DFT Calculations of Vibrational Frequencies

In the previous chapters, you have learned how to use DFT calculations to optimize the structures of molecules, bulk solids, and surfaces. In many ways these calculations are very satisfying since they can predict the properties of a wide variety of interesting materials. But everything you have seen so far also substantiates a common criticism that is directed toward DFT calculations namely that it is a zero temperature approach. What is meant by this is that the calculations tell us about the properties of a material in which the atoms are localized at equilibrium or minimum energy positions. In classical mechanics, this corresponds to a description of a material at 0 K. The implication of this criticism is that it may be interesting to know about how materials would appear at 0 K, but real life happens at finite temperatures. 

Much of the following chapters aim to show you how DFT calculations can give useful information about materials at nonzero temperatures. As a starting point, imagine a material that is cooled to 0 K. In the context of classical mechanics, the atoms in the material will relax to minimize the energy of the material. We will refer to the coordinates of the atoms in this state as the equilibrium positions. One of the simplest things that happens (again from 

Density Functional Theory A Practical Introduction. By David S. Sholl and Janice A. Steckel Copyright 2009 John Wiley Sons, Inc. 

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Since the charges on oxygen are —0.9 electron and on the i/t50-carbon Ci -1-0.5 electron, the dipolar forms are also expected to contribute significantly to the electronic structure of the anion. A certain similarity exists between the phenolate and enolate anions regarding the C—O distances. Quantum chemical calculations" of vibrational frequencies for free PhO in the ground state did show some discrepancies with experimental data3 5.366 frequencies determined using DFT methods compare reasonably... 

Extensive comparisons of experimental frequencies with HF, MP2 and DFT results have been reported [7-10]. Calculated harmonic vibrational frequencies generally overestimate the wavenumbers of the fundamental vibrations. Given the systematic nature of the errors, calculated raw frequencies are usually scaled uniformly by a scaling factor for comparison with the experimental data. 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

To calculate the vibrational frequency of CO using DFT, we first have to find the bond length that minimizes the molecule s energy. The only other piece of information we need to calculate is a = (d2E/db2)h hlj. Unfortunately, plane-wave DFT calculations do not routinely evaluate an analytical expression for the second derivatives of the energy with respect to atomic positions. However, we can obtain a good estimate of the second derivative using a finite-difference approximation ... 

Now that we understand how to get a well-converged vibrational frequency for CO from DFT, we can compare this result to experimental data. Experimentally, the stretching frequency of gas-phase CO is 2143 cm- This value is 20 cm-1 higher than our DFT result. This result is fairly typical of a wide range of vibrational frequencies calculated with DFT. The discrepancy between the DFT result and the true vibrational frequency arises in part because of our harmonic treatment of the vibrations, but is also due to the inexact nature of DFT in solving the Schrodinger equation. We return to this issue in the context of a more general discussion of the accuracy of DFT in Chapter 10. 

Hydrogen atoms on Cu( 111) can bind in two distinct threefold sites, the fee sites and hep sites. Use DFT calculations to calculate the classical energy difference between these two sites. Then calculate the vibrational frequencies of H in each site by assuming that the normal modes of the adsorbed H atom. How does the energy difference between the sites change once zero-point energies are included ... 

Calculations of 180 EIEs upon reactions of natural abundance O2 require the normal mode stretching frequencies for the 160—160 and 180—160 isotopologues (16 16j/ and 18 16, ). These values can often be obtained directly from the literature or estimated from known force constants. DFT calculations can be used to obtain full sets of vibrational frequencies for complex molecules. Such calculations are actually needed to satisfy the requirements of the Redlich-Teller product rule. In the event that the full set of frequencies is not employed, the oxygen isotope effects upon the partition functions change and are redistributed in a manner that does not produce a physically reasonable result. 

The DFT calculated temperature profiles are somewhat different for Cu-(tj1-02 i) and Co(i71-02 I). The maximum is predicted to occur at a lower temperature for the copper complex, which also exhibits the larger 180 EIE. An explanation for this behavior again can be found within the DFT calculations and the analysis of vibrational frequencies. Comparing the gas-phase structures and vibrational frequencies below 100 cm-1 indicates an isotope shift that is more than two times greater for Co(p1-02)Sal (7.7 cm ) than for Cu(rj1-02)TMG3Tren (3.0-3.4 cm-1). Therefore, the more temperature-dependent 180 EIE is associated with the greater isotope sensitivity of the low-frequency vibrational modes. This observation underscores the... 

The results of the DFT calculations for various stable C2H.V species and transitions states on Pt(lll) and Pt(211) are summarized in Table V, which also shows entropy changes for the various steps, as estimated from DFT calculations of the vibrational frequencies of the various adsorbed C2H species and transition states on 10-atom platinum clusters (55). Table V also includes estimates of the standard Gibbs free energy changes for the formation of stable C2H surface species and activated complexes responsible for C-C bond cleavage at 623 K. These estimates were made by combining... 

Density functional theory has also been applied to the Cope rearrangement. Nonlocal methods, such as BLYP and B3LYP, find a single transition state with approximately 2 A. The barrier height is in excellent agreement with experiment. These first DFT results were extremely encouraging because DFT computations are considerably less resonrce-intensive than MRPT. Moreover, analytical first and second derivatives are available for DFT, allowing for efficient optimization of stmc-tures (particularly transition states) and the computation of vibrational frequencies needed to characterize the nature of the stationary points. Analytical derivatives are not available for MRPT calculations, which means that there is a more difficult optimization procedure and the inability to fully characterize structures. 

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