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Symbolic programming package N Coperatorswith applications to theoretical atomicspectroscopyTo cite this article Rytis Juršnas and Gintaras Merkelis 2012 J Phys Conf Ser 402 012007

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Symbolic programming package NCoperators with

applications to theoretical atomic spectroscopy

Rytis Jursenas Gintaras MerkelisInstitute of Theoretical Physics and Astronomy of Vilnius University A Gostauto 12LT-01108 Vilnius Lithuania

E-mail RytisJursenastfaivult

Abstract A symbolic programming package NCoperators with applications to atomic physicsis introduced The package runs over Mathematica and it implements NCAlgebra thenoncommutative algebra package NCoperators features the algebra of irreducible tensoroperators the second quantization representation the angular momentum theory and theeffective operator approach exploited in many-body perturbation theory including Wickrsquostheorem The comprehensiveness is yet another characteristic feature of the present packageThe generation of expressions is performed in a way as if it were done by hand Althoughthe theoretical atomic spectroscopy is a direct target of NCoperators the package with minormodifications if any is believed to appropriate other areas of theoretical physics as well

1 IntroductionTo this day a number of Mathematica sources featuring the properties of creation andannihilation operator products are observed in literature Nostromo [1] SeQuant [2] Wick[3] Quantum [4] SNEG [5] etc Against the variety of areas of theoretical physics the packagessupply with none of them provide an opportunity to manage the Racah algebra [6ndash8] in afully compatible fashion the capability of constructing the irreducible tensor form of complexoperators that act on the basis of many-electron open-shell wave functions is advantageousin a special manner The package NCoperators is just what one needs to solve this kind oftasks [9] along with many other ones eg the summation of ClebschndashGordan products to the3njndashsymbols [10] Whilst the Racah package [11ndash13] based on Maple is known to handleseveral similar procedures such as the calculation of transformation matrices and recouplingcoefficients evaluation of many-particle matrix elements etc the advantages and distinctivefeatures of NCoperators are that (i) it manages the reduction of the product of more thantwo SU(2)ndashirreducible representations therefore giving an opportunity to calculate the matrixelements on the basis of many-open-shell wave functions (eg six open shells appear even in thesecond-order RayleighndashSchrodinger perturbation theory based on effective operator approach[14]) (ii) it is based on the algebraic manipulations of the quantities considered neither thediagrammatic representations such as the Jucys graphs exploited in the angular momentumtheory [15] though the diagrammatic visualization to some extent is implemented as well (iii)the output of expressions including ClebschndashGordan coefficients and 3njndashsymbols obtainedwithin Mathematica interface fits the standard text-based form (iv) Racah refers to Varshalovichet al [16] while NCoperators implements the sum rules given by Jucys et al [15 17] in many

IUPAP C20 Conference on Computational Physics (CCP 2011) IOP PublishingJournal of Physics Conference Series 402 (2012) 012007 doi1010881742-65964021012007

Published under licence by IOP Publishing Ltd 1

particular cases the difference is in the phase factor and different notions (eg the ClebschndashGordan coefficient versus the Wigner 3jndashsymbol) (v) at last but not least the packageNCoperators allows one to generate expansion terms of the RayleighndashSchrodinger perturbationtheory up to the third-order [18] and afterward express them in the irreducible tensor formTo calculate energy corrections such an operator representation puts into direct action thetechnique that was first proposed by Judd [19] and later extensively developed by Rudzikas etal [20ndash22]

At the present moment the package under consideration is used for private research purposesonly though the source files can be sent directly to anyone concerned with the subject

In this paper we review the package NCoperators from the perspectives of what it is likelyfor the application of techniques exploited in modern theoretical atomic spectroscopy

2 Effective operator approach in RSPTOur proposed formulation of the RayleighndashSchrodinger perturbation theory (RSPT) relies onthe following statement [9 Theorem 27] The nonzero terms of effective Hamiltonian H onthe model space P are generated by a maximum of eight types of the nndashbody parts of waveoperator Ω with respect to the single-electron states of the set In(αβ) for all n = 1

Several results immediately proceed from the present statement First it allows us to reducesignificantly the number of nndashbody parts of Ω denoted Ωn that are generated by using thegeneralized Bloch equation [18] That is to say only 8 of the 9n possible distributions of single-electron states (valence core excited) have non-zero contribution to the expansion series of H Second both approaches the coupled-cluster (CC) and the RSPT are treated similarly by meansof the tensor form of expansion terms ndash it is only the scalar multiplier (also known as the effectivematrix element calculated on the basis of single-particle wave functions) that separates CC andRSPT Therefore once the irreducible tensor form of H is constructed one can easily make thereplacements of routine multipliers depending on whether CC or RSPT approach is imposed onIn particular the package NCoperators generates the terms in the RSPT approximation

Let us clarify the above proposed statement in a more detail Let α and β denote the single-electron states the creation operator aα creates and the annihilation operator adagger

βannihilates

The single-electron states are valence (v) core (c) and excited or virtual (e) Then the modelspace P is constructed by declaring that (A) acP = 0 (B) adaggereP = 0 (C) avP 6= 0 and (D)adaggervP 6= 0 Here P denotes the projection operator that projects vectors in H the NndashelectronHilbert space onto the vectors in P one assumes that the initially given Hamiltonian H actson H whereas the effective Hamiltonian H acts on P That is the effective Hamiltonian reads

H = P HP +infinsumn=1

2summ=1

min(2m2n)sumξ=1

P VmΩnPξ (1)

which is obtained by applying the Wickrsquos theorem ( denotes the normal ordering ξ denotesthe ξndashpair contraction between Vm and Ωn) The quantities V1 and V2 denote the single-particleand two-particle interaction operators The wave operator Ωn is of the form

Ωn =sumIn(αβ)

Qaα1aα2 aαnminus1aαnadaggerβnadaggerβnminus1

adaggerβ2adaggerβ1P ωn(αβ) (2)

for Q is the orthogonal complement of P Here ωn(αβ) is the above mentioned effective nndashparticle matrix element with the energy denominator included The set In(αβ) = α1 α2

αnminus1 αn β1 β2 βnminus1 βn In agreement with items (A)-(D) Ωn is nonzero if and only if theαi states are valence andor excited while the βj states are valence andor core In addition Ωn


2

is zero if and only if the sum of SO(3)ndashirreducible representations lvi + lvi over all i = 1 2 nis even

nsumi=1

(lvi + lvi) equiv 0 mod 2 (3)

Eq (3) may be thought of as an additional parity selection rule that directly follows from items(C)-(D) A subsequent result is the initially proposed statement In agreement with the parityselection rule the wave operator Ωn eq (2) that distinguishes between single- two- three-four-particle effects (n = 1 2 3 4) reads

Ω1 =sumI

(1)1

aeadaggervωev +

sumI

(2)1

avadaggercωvc +

sumI

(3)1

aeadaggercωec (4a)

Ω2 =sumprime

I(458)2

aαaαprimeadaggerβprimeadaggerβωααprimeββprime +

sumI

(3)2

aeavadaggercadaggervωevvc +

sumprime

I(16)2

aeavadaggerβprimeadaggerβωevββprime

+sumprime

I(27)2

aαaαprimeadaggervadaggercωααprimecv (4b)

Ω3 =sumprime

I(14)3

aαaαprimeamicroadaggervprimeprimeadaggervprimeadaggervωααprimemicrovvprimevprimeprime +

sumprime

I(25)3

avavprimeavprimeprimeadaggercadaggerβprimeadaggervωvvprimevprimeprimevβprimec

+sumprime

I(3678)3

aαaαprimeamicroadaggercadaggerβprimeadaggervωααprimemicrovβprimec (4c)

Ω4 =sumprime

I(14)4

aeaαprimeavavprimeprimeadaggervprimeprimeprimea

daggervprimeprimeadaggervprimeadaggervωeαprimevvprimeprimevvprimevprimeprimevprimeprimeprime +

sumprime

I(25)4

avavprimeavprimeprimeavprimeprimeprimeadaggercadaggerβprimeprimeadaggervprimea

daggervωvvprimevprimeprimevprimeprimeprimevvprimeβprimeprimec

+sumprime

I(3678)4

aeaαprimeavavprimeprimeadaggercadaggerβprimeprimeadaggervprimea

daggervωeαprimevvprimeprimevvprimeβprimeprimec (4d)

Here and elsewhere it is assumed that the Greek letters denote all three types of single-electronstates The sums with primes denote the following operations

sumprime

I(458)2

equivsumI

(4)2

δαeδβv +sumI

(5)2

δαvδβc +sumI

(8)2

δαeδβc (5a)

sumprime

I(ab)x

equivsumI

(a)x

δβv +sumI

(b)x

δβc for x = 2 a = 1 b = 6 and x = 3 4 a = 2 b = 5 (5b)

sumprime

I(ab)x

equivsumI

(a)x

δαv +sumI

(b)x

δαe for x = 2 a = 2 b = 7 and x = 4 a = 1 b = 4 (5c)

sumprime

I(14)3

equivsumI

(1)3

δαvδmicroe +sumI

(4)3

δαeδmicrov (5d)

sumprime

I(3678)3

equivsumI

(3)3

δαvδβvδmicroe +sumI

(6)3

δαvδβcδmicroe +sumI

(7)3

δαeδβvδmicrov +sumI

(8)3

δαeδβcδmicrov (5e)


3

sumprime

I(3678)4

equivsumI

(3)4

δαvδβv +sumI

(6)4

δαvδβc +sumI

(7)4

δαeδβv +sumI

(8)4

δαeδβc (5f)

As seen from eq (5) no more than eight types of sets In(αβ) are observed Although each Ωn

contains more terms than those in eq (4) but only the displayed eight types of sets In(αβ) havenonzero contribution to the effective Hamiltonian H eq (1)

In particular one may express the third-order effective Hamiltonian H (3) by

H (3) =sum

Im+nminusξ

2summ=1

4sumn=1

min(2m2n)sumξ=1

P VmΩ(2)n Pξ (6)

where Ω(2)n is given by eq (4) with the effective matrix elements ωn(αβ) replaced by the second-

order effective matrix elements ω(2)n (αβ) Im+nminusξ denotes the set of all single-particle orbitals

Even if eq (3) is exploited there are hundreds of generated Ω(2)n and H (3) terms Therefore

the role of NCoperators becomes essential

3 The generation of expansion termsThe package NCoperators takes into consideration the anticommutation properties of creationand annihilation operators aαi a

daggerβj = δ(αi βj) aαi aβj = 0 adaggerαi a

daggerβj = 0 The realization

is illustrated in Fig 1

Figure 1 Manipulating anticommutation properties of creation and annihilation operators inNCoperators The outputs Out[60] and Out[62] are fully compatible with LATEX

As a representative example let us consider the single-particle operator P V1Ω(2)1 P1 The

interaction operator V1 =sum

αβ aαadaggerβvαβ where vαβ is the single-particle matrix element By

eq (4a) the wave operator

Ω(2)1 =

sumev

aeadaggervω

(2)ev +

sumvc

avadaggercω

(2)vc +

sumec

aeadaggercω

(2)ec (7)

The generation of P V1Ω(2)1 P1 terms is performed in Fig 2 In NCoperators the single-

electron states are designated by


4

ac cc1 cc2 etc adaggerc ca1 ca2 etcav vc1 vc2 etc adaggerv va1 va2 etcae ec1 ec2 etc adaggere ea1 ea2 etc

In a standard output (such as Out[4] in Fig 2) the notation of orbitals is simplified to

c a1 b1 c1 d1 e1 f1 c a2 b2 c2 d2 e2 f2

v m1 n1 p1 q1 k1 l1 v m2 n2 p2 q2 k2 l2e r1 s1 t1 u1 w1 x1 e r2 s2 t2 u2 w2 x2

One can recognize that the function OneContraction[2microν2vc1ca1] correspondsto the term at the sum

sumvc in eq (7) similarly the rest two functions

OneContraction[2microν2ec1ca1] and OneContraction[2microν2ec1va1] correspond tothe terms at

sumec and

sumev respectively In Out[4] the effective matrix element 〈x|veff2 |y〉

denotes ω(2)xy (εy minus εx) εx is the single-electron energy also observed in H0 =

sumα1aα1a

daggerα1εα1

(recall that the Hamiltonian H = H0 + V0 + V1 + V2) Other functions such as KronDelta[]NormalOrder[] MatrixEl[] should be obvious to identify by their names Therefore Out[4]shows us that the generated P V1Ω(2)

1 P1 terms are

Figure 2 The generation of P V1Ω(2)1 P1 terms with NCoperators

P V1Ω(2)1 P1=

sumev

avadaggervvveω

(2)ev minus

sumcv

avadaggervvcvω

(2)vc (8)

The next step is to obtain the scalars ω(2)

αβ This is the most time consuming process

In agreement with the generalized Bloch equation ω(2)

αβrepresents the sum of coefficients

ω(2)iji+jminus1(αβ) obtained from the terms RViΩ(1)

j PminusRΩ(1)j P ViPi+jminus1 R denotes the resolvent

Thus

ω(2)

αβ= ω

(2)111(αβ) + ω

(2)122(αβ) + ω

(2)212(αβ) + ω

(2)223(αβ)


5

In Figs 3-4 the algorithms for obtaining the multipliers ω(2)111 are displayed As seen ω(2)

111consists of nine terms in total

Figure 3 Generating the RV1Ω(1)1 P1 terms

ω(2)111(αβ)(εβ minus εα) =

summicro(αβ)

vαmicroω(1)

microβminussumν(αβ)

vνβω(1)αν

where ω(1)

αβ= vαβ(εβminusεα) is the effective matrix element drawn in Ω(1) If performing analogous

computations for the remaining coefficients ω(2)iji+jminus1 one finds that


sumc

summicro=ve

vcmicro

(ω

(1)

microαcβminus ω(1)

microαβc

)


sumc

summicro=ve

(vcαmicroβ minus vcαβmicro

)ω(1)microc


sumcve

(vcαve minus vcαev

)(ω

(1)

evcβminus ω(1)

evβc

)

where ω(1)αβmicroν = vαβmicroν(εmicro + εν minus εαminus εβ) vαβmicroν denotes a two-particle matrix element Finally

obtained coefficients ω(2)

αβare substituted in eq (8) The same procedure should be carried out

for the rest P VmΩ(2)n Pξ terms in order to establish all terms of H (3) (see eq (6)) The last

step is to represent obtained expressions in the irreducible tensor form


6

Figure 4 Generating the RΩ(1)1 P V1P1 terms

4 The irreducible tensor form of expansion termsAs first demonstrated by Judd [19] the operator aα represents the irreducible tensor operatorof SUJ(2) (in jjndashrepresentation) or SOL(3)times SUS(2) (in LSndashrepresentation) This operator isdenoted by aλα where λ equiv j or λ equiv l 12 Note that in the LSndashrepresentation aλα represents adouble tensor According to this we introduce a general form

P VmΩ(2)n Pξ=

sumΛ

+ΛsumM=minusΛ

sumΓ

OΛM ([w]κ) C(3)

mnξ(ΓΛ) (9)

for OΛ([w]κ) is the irreducible tensor operator of rank Λ this notation fits both representationsThe coefficient C(3)

mnξ(ΓΛ) is SU(2)ndashinvariant The letter Γ denotes additional numbers necessaryto obtain the above expression The letter w labels irreducible representations of S2(m+nminusξ)the symmetric group these are of the type [w] =

[2h21h1

] κ denotes additional numbers if

necessary in order to distinguish different reduction schemes of OΛ([w]κ) As one can easilyobserve the S2(m+nminusξ)ndashirreducible representations [w] label the irreducible tensor operators

OΛ([w]κ) associated with the (m + n minus ξ)ndashparticle operators P VmΩ(2)n Pξ For example

OΛ([12]) =[aλv times aλv

]Λ where aλv denotes the irreducible tensor operator obtained from thetransposed annihilation operator The complete classification of irreducible tensor operatorsobtained by reducing the products of creation and annihilation operators one can find in [10]

The matrix elements of irreducible tensor operators of rank Λ calculated on the basis ofmany-electron wave functions can be found for example in [20 22] For this reason the SU(2)ndashinvariants C(3)

mnξ(ΓΛ) are to be established to complete the task As it appears from eq (9) that

the structure of C(3)mnξ(ΓΛ) depends on ([w]κ) that is to say the reduction scheme In turn the


7

number of possible schemes depends on m+nminusξ Obviously for a fixed m+nminusξ all reductionschemes are equivalent and they are related by recoupling coefficients These coefficients canbe found by exploiting NCoperators

Figure 5 The computation of recoupling coefficient with NCoperators The phase factor canbe reduced by applying the triangle rule

For example the computation of recoupling coefficient


8

(α1α2(α12)α3(α123) α4α5(α45)α6(α456)α|α1α2(α12) α3α4(α34)(α1234) α5α6(α56)α

)is illustrated in Fig 5 The quantity

[j1 j2 jm1 m2 m

]denotes the ClebschndashGordan coefficient for the

tensor product j1 times j2 rarr ja b ed c f

is the 6jndashsymbol In Fig 5 the obtained coefficient must

be multiplied by (minus1)2α4+α5+α6+α56(2α45+1)(2α56+1)12α4 α5 α45α6 α456 α56

due to the recoupling

in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]

Let us give a brief examination of the example studied above in eq (8) For the irreducibletensor operator whose reduction scheme is labeled by the S2ndashirreducible representation [12] theassociated coefficient C(3)

111 is found by pulling out the SU(2)ndashinvariant parts from the following

products of matrix elements (see eq (8))sum

e vveω(2)plusmnev and

sumc vcvω

(2)plusmnvc where the superscript

plusmn indicates the sign of M in eq (9) The final result for the plus sign is

C(3)111(m0Λ) =(minus1)λvminusλv

radic2τ0 + 1

sumx

radic2x+ 1

[τ0 x Λm0 M minusm0 M

]times(

(minus1)Λsum

e

f(τ0λvλe)Ω(2)ev (x)

τ0 x Λλv λv λe

minus (minus1)x

sumc

f(τ0λcλv)Ω(2)vc (x)

τ0 x Λλv λv λc

)

where f(τiλαλβ) = minus(2λα + 1)(2λβ + 1)12[nαλαvτinβλβ] is proportional to the reducedmatrix element of interaction operator vτi i = 0 1 2 This matrix element is calculated on thebasis of spherical harmonics In particular i = 0 befits the interaction drawn in the secondquantized form of V1 in P V1Ω(2)

1 P1 The coefficients Ω(2)

αβ(x) are found from

ω(2)

αβ= (minus1)λβ+mβ

sumx

[λα λβ Λmα minusmβ M

]Ω(2)

αβ(x)

and they contain 27 Goldstone diagrams in total [9]

5 ConclusionWe have reviewed the symbolic programming package NCoperators concentrating on theapplications to many-body perturbation theory We have shown that the package is capableto generate expansion terms up to the third-order RSPT and represent them in the irreducibletensor form by exploiting the angular momentum theory On the basis that makes NCoperatorsbe multifaceted we hope that with minor modifications if such are necessary at all the packagecan be adapted to many other areas that apply the methods briefly considered in the presentpaper

References[1] Bochevarov A D and Sherrill C D 2004 J Chem Phys 121 3374[2] httpwwwfileschemvteduchem-deptvaleevsoftwaresequantsequanthtml

[3] Derevianko A 2010 J Phys B At Mol Opt Phys 43 074001[4] httphomepagecemitesmmxlgomezquantum

[5] Zitko R 2011 Comp Phys Commun 182 2259


9

[6] Racah G 1942 Phys Rev 61 186[7] Racah G 1942 Phys Rev 62 438[8] Fano U and Racah G 1959 Irreducible Tensorial Sets vol 4 (New York Academic Press)[9] Jursenas R and Merkelis G 2010 J Math Phys 51 123512

[10] Jursenas R and Merkelis G 2011 Cent Eur J Phys 9 751[11] Fritzsche S 1997 Comp Phys Commun 103 51[12] Gaigalas G Fritzsche S and Fricke B 2001 Comp Phys Commun 135 219[13] Fritzsche S Inghoff T Bastug T and Tomaselli M 2001 Comp Phys Commun 139 314[14] Jursenas R and Merkelis G 2011 At Data Nucl Data Tables 97 23[15] Jucys A P Levinson Y B and Vanagas V V 1960 Mathematical Apparatus of the Theory of

Angular Momentum [in Russian] vol 3 (Gospolitnauchizdat)[16] Varshalovich D A Moskalev A N and Khersonskii V K 1975 Quantum Theory of Angular

Momentum [in Russian] (Leningrad Nauka)[17] Jucys A P and Bandzaitis A A 1977 Theory of Angular Momentum in Quantum Mechanics

[in Russian] (Vilnius Mokslas)[18] Lindgren I and Morrison J 1982 Atomic Many-Body Theory vol 13 (Springer Series in

Chemical Physics)[19] Judd B R 1963 Operator Techniques in Atomic Spectroscopy (New York McGraw-Hill)[20] Rudzikas Z and Kaniauskas J 1984 Quasispin and Isospin in the Theory of Atom [in

Russian] (Vilnius Mokslas)[21] Kaniauskas J C Simonis V and Rudzikas Z B 1987 J Phys B At Mol Opt Phys 20

3267[22] Rudzikas Z 1997 Theoretical Atomic Spectroscopy (Cambridge Cambridge Univ Press)


10

Symbolic programming package NCoperators with

applications to theoretical atomic spectroscopy

Rytis Jursenas Gintaras MerkelisInstitute of Theoretical Physics and Astronomy of Vilnius University A Gostauto 12LT-01108 Vilnius Lithuania

E-mail RytisJursenastfaivult

Abstract A symbolic programming package NCoperators with applications to atomic physicsis introduced The package runs over Mathematica and it implements NCAlgebra thenoncommutative algebra package NCoperators features the algebra of irreducible tensoroperators the second quantization representation the angular momentum theory and theeffective operator approach exploited in many-body perturbation theory including Wickrsquostheorem The comprehensiveness is yet another characteristic feature of the present packageThe generation of expressions is performed in a way as if it were done by hand Althoughthe theoretical atomic spectroscopy is a direct target of NCoperators the package with minormodifications if any is believed to appropriate other areas of theoretical physics as well

1 IntroductionTo this day a number of Mathematica sources featuring the properties of creation andannihilation operator products are observed in literature Nostromo [1] SeQuant [2] Wick[3] Quantum [4] SNEG [5] etc Against the variety of areas of theoretical physics the packagessupply with none of them provide an opportunity to manage the Racah algebra [6ndash8] in afully compatible fashion the capability of constructing the irreducible tensor form of complexoperators that act on the basis of many-electron open-shell wave functions is advantageousin a special manner The package NCoperators is just what one needs to solve this kind oftasks [9] along with many other ones eg the summation of ClebschndashGordan products to the3njndashsymbols [10] Whilst the Racah package [11ndash13] based on Maple is known to handleseveral similar procedures such as the calculation of transformation matrices and recouplingcoefficients evaluation of many-particle matrix elements etc the advantages and distinctivefeatures of NCoperators are that (i) it manages the reduction of the product of more thantwo SU(2)ndashirreducible representations therefore giving an opportunity to calculate the matrixelements on the basis of many-open-shell wave functions (eg six open shells appear even in thesecond-order RayleighndashSchrodinger perturbation theory based on effective operator approach[14]) (ii) it is based on the algebraic manipulations of the quantities considered neither thediagrammatic representations such as the Jucys graphs exploited in the angular momentumtheory [15] though the diagrammatic visualization to some extent is implemented as well (iii)the output of expressions including ClebschndashGordan coefficients and 3njndashsymbols obtainedwithin Mathematica interface fits the standard text-based form (iv) Racah refers to Varshalovichet al [16] while NCoperators implements the sum rules given by Jucys et al [15 17] in many


Published under licence by IOP Publishing Ltd 1







βannihilates



2summ=1

min(2m2n)sumξ=1

P VmΩnPξ (1)


Ωn =sumIn(αβ)






2


nsumi=1



Ω1 =sumI

(1)1

aeadaggervωev +

sumI

(2)1

avadaggercωvc +

sumI

(3)1

aeadaggercωec (4a)

Ω2 =sumprime

I(458)2


sumI

(3)2


sumprime

I(16)2


+sumprime

I(27)2


Ω3 =sumprime

I(14)3


sumprime

I(25)3


+sumprime

I(3678)3


Ω4 =sumprime

I(14)4



sumprime

I(25)4



+sumprime

I(3678)4




sumprime

I(458)2

equivsumI

(4)2

δαeδβv +sumI

(5)2

δαvδβc +sumI

(8)2

δαeδβc (5a)

sumprime

I(ab)x

equivsumI

(a)x

δβv +sumI

(b)x


sumprime

I(ab)x

equivsumI

(a)x

δαv +sumI

(b)x


sumprime

I(14)3

equivsumI

(1)3

δαvδmicroe +sumI

(4)3

δαeδmicrov (5d)

sumprime

I(3678)3

equivsumI

(3)3


(6)3


(7)3


(8)3



3

sumprime

I(3678)4

equivsumI

(3)4

δαvδβv +sumI

(6)4

δαvδβc +sumI

(7)4

δαeδβv +sumI

(8)4

δαeδβc (5f)




H (3) =sum

Im+nminusξ

2summ=1

4sumn=1

min(2m2n)sumξ=1

P VmΩ(2)n Pξ (6)














Ω(2)1 =

sumev

aeadaggervω

(2)ev +

sumvc

avadaggercω

(2)vc +

sumec

aeadaggercω

(2)ec (7)




4



c a1 b1 c1 d1 e1 f1 c a2 b2 c2 d2 e2 f2





sumec and



sumα1aα1a

daggerα1εα1


1 P1 terms are


P V1Ω(2)1 P1=

sumev

avadaggervvveω

(2)ev minus

sumcv

avadaggervvcvω

(2)vc (8)







Thus

ω(2)

αβ= ω

(2)111(αβ) + ω

(2)122(αβ) + ω

(2)212(αβ) + ω

(2)223(αβ)


5





summicro(αβ)

vαmicroω(1)


vνβω(1)αν

where ω(1)




sumc

summicro=ve

vcmicro

(ω

(1)


microαβc

)


sumc

summicro=ve


)ω(1)microc


sumcve


)(ω

(1)

evcβminus ω(1)

evβc

)







6



P VmΩ(2)n Pξ=

sumΛ

+ΛsumM=minusΛ

sumΓ

OΛM ([w]κ) C(3)

mnξ(ΓΛ) (9)



[2h21h1










7





8



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10







βannihilates



2summ=1

min(2m2n)sumξ=1

P VmΩnPξ (1)


Ωn =sumIn(αβ)






2


nsumi=1



Ω1 =sumI

(1)1

aeadaggervωev +

sumI

(2)1

avadaggercωvc +

sumI

(3)1

aeadaggercωec (4a)

Ω2 =sumprime

I(458)2


sumI

(3)2


sumprime

I(16)2


+sumprime

I(27)2


Ω3 =sumprime

I(14)3


sumprime

I(25)3


+sumprime

I(3678)3


Ω4 =sumprime

I(14)4



sumprime

I(25)4



+sumprime

I(3678)4




sumprime

I(458)2

equivsumI

(4)2

δαeδβv +sumI

(5)2

δαvδβc +sumI

(8)2

δαeδβc (5a)

sumprime

I(ab)x

equivsumI

(a)x

δβv +sumI

(b)x


sumprime

I(ab)x

equivsumI

(a)x

δαv +sumI

(b)x


sumprime

I(14)3

equivsumI

(1)3

δαvδmicroe +sumI

(4)3

δαeδmicrov (5d)

sumprime

I(3678)3

equivsumI

(3)3


(6)3


(7)3


(8)3



3

sumprime

I(3678)4

equivsumI

(3)4

δαvδβv +sumI

(6)4

δαvδβc +sumI

(7)4

δαeδβv +sumI

(8)4

δαeδβc (5f)




H (3) =sum

Im+nminusξ

2summ=1

4sumn=1

min(2m2n)sumξ=1

P VmΩ(2)n Pξ (6)














Ω(2)1 =

sumev

aeadaggervω

(2)ev +

sumvc

avadaggercω

(2)vc +

sumec

aeadaggercω

(2)ec (7)




4



c a1 b1 c1 d1 e1 f1 c a2 b2 c2 d2 e2 f2





sumec and



sumα1aα1a

daggerα1εα1


1 P1 terms are


P V1Ω(2)1 P1=

sumev

avadaggervvveω

(2)ev minus

sumcv

avadaggervvcvω

(2)vc (8)







Thus

ω(2)

αβ= ω

(2)111(αβ) + ω

(2)122(αβ) + ω

(2)212(αβ) + ω

(2)223(αβ)


5





summicro(αβ)

vαmicroω(1)


vνβω(1)αν

where ω(1)




sumc

summicro=ve

vcmicro

(ω

(1)


microαβc

)


sumc

summicro=ve


)ω(1)microc


sumcve


)(ω

(1)

evcβminus ω(1)

evβc

)







6



P VmΩ(2)n Pξ=

sumΛ

+ΛsumM=minusΛ

sumΓ

OΛM ([w]κ) C(3)

mnξ(ΓΛ) (9)



[2h21h1










7





8



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10


nsumi=1



Ω1 =sumI

(1)1

aeadaggervωev +

sumI

(2)1

avadaggercωvc +

sumI

(3)1

aeadaggercωec (4a)

Ω2 =sumprime

I(458)2


sumI

(3)2


sumprime

I(16)2


+sumprime

I(27)2


Ω3 =sumprime

I(14)3


sumprime

I(25)3


+sumprime

I(3678)3


Ω4 =sumprime

I(14)4



sumprime

I(25)4



+sumprime

I(3678)4




sumprime

I(458)2

equivsumI

(4)2

δαeδβv +sumI

(5)2

δαvδβc +sumI

(8)2

δαeδβc (5a)

sumprime

I(ab)x

equivsumI

(a)x

δβv +sumI

(b)x


sumprime

I(ab)x

equivsumI

(a)x

δαv +sumI

(b)x


sumprime

I(14)3

equivsumI

(1)3

δαvδmicroe +sumI

(4)3

δαeδmicrov (5d)

sumprime

I(3678)3

equivsumI

(3)3


(6)3


(7)3


(8)3



3

sumprime

I(3678)4

equivsumI

(3)4

δαvδβv +sumI

(6)4

δαvδβc +sumI

(7)4

δαeδβv +sumI

(8)4

δαeδβc (5f)




H (3) =sum

Im+nminusξ

2summ=1

4sumn=1

min(2m2n)sumξ=1

P VmΩ(2)n Pξ (6)














Ω(2)1 =

sumev

aeadaggervω

(2)ev +

sumvc

avadaggercω

(2)vc +

sumec

aeadaggercω

(2)ec (7)




4



c a1 b1 c1 d1 e1 f1 c a2 b2 c2 d2 e2 f2





sumec and



sumα1aα1a

daggerα1εα1


1 P1 terms are


P V1Ω(2)1 P1=

sumev

avadaggervvveω

(2)ev minus

sumcv

avadaggervvcvω

(2)vc (8)







Thus

ω(2)

αβ= ω

(2)111(αβ) + ω

(2)122(αβ) + ω

(2)212(αβ) + ω

(2)223(αβ)


5





summicro(αβ)

vαmicroω(1)


vνβω(1)αν

where ω(1)




sumc

summicro=ve

vcmicro

(ω

(1)


microαβc

)


sumc

summicro=ve


)ω(1)microc


sumcve


)(ω

(1)

evcβminus ω(1)

evβc

)







6



P VmΩ(2)n Pξ=

sumΛ

+ΛsumM=minusΛ

sumΓ

OΛM ([w]κ) C(3)

mnξ(ΓΛ) (9)



[2h21h1










7





8



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10

sumprime

I(3678)4

equivsumI

(3)4

δαvδβv +sumI

(6)4

δαvδβc +sumI

(7)4

δαeδβv +sumI

(8)4

δαeδβc (5f)




H (3) =sum

Im+nminusξ

2summ=1

4sumn=1

min(2m2n)sumξ=1

P VmΩ(2)n Pξ (6)














Ω(2)1 =

sumev

aeadaggervω

(2)ev +

sumvc

avadaggercω

(2)vc +

sumec

aeadaggercω

(2)ec (7)




4



c a1 b1 c1 d1 e1 f1 c a2 b2 c2 d2 e2 f2





sumec and



sumα1aα1a

daggerα1εα1


1 P1 terms are


P V1Ω(2)1 P1=

sumev

avadaggervvveω

(2)ev minus

sumcv

avadaggervvcvω

(2)vc (8)







Thus

ω(2)

αβ= ω

(2)111(αβ) + ω

(2)122(αβ) + ω

(2)212(αβ) + ω

(2)223(αβ)


5





summicro(αβ)

vαmicroω(1)


vνβω(1)αν

where ω(1)




sumc

summicro=ve

vcmicro

(ω

(1)


microαβc

)


sumc

summicro=ve


)ω(1)microc


sumcve


)(ω

(1)

evcβminus ω(1)

evβc

)







6



P VmΩ(2)n Pξ=

sumΛ

+ΛsumM=minusΛ

sumΓ

OΛM ([w]κ) C(3)

mnξ(ΓΛ) (9)



[2h21h1










7





8



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10



c a1 b1 c1 d1 e1 f1 c a2 b2 c2 d2 e2 f2





sumec and



sumα1aα1a

daggerα1εα1


1 P1 terms are


P V1Ω(2)1 P1=

sumev

avadaggervvveω

(2)ev minus

sumcv

avadaggervvcvω

(2)vc (8)







Thus

ω(2)

αβ= ω

(2)111(αβ) + ω

(2)122(αβ) + ω

(2)212(αβ) + ω

(2)223(αβ)


5





summicro(αβ)

vαmicroω(1)


vνβω(1)αν

where ω(1)




sumc

summicro=ve

vcmicro

(ω

(1)


microαβc

)


sumc

summicro=ve


)ω(1)microc


sumcve


)(ω

(1)

evcβminus ω(1)

evβc

)







6



P VmΩ(2)n Pξ=

sumΛ

+ΛsumM=minusΛ

sumΓ

OΛM ([w]κ) C(3)

mnξ(ΓΛ) (9)



[2h21h1










7





8



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10





summicro(αβ)

vαmicroω(1)


vνβω(1)αν

where ω(1)




sumc

summicro=ve

vcmicro

(ω

(1)


microαβc

)


sumc

summicro=ve


)ω(1)microc


sumcve


)(ω

(1)

evcβminus ω(1)

evβc

)







6



P VmΩ(2)n Pξ=

sumΛ

+ΛsumM=minusΛ

sumΓ

OΛM ([w]κ) C(3)

mnξ(ΓΛ) (9)



[2h21h1










7





8



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10



P VmΩ(2)n Pξ=

sumΛ

+ΛsumM=minusΛ

sumΓ

OΛM ([w]κ) C(3)

mnξ(ΓΛ) (9)



[2h21h1










7





8



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10





8



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10



[j1 j2 jm1 m2 m






in[ α4 α5 α45β4 β5 β45

][ α45 α6 α456β45 β6 β456

]





sumc vcvω




radic2τ0 + 1

sumx

radic2x+ 1


]times(

(minus1)Λsum

e


τ0 x Λλv λv λe

minus (minus1)x

sumc


τ0 x Λλv λv λc

)




ω(2)


sumx


]Ω(2)

αβ(x)







9










10










10

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