Hamiltonian derivative

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

Apparently, a large number of successful relativistic configuration-interaction (RCI) and multi-reference Dirac-Hartree-Fock (MRDHF) calculations [27] reported over the last two decades are supposedly based on the DBC Hamiltonian. This apparent success seems to contradict the earlier claims of the CD. As shown by Sucher [18,28], in fact the RCI and MRDHF calculations are not based on the DBC Hamiltonian, but on an approximation to a more fundamental Hamiltonian based on QED which does not suffer from the CD. At this point, let us defer further discussion until we review the many-fermion Hamiltonians derived from QED. 

The electron coupled interaction of nuclear magnetic moments with themselves and also with an external magnetic field is responsible for NMR spectroscopy. Since the focus of this study is calculation of NMR spectra within the non-relativistic framework, we will take a closer look at the Hamiltonian derived from equation (76) to describe NMR processes. In this regard, we retain all the terms, which depend on nuclear magnetic moments of nuclei in the molecule and the external magnetic field through its vector potential in addition to the usual non-relativistic Hamiltonian. The result is... 

The corresponding Heisenberg Hamiltonian derives from the connectivity of the allyl structure and is shown in the scheme below the structure. This matrix... 

The standard transformation of the Laplacian from cartesian coordinates to polar coordinates leads to the following Hamiltonian, derived previously in chapter 2 ... 

If we compare these equations with the projection operator expansion given in equation (7.43), we find that the expressions are identical up to and including the X2 contribution but that the 7.3 term derived here corresponds not to the X3 term in the expansion (7.43) but to its symmetrised (Hermitian) form discussed at the end of section 7.2. Since the discrepancies that arise from these two different forms are of order Xs or higher, the effective Hamiltonians derived by the two methods are identical to order X3. In the literature the Van Vleck transformation is normally implemented by use of equations (7.67) to (7.70) although the X3 contribution (7.70) has often been ignored. 

There is a further term which should be included in the effective Hamiltonian, derived in chapter 7, describing the electron spin-nuclear rotation interaction. This may be written in the form... 

The g-factors also point to the problem of excited state mixing, the values of gL and gs in particular being too far from the free electron values for comfort. The single state effective Hamiltonian, derived by perturbation theory, may be inadequate in molecules where there are several close-lying electronic states which are appreciably mixed. 

It is clear from (A.8) and (A.9) that the gradient difference and derivative coupling in the adiabatic representation can be related to Hamiltonian derivatives in a quasidiabatic representation. In the two-level approximation used in Section 2, the crude adiabatic states are trivial diabatic states. In practice (see (A.9)), the fully frozen states at Qo are not convenient because the CSF basis set l Q) is not complete and the states may not be expanded in a CSF basis set evaluated at another value of Q (this would require an infinite number of states). However, generalized crude adiabatic states are introduced for multiconfiguration methods by freezing the expansion coefficients but letting the CSFs relax as in the adiabatic states ... 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

The usual form of the effective spin-spin Hamiltonian, derived by application of the Wigner-Eckart theorem with AS = AE = 0 (see Eq. (3.4.49) of Section 3.4.5), is... 

In the study of the vibronic spectrum of a doublet HCCS radical, Peric et al. calculated the spin-orbit coupling constant at the equilibrium geometry of the radical by using the two-component relativistic no-pair Hamiltonian derived by Samzow et al. In the calculation, truncated (8,8)MRDCI wave functions were used with orbitals optimized for the triplet state of the corresponding cation. The spin-orbit coupling constant of 261 cm agreed well with the experimental data. 

Finally, there are contributions arising not only from the electron correlation but also from the terms originating in the spin-spin Hamiltonians derived from the relativistic quantum mechanics of many-body systems. 

It can be shown by considering the stationary coordinate-dependent reaction flux for the spin-boson Hamiltonian derived within the Golden Rule (see Fig. 9.12) [114] that in the classical nonadiabatic limit, where the condition (9.32) is satisfied with... 

From Dirac to Schrodinger How Is the Non-Relativistic Hamiltonian Derived ... 

While the 0 -theory discussed in section 3.3 does not provide such averages it is essential that these can be performed in the framework of the MH model. With the effective Hamiltonian derived in section 2.4 it turns out that the moments correspond to the propagators of this theory with masses rk that reflect the fact that there is a distinct critical point associated to each moment, i.e. there is no multicritical point as in the spin models with finite numbers of components and as suggested by the d > 4 interpretation of the (j> polymer theory [39] in sect. 3.3. In extracting the scaling behavior of the moments gw or equivalently of the masses r the central quantities will be the terms linear in k in an expansion in A as suggested by Eq. (115). 

It one wants to treat an problem using a bielectronic effective Hamiltonian derived from exact calculations on H2, several questions arise concerning ... 

The peculiarity that is involved in the calculation of the DKH Hamiltonians derives from the fact that some terms in the Hamiltonians are of the form pV... Vp [compare Eq. (12.56)]. Hence, no momentum operators occur between the potential energy operators, and a new matrix representation would be needed for such terms. Even worse, the higher the order, the more complicated are the terms that arise. Hess solution to this problem was the introduction of a resolution of the identity (RI),... 

Spin-orbit coupling has also been introduced into relativistic ECPs [835, 836]. A spin-orbit pseudo-operator has been employed by Teichteil et al. to reproduce the results from an all-electron approach [789]. Effective spin-orbit Hamiltonians derived from the difference between /- and /-dependent relativistic ECPs have been proposed by, amongst others, Christiansen, Ross, Ermler and coworkers [819,821,837-842], by Dolg, Stoll, Preuss and coworkers [822,823,843,844] and are under constant development [845,846]). 

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