Cartesian coordinates Hamiltonian equations

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. 

Another apparent difference between the various free energy methods lies in the treatment of order parameters. In the original formulation of a number of methods, order parameters were dynamical variables - i.e., variables that can be expressed in terms of the Cartesian coordinates of the particles - whereas in others, they were parameters in the Hamiltonian. This implies a different treatment of the order parameter in the equations of motion. If one, however, applies the formalism of metadynamics, or extended dynamics, in which any parameter can be treated as a dynamical variable, most conceptual differences between these two cases vanish. 

Here Ha and Hb are the Hamiltonians of the isolated reactant molecules, Hso is the Hamiltonian of the pure solvent, and Vmt is the interaction energy between reactants and between reactant and solvent molecules, i.e., it contains the solute-solute as well as the solute-solvent interactions, qa and reactant molecules A and B, respectively, and pa and pb are the conjugated momenta. If there are na atoms in molecule A and tib atoms in molecule B, then there will be, respectively, 3ua coordinates c/a and 3rt j coordinates c/b Similarly, R are the coordinates for the solvent molecules and P are the conjugated momenta. In the second line of the equation, we have partitioned the Hamiltonians Hi into a kinetic energy part T) and a potential energy part V). 

In the special case that the generalized coordinates ft represent the Cartesian coordinates of n point masses and, furthermore, that momenta can be separated from coordinates in the Hamiltonian H, the Hamilton equations of motion reduce to the more familiar Newton s second law ... 

Since the dynamics of rigid bodies is based on the dynamics of particles, these rules must be related to the rules given in Chapter IV. For a discussion of a method of finding the wave equation for a system whose Hamiltonian is not expressed in Cartesian coordinates, see B. Podolsky, Phys. Rev. 32, 812 (1928), and for the specific application to the symmetrical top see the references below. 

Sir William Rowan Hamilton (1805-1865) devised an alternative form of Newton s equations of motion involving a function H, the Hamiltonian function for the system. For a system where the potential energy is a function of the coordinates only, the total energy remains constant with time that is, E is conserved. We shall restrict ourselves to such conservative systems. For conservative systems, the classical-mechanical Hamiltonian function turns out to be simply the total energy expressed in terms of coordinates and conjugate momenta. For Cartesian coordinates x, y, z, the conjugate momenta are the components of linear momentum in the x, y, and z directions p, Py, and p. ... 

The above example illustrates the development and solution to the singleparticle Liouville equation in cartesian coordinates. As discussed in Chap. 1, for structured particles, it is often more fruitful to work in generalized coordinates. Let s revisit the problem of a dipole in an external electric field. Example 1.1, and develop the generalized coordinates, conjugate momenta, hamiltonian, and associated Liouville equation for this system. 

When this Hamiltonian is used, the solutions are the wavefunctions for the H atom (0 s). In this specific case, the Hamiltonian can be directly substituted into H(p = 0, and the Cartesian coordinate system (x, y, and z) is converted to spherical coordinates (r, 6, and 0). The solutions to the resulting differential equations can be directly obtained and are found to have a radial function R(r) that is separate from an angular portion [ Y(0,0)]. The radial portions of several of these wavefunctions are given in the table shown to the right [the angular portion, V(ft0), is not shown]. 

The standard problem in theoretical quantum mechanics is to solve the Schrodinger equation to get energies and wavefunctions when aU you have to start with is the Hamiltonian operator. To write the Hamiltonian, we combine a kinetic energy operator K with the classical potential energy U. The kinetic energy operator is the sum over aU the Cartesian coordinates and all the particles in the system of the following one-particle, onedimensional operator ... 

For the most general case (see section on integrating classical equations of motion), T depends on both the momenta p and coordinates q. The index i in the equations above is the number of coordinates or conjugate momenta for the Hamiltonian. If Cartesian coordinates are used, this number is 3N, where N is the number of atoms. 

Write the Hamiltonian of a particle in terms of a) Cartesian coordinates and momenta, and b) spherical coordinates and their conjugate momenta. Show that Hamilton s equations of motion are invariant to the coordinate transformation from Cartesian to polar coordinates. 

In Equation 7.33 we have written out both the g-value and the zero-field coefficient of the basic S2 interaction term in the form of diagonal 3x3 matrices in which all off-diagonal elements are equal to zero. The diagonal elements were indexed with subscripts x, y, z, corresponding to the Cartesian axes of the molecular axes system. But how do we define a molecular axis system in a (bio)coordination complex that lacks symmetry The answer is that if we would have made a wrong choice, then the matrices would not be diagonal with zeros elsewhere. In other words, if the spin Hamiltonian would have been written out for a different axes system, then, for example, the g-matrix would not have three, but rather six, independent elements ... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

Cartesian and spherical coordinates, we can express ri2 in terms of ri, di, cf>i, V2,62, 2-Because of the e /47reori2 term, the Schrbdinger equation for helium cannot be separated in any coordinate system, and we must use approximation methods. The perturbation method separates the Hamiltonian (9.39) into two parts, and H, where is the Hamiltonian of an exactly solvable problem. If we choose... 

As discussed above in the chapter introduction, either Newton s or Hamiltonian s equations may be numerically integrated for direct dynamics simulations. There is also a choice of coordinate representation, such as Cartesian, internal, " or instantaneous normal modes. Though potential... 

---