Computing and Analyzing Vibrational Frequencies in ORCA Using Avogadro

Thermodynamic properties are derived from the partition function. Internal Energy: U = -NkT²(∂lnZ/∂T)_V,N. Entropy: S = Nk[ln(Z/N) + T(∂lnZ/∂T)_V,N + 1]. Gibbs Free Energy: G = -NkTln(Z/N) + PV. The Sackur-Tetrode Equation gives entropy for ideal monatomic gas: S = Nk[ln((V/N)(4πmU/3Nh²)^(3/2)) + 5/2]. These formulas connect microscopic partition functions to macroscopic thermodynamic properties, enabling calculation of system behavior from molecular-level information.

Different partition functions apply to different molecular motions: translational, rotational, and vibrational. Each motion contributes distinctively to thermodynamic properties. Entropy calculations require distinguishing between distinguishable and indistinguishable particles, with formulas S = Nk[ln(q/VN) + 1] and S = Nk[ln(q/(VN)) + 1] respectively. Helmholtz free energy (A = U - TS) and Gibbs free energy (G = H - TS) determine process spontaneity (ΔG < 0 indicates spontaneity). Pressure derives from Helmholtz free energy: P = - (∂A/∂V)T, yielding the ideal gas law P = NkT/V. These interconnected concepts form a unified framework connecting molecular energy distributions to macroscopic thermodynamic predictions.

Gibbs Free Energy change (ΔGrxn) is calculated using ΔG = ΔH - TΔS. When ΔGf° for a compound is unknown but ΔHf° and S° are available, ΔGf° = ΔHf° - TΔSf°. For example, with ΔHf°(CO) = -110.5 kJ/mol and S°(CO) = 197.7 J/K·mol, ΔGf°(CO) = -110.5 - 298 × 0.1977 = -110.5 + 59.0 = -51.5 kJ/mol. Temperature must be in Kelvin (298 K under standard conditions), and entropy must be converted to kJ/K·mol by dividing by 1000.

The vibrational partition function is a dimensionless quantity describing molecular distribution among vibrational energy levels. Two approaches exist: one excluding zero-point energy and another including it. The characteristic vibrational temperature θ_vib = hν/kB relates directly to frequency and inversely to reduced mass. This parameter determines how vibrational modes contribute to thermodynamic properties. Understanding these foundational concepts enables prediction of molecular behavior under various thermal conditions.

Enthalpy change calculations use: (1) mean bond enthalpy method (energy in = bonds broken, energy out = bonds formed); (2) calorimetry (Q = mcΔT, convert to kJ, divide by moles); (3) combustion data (ΔH = reactants - products); (4) formation data (ΔH = products - reactants). Enthalpy of neutralization for strong acid-strong alkali ≈ -57 kJ/mol. Entropy measures energy dispersal: ΔS = products - reactants. Gibbs equation: ΔG = ΔH - TΔS (convert ΔS to kJ/mol). Reaction feasible if ΔG < 0. Minimum temperature for feasibility: T = ΔH/ΔS. Even if ΔG < 0, reaction may not occur if activation energy is too high.

This final segment addresses critical post-processing steps for obtaining accurate infrared spectra. Users access scaling factor settings via option 14 in Multiwfn and apply the appropriate coefficient (0.96 for B3LYP/6-31G* calculations) sourced from NIST databases. The segment explains why scaling factors are necessary—computed frequencies typically require correction to match experimental values. After applying scaling factors, users generate the spectrum by entering 0. For publication-quality figures, data can be exported as graphical files or XY datasets for import into OriginLab, enabling customization of axis labels, formatting, and presentation style for research publications.

Different computational methods yield varying accuracy for vibrational frequencies: Hartree-Fock overestimates by ~10-11%; Configuration Interaction by ~5%; Coupled Cluster (CCSD) by 1-4%; CCSD(T) by 1-3%. These errors compare to experimentally deduced harmonic frequencies. Additional errors (~2-3%) arise from anharmonic effects in real molecules not captured by the harmonic oscillator approximation. Scaling factors (e.g., multiplying by 0.97) are commonly applied to bring computed frequencies closer to experimental values. However, scaling zero-point energies differently than frequencies may actually increase error, as shown in theoretical analysis comparing harmonic and fundamental frequency approximations.

This tutorial demonstrates the complete workflow for performing and plotting infrared (IR) spectra using Gaussian 09W/16 and Multiwfn software: (1) First optimize the molecular geometry using Gaussian with appropriate theory (B3LYP) and basis set (6-31G*), then save the optimized Cartesian coordinates; (2) Perform frequency calculation using the optimized coordinates; (3) Use Multiwfn to load the output file, select IR spectrum plotting (option 11), apply vibrational frequency scaling factor (typically 0.96 for B3LYP/6-31G* calculations), and generate the IR spectrum; (4) Alternatively, use GaussView to plot the IR spectrum directly from the frequency calculation output file. The scaling factor correction is essential because theoretical vibrational frequencies are systematically higher than experimental values.

This tutorial demonstrates how to simulate infrared (FTIR) absorption spectra using free computational chemistry software ORCA and Avogadro. The process involves drawing a molecule (nitrobenzene) in Avogadro, generating an input file with DFT calculations using the B3LYP functional and def2-TZVPP basis set, executing the calculation via command line, and analyzing the output to obtain vibrational frequencies. The calculated spectrum requires a scaling factor (0.963 for B3LYP/def2-TZVPP) to account for the harmonic oscillator approximation, and should be compared with experimental data from databases like SDBS for validation.

Four major factors influence vibrational frequencies in IR spectroscopy: (1) Coupling interaction splits single vibrations into symmetric (lower frequency) and asymmetric (higher frequency) modes when identical groups interact; (2) Hydrogen bonding decreases frequencies, producing broad bands for intermolecular bonding and sharp bands for intramolecular bonding; (3) Resonance increases bond length through electron delocalization, causing frequency decreases; (4) Inductive effects cause electron-releasing groups to decrease frequencies (e.g., CH3- shifts C=O from 1750 to 1745 cm⁻¹) while electron-withdrawing groups increase frequencies (e.g., Cl- shifts C=O from 1725 to 1714 cm⁻¹). These effects provide structural information about molecular substituents and bonding patterns.

This section presents advanced computational methods for calculating molecular spectra from simulation data. Ab initio molecular dynamics provides electronic structure on-the-fly, enabling dipole moment calculations through orbital localization techniques. For infrared spectroscopy, dipole moments are derived from Wannier function centers weighted by effective charges. For vibrational circular dichroism (VCD), the magnetic moment is computed using a continuity equation approach expressing current density as a conservative velocity field. The VCD spectrum emerges from Fourier transforming the cross-correlation between electric and magnetic moments. These methods enable direct comparison with experimental spectra, validating computational predictions. However, proper sampling of conformational space remains critical, as insufficient sampling produces inaccurate results even with accurate methodologies. The speaker demonstrates successful reproduction of experimental VCD spectra when including contributions from multiple cluster types, with hybrid functionals providing better agreement than pure GGA functionals.

Computational methods (quantum mechanics, density functional theory) can simulate CD spectra by calculating the electronic structure and transition properties of molecules. These simulations help interpret experimental CD data, assign absolute configurations, and understand the contributions of different molecular regions to the observed chirality. Current density analysis visualizes the flow of electrons during electronic transitions, revealing how different parts of a molecule contribute to CD signals. Matrix analysis decomposes CD signals into contributions from different molecular fragments, revealing how different chiral elements cooperate or interfere with each other. Vibrational circular dichroism (VCD) measures differential absorption of circularly polarized light in the infrared region. Unlike electronic CD, VCD probes vibrational modes and is sensitive to molecular conformation. VCD is particularly valuable for determining absolute configurations of molecules with multiple stereocenters, as it provides more detailed structural information than electronic CD. VCD analysis involves generating all possible stereoisomers, computing the lowest-energy conformations for each, simulating VCD spectra for each stereoisomer, and comparing with experimental data. The agreement between simulation and experiment allows assignment of absolute configurations. Initial VCD simulations often use harmonic approximations for vibrational frequencies, but for more accurate results, anharmonic corrections are applied to account for the true vibrational behavior of molecules.

Raman scattering is an inelastic scattering process where the frequency of scattered light changes. If the scattered light has higher frequency, it is called Stokes lines. If it has lower frequency, it is called anti-Stokes lines. This requires an incoherent light source.

Vibrational Circular Dichroism (VCD) measures differential absorption of circularly polarized light in the infrared region, corresponding to molecular vibrations. VCD is now preferred over electronic CD for absolute configuration determination because it provides many more spectral bands for analysis, does not require chromophores near the chiral center, and is less sensitive to conformational changes. VCD spectra of enantiomers are mirror images, and the technique can be combined with computational methods to assign absolute configurations with high confidence. VCD has become the standard method for stereochemical assignment in complex natural products.

This tutorial demonstrates the complete workflow for calculating and plotting harmonic VCD (Vibrational Circular Dichroism) spectra of chiral molecules like alpha-pinene using Gaussian 09W/16 and Multiwfn software. The process involves: (1) obtaining molecular structures from PubChem and converting them to Gaussian input format (.gjf), (2) performing geometry optimization using appropriate basis sets (e.g., B3LYP/6-31G*), (3) conducting VCD calculation with the keyword 'Freq=VCD' in the route section, (4) using Multiwfn to visualize the spectrum by selecting option 11 (Spectra plot) and entering the appropriate vibrational scaling factor (e.g., 0.96 for B3LYP/6-31G*), and (5) saving or exporting the plot for research purposes.