Schr ¨odinger equation in Jacobi coordinates

center of mass of the fragment, and its orientation with respect to the XYZsystem is defined using the three Euler anglesΩ= (γ₁,γ2,γ3).⁷⁰

• 3(N−¹)−⁶ internal coordinates Q = (Q₁, ...,Q_3N−9)are defined to describe the molecular fragment geometry.^§

To summarize, the nuclear configurations are completely described by the coordinate system {^R,Θ,Φ,Ω,Q}. The potential operatorsW_αβ are functions of the internal degrees of freedom only, and do not depend on how the molecule is oriented in space. Therefore, the potential is most conveniently constructed in a set of coordinates which includes the body-fixed polar angles(θ,φ), defined with respect to thexyzaxes, instead of the space-fixed angles(Θ,Φ).

3.2 Schr ¨odinger equation in Jacobi coordinates

Using the Jacobi system of coordinates, the nuclear kinetic energy operator can be defined as⁷² Tnuc =− ^¯h

whereµ_R is the reduced mass of the atom-fragment pair, Lis the angular momentum operator describing the orbital motion of the detached atom, andTQis the kinetic energy operator associated with the internal fragment motion.Trotis the kinetic energy operator for the rotational motion of the molecular fragment, which is conveniently expressed in terms of the body-fixed components of the fragment angular momentum operatorP = Px,Py,Pz

and the inertia tensor,^†

Trot= ¹

§For a diatomic fragment only the inter-atomic distance is necessary.

†The operatorsPiare defined in terms of the Jacobi angles as Px = i¯h

The elements of the inertia tensor I_jk(Q)depend on the internal coordinates Q. A simplification of Equation (3.2), useful in numerical applications, is obtained by assuming that the molecular fragment undergoes small distortions from its asymptotic equilibrium geometry,Q≈^Q0, so that the inertia tensor is approximated by a constant diagonal matrix,

I_jk(Q)≈ ^Ijk(Q₀) = I_jj(Q₀)δ_jk Trot ≈ ¹

2 Px²

I_x + Py²

I_y +Pz²

I_z

!

. (3.3)

Neglecting fragment vibrations in the kinetic energy operator leads to the neglect of the Coriolis coupling between vibrations and rotations. This is the approximation adopted in the subsequent derivations. The diabatic potentials and couplingsW_αβ can be expressed as functions of the body-fixed Jacobi coordinates{^R,θ,φ,Q}. The Hamiltonian operator depends, in addition, on the three Euler anglesΩ= (γ₁,γ₂,γ₃).

In order to the define the form of the nuclear wavefunction, one needs to consider the total angular momentum operatorJ =L+P which is associated to the rotation of the whole molecule as a rigid body. The triad {H^,J^2,JZ}^{, where} JZ is the projection of J on the space-fixed Z axis, is a set of commuting operators. As a consequence, the nuclear wavefunctionΨ^{J M}can be labelled with the quantum numbers J and Mwhich identify the eigenvalues ofJ^{2 and}JZ and are conserved during the dynamics driven by the HamiltonianH^.

The wavefunctionΨ^{J M} is typically represented via an expansion in a basis of angular functions Y^{J M}which are simultaneous eigenstates ofJ ^andJZ,⁷³

Ψ^{J M}(Q,R,Ω,Θ,Φ) =

∑

jlk

ψ^J_jlk(Q,R)Y_jlk^{J M}(Ω,Θ,Φ). (3.4) The angular functions are linear combinations of products of Wigner rotation matrices (for the anglesΩ= (γ₁,γ₂,γ₃)) and spherical harmonics (for the anglesΘandΦ),

Y_jlk^{J M}(Ω,Θ,Φ) =

∑

j mj=−^j

∑

l ml=−^l

h^jmjlm_l|^{J M}i

r2j+1 8π² D_m^j

jk(Ω)^∗Y_lml(Θ,Φ), (3.5)

where the quantum numbersj,landkare associated with the operatorsP^2,L^2andPz, respectively, andm_j andm_l are associated withPZ andLZ(i. e. the components of P ^andLalong the space-fixed Z axis). The basis set

q2j+1 8π² D^j_m

jk(Ω)^∗Y_lml(Θ,Φ)

consists in simultaneous eigenstates of the operators {P^2,PZ,L^2,LZ,Pz}, whereas the functions n

Y_jlk^{J M}(Ω,Θ,Φ)^o are simultaneous eigenstates of{J^2,JZ,P^2,L^2,Pz}. The orthogonal transformation between the two basis sets is defined by the Clebsch-Gordan coefficientsh^jmjlm_l|^{J M}i^.

3.2 s c h ro d i n g e r¨ e q uat i o n i n ja c o b i c o o r d i nat e s 23

The functions (3.5) form an orthonormal basis, Z _2π

0 dγ₁ Z _π

0 sinγ₂dγ₂ Z _2π

0 dγ₃ Z _π

0 sinΘdΘ^{Z 2π}

0 dΦY_j0^{J M}l⁰k⁰(Ω,Θ,Φ)^∗Y_jlk^{J M}(Ω,Θ,Φ) =δ_jj0δ_ll0δ_kk0. (3.6)

The matrix elements of the rotational kinetic energy operator, _2µ^L2

RR² +Trot, can be calculated analytically in the basisn

Y_jlk^{J M}(_Ω,Θ,Φ)^o.^‡

The nuclear wavefunctions associated with a diabatic electronic state can be written using space-fixed or body-fixed coordinates. The space-fixed form, in which the position of the detaching atom is expressed via the anglesΘandΦ, is useful for the calculation of observables related to the direction of polarization of the electric field, conventionally fixed along the Zaxis (an example is the anisotropy parameter^68,69). The use of the body-fixed frame, in which the angles (θ,φ)are used, is generally more appropriate in the solution of the Schr ¨odinger equation itself, because this is the coordinate system in which the potential is set up. The matrix elements of the potential functions are more conveniently evaluated in the body-fixed frame.

The spherical harmonics functions are transformed from the XYZ- to the xyz-system using Wigner rotation matrices,

Y_lml(Θ,Φ) =

∑

l m⁰_l=−^l

D_m^l

lm⁰_l(Ω)^∗Y_lm0

l(θ,φ). (3.7)

‡This is achieved by considering that the spherical harmonicsY_lml are eigenfunctions ofL²,

L^2Ylml =¯hl(l+1)Y_lml .

The matrix elements ofTrotcan be evaluated using the following formulas for the action ofPx,PyandPzon the rotation matrices:

PxD_mk^j (Ω)^∗ = ^h¯ 2

q

j(j+1)−^k(k+1)D^j_m,k+1(Ω)^∗+^h¯ 2

q

j(j+1)−^k(k−¹)D_m,k^j ₋₁(Ω)^∗ PyD_mk^j (Ω)^∗ = ^i¯h

2 q

j(j+1)−^k(k+1)D^j_m,k+1(Ω)^∗−^i¯h 2

q

j(j+1)−^k(k−1)D^j_m,k−1(Ω)^∗ PzD_mk^j (Ω)^∗ = ¯hkD_mk^j (Ω)^∗.

Inserting Equation (3.7) into Equation (3.5), the angular functions Y^{J M} can be expressed in the

According to Equation (3.8), the angular functionsY_jlk^{J M}are expanded in terms of products of (i) a spherical harmonicY_lm0

l(θ,φ), depending on the position of the detaching atomin the body-fixed frame, and (ii) a rotation matrix D^J_M,k+m0

l(Ω), describing the orientation of the body-fixed frame with respect to the space-fixed frame. The quantum numberm⁰_l is associated with the operatorLz

which is the component ofLalong the body-fixedzaxis.

While the rotational kinetic energy operatorTrotis easily expressed in the{Y_jlk^{J M}}basis, it is more convenient to evaluate the(θ,φ)-dependent potential functions in the body-fixed{^DJ_M,k+m0

lY_lm0

l}

basis. Obviously, in simulations the same basis has to be used for all the operators. The use of the latter basis is generally more advisable, because the potentials might not have an analytical expression and need to be evaluated by numerical quadrature over the(θ,φ)spherical angles. The

§The derivation of Equation (3.8) makes use of some identities which arise from the theory of the coupling of angular momenta. First, the Clebsch-Gordan coefficients are expressed using the Wigner 3j-symbols,

h^aαbβ|^c−γi= (−¹)^a−b−γ√

Then the following sum rule is used,

∑

These and other formulas, useful for the formal manipulation of equations involving angular momentum coupling, can be found on textbooks specialized on the quantum mechanical theory of angular momentum, as for example Ref.74.

Im Dokument Photodissociation dynamics and spectroscopy in the presence of conical intersections  (Seite 31-35)