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qp2/src/mo_optimization/diagonal_hessian_opt.irp.f

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add mo optimization
2023-04-18 13:56:30 +02:00
! Diagonal hessian
! The hessian of the CI energy with respects to the orbital rotation is :
! (C-c C-x C-l)
! \begin{align*}
! H_{pq,rs} &= \dfrac{\partial^2 E(x)}{\partial x_{pq}^2} \\
! &= \mathcal{P}_{pq} \mathcal{P}_{rs} [ \frac{1}{2} \sum_u [\delta_{qr}(h_p^u \gamma_u^s + h_u^s \gamma_p^u)
! + \delta_{ps}(h_r^u \gamma_u^q + h_u^q \gamma_r^u)]
! -(h_p^s \gamma_r^q + h_r^q \gamma_p^s) \\
! &+ \frac{1}{2} \sum_{tuv} [\delta_{qr}(v_{pt}^{uv} \Gamma_{uv}^{st} + v_{uv}^{st} \Gamma_{pt}^{uv})
! + \delta_{ps}(v_{uv}^{qt} \Gamma_{rt}^{uv} + v_{rt}^{uv}\Gamma_{uv}^{qt})] \\
! &+ \sum_{uv} (v_{pr}^{uv} \Gamma_{uv}^{qs} + v_{uv}^{qs} \Gamma_{pr}^{uv})
! - \sum_{tu} (v_{pu}^{st} \Gamma_{rt}^{qu}+v_{pu}^{tr} \Gamma_{tr}^{qu}+v_{rt}^{qu}\Gamma_{pu}^{st} + v_{tr}^{qu}\Gamma_{pu}^{ts})
! \end{align*}
! With pq a permutation operator :
! \begin{align*}
! \mathcal{P}_{pq}= 1 - (p \leftrightarrow q)
! \end{align*}
! \begin{align*}
! \mathcal{P}_{pq} \mathcal{P}_{rs} &= (1 - (p \leftrightarrow q))(1 - (r \leftrightarrow s)) \\
! &= 1 - (p \leftrightarrow q) - (r \leftrightarrow s) + (p \leftrightarrow q, r \leftrightarrow s)
! \end{align*}
! Where p,q,r,s,t,u,v are general spatial orbitals
! mo_num : the number of molecular orbitals
! $$h$$ : One electron integrals
! $$\gamma$$ : One body density matrix (state average in our case)
! $$v$$ : Two electron integrals
! $$\Gamma$$ : Two body density matrice (state average in our case)
! Source :
! Seniority-based coupled cluster theory
! J. Chem. Phys. 141, 244104 (2014); https://doi.org/10.1063/1.4904384
! Thomas M. Henderson, Ireneusz W. Bulik, Tamar Stein, and Gustavo E. Scuseria
! Here for the diagonal of the hessian it's a little more complicated
! than for the hessian. It's not just compute the diagonal terms of the
! hessian because of the permutations.
! The hessian is (p,q,r,s), so the diagonal terms are (p,q,p,q). But
! with the permutations : p <-> q, r <-> s, p <-> q and r <-> s, we have
! a diagonal term, if :
! p = r and q = s, => (p,q,p,q)
! or
! q = r and p = s, => (p,q,q,p)
! For that reason, we will use 2D temporary arrays to store the
! elements. One for the terms (p,q,p,q) and an other for the terms of
! kind (p,q,q,p). We will also use a 1D temporary array to store the
! terms of the kind (p,p,p,p) due to the kronoecker delta.
! *Compute the diagonal hessian of energy with respects to orbital
! rotations*
! By diagonal hessian we mean, diagonal elements of the hessian
! Provided:
! | mo_num | integer | number of MOs |
! | mo_one_e_integrals(mo_num,mo_num) | double precision | mono-electronic integrals |
! | one_e_dm_mo(mo_num,mo_num) | double precision | one e- density matrix (state average) |
! | two_e_dm_mo(mo_num,mo_num,mo_num) | double precision | two e- density matrix (state average) |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! Output:
! | H(n,n) | double precision | Hessian matrix |
! | h_tmpr(mo_num,mo_num,mo_num,mo_num) | double precision | Complete hessian matrix before the tranformation |
! | | | in n by n matrix |
! Internal:
! | hessian(mo_num,mo_num,mo_num,mo_num) | double precision | temporary array containing the hessian before |
! | | | the permutations |
! | p, q, r, s | integer | indexes of the hessian elements |
! | t, u, v | integer | indexes for the sums |
! | pq, rs | integer | indexes for the transformation of the hessian |
! | | | (4D -> 2D) |
! | t1,t2,t3 | double precision | time to compute the hessian |
! | t4,t5,t6 | double precision | time to compute the differ each element |
! | tmp_bi_int_3(mo_num,mo_num,mo_num) | double precision | 3 indexes temporary array for the bielectronic integrals (private) |
! | tmp_bi_int_3_shared(mo_num,mo_num,mo_num) | double precision | 3 indexes temporary array for the bielectronic integrals (shared) |
! | tmp_2rdm_3(mo_num,mo_num,mo_num) | double precision | 3 indexes temporary array for the 2 body density matrix (private) |
! | tmp_2rdm_3_shared(mo_num,mo_num,mo_num) | double precision | 3 indexes temporary array for the 2 body density matrix (shared) |
! | tmp_accu(mo_num,mo_num) | double precision | temporary array (private) |
! | tmp_accu_shared(mo_num,mo_num) | double precision | temporary array (shared) |
! | tmp_accu_1(mo_num) | double precision | temporary array (private) |
! | tmp_accu_1_shared(mo_num) | double precision | temporary array (shared) |
! | tmp_h_pppp(mo_num) | double precision | matrix containing the hessien elements hessian(p,p,p,p) |
! | tmp_h_pqpq(mo_num,mo_num) | double precision | matrix containing the hessien elements hessian(p,q,p,q) |
! | tmp_h_pqqp(mo_num,mo_num) | double precision | matrix containing the hessien elements hessian(p,q,q,p) |
! Function:
! | get_two_e_integral | double precision | bi-electronic integrals |
subroutine diag_hessian_opt(n,H)!, h_tmpr)
use omp_lib
include 'constants.h'
implicit none
! Variables
! in
integer, intent(in) :: n
! out
double precision, intent(out) :: H(n)!,n), h_tmpr(mo_num,mo_num,mo_num,mo_num)
! internal
!double precision, allocatable :: hessian(:,:,:,:)!, h_tmpr(:,:,:,:)
integer :: p,q,k
integer :: r,s,t,u,v
integer :: pq,rs
integer :: istate
double precision :: t1,t2,t3,t4,t5,t6
double precision, allocatable :: tmp_bi_int_3(:,:,:),tmp_bi_int_3_shared(:,:,:)
double precision, allocatable :: tmp_2rdm_3(:,:,:),tmp_2rdm_3_shared(:,:,:)
double precision, allocatable :: tmp_accu(:,:)
double precision, allocatable :: tmp_accu_shared(:,:), tmp_accu_1_shared(:)
double precision, allocatable :: tmp_h_pppp(:), tmp_h_pqpq(:,:), tmp_h_pqqp(:,:)
! Function
double precision :: get_two_e_integral
print*,''
print*,'--- diagonal hessian---'
print*,'Use the diagonal hessian'
! Allocation of shared arrays
!allocate(hessian(mo_num,mo_num,mo_num,mo_num))!,h_tmpr(mo_num,mo_num,mo_num,mo_num))
allocate(tmp_h_pppp(mo_num),tmp_h_pqpq(mo_num,mo_num),tmp_h_pqqp(mo_num,mo_num))
allocate(tmp_2rdm_3_shared(mo_num,mo_num,mo_num))
allocate(tmp_bi_int_3_shared(mo_num,mo_num,mo_num))
allocate(tmp_accu_1_shared(mo_num),tmp_accu_shared(mo_num,mo_num))
! OMP
call omp_set_max_active_levels(1)
!$OMP PARALLEL &
!$OMP PRIVATE( &
!$OMP p,q,r,s, tmp_accu,k, &
!$OMP u,v,t, tmp_bi_int_3, tmp_2rdm_3) &
!$OMP SHARED(H, tmp_h_pppp, tmp_h_pqpq, tmp_h_pqqp, &
!$OMP mo_num,n, mo_one_e_integrals, one_e_dm_mo, &
!$OMP tmp_bi_int_3_shared, tmp_2rdm_3_shared,tmp_accu_shared, &
!$OMP tmp_accu_1_shared,two_e_dm_mo,mo_integrals_map,t1,t2,t3,t4,t5,t6) &
!$OMP DEFAULT(NONE)
! Allocation of the private arrays
allocate(tmp_2rdm_3(mo_num,mo_num,mo_num),tmp_bi_int_3(mo_num,mo_num,mo_num))
allocate(tmp_accu(mo_num,mo_num))
! Initialization of the arrays
!!$OMP DO
!do s = 1,mo_num
! do r = 1, mo_num
! do q = 1, mo_num
! do p = 1, mo_num
! hessian(p,q,r,s) = 0d0
! enddo
! enddo
! enddo
!enddo
!!$OMP END DO
!$OMP DO
do p = 1, mo_num
tmp_h_pppp(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqpq(p,q) = 0d0
enddo
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqqp(p,q) = 0d0
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t1)
!$OMP END MASTER
! Line 1, term 1
! \begin{align*}
! \frac{1}{2} \sum_u \delta_{qr}(h_p^u \gamma_u^s + h_u^s \gamma_p^u)
! \end{align*}
! Without optimization :
! do p = 1, mo_num
! do q = 1, mo_num
! do r = 1, mo_num
! do s = 1, mo_num
! ! Permutations
! if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s))) then
! if (q==r) then
! do u = 1, mo_num
! hessian(p,q,r,s) = hessian(p,q,r,s) + 0.5d0 * ( &
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,s) &
! + mo_one_e_integrals(s,u) * one_e_dm_mo(p,u))
! enddo
! endif
! endif
! enddo
! enddo
! enddo
! enddo
! With optimization :
! *Part 1 : p=r and q=s and q=r*
! hessian(p,q,r,s) -> hessian(p,p,p,p)
! 0.5d0 * ( &
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,s) &
! + mo_one_e_integrals(s,u) * one_e_dm_mo(p,u))
! =
! 0.5d0 * ( &
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,p) &
! + mo_one_e_integrals(p,u) * one_e_dm_mo(p,u))
! =
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,p)
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do p = 1, mo_num
tmp_accu_1_shared(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do p = 1, mo_num
do u = 1, mo_num
tmp_accu_1_shared(p) = tmp_accu_1_shared(p) + mo_one_e_integrals(u,p) * one_e_dm_mo(u,p)
enddo
enddo
!$OMP END DO
!$OMP DO
do p = 1, mo_num
tmp_h_pppp(p) = tmp_h_pppp(p) + tmp_accu_1_shared(p)
enddo
!$OMP END DO
! *Part 2 : q=r and p=s and q=r*
! hessian(p,q,r,s) -> hessian(p,q,q,p)
! 0.5d0 * ( &
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,s) &
! + mo_one_e_integrals(s,u) * one_e_dm_mo(p,u))
! =
! 0.5d0 * ( &
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,p) &
! + mo_one_e_integrals(p,u) * one_e_dm_mo(p,u))
! =
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,p)
!$OMP DO
do p = 1, mo_num
tmp_accu_1_shared(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do p = 1, mo_num
do u = 1, mo_num
tmp_accu_1_shared(p) = tmp_accu_1_shared(p) + mo_one_e_integrals(u,p) * one_e_dm_mo(u,p)
enddo
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqqp(p,q) = tmp_h_pqqp(p,q) + tmp_accu_1_shared(p)
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6= t5-t4
print*,'l1 1',t6
!$OMP END MASTER
! Line 1, term 2
! \begin{align*}
! \frac{1}{2} \sum_u \delta_{ps}(h_r^u \gamma_u^q + h_u^q \gamma_r^u)
! \end{align*}
! Without optimization :
! do p = 1, mo_num
! do q = 1, mo_num
! do r = 1, mo_num
! do s = 1, mo_num
! ! Permutations
! if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s))) then
! if (p==s) then
! do u = 1, mo_num
! hessian(p,q,r,s) = hessian(p,q,r,s) + 0.5d0 * ( &
! mo_one_e_integrals(u,r) * one_e_dm_mo(u,q) &
! + mo_one_e_integrals(q,u) * one_e_dm_mo(r,u))
! enddo
! endif
! endif
! enddo
! enddo
! enddo
! enddo
! *Part 1 : p=r and q=s and p=s*
! hessian(p,q,r,s) -> hessian(p,p,p,p)
! 0.5d0 * (&
! mo_one_e_integrals(u,r) * one_e_dm_mo(u,q) &
! + mo_one_e_integrals(q,u) * one_e_dm_mo(r,u))
! =
! 0.5d0 * ( &
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,p) &
! + mo_one_e_integrals(p,u) * one_e_dm_mo(p,u))
! =
! mo_one_e_integrals(u,p) * one_e_dm_mo(u,p)
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do p = 1, mo_num
tmp_accu_1_shared(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do p = 1, mo_num
do u = 1, mo_num
tmp_accu_1_shared(p) = tmp_accu_1_shared(p) + mo_one_e_integrals(u,p) * one_e_dm_mo(u,p)
enddo
enddo
!$OMP END DO
!$OMP DO
do p = 1, mo_num
tmp_h_pppp(p) = tmp_h_pppp(p) + tmp_accu_1_shared(p)
enddo
!$OMP END DO
! *Part 2 : q=r and p=s and p=s*
! hessian(p,q,r,s) -> hessian(p,q,q,p)
! 0.5d0 * (&
! mo_one_e_integrals(u,r) * one_e_dm_mo(u,q) &
! + mo_one_e_integrals(q,u) * one_e_dm_mo(r,u))
! =
! 0.5d0 * ( &
! mo_one_e_integrals(u,q) * one_e_dm_mo(u,q) &
! + mo_one_e_integrals(q,u) * one_e_dm_mo(q,u))
! =
! mo_one_e_integrals(u,q) * one_e_dm_mo(u,q)
!$OMP DO
do p = 1, mo_num
tmp_accu_1_shared(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do u = 1, mo_num
tmp_accu_1_shared(q) = tmp_accu_1_shared(q) + mo_one_e_integrals(u,q) * one_e_dm_mo(u,q)
enddo
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqqp(p,q) = tmp_h_pqqp(p,q) + tmp_accu_1_shared(q)
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6= t5-t4
print*,'l1 2',t6
!$OMP END MASTER
! Line 1, term 3
! \begin{align*}
! -(h_p^s \gamma_r^q + h_r^q \gamma_p^s)
! \end{align*}
! Without optimization :
! do p = 1, mo_num
! do q = 1, mo_num
! do r = 1, mo_num
! do s = 1, mo_num
! ! Permutations
! if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s))) then
! hessian(p,q,r,s) = hessian(p,q,r,s) &
! - mo_one_e_integrals(s,p) * one_e_rdm_mo(r,q) &
! - mo_one_e_integrals(q,r) * one_e_rdm_mo(p,s)
! endif
! enddo
! enddo
! enddo
! enddo
! With optimization :
! *Part 1 : p=r and q=s*
! hessian(p,q,r,s) -> hessian(p,q,p,q)
! - mo_one_e_integrals(s,p) * one_e_dm_mo(r,q) &
! - mo_one_e_integrals(q,r) * one_e_dm_mo(p,s)
! =
! - mo_one_e_integrals(q,p) * one_e_dm_mo(p,q) &
! - mo_one_e_integrals(q,p) * one_e_dm_mo(p,q)
! =
! - 2d0 mo_one_e_integrals(q,p) * one_e_dm_mo(p,q)
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqpq(p,q) = tmp_h_pqpq(p,q) &
- 2d0 * mo_one_e_integrals(q,p) * one_e_dm_mo(p,q)
enddo
enddo
!$OMP END DO
! *Part 2 : q=r and p=s*
! hessian(p,q,r,s) -> hessian(p,q,p,q)
! - mo_one_e_integrals(s,p) * one_e_dm_mo(r,q) &
! - mo_one_e_integrals(q,r) * one_e_dm_mo(p,s)
! =
! - mo_one_e_integrals(q,p) * one_e_dm_mo(p,q) &
! - mo_one_e_integrals(q,p) * one_e_dm_mo(p,q)
! =
! - 2d0 mo_one_e_integrals(q,p) * one_e_dm_mo(p,q)
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqqp(p,q) = tmp_h_pqqp(p,q) &
- 2d0 * mo_one_e_integrals(p,p) * one_e_dm_mo(q,q)
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6= t5-t4
print*,'l1 3',t6
!$OMP END MASTER
! Line 2, term 1
! \begin{align*}
! \frac{1}{2} \sum_{tuv} \delta_{qr}(v_{pt}^{uv} \Gamma_{uv}^{st} + v_{uv}^{st} \Gamma_{pt}^{uv})
! \end{align*}
! Without optimization :
! do p = 1, mo_num
! do q = 1, mo_num
! do r = 1, mo_num
! do s = 1, mo_num
! ! Permutations
! if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s))) then
! if (q==r) then
! do t = 1, mo_num
! do u = 1, mo_num
! do v = 1, mo_num
! hessian(p,q,r,s) = hessian(p,q,r,s) + 0.5d0 * ( &
! get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,s,t) &
! + get_two_e_integral(s,t,u,v,mo_integrals_map) * two_e_dm_mo(p,t,u,v))
! enddo
! enddo
! enddo
! endif
! endif
! enddo
! enddo
! enddo
! enddo
! With optimization :
! *Part 1 : p=r and q=s and q=r*
! hessian(p,q,r,s) -> hessian(p,p,p,p)
! 0.5d0 * ( &
! get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,s,t) &
! + get_two_e_integral(s,t,u,v,mo_integrals_map) * two_e_dm_mo(p,t,u,v))
! =
! 0.5d0 * ( &
! get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,p,t) &
! + get_two_e_integral(p,t,u,v,mo_integrals_map) * two_e_dm_mo(p,t,u,v))
! =
! get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,p,t)
! Just re-order the index and use 3D temporary arrays for optimal memory
! accesses.
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do p = 1, mo_num
tmp_accu_1_shared(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do t = 1, mo_num
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,v,p) = get_two_e_integral(u,v,p,t,mo_integrals_map)
enddo
enddo
enddo
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,v,p) = two_e_dm_mo(u,v,p,t)
enddo
enddo
enddo
!$OMP CRITICAL
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_accu_1_shared(p) = tmp_accu_1_shared(p) &
+ tmp_bi_int_3(u,v,p) * tmp_2rdm_3(u,v,p)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
!$OMP END DO
!$OMP DO
do p =1, mo_num
tmp_h_pppp(p) = tmp_h_pppp(p) + tmp_accu_1_shared(p)
enddo
!$OMP END DO
! *Part 2 : q=r and p=s and q=r*
! hessian(p,q,r,s) -> hessian(p,q,q,p)
! 0.5d0 * ( &
! get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,s,t) &
! + get_two_e_integral(s,t,u,v,mo_integrals_map) * two_e_dm_mo(p,t,u,v))
! =
! 0.5d0 * ( &
! get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,p,t) &
! + get_two_e_integral(p,t,u,v,mo_integrals_map) * two_e_dm_mo(p,t,u,v))
! =
! get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,p,t)
! Just re-order the index and use 3D temporary arrays for optimal memory
! accesses.
!$OMP DO
do p = 1, mo_num
tmp_accu_1_shared(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do t = 1, mo_num
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,v,p) = get_two_e_integral(u,v,p,t,mo_integrals_map)
enddo
enddo
enddo
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,v,p) = two_e_dm_mo(u,v,p,t)
enddo
enddo
enddo
!$OMP CRITICAL
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_accu_1_shared(p) = tmp_accu_1_shared(p) + &
tmp_bi_int_3(u,v,p) * tmp_2rdm_3(u,v,p)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqqp(p,q) = tmp_h_pqqp(p,q) + tmp_accu_1_shared(p)
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6 = t5-t4
print*,'l2 1',t6
!$OMP END MASTER
! Line 2, term 2
! \begin{align*}
! \frac{1}{2} \sum_{tuv} \delta_{ps}(v_{uv}^{qt} \Gamma_{rt}^{uv} + v_{rt}^{uv}\Gamma_{uv}^{qt})
! \end{align*}
! Without optimization :
! do p = 1, mo_num
! do q = 1, mo_num
! do r = 1, mo_num
! do s = 1, mo_num
! ! Permutations
! if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s))) then
! if (p==s) then
! do t = 1, mo_num
! do u = 1, mo_num
! do v = 1, mo_num
! hessian(p,q,r,s) = hessian(p,q,r,s) + 0.5d0 * ( &
! get_two_e_integral(q,t,u,v,mo_integrals_map) * two_e_dm_mo(r,t,u,v) &
! + get_two_e_integral(u,v,r,t,mo_integrals_map) * two_e_dm_mo(u,v,q,t))
! enddo
! enddo
! enddo
! endif
! endif
! enddo
! enddo
! enddo
! enddo
! With optimization :
! *Part 1 : p=r and q=s and p=s*
! hessian(p,q,r,s) -> hessian(p,p,p,p)
! 0.5d0 * ( &
! get_two_e_integral(q,t,u,v,mo_integrals_map) * two_e_dm_mo(r,t,u,v) &
! + get_two_e_integral(u,v,r,t,mo_integrals_map) * two_e_dm_mo(u,v,q,t))
! =
! 0.5d0 * ( &
! get_two_e_integral(p,t,u,v,mo_integrals_map) * two_e_dm_mo(p,t,u,v) &
! + get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,p,t))
! =
! get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,p,t)
! Just re-order the index and use 3D temporary arrays for optimal memory
! accesses.
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do p = 1, mo_num
tmp_accu_1_shared(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do t = 1, mo_num
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,v,p) = get_two_e_integral(u,v,p,t,mo_integrals_map)
enddo
enddo
enddo
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,v,p) = two_e_dm_mo(u,v,p,t)
enddo
enddo
enddo
!$OMP CRITICAL
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_accu_1_shared(p) = tmp_accu_1_shared(p) +&
tmp_bi_int_3(u,v,p) * tmp_2rdm_3(u,v,p)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
!$OMP END DO
!$OMP DO
do p = 1, mo_num
tmp_h_pppp(p) = tmp_h_pppp(p) + tmp_accu_1_shared(p)
enddo
!$OMP END DO
! *Part 2 : q=r and p=s and p=s*
! hessian(p,q,r,s) -> hessian(p,q,q,p)
! 0.5d0 * ( &
! get_two_e_integral(q,t,u,v,mo_integrals_map) * two_e_dm_mo(r,t,u,v) &
! + get_two_e_integral(u,v,r,t,mo_integrals_map) * two_e_dm_mo(u,v,q,t))
! =
! 0.5d0 * ( &
! get_two_e_integral(q,t,u,v,mo_integrals_map) * two_e_dm_mo(q,t,u,v) &
! + get_two_e_integral(u,v,q,t,mo_integrals_map) * two_e_dm_mo(u,v,q,t))
! =
! get_two_e_integral(u,v,q,t,mo_integrals_map) * two_e_dm_mo(u,v,q,t)
! Just re-order the index and use 3D temporary arrays for optimal memory
! accesses.
!$OMP DO
do p = 1,mo_num
tmp_accu_1_shared(p) = 0d0
enddo
!$OMP END DO
!$OMP DO
do t = 1, mo_num
do q = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,v,q) = get_two_e_integral(u,v,q,t,mo_integrals_map)
enddo
enddo
enddo
do q = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,v,q) = two_e_dm_mo(u,v,q,t)
enddo
enddo
enddo
!$OMP CRITICAL
do q = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_accu_1_shared(q) = tmp_accu_1_shared(q) +&
tmp_bi_int_3(u,v,q) * tmp_2rdm_3(u,v,q)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqqp(p,q) = tmp_h_pqqp(p,q) + tmp_accu_1_shared(p)
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6 = t5-t4
print*,'l2 2',t6
!$OMP END MASTER
! Line 3, term 1
! \begin{align*}
! \sum_{uv} (v_{pr}^{uv} \Gamma_{uv}^{qs} + v_{uv}^{qs} \Gamma_{pr}^{uv})
! \end{align*}
! Without optimization :
! do p = 1, mo_num
! do q = 1, mo_num
! do r = 1, mo_num
! do s = 1, mo_num
! ! Permutations
! if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)))) then
! do u = 1, mo_num
! do v = 1, mo_num
! hessian(p,q,r,s) = hessian(p,q,r,s) &
! + get_two_e_integral(u,v,p,r,mo_integrals_map) * two_e_dm_mo(u,v,q,s) &
! + get_two_e_integral(q,s,u,v,mo_integrals_map) * two_e_dm_mo(p,r,u,v)
! enddo
! enddo
! endif
! enddo
! enddo
! enddo
! enddo
! With optimization
! *Part 1 : p=r and q=s*
! hessian(p,q,r,s) -> hessian(p,q,p,q)
! get_two_e_integral(u,v,p,r,mo_integrals_map) * two_e_dm_mo(u,v,q,s) &
! + get_two_e_integral(q,s,u,v,mo_integrals_map) * two_e_dm_mo(p,r,u,v)
! =
! get_two_e_integral(u,v,p,p,mo_integrals_map) * two_e_dm_mo(u,v,q,q) &
! + get_two_e_integral(q,q,u,v,mo_integrals_map) * two_e_dm_mo(p,p,u,v)
! =
! 2d0 * get_two_e_integral(u,v,p,p,mo_integrals_map) * two_e_dm_mo(u,v,q,q)
! Arrays of the kind (u,v,p,p) can be transform in 4D arrays (u,v,p).
! Using u,v as one variable a matrix multiplication appears.
! $$c_{p,q} = \sum_{uv} a_{p,uv} b_{uv,q}$$
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do q = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3_shared(u,v,q) = two_e_dm_mo(u,v,q,q)
enddo
enddo
enddo
!$OMP END DO
!$OMP DO
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3_shared(u,v,p) = get_two_e_integral(u,v,p,p,mo_integrals_map)
enddo
enddo
enddo
!$OMP END DO
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1d0, tmp_bi_int_3_shared,&
mo_num*mo_num, tmp_2rdm_3_shared, mo_num*mo_num, 0d0, tmp_accu, mo_num)
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqpq(p,q) = tmp_h_pqpq(p,q) + tmp_accu(p,q) + tmp_accu(q,p)
enddo
enddo
!$OMP END DO
! *Part 2 : q=r and p=s*
! hessian(p,q,r,s) -> hessian(p,q,q,p)
! get_two_e_integral(u,v,p,r,mo_integrals_map) * two_e_dm_mo(u,v,q,s) &
! + get_two_e_integral(q,s,u,v,mo_integrals_map) * two_e_dm_mo(p,r,u,v)
! =
! get_two_e_integral(u,v,p,q,mo_integrals_map) * two_e_dm_mo(u,v,q,p) &
! + get_two_e_integral(q,p,u,v,mo_integrals_map) * two_e_dm_mo(p,q,u,v)
! =
! 2d0 * get_two_e_integral(u,v,p,q,mo_integrals_map) * two_e_dm_mo(u,v,q,p)
! Just re-order the indexes and use 3D temporary arrays for optimal
! memory accesses.
!$OMP MASTER
call wall_time(t4)
!$OMP END MASTER
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,v,p) = 2d0 * get_two_e_integral(u,v,q,p,mo_integrals_map)
enddo
enddo
enddo
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,v,p) = two_e_dm_mo(u,v,p,q)
enddo
enddo
enddo
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_h_pqqp(p,q) = tmp_h_pqqp(p,q) &
+ tmp_bi_int_3(u,v,p) * tmp_2rdm_3(u,v,p)
enddo
enddo
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6= t5-t4
print*,'l3 1',t6
!$OMP END MASTER
! Line 3, term 2
! \begin{align*}
! - \sum_{tu} (v_{pu}^{st} \Gamma_{rt}^{qu}+v_{pu}^{tr} \Gamma_{tr}^{qu}+v_{rt}^{qu}\Gamma_{pu}^{st} + v_{tr}^{qu}\Gamma_{pu}^{ts})
! \end{align*}
! Without optimization :
! do p = 1, mo_num
! do q = 1, mo_num
! do r = 1, mo_num
! do s = 1, mo_num
! ! Permutations
! if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)) &
! .or. ((p==s) .and. (q==r))) then
! do t = 1, mo_num
! do u = 1, mo_num
! hessian(p,q,r,s) = hessian(p,q,r,s) &
! - get_two_e_integral(s,t,p,u,mo_integrals_map) * two_e_dm_mo(r,t,q,u) &
! - get_two_e_integral(t,s,p,u,mo_integrals_map) * two_e_dm_mo(t,r,q,u) &
! - get_two_e_integral(q,u,r,t,mo_integrals_map) * two_e_dm_mo(p,u,s,t) &
! - get_two_e_integral(q,u,t,r,mo_integrals_map) * two_e_dm_mo(p,u,t,s)
! enddo
! enddo
! endif
! enddo
! enddo
! enddo
! enddo
! With optimization :
! *Part 1 : p=r and q=s*
! hessian(p,q,r,s) -> hessian(p,q,p,q)
! - get_two_e_integral(s,t,p,u,mo_integrals_map) * two_e_dm_mo(r,t,q,u) &
! - get_two_e_integral(t,s,p,u,mo_integrals_map) * two_e_dm_mo(t,r,q,u) &
! - get_two_e_integral(q,u,r,t,mo_integrals_map) * two_e_dm_mo(p,u,s,t) &
! - get_two_e_integral(q,u,t,r,mo_integrals_map) * two_e_dm_mo(p,u,t,s)
! =
! - get_two_e_integral(q,t,p,u,mo_integrals_map) * two_e_dm_mo(p,t,q,u) &
! - get_two_e_integral(t,q,p,u,mo_integrals_map) * two_e_dm_mo(t,p,q,u) &
! - get_two_e_integral(q,u,p,t,mo_integrals_map) * two_e_dm_mo(p,u,q,t) &
! - get_two_e_integral(q,u,t,p,mo_integrals_map) * two_e_dm_mo(p,u,t,q)
! =
! - 2d0 * get_two_e_integral(q,t,p,u,mo_integrals_map) * two_e_dm_mo(p,t,q,u) &
! - 2d0 * get_two_e_integral(t,q,p,u,mo_integrals_map) * two_e_dm_mo(t,p,q,u)
! =
! - 2d0 * get_two_e_integral(q,u,p,t,mo_integrals_map) * two_e_dm_mo(q,u,p,t) &
! - 2d0 * get_two_e_integral(t,q,p,u,mo_integrals_map) * two_e_dm_mo(t,p,q,u)
! Just re-order the indexes and use 3D temporary arrays for optimal
! memory accesses.
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!----------
! Part 1.1
!----------
! - 2d0 * get_two_e_integral(q,u,p,t,mo_integrals_map) * two_e_dm_mo(q,u,p,t)
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_accu_shared(p,q) = 0d0
enddo
enddo
!$OMP END DO
!$OMP DO
do t = 1, mo_num
do p = 1, mo_num
do u = 1, mo_num
do q = 1, mo_num
tmp_bi_int_3(q,u,p) = 2d0 * get_two_e_integral(q,u,p,t,mo_integrals_map)
enddo
enddo
enddo
do p = 1, mo_num
do u = 1, mo_num
do q = 1, mo_num
tmp_2rdm_3(q,u,p) = two_e_dm_mo(q,u,p,t)
enddo
enddo
enddo
!$OMP CRITICAL
do p = 1, mo_num
do u = 1, mo_num
do q = 1, mo_num
tmp_accu_shared(p,q) = tmp_accu_shared(p,q) &
- tmp_bi_int_3(q,u,p) * tmp_2rdm_3(q,u,p)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqpq(p,q) = tmp_h_pqpq(p,q) + tmp_accu_shared(p,q)
enddo
enddo
!$OMP END DO
! Just re-order the indexes and use 3D temporary arrays for optimal
! memory accesses.
!--------
! Part 1.2
!--------
! - 2d0 * get_two_e_integral(t,q,p,u,mo_integrals_map) * two_e_dm_mo(t,p,q,u)
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_accu_shared(p,q) = 0d0
enddo
enddo
!$OMP END DO
!$OMP DO
do u = 1, mo_num
do p = 1, mo_num
do q = 1, mo_num
do t = 1, mo_num
tmp_bi_int_3(t,q,p) = 2d0*get_two_e_integral(t,q,p,u,mo_integrals_map)
enddo
enddo
enddo
do p= 1, mo_num
do q = 1, mo_num
do t = 1, mo_num
tmp_2rdm_3(t,q,p) = two_e_dm_mo(t,p,q,u)
enddo
enddo
enddo
!$OMP CRITICAL
do q = 1, mo_num
do p = 1, mo_num
do t = 1, mo_num
tmp_accu_shared(p,q) = tmp_accu_shared(p,q) &
- tmp_bi_int_3(t,q,p) * tmp_2rdm_3(t,q,p)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqpq(p,q) = tmp_h_pqpq(p,q) + tmp_accu_shared(p,q)
enddo
enddo
!$OMP END DO
! *Part 2 : q=r and p=s*
! hessian(p,q,r,s) -> hessian(p,q,p,q)
! - get_two_e_integral(s,t,p,u,mo_integrals_map) * two_e_dm_mo(r,t,q,u) &
! - get_two_e_integral(t,s,p,u,mo_integrals_map) * two_e_dm_mo(t,r,q,u) &
! - get_two_e_integral(q,u,r,t,mo_integrals_map) * two_e_dm_mo(p,u,s,t) &
! - get_two_e_integral(q,u,t,r,mo_integrals_map) * two_e_dm_mo(p,u,t,s)
! =
! - get_two_e_integral(p,t,p,u,mo_integrals_map) * two_e_dm_mo(q,t,q,u) &
! - get_two_e_integral(t,p,p,u,mo_integrals_map) * two_e_dm_mo(t,q,q,u) &
! - get_two_e_integral(q,u,q,t,mo_integrals_map) * two_e_dm_mo(p,u,p,t) &
! - get_two_e_integral(q,u,t,q,mo_integrals_map) * two_e_dm_mo(p,u,t,p)
! =
! - get_two_e_integral(p,t,p,u,mo_integrals_map) * two_e_dm_mo(q,t,q,u) &
! - get_two_e_integral(q,t,q,u,mo_integrals_map) * two_e_dm_mo(p,t,p,u) &
! - get_two_e_integral(t,u,p,p,mo_integrals_map) * two_e_dm_mo(t,q,q,u) &
! - get_two_e_integral(t,u,q,q,mo_integrals_map) * two_e_dm_mo(t,p,p,u)
! =
! - get_two_e_integral(t,p,u,p,mo_integrals_map) * two_e_dm_mo(t,q,u,q) &
! - get_two_e_integral(t,q,u,q,mo_integrals_map) * two_e_dm_mo(p,t,p,u) &
! - get_two_e_integral(t,u,p,p,mo_integrals_map) * two_e_dm_mo(q,u,t,q) &
! - get_two_e_integral(t,u,q,q,mo_integrals_map) * two_e_dm_mo(p,u,t,p)
! Arrays of the kind (t,p,u,p) can be transformed in 3D arrays. By doing
! so and using t,u as one variable, a matrix multiplication appears :
! $$c_{p,q} = \sum_{tu} a_{p,tu} b_{tu,q}$$
!----------
! Part 2.1
!----------
! - get_two_e_integral(t,p,u,p,mo_integrals_map) * two_e_dm_mo(t,q,u,q) &
! - get_two_e_integral(t,q,u,q,mo_integrals_map) * two_e_dm_mo(p,t,p,u)
!$OMP DO
do q = 1, mo_num
do u = 1, mo_num
do t = 1, mo_num
tmp_2rdm_3_shared(t,u,q) = two_e_dm_mo(t,q,u,q)
enddo
enddo
enddo
!$OMP END DO
!$OMP DO
do p = 1, mo_num
do u = 1, mo_num
do t = 1, mo_num
tmp_bi_int_3_shared(t,u,p) = get_two_e_integral(t,p,u,p,mo_integrals_map)
enddo
enddo
enddo
!$OMP END DO
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1d0, tmp_bi_int_3_shared,&
mo_num*mo_num, tmp_2rdm_3_shared, mo_num*mo_num, 0d0, tmp_accu, mo_num)
!$OMP DO
do p = 1, mo_num
do q = 1, mo_num
tmp_h_pqqp(q,p) = tmp_h_pqqp(q,p) - tmp_accu(q,p) - tmp_accu(p,q)
enddo
enddo
!$OMP END DO
! Arrays of the kind (t,u,p,p) can be transformed in 3D arrays. By doing
! so and using t,u as one variable, a matrix multiplication appears :
! $$c_{p,q} = \sum_{tu} a_{p,tu} b_{tu,q}$$
!--------
! Part 2.2
!--------
! - get_two_e_integral(t,u,p,p,mo_integrals_map) * two_e_dm_mo(q,u,t,q) &
! - get_two_e_integral(t,u,q,q,mo_integrals_map) * two_e_dm_mo(p,u,t,p)
!$OMP DO
do p = 1, mo_num
do u = 1, mo_num
do t = 1, mo_num
tmp_bi_int_3_shared(t,u,p) = get_two_e_integral(t,u,p,p,mo_integrals_map)
enddo
enddo
enddo
!$OMP END DO
!$OMP DO
do q = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3_shared(u,t,q) = two_e_dm_mo(q,u,t,q)
enddo
enddo
enddo
!$OMP END DO
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1d0, tmp_2rdm_3_shared,&
mo_num*mo_num, tmp_bi_int_3_shared, mo_num*mo_num, 0d0, tmp_accu, mo_num)
!$OMP DO
do q = 1, mo_num
do p = 1, mo_num
tmp_h_pqqp(p,q) = tmp_h_pqqp(p,q) - tmp_accu(p,q) - tmp_accu(q,p)
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6= t5-t4
print*,'l3 2',t6
!$OMP END MASTER
!$OMP MASTER
CALL wall_TIME(t2)
t2 = t2 - t1
print*, 'Time to compute the hessian :', t2
!$OMP END MASTER
! Deallocation of private arrays
! In the OMP section !
deallocate(tmp_2rdm_3,tmp_bi_int_3)
deallocate(tmp_accu)
! Permutations
! As we mentioned before there are two permutation operator in the
! formula :
! Hessian(p,q,r,s) = P_pq P_rs [...]
! => Hessian(p,q,r,s) = (p,q,r,s) - (q,p,r,s) - (p,q,s,r) + (q,p,s,r)
!!$OMP DO
!do p = 1, mo_num
! hessian(p,p,p,p) = hessian(p,p,p,p) + tmp_h_pppp(p)
!enddo
!!$OMP END DO
!!$OMP DO
!do q = 1, mo_num
! do p = 1, mo_num
! hessian(p,q,p,q) = hessian(p,q,p,q) + tmp_h_pqpq(p,q)
! enddo
!enddo
!!$OMP END DO
!
!!$OMP DO
!do q = 1, mo_num
! do p = 1, mo_num
! hessian(p,q,q,p) = hessian(p,q,q,p) + tmp_h_pqqp(p,q)
! enddo
!enddo
!!$OMP END DO
!!$OMP DO
!do s = 1, mo_num
! do r = 1, mo_num
! do q = 1, mo_num
! do p = 1, mo_num
! h_tmpr(p,q,r,s) = (hessian(p,q,r,s) - hessian(q,p,r,s) - hessian(p,q,s,r) + hessian(q,p,s,r))
! enddo
! enddo
! enddo
!enddo
!!$OMP END DO
! 4D -> 2D matrix
! We need a 2D matrix for the Newton method's. Since the Hessian is
! "antisymmetric" : $$H_{pq,rs} = -H_{rs,pq}$$
! We can write it as a 2D matrix, N by N, with N = mo_num(mo_num-1)/2
! with p<q and r<s
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do pq = 1, n
call vec_to_mat_index(pq,p,q)
do rs = 1, n
call vec_to_mat_index(rs,r,s)
!H(pq,rs) = h_tmpr(p,q,r,s)
if (pq == rs) then
k = pq
if (p == r) then
H(k) = tmp_h_pqpq(p,q) + tmp_h_pqpq(q,p) - tmp_h_pqqp(p,q) - tmp_h_pqqp(q,p)
elseif (p == s) then
H(k) = - tmp_h_pqpq(p,q) - tmp_h_pqpq(q,p) + tmp_h_pqqp(p,q) + tmp_h_pqqp(q,p)
endif
endif
enddo
enddo
!$OMP END DO
!!$OMP MASTER
!call wall_TIME(t5)
!t6 = t5-t4
!print*,'4D -> 2D :',t6
!!$OMP END MASTER
!$OMP END PARALLEL
call omp_set_max_active_levels(4)
! Display
!if (debug) then
! print*,'2D diag Hessian matrix'
! do pq = 1, n
! write(*,'(100(F10.5))') H(pq,:)
! enddo
!endif
! Deallocation of shared arrays, end
!deallocate(hessian)!,h_tmpr)
deallocate(tmp_h_pppp,tmp_h_pqpq,tmp_h_pqqp)
deallocate(tmp_accu_1_shared, tmp_accu_shared)
print*,'---diagonal_hessian'
end subroutine
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