A General Approach for Calculating Strongly Anharmonic Vibronic

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A General Approach for Calculating Strongly Anharmonic Vibronic Spectra with a High Density of States: The X2B1 0}

8

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Journal of Chemical Theory and Computation

In general, KI (He , x0 ) can be determined in the framework of iterative Krylov methods. In the Lanczos algorithm, the orthonormal basis {vn } for IKS is calculated according to the recurrence relationship

He vj = βj−1 vj−1 + αj vj + βj vj+1

(9)

v1 = x0

(10)

αj = vjT He vj

(11)

T βj = vj+1 He vj

(12)

The corresponding eigenvectors are generated by the diagonalization of the tridiagonal matrix with {αi } on the main diagonal, and {βi } occupying off-diagonal elements. In general, for an arbitrary number of Lanczos iterations l the parameter βl is an upper bound for the residual norm associated with an arbitrary approximate eigenvector. Thus, for a given accuracy level tr corresponding to the residual norm, the dimension of the IKS, I, is determined by the condition

βI < tr

(13)

Usually, this bound is too strong if only the convergence of the eigenvectors with the largest weights in x0 is required. In this case, the convergence of the residual norm rυn ,l corresponding to the eigenvector uυn ,l should be checked according to the criterion krυn ,l k = βl |eTl yυn ,l | < tr

(14)

where el is the l-th unit vector, and yυn ,l is the Ritz vector in the l-dimensional subspace of Lanczos vectors. Typically, the number of Lanczos iterations is significantly reduced if the convergence is required only for the states with the largest FCFs. However, even in this case the number of Lanczos vectors can be very large, up to ∼ 104 , due to a high density of 9



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states, which imposes significant requirements for the vector storage and computational cost for the vector orthogonalization if the dimension of the eigenvalue problem is ∼ 106 , as in the present study. To a large extent, these problems can be eliminated by the application of the RACE method which was recently developed in our group. 37 This algorithm is based on the minimum condition for the residual norm corresponding to the error in the calculated eigenvector. It is designed within the formalism of the so-called contracted invariant Krylov subspaces (CIKS) in which case the eigenvectors with the leading contributions to a given reference vector are to be calculated. In Ref. 37, the set of p target eigenvectors with the largest weights in x0 , Lp = Lp (A, x0 , ts , trel , ∆λ), is specified via 3 cut-off parameters. The parameter ∆λ provides the search range for the eigenvalues, and the values of ts and trel control the maximum total and relative weights, respectively, of the calculated eigenvectors in x0 . In the present study, the RACE method was employed for calculating vibronic transitions with the most significant FCFs. For this purpose, the search space within the CIKS was modified such that only the eigenvectors {uυn } for which wυn > tmin are to be calculated.

As analyzed in Ref. 34, in general, the accurate calculation of vibronic spectra for a given homogeneous linewidth parameter does not require precise determination of the contributing vibrational states. Within the RWF formalism, the expected relative error in the spectral ˜ L , q), can be controlled by the norm intensities corresponding to an approximate RWF, R(E of the residual vector, determined on the basis of the ISE, ˆ e (q) + εgi + iΓ R(E ˜ L , q) − iΨgi (q) r(EL , q) = EL − H

10


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˜ L , q) and r(EL , q) in a basis of Hartree products, Using the vector representation of R(E ˜ L , q) = R(E

X

˜ k (EL )Φk (q) R

(16)

k

r(EL , q) =

X

rk (EL )Φk (q)

(17)

k

one can employ an iterative subspace method in which the RWF is approximated as a linear combination of trial vectors {vk }: ˜ l (EL ) = R

l X

sk,l (EL )vk

(18)

k=1

The vector sl (EL ) of dimension l is the solution of a reduced linear problem obtained by the projection of the ISE on the trial vectors. We have shown in Ref.

34 that the iterative

solution of the ISE (eq. (4)) in a spectral range with the residual-based trivial preconditioner leads to an expansion basis {vk } of Lanczos type. In such a case the norm of the residual vector at each spectral point can be calculated as 34

krl (EL )k = |sl,l | βl

(19)

Thus, the Lanczos iterations can be carried out for accurate calculations of spectral envelopes, and the maximum number of iterations, lm , is determined based on the expected maximum relative error in the calculated intensities, r¯lm , as r¯lm = max krlm (EL )k2 < trwf EL

(20)

where trwf is a predefined accuracy parameter. For each spectral point, the vector sl (EL ) can be determined by solving the ISE in a subspace of Lanczos vectors. However, it is more

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efficient to express s(EL ) via the corresponding subspace Ritz vectors:

sl (EL ) = i

X n

eT1 yυn ,l yυ ,l EL − (ευn − εgi ) + iΓ n

(21)

According to eqs. (19)-(20), the global convergence of the RWF for all spectral points depends, in the first place, on the parameter βl , as in the case of the Lanczos-based solution of the eigenvalue problem. In addition, as follows from eq. (21) the interference effects caused by a high density of states and finite Γ may lead to a substantial reduction of |sl,l |, whereby the RWF typically converges at significantly lower number of iterations than the underlying vibrational states, especially if the corresponding FCFs are very small.

In the present study, both, the VCI FC and RWF calculations, are performed in the basis (e)

(e)

of the Hartree products {Φk (q) = Φk (q)} of one-dimensional modals {ϕim (qm )} obtained from the VSCF calculations for the vibrational ground state of the electronically excited PES, where im indicates the modal corresponding to normal coordinate m. The initial vibrational (g)

state Ψgi is calculated within the VCI Ansatz in terms of the Hartree products, {Φk (q)}, optimized in the VSCF calculations for the electronic ground state PES,

Ψgi (q) =

X

(g)

c0,k Φk (q)

(22)

k

Taking into account eq. (5) and eq. (6) the initial vector x0 can be calculated as

x0,k = hΦk |Ψgi i =

X n

cn Φk |Φn(g)

(23)

In order to simplify the calculation of expensive multidimensional overlap integral in eq. (23), the VSCF and VCI calculations of Ψgi (q) are performed in the same set of normal coordinates as for the electronically excited PES, which is achieved by the corresponding coordinate transformation of the vibrational Hamiltonian. 33,34 Hereafter, we introduce the so-called 12


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configuration-based Franck-Condon factors (CFCFs), ηk , which are related to the overlap integrals between Ψgi (q) and {Φk (q)}, ηk = |x0,k |2 = |hΦk |Ψgi i|2

(24)

Apparently, for any independent-mode approximation, including the VSCF one, the CFCFs and FCFs are identical. The notion of the configuration-based FCF is useful since it enables the understanding of some qualitative aspects of the vibronic intensity distribution. Typically, the number of configurations with significant CFCFs, Ncf cf , is relatively small (for the system under study it is ∼ 100). In general, strong multimode anharmonic effects can lead to a significant number of states, substantially larger than Ncf cf , to which the intensity-determining configurations with large CFCFs contribute. This makes the spectral envelope more diffuse, as compared to the independent-mode model. We assume that the compositions of the vibrational wave functions or RWF corresponding to the most intense transitions can be well described within a certain excitation level relative to the configurations with the largest CFCFs. In the most consistent treatment, such a configuration space would be defined within a multireference ansatz. However, in this study we employ a singlereference approach, in which the reference occupation numbers are determined in an average way according to the CFCFs values, as explained below. We define the modal-contracted (e)

Franck-Condon factor (MCFCF) for modal ϕim as ζ im =

X

(25)

ηk(im )

k(im )

(e)

where {k(im )} are the indices of the configurations involving modal ϕim . It is straightforward to show that if the configuration basis is complete then X

0 ≤ ζ im ≤ 1

ζ im = 1

i

13


(26)


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Assuming completeness of the vibrational modals, the MCFCFs can be expressed via the expansion coefficients {c0,k } of the initial vibrational state and the overlap integrals between the modals corresponding to the ground and electronically excited PESs:

ζ im =

X(m) k,l

E ED D (g) (e) (g) (e) ϕjm (l) |ϕim c0,k c0,l ϕjm (k) |ϕim

(27)

where superscript (m) means that the double summation is performed over k and l corresponding to the configurations whose occupation numbers may differ only in mode m, jm (k) indicates the modal for coordinate m which is occupied in configuration k. The modals with the largest MCFCFs effectively provide major contributions to the transition intensities. Accordingly, an optimal occupation number of mode m, n(m), in the reference vector used for the generation of the configuration basis can be calculated on the basis of the MCFCFs as

n(m) = Nint

X i

iζim

!

(28)

where Nint (x) is the nearest integer function of x. Alternatively, one can use n(m) which corresponds to the largest ζim . However, this is suboptimal if the MCFCFs are nearly uniformly distributed over a range of modals. As discussed in the results section, the calculations based on the reference vector calculated according to eq. (28) provide more consistent spectral envelopes in the whole range as compared to the trivial reference vector with {n(m) = 1}.

1.2 1.2.1

Computational details Electronic structure calculations

We used explicitly correlated coupled-cluster theory, 28 i.e. CCSD(T)-F12a and UCCSD(T)F12a, to optimize the structural parameters of CH2 F2 and CH2 F+• 2 . A correlation consistent basis set of triple-ζ quality, i.e. cc-pVTZ-F12, was used throughout. 38 cc-pVTZ/JKFIT, cc-

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pVTZ-F12/OPTRI and aug-cc-pVTZ/MP2FIT fitting bases were used as auxiliary bases for density fitting, etc. 39 Within the MP2-F12 step of the CCSD(T)-F12 calculations, the 3C(FIX) ansatz was used in all cases. 40 The Hartree-Fock reference energies were improved by addition of the complementary auxiliary basis set (CABS) singles correction. Symmetry was employed whenever possible. The same level of theory has been used to determine the harmonic frequencies and the normal coordinates for spanning the multi-dimensional potential energy surfaces. It has been shown in the past that this level of theory yields very excellent results, which are close to conventional coupled-cluster calculations in combination with a quintuple-ζ basis. 41 All electronic structure calculations have been performed with a development version of the Molpro suite of ab initio programs. 42 1.2.2

Vibrational structure calculations

A multimode expansion was used to represent the potential energy surface, which was truncated after the 3-mode (3D) or 4-mode (4D) coupling terms, respectively. These surfaces were determined in a fully automated fashion using the PES generator 26 as implemented in Molpro. 42 A multi-level scheme has been used, i.e. the 1D and 2D terms of the expansion were determined at the CCSD(T)-F12a/vtz-f12 level, while the 3D and 4D terms were determined at the CCSD(T)-F12a/vdz-f12 level. 43–45 Previous studies have shown that this multi-level combination yields excellent results in comparison to experimental data. 36 The resulting grid representation of the potential, which can be downloaded from our database of potential energy surfaces, 46 was fitted to a polynomial basis using our recently published Kronecker product approach. 47 Note that this representation of the PES consists of 45384 terms. Fitting errors (χ2 , sum of the squares of the errors at all grid points) were found to be as low as 10−17 E2h . From these polynomials we determined up to semi-quartic force constants in order to run 2nd order vibrational perturbation theory (VPT2) calculations. 48,49 Fermi resonances and Darling-Dennison resonances were explicitly accounted for.

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Variational calculations based on the n-mode representation of the potential were performed at the vibrational self-consistent field (VSCF) level 50–52 and subsequent state-specific configuration selective vibrational configuration interaction calculations (VCI). 53–57 As the potential energy surface was spanned in terms of normal coordinates, the Watson Hamiltonian has been used. 58 Vibrational angular momentum terms employing a constant µ-tensor were included in all VCI calculations. 59 20 distributed Gaussians were used to determine the VSCF modals and respective configuration basis in the VCI calculations. For the VCI calculations of the fundamental modes a correlation space generated by up to 6-tuple excitations and excitations up to the 8th root per mode was used. The sum of quanta within the n-tuple excitations was restricted to 24. This resulted in 2667521 (A1 ), 2579620 (A2 ), 2626148 (B1 ), 2626148 (B2 ) configurations in these calculations with respect to the individual irreps.

In addition to these state-specific configuration selective VCI calculations, the RACE method was employed for the automatic determination of states with non-negligible FCFs. In this case, the onvergence threshold was set to tr = 1 × 10−7 Eh , and the cutoff value for the FCFs was chosen to be tmin = 4 × 10−3 . The convergence threshold for the RWF calculations in the Lanczos basis was set to trwf = 10−3 . In the RACE and RWF calculations, the correlation space was increased as compared to the state-specific VCI calculation due to the need to account for many transitions simultaneously. In this case, the overall sum of the quanta within the 6-tuple excitations was restricted to 16, and the maximum excitation number per mode was set to 15. For comparison, different references, the ground VSCF and MCFCF-based configurations, were used for the generation of the correlation space. This resulted in up to ∼ 106 configurations of A1 symmetry which are relevant for calculating the spectra provided that Ψig is totally symmetric and non-Condon effects are neglected, as in the present study. The off-diagonal elements of the Hamiltonian matrices whose absolute values are larger than the cutoff parameter th = 10−6 Eh were kept in memory in a packed form, and the remaining small components were neglected, which had a negligible effect

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on the accuracy of the calculated spectra, as checked for lower values of th and smaller configuration bases.

2 2.1

Results and discussion Geometrical parameters

Equilibrium bond lengths and angles (re ) and VCI vibrationally averaged values (r0 ) thereof have been computed for difluoromethane and its radical cation in C2v symmetry. Within all calculations the fluorine atoms lie in the yz-plane, while the hydrogen atoms lie in the xyplane. Our results obtained from explicitly correlated coupled-cluster theory in combination with a triple-ζ basis are in excellent agreement with the experimental data of Hirota 60–62 (cf. Table 1) and the computed values of Tasinato et al., 63 who used conventional CCSD(T) calculations in combination with large basis sets and the inclusion of core correlation effects. The remaining differences to be seen in Table 1 are attributed to missing electron correlation Table 1: CCSD(T)-F12a/vtz-f12 Geometrical parameters of CH2 F2 .

C-H C-F HCH FCF HCF a b

Calc.a re r0 108.87 109.97 135.51 136.17 113.44 113.43 108.29 108.15 108.74 108.78

Exp.a,b re 108.4(3) 135.08(5) 112.8(3) 108.49(6)

rz 109.7(5) 136.01(14) [113.67] 108.11(16)

rs 109.34(30) 135.74(10) 113.67(17) 108.32(5)

Bond lengths are given in pm, angles in degree. Taken from Ref. 60.

contributions as arising from high-order coupled-cluster terms and core-valence correlation, relativistic effects, etc. However, the overall agreement is excellent, which must be expected for a closed-shell molecule of this size. To the best of our knowledge, experimental data for the difluoromethane radical cation have not yet been published. Our UCCSD(T)-F12a/vtzf12 parameters (cf. Table 2) are in nice agreement with the MP2(FC)/aug-cc-pVQZ values 17



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of Forysinski et al., 17 which supports the good performance of the latter approach in the equilibrium region. The change of the CH bond length upon vibrational averaging is more Table 2: UCCSD(T)-F12a/vtz-f12 Geometrical parameters of CH2 F+• 2 .

C-H C-F HCH FCF HCF a

Calc.a re r0 117.76 119.31 127.02 127.53 84.86 86.60 116.61 116.32 112.82 112.58

Bond lengths are given in pm, angles in degree.

pronounced for the radical cation than for the closed shell molecule. This is a result of • the stronger anharmonicity for this system arising from the low CH2 F+• −→ CHF+ 2 2 +H

fragmentation channel.

2.2

Vibrational frequencies

The fundamental modes of CH2 F2 have been computed at various levels of theory, cf. Table 3. The force constants needed within vibrational perturbation theory (VPT2) were retrieved from the multi-level potential as described in Ref. 48. The VPT2 results are in excellent agreement with the variational VCI calculations and the experimental data. In any case, the only slightly sensitive mode appears to be the B1 CH stretching mode, which of course is without relevance for the photoelectron spectrum. A result of a similar accuracy has been obtained by Luckhaus et al. from variational calculations 20 and by Tasinato et al. 63 using vibrational perturbation theory. Both groups based their calculations on conventional CCSD(T) calculations in combination with large basis sets.

The situation differs completely for the radical cation. First of all, we were not able to compute 4-mode coupling terms for this system due to severe convergence problems in 18


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Table 3: Fundamental vibrations of CH2 F2 . Values are given in cm−1 .

νi 11 21 31 41 51 61 71 81 91

Sym A1 A1 A1 A1 A2 B1 B1 B2 B2

a

b

Exp. Harm. VPT2 2947.9 3080.4 2951.1 1509.6 1550.5 1507.9 1111.6 1133.4 1111.1 528.3 533.9 527.7 1256.8 1286.4 1253.6 3014.5 3156.1 3009.7 1178.7 1198.5 1177.6 1435.5 1469.2 1434.4 1089.9 1118.2 1090.3

MAD MAX a b c

c

Calc. VCI(3D)c VCI(4D)c 2946.7 2948.1 1507.5 1507.6 1110.9 1110.8 527.3 527.3 1253.9 1253.6 3022.4 3010.2 1177.5 1177.3 1434.6 1434.5 1090.2 1090.2

1.8 4.8

2.0 7.9

1.6 4.3

Taken from Ref. 63. Obtained from CCSD(T)-F12a/vtz-f12 calculations. Based on a multi-level potential (see the computational details).

the Hartree-Fock or coupled-cluster calculations. For example, the CCSD-F12 iterations for grid points in the outer region of the 4-mode coupling surface of modes 1, 2, 6, and 8 did not converge and showed complete unreasonable norms and T1 diagnostics. This • is not a surprising result as the CH2 F+• −→ CHF+ 2 2 +H fragmentation channel leads to

a strong multiconfigurational character of the electronic wave function. Even if one was able to converge the Hartree-Fock calculations, the energies resulting from the subsequent coupled-cluster calculations must then be considered error prone. For example, CCSD(T) potential curves for a symmetric stretching of the CH bonds in ethene show substantial errors and a wrong slope for bond lengths beyond 200 pm. 64 In principle one would need to compute the entire surface at the multi-reference configuration interaction (MRCI) level, which is currently still out of range within an automated PES generation. Therefore, the VCI results are only available at the lower 3D level. Our results are compiled in Table 4. Secondly, the assignment of the ν1 transition energy is rather questionable. The VPT2 result appears to be completely unreasonable and must be considered a breakdown of VPT2 19



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−1 Table 4: Fundamental vibrations of CH2 F+• 2 . Values are given in cm .

νi 11 21 31 41 51 61 71 81 91 a b

Sym A1 A1 A1 A1 A2 B1 B1 B2 B2

Exp.

a

1246 969

1408

b

LFZS

1251 959 597 940 1761 523 1406 1045

Harm. 2491.8 1269.6 1089.5 602.3 983.1 2028.0 599.1 1470.5 1089.1

Calc. VPT2 VCI(3D) (1162.0) (2459.0) 1251.0 1249.8 946.6 999.1 596.9 595.4 937.0 935.7 1794.5 1781.9 520.9 539.4 1411.8 1407.0 1045.5 1041.4

Assignment sym. CH2 stretch sym. CF2 stretch HCH bending FCF bending CH2 twist asym. CH2 stretch CH2 rocking asym. CF2 stretch CH2 wagging

Experimental values taken from Ref. 17. Variational results of Luckhaus et al., taken from Ref. 20.

theory. Likewise, the weight of the 11 configuration, as obtained from the corresponding state-specific VCI calculation, was found to be as low as 15% and thus referring to it as a fundamental transition is doubtful. In total we found 9 bands between 2100 and 2630 cm−1 with nonnegligible contributions of the symmetric CH2 stretching mode, and thus this vibration is scattered over a multitude of states. Again, this problem surely arises from the fragmentation of the system as discussed above. Moreover, three of our values differ strongly from the results of Luckhaus et al. 20 These are the transitions involving the totally symmetric, 31 , and the two B1 modes, 61 and 71 . There is a strong resonance between the 2nd overtone of mode 7 (73 ) and the fundamental of 6 (61 ). Both configurations contribute to about 30% to the 61 state. The relative contributions would be very sensitive to subtle changes in the PES. The fundamental of mode 3 was found to show a strong Fermi resonance with the first overtone of 7. An analysis revealed, it is the 3-mode coupling potential involving normal coordinates 3, 6 and 7 which has the most significant influence on this resonance. Deleting this particular coupling surface from the PES representation lowers the transition energy of 71 to 522 cm−1 and that of 31 to 955 cm−1 , which agree nicely with the values presented by Luckhaus et al. 20 Note that the VPT2 results shown in Table 4 are also close to these values,

20


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which might indicate that the main effects of this particular 3-mode coupling surface arise from its outer regions. On the other hand, neglect of this surface led to a deterioration of the overall envelope of the photoelectron spectrum. Apparently, an accurate representation of the surface is very important and it is by no means clear whether the coupled-cluster theory is capable to fulfill this request. In any case, in all our calculations we have included this surface as we could not think of any scientific reasons for omitting individual coupling terms which have significant impact on the spectrum. 2.2.1

˜ 2 B1 ←− X ˜ 1 A1 photoelectron spectrum The X

The adiabatic ionization potential Ip = 12.725 eV has been determined from the PFI-ZEKE spectrum by Signorell and co-workers. 17 In our calculations, Ip = 12.734, which is very close to the value obtained from the anharmonic calculations of Luckhaus et al., 20 who computed Ip = 12.740 eV. We believe that the calculated value of Ip can still be improved if corevalence correlation, high-order coupled-cluster terms etc. are taken into account. However, we expect that the remaining corrections arise mainly from the electronic structure calculations rather than the difference of the zero point vibrational energies. In order to allow for direct comparison with experimental bandshapes, we applied a small shift to our computed spectrum to account for this remaining difference. We note that for all approximations analyzed below, the total integral over the Franck-Condon profile in the range 12.7-15.2 eV is larger than 0.999, as follows from the RWF calculations. Thus, all relevant transitions are covered in our analysis.

In a first step we performed a couple of tests with results shown in Fig. 2. Note that, in Fig. 2 a constant scaling factor for the intensity has been chosen such that the computed and the experimental intensities for the 0-0’ transition match best for the full anharmonic calculation. Moreover, in the 3 top graphs, i.e. Fig. 2a-2c, an anharmonic representation of the vibrational ground-state for the electronic ground-state PES has been chosen, while

21



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cm-1 10000

5000

15000

20000

0.08 0.06

a 0.04 0.02

0-0

0.08 0.06

b 0.04 FCF-Scaled Intensity

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0.02 0.08 0.06

c 0.04 0.02 0.08 0.06

d 0.04 0.02 0

12.7

13.2

13.7 14.2 Energy [eV]

14.7

15.2

Figure 2: The photoelectron spectrum of difluoromethane. The experimental spectrum is shown in black and has been reproduced with permission from Ref. 15. Computed spectra (red) have been obtained from different approximations: a) harmonic approximation for the PES of the ionized state, b) 1D approximation for the multimode expansion of the ionized state PES, c) VSCF approximation for the wave functions of the ionized state, but including 3D coupling terms in the potential, d) harmonic approximation for the electronic groundstate PES, but full anharmonic treatment for the ionized state. The spectra are calculated by the RWF method.

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the harmonic approximation has been chosen in Fig. 2d. Consequently, the tests shown in Fig. 2 do not include the most accurate full anharmonic calculation. Clearly, the harmonic approximation (Fig. 2a) is too crude to be able to explain this spectrum. In particular the high-energy region is completely wrong and the intensity is shifted way too much to high transitions. Note that the calculations of the harmonic spectrum includes Duschinsky effects, i.e. the discrepancy between the harmonic spectrum and the experimental one is not due to missing Duschinsky effects, but missing anharmonicity. Duschinsky effects were included in all calculations. Lifting the harmonic approximation by using 1D anharmonic potential curves (Fig. 2b), has significant impact on the spectrum, but does not improve the situation at all. The main effect is a narrowing of the peaks. This spectrum is dominated by relatively sharp vibronic bands, which is not consistent with a rather diffuse spectral envelope in the high-energy range. The VSCF based spectrum (Fig. 2c) is noticeably improved with respect to the intensity distribution in the low-energy part. In addition, the high-energy bands are splitted. Nevertheless, the neglect of vibration correlation effects still leads to strong deviations from the experimental spectrum, although a 3D expansion of the ionized state PES has been used. The best agreement is obtained, when accounting for vibration correlation effects in combination with a proper representation of the potential (Fig. 2d). In particular, the description of the high-energy bands has significantly been improved. However, even this spectrum shows distinct differences with respect to the experimental one, mainly in the region between 12.8 and 13.5 eV.

In a 2nd step, we have accounted for anharmonicity effects by increasing the modecoupling level within the multimode expansion of the potential (1D to 3D), while accounting for vibration correlation effects at the RWF level. The wave function of the electronic groundstate was obtained at the VCI level based on a 3D potential (in contrast to the calculation shown in Fig. 2d). Fig. 3 shows the spectra computed within the RWF formalism in dependence on the mode-coupling level. It is the inclusion of the 2D coupling terms in the

23



cm-1 0 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 FCF-Scaled Intensity

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5000

10000

15000

1D

0-0

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

20000

2D

3D

12.7

13.2

13.7 14.2 Energy [eV]

14.7

15.2

Figure 3: Experimental (black) and computed anharmonic (red) photoelectron spectrum of difluoromethane in dependence on the mode-coupling level of the multimode expansion of the PES corresponding to the ionized state. The experimental spectrum has been reproduced with permission from Ref. 15.

24


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representation of the potential, which lead to a substantial improvement of the simulation. Nevertheless, transitions above 13.5 eV are significantly off which are corrected by the inclusion of the 3D terms. The 3D spectrum shown in Fig. 3 is our best simulated spectrum and clearly is in the position to explain the main features of the photoelectron spectrum of difluoromethane. However, this spectrum still suffers from the fact, that the calculations are not fully converged with respect to the vibration correlation space. Systematic investigation has revealed that further increase of the correlation space does not lead to fully stable intensity patterns, which to a large extent occur due to the following reason: As long as the configuration basis spans the region within the bound potential well, an increase of the correlation space will always result in an improved description of the vibrational states. However, if the potential features a negative curvature in the outer region then the increase of the basis can lead to the appearance of formally highly excited configurations, which, however, can have very low energies, and thus mix with the states of interest. Indeed, this is what we have observed in the present study. The degree of such mixing tends to oscillate with the increase of the correlation space, but happens only after certain excitation levels, and is accompanied by the appearance of the vibrational energies below the one of the ground state. This, of course, is a disturbing result. Such a behavior can be related to the joint effects of the presence of low lying fragmentation channels, artifacts of the coupled-cluster theory applied to multireference problems, as well as the deficiencies of the polynomial representation of the potential function in the outer regions. The latter is one of the main problems. The polynomial representation is sufficiently accurate in the region where it was fitted to the ab initio grid points (the so called interpolation region, IR), whereas beyond this domain (the extrapolation region, ER) its deviations from the true potential can be substantial. These artifacts are particularly pronounced along dissociative coordinates for which the true potential approaches a plateau at larger distances from the equilibrium. In such a case, the change from the increasing polynomial potential function in the ER to the decreasing one and vice versa can occur within a narrow set of fitting settings. Thus, in the extrapolation

25



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region the potential can have an unphysical negative decay, which leads to the barrier-like form of the potential and artificial tunneling effects upon the increase of the VCI basis, as well as the appearance of the spurious states below the 0-0 transition. We note that this is a general problem of a polynomial representation of the potential which is common for all systems with low-lying dissociation channels. In principle, the problem can be shifted to larger VCI bases if one increases the IR. However, for larger extents of the PES the convergence problems of the single-reference CCSD(T)-F12a method become intractable. In addition, such a behavior can be due to the failure of the coupled-cluster theory for the description of dissociation channel. Robinson and Knowles have shown that CCSD(T) may even yield wrong negative slopes of the potential instead of the static behavior, 64 which would ensure the negative decay of the fitted potential in the extrapolation region. In particular, a relaxed scan calculation of the C-H bond length in CH2 F+• 2 at the CCSD(T)-F12a level has revealed that for the most part the corresponding potential energy curve looks reasonably accurate. However, a failure of the coupled-cluster method was observed in the vicinity of the C-H bond length of ∼2˚ A, where a shallow maximum is detected resulting in an effective barrier of ∼ 4 × 10−4 Eh relative to the static part of the potential at large C-H distance. Obviously, such artifacts by themselves can precondition a negative decay of the polynomial representation of the potential in the ER. In any case, solving this issue would require a global potential represented by other basis functions than polynomials, as well as very large vibrational correlation spaces. Within the framework of automated PES generation this is currently not possible with our program, but in our opinion the accuracy of the presented results justifies the use of the chosen approach. We also note that the Franck-Condon factors corresponding to the spurious vibrational eigenstates below the 0-0 transition are extremely small, which is explained by sufficient remoteness of the ER from the localization region of the initial vibrational state Ψgi .

Partly, this is also a problem of the single-reference approach for generating the VCI

26


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basis. In general, this method is most accurate for that particular state in which the leading term is represented by the reference configuration. For other states it is accurate only if the total excitation level and the number of simultaneously exited modes are sufficiently large. This in turn leads to a somewhat unbalanced correlation space, such that for making it accurate for all relevant states, it gets, in some directions, too much extended to unreliable outer parts of the potential. In this respect, it would be more helpful to employ a multireference approach based on many reference configurations selected according to the values of the CFCFs in order to describe the whole spectrum. For the single-reference approach employed here, we have increased the excitation level for generating the configuration basis only up to the point above which the states with the energies below the ground state start to occur. Thus, the calculated vibronic intensity distribution, as well as the composition of the underlying states, are not fully converged with respect to the size of the configuration space. First of all, this is manifested in some dependence of the calculated bandshape on the choice of the reference configuration, as illustrated in Fig. 4. The theoretical spectrum in the top graph of Fig. 4 corresponds to reference configuration 11 22 32 determined on the basis of the MCFCFs. Energetically, configuration 11 22 32 corresponds to the spectral point at ≈ 13.6 eV. As expected, this choice provides a uniform quality, over the entire range, of the predicted spectral envelope with respect to the experimental one. The calculated spectrum in the bottom graph of Fig. 4 was obtained with the ground reference configuration (unit modal occupation numbers). One can see that in this case the agreement for the low energy part of the spectrum is noticeably better than for the MCFCF-based reference, while in the region > 13.4 eV some substantial deviations from the experimental bandshape are observed. There is work in progress in order to overcome this problem. The PFI-Zeke spectrum of Signorell and co-workers shows significantly higher resolution than the spectrum of Pradeep and Shirley, which has been used so far. Therefore we have studied the low energy region in more detail and attempted an assignment of the most important transitions in that region. The spectrum based on the ground reference configuration

27



cm-1 0

2000

4000

6000

8000

10000

12000

0.08 0.07 0.06 0.05 0.04 0.03 FCF-Scaled Intensity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.02 0.01

0-0

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

12.7

13.2

13.7 Energy [eV]

14.2

Figure 4: Experimental (black) and computed anharmonic (red) photoelectron spectrum of difluoromethane in dependence on the choice of the reference configuration for the VCI basis. The calculated spectra correspond to the MCFCF-based 11 22 32 (top) and ground reference (bottom) configurations. The experimental spectrum has been reproduced with permission from Ref. 15.

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is shown in Fig. 5. This figure shows the same number of peaks and within a certain error cm 0

1000

-1

2000

3000

0.07

0.06

FCF-Scaled Intensity

Page 29 of 42

0.05

0.04

0.03

0.02

0-0

0.01

12.7

12.8

12.9 13.0 Energy [eV]

13.1

13.2

Figure 5: Experimental (black) and computed anharmonic (red) photoelectrum spectrum of difluoromethane for the low-energy region. The experimental spectrum has been reproduced with permission from Ref. 15. bar the same peak positions as shown in Fig. 5 in the paper of Signorell and co-workers. 17 It is mainly the intensity distribution, which differs, but this problem also showed up in the calculations of Luckhaus et al. 20 Nevertheless, we consider the overall agreement to be very good. In Table 5, we present the computed FCFs for those states in the low-energy region which are clearly defined by leading configurations and are strong enough to be identified in the experiment. The overall agreement with the experimental values as provided by Luckhaus et al. is excellent. However, our FCFs differ considerably from those reported in Ref. 20. Likewise, there are significant differences in the contribution of the individual configurations in the corresponding states, but this arises mainly from the fact that Luckhaus et al. work in a harmonic oscillator basis. As a consequence the assignments of individual transitions are also somewhat different, which results from very low weights of the underlying configurations. Note that our assignment does not reveal any state with a significant contribution of the fundamental of mode 1, which is supposed to be the main component in 29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 5: List of vibrational transitions of CH2 F+• 2 with significant Franck-Condon factors (FCF) calculated by the RACE method. Only those states are listed, which show a contribution of the leading configuration of at least 20%. Exp. ν Rel. Int. a

969 1137 1246 2100

s s s s

2257 2400 2491

s s s

Calc. ν a FCF 0 596 998 1135 1250 2100 2115 2262 2401b 2492 3169 3311

0.010 3e-4 0.012 0.019 0.013 0.025 0.010 0.010 0.014 0.006 0.027 0.017

Assignment ZPE 41 (0.99) 31 (0.72) 72 (0.24) 72 (0.64) 31 (0.25) 21 (0.92) 32 (0.48) 92 (0.21) 72 31 (0.10) 92 (0.73) 32 (0.12) 31 21 (0.37) 74 (0.18) 72 21 (0.33) 74 (0.29) 31 21 (0.12) 22 (0.12) 22 (0.78) 72 21 (0.11) 33 (0.28) 72 32 (0.17) 31 11 (0.14) 32 21 (0.22) 42 32 (0.12)

a

Wavenumbers are given in cm−1 . This band was found to be sensitive with respect to the reference configuration, which was set to be the vibrational ground state for all transitions in this table. At this level, this particular band was determined at 2424 cm−1 . Using the leading configuration in the corresponding state ( 72 21 ) as the reference for the VCI basis, this band was shifted to the provided value at 2401 cm−1 .

b

the fragmentation path of the molecule.

In a separable-mode approximation (harmonic, anharmonic (1D) or VSCF) the vibrational states are represented by single configurations, and the vibronic intensities are given by the values of the corresponding CFCFs. In this approximation, the main factors determining the intensity distribution are the values of the normal mode displacements corresponding to the shift of the electronically excited equilibrium geometry relative to the ground state ˜ 2 B1 ←− X ˜ 1 A1 transition in CH2 F2 there are only 3 totally symmetric norone. For the X mal coordinates with substantial shifts, 1, 2 and 3. Accordingly, for any separable-mode approximation, the vibrational configurations of the type 1n1 2n2 3n3 have the largest FCFs (equivalent to CFCFs). However, upon accurate account of the anharmonic correlation ef30


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fects, it was found that the majority of states with significant FCFs do not involve any leading contributions of the type 1n1 2n2 3n3 . To illustrate this behavior, we present in Table 6 the assignments for the strongest transitions in the spectrum. The compositions of the corresponding states indicate substantial involvement of non-totally symmetric modes, in particular mode 7, and increase of the multiconfigurational character upon going to higher energies. However, these modes do not make significant contributions to the spectral intensities. In such a case the determination of the wavefunction composition is not really reliable, but we show these results in Table 6 for illustrating the complexity of the problem. Thus, the substantial anharmonicities have a drastic impact on the composition of the vibronically active states, which gain the intensity mainly due to some small admixture of the configurations of the type 1n1 2n2 3n3 . This also explains the rather diffuse character of the spectral envelope.

2.3

Summary and conclusions

We have presented a computational approach for calculating vibronic spectra which feature strong anharmonic effects and an associated very high density of states. This method is based on the Raman wavefunction formalism in conjunction with VCI theory. It was ap˜ 2 B1 ←− X ˜ 1 A1 photoelectron spectrum in difluoromethane, plied to the calculation of the X which has been studied experimentally by a number of groups. The simulated spectrum corresponding to the electronic ground and excited state PESs, represented by up to 3-mode coupling terms, calculated using explicitly correlated coupled-cluster theory, is in very close agreement with the experimental data of Pradeep and Shirley and those of Signorell and co-workers. It was shown that any separable-mode approximation for the ionized PES, as in the harmonic, 1D or VSCF treatments, leads to a completely erroneous description of the bandshape, so that even the global spectral envelope is very poor. This fact indicates that the anharmonicities in the excited state PES have a great impact on the corresponding wavepacket dynamics even at initial times. It was shown that the account of the 2-mode 31



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 6: List of vibrational transitions of CH2 F+• 2 with significant Franck-Condon factors (FCF), as calculated by the RACE method for the correlation space based on reference configuration 11 22 32 determined according to the values of the MCFCFs. Only those states are listed whose leading configurations contribute at least 15%. ν and ν′ denote the relative (with respect to the ground state) and absolute transition energies. ν (cm−1 ) ν′(eV ) 0 998 1135 1250 2098 2114 2206 2248 2361 2492 3135 3156 3274 3289 3350 3361 3437 3493 4298 4535 4577 5491

12.734 12.858 12.875 12.889 12.994 12.997 13.008 13.013 13.027 13.043 13.123 13.125 13.140 13.142 13.149 13.151 13.160 13.167 13.267 13.296 13.302 13.415

FCF

Assignment

0.010 0.012 0.019 0.013 0.024 0.008 0.007 0.022 0.026 0.006 0.006 0.017 0.027 0.012 0.005 0.012 0.013 0.004 0.013 0.004 0.006 0.006

ZPE 31 (0.72) 72 (0.24) 72 (0.63) 31 (0.25) 21 (0.92) 32 (0.49) 92 (0.24) 92 (0.69) 32 (0.14) 74 (0.18) 11 (0.14) 21 72 (0.09) 21 31 (0.72) 21 72 (0.12) 21 72 (0.41) 21 31 (0.15) 22 (0.79) 21 72 (0.10) 72 92 (0.25) 21 52 (0.09) 33 (0.22) 32 72 (0.09) 21 32 (0.15) 31 74 (0.12) 72 92 (0.31) 31 92 (0.16) 21 92 (0.33) 31 74 (0.13) 21 32 (0.13) 21 92 (0.45) 21 32 (0.16) 21 74 (0.13) 76 (0.12) 22 31 (0.24) 31 41 52 (0.20) 21 41 81 91 (0.34) 81 93 (0.29) 32 81 91 (0.15) 22 32 (0.12) 22 41 81 91 (0.17)

32


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coupling terms in the ionized potential has the most prominent influence on the quality of the predicted bandshape. Although the inclusion of 3-mode couplings is less crucial, it yields some noticeable improvements and is responsible for a more diffuse intensity distribution in the high-energy part of the spectrum. Despite the presence of low-lying fragmentation channels in the ionized PES of difluoromethane, we have not found any significant indication that the spectral intensities arise to some extent from unbound states associated with the fragmentaion path. In particular, the increase of the vibrational configuration basis in a broad range does not lead to any noticeable variation in the qualitative composition and localization character of the vibrational states which have non-negligible FCFs. This was proved by the analysis of the eigenfunctions obtained by the RACE and Lanczos methods. Some admixture of highly delocalized configurations in the states with significant intensities occur for very large correlation spaces as soon as the corresponding basis functions span unreliable outer parts of the potential. Solving this issue would require an accurate global PES and a very large vibrational configuration basis. Even if the most significant transitions are not directly related to low-lying dissociation pathways, the latter are associated with a strong anharmonic behavior of a substantial part of the PES, including the region which most affects the spectral composition. In the harmonic approximation, the vibrational states involving excitations in the totally symmetric bending HCH and stretching, CH2 and CF2, modes have the largest intensities due to significant displacements along these coordinates of the excited state origin relative • to the ground state one. Since the low-lying fragmentation path CHF+ 2 +H is supposed to

have a large component along these modes, the corresponding vibrational states are strongly influenced by the anharmonic correlation effects. As analyzed in this study, this impact is so strong that the majority of states with significant FCFs does not involve any leading configurations which are important in the harmonic spectrum, and the corresponding intensity distribution is substantially shifted towards the lower energy part of the spectrum, as compared to the harmonic case.

33



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Acknowledgement Financial support by the Deutsche Forschungsgemeinschaft (Grant No. Ra 656/19-1) is kindly acknowledged. Further support was provided by the CMST COST Action CM1405 ”Molecules in Motion (MOLIM)”. This research was supported in part by the bwHPC initiative and the bwHPC-C5 project provided through associated compute services of the JUSTUS HPC facility at the University of Ulm. bwHPC and bwHPC-C5 are funded by the Ministry of Science, The experimental spectrum has been reprinted from J. Electron Spectrosc. Rel. Phenom. 66, T. Pradeep, D.A. Shirley, High resolution photoelectron spectroscopy of CH2 F2 , CH2 Cl2 and CF2 Cl2 using supersonic molecular beams, 125 (1993), with permission from Elsevier (License Number 4092931333953).

Supporting Information Available Supporting information concerning the potential energy surfaces of CH2 F2 and its radical cation is available. This comprises Cartesian coordinates of the equilibrium structure, displacements vectors for the normal coordinates and polynomial coefficients for the individual terms of the potential. This information is available free of charge via the Internet at http://pubs.acs.org

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Graphical TOC Entry 0.08

HARMONIC

0.07

ANHARMONIC

0.06

FCF-Scaled Intensity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.05

0.04

0.03

0.02

0.01

12.7

12.9

13.1

13.3

13.5

13.7

13.9

14.1

14.3

14.5

Energy [eV]

12.7

12.9

13.1

13.3

13.5

13.7

13.9

14.1

14.3

Energy [eV]

42


14.5

Page 42 of 42

A General Approach for Calculating Strongly Anharmonic Vibronic

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