Frequency calculations

This chapter discusses running frequency calculations using Gaussian. Frequency calculations can serve a number of different purposes  

4- To predict the IR and Raman spectra of molecules (frequencies and intensities). 

4 To identify the nature of stationary points on the potential energy surface. 

4 To compute zero-point vibration and thermal energy corrections to total energies as well as other thermodynamic quantities of interest such and the enthalpy and entropy of the system. 

Energy calculations and geometry optimizations ignore the vibrations in molecular systems. In this way, these computations use an idealized view of nuclear position. In reality, the nuclei in molecules are constantly in motion. In equilibrium states, these vibrations are regular and predictable, and molecules can be identified by their characteristic spectra. 

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Statistical mechanics computations are often tacked onto the end of ah initio vibrational frequency calculations for gas-phase properties at low pressure. For condensed-phase properties, often molecular dynamics or Monte Carlo calculations are necessary in order to obtain statistical data. The following are the principles that make this possible. 

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions. 

Complete a frequency calculation to verify that the geometry is correct. 

A number of types of calculations can be performed. These include optimization of geometry, transition structure optimization, frequency calculation, and IRC calculation. It is also possible to compute electronic excited states using the TDDFT method. Solvation effects can be included using the COSMO method. Electric fields and point charges may be included in the calculation. Relativistic density functional calculations can be run using the ZORA method or the Pauli Hamiltonian. The program authors recommend using the ZORA method. 

A number of molecular properties can be computed. These include ESR and NMR simulations. Hyperpolarizabilities and Raman intensities are computed using the TDDFT method. The population analysis algorithm breaks down the wave function by molecular fragments. IR intensities can be computed along with frequency calculations. 

The properties available include electrostatic charges, multipoles, polarizabilities, hyperpolarizabilities, and several population analysis schemes. Frequency correction factors can be applied automatically to computed vibrational frequencies. IR intensities may be computed along with frequency calculations. 

Near top speed, a fan may operate at a speed that is near or above the natural frequency of the wheel and shaft. Under such conditions, the fan can vibrate badly even when the wheel is clean and properly balanced. Whereas manufacturers often do not check the natural frequency of the wheel and shaft ia standard designs, many have suitable computer programs for such calculations. Frequency calculations should be made on large high speed fans. The first critical wheel and shaft speed of a fan that is subject to wheel deposits or out-of-balance wear should be about 25—50% above the normal operating speed. 

Computing the vibrational frequencies of molecules resulting from interatomic motion within the molecule. Frequencies depend on the second derivative of the energy with respect to atomic structure, and frequency calculations may also predict other properties which depend on second derivatives. Frequency calculations are not possible or practical for all computational chemistry methods. 

Including ReadFC is also useful whenever you already have performed a frequency calculation at a lower level of theory. When you have a difficult case and you have no previous frequency job, then CalcFC is a good first choice. CalcAII should be reserved for the most drastic circumstances. 

Gaussian can also predict some other properties dependent on the second and h er derivatives of the energy, such as the polarizabilities and hyperpolarizabilities. These depend on the second derivative with respect to an electric field, and are included automatically in every Hartree-Fock frequency calculation. 

Because of the nature of the computations involved, firequency calculations are valid only at stationary points on the potential energy surface. Thus, frequency calculations must be performed on optimized structures. For this reason, it is necessary to run a geometry optimization prior to doing a frequency calculation. The most convenient way of ensuring this is to include both Opt and Freq in the route section of the job, which requests a geometry optimization followed immediately by a firequency calculation. Alternatively, you can give an optimized geometry as the molecule specification section for a stand-alone frequency job. 

A frequency job must use the same theoretical model and basis set as produced the optimized geometry. Frequencies computed with a different basis set or procedure have no validity. We U be using the 6-31G(d) basis set for all of the examples and exercises in this chapter. This is the smallest basis set that gives satisfactory results for frequency calculations. 

All frequency calculations include thermochemical analysis of the system. By default, this analysis is carried out at 298.15 K and 1 atmosphere of pressure, using the principal isotope for each element type. Here is the start of the ermochemistry output for formaldehyde ... 

When comparing calculated results to thermodynamic quantities extrapolated to zero Kelvin, the zero point energy needs to be added to the total energy. As with the frequencies themselves, this predicted quantity is scaled to eliminate known systematic errors in frequency calculations. Accordingly, if you have not specified a scale factor via input to the Reodlsotopes option, you will need to multiply the values in the output by the appropriate scale factor (see page 64). 

Here is how the zero-point and thermal energy-corrected properties appear in the output from a frequency calculation ... 

Another use of frequency calculations is to determine the nature of a stationary point found by a geometry optimization. As we ve noted, geometry optimizations converge to a structure on the potential energy surface where the forces on the system are essentially zero. The final structure may correspond to a minimum on the potential energy surface, or it may represent a saddle point, which is a minimum with respect to some directions on the surface and a maximum in one or more others. First order saddle points—which are a maximum in exactly one direction and a minimum in all other orthogonal directions—correspond to transition state structures linking two minima. 

Imaginary frequencies are listed in the output of a frequency calculation as negative numbers. By definition, a structure which has n imaginary frequencies is an n order saddle point. Thus, ordinary transition structures are usually characterized by one imaginary frequency since they are first-order saddle points. 

If you were looking for... And the frequency calculation found... It means... So you should... 

Run frequenqf calculations on the two vinyl alcohol isomers we considered in the last chapter. Optimize the structures at the RHF level, using the 6-31G(d) basis set, and perform a frequency calculation on each optimized structure. Are both of the forms minima What effect does the change in structure (i.e., the position of hydrogen in the hydroxyl group) have on the frequencies ... 

Based on the results for propene, we might guess that the transition structure i. halfway between the two minima the structure with a C-C-O-H dihedral angle ul 90°. We would need to verify this with optimization and frequency calculations. 

The frequency job on this structure will confirm that it is a minimum. We ll consider more of the results of this frequency calculation in the next exercise. ... 

Perform frequency calculations on the members of the vinyl series listed below (lowest energy minima only) ... 

Perform frequency calculations for each of these strained hydrocarbon compounds ... 

Determine whether the structure on the right is the transition structure for this reaction based on an optimization and frequency calculation on it. What evidence can you provide for your conclusion ... 

Solution The optimization of 3-fluoropropene leads to a minimum on the PES, indicated by the fact that the frequency calculation results in no imaginary frequencies. 

The transition state optimization (Opt=(TS,CakFC)) of the structure on the right converges in 12 steps. The UHF frequency calculation finds one imaginary frequency. Here is the associated normal mode ... 

Zero-point and thermal energy corrections are usually computed with the same model chemistry as the geometry optimization. However, you may also choose to follow the common practice of always using the HF/6-31G(d) model chemistry for predicting zero-point and thermal energies (see page 149). Of course, such frequency calculations must follow a HF/6-31G(d) geometry optimization. 

Optimize the geometry of this system at the Hartree-Fock level, using the STO-3G minimal basis set and the 6-31G(d) basis set (augmented as appropriate). Run a frequency calculation following each optimization in order to confirm that you have found an equilibrium structure. 

In contrast, the other two frequency calculations determine the corresponding optimized structure to be a minimum. 

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