Refinement of the Hamiltonian

Let us assume the Born-Oppenheimer approximation (p. 229). Thus, the nuclei occupy some fixed positions in space, and in the electronic Hamiltonian (12.58) we have the electronic charges qj = —e and masses mj = mo = m (we skip the subscript 0 for the rest mass of the electron). Now, let us refine the Hamiltonian by adding the interaction of the particle magnetic moments (of the electrons and nuclei the moments result from the orbital motion of the electrons as well as from the spin of each particle) with themselves and with the external magnetic field. We have, therefore, a refined Hamiltonian of the system [the particular terms of 

Pieter Zeeman (1865-1943), Dutch physicist, professor at the University of Amsterdam. He became interested in the influence of a magnetic field on molecular spectra and discovered a field-induced splitting of the absorption lines in 1896. He shared the Nobel Prize with Hendrik Lorentz for their researches into the influence of magnetism upon radiation phenomensT in 1902. The Zeeman splitting of star spectra allows us to deter- 

Recalling the force lines of a magnet, we see that the magnetic field vector H produced by the nuclear magnetic moment jaIa should reside within the plane of and This means that A 

For closed-shell systems (the majority of molecules) the vector potential A i may be neglected, i.e. Aei(rj) = 0, because the magnetic fields of the electrons cancel out for a closed-shell molecule (singlet state). 

When such a vector potential A is inserted into Hi (just patiently make the square of the content of the parentheses) we immediately get 

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Choice of the Vector and Scalar Potentials Refinement of the Hamiltonian Effective NMR Hamiltonian (( ))... 

While direct diagonalization of the Hamiltonian matrix works well for situations in which there is a finite number of states, as in Fig. 9.1, it is clearly hopeless to try it in this case. A useful WKB approach was proposed by Edmonds12 and refined by Starace.13 Using the fact that azimuthal symmetry exists, Starace writes the wavefunction of the spinless Rydberg electron in cylindrical coordinates as13... 

A combination of the two techniques was shown to be a useful method for the determination of solution structures of weakly coupled dicopper(II) complexes (Fig. 9.4)[119]. The MM-EPR approach involves a conformational analysis of the dimeric structure, the simulation of the EPR spectrum with the geometric parameters resulting from the calculated structures and spin hamiltonian parameters derived from similar complexes, and the refinement of the structure by successive molecular mechanics calculation and EPR simulation cycles. This method was successfully tested with two dinuclear complexes with known X-ray structures and applied to the determination of a copper(II) dimer with unknown structure (Fig. 9.5 and Table 9.9)[119]. 

In view of the small number of resonant unperturbed modes in the observed spectra of CH3F [26,55] (Table A), usually used least-squares refinement of the six parameters of the model Hamiltonian for CX3Y molecules [4,26,28] is impractical. For estimating C-F bond energy in CH3F, only five parameters ( 1 - x,), 2, co 2, 1 ami A,2 are required. Furthermore, there are only five unperturbed stretching modes to calibrate it. Three of the five parameters are calculated with the equations obtained from Eqs. (11), (12), using the observed vi, V3, V4 [56] fundamentals ... 

The established methods of valence bonds and molecular orbitals (MO), including the method of linear combinations of atomic orbitals (LCAO), which have been so successful in the treatment of molecular systems, need further refinement when applied to crystals. The preference for the method of valence bonds in the case of solids is not accidental because it yields clearer results. In contrast, calculations dealing with the simplest molecules are currently tackled usually by the method of molecular orbitals, including the method of absolute, purely theoretical, quantiun-mechanical calculations, which is adopted in those cases when a sufficiently precise form of the Hamiltonian operator can be obtained for the system being considered. 

Some of these deficiences have been alleviated in the AMI and PM3 reparametrizations of the MNDO model, but many of them persist, indicating that they are due to features in the underlying model and can best be overcome by refinements of the model. Qualitative improvements in the results have recently indeed been achieved for (iii) by introducing d orbitals into the MNDO formalism, and for (i, v-vii, ix-xi) by including additional one-electron terms such as orthogonal-ization corrections into the core Hamiltonian. ... 

During the evolution of the current semi-empirical methods (AMI, PM3) a number of refinements were made to the core-repulsion function in order to improve, for example, the description of hydrogen-bonding [20, 21]. Such changes have in general led to increased flexibility within the modified semi-empirical Hamiltonians resulting in quite marked improvements in the accuracy of the parent methods. 

Average or effective Hamiltonian theory, as introduced to NMR spectroscopy by Waugh and coworkers [55] in the late 1960s, has in all respects been the most important design tool for development of dipolar recoupling experiments (and many other important experiments). In a very simple and transparent manner, this method facilitates delineation of the impact of advanced rf irradiation schemes on the internal nuclear spin Hamiltonians. This impact is evaluated in an ordered fashion, enabling direct focus on the most important terms and, in the refinement process, the less dominant albeit still important terms in a prioritized manner. 

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