Atomic orbital


In quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function describes an electron's charge distribution around the atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus.
Each orbital in an atom is characterized by a set of values of three quantum numbers,, and, which respectively correspond to an electron's energy, its orbital angular momentum, and its orbital angular momentum projected along a chosen axis. The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of and orbitals, and are often labeled using associated harmonic polynomials which describe their angular structure.
An orbital can be occupied by a maximum of two electrons, each with its own projection of spin. The simple names s orbital, p orbital, d orbital, and f orbital refer to orbitals with angular momentum quantum number and respectively. These names, together with their n values, are used to describe electron configurations of atoms. They are derived from description by early spectroscopists of certain series of alkali metal spectroscopic lines as sharp, principal, diffuse, and fundamental. Orbitals for continue alphabetically, omitting j because some languages do not distinguish between letters "i" and "j".
Atomic orbitals are basic building blocks of the atomic orbital model, a modern framework for visualizing submicroscopic behavior of electrons in matter. In this model, the electron cloud of an atom may be seen as being built up in an electron configuration that is a product of simpler hydrogen-like atomic orbitals. The repeating periodicity of 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 orbitals, respectively, though for higher values of quantum number, particularly when the atom bears a positive charge, energies of certain sub-shells become very similar and therefore, the order in which they are said to be populated by electrons can be rationalized only somewhat arbitrarily.

Electron properties

With the development of quantum mechanics and experimental findings, it was found that the electrons orbiting a nucleus could not be fully described as particles, but needed to be explained by wave–particle duality. In this sense, electrons have the following properties:
Wave-like properties:
  1. Electrons do not orbit a nucleus in the manner of a planet orbiting a star, but instead exist as standing waves. Thus the lowest possible energy an electron can take is similar to the fundamental frequency of a wave on a string. Higher energy states are similar to harmonics of that fundamental frequency.
  2. The electrons are never in a single point location, though the probability of interacting with the electron at a single point can be found from the electron's wave function. The electron's charge acts like it is smeared out in space in a continuous distribution, proportional at any point to the squared magnitude of the electron's wave function.
Particle-like properties:
  1. The number of electrons orbiting a nucleus can be only an integer.
  2. Electrons jump between orbitals like particles. For example, if one photon strikes the electrons, only one electron changes state as a result.
  3. Electrons retain particle-like properties such as: each wave state has the same electric charge as its electron particle. Each wave state has a single discrete spin depending on its superposition.
Thus, electrons cannot be described simply as solid particles. An analogy might be that of a large and often oddly shaped "atmosphere", distributed around a relatively tiny planet. Atomic orbitals exactly describe the shape of this "atmosphere" only when one electron is present. When more electrons are added, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection tends toward a generally spherical zone of probability describing the electron's location, because of the uncertainty principle.
One should remember that these orbital 'states', as described here, are merely eigenstates of an electron in its orbit. An actual electron exists in a superposition of states, which is like a weighted average, but with complex number weights. For instance, an electron could be in a pure eigenstate, or a mixed state + , or even the mixed state + . For each eigenstate, a property has an eigenvalue. Therefore, for the three states just mentioned, the value of is 2, and the value of is 1. For the second and third states, the value for is a superposition of 0 and 1. As a superposition of states, it is ambiguous—either exactly 0 or exactly 1—not an intermediate or average value like the fraction. A superposition of eigenstates and would have an ambiguous and, but would definitely be 1. Eigenstates make it easier to deal with the math. You can choose a different basis of eigenstates by superimposing eigenstates from any other basis.

Formal quantum mechanical definition

Atomic orbitals may be defined more precisely in formal quantum mechanical language. They are approximate solutions to the Schrödinger equation for the electrons bound to the atom by the electric field of the atom's nucleus. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, is approximated by an expansion into linear combinations of anti-symmetrized products of one-electron functions. The spatial components of these one-electron functions are called atomic 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.
In atomic physics, the atomic spectral lines correspond to transitions 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.
This notation means that the corresponding Slater determinants have a clear higher weight in the configuration interaction expansion. The atomic orbital concept is therefore a key concept for visualizing the excitation process associated with a given transition. For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are fermions ruled by the Pauli exclusion principle and cannot be distinguished from each other. Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when electron correlation is large.
Fundamentally, an atomic orbital is a one-electron wave function, even though many electrons are not in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital visualization heavily influenced by the Hartree–Fock approximation, which is one way to reduce the complexities of molecular orbital theory.

Types of orbital

Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the Schrödinger equation for a hydrogen-like "atom". Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The coordinate systems chosen for orbitals are usually spherical coordinates in atoms and Cartesian in polyatomic molecules. The advantage of spherical coordinates here is that an orbital wave function is a product of three factors each dependent on a single coordinate:. The angular factors of atomic orbitals generate s, p, d, etc. functions as real combinations of spherical harmonics . There are typically three mathematical forms for the radial functions which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons:
  1. The hydrogen-like orbitals are derived from the exact solutions of the Schrödinger equation for one electron and a nucleus, for a hydrogen-like atom. The part of the function that depends on distance r from the nucleus has radial nodes and decays as.
  2. The Slater-type orbital is a form without radial nodes but decays from the nucleus as does a hydrogen-like orbital.
  3. The form of the Gaussian type orbital has no radial nodes and decays as.
Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like orbitals. Gaussians are typically used in molecules with three or more atoms. Although not as accurate by themselves as STOs, combinations of many Gaussians can attain the accuracy of hydrogen-like orbitals.

History

The term orbital was introduced by Robert S. Mulliken in 1932 as short for one-electron orbital wave function. Niels Bohr explained around 1913 that electrons might revolve around a compact nucleus with definite angular momentum. Bohr's model was an improvement on the 1911 explanations of Ernest Rutherford, that of the electron moving around a nucleus. Japanese physicist Hantaro Nagaoka published an orbit-based hypothesis for electron behavior as early as 1904. These theories were each built upon new observations starting with simple understanding and becoming more correct and complex. Explaining the behavior of these electron "orbits" was one of the driving forces behind the development of quantum mechanics.