A. Quantum mechanical calculations
The linear absorption spectrum for the transition 11A2(πσ∗) ← X˜1A1(ππ) is calculated quantum mechanically. The calculations are performed using the molecular Hamiltonian of Eq. (1). Up to 12 vibrational modes are simultaneously included in the dynamics, in addition to the disappearing coordinates; the remaning degrees of freedom are frozen.
The initial state of the parent molecule, Ψ0(R,Q), is the ground vibrational state of the Hamiltonian ˆT +VX with energy E0; this Hamiltonian refers to the lowest locally diabatic electronic state. The wave function Ψ0(R,Q) is strongly localized near R = RFC ≈ 4.1a0, and the off-diagonal diabatic coupling matrix elements Vαβ can be safely neglected. The molecular state immediately after photoexcitation is given by
Φ(0) = µA2(R,Q)·ˆ
Ψ0(R,Q), (14)
where ˆ is the polarization vector of the electric field of the incident laser. At time t = 0, only the state A2 is populated.
The absorption spectra are calculated using the MCTDH code.58,59 First, the autocorre-lation function,
S(t) = hΦ(0)|exp
−iHtˆ
|Φ(0)i, (15)
24 is evaluated via a propagation on a discrete time grid. Next, the spectrum is calculated using the Fourier transform of S(t):
σ(Eph) = Eph
20c Z ∞
−∞
S(t)eiEphtdt . (16) The photon energy is measured relative to the energy E0 of the state Ψ0(R,Q). Averaging over the orientations of the electric field gives the total absorption spectrum
σtot(ω) = 1 3
X
=x,y,z
σ(Eph) (17)
The parameters of the quantum mechanical calculations discussed below are summarized in Table V. The active vibrational coordinates, the vertical excitation energy, the initial state and the intensity at the absorption maximum are indicated. Table VI reports the parameters of the MCTDH calculation.
B. Absorption spectrum as a convolution
The quantum mechanical calculations described in Sect. III A can be considerably sim-plified because the dissociation dynamics in the πσ∗ states is mainly direct, and the initial stages of the time evolution in the excited state reveal the shape of the absorption spec-trum. The N—H stretching frequency in the ground electronic state is large, ∼3915 cm−1, and the wave function Ψ0(R,Q) of the parent molecule is localized within a narrow inter-val ∆R ≈ 0.13 a0 around RFC. This has two consequences. First, Ψ0(R,Q) is accurately approximated by a product of an R- and aQ-dependent factor,
Ψ0(R,Q)≈ΨR(R)ΨQ(Q). (18)
Indeed, the Hessian matrix near RFC is approximately block diagonal, and the coupling between three coordinates of the dissociating H-atom on the one hand and the coordinates of the pyrrolyl unit on the other hand is vanishingly small. The photoexcited state Φ(0) with the TDM from the Herzberg-Teller expansion of Eq. (12) has the same product form,
Φ(0)≈FR(R)fQ(Q), (19)
(if µAQ2 vanishes) or is a sum of several such product terms (if µAR2 and µAQ2 are nonzero).
Second, the diabatic potential matrix WQ(Q|R) in Eq. (3) and, in particular, the functions
Included
normal modesTv [eV] Initial
state
TABLE V: Summary of the quantum mechanical calculations for the photodissociation of pyrrole.
Different models are denoted by the total number coordinates. The coordinates of the departing H atom (R, θ, φ) are part of all the calculations. For each case, the Table shows the list of included normal modes, the vertical excitation energy Tv of the resulting PES, the initial states Φs for the nonzero TDM components (s=x, y, z), and the maximum intensity of the calculated spectrum. In all the calculations, Ψ0 is the ground state of the N-dimensional Hamiltonian, withN = 6,11,15.
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καi(R) andγijα(R) in Eq. (4), can be fixed to their values at a distanceR ≈RFC chosen near the equilibrium of the state ˜X. The dynamics of the initial wave packet in the FC zone is therefore governed by the Hamiltonian
Hˆ0 ≈HˆR+ ˆHQ(Q|RFC), (20) represented as a sum of the commuting operators ˆHR = ˆTR+WR(R) and ˆHQ(Q|RFC) = TˆQ +WQ(Q|RFC). As a result, the vibrational motion of the ring is decoupled from the dissociative dynamics along R. In the locally diabatic representation, this separable repre-sentation is valid for any number of electronic states included in the dynamics: In the FC zone, the off-diagonal matrix elements of W(R,Q) vanish by construction.
26
Particle DVR type Ni, Nj, Nk nX, nA2 6D
R sine 98 5
(θ, φ) 2D Legendre 71, 21 5
Qb1(1,2,3) HO, HO, HO 17 4
11D
R sine 98 9
(θ, φ) 2D Legendre 71, 21 7
Qa1(1,2) HO, HO 37, 29 7
Qa1(3,4) HO, HO 21, 21 5
Qa1(5,6) HO, HO 25, 21 4
Qa1(7,8) HO, HO 21, 21 2
15D
R, Qa1(1) sine, HO 65, 37 23, 9
(θ, φ) 2D Legendre 61, 19 19, 6
Qa1(2,5) HO, HO 29, 25 16, 5
Qa2(1,2,3) HO, HO, HO 17, 17, 17 7, 4 Qb1(1,2,3) HO, HO, HO 17, 17, 17 4, 3 Qb2(1,3,5) HO, HO, HO 17, 17, 17 5, 3
TABLE VI: Computational details of the MCTDH calculations. The DVR type HO stands for the harmonic oscillator DVR.Ni, Nj, Nk are the number of primitive DVR functions used for each particle. nX and nA2 are the number of single-particle functions used for the ˜X and 1A2 states.
The 6D and 11D include only the 1A2 state.
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The separability of the dissociative and the vibrational dynamics, underlined in Eqs. (19) and (20), allows one to express the total absorption spectrum as a convolution of the spectra originating from theR- andQ-subspaces. Let us specifically consider the case of dissociation in the isolated state 11A2(πσ∗) photoexcited in az-polarized transition mediated by the TDM µAz2 depending on a singlea2-symmetric modeQa2(3). In fact, the convolution approximation can be extended to the TDMs obeying the general Herzberg-Teller expansion (Appendix A)
and to the situations in which off-diagonal coupling matrix elements are retained either in HˆR or in ˆHQ.
The autocorrelation functionS(t), defined in Eq. (15), is approximated by a product (the polarization index =z is omitted)
S(t)≈ hFR|exp
−iHˆRt
|FRiR · hfQ|exp
−iHˆQt
|fQiQ ≡sR(t)sQ(t), (21) where the spatial integration variables are explicitly indicated for each set of angular brack-ets. Next, the Fourier integral in Eq. (16) is transformed into a convolution of the Fourier integrals over the functions sR(t) and sQ(t) via a standard transformation (introduce an integration over the second time variable δ(t−τ) dτ, replace theδ-function with an integral exp [i(t−τ)ω] dω, and isolate the individual Fourier integrals). Defining ‘spectral functions’
without the energy prefactor,
¯
σR(E) = Z ∞
−∞
sR(t)eiEtdt ,
¯
σQ(E) = Z ∞
−∞
sQ(t)eiEtdt , (22)
the absorption cross section can be written as σ(Eph) = Eph
20c Z ∞
−∞
¯
σR(Eph−ω)¯σQ(ω)dω . (23) In the R-space, the motion of the wave packet is (directly or indirectly) dissociative, while the motion in the quadratic potentials of the Q-space is bound. Thus, the absorption spectrum in Eq. (23) consists of a series of excitations of the pyrrolyl ring broadened by the dissociation of the hydrogen atom. As shown in Appendix A, more convolution terms are needed to approximate the absorption spectrum of the A2 state if the transition is induced by a TDM in the Herzberg-Teller form of Eq. (12). The accuracy of the approximation is illustrated in Sect. V.
The convolution approach to the absorption spectrum can be considered as an extension of the familiar FC computations of bound−bound transitions60,61 to the case of dissociative spectra. This extension has several computational advantages. For example, the method reduces the amount of ab initio computations needed to evaluate ¯σR(E): While a 3D po-tential energy surface of the excited state in the disappearing modes (R, θ, φ) is required, the Qspace is described only using Hessians in the ground and the excited electronic states
28 at R = RFC, because the bound vibrational spectrum ¯σQ(E) is given by the FC overlap integrals,
¯
σQ(E) =X
m
|hϕm(Q)|fQ(Q)i|2δ(E−Em), (24) between the eigenfunctions ϕm(Q) (with energies Em) of the non-disappearing modes in the FC zone and the initial statefQ. Note that the harmonic stick spectrum in theQ-space can be efficiently calculated analytically using the techniques devloped for the FC factors in polyatomic molecules.62,63 The convolution calculations are further simplified if the πσ∗ state is purely repulsive. In this case, the absorption spectrum ¯σR(E) can be accurately reconstructed using the reflection principle64 which only requires the gradient of the 3D potential at the FC point. In the most optimistic scenario, a convolution calculation of the diffuse absorption spectrum becomes purely analytical, while the ab initio input refers to a single molecular geometry, namely the FC point.