Coordinates, generalized

These are called generalized coordinates and can be related to the Cartesian coordinates through a coordinate transformation 

The mathematical condition that Eqs. 3.15 and 3.16 have a solution is that the Jacobian determinant of these equations be different from zero (see for example W. F. Osgood s book in Further 

Considering only one particle, a set of generalized coordinates describing the position of this particle can be the spherical coordinates q = (/ , 6, )) (Fig. 3.3). These are related to the Cartesian coordinates through the following transformation 

Let us define the Lagrangian for a system of N particles as a function of the positions and velocities of the particles  

Differentiating the Lagrangian with respect to the positions yields 

In this chapter, we focus on the method of constraints and on ABF. Generalized coordinates are first described and some background material is provided to introduce the different free energy techniques properly. The central formula for practical calculations of the derivative of the free energy is given. Then the method of constraints and ABF are presented. A newly derived formula, which is simpler to implement in a molecular dynamics code, is given. A discussion of some alternative approaches (steered force molecular dynamics [35-37] and metadynamics [30-34]) is provided. Numerical examples illustrate some of the applications of these techniques. We finish with a discussion of parameterized Hamiltonian functions in the context of alchemical transformations. 

In this section, we discuss some of the equations used to calculate the derivative of the free energy. In a different form, those results will be used in ABF to both calculate d4/d and bias the system in an adaptive manner. 

To calculate d4/d , we need to evaluate partial derivatives, such as U- 4/7) , which measures the rate of change in energy with the order parameter. To do so we need to define generalized coordinates of the form ( , qi, , qN-1). Classical examples are spherical coordinates (r, 6, o), cylindrical coordinates (r, 0, z) or polar coordinates in 2D. Those coordinates are necessary to form a full set that determines 

Once an order parameter, , is specified we can define the free energy A as a function of through the relation A( ) = -kBT In P( ). In the canonical ensemble, the probability density function P(x, px) is equal to exp(—Jf (x, px)/kBT)/Q, where Q is the partition function  

The variables x and p are the positions and momenta of all the particles. With those definitions, it is possible to define, 4( ) in terms of an integral in the phase space 

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Luckhaus D 2000 6D vibrational quantum dynamics generalized coordinate discrete variable representation and (a)diabatic contraction J. Chem. Phys. 113 1329—47... 

Neuhauser D 1992 Reactive scattering with absorbing potentials in general coordinate systems Chem. 

A Hamiltonian version of the quaternionic description is also possible by viewing the quaternions as a set of generalized coordinates, introducing those variables into the rigid body Lagrangian (1), and finally determining the canonical momenta through the formula... 

Diagonalizing the K matrix converts arbitrary systems in generalized coordinate systems q... 

Fig. 2. General configurational—coordinate diagrams for (a) broad-band absorbers and emitters, and (b) narrow-band or line emitters. The ordinate represents the total energy of the activator center and the abscissa is a generalized coordinate representing the configuration of ions surrounding the...

To obtain thermodynamic perturbation or integration formulas for changing q, one must go back and forth between expressions of the configuration integral in Cartesian coordinates and in suitably chosen generalized coordinates [51]. This introduces Jacobian factors... 

For pedantic reasons, 1 am going to rewrite this energy expression in terms of so-called generalized coordinates, which in this simple case are exactly the Cartesian ones... 

General Coordination Chemistry, Coordination Numbers and Geometries 15... 

The general coordination modes of amidinate and guanidinate (R = NR 2) ligands are shown in Scheme 4. Both ligands display a rich coordination chemistry in which both chelating and bridging coordination modes can be achieved. By far the most common coordination mode is the chelating type A. 

The reaction between a Lewis acid R3M and a Lewis base ER3, usually resulting in the formation of a Lewis acid-base adduct R3M—ER3, is of fundamental interest in main group chemistry. Numerous experiments, in particular reactions of alane and gallane MH3 with amines and phosphines ER3, have been performed [14]. Several general coordination modes, as summarized in Fig. 2, have been identified by X-ray diffraction. 

Similar expressions can be derived for second spatial derivatives. The final form of the equations that result after a generalized coordinate transformation depends on the degree of differentiation by using the chain rule, i.e. on the treatment of the metrics x, x, and y. For more details we refer to the... 

If D is taken as a traceless tensor, Tr(/)) = Da = 0, there remain only two independent components for D (neglecting the three Euler angles for orientation in a general coordinate system). Usually, these are the parameters D and E, for the axial and rhombic contribution to the ZFS ... 

From the inspection of the data in Table 2.4, it is clear that NO changes its original molecular character after adsorption. In general, coordination of nitric oxide leads to a pronounced redistribution of the electron and spin densities, accompanied by modification of the N-0 bond order and its polarization. Thus, in the case of the (MNO 7 10 and ZnNO 11 species, slender shortening of the N-0 bond is observed, whereas for the MNO 6 and CuNO 11 complexes it is distinctly elongated. Interestingly, polarization of the bound nitric oxide assumes its extreme values in the complexes of the same formal electron count ( NiNO 10 and CuNO 10) exhibiting however different valence. 

Note that transformation by a general coordinate transformation matrix leaves the quadrupole matrix symmetrical, i.e., Py = Pp and with zero trace,... 

Let us assume that system 0 can be transformed to system 1 through the continuous change of some parameter A defined such that Ao and Ai correspond to systems 0 and 1, respectively. This parameter could be a macroscopic variable - viz. the temperature, a parameter that transforms J o to -A j, or a generalized coordinate (e.g., a torsional angle or an intermolecular distance) that allows the different structural states of the system to be distinguished. It follows that ... 

Fig. 2.5. Possible applications of a coupling parameter, A, in free energy calculations, (a) and (b) correspond, respectively, to simple and coupled modifications of torsional degrees of freedom, involved in the study of conformational equilibria (c) represents an intramolecular, end-to-end reaction coordinate that may be used, for instance, to model the folding of a short peptide (d) symbolizes the alteration of selected nonbonded interactions to estimate relative free energies, in the spirit of site-directed mutagenesis experiments (e) is a simple distance separating chemical species that can be employed in potential of mean force (PMF) calculations and (f) corresponds to the annihilation of selected nonbonded interactions for the estimation of e.g., free energies of solvation. In the examples (a), (b), and (e), the coupling parameter, A, is not independent of the Cartesian coordinates, x. Appropriate metric tensor correction should be considered through a relevant transformation into generalized coordinates...

The order parameter can be defined in two different ways. It can be either a function of atomic coordinates or just a parameter in the Hamiltonian. Examples of both types of order parameters are given in Sect. 2.8.1 in Chap. 2 and illustrated in Fig. 2.5. This distinction is theoretically important. In the first case, the order parameter is, in effect, a generalized coordinate, the evolution of which can be described by Newton s equations of motion. For example, in an association reaction between two molecules, we may choose as order parameter the distance between the two molecules. Ideally, we often would like to consider a reaction coordinate which measures the progress of a reaction. However, in many cases this coordinate is difficult to define, usually because it cannot be defined analytically and its numerical calculation is time consuming. This reaction coordinate is therefore often approximated by simpler order parameters. 

The delta function is not convenient to handle mathematically. However, if we define a set of generalized coordinates of the form ( , q, , cjn-i) and then-associated momenta -,pqAr x) then this integration simplifies to ... 

Using the formalism we just developed for generalized coordinates we can now derive an expression for the derivative of the free energy. Let us differentiate (4.4) with respect to ... 

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