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. ∂w ∂v ∂u ∂w ∂v ∂u ξ = − , η = − , ζ = − . (8) Quenching is achieved by directing the angular mo- ∂y ∂z ∂z ∂x ∂x ∂y mentum vectors of the electrons on elliptic orbits to- wards the corners of a regular tetrahedron. Angular For an electron of mass me the relationship between momentum vectors of the electrons on circular orbits vorticity and angular momentum follows from (2) as ξ · ( , , ) / are in turn directed to tetrahedral sites which com- dr x y z = Lx me, etc. plete the cube that also contains the corners of the The continuity equation first tetrahedron. For the carbon atom only the four ∂u ∂v ∂w equivalent elliptic orbits in tetrahedral array are oc- + + = 0 ∂x ∂y ∂z cupied. By analogy with the Lewis-Langmuir model of electron-pair bonds, this predicted arrangement was through (8), implies the Helmholtz relation argued with great effect, to account for the structure ∂ξ ∂η ∂ζ of aliphatic compounds. This proposition has turned + + = 0 ∂x ∂y ∂z out to be the most persistent legacy of the Sommerfeld model, although the sound principle of quenched angu- and hence2 a circulation potential ρ, which cor- lar momentum on which it was based, has been aban- responds to an angular momentum that satisfies doned in modern theories of chemical bonding. The Laplace’s equation.
1 The helium problem remained unsolved in the Laplace’s equation Sommerfeld scheme as quenching of the angular momentum of two electrons, both with quantum number n = 1 and together on the same Laplace’s equation, in cartesian coordinates, reads circular orbit (L = ±h¯), requires them to orbit in opposite sense z ∂2 ∂2 ∂2 on a collision course. A tentative solution pictures the electrons as 2 V V V ± / V = + + = 0. (9) moving in phase on elliptic orbits with Lz = h¯ 2, as shown in the ∂x2 ∂y2 ∂z2 diagram. This model, however crude, anticipated the description of two s-electrons with paired spins. 2Note that ∂ρ/∂x = ξ, etc. 376 J. C. A. Boeyens · Angular Momentum in Chemistry
It describes the situation in which the potential has the least mean gradient. For instance, the condition d2y/dx2 = 0 requires only that the curve described by the moving point shall have zero curvature, i. e. y = mx + c, a straight line. The equation may be solved by separation of the variables under the assumption that the potential may be written as a product function
V = X(x) ·Y(y) · Z(z) (10) in which X, Y and Z are functions of only one inde- pendent variable. On substitution of (10) into (9) and division by V, the equation becomes Fig. 3. Cartesian compo- nents of angular momen- tum L. 1 d2X 1 d2Y 1 d2Z + + = 0. (11) X dx2 Y dy2 Z dz2 remaining two must be complex quantities, giving so- lutions Whereas the value of each term in (11) is independent of the other two, each of them must be equal to a con- ±kx ±iky X = c1e , Y = c2e . stant, to give three equations of the type Solutions of (9) follow as d2V = KX. = ±k(x±iy), dx2 Vk c1e (14)
2 = + . Writing K as a squared quantity, K = ki , each equation Vz c2z c3 (15) 3 has a simple solution , such that These equations describe the components of angular + + momentum on an arbitrary scale that depends on the V(k ,k ,k )=ek1x k2y k3z, (12) 1 2 3 value of k. Non-zero ±k implies electronic rotation, in either direction, on a plane perpendicular to L.The k2 + k2 + k2 = 0. (13) 1 2 3 total angular momentum vector is directed at an arbi- These solutions describe both the distribution of charge trary angle, θ, with respect to Z and it may be con- density as well as the orbital angular momentum states sidered as precessing about Z as shown in Fig. 3. The of electronic charge on the atom. Without loss of gen- z-component, Lz, has a fixed value, but the components erality it is sufficient to consider only the one-electron Lx and Ly are variable and depend on the azimuthal an- case of hydrogen-like atoms. In this instance it makes gle ϕ. no sense to consider the electron as a point particle, but On writing the Laplacian in spherical polar coordi- rather as unit charge, symmetrically distributed over nates it consists of a radial part and an angular part, Λ 2, the available space that surrounds the nucleus [1, 7]. such that Eq. (13) shows that the three separation constants can- 1 ∂ ∂ not all be real numbers, except for three equal con- 2 = r2 +Λ 2 , (16) 2 ∂ ∂ stants of zero, which imply L = 0, the state of spherical r r r symmetry, quantum-mechanically known as an s-state. 1 ∂ ∂ 1 ∂2 The case of two zero constants will later be shown not Λ 2 = sinθ + . (17) to be allowed quantum-mechanically, and classically it sinθ ∂θ ∂θ sin2 ∂ϕ2 corresponds to the Bohr model of the H ground-state The angular momentum evidently depends on Λ 2 only. electron on a circular orbit with Lz =¯h. Separation of the variables is achieved by writing the Of particular interest is the case where one of the potential as the product function V = R(r) · Θ(θ) · constants (k3 say) is equal to zero. To satisfy (13) the Φ(ϕ), first multiplied by r2 sin2 θ and divided by V 3The roots of the auxiliary equation D2 − k2 =0are±k. Hence to isolate the azimuthal function, with separation con- X = cexp(±kx). stant m2. J. C. A. Boeyens · Angular Momentum in Chemistry 377
m Hence described by Yl and the projection Lz by (19). This surprising result shows that classical theory predicts sin2 θ d dR sinθ d dΘ r2 + sinθ = m2, (18) the angular momentum of an electron on an atom as R dr dr Θ dθ dθ restricted to discrete values depending on the integers l and m. Only the units are lacking and the success of the d2Φ = −m2Φ, Φ = ce±imϕ . (19) Bohr model of the atom consisted in correctly guessing dϕ2 the unit of angular momentum quantization ash ¯.The 2 − 2Λ 2 Single-valued solutions for (19) require that Φ(ϕ) = guess implies that L = h¯ and transforms eq. (17) Φ(ϕ + 2π) which implies integer m. Eq. (18) is next into the eigenvalue equation 2 θ divided by sin ,togive L2Y m(θ,ϕ)=l(l + 1)h¯2Y m(θ,ϕ) (23) l l 1 d dR r2 = c, (20) which defines the allowed values of L2 = l(l + 1)h¯2. R dr dr Likewise 1 1 d dΘ m2 m = m sinθ − + c = 0. (21) LzYl mhY¯ l (24) Θ sinθ dθ dθ sin2 θ defines the eigenvalues of the Z-component of angular Eq. (21) is recognized as Legendre’s associated equa- ( + ) momentum. tion, in which the separation constant c = l l 1 en- The fact that these values agree exactly with the sures well-behaved solutions for integer l [2]. Written quantum-mechanical results is hardly surprisingly see- in the form ing that Schr¨odinger’s amplitude equation 1 ∂ ∂Y 1 ∂2Y 2m sinθ + +l(l +1)Y = 0. 2ψ = − (E −V)ψ (25) sinθ ∂θ ∂θ sin2 θ ∂ϕ2 h¯2 (22) is the appropriate modification of Laplace’s equation Eq. (22) is the general differential equation of spheri- that leads to the correct quantization of the kinetic en- (T E −V) cal surface harmonics of integral order and finite over ergy = and angular momentum. the unit sphere. At the surface of a sphere of constant Hydrodynamic Model radius r = a,dV/dr = 0, the first term of the Laplacian (16) vanishes and the remainder reduces to (22). It remains to be demonstrated that the hydrody- Any solution Vl of Laplace’s equation of degree l is namic model correctly predicts the same numerical re- called a solid spherical harmonic, and has the form Vl = sults as the Bohr model. In order to produce a sta- l m(θ,ϕ) + r Yl . After separation of variables there are 2l ble atom it is necessary, as before, to assume that an m(θ,ϕ) Θ(θ) · Φ(ϕ) 1 independent functions Yl = = electrostatic force of attraction be balanced by some imϕ m( θ) m Ne Pl cos for each value of l.Forr =1,Vl = Yl other force, presumably of quantum-mechanical ori- m 4 so that Yl is the value of the solid harmonic at points gin . Whatever the nature of this force, it should be on the surface of the unit sphere defined by the coor- of the same magnitude as a classical centrifugal force. θ ϕ m dinates and , and hence Yl is called a surface har- Assuming the negative charge to be effectively concen- monic of degree l. Surface harmonics are orthogonal trated at a radial distance r from the nucleus, the force on the surface of the unit sphere. The associated Leg- balance requires that m( θ) − endre polynomials Pl cos have l m roots (zeros). 2 m v2 2 Each of them defines a nodal cone that intersects a con- e = e ϕ = p 2 (26) stant sphere in a circle, which defines a constant lati- 4πε0r r mer tude. These nodes are in the surface of the sphere and not at r = 0, as commonly assumed in the definition i. e. (dropping subscripts) of atomic orbitals. Surface harmonics are undefined at 1 me2 r =0. = . πε ( )2 A moment’s reflection on the geometry outlined in r 4 0 pr Fig. 3 suggests that the total angular momentum L2 is 4Identified in Bohmian mechanics as the quantum potential. 378 J. C. A. Boeyens · Angular Momentum in Chemistry
This r should not be interpreted to define a circular or- bit. Conservation of energy requires 1 e2 1 1 E = T +V = mv2 +V = − 2 4πε 2r r 0 (27) 4 = − me = − 1 . 2 2 Ry 2 2(4πε0) (pr) n The last equality is inferred from the spectroscopic Ry- dberg formula, 1 1 ∆E = Ry − , n1 n2 for integer ni. Noting that Planck’s constant h =2πh¯ has units of angular momentum (Js), an obvious way of introducing the quantum-mechanicalinteraction and establish the Rydberg formula is by substituting, with Bohr, pr = nh¯ in (27), to give me4 me4 1 E = − = − = −Ry (28) ( πε )2 2 2 ε 2 2 2 2 4 0 n h¯ 8 0n h n Fig. 4. Graphical illustration of two spherical surface har- monics on the unit sphere. The lower part of the diagram with the correct value of Rydberg’s constant. It is im- shows the mapping centred on a point at r = 0. In all cases portant to understand that the final expression (28) for 3D images require rotation about Z. total energy does not imply non-zero angular momen- ±1 tum of the ground state. It has a node along the equator. Y1 is on the right. In The dynamic model for hydrogen, assumed here, hydrodynamics surface oscillations are linked to a ve- is equivalent to the hydrodynamics of a liquid globe locity potential due to a point source at the origin [5]. with free surface, around a gravitational nucleus, used Electronic structure in the analysis of tidal waves [5]. During free motion of the system surface elements of fluid undergo sim- Resolution of the Bohr paradox depends on the as- ple harmonic oscillation along a normal coordinate. sumed nature of an electron. It is intuitively clear that These surface oscillations represent the infinite sum any such picture should be more detailed than the over all surface harmonics of integral order. They com- zero-dimensional point-particle of elementary-particle bine to form progressive waves that travel over the sur- physics, or the alternative notion of pure wave motion. face with no change in form. The motion of surface It seems self-evident that the definition of an entity, elements keep in step, passing simultaneously through which occurs in a void as either particle or wave, re- their equilibrium positions. Any free motion may be quires that it has substance. The alternative is that it simulated by superposition of the surface modes with a exists in a substantial medium. In this case the ulti- proper choice of amplitude and phase. One of the sim- mate description of matter, when analyzed beyond the plest solutions represents a system of standing waves, elementary-particle level, must correspond to a distor- which in the case of an electron, leads directly to the tion of, or an inhomogeneity, in the vacuum, assumed condition 2πr = nλ, n = 0,1,...that,combinedwith to be a continuous fluid, or aether. An object such as an De Broglie’s relation λ = h/p, produces the Bohr con- electron is therefore considered made up of the same jecture, pr = nh/2π. homogeneous stuff as the aether and owing its unique Oscillations at the surface of a sphere are prop- features to characteristic wave patterns that occur in its erly described by the spherical surface harmonics, interior or at its interface with the vacuum. m(θ,φ) Yl , i. e. those harmonics that satisfy Laplace’s A completely free electron, considered as a highly equation on the surface of a sphere. The surface har- compressible fluid, will presumably be of infinite ex- 0 / π θ monic Y1 = 3 4 cos is shown on the left in Fig. 4. tent and its most prominent property will be the wave J. C. A. Boeyens · Angular Momentum in Chemistry 379 motion. A closely confined electron resembles a me- the claim that the hydrodynamic model is in all re- chanical particle with less obvious wave properties. In spects equivalent to that of Schr¨odinger, Madelung [4] all intermediate situations both wave-like and particle- admits that a continuous distribution of electronic like properties may be observed. charge density presented an intractable problem with respect to the interaction between charge elements. Bohm interpretation Like the molecules that make up macroscopic flu- ids he seems to be looking for sub-electronic parti- A consistent quantum theory of matter and mo- cles to make up the electron. This is an unnecessary tion, based on Madelung’s hydrodynamic analogy, assumption. For an electron without parts quantum adapted by Bohm [6], has been developed. By substi- units of charge, mass and spin indivisibly belong to tuting a wave function in polar form, Ψ = Rexp(iS/h¯), the total electron [7], without internal self-interaction. Schr¨odinger’s equation is decomposed into Non-classical behaviour exists in the flexibility of the ∂R2 R2 S electron. + · = 0, (29) ∂t m Any fermion could be seen as a topological chiral knot in the fabric of space-time. It changes shape in ∂S ( S)2 h¯2 2R interaction with its environment and disperses in the + − +V = 0. (30) ∂t 2m 2m R form of a boson on contact with an enantiomer. The centre of mass follows the trajectory which Bohm as- 2 ρ( ) Madelung interpreted R as the density x of a con- signs to a particle. Bohm’s pilot wave is no more than / tinuous fluid, which has the stream velocity v = S m. harmonic undulation in the interior of the electron, ∂ρ/∂ + Eq. (29) becomes a continuity equation, t which undergoes perpetual rotation in spherical mode. (ρ ) div v = 0, and (30) determines changes of the ve- As suggested by Madelung electrons that come into locity potential S in terms of the classical potential V contact do not fuse together like bosons. They may ap- and a quantum potential pear to interpenetrate, but change their shape to avoid
one another; they never coincide and maintain individ- h¯2 2R h¯2 2ρ 1 ρ 2 = − = − − , ual integrity. Vq ρ ρ 2m R 4m 2 A proton, because of its different wave structure is more massive and less flexible than an electron. When arising in the effects of internal stress in the fluid. these particles meet the electron wraps itself around In his interpretation Bohm emphasized that in the the proton under the electrostatic attraction directed to- ( → ) classical limit Vq 0 eq. (30) reduces to the classi- wards the proton. This interaction modifies the elec- cal Hamilton-Jacobi equation. It is further argued that, tronic wave structure and effective size of what is now by analogy, eq. (30) defines an ensemble of possible described as an atom. In interstellar space where it ex- trajectories of a particulate electron, for which a def- ists as a rydberg atom it has dimensions orders of mag- inite trajectory is predictable if an initial position can nitude larger than, for instance, in a terrestrial hydro- be specified. This interpretation gives equal weight to gen plasma. At the effective surface the wave pattern particle and wave aspects. The electron moves like a follows the surface harmonics, consistent with the an- mechanical particle guided by a wave, but the associa- gular momentum and excitation of the electron. Orbital tion between particle and wave remains artificial. The angular momentum is observed in a magnetic field B main advantage is recognizing the role of the quan- where the electron is subject to a vector potential A in tum potential to emphasize the holistic nature of quan- addition to the scalar potential eφ, such that eq. (30) tum systems. The Bohm interpretation adds nothing to becomes [10] quantum formalism and all conclusions, although less obvious, still hold in terms of conventional theory. ∂S 1 e 2 h¯2 2R + S − A + eφ − = 0. ∂t 2m c R Madelung interpretation In hydrodynamic analogy the flow velocity, as in All models, from Bohr to Bohm and beyond, have a e problem with the assumed particle nature of electrons. mvv = S − A Madelung’s proposal, in its original form, has, for c the same reason, never been taken seriously. Despite is no longer irrotational; curlv = −(e/mc)B. Because 380 J. C. A. Boeyens · Angular Momentum in Chemistry of the wave pattern orbital angular momentum and en- 3 3 x ± iy Y ±1 = ∓ sinθe±iϕ = ∓ · (33) ergy of the electron assume only those discrete values 1 8π 8π r allowed by the integers associated with the wave pat- ,± tern. This behaviour of the electron is conveniently de- like (14) and (15) for k =0 1. scribed by Schr¨odinger’s equation. The chemical implications of the LL condition are as dramatic as those of the P-E and DB conditions. It Quantum model shows that only one of the real eigenfunctions Lx, Ly and Lz is defined at any instance. The common prac- The quantum-mechanical description of angular tice of using linear combinations of so-called “real or- momentum follows from substitution of the operator bitals” to describe “directed chemical bonds” is there- →− ∂/∂ equivalent of classical momentum, e.g. px hi¯ x fore forbidden by fundamantal quantum theory. Struc- into eqs. (1) – (4). As the order of multiplication is im- tural chemists will have to look for an alternative prin- portant in vector products, the operator components ciple that defines molecular shape. The quenching of orbital angular momentum, as pointed out by Sommer- h¯ ∂ ∂ Lˆ = y − z feld [3], is such an alternative. x i ∂z ∂y etc., do not commute amongst themselves. The com- Atomic Structure mutators, such as (L ,L ) = (L L − L L ), are non- x y x y y x All quantum-mechanical atomic models are based zero and have the values (L ,L ) = ihL¯ , (L ,L ) = ihL¯ , x y z y z x on solution of Schr¨odinger’s equation for an electron (L ,L ) = ihL¯ . It means that more than one component z x y in the field of a stationary proton, viz. eq. (25) with V = can never be measured simultaneously. The commuta- −e2/4πε r. The eigenvalues of total energy, L2 and L , tion properties of orbital angular momenta is a non- 0 z respectively, are ordered in terms of the quantum num- classical effect arising from the L´evy-Leblond (LL) bers n = 1,...; l = 0,1,..., (n − 1); m = −l, ..., +l. quantum condition [11], L =¯hm, which is also the l z Bound states are those with E < 0. The single electron basis of the Bohr conjecture. This condition is as fun- on the H atom is assumed to be at the lowest ground- damental as the Planck-Einstein and De Broglie con- state energy level, n =1,l = m = 0. The experimental ditions, E =¯hω, p =¯hk, which respectively specify l finding that this state was doubly degenerate meant that quantized energy and linear momentum. at least one more eigenfunction was needed to describe The quantum operators lead directly [2] to the eigen- the behaviour of the electron. value eqs. (23) and (24) for a central field. If Lz,the eigenvalue in a central field, is known, L and L re- x y Electron spin main unidentified within the limitation The so-called spin variable that accounts for the ob- 2 + 2 = 2 − 2 2 = ( + ) − 2 2 Lx Ly L m h¯ l l 1 m h¯ served degeneracy enters [8] the Schr¨odinger theory through the factor ih¯ in the time-dependent equation The maximum allowed value of m = l shows that, al- < + ways, Lz L. It is therefore impossible to have Lx ∂ −iEt/h¯ Ly = 0, a situation identified before as the Bohr model. ih¯ − H Ψ = 0 (Ψ = ψe ). (34) 2 ∂t The eigenfunctions of L and Lz are the spherical harmonics Y m(θ,ϕ). The first few of these, in normal- l On substituting from (23) it follows that ized form, are l = 0, s-state: 2 2 ∂Ψ h¯ − Ψ = − V ih¯ ∂ 1 2m t Y 0 = √ , (31) 0 π 4 or in terms of a squared Schr¨odinger operator for a free ( ) l = 0, p-state: particle V =0 , 2 2 ∂ 2 3 3 z 2 h¯ Y 0 = cosθ = · , (32) S Ψ = + ih¯ Ψ = 0. 1 4π 4π r 2m ∂t J. C. A. Boeyens · Angular Momentum in Chemistry 381
The complex square root of S2 does not exist and a both orbital and spin angular momenta and the con- linear form such as served quantity their vector sum J = L +S. ∂ S = 2mih¯ A − (ih¯ )+C Energy levels ∂t The spin quantum number s =1/2 for all one- becomes meaningful only if A and C are defined as electron states, while the magnetic spin quantum num- ± / square matrices. The immediate effect of such formu- ber has the possible values ms = 1 2, defining states lation [2] is that the wave function need to be defined of equal energy. The doubly degenerate ground state of as a row vector, called a spinor, the hydrogen atom therefore has the configuration de- noted by 1s and characterized by the quantum numbers ± / ψ+ n =1,l =0,ml =0,ms = 1 2. The first and second Ψ = , ( ψ− excited states are the doubly degenerate 2s state, n = 2,l =0,ml =0,ms = ±1/2) and the six-fold degenerate which represents different (spin) angular-momentum 2p state, (n =2,l =1,ml =0,±1,ms = ±1/2), a total states. Spin, like orbital angular momentum, is also of 8 electronic states with n = 2. Likewise, for n =3,4, described by two quantum numbers, but unlike l and etc. there are 18, 32 etc. possible states. ml these are half-integer numbers. Total spin has The reasonable expectation that the ground state of eigenvalues of s(s + 1)h¯2, S = 3/4¯h, and multi- all more-electron atoms should also be the 1s-state is plicity 2s + 1=2,fors =1/2, as found experimen- not confirmed by spectroscopic analysis. It is found tally. Spin components, e.g. Sx = msh¯ (ms = ±1/2) instead that no more than two electrons of any atom obey the same commutation rules as the components can share the 1s level. Likewise, a maximum of 6, of L. 10 or 14 electrons can share the degenerate p, d or f The first theoretical account of electron spin was levels of a given atom. Detailed analyses have shown provided by the relativistic wave equation of Dirac. that this distribution is a manifestation of a more gen- Consequently, it is often stated that spin, or intrinsic eral fundamental rule known as the exclusion princi- angular momentum, is a quantum and/or relativistic ple, which applies to all sub-atomic entities with non- property. This conclusion is not warranted. A more integral spin, known as fermions. It requires the total convincing physical model is based on the hydrody- wave function, with space and spin parts, to be anti- namic model outlined above. symmetric with respect to particle exchange. To understand the electronic configuration of an It remains to be explained how an inhomogeneity atom it is sufficient to note the equivalent formulation in the aether, said to constitute an electron, manages of the exclusion principle, that no more than two elec- to move about freely without getting entangled with trons on the same atom can have the same set of four the environment. One possible mechanism is through quantum numbers n,l,m ,m . In principle the elec- spherical rotation of the region that defines the elec- l s tronic energy for each set of allowed quantum numbers tron. The effect of axial rotation is to wind up the con- can therefore be calculated from Schr¨odinger’s radial necting medium until it shears and develops a surface equation: of discontinuity. During spherical rotation the inter- π connecting medium relaxes after every 4 cycle and d2ψ(r) 2m l(l + 1) does not shear. The rotating electron neither transmits + En −V(r) − ψ(r)=0. dr2 h¯2 2mr2 nor receives rotational energy, but is surrounded lo- cally by a medium that undergoes cyclical wave mo- For an atom of atomic number Z,thetermV(r) rep- tion. This undulating region constitutes the spin. The resents the potential energy of the electron in the field fact that the symmetry group of spherical rotation also of the nucleus and the Z − 1 other electrons. The cal- satisfies the spinor version of both Dirac’s equation [9] culation of V(r) is not a trivial operation and the inter- and Schr¨odinger’s equation [1] provides final support pretation of electronic energy and angular momentum for the proposed hydrodynamicformulation of electron no longer follows the hydrogen pattern, which only ap- spin. plies in an approximately central potential field. In the The appearance of electron spin indicates that the hydrodynamic model however, electrons at deep en- conservation of atomic angular momenta must involve ergy levels should be constrained in their motion and 382 J. C. A. Boeyens · Angular Momentum in Chemistry
ml = ±2 (say). Only the triplet state has antisymmetric space functions, which imply that the probability den- sity tends to zero as the electrons approach a common position. The Coulomb repulsion is therefore lower for the triplet state than for the singlet state, which has a wave function symmetric in space coordinates. The predicted spin ordering for all multiplets of a degenerate d-state, shown in Fig. 5, is consistent with Hund’s rules. Fig. 5. Spin orientation in d- multiplets with up to 10 elec- Molecular Structure trons in spherical atoms that obey the exclusion principle. It is known from chemical practice that atoms join The pair in brackets repre- up to form electrically neutral molecules consisting sent the unlikely alternative of positively charged nuclei dispersed in an electronic singlet state. The spin count fluid. Like the surface of a free electron or atom, the is defined as σ =2∑ m . s outer surface of a molecule will also tend to define a minimum gradient that satisfies Laplace’s equation, al- those at the highest (valence) level would still find beit in an environment of lower symmetry. Although themselves in a central potential field of an atomic the solutions are no longer the spherical harmonics of a core, which is equivalent to having an effective posi- central field, they still characterize a minimum angular tive charge at the nuclear position. The energy and an- momentum, commensurate with the highest symmetry gular momentum of only the valence shell are therefore allowed by intramolecular interaction. adequately described in terms of hydrogenic quantum To anticipate the geometrical distribution of atomic numbers. nuclei in a molecular interior it is necessary to con- Based on the previous conclusion it is possible to sider the realignment of spherical harmonic angular- rationalize the electronic structure of the first 20 ele- momentum vectors in the field of multiple nuclei. A ments and the general ordering of all elements of the closely related problem was considered by Jahn and periodic table. Teller [12] (JT) who formulated a theorem that clari- Spin ordering fies several issues around the prediction of molecular shapes. A conspicuous feature of the electronic configura- tion of atoms, not determined by energy calculations Jahn-Teller theorem based on n and l only, namely the relative spin orien- tation of valence-shell multiplets, is correctly specified JT investigated the effect of electronic orbital de- by Hund’s empirical rules: “The first (2l + 1)/2 elec- generacy on the symmetrical nuclear configurations of trons on a given energy sub level have the same spin polyatomic molecules. Any molecule has a continuous orientation. Addition of more electrons causes step- set of configurations consistent with a given symmetry. wise pairing of spins”. The same predictions are seen Among these one equilibrium configuration of mini- to follow from the assumption that all atoms are spher- mum energy is stable with respect to all totally sym- ically symmetrical and therefore have zero resultant metrical displacements, i. e. those which do not disturb angular momenta. This condition is satisfied if one the symmetry. However, this configuration is not nec- of an odd number of electrons that constitute a mul- essarily stable against all other types of nuclear dis- tiplet has ml = 0 and all others occur as pairs with placement, not if the electronic energy for neighbour- ±ml = 0. Given n, l and ml, the spin orientation ms ing conformations depends linearly upon any of the (up or down) is controlled by the exclusion principle, nuclear displacements. By analyzing the motion of a as demonstrated for the two-electron system in the d- single electron in the respective fields of three colinear multiplet l =2) showninFig.5.Inthiscasethereare nuclei and of four nuclei in square-planar array, it is two possibilities consistent with zero orbital angular demonstrated that linear and non-linear molecules re- momentum, for an even number of electrons, a singlet spond in fundamentally different ways to nuclear dis- state with paired spins at ml = 0 and a triplet state with placement. J. C. A. Boeyens · Angular Momentum in Chemistry 383
Fig. 6. Energy change on symmetrical and antisymmetrical Fig. 7. Illustration to demonstrate the rotation of an orbital distortion of trinuclear arrangement with angular momentum angular momentum vector. degeneracy.
The states of the electron are classified as σ, π or δ for orbital angular momentum projections of 0, ±1, ±2 (in units ofh ¯) along the polar axis, which is not in an arbitrary direction, but is fixed by symmetry. The σ states are non-degenerate, while the π, δ, etc. states are each two-fold degenerate, corresponding to either clockwise or anti-clockwise rotation of the electron about the polar axis. For the trinuclear molecule an un- Fig. 8. Energy changes on in-plane symmetry breaking in a symmetrical displacement can, without loss of gener- square-planar arrangement with electronic orbital degener- ality, be considered as displacement of the central nu- acy. cleus by a distance d perpendicular to the nuclear axis. multiplication by eimϕ . In the square-planar arrange- This displacement destroys the axial symmetry and re- ment four half-planes rotate together about the four- moves the degeneracy. Each degenerate state splits into fold symmetry axis, with −π/2 ≤ ϕ ≤ π/2form = ±1. two states, one symmetrical with respect to reflection The displacements shown in Fig. 8 may be regarded in the plane of the nuclei and the other antisymmetrical as positive and negative values of the same nuclear with respect to the same plane. These states have dif- displacement. This nuclear displacement reduces the ferent energies E and E . As the nuclear displacement s a four-fold symmetry to two-fold. The degenerate state is varied these states and their energies change contin- splits into two states Φσ and Φσ , the first with angular uously, but their symmetry remains. Displacements of momentum projected perpendicular to the symmetry d and −d result in the same state and energy, as shown plane σ and the second perpendicular to σ . Because in Fig. 6. The energies E and E are even functions s a the configurations I and II are equivalent the energies of d, and all that is required for stability with respect Eσ and Eσ are related, such that to nuclear displacement is that the function be positive in both cases. This condition is assured by the orbital Eσ (I)=Eσ (II) angular momentum, which is therefore responsible for stabilizing the linear arrangement. The effect of dis- and vice versa.AsseenfromFig.8theenergylev- placement is to change the alignment of the angular els cross at the energy E0 of the undisplaced configu- momentum vector Lz = mh¯ associated with the eigen- ration. There is no symmetry reason which precludes ± ϕ function e im , shown in Fig. 7. To change the direc- a linear dependence of the energy levels upon the nu- tion of Lz requires work against kinetic energy and con- clear displacement in the neighbourhood of E0,andthe sequently the angular momentum acts like a gyroscope square-planar configuration can in general not be a sta- to realign the vectors. ble equilibrium configuration. As a counter example the motion of a single elec- The results obtained from the two illustrative exam- tron in the field of a square-planar arrangement of four ples were generalized by examination of the pathways identical nuclei is considered. In the previous exam- of allowed symmetry breaking for all relevant sym- ple the total wave function was generated by rota- metry point groups, excluding those of complete axial tion of the angular-momentum eigenfunction in any symmetry. It was found that all non-linear nuclear con- half-plane (shaded in Fig. 7) about the polar axis, i. e. figurations with an orbitally degenerate one-electron 384 J. C. A. Boeyens · Angular Momentum in Chemistry
Fig. 10. Diagram to demon- strate the quenching of or- bital angular momentum in the methane molecule. Mag- netic moments cancel in projection along each possi- ble H-C polarization direc- tion, because of p-electron charges rotating in opposite sense, as shown.
Fig. 9. Diagram to demonstrate the stability of an electroni- principle of minimum orbital angular momemtum still cally non-degenerate square-planar system against distortion. implies that the favoured conformation will be of a symmetry that quenches the orbital angular momen- state are unstable against some normal displacement, tum. not the totally symmetrical. The symmetry groups considered by JT are all of the type that quenches the orbital angular momenta Non-degenerate states of electrons derived from free-atom energy levels that (± ) One aspect of Jahn and Teller’s theorem which is constitute degenerate pairs ml in a spherically sym- repeatedly emphasized in the chemical literature is the metrical environment. The only molecules excluded distortion of symmetrical molecules with one-electron from the set are those without any symmetry, i. e. orbital degeneracy. These cases are the exceptions to the chiral molecules. An important consideration is a general rule which is rarely emphasized nor applied that chiral molecules have residual orbital angular mo- to problems of molecular conformation. The rule ap- mentum in projection along a polar direction, e.g. an applied magnetic field. The JT distortions consist of plies wherever the pair of degenerate levels (±ml) ac- commodate two electrons, with total electronic energy spontaneous breaking of molecular symmetry by a nor- Eσ + Eσ as shown in Fig. 9. The total conformational mal displacement that introduces an interaction, which lowers the energy of the system, without affecting the energy in this case has a minimum at E0 which stabi- 5 lizes the symmetrical arrangement of nuclei. The rea- polar projection of the angular momentum . JT distor- son for this stabilization is hinted at, but not explicitly tion therefore does not cause optical activity. stated by JT. It relates to the electronic orbital angular momentum. Molecular shape Orbital angular momentum arises from the rotation The central JT theme describes the relationship of electronic charge about axes that depend on the between the symmetry and stability of non-linear environment, represented by either an applied mag- molecules, analyzed as a function of orbital degen- netic field or the polarization by ligand nuclei that sur- eracy. This, somewhat misleading terminology, refers round the reference atom. The total angular momen- specifically to degenerate states available for one single tum, which is the vector sum of the individual contri- electron. The enormous volume of more recent work butions, reduces to zero if the component charge rota- devoted to the study of JT effects [13] focusses ex- tions occur in opposite sense with respect to a single clusively on vibronic interactions at symmetry-related axis of rotation. The angular momentum is said to be degenerate intersections on potential energy surfaces. quenched and the corresponding minimum kinetic en- The more common and more important prediction of ergy of rotation stabilizes this symmetrical situation. symmetrical structures stabilized by non-degenerate In a free atom the electrons move in the symmetri- angular momentum states is largely forgotten. cal central Coulomb field of the nucleus and the angu- The first step in the formation of a molecule is to lar momenta of electrons at a degenerate pair (±m )of l consider a given atom as surrounded by a number of energy levels is quenched identically. For an atom in a non-interacting secondary atoms, or ligands, which is molecule the spherical symmetry is broken and the an- gular momentum vectors are redirected as demanded 5As for free atoms the unpaired electron has an effective mag- by the symmetry of the molecular environment. The netic quantum number ms =0. J. C. A. Boeyens · Angular Momentum in Chemistry 385 also the assumption of crystal field theory. The energy The significant new result is that the p-electron den- and angular momentum of the primary atom are con- sity should rotate in sets of parallel planes, perpendicu- served, but differently distributed, depending on the lar to the lines connecting H and C nuclei. For fixed nu- symmetry of the secondary shell of atoms. These quan- clei these planes intersect in points that define regular tities are still the eigenvalues of an atomic Schr¨odinger octahedra, shown in Fig. 10 for two different spacings equation, in a central field modified by interaction with with respect to the central carbon nucleus. The smaller the secondary shell. To first approximation it is cor- octahedron is defined by midpoints of the lines con- rect to assume that the orbital angular momentum is necting the hydrogen nuclei. The larger one contains conserved in magnitude, but not in orientation. The JT the hydrogen positions, on the black dots. Uncertainty approach is to modify the central-field description of associated with the circulation radius is sensitive to the energy and angular momentum eigenvalues in terms chemical nature and electronegativity of the ligands. of the representations of each symmetry group con- Qualitatively however, the prediction is well in line cerned. It is almost axiomatic to assume that the ar- with the various criteria for the definition of atomic rangement of identical ligands around a central atom shape based on charge-density distribution [15] and re- should be of the highest possible symmetry; like as- flects the statement of the author: suming a spherical shape for free atoms. According In reality, chemical bonding is a molecular to JT the resulting molecular structure automatically property, not a property of atomic pairs. ensures quenching of the orbital angular momentum. Introduction of unlike ligands lowers the symmetry, Many larger molecules resemble an assembly of un- which remains sufficient to quench the angular mo- saturated fragments consisting of the symmetrical cel- mentum, until no two ligands are alike. This prop- lular units of the methane type, after removal of one erty has been demonstrated [14] to produce a tetra- or more ligands, e.g. CH3,CH2, etc. When such frag- hedral structure for methane. It relies on the pres- ments combine to form saturated molecules their rel- ence of two p-electrons on the free carbon atom, ative orientation in the final product must ensure that with ms = ±1. the orbital angular momentum stays quenched.
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