Hamiltonian shell model

The model which we have developed is called the Fermion Dynamical Symmetry Model (FDSM)11 which is the subject matter of two recent preprints. The FDSM begins with a shell model Hamiltonian in one major valence shell. 

The microscopic derivation of a boson hamiltonian from a fermion one is basically a two step process. In the first step, one has to select the collective subspace of the shell model space. For the IBM this means truncating the shell model space to the space of collective S-D pairs. In the second step, this space has to be mapped onto the s-d boson space. 

To improve on this situation, we have used the Glasgow shell model code, rewritten to treat bosons [MOR80], to perform a least-squares fit to the excitation energies of a single nucleus using the IBM-2 Hamiltonian (1) without the Majorana term. In general, the model is applicable to only 10 to 12 experimentally known energy levels... 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

Meldner, H. Predictions of New Magic Regions and Masses for Super-Heavy Nuclei from Calculations with Realistic Shell Model Single Particle Hamiltonians . In Ref. [5], pp. 593-598. 

The earliest applications of the shell model, as with the Born model, were to analytical studies of phonon dispersion relations in solids.These early applications have been well reviewed elsewhere.In general, lattice dynamics applications of the shell model do not attempt to account for the dynamics of the nuclei and typically use analytical techniques to describe the statistical mechanics of the shells. Although the shell model continues to be used in this fashion, lattice dynamics applications are beyond the scope of this chapter. In recent decades, the shell model has come into widespread use as a model Hamiltonian for use in molecular dynamics simulations it is these applications of the shell model that are of interest to us here. 

Effective Hamiltonians and effective operators are used to provide a theoretical justification and, when necessary, corrections to the semi-empirical Hamiltonians and operators of many fields. In such applications, Hq may, but does not necessarily, correspond to a well defined model. For example. Freed and co-workers utilize ab initio DPT and QDPT calculations to study some semi-empirical theories of chemical bonding [27-29] and the Slater-Condon parameters of atomic physics [30]. Lindgren and his school employ a special case of DPT to analyze atomic hyperfine interaction model operators [31]. Ellis and Osnes [32] review the extensive body of work on the derivation of the nuclear shell model. Applications to other problems of nuclear physics, to solid state, and to statistical physics are given in reviews by Brandow [33, 34], while... 

Meldner, H. Predictions of new magic regions and masses for super-heavy nuclei from calculations with realistic shell model single particle Hamiltonians. Ark. Fys. 36, 593-598... 

We have not mentioned open shells of electrons in our general considerations but then we have not specifically mentioned closed shells either. Certainly our examples are all closed shell but this choice simply reflects our main area of interest valence theory. The derivations and considerations of constraints in the opening sections are independent of the numbers of electrons involved in the system and, in particular, are independent of the magnetic properties of the molecules concerned simply because the spin variable does not occur in our approximate Hamiltonian. Nevertheless, it is traditional to treat open and closed shells of electrons by separate techniques and it is of some interest to investigate the consequences of this dichotomy. The independent-electron model (UHF - no symmetry constraints) is the simplest one to investigate we give below an abbreviated discussion. 

Due to the integral approximations used in the MNDO model, closed-shell Pauli exchange repulsions are not represented in the Hamiltonian, but are only included indirectly, e.g., through the effective atom-pair correction terms to the core-core repulsions [12], To account for Pauli repulsions more properly, the NDDO-based OM1 and OM2 methods [23-25] incorporate orthogonalization terms into the one-center or the one- and two-center one-electron matrix elements, respectively. Similar correction terms have also been used at the INDO level [27-31] and probably contribute to the success of methods such as MSINDO [29-31],... 

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

With increasing the hole concentration the Fermi surface of the t-J model is transformed to a rhombus centered at Q [16], This result is in agreement with the Fermi surface observed in La2-xSrxCuC>4 [19] [however, to reproduce the experimental Fermi surface terms describing the hole transfer to more distant coordination shells have to be taken into account in the kinetic term of Hamiltonian (1)]. For such x another mechanism of the dip formation in the damping comes into effect. The... 

---