Lower bound method

The numerical efficiency of the viscosity lower bound method has allowed calculations on considerably longer chains. The long chain Hmit results for 12-arm stars without intramolecular interactions and with EV (up to 325 beads) and in the theta region (up to 145 beads) [164] are close to the previous estimates with shorter chains (the extrapolated ratio g obtained in this study is also included in Table 4). The lower bound method has also served to characterize globule-coil transitions of 12-arm star chains from intrinsic viscosity calculations [143], though finite size effects are considerably more important than in the characterization of this transition from the radius of gyration data (see Fig. 17). This is due to the noticeable increase in the solvent permeabiHty associated with the chain expansion in better solvent conditions. However, the permeability effects are smaller in the more compact star chains than in their linear counterparts. 

THE LOWER BOUND METHOD FOR DENSITY MATRICES AND SEMIDEFINITE PROGRAMMING... 

By replacing the wavefunction with a density matrix, the electronic structure problem is reduced in size to that for a two- or three-electron system. Rather than solve the Schrodinger equation to determine the wavefunction, the lower bound method is invoked to determine the density matrix this requires adjusting parameters so that the energy content of the density matrix is minimized. More precisely, the lower bound method requires finding a solution to the energy problem,... 

The promise of the early work on Be and He has recently been confirmed in the work of Nakatsuji and Mazziotti, which started to appear in 2001. This work showed that the lower bound method combined with second-order approximations yields accurate information for atoms and molecules. Nakatsuji and his co-workers [12] did a series of computational experiments where accuracies of between four and five figures were typically achieved. More precisely, they reported the correlation energy as a percentage of the exact correlation energy for a variety of atoms and molecules. They found these percentages ranged between 100% and 110% for atoms and diatomic molecules, and between 110% and 120% for triatomic molecules since these percentages are for lower bounds they never go below 100%. 

The central problem in electronic structure theory is to determine the ground state of a system of electrons, which is typically done variationally by minimizing the energy. The lower bound method can be invoked to achieve a feth-order approximation by replacing the variation minpgq5 (p,/ )g by the semidefinite program... 

The open circles slightly outside TZ in Fig. 2 are third-order estimates of boundary points obtained by the lower bound method and are accurate to three figures. These results are for three spin-up, spin-down pairs of electrons wandering on a ring with six lattice sites. 

It is now firmly established that the lower bound method can be relied on for accurate electronic structure calculations for atoms and molecules. The recent... 

We have focused on the lower bound method, but density matrix research has moved forward on a much broader front than that. In particular, work on the contracted Schrodinger equation played an important role in developments. A more complete picture can be found in Coleman and Yukalov s book [23]. It has taken 55 years and work by many scientists to fulfill Coleman s 1951 claim at Chalk River that except for a few details which would be easily overcome in a couple of weeks—the A-body problem has been reduced to a 2.5-body problem ... 

R. M. Erdahl, Two algorithms for the lower bound method of reduced density matrix theory. Reports Math. Phys. 15, 147-162 (1979). 

---