Approximate solution for electron correlation through the use of Schwinger probes

19 June 1998

Chemical Physics Letters 289 Ž1998. 419–427

Approximate solution for electron correlation through the use of Schwinger probes David A. Mazziotti Department of Chemistry, 12 Oxford Street, HarÕard UniÕersity, Cambridge, MA 02138, USA Received 30 December 1997; in final form 16 April 1998

Abstract Quantum energies and two-particle reduced density matrices Ž2-rdms. may be determined without using the N-particle wavefunction by combining the reconstruction of higher rdms from lower rdms and the contracted Schrodinger equation. In ¨ this letter we derive a systematic procedure for obtaining reconstruction functionals through the use of Schwinger probes. Previous functionals for the 3 and 4-rdms are generated as well as new functionals for higher rdms. Through a quasi-spin model we demonstrate that the 5-rdm, normalized to unity, may be reconstructed from lower rdms with the accuracy of its elements ranging from 10y4 for five fermions to 10y8 for fifty fermions. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction Because the interactions between electrons in an atom or molecule are modeled as pairwise within the Hamiltonian, the energy as well as other observables may be obtained exactly from a knowledge of the two-particle reduced density matrix Ž2-rdm.. Variational optimization of the ground-state 2-rdm by minimizing the energy has not been possible because simple conditions for guaranteeing that the 2-particle density matrix arises from an N-particle wavefunction have not been found Ž N-representability problem. w1–3x. As in density functional theory w4x, Nrepresentability research seeks to replace the N-particle wavefunction by a density or density matrix involving fewer particles to avoid the computational difficulties that arise for moderate to large N in traditional approaches to electron correlation w5–7x. The possibility for direct determination of the 2-rdm without the wavefunction has resurfaced in research on the contracted Schrodinger equation ŽCSchE., ¨

also known as the density equation w8–14x. Just as the Schrodinger equation relates the N-particle ¨ Hamiltonian to the wavefunction, the CSchE connects the 2-particle reduced Hamiltonian to the 2-rdm. By itself the CSchE is indeterminate because it also requires a knowledge of the 4-rdm w15x. N-representability conditions for the 4-rdm, if known, would give a unique solution to the CSchE. The development of reconstruction techniques for building higher rdms from the 2-rdm, however, have recently led to the solution of the CSchE for accurate atomic and molecular energies and 2-rdms. Two reconstruction methods, Ži. the functional approach where the higher rdms are expressed as functionals of the lower rdms and Žii. the ensemble representability method ŽERM. in which the higher rdm is constructed by forcing it to be Hermitian, antisymmetric, and positive semidefinite with correct contraction to the lower rdm, were explored in another paper w8x. Both of these strategies were illustrated through solution of the CSchE for a model

0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 7 0 - 9

D.A. Mazziottir Chemical Physics Letters 289 (1998) 419–427

420

fermionic system with as many as 40 particles where the resulting energies were as good as those from single-double configuration interaction ŽSDCI. while the resulting 2-rdms were better by an order of magnitude than those from SDCI. Similar accuracy has been achieved by Nakatsuji and Yasuda through the functional approach for atomic and molecular systems with between 4 and 14 electrons w13,14x. In this letter we wish to focus on the functional approach to reconstruction. Originally, Valdemoro derived the reconstruction functionals from the particle-hole equivalence and the commutation relations for second-quantized creation and annihilation operators w9x. Yasuda and Nakatsuji demonstrated how these functionals and additional corrections could be obtained through Feynman diagrams for Green’s functions w14x. Previously, we have shown how Nakatsuji’s correction for the 4-rdm may also be obtained from the perspective of particle-hole equivalence w8x. Here we present a fresh approach for deriving the functionals through an analogy with the methods for obtaining the connected and unconnected Green’s functions in quantum field theory w16–18x. Through the use of Schwinger probe variables we introduce generating functionals for connected and unconnected rdms and show how these notions are related to the reconstruction functionals.

ordering operator O allows us to write the generating functional as w19x G Ž J . s ² c
We first show that a generating functional for rdms may be written as

1

ž žÝ k

//


Ž 1.

where J is a vector of probe variables Jk which test different excitations and O is the operator which defines the order of the probe variables such that the probes Jk appear to the left of the conjugate probes Jk) in a series expansion of the exponential. For fermions the probes are special complex numbers which anticommute, known as Grassmann variables w16,17x. Even though the creation and annihilation operators in the exponentials do not commute, the

Ž 2.

l

E 2G

Dji11 s lim

J™0

Ž 3.

E Ji 1 E Jj)1 E

s lim ² c <

† k

žÝJ a / E = exp ž Ý J a / < c : Ž 4. EJ s lim ² c < a exp ž Ý J a / a exp ž Ý J a / < c : J™0

E Ji 1

exp

k

k

) l l

) j1

l

† i1

J™0

k

† k

k

) l l

j1

l

Ž 5. s ² c < a†i 1 a j1 < c : .

Ž 6.

In Eq. Ž4. above we employ the fact that since the derivative ErE Jj1 anticommutes with both Grassmann variables and annihilation and creation operators, it commutes with the product Jk a†k as well as the exponential of the product whose series expansion involves powers of Jk a†k . In general, the p-rdm is produced by the formula Dji11,,ij 22 ,, .. .. .. ,, ji pp s lim J™0

1 Jk a†k q Jk) a k

) l l

The elements of the 1-rdm may be formed by differentiating the generating functional GŽ J . with respect to Jj)1 and Ji1 and then taking the limit as all of the components of J approach zero

s G Ž J . s ² c < O exp

k

k

p

2. Theory

† k

ž Ý J a / exp ž Ý J a / < c : .

p!

1

E pG

p! E Ji p ... E Ji 2 E Ji 1 E Jj)1 ... E Jj)py 1 E J j)p

² c < a†i a†i ... a†i a j a j ... a j < c : . 1 2 p p py 1 1

Ž 7. Ž 8.

By differentiating the generating functional GŽ J . with respect to the probe variables and then taking the limit that all of these variables J vanish, we are extracting the coefficients of the Taylor series expansion of GŽ J . around the origin in terms of the Schwinger probe variables J. Hence, the rdms are the coefficients of a Taylor series expansion of the generating functional GŽ J .. Eq. Ž7. completely specifies the generating functional GŽ J ., and we will only require this information about GŽ J . when deriving the reconstruction functionals for rdms.

D.A. Mazziottir Chemical Physics Letters 289 (1998) 419–427

Formulas for the construction of Green’s functions are often characterized as having two parts, Ži. the unconnected part which may be written as products of lower-order Green’s functions and Žii. the connected part which may not be expressed as a simple product of lower-order functions w16–18x. We would like to adopt this terminology for the reconstruction of reduced density matrices. Furthermore, as in the case of Green’s functions, we will show that the unconnected formulas may be exactly determined. If we can define a generating functional W Ž J . for the connected rdms, we can obtain formulas for the unconnected parts by examining the difference functional GŽ J . y W Ž J .. By defining W Ž J . to be the generating functional for the connected p-rdms, denoted by pD, we are implicitly asserting that the connected p-rdms may be obtained from W Ž J . in a manner analogous to the method for procuring the complete p-rdms from GŽ J . in Eq. Ž7. p i 1 ,i 2 , . . . , i p D j1 , j 2 , . . . , j p

s lim J™0

computer algebra system w23x to differentiate Eq. Ž10. with respect to probe variables and then to take the limit as all of the probes J approach zero. The Hamiltonian of the system, we assume, conserves the number of particles, and thus, partial derivatives of W Ž J . involving an unequal number of J ) and J vanish. By expressing the derivatives of GŽ J . and W Ž J . as elements of the p-rdms pD and connected p-rdms pD through the use of Eqs. Ž7. and Ž9. respectively, we obtain a formula for the p-rdm in terms of the connected p-rdm pD and connected q-rdms qD where q - p. The terms involving simple products of the connected parts of the lower rdms qD constitute the unconnected portion of the p-rdm. The resulting formulas are exact ŽTable 1.. Assuming that the connected part pD of the p-rdm vanishes yields an approximate formula for the p-rdm in terms of lower rdms. Let us examine the exact formulas for fermionic 1 and 2-rdms 1

1

E pW

p! E Ji p ... E Ji 2 E Ji 1 E Jj)1 ... E Jj)py 1 E Jj)p

.

Ž 9.

421

Dji11 s1D ji11 ,

Ž 11 .

and 2

Dji11,,ij 22 s2D ji11 ,,ij22 q1D ji11 n1D ji22 .

Ž 12 .

Through both diagrammatic and algebraic approaches it has been shown that the functional for the full Green’s function is just the exponentional of the functional for the connected Green’s functions w16,20,21x. The zero-temperature Green’s functions become the reduced density matrices in the limit that the time of the creation and annihilation operators approach zero from the right and left respectively w16x. By time-ordering of the operators this produces the normal ordering for rdms in which the creation operators appear to the left of the annihilators as in Eq. Ž8.. Therefore, we find that an exponential relationship between GŽ J . and W Ž J . is valid for rdms,

Eq. Ž11. conveys the fact that the 1-rdm must be completely connected since there aren’t any lower rdms from which to build it. The second term in Eq. Ž12., the unconnected contribution to the 2-rdm, introduces us to the the Grassmann Žwedge. product w8,24–26x, denoted by the n symbol. The wedge product between two elements of the connected 1-rdm in the above equation may be explicitly evaluated by summing the distinct products arising from all antisymmetric permutations of the upper indices and all antisymmetric permutations of the lower indices

G Ž J . s exp Ž W Ž J . . .

Within the Hartree–Fock approximation the connected 2-rdm 2D vanishes, and the wedge product of the 1-rdms generates the 2-rdm. The elements of the wedge product may be expressed more generally as

Ž 10 .

In different terminology GŽ J . is said to generate the moments pD while W Ž J . generates the cumulants pD w22x. The fact that the functional W Ž J . creates the connected rdms has also been demonstrated through statistical arguments by Kubo in his article on cumulant expansions w19x. Approximate formulas for building higher rdms from lower rdms may now be obtained through the use of Eqs. Ž7., Ž9. and Ž10.. We use the Maple

1 i1 D j1 n1D ji22 s 12 1D ji11 1D ji22 y1D ji21 1D ji12

ž

pq 1 , . . . , i N s a ij11 ,i, j22 ,, .. .. .. ,, ijpp n bjipq 1 , . . . , jN

1

/.

Ž 13 .

2

ž /Ý N!

e Žp . e Ž s .

p ,s

pq 1 , . . . , i N , =ps a ij11 ,i, j22 ,, .. .. .. ,, ijpp bjipq 1 , . . . , jN

Ž 14 .

D.A. Mazziottir Chemical Physics Letters 289 (1998) 419–427

422

in which p represents all permutations of the upper indices and s represents all permutations of the lower indices while the function e Žp . returns q1 for an even number of transpositions and y1 for an odd number of transpositions. Since both the upper and the lower indices have N! permutations, there are Ž N!. 2 terms in the sum. Hence, normalization requires division by Ž N!. 2 . Wedge products between elements with an equal number of upper and lower indices possess an important commutative property pq 1 , . . . , i N a ij11 ,i, j22 ,, .. .. .. ,, ijpp n bjipq 1 , . . . , jN Ny p n a i Ny pq 1 ,i Ny pq 2 , . . . , i N s bji11,,......,, jiNy j Ny pq 1 , j Ny pq 2 , . . . , j N p

Ž 15 .

or anbsbna ,

Ž 16 .

where in the latter expression indices are not shown. For notational clarity we will supress the indices for the remainder of the paper. The exact functionals for the 1 and 2-rdms as well as higher rdms are summarized without indices in Table 1. Superscripts are employed to indicate the number of times that a matrix should be wedged with itself, i.e. 1D 3 s1D n1D n1D. In each of the formulas for the rdms the connected portion of the rdm is the first term, and all of the remaining terms are unconnected because they may be calculated completely from products of lower rdms. By setting the connected portions of the rdms to zero, we produce approximate formulas for building higher rdms from lower connected rdms qD. Since the connected rdms qD may be written in terms of rdms q D involving the same or fewer fermions, the higher

rdms may be built from lower rdms q D. For example, in Table 1 we express the unconnected part of the 5-rdm both as a functional of the lower connected rdms qD and as a functional of the lower rdms q D where q - 5. These approximate reconstruction formulas for the 5 and 6-rdms have not been previously reported in the literature on the contracted Schrodinger equation. The coefficients in these for¨ mulas agree with those expected from a cumulant expansion w22x as well as those from the appropriate time limit of the Feynman diagrams for the 5 and 6-particle Green’s functions w16x. The pairwise interactions between the N particles of the Hamiltonian lead to a fully correlated wavefunction or density matrix in which all N particles are linked to each other. When we build the p-rdm from unconnected products of lower rdms, antisymmetrized through the wedge product, we are accounting for the possible links among p y 1 of the particles, but links among all p particles which comprise the connected portion of the p-rdm are neglected. Historically, the idea of treating N indistinguishable particles as products of connected clusters of particles arose from the study of condensation phenomena in classical w27,28x and quantum statistical w29,30x mechanics. The importance of the connected correction will depend on the strength of the pairwise interactions in the Hamiltonian as well as the choice of p. For interactions in the range of atoms and molecules it has been shown that the unconnected terms alone provide accurate reconstructions for p s 4 w8,14x. In the next section we will also explore the possible accuracy for p s 5. The mean-field

Table 1 Reconstruction functionals for rdms are presented 1

D s1D

2

D s2D q1D n1D

3

D s3D q1D 3 q 3 2D n1D

4

D s4D q1D 4 q 6 2D n1D 2 q 4 3D n1D q 3 2D 2

5 D s5D q1D 5 q 10 2D n1D 3 q 10 3D n1D 2 q 15 2D 2 n1D q 5 4D n1D q 10 3D n2D w or 5D s5D y 24 1 D 5 q 60 2 D n1 D 3 y 20 3 D n1 D 2 y 30 2 D 2 n1 D q 5 4D n1 D q 10 3 D n2 D x 6

D s6D q1D 6 q 15 2D n1D 4 q 20 3D n1D 3 q 45 2D 2 n1D 2 q 15 4D n1D 2 q 60 3D n2D n1D q 15 2D 3 q 6 5D n1D q 15 4D n2D q 10 3D 2

D.A. Mazziottir Chemical Physics Letters 289 (1998) 419–427

ŽHartree–Fock. approximation appears in the unconnected portion of Eq. Ž12. which is the wedge product of the 1-rdm with itself. To move beyond this approximation for building the 2-rdm, we would have to add connected corrections. Correlation in the unconnected terms appears for the first time in the 3-rdm functional where links between pairs of particles are considered through the appearance of the connected 2-rdm. For atoms and molecules the 2norm of pD decreases as p increases. This explains why these formulas become more accurate as p increases. To generate the 3-rdm approximation with an accuracy equivalent to that of the 4-rdm formula in Table 1, Nakatsuji and Yasuda w13x introduce an approximation for the connected 3-rdm 3D. In another paper w8x we formed an improved 3-rdm 3 D impr by contracting the 4-rdm formula to generate a system of equations 3

D impr s

4 Ny3

L34 1 D 4 q 4 3D 3 D impr n1 D

ž

ž

/

q6 2D n1 D 2 q 3 2D n2D ,

/

Ž 17 .

whose solution would yield a 3-rdm 3 D impr with some 3-particle links. The contraction operator L denotes the summation required to map the 4-particle matrix to a 3-particle matrix. Valdemoro w9x originally obtained reconstruction functionals through particle-hole equivalence and the commutation relations for fermion creation and annihilation operators. Although Valdemoro’s reconstruction functionals agree with the unconnected parts of the 2 and 3-rdms, her formula for the 4-rdm is missing the term 3 2D 2 because it cancels with the corresponding hole correction in the commutation relation. Yasuda and Nakatsuji w13,14x supply the missing 4-rdm term through arguments involving schematic Green’s function diagrams. We previously presented a method for obtaining the 4-rdm correction from the particle-hole equivalence w8x. Valdemoro’s 5-rdm formula, which differs from the unconnected portion of the 5-rdm by the term 10 3D n2D, is 5

D Vald s 5 4D n1 D y 10 3 D n1 D 2 q 10 2 D n1 D 3 y 4 1 D 5 .

Ž 18 .

423

Here we have presented a systematic procedure for obtaining all of the unconnected terms for p-rdms when p ) 4.

3. Results and discussion To illustrate the accuracy of the above reconstruction technique, we employ a quasi-spin model, originally used by Lipkin as a benchmark to investigate fermionic correlation phenomena w31,32x. In the model there are N degenerate states with an energy of yer2 and another N degenerate states with an energy of qer2. Each state will have a unique pair of quantum numbers m and p. The quantum number m indicates whether the state is a member of the upper or lower energy level by assuming the value q1 or y1 respectively, and the quantum number p, ranging from 1 to N, denotes the position of the state in a given level. These energy levels are filled with N fermions where only one fermion can occupy a given state. For noninteracting fermions, the configuration of lowest energy yNer2 is achieved when each of the N fermions occupies one of the N states in the lower energy level yer2. Consider adding an interaction to the noninteracting system according to the Hamiltonian e H s Ý ma†m , p a m , p 2 m, p qV

Ý

a†m , p 1 a†m , p 2 aym , p 2 aym , p 1 ,

Ž 19 .

p1 , p 2 , m

where V is the interaction strength. By the nature of the V interaction term, the noninteracting groundstate configuration will only mix with configurations in which each fermion has a different p quantum number. Hence, for a nonzero value of the V parameter a total of 2 N configurations may contribute to the correlated ground state of the system. Because of the model’s symmetry, however, we can group the configurations into N q 1 classes. This reduction in basis size and additional computational details may be found in a previous paper w8x. In Figs. 1 and 2 we explore four different methods for reconstructing the 5-rdm from the lower rdms: Ži. using the exact 1,2,3,4-rdms in the 5-rdm formula in Table 1 with 5D s 0, Žii. building the

424

D.A. Mazziottir Chemical Physics Letters 289 (1998) 419–427

Fig. 1. Errors in four different reconstructions of the 5-rdm are compared as functions of the number of fermions N. The error in each approximate 5-rdm 5Dapprox is measured by the 2-norm of the matrix 5Dapprox y5Dexact .

5-rdm as in Ži. with the exact 4-rdm replaced by the formula in Table 1 with 4D s 0, Žiii. building the 5-rdm as in Žii. with the exact 3-rdm replaced by the approximation obtained by solving the system of equations in Ž17., and Živ. building the 5-rdm as in Žii. with the exact 3-rdm replaced by Valdemoro’s reconstruction functional which is equivalent to the 3-rdm formula in Table 1 with 3D s 0. These methods are listed in the order in which the accuracy is expected to decrease. In Fig. 1 the errors of these reconstruction techniques are compared by computing the 2-norms of 5Dapprox y5Dexact for N ranging from five to fifty. The 2-norm of a matrix corresponds to the magnitude of the eigenvalue with the largest absolute value. The interaction strength V is adjusted for each N to give a correlation energy whose percentage of the total energy is consistent with that predicted in w33x for atoms with an equivalent number of electrons. While all of the rdms in Table 1 are normalized to N!rŽ p!Ž N y p .!. in accordance with the second-quantized definition for the rdms in Eq. Ž8., we will renormalize the approximate and exact rdms to unity for all error calculations in this section. This N-independent normalization will facilitate the comparison of the reconstruction errors as functions of N. In general, the method Ži. is between one and two orders of magnitude better than method Žii. which neglects the connected part of the 4-rdm and between two and four orders of magnitude better than method Živ. which neglects the connected parts of both the 3-rdm and the 4-rdm.

Method Žiii. that employs the contraction approach to producing an improved 3-rdm is not much worse than Žii. which builds from a known 3-rdm. Fig. 2 compares the deviations of the approximate 5-rdms from being positive semidefinite, a necessary condition for N-representability. This deviation is measured by summing the squares of the negative eigenvalues and then taking the square root. Methods Žii. and Žiii. in Fig. 2 give nearly indistinguishable results for positive semidefiniteness. The reported range of N and the adjusted interaction strengths V are the same as those employed in determining the data for Fig. 1. While most work has focused on solving the 2,4-CSchE by building the 3 and 4-rdms from the 2-rdm, a 3,5-CSchE may also be derived whose indeterminacy may be removed by reconstructing the 4 and 5-rdms from the 3-rdm. One goal of the present letter is to provide an applicable reconstruction functional for building the 5-rdm from known 1, 2, and 3-rdms. This is achieved by method Žii. above. We would like to compare the accuracy of the 5-rdm from Žii. with the accuracy of the following two techniques for building the 4-rdm from known 1 and 2-rdms. Both techniques employ the 4D formula in Table 1 with 4D s 0, but they differ in the 3-rdm approximation: in method Ža. the 3-rdm is obtained from the system of equations in Ž17. and in method Žb. the 3-rdm is constructed from the 3 D formula in Table 1 with 3D s 0 which is equivalent to Valde-

Fig. 2. Deviations from positive semidefiniteness for four approximate reconstructions of the 5-rdm are compared as functions of the number of fermions N. The deviation in each approximate 5-rdm 5Dapprox is measured by adding the squares of the negative eigenvalues and then taking the square root of the result.

D.A. Mazziottir Chemical Physics Letters 289 (1998) 419–427

moro’s functional in w9x. The error in each of the reconstructed 4 and 5-rdms is measured by taking the 2-norm of the difference between the exact and the approximate rdm. The 2-norm errors of the reconstructed matrices are given in Fig. 3 as functions of the number of particles N where the interaction strengths V are adjusted just as in Figs. 1 and 2. As we emphasized in previous work w8x, method Ža. produces 4-rdms which are considerably more accurate than those from method Žb.. This figure, however, also shows that the reconstruction approach Žii. produces a 5-rdm that is comparable in accuracy to the 4-rdm by method Ža.. Similar results were also seen for N s 8 and N s 20 with the perturbation ranging from 0.01 to 0.1. For V larger than that expected in atoms and molecules the 5-rdm from Žii. appears to be more accurate than the 4-rdm from method Ža.. These results indicate that the solution of the 3,5-CSchE with the unconnected reconstruction approach of method Žii. will yield results at least as accurate as those from the 2,4-CSchE with a connected correction for the 3-rdm. Although several techniques have been considered for reconstructing the 5-rdm from the unconnected terms in Table 1, we have not compared the accuracy of this formula with Valdemoro’s functional in Eq. Ž18.. In Fig. 4 for a range of N we contrast the error from method Ži. for building the 5-rdm by using the exact 1,2,3,4-rdms in the unconnected portion of the 5 D formula in Table 1 with Ža. a Hartree–Fock reconstruction approximation where 5D HF s1 D 5 and

Fig. 3. Errors in two different reconstructions for the 4-rdm from a knowledge of the 2-rdm Žmethods Ža. and Žb.. are compared for a range of N with the error in method Žii. for building the 5-rdm from a knowledge of the 3-rdm.

425

Fig. 4. The error in the present unconnected reconstruction for the 5-rdm Ži. is compared with the errors in both the Hartree–Fock Ža. and Valdemoro Žb. 5-rdm reconstructions for a range of N.

Žb. Valdemoro’s functional with the exact 1,2,3,4rdms. In general, the unconnected functional Ži. is about an order of magnitude better than Valdemoro’s functional, and both methods Ži. and Žb. are several orders of magnitude better than the Hartree–Fock formula Ža.. In Fig. 5 we explore the accuracy of the reconstructions for interaction strengths larger than those expected in atoms and molecules. For N s 50 we report the errors in the approximate 5-rdms as V ranges from 0.003 to 0.03. Correlation for a 50-electron atom, which is about 0.05% of the total energy, is achieved when V is about 0.005; by V s 0.03 the correlation energy is nearly 8% of the total energy. All of the reconstruction formulas become less accurate as V increases, but the interesting result is that

Fig. 5. The error in the present unconnected reconstruction for the 5-rdm Ži. is compared at N s 50 with the errors in both the Hartree–Fock Ža. and Valdemoro Žb. 5-rdm reconstructions for a range of interaction strength V.

426

D.A. Mazziottir Chemical Physics Letters 289 (1998) 419–427

for large enough V Valdemoro’s approximation becomes better than the unconnected functional.

4. Summary Accurate energies and 2-rdms for atoms and molecules and other fermion systems have recently been calculated through the contracted Schrodinger ¨ equation ŽCSchE. without using the N-particle wavefunction. Although the integral form of the CSchE was presented in 1976 w34,35x, its solution has only become possible with the development of reconstruction formulas for building higher rdms from lower rdms w8–14x. In this letter we have discussed a systematic procedure for deriving reconstruction functionals for rdms through the use of Schwinger probes and generating functionals. The relation of the present approach to the generating functionals for Green’s functions in quantum field theory w16–18x allows us to introduce the new and useful terminology of connected and unconnected reduced density matrices where the unconnected part of an rdm may be written as wedge products of lower rdms. We employ the wedge product notation from Grassmann Žexterior. algebra w25,26x to simplify previous notation for the reconstruction formulas w8–10x. While the Schwinger-probe method produces functionals which agree with Valdemoro’s formula for the 3-rdm as well as her 4-rdm formula with Yasuda and Nakatsuji’s correction w9,14x, it also allows us to generate systematically the unconnected terms for the higher rdms. Here we present functionals for the 5 and 6-rdms, but formulas for higher rdms may be similarly obtained. For interactions within the Lipkin model w31x comparable to those in atoms and molecules we find that the accuracy of the 5-rdm, built from lower rdms through the unconnected terms, improves as the number of particles N increases and remains about an order of magnitude better than that of Valdemoro’s approximate 5-rdm as N ranges from 5 to 50 particles. Furthermore, reconstruction of the 5-rdm through wedge products of the 1, 2, and 3-rdms with each other matches the accuracy of reconstructing the 4-rdm from the 2-rdm by solving a system of linear equations. When the 2,4-CSchE and the 3,5-CSchE are compared for determining quantum energies, we must consider that the 5-rdm

need not be stored since it can be expressed as wedge products involving the 3-rdm and lower rdms and that the computational cost of solving a system of equations to build an accurate 4-rdm may be significant. For these reasons the development of the present reconstruction functional for the 5-rdm raises the possibility of an efficient method for solving the 3,5-CSchE to calculate energies and 3-rdms for atoms and molecules.

Acknowledgements The author wishes to express his gratitude to Professor Dudley R. Herschbach, Professor Paul C. Martin and Dr. Alexander R. Mazziotti for their advice and encouragement. The NSF is also acknowledged for supporting the work through a research fellowship.

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