Vibrational Numerics

Vibrational numerics converts a molecular energy surface into frequencies, normal modes, and nuclear wavefunctions. The calculation inherits electronic-surface errors and adds derivative, coordinate, and nuclear-basis errors. Accurate vibrations therefore require accurate changes in energy across nearby geometries, not just a converged energy at one point.

Helpful background. Born–Oppenheimer surfaces explains surface derivatives, and Rayleigh–Ritz explains variational nuclear bases.

Mass-weighted Hessians and rigid motions

At a stationary geometry, let $x$ be Cartesian nuclear displacements and $K$ the Hessian of the potential energy. The harmonic classical equations are $M\ddot x=-Kx$, with each nuclear mass repeated for its three coordinates. Set $q=M^{1/2}x$ to obtain

$$D=M^{-1/2}KM^{-1/2},\qquad D e_\alpha=\omega_\alpha^2 e_\alpha.$$

The same frequencies enter the quantum harmonic energies $\sum_\alpha\hbar\omega_\alpha(n_\alpha+1/2)$. The eigenvectors $e_\alpha$ are orthonormal in mass-weighted coordinates; Cartesian displacement patterns are $M^{-1/2}e_\alpha$, not the same vectors.

An isolated nonlinear molecule has three translations and three rotations; a linear molecule has only two independent rotations. Build these rigid-motion vectors at the geometry, orthonormalize them with a rank-aware procedure, and project them from the vibrational subspace. Small spurious frequencies often arise from incomplete geometry optimization or Hessian noise. A substantial negative eigenvalue of $D$ indicates an unstable direction of the represented surface, rather than a real oscillation frequency.

Numerical projection should not conceal a genuine internal instability. Report the unprojected rigid-mode defects and the stationarity residual before removing them.

A diatomic mass check

For a bond coordinate $r$ near equilibrium,

$$U(r)=U_0+\frac12k(r-r_0)^2,\qquad \mu=\frac{m_1m_2}{m_1+m_2},\qquad \omega=\sqrt{k/\mu}.$$

Changing an isotope changes $\mu$ while leaving the clamped-nuclei surface approximately unchanged within this model. If both masses double, $\mu$ doubles and $\omega$ decreases by $1/\sqrt2$. Confusing nuclear masses with electron-mass atomic units can produce a much larger discrepancy than any electronic approximation.

Angular frequency $\omega$, ordinary frequency $\nu=\omega/(2\pi)$, and spectroscopic wavenumber $\tilde\nu=\omega/(2\pi c)$ are distinct outputs. State which is reported.

Anharmonic eigenproblems

For a one-dimensional stretch with constant reduced mass, solve

$$\left[-\frac{\hbar^2}{2\mu}\frac{d^2}{dr^2}+U(r)\right]\chi_n =E_n\chi_n.$$

A DVR, finite-difference grid, or oscillator basis supplies the representation. Converge the coordinate range and resolution separately. Near dissociation, states extend far from equilibrium and a local harmonic basis or polynomial surface fit can fail badly. A radial problem uses its appropriate endpoint and, when included, centrifugal term; multidimensional curvilinear coordinates require the correct metric and kinetic-energy operator.

The Morse surface $U(x)=D(1-e^{-ax})^2$ on the full coordinate line has

$$E_n=\hbar\omega(n+\tfrac12) -\frac{[\hbar\omega(n+\tfrac12)]^2}{4D}, \qquad \omega=a\sqrt{2D/\mu},$$

for bound levels with $n+1/2<2D/(\hbar\omega)$. This benchmark checks decreasing level spacings and the finite number of bound states. A discretized finite box also produces continuum-like levels above dissociation; counting them as bound vibrations is incorrect.

Error assessment and observables

Compare analytic and finite-difference Hessians, vary displacement size and electronic thresholds, and refine the nuclear representation. Transition intensities also require dipole-moment surfaces and converged matrix elements; accurate energies do not guarantee accurate intensities. Multimode truncations should state which couplings are retained and test selected added couplings, especially near resonances.

Exercises

For $\hbar\omega=1$ and $D=5$ in consistent energy units, compute the Morse ground energy and first transition energy.

Solution

$E_0=0.5-0.25/20=0.4875$, and $E_1=1.5-2.25/20=1.3875$. The transition is $0.9$, below the harmonic spacing $1$. The reduction comes from anharmonicity, not numerical error.

References

• Light, J. C., I. P. Hamilton, and J. V. Lill, “Generalized discrete variable approximation in quantum mechanics,” Journal of Chemical Physics 82, 1400–1409 (1985), doi:10.1063/1.448462 — discrete-variable vibrational representations.
• Wilson, E. B., Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra, McGraw–Hill, 1955 — mass weighting and normal-coordinate analysis.