Choice of Coordinates

Given the first and second derivatives, the NR formula calculates the geometry step as the inverse of the Hessian times the gradient. 

In the coordinate system (x ) where the Hessian is diagonal, the step may be written as in eq. (12.21). 

Another way of removing the six translational and rotational degrees of freedom is to use a set of internal coordinates. For a simple acyclic system, these may be chosen as Aatom - 1 distances, - 2 angles and N tom - 3 torsional angles, as illustrated in the construction of Z-matrices in Appendix D. In internal coordinates the six translational and rotational modes are automatically removed (since only 3iVatoin - 6 coordinates are dehned), and the NR step can be formed straightforwardly. For cyclic systems, a choice of 3Aatom - 6 internal variables that span the whole optimization space may be somewhat more problematic, especially if synunetry is present. 

The transformation from a set of Cartesian coordinates to a set of internal coordinates, which may for example be distances, angles and torsional angles, is an example of a non-linear transformation. The internal coordinates are connected with the Cartesian coordinates by means of square root and trigonometric functions, not simple linear combinations. A non-hnear transformation will affect the convergence properties. This can be illustrate by considering a minimization of a Morse type function (eq. (2.5)) withD = a=l andx = AR. 

We will consider two other variables obtained by a non-linear transformation y = e and z = The minimum energy is at x = 0, corresponding to y = z = 1. Consider an NR optimization starting at x = -0.5, corresponding to y = 1.6587 and z = 0.6065. Table 12.1 shows that the NR procedure in the x-variable requires four iterations before x is less than lO . In the y-variable the optimization only requires one step to reach the 

Diagonalization of the Hessian is an example of a linear transformation the eigenvectors are just linear combinations of the original coordinates. A linear transformation does not change the convergence/divergence properties, or the rate of 

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As noted above, the coordinate system is now recognized as being of fimdamental importance for efficient geometry optimization indeed, most of the major advances in this area in the last ten years or so have been due to a better choice of coordinates. This topic is seldom discussed in the mathematical literature, as it is in general not possible to choose simple and efficient new coordinates for an abstract optimization problem. A nonlmear molecule with N atoms and no... 

It is obvious that CSP depends, as does TDSCF, on the choice of coordinates. As pointed out in Sec. 2.2, numerical convenience often limits the choice of the coordinates. CSP may, however, offer practical prospects for the choice of physically optimal modes. The deviation of the true potential from CSP separability is given by ... 

AVcorr can be evaluated readily from the classical MD simulation for any choice of coordinate system, and it may be possible to determine the modes that give the smallest AVcorr- These should be optimal CSP modes. Work along these lines is in ])rogress in our group. So far, however, the coordi-... 

The following three scalars remain independent of the choice of coordinate system in which the components of T are defined and hence are caUed the invariants of tensor T ... 

Figure 1 The course of energy minimization of a DNA duplex with different choices of coordinates. The rate of convergence is monitored by the decrease of the RMSD from the final local minimum structure, which was very similar in all three cases, with the number of gradient calls. The RMSD was normalized by its initial value. CC, IC, and SG stand for Cartesian coordinates, 3N internal coordinates, and standard geometry, respectively.

Theoreticians did little to improve their case by proposing yet more complicated and obviously unreUable parameter schemes. For example, it is usual to call the C2 axis of the water molecule the z-axis. The molecule doesn t care, it must have the same energy, electric dipole moment and enthalpy of formation no matter how we label the axes. I have to tell you that some of the more esoteric versions of extended Hiickel theory did not satisfy this simple criterion. It proved possible to calculate different physical properties depending on the arbitrary choice of coordinate system. 

To look ahead a little, there are properties that depend on the choice of coordinate system the electric dipole moment of a charged species is origin-dependent in a well-understood way. But not the charge density or the electronic energy Quantities that have the same value in any coordinate system are sometimes referred to as invariants, a term borrowed from the theory of relativity. 

These integrals can be terrifyingly difficult they involve the spatial coordinates of a pair of electrons and so are six-dimensional. They are singular, in the sense that the integrand becomes infinite as the distance between the electrons tends to zero. Each basis function could be centred on a different atom, and there is no obvious choice of coordinate origin in such a case. 

This definition is therefore independent of the choice of coordinate origin only when the charges sum to zero. The dipole moment of a charged species depends on the coordinate origin, which must be quoted when reporting such quantities. 

Just like the electric quadrupole moment, the electric field gradient matrix can be written in diagonal form for a suitable choice of coordinate axes. 

Magnetic properties should be independent of the choice of coordinate origin. The term choice of origin is often translated into choice of gauge, and so we say that physical properties should be gauge-invariant (for a discussion, see Hameka, 1965). 

The total molecular energy is invariant to all transformations involving basis orbitals, just as any physical event is invariant under any transformation of coordinates. But just as the proper choice of coordinates helps in visualizing physical events, so the choice of the proper orbital basis is helpful in visualizing molecular properties. 

With the above choice of coordinates, the internal Gqq), Coriolis (Gy ), and rotational (Coxb), parts of the metric are constant, linear, and quadratic functions... 

The transition from WA( ) = 1 to WA( ) = 0 as the distance from the nucleus A increases needs to be smooth enough such that numerical instabilities are avoided but at the same time also as abrupt as possible such that density peaks from nearby the nuclei are extinguished. The implementation of this concept in the three-dimensional space involves a special choice of coordinates - see Becke, 1988c, for details - but actually leads to a smoothened step function as schematically sketched in Figure 7-1 for the one-dimensional case. 

To provide a mathematical description of a particle in space it is essential to specify not only its mass, but also its position (perhaps with respect to an arbitrary origin), as well as its velocity (and hence its momentum). Its mass is constant and thus independent of its position and velocity, at least in the absence of relativistic effects. It is also independent of the system of coordinates used to locate it in space. Its position and velocity, on the other hand, which have direction as well as magnitude, are vector quantities. Their descriptions depend on the choice of coordinate system. In this chapter Heaviside s notation will be followed, viz. a scalar quantity is represented by a symbol in plain italics, while a vector is printed in bold-face italic type. 

Since chemical reactions usually show significant nonadiabaticity, there are naturally quantitative errors in the predictions of the vibrationally adiabatic model. Furthermore, there are ambiguities about how to apply the theory such as the optimal choice of coordinate system. Nevertheless, this simple picture seems to capture the essence of the resonance trapping mechanism for many systems. We also point out that the recent work of Truhlar and co-workers24,34 has demonstrated that the reaction dynamics is largely controlled by the quantized bottleneck states at the barrier maxima in a much more quantitative manner than expected. 

The preferred choice of internal coordinates is discipline dependent. Nevertheless, the conservation of solid mass will imply constraints on particular moments of the PSD. In general, given the relationship between the various choices of coordinates, it is possible (although not always practical) to rewrite the PBE in terms of any choice of internal coordinate. 

However, they represent the same physical quantities as in the first picture, and only our choice of coordinates has changed. We can define a new set of creation and annihilation operators c, Cj through... 

Symmetry Choice of Coordinate Axes Indices of Allowed dlmp... 

Each of the symmetry operations we have defined geometrically can be represented by a matrix. The elements of the matrices depend on the choice of coordinate system. Consider a water molecule and a coordinate system so oriented that the three atoms lie in the x-z plane, with the z—axis passing through the oxygen atom and bisecting the H-O-H angle, as shown in Figure 5.1. 

Stabilizers are usually used during the reduction of metal ions to stabilize the colloidal dispersions of fine metal particles. The coordination interaction is the main factor to stabilize the metal particles. Thus, polymers with coordinating groups are good stabilizers. The choice of coordinating groups should depend on the kind of metal. 

There are at least three other factors that have a smaller but important effect on the final choice of coordination number symmetry, the softness of the ions, and spatial constraints. 

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