Bohr’s molecular model, a century later

Despite the great power and scope of modern computers, there remains yearning for intuitive, pedagogically appealing ways to understand electronic structures in atoms, molecules, and more complicated systems. Niels Bohr’s century-old model, in which electrons parade around atomic nuclei much like planets orbit the Sun, retains such appeal. In its historical context, the Bohr model managed to elucidate the hydrogen-atom spectrum and thereby gave great impetus to the birth of quantum mechanics. ^{1 , 2 , 3 , 4} But it proved incompatible with quantum theory, which found that the ground state of the H atom was spherically symmetric and had zero orbital angular momentum, contrary to Bohr’s assumption of a planar, circular electron orbit with angular momentum h/2π, specified by Planck’s constant. Even so, today the Bohr model has valid roles in describing highly excited Rydberg atoms, ⁵ cavity quantum electrodynamics, ⁶ and quasi-Rydberg states in graphene. ⁷

From early on, Bohr’s model was considered to fall far short of his earnest aim to provide a physical basis for chemical bonding. Helge Kragh states, “Bohr’s original and most ambitious project of establishing a common theory for atoms and molecules—one that would be of equal significance to physics and chemistry—turned out to be a failure” (see the article by Kragh in Physics Today, May 2013, page 36 ; quote, page 41). A few years ago, however, redemption arrived. ^{8 , 9 , 10} Bohr’s model was found to emerge from the Schrödinger equation in the limit of infinitely many dimensions, where quantum mechanics morphs into classical mechanics. ¹¹

Here we revisit Bohr’s H-atom model to show how his postulates conform to quantum theory in its infinite-dimension limit. With that perspective, we take a fresh look at Bohr’s treatment of the H₂ molecule. Simple extensions of his molecular model, requiring just elementary algebra, turn out to give potential-energy curves, for both the H₂ ground electronic state and several excited states, that agree well with those obtained from highly accurate ab initio quantum mechanical computations done with supercomputers. Further examples illustrate that a rudimentary Bohr-like model also gives respectable results for electronic geometry of multielectron atoms. All told, at its centennial, his model deserves a recently bestowed accolade: ¹² “Bohr’n again.”

Bohr’s hydrogen atom

In his 1911 doctoral thesis dealing with the electron theory of metals, Bohr concluded that classical physics could not explain why electrons bound in atoms behaved differently from those carrying current. That led him to embark on a postdoctoral stint with J. J. Thomson, the discoverer of the electron, at Cambridge University in England. Naive and brash, Bohr was disillusioned when parts of his thesis critical of Thomson’s classical theory failed to arouse interest at Cambridge.

Bohr retreated to Manchester in March 1912 to work with Ernest Rutherford doing experiments that probed how much energy alpha particles lose as they traverse matter. That began an intense, mutually supportive, lifelong friendship. ¹³ (Figure 1 shows them at a sporting event.) Rutherford had published his discovery of the atomic nucleus nearly a year before, but it was not yet taken seriously; Thomson’s “plum-pudding” atomic model, which presumed positive tidbits were sprinkled among rings of electrons, was still in vogue. By May Bohr had switched to theory, prompted by his recognition of why a previous treatment of alpha-particle energy loss was inadequate. His thesis had prepared him to spot the fault: Electrons had been treated as free rather than bound in atoms. Hence, almost inadvertently, Bohr came to focus on finding an electronic architecture that would be compatible with the nuclear atom model.

In July Bohr submitted to Rutherford a seven-page manuscript, referred to since as the Manchester or Rutherford Memorandum. Precursor to the three long papers—dubbed the trilogy—that Bohr published a year later, the memorandum emphasized two key problems with the nuclear model. In classical mechanics, the centrifugal force on an orbiting electron is balanced by the coulombic attraction to the nucleus. But the classical orbits became unstable when populated with more than one electron. Moreover, nothing in classical physics determines orbital radii or frequencies. Bohr proposed to introduce a hypothesis, as yet speculative but involving Planck’s constant, from which he could determine specific allowed orbits. He frankly acknowledged that he considered it hopeless to try to justify such a hypothesis.

In part 1 of his trilogy, Bohr developed four versions of the hypothesis. His concluding and most familiar version invoked stationary circular orbits specified by quantized orbital angular momentum, L = nℏ, with n = 1, 2, 3, … . Introducing that quantum constraint into classical expressions, Bohr derived the radii ρ_n = (n²/Z)a₀ and energies E_n = −(Z²/2n²)(e²/a₀) of the allowed orbits for a one-electron atom. The quantity a₀ = ℏ²/me² was soon christened the Bohr radius; ℏ = h/2π is the reduced Planck constant, m is the electron mass, −e the electron charge, and Z the atomic number.

Bohr’s simple formula for the energy levels E_n was a spectacular success. It gave quantitative agreement with the H-atom spectral series seen in laboratory experiments and in stellar observations, and it predicted other spectral lines that were soon verified. It also identified the He⁺ ion from stellar spectral lines that had been mistakenly attributed to atomic hydrogen.

Superseded by quantum theory

Although impressed with Bohr’s success, many contemporary scientists were offended by his blithe introduction of amenable postulates. ^{1 , 3} Yet three of his key postulates were destined to become pervasive in quantum mechanics. Foremost was his concept of stationary states, although bereft of electron orbits. Another was his assertion that an electron in the lowest-energy stationary state, the ground state (n = 1), does not emit radiation. That was contrary to classical electrodynamics, which would require an orbiting electron to continuously radiate energy and spiral into the nucleus.

Likewise fundamental was Bohr’s attribution of spectra to radiation emitted or absorbed in transitions between stationary states, with spectral-line frequencies determined by the energy difference between states. That, too, contradicted classical electrodynamics by disconnecting the frequencies appearing in spectra from those of mechanical motions of either of the communicating states. As a consequence, Bohr was led to his correspondence principle, which says that the close approach of stationary states at high energies renders classical and quantum frequencies nearly the same at large n.

Bohr’s H-atom model enjoyed further success for about a dozen years. Especially impressive were extensions in 1916 by Arnold Sommerfeld. He included elliptical orbits and found the energy levels remained the same as those obtained by Bohr. He then added relativistic mechanics, which produced fine structure in agreement with observations. Also striking was the result of the celebrated 1922 experiment by Otto Stern and Walther Gerlach, undertaken to test the Bohr model (see the article by Bretislav Friedrich and Dudley Herschbach, Physics Today, December 2003, page 53 ). The experiment determined, to within 10% error, the magnetic moment of a silver atom. (That moment is due to the atom’s single valence electron.) The result, which agreed with that predicted from classical electrodynamics for an electron rotating in a circular orbit with angular momentum ℏ, was hailed as compelling evidence for the Bohr model.

By 1926 the advent of quantum mechanics and the discovery of spin had discredited Bohr’s model. Although the predicted H-atom spectrum, including the fine structure, was not altered, the principal quantum number was redefined to be n = l + k + 1, where l and k—the number of angular and radial nodes in the wavefunction—are both zero for the ground state. The Stern–Gerlach result had to be reinterpreted as being due to electron spin. The experiment actually demonstrated that a ground-state electron has zero orbital angular momentum (l = 0) and therefore an isotropic distribution, contrary to Bohr’s model.

However, for a Rydberg atom or molecule, wherein one electron is in an excited state with large n, the Bohr model applies nicely. ^{5 , 6 , 7} Such an atom can be treated as hydrogenic, because the lone excited electron is far away from its home base, which has a net effective charge of +1 due to shielding of the nucleus by the nearby electrons. For large n, especially for states with few radial nodes, the Schrödinger equation indeed gives results that agree with the Bohr picture: The excited electron moves in a planetary orbit with radius proportional to n² and energy proportional to n⁻².

Dimensional scaling

As applied to electronic structure, dimensional scaling is not at all exotic, although it is borrowed from quantum chromodynamics. (See the article by Edward Witten, Physics Today, July 1980, page 38 .) The procedure operates in a space in which all vectors have D Cartesian coordinates. It involves generalizing the Schrödinger equation to D spatial dimensions and rescaling coordinates and energy to absorb generic D-dependence. For the simple case of the H atom, the D-dependence is exactly known and the principal quantum number becomes n + (D − 3)/2. (See box 1.) Consequently, for large D the scaling of the electron radius becomes proportional to D² and scaling of energy proportional to D⁻². Thus going to large D for fixed n is equivalent to creating actual Rydberg states with large n at fixed D = 3.

Taking D to infinity with n = 1 converts the wave equation into the ground-state Bohr-model energy function. In particular, the centrifugal term representing Bohr’s orbiting electrons naturally appears in the infinite-D limit, even for states with no orbital angular momentum. That seeming paradox (which might have delighted Bohr!) is accommodated by D-scaling. In the large-D limit, the electrons are not orbiting but rather sitting still at fixed locations corresponding to the minimum of the energy function. That does not violate Heisenberg’s uncertainty principle. When D-scaling factors are introduced into the coordinates, the inverse factors enter into the conjugate momenta. Hence the coordinate–momentum products remain unchanged. Moreover, Bohr’s blithe assumption, considered outrageous in 1913, that an electron in the n = 1 state does not emit radiation while accelerating about the proton, is justified in the large-D limit. Since the electron configuration in that limit corresponds to the minimum of the energy function, energy conservation forbids the electron to emit a photon.

At first blush, the infinite-D limit may seem absurdly distant from the “real world” at D = 3. In typical dynamical problems, however, after taming any singularities by a suitable choice of scaling factors, the major dimension dependence of quantities of interest varies as 1/D. Hence the large-D limit, where 1/D → 0, is closer to the real world (1/D = 1/3) than is the oft-used D = 1 regime. Indeed, results obtained at large D usually resemble those for D = 3 and therefore provide a good first approximation. For the H atom, the D-dependence is known exactly, so incorporating it in the scaling factors enables the large-D limit to give the same results as D = 3 for the scaled energy and most probable electron location.

Bohr’s hydrogen molecule

In part 3 of his trilogy, Bohr presented chiefly calculations for the hydrogen molecule in its ground state. As depicted in configuration I of figures 2a and 3a, he assumed that the two electrons (labeled 1 and 2) each orbit the molecular axis with angular momentum ℏ in a common circular path midway between the protons (A and B). His calculation procedure was equivalent to determining the electronic energy and geometry, for a fixed internuclear spacing R, by evaluating the minimum of the energy function,

in atomic units. Here ρ₁ and ρ₂ are the electron orbit radii, r_ij is the distance between two particles i and j, and V is the Coulomb potential energy, which includes the attractive electron–nucleus interactions (1/r_1A, …) and the repulsive electron–electron (1/r₁₂) and nucleus–nucleus (Z²/R) interactions.

As with the H atom, Bohr’s E(R) energy function for H₂ corresponds to the Schrödinger equation in the large-D limit. Again, that is a redeeming grace. Even before the advent of quantum mechanics, a fatal conceptual flaw in the Bohr H₂ model was recognized. ¹⁴ Bohr had assumed the two electrons orbit synchronously around the molecular axis in the same direction. That would produce a large magnetic moment and cause the ground state of H₂ to be paramagnetic. In actuality, however, the ground state is weakly diamagnetic and has no electronic orbital angular momentum.

The conceptual quandary is banished at the large-D limit, where the electrons are motionless and therefore do not generate magnetic moments. From the D-scaling perspective, such orbits should be viewed not as tracing an electron trajectory but as the locus of the electron’s most probable location. Ironically, however, by virtue of the cylindrical symmetry about the molecular axis, those loci coincide with the Bohr orbits. Figure 2 compares, for two internuclear distances, the electron distributions of the Bohr orbits with those calculated using a good- quality wavefunction.

Bohr reported numerical results only for configuration I, at the internuclear distance that corresponds to the minimum of the E(R) function. That minimum occurs for R = 1.10a₀ and ρ₁ = ρ₂ = 0.95a₀, and it yields a bond dissociation energy of 0.105e²/a₀, or 2.86 eV, relative to the H + H asymptote. (We updated his result using the modern value of e²/a₀.) Accurate values for the equilibrium bond distance and bond energy are 1.40a₀ and 4.745 eV, respectively. ¹⁵ A comparison more fair to Bohr is with the results 1.51a₀ and 3.14 eV, obtained from an approximate wavefunction in 1927 by Walter Heitler and Fritz London, ¹⁶ ever since cited as the first a priori calculation of chemical bonding.

Minimizing Bohr’s energy function admits two other configurations, labeled II and III in figure 3a, in which the two electrons have separate orbits. Configurations I and II, in which the electrons remain inside the protons and on opposite sides of the molecular axis, constitute the singlet ground state ¹Σ_g⁺; configuration III, in which the electrons orbit outside the protons and on the same side of the molecular axis, corresponds to the triplet state, ³Σ_u⁺. Bohr recognized that configuration I would dissociate ultimately to 2H⁺ + 2e⁻, so he inferred, rather vaguely, that there must exist another configuration—configuration II—that dissociates to H + H. Configuration III, not recognized by Bohr, represents repulsively interacting H atoms.

Figure 3b plots the dependence of the electronic energy on R for the three configurations, as derived from Bohr’s E(R) function. Such curves, in accord with the Born–Oppenheimer approximation, specify the potential energy for molecular vibrations. Also shown are the results (blue points) from extensive quantum variational calculations for the ground singlet state and the lowest triplet state. Bohr’s model fairly accurately predicts the energy of the ground singlet state both at small R, where configuration I has the lowest energy, and at large R, where configuration II has the lowest energy. (Configuration II is the lowest-energy configuration for R > 1.20a₀; it merges with configuration I at small R.) The Bohr model accurately predicts the repulsive triplet state, configuration III, over the full range of R. But it does poorly at predicting the depth of the attractive well of the singlet state.

Simple enhancements of the Bohr model markedly improve results for the attractive well of the singlet state. (See box 2.) The most straightforward extension—the perturbation approach—is to expand the energy function about its minimum in powers of 1/D. Adding the first-order 1/D perturbation term gives the red curve in figure 3b, for which the equilibrium bond length and dissociation energy are 1.38a₀ and 4.50 eV, close to the accurate results. Similar results were obtained by two other methods—the switching-quantization approach and the constrained-Bohr approach. Those extensions have also yielded fairly accurate potential curves for some excited states of H₂ and for ground states of lithium hydride, beryllium hydride, diatomic lithium, and other simple diatomic combinations.

Bohr offered qualitative arguments indicating that the ground states of a helium hydride molecule and of He₂ would be entirely repulsive. Figure 4 shows remarkably accurate potential curves we obtained using his original formulation.

Multielectron atoms

In part 2 of his trilogy, Bohr discussed He and larger atoms, simply postulating that each electron circles the nucleus with orbital angular momentum of ℏ. For those atoms, unlike H, H₂, and Rydberg atoms and molecules, the model based on Bohr’s wishful postulate does not correspond to the Schrödinger equation in the large-D limit. Curious nonetheless to see what it would give, we examined a simplistic energy function:

where r_i is the distance from electron i to the nucleus. In analogy to Bohr’s hydrogenic function and to its large-D limit, ¹⁷ the multielectron energy function consists merely of centrifugal kinetic energy terms plus the coulombic potential energy. It is akin to a heuristic approximation employing D-scaled quantum numbers for excited electronic states of multielectron atoms. ^{10 , 18} The quantum numbers n_i for each of the N electrons are assigned using the familiar prescription for sequential filling of the atomic shells: n = 1, 2, 3, … , with 2n² the maximum occupation of each shell. At a crude level of approximation, minimizing the energy function gives the fixed locations that the electrons attain at the large-D limit in the D-scaled space, which for D = 3 represent the most probable configuration.

Figure 5 shows the electron configurations obtained for neutral atoms with 2 ≤ Z ≤ 6. The results agree remarkably well with the radius of maximum radial charge density in atomic shells and subshells, determined from D = 3 self-consistent field calculations using Roothaan-Hartree-Fock (RHF) wavefunctions. ¹⁵ For the n = 1 shell, the radii are within 1% of the RHF most probable radii. For the n = 2 shell in Li, the model result (3.85a₀) is much larger than the RHF value (3.09a₀), but for beryllium, boron, and carbon the agreement between the Bohr and RHF results (2.04a₀, 1.53a₀, and 1.21a₀, respectively) is good.

Not surprisingly, the net ground-state energies given by the model compare poorly with RHF results. However, a striking aspect is that simply minimizing the energy function predicts that in the n = 2 shell the valence electrons for Be, B, and C are disposed in linear, trigonal, and tetrahedral geometry, respectively. Those geometries correspond to the sp, sp², and sp³ modes of hybridization of the s and p orbitals invoked in myriad discussions of chemical bonding. Conventional wave-mechanics treatments of hybridization have to neglect differences in orbital energies to obtain those modes. In the simplistic Bohr-like scheme, they appear without any ado.

The Bohr model may acquire more progeny. The prototype examples described here and others presented elsewhere ^{8 , 9 , 10 , 11} provide fairly accurate results that support qualitative insights for a variety of electronic states of atoms and molecules, including excited electronic states and quasi-stationary states. Both the computations required and the guiding concepts are elementary and exemplary of Bohr’s enterprising spirit and acceptance of paradox.

Box 1. Relating Bohr to Schrödinger

For a one-electron atom with nuclear charge Z and orbital angular momentum ℓ in D-dimensional spherical coordinates, the radial Schrödinger equation is

in customary atomic units. (Distance is in units of the Bohr radius a₀; energy is in Hartree units e²/a₀.) Via the relationship ψ = r^{−(D − 1)/2}ϕ, the equation can be recast in terms of ϕ, the radial probability amplitude:

Thereby the sole D-dependence appears in the centrifugal potential, where Λ = ℓ + (D − 3)/2 replaces the orbital angular momentum quantum number, and the principal quantum number becomes n + (D − 3)/2.

The transcriptions for n and ℓ enable all properties of D-dimensional hydrogenic atoms to be readily related to those for D = 3. Shown here, for instance, are the radial probability distributions |ϕ|² for ground-state atoms with D between 2 and 100. Each curve is normalized with respect to its maximum, at r_max = (D − 1)²/4Z. The ground-state energy is E_D = −2[Z/(D − 1)]².

Scaling the radial coordinate via r → [(D − 1)/2]²r and the energy by E → [2/(D − 1)]²E transforms the ground-state wave equation into

In the limit D → ∞, the derivative term is quenched, and the wave equation reduces to

E(r) = 1/(2r²) − Z/r,

which is equivalent to the Bohr-model hydrogen-atom energy function. The minimum of E(r) determines the large-D limit for the scaled location and energy in the ground state: r = 1/Z and E = − Z²/2.

Thus far, only the radial wave equation has been considered. The angular part is separable, and for the ground state its wavefunction is spherically symmetrical. However, the D-dimensional volume element contains a factor (sin θ)^{D − 2}, so for D → ∞, the polar angle will be restricted to θ = 90°. Hence, in the large-D limit, the electron distribution corresponds to the planar circular orbit of the Bohr model.

For other atoms or molecules, D-scaling proceeds in essentially the same way. Via the ψ → ϕ transformation, the Schrödinger equation is recast so that the explicit D-dependence appears solely in the centrifugal energy, always as a simple, quadratic factor. ^{10 , 11}