Quadratic degree of freedom

Because of the freedom of motion in all three spatial directions we also speak of (quadratic) degrees of freedom. 

If we include the vibrational motion, there are two additional quadratic degrees of freedom in the Hamiltonian (for 8 and pr) adding Icb T per molecule to the energy of the system. 

At this point it is useful to prove the equipartition theorem. This theorem states that for each quadratic degree of freedom in the Hamiltonian, a... 

Unlike classical systems in which the Lagrangean is quadratic in the time derivatives of the degrees of freedom, the Lagrangeans of both quantum and fluid dynamics are linear in the time derivatives of the degrees of freedom. 

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). 

This sort of equation, known as a Lagrange equation of motion, is generally valid for systems with unconstrained degrees of freedom. If a system has n degrees of freedom, n second-order Lagrange equations will exist (functions of qj, qj, and time quadratic in qj). 

The exponential and logarithm functions have clear convex and concave characters, and do not do well in a nearly linear function. The last three functions provide minor corrections on the linear function, and do quite well. It should be pointed out that the first four functions have two arbitrary parameters each, the quadratic has three and the cubic has four. We expect that when the number of parameters increases, we can correlate ever more complicated data. For the cubic equation with three parameters and only six data points, there remain only three degrees of freedom. It is often said that Give me three parameters, and I can fit an elephant, so that it is a greater achievement to fit complicated data with as few parameters as possible, in the spirit of what is called the Occam s Razor. ... 

For the subplot analysis it appears that the effects due to A, and the interaction between B and Humidity are real, with some evidence of an interaction between B and Temperature. It is possible to split the two degrees of freedom for Temperature and Humidity into linear and quadratic contrasts and to construct a normal probability plot for the whole plot contrasts. This would reveal important effects due to the linear components of both Temperature and Humidity. 

Summary. An effective scheme for the laser control of wavepacket dynamics applicable to systems with many degrees of freedom is discussed. It is demonstrated that specially designed quadratically chirped pulses can be used to achieve fast and near-complete excitation of the wavepacket without significantly distorting its shape. The parameters of the laser pulse can be estimated analytically from the Zhu-Nakamura (ZN) theory of nonadiabatic transitions. The scheme is applicable to various processes, such as simple electronic excitations, pump-dumps, and selective bond-breaking, and, taking diatomic and triatomic molecules as examples, it is actually shown to work well. 

The latter of the quadratic form may be easier to evaluate, because the inverse of the covariance matrix. Since e has n1elements (n1equations), h will have a chi-square distribution with n1 degrees of freedom. Thus at specified significance level a... 

Fig. 1. The PNM (principal axes), Uu, in the system with two AIM populational degrees of freedom,, the associated charge sensitivities, and a separation of the pure CT and P components, dJf cT and d CP, of the equilibrium charge displacement d C= 7d/V, along the FF vector f - 6M f -= 0 F. Panel (a) refers to a general displacement dN > 0, while Panel (b) corresponds to d/V= 1. The hardness tensor ellipse with semiaxes of length , is defined by the quadratic surface diV d/V = h nl = = 1. As explicitly shown in Panel (a), the...

The nonregular nature of the fractional factorial design makes it possible to consider interaction effects as well as main effects see also Chapter 7. An interaction between two factors, each with three levels, has four degrees of freedom which can be decomposed into linear x linear, linear x quadratic, quadratic x quadratic, and quadratic x linear effects. The contrast coefficients for these effects are formed by multiplying the coefficients of the corresponding main effect contrasts. 

An objective judgment of the relative merits of Models 1 and 2 can be made by means of an F test, as discussed in Chapter XXI. This is a statistical test to determine if a decision to include the cT term is justified at the conventional 95 percent confidence level. For the present case, where the degrees of freedom Vj and V2 are 4 and 3 for the linear and quadratic models, respectively, the value is calculated as [see Eq. (XXI-38)]... 

The statistical average over the electronic degrees of freedom in Eq. [15] is equivalent, in the Drude model, to integration over the induced dipole moments pg and py. The Hamiltonian H, is quadratic in the induced dipoles, and the trace can be calculated exactly as a functional integral over the fluctuating fields pg and The resulting solute-solvent interaction energy... 

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