Interpretation of Semiclassical Transition Moments through Wave Function Expansion of Dipole Moment Functions with Applications to the OH Stretching Spectra of Simple Acids and Alcohols

Abstract

Semiclassical description of molecular vibrations has provided us with various computational approximations and enhanced our conceptual understanding of quantum mechanics. In this study, the transition moments of the OH stretching fundamental and overtone intensities (Δv = 1-6) of some alcohols and acids are calculated by three kinds of semiclassical methods, correspondence-principle (CP) approximation, quasiclassical approximation, and uniform WKB approximation, and their respective transition moments are compared to those by the quantum theory. On the basis of the local mode picture, the one-dimensional potential energy curves and the dipole moment functions (DMFs) were obtained by density functional theory calculations and then fitted to Morse functions and sixth-order polynomials, respectively. It was shown that both the transition energies and the absorption intensities derived in the semiclassical methods reproduced their respective quantum values. In particular, the CP approximation reproduces the quantum transition moments if the formula given by Naccache is used for the action integral value. On the basis of these semiclassical results, we present a picture to understand the small variance in the overtone intensities of these acids and alcohols. Another important result is the ratios of semiclassical-to-quantum transition moment are almost independent of the applied molecules even with a great molecular variance of the DMFs, and they depend only on the nature of the semiclassical approximations and the quantum number. The difference between the semiclassical and quantum transition moments was analyzed in terms of a hitherto unrecognized concept that the Fourier expansion of the time dependent DMF in the CP treatment is a kind of the wave function expansion method using trigonometric functions as the quotient functions. For a Morse oscillator, we derive the analytic and approximate expressions of the quotient functions in terms of the bond displace coordinate in both the CP and the quantum mechanical frameworks and discuss the methodological dependence of the calculated transition moments. As a byproduct, we have found a simple derivation of the DMF expression first derived by Timm and Mecke long time ago.

Graphical Transition Moment Decomposition and Conceptual Density Functional Theory Approaches to Study the Fundamental and Lower-Level Overtone Absorption Intensities of Some OH Stretching Vibrations

“…The lower-level transition moments of anharmonic oscillators are dominated by these terms as demonstrated in semiclassical analyses , for simple molecules and in model calculations for hydrogen-bonded systems . The conformational dependence of ⟨ v |μ z |0⟩ is essentially included in μ z because the potential energy functions V (Δ R ) for OH stretching vibrations are almost independent of the substituents or conformations .…”

“…These results suggest that the electron density In most isolated ROH molecules, for the displacement of ΔR, the DMF shows convex behavior, namely, M 2z < 0, as shown in Figure 3, while with hydrogen bond interactions with an acceptor, the DMF exhibits not only a large increase of the slope, but also a change of the shape from convex to concave, and typically, M 2z becomes positive; see, for example, ref 27. Using this DMF feature and the wave function expansion method, 46 it is easy to show that the fundamental v = 1 and the first-overtone v = 2 intensities are significantly strengthened/weakened in the hydrogen-bonded systems. In this section, we analyze the numerical origins of their M 1,x , M 1,z , M 2,x , and M 2,z values and the characteristics of the vector components of DMF parameters using linear response calculations.…”

“…If the time-dependent oscillation is considered through ΔR(t) = A sin ωt, the ρ (1) (r)ΔR and ρ (2) (r)ΔR 2 terms are responsible mostly for the fundamental and overtone absorption intensities, respectively. 46 If ρ (1) (r) and ρ (2) and lower-level overtone intensities will appear with various stereoelectronic effects. DFT, the electron density ρ(r) at position r is expressed as a quantity that is obtained by calculating the functional derivative of the energy E[N,v] = F[ρ] + ∫ v(r)ρ(r) dr with respect to the external potential v(r) while fixing the number of electrons N. 36 Here, F[ρ] is the sum of the kinetic energy functional and the electron−electron interaction energy functional.…”

“…If the time-dependent oscillation is considered through Δ R ( t ) = A sin ω t , the ρ (1) ( r )­Δ R and ρ (2) ( r )­Δ R 2 terms are responsible mostly for the fundamental and overtone absorption intensities, respectively . If ρ (1) ( r ) and ρ (2) ( r ) have their respective large values in different molecular regions, then different conformational dependences in the fundamental and lower-level overtone intensities will appear with various stereoelectronic effects.…”

Vibrational spectrum of a 1D oscillator: The quantum, the Wigner, and the classical ways

“…Methods mixing quantum wavefunction information and classical propagation, so‐called mixed quantum‐classical or semiclassical (SC) methods, have also been developed. [ 7–12 ]…”