Molecular Hamiltonians, nuclear magnetic

A systematic development of relativistic molecular Hamiltonians and various non-relativistic approximations are presented. Our starting point is the Dirac one-fermion Hamiltonian in the presence of an external electromagnetic field. The problems associated with generalizing Dirac s one-fermion theory smoothly to more than one fermion are discussed. The description of many-fermion systems within the framework of quantum electrodynamics (QED) will lead to Hamiltonians which do not suffer from the problems associated with the direct extension of Dirac s one-fermion theory to many-fermion system. An exhaustive discussion of the recent QED developments in the relevant area is not presented, except for cursory remarks for completeness. The non-relativistic form (NRF) of the many-electron relativistic Hamiltonian is developed as the working Hamiltonian. It is used to extract operators for the observables, which represent the response of a molecule to an external electromagnetic radiation field. In this study, our focus is mainly on the operators which eventually were used to calculate the nuclear magnetic resonance (NMR) chemical shifts and indirect nuclear spin-spin coupling constants. 

In order to complete our derivation of the molecular Hamiltonian we must consider the nuclear Hamiltonian in more detail. A thorough relativistic treatment analogous to that for the electron is not possible within the limitations of quantum mechanics, since nuclei are not Dirac particles and they can have large anomalous magnetic moments. However, the use of quantum electrodynamics [18] shows that we can derive the correct Hamiltonian to order 1 /c2 by taking the non-relativistic Hamiltonian ... 

The effective Hamiltonian used by Cecchi and Ramsey [63] to analyse the strong magnetic field spectrum was the sum of four terms, describing the molecular rotation, nuclear spin interactions, Stark interactions and Zeeman interactions. Specifically the Hamiltonian is the following,... 

The quantum-mechanical picture of hyperfine structures presented by the spin-spin nuclear magnetic resonance (NMR) and electron-spin resonance (ESR) spectra involves a variety of spin Hamiltonian parameters of molecular origin whose magnitude determines that of the coupling constants. In such an analysis, the most characteristic term arises from the Fermi -or contact -operator ... 

SBB The Molecular Hamiltonian 208 an The Molecular Wavefunction 214 an Covalent Bonds in Polyatomic Molecules 223 BIOSKETCH Douglas Groljahn 225 X Non-Covalent Bonds 231 an Nuclear Magnetic Resonance Spectroscopy 233... 

In a joint experimental and theoretical publication [1057], the results of nuclear magnetic resonance (NMR) measurements performed on polyisothianaphthene and a series of selected molecular model compounds were combined with quantum-chemical calculations carried out on oligomers within the Austin model 1 (AMI) and valence-effective Hamiltonian (VEH) methodologies in order to study the differences among planar aromatic, non-planar aromatic, and quinoid polythiophene and polyisothianaphthene. 

Exercise 5.7 Derive the first-order perturbation Hamiltonian, Eqs. (5.56) to (5.61), for the vector potential of a nuclear magnetic moment, fh, by inserting the vector potential, Eq. (5.55), in the general expression of the molecular Hamiltonian, Eq. (2.101), retaining the first-order term. 

For the derivation of the induced contribution we will start from the definition as second derivative of the electronic energy, Eq. (6.47) in the presence of a nuclear magnetic moment and the molecular rotation. The corresponding vector potentials, Eqs. (5.55) and (6.5), lead to two first-order perturbation Hamiltonians, Eqs. (5.56) and (6.19), and a new second-order-order-perturbation Hamiltonian [see Exercise 6.4]... 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

The most important examples of 2S states to be described in this book are CO+, where there is no nuclear hyperfine coupling in the main isotopomer, CN, which has 14N hyperfine interaction, and the Hj ion. A number of different 3E states are described, with and without hyperfine coupling. A particularly important and interesting example is N2 in its A 3ZU excited state, studied by De Santis, Lurio, Miller and Freund [19] using molecular beam magnetic resonance. The details are described in chapter 8 the only aspect to be mentioned here is that in a homonuclear molecule like N2, the individual nuclear spins (1 = 1 for 14N) are coupled to form a total spin, It, which in this case takes the values 2, 1 and 0. The hyperfine Hamiltonian terms are then written in terms of the appropriate value of h As we have already mentioned, the presence of one or more quadrupolar nuclei will give rise to electric quadrupole hyperfine interaction the theory is essentially the same as that already presented for1 + states. 

The analysis of molecular spectra requires the choice of an effective Hamiltonian, an appropriate basis set, and calculation of the eigenvalues and eigenvectors. The effective Hamiltonian will contain molecular parameters whose values are to be determined from the spectral analysis. The theory underlying these parameters requires detailed consideration of the ftmdamental electronic Hamiltonian, and the effects of applied magnetic or electrostatic fields. The additional complications arising from the presence of nuclear spins are often extremely important in high-resolution spectra, and we shall describe the theory underlying nuclear spin hyperfine interactions in chapter 4. The construction of effective Hamiltonians will then be described in chapter 7. 

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