Coupling operator

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

Battery couple Operating voltage range, V —wc/Ei— Energy density W-h/L... 

Brakes and clutches operate both dry and wet. In dry friction couples, the heat is removed by conduction to the surrounding air and stmctural members. Wet friction couples operate within a fluid, usually an oil, which absorbs the heat and maintains the couple at relatively low (below 200°C) temperatures. The fluid also traps the wear debris. 

By introducing the simplest semi-classical approximation to the propagators, in which the nuclear motion kinetic energy is assumed to commute with the anion and neutral potential energy functions and with the non BO coupling operators, one obtains... 

The Fermi coupling operator l is given by (90). Owing to the eigenvalue equation... 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

Vega, and coworkers as a sequence of well-placed ideal (i.e., infinitely strong) 7r-pulses serving to reintroduce the zero-quantum (ZQ) part of the homonuclear dipole-dipole coupling operator in a homonuclear two-spin system. The coherent averaging due to MAS is in the toggling frame of the n-pulses partially disrupted by a differential chemical shift term and thereby recoupling takes place. 

Nuclear transitions induced by an rf field B2(t) = B2cos cut applied along the x(/c = 1) or the y(ji = 2) direction in the laboratory frame eL are described by the coupling operator... 

In a spin system with anisotropic g and A tensors, the transition probability Wba between two nuclear spin states energy levels Eb and Ea may be calculated from the coupling operator given in (3.20). For a circularly polarized rf field, B2(t)R. in (3.20) has to be replaced by Bep(t)ft with the l.h. or r.h. rotating field B t) defined in (2,1). The nuclear transition probability is then given by... 

Indeed it is easy to see that, in general, the symmetry of the model will not be recovered by the variational solution since, if any one of the R departs from the symmetry of H, then the coupling operators Vrs will destroy the symmetry of the other departures from symmetry will quickly propogate throughout the model solution of the form (9) will have rather complicated behaviour in the variational process, for example each single-configuration approximation should show characteristic saddle point behaviour when variations 5 R are admitted. The minimum in the variational expression when the S R are constrained to have the correct symmetry should also be a local maximum with respect to symmetry-breaking variations S R. 

The NMR chemical shifts and indirect nuclear spin-spin coupling operators... 

THE NMR CHEMICAL SHIETS AND INDIRECT NUCLEAR SPIN-SPIN COUPLING OPERATORS... 

Note that this equation can be expressed in terms of the concentrations of Red and Ox by dividing the activities by the appropriate activity coefficients.) The pe of a solntion will therefore be poised at the pe determined by the dominant conple nntil that couple is exhausted. The pe of all other redox couples operating will tend to adjust to this electron activity. 

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

Evaluate required integrals over the electric and magnetic dipole moment and spin-orbit coupling operators as well as GIAO contributions, if needed. 

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 33], as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A Qiq) F( thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F( qk )- As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression... 

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