The First-Order Effective Hamiltonian

The H , = Hoo contribution to the effective Hamiltonian Hef / contains only-scaled isotropic chemical-shift terms. The first-order correction to the effective Hamiltonian requires the evaluation of commutators between DD elements, CSA elements and cross-terms DDx CSA. We should remind ourselves that the basic justification for using the van Vleck transformation is that the off-diagonal elements of the interactions are small with respect to the differences between the diagonal elements (see Eqs. 48a and 48b). When that is the case 

The first term is the dominant one in the expansion and must be considered first. Insertion of the explicit expressions for the dipole-dipole Hamiltonian results in 

In the non-spinning case these coefficients become zero when the dipolar Hamiltonian is symmetric, with and the n dependence is absent. How- 

The second-order terms of the effective Hamiltonian are proportional to the sd parameter to the third power, and in general, an order correction is proportional to The correction terms shift eigenvalues and hence the 

For a fixed spinning frequency and a characteristic RF frequency there are 

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Upon using a re-pulse centered in the middle of each rotor period t, and lasting one-third of a rotor period, the first-order effective Hamiltonian is... 

Upon entering the interaction frame of the rf irradiation for the CNvn or RN n sequences [cf. (14)] and taking the first-order effective Hamiltonians [cf. (17a) and (18a)], it is possible to establish the following selection rules for the averaging (and conversely recoupling) of the various interactions described in (45) as... 

By choosing C = a>r/4, the first-order effective Hamiltonian in a homonuclear two-spin system looks as follows ... 

Here we will restrict ourselves to discussing the first-order effective Hamiltonian of MSHOT3, and try to estimate changes in its Z -parameters. The time dependent part of the interaction Hamiltonian that is induced by the RF pulses determines the values of their coefficients. If each RF unit of length Tc/3 = 27r/3a c has a dipolar Hamiltonian with terms proportional to... 

For CqS in the MHz range dynamical effects on the single-quantum (SQ) and double-quantum (DQ) 14N coherences were compared. Hereby it was demonstrated that the DQ lineshape was not broadened as much as the SQ lineshape as the DQ transition is not affected by the first-order quadrupolar Hamiltonian. 

Before discussing the different decoupling pulse sequences, we should comment on the two following aspects. Let us first consider the zero-order effective Hamiltonian. Because fe// solely determined by the scaled isotropic chemical... 

If the states are degenerate rather than of different symmetry, the model Hamiltonian becomes the Jahn-Teller model Hamiltonian. For example, in many point groups D and so a doubly degenerate electronic state can interact with a doubly degenerate vibrational mode. In this, the x e Jahn-Teller effect the first-order Hamiltonian is then [65]... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

A nice derivation is given by Dr. Pascal Man, Directeur de recherche, CNRS, Universite Pierre et Marie Curie-Paris 6 at his web site http //www.pascal-man.coni/tensor-quadmpole-interaction/ V20-static.shtml. The Mathematica-5 script is also given and can be used for solving the first order Hamiltonian to explain quadmpole effects in high-field NMR spectra. 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

Using the properties of the Green s function (see Appendix B), the evaluation of the effect of distortion to transmission matrix elements can be greatly simplified. First, because of the continuity of the wavefunction and its derivative across the separation surface, only the multiplier of the wavefunctions at the separation surface is relevant. Second, in the first-order approximation, the effect of the distortion potential is additive [see Eq. (2.39)]. Thus, to evaluate the multiplier, a simpler undistorted Hamiltonian might be used instead of the accurate one. For example, the Green s function and the wavefunction of the vacuum can be used to evaluate the distortion multiplier. 

The fact that the phase transition in UO2 has the first-order character and the ordered magnetic moment of 1.74 /ab is considerably lower than the paramagnetic moment (about 3 /ab) is qualitatively consistent with the ratio of the strength of the bilinear and biquadratic parts of the effective spin Hamiltonian (8) of the 5f2-5f2... 

The change of the basic energy functional arises from the nonlinear nature of the effective Hamiltonian. This Hamiltonian has in fact an explicit dependence on the charge distribution of the solute, expressed in terms of (fM, which is the one-body contraction of I hf) ( hf > and lllus it is nonlinear. It must be added that this nonlinearity is of the first order, in the sense that the interaction operator depends only on the first power of ffM. 

The nonlinear nature of the Hamiltonian implies a nonlinear character of the Cl equations which must be solved through an iteration procedure, usually based on the two-step procedure described above. At each step of the iteration, the solvent-induced component of the effective Hamiltonian is computed by exploiting the first-order density matrix (i.e. the expansion Cl coefficients) of the preceding step. In addition, the dependence of the solvent reaction field on the solute wavefunction requires, for a correct application of this scheme, a separate calculation involving an iteration optimized on the specific state (ground or excited) of interest. This procedure has been adopted by several authors [17] (see also the contribution by Mennucci). 

The requirement needed to incorporate the solvent effects into a state-specific (multireference) method is fulfilled by using the effective Hamiltonian defined in Equation (1.159). The only specificity to take into account is that in order to calculate Va we have to know the density matrix of the electronic state of interest (see the contribution by Cammi for more details). Such nonlinear character of Va is generally solved through an iterative procedure [35] at each iteration the solvent-induced component of the effective Hamiltonian is computed by exploiting Equation (1.157) with the apparent charges determined from the standard ASC equation with the first order density matrix of the preceding step. At each iteration n the free energy of each state K is obtained as... 

If these variations are taken into account in the calculations on the QM part of the complex system, the effect of the MM system on the parameters of the effective Hamiltonian for the QM part turns out to be taken into account in the first order. It should be stressed that changes in the hybridization of the frontier atom due to participation of one orbital in the QM subsystem are not taken into account in any of the existing QM/MM schemes. This effect is not very large, so the first-order correction for taking it into account seems to be adequate. 

The last relation for the product xy is not an independent equation but it must be inserted into that for y and the system becomes one for x and y. Solving this system will be equivalent to solving the original 3x3 eigenvalue problem for the effective bond Hamiltonian. In a perturbative manner we get for the first order approximation ... 

Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

Since the Dirac equation is written for one electron, the real problem of ah initio methods for a many-electron system is an accurate treatment of the instantaneous electron-electron interaction, called electron correlation. The latter is of the order of magnitude of relativistic effects and may contribute to a very large extent to the binding energy and other properties. The DCB Hamiltonian (Equation 3) accounts for the correlation effects in the first order via the Vy term. Some higher order of magnitude correlation effects are taken into account by the configuration interaction (Cl), the many-body perturbation theory (MBPT) and by the presently most accurate coupled cluster (CC) technique. 

In the present treatment, we retain essentially all the diagonal matrix elements of X these are the first-order contributions to the effective electronic Hamiltonian. There are many possible off-diagonal matrix elements but we shall consider only those due to the terms in Xrot and X o here since these are the largest and provide readily observable effects. The appropriate part of the rotational Hamiltonian is —2hcB(R)(NxLx + NyLy). The matrix elements of this operator are comparatively sparse because they are subject to the selection rules AA = 1, A,Y=0 and AF=0. The spin-orbit coupling term, on the other hand, has a much more extensive set of matrix elements allowed... 

The first-order contribution to the effective Hamiltonian involves the diagonal matrix element of the operator in equation (7.80) ... 

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