Positional vector

Consider a polyatomic system consisting of N nuclei (where > 3) and elecbons. In the absence of any external fields, we can rigorously separate the motion of the center of mass G of the whole system as its potential energy function V is independent of the position vector of G (rg) in a laboratory-fixed frame with origin O. This separation introduces, besides rg, the Jacobi vectors R = (R , , R , .. , Rxk -1) = (fi I "21 I fvji) fot nuclei and electrons,... 

The fundamental method [22,24] represents a multidimensional nuclear wavepacket by a multivariate Gaussian with time-dependent width niaUix, A center position vector, R, momentum vector, p and phase, y,... 

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

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... 

If the position of the fluid particle at current time t is given by x t) then its motion is defined using the following position vector... 

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be expressed as... 

In a Cartesian (planar) coordinate system of (x, y, z) shown in Appendix Figure I the position vectors for points Pi and P2 are given as... 

Let jP be a vector whose components are functions of a scalar variable (e.g. time-dependent position vector of a point F in a three-dimensional domain)... 

Generalizing the Newton-Raphson method of optimization (Chapter 1) to a surface in many dimensions, the function to be optimized is expanded about the many-dimensional position vector of a point xq... 

Dyne, unit of force dyn Equilibrium position vector Ro... 

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... 

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. 

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.. 

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. 

I assume that you are familiar with the elementary ideas of vectors and vector algebra. Thus if a point P has position vector r (I will use bold letters to denote vectors) then we can write r in terms of the unit Cartesian vectors ex, Cy and as ... 

Suppose that the vector field u(f) is a continuous function of the scalar variable t. As t varies, so does u and if u denotes the position vector of a point P, then P moves along a continuous curve in space as t varies. For most of this book we will identify the variable t as time and so we will be interested in the trajectory of particles along curves in space. 

The physical significance of 1/ab is that it represents the work done in bringin up (2b from infinity to the point whose position vector is Tb, under the influenc of (2a which is fixed at Ta- Because of the symmetry of the expression, it als represents the work done in bringing up point charge (2a from infinity to th point with vector position Ta, under the influence of point charge Qb which fixed in space at position Tb-... 

The point in space has position vector r, and the field exists because of the presence of Qa- In order to measure the field at that point, we introduce a point lest charge Qb and measure the force exerted on it by Qa- The ratio F/Qb gives fhft field,... 

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). 

Adding these two equations gives the Verlet algorithm, which is used to advance the position vector r from its value at time t to time f + Sf... 

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... 

Notice that Rahman and Stiliinger make use of the concept of an effective pair potential. The potential energy of any substance may always be resolved systematically into pair, triplet, quadruplet,. .. contributions. If we consider N particles each with position vector R, then formally we can write... 

We need to be clear about the various coordinates, and about the difference between the various vector and scalar quantities. The electron has position vector r from the centre of mass, and the length of the vector is r. The scalar distance between the electron and nucleus A is rp, and the scalar distance between the electron and nucleus B is tb- I will write / ab for the scalar distance between the two nuclei A and B. The position vector for nucleus A is Ra and the position vector for nucleus B is Rb. The wavefunction for the molecule as a whole will therefore depend on the vector quantities r, Ra and Rb-It is an easy step to write down the Hamiltonian operator for the problem... 

We see from the figure that the position vector of the electron relative to the centre of is r — Fa, so we need a factor of... 

The product of these two exponentials is equal to a third GTO situated at point C along the line of centres AB, as shown in Figure 9.1. The resultant GTO Gc(rc- is situated at the point C with position vector fc such that... 

Secondly, it can be shown (Davidson, 1976) that the so-called nuclear cusp condition for nucleus A with position vector Ra gives... 

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... 

If the timestep is short relative to the velocity relaxation time, the solvent does not play much part in the motion. Indeed, if y = 0, there are no solvent effects at all. A simple algorithm for advancing the position vector r and velocity v has been given by van Gunsteren (van Gunsteren, Berendsen and Rullmaim, 1981) ... 

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