Computational quantum mechanics basis sets

The description of quantum-mechanical processes through a mixed quantum-classical (MQC) formulation has attracted considerable interest for more than seventy years. The reason for this is easy to understand As the numerical effort of a quantum-mechanical basis-set calculation increases exponentially with the number of nonseparable degrees of freedom (DoF), a straightforward quantum computation is restricted to only a few vibrational DoF of a polyatomic system. Classical mechanics, on the other hand. 

Fig. 9. Overtone bands of full Franck-Condon spectrum (inset) and selected bands, computed quantum mechanically (converged basis set calculation) and classically, using the Wigner method.

The configuration interaction calculation with all possible excitations is called a full Cl. The full Cl calculation using an infinitely large basis set will give an exact quantum mechanical result. However, full Cl calculations are very rarely done due to the immense amount of computer power required. 

Atoms defined in this way can be treated as quantum-mechanically distinct systems, and their properties may be computed by integrating over these atomic basins. The resulting properties are well-defined and are based on physical observables. This approach also contrasts with traditional methods for population analysis in that it is independent of calculation method and basis set. 

Computational chemists in the pharmaceutical industry also expanded from their academic upbringing by acquiring an interest in force field methods, QSAR, and statistics. Computational chemists with responsibility to work on pharmaceuticals came to appreciate the fact that it was too limiting to confine one s work to just one approach to a problem. To solve research problems in industry, one had to use the best available technique, and this did not mean going to a larger basis set or a higher level of quantum mechanical theory. It meant using molecular mechanics or QSAR or whatever. 

With quantum-mechanical methods, the second derivatives of the energy could be used directly for the FF and atomic polar tensors (APT) for the dipole derivatives. Both are standardly computed in most quantum-chemistry programs but for accurate results, moderately large basis sets and/or some accommodation for correlation interaction is needed. Until recently, this has restricted most ab initio studies to modest-sized molecules. 

Figure 21. The (So — S2) absorption spectrum of pyrazine for the reduced three-dimensional model using different spawning thresholds. Full line Exact quantum mechanical results. Dashed line Multiple spawning results for — 2.5, 5.0, 10, and 20. (All other computational details are as in Fig. 20.) As the spawning threshold is increased, the number of spawned basis functions decreases, the numerical effort decreases, and the accuracy of the result deteriorates (slowly). In this case, the final size of the basis set (at t — 0.5 ps) varies from 860 for 0 = 2.5 to 285 for 0 = 20.

Ab initio quantum mechanics is based on a rigorous treatment of the Schrodinger equation (or equivalent matrix methods)4-7 which is intellectually satisfying. While there are a number of approximations made, it relies on a set of equations and a few physical constants.8 The use of ab initio methods on large systems is limited if not impossible, even with the fastest computers available. Since the size of an ab initio calculation is defined by the number of basis functions in the system, ab initio calculations are extremely costly for anything past the second row in the periodic table, and for all systems with more than 20 or 30 total atoms. 

The study of the enantioselective hydrosilylation reaction was performed with a series of combined quantum mechanics/molecular mechanics (QM/MM) calculations [26, 30] within the computational scheme of ab initio (AIMD) (Car-Parrinello) [62] molecular dynamics. The AIMD approach has been described in a number of excellent reviews [63-66], AIMD as well as hybrid QM/MM-AIMD calculations [26, 47] were performed with the ab initio molecular dynamics program CPMD [67] based on a pseudopotential framework, a plane wave basis set, and periodic boundary conditions. We have recently developed an interface to the CPMD package in which the coupling with a molecular mechanics force field has been implemented [26, 68],... 

A review of the Journal of Physical Chemistry A, volume 110, issues 6 and 7, reveals that computational chemistry plays a major or supporting role in the majority of papers. Computational tools include use of large Gaussian basis sets and density functional theory, molecular mechanics, and molecular dynamics. There were quantum chemistry studies of complex reaction schemes to create detailed reaction potential energy surfaces/maps, molecular mechanics and molecular dynamics studies of larger chemical systems, and conformational analysis studies. Spectroscopic methods included photoelectron spectroscopy, microwave spectroscopy circular dichroism, IR, UV-vis, EPR, ENDOR, and ENDOR induced EPR. The kinetics papers focused on elucidation of complex mechanisms and potential energy reaction coordinate surfaces. 

When one refers to a quantum mechanical solvent model, the word model reverts to its usual sense in die context of QM methods it is the level of electronic structure theory used to describe die solvent. Thus, diere is no real distinction between the solvent and die solute in terms of computational technology - the wave function for the complete supersystem (or the DFT equivalent) is computed without resort to methodological approximations beyond those inherent to the level of electronic structure theory. To avoid problems with basis-set imbalances, one might expect calculations representing the solvent in a fully QM fashion to employ a common level of theory for all particles, but this does not have to be the case. 

The use of zeolite clusters in quantum chemical calculations has now progressed to quite a sophisticated level. Elementary steps of reaction mechanisms can now be characterized and the results used to distinguish which steps are the most plausible. Computational power is such that clusters and methods can avoid obvious pitfalls (too small a cluster, basis set, etc.). Several key concepts that have arisen from theoretical studies are illustrated in the preceding discussion. These include the following carbo-cations exist as parts of transition state structures, rather than as stable intermediates, and their stabilization is controlled by the zeolite lattice. The transition states are very different from the ground states to either side of them, and each different reaction has been shown to proceed via a different transition state. 

---