I N T E R N A T I O N A L J O U R N A L OF Q U A N T U M CHEMISTRY, V O L . XXIII , 399-404 (1983)

Approach to N Representability

HONG G U O A N D AU-CHIN TANG lnstirute of' Throrblical Chemistry, Kirin Unirersity, Chanxchun. China

Abstract

In this paper, some operator endomorphisms which givc rise to conditions for N representability of pth ordcr density matrices are prcscnted.

1. Introduction

The development of theory of N representability for reduced density matrices has been characterized by a number of distinct approaches [ 1-71. Recently, Kummer, Absar, and Coleman [8] have taken advantage of operator endomorphic behavior between convex sets and have given a treatment for density operators of order 2. In this paper, some operator endomorphisms which give rise to necessary or sufficient conditions for N representability are generalized in terms of the characteristic oper- ators presented in a previous paper [9] to the case of density operators of order p. However, we consider at most two-particle correlation.

2. Operator Endomorphisms and Characteristic Operators

Let us begin by considering a set of p-density operators J p and a set of N-repre- sentable p-density operators J R . According to the Kummer, Absar, and Coleman conventions [8], they may be defined, respectively, as

J P = ( D P E S P ( D P > , O , T r D P = 11 (0

(2)

and

J 5 = {LIP E L%(J")] ,

where LpN stands for contraction operator. Clearly,

J $ c J p . (3)

It is known that J p and JpN are involved in the evaluation of the minimum value bo of the symmetric Hermitian operator

by writing

400 GUO A N D T A N G

In a previous paper [9], the set consisting of all p-particle Hermitian operators SP, {ZIP}, to which J P and Jk belong, was decomposed into a sum of components corre- sponding to irreducible representations of the unitary group U ( r ) by means of the characteristic operators lF;p')

whcre

Fr ( r - 2q + l ) ( r - p - q)! (p - q) ! (q - t ) ! ( r - q - t + I ) !

u$ = (- 1 p - r t ! (7)

where r f denotes the expansion operator.

( T } proposed by Kummer, Absar, and Coleman [S] These characteristic operators can be used to study the operator endomorphisms

T : SP + S p , (9)

where the ( T } constitutes a (p + I)-dimensional commutative algebra A over the field R of real numbers. It is obvious that, by means of Eqs. (6) and (8), T may be formu- lated in terms of the characteristic operators (FCP[ to give

where

In the scheme of operator endomorphisms, it is not difficult to show that, just like J p and 15, T ( J p ) , and T ( J k ) are two unitarily invariant convex sets which belong to SP. Thus two kinds of operator endorriorphisms N% and S$ may be taken to give rise, respectively, to the necessary and sufficient conditions for N representability such that

Since the action of operator endomorphisms in N& and S% leaves the trace of density operators invariant, the relation

must be satisfied. The b&, that is, the inverse matrix elements of u p defined by Eq. (7), have been obtained in the previous paper [9]

N REPRESENTABILITY 401

(JJ - q ) ! ( r - q - t ) ! ( t - q ) ! ( r - p - q)!

bP = Iq

3. Necessary Conditions Induced by Operator Endomorphisms

It is clear from Eq. ( 1 3) that, for T E NpN, T ( D p ) 3 0 should be a necessary con- dition for N representability. By means of the Slater determinant

# = 1@'1@'2 ' .. @'N I ( 1 6 )

and of extreme AGP functions g N and gp arising from the spin-orbital space W: (+ i ; i = 1,2, . . . ,2s), necessary condition for T E NpN may be expressed as

= C Pt T r [ D f ( g P ) D f ( g N ) I 3 0, ( 1 8 ) f

where we have made use of FGPDp = 0:. Substituting #, g N , and gp into Eqs. (17) and ( 1 81, we obtain

+ c y [( 2 v f 1 2m ) [ 2 u f 1 2k )]-'p2"+' c ij

s - u - i - j - l s - u - i - j - l

(2j + 1)!(i!)2(u - i - J ) ! ( s - v - i - j - I ) ! ) ( s - k - i

where m = N/2 and k = p / 2 , and the p's are subject to the following relation:

402 GUO A N D TANG

When p = 2, Eqs. (1 9) and (20) are the necessary and sufficient conditions for T E

Nfh. Further, by solving the equalities given in Eqs. (19) and (20) withp = 2, and then using Eq. (1 1) and the inverse matrix of u p , we get the results obtained by Kummer, Absar, and Coleman [S]

where Id , Qfh, B ; , and C: represent four extreme points of Nfh, respectively.

of NpN as follows: Similarly, when p is an arbitrary positive integer, we can obtain three extreme points

ld = C F g p , 9

4. Sufficient Conditions Induced by Operator Endomorphisms

Let d( 1,2, . . . ,p) and #( 1,2, . . . , N ) be p-particle and N-particle functions, re- spectively. Since T E N%, we have

Tr[DP(d)TDP(I,b)] = Tr{[TDp(d)]Dp(I,b)) 0. (28)

Equation (28) gives us an equivalent definition of NpN,

where jpN is used to denote the corresponding polar of JpN [3]. Now let us introduce an operator endomorphism

which maps NG into SpN,

RpN(NpN) = S x c SpN,

N REPRESENTABILITY

where S$ is a convex subset of SpN. Indeed, as we know that

BP E SpN - I ' f B p = C br9FpBP > 0 , 4

we are led to define

D N = (b$)-' C b f q F y B p E J N , 4

and then obtain

DP = LpNDN = (b$- ' C bfqF$PBp E JpN 9

or

RpNBP E JpN,

where, in the last step of Eq. (34), we have made use of

LpNFy = F p p . 9

Since Eq. (35) holds good for any BP belonging to SpN, we immediately get

RpN(jP,) G JpN.

By means of the definition of Eq. (28) and NpN given in Eq. (29), Eq. (37) may be re- formulated as

RpNT(Jp) E J i for T E Ng. (38)

Clearly, Eq. (38) is valid for any T that belongs to NpN. Thus we obtain

RpN(NpN) = S$ c SpN. (39)

Since RpN is a linear mapping, RpN(T) should be an extreme point of S x , if T is an ex- treme point of NpN. As a direct result, we get four extreme points of S::

R; = C 9 4 (2) r ,? (3-l r 4 ' ) * F i 2 ,

Q; = R;Q; = c ( - 1 ) q (i) 4 ')-IF?,

(2) i' 4 ( 3 - l 1 4 2r1

(40)

(41)

U L = R;B; = c (-1)9 (:) [ 4 ')-' [ 1 - q (i + i) ( 1 - *)] F?, (42)

9

4

Y i = R&C& = C q q

X [I - q ( L + L ) ( l 2 r - N -*)I F?, (43)

404 GUO A N D TANG

and also we get three extreme points of S x :

(44)

It has been pointed out by Kummer, Absar, and Coleman [lo] that R& and Q$ are not only extreme points of Sx, but also extreme in SL.

Bibliography

[ I ] A. J. Coleman, Rev. Mod. Phys. 35,668 (1963). 121 C. Garrod and J. K. Percus, J. Math. Phys. 5, 1750 (1964). [3] H. Kummer, J. Math. Phys. 8 ,2063 (1967). [4] A. J . Coleman, Rep. Math. Phys. 4, 113 (1973). [5] R. M. Erdahl, J. Math. Phys. 13, 1608 (1972). [6] A. J . Colcman, Energy, Strzcture and Reacfiuity-Proceedings of the 1972 Boulder Seminar, D.

W. Smith and W. 6. McRae, Eds. (Wilcy, New York, 1973), p. 231. [7] J . E. Harriman, Phys. Rev. A 17, 1257 (1978). [8] H . Kummer, I. Absar. and A. J . Colcman, J . Math. Phys. 18, 329 (1977). [9] Tang Au-chin and Guo Hong (unpublished).

[ lo ] H. Kummer and I . Absar, J . Math. Phys. 18, 335 (1977).