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9.2 KiB
9.2 KiB
First diagonal hessian
subroutine first_diag_hessian_opt(n,H, h_tmpr)
include 'constants.h'
implicit none
!===========================================================================
! Compute the diagonal hessian of energy with respects to orbital rotations
!===========================================================================
!===========
! Variables
!===========
! in
integer, intent(in) :: n
! n : integer, n = mo_num*(mo_num-1)/2
! out
double precision, intent(out) :: H(n,n), h_tmpr(mo_num,mo_num,mo_num,mo_num)
! H : n by n double precision matrix containing the 2D hessian
! internal
double precision, allocatable :: hessian(:,:,:,:)
integer :: p,q
integer :: r,s,t,u,v
integer :: pq,rs
double precision :: t1,t2,t3
! hessian : mo_num 4D double precision matrix containing the hessian before the permutations
! h_tmpr : mo_num 4D double precision matrix containing the hessian after the permutations
! p,q,r,s : integer, indexes of the 4D hessian matrix
! t,u,v : integer, indexes to compute hessian elements
! pq,rs : integer, indexes for the conversion from 4D to 2D hessian matrix
! t1,t2,t3 : double precision, t3 = t2 - t1, time to compute the hessian
! Function
double precision :: get_two_e_integral
! get_two_e_integral : double precision function, two e integrals
! Provided :
! mo_one_e_integrals : mono e- integrals
! get_two_e_integral : two e- integrals
! one_e_dm_mo_alpha, one_e_dm_mo_beta : one body density matrix
! two_e_dm_mo : two body density matrix
!============
! Allocation
!============
allocate(hessian(mo_num,mo_num,mo_num,mo_num))!,h_tmpr(mo_num,mo_num,mo_num,mo_num))
!=============
! Calculation
!=============
if (debug) then
print*,'Enter in first_diag_hessien'
endif
! From Anderson et. al. (2014)
! The Journal of Chemical Physics 141, 244104 (2014); doi: 10.1063/1.4904384
! LaTeX formula :
!\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_u^r)]
!-(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_{ps}^{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*}
!================
! Initialization
!================
hessian = 0d0
CALL wall_time(t1)
!========================
! First line, first term
!========================
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
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
!=========================
! First line, second term
!=========================
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
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
!========================
! First line, third term
!========================
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
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)
endif
enddo
enddo
enddo
enddo
!=========================
! Second line, first term
!=========================
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
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
!==========================
! Second line, second term
!==========================
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
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
!========================
! Third line, first term
!========================
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 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
!=========================
! Third line, second term
!=========================
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
CALL wall_time(t2)
t2 = t2 - t1
print*, 'Time to compute the hessian :', t2
!==============
! Permutations
!==============
! Convert the hessian mo_num * mo_num * mo_num * mo_num matrix in a
! 2D n * n matrix (n = mo_num*(mo_num-1)/2)
! H(pq,rs) : p<q and r<s
do r = 1, mo_num
do s = 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
!========================
! 4D matrix -> 2D matrix
!========================
! Convert the hessian mo_num * mo_num * mo_num * mo_num matrix in a
! 2D n * n matrix (n = mo_num*(mo_num-1)/2)
! H(pq,rs) : p<q and r<s
! 4D mo_num matrix to 2D n matrix
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
! Display
if (debug) then
print*,'2D diag Hessian matrix'
do pq = 1, n
write(*,'(100(F10.5))') H(pq,:)
enddo
endif
!==============
! Deallocation
!==============
deallocate(hessian)
if (debug) then
print*,'Leave first_diag_hessien'
endif
end subroutine