25 KiB
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)
The hessian is a 4D matrix of size mo_num, p,q,r,s,t,u,v take all the values between 1 and mo_num (1 and mo_num include).
To do that we compute all the pairs (pq,rs)
Source : Seniority-based coupled cluster theory
- Chem. Phys. 141, 244104 (2014); https://doi.org/10.1063/1.4904384
Thomas M. Henderson, Ireneusz W. Bulik, Tamar Stein, and Gustavo E. Scuseria
Compute the hessian of energy with respects to orbital rotations
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 | t3 = t2 - t1, time to compute the hessian |
| t4,t5,t6 | double precision | t6 = t5 - t4, time to compute each element |
| tmp_bi_int_3(mo_num,mo_num,mo_num) | double precision | 3 indexes temporary array for the bielectronic integrals |
| tmp_2rdm_3(mo_num,mo_num,mo_num) | double precision | 3 indexes temporary array for the 2 body density matrix |
| ind_3(mo_num,mo_num,mo_num) | double precision | 3 indexes temporary array for matrix multiplication |
| tmp_accu(mo_num,mo_num) | double precision | temporary array |
| tmp_accu_sym(mo_num,mo_num) | double precision | temporary array |
Function:
| get_two_e_integral | double precision | bielectronic integrals |
subroutine 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(:,:,:,:)
double precision, allocatable :: H_test(:,:)
integer :: p,q
integer :: r,s,t,u,v,k
integer :: pq,rs
double precision :: t1,t2,t3,t4,t5,t6
! H_test : monum**2 by mo_num**2 double precision matrix to debug the H matrix
double precision, allocatable :: tmp_bi_int_3(:,:,:), tmp_2rdm_3(:,:,:), ind_3(:,:,:)
double precision, allocatable :: tmp_accu(:,:), tmp_accu_sym(:,:), tmp_accu_shared(:,:),tmp_accu_sym_shared(:,:)
! Function
double precision :: get_two_e_integral
print*,''
print*,'---hessian---'
print*,'Use the full 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_accu_shared(mo_num,mo_num),tmp_accu_sym_shared(mo_num,mo_num))
! Calculations
! OMP
call omp_set_max_active_levels(1)
!$OMP PARALLEL &
!$OMP PRIVATE( &
!$OMP p,q,r,s, tmp_accu, tmp_accu_sym, &
!$OMP u,v,t, tmp_bi_int_3, tmp_2rdm_3, ind_3) &
!$OMP SHARED(hessian,h_tmpr,H, mo_num,n, &
!$OMP mo_one_e_integrals, one_e_dm_mo, &
!$OMP two_e_dm_mo,mo_integrals_map,tmp_accu_sym_shared, tmp_accu_shared, &
!$OMP t1,t2,t3,t4,t5,t6)&
!$OMP DEFAULT(NONE)
! Allocation of private arrays
allocate(tmp_bi_int_3(mo_num,mo_num,mo_num))
allocate(tmp_2rdm_3(mo_num,mo_num,mo_num), ind_3(mo_num,mo_num,mo_num))
allocate(tmp_accu(mo_num,mo_num), tmp_accu_sym(mo_num,mo_num))Initialization of the arrays
!$OMP MASTER
do q = 1, mo_num
do p = 1, mo_num
tmp_accu_shared(p,q) = 0d0
enddo
enddo
!$OMP END MASTER
!$OMP MASTER
do q = 1, mo_num
do p = 1, mo_num
tmp_accu_sym(p,q) = 0d0
enddo
enddo
!$OMP END MASTER
!$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
!$OMP ENDDO
!$OMP MASTER
CALL wall_TIME(t1)
!$OMP END MASTERLine 1, term 1
Without optimization the term 1 of the line 1 is :
do p = 1, mo_num do q = 1, mo_num do r = 1, mo_num do s = 1, mo_num
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
enddo enddo enddo enddo
We can write the formula as matrix multiplication. $$c_{p,s} = \sum_u a_{p,u} b_{u,s}$$
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
call dgemm('T','N', mo_num, mo_num, mo_num, 1d0, mo_one_e_integrals,&
size(mo_one_e_integrals,1), one_e_dm_mo, size(one_e_dm_mo,1),&
0d0, tmp_accu_shared, size(tmp_accu_shared,1))
!$OMP DO
do s = 1, mo_num
do p = 1, mo_num
tmp_accu_sym_shared(p,s) = 0.5d0 * (tmp_accu_shared(p,s) + tmp_accu_shared(s,p))
enddo
enddo
!$OMP END DO
!$OMP DO
do s = 1, mo_num
do p = 1, mo_num
do r = 1, mo_num
hessian(p,r,r,s) = hessian(p,r,r,s) + tmp_accu_sym_shared(p,s)
enddo
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6=t5-t4
print*,'l1 1',t6
!$OMP END MASTERLine 1, term 2
do p = 1, mo_num do q = 1, mo_num do r = 1, mo_num do s = 1, mo_num
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
enddo enddo enddo enddo
We can write the formula as matrix multiplication. $$c_{r,q} = \sum_u a_{r,u} b_{u,q}$$
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
call dgemm('T','N', mo_num, mo_num, mo_num, 1d0, mo_one_e_integrals,&
size(mo_one_e_integrals,1), one_e_dm_mo, size(one_e_dm_mo,1),&
0d0, tmp_accu_shared, size(tmp_accu_shared,1))
!$OMP DO
do r = 1, mo_num
do q = 1, mo_num
tmp_accu_sym_shared(q,r) = 0.5d0 * (tmp_accu_shared(q,r) + tmp_accu_shared(r,q))
enddo
enddo
!OMP END DO
!$OMP DO
do r = 1, mo_num
do q = 1, mo_num
do s = 1, mo_num
hessian(s,q,r,s) = hessian(s,q,r,s) + tmp_accu_sym_shared(q,r)
enddo
enddo
enddo
!OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6=t5-t4
print*,'l1 2',t6
!$OMP END MASTERLine 1, term 3
Without optimization the third term is :
do p = 1, mo_num do q = 1, mo_num do r = 1, mo_num do s = 1, mo_num
hessian(p,q,r,s) = hessian(p,q,r,s) &
- mo_one_e_integrals(s,p) * one_e_dm_mo(r,q) &
- mo_one_e_integrals(q,r) * one_e_dm_mo(p,s))
enddo enddo enddo enddo
We can just re-order the indexes
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do s = 1, mo_num
do r = 1, mo_num
do q = 1, mo_num
do p = 1, mo_num
hessian(p,q,r,s) = hessian(p,q,r,s) &
- mo_one_e_integrals(s,p) * one_e_dm_mo(r,q)&
- mo_one_e_integrals(q,r) * one_e_dm_mo(p,s)
enddo
enddo
enddo
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6=t5-t4
print*,'l1 3',t6
!$OMP END MASTERLine 2, term 1
Without optimization the fourth term is :
do p = 1, mo_num do q = 1, mo_num do r = 1, mo_num do s = 1, mo_num
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
enddo enddo enddo enddo
Using bielectronic integral properties : get_two_e_integral(s,t,u,v,mo_integrals_map) = get_two_e_integral(u,v,s,t,mo_integrals_map)
Using the two electron density matrix properties : two_e_dm_mo(p,t,u,v) = two_e_dm_mo(u,v,p,t)
With t on the external loop, using temporary arrays for each t and by taking u,v as one variable a matrix multplication appears. $$c_{p,s} = \sum_{uv} a_{p,uv} b_{uv,s}$$
There is a kroenecker delta $$\delta_{qr}$$, so we juste compute the terms like : hessian(p,r,r,s)
!$OMP MASTER
call wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do t = 1, mo_num
do p = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,v,p) = get_two_e_integral(u,v,p,t,mo_integrals_map)
enddo
enddo
enddo
do p = 1, mo_num ! error, the p might be replace by a s
! it's a temporary array, the result by replacing p and s will be the same
do v = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,v,p) = two_e_dm_mo(u,v,p,t)
enddo
enddo
enddo
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1.d0, &
tmp_bi_int_3, mo_num*mo_num, tmp_2rdm_3, mo_num*mo_num, &
0.d0, tmp_accu, size(tmp_accu,1))
do p = 1, mo_num
do s = 1, mo_num
tmp_accu_sym(s,p) = 0.5d0 * (tmp_accu(p,s)+tmp_accu(s,p))
enddo
enddo
!$OMP CRITICAL
do s = 1, mo_num
do r = 1, mo_num
do p = 1, mo_num
hessian(p,r,r,s) = hessian(p,r,r,s) + tmp_accu_sym(p,s)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
!$OMP END DO
!$OMP MASTER
call wall_TIME(t5)
t6=t5-t4
print*,'l2 1', t6
!$OMP END MASTERLine 2, term 2
do p = 1, mo_num do q = 1, mo_num do r = 1, mo_num do s = 1, mo_num
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
enddo enddo enddo enddo
Using the two electron density matrix properties : get_two_e_integral(q,t,u,v,mo_integrals_map) = get_two_e_integral(u,v,q,t,mo_integrals_map)
Using the two electron density matrix properties : two_e_dm_mo(r,t,u,v) = two_e_dm_mo(u,v,r,t)
With t on the external loop, using temporary arrays for each t and by taking u,v as one variable a matrix multplication appears. $$c_{q,r} = \sum_uv a_{q,uv} b_{uv,r}$$
There is a kroenecker delta $$\delta_{ps}$$, so we juste compute the terms like : hessian(s,q,r,s)
!******************************
! Opt Second line, second term
!******************************
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do t = 1, mo_num
do q = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,v,q) = get_two_e_integral(u,v,q,t,mo_integrals_map)
enddo
enddo
enddo
do r = 1, mo_num
do v = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,v,r) = two_e_dm_mo(u,v,r,t)
enddo
enddo
enddo
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1.d0, &
tmp_bi_int_3 , mo_num*mo_num, tmp_2rdm_3, mo_num*mo_num, &
0.d0, tmp_accu, size(tmp_accu,1))
do r = 1, mo_num
do q = 1, mo_num
tmp_accu_sym(q,r) = 0.5d0 * (tmp_accu(q,r) + tmp_accu(r,q))
enddo
enddo
!$OMP CRITICAL
do r = 1, mo_num
do q = 1, mo_num
do s = 1, mo_num
hessian(s,q,r,s) = hessian(s,q,r,s) + tmp_accu_sym(q,r)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
!$OMP END DO
!$OMP MASTER
CALL wall_TIME(t5)
t6=t5-t4
print*,'l2 2',t6
!$OMP END MASTERLine 3, term 1
do p = 1, mo_num do q = 1, mo_num do r = 1, mo_num do s = 1, mo_num
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
enddo enddo enddo enddo
Using the two electron density matrix properties : get_two_e_integral(u,v,p,r,mo_integrals_map) = get_two_e_integral(p,r,u,v,mo_integrals_map)
Using the two electron density matrix properties : two_e_dm_mo(u,v,q,s) = two_e_dm_mo(q,s,u,v)
With v on the external loop, using temporary arrays for each v and by taking p,r and q,s as one dimension a matrix multplication appears. $$c_{pr,qs} = \sum_u a_{pr,u} b_{u,qs}$$
Part 1
!$OMP MASTER
call wall_TIME(t4)
!$OMP END MASTER
!--------
! part 1
! get_two_e_integral(u,v,p,r,mo_integrals_map) * two_e_dm_mo(u,v,q,s)
!--------
!$OMP DO
do v = 1, mo_num
do u = 1, mo_num
do r = 1, mo_num
do p = 1, mo_num
tmp_bi_int_3(p,r,u) = get_two_e_integral(p,r,u,v,mo_integrals_map)
enddo
enddo
enddo
do s = 1, mo_num
do q = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,q,s) = two_e_dm_mo(q,s,u,v)
enddo
enddo
enddo
do s = 1, mo_num
call dgemm('N','N',mo_num*mo_num, mo_num, mo_num, 1d0, tmp_bi_int_3,&
size(tmp_bi_int_3,1)*size(tmp_bi_int_3,2), tmp_2rdm_3(1,1,s),&
size(tmp_2rdm_3,1), 0d0, ind_3, size(ind_3,1) * size(ind_3,2))
!$OMP CRITICAL
do r = 1, mo_num
do q = 1, mo_num
do p = 1, mo_num
hessian(p,q,r,s) = hessian(p,q,r,s) + ind_3(p,r,q)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
enddo
!$OMP END DOWith v on the external loop, using temporary arrays for each v and by taking q,s and p,r as one dimension a matrix multplication appears. $$c_{qs,pr} = \sum_u a_{qs,u}*b_{u,pr}$$
Part 2
!--------
! part 2
! get_two_e_integral(q,s,u,v,mo_integrals_map) * two_e_dm_mo(p,r,u,v)
!--------
!$OMP DO
do v = 1, mo_num
do u = 1, mo_num
do s = 1, mo_num
do q = 1, mo_num
tmp_bi_int_3(q,s,u) = get_two_e_integral(q,s,u,v,mo_integrals_map)
enddo
enddo
enddo
do r = 1, mo_num
do p = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,p,r) = two_e_dm_mo(p,r,u,v)
enddo
enddo
enddo
do r = 1, mo_num
call dgemm('N','N', mo_num*mo_num, mo_num, mo_num, 1d0, tmp_bi_int_3,&
size(tmp_bi_int_3,1)*size(tmp_bi_int_3,2), tmp_2rdm_3(1,1,r),&
size(tmp_2rdm_3,1), 0d0, ind_3, size(ind_3,1) * size(ind_3,2))
!$OMP CRITICAL
do s = 1, mo_num
do q = 1, mo_num
do p = 1, mo_num
hessian(p,q,r,s) = hessian(p,q,r,s) + ind_3(q,s,p)
enddo
enddo
enddo
!$OMP END CRITICAL
enddo
enddo
!$OMP END DO
!$OMP MASTER
call wall_TIME(t5)
t6 = t5 - t4
print*,'l3 1', t6
!$OMP END MASTERLine 3, term 2
do p = 1, mo_num do q = 1, mo_num do r = 1, mo_num do s = 1, mo_num
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
enddo enddo enddo enddo
With q on the external loop, using temporary arrays for each p and q, and taking u,v as one variable, a matrix multiplication appears: $$c_{r,s} = \sum_{ut} a_{r,ut} b_{ut,s}$$
Part 1
!--------
! Part 1
! - get_two_e_integral(s,t,p,u,mo_integrals_map) * two_e_dm_mo(r,t,q,u)
!--------
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do q = 1, mo_num
do r = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,t,r) = two_e_dm_mo(q,u,r,t)
enddo
enddo
enddo
do p = 1, mo_num
do s = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,t,s) = - get_two_e_integral(u,s,t,p,mo_integrals_map)
enddo
enddo
enddo
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1d0, tmp_bi_int_3,&
mo_num*mo_num, tmp_2rdm_3, mo_num*mo_num, 0d0, tmp_accu, mo_num)
!$OMP CRITICAL
do s = 1, mo_num
do r = 1, mo_num
hessian(p,q,r,s) = hessian(p,q,r,s) + tmp_accu(s,r)
enddo
enddo
!$OMP END CRITICAL
enddo
enddo
!$OMP END DOWith q on the external loop, using temporary arrays for each p and q, and taking u,v as one variable, a matrix multiplication appears: $$c_{r,s} = \sum_{ut} a_{r,ut} b_{ut,s}$$
Part 2
!--------
! Part 2
!- get_two_e_integral(t,s,p,u,mo_integrals_map) * two_e_dm_mo(t,r,q,u)
!--------
!$OMP DO
do q = 1, mo_num
do r = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,t,r) = two_e_dm_mo(q,u,t,r)
enddo
enddo
enddo
do p = 1, mo_num
do s = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,t,s) = - get_two_e_integral(u,t,s,p,mo_integrals_map)
enddo
enddo
enddo
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1d0, tmp_bi_int_3,&
mo_num*mo_num, tmp_2rdm_3, mo_num*mo_num, 0d0, tmp_accu, mo_num)
!$OMP CRITICAL
do s = 1, mo_num
do r = 1, mo_num
hessian(p,q,r,s) = hessian(p,q,r,s) + tmp_accu(s,r)
enddo
enddo
!$OMP END CRITICAL
enddo
enddo
!$OMP END DOWith q on the external loop, using temporary arrays for each p and q, and taking u,v as one variable, a matrix multiplication appears: $$c_{r,s} = \sum_{ut} a_{r,ut} b_{ut,s}$$
Part 3
!--------
! Part 3
!- get_two_e_integral(q,u,r,t,mo_integrals_map) * two_e_dm_mo(p,u,s,t)
!--------
!$OMP DO
do q = 1, mo_num
do r = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,t,r) = - get_two_e_integral(u,q,t,r,mo_integrals_map)
enddo
enddo
enddo
do p = 1, mo_num
do s = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,t,s) = two_e_dm_mo(p,u,s,t)
enddo
enddo
enddo
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1d0, tmp_2rdm_3,&
mo_num*mo_num, tmp_bi_int_3, mo_num*mo_num, 0d0, tmp_accu, mo_num)
!$OMP CRITICAL
do s = 1, mo_num
do r = 1, mo_num
hessian(p,q,r,s) = hessian(p,q,r,s) + tmp_accu(s,r)
enddo
enddo
!$OMP END CRITICAL
enddo
enddo
!$OMP END DOWith q on the external loop, using temporary arrays for each p and q, and taking u,v as one variable, a matrix multiplication appears: $$c_{r,s} = \sum_{ut} a_{r,ut} b_{ut,s}$$
Part 4
!--------
! Part 4
! - get_two_e_integral(q,u,t,r,mo_integrals_map) * two_e_dm_mo(p,u,t,s)
!--------
!$OMP DO
do q = 1, mo_num
do r = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_bi_int_3(u,t,r) = - get_two_e_integral(u,t,r,q,mo_integrals_map)
enddo
enddo
enddo
do p = 1, mo_num
do s = 1, mo_num
do t = 1, mo_num
do u = 1, mo_num
tmp_2rdm_3(u,t,s) = two_e_dm_mo(p,u,t,s)
enddo
enddo
enddo
call dgemm('T','N', mo_num, mo_num, mo_num*mo_num, 1d0, tmp_2rdm_3,&
mo_num*mo_num, tmp_bi_int_3, mo_num*mo_num, 0d0, tmp_accu, mo_num)
!$OMP CRITICAL
do s = 1, mo_num
do r = 1, mo_num
hessian(p,q,r,s) = hessian(p,q,r,s) + tmp_accu(s,r)
enddo
enddo
!$OMP END CRITICAL
enddo
enddo
!$OMP END DO
!$OMP MASTER
call wall_TIME(t5)
t6 = t5-t4
print*,'l3 2',t6
!$OMP END MASTER
!$OMP MASTER
CALL wall_TIME(t2)
t3 = t2 -t1
print*,'Time to compute the hessian : ', t3
!$OMP END MASTERDeallocation of private arrays
In the omp section !
deallocate(tmp_bi_int_3, tmp_2rdm_3, tmp_accu, tmp_accu_sym, ind_3)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)
!$OMP MASTER
CALL wall_TIME(t4)
!$OMP END MASTER
!$OMP DO
do s = 1, mo_num
do r = 1, mo_num
do q = 1, mo_num
do p = 1, mo_num
h_tmpr(p,q,r,s) = (hessian(p,q,r,s) - hessian(q,p,r,s) - hessian(p,q,s,r) + hessian(q,p,s,r))
enddo
enddo
enddo
enddo
!$OMP END DO
!$OMP MASTER
call wall_TIME(t5)
t6 = t5-t4
print*,'Time for permutations :',t6
!$OMP END MASTER4D -> 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
!$OMP DO
do rs = 1, n
call vec_to_mat_index(rs,r,s)
do pq = 1, n
call vec_to_mat_index(pq,p,q)
H(pq,rs) = h_tmpr(p,q,r,s)
enddo
enddo
!$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 Hessian matrix'
do pq = 1, n
write(*,'(100(F10.5))') H(pq,:)
enddo
endifDeallocation of shared arrays, end
deallocate(hessian)!,h_tmpr)
! h_tmpr is intent out in order to debug the subroutine
! It's why we don't deallocate it
print*,'---End hessian---'
end subroutine