| The expansion of a complicated wavefunction in a basis of simple functions is one of the most common approximations in computational quantum chemistry. The most well-known example of this procedure is the use of Gaussian atomic orbitals for self-consistent field (SCF) wavefunctions, taken here to include Hartree-Fock (HF) and Density Functional Theory (DFT). The accuracy of energies and properties is closely tied to the size and quality of the basis. While correlated-electron energies depend more strongly on basis set size than SCF energies, a HF reference state is the pervasive starting point for nearly all common treatments of electron correlation.;Simply put, the small basis sets, such as the popular 6-31G* basis, are insufficient for accurate quantum chemistry. Unfortunately, extending the basis set beyond this range leads to a host of practical computational challenges. While linearly scaling techniques have been developed to successfully tackle the system-size dependence of SCF calculations, the formal quartic scaling with respect to the number of basis functions introduces an unmanageable prefactor that prevents access to the linear scaling regime. Furthermore, the sparsity necessary for linearly scaling calculations is reduced when spatially extended basis sets with high angular momentum are utilized, the very kind of basis sets required for accurate quantum chemical simulations.;In order to bridge this price/performance gap, a Dual-Basis (DB) ansatz for the SCF energy is developed. A perturbative correction in a large basis set is combined with a smaller basis set reference energy, denoted as (large) ← (small). This correction amounts to an approximate, single Roothaan step in the target basis set and accounts for density relaxation effects to first order. The computational cost of a HF/DFT energy calculation is reduced by roughly the number of SCF cycles in the target basis, typically a factor of ten. Speedups in the analytical gradient are more modest, due the need to solve an orbital response equation. This response equation may be formulated in the small basis set, however, and still leads to factor-of-three savings for geometry optimizations. The small-basis response formulation is also exploited in the bottleneck step of the analytical Hessian. Preliminary results indicate that DB-DFT frequencies are accurate, and a proposed algorithm again indicates possible 10-fold reduction in computational cost.;The resolution-of-the-identity (RI) approximation has drastically reduced the cost of perturbative correlation calculations (MP2), shifting the dominant cost contributor back to the underlying HF calculation. The DB-SCF method is well-suited for this situation and provides fast reference energies, combined with large basis set correlation energies. Bond-breaking energies, for example, are in error by less than 0.08 kcal/mol and orders of magnitude smaller than use of the smaller basis set alone. Analytical derivatives of this DB-MP2 method are presented. Dual-basis subsets for non-covalent interactions are straightforward and highly accurate, and a detailed demonstration on a prototype system, relevant to self-assembled monolayers, is presented.;Of course, even a factor-of-10 reduction in the computational prefactor cannot make all chemical systems computationally tractable with large basis sets. The aforementioned 6-31G* basis still benefits significantly from a DB-SCF scheme, and, when paired with a newly constructed 6-4G minimal basis set, provides a means to include polarization functions in otherwise less accurate calculations for large systems.;Thus, the work contained herein provides a unique means for fast and accurate characterization of the potential energy surface. A critical assessment is provided throughout and in the concluding chapter, along with several potential extensions for further development. |
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