Interactions spin Hamiltonian operator

Schematically, the double exchange interaction can be written in the form of a spin Hamiltonian operator (working on the system spin AB —for short, S) as...

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because... 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

We now notice that we could write a Hamiltonian operator that would give the same matrix elements we have here, but as a first-order result. Including the electron Zeeman interaction term, we have the resulting spin Hamiltonian ... 

In this section analytical expressions for ENDOR transition frequencies and intensities will be given, which allow an adequate description of ENDOR spectra of transition metal complexes. The formalism is based on operator transforms of the spin Hamiltonian under the most general symmetry conditions. The transparent first and second order formulae are expressed as compact quadratic and bilinear forms of simple equations. Second order contributions, and in particular cross-terms between hf interactions of different nuclei, will be discussed for spin systems possessing different symmetries. Finally, methods to determine relative and absolute signs of hf and quadrupole coupling constants will be summarized. 

H is the total Hamiltonian (in the angular frequency units) and L is the total Liouvillian, divided into three parts describing the nuclear spin system (Lj), the lattice (Ll) and the coupling between the two subsystems (L/l). The symbol x is the density operator for the whole system, expressible as the direct product of the density operators for spin (p) and lattice (a), x = p <8> ci. The Liouvillian (Lj) for the spin system is the commutator with the nuclear Zeeman Hamiltonian (we thus treat the nuclear spin system as an ensemble of non-interacting spins in a magnetic field). Ll will be defined later and Ljl... 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

Though the true electron spin operators were employed here as well as in the Breit-Pauli Hamiltonian, the phenomenological Spin Hamiltonian, in which the spin coupling is an exchange effect, is in sharp contrast to the Breit-Pauli Hamiltonian, that is including the (magnetic) spin-spin interactions. Since the exchange effect is an effect introduced by the Pauli principle imposed on the wave function, we may write the electron-electron interaction as an expectation value,... 

In molecules, the interaction of surrogate spins localized at the atomic centers is calculated describing a picture of spin-spin interaction of atoms. This picture became prominent for the description of the magnetic behavior of transition-metal clusters, where the coupling type (parallel or antiparallel) of surrogate spins localized at the metal centers is of interest. Once such a description is available it is possible to analyze any wave function with respect to the coupling type between the metal centers. Then, local spin operators can be employed in the Heisenberg Spin Hamiltonian. An overview over wave-function analyses for open-shell molecules with respect to local spins can be found in Ref. (118). 

The calculation of magnetic parameters such as the hyperfine coupling constants and g-factors for oligonuclear clusters is of fundamental importance as a tool for the evaluation of spectroscopic data from EPR and ENDOR experiments. The hyperfine interaction is experimentally interpreted with the spin Hamiltonian (SH) H = S - A-1, where S is the fictitious, electron spin operator related to the ground state of the cluster, A is the hyperfine tensor, and I is the nuclear spin operator. Consequently, it is... 

The 1.7% Co2+ ZnO DMS-QDs in Figure 34 were also examined by Zeeman spectroscopy in transmission mode. An average band-edge Zeeman shift of 53 cm 1/7 over the range of 0-7 T ( Fig. 35) was measured. The Zeeman data were analyzed in the mean-field approximation using Eq. 11, where x is the dopant mole fraction and (Sz) is the expectation value of the Sz operator of the spin Hamiltonian. 7/oa and iVop quantify the exchange interactions between the dopant and unpaired spins in the conduction (CB) and valence bands (VB), respectively (19, 159-161). 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

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