residual basis from the TDKS wavefunctions.

The residual basis vectors are defined as

\[ | d_n \rangle = | \psi_n \rangle - \sum_{i=1}^{N} | \phi_i \rangle \langle \phi_i | \psi_n \rangle \]

where \( | \psi_n \rangle \) are the TDKS wavefunctions, and \( | \phi_i \rangle \) are the adiabatic states.

The resulting residuals \( | d_n \rangle \) are stored in resd, a 3D array with shape (np, dim, nst).

Parameters

[in]	adiabatic_st	Adiabatic basis
[in]	ik	K-point index
[in]	st	Time-dependent wfcs
[in]	gr	Underlying mesh
[out]	conj_psi_phi	(i, j) = <\phi, d_j | \psi_i >
[out]	resd	Residuals

Definition at line 481 of file dm_propagation.F90.

construct_density_matrix()

subroutine dm_propagation_oct_m::construct_density_matrix ( integer, intent(in)  ik, type(states_elec_t), intent(in)  st, complex(real64), dimension(:, :), intent(in)  conj_psi_phi, complex(real64), dimension(:, :), intent(out)  rho_mat  )

private

Construct the full density matrix in the adiabatic and residual basis.

The density matrix elements in the combined basis are given by

\[ \rho_{ij} = \langle \chi_i | \left( \sum_l f_l\, | \psi_l \rangle \langle \psi_l | \right) | \chi_j \rangle \]

where \( \{ \chi_i \} = \{ \phi_1, \dots, \phi_N, d_1, \dots, d_N \} \) is the complete basis, consisting of adiabatic states \( \phi_i \) and residuals \( d_i \), and \( f_l \) is the occupation of state \( \psi_l \).

Parameters

[in]	ik	k-point index
[in]	st	Time-dependent wfcs
[in]	conj_psi_phi	(i, j) = <\phi, d_j | \psi_i >
[out]	rho_mat	The full density matrix

Definition at line 543 of file dm_propagation.F90.

population_in_adiabatic()

subroutine dm_propagation_oct_m::population_in_adiabatic ( integer, intent(in)  ik, type(states_elec_t), intent(in)  st, complex(real64), dimension(:, :), intent(in)  rho_mat, real(real64), intent(inout)  pop  )

private

Calculate population in adiabatic basis.

Number of electrons in the adiabatic basis is

\[ pop = \sum_{i,k} \rho_{ii} w_k \]

Parameters

[in]	ik	K-point index
[in]	st	Time-dependent wfcs
[in]	rho_mat	The full density matrix
[in,out]	pop	Population in the adiabatic basis

Definition at line 577 of file dm_propagation.F90.

update_st()

subroutine dm_propagation_oct_m::update_st ( type(dmp_t), intent(in)  dmp, integer, intent(in)  ik, type(grid_t), intent(in)  gr, complex(real64), dimension(:, :, :), intent(in)  resd, type(states_elec_t), intent(inout)  st, complex(real64), dimension(:, :), intent(inout)  rho_mat  )

private

Diagonalize the density matrix and update the wavefunction and occupation.

This routine solves the generalized eigenvalue problem:

\[ \rho \psi = \lambda B \psi, \]

where \( \rho \) is the density matrix, \( B \) is the overlap matrix, and \( \psi \) are the wavefunctions.

Definition at line 608 of file dm_propagation.F90.

update_wfc_occ()

subroutine dm_propagation_oct_m::update_wfc_occ ( type(states_elec_t), intent(in)  adiabatic_st, integer, intent(in)  ik, type(states_elec_t), intent(inout)  st, type(grid_t), intent(in)  gr, complex(real64), dimension(:, :, :), intent(in)  resd, complex(real64), dimension(:, :), intent(in)  rho_mat, real(real64), dimension(1:2*st%nst), intent(in)  occ  )

private

Transform the wavefunctions into real-space basis The wavefunctions in real-space basis are given by:

\[ \psi = \sum_{i=1}^{N} c_i \phi_i + c_{i+N} \, \mathrm{resd}_i, \]

where \( \phi_i \) are the adiabatic states and \( \mathrm{resd}_i \) are the residual basis vectors.

Definition at line 685 of file dm_propagation.F90.

update_wfc_occ_procrustes()

subroutine dm_propagation_oct_m::update_wfc_occ_procrustes ( type(states_elec_t), intent(in)  adiabatic_st, integer, intent(in)  ik, type(states_elec_t), intent(inout)  st, type(grid_t), intent(in)  gr, complex(real64), dimension(:, :, :), intent(in)  resd, complex(real64), dimension(:, :), intent(in)  rho_mat, real(real64), dimension(1:2*st%nst), intent(in)  occ  )

private

Make TDKS wavefunctions continuous by maximizing their overlap.

Applies a unitary transformation to \( \tilde{\Psi} \) such that the overlap with the input wavefunctions is maximized.

The resulting wavefunctions \( \psi_i \) are given by:

\[ \psi_i = \sum_{j=1}^{2N} \tilde{\psi}_j \, (U_{\mathrm{proj}})_{j i} = \sum_{j=1}^{2N} \sum_{l=1}^{2N} \chi_l \, (\rho_{\mathrm{mat}})_{l j} \, (U_{\mathrm{proj}})_{j i} \]

where \( U_{\mathrm{proj}} = U V^{\mathrm{T}} \), and \( U \) and \( V \) are obtained from the singular value decomposition (SVD) of the overlap matrix \( S \):

\[ S = U \Sigma V^{\mathrm{T}} \]

Here, \( \rho_{\mathrm{mat}} \) contains the eigenvectors of the density matrix, and \( \{ \chi_i \} = \{ \phi_1, \dots, \phi_N, d_1, \dots, d_N \} \) is the complete basis.

The occupation associated with state \( \psi_i \) is given by \( |\psi|^2 \), where:

\[ \psi = \sum_{j=1}^{2N} \sum_{l=1}^{2N} \chi_l \, (\rho_{\mathrm{mat}})_{l j} \, \sqrt{\mathrm{occ}_j} \, (U_{\mathrm{proj}})_{j i} \]

Definition at line 756 of file dm_propagation.F90.

dissipation()

subroutine dm_propagation_oct_m::dissipation ( type(dmp_t), intent(in)  dmp, type(states_elec_t), intent(in)  bst, type(states_elec_t), intent(in)  st, integer, intent(in)  ik, real(real64), intent(in)  dt, complex(real64), dimension(:,:), intent(inout)  nn  )

private

Apply dissipation to the density matrix This subroutine applies the dissipation operator to the density matrix using Taylor expansion of the matrix exponential upto order 4.

Note

The following equation is relevant to the implementation:

\[ \overrightarrow{D}^{\mathrm{inter}}_{ab;cd} = \sum_{k=1}^{k_{\mathrm{max}}}\sum_{\substack{m,n = 0 \\ m \neq n}}^{N-1}\ & \gamma^{m \to n}_{k}\Bigg( \delta_{c,(m;k)}\ \delta_{(n;k),a} \ \delta_{b,(n;k)}\ \delta_{(m;k),d}- \frac{1}{2}\big[ & \delta_{c,a}\ \delta_{b,(m;k)}\ \Delta_{l_1}\ \delta_{(m;k),d}+ \delta_{b,d}\ \\ & \delta_{c,(m;k)}\ \Delta_{l_1}\delta_{(m;k),a}\big]\Bigg) \]

where \( \overrightarrow{D}^{\mathrm{inter}}\) is the interband dissipator using vectorized Lindblad formalism, \( \gamma^{m \to n}_{k}\) is the damping factor for the transition from band \(m\) to band \(n\) at $\f k

Parameters

[in]	dmp	Density matrix propagation instance
[in]	bst	Adiabatic basis
[in]	st	Time-dependent wfcs
[in]	ik	K-point index
[in]	dt	Time step
[in,out]	nn	Density matrix

Definition at line 867 of file dm_propagation.F90.