Internal displacement coordinates

The procedure Split selects the internal displacement coordinates, q, and momenta, tt, (describing vibrations), the coordinates, r, and velocities, v, of the centers of molecular masses, angular velocities, a>, and directional unit vectors, e, of the molecules from the initial Cartesian coordinates, q, and from momenta, p. Thus, the staring values for algorithm loop are prepared. Step 1 Vibration... 

The procedure Merge transforms the internal displacement coordinates and momenta, the coordinates and velocities of centers of masses, and directional unit vectors of the molecules back to the Cartesian coordinates and momenta. Evolve with Hr = Hr(q) means only a shift of all momenta for a corresponding impulse of force (SISM requires only one force evaluation per integration step). 

It i usually convenient to work with a set of internal displacement coordinates, 5, as they have chemical significance. In the limit of small amplitudes of atomic displacements, the two sets of coordinates are linearly related. Thus,... 

The actual calculation consists of minimizing the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The potential-energy expressions derive from the force-field concept that features in vibrational spectroscopic analysis according to the G-F-matrix formalism [111]. The G-matrix contains as elements atomic masses suitably reduced to match the internal displacement coordinates (matrix D) in defining the vibrational kinetic energy T of a molecule ... 

In fact, the result of Equation 3.43 not only applies to internal displacement coordinates but also to Cartesian displacements. The kinetic energy in terms of Cartesian coordinates (Equation 3.11) can easily be transformed into an expression in terms of Cartesian momenta (Equation 3.28)... 

As a second set of internal displacement coordinates we may choose the increments or decrements to the three OCO angles. Before using this set to form a representation which will tell us the normal coordinates involving inplane OCO bends, we must be careful to note that all of the coordinates in the set are not independent. If all three of the angles were to increase by the same amount at the same time, the motion would have A symmetry. It is obviously impossible, however, for all three angles simultaneously to expand within the plane. Thus the A representation which we shall find when we have reduced the representation is to be discarded as spurious. 

The vibrational potential energy of a molecule can be expanded as a function of internal displacement coordinates, qjt in the following way ... 

The first step is to formulate the relationship between Cartesian displacement coordinates, dag, and internal displacement coordinates, qk k = 1, 2,... 3 N—6. For rigid molecules undergoing small amplitude vibrations we can assume that an expansion from the equilibrium configuration,... 

For rigid molecules it is convenient to express the potential energy by an expansion in powers of the internal displacement coordinates. A similar expansion can also be applied in the case of nonrigid molecules. However, the potential function is expanded only in the small amplitude coordinates, Rt t= 1, 2,... 3 N—717 36),... 

A nonlinear molecule of N atoms has 32V — 6 internal vibrational degrees of freedom, and therefore 3A — 6 normal modes of vibration (the three translational and three rotational degrees of freedom are not of vibrational spectroscopic relevance). Thus, there are 32V — 6 independent internal coordinates, each of which can be expressed in terms of Cartesian coordinates. To first order, we can write any internal displacement coordinate ry in the form... 

Consider an ideally infinite helical chain whose crystallographic repeat contains N chemical repeat units, each with P atoms. A screw symmetry operation transforms one chemical unit into the next, with a being the rotation about the helix axis and d the translation along the axis. Let r" denote the ith internal displacement coordinate associated with the nth chemical repeat unit. The potential energy, by analogy with Eq. (3), is given by... 

The potential energy is expanded as a power series in internal displacement coordinates R the coefficients are the force constants in the chosen coordinate representation. 

Linear transformations, including symmetry transformations in configuration space are analogous to those in Cartesian space (see Sections 1.2.2 and 1.2.3). For symmetrical reference structures, it is usually better to use not the internal coordinates themselves but to choose a new coordinate system in which the basis vectors are symmetry adapted linear combinations of the internal displacement coordinates with the special property that they transform according to the irreducible representations of the point group of the idealized, reference molecule (symmetry coordinates, see Chapter 2). 

To use this scheme we require then a linear transformation B between the internal displacement coordinates r (usually atom-atom distance changes in terms of which the potential is expanded to second order) and the cartesian displacement coordinates of the atoms x. However, the relation between these coordinates is not linear and therefore an approximation must be made at this point. Shimanouchi, Tsuboi, and Miyazawa (1961) expand the change in distance between atoms a and b, Tab, from its equilibrium value i ab to first order in the cartesian displacement of the atoms as given in (2.44) ... 

When molecular dynamics is applied to the understanding of chemical intramolecular phenomena the use of chemical internal displacement coordinates as initially defined by Wilson et al. [3] are much more useful and may have a direct chemical meaning (see below). 

In the second-order methods we have described, the choice of coordinate system was not made explicit. Prom a quantum-chemical perspective, analytical derivatives are most conveniently computed in Cartesian (or symmetry-adapted Cartesian) coordinates. Indeed, second-order methods are not particularly sensitive to the choice of coordinate system and second-order implementations based on Cartesian coordinates usually perform quite well. As we discussed above, however, if the Hessian is to be estimated empirically, a representation in which the Hessian is diagonal, or close to diagonal, is highly desirable. This is certainly not true for Cartesian coordinates some set of internal coordinates that better resemble normal coordinates would be required. Two related choices are popular. The first choice is the internal coordinates suggested by Wilson, Decius and Cross [25], which comprise bond stretches, bond angle bends, motion of a bond relative to a plane defined by several atoms, and torsional (dihedral) motion of two planes, each defined by a triplet of atoms. Commonly, the molecular geometry is specified in Cartesian coordinates, and a linear transformation between Cartesian displacement coordinates and internal displacement coordinates is either supplied by the user or generated automatically. Less often, the (curvilinear) transformation from Cartesian coordinates to internals may be computed. The second choice is Z-matrix coordinates, popularized by a number of semiempirical... 

Keywords Internal coordinates Reciprocal internal displacement coordinates Non-orthogonal B-matrix Pseudoinverse Sayvetz conditions GDIIS... 

Leonard et cd. [105] reported ab initio DMSs of H2CS computed using the functional HCTH and the TZ2P basis set. The x, y and z components of the ab initio dipole moment functions were approximated by third order polynomial expansions in internal displacement coordinates for the six degrees of freedom as follows ... 

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