Magnetic field perturbation

The sitnation is somewhat more cnmplir.ated when the perturbation is a magnefin field 

The vector potential is not uniquely defined since the gradient of any scalar function may be added (the curl of a derivative is always zero). It is convention to select it as 

The Hamilton operator in the presence of a magnetic field is given as 

The first term is identical to the usual kinetic energy operator. Inserting the expression for the vector potential (10.61) yields  

Here (r - Rc) (r - Rg) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r — Ro)(r — Rg) is a 3x3 matrix containing the ts of the x.y, 7 components, analogous to the quadrupole moment, eq. (10.4 ). Note that both the L and operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave function (for a singlet state) eqs. (I0.18)/(10.23). 

Let us exemplify some of the above generalizations for the case of an HF wave function. 

If the perturbation is a homogeneous electric field F (F = Ft), the perturbation operator Pi (eq. (10.21)) is the position vector r and P2 is zero. Assuming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, is given by the derivative formula (10.24) as shown in eq. (10.58) (since an HF wave function obeys the Hellmann-Feynman theorem). 

This is equivalent to the expression from first-order perturbation theory, (10.21). For non-variational wave functions the dipole moment calculated by the two approaches will be different, since the derivative of the wave function with respect to the field will not be zero. The second-order property, the dipole polarizability, is given by the derivative formula eq. (10.34) as shown in eq. (10.59). 

Although nuclei are often modelled as point charges in quantum chemistry, they do in fact have a finite size. The internal structure of the nucleus leads to a quadrupole moment for nuclei with spin larger than V2 (the dipole and octopole moments vanish by symmetry). This leads to an interaction term that is the product of the quadrupole moment with the field gradient (F = VF) created by the electron distribution. 

The situation is somewhat more complicated when the perturbation is a magnetic field. An electric field interacts directly with the charged particles (electrons and nuclei), and adds a potential energy term to the Hamiltonian operator. A magnetic field. 

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When an NMR experiment is performed, the application of a RFpulse orthogonal to the axis of the applied magnetic field perturbs the Boltzmann distribution, thereby producing an observable event that is governed by the Bloch equations [3]. Using a vector representation, the... 

Several chapters of this book show how magnetic field effects, as well as CIDEP and CIDNP spectral patterns, can be used to solve chemical problems. It should be noted that the study of how applied magnetic fields perturb chemical reactivity is a topic that is highly relevant to biological processes involving radical pairs, for example, photosynthesis. ... 

Fig. 16. Region of the electronic origins I, II, and III corresponding to the T, So transition of Pt(2-thpy)2 dissolved in n-octane. (a) Emission spectrum measured at T = 1.3 K and zero magnetic field, Aexc = 457.9 nm. (Compare Fig. 13.) (b) 20 K emission at B = 0 T. (Compare Fig. 14.) (c) Excitation spectrum measured at T = 1.3 K, B = 0 T, = 16,445 cm (A 17,163 cm (origin II) - 718 cm (vibration)), (d), (e), (f) Excitation spectra measured at T = 1.5 K for different magnetic field strengths, 718 cm vibrational satellite in the emission of the magnetic-field perturbed lowest triplet substate Ib- (Compare also Refs. [59,74])...

Derivative Techniques 240 10.4 Lagrangian Techniques 242 10.5 Coupled Perturbed Hartree-Fock 244 10.6 Electric Field Perturbation 247 10.7 Magnetic Field Perturbation 248 10.7.1 External Magnetic Field 248 13.1 Vibrational Normal Coordinates 312 13.2 Energy of a Slater Determinant 314 13.3 Energy of a Cl Wave Function 315 Reference 315 14 Optimization Techniques 316... 

Fa/dUp is the derivative of WG with respect to Hp when the molecule is perturbed by the uniform magnetic field perturbation ... 

The GIAO method is used in conjunction with analytical derivative theory in this approach the magnetic field perturbation is treated in an analogous way to the perturbation produced by changes in the nuclear coordinates. In this framework, the components of the nuclear magnetic shielding tensor are obtained as ... 

When 0 1/4 the kink-like oscillations persisted throughout the discharge, as well as the axial magnetic field perturbations. 

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