Initial conditions

2.4 QUASICLASSICAL TRAJECTORY CALCULATION

2.4.3 Initial conditions

In absence of any external field, any initial condition for the center of mass motion can be chosen. Here, PSi(t0) = P_S⁰

i = Si(t0) = S_i⁰ = 0 are taken. From the equations of motion for the center of mass it can be written that at any time t,PSi(t) =Si(t) = 0.

In order to simulate an average reactive attribute, a wide distribution of initial conditions must be taken into account, resulting in a large number of trajectory run.

To start a trajectory calculation, initial conditions such as q⁰_i, p⁰_i, Q⁰_i and P_i⁰ must be specified. Thus, an efficient choice of initial condition is essential.

In Figure 2.2 a schematic diagram of a bimolecular collision system is presented for simulating experimental situation. The origin of the coordinate system is at the

center of mass of BC. The distance vector from the center of mass of BC to A atom is located onyz plane and the relative velocity vector (ν) of A relative to BC is parallel to positive z-axis. Let us assume that BC diatom is in a well defined vibrational (v) and rotational (j) state. The orientation of BC diatom is given by spherical angles θ and φ. η represents the angle of rotation of internal angular momentum vector (J_r) of BC diatom. Here, ρis the distance from the center of mass of BC to A, and bis the impact parameter. Forj= 0, bis related to the total angular momentum as b=

√

J(J+1)~² µA,BC ν .

B Jr

z

C

x A

y

v θ

η

Ф

ρ b

Figure 2.2: Initial arrangement of A+BC collision system in Cartesian coordinates.

The initial values of positions and momenta for the relative motion between A and BC are now written as

• Q⁰₁= 0,

• Q⁰₂= b,

• Q⁰₃= −p

ρ²−b²,

• P₁⁰= 0,

• P₂⁰= 0,

• P₃⁰= µ_A,BC ν =p

2µ_A,BCEc .

At the beginning of the trajectory run, A and BC are kept well separated to minimize the interaction between A and BC.

The ro-vibrational energies of the reactant diatom are obtained from quantum mechanical calculations by employing the Colbert-Miller DVR method.⁴⁰ The turn- ing points of the diatom for a particular ro-vibrational state are calculated by equating the sum of potential (V_BC(q±)) and rotational energy (~²j(j+ 1)

2µBCq²_± ) to the ro-vibrational energy obtained from DVR calculation. Initial value for the diatomic distance can vary from q₊ to q− and the initial diatomic momentum can vary depending on the initial value of diatomic separation and the phase of the vibration. To start from all the pos- sible phases of the vibrating diatom random number ξ (0 ≤ ξ ≤ 1) are invoked for different trajectories. Before starting the integration of equations of motion for three body collision, the equations of motion for a rotating diatom (which is vibrating also) is solved for a time τ ξ orτ(ξ−0.5) for ξ < 0.5 or ξ ≥ 0.5 respectively. Here τ is the time period of vibrational motion, which is calculated by integrating the equations of motion for a rotating diatom starting from a turning point to another turning point. τ and q± are parameters which vary for different initial states and calculated once for a initial state specific calculation.

In order to randomize the initial condition for each trajectory, standard Monte Carlo sampling is considered. The initial values for θ, φ, η and ξ are determined from pseudo random numbers invoked by a computer program, which has a value in between 0 and 1. Hence, a new set of random variable θ⁰, φ⁰, η⁰ and ξ⁰ are to be declared, which are related to θ, φ, η andξ as

θ = cos⁻¹(1−2θ⁰), (2.72)

φ = 2πφ⁰, (2.73)

η = 2πη⁰, (2.74)

ξ = ξ⁰. (2.75)

As mentioned earlier, for a particular trajectory, at first the equations of motion for rotating oscillator are solved with a BC separation of q− if ξ < 0.5 and from q+

otherwise. In this case, the initial parameters are taken as

• q₁(0) =q±sinθ cosφ,

• q2(0) =q±sinθ sinφ,

• q3(0) =q±cosθ,

• p₁(0) = _q^Jr

±(sinφcosη−cosθcosφ sinη),

• p2(0) =−_q^Jr

±(cosφcosη+ cosθ sinφsinη),

• p3(0) = _q^Jr

±sinθsinη, with J_r = ~p

j(j+ 1). After integrating for due time span, the resulting values of q_i and p_i are taken as the initial value for solving the equations 2.66, 2.67, 2.69 and 2.70.

2.4.3.1 Sampling of initial conditions for j >0

The formalism proposed by Aoiz et al.⁶¹ is followed here to calculate trajectories for rotationally excited reactant states. For j >0, l takes the value in between (J +j) to

|J−j|for a particular set ofJ and j. The classical relationship between the moduli of J,land jcan be expressed by triangle rule

|J|² =|l|²+|j|²+ 2|l||j|cosθ_lj, (2.76)

where θ_lj is the angle between the vectors l and j. The space fixed axes (X, Y, Z) are chosen as reference frame, where the initial relative velocity vector is parallel toZ-axis.

l vector can be placed along X-axis without any loss of generality. Thus, l_X =|l| and lY =lZ= 0. In this reference frame, the Cartesian components of jcan be written as

j_X = J_X − |l|=|j|cosθ_lj, (2.77) jY = JY =|j|sinθ_lj sinα, (2.78) j_Z = J_Z=|j|sinθ_lj cosα. (2.79)

Here,αis the azimuthal angle ofjin theY Zplane and is chosen randomly as 0≤α≤2π.

The components ofjcan also be expressed in SF frame in terms of usual polar (θj) and azimuthal angle (φj) of jas

j_X = |j|sinθ_j cosφ_j, (2.80) j_Y = |j|sinθ_j sinφ_j, (2.81) jZ = |j|cosθ_j. (2.82)

θ_j and φ_j are determined from these set of equations. For the special case, when J = 0 and l and jare anti-parallel to each other, and cosθ_lj =−1 then if l lies on X-axis, θ_j =π/2 and φ_j =π.

Now the initial value of position and momentum coordinates for the reactant diatom to start the integration of equation of motion for rotating oscillator can be expressed as

• q1(0) =q±(cosθj cosφj cosηj−sinφj sinηj),

• q2(0) =q±(cosθj sinφj cosηj−cosφj sinηj),

• q₃(0) =−q_±sinθ_j cosη_j,

• p₁(0) = _q^jr

±(−cosθ_j cosφ_j sinη_j−sinφ_j cosη_j),

• p2(0) = _q^jr

±(−cosθ_j sinφj sinηj + cosφj cosηj),

• p3(0) = _q^j_±^rsinθj sinηj.