Atoms and Molecules

Contents

The many-electron problem
Self-consistent field theory
The Thomas–Fermi model
The periodic table
Angular-momentum coupling
The Born–Oppenheimer separation
The chemical bond
Molecular spectra
Intermolecular forces

The hydrogen atom of The Hydrogen Atom is the last atom that can be solved exactly. Adding a single electron adds an interaction term that couples all coordinates and destroys separability, and with it the closed-form spectrum; everything beyond helium is approximation plus computation. What is remarkable is how much survives. The shell structure that the exclusion principle imposes on the one-electron orbitals [Pauli:1925] reproduces the periodic table — the largest single body of chemical regularity ever assembled — from counting alone, and the self-consistent field of Hartree and Fock [Hartree:1928] [Fock:1930] turns the counting into numbers that match measured ionization energies. This chapter is where the quantum mechanics of Parts VIII and IX meets the evidence of chemistry.

Its second half joins nuclei to electrons. The Born–Oppenheimer separation [Born:1927] exploits the mass ratio to define a potential energy surface on which nuclei move, and that surface is what a chemical bond is: the electronic ground-state energy of $\mathrm{H}_{2}^{+}$ as a function of internuclear distance [Burrau:1927], or of $\mathrm{H}_{2}$ in the exchange treatment of Heitler and London [Heitler:1927]. Rotations, vibrations and electronic transitions on that surface are the molecular spectra [Herzberg:1950], and the residual attraction between closed-shell atoms with no bond at all is London's dispersion force [London:1930], whose retarded form [Casimir:1948a] is the bridge to the vacuum of Part XI. The chapter is placed after Identical Particles, whose antisymmetry postulate it uses on every page, and before the precision spectroscopy of Experiment: Precision Spectroscopy and Atomic Clocks, which measures what is computed here.

Derivation pending.

Atoms and Molecules: all derivations of this chapter are pending.

The many-electron problem

The Coulomb Hamiltonian

[Reserved: the nonrelativistic Hamiltonian of $Z$ electrons and a nucleus, with its one-body Coulomb attraction and its pairwise repulsion $\sum_{i<j}e^{2}/4\pi\epsilon_{0}\abs{\vect{r}_{i} -\vect{r}_{j}}$; why the repulsion term forbids separation of variables and hence any closed-form solution; the infinite-nuclear-mass approximation and the reduced-mass correction; the atomic units in which the literature is written, with the statement of their SI conversion required by the units axiom (Measurement, SI Units, and the Theory of Errors).]

Antisymmetry and Slater determinants

[Reserved: the exclusion principle in its original spectroscopic formulation [Pauli:1925] and its restatement as antisymmetry of the total state (Identical Particles); the Slater determinant as the antisymmetric product built from one-electron spin-orbitals [Slater:1929], and the Slater–Condon rules for matrix elements of one- and two-body operators between determinants; the exchange term as a consequence of antisymmetry rather than of any force.]

Helium: exchange and the ortho–para splitting

[Reserved: helium as the smallest instance of everything — Heisenberg's resonance treatment, which identified the singlet–triplet splitting as the exchange integral and explained why orthohelium and parahelium behave as two non-combining species [Heisenberg:1926]; first-order perturbation theory and the variational estimate of the ground-state energy (Approximation Methods); Hylleraas's explicitly correlated wave function [Hylleraas:1929], the first calculation to reach spectroscopic accuracy and still the template for benchmark work; the measured ionization energy $24.587\,\mathrm{eV}$ as the number to be reproduced.]

Self-consistent field theory

The Hartree method

[Reserved: Hartree's non-Coulomb central field, in which each electron moves in the spherically averaged field of the nucleus and of all the others, solved by iteration to self-consistency [Hartree:1928]; the resulting effective potential interpolating between $-Ze^{2}/r$ near the nucleus and $-e^{2}/r$ far away, which is what lifts the $\ell$-degeneracy of hydrogen and orders the subshells; the Hartree product's failure to be antisymmetric, and what that costs.]

Phenomenon 88.1 (The accidental degeneracy of hydrogen does not survive).

In hydrogen the energy depends on $n$ alone. In every other atom it depends on $n$ and $\ell$ separately, and always in the same direction: for fixed $n$, $E_{ns}<E_{np}<E_{nd}<E_{nf}$. The alkali spectra display this with no interpretation required [Rydberg:1890]. Their terms are described to high accuracy by

\begin{equation}\tag{88.1} E_{n\ell}=-\frac{hcR}{(n-\mu_{\ell})^{2}}\ec \end{equation}

with a quantum defect $\mu_{\ell}$ that is nearly independent of $n$ along a series, decreases rapidly with $\ell$, and is essentially zero for the highest $\ell$: in sodium the $s$ terms are displaced by more than one whole unit of $n$, while the $d$ and $f$ terms lie almost exactly on the hydrogen values. The ordering is severe enough to reorder shells — the $4s$ level lies below $3d$ — which is why the fourth period of the table is eighteen elements long and not eight.

Derivation. Model the field seen by one outer electron as a central potential $V(r)$ tending to $-Ze^{2}/4\pi\epsilon_{0}r$ as $r\to0$, where the nucleus is unscreened, and to $-e^{2}/4\pi\epsilon_{0}r$ as $r\to\infty$, where the other $Z-1$ electrons screen all but one unit of charge. Interpolating between those two limits is the whole content of the self-consistent field. Write

\begin{equation}\tag{88.2} \delta V(r)=V(r)+\frac{e^{2}}{4\pi\epsilon_{0}r} \end{equation}

for the departure from a hydrogenic problem of unit effective charge. Since $V$ is more attractive than the fully screened form everywhere and equals it only asymptotically, $\delta V(r)<0$ at every finite $r$, and it is concentrated at small $r$, falling off with the density of the core.

First-order perturbation theory (Approximation Methods) gives $\Delta E_{n\ell}=\int_{0}^{\infty}\dd r\, \abs{u_{n\ell}(r)}^{2}\,\delta V(r)$, negative for every $n\ell$. The dependence on $\ell$ follows from one fact about the radial functions: near the origin the regular solution behaves as $u_{n\ell}(r)\propto r^{\ell+1}$, because the centrifugal term $\hbar^{2}\ell(\ell+1)/2mr^{2}$ dominates there. So for fixed $n$ the weight $\abs{u_{n\ell}}^{2}$ carried into the region where $\delta V$ lives falls monotonically as $\ell$ grows, and with it the size of the shift,

\begin{equation*} \Delta E_{ns}<\Delta E_{np}<\Delta E_{nd}<\cdots<0\ec \end{equation*}

which is the observed ordering. Rewriting the shifted level in Rydberg form gives Equation (88.1) with $\mu_{\ell}>0$ decreasing in $\ell$; and $\mu_{\ell}$ is nearly independent of $n$ because the small-$r$ amplitude of $u_{n\ell}$ scales as $n^{-3/2}$ along a series, so that the shift and the level spacing scale together.

What this does not give is the order in which subshells fill across the table, which requires comparing different $n$ as well as different $\ell$; that ordering remains the empirical rule of Section 88.4.2, and the chapter says so there rather than pretending otherwise.

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Hartree–Fock

[Reserved: Fock's variational derivation from an antisymmetrized trial state [Fock:1930], independently obtained by Slater [Slater:1930], giving the Hartree–Fock equations with their nonlocal exchange operator; the self-consistency loop; Koopmans' theorem identifying the orbital energy with minus the ionization energy at frozen orbitals [Koopmans:1934], and its direct test in photoelectron spectroscopy; restricted and unrestricted forms, and the basis-set expansions that make the equations a matrix problem.]

Correlation energy

[Reserved: the correlation energy defined as the difference between the exact nonrelativistic energy and the Hartree–Fock limit [Loewdin:1955]; its magnitude — a fraction of a percent of the total energy, and of the same order as the chemical bond energies that must be predicted, which is why the definition matters; configuration interaction, and the systematic hierarchies that converge to the exact answer at a computational cost growing factorially.]

Density functional theory

[Reserved: the Hohenberg–Kohn theorems, that the ground-state density determines the external potential up to a constant and that a universal functional of the density is minimized by it [Hohenberg:1964]; the Kohn–Sham construction, which restores an orbital picture by mapping onto a fictitious non-interacting system with the same density [Kohn:1965]; the exchange–correlation functional as the one unknown, and the honest statement that its approximations are calibrated against experiment and computation rather than derived; the direct descent of this line from the Thomas–Fermi model of Section 88.3.]

The Thomas–Fermi model

[Reserved: the statistical treatment of the electron cloud as a degenerate Fermi gas at each point, due to Thomas [Thomas:1927] and Fermi [Fermi:1928], with the local relation between density and Fermi momentum taken from Quantum Statistics; the resulting nonlinear equation for the screened potential and its universal dimensionless solution; the predicted total energy scaling as $Z^{7/3}$, which the exact atomic energies do follow; the proof by Lieb and Simon that the model is the exact large-$Z$ limit [Lieb:1977]; and Teller's theorem that it cannot bind molecules [Teller:1962] — a model that is asymptotically exact for atoms and predicts no chemistry at all, which is the cleanest available lesson about the reach of a limit theorem.]

Phenomenon 88.2 (Atomic binding energies grow as $Z^{7/3}$).

The energy required to strip a neutral atom completely — the sum of all its ionization energies — is not proportional to $Z$, nor to $Z^{2}$ as a naive hydrogenic count of $Z$ electrons each bound by an energy of order $Z^{2}$ would suggest, but grows very nearly as $Z^{7/3}$ across the whole periodic table [Kramida:2023]. Equivalently, the bulk of the electron cloud contracts: the radius containing a fixed fraction of the charge falls as $Z^{-1/3}$. This is why atomic volumes vary by less than one order of magnitude from hydrogen to uranium while $Z$ varies by ninety, and hence why chemistry takes place on a shell whose radius is almost independent of the element.

Derivation. The Thomas–Fermi energy functional has three terms: the kinetic energy of a locally degenerate Fermi gas (Quantum Statistics), the attraction to the nucleus, and the electrostatic self-repulsion of the cloud,

\begin{equation}\tag{88.3} E[n]=C_{\mathrm{F}}\int n^{5/3}\dd^{3}r -\frac{Ze^{2}}{4\pi\epsilon_{0}} \int\frac{n(\vect{r})}{r}\dd^{3}r +\frac{e^{2}}{8\pi\epsilon_{0}}\iint \frac{n(\vect{r})\,n(\vect{r}')}{\abs{\vect{r}-\vect{r}'}} \dd^{3}r\,\dd^{3}r'\ec \end{equation}

$C_{\mathrm{F}}$ being a pure number times $\hbar^{2}/m$. Try the scaling

\begin{equation}\tag{88.4} n(\vect{r})=Z^{2}\,\nu\bigl(Z^{1/3}\vect{r}\bigr)\ec \end{equation}

which is the unique power-law choice preserving the electron count: substituting $\vect{u}=Z^{1/3}\vect{r}$ gives $\int n\,\dd^{3}r=Z\int\nu\,\dd^{3}u$, so $\nu$ is normalized once and for all. Each of the three terms then carries the same power of $Z$. The kinetic term collects $Z^{10/3}$ from $n^{5/3}$ and $Z^{-1}$ from the volume element: $Z^{7/3}$. The nuclear term collects $Z$ explicitly, $Z^{2}$ from $n$, $Z^{1/3}$ from $1/r$ and $Z^{-1}$ from the volume element: $Z^{7/3}$. The repulsion collects $Z^{4}$ from the two densities, $Z^{1/3}$ from the kernel and $Z^{-2}$ from the two volume elements: $Z^{7/3}$ again. Hence

\begin{equation}\tag{88.5} E[n]=Z^{7/3}\,\mathcal{E}[\nu]\ec \end{equation}

with $\mathcal{E}$ carrying no $Z$ at all: minimizing over $\nu$ once settles every atom, and the ground-state energy is a negative constant times $Z^{7/3}$. The same substitution makes the length scale $Z^{-1/3}$, which is the contraction stated above.

Two limits bound how far this may be pressed. It is not a fit but the exact leading behaviour of the true nonrelativistic Coulomb ground-state energy as $Z\to\infty$, which Lieb and Simon later proved as a theorem. And the same model, exact in that limit, cannot bind two atoms into a molecule at any $Z$ — Teller's theorem — so that a description asymptotically correct for atomic energies predicts no chemistry whatever. The pair of statements is the cleanest available warning that an asymptotic theorem constrains the quantity it is about and nothing else.

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The periodic table

The building-up principle

[Reserved: Bohr's building-up principle, in which the elements are constructed by adding electrons to successive orbits and the periodicity of chemical behaviour is read off the closing of groups [Bohr:1923]; Pauli's exclusion principle supplying the missing reason why a group closes at all [Pauli:1925], with the counting $2(2\ell+1)$ per subshell; the shell capacities 2, 8, 18, 32 and the observed period lengths; the noble gases and the alkali metals as the extreme cases of the same count.]

The Madelung rule and its exceptions

[Reserved: the empirical ordering of subshell filling by increasing $n+\ell$, and by increasing $n$ within equal $n+\ell$, stated by Madelung [Madelung:1936]; the partial rationalizations in terms of the effective potential of Section 88.2.1 [Wong:1979]; the roughly twenty elements that violate it, chromium and copper among them, and the fact that no derivation of the rule from the Schrödinger equation exists — stated here as an open item rather than glossed [Scerri:2007].]

Periodic trends and their evidence

[Reserved: first ionization energies across a period and down a group, with their sawtooth at the half-filled subshells; electron affinities; atomic and ionic radii; electronegativity as a derived rather than a primitive quantity [Pauling:1960]; Moseley's X-ray law, which established the atomic number as the ordering principle and closed the gaps in the table [Moseley:1913], with the pointer to Atomic Models and Spectra where the measurement sits. Each trend is to be given as measured data with uncertainties, not as a schematic arrow.]

Angular-momentum coupling

LS coupling and atomic terms

[Reserved: Russell–Saunders coupling, in which the orbital and spin angular momenta separately couple to $L$ and $S$ before spin–orbit combines them into $J$ [Russell:1925]; the term symbol $^{2S+1}L_{J}$; the addition of angular momenta of Angular Momentum and Spin applied to equivalent electrons, where antisymmetry eliminates most of the naive terms; the fine-structure interval rule and its measured departures; the alkali doublets as the simplest observed case.]

Hund's rules

[Reserved: Hund's empirical ordering of the terms of a configuration — maximum $S$, then maximum $L$, then minimum $J$ for a less-than-half-filled shell [Hund:1925]; their validity range, which is the ground configuration and not the excited ones; and the honest correction to the standard textbook explanation, since the detailed calculations show the first rule to be driven by reduced screening and increased electron–nucleus attraction rather than by reduced electron–electron repulsion [Boyd:1984].]

jj coupling and the intermediate regime

[Reserved: the opposite limit, in which the spin–orbit coupling — growing roughly as $Z^{4}$ for the inner electrons — dominates the residual electrostatic interaction, so that each electron's $j$ is good and the $j$ values couple to $J$; the intermediate coupling that describes most heavy atoms, with term mixing making $L$ and $S$ approximate labels; the observable consequence, namely intercombination lines that are strictly forbidden in pure LS coupling and are nonetheless measured, including the clock transitions used in Experiment: Precision Spectroscopy and Atomic Clocks.]

Atoms in external fields

[Reserved: the Zeeman effect in a many-electron atom, with the Landé $g$-factor derived from the LS coupling scheme [Lande:1921]; the Paschen–Back regime in which a strong field decouples $L$ and $S$ and the anomalous pattern reverts to the normal one [Paschen:1912]; the quadratic Stark effect for non-degenerate levels and the linear one for hydrogenic degeneracies (The Hydrogen Atom); the light shift, which is the same physics at optical frequency and is the systematic that dominates the error budgets of Experiment: Precision Spectroscopy and Atomic Clocks.]

Hyperfine structure

[Reserved: the nuclear-spin hypothesis introduced to explain the observed line satellites [Pauli:1924]; Fermi's contact interaction between the nuclear magnetic moment and the $s$-electron density at the nucleus [Fermi:1930], with the resulting splitting and the interval rule; the hydrogen ground-state splitting of $1420.4\,\mathrm{MHz}$ and its astronomical detection as the 21-centimetre line [Ewen:1951], which is the survey tool of Evidence-Based Cosmology; isotope shifts, both mass and field, as a probe of nuclear size, feeding Experiment: Precision Spectroscopy and Atomic Clocks.]

The Born–Oppenheimer separation

The adiabatic approximation

[Reserved: Born and Oppenheimer's expansion in the fourth root of the electron-to-nucleus mass ratio [Born:1927], giving the electronic problem at clamped nuclei, the potential energy surface as its eigenvalue, and the nuclear Schrödinger equation on that surface; the ordering of electronic, vibrational and rotational energy scales in the ratio $1:\sqrt{m/M}:m/M$, which is what makes a molecular spectrum look the way it does; the adiabatic theorem underlying it (Approximation Methods).]

Phenomenon 88.3 (Three separated spectral scales).

The spectrum of a molecule is not one ladder but three, cleanly separated: electronic transitions in the visible and ultraviolet, vibrational transitions in the infrared, rotational transitions in the microwave. Their spacings stand in the ratios

\begin{equation}\tag{88.6} E_{\text{el}}:E_{\text{vib}}:E_{\text{rot}} \approx1:\sqrt{m/M}:m/M\ec \end{equation}

with $m$ the electron mass and $M$ a nuclear mass, so that each scale is smaller than the one above it by a factor of order forty for a light molecule and more for a heavy one. This separation is the reason an electronic band carries a vibrational progression, why each vibrational band carries rotational fine structure, and why the three regions of a molecular spectrum are recorded with three different kinds of apparatus and were historically three different subjects.

Derivation. Let $a$ be the size of the molecule. An electron confined to that size has momentum of order $\hbar/a$ by the uncertainty relation of The Postulates of Quantum Mechanics, hence an energy scale

\begin{equation}\tag{88.7} E_{\text{el}}\sim\frac{\hbar^{2}}{ma^{2}}\ec \end{equation}

which fixes both the depth and the width of the potential energy surface, since both are electronic quantities. Displacing the nuclei by an amount of order $a$ therefore changes the electronic energy by an amount of order $E_{\text{el}}$, so the curvature of the surface at its minimum is $k\sim E_{\text{el}}/a^{2}$. Nuclei of mass $M$ vibrating in that curvature have

\begin{equation}\tag{88.8} E_{\text{vib}}=\hbar\omega\sim\hbar\sqrt{\frac{k}{M}} \sim\hbar\sqrt{\frac{E_{\text{el}}}{Ma^{2}}} =E_{\text{el}}\sqrt{\frac{\hbar^{2}}{Ma^{2}E_{\text{el}}}} =E_{\text{el}}\sqrt{\frac{m}{M}}\ec \end{equation}

the last step by Equation (88.7). Rotation of the whole molecule about its centre of mass has moment of inertia $I\sim Ma^{2}$ and level spacing

\begin{equation}\tag{88.9} E_{\text{rot}}\sim\frac{\hbar^{2}}{Ma^{2}} =E_{\text{el}}\,\frac{m}{M}\ec \end{equation}

again by Equation (88.7). Together these are Equation (88.6).

The same small parameter justifies the separation that produced the surface in the first place. The nuclear zero-point amplitude is $x_{0}=\sqrt{\hbar/M\omega}$, and dividing by $a$ and using Equations (88.7) and (88.8) gives $x_{0}/a\sim(m/M)^{1/4}$. That is why the Born–Oppenheimer expansion proceeds in the fourth root of the mass ratio and not in the ratio itself: what must be small is the excursion of the nuclei measured against the scale on which the electronic state changes.

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Where the surface fails

[Reserved: the Jahn–Teller theorem, that any non-linear molecule in an orbitally degenerate electronic state distorts to remove the degeneracy [Jahn:1937], with its observed signatures in the spectra of octahedral complexes; conical intersections, where two surfaces touch and the separation breaks down entirely, controlling radiationless transitions and photochemistry; the geometric phase acquired on a circuit around such an intersection, and its relation to the general Berry phase of Approximation Methods.]

The chemical bond

The one-electron bond

[Reserved: $\mathrm{H}_{2}^{+}$, the only molecule whose electronic problem separates exactly, in prolate spheroidal coordinates; Burrau's numerical solution [Burrau:1927], giving the bonding and antibonding curves, the equilibrium separation $106\,\mathrm{pm}$ and the dissociation energy $2.79\,\mathrm{eV}$; the bonding mechanism read off the exact solution — accumulation of electron density between the nuclei — with the virial analysis that attributes it correctly.]

Valence bond theory

[Reserved: Heitler and London's treatment of $\mathrm{H}_{2}$ [Heitler:1927], in which a symmetrized product of atomic orbitals binds and an antisymmetrized one does not, so that the covalent bond is exchange plus spin pairing and the singlet–triplet splitting is the bond; the resulting binding energy compared with the measured $4.478\,\mathrm{eV}$; Pauling's hybridization and resonance [Pauling:1931], and the systematization of directed valence [Pauling:1960].]

Phenomenon 88.4 (The covalent bond is spin-selected electrostatics).

Two hydrogen atoms in their ground states attract and form a stable molecule when their electron spins are opposed, and do not bind at all when the spins are parallel: the singlet potential curve has a deep minimum near an internuclear separation of $74\,\mathrm{pm}$, while the triplet curve is repulsive at every distance. The difference cannot be magnetic. The bond energy is several electronvolts, whereas the magnetic interaction of two electron moments that far apart is of order $10^{-4}\,\mathrm{eV}$, four to five orders of magnitude too small. The spin therefore does not act on anything; it selects, through the antisymmetry requirement of Identical Particles, which spatial state is permitted, and the two permitted spatial states have very different electrostatic energies.

Derivation pending.

The Heitler–London calculation: that the symmetrized and the antisymmetrized products of two hydrogenic ground-state orbitals have expectation energies differing by twice an exchange integral; that this integral has the sign, over the relevant range of internuclear distance, which puts the singlet below the separated atoms and the triplet above them; and the size of the discrepancy that remains between so simple a variational estimate and the measured dissociation energy

Molecular orbitals and LCAO

[Reserved: the alternative in which electrons occupy orbitals delocalized over the whole nuclear framework, developed independently by Hund [Hund:1928] and Mulliken [Mulliken:1928], with the linear-combination-of-atomic-orbitals approximation and the secular equations set out by Lennard-Jones [LennardJones:1929]; bonding and antibonding combinations, symmetry labels, correlation diagrams, and bond order; the prediction that $\mathrm{O}_{2}$ has a triplet ground state — confirmed by its paramagnetism — which valence bond theory in its simple form gets wrong.]

Phenomenon 88.5 (Molecular oxygen is paramagnetic).

Liquid oxygen poured between the poles of a magnet is held there. Gaseous oxygen has a magnetic susceptibility of the size characteristic of two unpaired electron spins per molecule [Curie:1895], and its microwave spectrum shows the fine structure of a triplet. Every straightforward pairing of the sixteen electrons of $\mathrm{O}_{2}$ into a closed-shell structure with a double bond predicts a diamagnetic singlet, and is wrong. This is one of the few questions on which two descriptions usually called equivalent give different answers — the simple valence-bond structure fails and the molecular-orbital picture succeeds — and it is settled by a magnet.

Derivation. Grant the molecular-orbital ordering, which puts the last two electrons of $\mathrm{O}_{2}$ into the antibonding $\pi_{g}$ pair. Those two orbitals are degenerate by the cylindrical symmetry of the molecule: they differ only by a rotation about the internuclear axis, so no electrostatic term can separate them. Call them $\phi_{1}$ and $\phi_{2}$. Two electrons in two degenerate orbitals, with the total state obliged to be antisymmetric, have exactly two possibilities: the symmetric spatial combination $\phi_{1}(1)\phi_{2}(2)+\phi_{2}(1)\phi_{1}(2)$ paired with the spin singlet, or the antisymmetric combination $\phi_{1}(1)\phi_{2}(2)-\phi_{2}(1)\phi_{1}(2)$ paired with the spin triplet. Their Coulomb expectations are $J\pm K$ with

\begin{equation}\tag{88.10} K=\frac{e^{2}}{4\pi\epsilon_{0}}\iint \frac{\phi_{1}^{*}(\vect{r}_{1})\phi_{2}^{*}(\vect{r}_{2})\, \phi_{2}(\vect{r}_{1})\phi_{1}(\vect{r}_{2})} {\abs{\vect{r}_{1}-\vect{r}_{2}}} \dd^{3}r_{1}\,\dd^{3}r_{2}\ec \end{equation}

the symmetric state at $J+K$ and the antisymmetric at $J-K$. The antisymmetric spatial function vanishes identically whenever $\vect{r}_{1}=\vect{r}_{2}$, so two electrons in it are never found at the same point, their mutual repulsion is the smaller, and $K>0$. The triplet therefore lies lower, $\mathrm{O}_{2}$ has $S=1$, and it is paramagnetic. Nothing magnetic entered the argument: the splitting is electrostatic, and the spin only labels which spatial state is allowed.

One qualification belongs with the result. The argument holds the orbitals fixed, and at that level it is exact. When the orbitals are allowed to relax the ordering survives, but its accounting changes — the dominant gain is then in the electron–nucleus attraction rather than in the electron–electron repulsion, which is the correction recorded in Section 88.5.2. The conclusion is robust; the usual one-line explanation of it is not.

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What the two pictures agree on

[Reserved: the demonstration that valence bond and molecular orbital descriptions of $\mathrm{H}_{2}$ are the same wave function in different bases once ionic and covalent terms are treated on an equal footing, and that they differ only in what they get right first — molecular orbital theory at equilibrium, valence bond theory at dissociation; the practical consequence for how molecular calculations are set up, and the standing reminder that a picture is a basis choice, not a physical claim [Pauling:1960] [Herzberg:1950].]

Molecular spectra

Rotational spectra

[Reserved: the rigid rotor, its $BJ(J+1)$ ladder and the equally spaced lines of a pure rotational spectrum in the microwave region, with centrifugal distortion as the first correction [Herzberg:1950]; the selection rule $\Delta J=\pm1$ and the requirement of a permanent dipole moment; bond lengths determined to four significant figures from measured rotational constants; the ammonia inversion doublet as the transition that ran the first maser [Gordon:1955] and opened the road to Experiment: Precision Spectroscopy and Atomic Clocks.]

Vibrational spectra

[Reserved: the harmonic approximation about the minimum of the potential energy surface, giving the normal modes of Oscillations and Mechanical Waves applied to a molecule; the Morse potential and its exactly solvable anharmonic spectrum [Morse:1929], with the convergence of levels to the dissociation limit and the Birge–Sponer extrapolation; overtones and combination bands; rotational fine structure and the P, Q and R branches of a vibration–rotation band [Herzberg:1950]; the greenhouse-active modes of $\mathrm{CO}_{2}$ and $\mathrm{H}_{2}\mathrm{O}$ as the worked case with the largest external stakes.]

Electronic transitions

[Reserved: band systems and their vibrational progressions; the Franck–Condon principle, argued classically from the slowness of nuclear motion [Franck:1926a] and given its quantum form as the overlap of vibrational wave functions on the two surfaces [Condon:1926], with the observed intensity distributions as its test; predissociation and photodissociation; molecular electronic spectra as the identification tool of interstellar chemistry (Stellar Structure and Nucleosynthesis).]

Raman scattering

[Reserved: Smekal's prediction of inelastic light scattering with shifted frequencies [Smekal:1923]; its observation in liquids by Raman and Krishnan [Raman:1928] and, independently and almost simultaneously, in quartz by Landsberg and Mandelstam [Landsberg:1928] — an attribution this treatise states in full; Stokes and anti-Stokes lines and the intensity ratio as a thermometer; the polarizability selection rule, complementary to the dipole rule, so that Raman and infrared spectroscopy see different modes of a centrosymmetric molecule; the scattering formalism itself belongs to Radiation and Scattering of Electromagnetic Waves.]

Phenomenon 88.6 (Light scattered with a molecular frequency shift).

Light scattered by a transparent liquid, gas or crystal contains, besides the unshifted Rayleigh line, weak lines displaced from it by fixed frequency intervals characteristic of the scattering substance and independent of the frequency of the illumination [Raman:1928] [Landsberg:1928]. The displacements coincide with vibrational and rotational intervals of the molecule — which is what distinguishes the effect from fluorescence, whose emitted frequency is fixed rather than its shift. The lines appear in pairs, one on each side of the exciting line, and the pair is not symmetric in intensity: the line displaced to lower frequency is much the stronger, and the ratio of the two intensities depends on the temperature alone, so that the two lines together are a thermometer requiring no calibration.

Derivation. The shift itself is conservation of energy. The scattering event leaves the molecule in a different vibrational or rotational state, so $h\nu_{\text{in}}=h\nu_{\text{out}}+\Delta E$ and the displacement $\nu_{\text{in}}-\nu_{\text{out}}=\Delta E/h$ is a property of the molecule, not of the illumination. A displacement downwards — the Stokes line — requires only that the molecule start in the lower state; a displacement upwards — the anti-Stokes line — requires that it already be in the upper one.

The intensity ratio follows from that asymmetry. By microscopic reversibility the two processes have the same squared matrix element, so their rates differ only through the populations of the initial states and through the frequency dependence of dipole radiation (Radiation and Scattering of Electromagnetic Waves), which supplies a factor $\nu_{\text{out}}^{4}$. In thermal equilibrium (Statistical Mechanics) the population ratio is the Boltzmann factor, so

\begin{equation}\tag{88.11} \frac{I_{\text{anti-Stokes}}}{I_{\text{Stokes}}} =\left(\frac{\nu_{\text{in}}+\Delta E/h} {\nu_{\text{in}}-\Delta E/h}\right)^{4} \ee^{-\Delta E/k_{\mathrm{B}}T}\ep \end{equation}

At room temperature and a vibrational interval of a few hundred reciprocal centimetres the exponential is small, which is why the anti-Stokes lines are the weak ones and why they vanish altogether in a cold sample. Inverting Equation (88.11) gives $T$; and because the two lines are recorded in one spectrum through one optical path, the detector response and the collection efficiency cancel from the ratio, which is what makes the thermometer self-calibrating.

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Intermolecular forces

London dispersion

[Reserved: second-order perturbation theory in the dipole–dipole interaction between two neutral, non-polar atoms, giving an attraction falling as $R^{-6}$ with a coefficient fixed by the atomic polarizabilities [Eisenschitz:1930] [London:1930]; that the force exists between closed shells with no bond, no permanent moments and no overlap, and that it is a correlation effect invisible to Hartree–Fock; the Lennard-Jones form combining it with exchange repulsion, and the van der Waals equation of state of Classical Thermodynamics as its macroscopic signature.]

Phenomenon 88.7 (Closed-shell atoms attract).

Two neutral atoms with no permanent electric moments, no unpaired electrons and no chemical affinity for one another nevertheless attract at distances well beyond the reach of their electron clouds, with an interaction energy falling as the inverse sixth power of the separation. The evidence is everywhere: the noble gases liquefy, the second virial coefficient of every real gas is negative at low temperature (Classical Thermodynamics), and even helium — the least polarizable atom there is — forms a dimer [Grisenti:2000]. The force is absent from any description in which each electron moves in the average field of the others, so the Hartree–Fock method of Section 88.2.2 returns exactly zero for it by construction. It is a correlation effect: an interaction between the fluctuations of two charge distributions, not between their means.

Derivation. Place the atoms at a separation $R$ large compared with their radii, so that the charge distributions do not overlap and the interaction may be expanded in multipoles. The leading term is dipole–dipole,

\begin{equation}\tag{88.12} W=\frac{1}{4\pi\epsilon_{0}R^{3}} \Bigl[\vect{d}_{1}\cdot\vect{d}_{2} -3(\vect{d}_{1}\cdot\hat{\vect{R}}) (\vect{d}_{2}\cdot\hat{\vect{R}})\Bigr]\ec \end{equation}

with $\vect{d}_{i}$ the dipole operators of the two atoms. Each atom in its ground state is spherically symmetric, so $\langle\vect{d}_{i}\rangle=0$ and the first-order energy $\langle W\rangle$ vanishes identically: on the average there is nothing there to interact. Second-order perturbation theory (Approximation Methods) does not vanish, and for the ground state it is negative term by term,

\begin{equation}\tag{88.13} \Delta E=-\sum_{k\neq0} \frac{\abs{\langle k|W|0\rangle}^{2}}{E_{k}-E_{0}} =-\frac{C_{6}}{R^{6}}\ec\qquad C_{6}>0\ec \end{equation}

the sixth power following immediately because $W\propto R^{-3}$ and the energy is quadratic in $W$, and $C_{6}$ being a double sum over the dipole transitions of the two atoms — that is, an integral over their polarizabilities. Every step used only that an atom has transitions with nonvanishing dipole matrix elements, which every atom has: the attraction is universal, and it is attractive for every pair.

Two consequences are worth recording. Being second order, the process excites both atoms virtually at once, which is precisely what a one-electron mean field cannot represent, and is why the effect is invisible to Hartree–Fock and is the standard benchmark for anything claiming to include correlation. And the derivation assumed the interaction to be instantaneous. Once $R$ exceeds the wavelength of the transitions doing the work, the light-crossing time is no longer negligible against the period of the fluctuation, the correlation is partly lost, and the attraction weakens to $R^{-7}$ — the retarded regime of Section 88.9.2, and the point at which the electromagnetic vacuum has to enter the account.

∎

Retardation and the Casimir–Polder force

[Reserved: Casimir and Polder's result that beyond a distance of order the transition wavelength, the finite speed of light weakens the interaction to $R^{-7}$ between two atoms and to $z^{-4}$ between an atom and a conducting wall [Casimir:1948a]; the direct measurement of the atom–surface potential in the retarded regime by deflection of a sodium beam through a micrometre-scale cavity [Sukenik:1993]; the pointer forward to the field-theoretic reading of the same result in Canonical Quantization of Fields.]

Weakly bound dimers

[Reserved: the helium dimer as the extreme test case — a molecule bound by dispersion alone, with a binding energy of order $100\,\mathrm{neV}$ and a mean internuclear separation of about $5\,\mathrm{nm}$, larger than the classical turning point, measured by diffraction of a helium beam from a transmission grating [Grisenti:2000]; why so weakly bound an object is the sharpest available check on a computed dispersion coefficient; Efimov states and the three-body physics that follows, noted here and left to the few-body literature.]