Field-free Hamiltonian

When discussing magnetic resonance phenomena, it is conventional to proceed along the lines of standard perturbation theory. If the field-free Hamiltonian is Hq then we write... 

This represents the total field-free Hamiltonian for a diatomic molecule, to order 1/c2 in the purely electronic terms and to order 1 /Mac2 in the nuclear terms, in a space-fixed axis system of arbitrary origin. We showed in chapter 2 that the solution of a Hamiltonian... 

Equation (9.106) is a field-free Hamiltonian, to which must be added the Zeeman terms the complete effective Hamiltonian for the problem is therefore... 

An important point to note is that B in the field-free Hamiltonian (11.7) is the rotational moment operator, and can have different values in (11.8) and (11.9). From these general expressions we can calculate the matrix elements involving the five primitive fimctions as shown below. In order to evaluate these matrix elements it is necessary to specify a value of L Freund and Miller pointed out that L = 2 for a d electron. This is the pure... 

This result may be re-expressed in terms of H°, the usual field-free Hamiltonian, and H an interaction Hamiltonian, as... 

Here // < is the Hamiltonian for the radiation field in vacuo, flmo the field-free Hamiltonian for molecule , and //m( is a term representing molecular interaction with the radiation. It is worth emphasising that the basic simplicity of Eq. (1) specifically results from adoption of the multipolar form of light-matter interaction. This is based on a well-known canonical transformation from the minimal-coupling interaction [17-21]. The procedure results in precise cancellation from the system Hamiltonian of all Coulombic terms, save those intrinsic to the Hamiltonian operators for the component molecules hence no terms involving intermolecular interactions appear in Eq. (1). 

Field-Free Hamiltonian The Form of Wavefunctions for Resonance States in the Context of Time- and of Energy-Dependent Theories... 

These states are formed inside the continuous spectra of the total Hamiltonian and are responsible for phenomena such as resonances in electron scattering from atoms or molecules, autoionization, predissociation, etc. Furthermore, in this work we also consider as unstable states those states that are constructs of the time-independent theory of the interaction of an atom (molecule) with an external field which is either static or periodic, in which case the effect of the interaction is obtained as an average over a cycle. In this framework, the "atom - - field" state is inside the continuous (ionization or dissociation) spectrum, and so certain features of the problem resemble those of the unstable states of the field-free Hamiltonian. The probability of decay of these field-induced unstable states corresponds either to tunneling or to ionization-dissociation by absorption of one or more photons. 

FIELD-FREE HAMILTONIAN THE FORM OF WAVEFUNCTIONS FOR RESONANCE STATES IN THE CONTEXT OF TIME- AND OF ENERGY-DEPENDENT THEORIES AND ITS USE FOR PHENOMENOLOGY AND COMPUTATION... 

In fhe general case, fhe contour for fhe complex variable z = E irj surrounds fhe specfrum of H counferclock-wise on the first Riemann sheet of E, in which case if is valid for f > 0 and for f < 0. However, in fhe case of fhe decay of an unsfable sfafe of a field-free Hamiltonian, rigor implies that the following physically consfraints musf be imposed on fhe infegrafion of Eq. (7) E > 0 and f > 0 [37,89]. 

Let us consider a thin crystal (e.g., a 2D sheet of graphite also referred to as graphene) possessing the reflection and rotation symmetries. Without lost of generality, the one-particle, field-free Hamiltonian can be written as... 

The energy bands (EBs) E(k) of the field-free Hamiltonian have transformed into the QEBs fi(k). It is, therefore, natural to inquire what are the symmetries of the QEBs and what is the relation between these symmetries and those of the field-free EBs. The answer lies in the application of the DS operators Pjv and on the FB eigenstates < e(k),k(r, t) which gives... 

Note that the field-free Hamiltonian is invariant under the 00th order (i.e., continuous) rotation Coo = —

[Pg.406]

Equations (2.25) and (2.21) do not tell us at which velocity the electron rotates to acquire kinetic and magnetic spin momenta. This is provided by another computation by Dirac [5]. He used a Heisenberg picture with a field-free Hamiltonian (but the conclusion would also hold with a field present) ... 

Fig. 13.1 Comparison of the convergence behavior between the FC functions generated by a magnetic field and the field-free Hamiltonians (S = 10 G,...

The operator K2(0) of Eq.(8.39) can be written as the sum of a zeroth-order Floquet Hamiltonian and a perturbation W. Here, the field-free Hamiltonian of the two-level system is considered as a perturbation of the field and the coupling... 

The TD approach is the natural candidate for the interpretation of data coming from pump-probe spectroscopies [13]. If both pump and probe fields are weak, the third-order time-dependent polarization can be computed propagating wave-packets with field-free Hamiltonians [14]. A nonperturbative approach can also be followed, numerically solving the TDSE with the external fields included in the Hamiltonian [15, 16]. Chapter 9 of this book presents the equation-of-motion phase-matching approach (EOM-PMA), which can be considered a mixed perturbative-nonperturbative method. 

By propagating it with the unperturbed evolution operator exp( - iHot/fi) (where Ho is the field-free Hamiltonian), one has... 

Born-Oppenheimer expressions for vibration-rotation energies of diatomic molecules have been derived several times (Herman and Asgharian, 1966 Watson, 1973 Bunker and Moss, 1977 Watson, 1980 Herman and Ogilvie, 1998). Here we will follow mainly the derivation by Brniker and Moss (Bunker and Moss, 1977). After separation of the translation of the whole molecule and transformation to nuclear centre of mass coordinates one can write the field-free Hamiltonian for the electronic ground state of symmetry of a diatomic molecule as... 

The methods that will be discussed in the following are all of the ab initio type. Given the molecular field free Hamiltonian in Eq. (2.9), with the nuclear coordinates and charges and the electronic mass given as parameters, in these methods all integrals over this Hamiltonian or parts of it are evaluated ab initio, i.e. by strict application of the appropriate mathematical rules and without using further data from experiment or otherwise. The emphasis will be in particular on the SCF and on so-called correlated methods. Semi-empirical or density functional theory (DFT) methods on the other hand will not be mentioned explicitly. However, most of what will be said about SCF-based methods for the calculation of properties will also apply to semi-empirical or DFT methods, because one can consider to a certain extent the semi-empirical and DFT methods as variants of SCF, just with a slightly different Hamiltonian. 

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