Maho Nakata

Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm

The ground-state fermion second-order reduced density matrix (2-RDM) is determined variationally using itself as a basic variable. As necessary conditions of the N-representability, we used the positive semidefiniteness conditions, P, Q, and G conditions that are described in terms of the 2-RDM. The variational calculations are performed by using recently developed semidefinite programming algorithm (SDPA). The calculated energies of various closed- and open-shell atoms and molecules are excellent, overshooting only slightly the full-CI energies. There was no case where convergence was not achieved. The calculated properties also reproduce well the full-CI results.

Density matrix variational theory: Application to the potential energy surfaces and strongly correlated systems

The density matrix variational theory (DMVT) algorithm developed previously [J. Chem. Phys. 114, 8282 (2001)] was utilized for calculations of the potential energy surfaces of molecules, H4, H2O, NH3, BH3, CO, N2, C2, and Be2. The DMVT(PQG), using the P, Q, and G conditions as subsidiary condition, reproduced the full-CI curves very accurately even up to the dissociation limit. The method described well the quasidegenerate states and the strongly correlated systems. On the other hand, the DMVT(PQ) was not satisfactory especially in the dissociation limit and its potential curves were always repulsive. The size consistency of the method was discussed and the G condition was found to be essential for the correct behavior of the potential curve. Further, we also examined the Weinhold–Wilson inequalities for the resultant 2-RDM of DMVT(PQG) calculations. Two linear inequalities were violated when the results were less accurate, suggesting that this inequality may provide a useful N-representability condition ...

Large-scale semidefinite programs in electronic structure calculation

It has been a long-time dream in electronic structure theory in physical chemistry/chemical physics to compute ground state energies of atomic and molecular systems by employing a variational approach in which the two-body reduced density matrix (RDM) is the unknown variable. Realization of the RDM approach has benefited greatly from recent developments in semidefinite programming (SDP). We present the actual state of this new application of SDP as well as the formulation of these SDPs, which can be arbitrarily large. Numerical results using parallel computation on high performance computers are given. The RDM method has several advantages including robustness and provision of high accuracy compared to traditional electronic structure methods, although its computational time and memory consumption are still extremely large.

Latest developments in the SDPA family for solving large-scale SDPs

The main purpose of this chapter is to introduce the latest developments in SDPA and its family. SDPA is designed to solve large-scale SemiDefinite Programs (SDPs) faster and over the course of 15 years of development, it has been expanded into a high-performance-oriented software package. We hope that this introduction to the latest developments of the SDPA Family will be beneficial to readers who wish to understand the inside of state-of-art software packages for solving SDPs.

Direct determination of second-order density matrix using density equation: Open-shell system and excited state

We formulated the density equation theory (DET) using the spin-dependent density matrix (SDM) as a basic variable and calculated the density matrices of the open-shell systems and excited states, as well as those of the closed-shell systems, without any use of the wave function. We calculated the open-shell systems, Be(3S), Be−(2S), B+(3S), B(2S), C2+(3S), C+(2S), N3+(3S), and N2+(2S), and the closed-shell systems, Be, Be2−, B+, B−, C2+, N3+, H2O, and HF. The new properties calculated are the transition energies and the spin densities at the nuclei. Generally speaking, the accuracy of the present results is slightly worse than that of the previous one using the spin-independent density matrix.

Direct determination of the density matrix using the density equation: potential energy curves of HF, CH4, BH3, NH3, and H2O

The density equation DE method was utilized for calculations of the potential energy curves of the molecules HF, CH , 4 BH , NH , and H O. The equilibrium geometries and the vibrational force constants of these molecules were determined by 33 2 the DE method without any use of the wavefunction. The calculated values are in close agreement with the results of the .

Algorithm 925: Parallel Solver for Semidefinite Programming Problem having Sparse Schur Complement Matrix

A SemiDefinite Programming (SDP) problem is one of the most central problems in mathematical optimization. SDP provides an effective computation framework for many research fields. Some applications, however, require solving a large-scale SDP whose size exceeds the capacity of a single processor both in terms of computation time and available memory. SDPARA (SemiDefinite Programming Algorithm paRAllel package) [Yamashita et al. 2003b] was designed to solve such large-scale SDPs. Its parallel performance is outstanding for general SDPs in most cases. However, the parallel implementation is less successful for some sparse SDPs obtained from applications such as Polynomial Optimization Problems (POPs) or Sensor Network Localization (SNL) problems, since this version of SDPARA cannot directly handle sparse Schur Complement Matrices (SCMs). In this article we improve SDPARA by focusing on the sparsity of the SCM and we propose a new parallel implementation using the formula-cost-based distribution along with a replacement of the dense Cholesky factorization. We verify numerically that these features are key to solving SDPs with sparse SCMs more quickly on parallel computing systems. The performance is further enhanced by multithreading and the new SDPARA attains considerable scalability in general. It also finds solutions for extremely large-scale SDPs arising from POPs which cannot be obtained by other solvers.

High-performance general solver for extremely large-scale semidefinite programming problems

Semidefinite programming (SDP) is one of the most important problems among optimization problems at present. It is relevant to a wide range of fields such as combinatorial optimization, structural optimization, control theory, economics, quantum chemistry, sensor network location and data mining. The capability to solve extremely large-scale SDP problems will have a significant effect on the current and future applications of SDP. In 1995, Fujisawa et al. started the SDPA(Semidefinite programming algorithm) Project aimed at solving large-scale SDP problems with high numerical stability and accuracy. SDPA is one of the main codes to solve general SDPs. SDPARA is a parallel version of SDPA on multiple processors with distributed memory, and it replaces two major bottleneck parts (the generation of the Schur complement matrix and its Cholesky factorization) of SDPA by their parallel implementation. In particular, it has been successfully applied to combinatorial optimization and truss topology optimization. The new version of SDPARA (7.5.0-G) on a large-scale supercomputer called TSUBAME 2.0 at the Tokyo Institute of Technology has successfully been used to solve the largest SDP problem (which has over 1.48 million constraints), and created a new world record. Our implementation has also achieved 533 TFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs.

On the size-consistency of the reduced-density-matrix method and the unitary invariant diagonal N-representability conditions

A promising variational approach for determining the ground state energy and its properties is by using the second-order reduced density matrix (2-RDM). However, the leading obstacle with this approach is the N-representability problem. By employing a subset of conditions (typically the P, Q, G, T1 and T2′ conditions) results comparable to those of CCSD(T) can be achieved. However, these conditions do not guarantee size-consistency. In this work, we show that size-consistency can be satisfied if the 2-RDM satisfies the following conditions: (i) the 2-RDM is unitary invariant diagonal N-representable; (ii) the 2-RDM corresponding to each (unspecified) subsystem is the eigenstate of the number of corresponding electrons; and (iii) the 2-RDM satisfies at least one of the P, Q, G, T1 and T2′ conditions. This is the first time that a computationally feasible (though demanding) sufficient condition for the RDM method that guarantees size-consistency in all chemical systems has been published in the literature.

Valence ionization spectra of 4π-electron molecules with low-lying satellites involving n–π* and π–π* transitions

The valence ionization spectra up to 25–30 eV of the 4π-electron molecules, butadiene, acrolein, glyoxal, methylenecyclopropene and methylenecyclopropane were investigated by the SAC-CI method. Accurate theoretical assignments of the spectra were given and further the natures of the low-lying satellites were examined. Acrolein and glyoxal have the low-lying satellites of n−1π-1π* and n-2π* states and the outermost satellites are lower than the π-2π* state of butadiene. However, their intensities are very small, since they do not effectively interact with the main peaks. The π-2π* state of methylenecyclopropene with constrained π-conjugation was calculated to be much higher than that of butadiene, though the first IP is lower. In these spectra, some split peaks were calculated at 15–16 eV and the continuous shake-up states were obtained in the region higher than ∼18 eV. §Dedicated to Professor Michael A. Robb on the occasion of his 60th birthday.