Angular Momentum, Chemistry tutorial

Introduction

Angular momentum is a possession of a physical system that is a steady of motion (is time-independent and well-defined) in 2 situations:

  • The system experiences a spherical symmetric potential field.
  • The system moves (in quantum mechanical sense) in isotropic space. In mutually cases the angular momentum operator commutes through the Hamiltonian of the system. By Heisenberg's uncertainty relation this means that the angular momentum can suppose a sharp value all together by the energy (eigenvalue of the Hamiltonian).

Definition of Angular Momentum Coupling

In quantum mechanics, the process of constructing eigenstates of whole angular momentum out of eigenstates of separate angular momenta is termed angular momentum coupling. For example, the orbit and spin of a single element can act together through spin-orbit interaction, in that case it  is  helpful  to  couple  the  spin  and  orbit  angular  momentum  of  the subdivision.  Or 2 charged particles, each by a well-defined angular momentum, might relate via Coulomb forces, in that case coupling of the 2 one-particle angular momenta to a sum angular momentum is a helpful step in the solution of the 2-particle Schrödinger equation. In both cases the divide angular momenta are no longer steady of motion, but the sum of the 2 angular momenta generally still is. Angular momentum coupling in atoms is of significance in atomic spectroscopy.

Angular momentum coupling of electron spins is of significance in quantum chemistry. As well in the nuclear shell form angular momentum coupling is ubiquitous.

Common Theory and Detailed Origin

Angular momentum is a property of a physical system that is a steady of motion (is time-independent and well-described) in 2 situations: (i) the system experience a spherical symmetric potential field. (ii) The system moves (in quantum mechanical sense) in isotropic space. In both cases the angular momentum operator commutes with the Hamiltonian of the system. By Heisenberg's uncertainty relation this means that the angular momentum can assume a sharp value concurrently through the energy (eigenvalue of the Hamiltonian). An instance of the 1st situation is an atom whose electrons only feel the Coulomb field of its nucleus. If we ignore the electron-electron interaction (and other small interactions such as spin-orbit coupling), the orbital angular momentum l of each electron commutes via the total Hamiltonian. In this model the atomic Hamiltonian is a addition of kinetic energies of the electrons and the spherical symmetric electron-nucleus interactions. The individual electron angular momenta l (i) commute through this Hamiltonian. That is, they are preserved properties of this approximate representation of the atom.

An instance of the 2nd situation is a stiff rotor moving in field-free space. A rigid rotor has a well-described, time-independent, angular momentum.

Such 2 situations originate in classical mechanics. The 3rd type of conserved angular momentum, associated with spin, doesn't contain a classical counterpart. Though, all rules of angular momentum coupling are relevant to spin too. In common the conservation of angular momentum implies full rotational symmetry (explained via the groups SO(3) and SU(2) and, equally, spherical symmetry implies conservation of angular momentum. If 2or more physical schemes have conserved angular momenta, it can be helpful to adjoin such momenta to a entire angular momentum of the joined system-a conserved property of the total system. The building of eigenstates of the total conserved angular momentum from the angular momentum eigenstates of the entity subsystems is termed to as angular momentum coupling.

Application of angular momentum coupling is useful when there is an interaction between subsystems that, without interaction, would have conserved angular momentum. By the very interaction the spherical symmetry of the subsystems is broken, but the angular momentum of the whole system continues a steady of motion. Utilize of the latter fact is helpful in the solution of the Schrödinger equation.

As an instance we consider 2 electrons, 1 and 2, in an atom (say the helium atom). If there is no electron-electron interaction, but only electron nucleus interaction, the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. Both operators, l(1) and l(2), are conserved. Though, if we switch on the electron-electron interaction depending on the distance d(1,2) between the electrons, then only a simultaneous and equal rotation of the two electrons will leave d(1,2) invariant. In such a case neither l(1) nor l(2) is a constant of motion but L =  l(1) + l(2) is. Given eigenstates of l(1)  and   l(2),  the  construction  of  eigenstates  of  L (that still  is conserved) is the coupling of the angular momenta of electron 1 and 2.

In quantum mechanics, coupling as well survives between angular momenta belonging to dissimilar Hilbert spaces of a single object, for instance its spin and its orbital angular momentum. Reiterating faintly another way the above: one expands the quantum conditions of composed systems (that is made of subunits like 2 hydrogen atoms or two electrons) in basis sets that are made of straight products of quantum states which in turn explain the subsystems separately. We assume that the locations of the subsystems can be chosen as eigenstates of their angular momentum operators (and of their component along any arbitrary z axis). The subsystems are therefore properly explained via a set of l, m quantum numbers (see angular momentum for details). When there is interaction between the subsystems, the total Hamiltonian contains terms that do not commute through the angular operators acting on the subsystems only. Though, such terms do commute by the total angular momentum operator. Sometimes one refers to the non-commuting contact terms in the Hamiltonian as angular momentum blending terms, because they demand the angular momentum coupling.

The methods in that the angular momenta connected through the orbital and spin motions in many-electron-atoms can be combined together are many and varied. In spite of this seeming complexity, the results are frequently readily computed for easy atom systems and are used to characterize the electronic states of atoms.

There are 2 principal coupling schemes utilized: Russell-Saunders (or L-S) coupling and j-j coupling. The interactions that can happen are of 3 kinds. Spin-spin coupling orbit-orbit coupling spin-orbit coupling.

Ls coupling

In light atoms (usually Z<30), electron spins is interact among themselves so they join to form a total spin angular momentum S. The similar happens via orbital angular momenta li, shaping a single orbital angular momentum L. The interaction between the quantum numbers L and S is termed Russell-Saunders Coupling or LS Coupling. Then S and L add jointly and form a total angular momentum J:

This is an approximation that is good as long as any external magnetic fields are weak. In superior magnetic fields, such 2 momenta decouple, giving increase to a different splitting pattern in the energy levels (the Paschen-Back effect.), and the size of LS coupling term becomes small. In the Russell Saunders scheme (named after Henry Norris Russell, 1877-1957 a  Princeton Astronomer and Frederick Albert Saunders, 1875-1963 a Harvard Physicist and published in Astrophysics Journal, 61, 38, 1925 ) it is supposed that: spin-spin coupling > orbit-orbit coupling > spin-orbit coupling. This is establish to provide a good approximation for first row transition series where J coupling can commonly be ignored, though for components through atomic number greater than thirty, spin-orbit coupling becomes more significant and the j-j coupling scheme is utilized.

Spin-spin coupling

Spin-spin coupling is the coupling of the intrinsic angular momentum (spin) of different particles. Such coupling between pairs of nuclear spins is a significant characteristic of nuclear magnetic resonance (NMR) spectroscopy as it can provide detailed information about the structure and conformation of molecules. Spin-spin coupling between nuclear spin and electronic spin is responsible for hyperfine structure in atomic spectra. S is the consequential spin quantum number for a system of electrons. The overall spin S arises from adding the individual ms mutually and is as a consequence of coupling of spin quantum numbers for the divide electrons.

Orbit-orbit coupling

L - The whole orbital angular momentum quantum number describes the energy state for a system of electrons. Such states or term letters are symbolized as follows (Lancashire, 2006):

Spin-orbit coupling

The behaviour of atoms and smaller particles is well described via the theory of quantum  mechanics, in that each particle has an intrinsic angular  momentum  termed  spin  and  precise  configurations  (of  for instance electrons in an atom) are described by a set of quantum  numbers. Collections of particles as well have angular momenta and corresponding quantum numbers, and under dissimilar circumstances the angular momenta of the parts add in different methods to form the angular momentum of the whole. Angular momentum coupling is a category including several of the techniques that subatomic particles can act together by each other.

Spin-orbit coupling as well recognized as spin-pairing describes a weak magnetic interaction, or coupling, of the particle spin and the orbital motion of this particle, for example the electron spin and its motion around an atomic nucleus. One of its effects is to divide the energy of internal states of the atom, for example spin-aligned and spin-ant aligned that would or else be identical in energy. This interaction is responsible for many of the factors of atomic structure.

In the macroscopic world of orbital mechanics, the term 'spin-orbit coupling' is sometimes utilized in the similar sense as spin-orbital resonance. Coupling happens between the consequential spin and orbital momenta of an electron that provides increase to J the total angular momentum quantum number. Multiplicity happens whenever numerous levels are close mutually and is agreed via the formula (2S+1).

The Russell Saunders expression symbol that consequences from such considerations is specified via:

(2S+1)L

JJ Coupling

In heavier atoms the situation is different. In atoms via bigger nuclear charges, spin-orbit interactions are often as large as or larger than spin-spin contacts or orbit-orbit interactions. In this situation, each orbital angular momentum li tends to join by each individual spin angular momentum si, originating entity total angular momenta ji. Such then  add  up  to  shape  the  total  angular momentum J This portrayal, facilitating computation of this type of interaction, is recognized as jj coupling.

Term symbols

Term symbols are utilized to symbolize the states and spectral conversions of atoms, they are establishing from coupling of angular momenta stated above. When the state of an atom has been specified through a term symbol, the permitted transitions can be found through selection rules via considering that transitions would conserve angular momentum. A photon has spin 1, and when there is a transition through emission or absorption of a photon the atom will require transforming state to conserve angular momentum.  The term symbol selection rules are.  ΔS=0,

ΔL=0,±1, Δl=±1, ΔJ=0,±1

Nuclear coupling

In atomic nuclei, the spin-orbit interaction is much stronger than for atomic electrons, and is incorporated directly into the nuclear shell model. In addition, unlike atomic-electron term symbols, the lowest energy state isn't L - S, but rather, l + s. All nuclear levels whose l value (orbital angular momentum) is greater than zero are consequently split in the shell model to generate states allocated via l + s and l - s. Due to the nature of the shell model, that assumes an average potential rather than a central Coulombic potential, the nucleons that go into the l + s and l - s nuclear states are considered degenerate within each orbital (for example The 2p3/2 contains four nucleons, all of the similar energy. Higher in energy is the 2p1/2 that encloses two equal-energy   nucleons).

Review of Quantum Numbers

Electrons in an atom reside in shells characterized by a particular value of n, the principal quantum number. Within each shell an electron can absorb an orbital that is further characterized via an orbital quantum number, l, where l can obtain all values in the range:

l = 0, 1, 2, 3, ... , (n-1),

traditionally termed s, p, d, f, etc. orbitals.

Each orbital has a feature shape reflecting the motion of the electron in that meticulous orbital, this motion being described via an angular momentum that reflects the angular velocity of the electron moving in its orbital. A quantum mechanics approach to determining the energy of electrons in an component or ion is depend on the outcome obtained through solving the Schrödinger wave equation for the H-atom. The different solutions for the different energy states are characterized via the 3 quantum numbers, n, l and ml.

ml is a subset of l, where the allowable values are: ml  = l, l-1, l-2, ..... 1, 0, -1,......., -(l-2), -(l-1), -l. There are thus (2l +1) values of ml for each l value, i.e. one s orbital (l = 0), three p orbitals (l = 1), five d orbitals (l = 2), etc. There is a fourth quantum number; ms that identifies the orientation of the spin of one electron relative to those of other electrons in the system. A single electron in free space has a basic property associated by it described spin, arising from the spinning of an asymmetrical charge distribution about its own axis. Like an electron moving in its orbital around a nucleus, the electron spinning about its axis has connected through its motion a well-described angular momentum. The value of ms is either + ½ or - ½.

In summary then, each electron in an orbital is characterized via 4 quantum numbers (Lancashire, 2006):

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