N-representability problem

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. 

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... 

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. 

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. 

The efforts by several very able quantum scientists in four countries in the period preceding 1972 had failed to obtain a complete solution of the N-representability problem. It was assumed that we would never find one. My announcement of the solution in June of that year at a Conference in Boulder was therefore greeted with incredulity except by Ernie Davidson who understood my argument immediately. 

Simply to look at the literature is to convince yourself of the importance that density functional theory (DFT) methods have attained in molecular calculations. But there is among the molecular physics community, it seems to me, a widespread sense of unease about their undoubted successes. To many it seems quite indecent that such a cheap and cheerful approach (to employ Peter Atkins s wonderful phrase) should work at all, let alone often work very well indeed. I think that no-one in the com-mimity any longer seriously doubts the Hohenberg-Kohn theo-rem(s) and anxiety about this is not the source of the unease. As Roy reminded us at the last meeting, the N— representability problem is still imsolved. This remains true and, even though the problem seems to be circumvented in DFT, it is done so by making use of a model system. He pointed out that the connection between the model system and the actual system remains obscure and in practice DFT, however successful, still appears to contain empirical elements And I think that is the source of our present unease. 

Elements of second order reduced density matrix of a fermion system are written in geminal basis. Matrix elements are pointed out to be scalar product of special vectors. Based on elementary vector operations inequalities are formulated relating the density matrix elements. While the inequalities are based only on the features of scalar product, not the full information is exploited carried by the vectors D. Recently there are two object of research. The first is theoretical investigation of inequalities, reducibility of the large system of them. Further work may have the chance for reaching deeper insight of the so-called N-representability problem. The second object is a practical one examine the possibility of computational applications, associate conditions above with known methods and conditions for calculating density matrices. 

Undoubtedly, the Hohenberg-Kohn theorem has spurred much activity in density functional theory. In fact, most of the developments in this field are based on its tenets. Nevertheless, the approximate nature of all such developments, renders them functionally" non-jV-representable. This simple means that all approximate methods based on the Hohenberg-Kohn theorem are not in a one to one correspondence with either the Schrodinger equation or with the variational principle from which this equation ensues [21, 22], Thus, the specter of the 2-matrix N-representability problem creeps back in density functional theory. Unfortunately, the immanence of such a problem has not been adequately appreciated. It has been mistakenly assumed that this 2-matrix /V-representability condition in density matrix theory may be translocated into /V-representability conditions on the one-particle density [22], As the latter problem is trivially solved [23, 24], it has been concluded that /V-representability is of no account in the Hohenberg-Kohn-based versions of density functional theory. As discused in detail elsewhere [22], this is far from being the case. Hence, the lack of functional. /V-representability occurring in all these approximate versions, introduces a very serious defect and leads to erroneous results. 

The number of density matrix elements 7" scales with m, i.e. it does not directly depend on n. However, since m should be choosen roughly proportional to n, there is a scaling n. If it were possible to take the 7p rather than the Cl coefficients as variational parameters, we would have got rid of the scaling problem of full Cl. Unfortunately the cannot be regarded as variational parameters, unless one can impose conditions which guarantee that a 7-matrix is derivable from an n-particle wave function. This n-representability problem has played a big role in the late sixties and has been most thoroughly been formulated by Coleman [88, 89]. Unfortunately a simple solution of the n-representability problem which allows one to replace the n-electron wave function by the two-particle density matrix in a variational approach has not been found. Note that 7 is not independent of 7", but can be derived from this by partial contraction... 

We have already introduced the notion of orbit in Sect. 2.6. Here we formalize this concept and show its implications for the functional N-representability problem. 

The historic development has been accounted for in many reviews (see Refs. [9,13]). Important theorems regarding fermionic behaviour was developed by Yang [14], Coleman [15] and Sasaki [16], for a recent review on reduced density matrices and the famous N-representability problem, see Ref. [17]. 

Kummer, H. N-Representability problem for reduced density matrices. 7. Math. Phys. 1967, 8, 2063-2081. 

To be able to calculate two-electron expectation values, knowledge of Y(1 10 is insufficient. To obtain the same simplification as for the one-electron expectation values, the two-matrix, a function of 1, T, 2, and 2, might be introduced, but this is not considered here. The two-matrix contains all necessary information, but unfortunately the two-matrix cannot easily be obtained without first calculating the wave function (the N-representability problem). 

Parameterization of the N-Atom Problem in Quantum Mechanics. II. Coupled-Angular-Momentum Spectral Representations for Four-Atom Systems. 

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. 

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 ... 

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... 

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... 

The importance of N-representability for pair-density functional theory was not fully appreciated probably because most research on pair-density theories has been performed by people from the density functional theory community, and there is no W-representability problem in conventional density functional theory. Perhaps this also explains why most work on the pair density has been performed in the first-quantized spatial representation (p2(xi,X2) = r2(xi,X2 xi,X2)) instead of the second-quantized orbital representation... 

There are two ways to fix this problem. First, one can attempt to derive N-representability conditions for the g-density in the spatial representation. This seems hard to do, although one constraint (basically a special case of the G condition for the density matrix) of this type is known, see Eq. (77). Deriving additional constraints is a priority for future work. 

The preceding analysis is just a transformation of one representation of the n-state problem to another representation. To be useful, the new representation must admit simplifying approximations not suggested by the original representation. One such approximation is to replace the frequency variable 03 in M( ) in (7.10) by a typical P space eigenfrequency, say We thereby obtain the frequency-independent effective operator... 

The use of this expression for a variational determination of T is a complex problem because of the /V-representability requirement [15, 16, 17], Nevertheless, there is a renewed interest in this problem and a number of methods, including so called cumulant-based approximations [18, 19] are being put forth as solutions to the representability problem. Although some advances can be obtained for special cases there appears to be no systematic scheme of approximating the density matrix with a well-defined measure of the N-representability error. Obviously, the variational determination of density matrices that are not guaranteed to correspond to an antisymmetric electronic wavefunction can lead to non-physical results. 

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

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