David P. Tew*^{a},
Christof Hättig^{b} and
Nora K. Graf^{b}
^{a}Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany. E-mail: d.tew@fkf.mpg.de
^{b}Quantum Chemistry Group, Ruhr-Universität Bochum, 44780 Bochum, Germany. E-mail: christof.haettig@rub.de
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 (^{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.
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 designed^{16} as approximation to CCSD with an (^{5}) scaling of the computational costs with system size instead of (^{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 (^{2}) and reduces disc storage demands to (^{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 variant^{21} 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 (^{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 denominators^{28} 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.
In coupled cluster theory, properties of an excited state f can be obtained as derivatives of the excited state quasienergy Lagrangian^{26}
(1) |
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:
(2) |
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
C^{x} = C^{0}U^{x} | (3) |
(4) |
(5) |
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, t^{x}_{μ}, and right eigenvectors, R^{f,y}_{ν}. 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 t^{y}_{ν} and eigenvectors R^{f,y}_{ν} for a coordinate y with first derivatives of the MO Fock matrix F^{x}_{pq} and two-electron integrals (pqrs)^{x} for a coordinate x. Here, as for all two-electron integrals in the correlation treatment, we employ the RI approximation
(6) |
(7) |
(8) |
(9) |
The one-electron densities D^{F,ex,y} require contractions of the doubles parts of the left eigenvectors and the derivatives of the right eigenvectors of the form and . For undifferentiated eigenvectors and amplitudes the RI approximation is sufficient to implement these contractions efficiently with only (^{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:
(10) |
(11) |
(12) |
K_{Qm,ai} = _{Q,ai}e^{(εi−εa)θm}, | (13) |
_{Qm,ai} = _{Q,ai}[R^{f}]e^{(εi−εa+ωf)θm}, | (14) |
We now turn to the evaluation of the expectation value Ĵ^{xy} of the second order Hamiltonian for an excited state. The contributions to Ĵ^{xy} 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:
(15) |
All other contributions to 〈Ĵ^{xy}〉^{ex} are rewritten as contractions of effective Fock matrices with derivatives of the overlap matrices and U^{x}. This is done to avoid the evaluations of two-electron integrals in the MO basis and with this any (^{5}) scaling steps that depend on two perturbations.
The second and third contribution to Ĵ^{xy} are combined by introducing a modified first-order Hamiltonian,^{27} , to:
(16) |
(17) |
(S^{[xy]},Ĥ) + ^{xy}((U^{y},S^{[x]}),Ĥ) = (S^{xy},Ĥ) | (18) |
(19) |
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 (^{2}) memory demands and strictly avoiding any operations that scale as (^{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 (^{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.
The implementation for ADC(2) is closely related to the implementation CIS(D_{∞}). Here the secular matrix is symmetrised:^{22,24}
(20) |
(21) |
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, T_{Lap} and T_{LRE}, 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
Δ^{MAD}_{Hess,CC2} ≤ 5T_{Lap} + 10T_{LRE} | (22) |
In this work we used a tight threshold for the response equations T_{LRE} = 10^{−10} and 11 grid points for the numerical Laplace integration, which corresponds to T_{Lap} 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.
We did not use the normal coordinates _{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
(23) |
The anharmonic force constants are used for the calculation of anharmonic frequencies, by setting up the vibrational Hamiltonian as:^{35}
H_{vib} = H^{0}_{vib} + H^{1}_{vib} + H^{2}_{vib} | (24) |
(25) |
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}
Spectroscopic studies reveal that para-difluorobenzene exhibits D_{2h} symmetry in both its ground and first excited state,^{45} and the CC2/cc-pVTZ optimised geometries indeed retain D_{2h} 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 D_{i} 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.
^{p}D_{i} | Sym. | Exp. | Harm. | VPT2 | Exp. | ||||
---|---|---|---|---|---|---|---|---|---|
S_{0} | TZ | QZ | 5Z | TZ | QZ | S_{1} | S_{1} | ||
20 | 30(b_{3u}) | 158 | 120 | 121 | 126 | 120 | 121 | 120^{47} | 120 |
14 | 8(a_{u}) | 422 | 148 | 147 | 161 | 169 | 168 | 175^{47} | 175 |
19 | 17(b_{2g}) | 374 | 271 | 276 | 280 | 307 | 305 | 274^{47} | 274 |
30 | 22(b_{2u}) | 348 | 352 | 353 | 352 | 349 | 350 | 352^{47} | 352 |
29 | 27(b_{3g}) | 446 | 388 | 388 | 392 | 380 | 381 | 403^{47} | 403 |
13 | 9(b_{1g}) | 800 | 404 | 406 | 411 | 453 | 476 | 588^{46} | * |
11 | 6(a_{g}) | 450 | 411 | 412 | 423 | 407 | 408 | 410^{47} | 410 |
18 | 29(b_{3u}) | 505 | 472 | 471 | 473 | 466 | 466 | 438^{47} | 438 |
12 | 7(a_{u}) | 945 | 486 | 490 | 503 | 612 | 676 | 583^{47} | * |
17 | 16(b_{2g}) | 692 | 527 | 546 | 539 | 651 | 647 | 528^{47} | * |
28 | 26(b_{3g}) | 635 | 558 | 558 | 558 | 541 | 542 | 558^{47} | 558 |
15 | 15(b_{2g}) | 838 | 610 | 616 | 623 | 700 | 658 | 670^{47} | 670 |
16 | 28(b_{3u}) | 928 | 614 | 621 | 624 | 636 | 813 | 619^{47} | * |
10 | 14(b_{1u}) | 740 | 716 | 716 | 713 | 702 | 703 | 666^{46} | 666 |
9 | 5(a_{g}) | 859 | 824 | 825 | 819 | 812 | 811 | 818^{47} | 818 |
8 | 13(b_{1u}) | 1014 | 960 | 962 | 961 | 949 | 951 | 937^{47} | 937 |
27 | 21(b_{2u}) | 1085 | 1025 | 1025 | 1022 | 1014 | 1006 | 1100^{48} | |
7 | 4(a_{g}) | 1140 | 1119 | 1117 | 1116 | 1100 | 1102 | 1116^{46} | 1116 |
6 | 12(b_{1u}) | 1228 | 1228 | 1222 | 1218 | 1203 | 1195 | 1015^{46} | |
26 | 25(b_{3g}) | 1285 | 1254 | 1257 | 1256 | 1241 | 1244 | 933^{46} | |
5 | 3(a_{g}) | 1257 | 1273 | 1265 | 1260 | 1241 | 1234 | 1251^{47} | 1251 |
25 | 20(b_{2u}) | 1306 | 1317 | 1319 | 1315 | 1297 | 1298 | 1591^{48} | |
4 | 11(b_{1u}) | 1514 | 1449 | 1444 | 1439 | 1412 | 1408 | 1335^{46} | 1335 |
23 | 24(b_{3g}) | 1595 | 1510 | 1509 | 1497 | 1475 | 1475 | 1516^{48} | 1516 |
3 | 2(a_{g}) | 1615 | 1557 | 1552 | 1540 | 1507 | 1503 | ||
24 | 19(b_{2u}) | 1437 | 1680 | 1677 | 1664 | 1635 | 1632 | 1591 | |
22 | 23(b_{3g}) | 3085 | 3253 | 3252 | 3246 | 3118 | 3116 | ||
2 | 10(b_{1u}) | 3073 | 3257 | 3255 | 3249 | 3118 | 3116 | ||
21 | 18(b_{2u}) | 3091 | 3267 | 3265 | 3258 | 3107 | 3103 | ||
1 | 1(a_{g}) | 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 (a_{u}, b_{1g}, b_{2g} and b_{3u}) 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 D_{6} and D_{27} were tentatively assigned to a b_{1u} band at 1015 cm^{−1} and a b_{2u} 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 b_{3g} band at 933 to the D_{26} fundamental can also be discarded. There is a somewhat larger than expected discrepancy between the predicted and observed D_{4} fundamental, which we cannot explain. Finally, we make a reassignment of the b_{2u} band at 1591 cm^{−1} to D_{24}. The problems encountered with the low frequency out of plane modes prevent meaningful comment of the assignments of modes D_{12}, D_{13}, D_{16} and D_{17}, 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}.
The CC2/cc-pVTZ optimised structures have C_{s} symmetry in both the ground and first (^{1}B_{2}) 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 M_{i} nomenclature of Gardner and Wright^{38} for the ring modes, and given pseudo C_{2v} symmetry labels, where the methyl group is treated as a single pseudo-atom.
M_{i} | Sym. | Harm. | VPT2 | Exp. | ||
---|---|---|---|---|---|---|
TZ | QZ | 5Z | TZ | S_{1} | ||
τ | 86 | 118 | 122 | 28 | ||
20 | b_{1} | 141 | 148 | 151 | 140 | 145 |
14 | a_{2} | 234 | 238 | 238 | 236 | 226 |
19 | b_{1} | 319 | 325 | 325 | 331 | 314 |
30 | b_{2} | 323 | 328 | 329 | 325 | 331 |
18 | b_{1} | 432 | 441 | 445 | 460 | |
11 | a_{1} | 449 | 450 | 451 | 442 | 457 |
29 | b_{2} | 524 | 524 | 526 | 512 | 532 |
13 | a_{2} | 539 | 544 | 560 | 563 | |
17 | b_{1} | 556 | 560 | 566 | 552 | |
16 | b_{1} | 639 | 651 | 669 | 734 | 697 |
12 | a_{2} | 658 | 667 | 686 | 736 | |
15 | b_{1} | 717 | 733 | 747 | 839 | |
10 | a_{1} | 751 | 749 | 749 | 737 | 753 |
28 | b_{2} | 932 | 932 | 932 | 915 | |
8 | a_{1} | 943 | 939 | 938 | 928 | 934 |
9 | a_{1} | 969 | 969 | 972 | 957 | 966 |
δ_{as}^{−} | 1003 | 1003 | 1002 | 972 | ||
δ_{as}^{−} | 1048 | 1047 | 1047 | 1024 | 1021 | |
27 | b_{2} | 1157 | 1155 | 1156 | 1137 | |
7 | a_{1} | 1162 | 1160 | 1160 | 1143 | |
6 | a_{1} | 1220 | 1214 | 1214 | 1196 | 1193 |
26 | b_{2} | 1302 | 1304 | 1305 | 1281 | |
24 | b_{2} | 1386 | 1385 | 1386 | 1364 | |
δ_{s}^{+} | 1393 | 1394 | 1395 | 1358 | ||
5 | a_{1} | 1426 | 1423 | 1424 | 1394 | |
δ_{as}^{+} | 1474 | 1472 | 1472 | 1432 | ||
δ_{as}^{+} | 1485 | 1484 | 1483 | 1443 | ||
23 | b_{2} | 1534 | 1526 | 1524 | 1482 | |
4 | a_{1} | 1557 | 1547 | 1545 | 1505 | |
25 | b_{2} | 1685 | 1668 | 1661 | 1635 | |
ν_{s} | 3024 | 3015 | 3013 | 2921 | 2893 | |
ν_{as} | 3098 | 3091 | 3088 | 2956 | 2956 | |
ν_{as} | 3142 | 3135 | 3133 | 3013 | 2988 | |
22 | b_{2} | 3209 | 3203 | 3202 | 3076 | 3048 |
3 | a_{1} | 3209 | 3204 | 3203 | 3077 | 3063 |
2 | a_{1} | 3218 | 3212 | 3210 | 3079 | 3077 |
21 | b_{2} | 3231 | 3224 | 3222 | 3087 | 3087 |
1 | a_{1} | 3240 | 3233 | 3231 | 3098 | 3097 |
Overtones | ||||||
20 | a_{1} | 281 | 296 | 302 | 281 | 290 |
14 | a_{1} | 469 | 476 | 476 | 471 | 452 |
19 | a_{1} | 638 | 650 | 650 | 666 | 629 |
8 | a_{1} | 1885 | 1878 | 1876 | 1854 | 1868 |
9 | a_{1} | 1939 | 1939 | 1945 | 1914 | 1929 |
Combination bands | ||||||
14 + 20 | b_{2} | 375 | 386 | 389 | 374 | 371 |
19 + 20 | a_{1} | 460 | 473 | 476 | 473 | 462 |
14 + 19 | b_{2} | 553 | 563 | 563 | 566 | 539 |
18 + 19 | a_{1} | 751 | 766 | 770 | 796 | 734 |
29 + 30 | a_{1} | 847 | 852 | 855 | 837 | 864 |
11 + 29 | b_{2} | 973 | 974 | 977 | 956 | 988 |
12 + 14 | a_{1} | 892 | 905 | 924 | 975 | 916 |
10 + 29 | b_{2} | 1275 | 1273 | 1275 | 1248 | 1263 |
8 + 11 | a_{1} | 1392 | 1389 | 1389 | 1370 | 1390 |
8 + 29 | b_{2} | 1467 | 1463 | 1464 | 1439 | 1463 |
9 + 11 | a_{1} | 1418 | 1419 | 1423 | 1399 | 1426 |
9 + 29 | b_{2} | 1493 | 1493 | 1498 | 1468 | 1494 |
8 + 9 | a_{1} | 1912 | 1908 | 1910 | 1885 | 1900 |
6 + 29 | b_{2} | 1743 | 1738 | 1740 | 1706 | 1727 |
23 + 25 | a_{1} | 3219 | 3194 | 3185 | 3121 | 3101 |
4 + 25 | b_{2} | 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 M_{12}, M_{15} and M_{16}.
Overall, the CC2/cc-pVTZ anharmonic frequencies agree very well with the experimental band centres for the fundamental transitions. The only outlier is mode M_{16}, 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 M_{12} and M_{15} 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 M_{1}, M_{2}, M_{3}, M_{21} and M_{22}, is complicated by the presence of Fermi resonances, which have not yet been untangled. Our calculations predict that M_{21} is in resonance with M_{4}M_{25} and that both M_{1} and M_{2} are in resonance with M_{23}M_{25}. The computed M_{4}M_{25} and M_{23}M_{25} 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 M_{11} and M_{19}M_{20} and we find no indication that mode M_{6} is in resonance with other states. Others, however, are identified by our calculations. The resonance between M_{10} and M_{18}M_{19} is found in both theory and experiment, but our calculations predict the fundamental M_{10} to be lower in energy than M_{18}M_{19}, which if reliable would interchanging Garderner et al.'s assignment of the resonant pair, which has M_{18}M_{19} lower in energy than M_{10}. Our calculations also verify the resonance between M_{10}M_{29} and M_{18}M_{19}M_{29}.
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.
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 S_{0} state can be transferred to guide assignment of the S_{1} state.^{62,63} Our calculations indicate that this assumption was flawed and we report a completely fresh assignment for the observed S_{1} 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 ^{o}D_{i} labeling for ortho-di-substituted benzene rings of ref. 40 and, following the convention of earlier works,^{64} use ν_{1}, δ_{1}, τ_{1} to refer to the motions of the hydrogen donating OH group, and ν_{2}, δ_{2}, τ_{2} to refer to the motions of the hydrogen accepting OH group.
^{o}D_{i} | C_{2v} | S_{0} | S_{1} | ||||
---|---|---|---|---|---|---|---|
Harm. | VPT2 | Exp. | Harm. | VPT2 | Exp. | ||
30 | a_{2} | 178 | 220 | 199 | 148 | 145 | |
τ_{2} | a_{2} | 209 | 254 | 475 | 425 | 461 | |
29 | b_{1} | 293 | 295 | 299 | 130 | 127 | 113 |
21 | a_{1} | 303 | 300 | 320 | 295 | 293 | 299 |
τ_{1} | b_{1} | 432 | 433 | 585 | 576 | 607 | |
20 | b_{2} | 438 | 434 | 449 | 399 | 390 | 395 |
28 | b_{1} | 456 | 454 | 456 | 369 | 348 | 317 |
19 | b_{2} | 552 | 545 | 542 | 517 | 486 | 472 |
27 | a_{2} | 561 | 590 | 582 | 342 | 347 | |
18 | a_{1} | 578 | 569 | 564 | 544 | 495 | 488 |
26 | a_{2} | 674 | 802 | 721 | 444 | 420 | |
25 | b_{1} | 743 | 744 | 741 | 495 | 508 | 502 |
17 | a_{1} | 772 | 760 | 768 | 732 | 714 | 735 |
24 | a_{2} | 820 | 846 | 851 | 595 | 560 | 588 |
16 | b_{2} | 854 | 840 | 859 | 829 | 812 | 840 |
23 | b_{1} | 897 | 924 | 916 | 761 | 732 | 746 |
22 | a_{2} | 930 | 988 | 963 | 837 | 836 | 863 |
15 | a_{1} | 1043 | 1028 | 1030 | 968 | 950 | 950 |
14 | b_{2} | 1098 | 1079 | 1092 | 1044 | 1028 | 1061 |
13/δ_{2} | a_{1} | 1163 | 1143 | 1151 | 1149 | 1122 | |
13/δ_{2} | a_{1} | 1168 | 1150 | 1151 | 1166 | 1159 | |
12 | b_{2} | 1210 | 1190 | 1195 | 1122 | 1104 | |
11 | b_{2} | 1268 | 1240 | 1251 | 1254 | 1220 | |
10 | a_{1} | 1309 | 1278 | 1263 | 1286 | 1252 | |
δ_{1} | b_{2} | 1367 | 1338 | 1365 | 1343 | 1312 | |
9 | a_{1} | 1458 | 1419 | 1324 | 1402 | 1368 | |
8 | b_{2} | 1494 | 1455 | 1479 | 1450 | 1415 | |
7 | a_{1} | 1540 | 1502 | 1504 | 1635 | 1581 | |
6 | b_{2} | 1646 | 1601 | 1607 | 1511 | 1466 | |
5 | a_{1} | 1651 | 1609 | 1616 | 1566 | 1517 | |
4 | b_{2} | 3186 | 3048 | 3051 | 3177 | 3046 | |
3 | a_{1} | 3211 | 3092 | 3051 | 3250 | 3102 | |
2 | b_{2} | 3224 | 3101 | 3060 | 3190 | 3065 | |
1 | a_{1} | 3235 | 3105 | 3081 | 3204 | 3065 | |
ν_{1} | b_{1} | 3750 | 3553 | 3605 | 3612 | 3390 | |
ν_{2} | a_{1} | 3813 | 3626 | 3663 | 3542 | 3252 |
The procedure adopted for assigning the vibrational modes of the S_{0} state was as follows: first the normal modes were analysed and categorised according to the ^{o}D_{i} nomenclature, resulting in approximate C_{2v} symmetry labels a_{1}, b_{2}, b_{1}, a_{2} with corresponding type A, B, C line shapes for the IR active bands a_{1}, b_{2}, b_{1}, respectively; these were then assigned in frequency order in effective C_{2v} 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 a_{1} and b_{2} 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 a_{2} 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 D_{26}. The only other significant outlier is mode D_{9}, which is the mode distorting from a aromatic ring to three localised double bonds.
Turning now to the S_{1} 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 D_{29} 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}.
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 S_{1} excited state than in the planar S_{0} 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.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp06952b |
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