PHYSICAL REVIEW A VOLUME 36, NUMBER 6 SEPTEMBER 15, 1987

Symmetry between model space and target space in effective-Hamiltonian theory

Florent Xavier GadeaLaboratoire de Physique Quantique, Uniuersite Paul Sabatier, 118, route de Ãarbonne, 31062 Toulouse Cedex, France

(Received 25 March 1987)

General relations concerning wave operators are proposed that exhibit symmetry between themodel space and the target space. The basic Bloch and des Cloizeaux effective Hamiltoniansoperating in the model space are shown to be equivalent to the exact Hamiltonian operating onwell-defined vectors belonging to the target space. For the Bloch formalism, a new perturbativedevelopment, involving the usual wave operator, is proposed. It has improved convergence prop-erties and makes possible the calculation, at each perturbation order, of an estimated error in theusual Bloch perturbative development. A numerical test on the simple Mathieu equation, per-formed at orders 2, 3, and 4, gives promising results.

INTRODUCTION

The general goal of effective Hamiltonian theories' isto establish a correspondence between two subspaces ofthe Hilbert space; while they have the same dimension,one is usually called the model since and is a priori select-ed, the other one is a stable subspace of the exact Hamil-tonian and may be called the target space. In practicalapplications, attention is concentrated on the model space.Usually the aim of the theory is to determine unknownexact eigenvalues and eigenvectors via the known vectorswhich define the model space. This is the case in many-body perturbation theory as developed by Brandow, VanVleck, Lowdin, Lindgren, Klein, ' from the basic for-malism of Bloch' and des Cloizeaux. Iterative methodscan also be used as developed by Durand and Mar-tensson. " A different application is made when some ex-actly (at least in a finite basis set) known solutions (thetarget space) are projected onto a model space, either todefine effective operators, supposed to be transferrable tolarger systems, for conceptual clarification or for numeri-cal convenience. For instance, in molecular dynamics cal-culations, adiabatic potential curves, which are solutionsof the total Born-Oppenheimer Hamiltonian, aretransformed into quasidiabatic functions which are bestsuited for further dynamical calculations as developed byLevy, ' Cimiraglia, ' and Spiegelmann. '

In all these cases the essential work is done in the mod-el space; one starts from an a priori definition of the mod-el space and determines the effective Hamiltonian whosespectrum is constrained to be a part of the spectrum ofthe exact Hamiltonian. Different possibilities exist for thechoice of the eigenvectors of the effective Hamiltonian andthis arbitrariness has led to different and related ver-sions. '

However, it is interesting to restore the intrinsic sym-metry of the global problem for conceptual and pedagogi-cal clarifications. This is the main motivation of this pa-per. This approach will lead us to introduce new opera-tors and to find new relations; beside their intrinsic in-terest these relations may be used for measuring the con-vergence of the perturbation series.

Let us make the analogy with the time-dependenttheory where there are two extreme representations.

The Schrodinger representation where the operators(Gs ) are independent of time and the vectors (P) are not.

The Heisenberg representation where the vectors (Po)are independent of time and the operators (GH ) are not.

These representations are equivalent because the matrixelements of Os in the P basis are identical to those of 6Hin the Po basis. In a similar way, usual effective Hamil-tonians (H' ) are defined in a fixed model space, whichcan be related to the Heisenberg representation with thefollowing formal correspondence go~model space,OH~H' . In this paper we search for an alternative rep-resentation, which can be compared to the Schrodingerrepresentation since the Hamiltonian operator H remainsunchanged and the matrix elements of H in the new rep-resentation are identical to those of H' in the modelspace.

Section I introduces the necessary notations and geome-trical properties of vectorial spaces, Sec. II is devoted tothe Bloch formalism, Sec. III is concerned with the desCloizeaux formalism, and Sec. IV gives two applicationsof this new approach, the second one being a numericaltest involving the Mathieu equation, and the first one aconceptual application.

I. NOTATIONS AND GEOMETRICAL PROPERTIES

Let us consider two nonorthogonal subspaces So and Sof a vectorial space 6 and their respective complementarysubspaces So and S . So and S are both of dimension nand are spanned by the orthonormal basis:

IR. & I I I li& 142& I P. &1.The projectors associated with So are

Po ——g iR)(R, i, go ——I Po-i =1

and those associated with S are P = g, , ~ g; ) ( g; ~,

Q = I P. By nonorthogo—nal subspaces we mean that

36 2557 1987 The American Physical Society

2558 FLORENT XAVIER GADEA 36

none of thelPog; ) (i = l, n) is orthogonal to all the vec-

tors of Sp, so that the vectorslPpg; ) (i = l, n) are linear-

ly independent as established by Bloch (1) and form abasis of So. Symmetrically the vectors

lPR; ) (i = 1,n )

form a basis of S.These new bases are neither orthogonal nor normalized.

Let us define two linear operators cop and co which trans-form them into their corresponding biorthogonal bases,

lPop & =coo

lPog & (i = l, n),

PR; ) =colPR; ) (i = l, n),

&Ppg;l Ppg&) =5~ (i = l, n; j= l, n),

& PR;l

PR ) = fi;~ (i = 1,n; j= 1,n ) .

(2)

(2')

In the following these biorthogonal bases will be notedwith an upper line. Let us also define two operatorsPo» P»

n

&o= Z IPoe;&&Poel

i =1

As noticed by Bloch (1) cop and co can be expressed as

n

o= g lPoP )&PotI

i =1n

co = g lPR; ) & PR;

l

i=1

It is clear from their definition that cop and co are self-adjoint operators.

As demonstrated first by Bloch (1) the biorthogonalvectors

l Pog ) have a very important property,

PlPop;)= lg;) (i =l, n),

Po lPR;)= lR;) (i =l, n),

(10)

(10')

and the usual wave operator is simply defined by

0lPog; ) =

l P;) (i = l, n),which is equivalent, using the operators P and cup definedabove, to

Q, =Peep . (12)

'PoQo =0 (4)

These operators give the covariant components of any vec-tor of 6', in the

lPop) basis for Po (

l

PR ) for P). Theyare, of course, different from Pp (P) which give the con-travariant coeKcients in the

lPop) basis (

l

PR ) ). Since

l Pop; ) ESp (i = 1,n ),

flolPR; ) =

lR; ) (i = 1,n),

Qp ——Ppcu .

For any vectorl

u ) belonging to 6,

lu &=Pplu)+Qp lu)

(12')

Let us now define an operator Ap which plays a sym-metric role with respect to the standard wave operator 0,

and

PQ=O. (4')

The vector Pp lu ) can be developed in the

lPpg) basis of

Sp,

Definition (1) only defines the action of cop on vectors be-longing to Sp, we specify its action on any vector belong-ing to 8 by setting

n

PoI

u &= g C IPof &

i =1

PpPPpropl

u )

(14)

and

~oQo =0 =PoPPorooPol

u ) +PoPPo~oQol

u )

=PpPPpropPpl

u ) from relation(5)

C PoPPproplPpg ) from ( 14) and using the

and we obtain

~o =~ooPo =Poroo

co =cc)P =PQ7

Since Pp can be written

linearity of these

operators

= g C;PpPlPop; ) from (1)

and

n n

Po= Z IPoO &&PoOi =1 i =1

= g C;Polg;) from (10)

=Ppl

u ) from (14) .

we have

i =1 i =1 This relation proves that

PoPPo~o=Po .

Pp =Ppccp p = copPp

P =Pcs =Q)P .Since all these operators are self-adjoint we immediatelyget

36 SYMMETRY BETWEEN MODEL SPACE AND TARGET SPACE IN . . ~ 2559

copPpPPp —P—p

and, respectively,

P =PP Pro

P =cuPPpP

(16)

(15')

(16')

copP =Ppci7,

coPp =Pcop )

(23)

(23')

which are very useful because they set up a correspon-dence between co and cop. They also imply that Ap andare adj oint operators,

If cop is considered as an operator acting on Sp relations(15) and (16) can be formally written

p=n

B=Qp

(24)

(24')

cop ——(PpPPp ) (17)

as done by Jdrgensen, '" Brandow, and Durand; in thesame way cu operating only in Sgives

Equation (22') can now be rewritten QSl =co. From (24),(24') and (6), (6') we can easily establish other useful rela-tions,

~ = (PPpP)

From these expressions a possible extension is

(17')Po~Pp =Po

PcopP =P .

(25)

(25')

~o=(PpPPp+Qp) ' =~p+Qp

co'=(PPpP+Q) '=co+Q .

(18)

(18')

Let us emphasize that starting from any relation using theoperators Pp P 0 Qp co and ~p we get directly anotherrelation by the correspondence

Let us note that relations (17) and (17') are consistentwith all the previous ones but they are not strictly neces-sary and we can restrict ourselves to relations (15), (16),(15'), and (16').

From relations (15), (16), (15'), (16') and (6), (6') we get

copPcop = cop (19)

With these definitions, relations (15), (16), (15'), and (16')remain unchanged but relations (6) and (6') are no longervalid and we only have

~pPp =Pp~o+~o

co P =PQ) +co

P~Pp

A~Do

CO~COp

PpA =Po

PA =A

APo ——A

QP =P

(26)

(27)

(28)

(29)

From the previous relations we can easily set up the usualrelations (see, for instance, Ref. 4)

coPpco =67 (19') As an example let us demonstrate the first one. From re-lations (12), (6), and (15) we get successively,

and we immediately deduce that 0 and O, p are projectionoperators PpQ =PpPcop =PpPPpcop =Pp

QQ =Pa) pPco p Pcs p 0, ,—— —— (20) This proves that 0, is a nonorthogonal projector on S.

Symmetrically we get the new relations

QpOp =PpcoP pcs =Ppco =Ap (20')

Since coo and co are self-adj oint operators, from thedefinitions (12) and (12') we have

PAo ——P

Ppnp =no

(26')

(27')

0 =a)pP,

0p ——coPp

(21)

(21')

We can now evaluate the expressions 0 0 and fLpQo using

(19) and (19'),

0 Q=cdpPPcop=ci)p

QpQp =coPoPpco =Q)

(22)

(22')

Relation (22) was previously established by Brandowand Durand, but relation (22') and more generally all therelations with a prime (') are new ones since the operatorsco and Ap have never been considered. From (15), (16),(15'), and (16') we get

FIG. l. Illustration of the correspondences introduced in Sec.I, between the subspaces So (model space) and S (target space).In thick lines the usual definition relations involving Po(Po

I0& =

IPOP&}»d & (t1

IPof& =

I@&}.

2560 FLORENT XAVIER GADEA 36

TABLE I. All products between two of the operators P, Pp,A, Ap, co, ct)p.

H is usually determined by its matrix representation inthe So subset,

Product

PPp

Qp

PPpPPAp

Ap

Pp

PPpPp

Pp

COp

Pp

COCOp

0p

PQ, p

Ap

COpCO

Ap

MpCO

Q)p

C063p

COpCOp

(R; lH lR, ) (i,j=l, n) .

This matrix is nonsymmetric since its eigenvectors are notorthogonal.

Our objective is now to find vectors (la ),

lp) ) which

satisfy

&~, lH lP, &=&R, lH'lR, & .

From relations (32), (30), (6'), and (31) we get

(R, lH lR, )=(R, lPpHQlR, )IlpP =Op,

QpPp Pp . ——(28')

(29')

Ao is a nonorthogonal projector on So.There is an ambiguity in the definition of the so-called

intermediate normalization; some authors (Lindgren, '

Brandow ) give the following definition:

IIl

R ) =Pcopl

R ) =Q)Ppl

R ) = cdlR ) =

l

PR ) . (30)

Thus the wave operator acting on the basis vectors of themodel space gives the biorthogonal complement of theirprojection on the target space. In the same way,

~IpI 0&=~pl 0&=

lPpg & . (30')

Figure 1 gives an illustration of the correspondences es-tablished in this section.

Others (Durand, Mukherjee' ) give an operatorialdefinition PpQPp=Pp if IIQp is not specified or PpQ=Ppwhen Qgp=0 is assumed.

Since we have assumed the vectorslP) to be orthonor-

mal, let us notice that we do not satisfy the first definition,but we satisfy the second one [relation (26)].

Table I gives all the products of any couple of operatorsQo P Po co coo. Now we can apply these operators to

any vector belonging to D. For instance, let us calculatenlR&,

=&PR, lH lPR, & . (34)

H =Ho+ V, (35)

Ho ——H —V,

Hp lR )=Ep lR ) (i=1 n) [Hp Pp]:0

(35')

(31')

The correspondence will be now

O,o~B,PO~P,

COO~Q7

Ho+ H,V~ —V,

and the effective Hamiltonian corresponding to Ho is

Therefore such vectors are the projection onto S of thebasis vectors of So and their biorthogonal complements.

To get the corresponding relation (34') we need tospecify So and to restore the symmetry between S and So.Let So be a subset formed by n eigenvectors of a zeroth-order Harniltonian Ho,

II. STUDY OF THE BLOCH FORMALISMFOR EFFECTIVE HAMILTONIANS

Let 6 be a Hilbert space and S a subset of 6 stable withrespect to the Hamiltonian H. Using the notations intro-duced in Sec. I we have

H o——PHo Ao

with eigenvalues Ep and eigenvectorslPR; ),

l

H p lPR; ) =Ep

lPR; )

and we get

&q, lHgl O, &=&Pp&, IH.

I Ppe, &

(32')

(33')

(34')

Hl g;) =E; ti;) (i = l, n), [H,P]=0 . (31)

The Bloch effective Hamiltonian is the well-known opera-tor [Hp, Q]Pp ———VAPp+QVQPp . (35")

Let us proceed with the generalized Bloch equation firstestablished by Lindgren, '

H =POHCY . (32) Using the correspondence described before we get

Hl

Ppr/i; ) =E; Ppp;) (i = l, n) . (33)

It is easy, from the definition of 0, to show thatlPpg; )

is an eigenvector of H associated with the eigenvalue E;,[H, Qp]P = VQpP QpVflpP . —(35"')

As usual we can introduce the reduced wave operator 7by

36 SYMMETRY BETWEEN MODEL SPACE AND TARGET SPACE IN. . . 2561

and the corresponding operator gp,

(36)I Pop; ) =coo '

IPog; ) (i = l, n) .

So, we have the relations between operators

(44)

np= l+&p (36')

Since P gives the complementary part when applied to

IPol&

1/2 1/2Cc)p COp =COp,

1/2 —1/2p up =Pp .

(4&)

(46)

XPoI 0& =Qo

I0&

Eq. (36) can be written

(37)Similarly we introduce the orthonormal basis of S,

IPR; ), i = l, n and the operators co'/ and co '/2, which

satisfy

[Il Ho]Po —Qo VfIPo —XPo VflPo

Respectively, we get

XoP IR)=Q R),[flo,H]P = QVQ, P—+XPVAoP .

(38)

(37')

(38')

Equation (38) is the starting point of the perturbation ex-pansion and 0, is usually developed in perturbation orders

IPR; ) =co' PR, ) (i = l, n),

IPR;) =co'

IPR;) (i =l,n),

IPR, )=co ' IPR;) (i =l,n),1/2 1/2

1/2 —1/2 p

(42')

(43')

(44')

(45')

(46')

n=p +n" j+n"'+ (39)

We can also formally develop Ap in perturbation orders,

n, =p +np'"+ n,[2'+ (39')

Equations (37'), (38'), and (39') are of limited interest inperturbation theory since P is not known. On the otherhand, relations involving simultaneously 0 and A, p or co

and coo, explicitly by (24), (24'), (30), (30') or implicitly by(34), are only valid at convergence.

As shown in Sec. IV they possibly offer a way ofmeasuring the convergence of the perturbation series ateach other. In iterative methods an error vector can bedefined with these relations allowing for the use ofconvergence-acceleration methods like the direct inversionin the iterative subspace (DIIS) introduced by Pulay. '

We can now easily show that the vectorsIPop) are

eigenvectors of H", using only the geometrical considera-tions and the relations established in Sec. I, without in-volving the complex space (as in Ref. 2).

Let us calculate the matrix element

(P,q, IH" IP,1(, &,

(Pog, Icoo

' QoHIIcoo '/ Pog; )

( P q1/2 —1/2~~I1 —1/2 1/2

IP q )

=(P,P, II1oHQ P,P, )

=E (Pof, ~1o tt';&

=E, (P,1(, P,q, &

III. STUDY OF THE DES CLOIZEAUXFORMALISM FOR EFFECTIVE HAMILTONIANS

The des Cloizeaux formulation of effective Hamiltonianuses an orthonormal basis of So, [ I Pop; ) i = 1,n ).Those vectors are eigenvectors of the Hermitian effectiveHamiltonian H

This operator has a complicated dependence upon 6,H"=(n'n) -'"n'Hn(n'n) -'" (40)

H dC —1/2~ zy ~ —1/2 (41)

As noted by des Cloizeaux, cop is related to the metric inthe

IPor/i) basis, and has strictly positive eigenvalues.

Therefore the operators cop and ~p ' are unambiguous-ly defined (this came from the nonorthogonality statementbetween S and So), and they have the following proper-ties:

[compare with Eq. (31)]; with our notations it can berewritten

Since theIPog ) from an orthonormal basis of So, the re-

sult is demonstrated.Similarly to Hp we can introduce an operator Hp

defined by

H' =co—'"nH n, co—'",

p =CO p pter

and whose eigenvectors in S areIPR; ) (i = l, n) and ei-

genvalues are Ep, or, equivalently, such thatI

(PR, IHodIPR/)=Eo5;~=(R;IHolR ) .

Taking benefit of the analogy with the Heisenberg-Schrodinger duality of representations mentioned in theIntroduction, let us search a representation similar to theSchrodinger one for the des Cloizeaux effective Hamiltoni-an; that means finding n vectors

I X; ) (i = l, n) such that

(X; IH IX))=(R; IH IR, ) .

Let us show that these vectors are theIPR; ) vectors by

proving that

IPoti/ & =~o '

IPof &

I

Potit') =~oIPof & (i = l, n),

(42)

(43) or, using (42') and (41),

(PR; H IPR;)=(R; IH IR&),

2562 FLORENT XAVIER GADEA 36

&R, lco' Hco' lR )=&R; leo' QoHAco ' lR ) .

0, is clearly a closed operator, and let us make the polardecomposition of A. We get

Respectively, we have

~'" lPR, ) =~'"~'" lPR, ) = lR, ) . (52')

Relations (52) and (52') give a new definition of f2 and Qo,Q=vl

l

sll

where

l

nl

=(n'n)'/2=~o'/2

(47)

(48)

1/2 1/2Ct) p

1/2 1/2p=COp CO

(53)

(53')

and 4' is a unitary operator. From Kato' we also haveor, using relations (46) and (46'),

0=l

fl"lVl, (49)

—1/2 1/2pCOp —CO p (54)

where

l

=(Ml )' =co'/'=lQo l

—1/2 1/2ppter

=COp

(48') Let us now define two operators as

(54')

Multiplying Eq. (47) by coo'/ at right and Eq. (49) by

co' at left& we get

Wp ——mp P .1/2

(55)

(55')ePp=n~o '"Pn=~-'/2n .

—1/2 —1/2~COp

Opto =cop Qp—1/2 —1/2

(50)

(50')

From Eqs. (50), (50'), (30), (30') and (44), (44'), we deduce

Multiplying the first equation at left by P, the second oneat right by Pp, and using PO, =APp ——A, we get W lPog;) = lP;) (i =l,n),

Wo lPR;)= lR;) (i =l, n) .

(56)

(56')

It is easy to prove that they are adjoint operators and rela-tions (52) and (52') demonstrate that they are "unitary" inthe sense that

They are, for the des Cloizeaux formalism, equivalent to0 and Oo in the Bloch formalism since relations (52) and(52') can be written

&R lH lR )=&R lQoco' Hco ' AlR)

=≺ lco' Hco ' lPR, )

=&PR, lH lPR, &, (51)

WWp ——P,WpO'=Pp,

Wp ——W

(57)

(57')

(58)

which proves the desired relation. We can similarly ob-tain

The des Cloizeaux effective Hamiltonian can now be castin a very simple form,

&0, IHo'

I 0, &= &Pot;

IHo

I Pot, & (51')H = WpHW (59)

Finally, we establish an interesting property of the trans-formation consisting in projection followed by symmetri-cal orthogonalization.

First we note that this operation transforms an ortho-normal set of n vectors into another orthonormal set of n

vectors. The correspondence is one to one; fromlt(/;)

one getslPop; ) by the operator coo Po and from

lR, )

one getslPR; ) by the operator co' P.

Let us now examine the action of co' on the vectors

l Pop; ) (the operator co', as co, implicitly involves a pro-jection on S). From relations (44), (30'), (50'), (45'), and(46') and Table I, we get

and

Hp ——WHp Wp . (59')

Equations (51) and (51') can be rewritten

&R,l

w Hw lR, &=&wR, lHlwR, ),

& Q, lWHo Wo gi ) = & WoiI/;

lHo

lWof ),

(60)

(60')

since from relations (56), (56'), (57), and (57') we immedi-ately get

1/2l

P P ) / —1 /2l

P

i/2 —1/2~l q )

1/2~ —i/2l q )

—1/2 fl —i/2l q )

—I/2 —i/2l q )

'',S

[P R)

IP R)

iP R)

I R)

IP.+)

(52)FICx. 2. Illustration of the action of the operators 8; 8'0,

, and (go introduced in Seg. III.

36 SYMMETRY BETWEEN MODEL SPACE AND TARGET SPACE IN . ~ . 2563

iPR, )=W iR, ) (i =l,n),

iPOQ;)=Wo i$;) (i =1,n) .

Relation (60) states the formal correspondence between Wand the evolution operator

—i [H(t —(o)]/A'VL.(t, t, =e

used in the time-dependent theory, but '9 is really unitary('MV/ = I) and W is only unitary in the sense of Eqs. (57),(57'), and (58).

Figure 2 gives an illustration of the action of the opera-tors W, 8'Q, 6)', coQ

IV. APPLICATIONS

B. Illustration of the convergence measure

1. General procedure

We have to calculate, at a given perturbation order, thetwo members of Eq. (34). Let us define '"'0 as the best ap-proximation obtained at order n —1 of the true 0 opera-tor, and which contributes to the nth-order total energy

'"'n =p +n"'+ n' '+0

0

For a single referenceiR o ) we directly have

( )g E +g(1)+. . . +g( )

We give now two illustrative applications of this formaldevelopment.

The first one is devoted to a case where the Blocheffective Hamiltonian is exactly determined. We use therelation (34) to give an interpretation of the results ob-tained by Joachim in a recent application of the Blocheffective Hamiltonian to a model time-evolution problem.

The second one is a numerical example. We want to il-lustrate how the relation (34) can be used to test the con-vergence of the usual Bloch perturbation expansion andalso to get accurate results. For this purpose we shallwork on the simple Mathieu equation.

A. Conceptual illustration

and

& R,i

("'H'iR, &

=("'E .

For a model space of dimension m, the diagonalization ofthe '"'H operator yields the ("'E; (i = l, m); the eigenvec-tors are the

iPo "'P; ) vectors. The projector P onto the

target space is approximated, at order n, byn

(n)p y i

(n)q ) &(n)q

i =1

From relation (30) we obtain the biorthogonal set of thei("(PR, ) (t =l,m),

Let us consider the results of Joachim who in a mod-el problem introduces an effective dynamics using theBloch effective Hamiltonian. Let us introduce, with hisnotations,

i s, ) andis2 ), two vectors of a Hilbert

space; they are of physical interest and they are noteigenvectors of the Hamiltonian H. Starting from theinitial state

is, ) at time t, , he was interested in the evo-

lution towards the final stateis2);

is() and

is2)

define the model space.iS() and

iSz), the two eigen-

vectors of H with maximum overlap withis, ) and

isz), represent the target space. Joachim found that

the real evolution was well approximated by an effectivecoupling V&z which comes from the Bloch effectiveHamiltonian, noted H. Joachim determines an effectiveevolution operator e ' '/" which couples

is() to

is2).

Using relation (34) this effective coupling becomes theexact coupling between two effective states iS(), iS2)belonging to the target space. The effective evolutionfrom

is, ) to

is2 ) is in fact the real true evolution from

iS, ) to iS, ),&s e'"'"is )=&S, ie '"'"iS,)

—iHt/Ri

)

Since we have introduced again the real evolution opera-tor, we immediately see that this approximation will givegood results when the vectors

is() and

isz) have no

strong overlap with any other eigenvector of H other than

i S( & andiS2).

and for the left-hand side,

So we essentially need to calculate the set of m vectors

'"'fliR; ) (i = 1,m ) .

(61)

The estimated error of the Bloch development is then thematrix

'"'e/ = &R;iPOH '" fl

iR ) —&'"'QR;

iH

i

"'AR ),which can be reduced to a number by taking, for example,

( )e (( 't

) (( ) ( ) t)2

2. Case op the Mathieu equation

The Mathieu equation, ( —d /d(9 +S cos 0)g(8)=EP(0), has four uncoupled groups of solutions. ' Bysetting HQ ———d /d 6 and V =S cos 0, the eigenfunc-tions of HQ belonging to the first group can be written

and thei

'" PR; ) vectors can be determined as thebiorthogonal set of the '"'Q

iR, ) vectors. Finally, at or-

der n, we have for the right-hand side of Eq. (34),

&R;i

'"'Hi R/) = &R;

iPOH '"(0

i R, )

2564 FLORENT XAVIER GADEA 36

TABLE II ~ Energy of the first eigenvalue for the S =2 value, at orders 2, 3, 4, from the usual Blochdeveloptnent and from this work [Eq. (61)]. The estimated error is the diff'erence between the Bloch de-velopment and this work, the error is the diff'erence between the Bloch development and the exact result.

S=2BlochThis workEstimated

errorError

Order 2

0.875 0000.878 788

—0.003 788—0.003 234

Order 3

0.875 0000.878 318

—0.003 318—0.003 234

Order 4

0.878 4180.878 235

0.000 1830.000 184

Exact

0.878 234

~

0) = —,t

n ) = —cosnO (n =2,4, 6, . . . )1 1

3/2~ '3/rr

where g0/a is the usual reduced resolvent

with eigenvalues Ep ——n . The nonzero coupling ele-n

ments between these functions are

Qo

~eS, (&00 —&0, 1'

(n~

V~

n) = — (n =0,2, 4, . . . ),S2

(0~ V ~2)=(2 V ~0&= v'8 '

Vln+2&=&n+2~ V ~n&= —(n =246 ) .S4

Since we are interested in the first eigenvalue, we take asmodel space the single reference

~R, ) =

~

0). In thiscase relation (61) becomes

We get in our case

nl0&=(P0+n"'1

~o) = ~»—

4&8

n 0) =(p +n' +n ")i0)

S43/8 2563/8

(4)n~

0) ((3)n+n(3))~

0)

( "'nR, H~

'"'nRi ) = Io&+— S 7S

4&8 40963/8

n" I= 'vp, ,a

n"I= 'v 'vp, ,'vp, vp, , —a a a

n =~'v~'v~'vppa a a

V VP0 VP0a a

(I"'noiH

i

'"'no)(0

~

'"'n""'n~

o&

As established by Bloch, ' the perturbative development ofA, up to the third order, is

n"'=Pp,

3

2256v'8 36 864V8

Numerical results at orders 2, 3, and 4 are shown inTable II for S =2 and in Table III for S =4.

It can be seen that expression (61) converges in a betterway than the usual Bloch development. At order n thenumerator of expression (61) involves a product of 2n —1

V terms and the denominator 2n —2, while the Bloch ex-pression involves only n products of V terms.

The value S =4 corresponds to the upper limit for con-vergence of the perturbation since the perturbation expan-sion can be exactly developed in powers of 5/4 (Ref. 21),

V—VP0 VP0 — VP0 V VP00 0 3 0

2

S 1 S 7 SE——2 2 4 128 4

429 S

2364 4

p+ 3 VPp VPp VPpa 68 687 S

18 874 368 4

8

TABLE III. Same as in Table II, for the S =4 value.

S=4BlochThis workEstimated

Order 2

1.50001.5555

Order 3

1.5000]..5488

Order 4

1.55471 ~ 5452

Exact

1.5448Order 6 Order 8

TABLE IV. Energy of the first eigenvalue at the sixth andeighth order of the Bloch expansion, for the values S =2 and 4.

errorError

—0.0555—0.0448

—0.0488—0.0448

0.00950.0099

S=2S=4

0.878 2211.5421

0.878 2351.5457

36 SYMMETRY BETWEEN MODEL SPACE AND TARGET SPACE IN. . . 2565

This development corresponds, of course, to the Bloch de-velopment since in the Bloch formalism 0 ' only contrib-utes to E'"+". Table IV gives the sixth and eighth orderfor the Bloch expansion for the values S =2 and 4. In thecase S =4, where the convergence is very slow, our resultat order 4 is better than the Bloch result at order 8. Inthe case S =2, the convergence is reached at order 4 forour series and at order 8 for the Bloch series. In bothcases, our result at order n can be compared to the Blochresult at order 2n. This is the reason why the estimatederror reproduces correctly the magnitude and the sign ofthe Bloch's error.

V. CONCLUSION

Since effective Hamiltonians are usually introduced forproblems where the exact solutions are not known, atten-tion is focused on the model space spanned by a set ofphysically selected vectors. However, the wave operatorestablishes a correspondence between this model spaceand an isodimensional stable subset of H. It may be im-portant to think about this stable subspace, called here thetarget space, when, for instance, intruder states generatesome ambiguity in the choice of that target space.

The present paper has taken advantage of the intrinsicsymmetry between the model and the target spaces, to de-velop a symmetric formalism which gives a general viewof the correspondences between these subspaces. For theBloch and des Cloizeaux formalism we have found the setof vectors such that the matrix elements of the exactHamiltonian between these vectors are the same as thematrix elements of the effective Hamiltonian in the modelspace. When the convergence is not achieved this relationbecomes approximate and can be used as a convergencetest. These formal correspondences lead for the Blocheffective Hamiltonian to a perturbative development whichhas improved convergence properties with respect to theusual Bloch one, so the convergence test can be used asan error estimate. Let us emphasize that this perturbativedevelopment can be applied to a multireference modelspace, degenerate or not. The test application to theMathieu equation gives satisfactory results.

ACKNOWLEDGMENT

The Laboratoire de Physique Quantique is "UniteAssociee au Centre National de la Recherche ScientifiqueNo. 505.

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Berlin, 1966), p. 335.C. Joachim (unpublished).

~t Handbook of Mathematical Functions, edited by M.Abramowitz and I. Stegun, (Dover, New York, 1968), p. 724.