$N$-representability of the Jastrow wave function pair density of the lowest-order

0.14 MB·English

by  Katsuhiko Higuchi

N-representability of the Jastrow wave function pair density of
the lowest-order
Katsuhiko Higuchi∗
Graduate School of Advanced Sciences of Matter,
8 Hiroshima University, Higashi-Hiroshima 739-8527, Japan
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0
2 Masahiko Higuchi†

n
a Department of Physics, Faculty of Science,
J

4 Shinshu University, Matsumoto 390-8621, Japan

] (Dated: February 2, 2008)
l
e
-
r Abstract
t
s
t. We have recently proposed a density functional scheme for calculating the ground-state pair
a
m
density (PD) within the Jastrow wave function PDs of the lowest-order (LO-Jastrow PDs) [M.
-
d
Higuchi and K. Higuchi, Phys. Rev. A 75, 042510 (2007)]. However, there remained an arguable
n
o problem on the N-representability of the LO-Jastrow PD. In this paper, the sufficient conditions
c
[
for the N-representability of the LO-Jastrow PD are derived. These conditions are used as the

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v
constraints on the correlation function of the Jastrow wave function. A concrete procedure to
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1
search the suitable correlation function is also presented.
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0
.
1 PACS numbers: 71.15.Mb , 31.15.Ew, 31.25.Eb
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:
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i
X
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I. INTRODUCTION
The pair density (PD) functional theory has been expected to be one of the promising
schemesbeyondthedensityfunctionaltheory.1 Recently, wehaveproposedthePDfunctional
theory that yields the best PD within the set of the Jastrow wave function PDs of the
lowest-order (LO-Jastrow PDs).2 The search region for the ground-state PD is substantially
extendedascomparedwiththeprevioustheory.3,4 Ontheotherhand,however, thereremains
a significant problem related to the N-representability of the LO-Jastrow PDs.2
Let us revisit the problem here. We shall consider an N -electron system. The Jastrow
0
wave function is given by
1
Ψ (x , ,x ) = f ( r r )Φ (x , ,x ), (1)
J 1 ··· N0 A | i − j| SSD 1 ··· N0
N0 1≤iY<j≤N0
p
where x denotes the coordinates including the spatial coordinate r and spin coordinate
i i
η , and where A , f ( r r ) and Φ (x , , x ) are the normalization constant,
i N0 | i − j| SSD 1 ··· N0
correlation function and single Slater determinant (SSD), respectively. The LO-Jastrow PD
is given by2,5,6
γ(2) (rr′;rr′) = f ( r r ) 2γ(2) (rr′;rr′) , (2)
LO | | i − j| | SSD N=N0
where γ(2) (rr′;rr′) is the PD calculated from the SSD. In the preceding paper,2
SSD N=N0
we have confirmed that Eq. (2) meets four kinds of necessary conditions for the N-
representability of the PD, and may become ”approximately N-representable”.7,8,9,10,11,12
However, the possibility of it being N-representable has not been discussed.2 This is an ar-
guable problem that is concerned with whether the reproduced PD is physically reasonable
or not.
The aim of this paper is to discuss the N-representability of Eq.(2) and to show the way
to search the suitable correlation function f ( r r ). The organization of this paper is as
i j
| − |
follows. For the convenience of the subsequent discussions, we first examine the properties
of the LO-Jastrow PD in Sec. II. The sufficient conditions for the N-representability of
Eq. (2), which are imposed on the correlation function, will be derived recursively in Sec.
III. Concrete steps for searching the correlation function that meets these conditions are
discussed in Sec. IV. Finally, concluding remarks are given in Sec. V.
2
II. PROPERTIES OF THE LO-JASTROW PD
In this section, we shall discuss the properties of the LO-Jastrow PD. To this aim, the
properties of PDs that are calculated from SSDs are investigated. The cofactor expansion
of Φ (x , ,x ) along the N th row leads to
SSD 1 ··· N0 0
1
Φ (x , ,x ) = ( 1)N0+1φ (x )Φ1 (x , ,x )
SSD 1 ··· N0 √N { − λ N0 SSD 1 ··· N0−1
0
+ ( 1)N0+2φ (x )Φ2 (x , ,x )
− µ N0 SSD 1 ··· N0−1
+ (3)
···
+ ( 1)2N0φ (x )ΦN0 (x , ,x ) ,
− ξ N0 SSD 1 ··· N0−1 }
whereφ (x ),φ (x ), andφ (x )aretheconstituent spinorbitalsofΦ (x , ,x ).
λ N0 µ N0 ··· ξ N0 SSD 1 ··· N0
In what follows, suppose that these spin orbitals are given as the solutions of simultaneous
equations of previous work,2 and therefore they are orthonormal to each other. Φi (x ,
SSD 1 ··
,x ) (1 i N ) in Eq. (3) denote (N 1)-electron SSDs that are defined as the minor
· N0−1 ≤ ≤ 0 0−
determinants multiplied by 1/ (N 1)!. The PD that is calculated from Eq. (3) is given
0
−
by p
1 N0
γ(2) (rr′;rr′) = γ(2) (rr′;rr′)i , (4)
SSD N=N0 N 2 SSD N=N0−1
0
− i=1
X
where γ(2) (rr′;rr′)i is the PD calculated from Φi (x , ,x ).
SSD N=N0−1 SSD 1 ··· N0−1
Likewise, thecofactorexpansionofeachΦi (x , ,x )yields(N 2)-electronSSDs,
SSD 1 ··· N0−1 0−
which are denoted by Φij (x , ,x ) (1 j N 1). Then, each γ(2) (rr′;rr′)i
SSD 1 ··· N0−2 ≤ ≤ 0− SSD N=N0−1
N0−1
is given by γ(2) (rr′;rr′)ij /(N 3), where γ(2) (rr′;rr′)ij is the PD calcu-
SSD N=N0−2 0 − SSD N=N0−2
j=1
lated from ΦPij (x , ,x ). Thus, γ(2) (rr′;rr′) can be expressed by the sum of
SSD 1 · · · N0−2 SSD N=N0
γ(2) (rr′;rr′)ij . By the repetition of this procedure, we arrive at
SSD N=N0−2
1 N0 N0−1 4 3
γ(2) (rr′;rr′) = γ(2) (rr′;rr′)ij···pq, (5)
SSD N=N0 (N 2)! ··· SSD N=2
0
− i=1 j=1 p=1 q=1
X X XX
where γ(2) (rr′;rr′)ij···pq (1 i N ,1 j N 1, ,1 p 4,1 q 3) are PDs calculated
SSD N=2 ≤ ≤ 0 ≤ ≤ 0− ··· ≤ ≤ ≤ ≤
from the two-electron SSDs. These two-electron SSDs, which are denoted by Φij···pq(x ,x ),
SSD 1 2
are obtained by the above-mentioned successive cofactor expansions.
3
Multiplying both sides of Eq. (5) by f ( r r′ ) 2, we finally get
| | − | |
1 N0 N0−1 4 3
f ( r r′ ) 2γ(2) (rr′;rr′) = f ( r r′ ) 2γ(2) (rr′;rr′)ij···pq.
| | − | | SSD N=N0 (N 2)! ··· | | − | | SSD N=2
0
− i=1 j=1 p=1 q=1
X X XX
(6)
This relation is the starting point to examine the N-representability of the LO-Jastrow PD.
III. SUFFICIENT CONDITIONS FOR THE N-REPRESENTABLITY OF THE
LO-JASTROW PD
We shall start with considering the N-representability of f ( r r′ ) 2γ(2) (rr′;rr′)ij···pq
| | − | | SSD N=2
that appears in the right-hand side of Eq. (6). Suppose that the two-electron wave function
that yields f ( r r′ ) 2γ(2) (rr′;rr′)ij···pq is given by the following Jastrow wave function;
| | − | | SSD N=2
1
Ψij···pq(x , x ) = f ( r r )Φij···pq(x ,x ), (7)
N=2 1 2 ij···pq | 1 − 2| SSD 1 2
A
2
q
where Aij···pq is the normalization constant. From Eq. (7), the PD is calculated as
2
f ( r r′ ) 2γ(2) (rr′;rr′)ij···pq/Aij···pq. Therefore, we get Aij···pq = 1 as the sufficient condi-
| | − | | SSD N=2 2 2
tion for the N-representability of f ( r r′ ) 2γ(2) (rr′;rr′)ij···pq. Hereafter, we assume that
| | − | | SSD N=2
such conditions hold for all values of i,j, ,p and q, i.e.,
···
Aij···pq = 1 for all values of i,j, ,p and q. (8)
2 ···
Note that the normalization constant Aij···pq is determined by both f ( r r ) and
2 | 1 − 2|
Φij···pq(x ,x ).5,6 Therefore, if they are given, we can calculate Aij···pq, and check whether
SSD 1 2 2
the conditions Eq. (8) are satisfied or not. As mentioned later, the conditions Eq. (8) are
the parts of sufficient conditions for the N-representability of Eq. (2).
Under the conditions Eq. (8), we have
f ( r r′ ) 2γ(2) (rr′;rr′)ij···pq = Ψij···pq γˆ(2)(rr′;rr′) Ψij···pq , (9)
| | − | | SSD N=2 N=2 N=2 N=2
where γˆ(2)(rr′;rr′)N=n denotes the PD operato(cid:10)r for a(cid:12)(cid:12)n n-electron syste(cid:12)(cid:12)m. Su(cid:11)bstituting Eq.
(9) into Eq. (6) and rearranging, we get
f ( r r′ ) 2γ(2) (rr′;rr′)
| | − | | SSD N=N0
N0 1 N0−1 1 N0−2 1 5 1 4 1 3
= Ψijk···opq γˆ(2)(rr′;rr′) Ψijk···opq
N 2 N 3 ···3 2 1 N=2 N=2 N=2
" 0 " 0 " " " #####
Xi=1 − Xj=1 − Xk=1 Xo=1 Xp=1 Xq=1D (cid:12) (cid:12) E
(cid:12) (cid:12)
(10)
(cid:12) (cid:12)
4
It should be noticed that the right-hand side of Eq. (10) has a characteristic form. Con-
cerning the N-representability of this form, the following theorem holds:
Theorem. If there exists the set of functions a (x ) (1 α n + 1) that satisfy the
α n+1
{ } ≤ ≤
conditions;
1
a∗α(xn+1)aα′(xn+1)dxn+1 = n+1δαα′, (11)
Z
n+1 n+1
σ a (x )Ψα (x , ,x ) = ( 1)σ¯ a (x )Ψα (x , ,x ), (12)
α n+1 N=n 1 ··· n − α n+1 N=n 1 ··· n
α=1 α=1
X X
then the following equations hold:
n+1
1
Ψα γˆ(2)(rr′;rr′) Ψα = Ψ γˆ(2)(rr′;rr′) Ψ (13)
n 1 h N=n| N=n| N=ni h N=n+1| N=n+1| N=n+1i
− α=1
X
with
n+1
Ψ (x , ,x ) = a (x )Ψα (x , ,x ). (14)
N=n+1 1 ··· n+1 α n+1 N=n 1 ··· n
α=1
X
Here Ψα (x , , x ) (1 α n + 1) denote the n-electron wave functions, and σ is a
N=n 1 · · · n ≤ ≤
permutation operator upon the electron coordinates, and σ¯ is the number of interchanges in
σ.
Proof. The left-hand side of Eq. (13) seems to be related to the average of γˆ(2)(rr′;rr′)
N=n
with respect to a density matrix for a mixed state. Indeed, if the density matrix for the
mixed state is given by
n+1
1
ρˆ = Ψα Ψα , (15)
n n+1 | N=nih N=n|
α=1
X
then the average of γˆ(2)(rr′;rr′) is calculated as
N=n
n+1
1
Tr[ρˆ γˆ(2)(rr′;rr′) ] = Ψα γˆ(2)(rr′;rr′) Ψα . (16)
n N=n n+1 h N=n| N=n| N=ni
α=1
X
By using Eq. (16), the left-hand side of Eq. (13) is rewritten as (n + 1)/(n
{ −
1) Tr[ρˆ γˆ(2)(rr′;rr′) ].
n N=n
}
On the other hand, it is expected that the average Tr[ρˆ γˆ(2)(rr′;rr′) ] may be given as
n N=n
the expectation value of γˆ(2)(rr′;rr′) with respect to a pure state for the whole system
N=n
that includes the n-electron system as a subsystem.13,14 We shall take an (n + 1)-electron
system as the whole system, and suppose that the wave function for such the (n+1)-electron
system is given by Eq. (14) with Eq. (11). Then we indeed get
Ψ γˆ(2)(rr′;rr′) Ψ = Tr[ρˆ γˆ(2)(rr′;rr′) ]. (17)
n+1 N=n n+1 n N=n
h | | i
5
Furthermore, if Ψ (x , ,x ) is antisymmetric, i.e., if Eq. (12) holds, then the
N=n+1 1 n+1
· · ·
expectation value of γˆ(2)(rr′;rr′) with respect to Ψ (x , ,x ) is given by
N=n+1 N=n+1 1 n+1
···
n+1
Ψ γˆ(2)(rr′;rr′) Ψ = Ψ γˆ(2)(rr′;rr′) Ψ . (18)
n+1 N=n+1 n+1 n+1 N=n n+1
h | | i n 1 h | | i
−
Consequently, Eq. (13) immediately follows from Eqs. (16), (17) and (18). This means
that Eq. (13) holds under the conditions that there exists the set of functions a (x )
α n+1
{ }
(1 α n+1) that satisfy Eqs. (11) and (12). Q.E.D.
≤ ≤
3
First, we apply the above theorem to the term 1 Ψijk···opq γˆ(2)(rr′;rr′) Ψijk···opq
1 N=2 N=2 N=2
q=1
D (cid:12) (cid:12) E
that appears in Eq. (10). According to the theoremP, this term c(cid:12)an be rewritten a(cid:12)s
(cid:12) (cid:12)
3
1
Ψijk···opq γˆ(2)(rr′;rr′) Ψijk···opq = Ψijk···op γˆ(2)(rr′;rr′) Ψijk···op (19)
1 N=2 N=2 N=2 N=3 N=3 N=3
Xq=1 D (cid:12) (cid:12) E D (cid:12) (cid:12) E
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
with
3
Ψijk···op(x , x ,x ) = aijk···op(x )Ψijk···opq(x , x ), (20)
N=3 1 2 3 q 3 N=2 1 2
q=1
X
if there exists the set of functions aijk···op(x ) (1 q 3) that satisfy the following condi-
q 3 ≤ ≤
tions: (cid:8) (cid:9)
1
aiqjk···op(x3)∗aiqj′k···op(x3)dx =3δq,q′, (21)
Z
3 3
σ aijk···op(x )Ψijk···opq(x ,x ) = ( 1)σ¯ aijk···op(x )Ψijk···opq(x ,x ). (22)
q 3 N=2 1 2 − q 3 N=2 1 2
q=1 q=1
X X
In addition to Eq. (8), the existence conditions for aijk···op(x ) (1 q 3), i.e., Eqs. (21)
q 3 ≤ ≤
and (22), are also the parts of sufficient conditions fo(cid:8)r the N-repr(cid:9)esentability of Eq. (2). We
assume that the set of functions aijk···op(x ) (1 q 3) is obtained. Substitution of Eq.
q 3 ≤ ≤
(20) into Eq. (10) leads to (cid:8) (cid:9)
f ( r r′ ) 2γ(2) (rr′;rr′)
| | − | | SSD N=N0
N0 1 N0−1 1 N0−2 1 5 1 4
= Ψijk···op γˆ(2)(rr′;rr′) Ψijk···op .(23)
N 2 N 3 ···3 2 N=3 N=3 N=3
" 0 " 0 " " ####
Xi=1 − Xj=1 − Xk=1 Xo=1 Xp=1D (cid:12) (cid:12) E
(cid:12) (cid:12)
(cid:12) (cid:12)
4
Next, we apply the theorem to the term 1 Ψijk···op γˆ(2)(rr′;rr′) Ψijk···op that
2 N=3 N=3 N=3
p=1
D (cid:12) (cid:12) E
appears in Eq. (23). Similarly to the above, wePobtain (cid:12) (cid:12)
(cid:12) (cid:12)
4
1
Ψijk···op γˆ(2)(rr′;rr′) Ψijk···op = Ψijk···o γˆ(2)(rr′;rr′) Ψijk···o (24)
2 N=3 N=3 N=3 N=4 N=4 N=4
Xp=1 D (cid:12) (cid:12) E D (cid:12) (cid:12) E
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
6
with
4
Ψijk···o(x , ,x ) = aijk···o(x )Ψijk···op(x , x , x ), (25)
N=4 1 ··· 4 p 4 N=3 1 2 3
p=1
X
ifthereexiststhesetoffunctions aijk···o(x ) (1 p 4)thatsatisfythefollowingconditions:
{ p 4 } ≤ ≤
1
aipjk···o(x4)∗aipj′k···o(x4)dx =4δp,p′, (26)
Z
4 4
σ aijk···o(x )Ψijk···op(x ,x , x ) = ( 1)σ¯ aijk···o(x )Ψijk···op(x ,x , x ). (27)
p 4 N=3 1 2 3 − p 4 N=3 1 2 3
p=1 p=1
X X
The existence conditions for aijk···o(x ) (1 p 4), i.e., Eqs. (26) and (27), are added to
{ p 4 } ≤ ≤
the set of sufficient conditions for the N-representability of Eq. (2). At this stage, Eq. (8)
and the existence conditions for aijk···op(x ) (1 q 3) and aijk···o(x ) (1 p 4) belong
q 3 ≤ ≤ { p 4 } ≤ ≤
to the set of sufficient conditions(cid:8). (cid:9)
Thus the theorem is applied repeatedly, so that further conditions are added to the set
of sufficient conditions. Continuing until the conditions for a (x ) (1 i N ), we
{ i N0 } ≤ ≤ 0
eventually obtain sufficient conditions for the N-representability of Eq. (2).
IV. CONCRETESTEPS FOR CONSTRUCTINGTHEN-REPRESENTABLE LO-
JASTROW PD
In the preceding section, the sufficient conditions for the N-representability of the LO-
Jastrow PD are derived. In this section, we consider the concrete steps for searching the
correlation function that meets these conditions or checking its existence.
1. First, we give a trial form of the correlation function. Using this, simultaneous equa-
tions for the N -electron system are solved in a self-consistent way.2
0
2. Let usconsider theSSD thatconsists oftheresultant spin orbitalsforthesimultaneous
equations. The SSD can generally be expanded using the cofactor. The SSD for the
N -electronsystemisexpandedalongtheN throw,thenwegettheN numberofSSDs
0 0 0
for the (N 1)-electron system, i.e., Φi (x , ,x ) (1 i N ). Successively,
0 − SSD 1 ··· N0−1 ≤ ≤ 0
each of the SSDs for the (N 1)-electron system is expanded along the (N 1)th
0 0
− −
row, and then the (N 1) number of SSDs for the (N 2)-electron system, i.e.,
0 0
− −
Φij (x , ,x ) (1 j N 1), can be obtained for each i. After that and
SSD 1 ··· N0−2 ≤ ≤ 0−
7
later, the cofactor expansions are likewise repeated, and we finally arrive at the SSDs
for the two-electron system, i.e.,Φijk····opq(x , x ). The number of two-electron SSDs
SSD 1 2
thus obtained is N !/2, since i, j, k, , o, p and q are integers such that 1 i N ,
0 0
···· ≤ ≤
1 j N 1, 1 k N 2, , 1 o 5, 1 p 4 and 1 q 3, respectively.
0 0
≤ ≤ − ≤ ≤ − ···· ≤ ≤ ≤ ≤ ≤ ≤
3. The two-electron SSD Φijk····opq(x , x ) and the correlation function determine the
SSD 1 2
normalization constant Aijk····opq.5,6 We check whether all of Aijk····opq are unity or not.
2 2
If no, we return to the step 1 and change the form of the correlation function. This
process proceeds until all of Aijk····opq become unity. Suppose that such the correlation
2
function is found, we get two-electron antisymmetric wave functions Ψijk····opq(x , x )
N=2 1 2
for all cases of i, j, k, , o, p and q through Eq. (7).
····
ijk····opq
4. By means of these Ψ (x , x ), three-electron wave functions are defined as Eq.
N=2 1 2
(20). We will check whether there exists the set of aijk····op(x ), q = 1, 2, 3 that are
q 3
satisfied with Eqs. (21) and (22). Note that the che(cid:8)ck has to be performed fo(cid:9)r all cases
ofi, j, k, , oandp. Ifsuitable aijk····op(x ) cannotbefound,weagainreturntothe
···· q 3
step1andmodifythecorrelation(cid:8)function. Su(cid:9)pposethesuitable aijk····op(x ) isfound
q 3
in this process, then three-electron antisymmetric wave functio(cid:8)ns Ψijk····op(x(cid:9), x ,x )
N=3 1 2 3
for all cases of i, j, k, , o and p can be constructed from Eq. (20).
····
5. We successively proceed the case of four-electron wave functions that are defined
as Eq. (25). In a similar way to the step 4, we check whether the set of
aijk····o(x ), p = 1, 2, 3, 4 meets the conditions of Eqs. (26) and (27). If no, we
p 4
r(cid:8)estart from the step 1 w(cid:9)ith the modified correlation function. Suppose that the
suitable aijk····o(x ) are found for all cases of i, j, k, and o, the four-electron
p 4 · · ··
ijk····o
antisymm(cid:8)etric wave(cid:9)functions Ψ (x , , x ) can be obtained from Eq. (25).
N=4 1 ·· 4
6. Likewise, we further proceed the problem constructing the antisymmetric wave func-
tions for more-electron systems. We search the set of a (x ), α = 1, 2, , n+1
α n+1
{ ···· }
that are satisfied with Eqs. (11) and (12), together with modifying the correlation
function. If we successfully find the correlation function that meets the conditions
(11) and (12) for any n ( N 1), the LO-Jastrow PD of the N -electron system
0 0
≤ −
becomes N-representable.
8
As easily inferred, the above steps are feasible only for the small-electron systems from
the practical viewpoint. However, it should be noted that the correlation function that
makes the LO-Jastrow PD N-representable may, in principle, be found along the above
steps, though there is a possibility that the suitable correlation function may not exist in
some system.
V. CONCLUDING REMARKS
In this paper, the sufficient conditions for the N-representability of the LO-Jastrow PD
arediscussed. Using the properties oftheLO-JastrowPD,we derive thesufficient conditions
that are imposed on the correlation function of the Jastrow wave function. As shown in Sec.
IV, additional steps to search the suitable correlation function, which satisfies the sufficient
conditions, are attached to the computational scheme proposed previously.2 Although the
number ofstepsrapidlyincreases withthatofelectrons, theconcretestepsthatarepresented
in Sec. IV are feasible for a small-electron system. Of course, there is a possibility that the
suitable correlation function cannot be found out. In this case, as mentioned in the previous
paper, LO-Jastrow PDs are approximately N-representable in a sense that they satisfy four
kinds of necessary conditions.2,7,8,9,10,11,12
Acknowledgments
ThisworkwaspartiallysupportedbyGrant-in-AidforScientificResearch(No. 19540399)
and for Scientific Research in Priority Areas (No. 17064006) of The Ministry of Education,
Culture, Sports, Science, and Technology, Japan.
∗ Electronic address: [email protected]
† Electronic address: [email protected]
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