DIM method

The most commonly used semiempirical for describing PES s is the diatomics-in-molecules (DIM) method. This method uses a Hamiltonian with parameters for describing atomic and diatomic fragments within a molecule. The functional form, which is covered in detail by Tully, allows it to be parameterized from either ah initio calculations or spectroscopic results. The parameters must be fitted carefully in order for the method to give a reasonable description of the entire PES. Most cases where DIM yielded completely unreasonable results can be attributed to a poor fitting of parameters. Other semiempirical methods for describing the PES, which are discussed in the reviews below, are LEPS, hyperbolic map functions, the method of Agmon and Levine, and the mole-cules-in-molecules (MIM) method. 

The LEPS and DIM methods become identical for s3 systems provided that orbital-overlap effects are disregarded.115... 

Kuntz et al [60] using the diatomics-in-molecule (DIM) method constructed three slightly dfferent DIM PESs for the two lowest states. All these DIM PESs exhibit no barrier to reaction (insertion or abstraction) on the lowest surface. 

For this reason, we have chosen to apply the dlatomlcs-inmolecules method to the study of the heavier Group lA trimers 3, Rbs, and CS3 and the Group IB trimers CU3, Ag3, and AU3. In this chapter, we will discuss the application of both ab initio and DIM methods to the study of the structure of small metal clusters, specifically clusters of the Group lA and Group IB atoms. These are, by choice, examples taken from our work but we have endeavored to comment on the general applicability of the various techniques. 

The Dlatomics-ln-Molecules Approach. The simple version of the DIM method that we employ is based on the Heitler-London approximation (28). In spirit, it is similar to the London-Eyring approach except that we use accurate diatomic potential curves (29a) rather than an approximate form for the diatomic triplet curve (e.g. the Sato parameter for LEPS surfaces) (29b). 

State. The first excited state lies at considerably higher energy for sodium and potassium decreasing to 1.39 eV for cesium ( ). Thus, the DIM method should be well-suited for studying clusters of sodium and potassium and of moderate utility for the study of rubidium and cesium. The application of DIM to clusters of Group IB atoms will be valid only if the ground atomic state d °s is well-separated from the first excited state,d s (for Ag,d °p is the first excited level isoenergetic with d s ). 

The interaction between an open shell atom with a closed shell atom or molecule can be modeled with the Diatomics-in-Molecule (DIM) method [11]. This is an approximate approach, however, with a sounel r hemir al mothntion. The total interaction energy between an open shell atom anel all the remaining closed shell atoms in the cduster can be expressed in a pair additive way as Vo/m = X) =i n where r is the raelius vector between the i-th rare gas atom anel the p-atom (open shell atom) anel is the interaction energy between a single rare gas atom... 

H H. In the same figure, we also present the results of Kimura [42] for the total electron capture obtained by the DIM method for H2. 

Thus the construction of the polyatomic Hamiltonian matrix is reduced in the ab initio DIM method to the construction of the corresponding matrices of the atomic and diatomic fragments. 

The DIM method is most commonly employed as a semi-empirical technique. The fragment Hamiltonian matrices are usually related to atomic and diatomic energies by making various approximations for the overlap matrices. Both the form of the DIM equation and the chosen set of PBF must be sufficient to account for all the qualitative features of the system being studied. Under such circumstances the approach may offer acceptable accuracy for modest computational effort. Given the input of experimental and accurate theoretical data for the fragments, it is not unreasonable to suppose that the method can yield results comparable to those from larger... 

Diatomics-in-Molecules (DIM) Method Semi-Empirical Valence-Bond Methods Approximate Pseudo-Potential Theories... 

The valence-bond approach plays a very important role in the qualitative discussion of chemical bonding. It provides the basis for the two most important semi-empirical methods of calculating potential energy surfaces (LEPS and DIM methods, see below), and is also the starting point for the semi-theoretical atoms-in-molecules method. This latter method attempts to use experimental atomic energies to correct for the known atomic errors in a molecular calculation. Despite its success as a qualitative theory the valence-bond method has been used only rarely in quantitative applications. The reason for this lies in the so-called non-orthogonality problem, which refers to the difficulty of calculating the Hamiltonian matrix elements between valence-bond structures. 

In order to apply the DIM method in the form discussed above, we have to know experimental potential energy curves for the states of the diatomics involved in the calculation, and we must also calculate overlap matrix elements between the composite functions. If experimental diatomic curves... 

A difficulty with this local approach to dynamical correlation is that, in Moller-Plesset theory, for example, the zero-order Fock operator is no longer diagonal in the space of the Slater determinants, making the application of such theories slightly more complicated than theories based on canonical orbitals. Currently, the development of local correlation methods is an active area of research [57-63]. The diatomics-inmolecules (DIM) method and the triatomics-in-molecules (TRIM) method, for instance, recover typically 95% and 99.7%, respectively, of the full MP2 correlation energy [63]. By means of a linear scaling local variant of the CCSDT method,... 

The calculation includes three diabatic potential energy surfaces obtained using the diatomic in molecules (DIM) method (36). Using this 2D quantum treatment in hyperspher-ical coordinates the nonadiabatic problem can be solved also for problems involving more than three diabatic surfaces. 

For the case of molecular targets, we show in Fig. 4 the results for the direct differential cross section for protons colliding with molecular hydrogen for proton energies of 0.5, 1.5, and 5.0 keV [18] averaged over all the target orientations. The experimental data are from Gao et al. [24]. Also, for comparison, we present the results of Kimura s DIM method (short-dashed line) [24]. 

Fig. 5. Probability for electron capture by a proton colliding with molecular hydrogen at 1 keV as obtained by END (solid Une). For comparison, we show the results from Kimura s DIM method (short-dotted line) [28] which have been scaled 0.1 times.

In the DIM method, originally proposed by Ellison and subsquently developed by Tully and Kuntz, the /th electronic state is first written as the expansion Cmi m which runs over some finite set of polyatomic... 

A list (not exhaustive) of global multi-sheeted potential energy surfaces for the relevant manifold of electronic states of the same symmetry of some polyatomic systems is given in Table 1. Space limitations prevent any discussion of such a work, with the reader being referred to the original papers d06,112-115,168,182-185 fQj, details. Much of the work published using the DIM method or some of its variants (Refs. 171,185-188, and references therein) has also been omitted. ... 

We are still in the situation where it is not possible to cover a polyatomic surface with ab initio points of sufficient accuracy that inaccuracies can be ignored the exceptions are sufficiently rare to prove the rule. For this reason potential functions which are based in part on empirical data will always be superior to those that are fully ab initio. However, it would be exceptional to derive a potential solely from empirical data. Spectroscopic and thermodynamic data generally only give information about minima on the surface often only the lowest minimum. Kinetic data generally only indicate barrier heights on reaction paths. Molecular beam scattering and equilibrium or transport gas phase data may provide sensitive tests of potentials but we cannot get the potential directly from the data. It is from the fusion of empirical and ab initio data that one obtains the best potential functions. This is, I believe, the reason for the success of LEPS fimctions [2], the DIM method [10] and the many-body expansion with empirical one and two-body terms [3]. 

The equations presented here suffice to construct the polyatomic hamiltonian in the smmetry-adapted basis, from the diatomic and atomic fragment matrices, tyj and respectively. The latter constitute the numerical input to the DIM method which, in addition to the mappings ... 

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