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module-SpinAngular.jl
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module-SpinAngular.jl
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#
# This module has mainly been contributed by Gediminas Gaigalas (gediminas.gaigalas@tfai.vu.lt)
"""
`module JAC.SpinAngular`
... a submodel of JAC that contains all methods for computing the spin-angular coefficients for one- and two-particle operators
of given rank and for symmetry-adapted CSF. At the present, these spin-angular can be obtained for all scalar (and symmetric)
one and two-particle interactions and they are sometimes refered to as 'pure angular coefficients' in order to distinguish
them from those used in GRASP (and its derivatives) where part of the `physical interaction' were included orginal
into the angular integrals. The spin-angular coefficients usally appear in the form:
<CSF_l | operator | CSF_r> = sum_{t} coeff_t (L,abcd) * X^L (abcd)
where CSF is a standard (jj-coupled) configuration state function, t a summation index, and where X^L(abcd) denotes the
effective interaction strength.
"""
module SpinAngular
using Printf, ..AngularMomentum, ..Basics, ..Defaults, ..ManyElectron, ..Radial
export Xronecker
"""
`abstract type SpinAngular.AbstractAngularType`
... defines an abstract type and a number of data types to work with one- and two-particle operators of given rank, see also:
+ struct OneParticleOperator ... to represent a single-particle operator with well-defined spherical tensor rank.
+ struct TwoParticleOperator ... to represent a two-particle operator with well-defined spherical tensor rank.
"""
abstract type AbstractAngularType end
"""
`struct SpinAngular.OneParticleOperator <: AbstractAngularType`
... a struct for defining the spherial tensor structure of a single-particle operator.
+ rank ::Int64 ... Rank of the operator.
+ parity ::Basics.Parity ... Parity of the operator (if needed ??)
+ sameOrbitalSet ::Bool ... True if orbitals for both CSF are taken from the same orbital set (if needed ??)
"""
struct OneParticleOperator <: AbstractAngularType
rank ::Int64
parity ::Basics.Parity
sameOrbitalSet ::Bool
end
"""
`SpinAngular.OneParticleOperator()` ... constructor for setting the default values.
"""
function OneParticleOperator()
OneParticleOperator( 0, Basics.plus, false)
end
# `Base.show(io::IO, op::OneParticleOperator)` ... prepares a proper printout of the op::OneParticleOperator.
function Base.show(io::IO, op::OneParticleOperator)
sa = "One-particle operator O^($(op.rank) $(op.parity))"
if op.sameOrbitalSet sa = sa * ", defined for CSF from the same orbital set."
else sa = sa * ", defined for CSF from different orbital sets."
end
println(io, sa)
end
"""
`struct SpinAngular.TwoParticleOperator`
... a struct for defining the spherial tensor structure of a two-particle operator.
+ rank ::Int64 ... Rank of the operator (if needed ??).
+ parity ::Basics.Parity ... Parity of the operator (if needed ??)
+ sameOrbitalSet ::Bool ... True if orbitals for both CSF are taken from the same orbital set (if needed ??)
"""
struct TwoParticleOperator
rank ::Int64
parity ::Basics.Parity
sameOrbitalSet ::Bool
end
"""
`SpinAngular.TwoParticleOperator()` ... constructor for setting the default values.
"""
function TwoParticleOperator()
TwoParticleOperator( 0, Basics.plus, false)
end
# `Base.show(io::IO, op::TwoParticleOperator)` ... prepares a proper printout of the op::TwoParticleOperator.
function Base.show(io::IO, op::TwoParticleOperator)
sa = "Two-particle operator O^($(op.rank) $(op.parity))"
if op.sameOrbitalSet sa = sa * ", defined for CSF from the same orbital set."
else sa = sa * ", defined for CSF from different orbital sets."
end
println(io, sa)
end
"""
`struct SpinAngular.Coefficient1p`
... a struct for defining a single spin-angular coefficient for a reduced one-particle matrix element <a || o^(L) || b>.
+ nu ::Int64 ... Rank (L or nu) of the single-particle interaction strength.
+ a ::Subshell ... Left-hand subshell (orbital).
+ b ::Subshell ... Right-hand subshell (orbital).
+ T ::Float64 ... (Value of) spin-angular coefficient.
"""
struct Coefficient1p
nu ::Int64
a ::Subshell
b ::Subshell
T ::Float64
end
# `Base.show(io::IO, coeff::Coefficient1p)` ... prepares a proper printout of the coeff::Coefficient1p.
function Base.show(io::IO, coeff::Coefficient1p)
# sa = "\n V^($(coeff.L)) [$(coeff.a), $(coeff.b)] = $(coeff.v)"
sa = " T^$(coeff.nu) [$(coeff.a), $(coeff.b)] = $(coeff.T)"
print(io, sa)
end
"""
`struct SpinAngular.Coefficient2p`
... a struct for defining a single spin-angular coefficient for a reduced two-particle matrix element <ab || o^(L) || cd>,
such as the Coulomb interaction strength.
+ nu ::Int64 ... Rank (L or nu) of the single-particle interaction strength.
+ a ::Subshell ... Left-hand subshell (orbital).
+ b ::Subshell ... Left-hand subshell (orbital).
+ c ::Subshell ... Right-hand subshell (orbital).
+ d ::Subshell ... Right-hand subshell (orbital).
+ V ::Float64 ... (Value of) spin-angular coefficient.
"""
struct Coefficient2p
nu ::Int64
a ::Subshell
b ::Subshell
c ::Subshell
d ::Subshell
V ::Float64
end
"""
`struct SpinAngular.QspaceTerm`
... a struct for defining a subshell term/state |j (nu) alpha Q J> == |j (nu) Q J Nr> for a subshell with well-defined j.
+ j ::AngularJ64 ... subshell j
+ Q ::AngularJ64 ... quasi-spin
+ J ::AngularJ64 ... total J of subshell term
+ Nr ::Int64 ... Additional quantum number Nr = 0,1,2.
+ min_odd ::Int64 ... the minimal limits of the subshell terms for odd number operators in second quantization
+ max_odd ::Int64 ... the maximal limits of the subshell terms for odd number operators in second quantization
+ min_even ::Int64 ... the minimal limits of the subshell terms for even number operators in second quantization
+ max_even ::Int64 ... the maximal limits of the subshell terms for even number operators in second quantization
"""
struct QspaceTerm
j ::AngularJ64
Q ::AngularJ64
J ::AngularJ64
Nr ::Int64
min_odd ::Int64
max_odd ::Int64
min_even ::Int64
max_even ::Int64
end
# `Base.show(io::IO, term::QspaceTerm)` ... prepares a proper printout of term::QspaceTerm.
function Base.show(io::IO, term::QspaceTerm)
if term.Nr == 0
sa = "|$(term.j) $(term.Q) $(term.J) min_odd=$(term.min_odd) max_odd=$(term.max_odd) " *
"min_even=$(term.min_even) max_even=$(term.max_even)>"
else
sa = "|$(term.j) $(term.Q) $(term.J); Nr=$(term.Nr) min_odd=$(term.min_odd) " *
"max_odd=$(term.max_odd) min_even=$(term.min_even) max_even=$(term.max_even) >"
end
println(io, sa)
end
# `Base.show(io::IO, coeff::Coefficient2p)` ... prepares a proper printout of the coeff::Coefficient2p.
function Base.show(io::IO, coeff::Coefficient2p)
sa = "\n V^($(coeff.nu)) [$(coeff.a), $(coeff.b)| $(coeff.c), $(coeff.d)] = $(coeff.V)"
print(io, sa)
end
"""
`struct SpinAngular.SchemeEta`
... to define various singleton structs in order to distinguish between different irreducible tensors
and matrix elements.
+ Eta_a ... a^(qj)_mq
+ Eta_W ... W^(k_12) = [a x a]
+ Eta_aW ... W^(k_12, k_2) = [a1 x [a2 x a3]]
+ Eta_Wa ... W^(k_12, k_2) = [[a1 x a2] x a3]
+ Eta_WW ... W^(kk0) = [[a1 x a2]^k x [a3 x a4]^k]^0
"""
struct SchemeEta_a end
struct SchemeEta_W end
struct SchemeEta_aW end
struct SchemeEta_Wa end
struct SchemeEta_WW end
"""
`struct SpinAngular.Diagram`
... to defines various singleton() structs in order to distinguish between different coupling schemes of the
matrix elements.
"""
struct DiagramC01 end
struct DiagramC02 end
struct DiagramC03 end
struct DiagramC04 end
struct DiagramC05 end
#======= This can likely be deleted soon.
"""
`struct SpinAngular.SubshellTerm`
... a struct for defining a subshell term/state |j^N (nu) alpha Q J> == |j^N (nu) Q J Nr> for a subshell with well-defined j.
+ j ::AngularJ64 ... subshell j
+ Q ::AngularJ64 ... quasi-spin
+ occupation ::Int64 ... occupation N
+ seniority ::Int64 ... seniority
+ J ::AngularJ64 ... total J of subshell term
+ Nr ::Int64 ... Additional quantum number Nr = 0,1,2.
"""
struct SubshellTerm
j ::AngularJ64
Q ::AngularJ64
occupation ::Int64
seniority ::Int64
J ::AngularJ64
Nr ::Int64
end
# `Base.show(io::IO, term::SubshellTerm)` ... prepares a proper printout of term::SubshellTerm.
function Base.show(io::IO, term::SubshellTerm)
if term.Nr == 0 sa = "|$(term.j)^($(term.occupation)) ($(term.seniority)) $(term.Q) $(term.J)>"
else sa = "|$(term.j)^($(term.occupation)) ($(term.seniority)) $(term.Q) $(term.J); Nr=$(term.Nr)>"
end
println(io, sa)
end
"""
`struct SpinAngular.Lambda`
... a struct for defining a subshell term/state |j (nu) alpha Q J> == |j (nu) Q J Nr> for a subshell with well-defined j.
+ Ji ::AngularJ64 ... J_i
+ Jj ::AngularJ64 ... J_j
+ Jip ::AngularJ64 ... J_i'
+ Jjp ::AngularJ64 ... J_j'
"""
struct Lambda
Ji ::AngularJ64
Jj ::AngularJ64
Jip ::AngularJ64
Jjp ::AngularJ64
end
# `Base.show(io::IO, lambda::Lambda)` ... prepares a proper printout of lambda::Lambda.
function Base.show(io::IO, lambda::Lambda)
sa = "Lambda($(lambda.Ji), $(lambda.Jj); lambda.Jip), $(lambda.Jjp))"
println(io, sa)
end ==================#
include("module-SpinAngular-inc-reducedcoeffs.jl")
#######################################################################################################################
#######################################################################################################################
"""
`SpinAngular.computeCoefficients(op::SpinAngular.OneParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})`
... computes the spin-angular coefficients for the reduced one-particle matrix element
<leftCsf || op^(k) || rightCsf >
if both CSF refer to the same list of subshells; a list of one-particle coefficients coeffs::Array{Coefficient1p,1} is returned.
leftCsf - is the bra confuguration state function in relativistic notation;
rightCsf - is the ket confuguration state function in relativistic notation;
subshells - is explicitly given relativistic subshell list if useStandardSubshells == false.
"""
function computeCoefficients(op::SpinAngular.OneParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})
if op.rank == 0 coeffs = computeCoefficientsScalar(op, leftCsf, rightCsf, subshells)
elseif op.rank > 0 coeffs = computeCoefficientsNonScalar(op, leftCsf, rightCsf, subshells)
else error("stop a")
end
return (coeffs)
end
"""
`SpinAngular.computeCoefficients(op::SpinAngular.TwoParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})`
... computes the spin-angular coefficients for the reduced two-particle matrix element
<leftCsf || op^(k) || rightCsf >
if both CSF refer to the same list of subshells; a Tuple of two lists with one- and two-particle coefficients tpl::Tuple{coeffs1p::Array{Coefficient1p,1}, coeffs2p::Array{Coefficient2p,1}} is returned.
leftCsf - is the bra confuguration state function in relativistic notation;
rightCsf - is the ket confuguration state function in relativistic notation;
subshells - is explicitly given relativistic subshell list if useStandardSubshells == false.
"""
function computeCoefficients(op::SpinAngular.TwoParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})
if op.rank == 0 coeffs = computeCoefficientsScalar(op, leftCsf, rightCsf, subshells)
else error("stop a")
end
return (coeffs)
end
"""
`SpinAngular.computeCoefficientsNonScalar(op::SpinAngular.OneParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})`
... computes the spin-angular coefficients for the reduced (scalar) one-particle matrix element
<leftCsf || op^(0) || rightCsf >
if both CSF refer to the same list of subshells; a list of one-particle coefficients coeffs::Array{Coefficient1p,1} is returned.
leftCsf - is the bra confuguration state function in relativistic notation;
rightCsf - is the ket confuguration state function in relativistic notation;
subshells - is explicitly given relativistic subshell list if useStandardSubshells == false.
"""
function computeCoefficientsNonScalar(op::SpinAngular.OneParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})
coeffs1p = Coefficient1p[]; kj = 2 * op.rank; kj = AngularJ64(kj//2)
if AngularMomentum.triangularDelta(leftCsf.J, kj, rightCsf.J) == 0 return( coeffs1p ) end
wa = 0.0; diff_occ = 0; count = 0; creation = 0; annihilation = 0
# Cycle through all subshells of the bra- and ket-CSF
for (i,shi) in enumerate(subshells)
diff = leftCsf.occupation[i]-rightCsf.occupation[i]
if abs(diff) > 1 return( coeffs1p )
elseif diff != 0
diff_occ = diff_occ + abs(diff); count = count + 1
end
if diff == 1
if creation == 0; creation = i; else return( coeffs1p ) end
elseif diff == -1
if annihilation == 0; annihilation = i; else return( coeffs1p ) end
end
end
if count == 0
for (i,shi) in enumerate(subshells)
Ni = leftCsf.occupation[i]; Nj = rightCsf.occupation[i]
if Nj != 0
ji = Basics.subshell_j(shi)
Ji = leftCsf.subshellJ[i]; Jj = rightCsf.subshellJ[i]
si = leftCsf.seniorityNr[i]; sj = rightCsf.seniorityNr[i]
Qi = qshellTermQ(ji,si); Qj = qshellTermQ(ji,sj)
aT = SpinAngular.getTermNumber(ji, Ni, Qi, Ji)
aTp = SpinAngular.getTermNumber(ji, Nj, Qj, Jj)
if recouplingCheck(leftCsf, rightCsf, kj, i, i, length(subshells)) != 0.0
wa = recoupling1p(leftCsf, rightCsf, kj, i, length(subshells))
if abs(wa) >= 2.0e-10
wa = wa * irreducibleTensor(SchemeEta_W(),aT, Ni, AngularM64(1//2), AngularM64(-1//2), op.rank, aTp, Nj)
if abs(wa) >= 2.0e-10
# Pure one-particle spin-angular coefficient
wa = -wa/sqrt(2. * op.rank + 1.)
# GRASP like spin-angular coefficient
wa = wa * sqrt(Basics.twice(ji) + 1.)
push!(coeffs1p, Coefficient1p(op.rank, shi, shi, wa))
end
end
end
end
end
elseif count == 2 && diff_occ == 2
if creation != annihilation
shii = subshells[creation]; ja = Basics.subshell_j(shii)
shjj = subshells[annihilation]; jb = Basics.subshell_j(shjj)
no_one = min(creation, annihilation); no_two = max(creation, annihilation)
if recouplingCheck(leftCsf, rightCsf, kj, no_one, no_two, length(subshells)) != 0.0
if creation == no_one && annihilation == no_two
j1 = ja; j2 = jb; wa = 1.0
elseif creation == no_two && annihilation == no_one
j1 = jb; j2 = ja
wa = (-1)^Int64((Basics.twice(ja)+Basics.twice(jb)-2 * op.rank + 2)/2)
else error("SpinAngular.computeCoefficientsNonScalar: stop a")
end
#
wa = wa * recoupling1p(leftCsf, rightCsf, j1, j2, kj, no_one, no_two, length(subshells))
if abs(wa) >= 2.0e-10
aN = leftCsf.occupation[creation]; aNp = rightCsf.occupation[creation]
Ji = leftCsf.subshellJ[creation]; Jj = rightCsf.subshellJ[creation]
si = leftCsf.seniorityNr[creation]; sj = rightCsf.seniorityNr[creation]
Qi = qshellTermQ(ja,si) ; Qj = qshellTermQ(ja,sj)
aT = SpinAngular.getTermNumber(ja, aN, Qi, Ji)
aTp = SpinAngular.getTermNumber(ja, aNp, Qj, Jj)
bN = leftCsf.occupation[annihilation]; bNp = rightCsf.occupation[annihilation]
Ji = leftCsf.subshellJ[annihilation]; Jj = rightCsf.subshellJ[annihilation]
si = leftCsf.seniorityNr[annihilation]; sj = rightCsf.seniorityNr[annihilation]
Qi = qshellTermQ(jb,si); Qj = qshellTermQ(jb,sj)
bT = SpinAngular.getTermNumber(jb, bN, Qi, Ji)
bTp = SpinAngular.getTermNumber(jb, bNp, Qj, Jj)
wa = wa * SpinAngular.irreducibleTensor(SchemeEta_a(),aT, aN, AngularM64(1//2), aTp, aNp) *
SpinAngular.irreducibleTensor(SchemeEta_a(),bT, bN,AngularM64(-1//2), bTp, bNp)
if abs(wa) >= 2.0e-10
occup = 0
for i = no_one:no_two-1 occup = occup + leftCsf.occupation[i] end
wa = (-1)^(occup+1) * wa
# Purer_one-particle spin-angular coefficient
wa = -wa/sqrt(2. * op.rank + 1.)
# GRASP like spin-angular coefficient
wa = wa * sqrt(Basics.twice(ja) + 1.)
push!(coeffs1p, Coefficient1p(op.rank, shii, shjj, wa))
end
end
end
end
end
return( coeffs1p )
end
"""
`SpinAngular.computeCoefficientsScalar(op::SpinAngular.OneParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})`
... computes the spin-angular coefficients for the reduced (scalar) one-particle matrix element
<leftCsf || op^(0) || rightCsf >
if both CSF refer to the same list of subshells; a list of one-particle coefficients coeffs::Array{Coefficient1p,1} is returned.
leftCsf - is the bra confuguration state function in relativistic notation;
rightCsf - is the ket confuguration state function in relativistic notation;
subshells - is explicitly given relativistic subshell list if useStandardSubshells == false.
"""
function computeCoefficientsScalar(op::SpinAngular.OneParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})
coeffs1p = Coefficient1p[]
if leftCsf.J != rightCsf.J return( coeffs1p ) end
wa = 0.0; diff_occ = 0; count = 0; creation = 0; annihilation = 0
# Cycle through all subshells of the bra- and ket-CSF
for (i,shi) in enumerate(subshells)
diff = leftCsf.occupation[i]-rightCsf.occupation[i]
if abs(diff) > 1 return( coeffs1p )
elseif diff != 0
diff_occ = diff_occ + abs(diff); count = count + 1
end
if diff == 1
if creation == 0; creation = i; else return( coeffs1p ) end
elseif diff == -1
if annihilation == 0; annihilation = i; else return( coeffs1p ) end
end
end
#
if count == 0
for (i,shi) in enumerate(subshells)
if recouplingCheck(leftCsf, rightCsf, i, i, i, i, length(subshells), 0) != 0.0
Ni = leftCsf.occupation[i]; Nj = rightCsf.occupation[i]
if Nj != 0
ji = Basics.subshell_j(shi)
Ji = leftCsf.subshellJ[i]; Jj = rightCsf.subshellJ[i]
si = leftCsf.seniorityNr[i]; sj = rightCsf.seniorityNr[i]
Qi = qshellTermQ(ji,si); Qj = qshellTermQ(ji,sj)
aT = SpinAngular.getTermNumber(ji, Ni, Qi, Ji)
aTp = SpinAngular.getTermNumber(ji, Nj, Qj, Jj)
if leftCsf == rightCsf
wa = irreducibleTensor(SchemeEta_W(),aT, Ni, AngularM64(1//2), AngularM64(-1//2), 0, aTp, Nj)
else
if recouplingCheck(leftCsf, rightCsf, i, i, i, i, length(subshells), 0) != 0.0
wa = irreducibleTensor(SchemeEta_W(),aT, Ni, AngularM64(1//2), AngularM64(-1//2), 0, aTp, Nj)
end
end
if abs(wa) >= 2.0e-11
# Pure one-particle spin-angular coefficient
wa = -wa/sqrt(Basics.twice(Ji) + 1.)
# GRASP like spin-angular coefficient
# wa = wa * sqrt(Basics.twice(ji) + 1.)
push!(coeffs1p, Coefficient1p(op.rank, shi, shi, wa))
end
end
end
end
elseif count == 2 && diff_occ == 2
if creation != annihilation
shii = subshells[creation]; ja = Basics.subshell_j(shii)
shjj = subshells[annihilation]; jb = Basics.subshell_j(shjj)
if ja == jb
no_one = min(creation, annihilation); no_two = max(creation, annihilation)
if recouplingCheck(leftCsf, rightCsf, no_one, no_two, no_two, no_two, length(subshells), 1) != 0.0
wa = recoupling1p(leftCsf, rightCsf, ja, jb, AngularJ64(0//2), no_one, no_two, length(subshells))
if abs(wa) >= 2.0e-11
aN = leftCsf.occupation[creation]; aNp = rightCsf.occupation[creation]
Ji = leftCsf.subshellJ[creation]; Jj = rightCsf.subshellJ[creation]
si = leftCsf.seniorityNr[creation]; sj = rightCsf.seniorityNr[creation]
Qi = qshellTermQ(ja,si); Qj = qshellTermQ(ja,sj)
aT = SpinAngular.getTermNumber(ja, aN, Qi, Ji)
aTp = SpinAngular.getTermNumber(ja, aNp, Qj, Jj)
bN = leftCsf.occupation[annihilation]; bNp = rightCsf.occupation[annihilation]
Ji = leftCsf.subshellJ[annihilation]; Jj = rightCsf.subshellJ[annihilation]
si = leftCsf.seniorityNr[annihilation]; sj = rightCsf.seniorityNr[annihilation]
Qi = qshellTermQ(jb,si); Qj = qshellTermQ(jb,sj)
bT = SpinAngular.getTermNumber(jb, bN, Qi, Ji)
bTp = SpinAngular.getTermNumber(jb, bNp, Qj, Jj)
wa = wa * SpinAngular.irreducibleTensor(SchemeEta_a(),aT, aN, AngularM64(1//2), aTp, aNp) *
SpinAngular.irreducibleTensor(SchemeEta_a(),bT, bN,AngularM64(-1//2), bTp, bNp)
if abs(wa) >= 2.0e-11
occup = 0
for i = no_one:no_two-1 occup = occup + leftCsf.occupation[i] end
wa = (-1)^(occup+1) * wa
# Purer_one-particle spin-angular coefficient
wa = -wa
# GRASP like spin-angular coefficient
# wa = wa * sqrt(Basics.twice(ja) + 1.)
push!(coeffs1p, Coefficient1p(op.rank, shii, shjj, wa))
end
end
end
end
end
end
return( coeffs1p )
end
"""
`SpinAngular.computeCoefficientsScalar(op::SpinAngular.TwoParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})`
... computes the spin-angular coefficients for the reduced (scalar) two-particle matrix element
<leftCsf || op^(0) || rightCsf >
a Tuple of two lists with one- and two-particle coefficients tpl::Tuple{coeffs1p::Array{Coefficient1p,1}, coeffs2p::Array{Coefficient2p,1}} is returned.
leftCsf - is the bra confuguration state function in relativistic notation;
rightCsf - is the ket confuguration state function in relativistic notation;
subshells - is explicitly given relativistic subshell list if useStandardSubshells == false.
"""
function computeCoefficientsScalar(op::SpinAngular.TwoParticleOperator, leftCsf::CsfR, rightCsf::CsfR, subshells::Array{Subshell,1})
coeffs2p = Coefficient2p[]
if leftCsf.J != rightCsf.J return( coeffs2p ) end
diff_occ = 0; count = 0
creation_one = 0; creation_two = 0
annihilation_one = 0; annihilation_two = 0
for (i,shi) in enumerate(subshells)
diff = leftCsf.occupation[i] - rightCsf.occupation[i]
if abs(diff) > 2 return( coeffs2p )
elseif diff != 0
diff_occ = diff_occ + abs(diff); count = count + 1
end
if diff == 1
if creation_one == 0; creation_one = i
elseif creation_two == 0; creation_two = i
else return( coeffs2p ) end
elseif diff == 2
if creation_one == 0; creation_one = i; creation_two = i
else return( coeffs2p ) end
elseif diff == -1
if annihilation_one == 0; annihilation_one = i
elseif annihilation_two == 0; annihilation_two = i
else return( coeffs2p ) end
elseif diff == -2
if annihilation_one == 0; annihilation_one = i; annihilation_two = i
else return( coeffs2p ) end
end
end
if count == 0
coeffs2p = twoParticleDiagonal(leftCsf, rightCsf, subshells)
elseif count == 2
if diff_occ == 2
coeffs2p = twoParticleDiffOcc2(leftCsf, rightCsf, creation_one, annihilation_one, subshells)
elseif diff_occ == 4
coeffs2p = twoParticle6(leftCsf, rightCsf, creation_one, annihilation_one, subshells)
end
elseif count == 3
coeffs2p = twoParticle15to18(leftCsf, rightCsf, creation_one,creation_two,annihilation_one,annihilation_two, subshells)
elseif count == 4
coeffs2p = twoParticle19to42(leftCsf, rightCsf, creation_one,creation_two,annihilation_one,annihilation_two, subshells)
end
return( coeffs2p )
end
"""
`SpinAngular.irreducibleTensor(eta::SchemeEta_a, aIndex::Int64, aN::Int64, mq::AngularM64, bIndex::Int64, bN::Int64)`
... computes the submatrix elements
(j^N alpha Q J || a^(qj)_{mq} || j'^N' alpha' Q' J')
as defined by Eq. (7) in Gaigalas et al. (CPC, 2001); a value::Float64 is returned.
eta - defines the irreducible tensoral form a^(qj)_mq;
aIndex - is the state number of the bra subshell in quasispin representation;
aN - is number of electrons in the bra subshell;
mq - is quasispin projection mq;
bIndex - is the state number of the ket subshell in quasispin representation;
aN - is number of electrons in the ket subshell.
"""
function irreducibleTensor(eta::SchemeEta_a, aIndex::Int64, aN::Int64, mq::AngularM64, bIndex::Int64, bN::Int64)
wa = 0.0
if aIndex < 64 && bIndex < 64
aT = qspaceTerms(aIndex); bT = qspaceTerms(bIndex)
if aT.min_odd != bT.min_even return(wa) end
if aN - bN - Basics.twice(mq) != 0 return(wa) end
if AngularMomentum.triangularDelta(aT.Q, AngularJ64(1//2), bT.Q) == 0 return(wa) end
if AngularMomentum.triangularDelta(aT.J, aT.j, bT.J) == 0 return(wa) end
aMQ = qshellTermM(aT.j, aN); bMQ = qshellTermM(bT.j, bN)
if qspacedelta(aT.Q, aMQ) == 0 return(wa) end
if qspacedelta(bT.Q, bMQ) == 0 return(wa) end
wa = - AngularMomentum.ClebschGordan_old(bT.Q, bMQ, AngularJ64(1//2), mq, aT.Q, aMQ)
wa = wa * completlyReducedCfpByIndices(aIndex, bIndex)/sqrt(Basics.twice(aT.Q)+1.)
elseif aIndex > 64 && bIndex > 64
aJ = Int64(round(mod(aIndex,1000)))
if mq.num == -1
j = Int64(round(mod(aIndex/(1000*1000),1000)))
# aQ = Int64(round(mod(aIndex/(1000),1000)))
bJ = Int64(round(mod(bIndex,1000)))
wa = (-1)^Int64((aJ-bJ-j+2*bN)/2) * sqrt(bN * (bJ + 1.0))
elseif mq.num == 1 wa = (-1)^aN * sqrt(aN * (aJ + 1.0))
end
end
return( wa )
end
"""
`SpinAngular.irreducibleTensor(eta::SchemeEta_W, aIndex::Int64, aN::Int64, mLeft::AngularM64, mRight::AngularM64, kj::Int64,
bIndex::Int64, bN::Int64)`
... computes the submatrix elements
(j^N alpha Q J || [a^(qj)_{mq1} x a^(qj)_{mq2}]^kj || j'^N' alpha' Q' J')
as defined by Eq. (8) in Gaigalas et al. (CPC, 2001); a value::Float64 is returned.
eta - defines the irreducible tensoral form [a^(qj)_{mq1} x a^(qj)_{mq2}]^kj;
aIndex - is the state number of the bra subshell in quasispin representation;
aN - is number of electrons in the bra subshell;
mLeft - is quasispin projection mq1;
mRight - is quasispin projection mq2;
bIndex - is the state number of the ket subshell in quasispin representation;
aN - is number of electrons in the ket subshell.
"""
function irreducibleTensor(eta::SchemeEta_W, aIndex::Int64, aN::Int64, mLeft::AngularM64, mRight::AngularM64, kj::Int64,
bIndex::Int64, bN::Int64)
wa = 0.0
if aIndex < 64 && bIndex < 64
aT = qspaceTerms(aIndex); bT = qspaceTerms(bIndex)
if aT.min_even != bT.min_even return( wa ) end
if AngularMomentum.triangularDelta(aT.J, AngularJ64(2*kj//2), bT.J) == 0 return(wa) end
aMQ = qshellTermM(aT.j, aN); bMQ = qshellTermM(bT.j, bN)
if qspacedelta(aT.Q, aMQ) == 0 return(wa) end
if qspacedelta(bT.Q, bMQ) == 0 return(wa) end
if aN - bN - Basics.twice(mLeft) - Basics.twice(mRight) != 0 return(wa) end
if mLeft == mRight
# cases a * a and a^+ * a^+
if (-1)^(kj) == 1
if AngularMomentum.triangularDelta(aT.Q, AngularJ64(2//2), bT.Q) == 0 return(wa) end
mtot_num = mLeft.num + mRight.num
mtot = AngularM64(mtot_num//mLeft.den)
wa = AngularMomentum.ClebschGordan_old(bT.Q, bMQ, AngularJ64(2//2), mtot, aT.Q, aMQ)
if abs(wa) < 1.0e-11 return( wa ) end
wa = wa * completelyReducedWkk(aIndex, bIndex, 1, kj)
if abs(wa) < 1.0e-11 return( wa ) end
wa = wa / sqrt(Basics.twice(aT.Q) + 1.)
end
else
# cases a * a^+ and a^+ * a
if kj == 0
if aIndex == bIndex
if mLeft == AngularM64(1//2)
wa = - aN
else
wa = Basics.twice(aT.j)+1. - aN
end
wa = wa * sqrt((Basics.twice(aT.J)+1.)/(Basics.twice(aT.j)+1.))
end
else
if (-1)^(kj) == 1 kq = 1
else kq = 0
end
if AngularMomentum.triangularDelta(aT.Q, AngularJ64((2*kq)//2), bT.Q) == 0 return(wa) end
mtot_num = mLeft.num + mRight.num
mtot = AngularM64(mtot_num//mLeft.den)
wa = AngularMomentum.ClebschGordan_old(bT.Q, bMQ, AngularJ64((2*kq)//2), mtot, aT.Q, aMQ)
if abs(wa) < 1.0e-11 return( wa ) end
wa = wa * completelyReducedWkk(aIndex, bIndex, kq, kj)
if abs(wa) < 1.0e-11 return( wa ) end
wa = wa / sqrt((Basics.twice(aT.Q) + 1.) * 2.)
if mLeft == AngularM64(-1//2) && (-1)^(kj) == -1 wa = -wa end
end
end
elseif aIndex > 64 && bIndex > 64
if aN > 2 error("SpinAngular.irreducibleTensor SchemeEta_W: stop a")
elseif bN > 2 error("SpinAngular.irreducibleTensor SchemeEta_W: stop b")
elseif mLeft.num + mRight.num + bN != aN error("SpinAngular.irreducibleTensor SchemeEta_W: stop c")
end
j = Int64(round(mod(aIndex/(1000*1000),1000)))
aJ = Int64(round(mod(aIndex,1000))); bJ = Int64(round(mod(bIndex,1000)))
rN = bN + mRight.num
if rN == 0 Q = Int64((j+1)/2); minRun = 0; maxRun = 0
elseif rN == 1 Q = Int64(((j+1)/2)-1); minRun = j; maxRun = j
elseif rN == 2 minRun = 0; maxRun = 2*j - 2
elseif error("SpinAngular.irreducibleTensor SchemeEta_W: stop d")
end
for runI = minRun:4:maxRun
rJ = AngularJ64(runI//2)
if rN == 2
if rJ == 0 Q = Int64((j+1)/2)
else Q = Int64(((j+1)/2)-2)
end
end
rIndex = ((j*1000) + Q)*1000 + runI
wa = wa + irreducibleTensor(SchemeEta_a(), aIndex, aN, mLeft, rIndex, rN) *
irreducibleTensor(SchemeEta_a(), rIndex, rN, mRight, bIndex, bN) *
AngularMomentum.Wigner_6j(AngularJ64(j//2), AngularJ64(j//2), kj, AngularJ64(bJ//2), AngularJ64(aJ//2), rJ)
end
wa = (-1)^Int64((aJ+bJ+2*kj)/2) * wa * sqrt(2.0*kj + 1.0)
end
return( wa )
end
"""
`SpinAngular.irreducibleTensor(eta::SchemeEta_WW, aIndex::Int64, aN::Int64, mLeft1::AngularM64, mRight1::AngularM64,
mLeft2::AngularM64, mRight2::AngularM64, kj::Int64, bIndex::Int64, bN::Int64)`
... computes the submatrix elements
( j^N QJ || [ W^(k_j) * W^(k_j)]^(0) || j^N' QJ)
for subshells with j = 1/2, 3/2, 5/2, 7/2, 9/2 and for j > 9/2 with N = 0, 1, 2; a value::Float64 is returned.
eta - defines the irreducible tensoral form [ W^(k_j) * W^(k_j)]^(0);
aIndex - is the state number of the bra subshell in quasispin representation;
aN - is number of electrons in the bra subshell;
mLeft1 - is quasispin projection mq1;
mRight1 - is quasispin projection mq2;
mLeft2 - is quasispin projection mq3;
mRight2 - is quasispin projection mq4;
bIndex - is the state number of the ket subshell in quasispin representation;
aN - is number of electrons in the ket subshell.
"""
function irreducibleTensor(eta::SchemeEta_WW, aIndex::Int64, aN::Int64, mLeft1::AngularM64, mRight1::AngularM64,
mLeft2::AngularM64, mRight2::AngularM64, kj::Int64, bIndex::Int64, bN::Int64)
wa = 0.0
if aIndex < 64 && bIndex < 64
aT = qspaceTerms(aIndex); bT = qspaceTerms(bIndex)
if aT.J != aT.J return( wa ) end
if aT.min_even != bT.min_even return( wa ) end
if aT.max_even != bT.max_even return( wa ) end
aMQ = qshellTermM(aT.j, aN); bMQ = qshellTermM(bT.j, bN)
if qspacedelta(aT.Q, aMQ) == 0 return( wa ) end
if qspacedelta(bT.Q, bMQ) == 0 return( wa ) end
if aN-bN-Basics.twice(mLeft1)-Basics.twice(mRight1)-Basics.twice(mLeft2)-Basics.twice(mRight2) != 0 return( wa ) end
for rIndex = aT.min_even:aT.max_even
rT = qspaceTerms(rIndex)
rN = Int64(bN+Basics.twice(mLeft2)+Basics.twice(mRight2))
rMQ = qshellTermM(rT.j, rN)
if qspacedelta(rT.Q, rMQ) != 0
wa6j = AngularMomentum.Wigner_6j(kj, kj, 0, bT.J, aT.J, rT.J)
if wa6j != 0.0
wairr = irreducibleTensor(SchemeEta_W(),aIndex, aN, mLeft1, mRight1, kj, rIndex, rN) *
irreducibleTensor(SchemeEta_W(),rIndex, rN, mLeft2, mRight2, kj, bIndex, bN)
wa = wa + (-1)^Int64((2*kj-Basics.twice(aT.J)+Basics.twice(rT.J))/2)* wairr
end
end
end
wa = wa/sqrt((2*kj + 1.)*(Basics.twice(aT.J) + 1.))
elseif aIndex > 64 && bIndex > 64
if aIndex != bIndex return( wa ) end
if aN > 2 error("SpinAngular.irreducibleTensor SchemeEta_WW: stop a")
elseif bN > 2 error("SpinAngular.irreducibleTensor SchemeEta_WW: stop b")
elseif mLeft1.num + mRight1.num + mLeft2.num + mRight2.num + bN != aN error("SpinAngular.irreducibleTensori SchemeEta_WW: stop c")
end
j = Int64(round(mod(aIndex/(1000*1000),1000)))
aJ = Int64(round(mod(aIndex,1000))); bJ = Int64(round(mod(bIndex,1000)))
rN = bN + mLeft2.num + mRight2.num
if rN == 0 Q = Int64((j+1)/2); minRun = 0; maxRun = 0
elseif rN == 1 Q = Int64(((j+1)/2)-1); minRun = j; maxRun = j
elseif rN == 2 minRun = 0; maxRun = 2*j - 2
elseif error("SpinAngular.irreducibleTensor SchemeEta_WW: stop d")
end
for runI = minRun:4:maxRun
rJ = AngularJ64(runI//2)
if rN == 2
if rJ == 0 Q = Int64((j+1)/2)
else Q = Int64(((j+1)/2)-2)
end
end
rIndex = ((j*1000) + Q)*1000 + runI
wa = wa + irreducibleTensor(SchemeEta_W(),aIndex, aN, mLeft1, mRight1, kj, rIndex, rN) *
irreducibleTensor(SchemeEta_W(),rIndex, rN, mLeft2, mRight2, kj, bIndex, bN) *
AngularMomentum.Wigner_6j(kj, kj, 0, AngularJ64(bJ//2), AngularJ64(aJ//2), rJ)
end
wa = (-1)^Int64((aJ+bJ)/2) * wa
end
return( wa )
end
"""
`SpinAngular.irreducibleTensor(eta::SchemeEta_aW,aIndex::Int64, aN::Int64, mLeft1::AngularM64, mLeft2::AngularM64,
mRight2::AngularM64, kj1::Int64, kj::AngularJ64, bIndex::Int64, bN::Int64)`
... computes the submatrix elements
( j^N QJ || [ a^(q j)_mq * W^(kj1)]^(kj) || j^N' QJ)
for subshells with j = 1/2, 3/2, 5/2, 7/2, 9/2 and for j > 9/2 with N = 0, 1, 2; a value::Float64 is returned.
eta - defines the irreducible tensoral form [ a^(q j)_mq * W^(kj1)]^(kj);
aIndex - is the state number of the bra subshell in quasispin representation;
aN - is number of electrons in the bra subshell;
mLeft1 - is quasispin projection mq;
mLeft2 - is quasispin projection mq3;
mRight2 - is quasispin projection mq4;
bIndex - is the state number of the ket subshell in quasispin representation;
aN - is number of electrons in the ket subshell.
"""
function irreducibleTensor(eta::SchemeEta_aW,aIndex::Int64, aN::Int64, mLeft1::AngularM64, mLeft2::AngularM64,
mRight2::AngularM64, kj1::Int64, kj::AngularJ64, bIndex::Int64, bN::Int64)
wa = 0.0; kkj1 = 2* kj1; kj1a =AngularJ64(kkj1//2)
if aIndex < 64 && bIndex < 64
aT = qspaceTerms(aIndex); bT = qspaceTerms(bIndex)
if AngularMomentum.triangularDelta(aT.J, kj, bT.J) == 0 return(wa) end
if aT.min_odd != bT.min_even return( wa ) end
if aT.max_odd != bT.max_even return( wa ) end
aMQ = qshellTermM(aT.j, aN); bMQ = qshellTermM(bT.j, bN)
if qspacedelta(aT.Q, aMQ) == 0 return( wa ) end
if qspacedelta(bT.Q, bMQ) == 0 return( wa ) end
if aN-bN-Basics.twice(mLeft1)-Basics.twice(mLeft2)-Basics.twice(mRight2) != 0 return( wa ) end
for rIndex=aT.min_odd:aT.max_odd
rT = qspaceTerms(rIndex)
rN = Int64(bN+Basics.twice(mLeft2)+Basics.twice(mRight2))
rMQ = qshellTermM(rT.j, rN)
if qspacedelta(rT.Q, rMQ) != 0
wa6j = AngularMomentum.Wigner_6j(aT.j, kj1a, kj, bT.J, aT.J, rT.J)
if wa6j != 0.0
wb = irreducibleTensor(SchemeEta_a(), aIndex, aN, mLeft1, rIndex, rN)
if abs(wb) >= 2.0e-10
wb = wb * wa6j * irreducibleTensor(SchemeEta_W(), rIndex, rN, mLeft2, mRight2, kj1, bIndex, bN)
wa = wa + wb
end
end
end
end
wa = wa * sqrt(Basics.twice(kj) + 1.0)
wa = (-1)^Int64((Basics.twice(kj) + Basics.twice(aT.J) + Basics.twice(bT.J))/2) * wa
elseif aIndex > 64 && bIndex > 64
if aN > 2 error("SpinAngular.irreducibleTensor SchemeEta_aW: stop a")
elseif bN > 2 error("SpinAngular.irreducibleTensor SchemeEta_aW: stop b")
elseif mLeft1.num + mLeft2.num + mRight2.num + bN != aN error("SpinAngular.irreducibleTensori SchemeEta_aW: stop c")
end
j = Int64(round(mod(aIndex/(1000*1000),1000)))
aJ = Int64(round(mod(aIndex,1000))); bJ = Int64(round(mod(bIndex,1000)))
rN = bN + mLeft2.num + mRight2.num
if rN == 0 Q = Int64((j+1)/2); minRun = 0; maxRun = 0
elseif rN == 1 Q = Int64(((j+1)/2)-1); minRun = j; maxRun = j
elseif rN == 2 minRun = 0; maxRun = 2*j - 2
elseif error("SpinAngular.irreducibleTensor SchemeEta_aW: stop d")
end
for runI = minRun:4:maxRun
rJ = AngularJ64(runI//2)
if rN == 2
if rJ == 0 Q = Int64((j+1)/2)
else Q = Int64(((j+1)/2)-2)
end
end
rIndex = ((j*1000) + Q)*1000 + runI
wa = wa + irreducibleTensor(SchemeEta_a(), aIndex, aN, mLeft1, rIndex, rN) *
irreducibleTensor(SchemeEta_W(),rIndex, rN, mLeft2, mRight2, kj1, bIndex, bN) *
AngularMomentum.Wigner_6j(AngularJ64(j//2), kj1a, kj, AngularJ64(bJ//2), AngularJ64(aJ//2), rJ)
end
wa = (-1)^Int64((aJ+bJ+Basics.twice(kj))/2) * sqrt(Basics.twice(kj) + 1.0) * wa
end
return( wa )
end
"""
`SpinAngular.irreducibleTensor(eta::SchemeEta_Wa, aIndex::Int64, aN::Int64, mLeft1::AngularM64, mRight1::AngularM64,
mLeft2::AngularM64, kj1::Int64, kj::AngularJ64, bIndex::Int64, bN::Int64)`
... computes the submatrix elements
( j^N QJ || [ W^(kj1) * a^(q j)_mq ]^(kj) || j^N' QJ)
for subshells with j = 1/2, 3/2, 5/2, 7/2, 9/2 and for j > 9/2 with N = 0, 1, 2; a value::Float64 is returned.
eta - defines the irreducible tensoral form [ W^(kj1) * a^(q j)_mq ]^(kj);
aIndex - is the state number of the bra subshell in quasispin representation;
aN - is number of electrons in the bra subshell;
mLeft1 - is quasispin projection mq1;
mRight1 - is quasispin projection mq2;
mRight2 - is quasispin projection mq;
bIndex - is the state number of the ket subshell in quasispin representation;
aN - is number of electrons in the ket subshell.
"""
function irreducibleTensor(eta::SchemeEta_Wa, aIndex::Int64, aN::Int64, mLeft1::AngularM64, mRight1::AngularM64,
mLeft2::AngularM64, kj1::Int64, kj::AngularJ64, bIndex::Int64, bN::Int64)
wa = 0.0; kkj1 = 2* kj1; kj1a =AngularJ64(kkj1//2)
if aIndex < 64 && bIndex < 64
aT = qspaceTerms(aIndex); bT = qspaceTerms(bIndex)
if AngularMomentum.triangularDelta(aT.J, kj, bT.J) == 0 return(wa) end
if aT.min_even != bT.min_odd return( wa ) end
if aT.max_even != bT.max_odd return( wa ) end
aMQ = qshellTermM(aT.j, aN); bMQ = qshellTermM(bT.j, bN)
if qspacedelta(aT.Q, aMQ) == 0 return( wa ) end
if qspacedelta(bT.Q, bMQ) == 0 return( wa ) end
if aN-bN-Basics.twice(mLeft1)-Basics.twice(mRight1)-Basics.twice(mLeft2) != 0 return( wa ) end
for rIndex = aT.min_even:aT.max_even
rT = qspaceTerms(rIndex)
rN = Int64(bN+Basics.twice(mLeft2))
rMQ = qshellTermM(rT.j, rN)
if qspacedelta(rT.Q, rMQ) != 0
wa6j = AngularMomentum.Wigner_6j(kj1a, aT.j, kj, bT.J, aT.J, rT.J)
if wa6j != 0.0
wb = irreducibleTensor(SchemeEta_a(), rIndex, rN, mLeft2, bIndex, bN)
if abs(wb) >= 2.0e-10
wb = wb * wa6j * irreducibleTensor(SchemeEta_W(), aIndex, aN, mLeft1, mRight1, kj1, rIndex, rN)
wa = wa + wb
end
end
end
end
wa = wa * sqrt(Basics.twice(kj) + 1.0)
wa = (-1)^Int64((Basics.twice(kj) + Basics.twice(aT.J) + Basics.twice(bT.J))/2) * wa
elseif aIndex > 64 && bIndex > 64
if aN > 2 error("SpinAngular.irreducibleTensor SchemeEta_Wa: stop a")
elseif bN > 2 error("SpinAngular.irreducibleTensor SchemeEta_Wa: stop b")
elseif mLeft1.num + mRight1.num + mLeft2.num + bN != aN error("SpinAngular.irreducibleTensor SchemeEta_Wa: stop c")
end
j = Int64(round(mod(aIndex/(1000*1000),1000)))
aJ = Int64(round(mod(aIndex,1000))); bJ = Int64(round(mod(bIndex,1000)))
rN = bN + mLeft2.num
if rN == 0 Q = Int64((j+1)/2); minRun = 0; maxRun = 0
elseif rN == 1 Q = Int64(((j+1)/2)-1); minRun = j; maxRun = j
elseif rN == 2 minRun = 0; maxRun = 2*j - 2
elseif error("SpinAngular.irreducibleTensor SchemeEta_Wa: stop d")
end
for runI = minRun:4:maxRun
rJ = AngularJ64(runI//2)
if rN == 2
if rJ == 0 Q = Int64((j+1)/2)
else Q = Int64(((j+1)/2)-2)
end
end
rIndex = ((j*1000) + Q)*1000 + runI
wa = wa + irreducibleTensor(SchemeEta_W(),aIndex, aN, mLeft1, mRight1, kj1, rIndex, rN) *
irreducibleTensor(SchemeEta_a(), rIndex, rN, mLeft2, bIndex, bN) *
AngularMomentum.Wigner_6j(kj1a, AngularJ64(j//2), kj, AngularJ64(bJ//2), AngularJ64(aJ//2), rJ)
end
wa = (-1)^Int64((aJ+bJ+Basics.twice(kj))/2) * sqrt(Basics.twice(kj) + 1.0) * wa
end
return( wa )
end
"""
`SpinAngular.normalForm(ia::Int64, ib::Int64, ic::Int64)`
... odering operators of second quantisation in norma form; a value::Int64 is returned.
ia, ib, ic - is the place of operators in the set of operators of second quantisatio.
"""
function normalForm(ia::Int64, ib::Int64, ic::Int64)
wa = Array{Int64}(undef,3); for i=1:3 wa[i] = (0) end
wa[1] = ia; wa[3] = ia
if wa[1] > ib wa[1] = ib end
if wa[3] < ib wa[3] = ib end
if wa[1] > ic wa[1] = ic end
if wa[3] < ic wa[3] = ic end
if ia > wa[1] && ia < wa[3] wa[2] = ia end
if ib > wa[1] && ib < wa[3] wa[2] = ib end
if ic > wa[1] && ic < wa[3] wa[2] = ic end
return( wa )
end
"""
`SpinAngular.normalPhase(ia::Int64, ib::Int64, ic::Int64, id::Int64)`
... compute the phase factor from odering operators of second quantisation in norma form; a wa::Int64 is returned.
ia, ib, ic, id - is the place of operators in the set of operators of second quantisatio.
"""
function normalPhase(ia::Int64, ib::Int64, ic::Int64, id::Int64)
wa = 1
if ia > ib wa = -wa end
if ia > ic wa = -wa end
if ia > id wa = -wa end
if ib > ic wa = -wa end
if ib > id wa = -wa end
if ic > id wa = -wa end
return( wa )
end
"""
`SpinAngular.qshellTermM(j::AngularJ64, N::Int64)`
... computes MQ quantum number; an M::Int64 is returned.
j - is angular momentum j for the subshell;
N - is number of electrons in the subshell;
"""
function qshellTermM(j::AngularJ64, N::Int64)
M = Int64(N-0.5*(Basics.twice(j)+1)); return( AngularM64(M//2) )
end
"""
`SpinAngular.qshellTermQ(j::AngularJ64, nu::Int64)`