Atomic orbital

In atomic theory and quantum mechanics, an atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term atomic orbital may also refer to the physical region or space where the electron can be calculated to be present, as defined by the particular mathematical form of the orbital.Drum mode u 01 {displaystyle u_{01}} Drum mode u 02 {displaystyle u_{02}} Drum mode u 03 {displaystyle u_{03}} Wave function of 1s orbital (real part, 2D-cut, r m a x = 2 a 0 {displaystyle r_{max}=2a_{0}} )Wave function of 2s orbital (real part, 2D-cut, r m a x = 10 a 0 {displaystyle r_{max}=10a_{0}} )Wave function of 3s orbital (real part, 2D-cut, r m a x = 20 a 0 {displaystyle r_{max}=20a_{0}} )Drum mode u 11 {displaystyle u_{11}} Drum mode u 12 {displaystyle u_{12}} Drum mode u 13 {displaystyle u_{13}} Wave function of 2p orbital (real part, 2D-cut, r m a x = 10 a 0 {displaystyle r_{max}=10a_{0}} )Wave function of 3p orbital (real part, 2D-cut, r m a x = 20 a 0 {displaystyle r_{max}=20a_{0}} )Wave function of 4p orbital (real part, 2D-cut, r m a x = 25 a 0 {displaystyle r_{max}=25a_{0}} )Mode u 21 {displaystyle u_{21}} (3d orbital)Mode u 22 {displaystyle u_{22}} (4d orbital)Mode u 23 {displaystyle u_{23}} (5d orbital) In atomic theory and quantum mechanics, an atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term atomic orbital may also refer to the physical region or space where the electron can be calculated to be present, as defined by the particular mathematical form of the orbital. Each orbital in an atom is characterized by a unique set of values of the three quantum numbers n, ℓ, and m, which respectively correspond to the electron's energy, angular momentum, and an angular momentum vector component (the magnetic quantum number). Each such orbital can be occupied by a maximum of two electrons, each with its own spin quantum number s. The simple names s orbital, p orbital, d orbital and f orbital refer to orbitals with angular momentum quantum number ℓ = 0, 1, 2 and 3 respectively. These names, together with the value of n, are used to describe the electron configurations of atoms. They are derived from the description by early spectroscopists of certain series of alkali metal spectroscopic lines as sharp, principal, diffuse, and fundamental. Orbitals for ℓ > 3 continue alphabetically, omitting j (g, h, i, k, ...) because some languages do not distinguish between the letters 'i' and 'j'. Atomic orbitals are the basic building blocks of the atomic orbital model (alternatively known as the electron cloud or wave mechanics model), a modern framework for visualizing the submicroscopic behavior of electrons in matter. In this model the electron cloud of a multi-electron atom may be seen as being built up (in approximation) in an electron configuration that is a product of simpler hydrogen-like atomic orbitals. The repeating periodicity of the blocks of 2, 6, 10, and 14 elements within sections of the periodic table arises naturally from the total number of electrons that occupy a complete set of s, p, d and f atomic orbitals, respectively, although for higher values of the quantum number n, particularly when the atom in question bears a positive charge, the energies of certain sub-shells become very similar and so the order in which they are said to be populated by electrons (e.g. Cr = 4s13d5 and Cr2+ = 3d4) can only be rationalized somewhat arbitrarily. With the development of quantum mechanics and experimental findings (such as the two slit diffraction of electrons), it was found that the orbiting electrons around a nucleus could not be fully described as particles, but needed to be explained by the wave-particle duality. In this sense, the electrons have the following properties: Wave-like properties: Particle-like properties: Thus, despite the popular analogy to planets revolving around the Sun, electrons cannot be described simply as solid particles. In addition, atomic orbitals do not closely resemble a planet's elliptical path in ordinary atoms. A more accurate analogy might be that of a large and often oddly shaped 'atmosphere' (the electron), distributed around a relatively tiny planet (the atomic nucleus). Atomic orbitals exactly describe the shape of this 'atmosphere' only when a single electron is present in an atom. When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection (sometimes termed the atom's 'electron cloud') tends toward a generally spherical zone of probability describing the electron's location, because of the uncertainty principle. Atomic orbitals may be defined more precisely in formal quantum mechanical language. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, is approximated by an expansion (see configuration interaction expansion and basis set) into linear combinations of anti-symmetrized products (Slater determinants) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their spin component, one speaks of atomic spin orbitals.) A state is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this independent-particle model of products of single electron wave functions. (The London dispersion force, for example, depends on the correlations of the motion of the electrons.) In atomic physics, the atomic spectral lines correspond to transitions (quantum leaps) between quantum states of an atom. These states are labeled by a set of quantum numbers summarized in the term symbol and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (for example, 1s2 2s2 2p6 for the ground state of neon—term symbol: 1S0).