Molecule Types Vibrational Structure

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Vibrational Energy Levels and Franck-Condon Factors

PGOPHER will simulate the vibrational structure associated with an electronic state or transition, including anharmonic and Renner-Teller effects and vibronic mixing. Ionization can also be simulated, providing spin effects can be ignored. Currently C₁, C_i, C₂, C_s, D₂, C_2v, C_2h, D_2h, C_∞v and D_∞h symmetry are explicitly supported. See the worked examples listed below for a quick start.

Note that when using this mode, rotational structure associated with the transition is not simulated. For compatibility with the rotational modes, J will often be displayed but is fixed at zero. (The J range is used to select the maximum vibrational angular momentum to include when simulating spectra.)

A harmonic oscillator basis set is used of the form:

where v_i are the standard vibrational quantum numbers and l_i the corresponding vibrational angular momenta (only required for degenerate modes). Λ and Σ are the electronic quantum numbers, normally only required for linear molecules. Anharmonic, Renner-Teller and other vibronic interactions are accounted for by off-diagonal matrix elements (expressed as perturbations in the language of PGOPHER) and a matrix diagonalisation if required.

To calculate the intensities of the vibrational transitions associated with an electronic transition multidimensional Franck-Condon factors are calculated. As the vibrational basis used for each state is different (the basis is determined by the normal modes) the normal modes in the two states (Q, Q') must be related by:

where K represents the change in equilibrium geometry and J accounts for mixing between modes (The Dushinsky effect). J and K can be input directly, or calculated from the l matrices from the two states.

Further Details

• Making a basic data file for simulating vibrational structure
• Worked Example - The Photoelectron Spectrum of H₂O
• Worked Example - Predicting the Photoelectron Spectrum of H₂O using ab initio Calculations
• Importing l matrices
• Manual entry of the l matrix
• The l Matrix Window
• The Dushinsky Transformation Window
• Molecule Settings
• Electronic State Settings
• Vibrational Modes
• Nuclear Co-ordinates
• Perturbations
• Multidimensional Franck Condon factors
• Vibrational Mode Changes
• Sample Data Files