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numerical simulation of the Hartree-Fock equations
Yvon Maday, Gabriel Turinici
To cite this version:
Yvon Maday, Gabriel Turinici. Error bars and quadratically convergent methods for the numerical
simulation of the Hartree-Fock equations. Numerische Mathematik, Springer Verlag, 2003, 94 (4),
pp.739-770. �hal-00798321�
Error bars and quadrati ally onvergent methods for the numeri al simulation of the Hartree-Fo k equations
Yvon Maday 1
and Gabriel Turini i 2 1
Laboratoired'Analyse Numerique, Universite Pierre etMarie Curie, 4,Pla e Jussieu,75252ParisCedex05,Fran e;e-mail:madayann.jussieu.fr
2
ASCI,UPR 9029,b^atiment 506, UniversiteParis-Sud,91405, Orsay Cedexet Laboratoire d'Analyse Numerique, Universite Pierre et Marie Curie, Fran e; e-mail:turini ias i.fr
Re eived:date/Revisedversion:date
Summary Thispaperpresentsanaposteriorierroranalysisofthe dis retizationmethodsusedin omputationalquantum hemistryon theHartree-Fo k equations. Upperand lower boundsforthe energy areobtainedfromanydis reteapproximationstrategyofthesolution andtheestimatorproposedisshowntopossessfurtherapproximation virtues.
1Introdu tion
Thepurposeofthispaperisto presentan aposteriori erroranalysis for the approximation of the Hartree-Fo k equations. This analysis isdesignedto quantitativelyassestheperforman eofan approxima-tionstrategyofasolutionoftheHartree-Fo kequationsobtainedby prior omputation.In agreement withthe generalparadigm ofthea posteriorianalysis of [13,15{17℄, an error bar for an output su h as theHartree-Fo k energy startingfrom theapproximatedsolutionat hand is proposed. As in the real laboratory experiments, numeri al omputationsdonotprovidetheexa tvalueofthesear hedquantity butratheranapproximationthatis tobequali edbythe useof the error bars in exa tly the same spirit as in the laboratory measure-ments. In addition we willshow that in some ases the a posteriori methodmayalsobe seenasan a eleratorof the onvergen eof the
The time independent S hrodinger equation that models the be-havior of a quantum mole ular system deals with state fun tions (x ), wherex denotesthe position ofthe parti les (nu lei and ele -trons) hen e is a variable that livesin R
3K
where K is the number of parti les
1
.This systemis fartoo large to be dire tly tra tableby numeri al simulationsfor mole ules larger than thehydrogen atom. The quantum hemists have thus introdu ed a series of simpli ed models.One ofthem (theBorn Oppenheimerapproximation)allows to separate the ele tron and the nu lei so as to onsider rst a sys-tem in whi h only the N ele trons of the mole ule move (thus are the only N variables of the state fun tion) and the nu lei are xed inx
j
(andappearasparameters).Forea h on guration(x 1
;:::;x m
) of themnu leia omplexele troni wavefun tion (x
1 ;:::;x N )2C, x i 2 R 3
, i = 1;:::;N is sought after that minimizes the energy of the system. This rst simpli ation is nevertheless not suÆ ient to make the resulting equations a essible for omputations for large mole ules;anothersimpli ationisthereforeintrodu edby onsider-ingthatthestate fun tionis aN dimensionaldeterminant ofsimple fun tions ofR
3
, alled Slater determinant:
(r 1 ;:::;r N )= 1 p N! det( i (r j )); where i
; i=1;:::;N arenowfun tionsofone variableinR 3
hosen orthogonal with respe t to the anoni al s alar produ t < ; > on L
2 (R
3 ).
Let usdenotebyK the subsetof(L 2 (R 2 )) N de ned by K =f( 1 ;:::; N )2(L 2 (R 2 )) N ;< i ; j >=Æ ij g: (1)
Assumingthatthemole uleisisolatedandonlyCoulombi for es arepresent,thedes riptionofthenon-relativisti ele tronswhere,for the sake of simpli itywe have negle ted the spindependen y,leads to thefollowingexpressionof theHartree-Fo kenergy :
E HF ( 1 ;:::; N )= N X i=1 Z R 3 jr i j 2 +V j i j 2 + 1 2 ZZ R 3 R 3 (x) (y) jx yj dxdy 1 2 ZZ R 3 R 3 j (x;y)j 2 jx yj dxdy; (2) 1
where the density matrix
(x;y), the ele troni density
(x) and thepotential V are given bytheformulaes:
(x;y)= N X i=1 i (x) i (y) (3) (x)= (x;x) V(x)= m X j=1 Z j jx x j j :
We have denoted herebyZ j
>0the harge ofthe j-thnu leo. In order to determine the ground state of the mole ule that, by de nition, minimizes the energy (2) under the onstraint (1), the EulerLagrangeequationsgive riseto theHartree-Fo k problem:
Find aL 2 (R 3 )-orthonormal system=f i g t i=1;N and an hermi-tian matrix=[ i;j ℄ i;j=1;N su h that 8i; 1iN; F ( i )= N X j=1 i;j j ; (4) whereF
istheFo koperator.Whena tingonanelement regular enoughof thevariable x2R
3
,thisoperatorasso iates thefollowing fun tionof thex2R 3 variable: F ( )(x)= +V(x)+( ? 1 jxj ) (x) Z R 3 (x;y) jx yj (y)dy: (5)
Here ?is the onvolution produ t
(f?g)(x)= Z
R 3
f(x y)g(y)dy:
Remark 1Itisstandardtonoti ethatthedensitymatrixisinvariant under unitary transforms, i.e. for any element U of the set of the N N unitarymatri es U(N) : 8(x;y)2R 3 ; (x;y)= U (x;y) (6)
Hen e itfollows thattheunitarytransformU an be hosen insu h a way that the hermitian matrix be ome diagonal: =[
i ℄
i=1;N . Thesolution =U=f(U)
i g
i=1;N
satis esindeedthemoresimple Hartree-Fo kproblem: 8i; 1iN; F ( i )= i i (7)
Thishighly nonlinearproblemis solvedthroughiterationsknown as Self Consistent Field approximation; we refer to [6℄ for a very re entand ompleteanalysison the onvergen eof someof these al-gorithms(Roothaanalgorithmand thelevelshiftingalgorithm).Itis stillaveryexpensiveproblemsin ethenonlinear ontributionhasa large omputational omplexity(werefer to [20,8℄forsome example of tailored te hniques to minimize this omplexity). The numeri al analysis of themethod used typi allybythe hemists ommunity is mostoftenanopenproblemandinany asewillnotprovidesound in-formationsin emostofthenumeri alapproximationsarevery often at the limitof the onvergen e. More interesting seems the on ept of a posteriori error estimators where, from the omputed solution, it is possible to derive reliable informationaboutthe validityof the omputationthathasbeendone.Thepurposeofthispaperisinthis dire tion. Denote by H=(H 1 (R 3 )) N
thenatural spa e forthe solutions of the Hartree-Fo k equations and by F
ij
the mapping F ij
: H 7! R de nedoveranyelement =(
i ) N i=1 by F ij ()=< i ; j > Æ ij :
Inallthat followsanyN-tupleelement=( i
) N i=1
willbe supposed to be a olumn (N 1) ve tor of H . Consider the minimization problem
inffE HF
();2H\K g (8)
Remark 2The analysis of problem(7) is not ompletely under on-trol: we an ite the partial results obtained in [10,11℄ about the existen eof agroundstate forpositiveorneutralmole ules andnon existen e results fornegative ions. The basi result of uniqueness of thedensitysolutionisstillanopenproblemofoutstandingdiÆ ulty. Under thehypothesis
m X j=1 Z j >N 1; (9)
it hasbeenproven in [11℄that a minimumof the problem(8) exists andanysu hminimumisasolutionoftheHartree-Fo kequation(4). Moreover, when this problem is written in the form (7) additional information is available on
i
, namely i
In order to make the presentation easy, we will assume in all that follows thattheele troni wavefun tionisreal and willwork onreal fun tion spa es; trivial adaptations allow the treatment of omplex valuedwavefun tions. 2Error de omposition 2.1Error metri s Let 0 =( 0i ) N i=1 2H\ Kbeaminimumof(8)and=( i ) N i=1 2H\ Kan approximationof 0
obtainedasthesolutionofaminimization problem: inffE HF ();2X N \K g (10)
whereX isa nitedimensional subspa eof H 1
(R 3
).
The a posteriorianalysis on theone hand studiesbounds forthe di eren e E HF ( 0 ) E HF
() and on the other hand proposes ex-pli ittrustintervalsonthedesired(butunknown)quantityE
HF (
0 ) using only the approximate solution at hand ; of ourse, due to thevariationalsetting,anupperboundonE
HF ( 0 )isE HF ()itself; the main fo us will therefore be pla ed on ndinglower bounds for E
HF (
0
), whi his anon-trivialproblemthat, toourknowledge,has notbeenaddressed intheliterature.
Beforedwellinginto theaposteriorianalysisof (8)itis ru ialto introdu etheproperde nitionfortheerrorbetweenaminimizer
0 andits approximation. Tothisendone hasto re alltheinvarian e propertyofthe Hartree-Fo kenergy:
E HF
( )=E HF
(U );8 2H\K ; 8U 2U(N) (11)
From(11)itfollowsthatif 0
isaminimizerof(8),thenforanyU 2 U(N), U
0
is also a minimizer and therefore a solutionof (4). The same onsiderationsremain truefortheproblem(10).It istherefore natural to onsider thedistan e between the sets fU
0
;U 2U(N)g and fV;V 2 U(N)g as themost appropriate de nition of the dis-tan ebetween
0
and.Forreasonsthatwillbemade learlateron, we willuse in fa tan equivalent form (see se tion 2.3) of the above de nition.For any 1 ; 2 2H let U 1 ; 2 =argminfkU 1 2 k 2 (L 2 (R 3 )) N ;U 2U(N)g: (12) For a given norm kk (kk (L 2 ) N , kk (H 1 ) N
...) we will measure the distan e between(setsrepresentedby)
1 and 2 as: k 1 2 k ? =kU ; 1 2 k=k 1 U ; 2 k;
thelast equalitybeing motivated by thefa tthat U 2 ; 1 =U t 1 ; 2 2 U(N).
Remark 3Note from (12) that U
2 ;
1
is intrinsi ally related to the normof(L 2 ) N ;whenkk=kk (L 2 ) N
were overthedistan ebetween thesets fU
1
;U 2U(N)g and fV 2
;V 2U(N)g.
The properties of this metri are losely related to the following de ompositionof H : H=A S ?? (13)
whereforany2H\K :
A =fC;C2R NN ;C t = Cg S =fS;S2R NN ;S t =Sg ?? =f =( i ) N i=1 2H ;< i ; j >=0;i;j =1;:::;Ng
We willdenote forany 1 ; 2 2(L 2 ) N : 1 ?? 2 ifforanyi;j=1;N: <( 1 ) i ;( 2 ) j >=0; then ?? an be de nedequivalently ?? =f 2H ; ??g: For any = ( i ) N i=1
2 H the de omposition (13) is obtained in thefollowingmanner: omputethematrixM =(M
ij ) N i;j=1 wherefor ea hi;j =1;:::;N:M ij =< i ; j
>.DenotebySthesymmetri part of M: S =
M+M t 2
and by C the antisymmetri part: C = M M
t 2
. Then S will be the omponent of in the spa e S
and C the omponent of in the spa e A
; in addition it is easy to see that ( S C)??, sothedi eren e S Cisin
?? .
Lemma 1Let ; 2H\K . Then the matrix U ;
solution of (12) has the properties
U ; 2S ?? ; U ; 2S U ; ?? : (14) In parti ular for = 0 , U 0 ; 0 =+S+W; S 2R NN :S t =S; W 2 ?? : (15)
Proof. Considerthede omposition
=C+S+W; C2A ; S2S ; W 2 ?? ; (16)
and denote M =C+S.Then we an write U ; =argminfkU k 2 (L 2 (R 3 )) N ;U 2U(N)g =argminfkU((Id N +M)+W) k 2 (L 2 (R 3 )) N ;U 2U(N)g =argminfk(U(Id N +M) Id N )k 2 (L 2 (R 3 )) N ;U 2U(N)g =argminfkU(Id N +M) Id N )k 2 R NN ;U 2U(N)g =argminfk(Id N +M) U t k 2 R NN ;U 2U(N)g (17)
Thetransformationfromthese ondtothethirdlineisa onsequen e of the fa t that W?? so therefore U(Id
N
+M)??W ; the next equalityis truebe ause 2K .
Foranyantisymmetri matrix ~ C2R
NN
onsiderthepathinU(N) given by t ! e
~ Ct
U ;
. The tangent at t =0 to thispath is ~ CU
; . Writingthe rstorder onditionsfortheminimalityin(17)weobtain:
0=<(Id N +M) U t ; ;U t ; ~ C t > R NN =<U ; (Id N +M) Id N ; ~ C t > R NN; 8 ~ C2R NN : ~ C t = ~ C;
whi hshowsthatU ;
(Id N
+M) isasymmetri matrix;and there-foreU ; 2S ??
.Toprovethese ondpartoftheequation (14) denote forany
1 ,
2 byC
1; 2
theantisymmetri matrixappearing in the de omposition 1 2 = C 1 ; 2 2 +S 1 ; 2 2 +W 1 ; 2 with C 1; 2 2 2A 2 ,S 1; 2 2 2S 2 andW 1; 2 2 ?? 2
;thenoneobtains bystraightforward omputations C 1 ; 2 = C 2 ; 1 . ut
Remark 4Inpra ti etherepresentativeofthe lassofisoenergy fun -tions fU
0
;U 2U(N)g istaken to be the one that solves equations (7), and the same is true for any of its approximations . It is not lear whether a norm for whi h this pra ti al hoi e gives optimal approximations in the sense of (12) exists and to what extent this hoi e isalso optimalintheL
2 norm.
2.2Order of the symmetri part of the error
Let ; 2 H\K and let us onsider the de omposition (16). We have seen that the antisymmetri part given by matrix C may be setto zeromodulosome appropriate\rotation" on ;itistherefore
Lemma 2Let ;2H\Kwithasso iatedde omposition(16).Then there exists onstants C
1 , C
2
depending onlyof N su h that:
kSk (L 2 (R 3 )) N C 1 k k 2 (L 2 (R 3 )) N (18) kSk H C 2 k k 2 H kk H (19)
Proof. Let us write W = D ~ W su h that < ~ W i ; ~ W j >= Æ ij , M = C+S.Denote =k k (L 2 (R 3 )) N = v u u t N X i;j=1 M 2 ij +D 2 ij Sin e 2K ,F ij ( )=0,i;j=1;:::;N. Forj =iwe obtain: 1=(1+M ii ) 2 + X j6=i M 2 ij + N X j=1 D 2 ij ; orequivalently: S ii =M ii = P N j=1 M 2 ij + P N j=1 D 2 ij 2 ;
whi h provesthat M ii 2 ,i=1;:::;N. Fori6=j one obtains: 0= X k6=i;k6=j M ik M jk +(M ii +1)M ji +M ij (M jj +1)+ N X k=1 D ki D kj ;
whi h gives after straightforward manipulationsS ij = M ij +M ji 2 2 ; this on ludes the proof of (18). For (19) one denotes rst that k k (L 2 (R 3 )) N k k H
and apply(18)to on ludethat S ij
k k
2 H
,i;j=1;:::;N.The on lusionfollowsthenbythede nition of thenorm kk H . ut 2.3Optimality in H 1 norm
Wehaveproposedinse tion2.1thatforanynormkktheerror 0 be omputed as kU 0; 0
k. Sin e the de nitionU 0;
is losely related to theL
2
norm it isnatural to ask whether thisde nitionis stillappropriatewhennormsotherthanL
2
Lemma 3Let =( 1
;:::; N
)2H\K and 2H\K anddenote
U 1 ; =argminfkU k H ;U 2U(N)g
There exists a onstant dependingonly of N and su h that
kU ; k H kU 1 ; k H kU ; k H
Proof. The inequality
kU ; k H kU 1 ; k H
follows asa onsequen e of thede nitionof U 1 ;
. DenotebyF thelinearspa egeneratedbyf
1 ;:::; N gandde ne: M =f 2H 1 (R 3 );<; > L 2 ;L 2=0; 8 2Fg: For any 2 H 1 (R 3 ) denote by F theL 2 proje tion of on F and M = F .We de neanorm kk d on H 1 (R 3 ) asfollows: k k 2 d =k F k 2 L 2 +k M k 2 H 1 (R 3 ) :
We willprove thatthis norm is equivalent to the anoni al norm of H
1 (R
3
) (with onstantsdependingonlyonN and ). Writeforany 2H 1 (R 3 ): k k H 1 (R 3 ) k F k H 1 (R 3 ) +k F k H 1 (R 3 ) k k d +k F k H 1 (R 3 ) Ck k d
wherewehaveusedthefa tthatthenormskk L 2 and kk H 1 (R 3 ) are equivalent onthe nite dimensionalspa e F.Itfollows thatthere existsa onstant C (dependingonlyon N and ) su h that forany 2H 1 (R 3 ) k k H 1 (R 3 ) Ck k d : Wewillprovenextthatthenormkk
H 1
(R 3
)
analsobelowerbounded bythe norm kk
d
modulo some onstant dependingonlyN and . Assume on the ontrary that this is not true. Then there exists a sequen e( n ) n1 H 1 (R 3 )su hthatk n k d =1andk n k H 1 (R 3 ) !0 as n ! 1. It follows that the sequen e
n
onverges to zero in L 2 andinparti ularthesequen e(
n F ) n1 ofL 2 proje tionstoF isalso onverging to zero: k nF k L
2 ! 0 (n ! 1); by the same argument asabovewe obtaink nF k H 1 (R 3 ) !0(n!1). Then k nM k H 1 (R 3 ) =k n nF k H 1 (R 3 ) k n k H 1 (R 3 ) +k nF k H 1 (R 3 )
and it follows that k n M k H 1 (R 3 ) ! 0 (n ! 1). Together with k nF k 2 ! 0 (n ! 1) we on lude that k n k d ! 0 (n ! 1), in
ontradi tion with theinitial assumption.We have therefore proved thatthere exists onstants ;C (dependingonlyN and ) su h that forany 2H 1 (R 3 ) k k d k k H 1 (R 3 ) Ck k d :
The above equivalen e imply that the anoni al norm k k d;N of (H 1 (R 3 );k k d ) N
is equivalent (with onstants depending only on N and ) to the anoni al norm ofH :
1 kk H kk d;N C 1 kk H ; 8 2H :
Sin e 2 K , the fun tions f 1
;:::; N
g are orthonormal with respe tto the s alarprodu tof L
2 (R
3
) and also with respe t to the s alarprodu t<;>
d
asso iatedwiththenormkk d .It followsby (17) that U ; =argminfkU k kk d;N ;U 2U(N)g;
asbothsolve thesame minimizationproblemon U(N).But then
kU 1 ; k H 1 C 1 kU 1 ; k d;N 1 C 1 kU ; k d;N 1 C 1 kU ; k H :
whi h on ludes theproof. ut
3Optimality onditions and oer ivity
We will begin this se tion with some elementary informationabout thegeometry ofthemanifolds K and H\K :
Lemma 4Let 2 H\K .The tangent spa e in to the manifold H\K isA ?? . Proof. Let(t):℄ ;[!H\K ,>0,(0)=beaC 1 pathin H\K . Considerthede omposition
0 (0)=S+C+W,S2S , C 2 A , W 2 ??
. By di erentiating the ondition F ij ((t)) =0 we obtain < i ; 0 j (0) > + < 0 i (0); j
>= 0 whi h proves that S
ij
=0. Sin e thisis true for any i;j = 1;:::;N we on lude S =0 i.e. 0 (0)2A ?? .
To prove that any =C+W 2A
??
may beseen as the tangent in of a C 1 path in H\K , hoose (t) :℄ ;[!H\K , 0 < <1, (t) = p 1 t 2 e Ct
+tW and note that 0
These ondorderoptimality onditionsfortheminimization prob-lem (8) will be seen to be very useful within our approa h. Let
0
2 H \K be a minimizer of (8) and 0
be the hermitian ma-trix orresponding to
0
in equations (4). We will writethe se ond order onditionsintheform:
D 2 E HF ( 0 )( ; )+< 0 ; > (L 2 (R 3 )) N0; 8 2A 0 ?? 0 :
Denote for any2H\K :
E ()=E HF ()+ N X i;j=1 ij F ij () (20) where ij =<F i ; j >, i;j=1;:::;N.
Remark 5The Hartree-Fo kequations (4) an be\symboli ally" de-rived as a orollary of lemma 4. Indeed, the rst order minimality onditionsasso iatedto (8)read
<DE HF ( 0 ); > (L 2 (R 3 )) N=0; 8 2A 0 ?? 0
whi histhesameaswritingDE HF ( 0 )=S 0 ,(Sbeingasymmetri matrix)whi hareexa tlyequations(4)sin eDE
HF ( 0 ) anbe iden-ti edwith(F 0 ;:::;F 0
).Moreover, withthede nition(20) we note that DE 0 0: (21) Denote bya
(;)thebilinearformD 2
E
()(;)and remarkthat
a 0 (;)=D 2 E HF ( 0 )(;)+< 0 ;> (L 2 (R 3 )) N :
Inorder to obtainan expli itformulafora
0
weneedthe expres-sion ofD 2 E HF ( 0 ).Let; 1 ; 2 2H\K . Then D 2 E HF ()( 1 ; 2 )=2 N X i=1 Z R 3 r 1 i r 2 i +V 1 i 2 i + 1 2 ZZ R 3 R 3 8 ; 1(x) ; 2(y)+4 1 ; 2(x) (y) jx yj dxdy 1 2 ZZ R 3 R 3 2 (x;y)( 1 ; 2(x;y)+ 1 ; 2(y;x)) jx yj dxdy 1 2 ZZ 3 3 +4 ; 1(x;y)( ; 2(x;y)+ ; 2(y;x)) jx yj dxdy;
withthe de nitions 1 ; 2(x;y)= P N i=1 1 i (x) 2 i (y); 1 ; 2(x)= 1 ; 2(x;x): We obtaintherefore: D 2 E HF ( 0 )( ; )=2 N X i=1 Z R 3 jr i j 2 +V 2 i + 1 2 ZZ R 3 R 3 8 0 ; (x) 0 ; (y)+4 (x) 0 (y) jx yj dxdy 1 2 ZZ R 3 R 3 4 0 (x;y) (x;y) jx yj dxdy 1 2 ZZ R 3 R 3 4 0 ; (x;y)( 0 ; (x;y)+ 0 ; (y;x)) jx yj dxdy:
We will study in the following the oer ivity properties of the bilinear form a
0
. Note that for any 2 H\K : E HF ( ) = E 0 ( ) and in addition a 0 = D 2 E 0 ( 0
). By di erentiating the invarian e property(11) weobtaininparti ular ( f. lemma4):
DE 0 ( )(C )=0;8 2H\K ; 8C 2A : (22) Di erentiating now (22)in = 0
and taking into a ount the fa t that 0 isa solutionof (4)we obtain: <D 2 E 0 ( 0 )(C 0 ; ~ C 0 +W)=0; 8C 0 ; ~ C 0 2A 0 ; 8W 2 ?? 0 :
Then it follows that a
0
vanishes on A
0
thus annot be oer ive there;the oer ivitypropertiesofa
0
aredes ribedbythefollowing two lemmata.
Lemma 5LetV 0
bethe losureofspanf 2A 0 ?? 0 :a 0 ( ; )= 0g with respe t to the anoni al topology of H . Then a
0 is null on V 0 V 0 . Proof. Let 1 ; 2 2A 0 ?? 0 besu hthata 0 ( i ; i )=0,i=1;2. Thensin ea 0 0onA 0 ?? 0
byastandardCau hy-S hwartz in-equalityforthepositivebilinearforma
0 weobtain2ja 0 ( 1 ; 2 )j a 0 ( 1 ; 1 )+a 0 ( 2 ; 2 ) and therefore a 0 ( 1 ; 2 ) = 0. It follows then that for any =
1 1 + 2 2 su h that 1 ; 2 2 R we have a 0
( ; ) = 0 whi h, together with the ontinuity of a
0
Proposition 1Let X 0 be a losed subspa eof ?? 0 (H ) su h that 8 2X 0 ; 6=0: a 0 ( ; )>0: Then a 0 is oer iveon X 0 .
The proof of this proposition makes use of the following auxiliary result
Lemma 6Themapping
7! 1 2 ZZ R 3 R 3 8 0 ; (x) 0 ; (y)+4 (x) 0 (y) jx yj dxdy 1 2 ZZ R 3 R 3 4 0 (x;y) (x;y) jx yj dxdy 1 2 ZZ R 3 R 3 4 0; (x;y)( 0; (x;y)+ 0; (y;x)) jx yj dxdy
issequentiallyweaklylowersemi ontinuouswithrespe ttothe anoni topology of H .
Proof. Letusre alltheHardyinequality(usedintheversionof[11℄ p.42) whi h holdsforall y2R
3 ,'2H 1 (R 3 ): Z R 3 j'(x)j 2 jx yj dxCk'k L 2 (R 3 ) kr'k L 2 (R 3 )
witha onstantCindependentofyand'.Notethatifu;v 2H 1 (R 3 ) u(x)v(y) p jx yj 2L 2 (R 3 R 3 ).Indeed: ZZ R 3 R 3 u 2 (x)v 2 (y) jx yj dxdy = Z R 3 Z R 3 u 2 (x) jx yj dx v 2 (y)dy Ckuk L 2 (R 3 ) kruk L 2 (R 3 ) Z R 3 v 2 (y)dyCkuk L 2 (R 3 ) kruk L 2 (R 3 ) kvk 2 L 2 (R 3 ) Let m
be a sequen e weakly onvergent in H to ; this sequen e is bounded inH;withoutloss ofgeneralityit an be supposed that k
m k
H 1.
Considera termof theform ZZ R 3 R 3 f(x)g(y) m i (x) m j (y) jx yj dxdy (23) wheref;g2f( 0 ) 1 ;:::;( 0 ) N
g.Wehaveseenthat
f(x)g(y) p jx yj , m i (x) m j (y) p jx yj 2 L 2 (R 3 R 3 ); sin ek m k H 1, it follows that m i (x) m j (y) p is weakly
onvergentinL 2 (R 3 R 3 )to 2 i(x) j(y) p jx yj
soanytermof theform (23)
is weakly ontinuous (so also lower weakly semi ontinuous), and of oursethesameistrueforanysumoftermsofthistype,inparti ular
0 ; m(x) 0 ; m(y) jx yj , 0 (x;y) m(x;y) jx yj , 0 ; m(x;y) 0 ; m(y;x) jx yj .
Theonlytermthatremains to be analyzed in(23)is
4 ZZ R 3 R 3 (x) 0 (y) 0 ; (x;y) 2 jx yj dxdy
We transformthe numeratorof theabove fra tion asfollows:
(x) 0 (y) ( 0; (x;y)) 2 = N X i=1 ( i ) 2 (x)( 0 ) 2 i (y) + X i<j ( i ) 2 (x)( 0 ) 2 j (y)+( j ) 2 (x)( 0 ) 2 i (y) N X i=1 ( i ) 2 (x)( 0 ) 2 i (y) X i<j i (x)( 0 ) i (y) j (x)( 0 ) j (y) = X i<j i (x)( 0 ) j (y) j (x)( 0 ) i (y) 2
Itiseasy toseefromthisequalitythat (x) 0 (y) ( 0 ; (x;y)) 2 is a onvexfun tionof andtherefore,bya lassi alfun tionalanalysis argument, issequentially weakly lowersemi ontinuous. ut
ProofofProposition1:Letuspro eedwiththeproofof propo-sition1.Supposeonthe ontrarythatthe on lusionisnottrue.Then there exists a sequen e f
m g m1 2X 0 su h thatk m k H =1,and lim m!1 a 0 ( m ; m
)=0;extra tingifne essaryasubsequen eout of it, we may suppose that f
m g
m1
is weakly onvergent in H to
2
Inorderto rigorouslyidentifytheweak limitoneusesappropriatetest fun -tions
p
jx yj (x) (y)1 1 forany ; 2L 2
(R 3
2X 0 .We rst write: a 0 ( m ; m )=2 N X i=1 Z R 3 jr m i j 2 + 2 N X i;j=1 0 ij Z R 3 m i m j +2 N X i=1 Z R 3 V( m i ) 2 + 1 2 ZZ R 3 R 3 8 0 ; m (x) 0 ; m(y) +4 m (x) 0 (y) jx yj dxdy 1 2 ZZ R 3 R 3 4 0 (x;y) m(x; y) jx yj dxdy 1 2 ZZ R 3 R 3 4 0 ; m (x;y)( 0 ; m(x; y)+ 0 ; m(y; x)) jx yj dxdy(24)
Re allthat([11℄p.42)that R
R 3
V 2
dxisweaklylowersemi ontinuous onH
1 (R
3
)([11℄p.42).Bythelemma6theintegralsonR 3
R 3
in(24) alsohaveweaklylowersemi ontinuityproperties.Sin ethematrix
0 has stri tly positiveeigenvalues(remark 2) the rst two termson the righthand sideof (24) de nea norm sothispart is also weakly lowersemi ontinuous ;we obtain
a 0 ( ; ) lim m!1 a 0 ( m ; m )=0
whi h together with (1) imply = 0. We will use now this infor-mation for a ner analysis of the sequen e a
0 ( m ; m ) ; by the argument above thereexistsa onstant
0
>0dependingon 0
su h thatforany 2H :
N X i=1 Z R 3 2jr i j 2 + N X i;j=1 0 ij Z R 3 i j 0 k k H :
Usingagain the lower semi ontinuityof the remaining terms we ob-tain: 0= lim m!1 a 0 ( m ; m )0 + liminf m!1 N X i=1 Z R 3 2jr m i j 2 + N X i;j=1 0 ij Z R 3 m i m j 0 liminf m!1 k m k H = 0 >0;
Motivated bythe above analysis, we will introdu ethe following hypothesis: 8 2 ?? 0 ; 6=0: a 0 ( ; )>0: (25)
whi h, by proposition1, ensures the existen e of a \ oer ivity on-stant" 0 >0 su h that 8 2 ?? 0 ; 6=0: a 0 ( ; ) 0 k k 2 H : (26)
Remark 6Using the lemma 5 a posteriori analysis may still be ar-riedoutwithoutthe hypothesis 25 ;some aspe ts of a more general analysis arepresentedinremark11.
4Error estimators, bounds and onvergen e a eleration
Let 0
;2H\Kbeasinse tion2.1: 0
aminimizerof(8)(whi his thusa solutionof(4))and 2H\Kagivendis reteapproximation of
0
obtainedbya previous omputation. Let us denote by = kU 0 ; 0 k H = kU ; 0 0 k H the distan e between and
0
. Even if the wavefun tion 0
may be intrinsi allyinteresting(e.g.when theformof themole ularorbitals is studied), the main result of a Hartree-Fo k omputation is the Hartree-Fo kenergy E
HF (
0 ).
We will suppose in all that follows that is lose enough to 0 su h that e.g. in the development of the error E
HF () E HF ( 0 ) with respe t to powers of : E
HF () E HF ( 0 ) = k k +o( k ) the se ondtermo(
k
)isindeedsmallerthan k
k
(duetotheasymptoti properties of the de ompositionthis is ertain to happen when is smallenough).
4.1Error estimators
The a posteriori analysis method presented in this se tion is on-ne ted to theworksof Babuska[1℄, Bernardi[4℄, Ladeveze[9℄, Oden [14℄,PousinandRappaz[18℄,Verfurth[21,22℄and isaimedatgiving quantitative indi ationson the form of theerror, throughbilateral estimates.Evenifthe onstantsarenotexpli itlyknown,thismethod may prove interestingwhen onlyrelative error estimates are needed (asinadaptative pro edures)orwhentheestimator isshownto
pos-Let us re all (see also (15)) that U ; 0 0 2 S 0 ?? 0 and denoteU ;0 0 =S 0 +W,S 0 2S 0 ,W 2 ?? 0
.Then one an write E HF () E HF ( 0 )=E HF (U ; 0 ) E HF ( 0 ) =E 0 ( 0 +S 0 +W) E 0 ( 0 ) =DE 0 ( 0 )(S 0 +W)+D 2 E 0 ( 0 )(S 0 +W;S 0 +W)+O( 3 ) =0+D 2 E 0 ( 0 )(W;W)+O( 3 )=a 0 (W;W)+O( 3 )
wherewe have used rstlythefa tthat 0
isthesolutionof(4)(see remark5equation 21)andse ondlythelemma 2for(U
; 0
; 0
)! ( ;).Fromthe ontinuityofa
0
and(26)one on ludesthatkWk 2 H isa thirdorder estimator oftheenergy errorE
HF () E HF ( 0 ).
Remark 7It easy to see by(19) thatkWk H
=+O( 2
).
Unfortunatelydire t omputationof W (andthenof kWk 2 H
) as-sumesknowledgeof
0
whi hisnotavailable.Howevergood approxi-mationsofkWk
2 H
thatrequireonlytheknowledgeof anbefound. Indeed, let us set F = DE
HF
, = U ;
0
and study the norm of F( ) inthedualspa e
?? of ?? kF( )k ?? = sup 2 ?? <DE HF ( );> kk H = sup 2 ?? <DE 0 ( ); > kk H = sup 2 ?? <DE 0 ( ) DE 0 ( 0 ); > kk H = sup 2 ?? D 2 E 0 ( 0 )( 0 ;) kk H +O( 2 )
We used inthe rst lineofthe equation above the de nition(20) of E
0
andtheidentityDF ij
( ;)0on ??
.Weshownowthatwe an repla eintheabovesupremumthespa e
?? =(U ; 0 ) ?? = ?? by ?? 0 .Let 2 ?? be writtenas =M 0 + ~ , ~ 2 ?? 0 .Notethat jM ij j=j< i ; 0j >j=j< i ; 0j j >j kk (L 2 (R 3 )) N k 0 k (L 2 (R 3 )) N (27)
soone an write
j a 0 ( 0 ;M 0 ) kk H j C 0 k 0 k H kk H k 0 k (L 2 (R 3 )) N kk H C 0 2 ;
where C
0
is the ontinuity onstant of a 0 . Sin e k ~ k H kk H =1+O() one on ludes that
kF( )k ?? = sup 2 ?? a 0 ( 0 ; ~ ) k ~ k H +O( 2 ) = sup ~ 2 ?? 0 a 0 ( 0 ; ~ ) k ~ k H +O( 2 )= sup ~ 2 ?? 0 a 0 (S 0 +W; ~ ) k ~ k H +O( 2 ) = sup ~ 2 ?? 0 a 0 (W; ~ ) k ~ k H +O( 2 )=kWk H +O( 2 ):
We have shownabove thatkF( )k
?? isa se ond order approx-imation of kWk H and therefore kF( )k 2 ??
will be a third order estimator of the energy error E
HF () E HF ( 0 ). We next prove that kF( )k ??
is invariant with respe tto themultipli ationof byunitary matri es and therefore equal to kF()k
??, soit an be omputed(aposteriori)usingonlyavailabledata(i.e.).Letus om-pute for inH\K thefun tion F(U), bythe de nitionof F this equals DE
HF
(U) whi h an bewritten:
DE HF (U)= F U ((U) i ) N i=1 = ( 1 2 +V)((U) i ) N i=1 + ( U ? 1 jxj )(U) i Z R 3 U (x;y) jx yj (U) i (y)dy N i=1 =U ( 1 2 +V)( i ) N i=1 +U ( ? 1 jxj ) i Z R 3 (x;y) jx yj i (y)dy N i=1 ;
where we have used the invarian e property (6). It was therefore proven that F(U)=UF();8 2H\ K ; and therefore kF( )k ?? =kF(U ; 0 )k ?? =kF()k ?? We will summarizetheresultsobtainedin thisse tion inthefollowing Theorem 1Let
0
be a minimizer of (8), 2 H\K a (given) dis rete approximation of
0
obtained by a previous omputation as des ribedinse tion2.1(10),anddenote=kU
0; 0 k H the quo-tient distan e between and
0
. Then, underthe assumption (25),
kDE HF ()k ?? =+O( 2 ):
Moreoverthere exists onstants 1 ; 2 dependingonlyon 0 su h that 1 kDE HF ()k 2 ?? +O( 3 )E HF () E HF ( 0 ) 2 kDE HF ()k 2 ?? +O( 3 ): (28)
Remark 8The onstants 1
; 2
in (28) are not known and therefore the quantity kDE
HF ()k
2
??
an be used to estimate the error in energybutnotto obtainpre ise errorbars.
4.2Expli it bounds for the Hartree-Fo k energy and onvergen e a eleration
The purpose of this se tion is to propose methods to nd expli it boundsfortheHartree-Fo kenergy.Themethodbelongstothemore generalparadigm[13,15{17℄of de nitionof expli itlowerandupper boundsforoutputsdependingonthesolutionofapartialdi erential equation.Theoutputofinterestwillbetaken tobetheHartree-Fo k energy;this hoi ewillbe seen( f.thm.2)to possesparti ularities that in fa t allow to design an improvement of the solution itself, althoughthisisnot expe ted to bethe ase forgeneraloutputs.
We will begin this se tion with some remarks on the oer ivity properties of thebilinearformsa
0 anda
.
Lemma 7Under the hypothesis (25) there exists a onstant > 0 depending only on
0
su h that for any U 2U(N) the bilinear form a U 0 is oer ive on (U 0 ) ?? = ?? 0
with oer ivity onstant .
Proof. Note that for any 1 2 H\ K , 2 2 H , U 2 U(N): a U 1 (U 2 ;U 2 )=a 1 ( 2 ; 2
),soby(25)andproposition1weobtain the on lusion. ut
Lemma 8Under the assumption (25) there exists a onstant >0 dependingonlyon
0
su hthat forall 2H\K withk 0
k H
the bilinear form a
is oer ive on ??
with a oer ivity onstant depending only of 0 . Proof. Let 2 ?? , kk H 1 be written as =M 0 + ~ , ~ 2 ?? 0 . We willgeneri allydenotebyC various onstantsdependingonlyon
0
. Re all that (by (27)) jM ij j kk (L 2 (R 3 )) Nk 0 k (L 2 (R 3 )) N, so fork 0 k H smallenough a (;)=a ( ~ +M 0 ; ~ +M 0 )a ( ~ ; ~ ) Ckk H k ~ k H k 0 k H : But fork 0 k H
smallenoughwe an also write
kk H k ~ k H k 0 k H kk H (kk H +kk H k 0 k H )k 0 k H Ckk 2 k 0 k H :
Sin ej ij 0 ij jCk 0 k H itfollowsthatja ( ~ ; ~ ) a 0 ( ~ ; ~ )j Ck ~ k 2 H k 0 k H soinfa t a (;)a 0 ( ~ ; ~ ) C(k ~ k 2 H +kk 2 H )k 0 k H k ~ k 2 H C(k ~ k 2 H +kk 2 H )k 0 k H :
ItsuÆ es now tousea lasttimejkk H k ~ k H jkk H k 0 k H to on lude. ut
In what follows we start the presentation of the onstru tion of (lower)boundsfortheHartree-Fo kenergy.Asitwasseeninlemma 7, under theassumption (25) we have uniform oer ivity properties forbilinearformsa
0
withrespe tto themultipli ationof 0
by uni-tarymatri es U 2U(N); forthisreasonwe an repla e
0
withany U
0
that tsbetterourneeds;wewillthereforesupposeinagreement withlemma 1 that
0 issu h that 0 =S+W 2S ?? . The onstru tion of (lower) bounds for the Hartree-Fo k energy isbased on thefollowingdevelopment:
E HF ( 0 ) E HF ()=E ( 0 ) E ()=E (+S+W) E () =DE ()(S+W)+ 1 2 D 2 E ()(S+W;S+W)+O( 3 )
Note rst that by theproperties of asdes ribed in se tion 2.1eq. (10)DE
()isnullonthedualspa eofthedis retizationspa esoin parti ular DE
()(S)=0 ;re all also the fa tthat S isof order
2
and W of order to obtain
E HF ( 0 ) E HF ()=DE ()(W)+ 1 2 D 2 E ()(W;W)+O( 3 ) (29)
Considernowtheproblem: ndthere onstru tederror ^ W 2 ?? su h that D 2 E ()( ^ W; )+DE ()( )=0; 8 2 ?? : (30) By the oer ivity of a
it follows that (30) has a unique solution ^
W 2 ??
.
Remark 9Note thatinorder to ompute ^
W onesolvesadire t(i.e. noteigenvalue)problemonthesolutionspa e;moreoveralloperators
Usingthe de nitionof ^
W one an rewrite (29):
E HF ( 0 )=E HF () D 2 E ()( ^ W;W)+ 1 2 D 2 E ()(W;W) +O( 3 )=E HF () 1 2 D 2 E ()( ^ W; ^ W) + 1 2 D 2 E ()(W ^ W;W ^ W)+O( 3 ): (31) Butsin ea ispositiveon ?? itfollowsthat 1 2 D 2 E ()(W ^ W;W ^
W) 0 so infa t we obtainan asymptoti expli it lower bound on the Hartree-Fo k energy:
E HF ( 0 )E HF () 1 2 D 2 E ()( ^ W; ^ W)+O( 3 ); (32)
whi h together with the inequality E HF ( 0 ) E HF () gives an in-tervalfortheexa t valueof theHartree-Fo k energy.
Remark 10Anaturalquestionistostudytheorderinofthelength oftheerrorbarfoundabove.Letusre allthattheerrorinenergyis oforder
2
;wewillprovethatthisintervalisoptimalinasensethat its lengthisalso of order
2
;indeedthedistan e betweentheupper and lowerboundis
1 2 D 2 E ()( ^ W; ^ W)+O( 3 ) whi his equivalentto k ^ Wk H
; all that remains to be proven is that k ^ Wk
H
C (with a onstant notdependingon
0 ). Indeed: k ^ Wk H CkDE ()k ?? CkDE () DE ( 0 )k ?? +CkDE ( 0 ) DE 0 ( 0 )k ?? C
wherewehave used thefa tthatDE 0 ( 0 ) is nullon ?? 0 . The nomination of ^
W as \re onstru ted error" is best explained bythefollowingproperty:
^
W =W +O( 2
): (33)
Inordertoprove(33)wewillprovethatW hasthefollowingproperty:
jD 2 E ()(W; )+DE ()( )jC 2 ; 8 2 ?? ; k k H =1:(34)
with a onstant C independent of , . Suppose (34) is true then jointlywith (30)one obtains:
jD 2 E ()(W ^ W; )jC 2 ;8 2 ?? ; k k =1:
Let = W ^ W kW ^ Wk H
;from the oer ivityof a =D 2 E ()we dedu e: 1 kW ^ Wk H kW ^ Wk 2 H C 2 ; and (33)follows.
Re all that, fromlemma 2,k 0
Wkis oforder 2
.Inorder to prove (34)itisthussuÆ ientto prove itfor
0 insteadofW: let uswrite DE ()( )=DE ( 0 )( )+D 2 E ( 0 )( 0 ; )+O( 2 ): Besideswe have jD 2 E ( 0 )( 0 ; ) D 2 E ()( 0 ; )jC 2 k k H ;
(with a onstant C dependingonlyof 0 ),so DE ()( )=DE ( 0 )( )+D 2 E ()( 0 ; )+O( 2 ) and therefore D 2 E ()( 0 ; )+DE ()( )=DE ( 0 )( )+O( 2 ):
It suÆ esnow to prove thatDE ( 0 )( )=O( 2 ).By thede nition of E , DE ( 0 )( )=DE 0 ( 0 )( )+ P N i;j=1 ( ij 0 ij )DF ij ( 0 )( ) =0+ P N i;j=1 ( ij 0 ij )DF ij ( 0 )( ):
Note rstlythat ij 0 ij C(C dependingonlyof 0 ).Moreover DF ij ( 0 )( )=< 0i ; j >+< 0j ; i > =< 0i i ; j >+< 0j j ; i > thus jDF ij ( 0
)( )j an be upperbounded by C (we used the fa t that 2
??
), whi h on ludestheproofof (33).
Combining(31)and (33)we an givea better versionof (32):
E HF ( 0 )=E HF () 1 2 D 2 E ()( ^ W; ^ W)+O( 3 ); (35)
so instead of a lower bound we have obtained an improvement of theHartree-Fo kenergy;notethatthisimprovementisofastri tly higherorderinsin ethebestapproximationknownbeforethe om-putationof
^
W wasE HF
()whi his exa t to theorder 2
.
pos-animprovementforthewavefun tionhasalso beenfound,namely ~
=+ ^
W. However we annot propose ~
as a legitimate solution of (4) sin eit is not ertain to be inK . We willsee inthe following thatitispossibleto nda orre tionto addto+
^
W whi hnotonly givesanadmissiblesolutionof(4)butalsoimproves withanother orderthe approximation (35)of theHartree-Fo k energyE
HF (
0 ). In order to improve even more the solution, remind the equality
0
=+W +S.Sin e both 0
and are inK we an write
Æ ij =< 0i ; 0j >=< i + N X k=1 S ik k +W i ; j + N X l =1 S jl l +W j > =Æ ij +<W i ;W j >+ N X k=1 S ik Æ kj + N X k=1 S jl Æ il +O( 4 ) (36)
be ausewe knowthat S ij =O( 2 ).We obtain 0=<W i ;W j >+S ij +S ji +O( 4 )=< ^ W i ; ^ W j >+S ij +S ji +O( 3 ) so denoting ~ S ij = 1 2 < ^ W i ; ^ W j >, we obtain that ~ S is a order 3 approximation ofS: ~ S=S+O( 3
). Notethatbyremark9 that the omputation of
~
S requiresknowledgeof only.
We will prove in the following that having an approximation ^ W of W to theorder 2 and an approximation ~ S of S to the order 3 is enough to have an approximation of theHartree-Fo k energy to the order 4 .Indeed,write E HF ( 0 ) E HF ()=E ( 0 ) E ()=E (+S+W) E () =DE ()(S+W)+ 1 2 D 2 E ()(S+W;S+W) + 1 3! D 3 E ()(S+W;S+W;S+W)+O( 4 ) =DE ()(W)+ 1 2 D 2 E ()(W;W)+D 2 E ()(S;W) + 1 3! D 3 E ()(W;W;W)+O( 4 ) = 1 2 D 2 E ()( ^ W; ^ W)+ 1 2 D 2 E ()(W ^ W;W ^ W)+ D 2 E ()( ~ S; ^ W)+ 1 3! D 3 E ()( ^ W; ^ W; ^ W)+O( 4 ) = 1 2 D 2 E ()( ^ W; ^ W)+D 2 E ()( ~ S; ^ W)+ 1 D 3 E ()( ^ W; ^ W; ^ W)+O( 4 );
sowe have obtained E HF ( 0 )=E HF () 1 2 D 2 E ()( ^ W; ^ W)+D 2 E ()( ~ S; ^ W)+ 1 3! D 3 E ()( ^ W; ^ W; ^ W)+O( 4 ):
wherealltermsinvolvedintherighthandside anbe omputedfrom .
One problemremainsthough,ourbestapproximationforthe so-lution 0 , namely ~ ~ = + ^ W + ~
S is still not ertain to be in K ; in fa t it an be proved that there exists an
^
S that depends only of that has the property
^ S =
~
S+O( 3
) and su h that ^
=+ ^ W +
^
S2K .Moreover, usingtheabovearguments,wewill also have E HF ( 0 )=E HF ( ^ )+O( 4
). Theexisten e and properties of
^
S follows by onsidering as in (36) the equations satis ed by ^ S. Denote byM thematrixwithentries <
^ W i ; ^ W j >then ^ S issolution of theequation (I+ ^ S) 2 =I M: (37)
Thisshowsthat ^ S isanO( 3 )approximationof ~ S.Thematrix ^ S an be omputed from equation 37 by taking the square root of I M whi h iswellde ned as
^
W is loseto W (andsmall). Note thatthis pro eduremaybe ostlyfornon-sparsematri esand anberepla ed inpra ti e withTaylor-like seriesexpansionformulas
I+ ^ S = p I M =I 1 2 M+ 1 8 M 2 1 16 M 3 +:::
We willsummarizetheresultsobtained inthisse tion inthe fol-lowingtheorem: Theorem 2Let 0 be a minimizer of (8), 2 H\K a (known) dis rete approximation of 0
obtained by a previous omputation as des ribed inse tion2.1(10).Then,undertheassumption (25),there exists an >0 su h that for any 2H\K withkU
0; 0 k there exists ^ W 2 ?? and ^ S2S
whose omputation requires only knowledge of su h that
^ =+
^ S+
^
W 2H\K hasthe following properties: k ^ 0 k H 1 k 0 k 2 H ; jE HF ( ^ ) E HF ( 0 )j 2 jE HF () E HF ( 0 )j 2 :
Error bars an be easily derived from the Thm. 2 above and the minimizationpropertiesof E HF ( 0 ):
Theorem 3Under the same assumptions and with the same nota-tionsasinThm. 2,thereexistsan ~>0su h that forany2H\K withkU 0 ; 0
k~ the following estimates hold:
2E HF ( ^ ) E HF ()E HF ( 0 )E HF ():
Remark 11The approa hdes ribed inthis se tion an be developed undermoregeneralassumptionsthan(25).DenotebyX
0
the losed subspa e of
?? 0
where (1) holds so that, in agreement with propo-sition 1 a 0 is oer ive on X 0
; using the same arguments as in lemma8 oneprovesfork
0 k
H
smallenough oer ivityfora on X 0 \ ??
;thisshowsthattheproblem(30)hasanuniquesolutionon X
0 \
??
andthissolutionisthenshowntopossesthesameproperty (33)as
^
W.A\re onstru tedsymmetri al"partisthen omputedby the same method as above and we obtainthus an improvement for theenergyandforthewavefun tion.Theonly omputational imped-iment to this program is that one annot really identify the spa e X
0 \
??
where problem (30) is to be solved ; one hooses then thelargest subspa ein
??
wherea
ispositive(therefore oer ive), whi h will ontain X
0
\ ??
, and proves that the solution of (30) on thisspa eis an order
2
approximation of thesolutionof (30)on X
0
\ ??
.Inpra ti e ( f.se tion5)therewasno needto implement thispro edure as(25) seemsto besatis ed.
Remark 12The numeri al omputationof ^
W involvestheresolution ofequation(30)overthedis retesubspa e
?? Æ of ?? ;the orrespond-ing solution ^ W Æ will be an approximation of ^ W whi h onverges to ^
W whenthedis retizationparameterÆ issu hthat ?? Æ onverges to thespa e ?? .
Remark 13Uponwritingthispaperweweremadeaware[5℄that(30) isequivalenttoadensitymatrixquadrati onvergen eequation(see for instan e [3℄ an referen es therein for an introdu tion). A study is being undertaken to further investigate the advantages that this equivalen emaybringat thenumeri al level.
5Numeri al simulations
at-Intheexperimentsofthe rst ategorywe he kedonsimple ases (hydrogenmole ule,helium)thatthemethodologyproposedaboveis oherent with availableresults when theproblem(30) thatprovides
^
W is solved on avery nedis retizationof H .
In a se ond stage more omplex mole ules were studied and the methodwasimplementedinaHartree-Fo kquantum hemistry ode. Beforepresenting theresultslet usremarkthatthepartial di er-entialequation(PDE) (30)is,forN large,very diÆ ultto dis retize with lassi al tools from the PDE equations ( nite elements, nite volumes, ...) due to the high dimensionality of the linear spa es in-volved. Moreover agooddis retization hasalso to take into a ount some spe i quantum hemi al e e ts as the singularities of the ele troni wavefun tion aroundnu lei; in on lusion,onlyverysmall quantumsystems arethusavailableforstudyusing lassi altools in solvingPDEs ;su hsystems are forexamplethe hydrogen mole ule (H
2
) and theheliumatom (He).
5.1Validation of the dis retization basis
We illustratein thisse tion howto useof the errorbars to validate thedis retizationbasisusedtosolvetheHartree-Fo kproblem.Error barsare omputedforseveralapproximationsoftheexa t wavefun -tion orrespondingtoseveraldis retizationbasisandtheexa t(best known) Hartree-Fo k energy is seen to fallwithin the error bars as indi ated by the theory. The size of the error bar an be therefore used to to asses the quality of the result and thus to validate the dis retisationbasisused.
For all the numeri al experiments we pla ed ourselves into the Restri ted ( losed) shellHartree-Fo k (Lewisele tron pair) approxi-mationthatstatesthatwhenthenumberofele tronsinamole uleis even,one angrouptogethertheele trons2by2;thetwoele tronsin ea hsu hpairwillsharea ommonspatialwavefun tionbutwillhave oppositespin. Withinthisapproximation,for a bi-ele troni system asthehydrogenmole uleorHeliumatom,thesear hoftheele troni wavefun tionofthesystemredu esto thesear h ofafun tionuof3 variablessu h that
u+Vu+ juj 2 ? 1 jxj +u=0 in R 3 :
The spa etobedis retizedisthereforeR 3
;infa tusing lassi al 3
the nu lei of the system ;in the ase of the Heliumatom thisbri k wastaken to bea ube entered aroundthenu leus.
WewillpresentinthefollowingtheresultsobtainedfortheHelium atom; ea h axis of a ube entered in the nu leus mentioned above wasdis retizedwith thesame numberof pointsthatvariedbetween 60 and 120 depending on the singularities of the initial solutions onsidered; pre ise results were obtained for about 100 points per dimensionand orrespondingve tors of size100
3 =10
6 . Several initial approximations
i
, i=2;3;4;5;6 of theele troni wavefun tion were onsidered; ea h orrespond to a quantum hem-i al omputation thatused spe i quantum basissets denominated as STOnG, n = 2;3;4;5;6 ; the larger the parameter n, the ner the basis used; in ea h ase the linear problem (30) was solved on the hosen grid as indi ated inRemark 12 and then the symmetri part of the errorwas re onstru ted as indi atedinprevious se tion. In order to solve (30) an iterative algorithm wasemployed, the ma-trixasso iated to D 2 E ()(;) (typi ally10 6 10 6
) beingtoo large fordire t inversion; nally inorder to take advantage of the tensor-produ t-like dis retization the omputation of onvolution produ ts wasdone bymeansof fastFouriertransforms.
The gure1showstheenergyoftheinitialwavefun tion (\Clas-si ally omputed energy"),thebestknownapproximationof the en-ergy Helium atom, the improved energy obtained as in thm. 2 and then the order
2
lower bound as des ribed in Thm. 3; agreement withthe theoreti alresults isobtained.
5.2Validation of the iterative resolution pro edure
The numeri al resolution of the Hartree-Fo k equations involves it-erative resolution of eigenvalue problems. The number of iterations ne essary is notknown in advan e and no natural stoping riterion exists.Wefoundthereforeimportantto illustratehowtheerrorbars presentedabove anbeusedtovalidatethenumberofiterationstobe undertakenbytheresolutionpro edure.Thistimeerrorbarsare om-putedforseveralapproximationsof theele troni wavefun tionea h orresponding to a di erent number of iterations, the dis retiza-tions basisbeingkept xed.Theerrorbargiveinthis aselower and upperboundsfortheHartree-Fo k energyofthe solutionof the Hartree-Fo k equations on the given dis retizations basis. The size of the error bar an be taken asa measure of theimprovement still possibleif iterations are arried on untill onvergen e (in the given
Lowerestimator Improvedenergy Bestknownenergy Classi ally omputedenergy
Estimatorbehaviour
BasissetSTOnG
Energy 6 5 4 3 2 -2.7 -2.75 -2.8 -2.85 -2.9 -2.95 -3 -3.05
Fig. 1. A posteriori improvements for the energyobtained withthe basis sets STOnG.
Motivated by the su ess of the rst series of experiments, this timethemole ules onsideredwerelarger,asisforinstan ethe ase of the arbyne mole uleCr(CO)
4
ClCH,with52 ele tron pairs(104 ele trons) ;themodel hosen wasagain theRestri ted HartreeFo k model;inthissetting theenergyto minimizeis
E HF ( 1 ;:::; N )= N X i=1 Z R 3 jr i j 2 +V j i j 2 + ZZ R 3 R 3 (x) (y) jx yj dxdy 1 2 ZZ R 3 R 3 j (x;y)j 2 jx yj dxdy
withthe same formalde nitions ( f. Eq.(3 ,4) for
(x),
(x;y)). TheEuler-Lagrangeequationsasso iatedtotheminimizationofE
HF on H\K are ompletely similar to (7) (only some multipli ative fa torsbeforethe lasttwo termsin(5) are hanged).
Dueto on ernsabout omputation omplexityandeÆ ien yand also forrealisti veri ationwehave hosen to implementthe a pos-teriori pro edure (and the \ onvergen e a eleration" version) in a quantum omputational hemistry odenamed Asterix[7,19,23℄. As a onsequen e, the evaluation of the performan es of the a
posteri- hemistryab initio odes. An introdu tionto the omplexity of the algorithmsusedis given inthefollowing.
One parti ularityof omputationalquantum hemistry odes (es-pe ially at the Hartree-Fo k level) is the presen e of very spe ial Galerkindis retizationbasis.Thisbasis ontainsingeneralfun tions on R
3
whi hare entered in thenu leiof thesystem andare sum of Gaussiantype fun tions;it isbeyond thes ope of thispaperto give arigorouspresentationofthebasisinvolved,letusjustsaythatthey allsatisfyanimportantrequirement:foranyelementsh
,h ,h and h Æ
ofthe dis retizationbasis,thequantity
( jj Æ) = ZZ R 3 R 3 h (x)h (x)h (y)h Æ (y) jx yj dxdy (38)
an be omputed inO(1)time 3
.
Letusdenotebynthenumberof basisfun tionsusedwhen om-putingthe Hartree-Fo k energyof a mole ule with N ele tron pairs (2N ele trons);ingeneral nistaken to dependlinearlyon N.
In order to solve the nonlinear eigenvalue equations (7) iterative (also named self onsistent - SCF) algorithms are used. The most straightforwardideaistostartfrom aninitialguess
1
forthe wave-fun tion and then, for any i1, onstru t the Fo k operator F
i = F i asso iated to i , diagonalize F i
and take its rst N eigenfun -tions as the next guess
i+1
for the wavefun tion (Roothaan algo-rithm) ;ideally this xed point algorithm will onverge and the so-lution will be the solution of equations (7). Numeri al reality does not however always validate this hoi e, we refer to [6℄ for a mathe-mati aldes ription of thephenomena involved. In order to ure the onvergen e de ien ies, various other methodshave beenproposed [6℄:thebasi level shiftmethod,DIIS,...
DuringtheSCFresolutionoftheHartree-Fo kequations,themost time onsumingpartisthe onstru tionoftheFo koperatorF
i ;we willseeinthefollowingthatthisisanO(N
4
)operation,oneorder of magnitudelargerthanthediagonalizationoftheFo koperatoritself (underassumptionthat nislinear inN).Let
B =fh
; =1;:::;ng
bea dis retizationbasisand =( P n =1 i h ) N i=1 be an element in the dis retized spa e X = (span(B))
N
and also in K . The matrix of theoperators and V take O(N
2
) timeto ompute, supposing 3
Using the fa tthatthe produ tof twogaussianfun tionsis also agaussian fun tion,analyti alformulasmaybeprovidedforthe omputationoftheintegral
that nite onstanttimeto ompute R 3 rh rh and R 3 Vh h is needed.The situationisverydi erent forthematri es ofthe opera-tors( ? 1 jxj )and 7! R R 3 (x;y) jx yj
(y)dy.Letustakeforinstan ethe lastoperator.To ompute thematrixofthisoperatoritis ne essary to omputeforall h
,h 2B: Z R 3 Z R 3 (x;y)h (y) jx yj dyh (x)dx= N X i=1 ZZ R 3 R 3 P n =1 i h (x) jx yj n X Æ=1 iÆ h Æ (y)h (y)h (x)dxdy = N X i=1 n X =1 n X Æ=1 i iÆ ( jj Æ):
Even ifformallythisisaO(N 5
) omputation(summationoverthree indi esforea hof theN
2
requiredterms),itis easy to seethat pre- omputinginO(N 3 ) forany ;Æ =1;:::;n: D ;Æ = P N i=1 i iÆ the omputation redu es to order N
4
; unfortunately no further redu -tionsarepossiblesothematrixoftheoperator 7!
R R 3 (x;y) jx yj (y)dy isobtainedby omputing(D ;Æ ) n ;Æ=1
,thenobtaininO(N 4 )the de-siredmatrix P n ;Æ=1 D ;Æ ( jj Æ) n ; =1
.The omputational
om-plexityofaSCFHartree-Fo k omputationisthereforeN I ?N 4 where N I
isthe numberof iterationsrequiredbytheSCFmethod,usually in the range 10 50. We shall apply the bound pro edure and the improvement strategyto qualifythe(known) solutionobtainedfrom thepreviousiterative pro edure farfrom onvergen e.
Letusnowpresent the omplexityissuesrelated tothe omputa-tionofthere onstru tederror
^
W.Theproblem(30)isapproximated on a produ t of N dimensional spa es so the solution will be an nN ve tor ( onsidering the same dis retization X of H as the one used to solve the Hartree-Fo k problem)
4
; we will denote by P thematrixof theproje tor from X to X\
??
;itis easy tosee that P is blo kdiagonal so proje ting an element = (
P n =1 i h ) n i=1 of X to X\ ?? will be an O(N 3
) operation. Let us denote by A thematrix of these ond di erential in of theenergy withrespe t to this dis retization, and by b
the \ve tor" orresponding to the 4
Sin eonlyonedis retization isusedfor the entire omputation,thebounds thusobtainedrefer totheenergyofthesolutionoftheHartree-Fo kproblemon dis rete spa eX.Whenthedis retization X is neenough,one an onsiderto obtain bounds forthe Hartree-Fo k energy.Inany situation,bounds are usefull e.g.asstopping riteriafortheiterativeSCFpro edure(andeventuallyto a el-erate onvergen e);then,inordertoobtain boundsontheHartree-Fo kenergy, orre tionneedtobesolvedonagrid neenoughtobe onsideredexa tasisthe
rst di erential in of the energy, interpretedas an element of the dualX
0
.The problem(30)hasthenthefollowingdis retization: nd w2R
nN
su hthat w=Pw and
(P t A P)w+(P t b )=0: (39) ThematrixA
ofthelinearsystem(39)isfullandimpossibleto om-pletely invert inpra ti e dueto the high omputational omplexity O(N
6
) required. However, using the same argument as above, ap-plying the matrix A
to a ve tor v 2 R nN an be done in O(N 4 ) operations. The problem (39) is then solved iteratively ; nally let usremarkthatthe total ostof there onstru tion ofthesymmetri partis an O(N
3
) pro ess.
The a posteriori method was tested in the omputation of the Hartree-Fo k energy of the arbyne Cr(CO)
4
ClCH mole ule. For ea hiterationstepoftheSCFalgorithmtheorder
4
exa tenergy es-timationswere onstru ted,andalsothe orrespondinglowerbounds asdes ribed inThm. 3.The onvergen e ofthe SCFmethod is pre-sentedinFig.2 and3.Remark thepresen eofquadrati ally onver-gen eperiods(iterations10-50),thepresen eof"jumps"(55-65)and slow onvergen e periods(70-90). In order to avoid the last regime, in pra ti e one only uses the SCF algorithm for a small number of iterations10-40andthenenlargesdis retizationbasis,ortriesto em-piri allyoptimize otherparameters (DIIS).
The resultsobtained by the a posteriori pro edure are presented inthe Fig. 4 and 5.Forsome approximatesolution obtainedduring the SCF iterations, the method des ribed in previous se tion was appliedtoimprovetheenergyandobtainalowerbound(initialdata orrespondingto more than60iterationsisinterpretedas onverged dueto numeri alround-o errors);wedonotatta h spe ialmeaning tothegoodpropertiesofthere onstru tederrorforN =30 ( f.Fig. 5).Astheresultsshow, themethodgivesnearly onvergedresultsas soonastheinitialapproximation isasgoodastheonefrom the10
th iterationof theSCFpro edure.
Remark 14Thenumberof iterationsrequiredto solve thelinear sys-tem (39) was of the order of 10, whi h makes this method more eÆ ient than the SCF y les; for instan e nding the improvement from the 10
th
SCF y le needs 10 iterations to solve (39) and is as good astheresult ofthe 60
th
SCFiteration.
Remark 15ApplyingthematrixA
toa ve torv 2R nN
in(39) re-4
Energy Energy onvergen e iterationnumber energy 30 25 20 15 10 5 -1920 -1930 -1940 -1950 -1960 -1970 -1980 -1990
Fig. 2. The onvergen e of the energy omputedby theSCFalgorithm inthe formusedbyChemists.ThenumberofSCF y les(iterations)rangesbetween1 and30.Noaposterioriimprovementsaremade.
Energy Energy onvergen e iterationnumber energy 90 80 70 60 50 40 30 20 -1982.5 -1983 -1983.5 -1984 -1984.5 -1985 -1985.5
Fig. 3. The onvergen e of the energy omputedby theSCFalgorithm inthe formusedbyChemists.ThenumberofSCF y les(iterations)rangesbetween15 and90.Noaposterioriimprovementsaremade.
Energyfor90iterations Lowerbound Initial(SCF)energy Improvement 30 25 20 15 10 -1970 -1975 -1980 -1985 -1990 -1995 -2000 -2005
Fig. 4. A posteriori error bounds and improvementsare omputed for the re-sultsoftheSCFpro edure.Inea h ase weplottheenergyoftheinitial(SCF) approximation, the energyofthe wavefun tionas omputedbythe aposteriori improvementpro edureandthelowerboundasdes ribedinThm.3.The refer-en e value of the energyis the result of the SCFalgorithm after 90 iterations. TheinitialapproximationstoimprovearetheresultsoftheSCFpro edurefora numberof y lesbetween7and30.
with the a priori introdu tion of further lo alization properties (as domainde ompositionmethods) oftheele troni wavefun tion asit is usually the ase when more eÆ ient Hartree-Fo k omputations are sear hed for [20℄, whi h results in the appli ation of the matrix A
being a O(N 3
) pro ess (or even less); ombining with lassi al onvergen e a eleration tools from the linear system solving (pre- onditioning...)andwiththeorem2,thismethod anbealsoseenas another approa h towards the design of Hartree-Fo k omputations of loweralgorithmi omplexity.
A knowledgements The authors wish to thank Marie-Madeleine Rohmer and Mar Benardfrom "LaboratoiredeChimie Quantique",Strasbourg,Fran e,for providingtheir odeAsterix[7,19,23℄ andfor supportingtheimplementationof theaposteriorimethodinAsterix.
Referen es
Energyfor90iterations Lowerbound Initial(SCF)energy Improvement 60 55 50 45 40 35 30 -1984 -1984.5 -1985 -1985.5 -1986 -1986.5 -1987
Fig. 5. See Fig. 4 for details. The initial approximations to improve are the resultsoftheSCFpro edureforanumberof y lesbetween30and60.
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