Calculations in Cartesian Coordinates

Take an N-atomic molecule with the nuclei each at their equilibrium internuclear position. Establish a Cartesian x, y, z coordinate system for each of the nuclei such that, for Xj with i = 1. 3N, xi is the Cartesian x displacement of nucleus 1, x2 is the Cartesian y displacement coordinate for nucleus 1, X3 is the Cartesian z displacement coordinate for nucleus 1. x3N is the Cartesian z displacement for nucleus N. Use of one or another quantum chemistry program yields a set of force constants I ij in Cartesian displacement coordinates 

One remembers that Eeiec is the isotope independent potential energy surface for vibration in the Born-Oppenheimer approximation, Eeiec = V. Note 

The classical kinetic energy T is given by the usual formula involving the time derivatives (velocities) of the Cartesian coordinates x , 

For convenience we introduce mass weighted coordinates (designated by a prime) 

for convenience, introduce a set of mass weighted force constants 

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The Stream Function Stream functions are defined for two-dimensional flow and for three-dimensional axial symmetric flow. The stream function can be used to plot the streamlines of the flow and find the velocity. For two-dimensional Bow the velocity components can be calculated in Cartesian coordinates by... 

Derivatives of the energy and molecular properties are most conveniently calculated in Cartesian coordinates. Due to translational and rotational symmetries these Cartesian derivatives are not independent. For example, for forces only 3N — m (m = 5, 6) Cartesian components are independent. The remaining five or six components can be calculated by taking into consideration the translational and rotational symmetries of the energy. This has two useful applications. First, the number of derivatives that must be calculated ab initio may be reduced. This saving is especially important for small molecules. Alternatively, one may calculate all derivatives ab initio and simply use translational and rotational symmetries as a simple check on the calculation. 

It is always possible to convert internal to Cartesian coordinates and vice versa. However, one coordinate system is usually preferred for a given application. Internal coordinates can usefully describe the relationship between the atoms in a single molecule, but Cartesian coordinates may be more appropriate when describing a collection of discrete molecules. Internal coordinates are commonly used as input to quantum mechanics programs, whereas calculations using molecular mechanics are usually done in Cartesian coordinates. The total number of coordinates that must be specified in the internal coordinate system is six fewer... 

Here we give the molecule specification in Cartesian coordinates. The route section specifies a single point energy calculation at the Hartree-Fock level, using the 6-31G(d) basis set. We ve specified a restricted Hartree-Fock calculation (via the R prepended to the HF procedure keyword) because this is a closed shell system. We ve also requested that information about the molecular orbitals be included in the output with Pop=Reg. 

This section displays positioning of the atoms in the molecule used by the program internally, in Cartesian coordinates. This orientation is chosen for maximum calculation efficiency, and corresponds to placing the center of nuclear charge for the molecule at the origin. Most molecular properties are reported with respect to the standard orientation. Note that this orientation usually does not correspond to the one used in the input molecule specification the latter is printed earlier in the output as the Z-matrix orientation. ... 

Force constant calculations are normally done in Cartesian coordinates. Suppose we have N atoms whose position vectors are Ri, R2,. .., Ra - Each of the atoms vibrates about its equilibrium position Ri g, Ri.e, , R v,e-The first step in our treatment is to define mass-weighted displacement coordinates... 

Essentially all force field calculations use Cartesian coordinates of the atoms as the variables in the energy expression. To obtain the distance between two atoms one need to calculate... 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

As we are particularly interested in surface reactions and catalysis, we will calculate the rate of collisions between a gas and a surface. For a surface of area A (see Fig. 3.8) the molecules that will be able to hit this surface must have a velocity component orthogonal to the surface v. All molecules with velocity Vx, given by the Max-well-Boltzmann distribution f(v ) in Cartesian coordinates, at a distance v At orthogonal to the surface will collide with the surface. The product VxAtA = V defines a volume and the number of molecules therein with velocity Vx is J vx) V Vx)p where p is the density of molecules. By integrating over all Vx from 0 to infinity we obtain the total number of collisions in time interval At on the area A. Since we are interested in the collision number per time and per area, we calculate... 

From quantum chemistry one obtains the force field of the molecule in Cartesian coordinates by taking the second derivative of the quantum chemically calculated ground state energy ... 

In the framework of the force field calculations described here we work with potential constants and Cartesian coordinates. The analytical form of the expression for the potential energy may be anything that seems physically reasonable and may involve as many constants as are deemed feasible. The force constants are now derived quantities with the following definition expressed in Cartesian coordinates (x ) ... 

Calculations in helical coordinates are very challenging. The procedure used in this text will be to unwrap the screw helix into Cartesian coordinates for the analysis. It is important to be able to calculate the helical length in the z direction at any radius r for the axial length 1 ... 

A. Phase Space. It will be useful here to anticipate a formulation that we will use in more detail in Section 3, namely, the solution of the classical equations of motion for the atoms of a molecule undergoing a chemical reaction. One starts with a molecule of defined geometry (say, in Cartesian coordinates) and with defined velocities for each of its atoms (expressible as components in the x, y, and z directions). The problem then is to solve Newton s second law of motion, F = mA, for each atom. The force, F, can be calculated as the first derivative of... 

When calculating FE energy states along particular directions in the BZ it is often convenient to work in Cartesian coordinates, that is to use the (e basis rather than the (b basis. The matrix representation of a reciprocal lattice vector bm is... 

An input structure (a molecular geometry) must be specified and submitted to calculation. The geometry can be specified in Cartesian coordinates (probably the usual way nowadays) or as bond lengths, angles and dihedrals (internal... 

Order parameters can also be calculated from an ensemble of structures given in Cartesian coordinates ... 

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