Vibrational frequencies clusters

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

In addition to the dependence of the intennolecular potential energy surface on monomer vibrational level, the red-shifting of the monomer absorption as a fiinction of the number of rare gas atoms in the cluster has been studied. The band origin for the Vppp = 1 -t— 0 vibration in a series of clusters Ar -HF, with 0 < n < 5, was measured and compared to the HF vibrational frequency in an Ar matrix (n = oo). The monomer vibrational frequency Vp p red shifts monotonically, but highly nonlinearly, towards the matrix value as sequential Ar atoms are added. Indeed, roughly 50% of the shift is already accounted for by n = 3. 

It is also possible to measure microwave spectra of some more strongly bound Van der Waals complexes in a gas cell ratlier tlian a molecular beam. Indeed, tire first microwave studies on molecular clusters were of this type, on carboxylic acid dimers [jd]. The resolution tliat can be achieved is not as high as in a molecular beam, but bulk gas studies have tire advantage tliat vibrational satellites, due to pure rotational transitions in complexes witli intennolecular bending and stretching modes excited, can often be identified. The frequencies of tire vibrational satellites contain infonnation on how the vibrationally averaged stmcture changes in tire excited states, while their intensities allow tire vibrational frequencies to be estimated. 

Table V. Calculated vibrational frequencies and IR intensities for the C4H6 - to C12H14+2 clusters. The frequencies are given in cm i and the intensities are relative to the most intense mode for each molecule. The types represent the following vibrations CCS = carbon carbon stretching, CHB = CH in plane bend, SCI = CH2 in plan bend...

Hobza, P., Bludsky, O. Suhai, S., 1999, Reliable Theoretical Treatment of Molecular Clusters Counterpoise-Corrected Potential Energy Surface and Anharmonic Vibrational Frequencies of the Water Dimer , Phys. Chem. Chem. Phys., 1, 3073. 

Besides water dimer, larger clusters of water molecules were extensively investigated by means of the DFT calculations87 111 114 127 128. Laasonen et al.113 studied the structure, the energies, and the vibrational frequencies of small water clusters (up to eight molecules)... 

DeLeo and Fowler (1985a) found that the total energy of the B—H complex was minimized for H at the BC site by cluster methods. Further, DeLeo and Fowler (1985b) reproduced the 1875 cm-1 vibrational frequency observed by Pankove et al. (1985) and also predicted an H-stretching frequency of 2220 cm-1 for the Al—H complex. Assali and Leite (1985,1986), also using cluster methods, found that the AB site for H in the B—H complex gave a minimum energy and were able to reproduce to observed vibrational frequency. 

Frequencies of the hydrogen stretching mode for H° at the bond-center site have been obtained from cluster calculations. Estreicher (1987) found the potential profile for displacements of H along the bond to be U-shaped and flat for small displacements (—0.1 A). Consequently, he found a very low vibrational frequency for such displacements ( that of single Si—H bonds). Other studies of the stretching mode have found 784 cm-1 (Deak et al., 1988) and 800 cm-1 (DeLeo et al., 1988). 

The requirements for Raman resonance that must be fulfilled are the following (120,121) (a) total symmetry of the vibrations with respect to the absorbing center, and (b) same molecular deformation induced by the electronic and vibrational excitations. Quantum chemical calculations (41) of the vibrational frequencies and the electronic structure of shell-3 cluster models allowed the assignment of the main vibrational features, as shown in Fig. 7. The 1125 cm-1 band is unequivocally assigned to the symmetric stretching of the Ti04 tetrahedron. 

Most of the calculations have been done for Cu since it has the least number of electrons of the metals of interest. The clusters represent the Cu(100) surface and the positions of the metal atoms are fixed by bulk fee geometry. The adsorption site metal atom is usually treated with all its electrons while the rest are treated with one 4s electron and a pseudopotential for the core electrons. Higher z metals can be studied by using pseudopotentials for all the metals in the cluster. The adsorbed molecule is treated with all its electrons and the equilibrium positions are determined by minimizing the SCF energy. The positions of the adsorbate atoms are varied around the equilibrium position and SCF energies at several points are fitted to a potential surface to obtain the interatomic force constants and the vibrational frequency. 

The ab initio SCF cluster wavefunction has been used to investigate the bonding of CO and CN- on Cu,0 (5,4,1), (5 surface layer, 4 second layer and 1 bottom layer atoms), and to calculate their field dependent vibrational frequency shifts in fields up to 5.2 x 107 V/cm(46). A schematic view of the Cu10 (5,4,l)CO cluster is shown in Figure 8. In order to assess the significance of Lambert s proposal, that the linear Stark effect is the dominant factor in the field dependent frequency shift, the effect of the field was calculated by three methods. One is by a fully variational approach (i.e., the adsorbate is allowed to relax under the influence of the applied field) in which the Hamiltonian for the cluster in a uniform electric field, F, is given by... 

As it is now very well known, accurate studies of the water-water interaction by means of ab-initio techniques require the use of larger and flexible basis sets and methods which consider correlation effects [85,94-96], Since high level ab-initio post-Hartree-Fock calculations are unfeasible because of their high computational cost for systems with many degrees of freedom, Density Functional Theory, more economical from the computational point of view, is being more and more considered as a viable alternative. Recently, we have presented [97] results of structural parameters and vibrational frequencies for the water clusters (H20) , n=2 to 8, using the DFT method with gradient corrected density functionals. 

Following the trends observed in some geometrical parameters, vibrational frequencies show themselves regular shifts, when passing from small to large clusters. 

Figure 10. Simulated IR spectrum of a liquid formed by the water clusters of Fig. 7, as resulting from DFT calculations. Dashed lines indicate vibrational frequencies of an isolated water molecule.

Figure 11, Density of states of a water sample, referring to two-, three-, tetra- and penta-coordinated 3D clusters and to the total of the sample, as resulting from MD simulation, T=305 K. Dotted lines indicate vibrational frequencies for a single water molecule in gas phase.

One way to include these local quantum chemical effects is to perform ab initio calculations on an HOD molecule in a cluster of water molecules, possibly in the field of the point charges of the water molecules surrounding the cluster. In 1991 Hermansson generated such clusters from a Monte Carlo simulation of the liquid, and for each one she determined the relevant Bom Oppenheimer potential and the vibrational frequencies. The transition-dipole-weigh ted histogram of frequencies was in rough agreement with the experimental IR spectrum for H0D/D20 [130],... 

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