|
Koopmans' Theorem
Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO). This theorem is named after Tjalling Koopmans, who published this result in 1934 for atoms. Koopmans' theorem is exact in the context of restricted Hartree–Fock theory if it is assumed that the orbitals of the ion are identical to those of the neutral molecule (the ''frozen orbital'' approximation). Ionization energies calculated this way are in qualitative agreement with experiment – the first ionization energy of small molecules is often calculated with an error of less than two electron volts. Therefore, the validity of Koopmans' theorem is intimately tied to the accuracy of the underlying Hartree–Fock wavefunction. The two main sources of error are orbital relaxation, which refers to the changes in the Fock operator and Hartree–Fock orbit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ionization Energy
In physics and chemistry, ionization energy (IE) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, Ion, positive ion, or molecule. The first ionization energy is quantitatively expressed as :X(g) + energy ⟶ X+(g) + e− where X is any atom or molecule, X+ is the resultant ion when the original atom was stripped of a single electron, and e− is the removed electron. Ionization energy is positive for neutral atoms, meaning that the ionization is an endothermic process. Roughly speaking, the closer the outermost electrons are to the atomic nucleus, nucleus of the atom, the higher the atom's ionization energy. In physics, ionization energy (IE) is usually expressed in electronvolts (eV) or joules (J). In chemistry, it is expressed as the energy to ionize a Mole (unit), mole of atoms or molecules, usually as Joule per mole, kilojoules per mole (kJ/mol) or Kilocalorie per mole, kilocalories per mole (kcal/mol). Comparison of ion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of Molecule, molecules, Material, materials, and solutions at the atomic level. These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed Wave function, wave functions as well as to observable properties such as structures, spectra, and Thermodynamics, thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics. Chemists rely heavily on spectroscopy through which information regarding the Quantization (physics), quantization of energy on a molecular scale can be obtained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fock Matrix
The Fock matrix is defined by the Fock operator. In its general form the Fock operator writes: :\hat F(i) = \hat h(i)+\sum_^ hat J_j(i)-\hat K_j(i)/math> Where ''i'' runs over the total ''N'' spin orbitals. In the closed-shell case, it can be simplified by considering only the spatial orbitals. Noting that the \hat J terms are duplicated and the exchange terms are null between different spins. For the restricted case which assumes closed-shell orbitals and single- determinantal wavefunctions, the Fock operator for the ''i''-th electron is given by:Levine, I.N. (1991) ''Quantum Chemistry'' (4th ed., Prentice-Hall), p.403 :\hat F(i) = \hat h(i)+\sum_^ \hat J_j(i)-\hat K_j(i)/math> where: :\hat F(i) is the Fock operator for the ''i''-th electron in the system, :(i) is the one-electron Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LUMO
In chemistry, HOMO and LUMO are types of molecular orbitals. The acronyms stand for ''highest occupied molecular orbital'' and ''lowest unoccupied molecular orbital'', respectively. HOMO and LUMO are sometimes collectively called the ''frontier orbitals'', such as in the frontier molecular orbital theory. Gap The energy difference between the HOMO and LUMO is ''the HOMO–LUMO gap''. Its size can be used to predict the strength and stability of transition metal Coordination complex, complexes, as well as the colors they produce in solution. As a rule of thumb, the smaller a compound's HOMO–LUMO gap, the less stable the compound. Recent quantum‐chemical analyses of over 700 compounds demonstrated that terrestrial secondary metabolites exhibit HOMO–LUMO gaps on average about 2 eV narrower than organic molecules found in carbonaceous meteorites, and that combining gap width with hydrophilicity creates a robust discriminator between biotic and abiotic chemistries. This sugges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron Affinity
The electron affinity (''E''ea) of an atom or molecule is defined as the amount of energy released when an electron attaches to a neutral atom or molecule in the gaseous state to form an anion. ::X(g) + e− → X−(g) + energy This differs by sign from the energy change of electron capture ionization. The electron affinity is positive when energy is released on electron capture. In solid state physics, the electron affinity for a surface is defined somewhat differently ( see below). Measurement and use of electron affinity This property is used to measure atoms and molecules in the gaseous state only, since in a solid or liquid state their energy levels would be changed by contact with other atoms or molecules. A list of the electron affinities was used by Robert S. Mulliken to develop an electronegativity scale for atoms, equal to the average of the electrons affinity and ionization potential. Other theoretical concepts that use electron affinity include electronic chem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Set (chemistry)
In theoretical chemistry, theoretical and computational chemistry, a basis set is a set of Function (mathematics), functions (called basis functions) that is used to represent the Wave function, electronic wave function in the Hartree–Fock method or Density functional theory, density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer. The use of basis sets is equivalent to the use of an approximate resolution of the identity: the Atomic orbital, orbitals , \psi_i\rangle are expanded within the basis set as a linear combination of the basis functions , \psi_i\rangle \approx \sum_\mu c_ , \mu\rangle, where the expansion coefficients c_ are given by c_ = \sum_\nu \langle \mu, \nu \rangle^ \langle \nu , \psi_i \rangle. The basis set can either be composed of atomic orbitals (yielding the linear combination of atomic orbitals approach), which is the usual choice within the qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronvolt
In physics, an electronvolt (symbol eV), also written electron-volt and electron volt, is the measure of an amount of kinetic energy gained by a single electron accelerating through an Voltage, electric potential difference of one volt in vacuum. When used as a Units of energy, unit of energy, the numerical value of 1 eV in joules (symbol J) is equal to the numerical value of the Electric charge, charge of an electron in coulombs (symbol C). Under the 2019 revision of the SI, this sets 1 eV equal to the exact value Historically, the electronvolt was devised as a standard unit of measure through its usefulness in Particle accelerator#Electrostatic particle accelerators, electrostatic particle accelerator sciences, because a particle with electric charge ''q'' gains an energy after passing through a voltage of ''V''. Definition and use An electronvolt is the amount of energy gained or lost by a single electron when it moves through an Voltage, electric potential differenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultraviolet Photoelectron Spectroscopy
Ultraviolet photoelectron spectroscopy (UPS) refers to the measurement of kinetic energy spectra of photoelectrons emitted by molecules that have absorbed ultraviolet photons, in order to determine molecular orbital energies in the valence region. Basic theory If Albert Einstein's photoelectric law is applied to a free molecule, the kinetic energy ( E_\text) of an emitted photoelectron is given by : E_\text = h\nu - I\,, where ''h'' is the Planck constant, ''ν'' is the frequency of the ionizing light, and ''I'' is an ionization energy for the formation of a singly charged ion in either the ground state or an excited state. According to Koopmans' theorem, each such ionization energy may be identified with the energy of an occupied molecular orbital. The ground-state ion is formed by removal of an electron from the highest occupied molecular orbital, while excited ions are formed by removal of an electron from a lower occupied orbital. History Before 1960, virtually all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Symmetry
In chemistry, molecular symmetry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule's chemical property, chemical properties, such as whether or not it has a molecular dipole moment, dipole moment, as well as its allowed spectroscopy, spectroscopic transitions. To do this it is necessary to use group theory. This involves classifying the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Symmetry is useful in the study of molecular orbitals, with applications to the Hückel method, to ligand field theory, and to the Woodward–Hoffmann rules. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry discuss symmetry. Another framework on a larger scale is the use of crystal sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HOMO/LUMO
In chemistry, HOMO and LUMO are types of molecular orbitals. The acronyms stand for ''highest occupied molecular orbital'' and ''lowest unoccupied molecular orbital'', respectively. HOMO and LUMO are sometimes collectively called the ''frontier orbitals'', such as in the frontier molecular orbital theory. Gap The energy difference between the HOMO and LUMO is ''the HOMO–LUMO gap''. Its size can be used to predict the strength and stability of transition metal complexes, as well as the colors they produce in solution. As a rule of thumb, the smaller a compound's HOMO–LUMO gap, the less stable the compound. Recent quantum‐chemical analyses of over 700 compounds demonstrated that terrestrial secondary metabolites exhibit HOMO–LUMO gaps on average about 2 eV narrower than organic molecules found in carbonaceous meteorites, and that combining gap width with hydrophilicity creates a robust discriminator between biotic and abiotic chemistries. This suggests that the HOMO–LU ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Functional Theory
Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals - that is, functions that accept a function as input and output a single real number. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry. DFT has been very popular for calculations in solid-state physics since the 1970s. However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
