quack/src/RPA/linear_response.f90

subroutine linear_response(ispin,dRPA,TDA,BSE,eta,nBas,nC,nO,nV,nR,nS,lambda,e,ERI,Omega_RPA,rho_RPA,EcRPA,Omega,XpY,XmY)

! Compute linear response

  implicit none
  include 'parameters.h'

! Input variables

  logical,intent(in)            :: dRPA,TDA,BSE
  double precision,intent(in)   :: eta
  integer,intent(in)            :: ispin,nBas,nC,nO,nV,nR,nS
  double precision,intent(in)   :: lambda
  double precision,intent(in)   :: e(nBas)
  double precision,intent(in)   :: ERI(nBas,nBas,nBas,nBas)

  double precision,intent(in)   :: Omega_RPA(nS)
  double precision,intent(in)   :: rho_RPA(nBas,nBas,nS)
  
! Local variables

  integer                       :: ia
  double precision              :: trace_matrix
  double precision,allocatable  :: A(:,:)
  double precision,allocatable  :: B(:,:)
  double precision,allocatable  :: ApB(:,:)
  double precision,allocatable  :: AmB(:,:)
  double precision,allocatable  :: AmBSq(:,:)
  double precision,allocatable  :: AmBIv(:,:)
  double precision,allocatable  :: Z(:,:)

! Output variables

  double precision,intent(out)  :: EcRPA
  double precision,intent(out)  :: Omega(nS)
  double precision,intent(out)  :: XpY(nS,nS)
  double precision,intent(out)  :: XmY(nS,nS)

! Memory allocation

  allocate(A(nS,nS),B(nS,nS),ApB(nS,nS),AmB(nS,nS),AmBSq(nS,nS),AmBIv(nS,nS),Z(nS,nS))

! Build A and B matrices 

  call linear_response_A_matrix(ispin,dRPA,nBas,nC,nO,nV,nR,nS,lambda,e,ERI,A)

  if(BSE) call Bethe_Salpeter_A_matrix(eta,nBas,nC,nO,nV,nR,nS,lambda,ERI,Omega_RPA,rho_RPA,A)

! Tamm-Dancoff approximation

  B = 0d0
  if(.not. TDA) then

    call linear_response_B_matrix(ispin,dRPA,nBas,nC,nO,nV,nR,nS,lambda,ERI,B)

    if(BSE) call Bethe_Salpeter_B_matrix(eta,nBas,nC,nO,nV,nR,nS,lambda,ERI,Omega_RPA,rho_RPA,B)

  end if

! Build A + B and A - B matrices 

  ApB = A + B
  AmB = A - B

! Diagonalize linear response matrix

  call diagonalize_matrix(nS,AmB,Omega)

  if(minval(Omega) < 0d0) &
    call print_warning('You may have instabilities in linear response: A-B is not positive definite!!')

  do ia=1,nS
    if(Omega(ia) < 0d0) Omega(ia) = 0d0
  end do

  call ADAt(nS,AmB,1d0*sqrt(Omega),AmBSq)
  call ADAt(nS,AmB,1d0/sqrt(Omega),AmBIv)

  Z = matmul(AmBSq,matmul(ApB,AmBSq))

  call diagonalize_matrix(nS,Z,Omega)

  if(minval(Omega) < 0d0) & 
    call print_warning('You may have instabilities in linear response: negative excitations!!')
 
  do ia=1,nS
    if(Omega(ia) < 0d0) Omega(ia) = 0d0
  end do

  Omega = sqrt(Omega)

  XpY = matmul(transpose(Z),AmBSq)
  call DA(nS,1d0/sqrt(Omega),XpY)

  XmY = matmul(transpose(Z),AmBIv)
  call DA(nS,1d0*sqrt(Omega),XmY)

! Compute the RPA correlation energy

  EcRPA = 0.5d0*(sum(Omega) - trace_matrix(nS,A))

end subroutine linear_response