Hamiltonian magnetic properties

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

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. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

As illustrated above, the microscopic explanation of observed magnetic properties hinges on the construction of an appropriate model. In most instances, simplifications have to be weighed and phenomenological models can be employed, such as the Heisenberg spin Hamiltonian. 

Here 0, is the isospin describing the C, orbital, Iy is the orbital exchange constant, and Jq is the Heisenberg spin-exchange constant. This Hamiltonian can describe many of the magnetic properties of TDAE-C60. 

To describe the electronic structure of carbon nanotubes the Hubbard model has been chosen as it can describe the electrical and magnetic properties and high temperature superconductivity effects also [11]. The model includes the terms of the electron jump energy in vicinity approach and the energy of Coulomb s repulsion of two electrons localized on the same point of unit cell. Hubbard Hamiltonian for the described system is following [11] ... 

T. Helgaker, P. Jorgensen, An electronic hamiltonian for origin independent calculations of magnetic properties, J. Chem. Phys. 95 (1991) 2595. 

The group of Zhao is studying a broad spectrum of clusters with a fixed set of methods. They use EA approaches on TB and empirical potentials, sometimes followed by GGA-DFT refinements electronic and magnetic properties are studied with an spd-band model Hamiltonian in the unrestricted Hartree-Fock (UHF) approximation. Among other systems, they have studied pure clusters of Ag [86],Rh [87],V [88] andCr [89], and mixed clusters of similar atom types, for example, Co/Cu [90] or V/Rh [91], for cluster sizes up to n=13-18. 

In the previous sections we have discussed the ligand-field theory from the point of view of quantum chemistry, and have presented an ab initio derivation of the ligand-field Hamiltonian (1-5). In principle this Hamiltonian can be constructed explicitly using the standard techniques of computational quantum chemistry, although in practice it is evident that this would be subject to the usual difficulties encountered with large molecules. Our concern in this section is with the use of Eq. (1-5) as the basis for a parameterisation scheme that permits the interpretation of the spectroscopic and magnetic properties of transition metal complexes in terms that are chemically intelligible. 

In the general case, magnetic properties will depend on the total angular momentum J, the magnetic moment will be fij, and the Hamiltonian for a single magnetic ion will be... 

The organization is transparent to the type of parameter provided that the derivative Fock operators are suitably constructed. The separation of terms in Eqn. (69) mentioned here corresponds to the idea of Takada, Dupuis, and King for skeleton Fock matrices [64]. A geometric derivative can be obtained with the same code as an electrical property, as can mixed derivatives. Furthermore, magnetic properties, which are unique because second derivatives of the Hamiltonian operator are not necessarily zero, fit into the DHF structure without modification. Open-shell wavefunctions can be em-... 

Nowadays these questions are in the focus of attention in the studies of the magnetic properties of the transition metal oxides and of the colossal magnetoresistance compounds in part. For the LaMnOa crystals the most consistent approach based on taking into account both the CJTE and the superexchange Hamiltonians was developed by Ishihara, Inoue, and Maekawa [35]. 

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