c
c
c     ###################################################
c     ##  COPYRIGHT (C)  1995  by  Jay William Ponder  ##
c     ##              All Rights Reserved              ##
c     ###################################################
c
c     #################################################################
c     ##                                                             ##
c     ##  subroutine eopbend2  --  out-of-plane bend Hessian; numer  ##
c     ##                                                             ##
c     #################################################################
c
c
c     "eopbend2" calculates second derivatives of the out-of-plane
c     bend energy via a Wilson-Decius-Cross angle bend for a single
c     atom using finite difference methods
c
c
      subroutine eopbend2 (i)
      implicit none
      include 'sizes.i'
      include 'angle.i'
      include 'atoms.i'
      include 'deriv.i'
      include 'group.i'
      include 'hessn.i'
      include 'opbend.i'
      integer i,j,k,iopbend
      integer ia,ib,ic,id
      real*8 eps,fgrp
      real*8 old,term
      real*8 d0(3,maxatm)
      logical proceed
c
c
c     set stepsize for derivatives and default group weight
c
      eps = 1.0d-7
      fgrp = 1.0d0
c
c     compute numerical out-of-plane Hessian for current atom
c
      do iopbend = 1, nopbend
         k = iopb(iopbend)
         ia = iang(1,k)
         ib = iang(2,k)
         ic = iang(3,k)
         id = iang(4,k)
c
c     decide whether to compute the current interaction
c
         proceed = .true.
         if (use_group)  call groups (proceed,fgrp,ia,ib,ic,id,0,0)
         if (proceed)  proceed = (i.eq.ia .or. i.eq.ib .or.
     &                              i.eq.ic .or. i.eq.id)
c
c     find first derivatives for the base structure
c
         if (proceed) then
            term = fgrp / eps
            call eopbend2a (iopbend)
            do j = 1, 3
               d0(j,ia) = deopb(j,ia)
               d0(j,ib) = deopb(j,ib)
               d0(j,ic) = deopb(j,ic)
               d0(j,id) = deopb(j,id)
            end do
c
c     find numerical x-components via perturbed structures
c
            old = x(i)
            x(i) = x(i) + eps
            call eopbend2a (iopbend)
            x(i) = old
            do j = 1, 3
               hessx(j,ia) = hessx(j,ia) + term*(deopb(j,ia)-d0(j,ia))
               hessx(j,ib) = hessx(j,ib) + term*(deopb(j,ib)-d0(j,ib))
               hessx(j,ic) = hessx(j,ic) + term*(deopb(j,ic)-d0(j,ic))
               hessx(j,id) = hessx(j,id) + term*(deopb(j,id)-d0(j,id))
            end do
c
c     find numerical y-components via perturbed structures
c
            old = y(i)
            y(i) = y(i) + eps
            call eopbend2a (iopbend)
            y(i) = old
            do j = 1, 3
               hessy(j,ia) = hessy(j,ia) + term*(deopb(j,ia)-d0(j,ia))
               hessy(j,ib) = hessy(j,ib) + term*(deopb(j,ib)-d0(j,ib))
               hessy(j,ic) = hessy(j,ic) + term*(deopb(j,ic)-d0(j,ic))
               hessy(j,id) = hessy(j,id) + term*(deopb(j,id)-d0(j,id))
            end do
c
c     find numerical z-components via perturbed structures
c
            old = z(i)
            z(i) = z(i) + eps
            call eopbend2a (iopbend)
            z(i) = old
            do j = 1, 3
               hessz(j,ia) = hessz(j,ia) + term*(deopb(j,ia)-d0(j,ia))
               hessz(j,ib) = hessz(j,ib) + term*(deopb(j,ib)-d0(j,ib))
               hessz(j,ic) = hessz(j,ic) + term*(deopb(j,ic)-d0(j,ic))
               hessz(j,id) = hessz(j,id) + term*(deopb(j,id)-d0(j,id))
            end do
         end if
      end do
      return
      end
c
c
c     ###############################################################
c     ##                                                           ##
c     ##  subroutine eopbend2a  --  out-of-plane bend derivatives  ##
c     ##                                                           ##
c     ###############################################################
c
c
c     "eopbend2a" calculates out-of-plane bending first derivatives
c     at a trigonal center via a Wilson-Decius-Cross angle bend;
c     used in computation of finite difference second derivatives
c
c
      subroutine eopbend2a (i)
      implicit none
      include 'sizes.i'
      include 'angle.i'
      include 'angpot.i'
      include 'atoms.i'
      include 'bound.i'
      include 'deriv.i'
      include 'math.i'
      include 'opbend.i'
      integer i,k
      integer ia,ib,ic,id
      real*8 angle,force
      real*8 dot,cosine
      real*8 cc,ee,bkk2,term
      real*8 deddt,dedcos
      real*8 dt,dt2,dt3,dt4
      real*8 xia,yia,zia
      real*8 xib,yib,zib
      real*8 xic,yic,zic
      real*8 xid,yid,zid
      real*8 xab,yab,zab
      real*8 xcb,ycb,zcb
      real*8 xdb,ydb,zdb
      real*8 xad,yad,zad
      real*8 xcd,ycd,zcd
      real*8 rdb2,rad2,rcd2
      real*8 dccdxia,dccdyia,dccdzia
      real*8 dccdxic,dccdyic,dccdzic
      real*8 dccdxid,dccdyid,dccdzid
      real*8 deedxia,deedyia,deedzia
      real*8 deedxic,deedyic,deedzic
      real*8 deedxid,deedyid,deedzid
      real*8 dedxia,dedyia,dedzia
      real*8 dedxib,dedyib,dedzib
      real*8 dedxic,dedyic,dedzic
      real*8 dedxid,dedyid,dedzid
c
c
c     set the atom numbers and parameters for this angle
c
      k = iopb(i)
      ia = iang(1,k)
      ib = iang(2,k)
      ic = iang(3,k)
      id = iang(4,k)
      force = kopb(i)
c
c     get the coordinates of the atoms in the angle
c
      xia = x(ia)
      yia = y(ia)
      zia = z(ia)
      xib = x(ib)
      yib = y(ib)
      zib = z(ib)
      xic = x(ic)
      yic = y(ic)
      zic = z(ic)
      xid = x(id)
      yid = y(id)
      zid = z(id)
c
c     zero out the first derivative components
c
      deopb(1,ia) = 0.0d0
      deopb(2,ia) = 0.0d0
      deopb(3,ia) = 0.0d0
      deopb(1,ib) = 0.0d0
      deopb(2,ib) = 0.0d0
      deopb(3,ib) = 0.0d0
      deopb(1,ic) = 0.0d0
      deopb(2,ic) = 0.0d0
      deopb(3,ic) = 0.0d0
      deopb(1,id) = 0.0d0
      deopb(2,id) = 0.0d0
      deopb(3,id) = 0.0d0
c
c     compute the out-of-plane bending angle
c
      xab = xia - xib
      yab = yia - yib
      zab = zia - zib
      xcb = xic - xib
      ycb = yic - yib
      zcb = zic - zib
      xdb = xid - xib
      ydb = yid - yib
      zdb = zid - zib
      xad = xia - xid
      yad = yia - yid
      zad = zia - zid
      xcd = xic - xid
      ycd = yic - yid
      zcd = zic - zid
      if (use_polymer) then
         call image (xab,yab,zab,0)
         call image (xcb,ycb,zcb,0)
         call image (xdb,ydb,zdb,0)
         call image (xad,yad,zad,0)
         call image (xcd,ycd,zcd,0)
      end if
      rdb2 = xdb*xdb + ydb*ydb + zdb*zdb
      rad2 = xad*xad + yad*yad + zad*zad
      rcd2 = xcd*xcd + ycd*ycd + zcd*zcd
      ee = xab*(ycb*zdb-zcb*ydb) +  yab*(zcb*xdb-xcb*zdb)
     &                + zab*(xcb*ydb-ycb*xdb)
      dot = xad*xcd + yad*ycd + zad*zcd
      cc = rad2*rcd2 - dot*dot
      if (rdb2.ne.0.0d0 .and. cc.ne.0.0d0) then
         bkk2 = rdb2 - ee*ee/cc
         cosine = sqrt(bkk2/rdb2)
         cosine = min(1.0d0,max(-1.0d0,cosine))
         angle = radian * acos(cosine)
c
c     get the out-of-plane bending master chain rule terms
c
         dt = angle
         dt2 = dt * dt
         dt3 = dt2 * dt
         dt4 = dt2 * dt2
         deddt = opbunit * force * dt * radian
     &              * (2.0d0 + 3.0d0*cang*dt + 4.0d0*qang*dt2
     &                  + 5.0d0*pang*dt3 + 6.0d0*sang*dt4)
         dedcos = -deddt * sign(1.0d0,ee) / sqrt(cc*bkk2)
c
c     chain rule terms for first derivative components
c
         term = ee / cc
         dccdxia = (xad*rcd2-xcd*dot) * term
         dccdyia = (yad*rcd2-ycd*dot) * term
         dccdzia = (zad*rcd2-zcd*dot) * term
         dccdxic = (xcd*rad2-xad*dot) * term
         dccdyic = (ycd*rad2-yad*dot) * term
         dccdzic = (zcd*rad2-zad*dot) * term
         dccdxid = -dccdxia - dccdxic
         dccdyid = -dccdyia - dccdyic
         dccdzid = -dccdzia - dccdzic
         term = ee / rdb2
         deedxia = ydb*zcb - zdb*ycb
         deedyia = zdb*xcb - xdb*zcb
         deedzia = xdb*ycb - ydb*xcb
         deedxic = yab*zdb - zab*ydb
         deedyic = zab*xdb - xab*zdb
         deedzic = xab*ydb - yab*xdb
         deedxid = ycb*zab - zcb*yab + xdb*term
         deedyid = zcb*xab - xcb*zab + ydb*term
         deedzid = xcb*yab - ycb*xab + zdb*term
c
c     compute first derivative components for this angle
c
         dedxia = dedcos * (dccdxia+deedxia)
         dedyia = dedcos * (dccdyia+deedyia)
         dedzia = dedcos * (dccdzia+deedzia)
         dedxic = dedcos * (dccdxic+deedxic)
         dedyic = dedcos * (dccdyic+deedyic)
         dedzic = dedcos * (dccdzic+deedzic)
         dedxid = dedcos * (dccdxid+deedxid)
         dedyid = dedcos * (dccdyid+deedyid)
         dedzid = dedcos * (dccdzid+deedzid)
         dedxib = -dedxia - dedxic - dedxid
         dedyib = -dedyia - dedyic - dedyid
         dedzib = -dedzia - dedzic - dedzid
c
c     set the out-of-plane bending derivatives
c
         deopb(1,ia) = dedxia
         deopb(2,ia) = dedyia
         deopb(3,ia) = dedzia
         deopb(1,ib) = dedxib
         deopb(2,ib) = dedyib
         deopb(3,ib) = dedzib
         deopb(1,ic) = dedxic
         deopb(2,ic) = dedyic
         deopb(3,ic) = dedzic
         deopb(1,id) = dedxid
         deopb(2,id) = dedyid
         deopb(3,id) = dedzid
      end if
      return
      end
