Spatial force

In order to improve parallelism and load balancing, a hybrid force-spatial decomposition scheme was adopted in NAMD 2. Rather than decomposing the nonbonded computation into regions of space or pairwise atomic interactions, the basic unit of work was chosen to be interactions between atoms... 

Spatial coordinate vector Thermodynamic force Spatial coordinate i of particle k... 

The most popular of the scanning probe tecimiques are STM and atomic force microscopy (AFM). STM and AFM provide images of the outemiost layer of a surface with atomic resolution. STM measures the spatial distribution of the surface electronic density by monitoring the tiumelling of electrons either from the sample to the tip or from the tip to the sample. This provides a map of the density of filled or empty electronic states, respectively. The variations in surface electron density are generally correlated with the atomic positions. 

AFM measures the spatial distribution of the forces between an ultrafme tip and the sample. This distribution of these forces is also highly correlated with the atomic structure. STM is able to image many semiconductor and metal surfaces with atomic resolution. AFM is necessary for insulating materials, however, as electron conduction is required for STM in order to achieve tiumelling. Note that there are many modes of operation for these instruments, and many variations in use. In addition, there are other types of scaiming probe microscopies under development. 

Feibelman P J 1987 Force and total-energy calculations for a spatially compact adsorbate on an extended, metallic crystal surface Phys. Rev. B 35 2626... 

The atomic force microscope (ATM) provides one approach to the measurement of friction in well defined systems. The ATM allows measurement of friction between a surface and a tip with a radius of the order of 5-10 nm figure C2.9.3 a)). It is the tme realization of a single asperity contact with a flat surface which, in its ultimate fonn, would measure friction between a single atom and a surface. The ATM allows friction measurements on surfaces that are well defined in tenns of both composition and stmcture. It is limited by the fact that the characteristics of the tip itself are often poorly understood. It is very difficult to detennine the radius, stmcture and composition of the tip however, these limitations are being resolved. The AFM has already allowed the spatial resolution of friction forces that exlribit atomic periodicity and chemical specificity [3, K), 13]. 

To obtain the force constant for constructing the equation of motion of the nuclear motion in the second-order perturbation, we need to know about the excited states, too. With the minimal basis set, the only excited-state spatial orbital for one electron is... 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

Fig. 1. Nonbonded force evaluation may be distributed among processors according to atomic coordinates, as in spatial decomposition (left), or according to the indices of the interacting atoms, as in force-matrix decomposition (right). Shades of gray indicate processors to which interactions are assigned.

Methods of decomposing the nonbonded force evaluation fall into two classes, spatial decomposition [15] in which atoms and their interactions are divided among processors based on their coordinates, and force-matrix decomposition [16] in which the calculation of the interaction between a pair of atoms is assigned to a processor without considering the location of either atom (Fig. 1). Spatial decomposition scales better to large numbers of... 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

UFF (universal force field) a molecular mechanics force field unrestricted (spin unrestricted) calculation in which particles of different spins are described by different spatial functions VTST (variational transition state theory) method for predicting rate constants... 

Induction heating is thought to cause vigorous convection because of the spatially varying average force field imposed along the melt surface. 

Molecular mechanics simulations can be readily mapped onto such kinds of machine architecture by using the spatial locaUty of the atoms to determine their allocation to processors. Short-range van der Waals forces can usually be accurately modeled with a cut-off distance of less than one nm, so interprocessor communication requirements can also be localized. 

The spatial Cauchy stress tensor s is defined at time by f = sn, where t(x, t, n) is a contact force vector acting on an element of area da = n da with unit normal i and magnitude da in the current configuration. The element of area... 

In the absence of an external force, the probability of moving to a new position is a spherically symmetrical Gaussian distribution (where we have assumed that the diffusion is spatially isotropic). 

In the second step, the spatial restraints and the CHARMM22 force field tenns enforcing proper stereochemistry [80,81] are combined into an objective function. The general form of the objective function is similar to that in molecular dynamics programs such as CHARMM22 [80]. The objective function depends on the Cartesian coordinates of —10,000 atoms (3D points) that form a system (one or more molecules) ... 

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