# From basis sets to force fields : improving methods for high-accuracy quantum chemical calculations of small molecules

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The first section of this work details a force field modeled on VSEPR theory. Previous studies¹ from Bartell et al. have validated the use of the following function to describe repulsion between outer atoms, X [subscript i], bonded to a shared center, A, in binary compounds of the form AX [subscript n]: V=K/r [superscript n above subscript ij]. Here, K and n are parameters and r [subscript ij] is the distance between repelled atoms. Bartell et al. fixed the bond distances A-X [subscript i] so that the atoms X [subscript i] are “points-on-a-sphere” around the central atom A. Our current work extends this POS force field to include flexibility in the bond distances A-X [subscript i]. The functional form for the energy of the bonds is that of the Morse oscillator: V [subscript m, sub-subscript i] = D [subscript e](1 -- exp[ -- α(r [subscript i] -- Re)])². Here, D [subscript e], α, and R [subscript e] are parameters of the force field and r [subscript i] is the distance between atoms A and X [subscript i]. This extended VSEPR force field is applied to PF₅. Parameters were optimized to minimize differences between the VSEPR and MP2/cc-pVTZ PF₅ diagonal quadratic force constants. Quadratic and cubic bending and stretching force constants are presented and compared between Morse-POS and MP2/cc-pVTZ methods. The second section of this dissertation focuses on the analytical transformation of force constants and molecular properties between isotopologues. Within the Born-Oppenheimer approximation, potential energy surfaces of isotopologues are identical. When beginning an exploration of a Born-Oppenheimer potential energy surface, a molecular geometry is chosen and the energy and derivatives are calculated. The most efficient choice of coordinates for these derivatives is normal coordinates, which are mass dependent. This thesis details the transformation of force constants, dipole moments, derivatives of dipole moments, rotational constants, and Coriolis constants from one isotopologue to another. Two alternative systems of coordinates for use in quantum chemical calculations are rectilinear and curvilinear internal coordinates. Rectilinear internal coordinates express the displacement of atoms in a molecule as changes in bond length, bond angle, and dihedral angle as linear combinations of Cartesian coordinates. Though other internal coordinates exist, the previously mentioned are the most commonly used. There are immediate problems with the general use of rectilinear internal coordinates. When one calculates the displacement in Cartesian coordinates needed to increase a bond angle, the bond angle does not scale linearly with the Cartesian coordinates. The following thesis provides the derivation of equations that allow for the analytical transformation from rectilinear to curvilinear coordinates of first through fourth derivatives of the energy. As shown here, these transformations may also be used in converting between the force fields of isotopologues. Additionally, the transformations used between force fields of isotopologues are generalized to the transformation of derivatives of the dipole moment vector. Because the dipole moment vector is rotationally variant, the transformation is extended to include rotational parameters that can successfully transform derivatives of vector quantities between isotopologues. Third, a benchmark comparison of atomic natural orbital (ANO) and Dunning's correlation consistent basis sets is presented²⁻⁴. ANO basis sets, made up of atomic natural orbitals of the CISD wavefunction, are used far less frequently than the correlation consistent basis sets of Dunning and coworkers. However, ANO basis sets, especially smaller truncations (like ANO0) are powerful tools in calculations that require a high-accuracy description of correlation energy as is necessary in second order vibrational perturbation theory (VPT2) calculations. A benchmark study comparing the ANOX (X=0,1,2) and cc-pVNZ (N=D,T,Q) families of basis sets is detailed here. Using VPT2, fundamental frequencies of a set of small molecules were calculated with each of these six basis sets and compared. Of the comparisons, ANO0 outperforms cc-pVDZ in this work, validating more development and interest in ANO-type basis sets. This thesis also details the construction and benchmarking of new ANO-type basis sets using atomic natural orbitals from coupled cluster wavefunctions with single and double excitations (CCSD) and single, double, and triple excitations (CCSDT). These are used to calculate the fundamental frequencies of a set of small molecules and are compared to the original ANO basis sets, which are constructed from the configuration interaction wavefunction with single and double excitations (CISD). The three ANO-type basis sets are comparable in error, though small systematic trends arise that validate the use of the original ANO-type basis sets over those constructed with CCSD or CCSDT wavefunctions. The final section discusses experimental and theoretical collaboration to characterize the rotational and vibrational spectra of isomers of dihydroxycarbene⁵⁻⁶. The first publication discussed here⁵ details measured rotational spectra of isotopomers of cis,trans-dihydroxycarbene. Using ab initio parameters and a fitting procedure, a high-accuracy semi-experimental equilibrium structure of cis,trans-dihydroxycarbene was determined. The semi-experimental equilibrium structure agrees with high-level ab initio calculations to two significant figures in the bond lengths and three significant figure in bond angles. This section also details a recent publication⁶ that uses ab initio calculations to aid in the characterization of vibrational spectra of cis,transand trans,trans-dihydroxycarbene in ⁴He nanodroplets. VPT2 fundamental frequencies are compared to experimental values, all matching to ~7 cm⁻¹. State-specific dipole moments and components of the dipole moment vector along inertial axes determined from ab initio calculations are also compared with experimental results.