Second energy derivatives, nuclear magnetic

The nuclear shielding is more precisely the second derivative of energy with respect to magnetic field (B) and nuclear magnetic moment (/u) (27). 

Another important extension of the theory concerns NMR chemical shift. Yamazaki et al. proposed a theory for computing the chemical shift of solvated molecules [17]. The nuclear magnetic shielding tensor o-x of a nucleus X can be represented as a mixed second derivative of the free energy A with respect to the magnetic field B and the nuclear magnetic moment mx ... 

The second derivative of the energy with respect to the magnetic field and the nuclear magnetic moment can be identified as the magnetic shielding... 

In fact, the quantum mechanical relationships for magnetizability, nuclear magnetic shielding, and nuclear spin-spin coupling arrived at from the derivatives of the second-order energy expressed in flie form (7.26)-(7.28) are the same as those obtained via RSPT [2-4],... 

Comparison with the spin Hamiltonian Eq. (5.76) shows that the identification of the property K S.k) in Eq. (5.64) as the reduced indirect nuclear spin-spin coupling tensor of nuclei K and L was correct and that the cartesian components of it can be defined as second derivative of the energy with respect to the components of the two nuclear magnetic moments and... 

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]... 

For example, in many systems, we found the GIAO-HF method to yield H-NMR chemical shifts with an accuracy of typically 0.2-0.5 For other nuclei the inclusion of correlation effects can become more impor-The computation at the GIAO-DFT level is closely related. NMR chemical shifts are calculated as second derivatives of the energy with respect to the external magnetic field B and the nuclear magnetic spin m of a nucleus N ... 

In the first, quantum mechanics can be used to calculate the NMR coupling constant between two nuclear spins in the molecular environment or the NMR shielding constant. Jensen [8] in Chapter 10 of the reference provides a comprehensive introduction to the methods for calculating these molecular properties. The NMR coupling constant, KAB, is related to the second derivative of the energy with respect to the internal magnetic moments, /u, arising from the nuclear spin of the two atoms, A and B. 

Equations (49), (50) and (51) can be differentiated with respect to external perturbations (e. g. electric or magnetic fields) and with respect to nuclear coordinates, allowing for the analytical computation of free energy gradients [103,104] and second derivatives [106,110] they are used for geometry optimisations in solution, and for the calculation of force constants, polarizabilities etc. 

There is an elegant way to single out the only necessary fi that give a contribution to the energy proportional to the product xi xj (no higher terms included), where jc and xj stand for some components of the magnetic field intensity H andfor of the nuclear spin I a (that cause perturbation of the molecule). As to the first-order correction ( diamagnetic ), we calculate the second derivative ( of the Hamiltonian with respect to the... 

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