DISEÑO CURRICULAR DE LA ERE

This section considers the case where the total energy, E, is higher than the dissociation energy of at least one of the coupled electronic-rotational states, that is, at least one chan- nel is open. Then, as long as V(R) cannot be blocked into any independent submatrices containing only closed-channels, all solutions vectors to Eq. (2.136), given by the rows of

χ(R), will contain an open-channel admixture. In this case, no bound radial-wavefunctions will be supported. Instead, the open channels provide a path to molecular dissociation so that the resultant radial-wavefunctions will extend to infinity and need not satisfy the previous outer boundary condition, limR→∞χ(R) = 0. A new asymptotic boundary con-

dition must then be imposed to correspond to the case of separated atoms. This scenario is reminiscent of a half-collision, and the formalism of scattering theory has been used by Mies (1980a, 1980b) [117, 118] to derive the correct asymptotic wavefunctions. The inner boundary condition, χ(R) = 0, is unaltered.

The two atomic products may be in the same or different, possibly excited, states and can be fully represented by the quantum numbers,

|iEJΠ_Ri.

Here, E is the combined internal-electronic and kinetic energies of the atoms; J is the total angular-momentum; Π is the total parity; R corresponds to the nuclear angular- momentum quantum number of Eq. (2.23) except at large R when the nuclei can no longer be considered a bound molecule, in this case_Rrelates directly to the partial-waves

§2.11 Solving the coupled Schr¨odinger equation 43

of scattering theory and is rigorously quantised; and i represents further degeneracies corresponding to all possible sublevels of orbital- and spin angular momenta internal to each atom, as well as the relative polarisation of these momenta between atoms. The variousicorrespond to different open-channels, and each results in a separate degenerate solution, enumerated by i= 1 to NO.

Physical radial-wavefunctions exist for any total energy because the kinetic energy is not quantised, and solutions to Eq. (2.136) will be continuous with respect to E. The above set of quantum numbers corresponds to Hund’s case (e) in molecular spectroscopy terms, which is the only Hund’s case with quantum numbers that are all rigorously defined for separated atoms.

For the cases studied in this thesis, dissociation is a two-step process, photoabsorp- tion followed by predissociation mediated by one or more open channels. Theoretical calculations of the absorption stage do not depend on the asymptotic wavefunction, but any experimentally observed dissociation products will require a theoretical description in terms of pure atomic states. This requirement restricts the form of the asymptotic interaction matrix in two ways: all off-diagonal elements must be zero, reflecting the non interaction of the remote atomic products; and the NO diagonal elements corresponding to the dissociation energies of the open-channel potential-energy curves must equal the combined electronic-energies of a pair of pure atomic states, plus the nuclear-rotational kinetic-energy defined about the nuclear centre-of-mass. Then, neglecting contributions from terms of order R−3 and below, the asymptotic interaction matrix will have elements

lim R→∞Vij(R) = ( limR→∞Ejel(R) + ~2_R j(Rj+1) 2µR2 i=j, 0 i₆=j. (2.147)

In this thesis, diabatic potential-energy curves and state interactions have been chosen which immediately conform to Eq. (2.147), otherwise, a diagonalisation of limR→∞V(R)

would be necessary.

The asymptotic wavefunctions approach the solutions ofNT×NT uncoupled Schr¨odinger equations. After substituting Eq. (2.147) into Eq. (2.132) these asymptotic equations are of the form −~2 2µ d2 dR2χij(R) +χij(R) ~2_{Rj(Rj + 1)} 2µR2 −k 2 j = 0, (2.148)

where the asymptotic wavenumber, kj, is real for open channels and imaginary for closed channels and is given by,

k_j2 = 2µ ~2 E₋ lim R→∞E el j (R) . (2.149)

The general solution to Eq. (2.148) is of the form, lim

R→∞χij(R) =Jj(R)Aij+Nj(R)Bij, (2.150)

where Jj(R) and Nj(R) are the well-known Bessel and Neumann functions [118] corre- sponding to spherical standing waves; the amplitudesAij andBij are dictated by the inner boundary condition and the structure ofV(R) at smallR. The inner boundary condition

χ(0) = 0 is necessary but the choice of _dRd χ(0) is arbitrary and each possible choice will result in a distinct set of Aij and Bij.

the asymptotic from of Eq. (2.150) is given by lim R→∞χij(R) =k −1₂ j sin(kjR−πR/2)Aij+k −1₂ j cos(kjR−πR/2)Bij. (2.151) For convenience, the set of asymptotic solutions in Eq. (2.150) may be conveniently written in matrix form,

lim

R→∞χ(R) =J(R)A+N(R)B, (2.152)

whereJ(R) andN(R) are diagonal with nonzero elements (J(R))_jj =Jj(R), and similarly forN(R). Any linear recombination of the rows of χij(R) will still solve the Eq. (2.132), and there is freedom to rearrange the calculated solutions with any unitary transformation, i.e., χnew(R) =χ(R)U.

For all closed channels,kj is imaginary and Eq. (2.151) will contain only exponentially convergent and divergent terms. The latter are physically unacceptable so the following outer boundary condition must be wilfully imposed,

lim

R→∞χij(R) = 0, if channelj is closed. (2.153)

Then, NC rows of A and B corresponding to closed channels will contain only zero ele- ments and limR→∞χ(R) will only spanNOlinearly-independent columns. It is then pos- sible to further treat the asymptotic solutions given by Eq. (2.152) following the reduction of matrices to dimension NO×NO by temporarily discardingNC rows and columns, and this is done hereafter. The rows corresponding to closed channels cannot be permanently discarded because they will contain nonzero elements inside their classical turning-points at smallR. The influence of these is not directly expressed in the asymptotic solution but is still implied due to their interaction at small R.

The reduction of the asymptotic solutions of Eq. (2.148) to matrices A and B will prove to be convenient in Sec. 2.11.5, but the outer-boundary condition for open channels has yet to be specified. This problem is discussed in Mies [118] and Schinke [137, pp. 42, 69] and briefly covered here. The desired functional form for each of the asymptotic solutions is,

lim R→∞χ

S

ij(R) =h+j (R)δij+h−j(R)Sij, (2.154) where h−_j (R) andh+_j (R) are complex-valued linear-combinations of Bessel and Neumann functions,

h+_j (R) = 2−12N_j(R) +iJ_j(R), (2.155)

and h−_j (R) = 2−12N_j(R)−iJ_j(R). (2.156)

Equation (2.154) will then have following the asymptotic form, after substitution of Eq. (2.151), lim R→∞χ S ij(R) = (2ki)− 1 2e+i(kiR−πR/2)δ_ij + (2k_j)− 1 2e−i(kjR−πR/2)S_ij. (2.157)

§2.11 Solving the coupled Schr¨odinger equation 45 is given by, lim R→∞ψi(r, R) =Ni   1 R(2ki) −1_2e+i(kiR−πR/2)_{φi(r) +} NO X j=1 1 R(2kj) −1_2e−i(kjR−πR/2)_φj(r)S ij  . (2.158) Equation (2.158) represents a physically-reasonable asymptotic solution and constitutes the necessary outer boundary condition. The first term is in the form of an outgoing spherical wave, with wavenumber ki, and represents a pair of nuclei dissociating via the

ith open channel. This form is then directly comparable with dissociation experiments, which observe atomic fragments in pure states. The remaining terms all have the form of incoming spherical waves and contain nonzero elements withi₆=jbecause of the channel mixing that occurs at small R. The radial argument of the electronic wavefunctions,

φi(r), has been dropped because any R-dependence at long range is purely the result of translation of the separated atoms, and it is simpler to consider the internal coordinates

r of the electrons to be relative to their respective nuclear-centres. The normalisation constant, _Ni, has yet to be determined.

Molecular dissociation is an irreversible process, but it is being treated here by the time independent Schr¨odinger equation and therefore the calculated wavefunctions correspond to a steady-state. Thus, the presence of incoming waves in Eq. (2.158) is necessary to describe an inwards flux of particles that precisely balances the outwards flux represented by the single outwardly-propagating wave. As further illustration, a rearrangement of the dissociation wavefunctions of Eq. (2.158) leads to the similar form,

lim R→∞ψi(r, R) =Ni   NO X j=1 (2kj)− 1 2 R e +i(kjR−πR/2)_S ijφj(r) + (2ki)− 1 2 R e −i(kiR−πR/2)_φ i(r)  , (2.159) which now describes incoming atoms in a pure state, and mixed-state outgoing atoms. This new formulation could be applied to studies of the elastic scattering of atoms.

It still remains to reshape the asymptotic radial solutions into the assumed form of Eq. (2.154). This may be done by adopting a matrix formulation in similar fashion to Eq. (2.152),

lim R→∞χ

S_{(R) =h}+_{(R) +h}−_{(R)S, (2.160)}

then finding a unitary transformation matrix, U, such that

χS(R) =χ(R)U, (2.161)

which, by substitution of Eq. (2.150) and Eq. (2.154) requires

h+(R) +h−(R)S =

J(R)A+N(R)BU. (2.162) Whereupon, after writing J and N in terms ofh− andh+, according to Eq. (2.155) and Eq. (2.156),

h−(R) +h+(R)S = 2−12 h+(R)(−iA+B) +h−(R)(iA+B)U. (2.163)

mation,

U = 212(−iA+B)−1, (2.164)

as well as the S matrix,

S = (iA+B)(−iA+B)−1. (2.165)

The total wavefunction of Eq. (2.127) must be spatially-normalised to unity. A nor- malisation procedure is described by Mies [118] in terms of the asymptotic wavefunction only. Then the correct value of the normalisation constant in Eq. (2.158) is

Ni =

~2_π

2µ . (2.166)