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properly added basis_correction
This commit is contained in:
Emmanuel Giner
parent
51cf96a506
commit
2fc294cd56
3
src/basis_correction/NEED
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3
src/basis_correction/NEED
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mu_of_r
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ecmd_utils
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dft_one_e
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27
src/basis_correction/README.rst
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27
src/basis_correction/README.rst
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================
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basis_correction
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================
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This module proposes the various flavours of the DFT-based basis set correction originally proposed in J. Chem. Phys. 149, 194301 (2018); https://doi.org/10.1063/1.5052714.
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This basis set correction relies mainy on :
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+) The definition of a range-separation function \mu(r) varying in space to mimic the incompleteness of the basis set used to represent the coulomb interaction. This procedure needs a two-body rdm representing qualitatively the spacial distribution of the opposite spin electron pairs.
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Two types of \mu(r) are proposed, according to the strength of correlation, through the keyword "mu_of_r_potential" in the module "mu_of_r":
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a) "mu_of_r_potential = hf" uses the two-body rdm of a HF-like wave function (i.e. a single Slater determinant developped with the MOs stored in the EZFIO folder).
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When HF is a qualitative representation of the electron pairs (i.e. weakly correlated systems), such an approach for \mu(r) is OK.
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See for instance JPCL, 10, 2931-2937 (2019) for typical flavours of the results.
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Thanks to the trivial nature of such a two-body rdm, the equation (22) of J. Chem. Phys. 149, 194301 (2018) can be rewritten in a very efficient way, and therefore the limiting factor of such an approach is the AO->MO four-index transformation of the two-electron integrals.
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b) "mu_of_r_potential = cas_ful" uses the two-body rdm of CAS-like wave function (i.e. linear combination of Slater determinants developped in an active space with the MOs stored in the EZFIO folder).
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If the CAS is properly chosen (i.e. the CAS-like wave function qualitatively represents the wave function of the systems), then such an approach is OK for \mu(r) even in the case of strong correlation.
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+) The use of DFT correlation functionals with multi-determinant reference (Ecmd). These functionals are originally defined in the RS-DFT framework (see for instance Theor. Chem. Acc.114, 305(2005)) and design to capture short-range correlation effects. A important quantity arising in the Ecmd is the exact on-top pair density of the system, and the main differences of approximated Ecmd relies on different approximations for the exact on-top pair density.
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The two main flavours of Ecmd depends on the strength of correlation in the system:
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a) for weakly correlated systems, the ECMD PBE-UEG functional based on the seminal work of in RSDFT (see JCP, 150, 084103 1-10 (2019)) and adapted for the basis set correction in JPCL, 10, 2931-2937 (2019) uses the exact on-top pair density of the UEG at large mu and the PBE correlation functional at mu = 0. As shown in JPCL, 10, 2931-2937 (2019), such a functional is more accurate than the ECMD LDA for weakly correlated systems.
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b) for strongly correlated systems, the ECMD PBE-OT, which uses the extrapolated on-top pair density of the CAS wave function thanks to the large \mu behaviour of the on-top pair density, is accurate, but suffers from S_z dependence (i.e. is not invariant with respect to S_z) because of the spin-polarization dependence of the PBE correlation functional entering at mu=0.
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An alternative is ECMD SU-PBE-OT which uses the same on-top pair density that ECMD PBE-OT but a ZERO spin-polarization to remove the S_z dependence. As shown in ???????????, this strategy is one of the more accurate and respects S_z invariance and size consistency if the CAS wave function is correctly chosen.
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1
src/basis_correction/TODO
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1
src/basis_correction/TODO
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change all correlation functionals with the pbe_on_top general
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27
src/basis_correction/basis_correction.irp.f
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27
src/basis_correction/basis_correction.irp.f
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program basis_correction
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implicit none
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BEGIN_DOC
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! TODO : Put the documentation of the program here
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END_DOC
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read_wf = .True.
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touch read_wf
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no_core_density = .True.
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touch no_core_density
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provide mo_two_e_integrals_in_map
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call print_basis_correction
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! call print_e_b
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end
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subroutine print_e_b
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implicit none
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print *, 'Hello world'
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print*,'ecmd_lda_mu_of_r = ',ecmd_lda_mu_of_r
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print*,'ecmd_pbe_ueg_mu_of_r = ',ecmd_pbe_ueg_mu_of_r
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print*,'ecmd_pbe_ueg_eff_xi_mu_of_r = ',ecmd_pbe_ueg_eff_xi_mu_of_r
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print*,''
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print*,'psi_energy + E^B_LDA = ',psi_energy + ecmd_lda_mu_of_r
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print*,'psi_energy + E^B_PBE_UEG = ',psi_energy + ecmd_pbe_ueg_mu_of_r
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print*,'psi_energy + E^B_PBE_UEG_Xi = ',psi_energy + ecmd_pbe_ueg_eff_xi_mu_of_r
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print*,''
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print*,'mu_average_prov = ',mu_average_prov
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end
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92
src/basis_correction/eff_xi_based_func.irp.f
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src/basis_correction/eff_xi_based_func.irp.f
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BEGIN_PROVIDER [double precision, ecmd_pbe_ueg_eff_xi_mu_of_r, (N_states)]
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BEGIN_DOC
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! ecmd_pbe_ueg_eff_xi_mu_of_r = multi-determinantal Ecmd within the PBE-UEG and effective spin polarization approximation with mu(r),
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!
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! see Eqs. 30 in ???????????
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!
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! Based on the PBE-on-top functional (see Eqs. 26, 27 of J. Chem. Phys.150, 084103 (2019); doi: 10.1063/1.5082638)
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!
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! and replaces the approximation of the exact on-top pair density by the exact on-top of the UEG
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!
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! !!!! BUT !!!! with an EFFECTIVE SPIN POLARIZATION DEPENDING ON THE ON-TOP PAIR DENSITY
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!
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! See P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995). for original Ref., and Eq. 29 in ???????????
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END_DOC
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implicit none
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double precision :: weight,density
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integer :: ipoint,istate
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double precision :: eps_c_md_PBE,mu,rho_a,rho_b,grad_rho_a(3),grad_rho_b(3),g0_UEG_mu_inf,on_top
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ecmd_pbe_ueg_eff_xi_mu_of_r = 0.d0
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print*,'Providing ecmd_pbe_ueg_eff_xi_mu_of_r ...'
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call wall_time(wall0)
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do istate = 1, N_states
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do ipoint = 1, n_points_final_grid
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weight=final_weight_at_r_vector(ipoint)
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mu = mu_of_r_prov(ipoint,istate)
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density = one_e_dm_and_grad_alpha_in_r(4,ipoint,istate) + one_e_dm_and_grad_beta_in_r(4,ipoint,istate)
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! We use the effective spin density to define rho_a/rho_b
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rho_a = 0.5d0 * (density + effective_spin_dm(ipoint,istate))
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rho_b = 0.5d0 * (density - effective_spin_dm(ipoint,istate))
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grad_rho_a(1:3) = one_e_dm_and_grad_alpha_in_r(1:3,ipoint,istate)
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grad_rho_b(1:3) = one_e_dm_and_grad_beta_in_r(1:3,ipoint,istate)
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! We take the on-top pair density of the UEG which is (1-zeta^2) rhoc^2 g0 = 4 rhoa * rhob * g0
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! with the effective rho_a and rho_b
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on_top = 4.d0 * rho_a * rho_b * g0_UEG_mu_inf(rho_a,rho_b)
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call ec_md_pbe_on_top_general(mu,rho_a,rho_b,grad_rho_a,grad_rho_b,on_top,eps_c_md_PBE)
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ecmd_pbe_ueg_eff_xi_mu_of_r(istate) += eps_c_md_PBE * weight
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enddo
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enddo
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double precision :: wall1, wall0
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call wall_time(wall1)
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print*,'Time for the ecmd_pbe_ueg_eff_xi_mu_of_r:',wall1-wall0
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END_PROVIDER
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BEGIN_PROVIDER [double precision, ecmd_lda_eff_xi_mu_of_r, (N_states)]
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BEGIN_DOC
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! ecmd_lda_eff_xi_mu_of_r = multi-determinantal Ecmd within the LDA and effective spin polarization approximation with mu(r),
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!
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! corresponds to equation 40 in J. Chem. Phys. 149, 194301 (2018); https://doi.org/10.1063/1.5052714
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!
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! !!!! BUT !!!! with an EFFECTIVE SPIN POLARIZATION DEPENDING ON THE ON-TOP PAIR DENSITY
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!
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! See P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995). for original Ref., and Eq. 29 in ???????????
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END_DOC
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implicit none
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integer :: ipoint,istate
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double precision :: rho_a, rho_b, ec
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logical :: dospin
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double precision :: wall0,wall1,weight,mu,density
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dospin = .true. ! JT dospin have to be set to true for open shell
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print*,'Providing ecmd_lda_eff_xi_mu_of_r ...'
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ecmd_lda_eff_xi_mu_of_r = 0.d0
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call wall_time(wall0)
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do istate = 1, N_states
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do ipoint = 1, n_points_final_grid
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mu = mu_of_r_prov(ipoint,istate)
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weight = final_weight_at_r_vector(ipoint)
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density = one_e_dm_and_grad_alpha_in_r(4,ipoint,istate) + one_e_dm_and_grad_beta_in_r(4,ipoint,istate)
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rho_a = 0.5d0 * (density + effective_spin_dm(ipoint,istate))
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rho_b = 0.5d0 * (density - effective_spin_dm(ipoint,istate))
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call ESRC_MD_LDAERF (mu,rho_a,rho_b,dospin,ec)
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if(isnan(ec))then
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print*,'ec is nan'
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stop
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endif
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ecmd_lda_eff_xi_mu_of_r(istate) += weight * ec
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enddo
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enddo
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call wall_time(wall1)
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print*,'Time for ecmd_lda_eff_xi_mu_of_r :',wall1-wall0
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END_PROVIDER
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121
src/basis_correction/pbe_on_top.irp.f
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121
src/basis_correction/pbe_on_top.irp.f
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BEGIN_PROVIDER [double precision, ecmd_pbe_on_top_mu_of_r, (N_states)]
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BEGIN_DOC
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!
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! Ecmd functional evaluated with mu(r) and depending on
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! +) the on-top pair density
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!
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! +) the total density, density gradients
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!
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! +) the spin density
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!
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! Defined originally in Eq. (25) of JCP, 150, 084103 1-10 (2019) for RS-DFT calculations, but evaluated with mu(r).
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!
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! Such a functional is built by interpolating between two regimes :
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!
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! +) the large mu behaviour in cst/(\mu^3) \int dr on-top(r) where on-top(r) is supposed to be the exact on-top of the system
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!
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! +) mu= 0 with the usal ec_pbe(rho_a,rho_b,grad_rho_a,grad_rho_b)
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!
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! Here the approximation to the exact on-top is done through the assymptotic expansion (in \mu) of the exact on-top pair density (see Eq. 29) but with a mu(r) instead of a constant mu
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!
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! Such an asymptotic expansion was introduced in P. Gori-Giorgi and A. Savin, Phys. Rev. A73, 032506 (2006)
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!
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END_DOC
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implicit none
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double precision :: weight
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double precision :: eps_c_md_on_top_PBE,on_top_extrap,mu_correction_of_on_top
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integer :: ipoint,istate
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double precision :: eps_c_md_PBE,mu,rho_a,rho_b,grad_rho_a(3),grad_rho_b(3),on_top
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ecmd_pbe_on_top_mu_of_r = 0.d0
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do istate = 1, N_states
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do ipoint = 1, n_points_final_grid
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weight = final_weight_at_r_vector(ipoint)
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mu = mu_of_r_prov(ipoint,istate)
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! depends on (rho_a, rho_b) <==> (rho_tot,spin_pol)
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rho_a = one_e_dm_and_grad_alpha_in_r(4,ipoint,istate)
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rho_b = one_e_dm_and_grad_beta_in_r(4,ipoint,istate)
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grad_rho_a(1:3) = one_e_dm_and_grad_alpha_in_r(1:3,ipoint,istate)
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grad_rho_b(1:3) = one_e_dm_and_grad_beta_in_r(1:3,ipoint,istate)
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if(mu_of_r_potential == "cas_ful")then
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! You take the on-top of the CAS wave function which is computed with mu(r)
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on_top = on_top_cas_mu_r(ipoint,istate)
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else
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! You take the on-top of the CAS wave function computed separately
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on_top = total_cas_on_top_density(ipoint,istate)
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endif
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! We take the extrapolated on-top pair density * 2 because of normalization
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on_top_extrap = 2.d0 * mu_correction_of_on_top(mu,on_top)
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call ec_md_pbe_on_top_general(mu,rho_a,rho_b,grad_rho_a,grad_rho_b,on_top_extrap,eps_c_md_on_top_PBE)
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ecmd_pbe_on_top_mu_of_r(istate) += eps_c_md_on_top_PBE * weight
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enddo
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enddo
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END_PROVIDER
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BEGIN_PROVIDER [double precision, ecmd_pbe_on_top_su_mu_of_r, (N_states)]
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BEGIN_DOC
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!
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! Ecmd functional evaluated with mu(r) and depending on
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! +) the on-top pair density
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!
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! +) the total density, density gradients
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!
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! +) !!!!! NO SPIN POLAIRIZATION !!!!!
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!
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! Defined originally in Eq. (25) of JCP, 150, 084103 1-10 (2019) for RS-DFT calculations, but evaluated with mu(r).
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!
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! Such a functional is built by interpolating between two regimes :
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!
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! +) the large mu behaviour in cst/(\mu^3) \int dr on-top(r) where on-top(r) is supposed to be the exact on-top of the system
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!
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! +) mu= 0 with the usal ec_pbe(rho_a,rho_b,grad_rho_a,grad_rho_b)
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!
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! Here the approximation to the exact on-top is done through the assymptotic expansion (in \mu) of the exact on-top pair density (see Eq. 29) but with a mu(r) instead of a constant mu
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!
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! Such an asymptotic expansion was introduced in P. Gori-Giorgi and A. Savin, Phys. Rev. A73, 032506 (2006)
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!
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END_DOC
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implicit none
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double precision :: weight
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double precision :: eps_c_md_on_top_PBE,on_top_extrap,mu_correction_of_on_top
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integer :: ipoint,istate
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double precision :: eps_c_md_PBE,mu,rho_a,rho_b,grad_rho_a(3),grad_rho_b(3),on_top,density
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ecmd_pbe_on_top_su_mu_of_r = 0.d0
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do istate = 1, N_states
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do ipoint = 1, n_points_final_grid
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weight = final_weight_at_r_vector(ipoint)
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mu = mu_of_r_prov(ipoint,istate)
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density = one_e_dm_and_grad_alpha_in_r(4,ipoint,istate) + one_e_dm_and_grad_beta_in_r(4,ipoint,istate)
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! rho_a = rho_b = rho_tot/2 ==> NO SPIN POLARIZATION
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rho_a = 0.5d0 * density
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rho_b = 0.5d0 * density
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grad_rho_a(1:3) = one_e_dm_and_grad_alpha_in_r(1:3,ipoint,istate)
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grad_rho_b(1:3) = one_e_dm_and_grad_beta_in_r(1:3,ipoint,istate)
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if(mu_of_r_potential == "cas_ful")then
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! You take the on-top of the CAS wave function which is computed with mu(r)
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on_top = on_top_cas_mu_r(ipoint,istate)
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else
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! You take the on-top of the CAS wave function computed separately
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on_top = total_cas_on_top_density(ipoint,istate)
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endif
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! We take the extrapolated on-top pair density * 2 because of normalization
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on_top_extrap = 2.d0 * mu_correction_of_on_top(mu,on_top)
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call ec_md_pbe_on_top_general(mu,rho_a,rho_b,grad_rho_a,grad_rho_b,on_top_extrap,eps_c_md_on_top_PBE)
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||||||
|
ecmd_pbe_on_top_su_mu_of_r(istate) += eps_c_md_on_top_PBE * weight
|
||||||
|
enddo
|
||||||
|
enddo
|
||||||
|
END_PROVIDER
|
||||||
|
|
||||||
|
|
||||||
82
src/basis_correction/print_routine.irp.f
Normal file
82
src/basis_correction/print_routine.irp.f
Normal file
@ -0,0 +1,82 @@
|
|||||||
|
subroutine print_basis_correction
|
||||||
|
implicit none
|
||||||
|
integer :: istate
|
||||||
|
provide mu_average_prov
|
||||||
|
if(mu_of_r_potential.EQ."hf")then
|
||||||
|
provide ecmd_lda_mu_of_r ecmd_pbe_ueg_mu_of_r
|
||||||
|
else if(mu_of_r_potential.EQ."cas_ful".or.mu_of_r_potential.EQ."cas_truncated")then
|
||||||
|
provide ecmd_lda_mu_of_r ecmd_pbe_ueg_mu_of_r
|
||||||
|
provide ecmd_pbe_on_top_mu_of_r ecmd_pbe_on_top_su_mu_of_r
|
||||||
|
endif
|
||||||
|
|
||||||
|
print*, ''
|
||||||
|
print*, ''
|
||||||
|
print*, '****************************************'
|
||||||
|
print*, '****************************************'
|
||||||
|
print*, 'Basis set correction for WFT using DFT Ecmd functionals'
|
||||||
|
print*, 'These functionals are accurate for short-range correlation'
|
||||||
|
print*, ''
|
||||||
|
print*, 'For more details look at Journal of Chemical Physics 149, 194301 1-15 (2018) '
|
||||||
|
print*, ' Journal of Physical Chemistry Letters 10, 2931-2937 (2019) '
|
||||||
|
print*, ' ???REF SC?'
|
||||||
|
print*, '****************************************'
|
||||||
|
print*, '****************************************'
|
||||||
|
print*, 'mu_of_r_potential = ',mu_of_r_potential
|
||||||
|
if(mu_of_r_potential.EQ."hf")then
|
||||||
|
print*, ''
|
||||||
|
print*,'Using a HF-like two-body density to define mu(r)'
|
||||||
|
print*,'This assumes that HF is a qualitative representation of the wave function '
|
||||||
|
print*,'********************************************'
|
||||||
|
print*,'Functionals more suited for weak correlation'
|
||||||
|
print*,'********************************************'
|
||||||
|
print*,'+) LDA Ecmd functional : purely based on the UEG (JCP,149,194301,1-15 (2018)) '
|
||||||
|
do istate = 1, N_states
|
||||||
|
write(*, '(A29,X,I3,X,A3,X,F16.10)') ' ECMD LDA , state ',istate,' = ',ecmd_lda_mu_of_r(istate)
|
||||||
|
enddo
|
||||||
|
print*,'+) PBE-UEG Ecmd functional : PBE at mu=0, UEG ontop pair density at large mu (JPCL, 10, 2931-2937 (2019))'
|
||||||
|
do istate = 1, N_states
|
||||||
|
write(*, '(A29,X,I3,X,A3,X,F16.10)') ' ECMD PBE-UEG , state ',istate,' = ',ecmd_pbe_ueg_mu_of_r(istate)
|
||||||
|
enddo
|
||||||
|
|
||||||
|
else if(mu_of_r_potential.EQ."cas_ful")then
|
||||||
|
print*, ''
|
||||||
|
print*,'Using a CAS-like two-body density to define mu(r)'
|
||||||
|
print*,'This assumes that the CAS is a qualitative representation of the wave function '
|
||||||
|
print*,'********************************************'
|
||||||
|
print*,'Functionals more suited for weak correlation'
|
||||||
|
print*,'********************************************'
|
||||||
|
print*,'+) LDA Ecmd functional : purely based on the UEG (JCP,149,194301,1-15 (2018)) '
|
||||||
|
do istate = 1, N_states
|
||||||
|
write(*, '(A29,X,I3,X,A3,X,F16.10)') ' ECMD LDA , state ',istate,' = ',ecmd_lda_mu_of_r(istate)
|
||||||
|
enddo
|
||||||
|
print*,'+) PBE-UEG Ecmd functional : PBE at mu=0, UEG ontop pair density at large mu (JPCL, 10, 2931-2937 (2019))'
|
||||||
|
do istate = 1, N_states
|
||||||
|
write(*, '(A29,X,I3,X,A3,X,F16.10)') ' ECMD PBE-UEG , state ',istate,' = ',ecmd_pbe_ueg_mu_of_r(istate)
|
||||||
|
enddo
|
||||||
|
print*,''
|
||||||
|
print*,'********************************************'
|
||||||
|
print*,'********************************************'
|
||||||
|
print*,'+) PBE-on-top Ecmd functional : (??????? REF-SCF ??????????)'
|
||||||
|
print*,'PBE at mu=0, extrapolated ontop pair density at large mu, usual spin-polarization'
|
||||||
|
do istate = 1, N_states
|
||||||
|
write(*, '(A29,X,I3,X,A3,X,F16.10)') ' ECMD PBE-OT , state ',istate,' = ',ecmd_pbe_on_top_mu_of_r(istate)
|
||||||
|
enddo
|
||||||
|
print*,''
|
||||||
|
print*,'********************************************'
|
||||||
|
print*,'+) PBE-on-top no spin polarization Ecmd functional : (??????? REF-SCF ??????????)'
|
||||||
|
print*,'PBE at mu=0, extrapolated ontop pair density at large mu, and ZERO SPIN POLARIZATION'
|
||||||
|
do istate = 1, N_states
|
||||||
|
write(*, '(A29,X,I3,X,A3,X,F16.10)') ' ECMD SU-PBE-OT , state ',istate,' = ',ecmd_pbe_on_top_su_mu_of_r(istate)
|
||||||
|
enddo
|
||||||
|
print*,''
|
||||||
|
|
||||||
|
endif
|
||||||
|
print*,''
|
||||||
|
print*,'**************'
|
||||||
|
do istate = 1, N_states
|
||||||
|
write(*, '(A29,X,I3,X,A3,X,F16.10)') ' Average mu(r) , state ',istate,' = ',mu_average_prov(istate)
|
||||||
|
enddo
|
||||||
|
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
83
src/basis_correction/weak_corr_func.irp.f
Normal file
83
src/basis_correction/weak_corr_func.irp.f
Normal file
@ -0,0 +1,83 @@
|
|||||||
|
|
||||||
|
BEGIN_PROVIDER [double precision, ecmd_lda_mu_of_r, (N_states)]
|
||||||
|
BEGIN_DOC
|
||||||
|
! ecmd_lda_mu_of_r = multi-determinantal Ecmd within the LDA approximation with mu(r) ,
|
||||||
|
!
|
||||||
|
! see equation 40 in J. Chem. Phys. 149, 194301 (2018); https://doi.org/10.1063/1.5052714
|
||||||
|
END_DOC
|
||||||
|
implicit none
|
||||||
|
integer :: ipoint,istate
|
||||||
|
double precision :: rho_a, rho_b, ec
|
||||||
|
logical :: dospin
|
||||||
|
double precision :: wall0,wall1,weight,mu
|
||||||
|
dospin = .true. ! JT dospin have to be set to true for open shell
|
||||||
|
print*,'Providing ecmd_lda_mu_of_r ...'
|
||||||
|
|
||||||
|
ecmd_lda_mu_of_r = 0.d0
|
||||||
|
call wall_time(wall0)
|
||||||
|
do istate = 1, N_states
|
||||||
|
do ipoint = 1, n_points_final_grid
|
||||||
|
! mu(r) defined by Eq. (37) of J. Chem. Phys. 149, 194301 (2018)
|
||||||
|
mu = mu_of_r_prov(ipoint,istate)
|
||||||
|
weight = final_weight_at_r_vector(ipoint)
|
||||||
|
rho_a = one_e_dm_and_grad_alpha_in_r(4,ipoint,istate)
|
||||||
|
rho_b = one_e_dm_and_grad_beta_in_r(4,ipoint,istate)
|
||||||
|
! Ecmd within the LDA approximation of PRB 73, 155111 (2006)
|
||||||
|
call ESRC_MD_LDAERF (mu,rho_a,rho_b,dospin,ec)
|
||||||
|
if(isnan(ec))then
|
||||||
|
print*,'ec is nan'
|
||||||
|
stop
|
||||||
|
endif
|
||||||
|
ecmd_lda_mu_of_r(istate) += weight * ec
|
||||||
|
enddo
|
||||||
|
enddo
|
||||||
|
call wall_time(wall1)
|
||||||
|
print*,'Time for ecmd_lda_mu_of_r :',wall1-wall0
|
||||||
|
END_PROVIDER
|
||||||
|
|
||||||
|
|
||||||
|
BEGIN_PROVIDER [double precision, ecmd_pbe_ueg_mu_of_r, (N_states)]
|
||||||
|
BEGIN_DOC
|
||||||
|
! ecmd_pbe_ueg_mu_of_r = multi-determinantal Ecmd within the PBE-UEG approximation with mu(r) ,
|
||||||
|
!
|
||||||
|
! see Eqs. 13-14b in Phys.Chem.Lett.2019, 10, 2931 2937; https://pubs.acs.org/doi/10.1021/acs.jpclett.9b01176
|
||||||
|
!
|
||||||
|
! Based on the PBE-on-top functional (see Eqs. 26, 27 of J. Chem. Phys.150, 084103 (2019); doi: 10.1063/1.5082638)
|
||||||
|
!
|
||||||
|
! but it the on-top pair density of the UEG as an approximation of the exact on-top pair density
|
||||||
|
END_DOC
|
||||||
|
implicit none
|
||||||
|
double precision :: weight
|
||||||
|
integer :: ipoint,istate
|
||||||
|
double precision :: eps_c_md_PBE,mu,rho_a,rho_b,grad_rho_a(3),grad_rho_b(3),on_top
|
||||||
|
double precision :: g0_UEG_mu_inf
|
||||||
|
|
||||||
|
ecmd_pbe_ueg_mu_of_r = 0.d0
|
||||||
|
|
||||||
|
print*,'Providing ecmd_pbe_ueg_mu_of_r ...'
|
||||||
|
call wall_time(wall0)
|
||||||
|
do istate = 1, N_states
|
||||||
|
do ipoint = 1, n_points_final_grid
|
||||||
|
weight=final_weight_at_r_vector(ipoint)
|
||||||
|
|
||||||
|
! mu(r) defined by Eq. (37) of J. Chem. Phys. 149, 194301 (2018)
|
||||||
|
mu = mu_of_r_prov(ipoint,istate)
|
||||||
|
|
||||||
|
rho_a = one_e_dm_and_grad_alpha_in_r(4,ipoint,istate)
|
||||||
|
rho_b = one_e_dm_and_grad_beta_in_r(4,ipoint,istate)
|
||||||
|
grad_rho_a(1:3) = one_e_dm_and_grad_alpha_in_r(1:3,ipoint,istate)
|
||||||
|
grad_rho_b(1:3) = one_e_dm_and_grad_beta_in_r(1:3,ipoint,istate)
|
||||||
|
|
||||||
|
! We take the on-top pair density of the UEG which is (1-zeta^2) rhoc^2 g0 = 4 rhoa * rhob * g0
|
||||||
|
on_top = 4.d0 * rho_a * rho_b * g0_UEG_mu_inf(rho_a,rho_b)
|
||||||
|
|
||||||
|
! The form of interpolated (mu=0 ---> mu=infinity) functional originally introduced in JCP, 150, 084103 1-10 (2019)
|
||||||
|
call ec_md_pbe_on_top_general(mu,rho_a,rho_b,grad_rho_a,grad_rho_b,on_top,eps_c_md_PBE)
|
||||||
|
ecmd_pbe_ueg_mu_of_r(istate) += eps_c_md_PBE * weight
|
||||||
|
enddo
|
||||||
|
enddo
|
||||||
|
double precision :: wall1, wall0
|
||||||
|
call wall_time(wall1)
|
||||||
|
print*,'Time for the ecmd_pbe_ueg_mu_of_r:',wall1-wall0
|
||||||
|
|
||||||
|
END_PROVIDER
|
||||||
1
src/ecmd_utils/NEED
Normal file
1
src/ecmd_utils/NEED
Normal file
@ -0,0 +1 @@
|
|||||||
|
|
||||||
4
src/ecmd_utils/README.rst
Normal file
4
src/ecmd_utils/README.rst
Normal file
@ -0,0 +1,4 @@
|
|||||||
|
==========
|
||||||
|
ecmd_utils
|
||||||
|
==========
|
||||||
|
|
||||||
152
src/ecmd_utils/ecmd_lda.irp.f
Normal file
152
src/ecmd_utils/ecmd_lda.irp.f
Normal file
@ -0,0 +1,152 @@
|
|||||||
|
!****************************************************************************
|
||||||
|
subroutine ESRC_MD_LDAERF (mu,rho_a,rho_b,dospin,e)
|
||||||
|
!*****************************************************************************
|
||||||
|
! Short-range spin-dependent LDA correlation functional with multideterminant reference
|
||||||
|
! for OEP calculations from Section V of
|
||||||
|
! Paziani, Moroni, Gori-Giorgi and Bachelet, PRB 73, 155111 (2006)
|
||||||
|
!
|
||||||
|
! Input: rhot : total density
|
||||||
|
! rhos : spin density
|
||||||
|
! mu : Interation parameter
|
||||||
|
! dospin : use spin density
|
||||||
|
!
|
||||||
|
! Ouput: e : energy
|
||||||
|
!
|
||||||
|
! Created: 26-08-11, J. Toulouse
|
||||||
|
!*****************************************************************************
|
||||||
|
implicit none
|
||||||
|
|
||||||
|
double precision, intent(in) :: rho_a,rho_b,mu
|
||||||
|
logical, intent(in) :: dospin
|
||||||
|
double precision, intent(out):: e
|
||||||
|
|
||||||
|
double precision :: e1
|
||||||
|
double precision :: rhoa,rhob
|
||||||
|
double precision :: rhot, rhos
|
||||||
|
rhoa=max(rho_a,1.0d-15)
|
||||||
|
rhob=max(rho_b,1.0d-15)
|
||||||
|
rhot = rhoa + rhob
|
||||||
|
rhos = rhoa - rhob
|
||||||
|
|
||||||
|
call ec_only_lda_sr(mu,rho_a,rho_b,e1)
|
||||||
|
if(isnan(e1))then
|
||||||
|
print*,'e1 is NaN'
|
||||||
|
print*,mu,rho_a,rho_b
|
||||||
|
stop
|
||||||
|
endif
|
||||||
|
call DELTA_LRSR_LDAERF (rhot,rhos,mu,dospin,e)
|
||||||
|
if(isnan(e))then
|
||||||
|
print*,'e is NaN'
|
||||||
|
print*,mu,rhot,rhos
|
||||||
|
stop
|
||||||
|
endif
|
||||||
|
e = e1 + e
|
||||||
|
|
||||||
|
end
|
||||||
|
|
||||||
|
!****************************************************************************
|
||||||
|
subroutine DELTA_LRSR_LDAERF (rhot,rhos,mu,dospin,e)
|
||||||
|
!*****************************************************************************
|
||||||
|
! LDA approximation to term Delta_(LR-SR) from Eq. 42 of
|
||||||
|
! Paziani, Moroni, Gori-Giorgi and Bachelet, PRB 73, 155111 (2006)
|
||||||
|
!
|
||||||
|
! Input: rhot : total density
|
||||||
|
! rhos : spin density
|
||||||
|
! mu : Interation parameter
|
||||||
|
! dospin : use spin density
|
||||||
|
!
|
||||||
|
! Ouput: e : energy
|
||||||
|
!
|
||||||
|
! Warning: not tested for z != 0
|
||||||
|
!
|
||||||
|
! Created: 26-08-11, J. Toulouse
|
||||||
|
!*****************************************************************************
|
||||||
|
implicit none
|
||||||
|
|
||||||
|
double precision rhot, rhos, mu
|
||||||
|
logical dospin
|
||||||
|
double precision e
|
||||||
|
|
||||||
|
double precision f13, f83, pi, rsfac, alpha2
|
||||||
|
double precision rs, rs2, rs3
|
||||||
|
|
||||||
|
double precision rhoa, rhob, z, z2, onepz, onemz, zp, zm, phi8
|
||||||
|
double precision g0f, g0s
|
||||||
|
double precision bd2, bd3
|
||||||
|
double precision c45, c4, c5
|
||||||
|
double precision bc2, bc4, bc3t, bc5t, d0
|
||||||
|
double precision delta2,delta3,delta4,delta5,delta6
|
||||||
|
double precision delta
|
||||||
|
|
||||||
|
parameter(f13 = 0.333333333333333d0)
|
||||||
|
parameter(f83 = 2.6666666666666665d0)
|
||||||
|
parameter(pi = 3.141592653589793d0)
|
||||||
|
parameter(rsfac = 0.620350490899400d0)
|
||||||
|
parameter(alpha2 = 0.2715053589826032d0)
|
||||||
|
|
||||||
|
rs = rsfac/(rhot**f13)
|
||||||
|
rs2 = rs*rs
|
||||||
|
rs3 = rs2*rs
|
||||||
|
|
||||||
|
! Spin-unpolarized case
|
||||||
|
if (.not.dospin) then
|
||||||
|
z = 0.d0
|
||||||
|
|
||||||
|
! Spin-polarized case
|
||||||
|
else
|
||||||
|
rhoa=max((rhot+rhos)*.5d0,1.0d-15)
|
||||||
|
rhob=max((rhot-rhos)*.5d0,1.0d-15)
|
||||||
|
z=min((rhoa-rhob)/(rhoa+rhob),0.9999999999d0)
|
||||||
|
endif
|
||||||
|
|
||||||
|
z2=z*z
|
||||||
|
|
||||||
|
bd2=dexp(-0.547d0*rs)*(-0.388d0*rs+0.676*rs2)/rs2
|
||||||
|
bd3=dexp(-0.31d0*rs)*(-4.95d0*rs+rs2)/rs3
|
||||||
|
|
||||||
|
onepz=1.d0+z
|
||||||
|
onemz=1.d0-z
|
||||||
|
phi8=0.5d0*(onepz**f83+onemz**f83)
|
||||||
|
|
||||||
|
zp=onepz/2.d0
|
||||||
|
zm=onemz/2.d0
|
||||||
|
c45=(zp**2)*g0s(rs*zp**(-f13))+(zm**2)*g0s(rs*zm**(-f13))
|
||||||
|
c4=c45+(1.d0-z2)*bd2-phi8/(5.d0*alpha2*rs2)
|
||||||
|
c5=c45+(1.d0-z2)*bd3
|
||||||
|
|
||||||
|
bc2=-3.d0*(1-z2)*(g0f(rs)-0.5d0)/(8.d0*rs3)
|
||||||
|
bc4=-9.d0*c4/(64.d0*rs3)
|
||||||
|
bc3t=-(1-z2)*g0f(rs)*(2.d0*dsqrt(2.d0)-1)/(2.d0*dsqrt(pi)*rs3)
|
||||||
|
bc5t = -3.d0*c5*(3.d0-dsqrt(2.d0))/(20.d0*dsqrt(2.d0*pi)*rs3)
|
||||||
|
|
||||||
|
d0=(0.70605d0+0.12927d0*z2)*rs
|
||||||
|
delta2=0.073867d0*(rs**(1.5d0))
|
||||||
|
delta3=4*(d0**6)*bc3t+(d0**8)*bc5t;
|
||||||
|
delta4=4*(d0**6)*bc2+(d0**8)*bc4;
|
||||||
|
delta5=(d0**8)*bc3t;
|
||||||
|
delta6=(d0**8)*bc2;
|
||||||
|
delta=(delta2*(mu**2)+delta3*(mu**3)+delta4*(mu**4)+delta5*(mu**5)+delta6*(mu**6))/((1+(d0**2)*(mu**2))**4)
|
||||||
|
|
||||||
|
|
||||||
|
! multiply by rhot to get energy density
|
||||||
|
e=delta*rhot
|
||||||
|
|
||||||
|
end
|
||||||
|
|
||||||
|
!*****************************************************************************
|
||||||
|
double precision function g0s(rs)
|
||||||
|
!*****************************************************************************
|
||||||
|
! g"(0,rs,z=1) from Eq. 32 of
|
||||||
|
! Paziani, Moroni, Gori-Giorgi and Bachelet, PRB 73, 155111 (2006)
|
||||||
|
!
|
||||||
|
! Created: 26-08-11, J. Toulouse
|
||||||
|
!*****************************************************************************
|
||||||
|
implicit none
|
||||||
|
double precision rs
|
||||||
|
double precision rs2, f53, alpha2
|
||||||
|
parameter(f53 = 1.6666666666666667d0)
|
||||||
|
parameter(alpha2 = 0.2715053589826032d0)
|
||||||
|
rs2=rs*rs
|
||||||
|
g0s=(2.d0**f53)*(1.d0-0.02267d0*rs)/((5.d0*alpha2*rs2)*(1.d0+0.4319d0*rs+0.04d0*rs2))
|
||||||
|
end
|
||||||
|
|
||||||
53
src/ecmd_utils/ecmd_pbe_general.irp.f
Normal file
53
src/ecmd_utils/ecmd_pbe_general.irp.f
Normal file
@ -0,0 +1,53 @@
|
|||||||
|
|
||||||
|
subroutine ec_md_pbe_on_top_general(mu,rho_a,rho_b,grad_rho_a,grad_rho_b,on_top,eps_c_md_on_top_PBE)
|
||||||
|
implicit none
|
||||||
|
BEGIN_DOC
|
||||||
|
!
|
||||||
|
! General e_cmd functional interpolating between :
|
||||||
|
!
|
||||||
|
! +) the large mu behaviour in cst/(\mu^3) on-top
|
||||||
|
!
|
||||||
|
! +) mu= 0 with the usal ec_pbe at
|
||||||
|
!
|
||||||
|
! Depends on : mu, the density (rho_a,rho_b), the square of the density gradient (grad_rho_a,grad_rho_b)
|
||||||
|
!
|
||||||
|
! the flavour of on-top densiyt (on_top) you fill in: in principle it should be the exact on-top
|
||||||
|
!
|
||||||
|
! The form of the functional was originally introduced in JCP, 150, 084103 1-10 (2019)
|
||||||
|
!
|
||||||
|
END_DOC
|
||||||
|
double precision, intent(in) :: mu,rho_a,rho_b,grad_rho_a(3),grad_rho_b(3),on_top
|
||||||
|
double precision, intent(out) :: eps_c_md_on_top_PBE
|
||||||
|
double precision :: pi, e_pbe,beta,denom
|
||||||
|
double precision :: grad_rho_a_2,grad_rho_b_2,grad_rho_a_b
|
||||||
|
double precision :: rhoc,rhoo,sigmacc,sigmaco,sigmaoo,vrhoc,vrhoo,vsigmacc,vsigmaco,vsigmaoo
|
||||||
|
integer :: m
|
||||||
|
|
||||||
|
pi = 4.d0 * datan(1.d0)
|
||||||
|
|
||||||
|
eps_c_md_on_top_PBE = 0.d0
|
||||||
|
grad_rho_a_2 = 0.d0
|
||||||
|
grad_rho_b_2 = 0.d0
|
||||||
|
grad_rho_a_b = 0.d0
|
||||||
|
do m = 1, 3
|
||||||
|
grad_rho_a_2 += grad_rho_a(m)*grad_rho_a(m)
|
||||||
|
grad_rho_b_2 += grad_rho_b(m)*grad_rho_b(m)
|
||||||
|
grad_rho_a_b += grad_rho_a(m)*grad_rho_b(m)
|
||||||
|
enddo
|
||||||
|
! convertion from (alpha,beta) formalism to (closed, open) formalism
|
||||||
|
call rho_ab_to_rho_oc(rho_a,rho_b,rhoo,rhoc)
|
||||||
|
call grad_rho_ab_to_grad_rho_oc(grad_rho_a_2,grad_rho_b_2,grad_rho_a_b,sigmaoo,sigmacc,sigmaco)
|
||||||
|
|
||||||
|
! usual PBE correlation energy using the density, spin polarization and density gradients for alpha/beta electrons
|
||||||
|
call ec_pbe_only(0.d0,rhoc,rhoo,sigmacc,sigmaco,sigmaoo,e_PBE)
|
||||||
|
denom = (-2.d0+sqrt(2d0))*sqrt(2.d0*pi)* on_top
|
||||||
|
if (dabs(denom) > 1.d-12) then
|
||||||
|
! quantity of Eq. (26)
|
||||||
|
beta = (3.d0*e_PBE)/denom
|
||||||
|
eps_c_md_on_top_PBE = e_PBE/(1.d0+beta*mu**3)
|
||||||
|
else
|
||||||
|
eps_c_md_on_top_PBE =0.d0
|
||||||
|
endif
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
72
src/ecmd_utils/ecmd_pbe_on_top.irp.f
Normal file
72
src/ecmd_utils/ecmd_pbe_on_top.irp.f
Normal file
@ -0,0 +1,72 @@
|
|||||||
|
|
||||||
|
|
||||||
|
subroutine ec_md_on_top_PBE_mu_corrected(mu,r,two_dm,eps_c_md_on_top_PBE)
|
||||||
|
implicit none
|
||||||
|
BEGIN_DOC
|
||||||
|
! enter with "r(3)", and "two_dm(N_states)" which is the on-top pair density at "r" for each states
|
||||||
|
!
|
||||||
|
! you get out with the energy density defined in J. Chem. Phys.150, 084103 (2019); doi: 10.1063/1.508263
|
||||||
|
!
|
||||||
|
! by Eq. (26), which includes the correction of the on-top pair density of Eq. (29).
|
||||||
|
END_DOC
|
||||||
|
double precision, intent(in) :: mu , r(3), two_dm
|
||||||
|
double precision, intent(out) :: eps_c_md_on_top_PBE(N_states)
|
||||||
|
double precision :: two_dm_in_r, pi, e_pbe(N_states),beta(N_states),mu_correction_of_on_top
|
||||||
|
double precision :: aos_array(ao_num), grad_aos_array(3,ao_num)
|
||||||
|
double precision :: rho_a(N_states),rho_b(N_states)
|
||||||
|
double precision :: grad_rho_a(3,N_states),grad_rho_b(3,N_states)
|
||||||
|
double precision :: grad_rho_a_2(N_states),grad_rho_b_2(N_states),grad_rho_a_b(N_states)
|
||||||
|
double precision :: rhoc,rhoo,sigmacc,sigmaco,sigmaoo,vrhoc,vrhoo,vsigmacc,vsigmaco,vsigmaoo,on_top_corrected
|
||||||
|
integer :: m, istate
|
||||||
|
|
||||||
|
pi = 4.d0 * datan(1.d0)
|
||||||
|
|
||||||
|
eps_c_md_on_top_PBE = 0.d0
|
||||||
|
call density_and_grad_alpha_beta_and_all_aos_and_grad_aos_at_r(r,rho_a,rho_b, grad_rho_a, grad_rho_b, aos_array, grad_aos_array)
|
||||||
|
grad_rho_a_2 = 0.d0
|
||||||
|
grad_rho_b_2 = 0.d0
|
||||||
|
grad_rho_a_b = 0.d0
|
||||||
|
do istate = 1, N_states
|
||||||
|
do m = 1, 3
|
||||||
|
grad_rho_a_2(istate) += grad_rho_a(m,istate)*grad_rho_a(m,istate)
|
||||||
|
grad_rho_b_2(istate) += grad_rho_b(m,istate)*grad_rho_b(m,istate)
|
||||||
|
grad_rho_a_b(istate) += grad_rho_a(m,istate)*grad_rho_b(m,istate)
|
||||||
|
enddo
|
||||||
|
enddo
|
||||||
|
do istate = 1, N_states
|
||||||
|
! convertion from (alpha,beta) formalism to (closed, open) formalism
|
||||||
|
call rho_ab_to_rho_oc(rho_a(istate),rho_b(istate),rhoo,rhoc)
|
||||||
|
call grad_rho_ab_to_grad_rho_oc(grad_rho_a_2(istate),grad_rho_b_2(istate),grad_rho_a_b(istate),sigmaoo,sigmacc,sigmaco)
|
||||||
|
|
||||||
|
! usual PBE correlation energy using the density, spin polarization and density gradients for alpha/beta electrons
|
||||||
|
call ec_pbe_only(0.d0,rhoc,rhoo,sigmacc,sigmaco,sigmaoo,e_PBE(istate))
|
||||||
|
|
||||||
|
! correction of the on-top pair density according to Eq. (29)
|
||||||
|
on_top_corrected = mu_correction_of_on_top(mu,two_dm)
|
||||||
|
|
||||||
|
! quantity of Eq. (27) with a factor two according to the difference of normalization
|
||||||
|
! between the on-top of the JCP paper and that of QP2
|
||||||
|
beta(istate) = (3.d0*e_PBE(istate))/( (-2.d0+sqrt(2d0))*sqrt(2.d0*pi)*2.d0* on_top_corrected)
|
||||||
|
|
||||||
|
! quantity of Eq. (26)
|
||||||
|
eps_c_md_on_top_PBE(istate)=e_PBE(istate)/(1.d0+beta(istate)*mu**3)
|
||||||
|
enddo
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
|
double precision function mu_correction_of_on_top(mu,on_top)
|
||||||
|
implicit none
|
||||||
|
BEGIN_DOC
|
||||||
|
! mu-based correction to the on-top pair density provided by the assymptotic expansion of
|
||||||
|
!
|
||||||
|
! P. Gori-Giorgi and A. Savin, Phys. Rev. A73, 032506 (2006)
|
||||||
|
!
|
||||||
|
! This is used in J. Chem. Phys.150, 084103 (2019); Eq. (29).
|
||||||
|
END_DOC
|
||||||
|
double precision, intent(in) :: mu,on_top
|
||||||
|
double precision :: pi
|
||||||
|
pi = 4.d0 * datan(1.d0)
|
||||||
|
mu_correction_of_on_top = on_top / ( 1.d0 + 2.d0/(dsqrt(pi)*mu) )
|
||||||
|
mu_correction_of_on_top = max(mu_correction_of_on_top ,1.d-15)
|
||||||
|
end
|
||||||
|
|
||||||
194
src/ecmd_utils/ecmd_pbe_ueg.irp.f
Normal file
194
src/ecmd_utils/ecmd_pbe_ueg.irp.f
Normal file
@ -0,0 +1,194 @@
|
|||||||
|
|
||||||
|
subroutine ecmd_pbe_ueg_at_r(mu,r,eps_c_md_PBE)
|
||||||
|
implicit none
|
||||||
|
BEGIN_DOC
|
||||||
|
! provides the integrand of Eq. (13) of Phys.Chem.Lett.2019, 10, 2931 2937
|
||||||
|
!
|
||||||
|
! !!! WARNING !!! This is the total integrand of Eq. (13), which is e_cmd * n
|
||||||
|
!
|
||||||
|
! such a function is based on the exact behaviour of the Ecmd at large mu
|
||||||
|
!
|
||||||
|
! but with the exact on-top estimated with that of the UEG
|
||||||
|
!
|
||||||
|
! You enter with r(3), you get out with eps_c_md_PBE(1:N_states)
|
||||||
|
END_DOC
|
||||||
|
double precision, intent(in) :: mu , r(3)
|
||||||
|
double precision, intent(out) :: eps_c_md_PBE(N_states)
|
||||||
|
double precision :: pi, e_PBE, beta
|
||||||
|
double precision :: aos_array(ao_num), grad_aos_array(3,ao_num)
|
||||||
|
double precision :: rho_a(N_states),rho_b(N_states)
|
||||||
|
double precision :: grad_rho_a(3,N_states),grad_rho_b(3,N_states)
|
||||||
|
double precision :: grad_rho_a_2(N_states),grad_rho_b_2(N_states),grad_rho_a_b(N_states)
|
||||||
|
double precision :: rhoc,rhoo,sigmacc,sigmaco,sigmaoo,vrhoc,vrhoo,vsigmacc,vsigmaco,vsigmaoo
|
||||||
|
double precision :: g0_UEG_mu_inf, denom
|
||||||
|
integer :: m, istate
|
||||||
|
|
||||||
|
pi = 4.d0 * datan(1.d0)
|
||||||
|
|
||||||
|
eps_c_md_PBE = 0.d0
|
||||||
|
call density_and_grad_alpha_beta_and_all_aos_and_grad_aos_at_r(r,rho_a,rho_b, grad_rho_a, grad_rho_b, aos_array, grad_aos_array)
|
||||||
|
grad_rho_a_2 = 0.d0
|
||||||
|
grad_rho_b_2 = 0.d0
|
||||||
|
grad_rho_a_b = 0.d0
|
||||||
|
do istate = 1, N_states
|
||||||
|
do m = 1, 3
|
||||||
|
grad_rho_a_2(istate) += grad_rho_a(m,istate)*grad_rho_a(m,istate)
|
||||||
|
grad_rho_b_2(istate) += grad_rho_b(m,istate)*grad_rho_b(m,istate)
|
||||||
|
grad_rho_a_b(istate) += grad_rho_a(m,istate)*grad_rho_b(m,istate)
|
||||||
|
enddo
|
||||||
|
enddo
|
||||||
|
do istate = 1, N_states
|
||||||
|
! convertion from (alpha,beta) formalism to (closed, open) formalism
|
||||||
|
call rho_ab_to_rho_oc(rho_a(istate),rho_b(istate),rhoo,rhoc)
|
||||||
|
call grad_rho_ab_to_grad_rho_oc(grad_rho_a_2(istate),grad_rho_b_2(istate),grad_rho_a_b(istate),sigmaoo,sigmacc,sigmaco)
|
||||||
|
call ec_pbe_only(0.d0,rhoc,rhoo,sigmacc,sigmaco,sigmaoo,e_PBE)
|
||||||
|
|
||||||
|
if(mu == 0.d0) then
|
||||||
|
eps_c_md_PBE(istate)=e_PBE
|
||||||
|
else
|
||||||
|
! note: the on-top pair density is (1-zeta^2) rhoc^2 g0 = 4 rhoa * rhob * g0
|
||||||
|
denom = (-2.d0+sqrt(2d0))*sqrt(2.d0*pi) * 4.d0*rho_a(istate)*rho_b(istate)*g0_UEG_mu_inf(rho_a(istate),rho_b(istate))
|
||||||
|
if (dabs(denom) > 1.d-12) then
|
||||||
|
beta = (3.d0*e_PBE)/denom
|
||||||
|
eps_c_md_PBE(istate)=e_PBE/(1.d0+beta*mu**3)
|
||||||
|
else
|
||||||
|
eps_c_md_PBE(istate)=0.d0
|
||||||
|
endif
|
||||||
|
endif
|
||||||
|
enddo
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
subroutine eps_c_md_PBE_from_density(mu,rho_a,rho_b, grad_rho_a, grad_rho_b,eps_c_md_PBE) ! EG
|
||||||
|
implicit none
|
||||||
|
BEGIN_DOC
|
||||||
|
! provides the integrand of Eq. (13) of Phys.Chem.Lett.2019, 10, 2931 2937
|
||||||
|
!
|
||||||
|
! !!! WARNING !!! This is the total integrand of Eq. (13), which is e_cmd * n
|
||||||
|
!
|
||||||
|
! such a function is based on the exact behaviour of the Ecmd at large mu
|
||||||
|
!
|
||||||
|
! but with the exact on-top estimated with that of the UEG
|
||||||
|
!
|
||||||
|
! You enter with the alpha/beta density and density gradients
|
||||||
|
!
|
||||||
|
! You get out with eps_c_md_PBE(1:N_states)
|
||||||
|
END_DOC
|
||||||
|
double precision, intent(in) :: mu(N_states) , rho_a(N_states),rho_b(N_states), grad_rho_a(3,N_states),grad_rho_b(3,N_states)
|
||||||
|
double precision, intent(out) :: eps_c_md_PBE(N_states)
|
||||||
|
double precision :: pi, e_PBE, beta
|
||||||
|
double precision :: aos_array(ao_num), grad_aos_array(3,ao_num)
|
||||||
|
double precision :: grad_rho_a_2(N_states),grad_rho_b_2(N_states),grad_rho_a_b(N_states)
|
||||||
|
double precision :: rhoc,rhoo,sigmacc,sigmaco,sigmaoo,vrhoc,vrhoo,vsigmacc,vsigmaco,vsigmaoo
|
||||||
|
double precision :: g0_UEG_mu_inf, denom
|
||||||
|
integer :: m, istate
|
||||||
|
|
||||||
|
pi = 4.d0 * datan(1.d0)
|
||||||
|
|
||||||
|
eps_c_md_PBE = 0.d0
|
||||||
|
grad_rho_a_2 = 0.d0
|
||||||
|
grad_rho_b_2 = 0.d0
|
||||||
|
grad_rho_a_b = 0.d0
|
||||||
|
do istate = 1, N_states
|
||||||
|
do m = 1, 3
|
||||||
|
grad_rho_a_2(istate) += grad_rho_a(m,istate)*grad_rho_a(m,istate)
|
||||||
|
grad_rho_b_2(istate) += grad_rho_b(m,istate)*grad_rho_b(m,istate)
|
||||||
|
grad_rho_a_b(istate) += grad_rho_a(m,istate)*grad_rho_b(m,istate)
|
||||||
|
enddo
|
||||||
|
enddo
|
||||||
|
do istate = 1, N_states
|
||||||
|
! convertion from (alpha,beta) formalism to (closed, open) formalism
|
||||||
|
call rho_ab_to_rho_oc(rho_a(istate),rho_b(istate),rhoo,rhoc)
|
||||||
|
call grad_rho_ab_to_grad_rho_oc(grad_rho_a_2(istate),grad_rho_b_2(istate),grad_rho_a_b(istate),sigmaoo,sigmacc,sigmaco)
|
||||||
|
call ec_pbe_only(0.d0,rhoc,rhoo,sigmacc,sigmaco,sigmaoo,e_PBE)
|
||||||
|
|
||||||
|
if(mu(istate) == 0.d0) then
|
||||||
|
eps_c_md_PBE(istate)=e_PBE
|
||||||
|
else
|
||||||
|
! note: the on-top pair density is (1-zeta^2) rhoc^2 g0 = 4 rhoa * rhob * g0
|
||||||
|
denom = (-2.d0+sqrt(2d0))*sqrt(2.d0*pi) * 4.d0*rho_a(istate)*rho_b(istate)*g0_UEG_mu_inf(rho_a(istate),rho_b(istate))
|
||||||
|
if (dabs(denom) > 1.d-12) then
|
||||||
|
beta = (3.d0*e_PBE)/denom
|
||||||
|
eps_c_md_PBE(istate)=e_PBE/(1.d0+beta*mu(istate)**3)
|
||||||
|
else
|
||||||
|
eps_c_md_PBE(istate)=0.d0
|
||||||
|
endif
|
||||||
|
endif
|
||||||
|
enddo
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
subroutine eps_c_md_PBE_at_grid_pt(mu,i_point,eps_c_md_PBE)
|
||||||
|
implicit none
|
||||||
|
BEGIN_DOC
|
||||||
|
! provides the integrand of Eq. (13) of Phys.Chem.Lett.2019, 10, 2931 2937
|
||||||
|
!
|
||||||
|
! !!! WARNING !!! This is the total integrand of Eq. (13), which is e_cmd * n
|
||||||
|
!
|
||||||
|
! such a function is based on the exact behaviour of the Ecmd at large mu
|
||||||
|
!
|
||||||
|
! but with the exact on-top estimated with that of the UEG
|
||||||
|
!
|
||||||
|
! You enter with the alpha/beta density and density gradients
|
||||||
|
!
|
||||||
|
! You get out with eps_c_md_PBE(1:N_states)
|
||||||
|
END_DOC
|
||||||
|
double precision, intent(in) :: mu
|
||||||
|
double precision, intent(out) :: eps_c_md_PBE(N_states)
|
||||||
|
integer, intent(in) :: i_point
|
||||||
|
double precision :: two_dm, pi, e_pbe,beta,mu_correction_of_on_top
|
||||||
|
double precision :: grad_rho_a(3),grad_rho_b(3)
|
||||||
|
double precision :: grad_rho_a_2,grad_rho_b_2,grad_rho_a_b
|
||||||
|
double precision :: rhoc,rhoo,ec_pbe_88
|
||||||
|
double precision :: delta,two_dm_corr,rho_a,rho_b
|
||||||
|
double precision :: grad_rho_2,denom,g0_UEG_mu_inf
|
||||||
|
double precision :: sigmacc,sigmaco,sigmaoo
|
||||||
|
integer :: m, istate
|
||||||
|
|
||||||
|
pi = 4.d0 * datan(1.d0)
|
||||||
|
|
||||||
|
eps_c_md_PBE = 0.d0
|
||||||
|
do istate = 1, N_states
|
||||||
|
! total and spin density
|
||||||
|
rhoc = one_e_dm_and_grad_alpha_in_r(4,i_point,istate) + one_e_dm_and_grad_beta_in_r(4,i_point,istate)
|
||||||
|
rhoo = one_e_dm_and_grad_alpha_in_r(4,i_point,istate) - one_e_dm_and_grad_beta_in_r(4,i_point,istate)
|
||||||
|
! gradients of the effective spin density
|
||||||
|
grad_rho_a_2 = 0.D0
|
||||||
|
grad_rho_b_2 = 0.D0
|
||||||
|
grad_rho_a_b = 0.D0
|
||||||
|
do m = 1, 3
|
||||||
|
grad_rho_a_2 += one_e_dm_and_grad_alpha_in_r(m,i_point,istate)**2.d0
|
||||||
|
grad_rho_b_2 += one_e_dm_and_grad_beta_in_r(m,i_point,istate) **2.d0
|
||||||
|
grad_rho_a_b += one_e_dm_and_grad_alpha_in_r(m,i_point,istate) * one_e_dm_and_grad_beta_in_r(m,i_point,istate)
|
||||||
|
enddo
|
||||||
|
sigmacc = grad_rho_a_2 + grad_rho_b_2 + 2.d0 * grad_rho_a_b
|
||||||
|
sigmaco = 0.d0
|
||||||
|
sigmaoo = 0.d0
|
||||||
|
rho_a = one_e_dm_and_grad_alpha_in_r(4,i_point,istate)
|
||||||
|
rho_b = one_e_dm_and_grad_beta_in_r(4,i_point,istate)
|
||||||
|
|
||||||
|
call ec_pbe_only(0.d0,rhoc,rhoo,sigmacc,sigmaco,sigmaoo,e_PBE)
|
||||||
|
if(e_PBE.gt.0.d0)then
|
||||||
|
print*,'PBE gt 0 with regular dens'
|
||||||
|
endif
|
||||||
|
if(mu == 0.d0) then
|
||||||
|
eps_c_md_PBE(istate)=e_PBE
|
||||||
|
else
|
||||||
|
! note: the on-top pair density is (1-zeta^2) rhoc^2 g0 = 4 rhoa * rhob * g0
|
||||||
|
denom = (-2.d0+dsqrt(2d0))*sqrt(2.d0*pi) * 4.d0*rho_a*rho_b*g0_UEG_mu_inf(rho_a,rho_b)
|
||||||
|
if (dabs(denom) > 1.d-12) then
|
||||||
|
beta = (3.d0*e_PBE)/denom
|
||||||
|
! Ecmd functional with the UEG ontop pair density when mu -> infty
|
||||||
|
! and the usual PBE correlation energy when mu = 0
|
||||||
|
eps_c_md_PBE(istate)=e_PBE/(1.d0+beta*mu**3)
|
||||||
|
else
|
||||||
|
eps_c_md_PBE(istate)=0.d0
|
||||||
|
endif
|
||||||
|
endif
|
||||||
|
enddo
|
||||||
|
end
|
||||||
|
|
||||||
94
src/ecmd_utils/on_top_from_ueg.irp.f
Normal file
94
src/ecmd_utils/on_top_from_ueg.irp.f
Normal file
@ -0,0 +1,94 @@
|
|||||||
|
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
|
||||||
|
double precision function correction_to_on_top_from_UEG(mu,r,istate)
|
||||||
|
implicit none
|
||||||
|
integer, intent(in) :: istate
|
||||||
|
double precision, intent(in) :: mu,r(3)
|
||||||
|
double precision :: rho_a(N_states),rho_b(N_states)
|
||||||
|
double precision :: g0_UEG_mu_inf, g0_UEG_mu
|
||||||
|
call dm_dft_alpha_beta_at_r(r,rho_a,rho_b)
|
||||||
|
|
||||||
|
correction_to_on_top_from_UEG = g0_UEG_mu_inf(rho_a(istate),rho_b(istate)) / g0_UEG_mu(mu,rho_a(istate),rho_b(istate))
|
||||||
|
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
double precision function g0_UEG_mu_inf(rho_a,rho_b)
|
||||||
|
BEGIN_DOC
|
||||||
|
! Pair distribution function g0(n_alpha,n_beta) of the Colombic UEG
|
||||||
|
!
|
||||||
|
! Taken from Eq. (46) P. Gori-Giorgi and A. Savin, Phys. Rev. A 73, 032506 (2006).
|
||||||
|
END_DOC
|
||||||
|
implicit none
|
||||||
|
double precision, intent(in) :: rho_a,rho_b
|
||||||
|
double precision :: rho,pi,x
|
||||||
|
double precision :: B, C, D, E, d2, rs, ahd
|
||||||
|
rho = rho_a+rho_b
|
||||||
|
pi = 4d0 * datan(1d0)
|
||||||
|
ahd = -0.36583d0
|
||||||
|
d2 = 0.7524d0
|
||||||
|
B = -2d0 * ahd - d2
|
||||||
|
C = 0.08193d0
|
||||||
|
D = -0.01277d0
|
||||||
|
E = 0.001859d0
|
||||||
|
if (dabs(rho) > 1.d-12) then
|
||||||
|
rs = (3d0 / (4d0*pi*rho))**(1d0/3d0) ! JT: serious bug fixed 20/03/19
|
||||||
|
x = -d2*rs
|
||||||
|
g0_UEG_mu_inf= 0.5d0 * (1d0- B*rs + C*rs**2 + D*rs**3 + E*rs**4)*exp(x)
|
||||||
|
else
|
||||||
|
g0_UEG_mu_inf= 0.d0
|
||||||
|
endif
|
||||||
|
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
double precision function g0_UEG_mu(mu,rho_a,rho_b)
|
||||||
|
implicit none
|
||||||
|
BEGIN_DOC
|
||||||
|
! Pair distribution function g0(n_alpha,n_beta) of the UEG interacting with the long range interaction erf(mu r12)/r12
|
||||||
|
!
|
||||||
|
! Taken from P. Gori-Giorgi and A. Savin, Phys. Rev. A 73, 032506 (2006).
|
||||||
|
END_DOC
|
||||||
|
double precision, intent(in) :: rho_a,rho_b,mu
|
||||||
|
double precision :: zeta,pi,rho,x,alpha
|
||||||
|
double precision :: B, C, D, E, d2, rs, ahd, h_func, kf
|
||||||
|
pi = 4d0 * datan(1d0)
|
||||||
|
rho = rho_a+rho_b
|
||||||
|
alpha = (4d0/(9d0*pi))**(1d0/3d0)
|
||||||
|
ahd = -0.36583d0
|
||||||
|
d2 = 0.7524d0
|
||||||
|
B = -2d0 * ahd - d2
|
||||||
|
C = 0.08193d0
|
||||||
|
D = -0.01277d0
|
||||||
|
E = 0.001859d0
|
||||||
|
rs = (3d0 / (4d0*pi*rho))**(1d0/3d0) ! JT: serious bug fixed 20/03/19
|
||||||
|
kf = (alpha*rs)**(-1d0)
|
||||||
|
zeta = mu / kf
|
||||||
|
x = -d2*rs*h_func(zeta)/ahd
|
||||||
|
g0_UEG_mu = (exp(x)/2d0) * (1d0- B*(h_func(zeta)/ahd)*rs + C*((h_func(zeta)**2d0)/(ahd**2d0))*(rs**2d0) + D*((h_func(zeta)**3d0)/(ahd**3d0))*(rs**3d0) + E*((h_func(zeta)**4d0)/(ahd**4d0))*(rs**4d0) )
|
||||||
|
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
double precision function h_func(zeta)
|
||||||
|
implicit none
|
||||||
|
double precision, intent(in) :: zeta
|
||||||
|
double precision :: pi
|
||||||
|
double precision :: a1, a2, b1, b2, b3, ahd, alpha
|
||||||
|
pi = 4d0 * datan(1d0)
|
||||||
|
ahd = -0.36583d0
|
||||||
|
alpha = (4d0/(9d0*pi))**(1d0/3d0)
|
||||||
|
a1 = -(6d0*alpha/pi)*(1d0-log(2d0))
|
||||||
|
b1 = 1.4919d0
|
||||||
|
b3 = 1.91528d0
|
||||||
|
a2 = ahd * b3
|
||||||
|
b2 = (a1 - (b3*alpha/sqrt(pi)))/ahd
|
||||||
|
|
||||||
|
h_func = (a1*zeta**2d0 + a2*zeta**3d0) / (1d0 + b1*zeta + b2*zeta**2d0 + b3*zeta**3d0)
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
Loading…
Reference in New Issue
Block a user