Harmonic Frequencies and Anharmonicity Constants

2 Harmonic Frequencies and Anharmonicity Constants Electronic Ground State. Harmonic frequencies of NH2 

A complete set of harmonic frequencies and anharmonicity constants of NH2, NH2, ND2, and NHD (in cm ) were derived from an ab initio-calculated general quartic 

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The resonance Raman spectrum of K4[Mo2C18] has been reinvestigated using 488.0 and 514.5 nm excitation. An enormous enhancement of the intensity of the Mo—Mo stretching mode relative to the intensity of other fundamentals was observed and an overtone progression in Vj to 5vj identified. From these data the harmonic frequency and anharmonicity constant X, were calculated as 347.1 + 0.5 cm -1... 

TABLE 2.1a. Harmonic Frequencies and Anharmonicity Constants of X2-Type Moiecuies (cm" ) ... 

Harmonic frequencies and anharmonicity constants Xj (i, k = 1 to4, iderived from fundamental and combination frequencies. The second set of values [16] refers to solution spectra in liquid Ar [11] and was evaluated by taking l-type resonance (see p. 200) into account ... 

The basis set dependence of the calculated harmonic frequencies and anharmonicity constants for triatomic and polyatomic molecules can be expected to resemble closely the patterns discussed above for diatomic molecules. As an example, the harmonic frequencies and anharmonicity constants of the well-studied HCN molecule are shown in Tables 20 and 21, respectively, from CCSD(T) calculations with the cc-pV/ Z basis sets. 

In principle, therefore, it is straightforward to generate the harmonic frequencies and anharmonic constants from electronic-structure calculations although the large number of quartic constants -for example, 3060 in a system of seven atoms - makes this approach cumbersome for laige molecules. 

It is less straightforward to extract the harmonic frequencies and anharmonic constants from experimental data since the force constants are not observables. Instead, the fewee constants are obtained from fits to the vibrational levels. From an initial set of force constants, the harmmiic frequencies o), and anharmonic constants Xjj are calculated, and the vibrational enCTgy levels are predicted from (15.6.9). The differences between the predicted and experimental eneigy levels are next used to modify the force constants. This procedure is repeated until convogence. 

The first accurate quartic force field for a molecule with more than five atoms was computed ab initio at the CCSD(T)/cc-pVTZ level by Martin and co-workers (MLTF). No complete set of experimental anharmonicity constants was available due to the size of the molecule, but comparison with all the available fundamentals for C2H4 and its deutero-isotopomers revealed that all fundamentals were reproduced to better than 10 cm This strongly suggested that the computed harmonic frequencies and anharmonicities were reliable despite the fairly large discrepancies between computed and experimentally derived harmonics (the latter involving extensive approximations), and that the computed harmonic force field, in particular, was more plausible than the experimentally derived one. 

Vibrational Spectra Many of the papers quoted below deal with the determination of vibrational spectra. The method of choice is B3-LYP density functional theory. In most cases, MP2 vibrational spectra are less accurate. In order to allow for a comparison between computed frequencies within the harmonic approximation and anharmonic experimental fundamentals, calculated frequencies should be scaled by an empirical factor. This procedure accounts for systematic errors and improves the results considerably. The easiest procedure is to scale all frequencies by the same factor, e.g., 0.963 for B3-LYP/6-31G computed frequencies [95JPC3093]. A more sophisticated but still pragmatic approach is the SQM method [83JA7073], in which the underlying force constants (in internal coordinates) are scaled by different scaling factors. 

The harmonic frequencies and the anharmonic constants may be obtained from experimental vibrational spectra, although their determination becomes difficult as the size of the system increases. In Table 1.10, we have listed experimental harmonic and anharmonic contributions to the AEs. These contributions may also be obtained from electronic-structure calculations of quadratic force fields (for harmonic frequencies) and cubic and quartic force fields (for anharmonic constants). For some of the larger molecules in Table 1.11, we have used ZPVEs calculated at the CCSD(T)/cc-pVTZ level or higher, see Ref. 12. In some cases, both experimental and theoretical ZPVEs are available and agree to within 0.3 kJ/mol [12, 57],... 

Here E0 is the uth vibrational energy level with wave function rn10 is an harmonic frequency, and A" is the anharmonicity constant. Under certain circumstances a system of this land, initially in its ground state, and driven by a cw field... 

Amos et al. " considered the same complex using comparable basis sets, and evaluated the anharmonic constants using standard second-order perturbation formulas, based upon third and fourth derivatives of the SCF energy. This treatment evaluates each vibrational frequency, Vj, in terms of a purely harmonic potential cOj, and anharmonic constants Xjj relating the various modes i and j (assuming all modes are nondegenerate). 

We have also calculated the equilibrium geometry r, harmonic vibrational frequency cdg, anharmonicity constant (o x, rotational constant B, rovibronic constant a, and centrifugal distortion constant D. is expressed in A and all other quantities are expressed in cm ... 

Often the amount of information concerning a molecule or ion is insufficient to justify the inclusion of anharmonic terms, or even rotational effects. The RRKM equation can nevertheless be successfully employed, and it can yield relatively accurate rate constants. We begin this section with the simplest use of the RRKM equation, in which we assume harmonic frequencies and assume that 7 = 0. 

The experimental vibrational harmonic frequencies cue for Au Cl (and Au Cl equal to 382.8 cm (and 373.9 cm , respectively) and anharmonicity constants coupled cluster procedure theory QCISD(T) which gives cue values of 369.5 cm (and 360.9 cm ) and weXe values of 1.32 cm (and 1.26 cm ), respectively, for Au Cl (and Au Cl) isotopomers. The estimated " dissociation energy of 3.0 0.7 eV and the value of 2.85 estimated at the QCISD level indicate that the AuCl dissociation energy should be below 3.5 eV and that the experimentally obtained value" of 3.5 0.1 eV is probably overestimated by about 0.5 eV. The vibrational-state dependencies of the molecular properties for Au Cl have been established (equations 77-79) ... 

Xu are the diagonal anharmonicity constants and Go is the polyatomic counterpart of the small Too Dunham constant [82] in diatomics. Consequently [50, 84, 90], the optimal scaling factor for ZPVEs is almost exactly midway between a 2(co) suitable for harmonic frequencies (as an approximate correction for systematic bias in the calculated frequencies) and a 2(v) suitable for fundamental frequencies (which additionally seeks to approximately corrects for anharmonicity). In fact, Alecu et al. [86] found for a large variety of basis sets and ab initio and DFT methods that 2((o)/2(ZPVE) = 1.014 0.002, which is almost exactly the ratio of 1.0143 found by Perdew and coworkers [87] between harmonic frequencies and ZPVEs derived from experimental anharmonic force fields. Note that the small uncertainty of 0.002 on a ZPVE of 140 kcal/mol still would translate to about 0.3 kcaFmol, and even that is probably optimistic for the uncertainty in an individual... 

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

In Equation 5.34 to is the harmonic frequency, v the vibrational quantum number, and xe and ye the first and second anharmonicity constants (mass dependent, co x /(coxe) = X /X = il/il, l, and i are vibrational reduced masses). The ZPE(v = 0) contribution to RPFR through first order is thus... 

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