Molecular Structure, Vibration and Rotation

7.1Introduction

A molecule is the first genuinely complicated system in this book. An atom has one heavy centre; a molecule has several that can move, and the electrons rearrange as they do.

The subject is made tractable by one observation: nuclei are thousands of times heavier than electrons and therefore move thousands of times more slowly. The electrons see the nuclei as frozen; the nuclei see only an averaged electronic energy. Making that precise is the Born–Oppenheimer approximation of Sec. 7.3, and it converts one intractable problem into two solvable ones.

Nothing in this chapter is new quantum mechanics. The electronic problem is the variational method of Chapter 5; molecular vibration is the harmonic oscillator of Chapter 2; molecular rotation is the angular momentum algebra of Chapter 3. That is precisely why those chapters came first.

To keep notation consistent with Chapters 8–11, we use the following labels throughout:

• electronic state: \(a,b\);

• vibrational quantum number: \(v\);

• rotational quantum number: \(J\) (with projection \(M_J\));

• spectroscopic wavenumber: \(\tilde\nu\) in cm\(^{-1}\).

The chapter has one practical goal: start from molecular structure and end with a level scheme that can be read spectroscopically. The sequence is Hamiltonian \(\to\) Born–Oppenheimer surfaces \(\to\) orbitals and labels \(\to\) vibrational/rotational ladders \(\to\) mode symmetry.

7.2The Molecular Hamiltonian

For a molecule of \(N\) electrons at positions \(\vec r=\{\vec r_i\}\) and \(M\) nuclei of charge \(Z_A\) and mass \(M_A\) at \(\vec R=\{\vec R_A\}\), the Hamiltonian in atomic units is

with

The notation \(\hat H_e(\vec r;\vec R)\) means that the nuclear positions enter as parameters: move the nuclei and the electrons feel a different potential, but \(\vec R\) carries no derivative in \(\hat H_e\).

The small parameter is the mass ratio. A proton is \(1836\) times an electron, a carbon nucleus \(22000\) times, so \(\hat K_N\) is smaller than the electronic kinetic energy by roughly that factor. This suggests solving the electrons first and letting the nuclei follow.

7.3The Born–Oppenheimer Separation

Step one: fix the nuclei. For each geometry \(\vec R\), solve the electronic problem

Its coordinate representation is \(\psi_i(\vec r;\vec R)=\braket{\vec r}{\psi_i(\vec R)}\). The eigenvalue is a function of the geometry, and the eigenfunctions at each \(\vec R\) form a complete orthonormal set in the electronic coordinates.

Step two: expand and project. Write the exact total state in the electronic basis,

which is exact because \(\{\ket{\psi_i(\vec R)}\}\) is complete at each \(\vec R\). In coordinate space this is

Now act with \(\hat H\) and see what the nuclear kinetic energy does.

By the product rule, for each nucleus,

The first term is what we want. The second and third involve derivatives of the electronic wave function with respect to nuclear position — they measure how much the electronic state changes as the nuclei move.

Projecting \(\hat H\ket{\Psi}=E\ket{\Psi}\) onto \(\bra{\psi_j(\vec R)}\) and using Eqs. (7.3) and (7.7),

where the non-adiabatic coupling operator collects the unwanted terms:

the brackets denoting integration over electronic coordinates only.

A molecular state now carries two labels: an electronic one \(i\) and a nuclear one \(\nu_i\). Energy can be stored in the electrons or in the motion of the nuclei, and the two ladders have very different rungs — the organising fact of molecular spectroscopy (Sec. 7.10).

7.4Molecular Orbitals: the LCAO Method

Solving Eq. (7.3) in practice uses the independent-electron picture of Chapter 5, with orbitals now spread over the whole molecule. The standard construction builds them from atomic orbitals.

Consider H\(_2^{+}\): one electron, two protons a distance \(R\) apart. Take the trial function

with coordinate representation \(\psi(\vec r)=c_A\phi_A(\vec r)+c_B\phi_B(\vec r)\), where \(\phi_A\) and \(\phi_B\) are hydrogen \(1s\) orbitals centred on the two nuclei. Three integrals appear:

called the overlap, Coulomb and resonance integrals. By symmetry \(H_{BB}=H_{AA}\).

7.5Molecular Orbital Labels

In an atom \(\hat L^{2}\) commutes with \(\hat H\) and \(l\) is a good quantum number. In a molecule it is not: the potential is not spherically symmetric, so \([\hat H_e,\hat L^{2}]\neq0\) and \(l\) labels nothing exactly.

What survives depends on what symmetry remains. For a linear molecule, two things do.

Rotation about the internuclear axis. Taking that axis as \(z\), the Hamiltonian is unchanged by rotation about it, so \([\hat H_e,\hat L_z]=0\) and the projection is conserved. Writing \(\lambda\) for one electron and \(\Lambda=|\sum\lambda|\) for the molecule, the letters are assigned by direct analogy with \(s,p,d\):

A \(\sigma\) orbital is cylindrically symmetric about the axis; a \(\pi\) orbital has one nodal plane containing it. Note that \(\pi\) orbitals come in pairs, \(\lambda=\pm1\), so a filled \(\pi\) shell holds four electrons.

Inversion, when there is a centre. A molecule symmetric about its midpoint — H\(_2\), CO\(_2\) — is unchanged by inverting every coordinate through that centre, so parity is a good quantum number exactly as in Chapter 3 and orbitals are labelled

Molecules with no inversion centre, such as CO or HCl, carry no \(g/u\) label at all — its presence tells you the molecule is centrosymmetric.

Since the dipole operator is odd (Chapter 3, Eq. (3.37)), Laporte's rule survives intact in centrosymmetric molecules: a one-photon transition must connect \(g\) to \(u\). This is the most useful selection rule in the remainder of the book.

The leading letter carries no physics: \(X\) means lowest, \(B\) means the third of that multiplicity. It is a name, and it is how every state in the literature is referred to.

Chapter 5's two shortcuts (Thm. 5.9.1) apply unchanged: a closed shell contributes \(S=0\), \(\Lambda=0\) and even parity, and a single hole behaves like a single electron.

7.5.1The Ground State of CO\(_2\)

7.5.2Ionization: Several Thresholds

In an atom one speaks of “the” ionization energy. A molecule has one for each orbital an electron can be removed from, and each leaves the ion in a different electronic state.

Two consequences shape Chapters 9 to 11.

Each threshold opens its own continuum. Above \(13.78\) eV, CO\(_2\) can exist as \(X^{2}\Pi_g\) plus a free electron of any energy. Above \(18.08\) eV there is a second, independent continuum built on \(B^{2}\Sigma_u^{+}\). A measurement that records the photoelectron's energy can distinguish them.

Excited neutral states can lie above a threshold. Nothing prevents a bound state of the neutral molecule from sitting at \(17.5\) eV, far above the first threshold at \(13.78\) eV. Such a state is bound yet degenerate with a continuum — the autoionization of Chapter 5, which in a molecule is the normal situation rather than an exotic one.

7.6Vibration

Now solve Eq. (7.10). For a diatomic the geometry is one number \(R\), and \(V(R)\) has a minimum at the equilibrium bond length \(R_e\) — that minimum is the chemical bond.

Expand about it:

The linear term vanishes because \(R_e\) is a minimum — which is exactly why a harmonic approximation is the natural first step at any stable geometry. Defining the force constant \(k=d^{2}V/dR^{2}|_{R_e}\), the nuclear equation becomes Chapter 2's harmonic oscillator with reduced mass \(\mu=M_AM_B/(M_A+M_B)\), so Theorem 2.7.2 gives

7.6.1Spectroscopic Constants and Units

Experiments usually report frequencies in wavenumbers rather than energies. Write

where \(\omega_e\) and \(\omega_ex_e\) are now in cm\(^{-1}\).

This compact form is useful because spectroscopy gives line positions directly in cm\(^{-1}\). The first term sets the fundamental spacing; the second sets how fast that spacing shrinks with \(v\).

7.6.2Anharmonicity

Equal spacing is an artefact of truncating Eq. (7.20) at second order. A real curve flattens towards the dissociation limit, so the true levels crowd together as \(v\) grows and eventually stop.

The Morse potential is one of the few anharmonic curves solvable in closed form, and its levels are

with \(x_e\) the anharmonicity constant. The spacing \(E_{v+1}-E_v=\hbar\omega_e[1-2x_e(v+1)]\) shrinks linearly with \(v\) and reaches zero at \(v_{\max}\simeq1/2x_e\), the last bound level.

7.7Rotation

Hold the bond length fixed at \(R_e\) and let the molecule tumble. The Hamiltonian is pure rotational kinetic energy,

with \(I\) the moment of inertia. But \(\hat J^{2}\) is the operator whose spectrum Chapter 3 obtained from commutators alone (Thm. 3.3.1), so the answer requires no new work:

\(B\) is the rotational constant. Unlike the vibrational ladder, the rotational one is not equally spaced: successive gaps are

growing linearly with \(J\). A rotational spectrum is therefore a sequence of lines evenly spaced by \(2B\) — one of the most recognisable patterns in spectroscopy, and the subject of Chapter 8.

7.7.1Vibration–Rotation Coupling

The rigid-rotor constant is not exactly constant across vibrational levels: when \(v\) increases, the bond length increases slightly, so \(I\) increases and \(B\) decreases. A standard first correction is

with \(\alpha_e>0\) the vibration–rotation coupling constant.

This single equation explains a common experimental feature: rovibrational lines are not perfectly symmetric around the band origin, because the upper and lower vibrational states have slightly different rotational constants.

7.8Polyatomic Molecules: Normal Modes

A molecule of \(N\) atoms has \(3N\) coordinates. Three describe translation of the whole molecule and two (linear) or three (nonlinear) describe rotation. What remains is vibration:

Expanding the potential to second order in all coordinates gives a quadratic form, and diagonalising it produces independent oscillators in collective coordinates — the normal modes, in which every atom moves in phase at one frequency. This is the eigenvalue problem of linear algebra, and it is set as the project.

CO\(_2\) is linear with \(N=3\), giving \(3(3)-5=4\) modes:

The bend counts twice — it can occur in either of two perpendicular planes — which is how three entries make four modes.

7.9Symmetry Workflow for Molecular Spectroscopy

For learning spectroscopy, symmetry is most useful when used as a procedure. For a linear molecule:

1. Assign geometry and symmetry (for CO\(_2\): linear, centrosymmetric).

2. Label each normal mode by its symmetry species.

3. Apply activity tests: IR active if the mode transforms like a dipole component; Raman active if it transforms like a quadratic form.

For CO\(_2\) in \(D_{\infty h}\), the mode labels are:

This pattern is the practical content of the mutual-exclusion rule for centrosymmetric molecules, developed in detail in Chapter 8.

7.10The Three Energy Scales

The most useful thing to carry out of this chapter is the hierarchy of the three ladders. Order-of-magnitude, for a small molecule:

The separations are factors of roughly a hundred, and they have two consequences. They make spectroscopy possible: each motion can be excited and observed nearly independently. And they justify Born–Oppenheimer itself — the electrons respond a hundred times faster than the nuclei, so they adjust essentially instantaneously.

The timescale column matters in its own right. A pulse of a few femtoseconds — the XUV pulse of Chapter 6 — is fast compared with nuclear motion. During such a pulse the molecule has no time to change shape. That observation licenses the fixed-nuclei treatment used in Chapter 11, and it is why the vibrational structure developed here can be set aside there.

7.11Summary

• The molecular Hamiltonian is \(\hat K_N+\hat V_{NN}+\hat H_e(\vec r;\vec R)\), with the nuclear positions entering \(\hat H_e\) as parameters.

• Expanding in the electronic basis and projecting gives Eq. (7.8); the Born–Oppenheimer approximation drops the non-adiabatic couplings Eq. (7.9), which carry an explicit \(1/M_A\). The nuclei then move on a single potential energy surface.

• LCAO for a homonuclear diatomic gives \(E_{\pm}=(H_{AA}\pm H_{AB})/(1\pm S)\) (Thm. 7.4.1); the bonding combination has no node between the nuclei.

• \(l\) is not a good quantum number in a molecule. Linear molecules retain \(\Lambda\) (\(\sigma,\pi,\delta\) / \(\Sigma,\Pi,\Delta\)) and, with an inversion centre, parity \(g/u\). A \(\pi\) shell holds four electrons. State parity is the product over occupied orbitals.

• Term symbols are \(^{2S+1}\Lambda^{\pm}_{g/u}\) with a leading letter \(X,A,B,C\). CO\(_2\) is \(X^{1}\Sigma_g^{+}\); ionizing each valence orbital gives \(X^{2}\Pi_g\) (13.78 eV), \(A^{2}\Pi_u\), \(B^{2}\Sigma_u^{+}\) (18.08 eV) and \(C^{2}\Sigma_g^{+}\), each opening its own continuum.

• Vibration is Chapter 2's oscillator, \(E_v=\hbar\omega_e(v+\tfrac12)\) with \(\omega_e=\sqrt{k/\mu}\); anharmonicity (Morse) crowds the levels and makes the ladder finite. In spectroscopic units this becomes \(\tilde G(v)=\omega_e(v+\tfrac12)-\omega_ex_e(v+\tfrac12)^2+\cdots\).

• Rotation is Chapter 3's \(\hat J^{2}\), \(E_J=BJ(J+1)\) with \(B=\hbar^{2}/2I\), lines spaced by \(2B\), giving the bond length directly. Vibration–rotation coupling shifts this to \(B_v=B_e-\alpha_e(v+\tfrac12)\).

• For spectroscopy, a symmetry workflow is essential: assign molecular symmetry, label each mode, then apply IR/Raman activity criteria. For CO\(_2\), \(\nu_1\) is Raman-active and IR-inactive, while \(\nu_2\) and \(\nu_3\) are IR-active.

• Electronic \(\gg\) vibrational \(\gg\) rotational by factors of about a hundred. A few-femtosecond pulse is fast compared with nuclear motion, which licenses fixed nuclei in Chapter 11.

7.12Exercises

1. The Born–Oppenheimer terms. Carry out Eq. (7.7) yourself and identify which term survives in Def. 7.3.2. Explain in two sentences why the neglected terms carry \(1/M_A\) and why that makes the approximation good.

2. The LCAO secular problem. Reproduce the proof of Theorem 7.4.1. Then, taking \(H_{AA}=-13.6\) eV, \(H_{AB}=-12.0\) eV and \(S=0.5\), compute \(E_{+}\) and \(E_{-}\) and verify that the antibonding level is destabilised more than the bonding level is stabilised. Explain what that implies for He\(_2\).

3. Term symbols. Derive the term symbol of the ion formed by removing one electron from each of \(1\pi_u\), \(3\sigma_u\) and \(4\sigma_g\) in Eq. (7.18), giving the reasoning for multiplicity, Greek letter and \(g/u\) subscript separately. Check against Example 7.5.2.

4. Anharmonicity.

   1. Expand Eq. (7.23) about \(R_e\) and confirm \(k=2D_ea^{2}\).

   2. Using Eq. (7.24), find the spacing \(E_{v+1}-E_v\) and hence \(v_{\max}\).

   3. For HCl, \(\hbar\omega_e=0.371\) eV and \(x_e=0.0174\). How many bound vibrational levels does it have, and what is the spacing between \(v=0\) and \(v=1\) compared with that between \(v=10\) and \(v=11\)?

5. Rotation. Repeat Example 7.7.1 for HCl, which has \(B=10.59\) cm\(^{-1}\) and \(\mu=1.627\times10^{-27}\) kg. Then compute the wavenumber of the \(J=0\to1\) and \(J=4\to5\) transitions and confirm they are spaced by \(2B\).

6. Counting modes. Use Eq. (7.29) for CO\(_2\), H\(_2\)O, NH\(_3\) (nonlinear, \(N=4\)) and benzene (nonlinear, \(N=12\)). For CO\(_2\), explain why the count is four when the table lists three entries.

7. Vibration–rotation coupling. A diatomic has \(B_e=1.95\) cm\(^{-1}\) and \(\alpha_e=1.8\times10^{-2}\) cm\(^{-1}\).

   1. Compute \(B_0\) and \(B_1\) from Eq. (7.28).

   2. Estimate how much the \(J=5\to6\) spacing changes between \(v=0\) and \(v=1\).

   3. Explain physically why the sign of the change is what it is.

8. Mode symmetry and activity in CO\(_2\). Using the labels in Sec. 7.9, state for each mode whether it is IR-active, Raman-active, both, or neither, and give a one-line symmetry reason.

7.13Project: A Potential Energy Curve and Its Levels

The problem. Take a realistic potential energy curve, find its vibrational and rotational levels numerically, and confront the harmonic and rigid-rotor approximations with what the real curve gives.

Use the Morse potential of Def. 7.6.1 with the H\(_2\) parameters of Example 7.6.1: \(D_e=4.75\) eV, \(R_e=0.741\) Å, \(a=1.94\) Å\(^{-1}\), \(\mu=m_p/2\). Convert everything to atomic units before starting.

1. On paper. Verify that Eq. (7.23) has its minimum at \(R_e\) with \(V=0\), that \(V\to D_e\) as \(R\to\infty\), and that \(k=2D_ea^{2}\). Predict \(\hbar\omega_e\) and \(x_e\).

2. On paper. Compute \(B\) from Eq. (7.26) and confirm it is roughly a hundred times smaller than \(\hbar\omega_e\) — the hierarchy of Sec. 7.10 for a real molecule.

3. On the computer. Solve the nuclear radial equation with the solver from Chapter 2, changing only the potential and the mass. Report the lowest six vibrational energies.

4. On the computer. Plot the six levels as horizontal lines inside the Morse curve. Fit \(E_v=\hbar\omega_e(v+\tfrac12)-\hbar\omega_ex_e(v+\tfrac12)^{2}\) to your numbers and compare \(\omega_e\) and \(x_e\) with the analytic values in Eq. (7.24).

5. On the computer. Add rotation by putting \(J(J+1)/2\mu R^{2}\) into the potential — Chapter 3's centrifugal barrier in molecular dress. Compute levels for \(J=0,10,20,30\) and show the well becoming shallower. At what \(J\) does the last bound state disappear?

6. What has to move. One animation: raise \(J\) from \(0\) and show the effective curve flattening while vibrational levels climb out of it and vanish one by one. That is centrifugal dissociation.

The check. Your \(v=0\) energy should sit close to \(\tfrac12\hbar\omega_e\) from (a): the harmonic approximation is good at the bottom of the well, which is where it should be. A large discrepancy almost always means a unit error — mixing eV with Å and atomic mass units is the usual cause, and converting everything to atomic units first is the cure.

Be ready to answer. Your spacings in (d) shrink as \(v\) increases, while Eq. (7.21) says they should be equal. Point at the step in Eq. (7.20) responsible, and say what physical fact about a real bond the harmonic approximation throws away.