N-representable

Harriman (20) has shown that this map is "onto" i.e. any element of comes from at least one element of S. Note that this jH-operty does not rule out the possibility that an element of can also come from operators not in. This "onto" property should be compared to the case that arises in the N-representability problem (28) where not every positive two-particle operator comes from a state in fj so the contraction map in that case does not have the onto property. 

Np The set of p-particle Trace Class operators that are N -representable. 

Note that by generalizing the concept of 7V-representability it can be said that the p - SRH are both N-representable and S-representable. 

However, to ensure that the electron density thus obtained is N-representable by a single determinant of N doubly occupied molecular orbitals it is necessary and sufficient that P be a normalized, hermitian, projector [1], i.e. 

The density functional (DF) method has been successful and quite useful in correlating experimental results when model densities are used in the calculations. In fact, the equations characteristic of the DF method can be derived from a variational approach as Kohn and Sham showed some time ago. In this approach, when model densities are introduced, it is not always possible to relate such densities to corresponding wave functions this is the N-representability problem. Fortunately, for any normalized well behaved density there exists a Slater single determinant this type of density is then N-representable. The problem of approximately N-representable density functional density matrices has been recently discussed by Soirat et al. [118], In spite... 

Soirat, A., Flocco, M. and Massa, L. Approximately N-representable density functionals density matrices, IntJ.Quantum Chem., 49 (1994), 291-298... 

Theorem 2. For a trial density p(r), satisfying the N-representability conditions, the trial ground-state energy Ev satisfies the relation... 

The partial-compliant matrix V of the 0 N, / -representation defines analogous interaction constants for the /-controlled (externally closed) molecules ... 

In the Hohenberg-Kohn formulation, the problem of the functional iV-representability has not been adequately treated, as it has been assumed that the 2-matrix IV-representability condition in density matrix theory only implies an N-representability condition on the one-particle density [21]. Because the latter can be trivially imposed [26, 27], the real problem has been effectively avoided. 

Notice too that in view of the one-to-one correspondence between densities and wavefunctions within an orbit, the variation of (f lpir) with respect to the one-particle density p(r) is equivalent to the variation of [ S, ] with respect to the wavefunction e This equivalence guarantees functional iV-represen-tability and illustrates the basic role of the notion of orbit in this respect. Note also that an approximate functional that does not satisfy Eq. (137) at all points of variation fails to comply with the unavoidable functional N-representability condition. 

The optimal energy for intra-orbit variation is attained at < [pop,(r) y ]. Because the functional N-representability condition is fulfilled, this value is an upper bound to (i.e., to the optimal energy within the Hohenberg-Kohn orbit as W/Hn, C ] =... 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

First consider a two-particle system. In this simplest case, the 2-RDM E is N-representable if it is positive semidefinite and the number of particles is fixed to two. It can easily be cast as the following SDP problem ... 

For the next step, we show how we consider the N-representability conditions for the 1-RDM y for a system with N particles that is all of its eigenvalues should be between zero and one [17]. In other words, this condition is equivalent to saying that y and / — y are positive semidefinite, where / is the identity matrix. Assuming that H is the one-body Hamiltonian, we have... 

The Ai-representability problem was defined in a remarkable paper by Coleman in 1963 [27]. This problem asks about the necessary and sufficient conditions that a matrix represented in a p-electron space must satisfy in order to be N-representable that is, the conditions that must be imposed to ensure that there exists an /-electron wavefunction from which this matrix may be obtained by integration over N-p electron variables. All the relations and properties that will now be described are the basis of a set of important necessary... 

It is true that one must work in a four- or in a three-electron space however, the reward is tantalizing to get an exact, not approximate solution. The difficulty is of course the high computational cost of introducing all the known N-representability conditions. The question whether one could relax the N-representability conditions to be imposed while keeping the procedure convergent is still open. 

Nakatsuji [37] in 1976 first proved that with the assumption of N-representability [3] a 2-RDM and a 4-RDM will satisfy the CSE if and only if they correspond to an A-particle wavefunction that satishes the corresponding Schrodinger equation. Just as the Schrodinger equation describes the relationship between the iV-particle Hamiltonian and its wavefunction (or density matrix D), the CSE connects the two-particle reduced Hamiltonian and the 2-RDM. However, because the CSE depends on not only the 2-RDM but also the 3- and 4-RDMs, it cannot be solved for the 2-RDM without additional constraints. Two additional types of constraints are required (i) formulas for building the 3- and 4-RDMs from the 2-RDM by a process known as reconstruction, and (ii) constraints on the A-representability of the 2-RDM, which are applied in a process known as purification. 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

This is necessary for pure-state n-representability. Together with... 

Any of the four conditions has an infinity of solutions. Actually, the energy is stationary for any eigenstate of the Hamiltonian, so one has to specify in which state one is interested. This will usually be done at the iteration start. Moreover, the stationarity conditions do not discriminate between pure states and ensemble states. The stationarity conditions are even independent of the particle statistics. One must hence explicitly take care that one describes an n-fermion state. The hope that by means of the CSE or one of the other sets of conditions the n-representability problem is automatically circumvented has, unfortunately, been premature. 

As we shall see, the stationarity conditions determine essentially the non-diagonal elements of y and the k, while the diagonal elements are determined by the specification of the considered state and the n-representability. 

We update y and the Ak by means of a unitary transformation and regard the latter as unknown. This has the big advantage that a unitary transformation preserves -representability. So it only matters to have an n-representable start. 

The n-particle density matrix of an w-particle state is pure-state n-representable if—for unit trace—it is idempotent. Since we normalize y as... 

If one could solve Eq. (203) exactly for exact energy— provided that the reference function is n-representable (e.g., is a normalized Slater determinant). The unitary transformation preserves the n-representability. Equation (203) is an infinite-order nonlinear set of equations and not easy to solve. However, the perturbation expansion terminates at any finite order. We have [6,12]... 

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

We may conclude that the 1-RDM and the functional 7/-representability problems are entirely different. The former is trivially solved since ONs sum up to the number of electrons N and lie between 0 and 1, assuring an N-representable 1-RDM. The latter refers to the conditions that guarantee the... 

Minimization of the functional (41) has to be performed under the orthonormality requirement in Eq. (4) for the NSOs, whereas the ONs conform to the N-representability conditions for D. Bounds on the ONs are enforced by setting rii = cos y, and varying y,- without constraints. The other two conditions may easily be taken into account by the method of Lagrange multipliers. 

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