Hamiltonian terms

The Hamiltonian term for the electron-electron dipolar interaction is ... 

We will see that a Hamiltonian term identical in form also arises from spin-orbit coupling, but first we will pause to see the effect of this Hamiltonian on the energy levels and ESR spectrum of a triplet-state molecule. The spin triplet wave functions can be written in the notation S,ms) ... 

We now will show that spin-orbit coupling can give a spin Hamiltonian term identical to that we obtained from the electron dipolar interaction. Consider the... 

Starting with the two ways of expressing the hyperfine Hamiltonian term, we equate the coefficients of Sx, Sy and Sz ... 

Assuming identical principal axes for A and P, the Hamiltonian term would have the form in the (x", y", z") coordinate system ... 

The preceding treatment of the spin Hamiltonian terms in bulk semiconductors, where they are relatively-well understood, will provide a basis for the discussion of the NMR of nanocrystalline semiconductors in Sect. 4, since as a group they present special considerations and many unanswered questions remain. Section 5 will provide some general conclusions and suggest future promising avenues of NMR research in semiconductors. 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

Density Functional Models. Methods in which the energy is evaluated as a function of the Electron Density. Electron Correlation is taken into account explicitly by incorporating into the Hamiltonian terms which derive from exact solutions of idealized many-electron systems. 

Hamiltonian term describing the interaction between the bridge state B ) and the acceptor state A)... 

The free Hamiltonian term quadratic in B i) must also be considered and is... 

Rapid molecular motions in solutions average to zero the dipolar and quadrupolar Hamiltonian terms. Hence, weak interactions (chemical shift and electron-coupled spin-spin couplings) are the main contributions to the Zeeman term. The chemical shift term (Hs) arises from the shielding effect of the fields produced by surrounding electrons on the nucleus ... 

The main purpose of the sequences is to obtain an averaged Hamiltonian H in which the dipolar term is very small compared with the chemical shift term. The zeroth-order of the average dipolar Hamiltonian term is given by the following equation ... 

Normally one does not worry about the free Hamiltonian term B2, but in the case of the B 3 field, we cannot afford this luxury. This term is written according to the field operators as... 

Introducing a p-region containing the molecule M and adjacent bath atoms, and a s-region including the remaining bath, the hamiltonian terms are regrouped into the form... 

If MM is chosen as the low level method in ONIOM, the approach falls into the general class of the QM/MM strategies (see refs [21,22] and references cited therein). However, in most of the QM/MM approaches, one calculation is run a QM method is applied to the core of the system and the rest (usually the largest part) is treated by a force field that contains intramolecular and intermolecular potential energy terms in the form of analytic functions. The interaction of the MM environment with the QM chromophore is represented by the hamiltonian term. 7/yM/MM will include the... 

It is apparent that the above matrix element is made of a sum of four terms, which are calculated independently (also consult Appendix 3.A.1). The calculation of each of these terms, for example, the first one (HI)), is quite analogous to the calculation of the overlap in Equation 3.17, except that the first monoelectronic overlap S in each product is replaced by a monoelectronic Hamiltonian term ... 

Abstract. Following a suggestion of Kostelecky et al. we have evaluated a test of CPT and Lorentz invariance from the microwave spectrosopy of muonium. Precise measurements have been reported for the transition frequencies U12 and 1/34 for ground state muonium in a magnetic field H of 1.7 T, both of which involve principally muon spin flip. These frequencies depend on both the hyperfine interaction and Zeeman effect. Hamiltonian terms beyond the standard model which violate CPT and Lorentz invariance would contribute shifts <5 12 and <5 34. The nonstandard theory indicates that P12 and 34 should oscillate with the earth s sidereal frequency and that 5v 2 and <5 34 would be anticorrelated. We find no time dependence in m2 — vza at the level of 20 Hz, which is used to set an upper limit on the size of CPT and Lorentz violating parameters. 

The conversion of muonium (y+e ) to its antiatom antimuonium (y e+) would be an example of a muon number violating process,2 and like neutrinoless double beta decay would involve ALe=2. The M-M system also bears some relation to the K°-K7r system, since the neutral atoms M and M are degenerate in the absence of an interaction which couples them. In Table III a four-Fermion Hamiltonian term coupling M and M is postulated, and the probability that M formed at time t=0 will decay from the M mode is given. Present experimental limits22 23 for the coupling constant G are indicated and are larger than the Fermi constant Gp. 

In Eq. (3.18.7), the off-diagonal core Hamiltonian terms H,v are given by the Mulliken approximation... 

We will present the effective Hamiltonian terms which describe the interactions considered, sometimes using cartesian methods but mainly using spherical tensor methods for describing the components. These subjects are discussed extensively in chapters 5 and 7, and at this stage we merely quote important results without justification. We will use the symbol T to denote a spherical tensor, with the particular operator involved shown in brackets. The rank of the tensor is indicated as a post-superscript, and the component as a post-subscript. For example, the electron spin vector A is a first-rank tensor, T1 (A), and its three spherical components are related to cartesian components in the following way ... 

The remaining important magnetic interactions to be considered are those which arise when a static magnetic field B is applied. The Zeeman interaction with a nuclear spin magnetic moment is represented by the Hamiltonian term... 

We deal first with molecules containing one unpaired electron (S= I /2) where magnetic nuclei are not present. The electron spin magnetic moment then interacts with the magnetic moment due to molecular rotation, the interaction being represented by the Hamiltonian term... 

The remaining important interactions which can occur for a 2 or3X molecule involve the presence of nuclear spin. Interactions between the electron spin and nuclear spin magnetic moments are called hyperfine interactions, and there are two important ones. The first is called the Fermi contact interaction, and if both nuclei have non-zero spin, each interaction is represented by the Hamiltonian term... 

The most important examples of 2S states to be described in this book are CO+, where there is no nuclear hyperfine coupling in the main isotopomer, CN, which has 14N hyperfine interaction, and the Hj ion. A number of different 3E states are described, with and without hyperfine coupling. A particularly important and interesting example is N2 in its A 3ZU excited state, studied by De Santis, Lurio, Miller and Freund [19] using molecular beam magnetic resonance. The details are described in chapter 8 the only aspect to be mentioned here is that in a homonuclear molecule like N2, the individual nuclear spins (1 = 1 for 14N) are coupled to form a total spin, It, which in this case takes the values 2, 1 and 0. The hyperfine Hamiltonian terms are then written in terms of the appropriate value of h As we have already mentioned, the presence of one or more quadrupolar nuclei will give rise to electric quadrupole hyperfine interaction the theory is essentially the same as that already presented for1 + states. 

Finally we note that the interaction with an applied magnetic field is important because of the large magnetic moment arising from the presence of electron spin (see (1.44)). The Zeeman interaction is represented by the Hamiltonian term... 

Questions arise immediately concerning the coupling of L, S and the nuclear rotation, R The possible coupling cases, first outlined by Hund, are discussed in detail in chapter 6. Here we will adopt case (a), which is the one most commonly encountered in practice. The most important characteristic of case (a) is that A, the component of L along the intemuclear axis, is indeed defined and we can use the labels , Id, A, etc., as described above. The spin-orbit coupling can be represented in a simplified form by the Hamiltonian term... 

Our objective is to replace the Hamiltonian terms in (8.394) by an effective Hamiltonian which operates only within the 2 n state, but which contains the molecular parameters p and q describing the admixture of excited 2E states. As we discussed in chapter 7, a suitable effective Hamiltonian for case (a), given by Brown and Merer [132] is... 

In complete analogy with this, for a nucleus with spin Io, interacting with a set of n equivalent other nuclei, the spin-Hamiltonian term is / Oplo (opI 1 + opI2+- -+opIn), and the same table applies to yield the spin-spin multiplet (see Ref. 36, p. 247, and Ref. 5, p. 26) for nuclear spins 1=1/2. 

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

To analyze the vibronic structures of the X, A and B electronic states Ph we constructed a vibronic Hamiltonian in a diabatic electronic basis which treats the nuclear motion in the X state adiabatically, and includes the nonadiabatic coupling between the A and B electronic states. The Hamiltonian terms of the dimensionless normal coordinates of the electronic ground state (XMi) of phenide anion is given by [19]... 

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