Basis convergence

The basis set (Bi) used in the present work is composed of all =l-6 C5+(nlm) states and the H(ls) initial state. In order to check basis convergence, we have performed calculations with the basic augmented by the H(2/w) states (B2), for three impact energies, 0.3, 10 and 300 keV/u. The... 

In the largest basis (cc-pCVQZ), the distribution of errors is very broad for Hartree-Fock but becomes sharply peaked at the CCSD(T) level. The MP2 and CCSD distributions are similar to each other and intermediate between those of Hartree-Fock and CCSD(T). It is important to note, however, that this progression of the n-electron models is not observed in the small cc-pCVDZ basis. Clearly, this basis is not sufficiently flexible for correlated calculations, providing too small virtual excitation spaces for these models to work properly—in particular, for reaction enthalpies. In the cc-pCVTZ basis, convergence is more satisfactory, especially for bond distances. 

Spatial orbitals are typically (but not necessarily) expanded in a basis set. The choice of the latter expansion is somewhat arbitrary, but the quality of the possible choices can be judged by considering completeness of the basis set and how quickly the basis converges to eigenfunctions of the Hamiltonian. Alternatives include plane-wave basis sets. Slater-type orbitals (STO), Gaussian-type orbitals (GTO), and numerical orbitals. 

It is our experience that from (say) a molecular mechanics-derived starting geometry, a trust-region-based second-order optimization will converge in five to seven steps, and only in rare cases will it require as many as nine. From a more accurate starting geometry, such as one obtained from an ab initio calculation in a smaller basis, convergence is typically obtained in two or three steps. It is quite likely that for well-behaved systems, and/or... 

Methfessel M, Rodriguez C O and Andersen O K 1989 Fast full-potential calculations with a converged basis of atom-centered linear muffIn-tIn orbitals structural and dynamic properties of silicon Phys. Rev. B 40 2009-12... 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, hamonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the bajectoiy, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

Abstract. This paper presents results from quantum molecular dynamics Simula tions applied to catalytic reactions, focusing on ethylene polymerization by metallocene catalysts. The entire reaction path could be monitored, showing the full molecular dynamics of the reaction. Detailed information on, e.g., the importance of the so-called agostic interaction could be obtained. Also presented are results of static simulations of the Car-Parrinello type, applied to orthorhombic crystalline polyethylene. These simulations for the first time led to a first principles value for the ultimate Young s modulus of a synthetic polymer with demonstrated basis set convergence, taking into account the full three-dimensional structure of the crystal. 

Secondly, the ultimate properties of polymers are of continuous interest. Ultimate properties are the properties of ideal, defect free, structures. So far, for polymer crystals the ultimate elastic modulus and the ultimate tensile strength have not been calculated at an appropriate level. In particular, convergence as a function of basis set size has not been demonstrated, and most calculations have been applied to a single isolated chain rather than a three-dimensional polymer crystal. Using the Car-Parrinello method, we have been able to achieve basis set convergence for the elastic modulus of a three-dimensional infinite polyethylene crystal. These results will also be fliscussed. 

Fig. 6. The calculated Young s modulus as a function of cut-off energy (basis set size). Convergence is basically reached for a cut-off of 54 Ry.

Regarding mechanical properties of polymers, the efficiency of the Car-Parrinello approach has enabled us to evaluate the ultimate Young s modulus of orthorhombic polyethylene, and demonstrate basis set convergence for that property. 

Truncating this series after the first derivative and integrating provides the basis for the hermodynamic integration approach. Moreover, if the Taylor series expansion is continued intil it converges then Equation (11.45) is equivalent to the thermodynamic perturbation brmula, so providing a link between the two approaches. In practice, it is always necessary... 

Changing the constants in the SCF equations can be done by using a dilferent basis set. Since a particular basis set is often chosen for a desired accuracy and speed, this is not generally the most practical solution to a convergence problem. Plots of results vs. constant values are the bifurcation diagrams that are found in many explanations of chaos theory. 

The ah initio module can run HF, MP2 (single point), and CIS calculations. A number of common basis sets are included. Some results, such as population analysis, are only written to the log file. One test calculation failed to achieve SCF convergence, but no messages indicating that fact were given. Thus, it is advisable to examine the iteration energies in the log file. 

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