Jacobi vectors

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,... 

In Eqs. (5) and (6), M is the total mass of the nuclei and is the mass of one electron. By using Eq. (2), the system s internal kinetic energy operator is given in terms of the mass-scaled Jacobi vectors by... 

Figure 1. Jacobi vectors for a three-nuclei, four-electron system. The nuclei are Pi, P2, P3, and the electrons are ei, 02, 63, 64,...

Consider a triatomic system with the three nuclei labeled A, Ap, and Ay. Let the arrangement channel -1- A A be called the X arrangement channel, where Xvk is a cyclic permutation of apy. Let Rx,r be the Jacobi vectors associated with this arrangement channel, where r is the vector from A to and the vector from the center of mass of AyA to A . Let R i, rx be the corresponding mass-scaled Jacobi coordinates defined by... 

We represent the four-atom problem in terms of diatom-diatom Jacobi coordinates R, the vector between the AB and CD centers of mass, and rj and r2, the AB and CD bond vectors. In a body-fixed coordinate system [19,20] with the z-axis chosen to R, only six coordinate variables need be considered, which we choose to be / , ri, and ra, the magnitudes of the Jacobi vectors, and the angles 01, 02, and (j). Here 0, denotes the usual polar angle of r, relative to the z-axis, and 4> is the difference between the azimuthal angles for ri and r2 (i.e., a torsion angle). 

Vector Parameterization of the N-Atom Problem in Quantum Mechanics. I. Jacobi Vectors. 

The hyperradial coinponciits are then transfonned to a new rcpiesentation with an internal cinantization axis along the Jacobi vector R. This transformation etm be written as a matrix... 

The vectors psi are related to the mass-weighted Jacobi vectors referred to a body frame R, p,, by... 

The kinetic energy K = Ym=i ps 2/2 in the Jacobi vectors can also be expressed in terms of the quantities referred to the body frame using the time derivative of Eq. (2) as... 

In PAHC the leftmost matrix R on the right-hand side of Eq. (15) identifies a body frame (the principal-axis frame). The Jacobi vectors referred to this body frame, p, and p2, are expressed as [cf. Eq. (2)]... 

Let the n 1 three-dimensional vectors zSi (i = 1,1) be the mass-weighted Jacobi vectors for a reference molecular configuration. The reference configuration is usually set to be a local equilibrium structure of the molecule oriented to a certain orientation. The Eckart subspace is defined as a (3n — 6)-dimensional subspace in the (3n — 3)-dimensional translation-reduced configuration space, which is parameterized by Jacobi vectors pf (/ = 1,..., m - 1) with three additional constraint conditions called the Eckart conditions,... 

Similarly to the three-atom system, we resume with the mass-weighted Jacobi vectors defined as... 

This definition of the Jacobi vectors implies that reaction channels are specified by multiples of ti/4 in the hyper-angles as will be shown later. 

According to the singular-value decomposition, the 3x3 matrix Ws, whose columns are the three mass-weighted Jacobi vectors, is decomposed as... 

In the three-body problem we can write down two Jacobi vectors, one (xq,) is the interparticle distance between two particles and the other (Xa) connects their center-of-mass to the third particle. So, the choice of the Jacobi vectors is not unique [1]. Here we will consider Xq, as the vector from the particle B to the particle C, and X as the vector from the particle A to the center-of-mass of the BC couple (see Figure 2). 

When the denominator of (6) vanishes, cos is zero. The interparticle distances from the Jacobi vectors can be calculated by ... 

Under proper mass-scaling the three possible Jacobi vectors sets can be related to each other by a planar rotation by an angle which depends only on the masses of A,... 

B, and C particles and is an extension of the so-called skewing angle concept [3]. For each set of the two Jacobi vectors the mass-scaling can be written... 

Besides this coordinate sets, other sets of orthogonal vectors have been considered in the literature. Kinematic Rotations by mass-dependent matrices allows to relate different particle couplings in the Jacobi scheme, and to build up alternative systems such as those based on the Radau-Smith vectors and hyperspherical coordinates [1,3]. The Radau-Smith vectors RSi, RS2 and the angle cos Urs (0 < )rs < n), showed in Figure 3 (the D point is defined by OD = OE x OA, where O is the center-of-mass of the BC couple, E is the center-of-mass of the three particles and A is the position of the A particle), can be calculated from the Jacobi vectors xa and X using ... 

The Jacobi vectors from the Radau-Smith vectors can be calculated by. 

Let s define three coordinates in terms of mass-scaled Jacobi vectors ... 

Therefore the asymmetric parametrization [2] can be expressed in terms of two coordinates referred to an internal system the angle da is the same encountered before as that formed by the two Jacobi vectors and the angle Xa is related to their ratio ... 

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