* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download A new approach for the two-electron cumulant in natural orbital
History of quantum field theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Hidden variable theory wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Quantum group wikipedia , lookup
Quantum state wikipedia , lookup
Hydrogen atom wikipedia , lookup
Coupled cluster wikipedia , lookup
Hartree–Fock method wikipedia , lookup
Path integral formulation wikipedia , lookup
Density matrix wikipedia , lookup
Molecular orbital wikipedia , lookup
Atomic orbital wikipedia , lookup
Tight binding wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
A New Approach for the Two-Electron Cumulant in Natural Orbital Functional Theory MARIO PIRIS Institute of Physical and Theoretical Chemistry, Friedrich-Alexander University Erlangen-Nuremberg, Egerlandstrasse 3, 91058 Erlangen, Germany Received 11 May 2005; accepted 29 August 2005 Published online 4 November 2005 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/qua.20858 ABSTRACT: The cumulant expansion gives rise to a useful decomposition of the two-matrix D in which the pair correlated matrix (cumulant) is disconnected from the antisymmetric product of the one-matrix ⌫. A new explicit antisymmetric approach for the two-particle cumulant matrix in terms of two symmetric matrices, ⌬ and ⌳, as functionals of the occupation numbers is proposed for singlet ground states of closedshell systems. It produces a natural orbital functional that reduces to the exact expression for the total energy in two-electron systems. The functional form of matrix ⌳ is readily generalized to any system with an even number of electrons. The diagonal elements of ⌬ equal the square of the occupation numbers, and the N-representability positivity necessary conditions of the two-matrix impose several bounds on the offdiagonal elements of matrix ⌬. The well-known mean value theorem and the partial sum rule obtained for the off-diagonal elements of ⌬ provide a prescription for deriving a practical functional. In particular, when the mean values { J i*} of the Coulomb interactions { Jij} for a given orbital i taking over all orbitals j ⫽ i are assumed to be equal {Kii/2}, a functional close to self-interaction-corrected GU functional is obtained, but the two-matrix fermionic antisymmetric holds. An additional term for the matrix elements of ⌳ between HF occupied orbitals is proposed to ensure a correct description of the occupation numbers for the lowest occupied levels. The functional is tested in fully variational finite basis set calculations of 57 molecules. It gives reasonable molecular energies at the equilibrium geometries. The calculated values of dipole moments are in good agreement with the available experimental data. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem 106: 1093–1104, 2006 Key words: natural orbital functional; cumulant; reduced density matrix; dipole moment; electron correlation energies Contract grant sponsor: Alexander von Humboldt (AvH) Foundation. International Journal of Quantum Chemistry, Vol 106, 1093–1104 (2006) © 2005 Wiley Periodicals, Inc. PIRIS 1. Introduction T he idea of a one-particle reduced density matrix (one-matrix) ⌫ functional appeared some decades ago [1]. More recently, several functionals of the natural orbitals and their occupation numbers have been proposed [2–11]. A major advantage of the method is that the kinetic energy and the exchange energy are explicitly defined using the one-matrix and do not require the construction of a functional. The unknown functional only needs to incorporate electron correlation. Moreover, the onematrix is a much simpler object than the N-particle wave function, and the ensemble N-representability conditions that have to be imposed on variations of ⌫ are well known [12]. Finally, the natural orbital functional (NOF) incorporates fractional occupation numbers in a natural way, which provides a correct description of both dynamical and nondynamical correlation. An NOF requires an expression of the two-particle reduced density matrix (two-matrix) D in terms of the one-matrix ⌫. Such reconstruction of the two-matrix can be achieved using the wellknown cumulant expansion of D [13]. In the present study, we propose an explicit antisymmetric form for the two-particle cumulant matrix. By employing an antisymmetric ansatz for the twomatrix, we can obtain a correct description of the pair density for parallel-spin electrons, which is poorly described by its predecessor [11]. Unlike density functional theory (DFT), density matrix functional theory affords an exact energy functional for two-electron systems [6, 14]. The ansatz presented here can be reduced to this exact expression providing the specific form of the cumulant matrix in the two-electron case. This functional form can be readily generalized to any N-electron system, but the off-diagonal elements of a symmetric matrix ⌬ remain unknown. In contrast, several constraints for these magnitudes can be achieved via some known positivity conditions. In the present work, we analyze the D, G, and Q conditions of two-matrix N-representability, and establish several bounds on the off-diagonal elements of matrix ⌬. To develop a practical method, a further approximation is assumed. Instead of the approach of off-diagonal elements of ⌬ considering constraints imposed by positivity conditions, we approximate the term that involves matrix ⌬, by employing the well-known mean value theorem and the partial 1094 sum rule obtained for this matrix. This leads to a total energy close to the self-interaction-corrected Hartree functional proposed by Goedecker and Umrigar (GU) [6]. The GU functional is not antisymmetric assuming a Hartree-product form for the opposite spin component of D, so it does not afford an N-representable two-matrix [15]. In contrast to this reconstruction, the current proposal can satisfy not only the hermiticity and particle permutation conditions, but also the trace relation and positivity conditions of the two-matrix. Besides, it appears to reproduce properly the occupation numbers for lower occupied levels. We begin with a presentation of the basic concepts and notations relevant to NOF theory (Section 2). We then present our ansatz for the two-electron cumulant matrix (Section 3). The two- and N-electron cases, as well as the N-representability of the functional, are discussed in detail. The following section is devoted to our further simplification to achieve a practical functional (Section 4). We end with a short presentation of the methodological details (Section 5) and some results for selected molecules (Section 6). 2. Basic Concepts and Notations 2.1. GENERAL FORMALISM We consider an N-electron Coulombic system described by the Hamiltonian Ĥ ⫽ 冘 h ⌫ˆ ⫹ 冘 具ij兩kl典D̂ ij ji ij kl,ij , (1) ijkl where hij denotes the one-electron matrix elements of the core-Hamiltonian, h ij ⫽ 冕冕 冋 1 dx i*共x兲 ⫺ ⵜ2 ⫺ 2 冘 兩r ⫺Z r 兩册 共x兲, I I I j (2) and 具ij兩kl典 denotes the two-electron matrix elements of the Coulomb interaction 具ij兩kl典 ⫽ 冕冕 ⫺1 dx 1dx2i*共x1兲j*共x2兲r12 k共x1兲l共x2兲. (3) Atomic units are used. Here and in the following x ⬅ (r, s) stands for the combined spatial and spin coordinates, r and s, respectively. The spin-orbitals VOL. 106, NO. 5 TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY {i(x)} constitute a complete orthonormal set of single-particle functions, 具 i兩 j典 ⫽ 冕 dx i*共x兲j共x兲 ⫽ ␦ij, ⌫ˆ ji ⫽ â †j â i (5) 冉冊 (6) and 1 † † â â â â 2 k l j i are constructed from the familiar creation and annihilation operators, {â†i } and {âi} [16], respectively, associated with the set of spin-orbitals {i(x)}. A quantum mechanical pure state of our N-particle system can be characterized by a normalized wave function ⌿ or a system density matrix ⌫N ⌫ N 共x⬘1, x⬘2, . . . , x⬘N; x1, x2, . . . , xN兲 ⫽ ⌿*共x⬘1, x⬘2, . . . , x⬘N兲⌿共x1, x2, . . . , xN兲. Tr D ⫽ (4) with the usual meaning of the Kronecker delta ␦ij. The one- and two-particle density matrix operators, D̂ kl,ij ⫽ and the trace of the two-matrix gives the number of electron pairs in the system (7) 冘 h ⌫ ⫹ 冘 具ij兩kl典D ij ji ij kl,ij , (8) ijkl where the one- and two-particle reduced density matrices, or briefly the one- and two-matrices, are defined as ⌫ ji ⫽ 具⌿兩⌫ˆ ji 兩⌿典 (9) D kl,ij ⫽ 具⌿兩D̂ kl,ij 兩⌿典. (10) ij,ij ij Tr ⌫ ⫽ 冘 ⌫ ⫽ N, ii i (11) 冉冊 N共N ⫺ 1兲 N ⫽ 2 . 2 D kl,ij ⫽ D *ij,kl (12) (13) and the antisymmetry, D kl,ij ⫽ ⫺Dlk,ij ⫽ ⫺Dkl, ji ⫽ Dlk, ji. (14) An important contraction relation between oneand two-matrices is in agreement with the previous normalization [Eqs. (11) and (12)]: ⌫ ji ⫽ 2 N⫺1 冘D jk,ik . (15) k This implies that the energy functional (8) refers only to the two-matrix, because D determines ⌫. Attempts to determine the energy by minimizing E [D] are complicated because of the lack of a simple set of necessary and sufficient conditions for ensuring that the two-matrix corresponds to an N-particle wave function (the N-representability problem) [12]. Alternatively, the last term in Eq. (8) can be replaced by an unknown functional of the onematrix [18]: E关⌫兴 ⫽ According to expression (8), the energy E of a state ⌿ is an exactly and explicitly known functional of ⌫ and D. The reduced density matrices satisfy important sum rules [17]. The trace (Tr) of the one-matrix equals the number of electrons, ⫽ Their diagonal elements in the coordinate-space representation are always non-negative, since ⌫(x1; x1) is related to the probability of finding one electron at x1, and D(x1, x2; x1, x2) is related to the probability of finding one electron at x1 and another at x2. D satisfies several relations that follow directly from the anticommutation rules and the hermiticity of the operators {â†i } and {âi}, namely, the hermiticity, The expectation value of the Hamiltonian (1) for the state ⌿ is then E⫽ 冘D 冘 h ⌫ ⫹ V 关⌫兴. ij ji ee (16) ij The functional Vee[⌫] is universal in the sense that it is independent of the external field. Its properties are well known [19]. However, it is very difficult to approximate because what we have done is to change the variational unknown from the complicated many-variable function ⌿ to a single one-matrix ⌫. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1095 PIRIS The one-matrix ⌫ can be diagonalized by a unitary transformation of the spin-orbitals {i(x)}, with the eigenvectors being the natural spin-orbitals and the eigenvalues {ni} representing the occupation numbers of the latter, ⌫ ji ⫽ n i ␦ ji . (17) In the following, all representations used are assumed to refer to this basis. Accordingly, the energy functional is determined by the natural orbitals and their occupation numbers, i.e., E[{ni, i}]. In addition, we assume all orbitals to be real. 冘D ␣ jk,ik ⫽ k N ␣␣ N ⌫ ⫽ n i ␦ ji . 4 ji 4 (23) It is readily demonstrated that the sum rules (22) and (23) are compatible with Eq. (15). The traces of these two-matrix components read Tr D ␣␣ ⫽ N共N ⫺ 2兲 , 8 Tr D␣ ⫽ N2 . 8 (24) Only singlet spin-compensated systems will be considered in the remainder of this study. 2.2. RESTRICTED FORMALISM The Hamiltonian (1) is spin independent, so only density matrix blocks that conserve the number of each spin type are nonvanishing. Specifically, the one-matrix has two nonzero blocks, an ␣ block (⌫␣) and a  block (⌫), ⌫ ␣␣ ji ⫽ 0, ⌫  ji ⫽ 0, (18) and the two-matrix has three nonzero blocks, an ␣␣ block (D␣␣), an ␣ block (D␣), and a  block (D), ␣␣ , ␣␣ ⫽ 0, D kl,ij ␣ , ␣ D kl,ij ⫽ 0,  ,  D kl,ij ⫽ 0. ␣i 共r兲 ⫽ i 共r兲 ⫽ i共r兲. 3.1. CUMULANT OF THE TWO-MATRIX It remains to find approximations for the unknown functional Vee[⌫]. To this end, we use a reconstructive functional D [⌫]; that is, we express elements Dkl,ij in terms of ⌫ji. We neglect any explicit dependence of D on the natural orbitals themselves because the energy functional (8) already has a strong dependence on the natural orbitals via the one- and two-electron integrals. Let us consider the following well-known [13] decomposition of D: (19) The set of spin-orbitals {i(x)} is split here into two subsets: {␣i (r)␣(s)} and {i (r)(s)}. In the case of spin compensated systems, the two blocks of the one-matrix are the same (⌫␣ ⫽ ⌫), i.e., n ␣i ⫽ n i ⫽ n i , 3. Two-Matrix (20) ␣␣ D kl,ij ⫽ 1 ␣ ␣ ␣␣ 共⌫ ⌫ ⫺ ⌫ kj␣ ⌫ ␣li ⫹ kl,ij 兲 2 ki lj n jn i ␣␣ 共 ␦ ki ␦ lj ⫺ ␦ kj ␦ li 兲 ⫹ kl,ij 2 (25) 1 ␣  1 ␣ ␣ 共⌫ ⌫ ⫹ kl,ij 兲 ⫽ 共n j n i ␦ ki ␦ lj ⫹ kl,ij 兲, 2 ki lj 2 (26) ⫽ ␣ ⫽ D kl,ij The trace of the one-matrix (11) becomes 2 冘 n ⫽ N. i (21) i For singlet states, the first and the last blocks of the two-matrix are also equal (D␣␣ ⫽ D), so in this work we deal only with D␣␣ and D␣. The parallel-spin component D␣␣ is antisymmetric, but D␣ possess no special symmetry. Each of these two-matrix blocks must contract to the appropriate one-matrix block: 冘D k 1096 ␣␣ jk,ik ⫽ 共N ⫺ 2兲 ␣␣ 共N ⫺ 2兲 ⌫ ji ⫽ n i ␦ ji 4 4 (22) where is the cumulant matrix. It should be noted that matrix elements of are nonvanishing only if all its labels refer to partially occupied natural orbitals with an occupation number different from 0 or 1 [13]. Taking into account the normalization condition for the one-matrix (21), it can easily be shown from Eqs. (22) and (23) that spin components of fulfill the following sum rules: 冘 ␣␣ jk,ik ⫽ n i 共n i ⫺ 1兲 ␦ ji (27) k 冘 ␣ jk,ik ⫽ 0. (28) k VOL. 106, NO. 5 TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY The first two terms on the right-hand side of Eq. (25) together satisfy property (14) of D␣␣. Therefore, the matrix ␣␣ should be antisymmetric as well. In general, the cumulant has a dependence of four indices, and it is not feasible to apply direct computation with such quantities to large systems. In this work, we consider the following parallel component of the cumulant matrix: ␣␣ kl,ij ⌬ji ⌬ji ⫽⫺ ␦ki␦lj ⫹ ␦kj␦li, 2 2 (29) where ⌬ is a symmetric matrix. The sum rule (27) and the approximate ansatz (29) imply the constraint 冘⬘ ⌬ ⫽ n 共1 ⫺ n 兲. ji i Taking into account that Lij ⫽ Kij for real orbitals, the expression (34) can be rewritten as E⫽2 冘 n h ⫹ 2 冘 共n n ⫺ ⌬ 兲J ⫺ 冘 ⌳ K . i ii j i i ji (35) NOF theory provides an exact energy functional for two-electron systems [6, 14]. In the weak correlation limit, the total energy is given by 冘 nh ⫹n K ⬁ E⫽2 i ii 1 冘 冑n n K ⬁ 11 ⫺ 2 i⫽1 1 i i1 i⫽2 冘 冑n n K . ⬁ ⫹ j i ij (36) i, j⫽2 The prime indicates that the i ⫽ j term is omitted. For ␣, we can achieve a suitable approximation if we replace the second term in Eq. (29) with the dependence obtained in the improved Bardeen– Cooper–Schrieffer (IBCS) method [20], i.e., ␣ kl,ij ⫽⫺ ⌬ji ⌸ki ␦ ␦ ⫹ ␦ ␦, 2 ki lj 2 kl ij (31) where we have introduced a new symmetric matrix ⌸. For purposes of convenience, as we see below, we define the matrix ⌸ in terms of a new symmetric matrix ⌳: ⌸ ki ⫽ n k n i ⫺ ⌬ ki ⫺ ⌳ ki . (32) Combining Eqs. (28), (30), (31), and (32) results in 2⌬ ii ⫹ ⌳ ii ⫽ 2n 2i ⫺ n i . 冘 n h ⫹ 冘 共n n ⫺ ⌬ 兲共2J ⫺ K 兲 ⫹ 冘 共n n ⫺ ⌬ ⫺ ⌳ 兲 L , i ii j i ji ij ij ij j i ji As can be seen from Eq. (36), the dependence of D on the natural occupations requires a distinction between HF occupied and virtual orbitals. Since D␣␣ ⫽ 0 for N ⫽ 2, one easily deduces from Eqs. (25) and (29) that ⌬ji ⫽ njni. Consequently, it is not difficult to see from Eqs. (26), (31), and (32) that D␣ nonzero elements have the form D␣ jj,ii ⫽ ⫺⌳ji/2. Thus, the total energy (35) turns into E⫽2 ji ij (34) ij with Jij ⫽ 具ij兩ij典, Kij ⫽ 具ij兩ji典, and Lij ⫽ 具ii兩jj典 [see Eq. (3) with i(x) replaced by i(r)]. Note that if ⌬ji ⫽ 0 and ⌳ji ⫽ njni (so ⌸ji ⫽ 0), the reconstruction proposed here yields the Hartree–Fock (HF) case, as expected. 冘 nh ⫺ 冘 ⌳ K . i ii i ji (37) ij ij From the requirement that for any two-electron system the expression (37) should yield Eq. (36), one has to set ⌳ ji ⫽ sign共⌳ji兲 冑njni, 再 ⫺1 ⫽ ⫹1 ⫺1 (33) Using Eqs. (17), (20), (25), (26), (29), (31), and (32), the energy in Eq. (8) reads as i ij 3.2. TWO-ELECTRON SYSTEMS j E⫽2 ji ij (30) i ij ij sign共⌳ji兲 if i ⫽ 1, j ⫽ 1 if i ⫽ 1, j ⱖ 2; if i ⱖ 2, j ⱖ 2 冎 i ⱖ 2, j ⫽ 1 (38) It is worth noting that the chosen ⌬ and ⌳ satisfy constraints (30) and (33). Moreover, the sign rule (38) also holds true for systems in which the largest occupation deviates significantly from one, indicating that it may possibly be valid for arbitrary correlation strengths [6]. 3.3. N-ELECTRON SYSTEMS Expression (38) suggests the following functional form for {⌳ji}: INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1097 PIRIS ⌳ ii ⫽ ⫺ni (39) ⌳ ji ⫽ sign共⌳ji兲 冑njni i ⫽ j, if (40) The so-called D, G, and Q conditions state that the two-electron density matrix, Eq. (10), the electron-hole density matrix G, G kl,ij ⫽ where sign共⌳ji兲 再 ⫹1 ⫽ ⫹1 ⫺1 if i ⱕ nco, j ⱕ nco if i ⱕ nco, j ⬎ nco; if i ⬎ nco, j ⬎ nco 冎 i ⬎ nco, 1 具⌿兩â k† â l â †j â i 兩⌿典, 2 (44) and the two-hole density matrix Q, j ⱕ nco , Q kl,ij ⫽ 1 具⌿兩â k â l â †j â †i 兩⌿典, 2 (45) (41) where the number of HF closed shells is denoted nco. Inserting Eq. (39) into the equality (33), affords ⌬ ii ⫽ n 2i . (42) By taking into account Eqs. (39) and (42), the energy functional (35) can be expressed as E⫽ 冘 n 共2h ⫹ K 兲 ⫹ 2 冘⬘ 共n n ⫺ ⌬ 兲J ⫺ 冘⬘ ⌳ K . i i ii ii j i ji must be positive semidefinite. There are also other two necessary conditions for the N-representability of a fermion two-matrix, which are called B and C conditions. A discussion of relations among these positivity conditions is given in Ref. [22]. Using the anticommutation relations for the creation and annihilation operators and definitions of the one- and two-matrices given in Eqs. (9) and (10), the spin component matrices of G and Q can be derived from the matrices D and ⌫ as follows: ij ij ji ij (43) ␣␣ G kl,ij ⫽ 1 ␣␣ ␦ ⌫ ␣ ⫺ D kj,il 2 lj ki (46) ␣ G kl,ij ⫽ 1 ␣ ␦ ⌫ ␣ ⫺ D kj,il 2 lj ki (47) ij Unfortunately, ⌬ji ⫽ njni, taken from the N ⫽ 2 case, violates the sum rule (30) in the general case of N ⬎ 2. This means that the functional form of nondiagonal elements of ⌬ remains unknown for N-electron systems. Nevertheless, some constraints can be achieved for these quantities, using known necessary conditions of two-matrix N-representability. 3.4. N-REPRESENTABILITY The one-matrix and the functional N-representability problems are completely different. Restriction of the occupation numbers {ni} to the range 0 ⱕ ni ⱕ 1 represents a necessary and sufficient condition for ensemble N-representability of the onematrix [12]. However, the functional N-representability refers to the conditions that guarantee the one-to-one correspondence between E[⌿] and E[⌫], a problem related to the N-representability of the two-matrix. Therefore, any approximation for Vee[{ni, i}] must comply at least with the known necessary conditions for the N-representability of the two-matrix [21]. 1098 ␣␣ Q kl,ij ⫽ 1 共 ␦ ␦ ⫺ ␦ ki ⌫ ␣jl ⫺ ␦ lj ⌫ ␣ik ⫺ ␦ kj ␦ li ⫹ ␦ kj ⌫ ␣il 2 ki lj ␣ Q kl,ij ⫽ ␣␣ ⫹ ␦ li ⌫ jk␣ 兲 ⫹ D ij,kl (48) 1 ␣ 共 ␦ ␦ ⫺ ␦ ki ⌫ jl ⫺ ␦ lj ⌫ ␣ik 兲 ⫹ D ij,kl . 2 ki lj (49) A matrix is positive semidefinite if, and only if, all its eigenvalues are non-negative. The solution of the eigenproblem for D␣␣ is readily carried out, yielding the following set of eigenvalues: d ␣␣ ⫽ 兵0, n j n i ⫺ ⌬ ji 其, j ⫽ i. (50) For more details, the reader can find an analogous derivation in Ref. [15]. D␣ consists of 1 ⫻ 1 blocks, along with a single R ⫻ R block, where R is the number of orbitals. The latter has elements D ␣ ii, jj ⫽ 1 共n n ⫺ ⌬ ji ⫺ ⌳ ji 兲. 2 j i (51) VOL. 106, NO. 5 TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY The 1 ⫻ 1 blocks have elements D␣ ij,ij, for j ⫽ i, and thus yield eigenvalues 1 d ␣ 共n n ⫺ ⌬ ji 兲. ji ⫽ 2 j i 再 ⍀ ji ⫺ ⌬ ji , ⫾ (52) To sum up, we have analytic expressions for all eigenvalues of D, except those arising from the single R ⫻ R block. Consequently, our reconstructive functional satisfies the D condition (d ⱖ 0) if ⌬ji ⱕ njni and the R ⫻ R block of D␣ is positive. Introducing the abbreviation ⍀ji ⫽ 1 ⫺ nj ⫺ ni ⫹ njni, and considering that Q has the same block structure as D, one has the following set of analytic eigenvalues: q ⫽ 0, n j ⫹ n i ⌬ ji ⫺ n j n i ⫹ 4 2 g ␣ ji ⫽ 冎 ⍀ ji ⫺ ⌬ ji , 2 j ⫽ i. (53) Accordingly, if one takes ⌬ji ⱕ ⍀ji, and the R ⫻ R block of Q␣ is positive, the Q condition is fulfilled. It is easy to verify that the D condition is more restrictive than the Q condition for ⌬ji between HF virtual orbitals (occupation numbers are close to zero), whereas for elements between HF occupied levels, the Q condition is predominant. Finally, we consider G. The spin component G␣␣ also contains a single block R ⫻ R, for which the eigenvalues have no analytic expression, and 1 ⫻ 1 blocks. These latter blocks contribute eigenvalues 1 4 冑 共n j ⫺ n i兲 2 ⫹ 4共n jn i ⫺ ⌬ ji ⫺ ⌳ ji兲 2. To ensure that g␣ ji ⱖ 0, the expression (56) gives rise to the inequality ⌬ ji ⱕ n j n i ⫹ n j n i ⫺ ⌳ 2ji . 2⌳ ji ⫺ n j ⫺ n i To obtain the NOF for a system of N electrons, one may attempt to approximate off-diagonal elements of ⌬ considering the sum rule (30), as well as the constraints imposed by the D, G, and Q conditions. However, it is not evident how to approach ⌬ji, for j ⫽ i, in terms of the occupation numbers. Therefore, let’s rewrite the energy term in Eq. (43), which involves ⌬ji as 冘⬘ ⌬ J ⫽ 冘 J * 冘⬘ ⌬ , ji ij i ij i j ⫽ i. (54) From Eq. (54), it is evident that g␣␣ ji ⱖ 0 if ⌬ji ⱖ ni(nj ⫺ 1). This inequality is easy to satisfy on the domain of allowed occupation numbers (nj ⱕ 1), if we consider non-negative ⌬ji. The opposite spin component consists entirely of 1 ⫻ 1 blocks G␣ ii,ii ⫽ 0, and 2 ⫻ 2 blocks 冉 G ␣ ij,ij G ␣ ji,ij 冊 G ␣ ij, ji . G ␣ ji, ji (55) (58) ji j where Ji* denotes the mean value of the Coulomb interactions Jij for a given orbital i taking over all orbitals j ⫽ i. From the property shown in Eq. (30) follows immediately 冘⬘ ⌬ J ⫽ 冘 n 共1 ⫺ n 兲J*. ji ij 1 共n ⫺ n j n i ⫹ ⌬ ji 兲, 2 i (57) 4. A Practical Functional i ij g ␣␣ ji ⫽ (56) i (59) i i Inserting this expression into Eq. (43), one obtains E⫽ 冘 共2n h ⫹ n K 兲 ⫹ 冘⬘ 共2n n J ⫺ ⌳ K 兲 ⫹ 冘 n 共1 ⫺ n 兲共K ⫺ 2J *兲. 2 i i ii ii j i i ij ji ij ij i i ii i (60) i A further simplification of our NOF is accomplished by setting Ji* ⬇ Kii/2, which produces E⫽ After some straightforward algebra, it can be shown that blocks (55) afford the eigenvalues INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 冘 共2n h ⫹ n K 兲 ⫹ 冘⬘ 共2n n J ⫺ ⌳ K 兲. i ii i 2 i ii j i ij ji ij ij (61) 1099 PIRIS TABLE I ______________________________________________________________________________________________ Total energies (Etotal) in Hartrees. Molecule AlCl AlF AlH BCl Be2 BeH2 BeO BeS BF BH C2H2 CH4 Cl2 CO CO2 CS F2 FCl FH H2CO H2N2 H2O H2O2 HBO HCF HCl HCN HCP HNO HOF HPO Li2 LiCl LiF LiH LIOH Mg2 MgO MgS N2 N2O Na2 NaCl NaF NaH NaOH NH3 O3 P2 PH3 1100 HFa CCSD(T)b NOFc B3LYPd ⫺701.446543 ⫺341.405383 ⫺242.438008 ⫺484.105019 ⫺29.122716 ⫺15.766705 ⫺89.406757 ⫺412.103398 ⫺124.101178 ⫺25.119105 ⫺76.820612 ⫺40.201379 ⫺918.908639 ⫺112.736756 ⫺187.631787 ⫺435.302218 ⫺198.669746 ⫺558.816103 ⫺100.009834 ⫺113.867947 ⫺109.997791 ⫺76.022615 ⫺150.769507 ⫺100.166126 ⫺137.753632 ⫺460.064368 ⫺92.875178 ⫺379.105181 ⫺129.783338 ⫺174.730223 ⫺416.121769 ⫺14.865878 ⫺467.006944 ⫺106.933139 ⫺7.981141 ⫺82.908772 ⫺399.187445 ⫺274.322470 ⫺597.077741 ⫺108.941801 ⫺183.675227 ⫺323.681228 ⫺621.397562 ⫺261.300190 ⫺162.372474 ⫺237.278603 ⫺56.194962 ⫺224.242821 ⫺681.421021 ⫺342.452229 ⫺701.645139 ⫺341.639667 ⫺242.506581 ⫺484.329332 ⫺29.224189 ⫺15.829649 ⫺89.648394 ⫺412.284575 ⫺124.351814 ⫺25.206100 ⫺77.106349 ⫺40.388310 ⫺919.198123 ⫺113.032821 ⫺188.113590 ⫺435.569554 ⫺199.045358 ⫺559.143280 ⫺100.198698 ⫺114.202886 ⫺110.357170 ⫺76.228954 ⫺151.166824 ⫺100.442361 ⫺138.060398 ⫺460.221565 ⫺93.181241 ⫺379.377798 ⫺130.149630 ⫺175.113429 ⫺416.441974 ⫺14.896032 ⫺467.157908 ⫺107.126788 ⫺8.008224 ⫺83.121379 ⫺399.256740 ⫺274.592547 ⫺597.256828 ⫺109.261984 ⫺184.203202 ⫺323.707095 ⫺621.546221 ⫺261.491762 ⫺162.400444 ⫺237.488190 ⫺56.399981 ⫺224.867269 ⫺681.667624 ⫺342.605974 ⫺701.723456 ⫺341.660582 ⫺242.525697 ⫺484.376709 ⫺29.200175 ⫺15.821505 ⫺89.609227 ⫺412.310091 ⫺124.346169 ⫺25.193976 ⫺77.069774 ⫺40.363928 ⫺919.345443 ⫺113.020010 ⫺188.134537 ⫺435.601712 ⫺199.109071 ⫺559.244622 ⫺100.178202 ⫺114.208902 ⫺110.370685 ⫺76.207940 ⫺151.202528 ⫺100.419037 ⫺138.086920 ⫺460.258382 ⫺93.145714 ⫺379.389201 ⫺130.185528 ⫺175.161571 ⫺416.509129 ⫺14.887910 ⫺467.192191 ⫺107.100782 ⫺8.000625 ⫺83.095776 ⫺399.282202 ⫺274.561446 ⫺597.302717 ⫺109.241787 ⫺184.233374 ⫺323.717745 ⫺621.586299 ⫺261.473972 ⫺162.396537 ⫺237.469800 ⫺56.378970 ⫺224.992524 ⫺681.742405 ⫺342.635555 ⫺702.682030 ⫺342.331111 ⫺242.981911 ⫺484.974255 ⫺29.343773 ⫺15.917811 ⫺89.898440 ⫺412.894462 ⫺124.656132 ⫺25.288104 ⫺77.327715 ⫺40.523216 ⫺920.341830 ⫺113.306694 ⫺188.577339 ⫺436.204876 ⫺199.495477 ⫺559.937606 ⫺100.425817 ⫺114.500848 ⫺110.639297 ⫺76.417892 ⫺151.537846 ⫺100.711017 ⫺138.398508 ⫺460.797390 ⫺93.421806 ⫺379.992926 ⫺130.467334 ⫺175.524723 ⫺417.135327 ⫺15.013967 ⫺467.792715 ⫺107.416662 ⫺8.082268 ⫺83.382938 ⫺400.156464 ⫺275.212883 ⫺598.249807 ⫺109.520563 ⫺184.656028 ⫺324.586846 ⫺622.556987 ⫺262.156146 ⫺162.852762 ⫺238.113220 ⫺56.556343 ⫺225.400708 ⫺682.683866 ⫺343.142663 (continued) VOL. 106, NO. 5 TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY TABLE I ______________________________________________________________________________________________ (Continued) HFa CCSD(T)b NOFc B3LYPd ⫺395.122081 ⫺398.673332 ⫺290.001640 ⫺291.229906 ⫺363.776369 ⫺686.438897 ⫺547.165010 ⫺395.425821 ⫺398.832641 ⫺290.109244 ⫺291.366504 ⫺364.055966 ⫺686.663532 ⫺547.686081 ⫺395.444920 ⫺398.871007 ⫺290.137664 ⫺291.394429 ⫺364.079292 ⫺686.742597 ⫺547.805959 ⫺396.057582 ⫺399.388424 ⫺290.613330 ⫺291.886230 ⫺364.716360 ⫺687.690102 ⫺548.579504 Molecule PN SH2 SiH2 SiH4 SiO SiS SO2 a Hartree–Fock total energies. CCSD(T) total energies. c Natural orbital functional total energies computed in this work. d B3LYP total energies. b Under these circumstances, the NOF (43) turns out to be identical to the self-interaction-corrected Hartree functional proposed by Goedecker and Umrigar [6], except for the choice of phases given by sign(⌳ji). The functional form (40) for the matrix elements of ⌳ between HF occupied orbitals gives a wrong description of the occupation numbers for the lowest occupied levels. They are identically equal to one. To ensure that these occupation numbers only are close to unity, we assume the form (see, e.g., Ref. [5]): ⌳ ji ⫽ ⌫ ⫽ j J ⫽ i ⫽ j. 冘 C 共r兲. (63) i The electronic energy (61) will then be a functional 冘 再冋2h ⌫ ⫹ 冘 n K ⌫ 册 2 i i ⫹ 冘⬘ [2n n J j j i ij i ⌫ ⫽ C iC i (65) j (66) j K ⫽ 冘 具兩典⌫ j ⌫ . (67) The orthonormality condition (4) reads as follows: 冘C S C i ⫽ ␦ ji , (68) (62) , 冘 具兩典⌫ j Let us apply the well-known procedure of taking molecular orbitals as linear combination of atomic orbitals (MO-LCAO), E关兵C i其, 兵ni其兴 ⫽ i i ⱕ nco, j ⱕ nco, i共r兲 ⫽ i i 冑 n jn i ⫹ 冑 共1 ⫺ n j兲共1 ⫺ n i兲 if 冘 n⌫ i i 冎 j i ⫺ ⌳jiK ]⌫ , (64) where it has been introduced the following matrices: where S ⫽ 具兩典 is the overlap matrix. 5. Methodological Details Direct minimization of the energy functional of Eq. (64) is required, subject to the following constraints: (i) the N-representability condition of the one-matrix (0 ⱕ ni ⱕ 1); (ii) the constant number of particles [Eq. (21)]; and (iii) the orthonormality condition [Eq. (68)]. One has to calculate the gradient of the functional both with respect to natural orbital coefficients {Ci} and the occupation numbers {ni}. Since the minimization with respect to occupations is much less expensive than with respect to the orbitals, one can decouple the variation of the occupation numbers from that of the natural orbitals, a procedure used by us in the improved Bardeen– Cooper–Schrieffer (IBCS) method [20]. In the inner INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1101 PIRIS loop, we find the optimal occupation numbers for a given set of orbitals under constraint 2. Bounds on the occupation numbers are enforced by setting ni ⫽ (sin ␥i)2 and varying the ␥i without constraints. In the outer loop, we minimize with respect to the orbital coefficients under the constraints 3 of mutual orthonormality. Both the inner- and outer-loop optimizations have been implemented using a sequential quadratic programming (SQP) method [23]. 6. Results In this section, calculations of total energies and dipole moments for selected molecules using contracted Gaussian basis sets 6-31G** [24] are presented. Computationally speaking, NOF theory in its current form is very demanding. Therefore, we have chosen a medium-size basis set for the calculations, and we have compared the results with those using other methods at the same level. We are aware of the fact that very large basis sets are required to estimate experimental values. Among the approaches compared are the coupled cluster technique, including all single and double excitations and a perturbational estimate of the connected triple excitations [CCSD(T)], as well as the Becke-3–Lee–Yang–Parr (B3LYP) density functional [25]. The CCSD(T) and B3LYP values were calculated with the Gaussian 94 system of programs [26], using the basis set keyword 5D. Table I presents the values obtained for the total energies of 57 molecules, employing the experimental geometry [27, 28]. For comparison, Table I includes the total energies calculated at the CCSD(T) and B3LYP levels. According to Table I, the values we have obtained are more like CCSD(T) calculations, which are very accurate results for the basisset correlation energies of these small molecules. The B3LYP values, as is well known, tend to be too low. We note that the percentage of the correlation energy obtained by CCSD(T) decreases as the number of electrons increases, whereas our functional keeps giving a slightly larger portion of the correlation energy (e.g., AlCl, SiS, P2, SO2, Cl2). For molecules with dipole moments () different from zero, we have also evaluated this property (Table II). For comparison, Table II includes the available experimental data [29] of and those calculated with the Gaussian 94 system of programs at the CCSD(T) and B3LYP levels. 1102 TABLE II ______________________________________ Dipole moments () in Debyes. Molecule HFa CCSD(T)b B3LYPc NOFd Expe AlCl AlF AlH BCl BeO BeS BF BH CO CS FCl FH H2CO H2O H2O2 HBO HCF HCl HCN HCP HNO HOF HPO LiCl LiF LiH LIOH MgO MgS N2O NaCl NaF NaH NaOH NH3 O3 PH3 PN SH2 SiH2 SiO SiS SO2 1.72 1.06 0.49 0.94 6.91 6.42 0.87 1.48 0.33f 1.26 1.26 1.98 2.75 2.20 1.86 3.11 1.45 1.48 3.23 0.71 2.02 2.24 2.93 7.43 6.20 5.89 4.27 7.81 9.03 0.60f 9.39 8.00 6.84 6.33 1.89 0.78 0.80 2.91 1.38 0.49 3.36 2.51 2.21 1.62 1.12 0.15 1.00 5.33 4.59 0.88 1.03 0.07 1.76 1.02 1.87 2.18 2.09 1.75 2.38 1.23 1.37 2.88 0.65 1.63 1.98 2.24 7.05 5.86 5.58 3.85 5.14 6.38 0.07f 8.97 7.52 6.19 5.85 1.81 0.49 0.80 2.46 1.30 0.31 2.59 1.53 1.78 1.57 0.99 0.28 1.21 5.58 5.14 1.10 1.32 0.10 1.57 0.91 1.82 2.17 2.04 1.72 2.45 1.34 1.43 2.88 0.71 1.61 1.96 2.01 6.91 5.62 5.58 3.65 6.36 7.12 0.01 8.61 7.05 5.96 5.48 1.79 0.62 0.96 2.45 1.41 0.52 2.60 1.74 1.68 1.60 1.09 0.23 1.00 6.16 5.38 0.96 1.28 0.07f 1.47 0.75 1.84 2.36 2.08 1.65 2.70 1.22 1.30 3.01 0.59 1.61 1.85 2.33 7.21 6.01 5.75 4.17 5.21 6.68 0.03 9.03 7.82 6.44 6.11 1.76 0.50 0.60 2.59 1.19 0.23 2.81 1.84 1.56 — 1.53 — — — — — — 0.11 1.98 0.88 1.82 2.33 1.85 2.20 — — 1.08 2.98 0.39 1.67 2.23 — 7.13 6.33 5.88 4.75 — — 0.17 9.0 8.16 — — 1.47 0.53 0.58 2.75 0.97 — 3.10 1.73 1.63 a Hartree–Fock dipole moments. CCSD(T) dipole moments. c B3LYP dipole moments. d Natural orbital functional dipole moments computed in this work. e Experimental dipole moments from Ref. [29]. f This value has an opposite sign relative to the experimental value. b VOL. 106, NO. 5 TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY The dipole moments obtained with the correlated methods are in good agreement with the experimental data considering the basis sets (6-31G**) used for these calculations. For the reported molecules, the correlated dipole moments are lower compared with HF , except for BCl, BF, and CS. For PH3 and SiH2, we have to mention that B3LYP dipole moments are increased with respect to the HF result, whereas the NOF and CCSD(T) values continue to be smaller than it. On the contrary, in the case of the AlF molecule, the B3LYP method decreases with respect to the HF value, whereas the NOF and CCSD(T) yield a higher dipole moment. Important cases are the CO and N2O molecules for which the HF approximation gives a dipole moment in the wrong direction, whereas correlation methods approach it to the experimental value. The quality of natural orbitals is critical to the accuracy of NOF theory. For example, in the case of the CO molecule, we can achieve the sign inversion of the dipole moment ( ⫽ 0.03 Debyes) by adding only one basis function (6-311G**). It is expected that better agreement can be obtained with further improvement of the basis sets. By considering the mean value theorem and the partial sum rule for matrix ⌬, we achieve a practical functional that is close to the self-interaction-corrected GU functional. Despite the previously reported N-representability violations of the GU functional, the cumulant expansion examined in the present work can correct these positivity problems, at the same time satisfying the trace relation of the two-matrix. In contrast, our ansatz (62) reproduces properly the occupation numbers for lower occupied levels. An improvement in our approach requires better approximations for the mean values { Ji*} of the Coulomb interactions. A representative set of 57 molecules is investigated. Comparison with other theoretical methods shows that the presented NOF provides total energies closer to accurate ab initio methods than to DFT energies. The agreement between theory and experiment for is satisfactory, considering the basis sets used. The accuracy of the dipole moments in all calculational approaches in the present study appears to be comparable to each other. In conclusion, the encouraging results obtained in this work demonstrate that our NOF can be used to predict other properties. References 7. Concluding Remarks Making use of the known cumulant expansion, a new reconstructive functional for the two-matrix is proposed. Its explicit antisymmetric form leads to a better description of the pair density for parallelspin electrons with respect to our previous proposal. The dependence obtained in the IBCS method was also considered for the opposite-spin component of the cumulant. The functional given in Eq. (35) can be reduced to the exact energy expression for singlet ground states of two-electron closed-shell systems such as H2 or He. This permits us to generalize its functional form for N-electron systems, except for the off-diagonal elements of a symmetric matrix ⌬. The fact that the N-representability conditions for the ensemble one-matrix are known is not sufficient to ensure the N-representability of the functional, a problem related to the N-representability of the two-matrix. To this end, the well-known necessary D, G, and Q conditions are discussed. The analytic determined eigenvalues provide rigorous bounds on the magnitudes {⌬ji} to guarantee that our reconstructed functional satisfies these positivity conditions. 1. (a) Gilbert, T. L. Phys Rev B 1975, 12, 2111; (b) Donnelly, R. A.; Parr, R. G. J Chem Phys 1978, 69, 4431; (c) Valone, S. M. J Chem Phys 1980, 73, 1344; (d) Valone, S. M. J Chem Phys 1980, 73, 4653; (e) Lieb, E. H. Int J Quantum Chem 1983, 24, 243; (f) Zumbach, G.; Maschke, K. J Chem Phys 1985, 82, 5604. 2. (a) Goedecker, S.; Umrigar, C. J Phys Rev Lett 1998, 81, 866; (b) Buijse, M. A.; Baerends, E. J Mol Phys 2002, 100, 401. 3. Holas, A. Phys Rev A 1999, 59, 3454. 4. Cioslowski, J.; Pernal, K. J Chem Phys 1999, 111, 3396. 5. Csanyi, G.; Arias, T. A. Phys Rev B 2000, 61, 7348. 6. Goedecker, S.; Umrigar, C. J. In Cioslowski, J., Ed.; ManyElectron Densities and Reduced Density Matrices; Kluwer/ Plenum: New York, 2000; p 165. 7. (a) Yasuda, K. Phys Rev A 2001, 63, 32517; (b) Yasuda, K. Phys Rev Lett 2002, 88, 053001. 8. Staroverov, V. N.; Scuseria, G. E. J Chem Phys 2002, 117, 2489. 9. Csanyi, G.; Goedecker, S.; Arias, T. A. Phys Rev A 2002, 65, 032510. 10. Cioslowski, J.; Buchowiecki, M.; Ziesche, P. J Chem Phys 2003, 119, 11570. 11. (a) Piris, M.; Otto, P. Int J Quantum Chem 2003, 94, 317; (b) Piris, M.; Martinez, A.; Otto, P. Int J Quantum Chem 2004, 97, 827; (c) Piris, M.; Otto, P. Int J Quantum Chem 2005, 102, 90; (d) Leiva, P.; Piris, M. J Mol Struct (Theochem) 2005, 719, 63. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1103 PIRIS 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. Coleman, A. J. Rev Mod Phys 1963, 35, 668. Kutzelnigg, W.; Mukherjee, D. J Chem Phys 1999, 110, 2800. Kutzelnigg, W. Theor Chim Acta 1963, 1, 327. Herbert, J. M.; Harriman, J. E. J Chem Phys 2003, 118, 10835. Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry; McGraw-Hill: New York, 1989. Davidson, E. R. Reduced Density Matrices in Quantum Chemistry; Academic Press: New York, 1976. Levy, M. Proc Natl Acad Sci USA 1979, 76, 6062. Levy, M. In Erdahl, R.; Smith, V. H., Jr., Eds.; Density Matrices and Density Functionals; Reidel: Dordrecht, the Netherlands, 1987; p 479. (a) Piris, M.; Cruz, R. Int J Quantum Chem 1995, 53, 353; (b) Piris, M. J Math Chem 1999, 25, 47. Coleman, A. J.; Yukalov, V. I. Reduced Density Matrices: Coulson’s Challenge; Lecture Notes in Chemistry; SpringerVerlag: Berlin, 2000. Kummer, H. Int J Quantum Chem 1977, 12, 1033. Fletcher, R. Practical Methods of Optimization; 2nd Ed.; John Wiley & Sons: New York, 1987. (a) Hariharan, P. C.; Pople, J. A. Theor Chim Acta 1973, 28, 1104 213; (b) Francl, M. M.; Petro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.; DeFrees, D. J.; Pople, J. A. J Chem Phys 1982, 77, 3654. 25. Levine, I. N. Quantum Chemistry, Prentice-Hall: Upper Saddle River, NJ, 2000. 26. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; AlLaham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Cioslowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; Head-Gordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94; Gaussian: Pittsburgh, PA, 1995. 27. Hubert, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure; Van Nostrand-Reinhold: New York, 1979; Vol 14. 28. (a) Bondybey, V. E. Chem Phys Lett 1984, 109, 436; (b) Stewart, J. J. P. J Comp Chem 1989, 10, 221. 29. Nelson, R. D., Jr.; Lide, D. R.; Maryott, A. A. Selected values of electric dipole moments for molecules in the gas phase; NSRDS-NBS 10, 1967. VOL. 106, NO. 5