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rotme_cart_tens.f90
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rotme_cart_tens.f90
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#define fwig 0
module rotme_cart_tens
use accuracy
use timer
#if (fwig>0)
use fwigxjpf
#endif
use moltype
implicit none
integer(ik), parameter :: maxnelem = 30
type kmat_type
real(rk), allocatable :: me(:,:,:,:)
end type kmat_type
type rotme_cart_tens_type
integer(ik) :: nelem = 0
integer(ik) :: nirrep = 0
character(cl) :: name = 'N/A'
character(cl), allocatable :: selem(:)
character(cl), allocatable :: sirrep(:)
complex(rk), allocatable :: tmat_s(:,:)
complex(rk), allocatable :: tmat_x(:,:)
logical :: sym_a = .true.
integer(ik) :: jmin = 0
integer(ik) :: jmax = 0
integer(ik), allocatable :: nktau(:)
integer(ik), allocatable :: ktau(:,:,:)
integer(ik), allocatable :: ktau_ind(:,:)
integer(ik) :: wang_ksign(2) = (/-1,1/)
type(kmat_type), allocatable :: mmat(:,:)
type(kmat_type), allocatable :: kmat(:,:)
integer(ik) :: dj = 1000
integer(ik) :: dm = 1000
integer(ik) :: dk = 1000
integer(ik) :: kmat_cmplx = 0
integer(ik), allocatable :: mmat_cmplx(:)
real(rk) :: zero_tol = 1.0d-14
procedure(prim_me_func), pointer, nopass :: func => null()
contains
procedure, public :: init => init_rotme_cart_tens_type
procedure, public :: wang_coefs => wang_jktau_coefs
end type rotme_cart_tens_type
abstract interface
subroutine prim_me_func(q1,q2,name,nelem,nirrep,mf,lf,sirrep,selem)
use accuracy
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nirrep, nelem
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
end subroutine prim_me_func
end interface
contains
!###################################################################################################
subroutine rotme_vzz_trace0(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: rank(2), isigma, sigma, irrep, j1, j2, k1, k2, m1, m2, nelem_s
complex(rk) :: tmat_s(6,6), tmat_x(6,6)
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = 'Vzz'
nelem = 6
nirrep = 1
nelem_s = 5
rank(1:nirrep) = (/2/)
if (present(sirrep)) then
sirrep(1:nirrep) = (/'T(2)'/)
endif
if (present(selem)) then
selem(1:nelem) = (/'xx','xy','xz','yy','yz','zz'/)
endif
! sigma = -2
tmat_s(1,1:nelem) = (/ cmplx( 0.5_rk/sqrt(6.0_rk), 0.0_rk ), &! xx
cmplx( 0.0_rk, -1.0_rk/sqrt(6.0_rk) ), &! xy
cmplx( 0.0_rk, 0.0_rk), &! xz
cmplx( -0.5_rk/sqrt(6.0_rk), 0.0_rk ), &! yy
cmplx( 0.0_rk, 0.0_rk), &! yz
cmplx( 0.0_rk, 0.0_rk) /) ! zz
! sigma = -1
tmat_s(2,1:nelem) = (/ cmplx(0.0_rk, 0.0_rk), &! xx
cmplx(0.0_rk, 0.0_rk), &! xy
cmplx(1.0_rk/sqrt(6.0_rk), 0.0_rk), &! xz
cmplx(0.0_rk, 0.0_rk), &! yy
cmplx(0.0_rk, -1.0_rk/sqrt(6.0_rk)), &! yz
cmplx(0.0_rk, 0.0_rk) /) ! zz
! sigma = 0
tmat_s(3,1:nelem) = (/ cmplx(0.0_rk, 0.0_rk), &! xx
cmplx(0.0_rk, 0.0_rk), &! xy
cmplx(0.0_rk, 0.0_rk), &! xz
cmplx(0.0_rk, 0.0_rk), &! yy
cmplx(0.0_rk, 0.0_rk), &! yz
cmplx(0.5_rk, 0.0_rk) /) ! zz
! sigma = +1
tmat_s(4,1:nelem) = (/ cmplx(0.0_rk, 0.0_rk), &! xx
cmplx(0.0_rk, 0.0_rk), &! xy
cmplx(-1.0_rk/sqrt(6.0_rk), 0.0_rk), &! xz
cmplx(0.0_rk, 0.0_rk), &! yy
cmplx(0.0_rk, -1.0_rk/sqrt(6.0_rk)), &! yz
cmplx(0.0_rk, 0.0_rk) /) ! zz
! sigma = +2
tmat_s(5,1:nelem) = (/ cmplx( 0.5_rk/sqrt(6.0_rk), 0.0_rk ), &! xx
cmplx( 0.0_rk, 1.0_rk/sqrt(6.0_rk) ), &! xy
cmplx( 0.0_rk, 0.0_rk), &! xz
cmplx( -0.5_rk/sqrt(6.0_rk), 0.0_rk ), &! yy
cmplx(0.0_rk, 0.0_rk), &! yz
cmplx(0.0_rk, 0.0_rk) /) ! zz
if (present(mf)) then
mf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
mf(1:nelem,irrep) = mf(1:nelem,irrep) + threej_symbol(j1,rank(irrep),j2,k1,sigma,-k2) * (-1)**(k2) * tmat_s(isigma,1:nelem)
enddo
enddo
endif
if (present(lf)) then
call pseudoinverse(nelem_s, nelem, tmat_s(1:nelem_s,1:nelem), tmat_x(1:nelem,1:nelem_s))
lf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
lf(1:nelem,irrep) = lf(1:nelem,irrep) + tmat_x(1:nelem,isigma) * threej_symbol(j1,rank(irrep),j2,m1,sigma,-m2)
enddo
enddo
lf = lf * (-1)**m2 * sqrt(real((2*j1+1)*(2*j2+1),rk))
endif
end subroutine rotme_vzz_trace0
!###################################################################################################
subroutine rotme_quad_trace0(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: rank(2), isigma, sigma, irrep, j1, j2, k1, k2, m1, m2, nelem_s
complex(rk) :: tmat_s(6,6), tmat_x(6,6)
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = 'QUAD'
nelem = 6
nirrep = 1
nelem_s = 5
rank(1:nirrep) = (/2/)
if (present(sirrep)) then
sirrep(1:nirrep) = (/'T(2)'/)
endif
if (present(selem)) then
selem(1:nelem) = (/'xx','xy','xz','yy','yz','zz'/)
endif
! sigma = -2
tmat_s(1,1:nelem) = (/ cmplx( 0.5_rk, 0.0_rk ), &! xx
cmplx( 0.0_rk, -1.0_rk ), &! xy
cmplx( 0.0_rk, 0.0_rk ), &! xz
cmplx( -0.5_rk, 0.0_rk ), &! yy
cmplx( 0.0_rk, 0.0_rk ), &! yz
cmplx( 0.0_rk, 0.0_rk ) /) ! zz
! sigma = -1
tmat_s(2,1:nelem) = (/ cmplx( 0.0_rk, 0.0_rk ), &! xx
cmplx( 0.0_rk, 0.0_rk ), &! xy
cmplx( 1.0_rk, 0.0_rk ), &! xz
cmplx( 0.0_rk, 0.0_rk ), &! yy
cmplx( 0.0_rk, -1.0_rk ), &! yz
cmplx( 0.0_rk, 0.0_rk ) /) ! zz
! sigma = 0
tmat_s(3,1:nelem) = (/ cmplx( 0.0_rk, 0.0_rk ), &! xx
cmplx( 0.0_rk, 0.0_rk ), &! xy
cmplx( 0.0_rk, 0.0_rk ), &! xz
cmplx( 0.0_rk, 0.0_rk ), &! yy
cmplx( 0.0_rk, 0.0_rk ), &! yz
cmplx( 3.0_rk/sqrt(6.0_rk), 0.0_rk ) /) ! zz
! sigma = 1
tmat_s(4,1:nelem) = (/ cmplx( 0.0_rk, 0.0_rk ), &! xx
cmplx( 0.0_rk, 0.0_rk ), &! xy
cmplx( -1.0_rk, 0.0_rk ), &! xz
cmplx( 0.0_rk, 0.0_rk ), &! yy
cmplx( 0.0_rk, -1.0_rk ), &! yz
cmplx( 0.0_rk, 0.0_rk ) /) ! zz
! sigma = 2
tmat_s(5,1:nelem) = (/ cmplx( 0.5_rk, 0.0_rk ), &! xx
cmplx( 0.0_rk, 1.0_rk ), &! xy
cmplx( 0.0_rk, 0.0_rk ), &! xz
cmplx( -0.5_rk, 0.0_rk ), &! yy
cmplx( 0.0_rk, 0.0_rk ), &! yz
cmplx( 0.0_rk, 0.0_rk ) /) ! zz
if (present(mf)) then
mf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
mf(1:nelem,irrep) = mf(1:nelem,irrep) + threej_symbol(j1,rank(irrep),j2,k1,sigma,-k2) &
* (-1)**(k2) * tmat_s(isigma,1:nelem)
enddo
enddo
endif
if (present(lf)) then
call pseudoinverse(nelem_s, nelem, tmat_s(1:nelem_s,1:nelem), tmat_x(1:nelem,1:nelem_s))
lf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
lf(1:nelem,irrep) = lf(1:nelem,irrep) + tmat_x(1:nelem,isigma) &
* threej_symbol(j1,rank(irrep),j2,m1,sigma,-m2)
enddo
enddo
lf = lf * (-1)**m2 * sqrt(real((2*j1+1)*(2*j2+1),rk))
endif
end subroutine rotme_quad_trace0
!###################################################################################################
subroutine rotme_alpha(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: rank(2), isigma, sigma, irrep, j1, j2, k1, k2, m1, m2, nelem_s
complex(rk) :: tmat_s(6,6), tmat_x(6,6)
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = 'ALPHA'
nelem = 6
nirrep = 2
nelem_s = 6
rank(1:nirrep) = (/0,2/)
if (present(sirrep)) then
sirrep(1:nirrep) = (/'T(0)','T(2)'/)
endif
if (present(selem)) then
selem(1:nelem) = (/'xx','xy','xz','yy','yz','zz'/)
endif
! Polarizability
! 0 {0: -sqrt(3)*xx/3 - sqrt(3)*yy/3 - sqrt(3)*zz/3}
! 1 {0: 0, 1: 0, -1: 0}
! 2 {0: -sqrt(6)*xx/6 - sqrt(6)*yy/6 + sqrt(6)*zz/3, 1: -xz - I*yz, 2: xx/2 + I*xy - yy/2, -1: xz - I*yz, -2: xx/2 - I*xy - yy/2}
! number of elements: 6
! order of Cartesian components: [xx, xy, xz, yy, yz, zz]
! order of spherical components: [[0, 0], [2, -2], [2, -1], [2, 0], [2, 1], [2, 2]]
! Cartesian to spherical transformation matrix:
tmat_s( 1 , 1 )= -1.0_rk/3.0_rk*sqrt(3.0_rk)
tmat_s( 1 , 2 )= 0
tmat_s( 1 , 3 )= 0
tmat_s( 1 , 4 )= -1.0_rk/3.0_rk*sqrt(3.0_rk)
tmat_s( 1 , 5 )= 0
tmat_s( 1 , 6 )= -1.0_rk/3.0_rk*sqrt(3.0_rk)
tmat_s( 2 , 1 )= 1.0_rk/2.0_rk
tmat_s( 2 , 2 )= cmplx(0,-1)
tmat_s( 2 , 3 )= 0
tmat_s( 2 , 4 )= -1.0_rk/2.0_rk
tmat_s( 2 , 5 )= 0
tmat_s( 2 , 6 )= 0
tmat_s( 3 , 1 )= 0
tmat_s( 3 , 2 )= 0
tmat_s( 3 , 3 )= 1
tmat_s( 3 , 4 )= 0
tmat_s( 3 , 5 )= cmplx(0,-1)
tmat_s( 3 , 6 )= 0
tmat_s( 4 , 1 )= -1.0_rk/6.0_rk*sqrt(6.0_rk)
tmat_s( 4 , 2 )= 0
tmat_s( 4 , 3 )= 0
tmat_s( 4 , 4 )= -1.0_rk/6.0_rk*sqrt(6.0_rk)
tmat_s( 4 , 5 )= 0
tmat_s( 4 , 6 )= (1.0_rk/3.0_rk)*sqrt(6.0_rk)
tmat_s( 5 , 1 )= 0
tmat_s( 5 , 2 )= 0
tmat_s( 5 , 3 )= -1
tmat_s( 5 , 4 )= 0
tmat_s( 5 , 5 )= cmplx(0,-1)
tmat_s( 5 , 6 )= 0
tmat_s( 6 , 1 )= 1.0_rk/2.0_rk
tmat_s( 6 , 2 )= cmplx(0,1)
tmat_s( 6 , 3 )= 0
tmat_s( 6 , 4 )= -1.0_rk/2.0_rk
tmat_s( 6 , 5 )= 0
tmat_s( 6 , 6 )= 0
if (present(mf)) then
mf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
mf(1:nelem,irrep) = mf(1:nelem,irrep) + threej_symbol(j1,rank(irrep),j2,k1,sigma,-k2) * (-1)**(k2) * tmat_s(isigma,1:nelem)
enddo
enddo
endif
if (present(lf)) then
call pseudoinverse(nelem_s, nelem, tmat_s(1:nelem_s,1:nelem), tmat_x(1:nelem,1:nelem_s))
lf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
lf(1:nelem,irrep) = lf(1:nelem,irrep) + tmat_x(1:nelem,isigma) * threej_symbol(j1,rank(irrep),j2,m1,sigma,-m2)
enddo
enddo
lf = lf * (-1)**m2 * sqrt(real((2*j1+1)*(2*j2+1),rk))
endif
end subroutine rotme_alpha
!###################################################################################################
subroutine rotme_beta(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: rank(2), isigma, sigma, irrep, j1, j2, k1, k2, m1, m2, nelem_s
complex(rk) :: tmat_s(10,10), tmat_x(10,10)
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = 'BETA'
nelem = 10
nirrep = 2
nelem_s = 10
rank(1:nirrep) = (/1, 3/)
if (present(sirrep)) then
sirrep(1:nirrep) = (/'T(1)','T(3)'/)
endif
if (present(selem)) then
selem(1:nelem) = (/'xxx', 'xxy', 'xxz', 'xyy', 'xyz', 'xzz', 'yyy', 'yyz', 'yzz', 'zzz'/)
endif
! Hyperpolarizability
! 0 {0: 0}
! 1 {0: -2*sqrt(15)*xxz/15 - 2*sqrt(15)*yyz/15 - 2*sqrt(15)*zzz/15, 1: sqrt(30)*xxx/15 + sqrt(30)*I*xxy/15 + sqrt(30)*xyy/15 + sqrt(30)*xzz/15 + sqrt(30)*I*yyy/15 + sqrt(30)*I*yzz/15, -1: -sqrt(30)*xxx/15 + sqrt(30)*I*xxy/15 - sqrt(30)*xyy/15 - sqrt(30)*xzz/15 + sqrt(30)*I*yyy/15 + sqrt(30)*I*yzz/15}
! 2 {0: 0, 1: 0, 2: 0, -1: 0, -2: 0}
! 3 {0: -3*sqrt(10)*xxz/10 - 3*sqrt(10)*yyz/10 + sqrt(10)*zzz/5, 1: sqrt(30)*xxx/20 + sqrt(30)*I*xxy/20 + sqrt(30)*xyy/20 - sqrt(30)*xzz/5 + sqrt(30)*I*yyy/20 - sqrt(30)*I*yzz/5, 2: sqrt(3)*xxz/2 + sqrt(3)*I*xyz - sqrt(3)*yyz/2, 3: -sqrt(2)*xxx/4 - 3*sqrt(2)*I*xxy/4 + 3*sqrt(2)*xyy/4 + sqrt(2)*I*yyy/4, -2: sqrt(3)*xxz/2 - sqrt(3)*I*xyz - sqrt(3)*yyz/2, -3: sqrt(2)*xxx/4 - 3*sqrt(2)*I*xxy/4 - 3*sqrt(2)*xyy/4 + sqrt(2)*I*yyy/4, -1: -sqrt(30)*xxx/20 + sqrt(30)*I*xxy/20 - sqrt(30)*xyy/20 + sqrt(30)*xzz/5 + sqrt(30)*I*yyy/20 - sqrt(30)*I*yzz/5}
! number of elements: 10
! order of Cartesian components: [xxx, xxy, xxz, xyy, xyz, xzz, yyy, yyz, yzz, zzz]
! order of spherical components: [[1, -1], [1, 0], [1, 1], [3, -3], [3, -2], [3, -1], [3, 0], [3, 1], [3, 2], [3, 3]]
! Cartesian to spherical transformation matrix:
tmat_s( 1 , 1 )= -1.0_rk/15.0_rk*sqrt(30.0_rk)
tmat_s( 1 , 2 )= cmplx(0,(1.0_rk/15.0_rk)*sqrt(30.0_rk))
tmat_s( 1 , 3 )= 0
tmat_s( 1 , 4 )= -1.0_rk/15.0_rk*sqrt(30.0_rk)
tmat_s( 1 , 5 )= 0
tmat_s( 1 , 6 )= -1.0_rk/15.0_rk*sqrt(30.0_rk)
tmat_s( 1 , 7 )= cmplx(0,(1.0_rk/15.0_rk)*sqrt(30.0_rk))
tmat_s( 1 , 8 )= 0
tmat_s( 1 , 9 )= cmplx(0,(1.0_rk/15.0_rk)*sqrt(30.0_rk))
tmat_s( 1 , 10 )= 0
tmat_s( 2 , 1 )= 0
tmat_s( 2 , 2 )= 0
tmat_s( 2 , 3 )= -2.0_rk/15.0_rk*sqrt(15.0_rk)
tmat_s( 2 , 4 )= 0
tmat_s( 2 , 5 )= 0
tmat_s( 2 , 6 )= 0
tmat_s( 2 , 7 )= 0
tmat_s( 2 , 8 )= -2.0_rk/15.0_rk*sqrt(15.0_rk)
tmat_s( 2 , 9 )= 0
tmat_s( 2 , 10 )= -2.0_rk/15.0_rk*sqrt(15.0_rk)
tmat_s( 3 , 1 )= (1.0_rk/15.0_rk)*sqrt(30.0_rk)
tmat_s( 3 , 2 )= cmplx(0,(1.0_rk/15.0_rk)*sqrt(30.0_rk))
tmat_s( 3 , 3 )= 0
tmat_s( 3 , 4 )= (1.0_rk/15.0_rk)*sqrt(30.0_rk)
tmat_s( 3 , 5 )= 0
tmat_s( 3 , 6 )= (1.0_rk/15.0_rk)*sqrt(30.0_rk)
tmat_s( 3 , 7 )= cmplx(0,(1.0_rk/15.0_rk)*sqrt(30.0_rk))
tmat_s( 3 , 8 )= 0
tmat_s( 3 , 9 )= cmplx(0,(1.0_rk/15.0_rk)*sqrt(30.0_rk))
tmat_s( 3 , 10 )= 0
tmat_s( 4 , 1 )= (1.0_rk/4.0_rk)*sqrt(2.0_rk)
tmat_s( 4 , 2 )= cmplx(0,-3.0_rk/4.0_rk*sqrt(2.0_rk))
tmat_s( 4 , 3 )= 0
tmat_s( 4 , 4 )= -3.0_rk/4.0_rk*sqrt(2.0_rk)
tmat_s( 4 , 5 )= 0
tmat_s( 4 , 6 )= 0
tmat_s( 4 , 7 )= cmplx(0,(1.0_rk/4.0_rk)*sqrt(2.0_rk))
tmat_s( 4 , 8 )= 0
tmat_s( 4 , 9 )= 0
tmat_s( 4 , 10 )= 0
tmat_s( 5 , 1 )= 0
tmat_s( 5 , 2 )= 0
tmat_s( 5 , 3 )= (1.0_rk/2.0_rk)*sqrt(3.0_rk)
tmat_s( 5 , 4 )= 0
tmat_s( 5 , 5 )= cmplx(0,-sqrt(3.0_rk))
tmat_s( 5 , 6 )= 0
tmat_s( 5 , 7 )= 0
tmat_s( 5 , 8 )= -1.0_rk/2.0_rk*sqrt(3.0_rk)
tmat_s( 5 , 9 )= 0
tmat_s( 5 , 10 )= 0
tmat_s( 6 , 1 )= -1.0_rk/20.0_rk*sqrt(30.0_rk)
tmat_s( 6 , 2 )= cmplx(0,(1.0_rk/20.0_rk)*sqrt(30.0_rk))
tmat_s( 6 , 3 )= 0
tmat_s( 6 , 4 )= -1.0_rk/20.0_rk*sqrt(30.0_rk)
tmat_s( 6 , 5 )= 0
tmat_s( 6 , 6 )= (1.0_rk/5.0_rk)*sqrt(30.0_rk)
tmat_s( 6 , 7 )= cmplx(0,(1.0_rk/20.0_rk)*sqrt(30.0_rk))
tmat_s( 6 , 8 )= 0
tmat_s( 6 , 9 )= cmplx(0,-1.0_rk/5.0_rk*sqrt(30.0_rk))
tmat_s( 6 , 10 )= 0
tmat_s( 7 , 1 )= 0
tmat_s( 7 , 2 )= 0
tmat_s( 7 , 3 )= -3.0_rk/10.0_rk*sqrt(10.0_rk)
tmat_s( 7 , 4 )= 0
tmat_s( 7 , 5 )= 0
tmat_s( 7 , 6 )= 0
tmat_s( 7 , 7 )= 0
tmat_s( 7 , 8 )= -3.0_rk/10.0_rk*sqrt(10.0_rk)
tmat_s( 7 , 9 )= 0
tmat_s( 7 , 10 )= (1.0_rk/5.0_rk)*sqrt(10.0_rk)
tmat_s( 8 , 1 )= (1.0_rk/20.0_rk)*sqrt(30.0_rk)
tmat_s( 8 , 2 )= cmplx(0,(1.0_rk/20.0_rk)*sqrt(30.0_rk))
tmat_s( 8 , 3 )= 0
tmat_s( 8 , 4 )= (1.0_rk/20.0_rk)*sqrt(30.0_rk)
tmat_s( 8 , 5 )= 0
tmat_s( 8 , 6 )= -1.0_rk/5.0_rk*sqrt(30.0_rk)
tmat_s( 8 , 7 )= cmplx(0,(1.0_rk/20.0_rk)*sqrt(30.0_rk))
tmat_s( 8 , 8 )= 0
tmat_s( 8 , 9 )= cmplx(0,-1.0_rk/5.0_rk*sqrt(30.0_rk))
tmat_s( 8 , 10 )= 0
tmat_s( 9 , 1 )= 0
tmat_s( 9 , 2 )= 0
tmat_s( 9 , 3 )= (1.0_rk/2.0_rk)*sqrt(3.0_rk)
tmat_s( 9 , 4 )= 0
tmat_s( 9 , 5 )= cmplx(0,sqrt(3.0_rk))
tmat_s( 9 , 6 )= 0
tmat_s( 9 , 7 )= 0
tmat_s( 9 , 8 )= -1.0_rk/2.0_rk*sqrt(3.0_rk)
tmat_s( 9 , 9 )= 0
tmat_s( 9 , 10 )= 0
tmat_s( 10 , 1 )= -1.0_rk/4.0_rk*sqrt(2.0_rk)
tmat_s( 10 , 2 )= cmplx(0,-3.0_rk/4.0_rk*sqrt(2.0_rk))
tmat_s( 10 , 3 )= 0
tmat_s( 10 , 4 )= (3.0_rk/4.0_rk)*sqrt(2.0_rk)
tmat_s( 10 , 5 )= 0
tmat_s( 10 , 6 )= 0
tmat_s( 10 , 7 )= cmplx(0,(1.0_rk/4.0_rk)*sqrt(2.0_rk))
tmat_s( 10 , 8 )= 0
tmat_s( 10 , 9 )= 0
tmat_s( 10 , 10 )= 0
if (present(mf)) then
mf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
mf(1:nelem,irrep) = mf(1:nelem,irrep) + threej_symbol(j1,rank(irrep),j2,k1,sigma,-k2) * (-1)**(k2) * tmat_s(isigma,1:nelem)
enddo
enddo
endif
if (present(lf)) then
call pseudoinverse(nelem_s, nelem, tmat_s(1:nelem_s,1:nelem), tmat_x(1:nelem,1:nelem_s))
lf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
lf(1:nelem,irrep) = lf(1:nelem,irrep) + tmat_x(1:nelem,isigma) * threej_symbol(j1,rank(irrep),j2,m1,sigma,-m2)
enddo
enddo
lf = lf * (-1)**m2 * sqrt(real((2*j1+1)*(2*j2+1),rk))
endif
end subroutine rotme_beta
!###################################################################################################
subroutine rotme_mu(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: rank(1), isigma, sigma, irrep, j1, j2, k1, k2, m1, m2, nelem_s
complex(rk) :: tmat_s(3,3), tmat_x(3,3)
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = 'MU'
nelem = 3
nirrep = 1
nelem_s = 3
rank(1:nirrep) = (/1/)
if (present(sirrep)) then
sirrep(1:nirrep) = (/'T(1)'/)
endif
if (present(selem)) then
selem(1:nelem) = (/'x','y','z'/)
endif
! Dipole moment
! 0 {0: 0}
! 1 {0: z, 1: sqrt(2)*(-x - I*y)/2, -1: sqrt(2)*(x - I*y)/2}
! number of elements: 3
! order of Cartesian components: [x, y, z]
! order of spherical components: [[1, -1], [1, 0], [1, 1]]
! Cartesian to spherical transformation matrix:
tmat_s( 1 , 1 )= (1.0_rk/2.0_rk)*sqrt(2.0_rk)
tmat_s( 1 , 2 )= cmplx(0,-1.0_rk/2.0_rk*sqrt(2.0_rk))
tmat_s( 1 , 3 )= 0
tmat_s( 2 , 1 )= 0
tmat_s( 2 , 2 )= 0
tmat_s( 2 , 3 )= 1
tmat_s( 3 , 1 )= -1.0_rk/2.0_rk*sqrt(2.0_rk)
tmat_s( 3 , 2 )= cmplx(0,-1.0_rk/2.0_rk*sqrt(2.0_rk))
tmat_s( 3 , 3 )= 0
if (present(mf)) then
mf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
mf(1:nelem,irrep) = mf(1:nelem,irrep) + threej_symbol(j1,rank(irrep),j2,k1,sigma,-k2) * (-1)**(k2) * tmat_s(isigma,1:nelem)
enddo
enddo
endif
if (present(lf)) then
call pseudoinverse(nelem_s, nelem, tmat_s(1:nelem_s,1:nelem), tmat_x(1:nelem,1:nelem_s))
lf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
lf(1:nelem,irrep) = lf(1:nelem,irrep) + tmat_x(1:nelem,isigma) * threej_symbol(j1,rank(irrep),j2,m1,sigma,-m2)
enddo
enddo
lf = lf * (-1)**m2 * sqrt(real((2*j1+1)*(2*j2+1),rk))
endif
end subroutine rotme_mu
!###################################################################################################
! <j',k',m'|cos(theta)|j,k,m> = <j',k',m'|d_{0,0}^{1}|j,k,m>
subroutine rotme_costheta(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: j1, j2, k1, k2, m1, m2
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = 'COSTHETA'
nelem = 1
nirrep = 1
if (present(sirrep)) then
sirrep(1:nirrep) = (/'T(0)'/)
endif
if (present(selem)) then
selem(1:nelem) = (/''/)
endif
if (present(mf)) then
if (k1==k2) then
mf = threej_symbol(j1,1,j2,k1,0,-k2) * (-1)**(k2)
else
mf = 0
endif
endif
if (present(lf)) then
if (m1==m2) then
lf = threej_symbol(j1,1,j2,m1,0,-m2) * (-1)**m2 * sqrt(real((2*j1+1)*(2*j2+1),rk))
else
lf = 0
endif
endif
end subroutine rotme_costheta
!###################################################################################################
! <j',k',m'|3cos^2(theta)-1|j,k,m> = 2 <j',k',m'|d_{0,0}^{2}|j,k,m>
subroutine rotme_3cos2theta_min1(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: j1, j2, k1, k2, m1, m2
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = '3COSTHETA2_MIN1'
nelem = 1
nirrep = 1
if (present(sirrep)) then
sirrep(1:nirrep) = (/'T(0)'/)
endif
if (present(selem)) then
selem(1:nelem) = (/''/)
endif
if (present(mf)) then
if (k1==k2) then
mf = threej_symbol(j1,2,j2,k1,0,-k2) * (-1)**(k2)
else
mf = 0
endif
endif
if (present(lf)) then
if (m1==m2) then
lf(1,1) = threej_symbol(j1,2,j2,m1,0,-m2) * (-1)**m2 * sqrt(real((2*j1+1)*(2*j2+1),rk))
else
lf = 0
endif
endif
end subroutine rotme_3cos2theta_min1
!###################################################################################################
subroutine rotme_mf(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: j1, j2, k1, k2, m1, m2
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = ''
nirrep = 1
nelem = 1
if (present(sirrep)) then
sirrep(1:nirrep) = 'T(0)'
endif
if (present(selem)) then
selem(1:nelem) = 'MF'
endif
if (present(mf)) then
mf(:,:) = 0.0
if (j1==j2 .and. k1==k2) then
mf = 1.0
endif
endif
if (present(lf)) then
lf(:,:) = 0.0
if (j1==j2 .and. m1==m2) then
lf = 1.0
endif
endif
end subroutine rotme_mf
!###################################################################################################
subroutine rotme_j(q1, q2, name, nelem, nirrep, mf, lf, sirrep, selem)
integer(ik), intent(in) :: q1(:), q2(:)
character(cl), intent(out) :: name
integer(ik), intent(inout) :: nelem, nirrep
complex(rk), intent(out), optional :: mf(:,:), lf(:,:)
character(cl), intent(out), optional :: sirrep(:), selem(:)
integer(ik) :: rank(1), isigma, sigma, irrep, j1, j2, k1, k2, m1, m2, nelem_s, ncoef(3), kval(2,3), fac, ialpha, icoef
real(ark) :: jm_coef, jp_coef
complex(rk) :: tmat_s(3,3), tmat_x(3,3), coef(2,3), cx1, cx2, cy1, cy2, cz1, res(3)
j1 = q1(1)
k1 = q1(2)
m1 = q1(3)
j2 = q2(1)
k2 = q2(2)
m2 = q2(3)
name = 'J'
nelem = 3
nirrep = 1
nelem_s = 3
rank(1:nirrep) = (/1/)
if (present(sirrep)) then
sirrep(1:nirrep) = (/'T(1)'/)
endif
if (present(selem)) then
selem(1:nelem) = (/'x','y','z'/)
endif
fac = j1*(j1+1)-k1*(k1-1)
if (fac<0) fac = 0
jp_coef = sqrt(real(fac,ark))
fac = j1*(j1+1)-k1*(k1+1)
if (fac<0) fac = 0
jm_coef = sqrt(real(fac,ark))
cx1 = jp_coef*0.5_ark
cx2 = jm_coef*0.5_ark
cy1 = jm_coef*cmplx(0.0_ark,0.5_ark)
cy2 = -jp_coef*cmplx(0.0_ark,0.5_ark)
cz1 = k1
ncoef(1:3) = (/2,2,1/)
coef(1:2,1) = (/cx1,cx2/); kval(1:2,1) = (/1,-1/)
coef(1:2,2) = (/cy1,cy2/); kval(1:2,2) = (/-1,1/)
coef(1:1,3) = (/cz1/); kval(1:1,3) = (/0/)
! 0 {0: 0}
! 1 {0: z, 1: sqrt(2)*(-x - I*y)/2, -1: sqrt(2)*(x - I*y)/2}
! number of elements: 3
! order of Cartesian components: [x, y, z]
! order of spherical components: [[1, -1], [1, 0], [1, 1]]
! Cartesian to spherical transformation matrix:
tmat_s( 1 , 1 )= (1.0_rk/2.0_rk)*sqrt(2.0_rk)
tmat_s( 1 , 2 )= cmplx(0,-1.0_rk/2.0_rk*sqrt(2.0_rk))
tmat_s( 1 , 3 )= 0
tmat_s( 2 , 1 )= 0
tmat_s( 2 , 2 )= 0
tmat_s( 2 , 3 )= 1
tmat_s( 3 , 1 )= -1.0_rk/2.0_rk*sqrt(2.0_rk)
tmat_s( 3 , 2 )= cmplx(0,-1.0_rk/2.0_rk*sqrt(2.0_rk))
tmat_s( 3 , 3 )= 0
if (present(mf)) then
mf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
res = 0
do ialpha=1, nelem
do icoef=1, ncoef(ialpha)
res(ialpha) = res(ialpha) + coef(icoef,ialpha) * threej_symbol(j1,rank(irrep),j2,k1-kval(icoef,ialpha),sigma,-k2)
enddo
enddo
mf(1:nelem,irrep) = mf(1:nelem,irrep) + res(1:nelem) * (-1)**(k2) * tmat_s(isigma,1:nelem)
enddo
enddo
endif
if (present(lf)) then
call pseudoinverse(nelem_s, nelem, tmat_s(1:nelem_s,1:nelem), tmat_x(1:nelem,1:nelem_s))
lf(:,:) = 0.0
isigma = 0
do irrep=1, nirrep
do sigma=-rank(irrep), rank(irrep)
isigma = isigma + 1
lf(1:nelem,irrep) = lf(1:nelem,irrep) + tmat_x(1:nelem,isigma) * threej_symbol(j1,rank(irrep),j2,m1,sigma,-m2)
enddo
enddo
lf = lf * (-1)**m2 * sqrt(real((2*j1+1)*(2*j2+1),rk))
endif
end subroutine rotme_j
!###################################################################################################
subroutine init_rotme_cart_tens_type(tens, jmin, jmax, dj, verbose)
class(rotme_cart_tens_type) :: tens
integer(ik), intent(in) :: jmin, jmax, dj
logical, optional, intent(in) :: verbose
integer(ik) :: info, j, ktau, tau_range(2), tau, nimg_j, j1, j2, nktau1, nktau2, nimg_k, ktau1, ktau2, ncoefs1, ncoefs2, icoef, jcoef, nimg_elem, &!
irrep, ielem, dk_max, dj_max, k, k1, k2, tau1, tau2, nelem, nirrep, dm_max, m1, m2, m1min, m1max, m2min, m2max, mmin, mmax, &!
nimg_elem_lf(maxnelem), nimg_k_lf(maxnelem), nimg_j_lf(maxnelem)
real(rk) :: zero_tol
complex(rk) :: coefs1(2), coefs2(2), me(maxnelem,maxnelem), dens, res(maxnelem,maxnelem)
logical :: verbose_, img_j((jmax+1)**2), img_k((2*jmax+1)**2), img_elem(maxnelem**2), img_elem_lf(maxnelem**2,maxnelem), &!
img_k_lf((2*jmax+1)**2,maxnelem), img_j_lf((jmax+1)**2,maxnelem)
character(cl) :: name
write(out, '(///a)') 'Compute rotational matrix elements of Cartesian tensor operator (rotme_cart_tens/init_rotme_cart_tens_type)'
verbose_ = .false.
if (present(verbose)) verbose_ = verbose
zero_tol = tens%zero_tol
tens%jmin = jmin
tens%jmax = jmax
write(out, '(/1x,a,2(1x,i4)/1x,a,1x,i4)') 'min and max J quanta:', jmin, jmax, "user-defined |j-j'| selection rules:", dj
call tens%func( (/0,0,0/), (/0,0,0/), name,nelem,nirrep)
tens%name = name
tens%nelem = nelem
tens%nirrep = nirrep
write(out, '(/1x,a,1x,a/1x,a,1x,i4/1x,a,1x,i4)') 'tensor:', tens%name, 'no. Cartesian components:', tens%nelem, 'no. irreps:', tens%nirrep
if (allocated(tens%sirrep)) deallocate(tens%sirrep)
if (allocated(tens%selem)) deallocate(tens%selem)
allocate(tens%sirrep(tens%nirrep),tens%selem(tens%nelem), stat=info)
if (info/=0) then
write(out, '(/a/a,10(1x,i6))') 'rotme_cart_tens/init_rotme_cart_tens_type error: failed to allocate tens%sirrep(tens%nirrep),tens%selem(tens%nelem)', &!
'tens%nelem, tens%nirrep =', tens%nelem, tens%nirrep
stop 'STOP'
endif
call tens%func( (/0,0,0/), (/0,0,0/), name,nelem,nirrep, sirrep=tens%sirrep(1:tens%nirrep), selem=tens%selem(1:tens%nelem))
write(out, '(1x,a,100(3x,a))') 'Cartesian components:', (trim(tens%selem(ielem)), ielem=1, tens%nelem)
write(out, '(1x,a,100(3x,a))') 'irreps:', (trim(tens%sirrep(irrep)), irrep=1, tens%nirrep)
! initialize rotational quanta K and tau
if (allocated(tens%nktau)) deallocate(tens%nktau)
if (allocated(tens%ktau)) deallocate(tens%ktau)
if (allocated(tens%ktau_ind)) deallocate(tens%ktau_ind)
allocate( tens%nktau(jmin:jmax), tens%ktau(2,2*jmax+1,jmin:jmax), tens%ktau_ind(0:jmax,0:1), stat=info )
if (info/=0) then
write(out, '(/a/a,10(1x,i6))') 'rotme_cart_tens/init_rotme_cart_tens_type error: failed to allocate tens%nktau(jmin:jmax), tens%ktau(2,2*jmax+1,jmin:jmax), , tens%ktau_ind(0:jmax,0:1)', &!
'jmin, jmax =', jmin, jmax
stop 'STOP'
endif
tens%nktau = 0
tens%ktau_ind = 0
do j=jmin, jmax
ktau = 0
do k=0, j
tau_range = (/0,1/)
if (k==0) tau_range = mod(j,2)
do tau=tau_range(1), tau_range(2)
ktau = ktau + 1
tens%ktau(1:2,ktau,j) = (/k,tau/)
tens%ktau_ind(k,tau) = ktau
enddo
enddo
tens%nktau(j) = ktau
enddo
tens%wang_ksign = (/1,-1/)
! init module "fwigxjpf" for computing 3j symbols
write(out, '(/1x,a,1x,i3)') 'initialize "fwigxjpf" for computing 3-j symbols with max angular momentum =', jmax
#if (fwig>0)
call fwig_table_init(int(2*(jmax+1),kind=4), int(3,kind=4))
call fwig_temp_init(int(2*(jmax+1),kind=4))
#endif
dj_max = 0
dk_max = 0
dm_max = 0
! compute rotaitonal matrix elements, molecule-fixed K-tensor
if (allocated(tens%kmat)) deallocate(tens%kmat)
allocate(tens%kmat(jmin:jmax,jmin:jmax), stat=info)
if (info/=0) then
write(out, '(/a/a,10(1x,i6))') 'rotme_cart_tens/init_rotme_cart_tens_type error: failed to allocate tens%kmat(jmin:jmax,jmin:jmax)', &!
'jmin, jmax =', jmin, jmax
stop 'STOP'
endif