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Chemical Physics 327 (2006) 427–433

Geometric spin frustration for isolated plaquettes of the lattices:An extended irreducible tensor operator method

Fan Wang, Zhida Chen *

State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and molecular engineering,

Peking University, Beijing 100871, China

Received 28 February 2006; accepted 17 May 2006Available online 25 May 2006

Abstract

A new strategy to search for the good quantum numbers for the corner-sharing spin systems, as archetypal plaquettes of the lattices,was suggested for the first time in order to study on geometric spin frustration. The calculations on energy spectra by using the irreduc-ible tensor operator method with the new strategy can be much reduced. As representative examples the energy spectra for the spin pen-tamer of the tetrahedron with a centered spin site and the spin heptamer of three corner-sharing equilateral-triangle were examined inorder to confirm efficiency of the new strategy. Through our code, with automatically searching for the good quantum numbers, theprojection operators Siz, Six and Siy matrices in the ground state space for the spin heptamer were reliably constructed.� 2006 Elsevier B.V. All rights reserved.

Keywords: Spin frustration; Irreducible tensor operator method; Good quantum number; Molecular magnetism; Energy spectrum

1. Introduction

Geometrically frustrated magnetic materials [1,2] havereceived much attention over many years since the pioneerwork of Toulouse on spin frustration [3]. Most characteris-tically, the spin frustrated systems remain a paramagneticphase to a freezing temperature TF, which is small on thescale set by the interaction strength, as measured via themagnitude of the Curie–Weiss constant Hcw [4–6]. Morerecently, the geometric spin frustration (GSF) [1,2] is oftendiscussed in the simplest site-sharing frustrated systems-usually triangular, Kagome and pyrochlore lattices (cf.Fig. 1) [2] that have the local spin s on each spin site. Whenspins are treated as collinear vectors in terms of a classicaldescription, it seems that spin on such lattices introducesthe degeneracy in the ground state of their spin arrange-ments [1,2,7,8]. When spin vectors are not forced to be col-

0301-0104/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2006.05.017

* Corresponding author. Tel.: +86 10 62751016; fax: +86 10 62751708.E-mail address: zdchen@pku.edu.cn (Z. Chen).

linear, it is possible to form the compromised noncollinearspin arrangements [1]. In the classical description theground state of an equilateral-triangle spin trimer (ET tri-mer) with antiferromagnetic (AF) interaction is degeneratefor a half-integer spin s, but is not degenerate for an integerspin s [7,10–12]. In contrast, the ground state of a regular-tetrahedron spin tetramer (RT tetramer) with AF interac-tion is degenerate regardless of whether the local spin s isan integer or a half-integer [7]. In the classical descriptioneach spin site of any plaquettes of the lattices always hasa nonzero spin vector. The above picture of GSF, arisingfrom the classical spin analysis, is not entirely consistentwith the picture from the quantum-mechanical analysissuggested by Kahn and other authors [7,9]. That is, anET trimer with the degenerate ground state has a nonzerospin moment at each spin site, but an RT tetramer doesnot, although its ground state is degenerate. Actually, themagnetic behavior of many other more complicated sys-tems with competing interactions in molecular magneticcommunity can be described in term of GSF describedabove. In the present work, we will show that the ground

(a)

(b)

Fig. 1. (a) Kagome lattice made up of corner-sharing equilateral-trianglespin trimers. (b) Pyrochlore lattice made up of corner-sharing regulartetrahedron spin tetramers.

428 F. Wang, Z. Chen / Chemical Physics 327 (2006) 427–433

states of the plaquettes with more complicated geometricstructures may exhibit more attractive spin structuresthrough the numerical and analysis results in the quan-tum-mechanical term.

It is important to provide a quantum-mechanicaldescription of the spin distribution in the plaquettes as wellas the dependence of the spin distribution on the localspins. In general, there is no simple relation between thedegeneracy and spin distribution of the ground state andthe local spins, especially when different local spin valuesappear in the plaquettes. However, a more extend researchof the features induced by GFS may show us the generalproperty of GFS. Moreover, there has been less work toexplain systematic differences between individual examplesof these magnetic systems. For instance, in the Heisenbergkagome antiferromagnet the thermal fluctuations give riseto entropic ground-state selection [13,14,22], known asorder by disorder [15–19], whereas this phenomenonappears to be absent for some related systems [20,21].Therefore, it is necessary to research in detail these arche-typal plaquettes of the lattices.

It is well known that the number of the spin basis func-tion necessary to describe a spin system of n spin sites is(2s + 1)n, so that the dimension of the associated spinHamiltonian matrix becomes (2s + 1)n · (2s + 1)n. As a

consequence, the diagonalization of the spin Hamiltonianmatrix quickly becomes impossible with increasing valuesof s and n. Hence, from the numerical point of view, thosesystems are quite challenging. In the present work, wesuggest a new strategy to search for the good quantumnumbers, where we partially resolve the problem ofnumerical calculations by the following two steps: thefirst, we use the irreducible tensor operators’ method toget a block diagonalized Hamiltonian matrix. The second,for further block diagonalized Hamiltonian matrix wefind out some good quantum numbers of the systems bya means that we have developed herein. The appropriategood quantum numbers that we choose are spin sumsof partial sites in those systems. This approach is specialeffective for the corner-sharing systems. In general, it isdifficult to probe the eigenvalue spectrum of a spin systemby the analytical means, however, we can get analyticalresults for smaller systems on the basis of the approachmentioned above.

This paper is organized as follows. In Section 2, weintroduce a new strategy to search for the good quantumnumbers and have coded it for further block diagonalizedHamiltonian matrix. In Section 3, we examine the natureof the ground states for some spin systems built up withthe plaquettes by corner sharing in order to testify effi-ciency of the strategy. In Section 4, the validity of thestrategy is further examined. Finally, the conclusion isgiven.

2. Irreducible tensor operator method and the good quantumnumbers

In molecular magnetism, the Hamiltonian of a system isusually described in terms of Heisenberg spin Hamiltonian,and in the present work we follow the tradition. Althoughwe use the Heisenberg Hamiltonian to describe the spinsystems, the methods mentioned herein can easily beextended to other models such as the XY model. In math-ematics, we can gain the energy spectrum of a spin systemby resolving the eigenvalues of the Hamiltonian matrix. Itis well known that there are three methods for eigenvaluesof the Heisenberg Hamiltonian. They are vector coupling(VC), irreducible tensor operators (ITO), and full matrixdiagonalization (FMD) [23].

In 1950, Kambe introduced the VC method [24]. This isthe simplest one in the three methods mentioned above,but limited in special symmetrical systems. However,FMD quickly becomes impossible with increasing the sizeof the system and the spin values s [25]. By considering thesymmetry of functors, ITO can effectively reduce the CPUcost with the block diagonalized Hamiltonian matrix [26].That is to say, the Hamiltonian matrix can be block diag-onalized by the good quantum numbers, in general, whichare the total spin S and the total spin projection Sz in theHeisenberg model. For instance, the Hamiltonian matrixfor a tetrahedron plaquette with s=1/2 spin momentumon each site is a (2s + 1)4 · (2s + 1)4 dimensional square

F. Wang, Z. Chen / Chemical Physics 327 (2006) 427–433 429

matrix, but under considering ITO and the property of theHeisenberg model the matrix can be divided into ninesquare matrices. In order to further reduce the dimensionof the square matrices in ITO we will introduce a newmeans to find out more good quantum numbers, besidethe total spin S and the total spin projection Sz in the Hei-senberg model.

The Heisenberg Hamiltonian of a system with n sites canbe written asbH ¼ �2

Xi;j

J ijSi � Sj: ð1Þ

We first suggest that if the Hamiltonian can be transformedinto

bH ¼Xi

�2Ai

Xk;k0

Sik � Si

k0

!þX

j

bT j �X

k

Sjk

!

¼X

i

�Ai ðX

k

SikÞ

2 �X

k

ðSikÞ

2

" #( )þX

j

ðbT j �X

k

SjkÞ;

ð2Þ

where k, k 0 = 1 . . . ti (t is an integer smaller than n), Ai is aconstant and bT j is a functor which is independent ofSj

kðk ¼ 1 . . . tÞ only, then the system has a set of the goodquantum number Si ¼

PkSi

kðk ¼ 1 . . . tÞ which are inter-changeable with the Hamiltonian bH . That is to say that

1. Interactions Jij(i, j 2 t)between the ti sites mentionedabove are equal to Ai.

2. If p is one site beyond the ti sites, and the interactionsJpj(p 62 ti, j 2 ti) between the site p and anyone of theti sites are equal, that is J pjðp 62 ti; j 2 tiÞ ¼ J pj0

ðp 62 ti; j0 2 tiÞ,

then, the good quantum numbers Si ¼P

kSkðk ¼ 1 . . .tiÞ and S ¼

PiSi can be gained.

As a representative example, we consider a spin hepta-mer made up of two corner-sharing regular-tetrahedronspin tetramers [Fig. 2]. According to Eq. (2) we have

1

5

6

7

2

3

4

Fig. 2. Spin heptamer resulting from sharing a corner between tworegular-tetrahedron spin tetramers.

the good quantum numbers, S234 = S2 + S3 + S4 andS567 = S5 + S6 + S7, where

A1 ¼ J 23 ¼ J 24 ¼ J 34 ð3ÞA2 ¼ J 56 ¼ J 57 ¼ J 67 ð4ÞcT 1 ¼ cT 2 ¼ �2J 1iS1ði ¼ 2; 3; . . . ; 7Þ ð5Þ

Thus, we can further reduce the size of the square matricesin ITO by the good quantum numbers.

The strategy mentioned above is essentially a transfor-mation of the basis of the full configuration interactionHilbert space, followed by reordering of the basis, inwhich the Hamiltonian matrix is further block diagonal-ized. Hence, we can get the Hamiltonian eigenvalues indi-cated by the full functor selection ð bH ; S; Sz; bGÞ, where bGare the functors corresponding to the selected good quan-tum numbers.

3. Energy eigenvalue spectrum of plaquettes

When the spin exchange interactions are described bythe Heisenberg spin HamiltonianbH ¼ �2

Xi;j

J ijSi � Sj; ð1Þ

where the spin sites i and j run over all the interacting pairsof the spin s.

For exhibiting the efficiency of our code of the extendedITO with the automatically selected good quantum numbermentioned above, we discuss a spin pentamer firstly inFig. 3 which is consisted of the tetrahedron with a centeredspin site. Assuming that its four vertexes have the 1/2 spinmoment, and the 1 spin moment for the centered site, andthe interaction strengths J and j are shown as in Fig. 3. Toour knowledge, such a plaquette has not been discussed yetin detail. From the strategy suggested above for the goodquantum number we can know that the total spin S andthe spin functor S 0 = S2 + S3 + S4 + S5, associated withthe four vertex sites of the plaquette, belong to the selectedgood quantum numbers, and any spin sum associated withthree or two among the four vertex sites also is the goodquantum numbers too. Thus, if we choose the basis, indi-cated by S23, S234, S2345, S, where S23 ¼

P3i¼2Si,

S234 ¼P4

i¼2Si, S2345 ¼P5

i¼2Si, and S ¼P5

i¼1Si, actually

35

2

4

J j

1

Fig. 3. A spin pentamer consisted of the tetrahedron with a centered spinsite.

Table 1The energy spectrum E and the corresponding states for the tetrahedronwith a centered spin site

(S23,S234,S2345,S) E

(0,1/2,0,1) 3J

(0,1/2,1,0) J + 4j

(0,1/2,1,1) J + 2j

(0,1/2,1,2) J � 2j

(1,1/2,0,1) 3J

(1,1/2,1,0) J + 4j

(1,1/2,1,1) J + 2j

(1,1/2,1,2) J � 2j

(1,3/2,1,0) J + 4j

(1,3/2,1,1) J + 2j

(1,3/2,1,2) J � 2j

(1,3/2,2,1) �3J + 6j

(1,3/2,2,2) �3J + 2j

(1,3/2,2,3) �3J � 4j

Table 2The energy spectrum E and the corresponding eigenstates of the spinheptamer of three corner-sharing equilateral-triangles spin trimers

(S23,S67,S2367,S) E

(0,0,0,3/2) E = 3J/2(0,0,0,1/2) E1 = 9J/2, E2 = 9J/2(1,0,1,5/2) E = �3J/2(1,0,1,3/2) E1 = 3J/2, E2 = �J/2, E3 = 7J/2(1,0,1,1/2) E1 = 5J/2, E2 = J/2, E3 = 9J/2(0,1,1,5/2) E = �3J/2(0,1,1,3/2) E1 = 3J/2, E2 = �J/2, E3 = 7J/2(0,1,1,1/2) E1 = 5J/2, E2 = J/2, E3 = 9J/2(1,1,0,3/2) E = �5J/2(1,1,0,1/2) E1 = J/2, E2 = J/2(1,1,1,5/2) E = �7J/2(1,1,1,3/2) E1 ¼ J=2;E2 ¼ �2�

ffiffiffi5p� �

J=2;E3 ¼ �2þffiffiffi5p� �

J=2(1,1,1,1/2) E1 ¼ J=2;E2 ¼ 1� 2

ffiffiffi2p� �

J=2;E3 ¼ 1þ 2ffiffiffi2p� �

J=2(1,1,2,7/2) E = �9/2J

(1,1,2,5/2) E1 ¼ J=2;E2 ¼ �3� 2ffiffiffi2p� �

J=2;E3 ¼ �3þ 2ffiffiffi2p� �

J=2(1,1,2,3/2) E1 ¼ J=2;E2 ¼ 2�

ffiffiffiffiffi13p� �

J=2;E3 ¼ 2þffiffiffiffiffi13p� �

J=2(1,1,2,1/2) E = J/2

430 F. Wang, Z. Chen / Chemical Physics 327 (2006) 427–433

the Hamiltonian matrix is diagonal already. The calculatedenergy spectrum and the corresponding states, indicated bythe quantum numbers, are shown in Table 1.

It is evident that the calculated energy spectrum is ana-lytical, rising from the function symbols. From the analyt-ical results we can find out the variety of the energyspectrum E depending on the interaction strength J and j,and can further obtain the magnetic susceptibility byreplacing the corresponding variables in the formula ofHeisenberg–Dirac–van Vleck (HDvV). Of course, as a sim-ple example, the eigenvalue of the pentamer mentionedabove can also be solved easily by vector coupling (VC)[24]. In succession we will put our attention to a more com-plicated system, a spin heptamer consisted of the three cor-ner-sharing equilateral-triangle spin trimers with s = 1/2(Fig. 4). It should be said that it is difficult to solve itseigenvalue spectrum by the coupling vector method [24].According to our strategy mentioned above, the goodquantum numbers are involved in S23, S67, S2367, S, whereS23 ¼

P3i¼2SI , S67 ¼

P7i¼6SI , S2367 ¼

Pi¼2;3;6;7Si, and

S ¼P7

i¼1Si, and thus the Hamiltonian matrix of the systemhas been block diagonalized. The energy spectrum and cor-responding eigenstates are show in Table 2.

For the spin heptamer with s = 1/2 and J < 0, the statedegeneracy of the ground state with the state energyE = 9J/2 is four: (0,1,1,1/2), (1,0,1,1/2) and double(0,0,0,1/2), which is consisted with the previous resultreported by Dai and Whangbo [9]. However, herein we

2

3

1 4

5

7

6

Fig. 4. A spin heptamer consisted of three corner-sharing equilateral-triangle spin trimers.

can probe the eigenvalue spectrum of the corner-sharingspin system by the analytical means and give the detailedcharacterization for the degenerate ground states. It isworth mentioning that to distinguish the double(0,0,0,1/2) states the additional operators are neededbecause the set of (S23,S67,S2367,S) is not a complete oper-ator set.

4. Local spin moments of states and the good quantum

numbers

The classical spin description shows that the presenceof GSF in a plaquette implies the degeneracy of theground state of its spin arrangements, and each spin sitehas a nonzero spin vector. But in the quantum-mechani-cal description the ground state of the spin system doesnot necessarily have a nonzero spin moment at its spinsite [9]. In the following, we will give the spin projectmatrix element, such as Sx, Sy, or Sz components ofevery site, in the space expended by the basis functions,indicated with the good quantum numbers mentionedabove.

A spherical tensor cT km is a set of 2k + 1 components

(m = �k, � k + 1, . . . ,k) that their transform under rota-tion of the coordinate system is in the same way as thespherical functions Ylm(#,u) [26,27]. The spherical ten-sors are also called the irreducible tensors or the irreduc-ible tensor operators of rank k of the rotation group. Infact the rank k of an irreducible tensor can be regardedas an angular momentum quantum number, while theindex m labeling the tensor components is the angularmomentum projection quantum number. A spherical ten-

sor of rank=1 has three components ðcT 11 ;cT 1

0 ;dT 1�1Þ, for

exampledT 1�1 ¼ �

1ffiffiffi2p ðSx � iSyÞ;cT 1

0 ¼ Sz: ð6Þ

F. Wang, Z. Chen / Chemical Physics 327 (2006) 427–433 431

Let wsLM � jsLMi be a spherical function corresponding tothe angular momentum L and its projection M, and then theindex s indicates the number of repeating representationsDL of the rotation group. These functions wsLM � jsLMiare eigenfunctions of the operatorscL2 and bLz . For example,for the function jS234 = 1/2, S2345 = 1,S = 1; S23 = 1,M23 = ±1,0i in the tetrahedron with a centered site men-tioned above, herein S234, S2345, S correspond s, and S23,M23 correspond L, M, respectively. First, let us consider ma-trix elements of the general form hjLM j T k

m j j0L0M 0i beforediscussing matrix elements of the spin projection. This nota-tion unites a set of (2L + 1)(2K + 1)(2L 0 + 1) matrix ele-ments. It is intuitively clear that there are some relationsresulting from the symmetry properties under simultaneoustransformation of wavefunctions and operators cT k

m whenthe coordinate system is rotated. These relations are estab-lished by Wigner–Eckart theorem, which is of fundamentalimportance in the group theory. The mathematical expres-sion of this theorem is as follows [26]

hkLM jcT km jk

0L0M 0i ¼ ð�1Þ2k

ð2Lþ 1Þ1=2hkLkcT kkk0L0ihLM jL0M 0kmi:

ð7Þ

The matrix element of the operator cT km includes two fac-

tors: a Wigner (Clebsch–Gordan) coefficient and the quan-

tity hkLkcT kkk0L0i, called the reduced matrix element. Itshould be noted that the reduced matrix element does notdepend on M, M 0 or m. In other words, this reduced matrixelement is same for the whole set of the matrix elements of

the operator cT km . The dependence on M, M 0 and m is con-

tained entirely in the Wigner coefficients. Hence, the matrixof any component of an irreducible tensor is obtainedfrom, on the one hand, the Wigner coefficientshLMjL 0M 0kmi = hL 0M 0kmjLMi and, on the other hand,the common factor outside the matrix. This characteristic,which expresses the essence of the Wigner–Eckart theorem,is particularly evident in the following matrix form:

cT km ¼

ð�1Þ2k

ð2Lþ 1Þ1=2hkLkcT kkk0L0iOk

m; ð8Þ

where Okm is a matrix composed from the Wigner elements

ðOkmÞMM 0 ¼ hLM jL0M 0kmi: ð9ÞIt follows from the properties of the Wigner coefficients

that a matrix element of the operator cT k is nonzero if thetriangle condition is satisfied, i.e., if jL � L 0j 6 k 6 L + L 0

and m = M �M 0. Such conditions are called selection

rules, since they can select out the nonzero matrix elements.We must point out that the selection rules only can judgethe doubtless vanishing matrix elements, however, the oth-ers are uncertain because the rules cannot give the practicalvalues of the reduced matrix element. Thus, one must cal-culate these values in some approach.

The selection rules can be applied to the total spin andalso the partial spins which all are the good quantum num-

bers. For example, in the system of the spin heptamer ofthe three corner-sharing equilateral-triangles spin trimers(Fig. 4), the matrix elements of Sz on the site 2 betweenthe degenerate ground states

jj ¼ 1 or 2; S23 ¼ 0; S67 ¼ 0; S2367 ¼ 0; S ¼ 1=2; MSi

and

j c ¼ 1; S23 ¼ 0; S67 ¼ 1; S2367 ¼ 1; S ¼ 1=2; MSi

are zero because S23 = 0 in the two ground states men-tioned above, that is to say, L = L 0 = 0 and k = 1 of Sz.According to the triangle condition jL � L 0j 6 k 6 L + L 0,the relation is not satisfied obviously. On the other hand,the Sz matrix elements between the two states jk = 1,S23 = 1, S67 = 0, S2367 = 1, S = 1/2, MSi (MS = �1/2)are nonzero because the triangle condition herein is satis-fied. The matrix of the spin projection Sz on the site 2in the space of the 8 degenerate ground states is showedbelow:

0 0 0 0 0:1443 0 0 0

0 0 0 0 �0:2500 0 0 0

0 0 0 0 0 �0:1443 0 0

0 0 0 0 0 0:2500 0 0

0:1443 �0:2500 0 0 �0:3333 0 0 0

0 0 �0:1443 0:2500 0 0:3333 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

;

where the order of the basis is

jj ¼ 1 or 2;S23 ¼ 0;S67 ¼ 0;S2367 ¼ 0;S ¼ 1=2;MS ¼ �1=2i;jj ¼ 1 or 2;S23 ¼ 0;S67 ¼ 0;S2367 ¼ 0;S ¼ 1=2;MS ¼ þ1=2i;jk ¼ 1;S23 ¼ 1;S67 ¼ 0;S2367 ¼ 1;S ¼ 1=2;MS ¼ �1=2i;jc ¼ 1;S23 ¼ 0;S67 ¼ 1;S2367 ¼ 1;S ¼ 1=2;MS ¼ �1=2i:

In the previous discussion, we have shown a method to ap-ply the selection rule to Sz matrix elements between statesindexed by the good quantum numbers.

It is shown from the above results that the heptamer sys-tem with J < 0 and S = 1/2 shown in Fig. 4 has the fourdegenerate ground states with the state energy E = 9J/2.They are (S23, S67, S2367, S) = (0,1,1,1/2), (1, 0,1,1/2) anddouble (0, 0,0,1/2). When additionally considering the SM

quantum number, there is a total degeneracy of eight forthe ground states: double (0, 0,0,1/2,SM = �1/2), double(0,0,0,1/2, SM = 1/2) and (1, 0,1,1/2, SM = �1/2),(0,1,1,1/2,SM = �1/2). For examining validity of thestrategy of the good quantum number, herein we directlycalculated each matrix element of Sz, Sx, Sy at the Site 1,2, 5, respectively, under the basis set mentioned above. Itis evident that the directly calculated matrices, shownbelow, are consistent with judgment from the selectionrules based on the good quantum number. The new strat-egy of searching for the good quantum number is validand practicable.

432 F. Wang, Z. Chen / Chemical Physics 327 (2006) 427–433

Site 1:

SZ ¼

0 0:2887 0 0 0 0 0 0

0:2887 �0:3333 0 0 0 0 0 0

0 0 0 �0:2887 0 0 0 0

0 0 �0:2887 0:3333 0 0 0 0

0 0 0 0 0:1667 0 0 0

0 0 0 0 0 �0:1667 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

SX ¼

0 0 0 �0:2887 0 0 0 0

0 0 �0:2887 0:3333 0 0 0 0

0 �0:2887 0 0 0 0 0 0

�0:2887 0:3333 0 0 0 0 0 0

0 0 0 0 0 �0:1667 0 0

0 0 0 0 �0:1667 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

SY

i¼

0 0 0 �0:2887 0 0 0 0

0 0 �0:2887 0:3333 0 0 0 0

0 0:2887 0 0 0 0 0 0

0:2887 �0:3333 0 0 0 0 0 0

0 0 0 0 0 �0:1667 0 0

0 0 0 0 0:1667 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Site2:

SZ ¼

0 0 0 0 0:1443 0 0 0

0 0 0 0 �0:2500 0 0 0

0 0 0 0 0 �0:1443 0 0

0 0 0 0 0 0:2500 0 0

0:1443 �0:2500 0 0 �0:3333 0 0 0

0 0 �0:1443 0:2500 0 0:3333 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

SX ¼

0 0 0 0 0 �0:1443 0 0

0 0 0 0 0 0:2500 0 0

0 0 0 0 �0:1443 0 0 0

0 0 0 0 0:2500 0 0 0

0 0 �0:1443 0:2500 0 0:3333 0 0

�0:1443 0:2500 0 0 0:3333 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

SY

i¼

0 0 0 0 0 �0:1443 0 0

0 0 0 0 0 0:2500 0 0

0 0 0 0 0:1443 0 0 0

0 0 0 0 �0:2500 0 0 0

0 0 �0:1443 0:2500 0 0:3333 0 0

0:1443 �0:2500 0 0 �0:3333 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Site 5:

SZ ¼

�0:5000 0 0 0 0 0 0 0

0 0:1667 0 0 0 0 0 0

0 0 0:5000 0 0 0 0 0

0 0 0 �0:1667 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

SX ¼

0 0 0:5000 0 0 0 0 0

0 0 0 �0:1667 0 0 0 0

0:5000 0 0 0 0 0 0 0

0 �0:1667 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

SY

i¼

0 0 0:5000 0 0 0 0 0

0 0 0 �0:1667 0 0 0 0

�0:5000 0 0 0 0 0 0 0

0 0:1667 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

where the basis functions are in the following order: double(0,0,0,1/2,SM = �1/2), double (0,0,0,1/2,SM = 1/2), and(1,0,1,1/2,SM = �1/2), (0, 1,1,1/2,SM = �1/2).

In molecular magnetism for the systems of the geometricspin frustration, the distribution and stability of the localspin moments in the degenerate ground states are still aopen subject. The further work is working in our group.

5. Concluding remarks

In the present work, a strategy of searching for the goodquantum numbers has been introduced for the first time inorder to reduce the calculations based on the irreducibletensor operator method. Meantime we have coded a pro-gram of the extended irreducible tensor operator methodwith automatically searching for the good quantum num-bers. For inspecting efficiency of the program, we expedi-ently obtained the energy eigenvalue spectrum andeigenstates of the tetrahedron with a centered spin siteand the three corner-sharing equilateral-triangles spin sys-tem. It is shown that the calculated state degeneracy ofthe later is consistent with the previous report [9]. For fur-ther examining the validity of the strategy for the goodquantum number, we directly calculated each matrix ele-ment of the spin projections Sz, Sx, Sy at the sites 1, 2,and 5 for the spin heptamer made up of three corner-shar-ing equilateral-triangles, and compared with the judgementfrom the selection rules based on the good quantum num-

F. Wang, Z. Chen / Chemical Physics 327 (2006) 427–433 433

ber. It is found the strategy is valid and practicable. In con-clusion, the new strategy of the good quantum numberscan predigest the tedious calculations on the eigenstatesand eigenvalues in the irreducible tensor operator method,and may be further extended in the more complicated core-sharing spin systems.

Acknowledgements

This work is supported by the National NaturalScience Foundation of China (Nos. 20490210 and20503001) and National Basic Research Program ofChina (2006CB601102).

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