Revista Mexicana de Física 23-(974) 243-256

KETS OF BROKEN SYMMETRY IN TIlE DERlVATIONOF MODEL IIAMILTONIANS'

D.]. Klein

Physics Depttrtment, Universíty 01 Texas

Ausli". Texas 787/2

(Recibido: septiembre 14, 1973)

243

ABSTRACT: Model Hamiltonians are ohen defined 00 a spacc providing a

simpJ e zcro.order picture of the sy.stem undcr investigation.

While the simplest basis kets of this zero-order eigenspace

may be of broken symmetry with respect to the iull group oi theperturbed Hamiltonian, the basis kets10Íten may be transformed

ioto one another by the action of appropriate group e1ements.

In such a case we describe a variational developmcot oC the

model f1amiltonian. This scheme is basedprimarily 00 a singlebasis ker oC broken symmetry; the schone provides an alternare

to [he usual penurbative devdopmmr and is, in principIe, exacto

Particular reference to the Heiscnberg exchange Hamiltonianis made •

•Research supponed by Roben A. Welch Fouodation, Houston, Texas.

244

l. INTROIJlJCTION

Klein

In computational work rhe vecroe space on wh ¡eh a lIamiltonian isdefined may be truncated to a manageable size through rhe use of physicalintuirion and experience. Sometirncs rhe calculations afe rcsrricted ro aspace'which may be generated froro a single "primitivc" ket through rhe aetionof the elemenes of a group which cornmutes with rhe Hamiltonian. Indeed insimple treatments of Van dee Waals interactions, Frenkel excitoos, mínima!

basis lIartree-Fock theoey, crystal {¡cId (heoe)', atomic shell theoey and thetheory of intcratomic exchange interactions, one often cncounters su eh aresrricrcd "zero-order" space. The typicaJ trcatment of rhese restrictedproble~s ¡ovokes rhe extensively studied methods oC symmetry adaptation(see, for instance, references 1,2 or 3) and then diagonalization in the symme-try subspaces. lo exteoding the treatment beyond this zero-order space onemal' extend the space and merely symmetry adapt again. Another con"en-tional alternative is to employ degenerate perturbation theory'" to constructan effeccive Hamiltonian on che zera-arder spacc and again symmetry adaptand diagonalize.

Here we describe yec another alternative which uses group theorr toavoid symmetry adaptation, in the sense chat che desired eigenvalues are ob-tained through a computation of a ket which is oC broken symmetry and whichresembles the single primitive ket referred to above. lndeed this singleprimitive keE may be regarded as a zero-order descripcion of che muitistateket which we wish to compute.

Although multistate kees ha ve arisen in a vasC number of wide-rangingapplicacions as an approximate concept, the realization of a more exact conceptin mis regard has only inErequentIy beeo found in che litcracurc. One cadyrcalization oE this general concept is found in papers 'by Koster5 and Parzen 6

where they propose the use oC localized Wannicr-cype lIartree-Fock orbitalslor band theory calculacions rather than the common symmetry adapted de-localized Bloch-type Hartree-Fock orbitals. They poinc out that these local-ized orbitals are physically appealing and may be obtained by minimizingexpectation values of the une-electron lIartrec.Fock Hamihonian subject tosymmcuy constraints requiring the kcts to be oC mixed space group syrnmcuy.The group-theoretic aspects arising in this application were further developedby des Cloizeaux 7 and Ruch and Schonholer8• Other schemcs not invokingthe use ol group-theoretic ideas have also beeo described9 to directly com-pute these localized Harcree-Fock orbitals.

Anothcr realization of the general concept of a refined multistate ketoccurs in the work of lIerriog10 where he proposes a perturbative ueatmenC

Kf'ls.ol broketJ symmelry ... 245

for che multistate kec representing a collection oí weakly interacting uex_change-couplcd" atoms or moIecules. In this case the zero.order multistateket is a simple product of atomic kets and is of mixcd permutational symmetry.Several computations using lIerring's method have been madpll _ Differentperturbar ion expansions of this multistare ket were suggestcd by Hirschlelderand Silbey12, Kirtman13 and orhersJ4• These perturbative schemes have beeo1S

carried out 00 simple atoros and molecules. Kleio16 suggested a variatiooalscherne for rhis multistate kct, and applications exhibitiog computationaladvantages have be en madeI7•18• Adams19 has formulated this multistatevariarional scheme io a di((erent manner. In fact the formulations ol Kleinand Adaros turn out to be formally similar 'to the group.theoretics.8 and noo-group-theoretic9 procedures earlier describen {or localized lIartree.Fock orbit-als.

lo rhe following we shall consider the group-theoretic multistate ketvariational scheme indicating sorne potential computational advantages. Anapplication in deriving the flcisenberg spin flarniltonian i5 discussed too.

2. GE:-'¡ERAL TIlEORY

We consider a Hamiltonian JI which commutes with a group C¡,

[u,cJ=o, (2.1)

Wc let lo> be a zero'"Order primitive ket from which we may generate a serof kets

(2.2)

which span a space contalnlOg zero-order approximations to the eigenkcrs ofinterest. We assume rhis spanning set (2.2) is linearly independent, althoughthe forma!ism goes through 16 wirh on!y slight modificarion if it is symmetryadapted to a suhgroup of Q.. If G is a "complicated" group such as {-hesyrnmetric group JJN wirh N sufficiently farge, then diagonalization ol 11 jUStin the simple zero-order space of (2.2) can pose a difficult problem. If onesymmetry adapts with respect ro an irreducible representation.a of G, [heflamihonian matrix still has [he dimension fa of Q. Further evaluarionof each individual marrix element on me symmetry adapted basis ca~ be diffi:-

246 KI ein

cult, ir they are lO be constructcd (rom (¡rsr computed primitive ffi<1.1rix ell."fTIcnts,

as < O I GlIl O> , G EG. Extension of (he vector space lO obtain a more ac-

curare result merely compounds thcsc difficulties.Sorne oí these problcms can hopefully be eascd ir wc vacy the mu/ti.

stale kc( I tjJ > lO minimizc the Ifamiltonian expectation value

(2.3)

subjecl ro a ser of predctermined symmctry consrraints

(2.4)

Heee e,as IS a matric basis elerncnt (oc Wigner elcmcnt) oí the group algebra

of e¡,

= (fa/g) ~".e¡I

a[",1] " (2.5)

where g i5 the ordcr of G and

ducible representadon oí G.

a[G"] is ,he (s,r),h ciernen' of ,he ath irte'

."In addition lO the mulriplication property

(2.6)

the marcie basis elemcnts also salisfy the relatioo

(2.7)

ir rhe repre' ..entarioo matrices arc unitary. The constrained rnlnirnizationproblem is conveniently expressed in terros oC Lagrange multipliers Carsassociated w¡th each oC the constraims in (2."1),

s < .1.111 - ¿ i:' e a 1.1• >'P ar5 '5 'Pars

o (2.8)

Kets 01 broJcen sy",,,,etry ...

v vLeuing [H] and [e~.J be ,he ,ep,esen'ations of H and e,a,.ayer which 11./J > is varied, we theo abtain

The term

247

\Jv

in the space

(2.9)

(2.10)

'Nhich occurs in the multistate ket equation (2.9) may be interpreted as aneffective potential or Hamiltonian, which produces the same effect when

vae,ing on 1.;.> as does [H] • We henee expee"hese Lag,ange multipliersEa.,s to contain information concerning some of me eigenvalues al H.

To hel~ identify this eigenvalue informatian, we define the matrixE" wi,h (r, s)t elemen' 2",s. Then no,ing tha, (2.4) and (2.7) imply ,ha,Ea. is Hermitean, we let Ua. be a unitary transformation which diagonalizes

E" '

tl [U"J. 2",. [U,,] • ='5', S S

(2.11)

Further, defining transformed matric basis elements

e~." l [U,,]. e~s'5'5 "

(2.12)

we find ,ha, the multis,a,e ke, equation (2.9) beeomes

([11(- ~2.,;'[e~J)I.;.>=o (2.13)0., , ,

vShould [he space \) .00 which [he calculadon is to be carrled out. be invari.

v.10' 'u G, ,he n we may apply [e'::.] to (2.13) and obtain

"

248

o .

Kl ein

(2.14)

flence [he Ea.; are eigenvalues ro [H]v and [he [e~....Jvly;> are eigenkets,v "at lease when U is invarianr lO Q.

In practice we might construct an approximate but accurate multistateket by variacion in a space DOr invarianr ro y.. In (his case we still define

che Lagrange multípliee macrix Ea and cake ¡es eigenvalues to be approxi.v

matioos to the (me eigenvalues. Using a space \) which is DOr invariantto G can introduce significant computacional savings if ir is of significantly

vsmaller dimension than [he invariant space induced {rom U . Also because

vlJ is not necessarily invariant undee (ji sorne imegrals v.bich would otherwiseappear need 00[. A particular case of interese is when G is caken as (he.

vsymmetric group éJN acting 00 electronic índices and when the basis of Uis taken to consist prirnarily of products of atornic kets with given electronsassociated with given atorns; in this case multicenter exchange integral smay be avoided. This variational scherne has beeo successfully carried out17

for the lowest 1:£+ and 3:£+ states of the H molecule with G =.!J and theg u 2 2avoidance of most of the two.center exchange integrals. These H 2 calcu.latioos illustra[e [he me[hod and sorne of ics possibilicies, buc che full po.[encial is noc realized £ill larger syscerns are considered.

3. APPLlCATlON TO THE DERIVATlON OF THE

HEISENBERG SPIN IIAMIL TONIAN

In chis section we consider a regular lartice of N equivalent siceseach wich one unpaired electron for che isolated site limito A simple Hubbardmodel is treated which takes into account only one orbital (possibly rnultiplyoccupied) 00 each site, and che limic N -00 is ultimacely assumed. The groupG is rhe N-electron symmerric group ~N. Leuiog epi be che orchooormal orbiral00 che ith sire, we see rhar io order lor che mu1tisrace ker It/J> co approachche simple producc form

I O> " 'f' (I)@'f'(2) 10 ••• lO,!, (N)I 2 N (3.1)

Kets 01 broJcen symmetry ...

in the isolated site limit, a reasonable choice for Qa.rs is

11 =<OleaIO>=rliars rs N! rs (3.2)

249

lIence we take this choice.1Rather than working with me rnatric basis, which we found useful to

prove sorne general characteristics of the scheme, we wish now to ernploy(he group basis of (he group aigeb(a. Hence in (erms of (he group basis (hesymme,ry consrrain(s oC (3.2) become

The multistate equation is

1i<.pIIl-UI.p>=0 ,

with

(3.3)

(3.4)

corresponding (o (2.9) and (2.10).Nex( 'o describe ,he Hubbard model 11 and (he mu1tis(a(e ke( an 5111%

we introduce sorne notation

(3.6)

The commu[ation properties

(3.7)

250 Klein

of [hese X'"'Operators show mar [hey span a Lie algebra associated1 with meII'-Iold direet produce 01 ehe uni,a,y group \J.(II'). Furrhe, p,oducrs 01 <heseX-operators may he diagrarnmarically indicated by ordering the arrows suchthar lower-lying arrows correspond to operators further to (he right in aproduce In terms of (hese X-operators (he l:Iubbard model is

11 ; T:S :s X".j ".-n

+ I :s ~ X~" X~"i<¡". J 1

(3.8)

where m '" n indicates sites m and n are nearest neighbors. As is ......ell .•.known20

,his mode! embodies ,he simples< desc,ip,ion 01 a me'al ( I TI» 1). ,hesimplest description of an array of isolated atoms (1 T 1« 1), and a varictyof interesting situations intermediare between (hese [Wo extreme limits.

The variational ansalz lar [he unnormalized muhistate ket 11./J> istaken [O be of [he forro

s' = .. -.s'••• (3.9)

• • Ilitre S", and S",,, are 1- and 2-partic1e excitations hom O>,

- ..S'" =: x :£. X"'"•• •••

s' _••• x"'" X"'''Y •• •

(3.10)

This alsalz is similar to that airead)' described elsewhere21 for antis)'rn-metrizt>d single-stafe kets. The particle excitarion operarors alt commurewirh one another, although the)' do not commute wirh thetr adjoints. denoted,for instance, by S :::;(S+)t.

Since there are just tWo variational parameters, x and Y. in thl' ansalzof (3.9), we expect nor ro be able ro simulraneouls)' sarisfy all me consrrainrsof (3.3). lIo\\'e\'er, {or rhe variational alsaJz ••••..e have taken, •••...e do see that

251

rhese consuaincs will in general be very closely satisfied foc permutaüonsPE élNI which transfer elecrrons amoog distant sÚes oc which transfer a geearnumber of electrons between oeae sires. Hence, we see mar che most imponamconstraints ro consider are those foc which P is che idenrity oc a nearestneighbor 'transposition. Further, hecause of che space group symmetry ¡m-posed 00 oue ansalz, che 'constraints foc aH nearest neighhor transpositionswill be satisfied iiany ane is satisfied. Hene e all che neacest De ighbortransposition Lagrange multípliees may be caken as equal. Considering onlyche constraints foc [hese nearest neighbor transp_ositions, we are (hen rominimize

E = <1jJIH- ~ L (ij)I1jJ>/<1jJI1jJ>i - j

subject to me single constraint

<1jJI(ij}I1jJ> = O

The resulting effective Hamiltonian

(3.1I)

(3.12)

(3.13)

is seeo to be of the form of the convencional nearest neighbor Heisenbergspin Harniltonian. This present derivation of che Heisenberg spin Hamiltoniandiffers markedly from ,he usual general deriva,ionslO• 22.>1 which ha ve pre-viously beeo oí a perturbative nature.

The evaluatlon of the Harniltonian rnatrix elernent is straightforwardto obtain the result,

E= <01 {xT ~ +xyT ;:----;m-n m n m n

+ x2¡ '~}IO>A( )+m n "In

'" ,¡ p+ ~ <01 {x'IO +2x2)m-n - p

:Qb: +x2y'¡f-..c

p---.'" .= N {n_(xT + xyT +":>1) A +n (x' +6x. + '?y2 + x'yll A }

(12) /\ (123) (3.14)

252 Klein

\\'her~ we have lefe out che electron-index labels on [he arrows, sine e (hefeis onIy ane allowed choice. The graph.theoretic numbers n .•.•and n", whichdepend onIr on (he lauice25, aTe i che number of si tes bonded (O a givensirc and % the number of triples of bonded sites involving a given sire. Thequantities A,( ) are residual over/ap marrix elemenrs,"]m2" " ma

A " < O lexp (S("" ... ",) exp (S("'.I"" "'a»)' O>("' "'"'al <,pl,p> (3.15)

Recurrence relatioos foc rados of these residual overlaps are easily obtained,

[A] _1(m ... mn)

A(",:", ",:np) (3.16)

where p""'vn and IlIP =t- mI"" ,ma.!terating chis recurrence celacioo severaltimes [hen approximaring aH che remaining rados, say by 1, gives aD approxi-mariao to [he desired ratio 00 [he Iefe of (3.16); [he more iterations carriedout, (he more accura(e is (he resule. Convergence appears (o be quite rapidfor values o( x2n .. and y2n .. less than one, as occurs for me values n~.1 T j« Jof (he Hamiltonian parame(ers.

For (he case of a linear chaln an anal ytic solution (O (hese recurrencerela(ions may be ob(ained 18. Carrying out numerical compu(ations for sucha linear chain, we ob(ain values for (he "exchange" Lagrange multiplier ~.The solu(ion (o (he Heisenberg rnodel is expec(ed (O be simpler24 (han ro (hefull Hubbard rnodeI. Since rhe ground s(are energy of rhe linear Heisenbergrnodel is known 26, we obtain a numerical prediction fex rhe ground state energyof the linear Hubbard rnode!' This multistate ket predicred energy 1S dis-played in figure 1, where a perturbarlon result"'3, 24 and the exact27 Hubbardrnodel ground srare energy are also shown. The presenr simple ansatz isseen to improve significant1y on the perturbarive result, at least for sufficientlylarge I TI /1 .

Kets 01 broken symmet,y •••

0.0

-0.8

-1.6

-2.4

E/I

0.6

2-order

1.2TII

253

Fig. 1. Comparison oí [he ground ~i[ate energics of the linear half"{illedlIubbard rnodel as computed by (a) a second-order perturba[iveme[hod, (b) by the currcnt multistatc ket ansat%, and (e) an exac[[rea[ment.

254

4. CO:-.lCLUSIO:-.I

Klein

lt appears [har [he multistate ket variational scheme has promise asa computacional alternative to more conventional schemes. Incleed its charac-rerisrics, relaring to avoidance of multicenter exchange integrals in molecularcalculations and relaring to lIeisenberg spin Hamiltonian der¡valioos,may be turned ro distincr advantage in treatments of suitable problems.Explorarían of other characteristics and problems may also be ol interest.

REFERE:-.ICES

1. E. P. Wigner, Group Theory aud Its t1pplicaliou lO Ihe Quanlum.\1echQ1JÍcs oi .llomic Speclra (Aeademie Press, 1959).

2. ~1. ~1oshinsky, Group Theor)' and Ihe Many-Body Problem(Gordon and !lreaeh, 1968).

3. F. A. ~latsen, Adv. in Quantum Chem. 1, p. 60, ed. P.O. Lowdin(Aeademie Press, 1964).

4. S(.'e foc insranc(",(a) ).11. Van Vleek, Phys. Rev. 33 (929) 467.(h) 11. Primas, lIelv. Phys. Acta 34 096Il 331.(e)). des Cloizeaux, Nue!. Phys. 20 (960) 321.(d) L.:-.I. Buleavski, Zh. Eksp. Teor. Fiz. 51 (1966) 230;

[English trans.: ) ETP 24 (967) 154] .5. G. F. Koster, Phys. Rev. 89(953) 67.6. G. Parzen, Phys. Rev. 89 (953) 237.7. ). des Cloizeaux, Phys. Rev. 129 (963) 554.8. E. Rueh and A. Sehonhofer, Theoret. Chim. Acta 2 (962) 332.9. See foc instance,

(a) W.H. Adams,). Chem. Phys. 34 (961) 89; 37(962) 2009.(b) T. L. Gilbert, Molecular Orbilals in ehemislry, Physies, and

Biolog)', ed. P.O. Lowdin, p. 405 (Aeademie Press, 1964).(e) A. B. Kunz, Phys. Stat. So!. 36(969) 301.(d) J.D. Weeks, P.W. Anderson, and A.G.H Davidson,

J. Chem. Phys. 58 (1973) 1388.10. (a) C. Herring, Rev. ~Iod. Phys. 34 (1962) 631.

(b) C. lIerring, ,lfagnelism /lB, p. 1, ed. G. T. Rado and 11. Suhl(Aeademie Press, 1963).

Kels 01 b,oken symme/ry ... 255

11. (a) e. lIerring and M. Flieker, Phys. Rev. 134A (964) 362.(b) B. M. Smirnov, Zh. Eksp. Teor. Fiz. 46 (1964) 1017;

[English trans.: JETP 20 (1965) 156].(e) B.M. Smirnov and O.B. Firsov, Zh. Eksp. Teor. Fiz. 477 (1964)233;

[English trans.: JETP 20 (1965) 156].(d) M. Y. Ovehinnikova, Zh. Eksp. Teor. Fiz. 49 (1965) 275;

[English trans.: JETP 27 (1966) 194].12. J. O. lIirsehfelder and R. Silbey, J. Chem. Phys. 45 (1966) 2188.13. B. Kirtman, Chem. Phys. Le"ers 1(1968) 631.14. (a) J .0. Ilirsehfelder, Chem. Phys. Lecters I (967) 363.

(b) P.R. Certain and J.O.llirsehfelder, J. Chem. Phys. 52 (970)5980, 5992.

(e) D.J. Klein, Inri. J. Quan,um Chem. 4S(971) 271.(d) J.O.llirsehfelder and D.\1. Chpman, Chem. Phys. Le"ers

14 (1972) 293.15. (a) W. A. Saunders, J. Chen!. Phys. 51 (1969) 491.

(b) P. R. Certain, J .0. lIirsehfelder, W. Kolos, and L. Wolniewiez,J. Chem. Phys. 49 (1968) 24.

(e) B. Kirtman and R.L. \Iowery, J. Chem. Phys. 55(971) 1447.(d)13. Kirtman and U. Kaldor, Phys. Rev. 3A (1971) 1295.(e) P. R. Certain, D.R. Dion and J .0. Ilirsehfelder,

J. Chem. Phys. 52 (1970) 5988.(O B. Kirtman, S. Y. Chang and W.R. Seo", J. Olem. Phys. 56 (1972) 1685.(g) T.J. Venanzi and B. Kirtman, J. Chem. Phys. 59 (1973) 523.

16. D.J. Klein, Phys. Rev. 3A (971) 528.17. (a) D.e. Foy' and D.J. Klein, Phys. Rev. (in press)

(b) W.II. Adams, InrI. J. Quantum Chem. (in press)18. D. J. Klein and D. C. Foy', Phys. Rev. (in press)19. W.II. Adams, Chem. Phys. Le"ers II (971) 441.20. See, for instance,

(a) J. lIubbard, Proe. Roy. Soe. A276 (963) 238.(b) several artieles in Rev .. 1I0d. Phys. 40 (964) 790-811.(e) e. lIerring, Maglletism IV, ed. G. T. Rado and 11. Shul

(Academic Press, 19ó6).21. (a) O. Sinano!!lu, J. Chem. Phys. 36 (1962) 706, 3198.

(b) R. Brou" Phys. Rev. III (958) 1324.(e) 11. Primas, .Itadem Qualltum Chemislry 2, p. 45, ed. O. Sinano!!lu

(Aeademie Press, 1965).(d) J. Cizek, J. Chem. Phys. 45 (966) 4256.

256 Klein

22. (a) W. lIeisenberg, Z. Physik 49 (1928) 619.(b) P. A.M. Dirae, Proe. Roy. Soe. A123 (929) 714.(e) ) .11. Van Vleek, Phys. Rev. 45 (1934) 405.

23. P. W. Anderson, Phys. Rev. 115(959) 2.24. D.). Klein and W.A. Seitz, Phys. Rev. 8Il (973) 2236.25. See, fot instanee, C. Domb, Adv. Phys. 9 (1960) 149.26. (a) 11. Ilethe, Z. Physik 71 (931),205.

(b) L. lIulthen, Arkiv. MaL Astron. Fysik 26A (938) 1.27. E.H. Lieb and F. Y. Wu, Phys. Rev. Letters.20 (968) 1445.

RESU\lEN

Los hamiltonianos modelo se definen frecuentemente en un espac iDque proporciona una imagen simple, de orden cero, del sistema investigado.\1icoteas que los kets de la base, más simples de este eigenespacio de ordencero, pueden tener simetría rota con respecto a rodo el grupo del hamiltonia ...no perturbado. los kets de la base pueden frecuentemente transformarse uno en

otro, por la acción de los elementos ':lpropiados del grupo. En tal caso des-cribimos un desarrollo variacional del hamilroniano modelo. Este esque-ma se hasa principalmente en un solo ket básico de simetría rota; el esquemaproporciona una alternativa al desarrollo perturbativo usual, y es, en principio,('xacto. S(, hace particular referencia al hamiltoniano de intercambio deHeiscnherg.