Order coupling

Higher Order Coupling in Some Jahn-Teller and Renner-Teller Effects... 

The method shown affords easy generalization to higher order coupling in the important case where a single mode is engaged, that is, G i = g i = (l/i /2) e . Then the two off-diagonal terms derived above are, after physics-based constant coefficients have been affixed, in the upper right comer... 

In a similar fashion, a fraction of the velocity of the molecules with first-order coupling is transmitted to other molecules entangled with the latter. This is called second-order coupling (subscript 2). Still higher orders of effect radiate from the original molecule in the manner suggested by Fig. 2.13. Because of the... 

Equation (10.42) are the first-order Coupled Perturbed Hartree-Fock (CPHF) equations." The perturbed MO coefficients are given in terms of unperturbed quantities and the first-order Fock, Lagrange (a) and overlap matrices. The F term is given as (eq. (3.52)). 

Finally, we show that the second-order coupling between direct tunneling transitions is subdominant to the already computed quantities. Consider an interaction of the form yy l,-2 )(2,l + H.C. If one repeats simple-mindedly the steps leading to Eq. (75), one obtains the following simple expression for the free energy correction due to interaction between the underlying structural transitions ... 

For example, the double triplet (6 6.16) was obscured due to the poor signal-to-noise and the severe second order coupling at 100 MHz. Simulations (7,8) of the olefinic protons were critical to these assignments (Figure 3). 

In an early work by Mertz and Pettitt, an open system was devised, in which an extended variable, representing the extent of protonation, was used to couple the system to a chemical potential reservoir [67], This method was demonstrated in the simulation of the acid-base reaction of acetic acid with water [67], Recently, PHMD methods based on continuous protonation states have been developed, in which a set of continuous titration coordinates, A, bound between 0 and 1, is propagated simultaneously with the conformational degrees of freedom in explicit or continuum solvent MD simulations. In the acidostat method developed by Borjesson and Hiinenberger for explicit solvent simulations [13], A. is relaxed towards the equilibrium value via a first-order coupling scheme in analogy to Berendsen s thermostat [10]. However, the theoretical basis for the equilibrium condition used in the derivation seems unclear [3], A test using the pKa calculation for several small amines did not yield HH titration behavior [13],... 

Complex representations, multidegenerate nonlinear coupling, higher order coupling, 243 -244... 

Nonlinear coupling, multidegenerate conditions higher order coupling, complex representations, 243-244 molecular systems, 233-249 adiabatic-to-diabatic transformation, 241— 242... 

Coupling between adjacent single protons, (a) First-order coupling, AX. (b) Second order coupling, AB -> A2. 

The correlation between the X-ray crystallographic data of the W—CN bond lengths and the first-order coupling constants between the 183W and 13C in the equatorial cyano ligands, as predicted by the Fermi-contact term (51), is fairly good (see Table III). This illustrates the increase in cis W—CN bond length induced by increased donor... 

Proton Chemical Shifts and First-Order Coupling-Constants for Selected KDO Derivatives0 (Ref. 81)... 

Compound number Chemical shift (6, p.p.m.) First-order coupling-constants (Hz) ... 

The system Alq3 DCM was successfully incorporated into a DFB structure by different research groups and optically pumped [194, 195]. The materials were deposited on embossed flexible substrates with grating periods for second-and third-order coupling (400-600 nm). By varying the him thickness, the wavelength of the lasers can be tuned over 44 nm [196]. 

Ep shifts with the scan rate v at slow scan rates (where the chemical reaction is dominating), 9Ep/9 log v = 30/v mV for a first-order coupled chemical reaction... 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

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