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345 lines
9.1 KiB
Fortran
345 lines
9.1 KiB
Fortran
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subroutine first_diag_hessian_opt(n,H, h_tmpr)
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include 'constants.h'
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implicit none
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!===========================================================================
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! Compute the diagonal hessian of energy with respects to orbital rotations
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!===========================================================================
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!===========
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! Variables
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!===========
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! in
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integer, intent(in) :: n
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! n : integer, n = mo_num*(mo_num-1)/2
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! out
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double precision, intent(out) :: H(n,n), h_tmpr(mo_num,mo_num,mo_num,mo_num)
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! H : n by n double precision matrix containing the 2D hessian
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! internal
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double precision, allocatable :: hessian(:,:,:,:)
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integer :: p,q
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integer :: r,s,t,u,v
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integer :: pq,rs
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double precision :: t1,t2,t3
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! hessian : mo_num 4D double precision matrix containing the hessian before the permutations
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! h_tmpr : mo_num 4D double precision matrix containing the hessian after the permutations
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! p,q,r,s : integer, indexes of the 4D hessian matrix
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! t,u,v : integer, indexes to compute hessian elements
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! pq,rs : integer, indexes for the conversion from 4D to 2D hessian matrix
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! t1,t2,t3 : double precision, t3 = t2 - t1, time to compute the hessian
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! Function
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double precision :: get_two_e_integral
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! get_two_e_integral : double precision function, two e integrals
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! Provided :
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! mo_one_e_integrals : mono e- integrals
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! get_two_e_integral : two e- integrals
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! one_e_dm_mo_alpha, one_e_dm_mo_beta : one body density matrix
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! two_e_dm_mo : two body density matrix
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!============
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! Allocation
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!============
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allocate(hessian(mo_num,mo_num,mo_num,mo_num))!,h_tmpr(mo_num,mo_num,mo_num,mo_num))
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!=============
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! Calculation
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!=============
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if (debug) then
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print*,'Enter in first_diag_hessien'
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endif
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! From Anderson et. al. (2014)
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! The Journal of Chemical Physics 141, 244104 (2014); doi: 10.1063/1.4904384
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! LaTeX formula :
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!\begin{align*}
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!H_{pq,rs} &= \dfrac{\partial^2 E(x)}{\partial x_{pq}^2} \\
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!&= \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)
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!+ \delta_{ps}(h_r^u \gamma_u^q + h_u^q \gamma_u^r)]
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!-(h_p^s \gamma_r^q + h_r^q \gamma_p^s) \\
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!&+ \frac{1}{2} \sum_{tuv} [\delta_{qr}(v_{pt}^{uv} \Gamma_{uv}^{st} +v_{uv}^{st} \Gamma_{pt}^{uv})
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!+ \delta_{ps}(v_{uv}^{qt} \Gamma_{rt}^{uv} + v_{rt}^{uv}\Gamma_{uv}^{qt})] \\
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!&+ \sum_{uv} (v_{pr}^{uv} \Gamma_{uv}^{qs} + v_{uv}^{qs} \Gamma_{ps}^{uv}) \\
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!&- \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})
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!\end{align*}
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!================
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! Initialization
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!================
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hessian = 0d0
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CALL wall_time(t1)
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!========================
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! First line, first term
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!========================
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do p = 1, mo_num
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do q = 1, mo_num
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do r = 1, mo_num
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do s = 1, mo_num
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! Permutations
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if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)) &
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.or. ((p==s) .and. (q==r))) then
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if (q==r) then
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do u = 1, mo_num
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hessian(p,q,r,s) = hessian(p,q,r,s) + 0.5d0 * ( &
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mo_one_e_integrals(u,p) * one_e_dm_mo(u,s) &
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+ mo_one_e_integrals(s,u) * one_e_dm_mo(p,u))
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enddo
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endif
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endif
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enddo
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enddo
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enddo
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enddo
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!=========================
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! First line, second term
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!=========================
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do p = 1, mo_num
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do q = 1, mo_num
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do r = 1, mo_num
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do s = 1, mo_num
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! Permutations
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if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)) &
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.or. ((p==s) .and. (q==r))) then
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if (p==s) then
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do u = 1, mo_num
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hessian(p,q,r,s) = hessian(p,q,r,s) + 0.5d0 * ( &
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mo_one_e_integrals(u,r) * one_e_dm_mo(u,q) &
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+ mo_one_e_integrals(q,u) * one_e_dm_mo(r,u))
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enddo
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endif
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endif
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enddo
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enddo
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enddo
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enddo
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!========================
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! First line, third term
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!========================
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do p = 1, mo_num
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do q = 1, mo_num
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do r = 1, mo_num
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do s = 1, mo_num
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! Permutations
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if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)) &
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.or. ((p==s) .and. (q==r))) then
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hessian(p,q,r,s) = hessian(p,q,r,s) &
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- mo_one_e_integrals(s,p) * one_e_dm_mo(r,q) &
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- mo_one_e_integrals(q,r) * one_e_dm_mo(p,s)
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endif
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enddo
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enddo
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enddo
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enddo
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!=========================
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! Second line, first term
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!=========================
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do p = 1, mo_num
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do q = 1, mo_num
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do r = 1, mo_num
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do s = 1, mo_num
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! Permutations
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if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)) &
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.or. ((p==s) .and. (q==r))) then
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if (q==r) then
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do t = 1, mo_num
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do u = 1, mo_num
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do v = 1, mo_num
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hessian(p,q,r,s) = hessian(p,q,r,s) + 0.5d0 * ( &
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get_two_e_integral(u,v,p,t,mo_integrals_map) * two_e_dm_mo(u,v,s,t) &
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+ get_two_e_integral(s,t,u,v,mo_integrals_map) * two_e_dm_mo(p,t,u,v))
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enddo
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enddo
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enddo
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endif
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endif
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enddo
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enddo
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enddo
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enddo
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!==========================
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! Second line, second term
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!==========================
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do p = 1, mo_num
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do q = 1, mo_num
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do r = 1, mo_num
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do s = 1, mo_num
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! Permutations
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if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)) &
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.or. ((p==s) .and. (q==r))) then
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if (p==s) then
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do t = 1, mo_num
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do u = 1, mo_num
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do v = 1, mo_num
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hessian(p,q,r,s) = hessian(p,q,r,s) + 0.5d0 * ( &
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get_two_e_integral(q,t,u,v,mo_integrals_map) * two_e_dm_mo(r,t,u,v) &
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+ get_two_e_integral(u,v,r,t,mo_integrals_map) * two_e_dm_mo(u,v,q,t))
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enddo
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enddo
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enddo
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endif
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endif
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enddo
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enddo
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enddo
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enddo
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!========================
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! Third line, first term
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!========================
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do p = 1, mo_num
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do q = 1, mo_num
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do r = 1, mo_num
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do s = 1, mo_num
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! Permutations
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if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)) &
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.or. ((p==s) .and. (q==r))) then
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do u = 1, mo_num
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do v = 1, mo_num
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hessian(p,q,r,s) = hessian(p,q,r,s) &
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+ get_two_e_integral(u,v,p,r,mo_integrals_map) * two_e_dm_mo(u,v,q,s) &
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+ get_two_e_integral(q,s,u,v,mo_integrals_map) * two_e_dm_mo(p,r,u,v)
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enddo
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enddo
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endif
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enddo
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enddo
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enddo
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enddo
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!=========================
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! Third line, second term
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!=========================
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do p = 1, mo_num
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do q = 1, mo_num
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do r = 1, mo_num
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do s = 1, mo_num
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! Permutations
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if (((p==r) .and. (q==s)) .or. ((q==r) .and. (p==s)) &
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.or. ((p==s) .and. (q==r))) then
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do t = 1, mo_num
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do u = 1, mo_num
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hessian(p,q,r,s) = hessian(p,q,r,s) &
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- get_two_e_integral(s,t,p,u,mo_integrals_map) * two_e_dm_mo(r,t,q,u) &
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- get_two_e_integral(t,s,p,u,mo_integrals_map) * two_e_dm_mo(t,r,q,u) &
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- get_two_e_integral(q,u,r,t,mo_integrals_map) * two_e_dm_mo(p,u,s,t) &
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- get_two_e_integral(q,u,t,r,mo_integrals_map) * two_e_dm_mo(p,u,t,s)
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enddo
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enddo
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endif
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enddo
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enddo
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enddo
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enddo
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CALL wall_time(t2)
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t2 = t2 - t1
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print*, 'Time to compute the hessian :', t2
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!==============
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! Permutations
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!==============
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! Convert the hessian mo_num * mo_num * mo_num * mo_num matrix in a
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! 2D n * n matrix (n = mo_num*(mo_num-1)/2)
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! H(pq,rs) : p<q and r<s
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do r = 1, mo_num
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do s = 1, mo_num
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do q = 1, mo_num
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do p = 1, mo_num
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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))
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enddo
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enddo
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enddo
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enddo
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!========================
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! 4D matrix -> 2D matrix
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!========================
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! Convert the hessian mo_num * mo_num * mo_num * mo_num matrix in a
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! 2D n * n matrix (n = mo_num*(mo_num-1)/2)
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! H(pq,rs) : p<q and r<s
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! 4D mo_num matrix to 2D n matrix
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do rs = 1, n
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call vec_to_mat_index(rs,r,s)
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do pq = 1, n
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call vec_to_mat_index(pq,p,q)
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H(pq,rs) = h_tmpr(p,q,r,s)
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enddo
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enddo
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! Display
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if (debug) then
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print*,'2D diag Hessian matrix'
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do pq = 1, n
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write(*,'(100(F10.5))') H(pq,:)
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enddo
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endif
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!==============
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! Deallocation
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!==============
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deallocate(hessian)
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if (debug) then
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print*,'Leave first_diag_hessien'
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endif
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end subroutine
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