Hamiltonian exchange term

Therefore, considering Bloch states, a positive exchange term coupling opposite spins is added to the Hamiltonian (11) ... 

If electrons are localized in different parts of a molecule, exchange terms between these electrons are small. In this localized state several different permutation states may be degenerate or near degenerate in zero order. Consequently, these states are highly mixed under the full Hamiltonian. 

Since f o and F are not, in general, eigenfunctions of the respective hamiltonians, an additional term appears in the perturbational treatment, known as the zeroth-order exchange energy or complementary exchange term,61 A r). In regions of small orbital overlap, A may be written as62... 

Within the DFT framework, the molecular Kohn Sham (KS) operator for a molecular solute becomes a sum of the core Hamiltonian h, a Coulomb and (scaled) exchange term, the exchange-correlation (XC) potential Vxc and the solvent reaction operator VPCM of Eq. (7-1), namely ... 

The system is described by a spin density matrix, p(t). The electron spins are then allowed to evolve under the spin Hamiltonian (Equation 8.13 with the exchange term removed) by application of the Liouville-Von Neumann equahon. 

Calculate all of the integrals necessary to describe the core Hamiltonian, the Coulomb and exchange terms, and the overlap matrix. 

Although the low field effects on chemical reactions through radical pairs had been explained by the LCM, Timmel et al. [14] proposed that the so-called low field effects arose also fi om coherent superpositions of degenerate electron-nuclear spin states in a radical pair in zero field. They made some model calculations for their mechanism. At first, let us consider the case of a radical pair with a single spin-1/2 nucleus, e.g., a proton. When the exchange term is not included (J= 0 J), its spin Hamiltonian (H) can be expressed from Eq. (3-3) as... 

The third term in the Hamiltonian is the exchange term and represents the quantum mechanical exchange interaction. The quantity Jij is, in fact, the exchange integral. Finally, the last term is a dipole interaction term and represents the magnetic dipole-dipole interaction of two spins in close proximity. These last two terms, in contrast to the first two, do not depend upon orientation relative to an external coordinate system but rather depend upon the relative orientation of more than one spin... 

Linear transformations of the integrals previously listed, in order to prepare for the determination of the effective Hamiltonian and take advantage of the symmetry properties of the molecule. The computing time needed for this increases at least as w even if the molecule has no symmetry at all, the presence of exchange terms in the Hartree-Fock theory makes a part of the transformation still necessary, namely the construction of the integral list. 

The magnitude of Slater s exchange term in the Hamiltonian was questioned three years later by Caspar [10] who, through a different derivation, obtained the same form, but with a factor of 2/3. This new value was confirmed later by Kohn and Sham. The factor under debate, then named a, was used later on as an atom-dependent parameter [11] in the so-called Xa methodologies. The first among these was derived from the original muffin-tin approximation of Slater [8] and generalized by Johnson [12,... 

A satisfactory description of the magnetic anisotropy in R2Fe14B requires a Hamiltonian which consists of crystal-field terms as well as of exchange terms. For a given rare earth atom this Hamiltonian may be expressed as... 

In the full Hamiltonian the electron repulsion is exact , it is simply the sum of all the inter-particle repulsions. In the HF Hamiltonian this is replaced by Coulomb and exchange terms. The Coulomb term is simply the net average repulsion field due to all the electrons in the molecule and the exchange term removes the self-interaction term included in this average sum plus some further small corrections. 

The second interaction originates from the electrostatic Heisenberg exchange term in quantum mechanics. The magnitude of the interaction is conventionally denoted by the symbol J. The conventions for the sign of J differ between treatises [7]. In this work the sum of the zero-field and exchange interactions is expressed with a Hamiltonian of the type ... 

We now discuss the question of possible anisotropy in the two-ion exchange terms in the hamiltonian. It may be shown (Mackintosh and Bjerrum-M0ller, 1972 Nicklow, 1971 Nicklow et al., 1971) that tu, is not a symmetric function of q for wave vectors along the c-axis of a cone-shaped magnetic structure. It may also be shown (Nicklow et al., 1971) that both single-ion and two ion (pure axial) anisotropy terms cancel in the expression for the difference in the spin-wave frequencies for +q and -q,... 

The magnetic anisotropy of RRh2Si2 (R = Nd, Tb, Ho, Er) which has an API type magnetic structure can be explained by using a R ion Hamiltonian which includes the crystal field term and the isotropic exchange term under the two-dimensional molecular field approximation, when the appropriate crystal field parameters are adopted. The crystal field parameters calculated are listed in table 20 (Takano et al. 1987a). 

In the above Hamiltonian the quantities and are fermion operators in the momentum state k and spin state a. The electron spin operators are denoted by s and s. The direct interaction term Vq is independent of the spin, while the exchange term is explicitly spin dependent. The sign of is chosen so that parallel spins are favored. In principle, the potentials are functions of the momenta k,k and q, but for the purposes of illustrating the physics, it is sufficient to restrict ourselves to the... 

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