Modelling he Pauli Po en ial in he Pai Densi y
Func ional Theo y
C. Amo illiaand ´
A. Nagyb
aDipa imen o di Chimica e Chimica Indus iale,
Uni e si `a di Pisa, Via Riso gimen o 35, 56126 Pisa, I aly
bDepa men o Theo e ical Physics, Uni e si y o Deb ecen,
H–4010 Deb ecen, Hunga y
Oc obe 22, 2008
Abs ac
In he g ound s a e he pai densi y can be de e mined by sol ing a single
auxilia y equa ion o a wo-pa icle p oblem. A no el me hod o de e mining
he Pauli po en ial en e ing his equa ion is p esen ed and, s a ing om a
eliable desc ip ion o he pai densi y, an analy ical exp ession is de i ed o
a omic sys ems. Tes calcula ions a e p esen ed o Be and isoelec onic C2+
and O4+ ions.
1 In oduc ion
Gene alized densi y unc ional heo ies ha e ecei ed a g owing impo ance in ecen
yea s. Fo elec on sys ems, he in e es has been posed on he pai densi y as he
undamen al a iable ins ead o he one pa icle densi y. I u ned ou ha he e
exis a a ia ional p inciple o he pai densi y (analogous o he Hohenbe g-Kohn
heo ems o he densi y unc ional heo y). I has been shown ha - ins ead o
Kohn-Sham equa ions - in he pai densi y unc ional heo y [1–4] he g ound s a e
1
p oblem o an a bi a y sys em is educed o a wo-pa icle p oblem. The wo-
pa icle equa ion is w i en [1–4] as
−1
2∇2
1−1
2∇2
2+ ( 1) + ( 2) + N−1
+ P( 1, 2)χ( 1, 2)
=µχ( 1, 2),(1)
whe e is he ex e nal po en ial, Nis he numbe o elec ons and he no a ion
=| 1− 2|is used. The g ound-s a e eigen unc ion o his equa ion, ˜χ0say,
co esponds o he pai densi y ampli ude and is ela ed o he pai densi y no he
eal sys em as
n=N(N−1)
2|˜χ0|2.(2)
Eq. (1) con ains an unknown e m, P, o comple ely kine ic o igin. A e a densi y
unc ional analogy Pis called Pauli po en ial. Eq. (1) is analogous o he densi y
unc ional equa ion o he squa e oo o he densi y (which da es back o Thomas
and Fe mi [5] and is analyzed by Le y, Pe dew and Sahni [6]).
The pai densi y can be nume ically calcula ed ei he on he Ha ee-Fock le el
o on highly co ela ed le el. The pai densi y can also be de e mined om Eq. (1)
in a a he s aigh o wa d way i he Pauli po en ial is known. Howe e , he e
a e no da a o he Pauli po en ial in he li e a u e, ye . Al hough ecen ly, he
elec on-elec on cusp condi ion and asymp o ic beha iou o he Pauli po en ial
ha e been de i ed [7], much wo k is called o o comple ely unde s and how such a
po en ial could be modellized in analogy wi h he Kohn-Sham po en ial in o dina y
densi y unc ional heo y (DFT). Conside ing he p esen knowledge, his kind o
wo k appea s ex emely di icul . We belie e ha an impo an i s s ep, in o de
o gain mo e insigh in his di ec ion, is he econs uc ion o he Pauli po en ial
2
om a eliable pai densi y o m o some ealis ic ac able elec on sys em. Fo
hese cases, a six a iable unc ional o m o Pshould be, in p inciple, ob ained.
This unc ion, a his poin , should be iewed as a sou ce o in o ma ion expecially
wi h he aim o inding hose p ope ies ha can be ans e ed o o he sys ems o
which he pai densi y is unknowm.
Mo i a ed by he abo e conside a ion, in his pape we p esen model Pauli
po en ial o he Be a om and isoelec onic a omic ions C2+ and O4+. The me hod,
which is in ended o cap u e he main ea u es o P, is based on an ansa z on
he o m o he pai densi y ampli ude. We gene alize he me hod o Amo illi e
al. [8]. The o iginal me hod was used o ob ain he exac Hamil onian o an analy ic
g ound-s a e wa e unc ion o He-like ions. He e, a gene aliza ion is p esen ed o
p oducing he Pauli po en ial om a model pai densi y ampli ude.
The pape is o ganized as ollows: In sec ion 2 he pai densi y unc ional he-
o y is e iewed. In sec ion 3 a model pai densi y ampli ude and he co esponding
po en ial is p esen ed. Sec ion 4 desc ibes nume ical examples: he Be and some
isoelec onic a omic ions. The las sec ion is de o ed o discussion.
2 The pai densi y unc ional heo y
Fi s , he pai -densi y unc ional heo y [1–3] is summa ized. Conside he many
elec on Hamil onian H,
ˆ
H=ˆ
T+ˆ
Vee +
N
X
i=1
( i),(3)
3
whe e
ˆ
T=
N
X
i=1
(−1
2∇2
i) (4)
is he kine ic ene gy ope a o ,
ˆ
Vee =
N
X
i<j
1
| i− j|(5)
is he elec on-elec on epulsion ene gy ope a o and ( ) is a local ex e nal po en-
ial. Fo con enience we conside an e en numbe o pa icles. The second-o de ed
educed densi y ma ix is de ined as
n2(x1,x2;x0
1,x0
2) = N(N−1)
2ZΨ(x1,x2,x3..., xN)Ψ∗(x0
1,x0
2,x3..., xN)dx3...dxN,(6)
whe e xis ands o he spa ial and he spin coo dina es: i, σiand he in eg al sym-
bol when e e ed o spin deno es summa ion. The diagonal o he spin-independen
second-o de ed densi y ma ix
n( 1, 2) = X
σ1,σ2
n2( 1, σ1, 2, σ2) (7)
also called pai densi y is he key quan i y.
I is con enien o in oduce new posi ion a iables
qJ= ( j, j0) = (qJ1, qJ2, qJ3, qJ4, qJ5, qJ6),(8)
i.e. he pai s will be deno ed by capi al indices while he pa icles in each pai will
be iden i ied by he co esponding unp imed and p imed le e s. Wi h he abo e
no a ion he numbe o a iables is he same as ha o he ini ial sys em. Each
pa icle is associa ed wi h a single pai , i.e. he numbe o indices Jis N/2. The
’in e nal’ po en ial o he pa icles in pai Jis gi en by
˜ (qj) = ˜ ( j, j0) = 1
| j− j0|.(9)
4
while he in e ac ion be ween pai s Iand Jis
WIJ =W(qI,qJ) = W( i, i0; j, j0) = 1
| i− j|+1
| i− j0|+1
| i0− j|+1
| i0− j0|.(10)
The ene gy o he pai s due o he ex e nal po en ial is
ˆ
U=
M
X
I=1
u(qJ) =
N
X
I=1
( ( j) + ( j0)) .(11)
De ining he ope a o ˆ
L ep esen ing he in e nal ene gy o pai s
ˆ
L=
M
X
I=1
(−1
2∇2
I+ ˜ (qI)) ,(12)
he ini ial Hamil onian can be exp essed as
ˆ
H=ˆ
L+ˆ
W+ˆ
U , (13)
whe e
ˆ
W=1
2
M
X
I6=J
WIJ (14)
is he in e ac ion ene gy be ween di e en pai s (M=N/2).
ˆ
His he same as he ini ial Hamil onian, bu now i is w i en in e ms o pai s
o pa icles, wi h ˆ
L+ˆ
U ep esen ing he Hamil onian o independen (nonin e ac ing
wi h each o he ) pai s and ˆ
W ep esen ing he in e pai in e ac ion.
The Laplacian in he kine ic ene gy ope a o can also be w i en as
∇2
I=∇2
qI=∇2
i+∇2
i0=
6
X
α=1
∂2
∂q2
Iα
.(15)
The ene gy o he independen pai s has he o m
Q[n] = min
Ψ→nhΨ|ˆ
L+ˆ
W|Ψi.(16)
5

The sea ch o he minimum is o e all an isymme ic wa e unc ions Ψ which yield
he gi en n. Then he g ound s a e ene gy can be w i en as
E= min
n1
N−1Zu( 1, 2)n( 1, 2)d 1d 2+Q[n].(17)
The ac o 1/(N−1) comes om he no maliza ion o n. The densi y o pai I
n(qI) = n( i, i0) = X
σi,σi0
n2( i, σi, i0, σi0) (18)
is he pai densi y in he o iginal space.
The Hohenbe g-Kohn heo ems [10] ha e been gene alized o he pai densi y
[11,12] o he o iginal space. The g ound s a e inequali y is
1
N−1Zn(q)u(q) + Q[n]≥E0.(19)
whe e E0and n0a e he g ound-s a e ene gy and he diagonal o he spin indepen-
den second-o de densi y ma ix, espec i ely.
In he pai densi y unc ional heo y he adiaba ic connec ion is de ined by he
pa ame ized Hamil onian
ˆ
Hα=ˆ
L+αˆ
W+ˆ
Uα,(20)
whe e ˆ
Uα=PIuα
I(q) is gi en by he condi ion ha he pai densi y n(q), o
he o iginal space keeps being independen o α. Fo α= 0 he ’non-in e ac ing
Hamil onian’
ˆ
Hα=0 =ˆ
L+ˆ
Uα=0 =X
I=1
hα=0
I(21)
is ob ained. In his auxilia y sys em he in e ac ion be ween he pai s is ze o and
he auxilia y equa ions ha e he o m
ˆ
H0Ψ0=E0Ψ0.(22)
6
The wa e unc ion in his auxilia y sys em can be w i en as a symme ized
exp ession o an isymme ic wo-pa icle unc ions χI:
Φ0(x1, ..., xN) = ˆ
S(χ1(x1,x2)...χM(xN−1,xN)) .(23)
ˆ
S=1
N!X
P
ˆ
P(24)
is he symme ize ope a o . Pis he pe mu a ion ope a o and he sum is o e all
pe mu a ion o he elec on pai s. This wa e unc ion is an isymme ic wi h espec
o he exchange o he a iables o a single pai and symme ic wi h espec o he
exchange o he pai s. The disad an age o he p esen no a ion is ha i does no
allow ansposi ion o a iables belonging o wo di e en pai s. In he g ound s a e
n(q) = NN−1
2X
σ
|χ0(x1,x2)|2=NN−1
2|˜χ0(q)|2,(25)
whe e he wo-pa icle unc ion ˜χ0sa is y he eigen alue equa ion
h0(q)˜χ0(q) = −1
2∇2
q+ e (q)˜χ0(q) = ε0˜χ0(q),(26)
We men ion in passing ha i is possible o w i e he pai densi y in e ms o
geminals. The p esen e sion o he pai densi y e sion o heo y has he ad an age
ha he calcula ion o nis always educed o he solu ion o a wo-pa icle equa ion
ha is he N-body p oblem can be educed o a wo-body p oblem. I has been
p o ed [1] ha he auxilia y po en ial is uniquely de e mined by he diagonal o m
o he spin independen second-o de densi y ma ix and he e ec i e po en ial is o
he o m
e (q) = ( 1) + ( 2) + N−1
12
+ p,(27)
7
whe e
p= (N−1)δTP
δn (28)
and
TP=T−T0(29)
is he di e ence o he kine ic ene gies o he eal sys em (T=hΨ|ˆ
T|Ψi) and he
auxilia y sys em
T0=
M
X
I=1 Zχ∗
I(x1,x2)[−1
2∇2
q]χI(x1,x2).(30)
By a densi y unc ional analogy he unc ional TP[n] is called Pauli ene gy. The
o al ene gy has he o m
E[n] = T0[n] + TP[n] + Zn(q)
12
dq+1
N−1Zn(q)u(q)dq.(31)
The disad an age o he p esen ea men is ha i is ha d o cap u e he
e mionic s uc u e o an elec onic sys em wi h a single e ec i e po en ial. Howe e ,
we ha e always a wo-pa icle p oblem o sol e independen ly o he numbe o
elec ons. I is wo h o make e o s o ind adequa e app oxima ion o he Pauli
po en ial in o de o u ilize his bene i . The p esen s udy is a s ep in his di ec ion.
The auxilia y equa ions can also be de i ed by cons ained sea ch [1,13]. The
wo-pa icle equa ion (26) was la e de i ed [14] in a di e en way which is no
es ic ed o e en numbe o elec ons.
8
3 A model pai densi y ampli ude and he co e-
sponding po en ial
As i was shown in he i s pape [1] he Pauli po en ial is uniquely de e mined by
he pai densi y. Tha is, om he knowledge o n, Pcan be gi en by in e ing Eq.
(1)
P( 1, 2) = −Kloc( 1, 2)−w( 1, 2),(32)
whe e
Kloc( 1, 2) = −1
2˜χ0( 1, 2)h∇2
1+∇2
2i˜χ0( 1, 2) (33)
and
w( 1, 2) = ( 1) + ( 2) + N−1
−µ. (34)
We ha e ecen ly p o ed [7] ha he Pauli po en ial asymp o ically beha es
as
P→(N−2) 1
1
+1
2
−1
(35)
when 1→ ∞, 2→ ∞ and → ∞. The elec on-elec on cusp condi ion has he
o m:
P=2−N
(36)
as →0.
Wi h he aim o econs uc Pin some unc ional o m, we s a ou om a
model unno malized pai densi y ampli ude in he o m
χ=χHF (λ 1, λ 2)(1 + g( )) ,(37)
9
a λ E Tp< −2> < −1> < > < 2> < 3>
4 0.9835 –14.660(4) 0.878(3) 9.319 4.279 15.537 54.485 232.96
5 0.9847 –14.669(4) 0.864(3) 9.541 4.320 15.469 54.107 230.88
6 0.9860 –14.669(4) 0.855(3) 9.701 4.349 15.418 53.815 229.23
HF(a)–14.573 1.005 10.536 4.489 15.120 51.956 218.11
co (b)–14.667 — 9.536 4.337 15.272 52.854 222.48
(a)In he Tpcolumn is epo ed he di e ence Tex −T(1)
wand momen s
a e om e s. [16,18].
(b)Momen s om [18].
Table 2: To al ene gy (E), Pauli kine ic ene gy (Tp) and some momen s < k>
o Be a om o di e en choices o he co ela ion unc ion pa ame e aand he
scaling cons an λcalcula ed in his wo k and compa ison wi h HF and co ela ed
li e a u e da a. Da a a e in a omic uni s.
a λ E Tp< −2> < −1> < > < 2> < 3>
5 1.00090 –36.538(9) 3.253(8) 25.37 7.540 8.013 14.031 29.466
6 1.00063 –36.538(9) 3.226(8) 25.35 7.577 7.997 13.991 29.369
7 1.00055 –36.539(9) 3.204(8) 25.94 7.604 7.985 13.961 29.296
HF(a)–36.408 3.475 27.06 7.716 7.945 13.863 29.06
co (b)–36.534 — 25.50 7.548 8.118 14.502 31.14
(a)In he Tpcolumn is epo ed he di e ence Tex −T(1)
wand momen s
a e om e s. [16,18].
(b)Momen s om [16].
Table 3: To al ene gy (E), Pauli kine ic ene gy (Tp) and some momen s < k> o
C2+ a omic ion o di e en choices o he co ela ion unc ion pa ame e aand he
scaling cons an λcalcula ed in his wo k and compa ison wi h HF and co ela ed
li e a u e da a. Da a a e in a omic uni s.
ampli ude as hey esul om a i ing o he same accu a e unc ion. The comple e
de ini ion o ˜χ0depends a his poin by he pa ame e aen e ing he co ela ion
unc ion g( ) and he scaling ac o λ. We made di e en choices o such pa ame e s
o all he h ee cases and he inal esul s a e collec ed in Tabs. 2,3,4.
The main p oblem encoun e ed in he calcula ion o he o al ene gy by Mon e
Ca lo me hod has been ela ed o he high a iance o he unc ion o be a e aged
16

a λ E Tp< −2> < −1> < > < 2> < 3>
7 1.00005 -68.40(1) 6.91(1) 49.11 10.712 5.492 6.534 9.280
8 1.00003 -68.40(1) 6.91(1) 49.45 10.741 5.486 6.522 9.260
9 1.00003 -68.41(1) 6.90(1) 49.73 10.763 5.481 6.513 9.244
HF(a)-68.257 7.374 51.8 10.887 5.455 6.469 9.167
co (b)-68.411 — 49.17 10.694 5.570 6.769 9.843
(a)In he Tpcolumn is epo ed he di e ence Tex −T(1)
wand momen s
a e om e s. [16,18].
(b)Momen s om [16].
Table 4: To al ene gy (E), Pauli kine ic ene gy (Tp) and some momen s < k> o
O4+ a omic ion o di e en choices o he co ela ion unc ion pa ame e aand he
scaling cons an λcalcula ed in his wo k and compa ison wi h HF and co ela ed
li e a u e da a. Da a a e in a omic uni s.
which is de ined in Eq. (59). This equi es a long simula ion o achie e an ene gy
mean alue wi h an accu acy o he o de o some mHa ee. The same occu s o
Tp.
Looking a he esul s o Tabs. 2,3,4, i is e iden ha he op imal alues o
he pa ame e s aand λmus be ound by sea ching o a comp omise be ween he
need o ge ing eliable alues o he momen s < k>and he bes ene gy. The
esul s show also ha a conside able ac ion o co ela ion ene gy has been aken
in o accoun . I is also impo an o no ice ha he a iance becomes la ge when
he nuclea cha ge inc eases bu also ha ou app oxima ion, mainly based on sho
ange co ela ion, should wo ks be e in such cases.
I is also in e es ing o look a he alues o Tp. F om he bounds on he
gene alized Weizs¨acke - ype kine ic ene gy in oduced by Aye s [15] i ollows ha
0≤Tp≤Tex −T(1)
w.(63)
Looking a ou esul s, his inequali y is sa is ied in he ange o a alues consid-
17
e ed he e. De ia ions om his beha io mus lead o conside a ions ela ed o
N- ep esen abili y.
Finally, i is wo hwhile o look a he plo s o he e ec i e po en ial de i ed by
he app oxima e pai densi y ampli ude. This has been done o he con ibu ions
V1( 1, 2) and V2( ) while V3( 1, 2, ) canno be easily shown being dependen on
h ee independen a iables. Fo his pu pose, and only o Be, we plo V1( 1, 2) in
Fig.1 and V2( ) in Fig.2. F om Fig.1, i is clea ha V1is domina ed by he ex e nal
nuclea po en ial when 1o 2 ends o 0 while is abou cons an o bo h la ge 1
and 2, being −µ he limi in his case. The ipples o he wo dimensional su ace o
Fig.1 a e ins ead a consequence o he exchange in e ac ion and de e mine he shell
s uc u e o he one pa icle densi y o Be a om. Finally, V2, shown in Fig.2, is always
epulsi e. Fo small , i beha es as he elec on-elec on in e ac ion po en ial while
i goes o ze o mo e apidly o la ge . The ipples o V1, he long ange beha io
o V2and he con ibu ion V3a e special ea u es o P.
5 Conclusions
In his wo k, we ha e illus a ed a me hod o econs uc he Pauli po en ial o pai
densi y unc ional heo y o ou elec on a omic ions. The po en ial is de i ed by
in e ing he e ec i e wo elec on equa ion in ol ing he pai densi y ampli ude
assuming ha he pai densi y i sel can be w i en in an analy ical ac able o m.
Cusp and asymp o ic condi ions ha e been sa is ied and app op ia e adjus able pa-
ame e s ha e been used in o de o ep oduce, wi hin a easonable accu acy, he
o al ene gy and some lowe momen o he in acule densi y. Some in e es ing ea-
18
u es o he Pauli po en ial ha e been ound o he sys ems ea ed he e. These
ea u es a e con ained in he exp essions (40), (44) and (45) o V1,V2and V3.V1and
V2include also he nuclea and he elec on-elec on elec os a ic po en ial ene gies.
Some illus a ions a e gi en also in Figu es 1 and 2.
We would like o emphasize ha he p esen me hod is no es ic ed o ou -
elec on sy ems. In he he pai densi y heo y one has o sol e an e ec i e wo
elec on equa ion independen ly on he numbe o elec ons. Tha is, he no el
me hod in oduced he e o in e he e ec i e wo elec on equa ion can always be
applied i he he pai densi y (o he he pai densi y ampli ude) is a ailable.
Fo he u u e, i will be in e es ing o analyze in de ails each indi idual e m in
o de o ind a gene aliza ion o he abo e exp essions o all polyelec onic sys ems
in a o m which does no equi e he in e sion o he e ec i e wo elec on equa ion
wo ked he e.
Abou V2, we would like o e e b ie ly o he ’a e age-pai -densi y heo y’
o Go i-Gio gi and Sa in. In his heo y he sphe ically and sys em-a e aged pai
densi y ( ) is de e mined by simple adial equa ions conjec u ed by Go i-Gio gi
and Sa in [21]:
h−∇2
+we ( )iφi( ) = iφi( ),(64)
he solu ions o which gi e ( ) as
X
i
θi|φi( )|2= ( ),(65)
ha is, ( ) is gi en by a weigh ed sum o he squa e o some o hogonal ’e ec i e’
geminals φiwi h weigh ing ac o s o ’occupancy’ θi. The po en ial we ( ) in Eq.
19
(64) was app oxima ed as
we ( ) = w(0)
e ( ) + wc
e ( ),(66)
whe e
w(0)
e ( ) = ∇2 1/2
KS
1/2
KS
(67)
and
wc
e ( ) = 1
+ 2
2 3
s
−3
2 s!θ( s− ).(68)
θ( s− ) is he Hea iside s ep unc ion and
s=4π
3%−1/3
,(69)
whe e %is he a e age elec on densi y. The co ela ion po en ial wc
e ( ) o iginally
p oposed by O e hause [22], has been used o sol e Eq. (64) o he uni o m elec on
gas [21, 23]. I leads o an accu a e desc ip ion o he sho - ange pa o . Ou
po en ial V2( 1, 2) (44), using he exp ession (51) o g, has he o m
V2( 1, 2) = 1
(1 + a )2(1 + (a+ 1/2) ).(70)
We immedia elly no ice ha he dominan e m in (70) o small is 1/ . I is
he same as he i s e m in he O e hause po en ial, which is also he dominan
pa o he O e hause po en ial o small . Thus he po en ial V2( 1, 2) has some
esemblance o he O e hause po en ial. The 1/ e m in he O e hause po en ial
comes om he cusp condi ion on ( ) [21]. The dominan e m in (70) o small
has he same o igin.
We also men ion in passing ha i was de i ed ia a double adiaba ic connec-
ion by one o he p esen au ho s [4] ha he squa e oo o he sphe ically and
20
sys em a e aged pai densi y is he solu ion o a simple adial equa ion, ha is, con-
a y o he heo y o Go i-Gio gi and Sa in, i is possible o ob ain ( ) h ough
a solu ion o a single equa ion. I a single geminal is used he e mionic cha ac e
should be e lec ed in he po en ial which is consequen ly mo e complica ed. I mo e
han one geminals a e used he sphe ically and sys em a e aged pai densi y has a
mo e complica ed o m bu he po en ial can be mo e easily app oxima ed. The
numbe o geminals N(N−1)/2 depends on he numbe o elec ons. The e o e a
single geminal app oach migh gain an impo an ole as he numbe o elec ons
inc eases.
In he densi y unc ional heo y he e has been a g owing in e es in de e -
mining he exac exchange, exchange-co ela ion and Kohn-Sham po en ials in he
knowledge o he densi y. Se e al me hods ha e been wo ked ou [24–29]. The ex-
ac po en ials a e e y use ul, o example o check he accu acy o app oxima e
me hods. An analogous p oblem in he pai densi y unc ional heo y is o ob ain
he Pauli po en ial in he knowledge o he pai densi y as he e he elec on-elec on
i e ac ion is exac ly ea ed, bu he kine ic ene gy unc ional is unknown. The p ob-
lem he e is mo e complica ed in he sense ha a wo-pa icle po en ial Pshould be
calcula ed, ins ead o a one-body exchange-co ela ion po en ial o he densi y unc-
ional heo y. On he o he hand, i is also simple as only a single equa ion has o
be in e ed ins ead o se e al Kohn-Sham egua ions in he densi y unc ional heo y.
The accu a e o m o he Pauli po en ial ob ained by he p esen me hod can be
used la e o ind app oxima e exp essions o i . One has o be, howe e , ex emely
ca e ul in he cons uc ion because o he N- ep esen abili y p oblem [11,15,30–43].
Dal Ri e al. [19] de i ed densi y ma ices om Jas ow- ype ial wa e unc-
21

ions. The pai densi y used in his wo k can be conside ed as he lowes o de
app oxima ion o he gene al, N- ep esen able pai densi y p esen ed by Dal Ri e
al. Consequen ly, ou pai densi y is, a leas app oxima ely , N- ep esen able.
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Acknowledgemen s
This pape was w i en in he ame o he Bila e al Scien i ic Coope a ion be ween
I aly and Hunga y sponso ed by Consiglio Nazionale delle Rice che and he Hun-
ga ian Academy o Sciences.
G an OTKA No. T 029469 is g a e ully acknowledged.
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