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Anharmonic excited state frequencies of para-difluorobenzene, toluene and catechol using analytic RI-CC2 second derivatives

David P. Tew*a, Christof Hättigb and Nora K. Grafb
aMax Planck Institute for Solid State Research, 70569 Stuttgart, Germany. E-mail:
bQuantum Chemistry Group, Ruhr-Universität Bochum, 44780 Bochum, Germany. E-mail:

Received 8th November 2018 , Accepted 7th January 2019

First published on 7th January 2019

Analytic second nuclear derivatives for excited electronic state energies have been implemented for the resolution-of-the-identity accelerated CC2, CIS(D) and ADC(2) models. Our efficient implementation with [scr O, script letter O]([scr N, script letter N]2) memory demands enables the treatment of medium sized molecules with large basis sets and high numerical precision and thereby paves the way for semi-numerical evaluation of the higher-order derivatives required for anharmonic corrections to excited state vibrational frequencies. We compare CC2 harmonic and anharmonic excited state frequencies with experimental values for para-difluorobenzene, toluene and catechol. Basis set problems occur for out-of-plane bending vibrations due to intramolecular basis set superposition error. For non-planar molecules and in plane modes of planar molecules, the agreement between theory and experiment is better than 30 cm−1 on average and we reassign a number of experimental bands on the basis of the ab initio predictions.

1 Introduction

The characterisation of molecules in excited electronic states remains a challenge, both for experimental and theoretical chemistry. Electronic excitation is often accompanied by significant structural change and complex intramolecular vibrational energy redistribution processes,1–3 resulting in rich spectra that are sometimes difficult to interpret. Better theoretical treatments of electronic excited states are key to understanding photochemical phenomena and ultimately to harnessing photochemistry as a route to controlling molecular bond fission processes.4–7

Coupled cluster methods are among the most accurate ab initio electronic structure methods.8 Coupled cluster methods for excited state properties have been developed extensively by Stanton and Gauss in the framework of equation of motion coupled cluster (EOM-CC) theory.9–11 Benchmark studies,12–15 found that EOM-CCSD ground and excited state harmonic frequencies agree with values derived from experiment with a root mean squared deviation (RMSD) of 20–30 cm−1.

CC2 was designed16 as approximation to CCSD with an [scr O, script letter O]([scr N, script letter N]5) scaling of the computational costs with system size [scr N, script letter N] instead of [scr O, script letter O]([scr N, script letter N]6). CC2 conserves the order in the fluctuation potential through which single excitation dominated transitions are described correctly in CCSD theory. In the original implementations with exact four-index electron repulsion integrals the high prefactor of the computational costs and the high memory demands for transforming and storing the two-electron integrals severely limited system and basis set sizes for which calculations could be performed on commonly available computer hardware. This bottleneck is removed by combining CC2 with the resolution of the identity (RI) approximation for the two electron integrals.17,18 The RI approximation accelerates the calculations by one to two orders of magnitude, depending on the basis set, reduces memory demands to [scr O, script letter O]([scr N, script letter N]2) and reduces disc storage demands to [scr O, script letter O]([scr N, script letter N]3). These computational savings make it possible to study large molecules such as chlorophylls,19 which previously could only be studied with CIS or TDDFT.

The same technique can be used to accelerate other second-order methods for excited states, e.g. configuration interaction singles with a perturbative doubles correction,20 CIS(D), and its iterative variant21 CIS(D), and the algebraic diagrammatic construction through second-order,22 ADC(2), with similarly pivotal efficiency savings. Compared to the non-iterative perturbative CIS(D) approximation, the iterative methods CIS(D), ADC(2), and CC2 have the advantage that they provide a consistent description of excited state potential energy surfaces (PES), even in the region of avoided crossings, and can thus be used more straightforwardly for searching and characterizing stationary points on the excited state PES.

Implementations of analytic excited state gradients of these approximate second order methods have been reported in ref. 23 and 24, where it is demonstrated that memory demands for first derivatives scale at most as [scr O, script letter O]([scr N, script letter N]2) when using the RI approximation. Recently, second-order electronic response properties for ground and excited states and ground state nuclear Hessians as analytic second derivatives of the energy have been implemented.25–27 To preserve both the high computational efficiency and the low storage demands for second derivatives the RI approximation is combined with a numerical Laplace transformation of orbital energy denominators28 for the contribution of double excitation amplitudes to first derivatives of the density matrices.

In the current work, we extend the theory and implementation to analytic geometrical second derivatives for CC2, CIS(D) and ADC(2) excited state energies. Analytic second derivatives are particularly important for obtaining the high numerical accuracy required to calculate semi-numerical third and fourth derivatives for polyatomic molecules. Only recently has a similar route has been pursued for TDDFT.29,30 With the implementation of analytic Hessians for CC2 and ADC(2) it becomes possible to compute anharmonic vibrational spectra of polyatomic molecules with a correlated ab initio wavefunction method. We demonstrate the applicability of our implementation by computing harmonic and anharmonic excited state frequencies for medium sized molecules, which we compare to experimentally observed band centres.

2 Theory

2.1 Excited state Hessian for RI-CC2

The theory and implementation of orbital-relaxed electric second-order response properties for excited states at the RI-CC2 level has been presented in ref. 26. In the current work we focus on the additional theory required for geometric second derivatives. We use identical notation to that of ref. 26 and 27 and, rather than repeat them here, refer the reader to that work for the full definition of all of the terms and intermediates.

In coupled cluster theory, properties of an excited state f can be obtained as derivatives of the excited state quasienergy Lagrangian26

image file: c8cp06952b-t1.tif(1)
The Lagrangian is composed of the ground state energy ECC, the vector functions Ωμ for the ground state cluster amplitudes tμ, with Lagrange multipliers [t with combining macron]fμ, and the excitation energies image file: c8cp06952b-t2.tif. The fourth term is the orbital-rotation constraint, which imposes vanishing Fock matrix elements Fpq for the relevant orbital pairs pqμ0. The last and second last terms determine the phase and normalisation of the left and right eigenvectors of the Jacobi matrix A, by coupling them to the eigenvectors Lf,(0) and Rf,(0) of the unperturbed limit.26

The hessian of the excited state is obtained by differentiating the Lagrangian twice with respect to the nuclear positions. The result can be obtained in the same way as for excited state polarisabilities, presented in ref. 26:

image file: c8cp06952b-t3.tif(2)
Ĵxyex is the expectation value of an effective second order Hamiltonian (vide infra) evaluated with the density for the excited state. The second and third terms depend on first derivatives of integrals and first derivatives of the cluster amplitudes (tyν) and right eigenvectors Rf,yν contained in the derivative one- and two-particle densities. The final two terms collect contributions that are bilinear in first derivatives of the cluster amplitudes and eigenvectors.

The only difference to polarizabilities are that for derivatives with respect to nuclear coordinates the AO basis and consequently all AO one- and two-electron integrals depend on the perturbation. This also gives rise to additional contributions to the derivatives of the MO coefficients related to the changes in the AO overlap. We write the first derivative of the molecular orbital (MO) coefficients with respect to nuclear displacements as

Cx = C0Ux (3)
where Ux contains the derivative of the orbital rotation parameters κx and the derivative of the overlap matrix S[x] in the basis of the unmodified molecular orbitals (UMOs).
image file: c8cp06952b-t4.tif(4)
Here and throughout, a square bracket indicates a derivative of the UMO integrals, which includes the derivative of the AO integrals, but not the derivative of the MO coefficients.31,32 The first term in eqn (2) is the expectation value of an effective second-order Hamiltonian which collects several contributions to the second geometric derivatives of the Hamiltonian in the MO basis:
image file: c8cp06952b-t5.tif(5)
For electric response properties only the second and third terms are present and H[y] has no two-electron part. The round brackets indicate a one-index transformation of the integrals.32 The evaluation of the expectation value 〈Ĵxyex with [scr O, script letter O]([scr N, script letter N]2) memory demands will be described below.

The last two terms in eqn (2) depend only indirectly on the derivatives of the one- and two-electron integrals, through the derivatives of the cluster amplitudes, txμ, and right eigenvectors, Rf,yν. These are evaluated as described in ref. 26 for electronic derivatives, with the difference that now the first derivatives of the AO two-electron integrals have to be included in the calculation of the amplitude and eigenvector derivatives. Detailed expressions for the derivatives of the cluster amplitudes are given in ref. 27 and the changes to the expressions for the derivatives of the eigenvectors are obtained in an analogous way.

The second and third terms in eqn (2) combine (via the densities) first derivatives of the amplitudes tyν and eigenvectors Rf,yν for a coordinate y with first derivatives of the MO Fock matrix Fxpq and two-electron integrals (pq[| with combining circumflex]rs)x for a coordinate x. Here, as for all two-electron integrals in the correlation treatment, we employ the RI approximation

image file: c8cp06952b-t6.tif(6)
with three index intermediates image file: c8cp06952b-t7.tif composed of two-index VPQ = (P|Q) and three-index (pq|P) electron repulsion integrals. The indices P and Q denote functions from an auxiliary basis set. For an efficient evaluation we rewrite the two-particle term as:
image file: c8cp06952b-t8.tif(7)
where Feff,ypq is an effective Fock matrix (cf. ref. 26 and 27) and ΔQ,yαβ and γyPQ are, respectively, a three-index two-particle density in the AO basis (indices α, β) and a two-index two-particle density:
image file: c8cp06952b-t9.tif(8)
image file: c8cp06952b-t10.tif(9)
We highlight that ΔQ,yαβ is evaluated without ever building the four-index two-particle density [d with combining circumflex]nsep,ypqrs. Full working expressions for the effective Fock matrix and for the two- and three-index densities are provided in the ESI.

The one-electron densities DF,ex,y require contractions of the doubles parts of the left eigenvectors and the derivatives of the right eigenvectors of the form image file: c8cp06952b-t11.tif and image file: c8cp06952b-t12.tif. For undifferentiated eigenvectors and amplitudes the RI approximation is sufficient to implement these contractions efficiently with only [scr O, script letter O]([scr N, script letter N]2) memory demands and without storing doubles vectors as four-index quantities on disc. For differentiated eigenvectors, however, this is not the case. There are additional terms that are not simply dressed two-electron integrals, but come from contractions of the undifferentiated eigenvectors with derivatives of the Fock matrix:

image file: c8cp06952b-t13.tif(10)
For full definitions of the intermediates see ref. 26. To convert the last three terms into an expression that is separable in the index pairs ai and bj we apply a numerical Laplace transformation of the orbital energy denominators
image file: c8cp06952b-t14.tif(11)
where θm and wm are the integration points and weights, respectively. With this approximation the eigenvector elements can be written as
image file: c8cp06952b-t15.tif(12)
KQm,ai = [B with combining circumflex]Q,aie(εiεa)θm, (13)
[K with combining macron]Qm,ai = [B with combining macron]Q,ai[Rf]e(εiεa+ωf)θm, (14)
and the transformations with the derivatives of the Fock matrix can be performed on the three-index intermediates. This makes it possible to evaluate the contributions from Rf to the right hand side of Rf,y within the same loops as the contributions from the dressed integrals, just with an additional summation over the Laplace grid points.

We now turn to the evaluation of the expectation value Ĵxy of the second order Hamiltonian for an excited state. The contributions to Ĵxy are grouped into three terms. The first term contains the second derivatives of the AO integrals for the Hamiltonian. It is evaluated in the same manner in the AO basis as the respective contribution from H[x] to the gradient described in ref. 23:

image file: c8cp06952b-t16.tif(15)
Deff,ex is the effective orbital-relaxed excited state and DSCF the SCF one-particle density. h[xy]pq and ACPHF,[xy]pqrs are, respectively, the core Hamiltonian and CPHF matrices computed from the second derivatives of the AO integrals. ΔQαβ and ΔQ,[x]αβ are auxiliary three-index and γPQ and γ[x]PQ two-index two-particle densities defined in the ESI.

All other contributions to 〈Ĵxyex are rewritten as contractions of effective Fock matrices with derivatives of the overlap matrices and Ux. This is done to avoid the evaluations of two-electron integrals in the MO basis and with this any [scr O, script letter O]([scr N, script letter N]5) scaling steps that depend on two perturbations.

The second and third contribution to Ĵxy are combined by introducing a modified first-order Hamiltonian,27 image file: c8cp06952b-t17.tif, to:

image file: c8cp06952b-t18.tif(16)
Such expectation values of one-index transformed Hamiltonians can be evaluated as contraction of the transformation matrices Ux with effective Fock matrices:
image file: c8cp06952b-t19.tif(17)
The definition and calculation of the effective Fock matrices for the modified first-order Hamiltonian image file: c8cp06952b-t20.tif have been given in ref. 26 and 27. The last two contributions to Ĵxy are combined by introducing the second derivative of the overlap matrix Sxy = S[xy] + [P with combining circumflex]xy(Uy,S[x]) to
(S[xy],Ĥ) + [P with combining circumflex]xy((Uy,S[x]),Ĥ) = (Sxy,Ĥ) (18)
and is evaluated as:
image file: c8cp06952b-t21.tif(19)
where Feff,expq is the (unperturbed) excited state effective Fock matrix, also known as energy weighted density matrix. Its implementation for RI-CC2 has been described in ref. 23.

The above formulas have been implemented in the development version of the ricc2 program of the TURBOMOLE package.33 All contributions are evaluated using integral-direct algorithms in the AO basis with [scr O, script letter O]([scr N, script letter N]2) memory demands and strictly avoiding any operations that scale as [scr O, script letter O]([scr N, script letter N]5) with the basis set size and at the same time quadratically with the number of perturbations, so that the computation of the full Hessian still scales only as [scr O, script letter O]([scr N, script letter N]6) with the system size. The time-determining steps are the solution of the linear equations for the first derivatives of the amplitudes and eigenvectors and the calculation of the first-order density matrices that have to be done for each perturbation.

The implementation of the excited state hessians and the excited state polarizabilities enables also the calculation of the derivatives of the (excited state) dipole moment as mixed derivatives by differentiating once with respect to the strength of an electric field and once with respect to the nuclear coordinates.

2.2 CIS(D) and ADC(2)

With only few modifications the implementation is easily adapted to CIS(D) and ADC(2).21,22 In both methods the coupled cluster ground state amplitudes are replaced by the amplitudes from first-order Møller–Plesset perturbation theory (MPPT). We assume that the Brillouin condition is fulfilled and thus the singles ground state amplitudes t(0) and their derivatives tx vanish in first order MPPT, as do the singles Lagrange multipliers [t with combining macron]f,(0). Discarding all contributions from these singles parameters turns the CC2 code into a CIS(D) code. As a side effect, the equations for the derivatives of the cluster amplitudes can be inverted directly, in the canonical implementation, bypassing the iterative solution procedure.

The implementation for ADC(2) is closely related to the implementation CIS(D). Here the secular matrix is symmetrised:22,24

image file: c8cp06952b-t22.tif(20)
This only requires some small modifications in the singles-singles block of the Jacobian to symmetrise the contributions from 〈μ1|[[H,T(1)2],τν1]|HF〉 and the corresponding contributions to the right hand side of the equation for Rf,y and to the derivatives of the Jacobian. Due to the symmetric secular matrix, the left eigenvectors are identical with the right eigenvectors and only one set of eigenvalue equations has to be solved.

2.3 Thresholds and numerical accuracy

The numerical accuracy of the excited state Hessians, and also the computational costs, depend mainly on two thresholds: TLap and TLRE. TLap controls the numerical Laplace transformation such that
image file: c8cp06952b-t23.tif(21)
TLRE is the threshold for the residual of the equations for the first derivatives of the cluster amplitudes and eigenvectors. The grid points for the Laplace transformation were optimised for the interval from εmin = 2(εLUMOεHOMO) − ωf and εmax = 2(εvir,maxεocc,max), where ωf is the excitation energy, by minimizing FLap with respect to θm and wm.

We investigated the dependence of the results on the thresholds for a test set composed of the lowest excited states of glyoxal, methanethial, propinal, benzene and naphthalene at their equilibrium structures, the lowest two excited states of water and formaldehyde at the ground and the excited state equilibrium structure, and the third excited state of formaldehyde and the lowest two excited states of thiophene again at their equilibrium structures. For all 16 cases we computed the Hessian for the cc-pVDZ and the aug-cc-pVDZ basis sets with different values for both thresholds, TLap and TLRE, and evaluated the deviations in the elements of the Hessian from a reference calculation with very tight thresholds. For the aug-cc-pVDZ result the mean absolute deviations of the Hessian elements follow roughly the relation

ΔMADHess,CC2 ≤ 5TLap + 10TLRE (22)
with the maximum deviations 20–50 times larger. For the cc-pVDZ basis set the deviations are between one and two orders of magnitude smaller. In general, we observe that, not unexpectedly, the thresholds have to be set about two orders of magnitude tighter than for ground state calculations and that tighter thresholds are needed if other states are close by.

In this work we used a tight threshold for the response equations TLRE = 10−10 and 11 grid points for the numerical Laplace integration, which corresponds to TLap between 2 × 10−5 and 2 × 10−10. These tight values were required because the third and fourth derivatives for VPT2 theory are obtained by numerical differentiation of the hessian. We kept the number of Laplace points the same for all hessians used in a finite difference formula to prevent any numerical noise from changes of the grid points.

2.4 Calculation of anharmonic corrections and frequencies

The analytical implementation of the hessians enables a semi-numerical evaluation of third and fourth derivatives of the energy, which are sufficiently accurate to be used as cubic and quartic force constants for the calculation of anharmonic frequencies. The third and fourth derivatives are calculated by central finite differences of hessians. Since only semi-diagonal quartic derivatives are required for VPT2 theory, the total number of hessians required is Nacc·(3N − 6) + 1, where Nacc is either two or four. Nacc = 4 is required when four- and five-point formulas are used for cubic and quartic derivatives to reduce the error to [scr O, script letter O](δ4) in the displacement δ.34

We did not use the normal coordinates [l with combining right harpoon above (vector)]m of the equilibrium structure for the displacement vectors since they are normalized in the mass weighted coordinate system, which leads to unbalanced step sizes. Instead, we used rescaled coordinates

image file: c8cp06952b-t24.tif(23)
which ensures that the displacements become nearly independent of the atomic masses. Here, the vector [l with combining right harpoon above (vector)]m denotes the mth eigenvector of the mass weighted hessian M−1/2FM−1/2 and M is the diagonal matrix of the atomic masses, and image file: c8cp06952b-t25.tif is the reduced mass of the normal mode m.

The anharmonic force constants are used for the calculation of anharmonic frequencies, by setting up the vibrational Hamiltonian as:35

Hvib = H0vib + H1vib + H2vib (24)
image file: c8cp06952b-t26.tif(25)
The zeroth order term H0vib is a quantum harmonic oscillator in reduced normal coordinates. The first-order correction H1vib consists of the cubic force constants ϕrst. The second-order correction H2vib contains the quartic force constants ϕrstu and a Coriolos term that depends on the equilibrium rotational constants Beα and the vibrational angular momenta jα.

The anharmonic frequencies are calculated using second order vibrational perturbation theory (VPT2), as it is implemented in the program DYNAMOL.36 Therein, the VPT2 equations and treatment of resonance effects follow the formulation described in the paper of Amos et al.37

3 Results

We have used our new efficient CC2 excited state hessian implementation to construct harmonic and quartic force fields of toluene, para-difluorobenzene and catechol in their first excited electronic state, from which we obtain the band centres of vibrational transitions using VPT2. These medium sized molecules have been studied experimentally and we compare the computed and experimental wavenumbers, assessing the accuracy of the CC2 predictions and correcting assignments where appropriate. The nomenclature of the normal modes is adopted from the systematic studies of the Wright group for substituted benzenes,38–40 which provides a more unique assignment than the Mulliken41 or Wilson42 nomenclature. For modes localised on the substituents, the common spectroscopic notation for stretches (ν), bends (δ) and torsions (τ) is used. Outputs from all of our VPT2 calculations are provided in the ESI. These contain a full list of fundamentals, overtones, combination bands, all Fermi and Darling–Dennyson resonances, and all effective Hamiltonians for the polyads used to treat these resonances variationally.

3.1 para-Difluorobenzene

The molecule para-difluorobenzene has been the focus of a considerable number of spectroscopic explorations to determine its ground and excited state structure and dynamics43–48 including the intramolecular vibrational redistribution pathways in the excited state, which are mediated by rotational coupling and through Fermi resonance.49,50 Here we examine the fundamental vibrational transitions of para-difluorobenzene in its first excited state 11B2u using CC2 theory.

Spectroscopic studies reveal that para-difluorobenzene exhibits D2h symmetry in both its ground and first excited state,45 and the CC2/cc-pVTZ optimised geometries indeed retain D2h symmetry. The optimised structures are listed in the ESI. The computed 0–0 transition is 4.67 eV, which is close to the experimentally determined energy of 4.57 eV.46

Table 1 reports harmonic and anharmonic wavenumbers for the fundamental vibrational transitions of para-difluorobenzene in its first excited state, computed using CC2/cc-pVTZ and CC2/cc-pVQZ levels of theory using the optimised CC2/cc-pVTZ excited state geometry. The harmonic wavenumbers computed using the cc-pV5Z basis set are also listed. Using the normal mode vectors, each transition has been assigned using the Di labeling convention of ref. 39 for para-di-substituted benzenes. For ease of reference, the irreducible representation, Mulliken labeling and ground state experimental wavenumbers have also been included.

Table 1 CC2 excited state vibrational wavenumbers for para-difluorobenzene
pDi Sym. Exp. Harm. VPT2 Exp.
20 30(b3u) 158 120 121 126 120 121 12047 120
14 8(au) 422 148 147 161 169 168 17547 175
19 17(b2g) 374 271 276 280 307 305 27447 274
30 22(b2u) 348 352 353 352 349 350 35247 352
29 27(b3g) 446 388 388 392 380 381 40347 403
13 9(b1g) 800 404 406 411 453 476 58846 *
11 6(ag) 450 411 412 423 407 408 41047 410
18 29(b3u) 505 472 471 473 466 466 43847 438
12 7(au) 945 486 490 503 612 676 58347 *
17 16(b2g) 692 527 546 539 651 647 52847 *
28 26(b3g) 635 558 558 558 541 542 55847 558
15 15(b2g) 838 610 616 623 700 658 67047 670
16 28(b3u) 928 614 621 624 636 813 61947 *
10 14(b1u) 740 716 716 713 702 703 66646 666
9 5(ag) 859 824 825 819 812 811 81847 818
8 13(b1u) 1014 960 962 961 949 951 93747 937
27 21(b2u) 1085 1025 1025 1022 1014 1006 110048  
7 4(ag) 1140 1119 1117 1116 1100 1102 111646 1116
6 12(b1u) 1228 1228 1222 1218 1203 1195 101546  
26 25(b3g) 1285 1254 1257 1256 1241 1244 93346  
5 3(ag) 1257 1273 1265 1260 1241 1234 125147 1251
25 20(b2u) 1306 1317 1319 1315 1297 1298 159148  
4 11(b1u) 1514 1449 1444 1439 1412 1408 133546 1335
23 24(b3g) 1595 1510 1509 1497 1475 1475 151648 1516
3 2(ag) 1615 1557 1552 1540 1507 1503    
24 19(b2u) 1437 1680 1677 1664 1635 1632   1591
22 23(b3g) 3085 3253 3252 3246 3118 3116    
2 10(b1u) 3073 3257 3255 3249 3118 3116    
21 18(b2u) 3091 3267 3265 3258 3107 3103    
1 1(ag) 3088 3271 3268 3261 3125 3123    

Examining the basis set convergence of the harmonic frequencies, we find that while the in plane modes converge rapidly with the basis set, the out of plane modes (au, b1g, b2g and b3u) converge slowly, with deviations of more than 10 cm−1 between cc-pVTZ and cc-pV5Z values for the low frequency vibrations. As highlighted by Martin, Taylor and Lee for benzene and acetylene,51,52 modes that break planarity suffer from basis set inconsistency errors that artificially lower the frequency and overestimate anharmonic terms in the quartic force field.

This artificial enhancement of the anharmonic couplings to out-of-plane vibrations complicates ab initio assignment the experimental bands. These difficulties notwithstanding, our calculations confirm the majority of the assignments of the 24 fundamental bands observed experimentally. The rightmost two columns of Table 1 list the previous assignment as collated in ref. 47 and our own ab initio assignment, respectively.

The fundamentals D6 and D27 were tentatively assigned to a b1u band at 1015 cm−1 and a b2u band at 1100 cm−1, respectively, in gas phase two-photon spectroscopy measurements.48 These assignments can be confidently discarded on the basis of our calculations and it is likely that these bands do not correspond to fundamental transitions. Similarly, Knight and Kable's tentative assignment of the b3g band at 933 to the D26 fundamental can also be discarded. There is a somewhat larger than expected discrepancy between the predicted and observed D4 fundamental, which we cannot explain. Finally, we make a reassignment of the b2u band at 1591 cm−1 to D24. The problems encountered with the low frequency out of plane modes prevent meaningful comment of the assignments of modes D12, D13, D16 and D17, which is indicated by an asterix in the table. For the in plane modes, the overall agreement between anharmonic CC2/cc-pVQZ and experimentally observed transitions is very good, with an RMSD of 26 cm−1.

3.2 Toluene

The excited state vibrational frequencies of toluene have been studied using dispersed fluorescence spectroscopy,53 UV-IR double resonance spectroscopy,54 one-colour resonance-enhanced multiphoton ionization and two-colour zero kinetic energy spectroscopy.55,56 21 of the 39 fundaments have been assigned, as have the methyl group internal rotor bands, five further overtone bands and several combination bands in the range 200–1500 cm−1.

The CC2/cc-pVTZ optimised structures have Cs symmetry in both the ground and first (1B2) excited states. In the ground state the carbon atoms fall in a plane, one hydrogen of the methyl group orientated perpendicular to the plane. In the excited state, the methyl group moves slightly out of plane and has a dihedral angle of 4 degrees with the benzene ring. The 0–0 transition energy for the first excited state is at 4.65 eV,53 which is closely reproduced by CC2/cc-pVTZ theory, which yields 4.86 eV.

Table 2 summarises the calculated and experimental frequencies of toluene in its first excited state. We list harmonic frequencies in cc-pVTZ, cc-pVQZ and cc-pV5Z basis sets, computed at the optimised cc-pVTZ structure, and VPT2 anharmonic frequencies using the cc-pVTZ basis set. The normal mode vectors were analysed and classified using the Mi nomenclature of Gardner and Wright38 for the ring modes, and given pseudo C2v symmetry labels, where the methyl group is treated as a single pseudo-atom.

Table 2 CC2 excited state vibrational wavenumbers for toluene
Mi Sym. Harm. VPT2 Exp.
τ   86 118 122 28  
20 b1 141 148 151 140 145
14 a2 234 238 238 236 226
19 b1 319 325 325 331 314
30 b2 323 328 329 325 331
18 b1 432 441 445 460  
11 a1 449 450 451 442 457
29 b2 524 524 526 512 532
13 a2 539 544 560 563  
17 b1 556 560 566 552  
16 b1 639 651 669 734 697
12 a2 658 667 686 736  
15 b1 717 733 747 839  
10 a1 751 749 749 737 753
28 b2 932 932 932 915  
8 a1 943 939 938 928 934
9 a1 969 969 972 957 966
δas   1003 1003 1002 972  
δas   1048 1047 1047 1024 1021
27 b2 1157 1155 1156 1137  
7 a1 1162 1160 1160 1143  
6 a1 1220 1214 1214 1196 1193
26 b2 1302 1304 1305 1281  
24 b2 1386 1385 1386 1364  
δs+   1393 1394 1395 1358  
5 a1 1426 1423 1424 1394  
δas+   1474 1472 1472 1432  
δas+   1485 1484 1483 1443  
23 b2 1534 1526 1524 1482  
4 a1 1557 1547 1545 1505  
25 b2 1685 1668 1661 1635  
νs   3024 3015 3013 2921 2893
νas   3098 3091 3088 2956 2956
νas   3142 3135 3133 3013 2988
22 b2 3209 3203 3202 3076 3048
3 a1 3209 3204 3203 3077 3063
2 a1 3218 3212 3210 3079 3077
21 b2 3231 3224 3222 3087 3087
1 a1 3240 3233 3231 3098 3097
20 a1 281 296 302 281 290
14 a1 469 476 476 471 452
19 a1 638 650 650 666 629
8 a1 1885 1878 1876 1854 1868
9 a1 1939 1939 1945 1914 1929
Combination bands
14 + 20 b2 375 386 389 374 371
19 + 20 a1 460 473 476 473 462
14 + 19 b2 553 563 563 566 539
18 + 19 a1 751 766 770 796 734
29 + 30 a1 847 852 855 837 864
11 + 29 b2 973 974 977 956 988
12 + 14 a1 892 905 924 975 916
10 + 29 b2 1275 1273 1275 1248 1263
8 + 11 a1 1392 1389 1389 1370 1390
8 + 29 b2 1467 1463 1464 1439 1463
9 + 11 a1 1418 1419 1423 1399 1426
9 + 29 b2 1493 1493 1498 1468 1494
8 + 9 a1 1912 1908 1910 1885 1900
6 + 29 b2 1743 1738 1740 1706 1727
23 + 25 a1 3219 3194 3185 3121 3101
4 + 25 b2 3241 3215 3206 3140  

Concerning the basis set convergence of the harmonic wavenumbers, we find a similar pattern as for para-difluorobenzene. The in plane modes are converged to within 10 cm−1 with the cc-pVTZ basis, which is well below the intrinsic error bar of 30 cm−1 commonly ascribed to CC2 theory (for ground state frequencies) due to missing higher order correlation effects. As for para-difluorobenzene, the out-of-plane modes display a much slower basis set convergence, with differences of more than 30 cm−1 between cc-pVTZ and cc-pV5Z values for the torsion and for modes M12, M15 and M16.

Overall, the CC2/cc-pVTZ anharmonic frequencies agree very well with the experimental band centres for the fundamental transitions. The only outlier is mode M16, which appears to have artificially enhanced positive anharmonic corrections due to the basis set incompleteness errors. By the same token, we expect the predicted fundamentals for modes M12 and M15 to also lie above the experimental bands, should they be measured in the future. The methyl internal rotation is not expected to be well described through VPT2 theory and requires a more advanced vibrational treatment using a potential energy surface that exhibits the three equivalent minima, see ref. 57–59 for examples of such theories. The experimental assignment of the five aromatic C–H stretching frequencies M1, M2, M3, M21 and M22, is complicated by the presence of Fermi resonances, which have not yet been untangled. Our calculations predict that M21 is in resonance with M4M25 and that both M1 and M2 are in resonance with M23M25. The computed M4M25 and M23M25 combination bands are listed in Table 2 and lie at higher frequency than the principle C–H stretches. Since we lack predicted intensities or experimental band shape information, we pragmatically assign the experimental bands in order of increasing frequency, which results in excellent agreement between computed and experimental values.

In Table 2 we also list selected overtones and combination bands for comparison with those determined experimentally, collected from ref. 53, 54 and 56. The overtone and combination bands agree with experiment to within the error bar expected from the agreement found for the fundamentals. Some Fermi resonances observed in the spectra are not reproduced by our VPT2 calculations. Specifically, we do not find a resonance between M11 and M19M20 and we find no indication that mode M6 is in resonance with other states. Others, however, are identified by our calculations. The resonance between M10 and M18M19 is found in both theory and experiment, but our calculations predict the fundamental M10 to be lower in energy than M18M19, which if reliable would interchanging Garderner et al.'s assignment of the resonant pair, which has M18M19 lower in energy than M10. Our calculations also verify the resonance between M10M29 and M18M19M29.

Overall, the RMSD between observed and predicted band centres using VPT2 theory with a CC2/cc-pVTZ force field is 16 cm−1 for the fundamental transitions in the excited electronic state.

3.3 Catechol

Catechol (l,2-dihydroxybenzene) is biochemically important since the catecholamines adrenaline, noradrenaline, and dopamine are active in neurotransmission. It has been the subject of extensive spectroscopic studies, many of which have focused on the structure and dynamics of the low energy rotamers formed through changing the relative orientations of the hydroxyl groups, and the resulting differing levels of intra- and intermolecular hydrogen bonding.60 The vibrational frequencies of the excited state have been probed using resonant two-photon ionization, fluorescence emission techniques, and molecular-beam hole-burning experiments.61,62

Catechol is planar in the lowest energy isomer of the ground electronic state and the hydroxyl groups form an intermolecular hydrogen bond, but the structure of the excited state is assumed to be slightly distorted out of plane.62 Our structural investigations, using CC2/cc-pVTZ theory to optimise geometries of the ground and excited states, confirm that the ground state is planar with an intermolecular hydrogen bond. We find that the excited state retains the intermolecular hydrogen bond, but is significantly distorted from planarity, puckering at the carbons with the hydroxyl groups. The optimised structures are reported in the ESI. CC2/cc-pVTZ predicts the 0–0 transition at 4.53 eV which is in line with the experimentally determined 0–0 excitation energy of 4.42 eV.62

The calculated and experimental frequencies of the ground and excited states are compiled in Table 3. Only 16 fundamentals of the excited state have been reported. The previous assignment was predicated on the assumption that the selection rules and band shapes for the planar S0 state can be transferred to guide assignment of the S1 state.62,63 Our calculations indicate that this assumption was flawed and we report a completely fresh assignment for the observed S1 vibrational bands. We therefore present frequencies also for the ground state, comparing our purely ab initio assignment procedure to the more comprehensive experimental assignments available for this state.64–67 We use the oDi labeling for ortho-di-substituted benzene rings of ref. 40 and, following the convention of earlier works,64 use ν1, δ1, τ1 to refer to the motions of the hydrogen donating OH group, and ν2, δ2, τ2 to refer to the motions of the hydrogen accepting OH group.

Table 3 CC2/cc-pVTZ vibrational wavenumbers for catechol in the S0 and S1 states
oDi C2v S0 S1
Harm. VPT2 Exp. Harm. VPT2 Exp.
30 a2 178 220 199 148 145  
τ2 a2 209 254   475 425 461
29 b1 293 295 299 130 127 113
21 a1 303 300 320 295 293 299
τ1 b1 432 433   585 576 607
20 b2 438 434 449 399 390 395
28 b1 456 454 456 369 348 317
19 b2 552 545 542 517 486 472
27 a2 561 590 582 342 347  
18 a1 578 569 564 544 495 488
26 a2 674 802 721 444 420  
25 b1 743 744 741 495 508 502
17 a1 772 760 768 732 714 735
24 a2 820 846 851 595 560 588
16 b2 854 840 859 829 812 840
23 b1 897 924 916 761 732 746
22 a2 930 988 963 837 836 863
15 a1 1043 1028 1030 968 950 950
14 b2 1098 1079 1092 1044 1028 1061
13/δ2 a1 1163 1143 1151 1149 1122  
13/δ2 a1 1168 1150 1151 1166 1159  
12 b2 1210 1190 1195 1122 1104  
11 b2 1268 1240 1251 1254 1220  
10 a1 1309 1278 1263 1286 1252  
δ1 b2 1367 1338 1365 1343 1312  
9 a1 1458 1419 1324 1402 1368  
8 b2 1494 1455 1479 1450 1415  
7 a1 1540 1502 1504 1635 1581  
6 b2 1646 1601 1607 1511 1466  
5 a1 1651 1609 1616 1566 1517  
4 b2 3186 3048 3051 3177 3046  
3 a1 3211 3092 3051 3250 3102  
2 b2 3224 3101 3060 3190 3065  
1 a1 3235 3105 3081 3204 3065  
ν1 b1 3750 3553 3605 3612 3390  
ν2 a1 3813 3626 3663 3542 3252  

The procedure adopted for assigning the vibrational modes of the S0 state was as follows: first the normal modes were analysed and categorised according to the oDi nomenclature, resulting in approximate C2v symmetry labels a1, b2, b1, a2 with corresponding type A, B, C line shapes for the IR active bands a1, b2, b1, respectively; these were then assigned in frequency order in effective C2v symmetry blocks to the bands observed by Wilson for vapour phase catechol;64 the only exception to this are the four α(CH) bending modes, where the a1 and b2 labels are swapped compared to Wilson's and where recent work casts doubt on the original assignment.40 Where additional or more precise measurements are available, Wilson's band centres are replaced or supplemented by the modern values and the Raman active a2 modes are assigned in frequency order using measurements from solid state catechol; the bands assigned to the OH bending (δ), stretching (ν) and torsional (τ) modes are identified from examination of the normal mode coordinates.

The overall agreement with experiment and CC2/cc-pVTZ theory is excellent, with an RMSD of 29 cm−1. The difficulties associated with out-of-plane vibrations are much less pronounced here and there is only one outlier of this type, the ring puckering mode D26. The only other significant outlier is mode D9, which is the mode distorting from a aromatic ring to three localised double bonds.

Turning now to the S1 state, comparison with experiment is problematic. Only 16 frequencies have been assigned and the assignment appears flawed. A vibronic progression with a spacing of 113 cm−1 was observed in resonant two-photon ionisation spectra of catechol and was assigned to OH torsional overtones.62 Our calculations do not predict low frequency torsional modes, but in fact predict a significant redshift of the torsional frequencies upon electronic excitation. The more tetrahedral arrangement at the oxygen centres in the excited state structure leads to a stronger intermolecular hydrogen bond, higher torsional frequencies and lower OH stretching frequencies. Instead, our calculations suggest assigning the progression of 113 cm−1 to the low frequency D29 ring bend. Note that the selection rules based on spatial symmetry do not rigorously apply since the excited state structure is significantly distorted out of plane. Having discounted the presence of low-frequency torsional modes, many of the spectral features observed in the experimental works must be reassigned. Proceeding to match computed and experimental values, accounting for band shape information where available, we report a fresh ab initio assignment in Table 3. The RMSD between VPT2 fundamentals using a CC2/cc-pVTZ force field and the observed bands with our fresh assignment is 22 cm−1.

4 Conclusion

Excited state hessians have been implemented for CC2, with a focus on keeping the scaling of the main memory demands to at most [scr O, script letter O]([scr N, script letter N]2). This has been realized by exploiting the RI approximation for the two electron integrals and by choosing a Laplace decomposition of orbital energy denominators in the calculation of the first-order density matrices. The implementation is an extension of that for excited state polarizabilities and was extended straightforwardly to excited state hessians for the CIS(D) and ADC(2) methods. The code is parallelized with OpenMP and MPI to make use of modern computer hardware. The analytic implementation enables the semi-numerical calculation of third and fourth derivatives for anharmonic corrections.

We applied VPT2 theory based on CC2 quartic force fields to para-difluorobenzene, toluene and catechol and compared the computed frequencies to experimentally observed vibrational bands of the first excited states. In contrast to previous benchmark studies, we find that the accuracy of CC2 frequencies for excited electronic states is comparable to MP2 or CC2 frequencies for the ground electronic state, with typical average deviations between theory and experiment of less than 30 cm−1 for fundamental transitions when using a cc-pVTZ basis. However, we find that out-of-plane modes carry a much larger uncertainty due to internal basis set superposition errors, which leads to a strong basis set dependence of the force field terms, and unphysically large anharmonic corrections to typically underestimated harmonic frequencies.

In addition to assessing the accuracy of CC2 theory, our calculations have revealed some anomalies in the assignment of some of the experimentally observed bands. Our calculations discount some of the more tentative assignments in the spectrum of para-difluorobenzene. More significantly, our calculations indicate that the assignment of the lowest frequency features in the vibrational bands of catechol to OH torsions is incorrect, and that these lie much higher in energy due to the stronger hydrogen bond in the distorted S1 excited state than in the planar S0 state. Instead, we assign these features to ring modes, which become symmetry allowed transitions due to the low symmetry of the relaxed excited state structure. Due to this re-interpretation of the experimental bands it was necessary to perform a completely fresh assignment, and our new assignment can be considered a pure ab initio assignment.

Conflicts of interest

There are no conflicts to declare.


Open Access funding provided by the Max Planck Society.


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Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp06952b

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