Variable atomic coordinates

The quaniity, (R). the sum of the electronic energy computed 111 a wave funciion calculation and the nuclear-nuclear coulomb interaciion .(R.R), constitutes a potential energy surface having 15X independent variables (the coordinates R j. The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

The most straightforward fype of lattice minimisation is performed at constant volume, where the dimensions of the basic imit cell do not change. A more advanced type of calculation is one performed at constant pressure, in which case there are forces on both the atoms and the unit cell as a whole. The lattice vectors are considered as additional variables along with the atomic coordinates. The laws of elasticify describe the behaviour of a material when... 

Figure 5 Time dependence of RMSD of atomic coordinates from canonical A- and B-DNA forms m two trajectories of a partially hydrated dodecamer duplex. The A and B (A and B coiTespond to A and B forms) trajectories started from the same state and were computed with internal and Cartesian coordinates as independent variables, respectively. (From Ref. 54.)...

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

In the A-dynamics method,1 both X variables (coupling parameters) and atomic coordinates are propagated using molecular dynamics (MD). The dynamics of the system is generated from an extended Hamiltonian67 68... 

Tx and Tx are the kinetic energies of the atomic coordinates and X variables, respectively. The As are treated as volumeless particles with mass mx. Since the X variables are associated with the chemical reaction coordinates , the A-dynamics method can utilize the power of specific biasing potentials in the umbrella sampling method to overcome sampling problems that require conventional FEP calculations to be performed in multiple steps. 

Instead of using MD, the X variables may also be sampled stochastically. In the hybrid CMC/MD approach, Metropolis Monte Carlo69 is used to evolve the X variables and molecular dynamics is used to evolve the atomic coordinates. The Metropolis Monte Carlo criteria leads to the generation of a canonical ensemble of the ligands when the following transition probability is used... 

Since A is treated as a dynamic variable, just as the atomic coordinates, we use XToi to denote the phase space that encompasses X, X, and x. Thus the hybrid potential in Equation 14 can be rewritten as... 

A crystallographic example of optimization would be the minimization of a least-squares or a negative log-likelihood residual as the objective function, using fractional or orthogonal atomic coordinates as the variables. The values of the variables that optimize this objective function constitute the final crystallographic model. However, due to the... 

The atomic dipole moment is dependent on eight variables the net charge of the atom, derived from Pvab the k parameter, the atomic coordinates, and the three population parameters P10, P11 + and Pu. If we are interested in the error in the magnitude of the molecular dipole moment /r,otal, and we omit columns of D which contain only zero s, D will be a 1 x 8N matrix, where N is the number of... 

The hydrogen atom orbitals are functions of three variables the coordinates of the electron. Their physical interpretation is that the square of the amplitude of the wave function at any point is proportional to the probability of finding a particle at that point. Mathematically, the electron density distribution is equal to the square of the absolute value of the wave function ... 

When the general arrangement is known it is then necessary to determine precise atomic coordinates. Sometimes the positions of certain atoms are invariant—they are fixed by symmetry considerations—but in complex crystals most of the atoms are in general5 positions not restricted in any way by symmetry. The variable parameters must be determined by successive approximations here the work of calculating structure amplitudes for postulated atomic positions can be much shortened by the use of graphical methods, to be described later in this chapter. It cannot be denied, however, that the complete determination of a complex structure is a task not to be undertaken lightly the time taken must usually be reckoned in months. 

The initial description of the model is simple, as shown in Figure 3. The atomic coordinates of any suitable structure can serve as the input trial structure, even including a wrong monomer residue. The polar coordinates are calculated from the trial structure, adjusted and modified as necessary, and then subjected to refinement in accordance with the selected list of variables, limits and constraints. Any set of standard values and nonbonded potential function parameters can be used. Hydrogen bonds can be defined as desired, variables can be coupled, and the positions of solvent molecules can be individually refined. Single and multiple helices are equally easily handled, as are a variety of space groups. 

B, J, C, N) performs the desired calculation I and J are singly dimensional integer arrays, the contents of which point to elements of the A and B arrays respectively. For the present purpose A and B would be identical and would contain the atomic coordinates. The integer variable N determines the number of indexed subtractions performed. The overall operation performed... 

The intrinsic disorder of the continuous network is less easily classified in terms of defects. The network has many different configurations, but provided the atomic coordination is the same, all these structures are equivalent and represent the natural variability of the material. Since there is no correct position of an atom, one cannot say whether a specific structure is a defect or not. Instead the long range disorder is intrinsic to the amorphous material and is described by a randomly varying disorder potential, whose effect on the electronic structure is summarized in Section 1.2.5. 

Static simulations are generally based on an energy minimization procedure that is, the energy of the system is written as a function of structural variables that include atomic coordinates and cell dimensions. The... 

Now let us ask what is needed to numerically evaluate this expression for the Fourier transform. The diffraction vector s = k — ko, as well as X, are experimental variables that are chosen, and the Zj are known for each atom as well. The only remaining variables are the xj, and these can be generated from the atomic coordinates xj, yj, zj. Thus all we really need to compute the resultant waves making up the diffraction pattern, for any array of scattering points, are their relative positions in space. 

We know that Fcaic-hu = fj exP Xj) from the structure factor expression that we derived in Chapter 5, where h Xj = hxj + kyj + hj (we have left out the Bj factor for simplicity, but we could just as well have included it). As before, we would like to find the values for xj, yj, Zj, the atomic coordinates, that make Fcaic most like Fobs for the entire set of data. The problem is, however, that Fcaic is not a linear function of xj, yj, and Zj. The variables are related to Fcaic by a transcendental function exp i2jt(h xj). The least squares procedure does not work in an exact sense for such functions. 

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