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David A Mazziotti - One of the best experts on this subject based on the ideXlab platform.
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open shell molecular electronic states from the parametric two electron reduced density matrix method
Journal of Chemical Physics, 2009Co-Authors: Eugene A Deprince, David A MazziottiAbstract:The parametric variational two-electron reduced-density-matrix (2-RDM) method, developed from an analysis of positivity (N-Representability) constraints on the 2-RDM, is extended to treat both closed- and open-shell molecules in singlet, doublet, and triplet spin states. The parametric 2-RDM method can be viewed as using N-Representability conditions to modify the 2-RDM from a configuration interaction singles-doubles wave function to make the energy size extensive while keeping the 2-RDM approximately N-representable [J. Kollmar, Chem. Phys. 125, 084108 (2006); A. E. DePrince and D. A. Mazziotti, Phys. Rev. A 76, 049903 (2007)]. Vertical excitation energies between triplet and singlet states are computed in a polarized valence triple-zeta basis set. In comparison to traditional single-reference wave function methods, the parametric 2-RDM method recovers a larger percentage of the multireference correlation in the singlet excited states, which improves the accuracy of the vertical excitation energies. Fur...
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parametric approach to variational two electron reduced density matrix theory
Physical Review A, 2007Co-Authors: Eugene A Deprince, David A MazziottiAbstract:Two general variational paradigms for computing ground-state energies and properties of molecular quantum systems are (i) the parametrization of the N-particle wave function, as in truncated configuration interaction, which yields an upper bound on the energy in a given basis set and (ii) the constraint of the two-electron reduced-density matrix (2-RDM) by necessary N-Representability conditions (without using the wave function) which yields a lower bound on the energy in a given basis set [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. In this paper we synthesize these two directions in a class of techniques which we call parametric variational 2-RDM methods. The 2-RDM in these methods is parametrized to be size consistent while approximately satisfying the N-Representability conditions. We extend an energy functional of Kollmar [C. Kollmar, J. Chem. Phys. 125, 084108 (2006)], which modifies configuration interaction with double excitations to be size consistent, by including not only double but also single excitations explicitly. Using the 2-RDM parametrization, we calculate ground-state energies at both equilibrium and nonequilibrium geometries in correlation-consistent polarized valance double-zeta (cc-pVDZ) basis sets. Energies as well as properties from the parametric variational 2-RDM method, particularly at nonequilibrium geometries, are better in accuracy than those obtained from more » coupled cluster with single and double excitations. The present work shows clearly that, except in the dissociation of N{sub 2}, the deviation of the 2-RDM from the well-known N-Representability conditions, such as the D, Q, and G conditions, is negligible. Furthermore, calculations with helium atoms demonstrate the size consistency of the method. The computational results on N Representability and size consistency are especially important because they legitimatize the selected parametrization of the 2-RDM. « less
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variational reduced density matrix method using three particle n Representability conditions with application to many electron molecules
Physical Review A, 2006Co-Authors: David A MazziottiAbstract:Molecular two-electron reduced density matrices (2-RDMs) are computed variationally without the many-electron wave function by constraining the 2-RDM with a set of three-particle N-Representability conditions known as three-positivity conditions. These constraints restrict four distinct three-particle probability distributions, which can be defined for any N-particle system, to be nonnegative. The variational calculation of the 2-RDM with full three-positivity conditions is implemented with the first-order semidefinite programming algorithm [D.A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We also derive and implement a generalization of the T{sub 2} Representability condition, which is a subset of the three-positivity conditions. Energies and 2-RDMs are computed for several molecules as well as the nitrogen molecule at both equilibrium and nonequilibrium geometries. The ground-state energies for nitrogen are within 0.0015 a.u. of the values from full configuration interaction at all internuclear distances.
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variational reduced density matrix theory strength of hamiltonian dependent positivity conditions
Chemical Physics Letters, 2004Co-Authors: Gergely Gidofalvi, David A MazziottiAbstract:Abstract Variational reduced-density-matrix (RDM) theory, employing the 2-RDM as the primary variable, has the potential to overcome the scaling limitations of configuration interaction to provide accurate electronic ground-state energies. A significant aspect of variational RDM theory is the inclusion of N-Representability conditions which ensure that the 2-RDM corresponds to the N-particle wavefunction. Recent implementations of the method have mainly considered Hamiltonian-independent positivity conditions. In this Letter, we evaluate the strength of two Hamiltonian-dependent conditions. While one of the conditions is proven inactive, the positivity of the matrix S H j i = 〈 ψ | C ˆ i † [ H ˆ , C ˆ j ] | ψ 〉 provides additional N-Representability conditions that may be beneficial in future RDM calculations.
Eugene A Deprince - One of the best experts on this subject based on the ideXlab platform.
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variational optimization of the two electron reduced density matrix under pure state n Representability conditions
Journal of Chemical Physics, 2016Co-Authors: Eugene A DeprinceAbstract:The direct variational optimization of the ground-state two-electron reduced-density matrix (2-RDM) is typically performed under ensemble N-Representability conditions. Accordingly, variationally obtained 2-RDMs for degenerate ground states may not represent a pure state. When considering only ground-state energetics, the ensemble nature of the 2-RDM is of little consequence. However, the use of ensemble densities within an extended random phase approximation (ERPA) yields astonishingly poor estimates of excitation energies, even for simple atomic systems [H. van Aggelen et al., Comput. Theor. Chem. 1003, 50–54 (2013)]. Here, we outline an approach for the direct variational optimization of ground-state 2-RDMs that satisfy pure-state N-Representability known as generalized Pauli constraints. Within the ERPA, 2-RDMs that satisfy both ensemble conditions and the generalized Pauli constraints yield much more reliable estimates of excitation energies than those that satisfy only ensemble conditions.
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n Representability driven reconstruction of the two electron reduced density matrix for a real time time dependent electronic structure method
Journal of Chemical Physics, 2014Co-Authors: David Jeffcoat, Eugene A DeprinceAbstract:Propagating the equations of motion (EOM) for the one-electron reduced-density matrix (1-RDM) requires knowledge of the corresponding two-electron RDM (2-RDM). We show that the indeterminacy of this expression can be removed through a constrained optimization that resembles the variational optimization of the ground-state 2-RDM subject to a set of known N-Representability conditions. Electronic excitation energies can then be obtained by propagating the EOM for the 1-RDM and following the dipole moment after the system interacts with an oscillating external electric field. For simple systems with well-separated excited states whose symmetry differs from that of the ground state, excitation energies obtained from this method are comparable to those obtained from full configuration interaction computations. Although the optimized 2-RDM satisfies necessary N-Representability conditions, the procedure cannot guarantee a unique mapping from the 1-RDM to the 2-RDM. This deficiency is evident in the mean-field-quality description of transitions to states of the same symmetry as the ground state, as well as in the inability of the method to describe Rabi oscillations.
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open shell molecular electronic states from the parametric two electron reduced density matrix method
Journal of Chemical Physics, 2009Co-Authors: Eugene A Deprince, David A MazziottiAbstract:The parametric variational two-electron reduced-density-matrix (2-RDM) method, developed from an analysis of positivity (N-Representability) constraints on the 2-RDM, is extended to treat both closed- and open-shell molecules in singlet, doublet, and triplet spin states. The parametric 2-RDM method can be viewed as using N-Representability conditions to modify the 2-RDM from a configuration interaction singles-doubles wave function to make the energy size extensive while keeping the 2-RDM approximately N-representable [J. Kollmar, Chem. Phys. 125, 084108 (2006); A. E. DePrince and D. A. Mazziotti, Phys. Rev. A 76, 049903 (2007)]. Vertical excitation energies between triplet and singlet states are computed in a polarized valence triple-zeta basis set. In comparison to traditional single-reference wave function methods, the parametric 2-RDM method recovers a larger percentage of the multireference correlation in the singlet excited states, which improves the accuracy of the vertical excitation energies. Fur...
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parametric approach to variational two electron reduced density matrix theory
Physical Review A, 2007Co-Authors: Eugene A Deprince, David A MazziottiAbstract:Two general variational paradigms for computing ground-state energies and properties of molecular quantum systems are (i) the parametrization of the N-particle wave function, as in truncated configuration interaction, which yields an upper bound on the energy in a given basis set and (ii) the constraint of the two-electron reduced-density matrix (2-RDM) by necessary N-Representability conditions (without using the wave function) which yields a lower bound on the energy in a given basis set [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. In this paper we synthesize these two directions in a class of techniques which we call parametric variational 2-RDM methods. The 2-RDM in these methods is parametrized to be size consistent while approximately satisfying the N-Representability conditions. We extend an energy functional of Kollmar [C. Kollmar, J. Chem. Phys. 125, 084108 (2006)], which modifies configuration interaction with double excitations to be size consistent, by including not only double but also single excitations explicitly. Using the 2-RDM parametrization, we calculate ground-state energies at both equilibrium and nonequilibrium geometries in correlation-consistent polarized valance double-zeta (cc-pVDZ) basis sets. Energies as well as properties from the parametric variational 2-RDM method, particularly at nonequilibrium geometries, are better in accuracy than those obtained from more » coupled cluster with single and double excitations. The present work shows clearly that, except in the dissociation of N{sub 2}, the deviation of the 2-RDM from the well-known N-Representability conditions, such as the D, Q, and G conditions, is negligible. Furthermore, calculations with helium atoms demonstrate the size consistency of the method. The computational results on N Representability and size consistency are especially important because they legitimatize the selected parametrization of the 2-RDM. « less
James D Whitfield - One of the best experts on this subject based on the ideXlab platform.
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solver for the electronic v representation problem of time dependent density functional theory
Journal of Chemical Theory and Computation, 2020Co-Authors: James Brown, Jun Yang, James D WhitfieldAbstract:One route to numerically propagating quantum systems is time-dependent density functional theory (TDDFT). The application of TDDFT to a particular system's time evolution is predicated on V-Representability, which we have analyzed in a previous publication. Here, we describe a newly developed solver for the scalar time-dependent Kohn-Sham potential. We present and interpret the force-balance equation central to our numerical method, describe details of its implementation, and present illustrative numerical results for one- and two-electron systems in both one-dimensional and three-dimensional grids. Innovations of our numerical implementation include the use of preconditioning when inverting the force-balance matrix and an improved propagation method obtaining the Kohn-Sham potential self-consistently at each step of the propagation. A new characterization of V-Representability for one-electron systems is also included, along with possible improvements and future directions.
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explicit solver for the electronic v representation problem
arXiv: Chemical Physics, 2015Co-Authors: James D WhitfieldAbstract:One route to numerically propagating quantum systems is time dependent density functional theory (TDDFT). The application of TDDFT to a particular system's time evolution is predicated on V-Representability which we have analyzed in a previous publication. In this work, we provide new insights concerning lattice V-Representability using an newly developed explicit solver for the time-dependent Kohn-Sham potential which contrast with implicit solvers studied in the past few years. We present and interpret the force-balance equation central to our numerical method, describe details of its implementation, and present illustrative numerical results. A new characterization of V-Representability for one-electron systems is also included. Taken together, the results here open the door to deeper theoretical and numerical investigations of the foundations of TDDFT.
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computational complexity of time dependent density functional theory
New Journal of Physics, 2014Co-Authors: James D Whitfield, Manhong Yung, David G Tempel, Sergio Boixo, Alan AspuruguzikAbstract:Time-dependent density functional theory (TDDFT) is rapidly emerging as a premier method for solving dynamical many-body problems in physics and chemistry. The mathematical foundations of TDDFT are established through the formal existence of a fictitious non-interacting system (known as the Kohn–Sham system), which can reproduce the one-electron reduced probability density of the actual system. We build upon these works and show that on the interior of the domain of existence, the Kohn–Sham system can be efficiently obtained given the time-dependent density. We introduce a V-Representability parameter which diverges at the boundary of the existence domain and serves to quantify the numerical difficulty of constructing the Kohn–Sham potential. For bounded values of V-Representability, we present a polynomial time quantum algorithm to generate the time-dependent Kohn–Sham potential with controllable error bounds.
Werner Kutzelnigg - One of the best experts on this subject based on the ideXlab platform.
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density cumulant functional theory from a unitary transformation n Representability three particle correlation effects and application to o4
Journal of Chemical Physics, 2014Co-Authors: Alexander Yu Sokolov, Henry F Schaefer, Werner KutzelniggAbstract:A new approach to density cumulant functional theory is developed that derives density cumulant N-Representability conditions from an approximate Fock space unitary transformation. We present explicit equations for the third- and fourth-order two-particle cumulant N-Representability, as well as the second-order contributions that depend on the connected three-particle density cumulant. These conditions are used to formulate the ODC-13 method and the non-iterative (λ3) correction that employ an incomplete description of the fourth-order two-particle cumulant N-Representability and the second-order three-particle correlation effects, respectively. We perform an analysis of the ODC-13 N-Representability description for the dissociation of H2 and apply the ODC-13 method and the (λ3) correction to diatomic molecules with multiple bond character and the symmetry-breaking tetraoxygen cation (O4+). For the O4+ molecule, the vibrational frequencies of the ODC-13(λ3) method do not exhibit spatial symmetry breaking and are in a good agreement with the recent infrared photodissociation experiment. We report the O4+ equilibrium structure, harmonic frequencies, and dissociation energy computed using ODC-13(λ3) with a diffuse, core-correlated aug-cc-pCVTZ basis set.A new approach to density cumulant functional theory is developed that derives density cumulant N-Representability conditions from an approximate Fock space unitary transformation. We present explicit equations for the third- and fourth-order two-particle cumulant N-Representability, as well as the second-order contributions that depend on the connected three-particle density cumulant. These conditions are used to formulate the ODC-13 method and the non-iterative (λ3) correction that employ an incomplete description of the fourth-order two-particle cumulant N-Representability and the second-order three-particle correlation effects, respectively. We perform an analysis of the ODC-13 N-Representability description for the dissociation of H2 and apply the ODC-13 method and the (λ3) correction to diatomic molecules with multiple bond character and the symmetry-breaking tetraoxygen cation (O4+). For the O4+ molecule, the vibrational frequencies of the ODC-13(λ3) method do not exhibit spatial symmetry breaking ...
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density cumulant functional theory from a unitary transformation n Representability three particle correlation effects and application to o4
Journal of Chemical Physics, 2014Co-Authors: Alexander Yu Sokolov, Henry F Schaefer, Werner KutzelniggAbstract:A new approach to density cumulant functional theory is developed that derives density cumulant N-Representability conditions from an approximate Fock space unitary transformation. We present explicit equations for the third- and fourth-order two-particle cumulant N-Representability, as well as the second-order contributions that depend on the connected three-particle density cumulant. These conditions are used to formulate the ODC-13 method and the non-iterative (λ3) correction that employ an incomplete description of the fourth-order two-particle cumulant N-Representability and the second-order three-particle correlation effects, respectively. We perform an analysis of the ODC-13 N-Representability description for the dissociation of H2 and apply the ODC-13 method and the (λ3) correction to diatomic molecules with multiple bond character and the symmetry-breaking tetraoxygen cation ( O 4 + ). For the O 4 + molecule, the vibrational frequencies of the ODC-13(λ3) method do not exhibit spatial symmetry breaking and are in a good agreement with the recent infrared photodissociation experiment. We report the O 4 + equilibrium structure, harmonic frequencies, and dissociation energy computed using ODC-13(λ3) with a diffuse, core-correlated aug-cc-pCVTZ basis set.
Juan Pablo Vielma - One of the best experts on this subject based on the ideXlab platform.
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mixed integer convex Representability
Mathematics of Operations Research, 2021Co-Authors: Miles Lubin, Juan Pablo Vielma, Ilias ZadikAbstract:Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible region...
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mixed integer convex Representability
Integer Programming and Combinatorial Optimization, 2017Co-Authors: Miles Lubin, Ilias Zadik, Juan Pablo VielmaAbstract:We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for Representability which we use to prove the first known negative results. Finally, we study Representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.
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mixed integer convex Representability
arXiv: Optimization and Control, 2017Co-Authors: Miles Lubin, Juan Pablo Vielma, Ilias ZadikAbstract:Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first non-Representability results for various non-convex sets such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear Representability by Jeroslow and Lowe, we study the Representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets.
