Metallic clusters harmonic

The closest forerunner to TSH theory was due to Hoffmann, Ruedenberg, and Verkade. These authors used spherical harmonics at the center of the cluster to generate linear combinations of orbitals with particular symmetry and nodal characteristics. The energy of these orbitals could then be estimated by the number of angular nodes in the parent spherical harmonic. For aUcah metal clusters, one could actually produce the same answers for the symmetries of... 

Transition metal clusters also have Axy and atomic orbitals, which are classified as 5-type in TSH theory. To represent the transformation properties of these orbitals, we use second derivatives of the spherical harmonics, that is, tensor spherical harmonics - hence the name of the theory. As for the vector surface harmonics, there are again both odd and even 5 cluster orbitals, denoted by L and L, respectively. Usually, both sets are completely filled in transition metal clusters, and we will not consider their properties in any detail in this review. However, the cases of partial occupation are important and have been described in previous articles. ... 

To study the bonding in transition metal cluster compounds, a new type of Spherical Harmonic, with tensor properties, is required. This is because the metal d orbitals have (in addition to 1 o and 2 jt components) 2 8 components (d and dx2 y2) which are doubly noded in the plane perpendicular to the radial vector (see Fig. 16c) and which, therefore, behave as tensors. Two Tensor Surface Harmonic functions may be obtained from each Scalar Spherical Harmonic as follows146 ... 

By employing the Scalar, Vector and Tensor Harmonics, Stone derived the (7 n + 1) cluster valence MO count for deltahedral transition metal clusters and later developed the model to provide an alternative derivation of Mingos electron counting rules for condensed metal clusters1456. ... 

Reproduction of metal-cluster magic numbers using a -deformed, 3-dimensional, harmonic oscillator model... 

Recently it has been shown [21] that the spectrum of this g-deformed, 3-dimensional harmonic oscillator (Q30) reproduces very well that of the modified harmonic oscillator introduced by Nilsson [12, 15] without the spin-orbit coupling term. Since the Nilsson model without spin-orbit coupling is essentially the Nilsson-Clemenger model used for the description of metal clusters [11], it is worth examining whether the Q30 model can be used to reproduce the magic numbers and some other properties of simple metal clusters. 

Indeed it has been found [22] that Q30 (which follows ug(3) D sog(3) symmetry) yields correctly all magic numbers experimentally observed for alkali metal clusters up to 1500, the expected limit of validity for theories based on the filling of electronic shells [4], This indicates that u,(3), which is a nonlinear deformation of the u(3) symmetry describing the 3-dimensional isotropic (spherical) harmonic oscillator, is a good candidate for describing the symmetry of alkali metal clusters. 

We have shown how to derive potentials which, when put into the standard Schrodinger equation, provide approximately the same spectrum as the q-deformed, 3-dimensional harmonic oscillator. In the present work, we have also found prescriptions for choosing the model parameters r and A, thus closing the procedure for obtaining magic numbers. This could be very useful in further investigations on other properties of metal clusters. 

The Mie solution given here assumes a linear response if the classical oscillator is driven hard, anharmonic terms appear which give rise to a nonlinear Mie scattering, similar to effects considered in chapter 9. The perturbative theory of nonlinear Mie scattering has been given for spherical metal clusters, and gives rise to second and third harmonic generation [699]. 

The improved numerical stability of the new deMon2K version also opened the possibility for accurate harmonic Franck-Condon factor calculations. Based on the combination of such calculations with experimental data from pulsed-field ionization zero-electron-kinetic energy (PFl-ZEKE) photoelectron spectroscopy, the ground state stmcture of V3 could be determined [272]. Very recently, this work has been extended to the simulation of vibrationaUy resolved negative ion photoelectron spectra [273]. In both works the use of newly developed basis sets for gradient corrected functionals was the key to success for the ground state stmcture determination. These basis sets have now been developed for aU 3d transition metal elements. With the simulation of vibrationaUy resolved photoelectron spectra of small transition metal clusters reliable stmcture and... 

DEFORMED HARMONIC OSCILLATORS FOR METAL CLUSTERS AND BALIAN-BLOCH THEORY... 

Deformed harmonic oscillators for metal clusters and Balian-Bloch theory 409... 

---