| The solution of all of quantum chemistry is based on the many-body Schrodinger equation. This solution consists of the eigen-value(the energies typically obtained from ab-initio calculations and eigenfunction the wave-function, where ?(r)??*(r) = ?(r) and is a quantum mechanical observable. It is important that we include the eigen-functions, since they contain so much information which otherwise would be lost and because the charge density ?(r) is an experimental observable. Therefore, in this thesis the basic idea is to understand that we can also consider the eigen-vector part. This observation using the concept that quantum theory of atoms in molecules(QTAIM) can have a scalar as well as vector character. The normal and pervading use of QTAIM currently is to consider only the eigen-values of the Hessian of the total charge density distribution ?(r) and neglect the eigen-vectors. In addition in this thesis, we don't assume that QTAIM is the only atomic partitioning scheme that exists and so we undertake the first ever calculation of a full molecular graph for the Ehrenfest Force partitioning, based on the stress tensor, for a range of chemical environments. Starting from the early topological origins of the scalar aspects of QTAIM we have replaced the use of the nuclear skeleton and Euclidean geometry traditionally used to describe molecular morphology and replaced with the new mathematically rigorous approach. This new scalar approach has yielded a new non-Euclidean geometry; Quantum Geometry to quantify the dimensionality of a given molecular graph, define missing, topologically unstable and forbidden solutions of the Poincare-Hopf sum rules and a new description for atomic site reactivity NNRCP. In addition we explore the solution set of the Poincare-Hopf sum rules where we have degenerate solutions such as occur for non-nuclear attractors(NNAs).For the vector-based character of QTAIM and the stress tensor within the QTAIM partitioning we introduce the response b for bond torsion, applications include the accurate prediction of peptidefolding, the role of the controversial H---H bonding interactions in biphenyl and explaining the fast cis- photo-isomerization of the 11-cis retinal reaction as well as a new measure of symmetry breaking. For the Ehrenfest Force partitioning we introduce a new measure of covalent character based on the alignment of the most facile directions of the Hessian of the Ehrenfest Force partitioning for adjacent covalent and hydrogen bonds in water clusters.Within the QTAIM partitioning scheme we introduced a new scalar measure of site reactivity equivalent to the atomic coordination number based only on the electronic structure. Very goodcorrelation was found for the number nearest neighbor of ring critical points(NNRCPs) positioned on the boundary of the atomic basin of the dopant(Pt) nucleus with the relative zero point energy(ZPE) corrected energies. A weaker condition than the number of associated bond paths for the association of the dopant Pt nucleus with the Au6 Pt molecular graph was found for NNRCP = 0. The need to make more quantitative use of the total electronic charge density distribution ?(r) was demonstrated in this thesis. This is framed in the perspective of the ground breaking early work of Bader and coworkers, along with mathematicians who captured the essential nature of a molecule in a suitably compact form in real space in the form of quantum topology phase diagrams(QTPDs). Quantum topology was then used to create a topology phase diagram for both molecules and solids.The known potential energy surface of Si6Li6 within the QTAIM formalism was explored and expanded to include seventeen new unique topologies. To accommodate the NNAs that exist for isomers of Si6Li6, two types of 3-D quantum topology phase diagram were created to ensure unique solutions of the Poincaré-Hopf relation. The most energetically stable isomer was positioned differently on the QTPD to that of earlier published work for a range of isomer sets. A topologically unstable molecular graph containing a cage critical point(CCP) violating the condition of that normally involves enclosing topological features was found.We presented the QTPD of a total of seventeen all new QTAIM topologies of the cis- and transisomers of the cyclic contryphan-Sm peptide. The resultant QTPD consists of two distinct regions for the cis- and trans-isomers that only overlaped for topologies associated with the lowest energy minima of the cis and trans-isomers. The QTAIM topologies of twenty-nine ‘missing' isomers were determined. A contracted formulation of the QTPD was presented, this formulation included the inter-amino acid bond critical point(BCP) that link together the amino acid units, the disulphide bridge ‘pivot' BCP and side chain bonding interactions. Seven inter-amino acid BCPs that link the amino acid units coincide with the peptide backbone and replace the conventional qualitative approach to reduce the complexity of the peptide. We expanded the theoretical interpretation of ellipticity ? to include the associated eigen-vectors and found higher values of the ellipticity ? were associated with a greater preference to conserve folding states. We quantified previous findings that suggested the S--S disulfide bond is central to the folding behavior of the cyclic contryphan-Sm peptide. In addition, we explained why the cis-isomer was the major form of the cyclic contryphan-Sm peptide.Much of excited state phenomena relates to the redistribution of charge density within a molecule, typically after UV irradiation. Photo-induced phenomena were quantitatively understood in terms of well-developed physics-based descriptions of potential energy surfaces. However, the chemistry treatment of excited state phenomena reduces an entire molecule to a Cartesian geometry and a non-quantitative orbital-based bonding description. Detailed information about each bond in a molecule that could be provided by a quantum mechanics based theory of the charge density is lost because the total charge density, that is the eigen-vector portion of the solution to the Schr?dinger equation, is used in a non-quantitative fashion in visualization tasks. The consequences of this focus on energetics is that it is very difficult to find the molecular mechanisms responsible for excited state processes without the detailed bond-by-bond chemistry and physics information of molecules. The local interpretation that is possible due to the partitioning created in QTAIM enables the creation of a consistent and flexible framework of new bond-by-bond descriptors that led us to create an understanding of the range of novel areas of application of QTAIM.One weakness of QTAIM is that QTAIM properties become degenerate close to the conical intersection(CI) seam. This weakness can be addressed by using a higher resolution atoms-inmolecules partitioning based on the stress tensor. The stress tensor approach enabled a closer approach to the CI before there is degeneracy in the values of the stress tensor partitioning properties.After photo-excitation, molecules may undergo a variety of different processes. The excited states were characterized by a significant shift in electron density, which may lead to ionic or diffuse bonding character. When a molecule absorbs UV light, it is promoted to an electronically excited state and the bond with the hydrogen atom shifts from one electronegative atom to the other. Excess vibrational energy is then redistributed before the molecule decays back to the ground state. The process of photo-excitation also induces the distortion of the molecule and changes in the electronic charge density distribution and this leads to the conical intersection(CI) seam. Therefore our previous work on QTAIM regarding the coupling of normal modes and the electronic charge density shows promise for probing the properties of CI seams. The role of CI has been firmly established in the last two decades, important examples including double bond isomerization of retinal chromophore photo-functional materials like organic photo-protecting additives(TinuvinP), which go through ESIPT. The retinal chromophore is a so-called polyene with several carbon- carbon double bonds. In principle, all those bonds could isomerize in the excited state but only of them do. This isomerization initiates the transmission of the signal that is part of the vision process. QTAIM could help us to understand why a specific double bond is isomerizing and not the others.For some simple systems there were models of the appearance of the CI on the potential energy surface, but we want to generalize these models to more complicated cases including ESIPT and double bond isomerization. QTAIM is important for excited states because prior to our recent QTAIM work there were no such models for the excited state, which were vital for computational photochemistry. In this work we have developed QTAIM analysis including the new stress-tensor partitioning for a variety of excited state phenomena. The stress-tensor atoms-in-molecules partitioning contains fourth order derivatives of the charge density as compared with QTAIM, which only contains second order derivatives. The stress tensor version partitioning has a much more complex topology than that derived from the charge density. Consequently the stress tensor partitioning revealed topological features that were not described within QTAIM.Investigations of the photo-isomerization of the 11-cis retinal photo-isomerization reaction QTAIM and stress tensor framework was undertaken to gain insights into the speed of the 11-cisretinal photo-isomerization reaction. The excited state cis- reactant S1-PSB11 decays through a barrierless minimum energy path(MEP) towards a conical intersection(CI) seam to reach the ground state trans- photo-product S0-PSBT. The QTAIM and stress tensor analysis allowed us to provide an explanation of the isomerization mechanism from the perspective of the topology of the total electronic charge density r(r) for both ground and excited states. Close to the CI seam the stress tensor was found to be more reliable than QTAIM and better at separating out the properties of the excited and ground states. We undertook an investigation of the two reaction coordinates, the two torsion directions, about the C11-C12 bond mid-point and find that the PES was steeper for the ‘clockwise' direction. This suggested a vector-based analysis in addition to the usual and scalar analysis which was then undertaken. The response b of the total electronic charge density to the external bond torsion ? is found to predict a preferred configuration at the CI seam since the torsional bond is freer to rotate in a ‘clockwise' direction. A QTAIM vector interpretation of bond breaking is discovered that is used to follow the isomerization reaction and explain why the 11-cis-retinal photo-isomerization reaction is one of the fastest in nature.We used QTAIM and the stress tensor to undertake a topology focused analysis of the effects of the torsion ? of the central C-C bond of the biphenyl molecule on a bond-by-bond basis using both a scalar and vector-based analysis. Using the local total energy density H(rb) we show that an exchange of chemical character in the biphenyl bonding network leads to favorable conditions for the formation of the controversial H---H bonding interactions for a planar biphenyl geometry. The H---H bonding interactions were found to be topologically unstable for any torsion of the C-C bond away from the planar biphenyl geometry. Conversely, we demonstrate that any value of ? > 0.0° results in an increase in the topological stability of the C nuclei comprising the torsional C-C bond. Evidence of the effect of the H---H bonding interactions on the torsion ? of the central C-C bond of the biphenyl molecule in the form of the QTAIM response ?. Using a vector-based treatment of QTAIM we find an interpretation of the hyper-conjugation effect. The stress tensor is found to be a new tool for use in the detection of symmetry breaking that is found to be relevant for the incommensurate gas to solid phase transition occurring in biphenyl for a value of the torsion reaction coordinate ? ? 5.0°.Eigen-vectors of the electronic stress tensor, which is directly related to the Ehrenfest force, at critical points of the electron density provide insight into the ‘normal electronic modes' that accompany structural dynamics and rearrangements. Eigen-vectors of the Hessian matrix of the electron density emerge as effective approximations to the eigen-vectors of the stress tensor. The stress tensor has been identified as the key quantity whereby the ‘mechanics' of an atom in a molecule can be determined. This thesis provides strong support for conceptual tools based on the stress tensor and can be viewed as building on that foundation to create a stress-tensor and hence Ehrenfest force atoms-in-molecules partitioning as a natural and timely successor to the original QTAIM theory. Historically, atomic basis sets lacked the quality to undertake a partitioning based on higher order derivatives of the charge density, such as the stress tensor, but this is no longer the case with the ability to use large basis sets e.g. the ANO-RCC basis sets. Adapted ANO-RCC basis sets were used to determine twelve molecular graphs of the Ehrenfest force for H2, CH3 NO, CH4, CH2 O, C2H2, C2H4, C3H3 NO, N4H4, H2 O,(H2O)2,(H2O)4 and(H2O)6. The molecular graphs with a mix of various bonding types were chosen to include sigma-, ?- and hydrogenbonding and all four types of topological critical points. We then compared a wide range of BCP properties: trace of the Hessian, eigen-values, ellipticity ?, stiffness, total local energy H(rb) and the eigen-vectors were calculated at the BCPs and compared for the Ehrenfest, QTAIM and stress tensor schemes. QTAIM was found to be the only partitioning scheme that differentiated between the shared- and closed-shell chemical bond types. However, only the results from the Ehrenfest force partitioning, were demonstrated to be physically intuitive. This was demonstrated for the water molecule, the water-dimer and the water clusters(H2O)4 and(H2O)6. In particular, both the stiffness S and the trace of the Hessians of the appropriate quantities of the sigma-bond BCPs for the water clusters were found to depend on the quantum topology dimension of the molecular graph. The behavior of all the stress tensor BCP properties were found to be erratic. This was explained by the ambiguity in the theoretical definition of the stress tensor. As a complementary approach the Ehrenfest force provides a new indicator of the mixed chemical character of the hydrogen-bond BCP, which arises from the collinear donor sigma-bond donating a degree of covalent character to the hydrogen-bond. The indicator is the relative orientation of the shallowest direction of the Ehrenfest potential of the hydrogen-bond BCPs and the associated direction of the collinear sigma-bond BCP. |
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