Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.Nghiên cứu cấu trúc và tính thơm của một số cluster boron bằng phương pháp hóa học lượng tử.
Research introduction
Since F A Cotton first defined "metal atom cluster" in 1964 as a finite group of metal atoms primarily bonded to each other, the study of clusters has significantly expanded This definition has paved the way for extensive experimental and theoretical research, establishing atomic clusters as a vibrant multidisciplinary scientific field.
Cluster science focuses on identifying the stable structures of atom groups and understanding their stability, physicochemical properties, and potential applications The concept of aromaticity has emerged as a significant topic within this field, closely linked to thermodynamic stability Despite being a cornerstone of modern chemistry, aromaticity lacks a precise definition due to various conflicting qualitative and quantitative models The Hückel model, initially designed for planar hydrocarbons, is the most recognized of these models Proper application of the Hückel model requires solving the secular equation for specific structures to ascertain the number of electrons involved However, this requirement is often overlooked, leading to a reliance on the qualitative (4n + 2) electron counting rule, which is applied not only to planar circular molecules but also to non-planar and three-dimensional structures This overuse of the (4n + 2) rule has obscured its fundamental essence and origin.
This dissertation seeks to clarify common misconceptions about aromaticity by developing precise models based on geometric shapes We will introduce the circular disk model, ribbon model, and hollow cylinder model to highlight the variations and commonalities in electron counting rules, particularly when the geometries of the studied species deviate from the typical planar structures of organic hydrocarbons.
Objectives and scope of the research
This research aims to identify the geometrical structures, electronic configurations, and thermodynamic stability of various boron and doped boron clusters By analyzing the distinct geometries of the clusters, we propose aromaticity models to elucidate their stability.
Research scopes: The boron and doped boron clusters surveyed in the dissertation include B2Si3 q and B3Si2 p in different charged states, the neutral and dianionic B70 0/2-
The B12Lin (n = 0 – 14) and B14FeLi2 structures are analyzed using the ribbon model in conjunction with the Hückel model to clarify the characteristics of B2Si3 q and B3Si2 p clusters Stability insights into the quasi-planar isomer of B70 0/2- and the cone-like B12Li4 are effectively explained through the disk model Additionally, the hollow cylinder model aids in understanding the properties of these boron clusters.
Novelty and scientific significance
• This dissertation aims to clarify the need to distinguish the classical Hückel model from the ribbon model and extend the basic concepts of the ribbon model
Benchmark calculations confirm that the density functional TPSSh is effective for optimizing structures with both boron (B) and silicon (Si) atoms Additionally, it is suitable for simulating results in photoelectron spectroscopy and resonant infrared-ultraviolet two-color experiments.
3 ionization spectroscopy, the B3LYP functional provides values closer to experimental data
This dissertation reveals the discovery of a triplet ground state in a quasi-planar B70 cluster, identified through the topological leapfrog principle This specific isomer is anticipated to demonstrate significant thermodynamic stability in its dianion form To analyze the structure and stability of both the neutral and dianionic states of this quasi-planar configuration, a disk model has been utilized, along with a proposed new electron count for circular disk species.
A comprehensive study of lithium-doped boron clusters B12Lin (where n ranges from 1 to 14) has been conducted to explore the growth mechanisms of Li doping in boron clusters, which could have significant applications in hydrogen storage materials and Li-ion batteries The findings indicate that B12Li8 stands out as the most promising candidate for future experimental investigations as a hydrogen storage material.
B12Li4 is a stable cone-shaped cluster similar to B13Li, and a disk-cone model is proposed based on this study
This dissertation emphasizes the importance of differentiating the hollow cylinder model (HCM) from the Hückel model Understanding the HCM is crucial for rationalizing the thermodynamic stability of tubular clusters and predicting the formation of new stable clusters Additionally, the stability of B14FeLi2 is analyzed through the lens of the HCM.
This dissertation presents a unique approach and groundbreaking findings through the development of electron count rules that assess the aromatic character of atomic clusters These rules are derived from comprehensive solutions to wave equations specifically adapted to the geometric structures of the clusters.
DISSERTATION OVERVIEW
Overview of the research
Recent advancements in materials science have propelled cluster science, which investigates atomic clusters composed of a few to several hundred atoms, to significant progress This field has moved beyond theoretical frameworks to practical applications, particularly in understanding catalytic processes where clusters serve as model systems to explore catalyst reactivity and selectivity Notably, atomic clusters form the foundation for single-atom catalytic processes in chemical reactions Tiny clusters like C60 demonstrate quantum confinement effects, enabling them to absorb and emit light at specific wavelengths, making them suitable for photovoltaic applications Additionally, coinage metal clusters exhibit unique luminescence properties, leading to the development of various sensors For instance, gold clusters enhance surface-enhanced Raman spectroscopy, allowing for the sensitive detection of chemicals at extremely low concentrations, including pollutants and biomarkers Furthermore, iron oxide clusters, or superparamagnetic iron oxide nanoparticles (SPIONs), are utilized as contrast agents in magnetic resonance imaging, improving the visibility of tissues and aiding in the diagnosis of diseases such as cancer.
Boron clusters are fascinating atomic structures that present intriguing challenges for researchers due to their electron-deficient nature Their diverse arrangements and unique electronic properties make them a compelling subject of study, particularly because they possess fewer valence electrons compared to other clusters.
Boron atoms uniquely form clusters with unconventional bonding patterns and diverse geometries, such as planar, quasi-planar, icosahedral, cage-like, tubular, and fullerene structures These size-dependent electronic properties of boron clusters present opportunities for investigating novel electronic phenomena and understanding size-dependent effects However, synthesizing and characterizing boron clusters poses challenges, necessitating specialized techniques and precise control over their reactivity and stability Despite these difficulties, the intriguing characteristics of boron clusters position them as a promising research area with potential applications in catalysis, drug delivery, electronics, and energy storage.
[19] Understanding and harnessing the full potential of boron clusters pave the way for advancements in cluster science, and thereby in materials chemistry
The stability of boron clusters is intricately linked to aromaticity, which is explained by various models The Wade-Mingos rule indicates that boranes with (2n+2) skeletal electron pairs are aromatic and thus more stable Additionally, the Hückel and Baird rules further elucidate the stability and electronic structure of boron clusters Moreover, the Mӧbius electron counting rule, typically used in organometallic chemistry, has also been applied to predict bonding electron availability in these clusters.
The stability of atomic clusters is significantly affected by both the number of atoms and their charge state For instance, the B12 cluster, which typically exists in a stable quasi-planar configuration, can transition to a fullerene-like structure when doped with two silicon atoms.
[25] This results in the formation of the stable B12Si2 doped structure, as elucidated
The modeling approach of the cylinder model reveals that the addition of two extra electrons transforms the B12Si2 skeleton into a ribbon-like configuration in the B12Si2 2- dianion A developed ribbon model explains the robustness of the B12Si2 2- cluster, highlighting the crucial role of charge state and atom arrangement in determining the cluster's stability and structural characteristics.
This doctoral study examines the stability of pure and doped boron clusters, providing insights into their geometries through the lens of aromaticity models Notably, the findings demonstrate significant correlations with the Hückel and Baird rules, enhancing our understanding of these clusters' properties.
Objectives of the research
Geometrical and electronic structures of the pure boron and doped boron clusters: including the neutral and dianionic B70, the mixed lithium boron B12Lin with n = 0 –
14, the mixed B2Si3 q and B3Si2 p , and the multiply doped B14FeLi2 boron cluster.
Research content
Aromaticity models, such as the traditional Hückel and Baird rules, alongside newer approaches like the disk, ribbon, and hollow cylinder models, are essential for understanding the chemical properties and parameters associated with aromaticity These models help rationalize the thermodynamic stability of various structures under investigation.
Research methodology
1.4.1 Search for lower-lying isomers
The stochastic search algorithm [27, 28] combining a random kick [29] and a genetic algorithm (GA) [30, 31] (cf Figure 1.1) is used to generate a massive
The initial geometries of the studied cluster are optimized using density functional theory with the TPSSh functional and the dp-polarization 6-31G(d) basis set, excluding harmonic vibrational frequency calculations Structures with relative energies within approximately 2 eV of the lowest isomer energy are then re-optimized with the same TPSSh functional, utilizing a larger 6-311+G(d) basis set.
[34, 35], and following by harmonic vibrational frequency calculations to ensure that the found structures are true minima Structure optimization and vibrational computations are performed employing the Gaussian 16 program [36]
Figure 1.1 Illustration of a genetic algorithm (GA) procedure ([31])
1.4.2 ELF – The electron localization function
The electron localization function (ELF) is a valuable tool for analyzing the chemical bonding phenomena in clusters, particularly in the context of topological bifurcation of electron density ELF serves as a local measure of the Pauli repulsion between electrons, reflecting the exclusion principle in three-dimensional space The mathematical definition of ELF, denoted as 𝜂(𝐫), is expressed through a specific equation.
The local kinetic energy densities due to the Pauli exclusion principle (𝐷 P) and Thomas–Fermi (𝐷 h), along with electron density (𝜌), can be evaluated using Hartree–Fock or Kohn–Sham orbitals The total electron localization function (ELF) can be divided into components for σ and π electrons (ELFσ and ELFπ), allowing for a detailed assessment of aromatic character Aromatic species exhibit high bifurcation values for ELFσ or ELFπ, while antiaromatic systems show low values In certain structures, σ localized electrons can be distinguished from delocalized electrons, enabling the calculation of ELFσl and ELFσd for a clearer understanding of electron localization The total and partial ELF calculations are performed using Dgrid-5.0 software, with isosurfaces visualized through Gopenmol or ChimeraX software.
The ring current methodology is an effective approach for analyzing and comprehending the aromatic character of molecules This study employs the SYSMOIC program, which utilizes the CTOCD-DZ2 method to calculate and visualize magnetically induced current density SYSMOIC computes the current density tensor for restricted Hartree-Fock (HF) and density functional theory (DFT) singlet wavefunctions.
In the orbital approximation for a closed-shell ground state, the n-electron wavefunction is a Slater determinant of n/2 doubly occupied spatial orbitals, chosen to be real: Ψ(1,2, ⋯ , 𝑛) = 1
√𝑛!det[𝜓 1 𝛼, 𝜓 1 𝛽, 𝜓 2 𝛼, 𝜓 2 𝛽, ⋯ , 𝜓 𝑛/2 𝛼, 𝜓 𝑛/2 𝛽] (1.2) and the current density tensor is a sum of orbital contributions:
The anisotropy of the induced current density (ACID) is a valuable method for assessing the aromaticity of a species, akin to the ring current analysis in SYSMOIC This mapping technique reveals aromaticity through clockwise ring currents, while indicating antiaromaticity when electron flow occurs in opposing directions.
Figure 1.2 The current density maps of π electron contribution of a) C4H4 and b)
C6H6 plotted by both SYSMOIC and ACID packages
Figure 1.2 illustrates the π electron contributions to the current density maps of cyclobutadiene and benzene, generated using the SYSMOIC and ACID packages The analysis reveals the presence of counter-clockwise ring currents (paratropic) in these compounds.
C4H4 and the clockwise ring currents (diatropic) of C6H6 indicate that while C4H4 is a π-antiaromatic species, C6H6 is a π-aromatic species
1.4.4 Bond order and net atomic charge
The net atomic charge (NAC) and bond order for each cluster are determined using the DDEC6 atomic population analysis, a density partitioning method that accurately computes NACs and bond orders based on electron and spin density distributions Benchmarking computations have shown that the DDEC6 package offers a remarkable combination of high precision and efficient resource utilization compared to other methodologies Additionally, the open-source nature of DDEC6 provides the advantage of unrestricted accessibility for users.
THEORETICAL BACKGROUNDS AND
Theoretical backgrounds of computational quantum chemistry
The Schrửdinger equation is a fundamental equation of quantum mechanics, describing the status of a quantum system, and it is represented as:
Here, 𝐻̂ is the Hamiltonian operator, 𝜓 is the wave function, and 𝐸 is the energy Equation (2.1) provides two important quantities: the eigenvalue of energy 𝐸 and the eigenvector 𝜓
The Schrửdinger equation for a system with multiple electrons is written as:
In a many-electron system with M nuclei and N electrons, the Hamiltonian operator (H) encompasses various energy components: the kinetic energy of the nuclei (𝑇̂ 𝑛), the kinetic energy of the electrons (𝑇̂ 𝑒), the potential energy of interactions between nuclei (𝑉̂ 𝑛𝑛), the potential energy of interactions between nuclei and electrons (𝑉̂ 𝑛𝑒), and the potential energy of interactions among electrons (𝑉̂ 𝑒𝑒) This comprehensive representation captures the complex interactions within the system.
Solving the Schrödinger equation for many-electron systems is a challenging task due to its complexity To simplify this process, the Born-Oppenheimer model was introduced, which separates the motion of nuclei from that of electrons This separation is feasible because electrons, having a much smaller mass, move significantly faster than nuclei, allowing for a more manageable approach to understanding their interactions.
In a stationary state concerning electron motion, the kinetic energy of the nuclei is zero, and the potential energy of interaction between the nuclei becomes a constant Consequently, the Hamiltonian operator can be expressed in a simplified form.
The Born-Oppenheimer approximation allows the wave function of a system to be expressed as a product of two components, with the first component representing the motion of electrons surrounding stationary nuclei.
𝜓 𝑒 (𝑟⃗, 𝑅⃗⃗), and the second component describes the motion of the nuclei 𝜓 𝑛 (𝑅⃗⃗):
The Born-Oppenheimer approximation allows for the separation of nuclear and electronic motion, resulting in a Schrödinger equation that specifically describes the behavior of electrons within a system.
The total energy of the system is the sum of the electronic energy and the potential energy of the nuclei:
In the Schrödinger equation, electronic energy serves as potential energy for nuclear motion As nuclei move during a chemical transformation, electrons create a potential energy surface that reflects corresponding electronic energies This surface is fundamental to quantum chemical methods in molecular studies.
13 geometries and spectroscopic properties, as well as reactivities and kinetics of the system considered
The Born-Oppenheimer approximation posits that nuclei are stationary, but in reality, they undergo vibrations and motions around their equilibrium positions within a molecule Besides the electronic and nuclear repulsion energies, a molecule also contains energy derived from its vibrational and rotational motions Therefore, the total energy of a molecule encompasses these various energy contributions.
In statistical mechanics, the energies associated with vibrational and rotational motions are typically estimated using simplified models, like harmonic oscillators for vibrations and rigid rotors for rotations, due to their significantly lower energy values compared to electron energies and nuclear repulsion energies.
Integrating the Schrödinger equation is performed analytically, utilizing the Hamiltonian operator and molecular wave function derived from fundamental quantum equations The ab initio computational method is recognized as the most advanced approach available today However, when dealing with multi-electron systems, the extensive number of required integrals demands significant memory and computational speed Established ab initio methods based on wave functions, such as Hartree-Fock and Roothaan, do not account for correlation energy In contrast, methods that partially address correlation energy include perturbation theory, particularly the widely used Møller-Plesset perturbation (MPn), as well as Configuration Interaction (CI), Multi-configuration Self-Consistent Field (MCSCF), Multi-reference Configuration Interaction (MRCI), and Coupled-Cluster (CC) methods.
The Hartree-Fock (HF) method is a key technique in quantum chemistry for analyzing the electronic structure of atoms and molecules Originating from the work of Douglas Hartree and Vladimir Fock in the 1920s and 1930s, the HF method serves as a foundational tool for more sophisticated computational approaches that account for electron correlation effects.
The Hartree–Fock method makes five major simplifications in order to deal with this task:
The Hartree–Fock method inherently relies on the Born–Oppenheimer approximation, which distinguishes between electronic and nuclear motions in molecules This approximation allows for the electronic motion to be treated quantum mechanically while addressing nuclear motion either classically or quantum mechanically As a result, the complete molecular wave function is represented as a function of the electrons' coordinates, calculated at fixed nuclear positions.
The Hartree–Fock method typically overlooks relativistic effects, treating the momentum operator as entirely non-relativistic This approach disregards important relativistic corrections to electron kinetic energy and other related phenomena Although relativistic Hartree–Fock methods are available, the conventional practice remains non-relativistic.
The Hartree–Fock method employs a variational solution to calculate electronic energy by approximating the wave function as a linear combination of a finite number of orthogonal basis functions While this finite basis set simplifies calculations, it is an approximation of a complete basis set, which can impact the accuracy of the results.
In the Hartree-Fock method, each energy eigenfunction is represented by a single Slater determinant, which is an antisymmetrized product of one-electron wave functions This approach ensures the proper treatment of electron exchange and correlation, fundamental aspects in quantum mechanics.
15 or orbitals, accounting for the exchange symmetry of identical particles This assumption simplifies the calculations but neglects the effects of electron correlation beyond mean-field approximation
The Hartree-Fock (HF) method utilizes the mean-field approximation, which overlooks the effects of electron correlation, including both Coulomb and Fermi correlations While the HF method effectively captures electron exchange, it fails to account for Coulomb correlation, such as London dispersion forces As a result, the HF method cannot fully represent dispersion interactions.
Aromaticity models in boron clusters
Aromaticity models play a crucial role in cluster science by offering valuable insights into the stability, magnetic properties, and other characteristics of molecules As of 2017, a total of 45 aromaticity models have been proposed This dissertation will focus solely on the models that have been utilized in the research.
2.2.1 The Hückel and Baird rules
The concept of aromaticity originated as a way to explain the unique stability of benzene (C6H6), a six-carbon ring that serves as the simplest and foundational hydrocarbon for many significant aromatic compounds Discovered by Faraday, benzene's distinctive properties have led to its importance in organic chemistry.
Since 1825, various chemical models have been created to explain the unique properties of benzene, with the Hückel model for aromaticity ultimately emerging as the most effective and widely adopted among chemists.
The Hückel theory describes the structure of cyclic CnHn annulenes, which are composed of n carbon atoms, by dividing them into two distinct groups The first group includes n sp²-hybridized carbon atoms that create a σ-framework consisting of C–C and C–H bonds.
The π system in a molecule comprises electrons that move within the effective field created by a rigid σ-structure, where all carbon atoms are equivalent and the electrons act as independent particles The molecular orbitals (MOs) of this π-system are formed through a linear combination of unhybridized p-atomic orbitals (AOs) When the molecular plane (xy) establishes the σ-framework of the annulene, the π-system is characterized by n unhybridized p z AOs, which combine to generate an ensemble of π-MOs.
(2.12) where 𝜙 𝑖 is p z AO of atom i, 𝑐 𝑖 the contributing coefficient The wave function (2.12) is now substituted in the Schrửdinger equation:
The expectation value of the Hamiltonian operator gives the energy of the system:
We now substitute wave function (2.12) into equation (2.15) and get the energy:
Let us suppose that we use real AOs with the real coefficients, the equation (2.16) becomes:
We define the Hamiltonian matrix elements (𝐻 𝑖𝑗 ) and the overlap integrals (𝑆 𝑖𝑗 ) as:
𝑆 𝑖𝑗 = 𝑆 𝑗𝑖 = ∫ 𝜙 𝑗 𝜙 𝑖 𝑑𝜏 (2.19) The energy expression in terms of these matrix elements now becomes:
The variational principle states that the optimal wavefunction approximation is achieved by minimizing the system's energy To accomplish this, we must minimize the energy E concerning the coefficients ci This leads us to express equation (2.20) in a more manageable form.
Taking the partial derivative of the above with respect to the coefficients 𝑐 𝑖 and using product rule on the left-hand side, we have:
Note that the derivative of a double summation returns to a single summation One can imagine this by thinking about the term-wise differentiation in the double summation
𝜕𝑐 𝑖 = 0 in the above equation to obtain the coefficients with which the energy of the system is minimized Thus, equation (2.22) becomes:
𝐻 𝑖𝑗 (2.23) which can be equivalently written as:
𝑐 𝑗 = 0 (2.24) or, in the matrix form
For a simplification, the matrix elements in the secular equations can be written in terms of parameters 𝛼 and 𝛽 where:
Equation (2.25), when written out in full, now has the form:
To obtain non-trivial solutions of linear combinations of atomic orbitals of a system, we set the secular determinants to zero, viz.,
The process yields N eigenvalue solutions (E), which can be substituted into equation (2.15) to derive the corresponding coefficients (eigenvectors) for the linear combination of atomic orbitals (LCAO) related to the energy These coefficients represent negative parameters that approximate the energy of orbital i and the interaction energy between adjacent orbitals i and j.
To simplify further the solutions, the Hückel approximations assumes that:
(1) the overlap between orbitals is neglected, 𝑆 𝑖𝑗 = 0,
(2) the atomic orbitals are normalized, 𝑆 𝑖𝑗 = 1, and
(3) only adjacent orbitals have interactions, 𝐻 𝑖𝑗 ≠ 0 only if i and j are adjacent to each other
Equation (2.29) when written out in full, now has the form:
The secular determinant for benzene is now available:
If both sides of (2.31) are divided by 𝛽 6 and a new variable 𝑥 is defined as:
𝛽 (2.32) the secular determinant for benzene becomes:
= 0 (2.33) with the six roots 𝑥 = ±2, ±1, ±1 This corresponds to the following energies (ordered from the most stable to the least since 𝛽 < 0 ): o 𝐸 1 = 𝛼 + 2𝛽 o 𝐸 2 = 𝛼 + 𝛽 o 𝐸 3 = 𝛼 + 𝛽 o 𝐸 4 = 𝛼 − 𝛽 o 𝐸 5 = 𝛼 − 𝛽 o 𝐸 6 = 𝛼 − 2𝛽
The two pairs of 𝐸 = 𝛼 ± 𝛽 energy levels are two-fold degenerate (Figure 2.1):
According to Hückel theory, the π molecular orbitals of benzene are depicted in Figure 2.1, where the dashed line indicates the energy level of an isolated p orbital Orbitals situated below this line are classified as bonding, while those above it are identified as antibonding.
In benzene, each carbon atom contributes one electron to the π-bonding framework, resulting in all bonding molecular orbitals being fully occupied Consequently, benzene exhibits an electron configuration of π1 2 π2 4.
Figure 2.2 MO energy diagrams of C4H4 (in both singlet and triplet states), C6H6,
C8H8 (in both singlet and triplet states), and C10H8 The blue/red labels indicate the aromatic/antiaromatic species
The MO energy diagrams for hydrocarbon compounds with 4, 6, 8, and 10-membered rings illustrate key differences in electron configurations Benzene, containing 6 π-electrons, and naphthalene, with 10 π-electrons, adhere to Hückel's rule of aromaticity (4n + 2), where n equals 1 for benzene and 2 for naphthalene Both compounds are recognized as aromatic species, confirming their stability and unique chemical properties.
The singlet states of both C4H4 and C8H8 have two π-electrons filling one
The two-fold degeneracy of MO structures is disrupted by the Jahn-Teller effect, resulting in reduced symmetry and stabilized configurations The singlet isomers C4H4 and C8H8, containing 4 and 8 π-electrons respectively, adhere to the electronic counting rule 4n, classifying them as antiaromatic species.
The triplet ground state of the planar C5H5 + cation which owns 4 π electrons
[86, 87] supports the Baird rule of a 4n triplet aromatic [22] The Baird rule turns the triplet states of C4H4 and C8H8 into aromatic species
The Hückel rule states that a cyclic, planar, and fully conjugated molecule is considered aromatic if it contains (4n + 2) π electrons within a closed-shell system Conversely, a structure with 4n π electrons in a closed-shell configuration is classified as antiaromatic, resulting in significant instability.
The Baird rule is essential for analyzing cyclic, planar, and fully conjugated systems in their triplet state According to this rule, open-shell systems with 4n π electrons exhibit aromatic characteristics, while those with (4n + 2) π electrons are identified as antiaromatic.
The Hückel and Baird rules are essential for predicting the chemical and physical properties of both organic and inorganic compounds, including the electrical conductivity of conducting polymers and the magnetic properties of organic compounds Initially focused on π electrons in planar cyclic hydrocarbons, these rules have since been widely applied to various two-dimensional and three-dimensional structures, facilitating research into new organic compounds with enhanced electrical and magnetic characteristics.
31 only for π electron systems but also for σ and δ electron systems mostly in atomic clusters [95–97]
The ribbon structure of boron derivatives has attracted much interest in the last decade because of its exceptional stability A double chain of boron atoms in the
The B22H2 2-dianion exhibits an elongation of up to 17.0 Å, with a minimal interchain distance of approximately 1.5 Å Research on ribbon structures has revealed that those with exceptional stability typically have electron configurations that adhere to π 2(n+1) σ 2n An illustration of the assignment of π and σ electrons can be found in Figure 2.4.
Figure 2.3 Calculated curves as a function of size n for (a) adiabatic detachment energies of Li2BnH2 - (n = 6–22) ribbon clusters, and (b) Ionization energies of
RESULTS AND DISCUSSION
The Hückel rule and the ribbon model: The cases of B 2 Si 3 q and
The main content in this section is taken from the published paper entitled "Boron Silicon B2Si3 q and B3Si2 p Clusters: The Smallest Aromatic Ribbons", by Long Van
Duong, Nguyen Ngoc Tri, Nguyen Phi Hung, and Minh Tho Nguyen, in the Journal of Physical Chemistry A, vol 126, no 20, pp 3101–3109, May 2022
The selection of appropriate functionals and basis sets for studying specific chemical properties often exceeds the knowledge of practical computational quantum researchers To address this, statistical benchmark studies have proven effective, creating a database that aids scientists in developing accessible methods for enhancing theoretical approaches A robust benchmark should start with small structures and, when possible, include experimental data to ensure reliable conclusions In this context, research on boron and boron-doped clusters revealed a discrepancy with results reported by Lu and colleagues regarding the B2Si3 isomers, as illustrated in Figure 3.1, where a previously missed isomer was properly identified.
51 emerges as the global minimum, whereas the reported isomer is higher than the missing isomer by ~2.0 kcal/mol in relative energy
Figure 3.1 Photoelectron spectra of B2Si3 - clusters recorded with 266 nm photons
The IR-UV2CI spectrum of B2Si3 is compared with calculated IR absorption spectra for low-energy structures 3.2a-e The relaxed isomer 3.2.a was obtained through the CCSD method and various DFT functionals.
The neutral state resonant infrared-ultraviolet two-color ionization (IR-UV2CI) spectroscopy of B2Si3, as detailed by Truong et al., provides a valuable benchmark for calculations A benchmarking survey by Koukaras indicates that the most stable isomer, 3.2.a, identified by Truong, exhibits a perfect planar structure through CCSD computations and specific DFT functionals In contrast, other DFT functionals suggest a quasi-planar structure, highlighting the variability in computational predictions.
The isomer B2Si3.a, illustrated in Figure 3.2.a, possesses 2 π electrons and 2 σ delocalized electrons, indicating its potential for double aromaticity according to Hückel's (4n + 2) rule with n = 0 In contrast, another isomer, B2Si3.b, presents distinct characteristics that warrant further exploration.
Figure 3.3) with higher symmetry, has the same electron configuration as the
Replacing a silicon atom with a boron atom in the B3Si2 structure yields a similar electron configuration Various functional methods can optimize all three structures, achieving either a perfect planar or quasi-planar conformation, as illustrated in Figure 3.3 Typically, structures with double aromaticity demonstrate high thermal stability, symmetry, and a planar shape, particularly in small molecules Additionally, Figure 3.3 presents examples of B3Si and B4Si structures.
The structures B2Si3.a, B2Si3.b, and B3Si2 raise questions about their classification as double aromatic compounds due to their consistent planar conformation across various optimization methods To address this uncertainty, a systematic investigation is conducted on B2Si3 with charge q ranging from -2 to 2, and B3Si2 with charge p varying from -3 to 1.
Figure 3.3 An illustration of clusters with 2 π electrons and 2 σ delocalized electrons
Figures 3.4 and 3.5 illustrate the global equilibrium structures and various lower-lying isomers of B2Si3 with charges ranging from -2 to 2 and B3Si2 with charges from -3 to 1 The relative energies presented are derived from single-point electronic energy calculations at the (U)CCSD(T)/CBS level, utilizing TPSSh/6-311+G(d) optimized geometries, all adjusted for zero-point energy (ZPE) corrections based on TPSSh/6-311+G(d) harmonic vibrational frequencies Relative energies from TPSSh/6-311+G(d) + ZPE are indicated in parentheses, while those from single-point CASSCF/CASPT2 computations are shown in brackets For conciseness, the (U)CCSD(T)/CBS energy is referred to as CBS energy The relative energies between isomers discussed in subsequent sections are derived from CBS + ZPE computations, unless stated otherwise, with isomers labeled as X.A, where X represents the increasing order of relative energy and A denotes the cluster name DFT calculations were performed using TPSSh and other specified functionals.
The Gaussian 16 program was employed for computations, while the ORCA package was used to calculate the CBS energy, extrapolated from the aug-cc-pVxZ basis sets (where x = D, T, and Q) Additionally, multi-configurational perturbation theory calculations were conducted using a completely active space wave function as references (CASSCF/CASPT2) with the ORCA program This approach was applied to species with both even and odd characteristics.
The low-lying isomers of B2Si3 q clusters, ranging from q = -2 to +2, are depicted in Figure 3.4 Geometry optimizations were performed utilizing the TPSSh/6-311+G(d) theoretical level The relative energies, measured in kcal/mol, are calculated using three distinct methods, which will be detailed in the following sections.
55 electron numbers, the CASSCF(12,12) and CASSCF(11,12) wavefunctions are constructed, respectively
Figure 3.5 illustrates the geometries of low-lying isomers of B3Si2 p clusters as the value of p varies from -3 to +1 Geometry optimizations were performed using the TPSSh/6-311+G(d) theoretical level, and the relative energies (in kcal/mol) were calculated through three distinct methods, which will be discussed in detail in the following text.
The study conducts a benchmark calculation to validate the effectiveness of the TPSSh/6-311+G(d) method for optimizing structures that incorporate boron and silicon Each charged state of the B2Si3 q and B3Si2 p series, except for the high spin contamination trianion B3Si2 3-, is reoptimized using DFT with various popular functionals and either the 6-311+G(d) or def2-QZVPP basis set Notably, the HSE06 functional is among those utilized for these optimizations.
56 and PBE0 [62] which are the two best functionals according to Koukaras’ study
In studies utilizing experimental results, various DFT functionals such as B3LYP, TPSS-D3, and the hybrid functional TPSSh have been employed The geometries generated by each functional are assessed through single-point electronic energy calculations at these optimized structures using the highly accurate coupled-cluster (U)CCSD(T)/CBS method The lowest CBS energy obtained serves as the reference energy, denoted as Eref, while the deviation from Eref is represented as δE, as detailed in Table 3.1 Additionally, the root mean square (RMS) can be calculated from these deviations.
The RMS values presented in Table 3.1 serve as a universal metric for evaluating the accuracy of optimized geometries across nine isomers Although the 6-311+G(d) basis set is less time-consuming than the def2-QZVPP, it consistently yields significantly better results, with RMS values approaching zero when using the PBE0, HSE06, or TPSSh functionals Therefore, the combination of the 6-311+G(d) basis set with the functionals PBE0, HSE06, and TPSSh proves to be a reliable method for investigating the geometries of stable mixed B−Si clusters in various charge states.
Table 3.1 presents the deviations in total energies of various structures calculated using single-point (U)CCSD(T)/CBS methods, based on geometries optimized at different DFT levels The results for the optimized geometry B3LYP/6-311+G(d) show deviations of 0.06 for B3Si2 2-, 0.14 for B3Si2 -, 0.04 for B3Si2, 0.06 for B3Si2 +, 0.44 for B2Si3 2-, 0.39 for B2Si3 -, 0.22 for B2Si3, 0.19 for B2Si3 +, 0.53 for B2Si3 2+, and 0.29 for RSM.
Table 3.2 presents a comparison of the two PES peaks of B2Si3 - ([103]), utilized to calculate Vertical Detachment Energies (VDEs) and Adiabatic Detachment Energies (ADEs) in electronvolts (eV) This analysis employs various functionals alongside the 6-311+G(d) basis set and CCSD(T)/aug-cc-pVTZ for both anionic isomers, I.B2Si3 - and II.B2Si3 -.
Isomer B3LYP PBE0 HSE06 TPSSh TPSS-D3 CCSD(T) Expt a
ADE 2.33 2.49 2.48 2.50 2.50 2.35 a The VDEs are taken from ref [103]
The disk aromaticity on the quasi-planar boron cluster B 70 0/2-
This section summarizes key findings from the published research paper "A Topological Path to the Formation of the Quasi-Planar B70 Boron Cluster and Its Dianion" by Pinaki Saha, Fernando Buendia Zamudio, Long Van Duong, and Minh Tho Nguyen, featured in Physical Chemistry Chemical Physics, Advance Article, 2023 The study explores the topological pathways involved in the formation of the quasi-planar B70 boron cluster and its dianion, providing insights into its structural properties and potential applications in material science.
Topological principles are crucial in the search, design, and study of nanostructures, as they provide a mathematical framework for understanding the connectivity and spatial arrangement of atoms within molecular systems By leveraging these principles, researchers can precisely manipulate nanostructures, leading to the development of new materials with unique properties and functions Additionally, computational methods like density functional theory (DFT) enable the prediction and analysis of nanostructure properties, facilitating exploration of their applications in various fields, including catalysis, electronics, energy, and medicine.
The topological leapfrog principle is a method for exploring new molecular shapes and designing specific chemical structures by focusing on topology This process begins with a small, known structure and employs a series of operations to systematically add or remove atoms, allowing for the controlled generation of new molecular configurations.
The leapfrog search method for generating structures involves three key operations: dual, capping, and omni-capping Initially, a dual operation is applied to the initial guess cluster structure, which includes swapping faces and vertices, followed by a perpendicular rotation at each edge of the parent geometry Next, a capping operation is performed to cap all new hexagons Finally, the omni-capping operation converts the structure into a triangular tessellated geometry Geometric optimizations are then executed on this generated structure to achieve the final cluster configuration This approach aims to develop new nanostructures with unique properties and functionalities, leveraging the original structure's topology.
75 planar B50 [113] and B56 [114] boron clusters were established using a topological leapfrog approach from the stable elongated B10 2- and B12, respectively
This study employs the topological leapfrog algorithm to investigate the formation of the B70 quasi-planar structure from a B16 configuration with 13 vertices, as illustrated in Figure 3.14 Additionally, the research utilizes the Mexican Enhanced Genetic Algorithm (MEGA), integrated within the Vienna ab initio simulation package (VASP), to explore and generate various isomers of B70.
Figure 3.14 A quasi-planar structure consisting of 70 boron atoms was generated using the topological leapfrog algorithm starting from an initial B16 form with 13 vertices (the atom with yellow glow)
The quasi-planar shape of B70 0/2- is ideal for understanding cluster stability due to its electron configuration adhering to the disk model This research aims to develop a comprehensive electron counting rule that integrates both the Hückel and Barid rules.
Figure 3.15 illustrates the geometric structures of the lowest-energy isomers, featuring several quasi-planar (QP) isomers labeled QP.n, where n = 1, 2, 3, etc., differing by the arrangement of hexagonal holes Among these, QP.1 stands out as the lowest-energy quasi-planar isomer with a triplet ground state, while the others exhibit a singlet ground state The global minimum structure of B70 has previously been identified as the bilayer structure 3D.1, which is corroborated by this study's calculations showing its energy is 1 kcal/mol lower than that of the tubular structure 3D.2 Nonetheless, under specific growth conditions, the tubular structure 3D.2 could potentially be realized experimentally.
In order to gain a better understanding of the quasi-planar isomer QP.1 shown in Figure 3.15, which is marginally less stable than its bilayer 3D.1 and tube
A detailed analysis of QP.2 counterparts will be conducted to gain a deeper understanding of their structure The planar structure of QP.2 is derived from the leapfrog algorithm, which is applied to an initial B16 unit, as illustrated in Figure 3.11 This initial guess structure is formed by combining three hexagonal units, providing a foundation for further examination.
The B6 structures, illustrated in Figure 3.11, undergo dual operations followed by successive capping and omni-capping, resulting in a final structure comprising 61 boron atoms Unlike carbon fullerenes, which utilize the leapfrog process with omni-capping and dual operations, boron systems require an additional boron cap due to the electron deficit of boron atoms This process effectively eliminates all dangling bonds, leading to a planar B70 structure Geometry optimization of this structure subsequently results in the quasi-planar structure QP.1, as shown in Figure 3.15.
The selection of energetically favorable isomers of B70 includes both three-dimensional (3D) and quasi-planar (QP) isomers The notation "n = 1, 2, 3, …" signifies the relative energy ranking of each isomer within these categories.
The lowest triplet state of QP.1 is approximately 2 kcal/mol more stable than its closed-shell singlet form, indicating that the dianionic quasi-planar B70 2- structure could stabilize into a closed-shell configuration Anionic boron cluster species are frequently identified in experimental studies utilizing photoelectron spectroscopy (PES) Consequently, the B70 size may be experimentally detectable via its anion and dianion structures through PES analysis.
The quasi-planar isomer QP.1 exhibits a notably low vertical ionization energy of 5.3 eV and a significant vertical two-electron affinity of approximately 5.6 eV This indicates that the dianionic state of QP.1 is likely to be thermodynamically more stable than its neutral form.
Our calculations reveal that the quasi-planar dianion QP.1 of the dianionic B70 2- isomer is more stable, exhibiting an energy state approximately 3 kcal/mol lower than the bilayer dianion 3D.1 Conversely, the tubular dianion 3D.2 is significantly higher in energy.
3.2.3 Disk model and electron count rule
The π molecular orbitals (MOs) of B70 2- can be analyzed using the disk aromaticity model (DM), as illustrated in Figure 3.16 This model derives its wave functions from the Schrödinger equation, which describes a particle in a disk, characterized by two quantum numbers: the radial quantum number (𝑛 = 1, 2, 3, …) and the rotational quantum number (𝑙) The eigenstates are organized in ascending order as 1𝜎, 1𝜋, 1𝛿, 2𝜎, etc Eigenstates with a rotational quantum number of zero (𝑙 = 0) are non-degenerate, while those with non-zero values (𝑙 ≠ 0) are doubly degenerate Consequently, the electron count in the DM adheres to the (4N + 2M) rule, where N represents the number of energy levels with 𝑙 ≠ 0, and M denotes the number of non-degenerate energy levels (𝑙 = 0) The ground eigenstate is the 1𝜎-orbital, indicating that 𝑁 must be at least 1 In smaller systems, the number of non-degenerate orbitals decreases, causing the DM rule to align with the classical 4N + 2 Hückel rule when M equals 1.
The B70 2-dianion contains 50 π electrons, adhering to the DM rule for N = 11 and M = 3 The π molecular orbitals (MOs), labeled as MO122, MO130, and MO163, correspond to the energy levels 1σ, 2σ, and 1σ, respectively Additionally, there is a notable separation from double degeneracy into two pseudo-degenerate states.
Binary boron lithium clusters B 12 Li n with n = 1–14: the disk-cone
The main content in this section is reproduced from the published paper entitled
The study titled "The binary boron lithium clusters B12Lin with n = 1–14: in search for hydrogen storage materials," authored by Long Van Duong, Nguyen Thanh Si, Nguyen Phi Hung, and Minh Tho Nguyen, explores the potential of boron-lithium clusters as innovative hydrogen storage materials Published in Physical Chemistry Chemical Physics, this research investigates the structural and energetic properties of B12Lin clusters to assess their viability for efficient hydrogen storage applications The findings contribute to the ongoing quest for sustainable energy solutions through advanced material design.
Global warming, primarily driven by fossil fuel consumption and the depletion of these resources, is a major concern for humanity Investigating green energy sources not only tackles the challenge of energy scarcity but also promotes sustainable solutions for the future.
Hydrogen energy is gaining prominence as a sustainable solution to combat global warming, primarily due to its reaction that produces water while releasing substantial heat However, the high heat generation associated with hydrogen usage raises safety concerns As a result, research into safe hydrogen storage methods is increasingly important and continues to draw significant attention.
High molecular hydrogen adsorption capacity is essential for designing effective hydrogen storage materials, as it determines the amount of hydrogen gas a material can hold relative to its mass or volume Materials with higher hydrogen storage capacities are more efficient for practical applications, requiring numerous adsorption centers for hydrogen molecules to physically adhere to the surface, a process known as physisorption Additionally, strong adsorption strength is crucial, ensuring that hydrogen molecules are effectively retained even under moderate pressures and temperatures This study explores mixed-phase Li-B clusters as promising candidates, given that both lithium and boron are lightweight elements, with lithium demonstrating significant potential for effective hydrogen adsorption.
The B12 cluster is notable for its geometric transformations, with the octahedral B12 serving as the foundational building block for solid boron In its cluster form, B12 adopts a quasi-planar structure that exhibits high thermal stability Recent research by Dong et al demonstrated that introducing one to three lithium atoms into B12 alters its shape, transitioning from quasi-planar to tubular and eventually to cage-like structures The B12Li3 cage form has garnered attention for its hydrogen storage potential; however, its thermal stability is limited, primarily due to its open-shell configuration This study aims to systematically investigate these characteristics.
84 stability of B12Lin clusters with n = 0 – 14 to identify the most promising candidate for hydrogen storage materials among these clusters
The study's findings reveal that B12Li8 is a highly promising hydrogen storage material, meeting multiple essential criteria Furthermore, the cone-shaped B12Li4 structure demonstrates remarkable thermal stability This dissertation places significant emphasis on elucidating the stability of the B12Li4 structure through the disk model approach.
3.3.2 The growth pattern of B 12 Li n with n = 0 – 14
The optimized geometries of the most stable B12Lin clusters, with n ranging from 0 to 14, are illustrated in Figures 3.19 and 3.20 These structures were analyzed using the TPSSh/6-311+G(d) theoretical level, and single-point energy calculations (U)CCSD(T)/cc-pVTZ + ZPE were conducted to confirm the accuracy of the DFT results A benchmark study was performed to validate the effectiveness of the TPSSh theoretical level for assessing the stability of mixed B-Li clusters, although this analysis is not included here as it is not the primary focus of the dissertation; a similar evaluation is already discussed in section 3.1.2.
In this article, isomer structures are designated using the convention nX, where n represents the number of lithium atoms and X denotes an increasing order of relative energy The isomer labeled as nA consistently refers to the lowest-energy isomer for that specific size Additionally, the figures highlight the lowest-lying isomer(s) of each size, which fall within a 3 kcal/mol range, as determined at the (U)CCSD(T)/cc-pVTZ + ZPE level.
The optimized shapes of the B12Lin clusters with n = 0 – 6 are displayed in Figure 3.19 Firstly, the thermodynamically stable quasi-planar pure B12 (n = 0) [15,
The study presents a new isomer of B12Li3, designated as 3A, which features a quasi-3D structure This finding contrasts with previously documented isomers of B12Li1 (1A) and B12Li2 (2A) as outlined in reference [120] The sizes and characteristics of these isomers have been thoroughly described, with the new isomer of B12Li3 offering significant insights into its structural properties.
An elongated B11 structure features a B atom on one side, while a Li atom coordinates the opposite side, accompanied by two additional Li atoms positioned in a different plane This isomer demonstrates greater stability compared to the cage-like structure 3B identified by Dong and colleagues, with a stability difference of approximately 5 kcal/mol.
DFT calculations, UCCSD(T) results show that both structures, along with three other structures, have relative energies smaller than 3 kcal/mol, still in favour of 3A
Figure 3.19 Geometry, point group and relative energy (kcal/mol,
(U)CCSD(T)/cc-PVTZ + ZPE) of B12Lin with n = 0 – 6 Relative energies at TPSSh/6-311+G(d) + ZPE are given in parentheses TPSSh/6-311+G(d) optimized geometries are used
The energy ordering among the 3A to 3E isomers is influenced by the computational methods used, leading to minor absolute differences in their energy levels Consequently, these isomers can be regarded as quasi-degenerate in terms of energy.
The global minimum structure of B12Li4, denoted as 4A, features a pyramidal shape with three lithium (Li) atoms positioned around the apex and one Li atom located inside the pyramid Isomers 4B and 4D are generated through the displacement of the Li atom in the 4A structure The first cage structure, referred to as 4C, emerges from the B12 framework, exhibiting an energy increase of approximately 9 kcal/mol compared to 4A This cage results from a distortion of the T_d symmetry structure 4C-T_d, which displays three degenerate negative frequencies.
The B12Li5 size (n = 5) exhibits an energy ordering reversal between the isomers 5A and 5B, as determined by DFT and UCCSD(T) calculations In this context, isomer 5B features a coordination of lithium atoms surrounding a quasi-planar structure of B12, while isomer 5A presents a caged B12 framework Notably, both isomers are nearly energetically equivalent, with a relative energy difference of less than 3 kcal/mol.
The three lower-lying isomers of B12Li6, specifically 6A, 6B, and 6C, are in competition to establish the global minimum configuration These structures feature lithium atoms arranged around a quasi-planar B12 framework, characterized by a B4 rhombus that is slightly deviated from the plane.
The optimized shapes and characteristics of the following B12Lin series with n
The global minimum of B12Li7 is anticipated to be one of the four isomers: 7A, 7B, 7C, or 7D, which all have similar energy levels Isomers 7A, 7B, and 7C are formed by adding a lithium atom to the B12Li6 structures 6A, 6B, and 6C, while isomer 7D features a three-dimensional boron framework.
B 14 FeLi 2 and the hollow cylinder model
The main content in this section is reproduced from the published paper intitled
"The teetotum cluster Li2FeB14 and its possible use for constructing boron nanowires", by Ehsan Shakerzadeh, Long Van Duong, My Phuong Pham-Ho,
Elham Tahmasebi and Minh Tho Nguyen in Physical Chemistry Chemical Physics, vol 22, no 26, pp 15013–15021 (2020)
In 2014, Tam and colleagues reported the stability of the iron-doped boron cluster B14Fe in a triplet state, featuring a tubular arrangement with the Fe atom at the center This raised questions about whether the magnetic properties of iron could be entirely suppressed through doping The current study demonstrates that introducing two lithium atoms effectively quenches the magnetic properties of iron while maintaining the stability of the B14 cluster in its double ring form Additionally, the hollow cylinder model provides a crucial explanation for the stability of the B14FeLi2 structure.
Figure 3.31 The lowest-lying isomer of B14Fe [145]
3.4.2 Stability of B 14 FeLi 2 and its potential applications
Calculations at the TPSSh/def2-TZVP + ZPE level indicate that the low-spin teetotum Li2B14Fe structure, characterized by D7d symmetry, is the ground state This structure features two B7 strings that encapsulate the Fe atom, with two Li atoms symmetrically positioned on either side of the Fe along the axis The B-B bond length within each B7 string measures 1.64 Å, while the distance between the two strings is 1.81 Å Previously, the most stable isomer of B14Fe was identified as a double ring (DR) configuration composed of two B7 strings.
103 seven-membered rings disposed in an anti-prism form and doped by the Fe atom at the centre of the cylinder
Figure 3.32 Optimized structures of lower-energy isomers of B14FeLi2;E values are in kcal/mol from TPSSh/def2-TZVP energies with ZPE corrections
The high symmetry (D7d) and high spin double resonance (DR) structure of B14Fe in a triplet state is approximately 2 kcal/mol more stable than its low-spin counterpart When two lithium (Li) atoms are incorporated into the B14Fe DR framework, the high symmetry is maintained, but the B14FeLi2 in a low-spin singlet state becomes significantly more stable than the triplet structure by about 19 kcal/mol The bond lengths between boron (B) atoms within each string and between the two B7 strings are measured at 1.62 Å and 1.76 Å, respectively, indicating that the addition of Li atoms has minimal impact on the inter-ring distance, as the two B7 strings remain relatively distant from each other.
~0.05 Å Also, the B14 cylinder is slightly compressed upon approach of the Fe
104 atom The Fe-B length amounts to 2.12 Å in B14Fe, while it is about 2.05 Å in
The molecular orbitals (MOs) of B14FeLi2 are formed from the MOs of the singlet B14 skeleton, with an additional contribution from the d atomic orbitals (AOs) of the iron (Fe) atom Certain MOs of the singlet B14 skeleton are identified using a hollow cylinder model.
The calculated HOMO, LUMO, and energy gap for the teetotum structure are –5.1 eV, –3.0 eV, and 2.1 eV, respectively In comparison, the stable high spin tubular B14Fe exhibits a SOMO-LUMO gap of 0.9 eV, determined using the TPSSh/def2-TZVP method Notably, doping Li atoms into B14Fe results in an increase in the frontier orbitals gap Additionally, the vertical ionization energy of B14FeLi2 is calculated to be 6.5 eV, derived from the energy difference between the two teetotum configurations.
105 forms in the neutral and cationic states The IE of B14Fe is IE(B14Fe) = 7.5 eV, and thus addition of Li atoms reduces the IE by up to 1 eV
Figure 3.33 indicates the formation of MOs of B14FeLi2 from the singlet B14 skeleton and a distribution from d-AO of the Fe 2+ ion Within this point of view, the
The Fe atom gains two electrons from two Li atoms, resulting in a more stable triplet DR B14 skeleton, which is favored over the singlet by 2 kcal/mol at the TPSSh/def2-TZVP level Additionally, the triplet Li-Fe-Li linear unit exhibits lower energy compared to the quintet and singlet states by 2 and 86 kcal/mol, respectively Consequently, the Li-(Fe@B14)-Li teetotum forms from the interaction between the triplet B14 skeleton and the triplet Li-Fe-Li linear unit.
B14 singlet skeleton with only 2 kcal/mol higher energy than the triplet state can equally be used to have a better look for the formation of MOs of B14FeLi2
The molecular orbitals (MOs) of the singlet B14 skeleton can be analyzed using the hollow cylinder model (HCM) Similar to B14Ni, the transition metal contributes significantly, with 50% of the 3d xz and 3d yz orbitals influencing the (1 ±2 2)-orbitals, which correspond to the LUMO and LUMO' of the B14 skeleton, forming the HOMO – 3 and HOMO – 3' of B14FeLi2 Additionally, the HOMO – 2 and HOMO – 2' are derived from the (2 ±1 2)-orbitals, incorporating 51% of the 3d xy and 3d x²-y² orbitals The hybridization of the (3 0 1)-orbital (HOMO – 3) of the B14 skeleton with the 3d z² orbital leads to the formation of a bonding HOMO – 9 and an antibonding HOMO – 1 Furthermore, the s-AO and p-AO of Fe contribute to the MOs of Li2FeB14, resulting in an electron configuration for Fe of [4s 0.1 3d 8.36 4p 0.51 5s 0.26 4d 0.04 5p 1.16].
B14FeLi2 is the (2 0 2)-orbital of HCM which can make a structure becoming highly thermodynamically stabilized such as the cases of Ni@B14, Ni2@B20 2-, and
Ni2@B22 [143] Insertion of a Fe atom inside the DR B14 expands both peripheral B-
B bonds and B-B bonds between two strings which results in a weakening of all B-
B bonds The (2 0 2)-orbital of HCM plays a role of shortening the peripheral B-B bonds which amount to 1.63 Å, whereas the B-B bonds between two rings are now 1.79 Å
Figure 3.34 ACID map of Li2FeB14 from a) top view and b) side view
Figure 3.34 illustrates the anisotropy of the induced current density (ACID) maps for B14FeLi2 at an isosurface value of 0.05 The clockwise arrows on the ACID isosurface represent diatropic ring currents, while the anti-clockwise arrows indicate paratropic ring currents An external magnetic field is oriented along the Oz axis, directed out of the paper plane (Z+) Clockwise current density vectors are highlighted with red-glowing arrows, whereas anti-clockwise vectors are shown with orange-glowing arrows The right-side figures provide a view of the left-side figures after an 80º rotation about the Ox axis, revealing intriguing details about the current density vectors.
B14FeLi2 exhibits weak diatropic current flow within the B14 border, while demonstrating strong diatropic currents surrounding the Fe atom and significant paratropic currents at each highlighted B atom The contributions from three sets of molecular orbitals, as defined by the hollow cylinder model (HCM), are illustrated in Figure 3.35, providing insights into the ACID maps for boron DR clusters Notably, the radial set (π set) indicates that Li2FeB14 is classified as a π-aromatic species, as evidenced by the clockwise arrows around the B atoms.
Fe atoms The tangential set (σ set) just shows the clockwise arrows around Fe
107 atoms, whereas the localized set (s-MOs set in HCM) causes the anti-clockwise arrows around each B atom
Figure 3.35 ACID isosurface (isovalue = 0.05) of three valence MOs sets of
B14FeLi2 on the view from Li-Fe-Li axis (Oz axis) including a) localized set, b) tangential set and c) radial set
The time-dependent density functional theory (TD-DFT) method, specifically TPSSh/def2-TZVP, is employed to predict the optical absorption spectrum of Li2FeB14, revealing approximately 50 lower-lying excited states The high symmetry of Li2FeB14 results in several forbidden transitions, with a frontier energy gap of around 2.1 eV Notably, the UV-Vis spectrum exhibits a significant peak at 3.7 eV (~337 nm) corresponding to the transition from HOMO – 1 to LUMO + 2, and another peak at 4.2 eV (~298 nm) linked to the transition from HOMO to LUMO + 6 These major peaks, along with additional minor peaks at longer wavelengths, indicate that Li2FeB14 can absorb UV light while remaining completely transparent in other regions.
108 respect to visible light Accordingly, Li2FeB14 can be regarded as a candidate material for visible-inert optoelectronic devices
Figure 3.36 Predicted electronic absorption spectrum of the teetotum B14FeLi2
B14FeLi2 introduces a groundbreaking feature by enabling the creation of new boron-based nanowires Utilizing stable B14Fe cylinders and B14FeLi2 teetotum motifs, a nanowire can be constructed in the form of [Li-B14Fe-Li]-[B14Fe]-[Li-B14Fe-Li] The relaxed structure of this nanowire is determined at the TPSSh/6-31+G(d) level, resulting in linear forms optimized as equilibrium structures The calculated HOMO-LUMO gap for this configuration is 0.3 eV, significantly lower than that of isolated Li2FeB14 (2.2 eV) or B14Fe (0.9 eV), indicating a fully metallic character Additionally, an alternative design of the nanowire as [B14Fe]-[Li-B14Fe-Li]-[B14Fe] was explored, but it did not yield a true minimum.
More interestingly, another wire is also predicted using the magnesium atom as linkage The true energy minimum structures of B28Fe2Li2Mg, B42Fe3Li2Mg2,
The compounds B56Fe4Li2Mg3 and B70Fe4Mg4Li2, analyzed at the TPSSh/6-31+G(d) level, reveal two distinct nanowire structures The first structure exhibits an antiprism shape, while the second structure takes on a prism shape formed by two adjacent B14 units.
The B28Fe2Li2Mg nanowire is constructed from two Li2FeB14 motifs, where the two central lithium atoms are substituted with a single magnesium atom Additionally, the B42Fe3Mg2Li2 wire can be formed by combining three motifs, resulting in energetically degenerate structures.
The innovative strategy presented involves constructing boron-based wires by linking various quantities of B14FeLi2 units, while substituting every two central lithium atoms with magnesium This approach enables the creation of longer wires, enhancing the potential applications of boron-based materials in advanced technology.
𝑛Li 2 FeB 14 + (𝑛 − 1)Mg → Li 2 Fe 𝑛 Mg 𝑛−1 𝐵 14𝑛 + (2𝑛 − 2)Li 𝑛 ≥ 2 (3.6)
This strategy offers a significant advantage by maintaining the (N 0 2)-orbital of the HCM as the highest occupied molecular orbital (HOMO) in wires featuring N B7 rings, which is anticipated to enhance the stability of the wire in a high-symmetry environment Additionally, the HOMO-LUMO gaps of B28Fe2Li2Mg are a crucial factor in this context.
GENERAL CONCLUSIONS AND FUTURE DIRECTIONS
General Conclusions
This theoretical study utilized quantum chemical calculations to investigate the geometries, electronic structures, and bonding characteristics of various pure and doped boron clusters with different impurities Key findings are summarized from Chapter 3, highlighting the introduction of different aromaticity models tailored to each structure's geometry for assessing thermodynamic stability and relevant physicochemical properties Notably, the TPSSh functional proved reliable for optimizing geometric structures of boron-containing clusters, while the B3LYP functional showed better alignment with experimental values for vertical detachment energies (VDEs) and harmonic vibrational frequencies Furthermore, the research differentiated between the Hückel rule and the ribbon model through studies on B2Si3 and B3Si2 clusters, establishing a framework for evaluating cluster stability The ribbon aromaticity model was classified into sub-classes, including aromaticity, semi-aromaticity, antiaromaticity, and triplet-aromaticity, based on specific electronic configurations, with the identification of a self-lock phenomenon being essential for classification as a ribbon structure.
An alternating distribution between π and σ delocalized electrons will subsequently be found in the resulting aromatic ribbon structure
The investigation of the disk model revealed that the QP.1 structure in the triplet state is the most stable quasi-planar form of B70, aligning with the topological leapfrog principle The stability of the B70 2- dianion's QP form increases upon adding two electrons to the neutral QP.1 structure, filling its open-shell SOMO levels Both ring current maps for the QP.1 in triplet neutral and singlet dianionic states indicate aromatic characteristics, leading to proposed generalized electron count rules for the disk model The disk-cone model emerged from studying the stability of cone-like structures B13Li and B12Li4, which exhibit double aromaticity based on their electron configurations Notably, B12Li8 shows promise for H2 storage with a gravimetric weight ratio of hydrogen reaching 30 wt% and interaction energies suggesting a physisorption-like behavior Lastly, the hollow cylinder model highlights the stability of the B14FeLi2 structure, which does not absorb visible light but can extend into conductive nanowires with a band gap of approximately 0.8 eV, indicating its potential for photovoltaic applications The stability of B14FeLi2 is attributed to the effective hybridization of molecular orbitals within its framework.
This doctoral study has introduced several innovative models that enhance the understanding of aromaticity, a key concept in modern chemistry These new models and electron count rules were developed through comprehensive mathematical analysis, utilizing wave equation solutions tailored to specific geometries So far, these models have proven effective across diverse geometrical forms Additionally, the classical electron counts that define aromatic character have been identified as the most straightforward instances within these new frameworks.
Future Directions
While various aromatic models, including the ribbon, disk, and hollow cylinder models, have gained recognition in the scientific community through numerous citations in esteemed international chemistry and physics journals, their application must be expanded to a broader range of atomic clusters and chemical compounds to validate their effectiveness Additionally, established aromaticity models such as spherical aromaticity, the jellium model, and the elongated model require thorough evaluation to ascertain their appropriateness and complementary roles for each specific structure analyzed.
The ultimate goal is to enable chemists to routinely utilize these models for a deeper understanding of their compounds and materials My future research will focus on exploring various classes of boron clusters and other elemental clusters to enhance our understanding of the relationship between shape, structure, and bonding This research aims to uncover detailed properties of these clusters and investigate the influence of aromaticity on their thermodynamic stability.
Furthermore, the current and potential applications of atomic clusters should also be considered and extended From this doctoral study, the B12Li8 cluster has
Recent studies indicate that atomic clusters exhibit exceptional hydrogen adsorption capacities, surpassing many previously reported materials Ongoing research will explore various applications, including their use as building blocks for advanced materials, catalysts for chemical reactions, and drug carriers in medical treatments Overall, the future of atomic clusters appears promising, with theoretical predictions suggesting significant potential in multiple fields.
LIST OF PUBLICATIONS CONTRIBUTING TO THE
1) Boron Silicon B2Si3 q and B3Si2 p Clusters: The Smallest Aromatic Ribbons
Long Van Duong, Nguyen Ngoc Tri, Nguyen Phi Hung, and Minh Tho
Nguyen, J Phys Chem A, vol 126, no 20, pp 3101–3109, May 2022
2) A topological path to the formation of the quasi-planar B70 boron cluster and its dianion
Pinaki Saha, Fernando Buendia Zamudio, Long Van Duong, and Minh Tho Nguyen, Phys Chem Chem Phys., Advance Article, 2023
3) The binary boron lithium clusters B12Lin with n = 1–14: in search for hydrogen storage materials
Long Van Duong, Nguyen Thanh Si, Nguyen Phi Hung, and Minh Tho
Nguyen, Phys Chem Chem Phys., vol 23, no 43, pp 24866–24877, 2021
4) The teetotum cluster Li2FeB14 and its possible use for constructing boron nanowires
Ehsan Shakerzadeh, Long Van Duong, My Phuong Pham-Ho, Elham
Tahmasebi, and Minh Tho Nguyen, Phys Chem Chem Phys., vol 22, no
The Asia Pacific Association of Theoretical and Computational Chemistry (APATCC-10) at the International Centre for Interdisciplinary Science and Education (ICISE), Quy Nhon – Vietnam, February 19 th – 23 rd , 2023
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Cartesian coordinates of some optimized structures computed at the TPSSh/6- 311+G(d) level The sum of bond orders (SBOs) and net atomic charges (NACs) of each atom are also given