V-representation

The u and v representations are sometimes distinguished as the Schrodinger and the Heisenberg representation. For stationary operators P, then, the Heisenberg equation of motion is... 

Levy, M., 1979, Universal Variational Functionals of Electron Densities, First Order Density Matrices, and Natural Spin Orbitals and Solution of the v-Representability Problem , Proc. Natl. Acad. Sci. USA, 16, 6062. 

Scheme V. Representation of the catalytic p-type Si photocathode for Ht evolution prepared by derivatizing the surface first with Reagent III followed by deposition of approximately an equimolar amount of Pd(0) by electrochemical deposition. The Auger/depth profile analysis for Pd, Si, C, and O is typical of such interfaces (49) for coverages of approximately 10 8 mol PQ2 /cm2.

However, it is indeed fortunate that the IV-representability problem for the electron density p(r) greatly simplifies itself. In fact, the necessary and sufficient conditions that a given p(r) be /V-representable are actually given by Equation 4.5 above. Nevertheless, question remains Can the single-particle density contain all information about a many-electron system, at least in its ground state An affirmative answer to this question can be given from Kato s cusp condition for a nuclear site in the ground state of any atom, molecule, or solid, viz.,... 

At this point, it is necessary to say a few words about the v-representability of the electron density. An electron density is said to be v-representable if it is associated with the antisymmetric wave function of the ground state, corresponding to an external potential v(r), which may or may not be a Coulomb potential. Not all densities are v-representable. Furthermore, the necessary and sufficient conditions for the v-representability of an electron density are unknown. Fortunately, since the /V-representability (see Section 4.2) of the electron density is a weaker condition than v-representability, one needs to formulate DFT only in terms of /V-representable densities without unduly worrying about v-representability. 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

J. R. Hammond and D. A. Mazziotti, Variational two-electron reduced-density-matrix theory partial 3-positivity conditions for V-representability. Phys. Rev. A 71, 062503 (2005). 

F. Weinhold and E. Bright Wilson, Jr., Reduced density matrices of atoms and molecules. 11. On the V-representability problem. J. Chem. Phys. 47, 2298 (1967). 

Properties of the 2-RDM and the V-Representability Ih-oblem The Matrix Contracting Mapping The Contracted Schrodinger Equation... 

C. Valdemoro, L. M. Tel, and E. Perez-Romero, V-representability problem within the framework of the contracted Schrodinger equation. Phys. Rev. A 61, 032507 (2000). 

G. Gidofalvi and D. A. Mazziotti, Boson correlation energies via variational minimization with the two-particle reduced density matrix exact V-representability conditions for harmonic interactions. Phys. Rev. A 69, 042511 (2004). 

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. 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

The 2-RDM formulation, Eq. (38), allows us to generalize the constrained search to approximately V-representable sets of 2-RDMs. In order to approximate the unknown functional Eee[V, D], we use here a reconstructive functional D[ D] that is, we express the elements D h in terms of the We neglect any explicit dependence of on the NOs themselves because the energy functional already has a strong dependence on the NOs via the one- and two-electron integrals. 

M. Levy, Universal variational functionals of electron-densities, Ist-order density-matrices, and natural spin-orbitals and solution of the v-representability problem. Pmc. Natl. Acad. Sci. U.S.A. 76(12), 6062-6065 (1979). 

Necessary and Sufficient Conditions for V-Representability Linear Inequalities from the Orbital Representation... 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

Since it is obviously impossible to require that Tr[Pg AfFg] > 0 for every choice of Pq, one imposes this constraint only for a few operators. Moreover, because one needs to be able to prove that the operators are positive semidefinite, the operators that are selected for use as constraints are typically much simpler than a molecular Hamiltonian. This is unfortunate, because if one could ensure that Tr[Hg Fg] > Egs Hff) for the Hamiltonian of interest, then the computational procedure would be exact. Future research in V-representability might focus on developing constraints based on molecular considerations. 

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 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). 

Mitchell, T., Steinberg, L., Reid, G., Schooley, P., Jacobs, H., and Kelly, V., Representations for reasoning about digital circuits. Proc. IJCAI-8 (1981). 

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