AB INITIO MOLECULAR HAMILTONIAN

A. The form of the molecular Hamiltonian

(1) The molecular Hamiltonian,

is set in the basis of three locally diabatic electronic states ˜XA1 (abbreviated as X orA1), A2, and B1; the state symmetry labels refer to C2v geometries. Pyrrole is described using (a) three Jacobi coordinates R≡(R, θ, φ) of the dissociating H-atom relative to the center of mass of the pyrrolyl fragment (the so-called ‘disappearing modes’; see Fig. S1) and (b) 21 dimensionless normal modes Qof pyrrolyl, calculated at the equilibrium geometry of the fragment (the so-called ‘non-disappearing modes’; examples of them are given in Fig. S2).

The normal modes Q are partitioned into four blocks according to the irreps Γ of the C_2v symmetry group, Q={Qa1,Qa2,Qb1,Qb2}.

(2) The kinetic energy operator in Eq. (1) is set in the body-fixed principal axis frame for the zero total angular momentum of pyrrole (atomic units are used hereafter):

Tˆ=− 1 The first term is the kinetic energy of the relative motion of the H-atom and pyrrolyl;

µ is the corresponding reduced mass. The second term refers to pyrrolyl vibrations; the

∗ Electronic mail: Sergy.Grebenshchikov@ch.tum.de

FIG. S1: Body-fixed pyrrolyl axes and Jacobi coordinates (R, θ, φ) for the detaching H atom in pyrrole (H - gray; C - orange; N - blue).

Q_a1(5) (ω = 1558 cm⁻¹) Q_a2(1) (ω = 533 cm⁻¹) Q_a2(3) (ω = 932 cm⁻¹) FIG. S2: Pyrrolyl normal modes used in the quantum mechanical calculations. The a₁ modes are the totally symmetric in-plane distorsions. The out-of-planea₂ modes are antisymmetric with respect to the reflection planesσ_v,σ⁰_v and symmetric with respect to rotations about the C₂ axis.

sum PΓ

i runs over the vibrational modes i belonging to the irrep Γ. The last two terms describe the orbital motion of the H-atom and the rotational motion of the rigid pyrrolyl ring;j = (jx, jy, jz) is the pyrrolyl angular momentum operator and the inertia constantsIx, Iy and Iz are evaluated at the fragment equilibrium geometry; the C2v symmetric pyrrolyl ring lies in the yz-plane, withz being the C₂ axis (see Fig. S1).

(3) The elements of the diabatic potential energy matrix in Eq. (1) are constructed as sums of two groups of terms,

V(R,Q) = VR(R) +VQ(Q|R) (3)

FunctionsVR(R) of the three disappearing modes are constructed using spline interpolations of the ab initio points calculated on a dense coordinate grid (R, θ, φ). Functions VQ(Q|R),

3 depending on the 21 non-disappearing modesQ, are constructed in the spirit of the vibronic coupling model,¹ with pyrrolyl treated as a semirigid ring using quadratic Hamiltonians, and parameters depending on the interfragment distance R.

(4) The diagonal elements V^α have the form (α=X, A₂, B₁):

V^α(R, θ, φ,Q) = V_R^α(R, θ, φ) +X

i a1

κ^α_i(R)Qa1(i) + 1 2

X

Γ

X

i,j

Γγ_Γ,ij^α (R)QΓ(i)QΓ(j). (4)

where κ^α_i(R) are the R-dependent gradients (vanishing for all modes but a₁) and γ_ij^α(R) are the R-dependent normal mode Hessians matrices evaluated at Q = 0; matrices γ^α are four-block diagonal: γ^α =γ_a^α₁ ⊕γ_a^α₂ ⊕γ_b^α₁ ⊕γ_b^α₂.

(5) The off-diagonal diabatic couplings V^αβ are V^αβ(R, θ, φ,Q) = V_R^αβ(R, θ, φ) +X

i

Γα×Γβ

λ^αβ_i (R)QΓα×Γβ(i). (5) The functions V_R^αβ(R, θ, φ) are the coupling elements between the regularized quasi-diabatic states constructed on the ab initio coordinate grid as suggested by K¨oppel et al.² Although they do not have a simple analytical representation, they are forced to follow the lowest al-lowed orders in the symmetry-adapted spherical harmonics near conical intersections, namely V_R^A1A2 ∼sin²θsin(2φ) and V_R^A1B1 ∼sinθcosφ. The Q-dependent coupling terms for X/A2

and X/B1 are linear in the vibrational modes of a2 and b1 symmetry, respectively. The coupling matrix element V^A2B1 between the states A₂/B₁ is not included in the quantum mechanical calculations.

(6) The quasi-diabatic representation of the Hamiltonian of Eq. (1) is local, and a given off-diagonal matrix element is non-zero only in the vicinity of the respective conical intersection.

For the second term in Eq. (5), this is achieved by using the following functional form for the coupling strength λ^αβ_i :

λ^αβ_i (R) = λ^αβ_CI,iexp −

R−R^αβ_CI

∆

n!

, (6)

where R^αβ_CI is the position of a conical intersection between α/β = X/A2 or X/B1. The parameters λ^αβ_CI,i, ∆ and n are tuned ‘by eye’ in order to obtain smooth diabatic Hessians for the coupled states. For the first term in Eq. (5), similar attenuation functions are applied to the (R, θ, φ)-dependent mixing angles of the regularized adiabatic-to-diabatic transformation.

B. Ab initio parameters of the molecular Hamiltonian

(1) The matrix elements of the molecular Hamiltonian are found from the electronic structure calculations performed along the minimum energy path (MEP) for the hydrogen detachment from the NH group. The MEP is evaluated on a grid in the Jacobi coordinateRin the lowest excited state ¹A2(πσ^∗). The grid consists of 37 points chosen between Rmin = 3.5a0 and R_max = 8.3a₀ (at short distances the spacing is ∼ 0.1a₀, in the asymptotic region it is

∼0.4a0). The resulting energy profiles are shown in Fig. 1 of the main paper. Along MEP, the molecule is constrained to C2v geometries (i.e., θ, φ = 0). Aug-cc-pVTZ (AVTZ) basis set of Dunning³ is used and further supplemented with the diffuse s and pfunctions added to the N and H atoms of the dissociating bond (one set of s and p functions for N and two sets for H). The exponents of these functions are derived in an even temprered manner from the most diffuse s and p functions of the AVTZ basis by dividing the exponents successively by a factor of 3.0.⁴ This extension is necessary to correctly describe the Rydberg character of the A2 and B1 states, and the resulting basis set is referred to as AVTZ+.

(2) Most calculations are performed at the CASPT2 level of theory. The reference wave-functions are obtained from the state-averaged CASSCF calculations including the 1A1, 1A2

and 1B1 states. The active space (eight electrons in seven orbitals) comprises fiveπ valence molecular orbitals, three of b₁ and two of a₂ symmety, the 9a₁(σ) and the 10a₁(3s/σ^∗) or-bitals. Electronic structure calculations are performed using the highest possible symmetry.

In particular, the a1, a2, b1 and b2 blocks of the Hessian matrices are calculated separately using C2v, C2, Cs and C_s⁰ symmetries, respectively. Coordinate dependent transition dipole moment (TDM) functions, necessary to properly describe the optical excitation of the πσ^∗ states from the ground electronic state, are calculated at the CASSCF level.

(3) Table I summarizes the characteristic features of the three calculated electronic states of pyrrole. The ordering of the electronic states at the Franck-Condon point is correctly reproduced by the present CASPT2 calculations. Vertical excitation energies of the πσ^∗ states are underestimated by 0.4 eV-0.6 eV compared to the CCSD and MRCI results; the agreement with the known experimental values is within similar bounds. The accuracy of the calculated dissociation thresholds is slightly better. Tables II and III provide further characterization of the molecular Hamiltonian and give the dimensionless coupling strength parameters κc = λ/ω for the coupling modes of a2 and b1 symmetries, respectively. The

5

TABLE I: Characteristic data of the ab initio potential energy surfaces of the three lowest electronic states of pyrrole: Vertical excitation energy T_v (in eV); band origin T₀ (in eV), which includes ZPEs of the ground and the excited electronic states; quantum mechanical thresholds D0 for the electronic channels diabatically correlating with the calculated states (in eV). Available theoretical and experimental results are shown for comparison.

Diabatic state T_v T₀ T₀ (exp) Dissociation channel D₀^a D₀ (exp.) X˜¹A₁(ππ) 0.0 0.0 0.0 H(¹S)/pyrrolyl(1²A₁) 5.09 −

4.80^a 4.32 <4.88^d H(¹S)/pyrrolyl(1²A₂) 3.40 4.07^d 1¹A₂(πσ^∗) 5.17^b

5.59^c

5.45^a 5.30 5.86^e H(¹S)/pyrrolyl(1²B1) 3.96 4.62−4.67^f 2¹B₁(πσ^∗) 5.88^b

5.84^c

aThis work; CASPT2.

bRef. 5; CCSD.

cRef. 6; MRCI.

dRef. 7.

eRef. 8.

fDFT⁹ and MRCI¹⁰ methods estimate the difference in the threshold energiesD0(1²B1)−D0(1²A2) to be in the range of 0.55 eV—0.60 eV.

frequency ω is taken to be the pyrrolyl frequency at the respective intersection.

(4) The potential energies of the states ˜X,A2, andB1as functions of the disappearing modes are found on a three-dimensional grid (Ri, θj, φk), with the nodesRibeing grid points on the MEP; the CASPT2 method and the C1 symmetry are used. The polar angles θj cover the range [0^◦,90^◦] with a step of 5^◦; energies for θ > 90^◦ are extrapolated. The grid in the az-imuthal angleφranges from 0^◦ (H-atom in theσ⁰_vplane) to 90^◦(H-atom in theσv plane) with a step of 15^◦; energies for largerφ are reconstructed usingC2v symmetry of the pyrrolyl ring.

The quasi-diabatic representation near theX/B1conical intersection is constructed from the adiabatic energies using the regularized adiabatic-to-diabatic transformation (ADT) method of K¨oppel et al.² In the next step, all matrix elements set on the grid are interpolated with cubic splines. The angular modes do not couple the states X and A2 at either φ = 0^◦

TABLE II: Vibrational frequencies ω(i) (in cm⁻¹), coupling strengths λ_i (in cm⁻¹), and the di-mensionless coupling strength parametersκ_c=λ_c/ω_cfor the vibrational modes ofa₂ symmetry at theX/A₂ conical intersection.

Mode λ_c ω_c κ_c

Q_a2(1) 574.0 551.8 1.04

Qa2(2) 759.0 895.7 0.85

Q_a2(3) 213.0 940.3 0.23

TABLE III: Vibrational frequencies ω(i) (in cm⁻¹), coupling strengths λ_i (in cm⁻¹), and the dimensionless coupling strength parameters κ_c=λ_c/ω_cfor the vibrational modes of b₁ symmetry at theX/B₁ conical intersection. For the angleθ,λ_θ is evaluated as a gradient∂V /∂(sinθ) at the intersection.

Mode λ_c ω_c κ_c

θ 19751.00 854.00 23.12

Q_b1(1) 1151.00 823.6 1.40

Q_b1(2) 600.00 765.2 0.78

Q_b1(3) 265.00 875.3 0.30

or 90^◦, and the small coupling matrix element V^XA2(R, θ, φ) is set as an analytic function V^XA2(R, θ, φ) = λ^XA_ang²(R) sin²θsin(2φ), with λ^XA_ang²(R) = c[1−atan ((R−R0)/∆R)]. The parameters in this function are chosen ‘by eye’ in order to give smooth diabatic curves for φ 6= 0^◦,90^◦.

(5) The parameters of the Q-dependent part of the Hamiltonian are calculated as first and second derivatives with respect to deviations from the MEP. To this end, the dimension-less normal modes Q are constructed using the Hessian matrix at the equilibrium pyrrolyl geometry via

Qi =X

r

rωiMr

~ UirXr=X

r

LirXr , i= 1,21 andr= 1,27 (7) where {Xr} are the 27 Cartesian coordinates of all atoms; Mr is the mass of the atom associated with the coordinate Xr; ωi is the frequency of the normal mode Qi; {Uir} is the matrix of eigenvectors of the mass-weighted Cartesian Hessian, after the removal of the

7 rows corresponding to the global translations and rotations. Next, the Cartesian gradient and the Cartesian Hessian for the pyrrolyl ring are calculated along the MEPQ=Q_min(R) and transformed to the basis of the dimensionless normal modes giving the gradient vector g^α(R) and the Hessian matrix γ^α(R). The gradient g^α(R) vanishes only for the state A₂; for the states ˜X and B1, it differs from zero. For these states, the gradientκ^α(R) atQ=0 is reconstructed using the gradient g^α(R) and the Hessian γ^α(R):

κ^α(R) =g^α(R)−γ^α(R)Qmin(R) , (8) For some blocks of the adiabatic Hessian, a local quasi-diabatic representation is constructed:

The Jacobi distanceRis the tuning mode for the conical intersections ˜X/A2 and ˜X/B1, and the blocksa2 andb1 diverge asRapproaches an intersection. The diabatic matrix elements, smoothly depending on R, are found using a transformation akin to the ADT method of K¨oppel et al.². For example, for the ˜X/A2 pair, one finds

γ_ij^X(R)

dia = ˜γ_ij^X(R)

adia+ 2 λ^XA_i ²(R)λ^XA_j ²(R) V_R^X(R,0,0)−V_R^A2(R,0,0) γ_ij^A2(R)

dia= ˜γ_ij^A2(R)

adia−2 λ^XA_i ²(R)λ^XA_j ²(R)

V_R^X(R,0,0)−V_R^A2(R,0,0) , Γ_i = Γ_j =a₂ . (9) Similar equations hold for the ˜X/B1 pair of states. The local character of the diabatic functions is guaranteed via the R-dependence of the coupling coefficients λ_i defined in Eq.

(6).

(6) The photodissociation dynamics in the ¹πσ^∗ states is sensitive to the coordinate depen-dence of the TDMs expressed as a Herzberg-Teller expansion,

µ(R,Q)≈µR(R) +µQ(Q), (10)

in which only linear (and therefore additive) deviations from the Franck-Condon geometry are taken into account. The expansion coefficients are calculated at the CASSCF level of theory as numerical first derivatives of the TDMs with respect to nuclear displacements along the normal modes.

Im Dokument Photodissociation dynamics and spectroscopy in the presence of conical intersections  (Seite 189-195)