Van der Waals force
In molecular physics, the van der Waals force, named after Dutch scientist Johannes Diderik van der Waals, is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; the Van der Waals force vanishes at longer distances between interacting molecules. Van der Waals force plays a fundamental role in fields as diverse as supramolecular chemistry, structural biology, polymer science, surface science, condensed matter physics, it underlies many properties of organic compounds and molecular solids, including their solubility in polar and non-polar media. If no other force is present, the distance between atoms at which the force becomes repulsive rather than attractive as the atoms approach one another is called the van der Waals contact distance; the van der Waals force has the same origin as the Casimir effect, arising from quantum interactions with the zero-point field. The term van der; the term always includes the London dispersion force between instantaneously induced dipoles.
It is sometimes applied to the Debye force between a permanent dipole and a corresponding induced dipole or to the Keesom force between permanent molecular dipoles. Van der Waals forces include attraction and repulsions between atoms and surfaces, as well as other intermolecular forces, they differ from covalent and ionic bonding in that they are caused by correlations in the fluctuating polarizations of nearby particles. Being the weakest of the weakest chemical forces, with a strength between 0.4 and 4kJ/mol, they may still support an integral structural load when multitudes of such interactions are present. Such a force results from a transient shift in electron density; the electron density may temporarily shift more to one side of the nucleus. This generates a transient charge to which a nearby atom can be either repelled; when the interatomic distance of two atoms is greater than 0.6 nm the force is not strong enough to be observed. In the same vein, when the interatomic distance is below 0.4 nm the force becomes repulsive.
Intermolecular forces have four major contributions: A repulsive component resulting from the Pauli exclusion principle that prevents the collapse of molecules. Attractive or repulsive electrostatic interactions between permanent charges, quadrupoles, in general between permanent multipoles; the electrostatic interaction is sometimes called the Keesom interaction or Keesom force after Willem Hendrik Keesom. Induction, the attractive interaction between a permanent multipole on one molecule with an induced multipole on another; this interaction is sometimes called Debye force after Peter J. W. Debye. Dispersion, the attractive interaction between any pair of molecules, including non-polar atoms, arising from the interactions of instantaneous multipoles. Returning to nomenclature, different texts refer to different things using the term "van der Waals force"; some texts describe the van der Waals force as the totality of forces. All intermolecular/van der Waals forces are anisotropic, which means that they depend on the relative orientation of the molecules.
The induction and dispersion interactions are always attractive, irrespective of orientation, but the electrostatic interaction changes sign upon rotation of the molecules. That is, the electrostatic force can be attractive or repulsive, depending on the mutual orientation of the molecules; when molecules are in thermal motion, as they are in the gas and liquid phase, the electrostatic force is averaged out to a large extent, because the molecules thermally rotate and thus probe both repulsive and attractive parts of the electrostatic force. Sometimes this effect is expressed by the statement that "random thermal motion around room temperature can overcome or disrupt them"; the thermal averaging effect is much less pronounced for the attractive induction and dispersion forces. The Lennard-Jones potential is used as an approximate model for the isotropic part of a total van der Waals force as a function of distance. Van der Waals forces are responsible for certain cases of pressure broadening of spectral lines and the formation of van der Waals molecules.
The London-van der Waals forces are related to the Casimir effect for dielectric media, the former being the microscopic description of the latter bulk property. The first detailed calculations of this were done in 1955 by E. M. Lifshitz. A more general theory of van der Waals forces has been developed; the main characteristics of van der Waals forces are: They are weaker than normal covalent and ionic bonds. Van der Waals forces can not be saturated, they have no directional characteristic. They are all short-range forces and hence only interactions between the nearest particles need to be considered. Van der Waals attraction is greater. Van der Waals forces are independent
Geckos are lizards belonging to the infraorder Gekkota, found in warm climates throughout the world. They range from 1.6 to 60 cm. Most geckos cannot blink, but they lick their eyes to keep them clean and moist, they have a fixed lens within each iris. Geckos are unique among lizards in their vocalizations, they use chirping or clicking sounds in their social interactions, sometimes when alarmed. They are the most species-rich group of lizards, with about 1,500 different species worldwide; the New Latin gekko and English "gecko" stem from the Indonesian-Malay gēkoq, imitative of sounds that some species make. All geckos except species in the family Eublepharidae lack eyelids. Species without eyelids lick their own corneas when they need to clear them of dust and dirt. Nocturnal species have excellent night vision; the nocturnal geckos evolved from diurnal species. The gecko eye therefore modified its cones that increased in size into different types both single and double. Three different photopigments have been retained and are sensitive to UV, green.
They use a multifocal optical system that allows them to generate a sharp image for at least two different depths. Most gecko species can lose their tails in defense, a process called autotomy. Many species are well known for their specialised toe pads that enable them to climb smooth and vertical surfaces, cross indoor ceilings with ease. Geckos are well known to people who live in warm regions of the world, where several species of geckos make their home inside human habitations; these become part of the indoor menagerie and are welcomed, as they feed on insects, including moths and mosquitoes. Unlike most lizards, geckos are nocturnal; the largest species, the kawekaweau, is only known from a single, stuffed specimen found in the basement of a museum in Marseille, France. This gecko was 60 cm long and it was endemic to New Zealand, where it lived in native forests, it was wiped out along with much of the native fauna of these islands in the late 19th century, when new invasive species such as rats and stoats were introduced to the country during European colonization.
The smallest gecko, the Jaragua sphaero, is a mere 1.6 cm long and was discovered in 2001 on a small island off the coast of the Dominican Republic. Like other reptiles, geckos are ectothermic, producing little metabolic heat. A gecko's body temperature is dependent on its environment. In order to accomplish their main functions—such as locomotion, reproduction, etc.—geckos must have a elevated temperature. All geckos shed their skin at regular intervals, with species differing in timing and method. Leopard geckos will shed at about two- to four-week intervals; the presence of moisture aids in the shedding. When shedding begins, the gecko will speed the process by detaching the loose skin from its body and eating it. For young geckos, shedding will occur more at once every week, but when they grow, they shed once every one or two months. About 60% of gecko species have adhesive toe pads that allow them to adhere to most surfaces without the use of liquids or surface tension; such pads have been gained and lost over the course of gecko evolution.
Adhesive toepads evolved independently in about 11 different gecko lineages and were lost in at least 9 lineages. The spatula-shaped setae arranged in lamellae on gecko footpads enable attractive van der Waals' forces between the β-keratin lamellae/setae/spatulae structures and the surface; these van der Waals interactions involve no fluids. A recent study has however shown that gecko adhesion is in fact determined by electrostatic interaction, not van der Waals or capillary forces; the setae on the feet of geckos are self-cleaning and will remove any clogging dirt within a few steps. Teflon, which has low surface energy, is more difficult for geckos to adhere to than many other surfaces. Gecko adhesion is improved by higher humidity on hydrophobic surfaces, yet is reduced under conditions of complete immersion in water; the role of water in that system is under discussion, yet recent experiments agree that the presence of molecular water layers on the setae as well as on the surface increase the surface energy of both, therefore the energy gain in getting these surfaces in contact is enlarged, which results in an increased gecko adhesion force.
Moreover, the elastic properties of the b-keratin change with water uptake. Gecko toes seem to be "double jointed", but this is a misnomer and is properly called digital hyperextension. Gecko toes can hyperextend in the opposite direction from human toes; this allows them to overcome the van der Waals force by peeling their toes off surfaces from the tips inward. In essence, by this peeling action, the gecko separates spatula by spatula from the surface, so for each spatula separation, only some nN are necessary. Geckos' toes operate well below their full attractive capabilities most of the time, because the margin for error is great depending upon the surface roughness, the
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermions cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, extended to all fermions with his spin–statistics theorem of 1940. In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers: n, the principal quantum number, ℓ, the angular momentum quantum number, mℓ, the magnetic quantum number, ms, the spin quantum number. For example, if two electrons reside in the same orbital, if their n, ℓ, mℓ values are the same their ms must be different, thus the electrons must have opposite half-integer spin projections of 1/2 and −1/2. Particles with an integer spin, or bosons, are not subject to the Pauli exclusion principle: any number of identical bosons can occupy the same quantum state, as with, for instance, photons produced by a laser or atoms in a Bose–Einstein condensate.
A more rigorous statement is that with respect to exchange of two identical particles the total wave function is antisymmetric for fermions, symmetric for bosons. This means that if the space and spin co-ordinates of two identical particles are interchanged the wave function changes its sign for fermions and does not change for bosons; the Pauli exclusion principle describes the behavior of all fermions, while bosons are subject to other principles. Fermions include elementary particles such as quarks and neutrinos. Additionally, baryons such as protons and neutrons and some atoms are fermions, are therefore described by the Pauli exclusion principle as well. Atoms can have different overall "spin", which determines whether they are fermions or bosons — for example helium-3 has spin 1/2 and is therefore a fermion, in contrast to helium-4 which has spin 0 and is a boson; as such, the Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability, to the chemical behavior of atoms.
"Half-integer spin" means that the intrinsic angular momentum value of fermions is ℏ = h / 2 π times a half-integer. In the theory of quantum mechanics fermions are described by antisymmetric states. In contrast, particles with integer spin have symmetric wave functions. Bosons include the photon, the Cooper pairs which are responsible for superconductivity, the W and Z bosons. In the early 20th century it became evident that atoms and molecules with numbers of electrons are more chemically stable than those with odd numbers of electrons. In the 1916 article "The Atom and the Molecule" by Gilbert N. Lewis, for example, the third of his six postulates of chemical behavior states that the atom tends to hold an number of electrons in any given shell, to hold eight electrons which are arranged symmetrically at the eight corners of a cube. In 1919 chemist Irving Langmuir suggested that the periodic table could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set of electron shells around the nucleus.
In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons corresponded to stable "closed shells". Pauli looked for an explanation for these numbers. At the same time he was trying to explain experimental results of the Zeeman effect in atomic spectroscopy and in ferromagnetism, he found an essential clue in a 1924 paper by Edmund C. Stoner, which pointed out that, for a given value of the principal quantum number, the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the closed shell of the noble gases for the same value of n; this led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state, if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as electron spin.
The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric with respect to exchange. An antisymmetric two-particle state is represented as a sum of states in which one particle is in state | x ⟩ and the other in state | y ⟩, is given by: | ψ ⟩ = ∑ x, y A | x, y ⟩, antisymmetry under exchange means that A = −A; this implies A = 0 when x = y, Pauli exclusion. It is true in any basis. Conver
Chemisorption is a kind of adsorption which involves a chemical reaction between the surface and the adsorbate. New chemical bonds are generated at the adsorbant surface. Examples include macroscopic phenomena that can be obvious, like corrosion, subtler effects associated with heterogeneous catalysis; the strong interaction between the adsorbate and the substrate surface creates new types of electronic bonds. In contrast with chemisorption is physisorption, which leaves the chemical species of the adsorbate and surface intact, it is conventionally accepted that the energetic threshold separating the binding energy of "physisorption" from that of "chemisorption" is about 0.5 eV per adsorbed species. Due to specificity, the nature of chemisorption can differ, depending on the chemical identity and the surface structure. An important example of chemisorption is in heterogeneous catalysis which involves molecules reacting with each other via the formation of chemisorbed intermediates. After the chemisorbed species combine the product desorbs from the surface.
Self-Assembled monolayers are formed by chemisorbing reactive reagents with metal surfaces. A famous example involves thiols adsorbing onto the surface of gold; this process forms strong Au-SR bonds and releases H2. The densely packed SR groups protect the surface; as an instance of adsorption, chemisorption follows the adsorption process. The first stage is for the adsorbate particle to come into contact with the surface; the particle needs to be trapped onto the surface by not possessing enough energy to leave the gas-surface potential well. If it elastically collides with the surface it would return to the bulk gas. If it loses enough momentum through an inelastic collision it "sticks" onto the surface, forming a precursor state bonded to the surface by weak forces, similar to physisorption; the particle diffuses on the surface. It reacts with the surface or desorbs after enough energy and time; the reaction with the surface is dependent on the chemical species involved. Applying the Gibbs energy equation for reactions: Δ G = Δ H − T Δ S General thermodynamics states that for spontaneous reactions at constant temperature and pressure, the change in free energy should be negative.
Since a free particle is restrained to a surface, unless the surface atom is mobile, entropy is lowered. This means. Figure 1 is a graph of chemisorption energy curves of tungsten and oxygen. Physisorption is given as a Lennard-Jones potential and chemisorption is given as a Morse potential. There exists a point of crossover between the physisorption and chemisorption, meaning a point of transfer, it can occur above or below the zero-energy line, representing an activation energy requirement or lack of. Most simple gases on clean metal surfaces lack the activation energy requirement. For experimental setups of chemisorption, the amount of adsorption of a particular system is quantified by a sticking probability value. However, chemisorption is difficult to theorize. A multidimensional potential energy surface derived from effective medium theory is used to describe the effect of the surface on absorption, but only certain parts of it are used depending on what is to be studied. A simple example of a PES, which takes the total of the energy as a function of location: E = E e l + V ion-ion where E e l is the energy eigenvalue of the Schrödinger equation for the electronic degrees of freedom and V i o n − i o n is the ion interactions.
This expression is without translational energy, rotational energy, vibrational excitations, other such considerations. There exist several models to describe surface reactions: the Langmuir-Hinschelwood mechanism in which both reacting species are adsorbed, the Eley-Rideal mechanism in which one is adsorbed and the other reacts with it. Real systems have many irregularities, making theoretical calculations more difficult: Solid surfaces are not at equilibrium, they may be perturbed and irregular and such. Distribution of adsorption energies and odd adsorption sites. Bonds formed between the adsorbates. Compared to physisorption where adsorbates are sitting on the surface, the adsorbates can change the surface, along with its structure; the structure can go through relaxation, where the first few layers change interplanar distances without changing the surface structure, or reconstruction where the surface structure is changed. For example, oxygen can form strong bonds with metals, such as Cu.
This comes with the breaking apart of surface bonds in forming surface-adsorbate bonds. A large restructuring occurs by missing row as seen in Figure 2. A particular brand of gas-surface chemisorption is the dissociation of diatomic gas molecules, such as hydrogen and nitrogen. One model used to describe the process is precursor-mediation; the absorbed molecule is adsorbed onto a surface into a precu
In computational physics and chemistry, the Hartree–Fock method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Hartree–Fock method assumes that the exact, N-body wave function of the system can be approximated by a single Slater determinant or by a single permanent of N spin-orbitals. By invoking the variational method, one can derive a set of N-coupled equations for the N spin orbitals. A solution of these equations yields energy of the system. In the older literature, the Hartree–Fock method is called the self-consistent field method. In deriving what is now called the Hartree equation as an approximate solution of the Schrödinger equation, Hartree required the final field as computed from the charge distribution to be "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution; the solutions to the non-linear Hartree–Fock equations behave as if each particle is subjected to the mean field created by all other particles and hence, the terminology continued.
The equations are universally solved by means of an iterative method, although the fixed-point iteration algorithm does not always converge. This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method; the Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules and solids but it has found widespread use in nuclear physics.. In atomic structure theory, calculations may be for a spectrum with many excited energy levels and the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not the ground state. For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately; the rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case.
The discussion here is only for the Restricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals doubly occupied. Open-shell systems, where some of the electrons are not paired, can be dealt with by one of two Hartree–Fock methods: Restricted open-shell Hartree–Fock Unrestricted Hartree–Fock The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the Schrödinger equation in 1926. In 1927, D. R. Hartree introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions. Hartree was guided by some earlier, semi-empirical methods of the early 1920s set in the old quantum theory of Bohr. In the Bohr model of the atom, the energy of a state with principal quantum number n is given in atomic units as E = − 1 / n 2, it was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula.
By introducing the quantum defect d as an empirical parameter, the energy levels of a generic atom were well approximated by the formula E = − 1 / 2, in the sense that one could reproduce well the observed transitions levels observed in the X-ray region. The existence of a non-zero quantum defect was attributed to electron-electron repulsion, which does not exist in the isolated hydrogen atom; this repulsion resulted in partial screening of the bare nuclear charge. These early researchers introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data. Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e. ab initio. His first proposed method of solution became known as the Hartree Hartree product. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, its connection to the solution of the many-body Schrödinger equation was unclear.
However, in 1928 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz as a product of single-particle functions. In 1930, Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function; the Hartree method used the Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of quantum statistics, it was shown that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable ansatz for applying the variational principle. The original Hartree metho