Angular Momentum in Chemistry Jan C. A. Boeyens Unit for Advanced Studies, University of Pretoria, South Africa Reprint requests to J. C. A. Boeyens. E-mail: [email protected] Z. Naturforsch. 2007, 62b, 373 – 385; received October 19, 2006 Dedicated to Prof. Helgard G. Raubenheimer on the occassion of his 65th birthday Noting that current chemical theory is based almost exclusively on electronic energy and spin variables the equal importance of orbital angular momentum is explored in this paper. From its clas- sical definition the angular momentum of electrons in an atom is shown to obey Laplace’s equation, which automatically leads to discrete values in terms of spherical harmonics. This analysis assumes a continuous distribution of electronic charge, which resembles a fluid at equilibrium. It serves to elucidate the success and failure of Bohr’s conjecture and the origin of wave-particle duality. Applied to atoms, minimization of orbital angular momentum leads to Hund’s rules. The orientation of angu- lar momenta in lower-symmetry molecular environments follows from the well-known Jahn-Teller theorem. Key words: Bohr Conjecture, Laplace’s Equation Introduction chemical bonding that gained universal recognition in chemistry. Study of the physical world is made possible by the recognition of several fundamental symmetries. Ac- cording to Noether’s theorem each symmetry leads to Angular Momentum the recognition of a conservation law [1]. In dynamical Angular momentum in classical mechanics is a vec- systems conservation of mass, energy, momentum and tor quantity that describes rotational motion and must angular momentum derive from the assumed homo- be of obvious importance for the understanding of geneity and isotropy of space-time. Together with the electronic motion in atoms and molecules. The angular conservation of electronic charge, which arises from momentum vector that quantifies circular rotation of a the internal symmetry of the electromagnetic field, all particle about an axis is defined as the moment of the of these conservation laws are of basic importance in linear momentum, L = r × p, i. e. the vector product of the theoretical understanding of chemistry. the linear momentum and the radius vector from the The conservation laws of mass and energy are the point of rotation. It is directed along the axis of rota- cornerstones of the theories of chemical composition tion, perpendicular to the plane of rotation, and is pos- and thermodynamics. The kinetic theory of gases re- itive in the direction that drives a right-handed screw lies on the conservation of linear momentum and the (Fig. 1). conservation of electronic charge underpins all mod- els of chemical transformation. In contrast, the con- servation of angular momentum is largely ignored, except by reference to electron spin, which is only one quantum-mechanical aspect of the total angular momentum of an electron. This omission leaves an unbridgeable gap in the theory of molecular struc- ture and conformation, which has been a source of endless frustration for almost a century. The reason for this state of affairs can be traced to the uncrit- Fig. 1. Definition of the angular ical embrace of an unreasonable model of directed momentum vector. 0932–0776 / 07 / 0300–0373 $ 06.00 © 2007 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com 374 J. C. A. Boeyens · Angular Momentum in Chemistry The classical angular momentum vector decom- The mystery of Bohr’s conjecture and the calcula- poses into cartesian components. From the definition tion of Rydberg’s constant, based on this conjecture, of has never been resolved satisfactorily. The mystery ( ) consists therein that the quantum number n ,which iijjjkk quantifies the angular momentum L = r × p = xyz = rpsinα (1) px py pz mvr = nh¯ (6) simple vector algebra [2] defines the components of L is combined with the Coulomb potential V = eφ = 2 as −e /4πε0r to yield the total energy L = yp − zp , (2) Ry x z y E = − (7) n2 Ly = zpx − xpz, (3) of the hydrogen ground-state electron in rydberg units. Lz = xpy − ypx (4) In reality however, the ground state requires values of n =0andn = 1 in eq. (6) and (7) respectively. The and confusion arises from the fact that the quantum num- L · L = L2 = L2 + L2 + L2. (5) ber of eq. (6) is the azimuthal quantum number ml, x y z whereas n of eq. (7) is the principal quantum number, The moment of the force, or torque associated with and both of these are integers. The wrong assumption temporal change of angular momentum, N = ∂L/∂t,is is that (6) refers to the hydrogen ground state. In fact, also defined as a vector product, N = r × (dp/dt),in it simply defines the smallest unit of non-zero orbital / the same way as the angular momentum. angular momentum,h ¯ = mvr ml, as equivalent to the On analyzing the angular momenta of electrons in angular momentum of a point particle of mass m and an atom another important factor comes into play. With linear velocity v, in a circular orbit of radius r, about the total positive charge concentrated at the point-like a fixed point. The energy of such a hypothetical point nucleus the electron density is considered to be in a particle on a stable orbit defines the corresponding ryd- central Coulomb field with rotation about a fixed point berg unit of energy, rather than an axis. In the absence of external torque, me4 me4 e. g. a magnetic field, the electronic charge on an atom Ry = = . ε2 2 ( πε )2 2 constitutes a stable mechanical system. With all forces 8 0 h 2 4 0 h¯ and torques in balance the angular momentum L there- The Bohr model introduced the enduring concept of fore satisfies an equation like that of Laplace. Allowed stationary quantum states and first defined fundamental rotational modes appear as the solutions to the angular- concepts such as the bohr magneton, µ = eh¯/2mc and dependent part of the equation, the precise details of B the fine-structure constant, α = v /c = e2/hc¯ 1/137, which depend on the assumed nature of the extranu- 1 still in common use. However, inability to extend the clear electrons. description to more complex atoms made it unaccept- able as a chemical model. The unphysical definition The Bohr conjecture of the stationary state in terms of an accelerated elec- The most advanced model, that of quantum field the- tron that fails to radiate, remained the major objection ory, defines the electron as a zero-dimensional point against the model. particle. Its properties of mass, charge and spin are Although this major problem could not be resolved considered to be of mathematical significance only. Ef- Sommerfeld [3] extended the Bohr treatment to give forts to form a physical picture of the electrons are con- reasonable accounts of atomic structure in general, the sidered meaningless. Nevertheless, the first successful formation of chemical bonds and the periodic table of account of atomic structure was based on such a parti- the elements. This was achieved by the introduction cle model and Bohr’s conjecture that the angular mo- of elliptic orbits in line with Kepler’s laws of plane- mentum of an orbiting electron is quantized in units tary motion and widening of Bohr’s conjecture by the ofh ¯. formulation of quantum rules for periodic systems in J. C. A. Boeyens · Angular Momentum in Chemistry 375 idea of defining tetrahedral orbitals by hybridization is an obvious attempt to emulate the attractive Sommer- feld construct, albeit at the expense of the fundamental principle of angular momentum conservation. Fig. 2. Sommerfeld model of the The need to introduce half-integer quantum num- neon atom in terms of four elliptic bers can now be seen to have arisen from inappropriate and four circular orbits, both types use of Bohr’s conjecture, assuming non-zero angular in tetrahedral array. momentum for electrons in s-states. This dilemma sig- nalled the ultimate failure of Sommerfeld’s model. terms of action integrals such as pkdqk = nkh, k = 1,2,.., n = integer, where q and p are generalized coor- The unsuccessful Bohr-Sommerfeld model was dinates and their canonically conjugate momenta. Use based on the assumption that an orbiting particle pro- of these rules enabled the assignment of electronic, vi- vided the only possible source of electronic orbital an- brational and rotational molecular spectra and predicts gular momentum in an atom. A valid alternative is de- Bragg’s law for the interpretation of diffraction phe- fined by a hydrodynamic model that pictures the elec- nomena. tron as an ideal compressible fluid that surrounds the Success of the Sommerfeld model was due to the nucleus. At equilibrium, in a stationary state, the fluid introduction of two additional quantum numbers, re- appears incompressible, in accord with one of the pi- quired to describe elliptic orbits. The scheme is illus- oneering interpretations [4] of the Schr¨odinger func- trated by the electronic structure that it proposed for tion, which has never been fully appreciated. In this the neon atom. Ten electrons, as shown in Fig. 2, are model angular momentum derives from the hydrody- distributed over two shells, or energy levels, on sets namic circulation, or vorticity, of the fluid, which is of two and eight orbits respectively. The inner shell subject to the central Coulombic attraction [5] and be- accommodates two electrons1 while the second shell comes observable in an applied magnetic field. The consists of four elliptic and four circular orbits, ar- components of vorticity, at equilibrium, are defined in ranged in space, such that the total electronic angular terms of the components of flux, u, v, w as: momentum is quenched.