Simulation of the Raman spectra of CO2: Bridging the gap between algebraic models and experimental spectra

Abstract

The carbon dioxide Raman spectrum is simulated within an algebraic approach based on curvilinear coordinates in a local representation. The two main advantages of the present algebraic approach are a possible connection with configuration space and the correct description of systems with either local or normal mode character. The system Hamiltonian and polarizability tensor are expanded in terms of curvilinear coordinates. The curvilinear coordinates are in turn expanded into normal coordinates, obtaining an algebraic representation in terms of normal bosonic operators. A canonical transformation maps the operators into a local algebraic representation. The final step is an anharmonization procedure to local operators. The Raman spectrum of CO2 has been simulated, obtaining results close to experimental accuracy, and polarizability transition moments for the Raman spectral lines between 1150 cm(-1) and 1500 cm(-1) are reported. The comparison between experimental and simulated spectra has provided six new CO2 experimental vibrational terms.

An algebraic alternative for the accurate simulation of CO2 Raman spectra

“…Because our methodology to establish the vibrational Hamiltonian has already described in detail in Ref., here, we present only the salient ingredients of the model. The vibrational degrees of freedom of CO 2 are described in terms of curvilinear internal displacement coordinates.…”

“…The procedure to follow consists in the expansion of the G(S) matrix and the potential V ( S ) as a Taylor series in terms of the S variables around the equilibrium configuration, truncating the expansion up to sixth order, which turns out to be enough to obtain high quality results. On the other hand, as explained in Ref., the curvilinear coordinates ( S 1 , S 3 , S 2 a , and S 2 b ) may be expanded in terms of rectilinear symmetry coordinates (normal coordinates) Q α , ( ). Therefore, the Hamiltonian is transformed to where p are the conjugate momenta of Q .…”

“…Establishing the mapping | j v i ⟩→| j n i ⟩, in our approach, we take the matrix elements where n i is the vibrational number of quanta, n i =0,1,2,…, j −1, and k s =2 j +1 is related to the depth of the internal bond stretching coordinates potential. Whereas the anharmonization is valid also for 1D bending degrees of freedom in semi‐rigid bent molecules, the algebraic modeling of vibrational bending degrees of freedom in linear and non‐rigid molecules is based on the U (3) Lie algebra in order to encompass the coupling between rotational and vibrational degrees of freedom that occurs in degenerate and large amplitude bending modes . In this latter case, we introduce the anharmonization procedure equivalent to Equation through the mapping with matrix elements …”

“…We compare simulated and experimental results to reckon which polyad scheme is more suitable. In a previous work, polyad P 212 was considered to generate a Raman spectrum simulation based on a spectroscopic description including 101 vibrational levels up to 9,600 cm −1 with a deviation of root mean square ( rms ) = 0.53 cm −1 and involving 9 fitted transition moments . The present simulated Raman spectrum involves the 178 known experimental term energies with deviations of rms = 0.14−0.20 cm −1 up to 26,600 cm −1 based on the same 9 fitted transition moments for three different polyads.…”

Fidelity, entropy, and Poincaré sections as tools to study the polyad breaking phenomenon

“…Because P L is only preserved in H 2 O, the transition to CO 2 involves a polyad breaking process with conspicuous changes in the molecular properties. A fundamental ingredient for this analysis is the parameterization used to connect the systems, motivated from previous works on the description carbon CO 2 using an algebraic local model [24]. The parameterization comes from the analysis of two interacting harmonic oscillators up to quadratic terms, whose local description in second quantization takes the form…”