Arrow diagram method based on overlapping electronic groups: corrections to the linked AD theorem

0.34 MB·English

by  Yu Wang

Arrow diagram method based on overlapping ele
troni
groups:

orre
tions to the linked AD theorem
5 ∗
0 Yu Wang and Lev Kantorovi
h
0
2 2nd February 2008

n
a
J Department of Physi
s, King's College London, Strand, London, WC2R 2LS, United Kingdom

6 ∗
1 E-mailaddress:lev.kantorovit
hk
l.a
.uk

] Abstra
t
h
Arrow diagram (AD) method (L. Kantorovi
h and B. Zapol, J. Chem. Phys. 96, 8420 (1992); ibid,
p hΨ|O|Ψi
8427) provides a
onvenient means of systemati

al
ulation of arbitrary matrix elements, , of
- O Ψ
m symmetri
al operators, , in quantum
hemistry when the total system waΦvefun
tion is represented as
A b
e an antisyAmmetrised produΨ
t=ofAoverlapΦping many-ele
trongroup fun
tions, ,
orresponding to ea
h part
h (group) of the systemb: A A. For extended (e.g. in(cid:28)nite) systems the
al
ulation is somewhat
s.c dbie(cid:30)d
iuvlitd,ehdobwyevtehre,naosrmmaealinsavtiaolnubeisQntoefgrtahleSop=erhaΨto|rΨsir,ewquhiir
ehtihsagtiveean
hbytearmn AoDf tehxepdainasgioranmasexwpealln.sAionlinisketdo
c AD theorem suggested previously (L. Kantorovi
h, Int. J. Quant. Chem. 76, 511 (2000)) to deal with
si this problem is reexamined in this paper using a simple Hartree-Fo
k problem of a one-dimensional ring of
y in(cid:28)nite size whi
h is found to be analyti
ally solvable. We (cid:28)nd that
orre
tions to the linked AD theorem
h are ne
essary in a general
ase of a (cid:28)nite overlap between di(cid:27)erent ele
troni
groups. A general method of
p
onstru
ting these
orre
tions in a formof a power series expansion with respe
t to overlapis suggested. It
[
is illustrated onthe ring model system.

2
v
9 1 Introdu
tion
7
0 For a wide range of quantum-me
hani
al systems, the entire ele
troni
system
an
onveniently be split into
2 a set of ele
tron groups (EG) su
h as
ore and valen
e ele
trons in mole
ules or
rystals, ele
trons on atoms
1
or ions in atomi
or ioni
solids,
ore and bond ele
trons in strongly
ovalent materials, et
. [1, 2, 3℄. Similar
4
partitioning ideas
an also be applied to separate ele
trons in a
luster and environment regions to derive a
0
/ parti
ular embedding potential for the quantum
luster [3, 4, 5℄. Provided that the partition s
heme applied
s
c to the given system is physi
ally (or
hemi
ally) appropriate, one
an assume that ele
trons (cid:28)xed to the given
i groupspendmostoftheirtimeatthisgroup. Therefore,toagoodapproximation,thewavefun
tionofthewhole
s I
y system
onsisting of ele
troni
groups
an be represented as an antisymmetrised produ
t of wavefun
tions
Φ (X )
I I
h of every individual group:
M
p
Ψ(X , ,X )=A Φ (X )
v: 1 ··· M I I (1)
I=1
i Y
X M I b NI
where is the total nuXmb=er(xof,E.G..'xs. H)ere the -th group is asso
iated with elxe
=tro(rn,sσw)hi
h
oordinates
ar are
olle
tedσinto a set I 1 NI . Note that the single ele
tron
oordinates in
lude the spin

oordinate as well. The antisymmetrisation operator is de(cid:28)ned via
1
A= ǫ P
P
N! (2)
PX∈SN
b b
1
N! P S
N
where thesum runs overall permutations ofthe
omplete groupofpermutations , thetotalnumberof
N =N +N +...+N ǫ = 1
1 2 M P
ele
tronsin the wholesystem being andthe fa
tor ± a

ordingto theparityof
P b ΦI(XI)
thepermutation . Itisassumedherethatthegroupfun
tions arealreadyindividuallyantisymmetri
.
A
By applying expression (2) for the operator in Eq. (1), one obtains an expansion of the wavefun
tion via
b Ψ O Ψ
variousfun
tions-produ
ts. Whenusedin
al
ulatingmatrixelementsh | | iofsymmetri
al(withrespe
tto
bO O(x ,...,x )
1 N
permutationsofele
troni

oordinates)operators ≡ (e.g. an external(cid:28)eldorele
tron-ele
tron
b
intera
tion), a
orresponding expansion of matrix elements between various fun
tions-produ
ts is obtained.
b b
Earlier attempts (see, e.g. [6, 7, 8, 9, 10, 11, 12℄) to simplify the expansion and asso
iate ea
h term with
a transparent diagram have been generalised in Refs. [13, 14℄ where the Arrow Diagram (AD) theory was
S
N
developed. Firstly, by exploiting a double-
oset de
omposition of the group , many terms in the expansion
werefoundtobeidenti
alandthe
orrespondingrulestonumberalldistin
ttermswereworkedoutinageneral

ase. Se
ondly, ea
h distin
t term in the expansion was given by a well-de(cid:28)ned pi
ture (arrow diagram) whi
h
allowed to
onstru
t simple rules to asso
iate an analyti
al expression with any of the AD's. The theory was
formulated in a very general way for arbitrary number of ele
tron groups and expli
it AD expansions were

onstru
ted for redu
ed density matri
es (RDM) [1, 2℄ of orders one and two (RDM-1 and RDM-2). These
enable one to
al
ulate matrix elements of arbitrary symmetri
al one- and two-parti
le operators and are thus
su(cid:30)
ient in most
ases relevant to quantum
hemistry of mole
ules and solids (generalisation to higher order
RDM's is
umbersome but straightforward).
Ingeneral,thesu

essofthiste
hniquedepends
ru
iallyonthevalueoftheoverlapbetweendi(cid:27)erentgroup
Φ (X )
I I
fun
tions and thus their lo
alisation in
ertain regions of spa
e (also known as stru
ture elements [3℄).
This is be
ause greater lo
alisation of the group fun
tions results in better
onvergen
e of the AD expansion,
i.e. smaller number of terms in the expansion are to be retained. That is why the
hoi
e of ele
troni
groups
and appropriate regions of their lo
alisation [15, 16℄ is so important for this method to work.
Although theADtheoryformulatedin [13, 14℄ isappli
ableto anysystem
onsistingofarbitrarynumberof
overlappingele
trongroups,one
annot dire
tly use it to des
ribean extended (in parti
ular, in(cid:28)nite) system.
Indeed, to
al
ulate an observable
Ψ O Ψ
O= h | | i
Ψ Ψ (3)
h | i
b
O
of the operator , in the
ase of an extended system one has to
al
ulate the ratio of two AD expansions, one
Ψ O Ψ S = Ψ Ψ
arising from the matrix element h | | i and another - from the normalisation integral h | i. Both
b
expansions
ontain very many terms; in fa
t, the number of terms is in(cid:28)nite in the
ase of an in(cid:28)nite system
b
(e.g. a solid).
S
Itwasarguedin[17℄thatthenormalisationintegral tendstoin(cid:28)nity withthenumberofele
troni
groups
M
. This assertionmade it possibleto provethe so-
alledlinked AD theorem[17℄. It states that the observable
O = Ψ O Ψ Ψ O Ψ O
h | | ic, i.e. it is equal to the AD expansion for the matrix element h | | i of the operator ,
in whi
h only linked (
onne
ted) ADs are retained (subs
ript (cid:16)
(cid:17)), and the normalisation integral should be
b b b
dropped altogether. This expression is very attra
tive as it allows one to
onsider a single AD expansion for
arbitrary matrix elements of operators.
It is the main obje
tive of this paper to show that the linked AD theorem is
orre
t only approximately.
When the overlapbetweendi(cid:27)erentgroup fun
tions be
omes larger,
orre
tionsareto be introdu
ed. We show
this by (cid:28)rst
onsidering in detail a simple (cid:16)toy(cid:17) model of a 1D ring for whi
h the exa
t analyti
al solution is
possible. ThisallowsustoinvestigateindetailthelimitingbehaviouroftheADexpansionforthissystemwhen
the number of groups in the ring tends to in(cid:28)nity. Then, we suggest a general method of
al
ulating the mean
values, Eq. (3), in a form of
orre
tions to the linked AD theorem.
The paper is organised as follows. In the subsequent se
tion we shall brie(cid:29)y review the AD theory and
introdu
e the linked AD theorem for the normalisationintegral and RDM-1. The
onsiderationof higher order
RDM'sisanalogous. Inse
tion3,we
onsiderindetailthetoymodelandobtainanexa
texpressionforRDM-1
in thelimitofanin(cid:28)nite ringsize. Thisexa
tresultswillthenbe
omparedwiththatobtainedusingthelinked
2
AD theorem. A systemati
way of
onstru
ting
orre
tions to the linked AD theorem is then suggested in
se
tion 4.
2 The Arrow Diagrams theory
S I
Firstwe
onsiderasymmetri
group NI. Itselementspermuteele
troni

oordinatesbelongingonlytothe -th
S =S S S S
group. Joiningtogether all su
h groupswe obtain a subgroup 0 N1∪ N2∪···∪ NM of N. Any element
S
0
of only inter
hanges ele
troni

oordinates within the groups, i.e. performs only intra-group permutations.
The algebrai
foundation of the AD theory is based on the double
oset (DC) de
omposition of the
omplete
S S
N 0
symmetri
group with respe
t to , whi
h enables to single out all inequivalent intergroup permutations
P
qT
[13℄:
1
b S = µ S P S
N qT 0 qT 0
N (4)
0
qT
X
b
q
Here the sum runs over all distin
t types, , of operations for intergroup permutations as well as all distin
t
T N = N !N ! N !
0 1 2 M
ways, , of labelling a
tual groups (EG's) involved in it, and ······ is a numeri
al fa
tor.
P
qT
is a DC generator, whi
h, involves only inter-group permutations and, in general,
an be
onstru
ted as a
produ
t of some primitive
y
les ea
h involving no more than one ele
tron from ea
h group.
b
Ea
h su
h a
y
le is represented as a dire
ted
losed loop
onne
ting all groups involved in it. Thus,
P
qT
in general, ea
h permutation
an be drawn as an arrow diagram
ontaining a
olle
tion of
losed dire
ted
loops. Theseloopsmaypassthroughthegivengroupseveraltimes(dependingonthenumberofele
tronsofthe
b P
qT
group whi
h are involved in the permutation ). If the groups involved in the diagram
annot be separated
without destroying the dire
ted loops, the AD is
alled linked or
onne
ted. If, however, this separation is
b
possiblethanthediagramis
allednon-linked ordis
onne
ted andit
anberepresentedasa
olle
tionoflinked
µ
qT
parts. The de
omposition
oe(cid:30)
ients
an be
al
ulated merely by
ounting arrows entering and leaving
ea
h group in the AD.
A
By writing the antisymmetriser of Eq. (2) using the DC generators, it is possible to obtain the diagram
expansion of the normalisation integral [14℄:
b
S = AΦ AΦ =Λ ǫ µ Φ P Φ
q qT qT
(5)
D (cid:12) E XqT D (cid:12) (cid:12) E
Λ =N /N! Φ = Φ (X ) b (cid:12)(cid:12) b (cid:12)(cid:12)b (cid:12)(cid:12) S
where 0 and I I I is a produ
t of all group fun
tions. Eq. (5) gives an expansion of in
S Φ P Φ
Q N qT
termsofdiagramsidenti
altothoseforthepermutationgroup ofEq. (4). Thematrixelements
P D (cid:12) (cid:12) E
are represented as a produ
t of RDM's of ele
troni
groups involveΦd in the permutation qT integra(cid:12)(cid:12)tbed (cid:12)(cid:12)over
I

orrespondingele
troni

oordinates. If ea
hofthegroupfun
tions isgivenasasum ofSlaterdeterminants
b Φ P Φ
qT
withmole
ularorbitalsexpandedviasomeatomi
orbitals(AO)basisset,thenthematrixelements
D (cid:12) (cid:12) E
areeventuallyexpressedasasumofprodu
tsofsimpleoverlapintegralsbetweentheAO'sofdi(cid:27)erente(cid:12)(cid:12)leb
tr(cid:12)(cid:12)oni
groups(see,.e.g. Ref. [17℄). It is important to note here that a total
ontribution of a non-linked AD is exa
tly
equal to the produ
t of
ontributions asso
iated with ea
h of its linked parts [14℄.
Forthe following, it is
onvenientto sortout all the terms in the diagrammati
expansionby the number of
Λ
groups involved in the AD's. Getting rid of the
ommon fa
tor , we
an write:
M
S
S = =1+ S (A ,A , ,A )
K 1 2 K
Λ ··· (6)
KX=2 A1<X···<AK
e
3
S (A ,A , ,A )
K 1 2 K
where ··· is the
ontribution of all AD's that
orrespond to intergroup permutations among
K A ,A ,...,A
1 2 K
di(cid:27)erent groups with the parti
ular labelling . The unity in equation above
orresponds to
the trivial permutation.
For
onvenien
e, the expansion in Eq. (6) will be referred to as the normalisation-integral expansion. In
prin
iple, in this expansion all groups of the entire system parti
ipate. It is also found
onvenient to introdu
e
S(T)
a derivative obje
t, , whi
h is obtained from the above expansion by retaining all the AD's whi
h are
T = A ,...,A
1 K
asso
iated only with the groups from a (cid:28)nite manifold { } (i.e.
ontaining groups with labels
A ,...,A T e S
1 K
). If
omprises the whole system, we arrive at of Eq. (6). We shall also introdu
e a manifold
[T]
, i.e. an arti(cid:28)
ial system whi
h is obtained by (cid:16)removing(cid:17) from the entire system all groups
omprising the
T S([T]) S[Te]
set . Then, (we shall also use a simpler notation when
onvenient) is obtained from Eq. (6) by
T S([T])
retainingonlyAD'sin whi
hanyofthegroupsbelongingto isnot present; in otherwords, isobtained
S e e T
from by assuming that in any of the AD's all the overlapintegralsinvolving groups from the manifold are
S S[T] eT
equal to zero. Obviously,
an be
onsidered as a parti
ular
ase of when the manifold is empty. We
e S[A ,...,A ] S[T] T = A ,...,A
1 K 1 K
shallo

asionallyalsousethenotation for , where { }, if wewanttoindi
ate
e e
expli
itly whi
h parti
ular groups are ex
luded.
e e
The unnormalised RDM-1 of the whole system,
ρ(x;x′)=N Ψ(x,x ,...,x )Ψ∗(x′,x ,...,x ) x ... x
2 N 2 N 2 N
d d (7)
Z

an also be written as a matrix element of a
ertain symmetri
al one-parti
le operator [1, 2, 14℄. Therefore, as
A
in the
ase of the normalisation integral, by inserting the AD expansion for the operator , one obtains the
S

orresponding AD expansion for the RDM-1 [14℄. It
an be
onstru
ted by
onsidering the AD expansion of
(bx,x′)
and then modifying ea
h diagram by pla
ing a small open
ir
le, representing the variables , in either of
the three following ways: (i) on a group not involved in the diagram; (ii) on a group involved in the diagram,
S
and, (cid:28)nally, (iii) on an arrow. Thus, ea
h AD in the expansion serves as a referen
e in building up the AD
expansion for the RDM-1. Sin
e the
ontribution of any diagram,
onsisting of non-linked parts, is equal to
the produ
t of
ontributions
orresponding to ea
h of the parts, it
an be shown [17℄ that, in general, the AD
expansion of the RDM-1
an be represented as
ρ(x;x′) M
ρ(x;x′)= = S[A ,...,A ] ρt (A ,...,A x;x′)
Λ 1 K K 1 K k (8)
KX=1 A1<X...<AK Xt
e e
ρt (A ,...,A x;x′)
where K 1 K k is the sum of
ontributions of all AD's with an open
ir
le whi
h are
onstru
ted
A ,A , ,A t
1 2 K
using the parti
ular group labelling ··· . The sum over takes a

ount of the two positions of the
open
ir
le on the referen
e AD: on an arrow and on a group. Note that ea
h AD with the open
ir
le is
ne
essarily a linked (
onne
ted) AD. One
an see from the above equation that ea
h AD with an open
ir
le,
T = A ,...,A
1 K

onstru
tedfromgroups { }, ismultipliedbythesumofallpossiblenormalisation-integralAD's
S[A ,...,A ]=S[T] [T]
1 K
(i.e. AD's without the
ir
le), ,
onstru
ted using the rest of the system .
TheRDM-1introdu
edbyEq. (7)isnotnormalisedtothetotalnumberofele
tronsinthesystemsin
ethe
Ψ e e ΦI S = Ψ Ψ =1
wavefun
tion
onstru
tedusing group fun
tions, , is, in general, not normalised to unity: h | i6 .
Therefore, the true RDM-1 should be
al
ulated a

ording to Eq. (3) as [17℄:
ρ(x;x′) M
ρ(x;x′)= = f ρt (A ,...,A x;x′)
Ψ Ψ [T] K 1 K k (9)
h | i KX=1 A1<X...<AK Xt
where
S[A ,...,A ]
1 K
f =f =
[T] [A1,...,AK] S (10)
e
e
4
ψ ψM ψ1 ψ ψM ψ1
M−1 ψ M−1 ψ
M 1 2 M 1 2
M−1 2 M−1 2
... ψ ... ψ
3 3
3 3
i+1 i+1
ψ i ... ψ i ...
i+1 i−1 i+1 i−1
ψ ψ ψ ψ
i i−1 i i−1
(a) (b)
ψ ψ ... ψ
1 2 M
Figure1: The 1D'toy'model: (a) aHF ring
onsistingofequidistantone-ele
tronorbitals , , , and
(b) the
orresponding
hain model in whi
h there is no overlap between the (cid:28)rst and the last orbitals in the
ring.
T
are numeri
al pre-fa
tors. These depend only on the
hosen set of groups . Sin
e the normalisation integral
S
is represented via an AD expansion, Eq. (5), one
an see that the pre-fa
tors are given as a ratio of two
AD expansions, ea
h
ontaining a very large (in(cid:28)nite) number of terms for a large (in(cid:28)nite) system. Thus, it
e
follows from theρltas(tApa,.ss.a.,gAe in Exq;.x(′9)), that the true RDM-1 is representedfas a sum of all linked AD's with
the open
ir
le, K 1 K k , multiplied by numeri
al pre-fa
tors, [A1,...,AK]. The
al
ulation of the
pre-fa
torsposesthemainprobleminapplyingtheADtheorytoin(cid:28)niteorevenlargesystem(forsmallsystems
ρ
all AD's
an be a

ounted for expli
itly and thus
an easily be
al
ulated, at least in prin
iple).
f
[T]
It wasarguedin [17℄ that in the limit of an in(cid:28)nite system the pre-fa
tors tend exa
tly to unity forany
T = A ,...,A
1 K

hoi
e of the groups { }. As a result, it was argued that the true RDM-1
an be represented as
asingle AD expansion
ontainingonlyAD's with theopen
ir
lewhi
h, aswasmentionedearlier,areall linked
S[A ,...,A ]
1 K
AD's. The mainargumentput forwardto provethis so-
alled'linked-AD theorem' wasthat any
S M
(and, in parti
ular, ) tends to in(cid:28)nity when the number of ele
troni
groups tends to in(cid:28)nity.
e
The mainobje
tiveofthis paperisto showthatthesituation ismoresubtle. Although theargumentabout
e S[T]
the limiting behaviour of put forwardin [17℄ may seem quite plausible, in some
ases, as will be shown in
S[T]
the next se
tion by
onsidering an exa
tly solvable 'toy' model, may have a zero limit. The main reason
e
for this is that
ontributions of di(cid:27)erent AD's in the expansion have alternating signs, the point overlooked
f e
[T]
previously. As a result, the pre-fa
tors do not ne
essarily tend to unity, but may take a di(cid:27)erent value
depending on the given system.
3 1D toy model: a Hartree-Fo
k ring
M
Let us
onsideraring of equally spa
edone-ele
trongroupsasshownin Fig. 1(a). Ea
hgroupis des
ribed
s ψ (x) i = 1,2,...,M
i
by a single real -type normalised wavefun
tion ( ) lo
alised around the group
entre. All
ψ (x)
i
groups are identi
al, i.e. every lo
alised fun
tion
an be obtained by a
orresponding translation of the
N = M
neighbouring fun
tion. Note that for this model and the ring wavefun
tion of Eq. (1) represents
Ψ det ψ (x ) ψ (x )
1 1 M M
essentially a single Slater determinant ∝ { ··· }. Hen
e, this theory should be equivalent
to the Hartree-Fo
k (HF) method.
Weassumethatonlyneighbouringfun
tionsoverlap. Duetothesymmetryofthemodel,onlyoneparameter,
5
σ = ψ (x)ψ (x) x
i i+1
namelytheoverlapintegral d betweenanyneighbouringgroups,isrequiredto
hara
terise
ψ (x)
i
the system; the a
tual spatial forRm of is not a
tually needed.
M
We shall show that the RDM-1 of this system
an be
al
ulated analyti
ally for any . Most remarkably,
M
it will be shown that the → ∞ limit is also analyti
ally attainable. Using this result, we will analyse the
linked-AD theorem. We shall (cid:28)nd that some
orre
tions are required.
Two independent methods will be used to
onsider the toy model.
3.1 Finite ring: method based on the inverse of the overlap matrix
It is (cid:28)rst instru
tive to perform the
al
ulation using a method whi
h is
ompletely independent of the AD
theory. This is indeed possible sin
e an expression for the RDM-1 of any HF system
an be written expli
itly
via orbitals and their overlap matrix [1℄. Taking into a

ount that only overlap between nearest neighbours
exists, we obtain for the ring ele
tron density (the diagonal elements of the RDM-1):
N
ρ(x)= ψ (x)(S−1) ψ (x)
i ij j
i,j=1
X
N
= ψ (x)(S−1) ψ (x)+2ψ (x)(S−1) ψ (x)
i ii i i i,i+1 i+1
(11)
i=1
X(cid:2) (cid:3)
S−1
Here is the inverse of the overlap matrix
1 σ σ
σ 1 σ
 
 ... ... ... 
S= 
 ... ... ...  (12)
 
 ... ... ... 
 σ 1 σ 
 
 σ σ 1 
 N×N
 
betweentheringorbitals. Notethatalltheomittedelementsintheaboveexpressionarezeros. This
onvention
is adopted hereafter. Then, the normalisationintegral:
GRing = S
N det (13)
S
is simply given by the determinant of the matrix of Eq. (12). To avoiGdRainpgossible
onfusion with the overlap
matrix, we use for the normalisation integral a di(cid:27)erent notation here, N .
For the
onvenien
e of the following
al
ulation, we also introdu
e a similar
hain system shown in Fig. 1
1 M
(b) in whi
hthe overlapbetween the groups and isequal to zero. The normalisationintegralfor the
hain
is
1 σ
σ 1 σ
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) ... ... ... (cid:12)(cid:12)
GChain =(cid:12) (cid:12)
N (cid:12)(cid:12) ... ... ... (cid:12)(cid:12) (14)
(cid:12) (cid:12)
(cid:12)(cid:12) ... ... ... (cid:12)(cid:12)
(cid:12) σ 1 σ (cid:12)
(cid:12) (cid:12)
(cid:12) σ 1 (cid:12)
(cid:12) (cid:12)N×N
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6
Owingtothesymmetryoftheaboveintrodu
eddeterminants,itispossibletoderivethefollowingre
urren
e
relations: GRing =GChain 2σ2GChain+2( 1)N+1σN
N N−1 − N−2 − (15)
GChain =GChain σ2GChain
N N−1 − N−2 (16)
The (cid:28)rst one is obtained by opening the determinant of the matrix (12) along its (cid:28)rst row (or
olumn), while
GthReinseg
=onGdCrehlaaitnio=n1is obtaGinCehdaiinn=th1e saσm2eway from Eq. (14). Combined with the obviouGsC'hinaiintial' exGpRreisnsgions
1 1 and 2 − , the aboverelationsallowone to
al
ulate the N and N for
N
any value of . For instan
e, we obtain,
GRing =1 6σ2+9σ4 4σ6
6 − − (17)
GRing =1 7σ2+14σ4 7σ6+2σ7
7 − − (18)
GChain =1 5σ2+6σ4 σ6
6 − − (19)
GChain =1 6σ2+10σ4 4σ6
7 − − (20)
Note thealternatingsignin everynext termin theexpressionsabove. Also, in agreementwith[17℄, one
ansee
N
that numeri
al pre-fa
tors in ea
h term growwith in
rease of .
Itisnowstraightforwardtoinvestigatethelimitingbehaviourofthesequantitieswhenthenumberofgroups
N
tends to in(cid:28)nity (the → ∞ limit). We (cid:28)nd, using a simple numeri
al
al
ulation, that both of them tend
N σ < 0.5
tσo zer0o.5with in
rease oGf Chafionr any GvaRluinegof (it will be
lear later on that the limit does not exist for
≥ ). Sin
e both N and N
orrespond to the normalisation integrals for the two systems, one

an see that this limiting behaviour is quite di(cid:27)erent from whatlismhould hGaRveinbgeen elximpe
ted fGroCmhatinhe analysis
performed in Ref. [17℄. This is explained by the fa
t that both N→∞ N and N→∞ N are given
by in(cid:28)nite series with alternating terms. The limitσinngnb=eh2a,v3io,u..r.of ea
h of the series is nNot straightforward in
spite of the fa
t that the pre-fa
tors to ea
h term ( ) tend to in(cid:28)nity when →∞.
Inorderto
al
ulatetheRDM-1(11), onealsonSeedthe
orrespondingelements oftheinverseoftheoverlap
matrix. Using the expli
it stru
ture of the matrix given in Eq. (12), one obtains:
C GChain
S−1 = i,i = N−1
i,i S GRing (21)
det N
(cid:0) (cid:1)
C σGChain+( 1)N+1σN−1
S−1 = S−1 = i,i+1 = − N−2 −
i,i+1 i+1,i S GRing (22)
det N
(cid:0) (cid:1) (cid:0) (cid:1)
C C
i,i i,i+1 i,i i+1,i
where and are
ofa
tors of S and S respe
tively.
Using Eqs. (21) and (22), we obtain for the ele
tron density (11):
GChain N
ρ(x)= N−1 ψ2(x)
GRing ! i
N i=1
X
GChain N 2 ( σ)N N
2σ N−2 ψ (x)ψ (x) − ψ (x)ψ (x)
− GRing ! i i+1 − σ GRing ! i i+1 (23)
N i=1 N i=1
X X
Theaboveexpressionallowsoneto
al
ulatetheele
trondensityoftheringwitharbitrarynumberofgroups.
N
The →∞ limit will be
onsidered separately in se
tion 3.3.
7
...
GRing= 1+ + + +...+ +
N ...
S N
Figure 2: The AD expansion for the ring normalisation integral, . The ring
onsists of a (cid:28)nite number of
one-ele
tron groups.
e
...
ρ
(x)= 1+ + +... + + 1+ + +... + +
...
ρ(x) ρ(x;x)
Figure 3: The AD expansion for the diagonal elements of the ring RDM-1, ≡ , Eq. (8).
e e
3.2 Finite ring: method based on the AD theory
Weshallshowinthisse
tionthatthesameexpressionsasderivedintheprevioussubse
tion
analsobeobtained
using a mu
h simpler algebra of the AD theory. S GRing
To this end, we (cid:28)rst
onsider the normalisation integral ≡ N of the ring. Sin
e all groups
ontain
a single ele
tron and only nearest groups have non-zero overlap, the AD expansion
ontains essentially only
e
bubble-likeAD'sbetweentwoadja
entgroups(in
luding all non-linkedAD's
onstru
tedoutofthem)and two
N
AD's
onne
ting all grGoCuphasin(a -vertex polygon) with opposite dire
tion of arrows, as depi
ted in Fig. 2. An
AD expansion for the N looks similarly. There are two di(cid:27)eren
es: (i) bubble AD's
ontaining the groups
1 N
and aremissing,and(ii)itdoesnot
ontainthetwopolygondiagrams. Re
allthatthereisno(cid:16)
onne
tion(cid:17)
N
between the groups 1 and in the
hain.
AnADexpansionfortheele
trondensity
analsobewrittenexpli
itlyasshowninFig. 3. It
ontainsthree
i
terms. The(cid:28)rsttermisasso
iatedwithanopen
ir
leAD,asso
iatedwith,say,group ,multipliedbyallbubble
N 1 1,...,i 1,i+1,...,N
AD's(showninthesquarebra
kets)whi
hare
onstru
tedoutoftherest − groups − .
ρ(◦)(x)=ψ2(x)
TGhCeha
ionntributionoftheopen
ir
lei i i , whilethebubble AD'sintheNsqua1rebra
ketsallamountto
N−1 (andarethesameforea
h ),sin
etheyrepresentallAD'sfora
hainwith − groups. Finally,onehas
i GChain N ρ(◦)(x)
tosumoverallvaluesof (allpositionsoftheopen
ir
le). Hen
e,the(cid:28)rstterm
ontributes N−1 i=1 i
ρ(x)
to the density . P
These
ondterminFig. 3
ontainstwobubblediagramswithanopen
ir
leonbothofitsarrows,multiplied
N 2
e
by all possible bubble ADi's
onist+ru1
ted out of the rest − gρr(obuubpblse.)(xA)s=sumiσnψg (txh)aψt th(ex)bubble AD is
taken between the groups and , we obtain the
ontribution i,i+1 − i i+1 for it. Sin
e
1,...,i 1,i+2,...,N
GthCeh
aoinntribution of all groups −i i+1 in the square bra
kets in the se
ond term is equal to
N−2 (whi
h is the same for any pair , ), the (cid:28)nal
ontribution of the se
ond term in Fig. 3 be
omes
2GChain N ρ(bubble)(x)
N−2 i=1 i,i+1 , wherewehavealsosummedoverallpairs,i.e. allpossiblepositionsoftheopen
ir
le
on the buPbble AD's (re
all, that the bubble AD's
an be taken only between nearestneighbours in our model).
8
Finally, tρh(epollay)st=tw(o tσe)rNm−s1iψn F(xig)ψ. 3 a(rxe)represented by polygons with an open
ir
le. Ea
h su
h a diagrami

ontributes i,i+1 − i i+1 if the open
ir
le is positioned on the arrow
onne
ting groups
i+1
and . Thetotal
ontributionofthepolygonsisobtainedbysummingoverallpossiblepositionsofthe
ir
le
and multiplying by a fa
tor of two sin
e ea
h AD a

epts two dire
tions of the arrows.
Summingallthree
ontributions,wearriveatthefollowingexpressionfortheunnormalisedele
trondensity:
N N N
ρ(x)=GChain ρ(◦)(x)+2GChain ρ(bubble)(x)+2 ρ(poly)(x)
N−1 i N−2 i,i+1 i,i+1 (24)
i=1 i=1 i=1
X X X
e
GRing ρ(x)
Dividing this expression by N , one obtains the normalised ele
tron density , whi
h, as
an easily be
seen, appears to be identi
al to that of Eq. (23) obtained in the previous se
tion using Slater determinants.
To(cid:28)nish the proofofthe equivalen
eof the twomethods, weonly need to re-derivethe re
urren
erelations
(15) and (16) using ex
lusively the AGDChtahienory.
Consider (cid:28)rst the expansion for N whi
h
ontains only zero, one, two, et
. bubble diagrams (see the
N
bubble-
onGtaCihnainingterms in Fig. 2). Re
allthatthe
hainisbrokenbetweengroups1and . Letus (cid:16)(cid:28)x(cid:17) group
1. Then, N
an be written as the sum of two
ontributions: (i) due to all diagrams involving group 1
and (ii) due to the rest of them whi
h do not involve it. The (cid:28)rst
ontribution is simply a single bubble AD
σ2
between groups 1 aσn2dG2,Cehqaiunal to − , times all the bubble AD's made of all the other gr1oups, i.e. the whole

ontribution is − N−2 . The
ontribution of all AD's whi
h do not involve group is simply equal to
GChain
N−1 . One
a(cid:0)n imm(cid:1)ediately re
ognise the re
urren
e relation (16).
Eq. (15) is proven in a similar way. Again, wGe C(cid:28)rhsatin(cid:16)(cid:28)x(cid:17) group 1. Then, the sum of all AD's show1n in Fig.
2 is equal to the sum of three
ontributions: (i) N−1 due to all AD's whi
h do not involve group ; (ii) all
2 σ2 GChain
thebubble AD'sinvolvingit, givenby − N−2 analogouslytothe
aseofthe
haindis
ussedabove(the
1
fa
toroftwoarisesduetothefa
tthato(cid:0)ne
a(cid:1)n
onstru
ttNwobubbleAD'swithgroup ,namelythoseinvolving
1,N 1,2 2( σ)
group pairs and ); (iii) the
ontribution − − of the two polygon AD's. Summing all three terms,
one arrives exa
tly at Eq. (15), as required.
3.3 In(cid:28)nite ring
Our task now is to
al
ulate the pre-fa
tors (in the round bra
kets) in the expression for the density of Eq.
(23). To do this, it was found
onvenient to introdu
e the following quantity:
GChain
P =σ2 N−1
N GCNhain! (25)
GChain GChain P =σ2/ 1 σ2
Using exGplCi
hiatinexpressionsfor the 2 and 1 ,Pwe get 2 − . Using the re
urren
e relation
(16) for N , a very simple re
urren
e relation for N
an also be de(cid:0)rived: (cid:1)
σ2
P =
N
1 PN−1 (26)
−
The introdu
ed quantity is very useful in two respe
ts. Firstly, the (cid:28)rst two pre-fa
tors in Eq. (23)
an be
dire
tly expressed via it:
GChain 1
N−1 =
GRNing 1−2PN−1+2σχN−1 (27)
GChain σ−2P
N−2 = N−1
GRNing 1−2PN−1+2σχN−1 (28)
9
where ( σ)N
χ = −
N GChain (29)
N
N
Se
ondly, it has a (cid:28)nite → ∞ limit. Indeed, assuming that su
h a limit exists, we obtain from Eq. (26)
P (1 P ) = σ2 P = lim P
∞ ∞ ∞ N→∞ N
a1 s1imp√le1qua4dσr2ati
equation − , where σ =0 . TheGrCoohtasino=f t1his equatNion are
2 ± − . To
hoose the
orre
t sign, we
onsider the
ase of , when N for any , and
P =0
N
th(cid:0)us . On(cid:1)ly the root with the minus sign,
1 √1 4σ2
P = − −
∞
2 (30)
σ < 1
satis(cid:28)es this
ondition. It also appears that a de(cid:28)nite real limit exists only for 2 whi
h
orresponds to a
notverylargeoverlap. Thisrestri
tionisa
onsequen
eofthenearest-neighbourapproximationadoptedin our
P >0 N
N
toy model. Note in passing that for any .
χ
N
Finally, to (cid:28)nish the
al
ulation, we have to
onsider the limiting behaviour of the ratios and
( σ)N σχ
N−1
ξ = − = −
N GRNing 1−2PN−1+2σχN−1 (31)
lim χ
N→∞ N
whi
h enter Eqs. (27), (28) and (23), respe
tively. To
al
ulate the , we use the de(cid:28)nition (25) of
P
N
to write:
σ2 N−3 σ2 N−3
GChain = GChain > GChain
N−1 P P P · 2 P · 2
N−(cid:0)1 N(cid:1)−2··· 3 (cid:18) ∞(cid:19)
P < P N 0 σ < 1 GChain = 1 σ2 > 0
sin
e N ∞ for any (cid:28)nite value of and any ≤ 2. Sin
e 2 − and any of the
P
K
quantities are positive, one
an write
σN σ3 P N−3
∞
0< <
GChain GChain · σ
N−1 2 (cid:18) (cid:19)
P∞ <1 0 σ < 1 lim P∞ N−3 =0
Be
ause σ for any ≤ 2, we obtain N→∞ σ and then
(cid:0) (cid:1)
σN
lim =0
N→∞ GCNh−a1in! (32)
χ ξ 0
N N
Hen
e, of Eq. (29) has a zero limit and disappears in the ratios (27) and (28). Consequently, → too
(see Eq. (31)), sothat the third termin therighthand side ofEq. (23) alsodoes not
ontribute when the limit
of an in(cid:28)nite ring is taken.
Finally,
ombining all
ontributions, we obtain the following exa
t result for the in(cid:28)nite ring:
N N
ρ(x)= f ρ(◦)(x)+2 f ρ(bubble)(x)
[i] i [i,i+1] i,i+1 (33)
i=1 i=1
X X
where the
orresponding expressions for the two pre-fa
tors are:
1
f =
[i] √1 4σ2 (34)
−
1 1
f = 1
[i,i+1] 2σ2 √1 4σ2 − (35)
(cid:18) − (cid:19)
10