296 OLSON, KANAZAWA, AND TAUBE

the reasonab]eness of the two magnetic reJaxationmechanisms discussed. It is especially important toexamine the p]ausibility of important sca]ar contribu-tions because of the implications of this mechanism forthe use of Nl\1R line broadening to eJucidate the natureof both outer-sphere metal-solvent and metal-ioninteractions. Two factors that make the scalarmechanism more competitive in 170studies than it is inproton X~IR are generally greater values of the scalarcoupling constant, .4, and the fact that 'Yis seven timesgreater for protons than for 170nudei. The disparity in

1'11 E JOUR~AL OF CJlE:lIICAL PJI\'SICS

'Yvalues is quite significant, since the ratio of dipolar toscalar broadening goes as 1'2.

ACKNOWLEDGMENT

Financial support for this research by the AtomieEnergy Commission, Grant Nos. At 04 3 326 P.A. 6,Mod.5 and 6, to the National Science Foundation, forwhom !\1. Olson has been a Fellow from 1965 to thepresent, and to the above Foundation for componentsof the wide-line instrument purchased under GrantNo. GP 5430, is gratefully acknowledged.

VOLUME 51, :\UMDER 1 1 JULY 1969

First- and Second-Order Density Matrices of Symmetry-ProjectedSingle-Determinant Wavefunctions*

JACK Sn.lO:-:st A.'."DJOII:-: E. HARRI!dANt

Theorelieal Chemistr)' Jnslitllte, U11i:'ersit)' oj Jrisco11si11, Jladison, WiseO1/si" 53706

(Received 14 Februar)' 1969)

We derh'e the first- and second-order densit). matrices of s)'mmetr)'-projected single-determinant wave-functions, in the case of finite groups. We also givc Ihe first-order density matrices for functions projectedin the axial-rotation group. A method i5presented for extracting from a density matrix its totally symmetriccomponent, and Ihe eigenfunctions of this component are shown to be the same after projection as before.

1. INTRODUCTION

It has been found that the effects of applying aspin projection operator to a determinental wavcfunc-tion im.oh'ing different orbitaIs for different spins canbe conveniently summarized by considering the first-and second-order reduced density matrices.1.2In thispaper we will be concerned with the density matricesof waYefunctions obtained by applying to a single-determinant projection operators of point groups or theaxial-rotation group.

In Sec. II we consider the effects of point-groupsymmetry projection and obtain e:-.-pressionsfor theelements of the first- and second-order density matricesof the projected function. An example is considered inSec. III. In Secs. IV and V we treat the axial-rotationgroup and an example of this group.

It is a]so of interest to obtain the totany symmetnccomponents of density matrices since only these com-ponents contribute to the e:-.-pectationvalues of sym-metric operators. We consider this problem in Sec.VI. In Sec. VII, we consider the effect of projectionon the eigenfunctions of the totally symmetric com-ponent of the first-order density matrix. We find that

* This work was supported by the National Aeronautics andSpace Administration Grant NGL 50-002-001.

t XSF Graduate FeIlow.t Alfred P. Sloan FeIlow.I J. E. Harriman, J. Chem. Phys. 40,2827 (1964).2A. Hardisson and J. E. Harriman, J. Chem. Phys. 46, 3639

(1967).

thesc cigenfunctions can be taken to be the same afterprojection as before.

II. DERIVATION OF THE DENSITY MATRICESFOR POINT-GRO UP PROJECTION

WCstart with a function ~ which we wish to adaptto the symmctr)' of a particular group:

~(1. .. .V) =G4Jl( 1). . .q,.v(;V), (1)

where et is the .V-particle antis)'mmetrizer, q,l.. .q,.vare orthonormaI spin-orbitals without any particularsymmetry properties, and the arguments 1 Y referto the space and spin coordinates of particles 1 through.V, respectively. The normalized, symmetry-adaptedwavefunction can be obtained by projection as

\}f.=",.-ll2e.~, (2)where

"'.= f (e.~)*(e.~)dT. (3)

The projection operator e. can be e}.-pressedas3

e.= L: C.(R)R,Il

(4a)

where the sum is over aU operations in the group. The

. E. Wigner, Group TheoTY and its App/ications to t/ze Quantum.\lechanics oj Atoll/ie Spectra (Academic Press Inc., New York,1964).

FIRST- AND SECOND-ORDER DENSITY MATRICES 297

index II refers to that irreduciblerepresentationwhichcharacterizes '1'.. The coefficients C.(R) are dcfined intenns of the characters x.(R), the dimension n. ofirreducible representation II, and the order of thegroup g:

C.(R) =(n.fg)x.*(R).

Since we are dealing with a Olany-particIe system, thegroup operators R will be of the form

R(l. ..N) =R(1)R(2).. .R(,r),

where R(k) operates only on the coordinates of thekth particIe. For convenience, we adopt a specialnotation for the effect of a symmetry operation on aspin-orbi tal:

R(k)cpi(k) =cpiR(k).

Then '1'. is seen to be

'1'.( 1. . :X) =CAI.-l/2a L: ('.(R) 4>/1( 1) . . .cp:VR(X).H

We first cvaluatc thc normalization constant CAl"

making use of the facts that 0. is sclf-adjoint andidempotent,. and that a is self-adjoint an(1has

a2= L: (-l)I'P,P

the sum of all X! signcd pcrl1111tations of X particlcs.Then ~:.

, . ..I' _o \.CAI,= L: ('.(R) L: (-1)1' ~o,o r

RP'

X f 411*(1).. '4>~(~yr(j,/ll~,~rt ~ .~~.,.R(.r)dT, (9)l-'""

herc Pi is thc indcx produccd from i by p-I. The in-tcgral in (9) is a prodllct of onc-clectron intcgrab of theform

f 4>i*(i)4>I'iR(i)dTi=f cpi*(i)R(i)cp/'i(i) dTi. (10)

Thc exprcssion for CAl.thercforc takcs the form

CAI.=L: C.(R) L: (-l)pRl,I'I' ..R.V,I'.Y.H P

= E C.(R) dctR.H

( 11)

Notice that the dcterminants, det R, are to involvc onlythe spin-orbitals appearing in~. They are determinantsof NXN matrices. Thus if we know the matrix elcmentsof the group operations in thc spin-orbital basis, wecan calculatc CAl..We will assume that these matrixelementsare known. >

We now proceed to the calculation of the first-orderdensity matrix. Given a symmetric, one-pa.rticleoperator n, we can write its expectation valllc (n)"

with respect to '1'. in two ways:

(n).= f 'l'.*!Nr.dT=f' n(lh.(lll')dTl' (12)

(4b) The first expression is the usual definition of the ex-pectation value, and the second is the density matrixequivalent. The prime on the integral means thatn(l) first operales on the unprimed coordinates onIy,then the priOled coordinates are set equal to the un-primed, and integration is carried out. Our method willbe to calculate (n). using the first expression, and thento identify "(.as that which will correctly reproduce thesecond expression.

Therefore, we must calculate

(n).=CAI,-1f ~*0.n0.~dT. o

(5)

(6) ( 13)

First we examine the operator 0,{10.:( i) .Y

e,ne.= E ('.(R)R E n(i)0.Il i-I

.v= E C.(R) E R(i)n(i) R-l(i) R0..

H i-I(1-1)

(8) Substituting (1-1) into (13) and noting that R and 0.commute with a, we obtain

(d);=CAI.-1f C.(R) Iacpl*(l).. '4>.\"*(X)

.\" .X E R(i)n(i)R-I(i)aR0.4>1(1).. 'q,.V(.\")dT

.." i-I

" . 't v

,. '000 =CAI.-I EC.(R)C.(S) L E (-1)1'H,S i-I P

X f q,1*(l).. '4>x*(X) R(i)n(i) R-l (i) q,Pl'IR,';

X(l)"'4>p.vRS(.r)clT. (15)

Since the symmctry operators are unit;\ry,

f 4>i*(i)R(i)1!(i)R-l(i)4>I'iRSdTi

= f q,rl*(i)f!(i)cp/.,S(i)dTi. (16)

If we now let T=RS,so that R=TS-l, we haveN

(n).=CAI.-lEC.(TS-I)C.(S) E T(i UJS.T i,j-l

X f CPiSJ-I*(l)f!(l)q,f(l)dn, (17)

where T( i Ij) is thei, j cobctor of T.

298 J. SIMONS AND J, E, HARRIMAN

"re now assume thc cxislence of an orthonormaI setof spin-orbitals q,I" 'q,N,q,N+I'. 'q,.\f,such th3otq,I" 'q,xare the spin-orbitals occurring in <I>and that for any Rin the group, Rq" can be expressed as

Af

Rq" = L Rk,q,k,k-I i=l, 2, ,..M.

We can then write

f q".s:r-I*(l)!2(1)q,/(1)dTI

lit

f= L (ST-I)k,*S/i q,k*(l)n(l)q,/(l)dTIk,/-I

JI(

= L (TS-I),kSI/JkI;k,/~I

and thusx

<n).=",.-I L C.(TS-I)C,(S) L TU jj)S,T i.j-I

.It

X L (T5~I)ikS/iQk/k,/-I

M

= L [",.-1 L C,(TS-I)C,(S)k,/-I S,T

x

X L TU U) (TS-I),kS/;J2k/.,,;-1

Therefore, we may identify "(. as

M

')'.(111/) = L ')'(')k/q,/(1)q,k*(I/),k,/-1

where

')'(')kl=",.-I L C,(TS-I)C.(S)S,T

1>1

X L T(i U)(TS-I) ikS/i',,;-1

In a similar fashion, we find the second-order densitymatrix to be given as

f.(1211/2/)Af

L f(') pqrt<P"(1) q,q ( 2) q,,*( 1/) q,e*(2/)",q,r,t-I

with

N 1>1

f (')P9"C=",.-1 L C.(TS-I)C.(S) L L TUj Iki)T.S '<i-I k<I-1

X SpkSq/( T S-I) ,.( T S-I) ie,

where T(ij Ikl) is the ij; ki cofactor of 1'.4

'P.-O. Lowdin, Rev. Mod. Phys. 39,259 (1967).

( 18)

Equations (21), (22), (23), and (24) arc the finalcxprcssions for the desired 11rst-and second-order den.sity matrices in the case of finite grollps. Notice thatwe onI)' need know lhe malrices of lhe group opcrationsin thc spin-orbital basis, and lhe appropriale characlertable (the grolIp Jl1ultiplicalion labIe must also beknown, but can be delermined if necessary from lheorbital-basis reprcsentation). Once we have thisinformation, calculation of "(. and r. is straightfor-ward.

III. EXAMPLE FOR FINITE POINT GROUPS

As an examplc we considcr a wavcfunclion for lhcwaler molcculc

(19) <l>= a[q,la(1 )t/JIP(2), , '4>~a(i)q,l( 8)9;,a(9)q,;,P(10) J.

where the orbilals arc

411= (2+ 2~)-1:2( lSA + 1Su), ~= f IS.dSlIdT,

q,~=ISo, 413= 2So, q,~ = 2p:.o,

41;,= 2-1!2( 2pr.o+ 2p",o),

q,r.=2-112(2pr,o- 2pll,o).

(20)In what fo))ows we will neglect the overlap bctwccnq,l!q,.~and 413,in order to prcscrvc thc simplicity of thecxamplc. The molecule is assumed lo ha\-e C2. sym-metry and to lie in the )'Z pIane. Subscripls A and Brder to thc lwo hydrogen atoms and O to the oxygcnatom. OrbitaIs q,1 through q,~ are thus symmctryadapted, but 9;, and q,6are not. The last orbital is

. added to give a complete sct in the sensc of Eq. (18),In this basis, the matrices of the symmetry operators

decompose into pairs of 6X6 matrices, a set of malricesim'oh-ing a spin, and an idcntical sct involving /3spin.These matrices are as follows: E is the 6X6 unitmatrix. Each of C2, d, = dr:, and d,' = d"z has diagonaldements 1,1 through 4,4 equal to 1. In addition,(C2h.5= (C2)6,6= -1, (lT.h.6= (1T,)6...=1, and (lT,'h,6=(lT.'h.5= -1. Other elements are zero.

We will iIIustrale the formulas developed in theprevious section by obtaining the density matrices forthe A2 component of <1>.Using Eq. (11), we find that

(21)

(22)

(23)"'A2= 1[delE+det C2- detd.- detd.'J

=t[1+1+0+0]=l.

(24)

Remember that in Eq. (11) the determinants are overthe spin-orbitals appearing in <1>;so only the pairs of5X 5 submatrices of our operator matrices are to beused to calculate "'A2'We also find it helpful to con-struct a table of the various terms appearing in theexpressions for "(. and r. (Table I),

FIRST- AND SECOND-ORDER DENSITY MATRICES

Using this information and Eq. (22) we obtain4

"(.\:(11 t/)= :E-t>;{t)cpi*(l/)[aa*+,8,8*Ji-I

+ ~cp;;(1)41;;*(1') [aa*+,8,8*J

+lcp6( 1)416*(1') [aa*+,8,8*].

Similarly, by using Eq. (24), we find that

rA2(t21 t/2')4

= l :E II iaja 12+I iaj,8 12+I i,8ja 12+1i,8j,8 12}i.i-I

4

+l L: \I ia5a 12+1 iaS,8 12+1 i,85a 12+] i,85,8 !2}i-I

4

+ l :E II ia6a 12+I ia6,8 12+1i,86a 12+I i,86,812}i-I

+ 111 5a5,8 12+ I 5,85a 12+ 16a6P. 12+ I 6,86a 12}

- l[ (RqJ;;a(l) 41;;11( 2) J[ &,'q,6a( l' )q,l( 2') J*

- l[ fficp6a(l)cpl( 2) J[ &,'cp"a(1/)cpl(2/) ]*,where

&,= 2-1!2( 1- P12) , &,'= 2-1:2(1- Pl'~' ) ,

and

I iaj,8 J2= [<Bcpja(1)41/(2) J[ <B'cpja(1')q,/(2') J*.

Equivalent results are of course obtainable moredirectly. The advantage of the present method in-creases with the number. of nonsymmetry-adaptedorbitaIs, and is much more apparent for non-Abeliangroups.

IV. THE AXIAL-ROTATION GROUP

To extend the above treatment for the first-orderdensity matrix to the axial-rotation group, we needonly change the formula for the projection operator.The proper projection operator iS5

1

120'

{1m= - da exp( -ima)Ra.2/1' o

In this expression Ra is an operator which rotatesthrough angle a. Ii we define

(25)

Qn.(iklj Ia)

= (211")-2exp( -ima) tO' dcp(R",) ik(Ra-4) Ij, (26)O

then we obtain for the coefficients

120' N

"(m)/k="'m-I da:E Ra(i !j)Qm(iklj Ia), (2i)O i,i-I

\

where Ra( i Ij) is the i, j cofactor of Ra. The normaliza-6 M. Hamermesh, Group T/leory and ils ApplicaJions to Pllysical

Probiellis (Addison-WesleyPub!. Co., Inc., Reading, Mass.,196l).

299

T ABLE1. Combinations ol group operations usedto ca1culate density matnces.

T

E

C:

IT.

<T,'

tion integral is given by

1 120'

"'m= - da exp(-ima) detRa.2/1' o

The evaluation of the second-order density matrix issomewhat more difficult and will not be presented here.

(2ia)

V. EXAMPLE FOR AXIAL-ROTATION GROUP

As an example in this case we will calculate thefirst-order density matrix of the Lz= 1 component of anatomie function (we will not consider the L2 behavior).\\' e take

<I>=a[lsa( 1) lsll(2) 2sa( 3) 2s11(4) 2p,ra(5) J.Since 2Pz=2-1/2(2P+I+2P_1) and the SorbitaIs aresymmetric, the effect of a rotation about the Z axis byangle ~can be written as

R~<I>=a[lsa( 1) lsll(2) 2sa(3) 2sll(4)

X {2-1/2(eiE2p+I+e-iE2p-l) }a(5)].

From this and Eq. (2i),

"(1(111/) =[ls(l) ls*(1/)+2s(1)2s*(1/)J(aa*+,8,8*)

+![q,3( 1)413*( 1') +414 ( 1)414*( 1')

+CP3(1)414*( 1') +414( 1)413*( 1') Jaa*,where

413 = 2-1/2(2P+1+2P-l) =2pzand

CP4=2-1/2(2P+1-2p-l) =i2PII,

S-l TS-l XA,(S) XA,(TS-l)

E E 1 1

C: C: 1 1

IT. IT. -1 -1

<T.' <T.' -1 -1E C: 1 1

C: E 1 1

ITp ITp' -1 -1

<Tp' IT. -1 -1

E ITp 1 -1

C: <T" 1 -1

<Tp E -1 1

IT.' C: -1 1

E <T,' 1 -1

C: <Tp 1 -1

<T. C: -1 1

IT,' E -1 1

300 J. SIMON S AND J. E. HARRIMAN

but this can be written as

'Yl(111') =[1s(1)1s*(1')+2s(1)2s*(1')](aa*+pi3*)

+2p+l(1) 2PH*(I')aa*.

This result is of course expected because the L. =1component of q>is given by

'l'l =(1[lsa(1) lsP(2) 2sa(3) 2sP(4) 2p+la( 5)],

of the C.(R) , we obtain

('Y.O)ii = elCJJ,-1 L: C.(TS-l)C.(S)R,S,T

N

X L T(k Il)R,,(TR:-lhi10,1-1

N

=elCJJ.-l L C.(T) L: T(k Il)Riz(TR:-lhi'R,T 1:,1-1

and the first-order density matrix above followsdirectly. (33For a more complicated function, however, the density- )matrix method can be simpler than the direct method. In a similar way we obtain the matrix elements of the

symmetric component of the second-order densitymatrixVI. THE TOTALLY SYMMETRIC COMPONENTS

OF ,... AND r.6

Tbe totally symmetric component r,o of r, can beobtained as

r.o= Ci L Rr.R-lR

=el L R(12)R(I'2')r.(12 I1'2')R

and

"(.o=el L R(1)R(1'h.(111').R

We will consider "(.°first:

'Y,°(1 I1')]f

=el L L 'Y(.\/Q(1)<Pk(1)Q(1')<PI*(l')Q 1:,1-1

}.{ }.{

= L [el L L 'Y(.)kzQikQ-I/;]<Pi(1)<Pi*(1')i,i-l Q 1:,/-1

}.{

= L (-y.O)il/>;(1)<P/(1'),i,i-I

wbereAl

('hO)ii=el L L 'Y(V)kzQiJ:!:\"Q 1:,1-1

butM

L Qik'Y(.)kQ-IZi1:,/-1

M . N

= L CJJ.-l L C.(TS-I)C.(S) L T(p Iq)(TS-I)pl1:,1-1 S,T p,v-I

X SkoQ-I/)QikN

=CJJ,-l L C.(TS-I)C.(S) L T(p Iq)S,T p,o-l

X(TS-IQ-I)pi(QS)iO' (32)

Letting R=QS, and using the orthogonality properties

. R. McWeenyand W. Kutzelnigg, Intern. J. Quantum Chem.2, 186 (1968).

(r,O)..vq=elCJJ,-l L C,(T)R,T

N

X L T(ii Ikl)R..iR.i(TR:-lhp(TR:-l) lo' (34)i<j-I,k<I-1

(28) Equations (33) and (34) are the final expressions fortbe coefficients in tbe e~.:pansionof "(,° and r,o, re-spectively. Again a knowledge of the matrices of thegroup operations and the appropriate character tableis all tbat is needed to perform the calculations. Be-cause Eqs. (33) and (34) are slightly simpler thanEqs. (22) and (24); the totalIy synunetric componentsshould prove more convenient for perfomling numericalcalculations involving symmetric operators.

(29)

(30)

VII. TOTALLY SYMMETRIC COMPONENTS OFFIRST-ORDER DENSITY MATRICESBEFORE AND AFTER PROJECTION

It has been noticed in spin-projection calculationsthat tbe natural orbitais of the charge density matrixare not changed by projection.l In what follows, we willexamine the corresponding problem in the point-groupcase.

Let tbe totally symmetric component of the first-order density matrix before projection be denoted byrl. With ,...°from Eq, (33), we have for an element ofthe commutator of rl and ,...°

M

[y.O, p°];j= L {(-y.O)ik(pOhj- (pO)ikr 'Y,O)kj}l-I

(31)

N

=eICJJ.-I L C.(T) L T(m In)R,T m,n-l

X {Rin(TR-IpO)mi- (pOR)in(TR:-l)mi}. (35)

Because both rl and "(,° are by definition totallysymmetric, they will commute with all symmetryoperations and so will their commutator. We cantransform to a symmetry adapted basis {7Ja..;)relatedto the original spin-orbital basis {4>;}by a unitarytransformation

J.. ....<Pi= L L L U-l(aai),j 7Ja..i.

a -1 i-l(36)

,""I

FIRST- AND SECOND-ORDER DENSITY MATRICES 301

Thc sct aai can bc thought of as constutituing a singlcindcx labcling 7] or as a sct of indicics. Thc irrcduciblcrcprcsentation is idcntificd by a and furthcr labeled bya if J a, thc numbcr of timcs thc ath irrcduciblc rcprc-scntation occurs in thc rcduction of thc {a.i} rcprcscn-tation, is grcatcr than 1. Thc functions transformingamong thcmsclvcs within a given irreduciblc reprc-sentation arc indcxcd by i.

In this symmctry-adaptcd basis, symmctnc opcr-ators will havc zcro matnx elcmcnts bctwccn functionsbclonging to diffcrent irrcduciblc rcprcsentations.Furthcr, thc block associatcd with a givcn irrcduciblercprcscntation (a and a) must bc a multiplc of thcnaXna unit matrix

(7]aaiInoI7]aaj)=8;jHaa) I

wherc no can bc ,l, .,,°, or thcir commutator and ~(aa), depends on thc operator, a and a, but not on i or j.

Thc matnccs of the symmctry operations themsclveswill bc zero outside each naX lIa block labcled by aand ~a.

lising Eqs. (35) and (36), and the fact that flcommutes with all symmetry opcrations, we obtain

.V['Y.0,p°];j=C1",.-ILC.(T) L T(m[lZ)

R.T ""n-I

x L L Il:i.CGaq)(lI"aqIR 11I"ar)a.G,q,r /J,b,/,'

XF-IC"".),nc.:...,(iJb.)(~bo I pOTR-11 ~bl)

X ("-I(fJbI) ,j-U ;,(",'q) (lI"aq I RpO 11I"a,)(;"-ICa',.) ,nl.i CfJ(,,)

X(~bo I TR:-I I ~b/)U-I(,Jb/),j}.

Therefore, wc may idcntify thc matrix element in thcsymmctry-adapted basis as

.\"

(lIaaq I ['Y.o,pO]I~bl)=CI",.-1 L C,(T) L T(m Ile)R,T ...,n-I

x L I (lI"aqI R l7]aa')(~b.1 pOTR:-II ~bl)','

- ('Iaaq I RpO 17]"a')(~bo! TR-II ~bl>} U-IC"a,),nU...,(/Jb.).

(39)From Eq. (3i) we h.we

(7]"aqI Rpo 17]"a,)= L ('I"aq IR 1 17f...) (17f... 1 po I'I"a,)f,',"

= (17a..qI R l17a,,')('I"or I pO 17]"a,),

wherc (17"a,I pO17]"ar)is in fact independent of r.

(40)

Bccause ['Y.o,pO] is anti-Hermitian, it can be di-agonalized by a unitary transformation. Thc set{7]aa;Jhas not yet been complctely specified, 50 we nowusc our freedom to choose it 50that ['Y.O,pO]is diagonal.Equation (39) then bccomes

('I"aqI ['Y.o,pO]1 Tfbt)=8"~ab8qICI",.-1 L C.(T)R,T

N

X :E T(m I n) :E (7]aGqI R l7]aGr)(7]aGlI TR:-I I 7]aGq)",,_1 r,'

x {(7]"a.1 pO I 7]"a.)- (7]"ar I pO! 17"ar)} U-IC"ar),nU...,C"a,).

(41)

(3i) Bccausc (7]"arI pOi7]"..,) does not depend on r,

(7]"aqI ['Y.O,pO] I ~bl) =0 (42)

~nd from this we deduce that the eigenfunctions of thetotally symmetric component of the first-order densitymatrix will not bc changed by projection.

vm. DISCUSSION OF RESULTS

(38)

In this papcr we have shown how to calculate thefirst- and second-order density matrices of symmetry-projccted wavefunctions. Our technique does notrequire the explicit calculation of the projected wave-function. The method should prove very useful whenapplied to point-group projection in polyatomicmolecules and L: projection in diatomics.

We have also shown how to project irom a givendensity matrix the totally symmetriccomponent. Thiscomponent is all that is necded in the calculation ofexpectation values of symmetric operators. Finally,we have shown that the KSO's of the totally symmetriccomponcnt of the first-order density matrix arc un-changed by projections.

Thcsc results are of potcntial use in two ways:

(1) As a practical method of obtaining expectationvalues of symmctric operators for the component of aparticular symmctry, projected from a nonsymmetry-adapted single-determinant wavefunction.

(2) As the first step in obtaining a varh~tionallyuseful cncrgy expression for an extended Hartree-Fock calculation in which nonsymmetry-adaptedorbitais are used to obtain a lowcr energy withoutsacrificing the symmetry properties of the total wavc-function.