Dimensionless internal coordinate

To describe the turbulent flow in the wall region, one introduces the so-called friction velocity [/ and the dimensionless internal coordinate y+ according to the formulas... 

For the description of heat and mass transfer in the wall layer, it is common to introduce the friction velocity / , friction temperature 0, and dimensionless internal coordinate y+ by the relations... 

Vibrational anharmonicity constant Vibrational coordinates Internal coordinates Normal coordinates, dimensionless Mass adjusted Vibrational force constants "eAe A,s get Ri, r 0J, etc. Qr m-i ... 

Fig. 32. Computed spatiotemporal concentration profile of the stored NOx for the NEDC driving cycle aged NSRC. The z/L stands for the dimensionless spatial coordinate along the monolith 0 is at the inlet, 1 at the outlet (Giithenke et al., 2007b). Reprinted with permission from SAE Paper 2007-01-1117 2007 SAE International.

Equation (7.143) can be made dimensionless by a proper choice of characteristic length and time scales. It is easy for a specific system to identify a characteristic length L and a characteristic velocity 1/ resulting in a characteristic advection time tm = L/ Z/. Once the steady state has been reached, it is straightforward to calculate average values for the internal coordinates ... 

By repeated application of the chain rule for derivatives, this potential energy can be transformed through quartic terms to the representation in dimensionless normal coordinates of eq. (28) in the standard way.Since the internal coordinates are curvilinear while the normal coordinates are not, this transformation is necessarily non-1 inear. 

The transformation from the internal coordinate force constants K3(s), Ka 3(s), etc. to the dimensionless normal coordinate force constants ki(s), kij(s), etc. can be accomplished through a set of several steps. For this purpose it is convenient to employ a dual notation for the atomic cartesian coordinates. Let be an index such that... 

Having obtained the dimensionless normal coordinate force constants from those in internal coordinates, eqs. (29)-(31) can be used to obtain the perturbation theory approximation to the vibrational energy l( .vels. 

The general expression in internal coordinates (x, i = 1 to 3) of the intramolecular potential function is V = V2 Sfjj Xj Xj + VsiSfjjkXj Xj x +(higher terms), with independent summation over each repeated index. The expression in (dimensionless) normal coordinates qj is V = V22o)jqf+ 2kjjk qjqjqk +(higher terms), with the constraint i[Pg.31]

The kinetic energy of a molecule is most easily expressed in (mass-adjusted) Cartesian coordinates. The potential energy has its most clear physical interpretation in (curvilinear) internal coordinates. When force constants are correlated to spectroscopic observables, mostly (dimensionless) normal... 

In MMPT, the interaction within a general D-H A motif is described by a parameterized three-dimensional PES V(R, r,0). R is the distance between the donor (D) and the acceptor (A), r is the distance between the D and H atom, and 9 is the angle between R and f (Figure 10.1). To facilitate the parameterization of MMPT potentials, the internal coordinate r is replaced by a dimensionless coordinate... 

One can appreciate in (11.68) the intrinsic physical significance that the coexistence coordinate a possesses in an actual phase diagram, even though it involves simultaneous changes of coordinates T, P having different units (so that units of cr would be difficult to discuss). If T) and P) were measured in different units, the numerical value of the coefficient y(r would readjust to represent the same physical coordinate, so that no mathematical or physical inconsistency could result. As remarked previously, lengths in Ms depend (consistently) on the chosen units of measured response functions, whereas internal angles are dimensionless invariants of the system. 

Up to now we considered number AT of the quantum states or cells in a multidimensional configuration or phase space, formed by coordinates qi and momenta p, (/ = 1,2.../) where I is the number of degrees of fi-eedom. Each cell had volume of h h = 2nh is Planck constant). In general, these states include all possible degrees of freedom, such as translational and rotational motion of all molecules, their internal (atomic) motion, interactions with other molecules, etc. Now, in the classical limit, instead of AT we introduce a volume in the phase space ApAq, in which a subsystem evolves in time. Additionally, to have the absolute value of the entropy, we introduce the volume of the elementary cell in the phase space 2%Kf and write the dimensionless entropy in the form... 

In analogy to Equation (5.43), develop an expression for the internal energy departure function in dimensionless coordinates ... 

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