The phenomenology of an atom in a molecule

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Chapter 2: Theoretical background and developments

Introduction

The primary novel development of this work is the derivation and implementation of the Fragment, Atomic, Localized, Delocalized and Interatomic (FALDI) density decomposition scheme.^1-4 The FALDI scheme, at its current development, has many different aspects related to calculation and decomposition of either deformation densities or static densities. Each of the aspects of FALDI and its applications are described separately, in publication form, in the various chapters of this thesis. This chapter serves as a consolidation of FALDI in its entirety. In addition, FALDI has gone through several developments as each aspect was published, with later developments also serving as an improvement of earlier aspects. This chapter will lay down a logical overview of FALDI in its current form, regardless of the historical timeline of its development.
The FALDI scheme is rooted in the first- and second-order density matrices, domain-averaged exchange-correlation electron holes, electron (de)localization, deformation densities and atoms-in-molecules atomic volumes. Section 2.1 of this chapter serves as a background to the reader, and will cover a brief background of the topics relevant to the discussion of FALDI. Specifically, the topics which will be covered in Section 2.1. are: (i) The electron density, (ii) The pair-density and electron holes, (iii) The Quantum Theory of Atoms in Molecules, (iv) Extended population analysis within QTAIM and (v) Domain Averaged Fermi Holes. Note that foundational quantum chemical theory, and specifically methods for approximating and calculating electronic structures will not be covered in this thesis. For interested readers we recommend Szabo and Ostlund⁵ as very good reference material.
Section 2.2 will cover the derivation, implementation and implications of FALDI itself. Specifically, the topics which will be covered in Section 2.2 are, in the discussed order: (i) the general FALDI decomposition, (ii) FALDI exclusive (de)localization indices and distributions, a classification scheme for identifying FALDI components as bonding, non-bonding and anti-bonding, (iv) the CP(r) function, a tool for investigating the origins of atomic interaction lines, and (v) FALDI deformation densities.

Theoretical background

The electron density

The first-order electron density is a scalar distribution describing the probability of finding any of N electrons with arbitrary spin at a given coordinate in real-space, normalized to N. It arises from the probability interpretation of the square of the wavefunction, and the most general manner in which it can be calculated is by integrating the square of the wavefunction over all spin coordinates and all but one spatial coordinate and then normalizing to N Eq. 1 is useful in general and illustrates the interpretation of the electron density (ED). In general though, the N-electron wavefunction is usually expressed in terms of a number (N) of 1-electron functions known as spin-orbitals (also called molecular orbitals, MO), which themselves are expressed as linear combinations of multiple basis functions. Spin-orbitals can take a variety of forms, from canonical MOs expressing the wavefunction as a single Slater determinant through Kohn-Sham orbitals used in Density Functional Theory (DFT), to MOs expressed as natural orbitals for wavefunctions containing multiple Slater determinants. Using the Müller approximation,⁶ the ED can be expressed in terms of weighted spin-orbitals
The ED is an extremely critical element of quantum chemistry. It provides a spin-independent description of the average electron distribution in a molecule and can therefore be associated with a very large range of chemical phenomena. The ED is comparable to the wavefunction itself in that many physical properties of a specific state are inherently described by the ED alone. In addition, the ED is a quantum mechanical observable and can therefore be experimentally determined through, e.g. X-ray diffraction.

The pair density and electron holes

One of the biggest complexities in quantum chemistry is the particle-wave duality, as it requires that we have to treat all the particles in a chemical system (specifically electrons within the Born-Oppenheimer approximation) as both particles and waves. This requirement excludes the consideration of an independent particle model, which would’ve greatly simplified any chemical modelling, but rather necessitates explicit inclusion of correlation of electron positions and momenta. Inclusion of electron correlation in electron structure calculations is well-documented, and is the raison d’être for the Hartree-Fock (HF) approximation, DFT as well as post-SCF methods. Here, we are specifically interested in exploring how electron correlation affects the distribution of electrons throughout a chemical system.
The ED in Eqs. 1 and 2 is a first-order scalar field, which implicitly rather than explicitly depends on the correlated movement of electrons, i.e. if the wavefunction is correlated, the ED will represent the probability distribution as affected by averaged correlation effects. A correlated wavefunction (regardless of the degree of approximation) depends on all spatial and spin coordinates of all electrons simultaneously, and the position or momentum of a single electron will therefore simultaneously affect the distribution of all other electrons – an effect which is not immediately apparent in the ED.
This pseudo 2^nd-order distribution is known as the electron hole function, or more commonly as the exchange-correlation (XC) hole. The XC-hole describes how the ED is diminished or increased at r1 due to the presence of a reference electron at r2. Since electron correlation usually leads to decreased ED in the vicinity of the reference electron, the XC-hole commonly describes how an electron is excluded at r1 due to the reference electron. In addition, the XC-hole results in exactly –1 when integrated over all space – that is to say, if an electron is known to be at r2, it cannot be found anywhere else and a single electron is entirely excluded from the remainder of the molecule. Wavefunctions built from spin-orbitals generally need to account for two types of electron correlation. The first is Fermi or exchange electron correlation that is a spin-dependent property arising from the antisymmetry of a wavefunction. Specifically, and very importantly, Fermi correlation applies only to electrons of the same spin. Exchange electron correlation is a purely quantum mechanical term, and can best be understood by investigating the PD in a 2 electron, antisymmetric, single determinant wavefunction. We consider first the case of a single determinant (SD) wavefunction containing two electrons of opposite spin:
Where Fermi correlation arises from the spins of electrons, Coulomb correlation arises from the instantaneous electrostatic repulsion between electrons. Spin-orbitals are inherently independent from each other, and as such, the electron-electron repulsion is calculated as an average between all spin-orbitals. However, in reality, electrons (regardless of spin) will experience both greater and weaker electron-electron repulsion than the average as they move around through space. Such instantaneous repulsions will change the PD – the probability of finding two electrons simultaneously at r1 and r2 will be much less than what the average would suggest if r1 and r2 were close together, independent of their spin. HF wavefunctions do not account for this effect at all, and only treat electron-electron repulsion in an averaged manner. The density distributions for HF wavefunctions are therefore entirely independent in terms of Coulomb correlation, whereas post-SCF corrections and multi-determinant wavefunctions account for Coulomb correlation in varying degrees.
very large. Therefore, it is only the exchange-correlation hole, as well as the total exchange-correlation energy, which is sensible to analyse. In general however, the Fermi hole dominates the total XC-hole, and since the Fermi and Coulomb holes integrate to –1 and 0, respectively, the total XC-hole also integrates to –1. As a result, the total XC-hole always reduces the total electron-electron repulsion, and reduces the total molecular energy.
Finally, it is important to note that the XC-hole can be seen as a measure of electron delocalization. Since the XC-hole in Eq. 6 provides the origin of the excluded electron at r1, i.e. ρ(r1) is reduced by the probability of an electron in volume element dr2 being found at r1. Therefore, by keeping r2 constant but varying r1, the XC-hole can give a pseudo-dynamic

The Quantum Theory of Atoms in Molecules

Richard Bader developed the Quantum Theory of Atoms in Molecules (QTAIM)⁷ together with various co-workers over the course of three decades. QTAIM is an example of a theory of an atom-in-a-molecule; while there have been many attempts towards defining atoms in molecules, QTAIM has been proven to be robust, powerful and general. Bader has shown⁷ that the atomic definition within QTAIM is a form of a generalized quantum mechanics of open systems, of which the molecule as a whole is a special case. While we will review a few critical concepts from QTAIM in this section, interested readers can further consult Bader’s book⁷ or a work edited by Matta et al.⁸ QTAIM defines atoms within molecules through the topology of the ED, and QTAIM has opened up the field of Quantum Chemical Topology (QCT).⁹ While the topology of the ED has many subtleties and fine structure, we are principally concerned with critical points (local maxima or minima) within the ED distribution. A critical point (CP) in the electron density at a coordinate rc is a local maximum, minimum or a saddle point where the first derivative (and each of its three components) vanishes

Foreword
Abstract
Outputs of this work
List of Publications
List of conference presentations
Table of Contents
List of Figures
List of Tables
List of Acronyms
Chapter 1. Introduction
The phenomenology of an atom in a molecule
On the nature of chemical bonding
Theoretical descriptions of atoms in molecules
Problem statement
Aims and general approach
Overview of this thesis
A note on the historical development of FALDI
References
Chapter 2. Theoretical Background and Developments.
Theoretical Background
The electron density
The pair density and electron holes
The Quantum Theory of Atoms in Molecules
QTAIM atomic overlap matrices, localization and delocalization indices
Domain Averaged Fermi Holes
Novel Theoretical Developments
The FALDI density decomposition scheme
FALDI localized and delocalized natural density functions
FALDI exclusive localization and delocalization indices
Local bonding, nonbonding and antibonding classification scheme of
FALDI components
FALDI decomposition of the gradient of the total ED
FALDI deformation densities
References
Chapter 3. Exact and Exclusive Electron Localization Indices within QTAIM Atomic Basins
Introduction
Theoretical Basis
Computational Details
Results and Discussion
The H2 molecule
The N2 molecule
The ethene molecule
The ethylene molecule
Formamide
Benzene
Conclusions
Acknowledgment
References
Chapter 4. FALDI‐based decomposition of an atomic interaction line leads to 3D representation of the multicenter nature of interactions
Introduction
Computational Details
Theoretical Background and Development
Domain averaged Fermi holes
The fragment, atom, localized, delocalized and interaction density
decomposition
Partial second derivatives of deloc–ED distributions
Results and Discussion
Intramolecular H-bonding interaction in β-alanine
Multicenter bonding nature of boron-hydrogen interaction in diborane
Typical carbon-carbon covalent bonding interaction in “linear” n-butane
Comparison of two different M–C bonding interactions in carbene
complexes
Comparitive Analysis
Conclusions
Acknowledgment
References
chapter  5. FALDI-Based Criterion for and the Origin of an Electron Density Bridge with an Associated (3,–1) Critical Point on Bader’s Molecular Graph
Introduction
Theoretical Background
The FALDI density decomposition scheme
Classification scheme for ED components
The decomposition of the gradient in bonding, nonbonding and
antibonding terms.
Chapter 6. Toward deformation densities for intramolecular interactions without radical reference states using the fragment, atom, localized, delocalized, and interatomic (FALDI) charge density decomposition scheme
Introduction
Theoretical Development
Framework for conformational deformation densities
DAFH-based density decomposition
General properties of the DAFH
Introducing the FALDI density decomposition scheme
Decomposing atomic-ED distributions into localized, delocalized and
interatomic contributions
Conformational deformation densities using the FALDI-DD
decomposition
Computational Details
Results and Discussion
Total deformation density from orthodox Δρ(r) and FALDI Δ ρc(r)
Atomic FALDI deformation densities
Fragment FALDI deformation densities
Diatomic and Intrafragment interactions from the FALDI perspective
Conclusions
References
Chapter 7. Exploring fundamental differences between red-and blueshifted intramolecular hydrogen bonds using FAMSEC, FALDI, IQA and QTAIM
Introduction
Theoretical Background
Computational Details
Results and Discussion
Exploring the changes of the atomic electron population and bond charge
polarization
FALDI-based atomic deformation densities of the W–X–H⋅⋅⋅Y–Z fragment
FALDI-based interatomic delocalized deformation densities of the X–
H⋅⋅⋅Y fragment
1D FALDI cross-sections of X–H⋅⋅⋅Y interactions
Chapter 8. Conclusions
Summary
Implications
Future Work
References
Appendix
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