Fragment Hamiltonians

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... 

The intennolecular Hamiltonian of the product fragments is used to calculate the sum of states of the transitional modes, when they are treated as rotations. The resulting model [28] is nearly identical to phase space theory [29],... 

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. 

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

SRPA has been already applied for atomic nuclei and clusters, both spherical and deformed. To study dynamics of valence electrons in atomic clusters, the Konh-Sham functional [14,15]was exploited [7,8,16,17], in some cases together with pseudopotential and pseudo-Hamiltonian schemes [16]. Excellent agreement with the experimental data [18] for the dipole plasmon was obtained. Quite recently SRPA was used to demonstrate a non-trivial interplay between Landau fragmentation, deformation splitting and shape isomers in forming a profile of the dipole plasmon in deformed clusters [17]. 

The starting point for Lowdin s PT [1-6] and Eeshbach s projection formalism [7-9] is the fragmentation of the Hilbert space H = Q V, of a given time-independent Hamiltonian H, into subspaces Q and V by the action of projection operators Q and P, respectively. The projection operators satisfy the following conditions ... 

Thus, the electronic coupling is equal to the Hamiltonian matrix element Hda between donor and acceptor states. Recently, this relation was employed to estimate the coupling between nucleobases in DNA fragments [32]. Our estimates showed that both terms in Eq. 5, Hda and Sda Hdd+Haa)> are of the same order of magnitude for the coupling between nucleobases. Thus, Eq. 6 is a rather crude and unnecessary approximation of Eq. 5. 

We applied the effective Hamiltonian approach to DNA fragments of standard geometry with d=a=GGG, determined the effective couphng from Eq. 18, and analyzed the coupling following Eq. 24. Now, we will discuss the results for various bridge compositions (Table 7) [51]. 

Obviously, the situation will be more difficult when additional approximations have to be employed, e.g., the divide-and-conquer scheme, an effective Hamiltonian or a perturbation approach. A central open question is which electronic states of the fragments have to be included so that reliable results, as compared with supermolecular calculations, can be provided. In any case, accounting for just one state per base pair will yield only semi-quantitative results. A careful analysis of this point is highly desirable for DNA-related systems. 

Fig. 22 Schematic view (left) of a fragment of poly(dG)-poly(dC) DNA molecule each GC base pair is attached to sugar and phosphate groups forming the molecule backbone. On the right side, the diagram of the lattice adopted in building our model, with the r-stack connected to the isolated states denoted as -edges. The total Hamiltonian...

Consider the excitation of a molecule with the energy-level scheme described. Let ifiE be an eigenfunction of the complete Hamiltonian, H, and zeroth order Hamiltonian, H0, corresponding to the single discrete state, the set of vibronic states of another electronic state, and the fragmentation continuum, respectively. We write,... 

Second Quantization Photodissociation Hamiltonian, if we consider a system containing many molecules and fragments, it is convenient to use second quantization formalism. We have introduced above the matrix element for photodissociation (see eqs. 50 and 53-57). Based on it, one can write the total... 

Fragments A and B, and p = RB - RA is the distance between their centers of mass. In order to obtain the expression for the nuclear wavefunction D(p,q, qB), one solves the Schroedinger equation associated with the Hamiltonian (66). 

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