Body-frame

Neuhauser D, Baer M, Judson R S and Kouri D J 1989 Time-dependent three-dimensional body frame quantal wavepacket treatment of the atomic hydrogen + molecular hydrogen exchange reaction on the Liu-Siegbahn-Truhlar-Horowitz (LSTH) surfaced. Chem. Phys. 90 5882... 

The quaternions obey coupled differential equations involving the angular velocities tJi, tangular velocity about the first axis of inertia, etc.). These differential equations take the form... 

Figure 1. Body-frame expansion coefficients (in hartrees) of eq. (3) for rigid-rotator HF-HF collisions as functions of the collisional separation coordinate r (in bohrs) for = fx...

BMI (and percentile of BMI) is only one factor in determining a person s weight-related health risk. Having a BMI in the healthy range does not necessarily mean that a person is fit and healthy. BMI does not take into account lean body mass or body frame. A muscular, large-framed person s BMI could indicate that he or she is obese, but other factors would show that this is not the case. 

Launching launch frame has rail to accept tee shaped launch lug on missile body, frame is adjustable for height and has top mounted handle for carrying. 

Figure 2 depicts a somewhat different case. Excitation of an atomic chromophore results in short-lived molecular states that have specific electronic character, i.e., selectively oriented p-orbitals in the body frame of the complex [21]. The system then reacts, and the imparted orbital specificity can be carried through to products, being manifest as chemical selectivity and/or biased product E, V, R, Texcitations [22]. 

The propagation of the solutions to the CC equations are carried out entirely in the fully-uncoupled, body-frame basis of eq. 2. At the end of the propagation, but before extraction of the S matrix, we transform the log-derivative matrix into the partially coupled basis discussed above. Here, the atomic states in the reactant arrangement are labeled by the total electronic angular momentum of the atom, ja, and... 

Keuhauser. D.. Baer, M., Judson. R.S. and Kouri. D.J. (1989) Time-dependent three-dimensional body frame tjnantal wave packet treatment of the H + H2 exchange reaction on the Liu-Si< gbahn-Truhlar-Horovi itz (LSTH) surface,. 7. Chem. Phys. 90, 5882-5884. 

In this contribution, we will describe the essential features of the hyperspherical body-frame method (section 2) and of its numerical implementation (section 3). We will then present in section 4 selected recent results obtained for the three insertion reactions 0( D) + Ho, N( D) + H2 and C( D) + Ho. Our tiine-independent quariturn-dynamical results will be compared with experiment. 

Mass-scaled Jacobi coordinates associated to a generic arrangement X — a for A -I- BC, /I for B + CA and ) for C + AB) cU c denoted by r (diatom vector) and R (atom-molecule vector). They are used in the definition of hyperspherical coordinates which parametrize the nuclear motion of the system, namely the principal axis body frame hyperspherical coordinates [3, 4, 5]. These coordinates are ... 

The hainiltonian of the ABC system in body-frame hyperspherical democratic coordinates is written as [4] ... 

The expression (10) is not suitable to determine collisional wavcfunctions because their asymptotic part is determined in Jacobi space-fixed coordinates specific for each arrangement A. The transformation from body-frame hypcrsphcrical democratic coordinates (p,6,(j),S) to space-fixed (Rx,rx) Jacobi coordinates is performed in several steps. It necessitates the introduction of Fock internal hvjier-spherical coordinates p,oJx., ii ) to analyze the rovibrational character of the surface functions in the fragmentation region at huge p. 

For a given arrangement A, the two Fock angles cox.rjx) are the hyperradial correlation angle lux — tim rx/Bx) and the bending angle r]x — cos (lix.rx). The expression of the hamiltonian Ti in Fock Body-Frame coordinates is... 

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... 

This expression of K is gauge-invariant, and both of the first and the second terms on the right-hand side of Eq. (10) are independent of the choice of body frame. The former vanishes if and only if the total angular momentum L is zero [cf. Eq. (5)]. The metric tensor g in Eq. (11) is also gauge-invariant and is the true metric appropriate for the description of internal motions in polyatomic molecules. 

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)]... 

The gauge-invariant metric tensor of internal space and classical equations of internal motion in terms of the PAHC are given explicitly as follows, from which the general properties of a metric force associated with this coordinate system will be deduced. By applying Eq. (20) to Eq. (6), we obtain for the moment of inertia tensor referred to the body frame as... 

We define the order of the singular values as a > a2 > 31. The planar and collinear configurations give a3 0 and a2 a3 = 0, respectively. Furthermore, we let the sign of a3 specify the permutational isomers of the cluster [14]. That is, if (det Ws) = psl (ps2 x ps3) > 0, which is the case for isomer (A) in Fig. 12, fl3 >0. Otherwise, a3 < 0. Eigenvectors ea(a = 1,2,3) coincide with the principal axes of instantaneous moment of inertia tensor of the four-body system. We thereby refer to the principal-axis frame as a body frame. On the other hand, the triplet of axes (u1,u2,u3) or an SO(3) matrix U constitutes an internal frame. Rotation of the internal frame in a three-dimensional space, which is the democratic rotation in the four-body system, is parameterized by three... 

Each choice of the body-axes specifies a reference orientation, in which the body frame coincides with the space-fixed frame. Let us choose one of the body axes, say u 3, to be in the direction of... 

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