Electric dipole and polarizability surfaces are developed for the methanol molecule using {\it ab initio} electronic structure data, computed at the CCSD/aug-cc-pVTZ level of theory, and equivariant neural networks. These property surfaces are used to compute vibrational infrared and Raman intensities up to the OH stretching fundamental vibration. The intensity computations use the vibrational energies and wave functions obtained in continued variational vibrational computations from earlier work [J. Chem. Phys. 163, 064101 (2025)]. The vibrational representation accounts for the large-amplitude torsion and uses curvilinear normal coordinates for the small-amplitude modes, allowing truncation of the vibrational basis set and the integration grid.
Alongside the (ro-)vibrational energy intervals, the peak intensity is an important element for understanding molecular spectra and dynamics. To compute infrared (IR) and Raman intensities, we need high-quality representations of not only the potential-energy, but also the electric dipole and polarizability surfaces.
It has long been recognised that the permutational invariance of atomic nuclei is an essential feature of a mathematical representation of the potential energy surface (PES), [1][2][3][4][5][6][7][8] and also the dipole moment surface (DMS). 2,9 Traditionally, the polynomial expansion has been employed for PES 1,2,5,7,8 and DMS. 2,[9][10][11][12] Recent advances in equivariant representation techniques [13][14][15][16] enable the construction of functions that are both permutationally invariant and correctly preserve spatial rotation-inversion properties for tensors. Recently, machine-learning-based fitting techniques such as high-dimensional neural network potential, 17,18 Gaussian approximation potential, 13,19 atomic cluster expansion, 20 graph neural network 15 have been developed.
Methanol is one of the smallest prototypes for a polyatomic molecule with one large-amplitude motion. The coupling of small-amplitude vibration, torsion, and rotation has been long studied by high-resolution spectroscopy. [21][22][23][24][25][26][27][28] Methanol has been proposed as a sensitive probe, exploiting internal rotation, to detect variations in the protonto-electron mass ratio. [29][30][31][32] Recently, precision spectroscopy studies of methanol have been conducted using frequency combs. 33,34 In parallel, a theoretical proposal for detecting parity-violation in molecules targeted halogensubstituted methanol molecules and identified the most promising candidate from this family. 35 The transition intensities of methanol (and isotopologues) have been proposed to probe the physical conditions in outer space, such as temperature. 36,37 Besides optical and astrophysical uses, a comprehensive vibrational dataset (including transition properties computed in this work) may help us understand positron annihilation spectra of methanol. 38 Mode combination and overtone vibrations-usually faint in regular IR spectra-have been recently proposed as key contributors to positron annihilation spectra via vibrational Feshbach resonances. 39,40 Developments in exact quantum dynamics have focused on polyatomic systems with large-amplitude motions. Recent work extended molecular complexity up to malonaldehyde, 41 handled non-Abelian symmetry, such as in acetonitrile, 42 and examined intra-and intermolecular quantum dynamics in complexes. [43][44][45][46] Further methodological advances elaborated tensor-train and monomer-based contractions, [42][43][44][45] truncated basis and grid representations, 41 novel collocation techniques, 47 and extension of the n-mode representation beyond normal coordinates. 48 This work is the third piece in a recent series on the methanol molecule; first, the coordinate definition and pruned variational vibrational computations were elaborated; 49 second, a new potential energy surface was developed. 50 Third, i.e., this work is about the development of property (dipole and electric polarizability) surfaces and their first application (and assessment) in vibrational transition computations.
We solve the vibrational Schrödinger equation
where the Hamiltonian,
is the sum of the Tvib vibrational kinetic energy operator and the potential energy surface (PES). For the potential energy part, we use the methanol PES of Ref. 50 (PES2025), computed at the F12b-CCSD(T)/cc-pVTZ-F12 level and fitted with permutationally invariant polynomials. As to the vibrational kinetic energy operator, it is first necessary to define physically-motivated internal (vibrational) coordinates, ρ = (ρ 1 , ρ 2 , . . . , ρ D ) with D = 3N -6 for an N -atomic molecule. The coordinate transformation from the laboratory Cartesian coordinates to centre-of-mass translational, orientational, and vibrational coordinates is characterised by the (mass-weighted) metric tensor, g.
Following earlier work, [51][52][53] we write the vibrational kinetic energy operator in the following general form,
with
and
and D = 12 (for a six-atomic molecule), g = det g and G kl = (g -1 ) kl . We use the GENIUSH implementation of the numerical kinetic energy operator approach, and rely on the ’t-vector formalism’ 53,54 to construct the kinetic energy coefficients (over the integration grid).
The vibrational coordinates of CH 3 OH are defined as in previous work, 49,50 following Refs. 55-57. In short, we reiterate the coordinate definition for completeness. First, ‘primitive’ internal coordinates are defined similarly to the Z-matrix approach (Fig. 1), Then, these Cartesian coordinates are shifted so that the centre of mass is at the origin of the coordinate system, which completes the definition of the ‘primitive’ body-fixed (pBF) frame used
This content is AI-processed based on open access ArXiv data.