520 lines
15 KiB
Fortran
520 lines
15 KiB
Fortran
BEGIN_PROVIDER [ double precision, expo_j_xmu_1gauss ]
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&BEGIN_PROVIDER [ double precision, coef_j_xmu_1gauss ]
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implicit none
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BEGIN_DOC
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! Upper bound long range fit of F(x) = x * (1 - erf(x)) - 1/sqrt(pi) * exp(-x**2)
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!
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! with a single gaussian.
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!
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! Such a function can be used to screen integrals with F(x).
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END_DOC
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expo_j_xmu_1gauss = 0.5d0
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coef_j_xmu_1gauss = 1.d0
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END_PROVIDER
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! ---
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BEGIN_PROVIDER [ double precision, expo_erfc_gauss ]
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implicit none
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expo_erfc_gauss = 1.41211d0
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END_PROVIDER
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BEGIN_PROVIDER [ double precision, expo_erfc_mu_gauss ]
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implicit none
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expo_erfc_mu_gauss = expo_erfc_gauss * mu_erf * mu_erf
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END_PROVIDER
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BEGIN_PROVIDER [ double precision, expo_good_j_mu_1gauss ]
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&BEGIN_PROVIDER [ double precision, coef_good_j_mu_1gauss ]
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implicit none
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BEGIN_DOC
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! exponent of Gaussian in order to obtain an upper bound of J(r12,mu)
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!
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! Can be used to scree integrals with J(r12,mu)
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END_DOC
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expo_good_j_mu_1gauss = 2.D0 * mu_erf * expo_j_xmu_1gauss
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coef_good_j_mu_1gauss = 0.5d0/mu_erf * coef_j_xmu_1gauss
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END_PROVIDER
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BEGIN_PROVIDER [ double precision, expo_j_xmu, (n_fit_1_erf_x) ]
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BEGIN_DOC
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! F(x) = x * (1 - erf(x)) - 1/sqrt(pi) * exp(-x**2) is fitted with a gaussian and a Slater
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!
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! \approx - 1/sqrt(pi) * exp(-alpha * x ) exp(-beta * x**2)
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!
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! where alpha = expo_j_xmu(1) and beta = expo_j_xmu(2)
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END_DOC
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implicit none
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!expo_j_xmu(1) = 1.7477d0
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!expo_j_xmu(2) = 0.668662d0
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!expo_j_xmu(1) = 1.74766377595541d0
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!expo_j_xmu(2) = 0.668719925486403d0
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expo_j_xmu(1) = 1.74770446934522d0
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expo_j_xmu(2) = 0.668659706559979d0
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END_PROVIDER
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! ---
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BEGIN_PROVIDER [double precision, expo_gauss_j_mu_x, (ng_fit_jast)]
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&BEGIN_PROVIDER [double precision, coef_gauss_j_mu_x, (ng_fit_jast)]
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BEGIN_DOC
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!
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! J(mu,r12) = 1/2 r12 * (1 - erf(mu*r12)) - 1/(2 sqrt(pi)*mu) exp(-(mu*r12)^2) is expressed as
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!
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! J(mu,r12) = 0.5/mu * F(r12*mu) where F(x) = x * (1 - erf(x)) - 1/sqrt(pi) * exp(-x**2)
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!
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! F(x) is fitted by - 1/sqrt(pi) * exp(-alpha * x) exp(-beta * x^2) (see expo_j_xmu)
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!
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! The slater function exp(-alpha * x) is fitted with n_max_fit_slat gaussians
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!
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! See Appendix 2 of JCP 154, 084119 (2021)
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!
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END_DOC
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implicit none
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integer :: i
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double precision :: tmp
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double precision :: expos(ng_fit_jast), alpha, beta
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if(ng_fit_jast .eq. 1) then
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coef_gauss_j_mu_x = (/ -0.47947881d0 /)
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expo_gauss_j_mu_x = (/ 3.4987848d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x(i) = tmp * expo_gauss_j_mu_x(i)
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enddo
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elseif(ng_fit_jast .eq. 2) then
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coef_gauss_j_mu_x = (/ -0.18390742d0, -0.35512656d0 /)
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expo_gauss_j_mu_x = (/ 31.9279947d0 , 2.11428789d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x(i) = tmp * expo_gauss_j_mu_x(i)
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enddo
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elseif(ng_fit_jast .eq. 3) then
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coef_gauss_j_mu_x = (/ -0.07501725d0, -0.28499012d0, -0.1953932d0 /)
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expo_gauss_j_mu_x = (/ 206.74058566d0, 1.72974157d0, 11.18735164d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x(i) = tmp * expo_gauss_j_mu_x(i)
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enddo
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elseif(ng_fit_jast .eq. 5) then
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coef_gauss_j_mu_x = (/ -0.01832955d0 , -0.10188952d0 , -0.20710858d0 , -0.18975032d0 , -0.04641657d0 /)
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expo_gauss_j_mu_x = (/ 4.33116687d+03, 2.61292842d+01, 1.43447161d+00, 4.92767426d+00, 2.10654699d+02 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x(i) = tmp * expo_gauss_j_mu_x(i)
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enddo
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elseif(ng_fit_jast .eq. 6) then
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coef_gauss_j_mu_x = (/ -0.08783664d0 , -0.16088711d0 , -0.18464486d0 , -0.0368509d0 , -0.08130028d0 , -0.0126972d0 /)
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expo_gauss_j_mu_x = (/ 4.09729729d+01, 7.11620618d+00, 2.03692338d+00, 4.10831731d+02, 1.12480198d+00, 1.00000000d+04 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x(i) = tmp * expo_gauss_j_mu_x(i)
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enddo
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elseif(ng_fit_jast .eq. 7) then
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coef_gauss_j_mu_x = (/ -0.01756495d0 , -0.01023623d0 , -0.06548959d0 , -0.03539446d0 , -0.17150646d0 , -0.15071096d0 , -0.11326834d0 /)
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expo_gauss_j_mu_x = (/ 9.88572565d+02, 1.21363371d+04, 3.69794870d+01, 1.67364529d+02, 3.03962934d+00, 1.27854005d+00, 9.76383343d+00 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x(i) = tmp * expo_gauss_j_mu_x(i)
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enddo
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elseif(ng_fit_jast .eq. 8) then
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coef_gauss_j_mu_x = (/ -0.11489205d0 , -0.16008968d0 , -0.12892456d0 , -0.04250838d0 , -0.0718451d0 , -0.02394051d0 , -0.00913353d0 , -0.01285182d0 /)
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expo_gauss_j_mu_x = (/ 6.97632442d+00, 2.56010878d+00, 1.22760977d+00, 7.47697124d+01, 2.16104215d+01, 2.96549728d+02, 1.40773328d+04, 1.43335159d+03 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x(i) = tmp * expo_gauss_j_mu_x(i)
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enddo
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!elseif(ng_fit_jast .eq. 9) then
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! coef_gauss_j_mu_x = (/ /)
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! expo_gauss_j_mu_x = (/ /)
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! tmp = mu_erf * mu_erf
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! do i = 1, ng_fit_jast
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! expo_gauss_j_mu_x(i) = tmp * expo_gauss_j_mu_x(i)
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! enddo
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elseif(ng_fit_jast .eq. 20) then
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ASSERT(n_max_fit_slat == 20)
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alpha = expo_j_xmu(1) * mu_erf
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call expo_fit_slater_gam(alpha, expos)
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beta = expo_j_xmu(2) * mu_erf * mu_erf
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tmp = -1.0d0 / sqrt(dacos(-1.d0))
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x(i) = expos(i) + beta
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coef_gauss_j_mu_x(i) = tmp * coef_fit_slat_gauss(i)
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enddo
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else
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print *, ' not implemented yet'
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stop
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endif
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tmp = 0.5d0 / mu_erf
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do i = 1, ng_fit_jast
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coef_gauss_j_mu_x(i) = tmp * coef_gauss_j_mu_x(i)
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enddo
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END_PROVIDER
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! ---
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BEGIN_PROVIDER [double precision, expo_gauss_j_mu_x_2, (ng_fit_jast)]
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&BEGIN_PROVIDER [double precision, coef_gauss_j_mu_x_2, (ng_fit_jast)]
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BEGIN_DOC
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!
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! J(mu,r12)^2 = 0.25/mu^2 F(r12*mu)^2
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!
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! F(x)^2 = 1/pi * exp(-2 * alpha * x) exp(-2 * beta * x^2)
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!
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! The slater function exp(-2 * alpha * x) is fitted with n_max_fit_slat gaussians
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!
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! See Appendix 2 of JCP 154, 084119 (2021)
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!
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END_DOC
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implicit none
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integer :: i
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double precision :: tmp
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double precision :: expos(ng_fit_jast), alpha, beta
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double precision :: alpha_opt, beta_opt
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if(ng_fit_jast .eq. 1) then
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coef_gauss_j_mu_x_2 = (/ 0.26699573d0 /)
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expo_gauss_j_mu_x_2 = (/ 11.71029824d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x_2(i) = tmp * expo_gauss_j_mu_x_2(i)
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enddo
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elseif(ng_fit_jast .eq. 2) then
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coef_gauss_j_mu_x_2 = (/ 0.11627934d0 , 0.18708824d0 /)
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expo_gauss_j_mu_x_2 = (/ 102.41386863d0, 6.36239771d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x_2(i) = tmp * expo_gauss_j_mu_x_2(i)
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enddo
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elseif(ng_fit_jast .eq. 3) then
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coef_gauss_j_mu_x_2 = (/ 0.04947216d0 , 0.14116238d0, 0.12276501d0 /)
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expo_gauss_j_mu_x_2 = (/ 635.29701766d0, 4.87696954d0, 33.36745891d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x_2(i) = tmp * expo_gauss_j_mu_x_2(i)
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enddo
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elseif(ng_fit_jast .eq. 5) then
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coef_gauss_j_mu_x_2 = (/ 0.01461527d0 , 0.03257147d0 , 0.08831354d0 , 0.11411794d0 , 0.06858783d0 /)
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expo_gauss_j_mu_x_2 = (/ 8.76554470d+03, 4.90224577d+02, 3.68267125d+00, 1.29663940d+01, 6.58240931d+01 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x_2(i) = tmp * expo_gauss_j_mu_x_2(i)
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enddo
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elseif(ng_fit_jast .eq. 6) then
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coef_gauss_j_mu_x_2 = (/ 0.01347632d0 , 0.03929124d0 , 0.06289468d0 , 0.10702493d0 , 0.06999865d0 , 0.02558191d0 /)
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expo_gauss_j_mu_x_2 = (/ 1.00000000d+04, 1.20900717d+02, 3.20346191d+00, 8.92157196d+00, 3.28119120d+01, 6.49045808d+02 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x_2(i) = tmp * expo_gauss_j_mu_x_2(i)
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enddo
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elseif(ng_fit_jast .eq. 7) then
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coef_gauss_j_mu_x_2 = (/ 0.05202849d0 , 0.01031081d0 , 0.04699157d0 , 0.01451002d0 , 0.07442576d0 , 0.02692033d0 , 0.09311842d0 /)
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expo_gauss_j_mu_x_2 = (/ 3.04469415d+00, 1.40682034d+04, 7.45960945d+01, 1.43067466d+03, 2.16815661d+01, 2.95750306d+02, 7.23471236d+00 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x_2(i) = tmp * expo_gauss_j_mu_x_2(i)
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enddo
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elseif(ng_fit_jast .eq. 8) then
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coef_gauss_j_mu_x_2 = (/ 0.00942115d0 , 0.07332421d0 , 0.0508308d0 , 0.08204949d0 , 0.0404099d0 , 0.03201288d0 , 0.01911313d0 , 0.01114732d0 /)
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expo_gauss_j_mu_x_2 = (/ 1.56957321d+04, 1.52867810d+01, 4.36016903d+01, 5.96818956d+00, 2.85535269d+00, 1.36064008d+02, 4.71968910d+02, 1.92022350d+03 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x_2(i) = tmp * expo_gauss_j_mu_x_2(i)
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enddo
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!elseif(ng_fit_jast .eq. 9) then
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! coef_gauss_j_mu_x_2 = (/ /)
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! expo_gauss_j_mu_x_2 = (/ /)
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!
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! tmp = mu_erf * mu_erf
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! do i = 1, ng_fit_jast
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! expo_gauss_j_mu_x_2(i) = tmp * expo_gauss_j_mu_x_2(i)
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! enddo
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elseif(ng_fit_jast .eq. 20) then
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ASSERT(n_max_fit_slat == 20)
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!alpha_opt = 2.d0 * expo_j_xmu(1)
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!beta_opt = 2.d0 * expo_j_xmu(2)
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! direct opt
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alpha_opt = 3.52751759d0
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beta_opt = 1.26214809d0
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alpha = alpha_opt * mu_erf
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call expo_fit_slater_gam(alpha, expos)
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beta = beta_opt * mu_erf * mu_erf
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tmp = 1.d0 / dacos(-1.d0)
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_x_2(i) = expos(i) + beta
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coef_gauss_j_mu_x_2(i) = tmp * coef_fit_slat_gauss(i)
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enddo
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else
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print *, ' not implemented yet'
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stop
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endif
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tmp = 0.25d0 / (mu_erf * mu_erf)
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do i = 1, ng_fit_jast
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coef_gauss_j_mu_x_2(i) = tmp * coef_gauss_j_mu_x_2(i)
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enddo
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END_PROVIDER
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! ---
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BEGIN_PROVIDER [double precision, expo_gauss_j_mu_1_erf, (ng_fit_jast)]
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&BEGIN_PROVIDER [double precision, coef_gauss_j_mu_1_erf, (ng_fit_jast)]
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BEGIN_DOC
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!
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! J(mu,r12) x \frac{1 - erf(mu * r12)}{2} =
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!
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! - \frac{1}{4 \sqrt{\pi} \mu} \exp(-(alpha1 + alpha2) * mu * r12 - (beta1 + beta2) * mu^2 * r12^2)
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!
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END_DOC
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implicit none
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integer :: i
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double precision :: tmp
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double precision :: expos(ng_fit_jast), alpha, beta
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double precision :: alpha_opt, beta_opt
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if(ng_fit_jast .eq. 1) then
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coef_gauss_j_mu_1_erf = (/ -0.47742461d0 /)
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expo_gauss_j_mu_1_erf = (/ 8.72255696d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_1_erf(i) = tmp * expo_gauss_j_mu_1_erf(i)
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enddo
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elseif(ng_fit_jast .eq. 2) then
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coef_gauss_j_mu_1_erf = (/ -0.19342649d0, -0.34563835d0 /)
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expo_gauss_j_mu_1_erf = (/ 78.66099999d0, 5.04324363d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_1_erf(i) = tmp * expo_gauss_j_mu_1_erf(i)
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enddo
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elseif(ng_fit_jast .eq. 3) then
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coef_gauss_j_mu_1_erf = (/ -0.0802541d0 , -0.27019258d0, -0.20546681d0 /)
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expo_gauss_j_mu_1_erf = (/ 504.53350764d0, 4.01408169d0, 26.5758329d0 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_1_erf(i) = tmp * expo_gauss_j_mu_1_erf(i)
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enddo
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elseif(ng_fit_jast .eq. 5) then
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coef_gauss_j_mu_1_erf = (/ -0.02330531d0 , -0.11888176d0 , -0.16476192d0 , -0.19874713d0 , -0.05889174d0 /)
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expo_gauss_j_mu_1_erf = (/ 1.00000000d+04, 4.66067922d+01, 3.04359857d+00, 9.54726649d+00, 3.59796835d+02 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_1_erf(i) = tmp * expo_gauss_j_mu_1_erf(i)
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enddo
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elseif(ng_fit_jast .eq. 6) then
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coef_gauss_j_mu_1_erf = (/ -0.01865654d0 , -0.18319251d0 , -0.06543196d0 , -0.11522778d0 , -0.14825793d0 , -0.03327101d0 /)
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expo_gauss_j_mu_1_erf = (/ 1.00000000d+04, 8.05593848d+00, 1.27986190d+02, 2.92674319d+01, 2.93583623d+00, 7.65609148d+02 /)
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_1_erf(i) = tmp * expo_gauss_j_mu_1_erf(i)
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enddo
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elseif(ng_fit_jast .eq. 7) then
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coef_gauss_j_mu_1_erf = (/ -0.11853067d0 , -0.01522824d0 , -0.07419098d0 , -0.022202d0 , -0.12242283d0 , -0.04177571d0 , -0.16983107d0 /)
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expo_gauss_j_mu_1_erf = (/ 2.74057056d+00, 1.37626591d+04, 6.65578663d+01, 1.34693031d+03, 1.90547699d+01, 2.69445390d+02, 6.31845879d+00/)
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|
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tmp = mu_erf * mu_erf
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do i = 1, ng_fit_jast
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expo_gauss_j_mu_1_erf(i) = tmp * expo_gauss_j_mu_1_erf(i)
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enddo
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|
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elseif(ng_fit_jast .eq. 8) then
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|
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coef_gauss_j_mu_1_erf = (/ -0.12263328d0 , -0.04965255d0 , -0.15463564d0 , -0.09675781d0 , -0.0807023d0 , -0.02923298d0 , -0.01381381d0 , -0.01675923d0 /)
|
|
expo_gauss_j_mu_1_erf = (/ 1.36101994d+01, 1.24908367d+02, 5.29061388d+00, 2.60692516d+00, 3.93396935d+01, 4.43071610d+02, 1.54902240d+04, 1.85170446d+03 /)
|
|
|
|
tmp = mu_erf * mu_erf
|
|
do i = 1, ng_fit_jast
|
|
expo_gauss_j_mu_1_erf(i) = tmp * expo_gauss_j_mu_1_erf(i)
|
|
enddo
|
|
|
|
!elseif(ng_fit_jast .eq. 9) then
|
|
|
|
! coef_gauss_j_mu_1_erf = (/ /)
|
|
! expo_gauss_j_mu_1_erf = (/ /)
|
|
|
|
! tmp = mu_erf * mu_erf
|
|
! do i = 1, ng_fit_jast
|
|
! expo_gauss_j_mu_1_erf(i) = tmp * expo_gauss_j_mu_1_erf(i)
|
|
! enddo
|
|
|
|
elseif(ng_fit_jast .eq. 20) then
|
|
|
|
ASSERT(n_max_fit_slat == 20)
|
|
|
|
!alpha_opt = expo_j_xmu(1) + expo_gauss_1_erf_x(1)
|
|
!beta_opt = expo_j_xmu(2) + expo_gauss_1_erf_x(2)
|
|
|
|
! direct opt
|
|
alpha_opt = 2.87875632d0
|
|
beta_opt = 1.34801003d0
|
|
|
|
alpha = alpha_opt * mu_erf
|
|
call expo_fit_slater_gam(alpha, expos)
|
|
beta = beta_opt * mu_erf * mu_erf
|
|
|
|
tmp = -1.d0 / dsqrt(dacos(-1.d0))
|
|
do i = 1, ng_fit_jast
|
|
expo_gauss_j_mu_1_erf(i) = expos(i) + beta
|
|
coef_gauss_j_mu_1_erf(i) = tmp * coef_fit_slat_gauss(i)
|
|
enddo
|
|
|
|
else
|
|
|
|
print *, ' not implemented yet'
|
|
stop
|
|
|
|
endif
|
|
|
|
tmp = 0.25d0 / mu_erf
|
|
do i = 1, ng_fit_jast
|
|
coef_gauss_j_mu_1_erf(i) = tmp * coef_gauss_j_mu_1_erf(i)
|
|
enddo
|
|
|
|
END_PROVIDER
|
|
|
|
! ---
|
|
|
|
double precision function F_x_j(x)
|
|
implicit none
|
|
BEGIN_DOC
|
|
! F_x_j(x) = dimension-less correlation factor = x (1 - erf(x)) - 1/sqrt(pi) exp(-x^2)
|
|
END_DOC
|
|
double precision, intent(in) :: x
|
|
F_x_j = x * (1.d0 - derf(x)) - 1/dsqrt(dacos(-1.d0)) * dexp(-x**2)
|
|
|
|
end
|
|
|
|
double precision function j_mu_F_x_j(x)
|
|
implicit none
|
|
BEGIN_DOC
|
|
! j_mu_F_x_j(x) = correlation factor = 1/2 r12 * (1 - erf(mu*r12)) - 1/(2 sqrt(pi)*mu) exp(-(mu*r12)^2)
|
|
!
|
|
! = 1/(2*mu) * F_x_j(mu*x)
|
|
END_DOC
|
|
double precision :: F_x_j
|
|
double precision, intent(in) :: x
|
|
j_mu_F_x_j = 0.5d0/mu_erf * F_x_j(x*mu_erf)
|
|
end
|
|
|
|
double precision function j_mu(x)
|
|
implicit none
|
|
double precision, intent(in) :: x
|
|
BEGIN_DOC
|
|
! j_mu(x) = correlation factor = 1/2 r12 * (1 - erf(mu*r12)) - 1/(2 sqrt(pi)*mu) exp(-(mu*r12)^2)
|
|
END_DOC
|
|
j_mu = 0.5d0* x * (1.d0 - derf(mu_erf*x)) - 0.5d0/( dsqrt(dacos(-1.d0))*mu_erf) * dexp(-(mu_erf*x)*(mu_erf*x))
|
|
|
|
end
|
|
|
|
double precision function j_mu_fit_gauss(x)
|
|
implicit none
|
|
BEGIN_DOC
|
|
! j_mu_fit_gauss(x) = correlation factor = 1/2 r12 * (1 - erf(mu*r12)) - 1/(2 sqrt(pi)*mu) exp(-(mu*r12)^2)
|
|
!
|
|
! but fitted with gaussians
|
|
END_DOC
|
|
double precision, intent(in) :: x
|
|
integer :: i
|
|
double precision :: alpha,coef
|
|
j_mu_fit_gauss = 0.d0
|
|
do i = 1, n_max_fit_slat
|
|
alpha = expo_gauss_j_mu_x(i)
|
|
coef = coef_gauss_j_mu_x(i)
|
|
j_mu_fit_gauss += coef * dexp(-alpha*x*x)
|
|
enddo
|
|
|
|
end
|
|
|
|
! ---
|
|
|