DIMITRIS SKOUTERIS

Department of Mathematics & Informatics

University of Perugia

The problemThe aim is to resolve the time independent

Schrödinger equation for scattering systems of the form A + BC (yes, that’s why).

A triatomic scattering system can undergo elastic, inelastic or (the interesting part) reactive processes.

Generally, coordinates which do well in describing the A + BC channel are awful for other arrangements such as AB+C.

If we want ‘equal footing’ for all three processes, a near-universal set of coordinates is needed.

Jacobi coordinatesThe typical coordinates used for describing the A+BC

system are Jacobian coordinates. These consist of:

1.The B – C distance.

2.The distance between A and the BC centre of mass.

3.The angle between the two vectors.

The advantage of the Jacobi coordinates is that the Hamiltonian is expressed very simply in terms of them.

HOWEVER…

The coordinate problemThe Jacobian coordinates are very much dependent on

the arrangement. Thus, the Jacobian coordinates for the A+BC arrangement do not do well in describing the AB+C arrangement.

For example, at large AB – C distances, the Jacobi angle for the A + BC arrangement can vary within a very narrow range. Huge angular grids are needed.

Therefore, we turn our attention to a more universal set of coordinates…

Hyperspherical coordinatesEven though they do not completely resolve the

problem, these coordinates alleviate it considerably.The system is imagined in spherical polar

coordinates. The radius ρ corresponds to the ‘extent’ of the system (in the sense that ρ=0 means all three atoms at the same point in space, whereas infinite ρ implies infinite separation of a pair of atoms).

The hyperangle θ serves to indicate a continuous passage between ‘reactants’ and ‘products’.

The Schrödinger equationThe equation has the usual familiar eigenvalue form:

H ψ = E ψ

where H is the Hamiltonian operator, ψ is the wavefunction and E is the total energy we are interested in. We know H, we choose E as a parameter and we need to calculate ψ. Actually, we do not even need the whole of ψ…

Boundary conditionsAt infinite ρ, the wavefunction has a well defined

behaviour. It is a linear combination of components, each of which corresponds to a particular state. For our (chosen) initial state, the wavefunction has the form:

ψ→ e -ikR – Sel e ikR (as ρ→∞)

consisting of an incoming wave (prepared by us) and an outgoing wave (the result of elastic scattering).

Boundary conditionsOn the other hand, for a state which is not our initial state

(and is, therefore, the result of inelastic or reactive scattering) the relevant component of the wavefunction has the form at infinity:

ψ→ – S e ikR (as ρ→∞)i.e. there is no incoming wave.

It is the coefficient S which is the ‘holy grail’ of any scattering calculation – it is called the ‘S matrix element’ and its square amplitude denotes the probability of scattering to a particular state.

Boundary conditionsAt ρ = 0, on the other hand, the wavefunction is equal to

0. Every single component of it has to vanish at the origin.

This provides us with a starting point for our propagation – our problem is an initial value problem.

The unfortunate fact is that we know the value of the wavefunction precisely on the opposite part of where we want it!

The HamiltonianThe Hamiltonian for a triatomic system, in hyperspherical

coordinates, can be written as a sum of terms:

H = Tρ + Tθ + T + TJ + V(ρ,θ,)

In order of appearance, these terms correspond to the kinetic energy along the hyperradius, the hyperangle, diatomic rotation, overall rotation and potential energy.

It is this Hamiltonian which is used to solve the Schrödinger equation.

PropagationThus, we need to propagate the wavefunction from ρ=0

up to infinity, according to the Schrödinger equation. There is only a small obstacle – the equation is second order with respect to ρ. In other words, the value of the wavefunction is not enough to start – one has to know its derivative as well! The system is, as yet, under-determined.

On the other hand, some kind of under-determinacy is to be expected – after all, there has been no mention of the initial state in which our triatomic system finds itself at the start of the scattering process.

Close-coupled representation

At each point of the propagation, the wavefunction is represented as a linear combination of basis functions:

Ψ = ∑ ci (ρ) ψi (r ; ρ)

where r represents the internal coordinates of the system. The basis functions are adiabatic eigenfunctions of the internal Hamiltonian at specific points (‘sectors’) of ρ. Thus the wavefunction is represented as a column vector.

At ρ=0 (the start of the propagation) all coefficients ci are identically zero. But we know nothing about their derivatives…

Matrix propagationTherefore we resort to a trick: instead of propagating a

single Nx1 column vector, we propagate a NxN matrix of vectors (denoted by C), whose derivative vectors are linearly independent. This way, effectively, we propagate all possible wavefunctions at once. Here N denotes the number of basis functions in our set (pre-specified by us).

We do not know which column vector is the correct one…

… but IGNORANCE IS STRENGTH !

Sector-by-sectorAt each value of ρ along a pre-specified grid, the internal

Hamiltonian is diagonalised, producing the appropriate ψi basis function for the current sector. (This is the slow step of the calculation, scaling as N3).

Our NxN C matrix undergoes a linear transformation (to adapt to the new basis set) and is then propagated along the sector. After that, it is ready for the next one (linear transformations and multiplications scale much more favourably).

The energy factorAs you may remember, the total energy (E) enters

explicitly into the Schrödinger equation. This implies that a separate propagation needs to be carried out (within each sector) for each energy value.

Thus, the CPU time needed for a calculation is proportional to the energy values needed.

The propagation goes on and on and on until we reach…

… The Asymptotic Region

Asymptotic matchingThere is no strict definition of the asymptotic region – it is

simply the region where the various arrangement channels are separated by such enormous barriers that any communication between them is precluded.

As soon as this region is reached, we can use the form that we want our wavefunction and its derivative to have asymptotically. Since we have propagated the most general form of the wavefunction, we can match it now to an asymptotic function corresponding to our desired initial state.

Asymptotic matchingMatching our propagated matrix to the form

ψ→ e ikR – S e –ikR (as ρ→∞)

we can calculate:

1.The S matrix elements (whose square amplitudes give the reaction probabilities).

2.The column vector of coefficients relating the propagated NxN matrix to the actual column vector for our initial state.

But we do not really need 2.!!! Isn’t there a way we can lose this piece of information, simplifying the calculation ?

THERE IS !

The log-derivative matrixInstead of propagating the NxN matrix C(ρ) and its

derivative, we propagate another matrix Y defined as

Y = C’ (ρ) x C-1 (ρ)

where the first matrix is the derivative w.r.t. ρ of C and the second one is its inverse matrix.

This is known as the log-derivative matrix. On asymptotic matching, we obtain only the S matrix elements. Thus, we lose a (useless) piece of information and we only need to propagate one matrix instead of two.

ResultsOnce obtained the S matrix elements, one can take their

square amplitude in order to obtain detailed state-to-state probabilities. Alternatively, one can take coherent sums of S matrix elements to derive differential cross sections.

From the reaction probabilities one can obtain:

1. Integral cross sections.

2.Cumulative reaction probabilities (summing over reactant states).

3.Rate constants (what really interests a chemist!)

AcknowledgementsMy bosses, Prof. A. Laganà and Prof. O. Gervasi

Alessandro Costantini

Nadia Balucani

Stefano Crocchianti

The University of Perugia (money!)