Vector atomic position

Here and below, T , 1, , and e, i, j = 1,. . . , 5, denote atomic position vectors, atom-atom distances, and the corresponding unit vectors, respectively. In order to construct a correctly closed conformation, variables qi,. . . , q4 are considered independent, and the last valence angle q is computed from Eq. (7) as follows. Variables qi,.. ., q4 determine the orientation of the plane of q specified by vector 634 and an in-plane unit vector 6345 orthogonal to it. In the basis of these two vectors, condition (7) results in... 

A procedure commonly used to extract dynamic data directly from experimental incoherent neutron scattering profiles is described in Ref. 17. It is assumed that the atomic position vectors can be decomposed into two contributions, one due to diffusive motion, fi /t), and the other from vibrations, Uijt), i.e.. 

Despite the success of direct methods, there are still certain structures that are not readily solved, if at all, by these methods. This may be due to a breakdown in the certainty with which the triplet relationships are developed, or to a breaJidown of the assumption that the atomic position vectors form a set of random variables, so that the probability functions that are derived may no longer be strictly valid. Some new developments for dealing with these problems have focused on improving the techniques for calculating the triplets and including in the direct methods procedure as much structural information as possible, such as... 

Geometric algebra approach offers some advantages over other methods presented in the literature. First of all, atomic position vectors themselves are manipulated instead of their components, and hence all expressions are simple at each stage of derivation. This is not the case when Cartesian components and back substitutions are used to obtain contravariant measuring vectors [57]. As a... 

Lattice modes, or external modes, as well as internal modes produce harmonic displacements of the atoms. The internal modes result in local molecular deformation, whilst external modes are supposed to entrain the atoms when a molecule is rigidly displaced from its equilibrium position in the lattice. The time-dependent atomic position vector r can be expressed in terms of the internal displacement vector u nx, taken with respect to the molecular centre of mass, and the external displacement vector, Hext the displacement vectors have units of length. A, Eq. (A2.52). 

If r t) is the time dependent atomic position vector, taken with respect to the origin of the crystallographic cell, it can be expressed as the sum of two terms, the molecular centre of mass, this is a time dependent vector, (0ext> that allows a description of the vibrations of the crystal, the phonons. The second term is the position vector of the atom, H(0int, given by a Cartesian coordinate system with its origin at the molecular centre of mass. 

Figure 2. Diatomic molecule in space, = position vector for the molecular center of mass from the origin of coo-dinates 0. j = atomic position vectors from r (o = 1,2). = atomic position vectors from O. R - = molecular quantum centroid position vector from O.

ACj, change in elastic constant R, atom position vector... 

The integrand in this expression will have a large value at a point r if p(r) and p(r+s) are both large, and P s) will be large if this condition is satisfied systematically over all space. It is therefore a self- or autocorrelation fiinction of p(r). If p(r) is periodic, as m a crystal, F(s) will also be periodic, with a large peak when s is a vector of the lattice and also will have a peak when s is a vector between any two atomic positions. The fiinction F(s) is known as the Patterson function, after A L Patterson [14], who introduced its application to the problem of crystal structure detemiination. 

The most well-known and at the same time the earliest computer model for a molecular structure representation is a wire frame model (Figure 2-123a). This model is also known under other names such as line model or Drciding model [199]. It shows the individual bonds and the angles formed between these bonds. The bonds of a molecule are represented by colored vector lines and the color is derived from the atom type definition. This simple method does not display atoms, but atom positions can be derived from the end and branching points of the wire frame model. In addition, the bond orders between two atoms can be expressed by the number of lines. 

Equation (3.85) T is a translation vector that maps each position into an equivalent ition in a neighbouring cell, r is a general positional vector and k is the wavevector ich characterises the wavefunction. k has components k, and ky in two dimensions and quivalent to the parameter k in the one-dimensional system. For the two-dimensional lare lattice the Schrodinger equation can be expressed in terms of separate wavefunctions ng the X- and y-directions. This results in various combinations of the atomic Is orbitals, ne of which are shown in Figure 3.13. These combinations have different energies. The /est-energy solution corresponds to (k =0, ky = 0) and is a straightforward linear... 

The X-ray and neutron scattering processes provide relatively direct spatial information on atomic motions via detennination of the wave vector transferred between the photon/neutron and the sample this is a Fourier transfonn relationship between wave vectors in reciprocal space and position vectors in real space. Neutrons, by virtue of the possibility of resolving their energy transfers, can also give infonnation on the time dependence of the motions involved. 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

Here is the position operator of atom j, or, if the correlation function is calculated classically as in an MD simulation, is a position vector N is the number of scatterers (i.e., H atoms) and the angular brackets denote an ensemble average. Note that in Eq. (3) we left out a factor equal to the square of the scattering length. This is convenient in the case of a single dominant scatterer because it gives 7(Q, 0) = 1 and 6 u,c(Q, CO) normalized to unity. 

The state of any particle at any instant is given by its position vector q and its linear momentum vector p, and we say that the state of a particle can be described by giving its location in phase space. For a system of N atoms, this space has 6iV dimensions three components of p and the three components of q for each atom. If we use the symbol F to denote a particular point in this six-dimensional phase space (just as we would use the vector r to denote a point in three-dimensional coordinate space) then the value of a particular property A (such as the mutual potential energy, the pressure and so on) will be a function of r and is often written as A(F). As the system evolves in time then F will change and so will A(F). 

A molecular dynamics (MD) calculation collects statistical information as it progresses. So, for example, if the calculated position vector of atom A at times ti,t2, -, tn is rA(f2), .., t-A(tn), then the statistical-mechanical average is... 

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... 

A search for intermolecular bonds resulted in one possible hydrogen bond between hydroxyl 013 and lactone carbonyl Ol. The distance between 01 and 013 is 2.85 A, a value well within the range expected for OH-O hydrogen bonds (25). The hydrogen atom position for hydroxyl 013 was chosen to be along the 013-01 vector. The hydrogen position was not evident in the difference electron density map, presumably due to problems modeling the 013 position. 

In order to apply quantum-mechanical theory to the hydrogen atom, we first need to find the appropriate Hamiltonian operator and Schrodinger equation. As preparation for establishing the Hamiltonian operator, we consider a classical system of two interacting point particles with masses mi and m2 and instantaneous positions ri and V2 as shown in Figure 6.1. In terms of their cartesian components, these position vectors are... 

Minimize the Gibbs energy Gk of each supercell at temperature T with respect to all the lattice vectors and atom positions. 

Here Slater functions (a3 2/7r 1/2) exp( —a r — ra ) with the atom being centered at position vectors ra. The overlap between these functions is given by S. After an FT and integrating over momentum coordinates of one particle, the EMD of H2 molecule within VB and MO theory are derived as... 

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