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Jet-Cooled Phosphorescence Excitation Spectrum of the T1(n,π*) ← S0 Transition of 4H-Pyran-4-one
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A: Structure, Spectroscopy, and Reactivity of Molecules and Clusters

Jet-Cooled Phosphorescence Excitation Spectrum of the T1(n,π*) ← S0 Transition of 4H-Pyran-4-one
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  • Sean W. Parsons
    Sean W. Parsons
    Department of Chemistry and Biochemistry, University of Wisconsin-Eau Claire, 105 Garfield Avenue, Eau Claire, Wisconsin 54701, United States
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  • Devon G. Hucek
    Devon G. Hucek
    Department of Chemistry and Biochemistry, University of Wisconsin-Eau Claire, 105 Garfield Avenue, Eau Claire, Wisconsin 54701, United States
    More by Devon G. Hucek
  • Piyush Mishra
    Piyush Mishra
    Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, Indiana 47907, United States
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  • David F. Plusquellic
    David F. Plusquellic
    Applied Physics Division, National Institute of Standards and Technology, 325 Broadway Avenue, Boulder, Colorado 80305, United States
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  • Timothy S. Zwier
    Timothy S. Zwier
    Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, Indiana 47907, United States
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    Orcidhttps://orcid.org/0000-0002-4468-5748
  • Stephen Drucker*
    Stephen Drucker
    Department of Chemistry and Biochemistry, University of Wisconsin-Eau Claire, 105 Garfield Avenue, Eau Claire, Wisconsin 54701, United States
    *E-mail: [email protected]. Phone: (715) 836-5390.
    More by Stephen Drucker
    Orcidhttps://orcid.org/0000-0003-3474-7930
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The Journal of Physical Chemistry A

Cite this: J. Phys. Chem. A 2023, 127, 16, 3636–3647
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https://doi.org/10.1021/acs.jpca.3c01059
Published April 17, 2023

Copyright © 2023 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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The 4H-pyran-4-one (4PN) molecule is a cyclic conjugated enone with spectroscopically accessible singlet and triplet (n,π*)excited states. Vibronic spectra of 4PN provide a stringent test of electronic-structure calculations, through comparison of predicted vs measured vibrational frequencies in the excited state. We report here the T1(n,π*) ← S0 phosphorescence excitation spectrum of 4PN, recorded under the cooling conditions of a supersonic free-jet expansion. The jet cooling has eliminated congestion appearing in previous room-temperature measurements of the T1 ← S0 band system and has enabled us to determine precise fundamental frequencies for seven vibrational modes of the molecule in its T1(n,π*) state. We have also analyzed the rotational contour of the 000 band, obtaining experimental values for spin–spin and spin-rotation constants of the T1(n,π*) state. We used the experimental results to test predictions from two commonly used computational methods, equation-of-motion excitation energies with dynamical correlation incorporated at the level of coupled cluster singles doubles (EOM-EE-CCSD) and time-dependent density functional theory (TDDFT). We find that each method predicts harmonic frequencies within a few percent of observed fundamentals, for in-plane vibrational modes. However, for out-of-plane modes, each method has specific liabilities that result in frequency errors on the order of 20–30%. The calculations have helped to identify a perturbation from the T2(π,π*) state that leads to unexpected features observed in the T1(n,π*) ← S0 origin band rotational contour.

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Introduction

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In this paper, we report the vibrationally resolved T1(n,π*) ← S0 phosphorescence excitation spectrum of 4H-pyran-4-one (4PN, Figure 1), recorded under the cooling conditions of a supersonic free-jet expansion. The 4PN molecule is a prototypical cyclic enone and serves as a model for investigating the π* ← n chromophore in α,β-unsaturated carbonyls. Triplet excited states of unsaturated molecules can play key roles in photochemistry, (1) combustion, (2) and energy storage. (3) Triplet excited states are generally long-lived, enhancing their abiliity to participate in these processes. The triplet-state lifetimes are long because radiative decay to the singlet ground state is spin-forbidden. However, the attribute of spin-forbiddenness makes it difficult to study triplet excited states spectroscopically when starting with a molecule in its singlet ground state.

Figure 1

Figure 1. Structural formula of 4H-pyran-4-one (4PN), along with the molecule-fixed coordinate system used here. The molecule has a C2v planar equilibrium geometry in its ground and T1(n,π*) states.

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In previous work, (4,5) we have overcome this limitation by using the highly sensitive cavity ringdown (6) (CRD) absorption technique at room temperature or phosphorescence excitation (PE) spectroscopy under jet-cooled conditions (7) to measure singlet–triplet spectra. The jet-cooled PE technique was pioneered in the 1980s to study triplet excited states of small- (8) and medium-sized (9) molecules.
Recently, we measured the T1 ← S0 vibronic band system of 4PN vapor using CRD spectroscopy. (4) Under the ambient conditions of that experiment, the 4PN spectrum is congested with vibronic hot bands (v″ > 0) associated with both the T1(n,π*) ← S0 and the higher-energy S1(n,π*) ← S0 transition. The present jet-cooled PE approach has suppressed nearly all of the vibronic hot bands and significantly narrowed the rotational contours of the observed T1 ← S0 cold bands (v″ = 0). These simplifications have allowed us to confirm (or correct, in certain cases) vibronic assignments we made previously. (4) We observe effective rotational temperatures as on the order of 5 K in the jet expansion, and this has permitted us to analyze the rotational contour of the T1 ← S0 origin (000) band. The analysis provides estimates for spin–spin and spin–rotation constants (10) of the T1(n,π*) state of 4PN.
We have used the vibronic assignments in the jet-cooled PE spectrum to ascertain fundamental frequencies for most of ring vibrational modes in the T1(n,π*) state of 4PN. The measured frequencies have allowed us to evaluate computational methods for treating excited states. Two accessible and generally reliable techniques are equation-of-motion excitation energies with dynamical correlation incorporated at the level of coupled cluster singles doubles (EOM-EE-CCSD) (11,12) and time-dependent density functional theory (TDDFT). (13) At a moderate computational cost, the EOM-EE-CCSD ab initio method can perform nearly as well (14−16) as “gold-standard” methods such as complete active-space second-order perturbation theory (CASPT2) (17) or closed shell coupled cluster singles and doubles with perturbative triples (CC3). (18) The gold standards are too expensive for treating molecules having more than a few heavy atoms (such as 4PN), but EOM-EE-CCSD is feasible for excited-state geometry optimization and harmonic frequency calculations of monocylic molecules. (4,19)
For even larger molecules, the very inexpensive TDDFT method is in widespread use. For example, accurate TDDFT calculations of vibronically resolved spectra are available for molecules such as conjugated dyes containing several fused rings. (20)
Spectroscopic examination of 4PN─particularly its T1(n,π*) excited state─offers a rigorous test of the EOM-EE-CCSD and TDDFT methods. The 4PN molecule has several different functional groups, along with conjugation, and these characteristics can affect the molecular orbitals in subtle ways. Moreover, configuration interaction between T1(n,π*) and other states in the triplet manifold is important, as evidenced by significant differences in T1(n,π*) vs S1(n,π*) fundamental frequencies (4,21) measured for certain corresponding modes. TDDFT does not treat static correlation explicitly and is not expected to reproduce these frequency differences as well as EOM-EE-CCSD can. Though 4PN is near the upper size limit for EOM-EE-CCSD frequency calculations (with triple-ζ quality basis sets), it is possible to carry out these calculations within a few days by using modern parallelizable algorithms. (22)
In a prior investigation of 4PN, (4) we measured the CRD absorption spectrum of the T1(n,π*) ← S0 transition at room temperature. Most of the CRD assignments involved low-frequency, nontotally symmetric ring modes. We inferred their upper-level fundamental frequencies via observation of symmetry-allowed sequence bands such as N11, considered along with known ground-state fundamentals. Now, under jet-cooled conditions, we observe several exceedingly weak, Franck–Condon forbidden, N01 transitions that provide excited-state fundamentals more directly, without relying on combination differences involving ground-state vibrational levels. The ground-state frequencies are uncertain for some modes because they have only been measured in condensed-phase samples. Thus, detection of N01 transitions in the present work has confirmed earlier sequence assignments and has improved the precision of the upper-state fundamentals for several nontotally symmetric modes.
In addition to this specific kind of improvement, the wavenumber assignments we present here are generally better in terms of accuracy and precision than in the room-temperature T1 ← S0 CRD spectrum because of the extreme narrowing of the rotational contours of the jet-cooled vibronic bands.
In the present study, we measured jet-cooled T1(n,π*) ← S0 vibronic transitions in a region up to about +900 cm–1 with respect to the origin band. In the room-temperature CRD spectrum, assignments in the higher wavenumber region are difficult, because T1 ← S0 vibronic features become submerged by hot bands associated with the S1(n,π*) ← S0 system. The latter has a 000 origin band that is about 1000 cm–1 above that of the T1 ← S0 system. At room temperature, the spin-allowed S1← S0 hot bands become much more intense than the spin-forbidden T1 ← S0 transitions, starting at about −400 cm–1 with respect to the S1← S0 origin, or +600 cm–1 relative to the T1 ← S0 origin band. Jet cooling suppresses the S1← S0 hot bands in this region, and has revealed previously indetectable T1 ← S0 bands.
With the experimental advances outlined above, we have been able to use spectroscopic measurements of 4PN to test T1(n,π*) frequency predictions of the EOM-EE-CCSD and TDDFT computational methods comprehensively. This analysis has revealed unexpected shortcomings of the more expensive EOM-EE-CCSD method in this application.
Finally, we provide insights from the analysis of the T1(n,π*) ← S0 origin band rotational contour of 4PN. The overall contour has a unique shape that is amenable to simulation, even though the resolution of the experiment (approximately 0.3 cm–1) does not permit assignment of individual rotational lines. We find that T1(n,π*) inertial constants obtained from both TDDFT and EOM-EE-CCSD calculations can produce acceptable band-contour simulations. However, agreement between simulated and observed contours requires careful choices of spin–spin and spin–rotation parameters (10) for the T1(n,π*) upper state. We have narrowed down a range of spin constants that will qualitatively optimize the contour simulations, given plausible (i.e., TDDFT or EOM-EE-CCSD) inertial constants. For medium-sized molecules such as 4PN in the gas phase, the literature contains very few experimental determinations of triplet-state spin constants. (7,23) Continued experimental investigations along these lines could stimulate the refinement of computational approaches for evaluating spin interactions in rotating molecules.

Experimental and Computational Details

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Experiment

Spectra reported here were measured at the University of Wisconsin-Eau Claire (UW-Eau Claire) using the apparatus described below. At Purdue University, we conducted preliminary experiments to optimize jet expansion conditions and record initial spectra of the T1(n,π*) ← S0 band system of 4PN.
We conduct PE experiments at UW-Eau Claire Claire using a vacuum chamber consisting of a 6-way cross with ISO160 flanges and pumped by an Edwards Diffstak 100 diffusion pump. (24) A pulsed nozzle (Parker Series 9), with an orifice of 1.0 mm, supplies gas for the free-jet expansion. The pressure of helium buffer gas in the stagnation chamber ranges from 1.0 to 5.0 atm (1.0 × 105 to 5.0 × 105 Pa), depending on the spectral features we are investigating. The pulsed nozzle operates at 10 Hz with a pulse width of about 220 μs. The jet expansion is directed toward the throat of the diffusion pump. Under these conditions, the time-averaged chamber pressure is below 1.0 × 10–5 Torr (1.3 × 10–8 Pa) when the nozzle is operating.
To record the spectra reported in this paper, we loaded typically 200 μL of liquid 4PN (Sigma-Aldrich) onto a glass-wool plug and located the plug in the pressurized helium tube leading to the pulsed nozzle. The temperature of the tube was maintained at about 90 °C with a resistive heating rope.
A tunable dye laser (Lambda-Physik Scanmate 2), equipped with a second harmonic generator (SHG), produces ultraviolet light for sample excitation. The dye laser is pumped by a frequency-doubled Nd:YAG laser (Continuum Surelite II), operating at 10 Hz. We used LDS 698 and LDS 722 dyes (Exciton) for the present experiments. The fundamental wavelength (vacuum) of the dye laser is calibrated to an uncertainty of ±0.002 nm using a wavemeter (Burleigh WA-4550). The nominal bandwidth of the dye-laser fundamental output is 0.1 cm–1 at 725 nm. The Nd:YAG pump laser is set to produce 150-mJ pulses at 532 nm, leading to maxiumum dye-laser output of about 30 mJ per pulse at 725 nm and 5 mJ/pulse of frequency-doubled (ultraviolet) light after the visible light passes through an angle-tuned β-(barium borate) SHG crystal.
To measure the T1(n,π*) ← S0 band system of 4PN, we attenuated the dye laser output to obtain ultraviolet pulse energies around 2.5 mJ throughout the interval between 353 and 367 nm. The ultraviolet laser beam passes through a biconvex lens with a focal length of 50 cm and then enters the vacuum chamber perpendicular to the jet expansion, so that the laser’s focal point is about 5 cm beyond the point where the laser and gas expansion cross. This crossing point is 13 mm (i.e., 13 nozzle diameters) downstream of the nozzle orifice.
After the laser excites the expanding gas sample, the isotropic emission is imaged onto a photomultiplier mounted outside the chamber and facing perpendicularly to the jet expansion. The apparatus for light collection includes a 2-in. (51 mm) diameter biconvex lens installed in an f/1 arrangement. The collection efficiency was improved by locating a spherical mirror (90 mm radius of curvature) across from the collection lens, so that rays emanating away from the collection lens reflect directly back onto themselves, enter the lens, and are focused onto the photomultiplier.
The present PE measurements of spin-forbidden T1(n,π*) ← S0 transitions require ca. 10 times greater laser fluence than typical fluorescence–excitation studies of spin-allowed transitions. To attenuate scattered laser light, a long-pass edge filter, with a cutoff wavelength of 375 nm, is located in front of the photomultiplier sensor. Even with the long-pass filter in place, the intense residual laser scatter tends to destabilize the photomultiplier, leading to artifacts in the emission decay curve at long times after the laser pulse. To eliminate this problem, we use a photomultiplier module equipped with a gating circuit (Hamamatsu H11706-40). The gate is kept closed during the laser pulse and opens 300 ns later. We record the photomultiplier signal over a time interval between 1.0 and 3.0 μs after the excitation. The phosphorescence decay signal drops to zero after approximately 8 μs. The decay function is a convolution of the long-lived radiative relaxation and the dropoff due to traversal of excited-state molecules out of the detector’s viewing region.
We record the phosphorescence decay curve by using a PC-based waveform digitizer (AlazarTech 9130) having 12 bits of vertical resolution and a 50 Ms/sec acquisition rate. At a given wavelength, we download and average the decay curves from typically 80 laser shots. From the averaged curve, we calculate and plot the average voltage over the selected time interval. We repeat this process at UV laser wavelength intervals of typically 0.01 nm to record an entire spectrum.

Computational Methods

We carried out quantum-chemical calculations on the 4PN molecule as described previously. (4) In brief, we used the Q-chem 5.2 (25) computational chemistry package for TDDFT calculations within the Tamm–Dancoff approximation. We performed a TDDFT geometry optimization of the T1(n,π*) state, followed by a harmonic-frequency calculation using the Perdew, Burke, and Enzerhof hybrid functional without the adjustable parameters (PBE0) (26) XC functional and def2-TZVP basis set. We also used Q-chem 5.2 to carry out EOM-EE-CCSD geometry optimizations of the T1(n,π*) state, employing the cc-pVTZ, ANO1, or 6-311G(2pd,2df) basis set. The geometry optimizations were followed by EOM-EE-CCSD harmonic-frequency calculations using the CFOUR 2.1 (27) package. We used the frozen-core approximation for all EOM-EE-CCSD calculations.
We employed the WebMO graphical interface program (28) to visualize and arrive at descriptions of the 4PN normal modes, and to render isosurfaces of the molecule’s canonical molecular orbitals.

Results

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Vibronic Analysis

Figure 2 is an overview of the 4PN spectral region we measured via jet-cooled PE spectroscopy, using a helium backing pressure of 5 atm. We used rotational contours to confirm that the vibronic bands in this region belong to the T1(n,π*) ← S0 system. Later, we present a detailed analysis of the rotational structure; but in brief, the selection rules can be formulated using an angular-momentum coupling scheme analogous to Hund’s case (b) for linear molecules. (10,29) A pattern-forming quantum number is N, which represents the rotation of the molecular framework in space. In 4PN, the T1(n,π*) ← S0 origin band shows a distinctive three-peak contour (Figure 2 inset), corresponding to intense Q-, R-, and S-form branches (ΔN = 0, +1, and +2, respectively) that are present within overlapping ΔKa = 0 subbands.

Figure 2

Figure 2. Overview of the T1(n,π*) ← S0 phosphorescence excitation spectrum of 4PN, recorded using a supersonic free-jet expansion. The buffer gas was 5 atm of helium. The maximum of the 000 origin band is located at 27293.2 cm–1. Inset shows the origin band on an expanded horizontal scale.

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Under the present jet-cooled conditions, the maximum of the 000 band is at 27 293.2 cm–1. This value is slightly different from the location of the origin-band maximum in the room-temperature CRD spectrum, (4) 27 290.2 cm–1. In the latter case, the maximum occurs within the Q-form branch, whereas in the jet-cooled spectrum, the maximum is in the R-form branch.
Spin–orbit mixing between the T1(n,π*) (A2) state of 4PN and nearby singlet excited states provides the oscillator strength for the T1(n,π*) ← S0 electronic transition. The three-peak contour of the origin band is associated with a transition dipole moment that lies mainly in the z direction (Figure 1). (29) This indicates that the dominant contribution to oscillator strength comes from the S1(π,π*) (A1) state.
Other assigned bands within the T1 ← S0 system have the same three-peak rotational contour as the origin band, as long as the vibrational wave functions in the ground and excited state belong to the same irreducible representation in the C2v point group, so that the Franck–Condon (FC) factor is nonzero. In other cases (for example, the 1801 (b1) out-of-plane ring-bending fundamental), the FC factor vanishes by symmetry, but the transition is made very slightly allowed through vibronic interaction between T1(n,π*) and another triplet excited state. (The electronic symmetry of the T1(n,π*) state is A2, and so the overall vibronic symmetry of a b1 vibrational state in T1 is (b1 × A2) = B2. These vibronic states can interact with the T3(π,π*) B2 electronic state, which itself gains oscillator strength via spin–orbit coupling with the S2(π,π*) A1 state. The spin–orbit coupling is mediated by the |y⟩ (B2) triplet spin component.) Fundamental bands in this category, such as 1801 and 1701 (b1 out-of-plane carbonyl wag), are extremely weak, but we predicted their locations within 1 cm–1 by using sequence band positions (e.g., 1811) available from the room-temperature CRD spectrum, (4) along with known (30) ground-state fundamentals. Figure 3 shows the 1801 and 1701 band assignments in the jet-cooled spectrum.

Figure 3

Figure 3. Low-wavenumber region of the spectrum in Figure 2. Blue tie lines show assignments of vibronic cold bands (v″ = 0), and red tie lies show assignments of hot bands (v″ > 0). Franck–Condon factors (FCF) were computed using a procedure described in the text. Assignment labels without tie lines, shown above the spectrum, refer to FC-forbidden bands.

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Higher-frequency band assignments, both FC-allowed and forbidden, are indicated in Figures 3, 4, and 5. To facilitate the assignments, we used a TDDFT (PBE0) harmonic-frequency calculation of the T1(n,π*) state, reported previously in conjunction with our room-temperature CRD investigation of 4PN. (4) The TDDFT calculation provided normal-mode descriptions listed in Table 1.

Figure 4

Figure 4. Continuation of the spectrum in Figure 3.

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Figure 5

Figure 5. Continuation of the spectrum in Figure 4. The black trace is plotted using the same vertical scale as in Figures 3 and 4. The green trace is plotted on a contracted (× 1/4) vertical scale in order to show peaks near 800 and 900 cm–1, which have maximum intensity.

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Table 1. Calculated Normal Modes for the T1(n,π*) State of 4PN
Mode numberSymmetryDescription
27b2in-plane carbonyl wag
26b2in-plane ring bend
18b1out-of-plane ring bend
17b1out-of-plane carbonyl wag + ring bend
16b1ring inversion
15b1out-of-plane C–H wag + ring bend
13a2ring twist
10a1ring breathe
9a1ring breathe + carbonyl stretch
8a1ring breathe + C–O–C symmetric stretch
We used the normal-mode vectors from this calculation, along with normal modes of the ground state from a DFT (PBE0) calculation, to estimate the FC factors for vibronic bands in the T1(n,π*) ← S0 spectrum. We calculated the relative intensity of each band as (FC factor × Boltzmann factor). This product is represented in Figures 3–5 by the length of a tie line attached to an observed vibronic band. The Boltzmann factors were determined using a nominal vibrational temperature of 150 K, along with experimental (30) ground-state vibrational frequencies. The entire observed spectrum was scaled vertically to make the origin-band maximum numerically equal to its calculated FC factor. Thus, for a given vibronic band, the tie line representing the calculated intensity (FC factor × Boltzmann factor) will match the peak height if the assignment is correct and the relative intensity is quantitatively accurate. This condition helped us finalize assignments of FC-allowed vibronic bands.
Intensity predictions are not readily available for FC-forbidden bands (those made allowed by triplet–triplet vibronic interaction). These bands have distinctive rotational contours containing a single narrow maximum. In these cases, for example the 1501 and 1601 fundamentals, assignments are based simply on proximity to the band positions predicted by the TDDFT calculation.
In Figures 3–5, hot-band assignments are indicated in red. Hot-band sequences 1011, 1311, and 2611 are prominent, despite the jet cooling. The FC factors for these bands are large and nearly the same as that of the origin band, because the ring geometry does not change significantly upon electronic excitation. In the highest-wavenumber region of the spectrum (Figure 5), the number of T1(n,π*) ← S0 assignments is limited because of overlapping hot bands belonging to the S1(n,π*) ← S0 system. Figure 6 shows the result of reducing the helium pressure to produce a warmer jet expansion in this region of the spectrum. The intensity of S1(n,π*) ← S0 vibronic hot bands increases under these conditions. An example is the peak at 27 971 cm–1, assigned to the 1310 band of the S1← S0 transition. The origin of this band system (21) is at 28 365 cm–1, and the −394 cm–1 shift corresponds to the experimentally known (30) ground-state fundamental for ν13.

Figure 6

Figure 6. Region of T1(n,π*) ← S0 phosphorescence excitation spectrum of 4PN, about 700 cm–1 above the origin band. These traces show the suppression of S1(n,π*) ← S0 hot bands under colder helium expansion conditions.

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Table 2 lists the observed vibronic band positions, along with the assignments shown in Figures 3–5. The locations of combination bands and overtones are nearly equal to those inferred from observed fundamentals (see second column of Table 2), with discrepancies attributable to anharmonicity. Mode 17 (out-of-plane carbonyl wag) is particularly anharmonic, as we have discussed previously. (4)
Table 2. Vibronic Band Positionsa in the T1(n,π*) ← S0 PE Spectrum of Jet-Cooled 4PN
Observed pos. (cm–1)Inferred pos. (cm–1)AssignmentInference (harmonic approx.) based on fundamental frequenciesb (cm–1)Prev. assign.c
–30.3d–431011463.3 – 5041011
–23.1–25.81811123.0 – 148.761811
0 000 000
11.7 1311  
22.3 2611 1311
110.1107.716011710531.6 – 423.9216011710
123.0 1801  
206.2 1701  
253.0246.018022(123.0) 
260.6194.717133(206.2) – 423.92 
322.4313.916011711531.6 + 206.2 – 423.92 
336.1329.217011801206.2 + 123.017011801
430.9412.417022(206.2)1702
463.3 1001 1001
473.5452.21701 1802206.2 + 2(123.0) 
531.6 1601  
602.9 1501  
656.0654.61601 1801531.6 + 123.0 
667.766227022(331)e 
748.3737.81601 1701531.6 + 206.2 
770.6 801 2711 8012711
786.3 901 901
894.8 801 801
a

Listed band positions are maxima relative to the 000 origin-band maximum, observed at 27 293.2 cm–1.

b

Excited-state fundamentals are from N01 band positions in this work; ground-state fundamentals are from refs (30) and (31).

c

Previous assignments from ref (4) are listed for bands observed both at room temperature (ref (4)) and in the present jet-cooled spectrum.

d

Uncertainty in a band maximum, due to noise at the tops of the peaks, is ±0.5 cm–1.

e

The fundamental frequency of ν27 in the T1(n,π*) state was estimated by using the difference in 8012711 and 801 band positions, along with the ν27 ground-state fundamental from ref (30).

The vibronic band assignments in this study of jet-cooled 4PN are a superset of those we offered for the room-temperature CRD spectrum. (4) Many of the CRD assignments were tentative because T1 ← S0 peaks sat atop a broad, uneven baseline containing hot bands belonging to both T1 ← S0 and S1← S0 transitions. In the present jet-cooled work, all bands are observed as distinct peaks emerging from a nearly flat baseline. Table 2 shows that the assignments preferred in the present jet-cooled investigation match those of the room-temperature CRD spectrum, with the exception of just the 2611 band. This band was misassigned (4) as 1311 in the room-temperature spectrum. We now find that the 1311 band is located at a position close to the 000 band, where it would have been submerged at room temperature by the very intense and broad origin peak.
We used the spectroscopic assignments discussed above to ascertain fundamental vibrational frequencies for the T1(n,π*) state of 4PN. These results are listed in Table 3. For modes of a1 and b1 symmetry, the fundamental frequency in the T1(n,π*) state was taken as the difference between the locations of the N01 and 000 band maxima. For these modes, experimental uncertainty in the T1(n,π*) frequency is less than 1 cm–1 and is due mainly to noise at the top of observed peaks in the PE spectrum. For modes of a2 and b2 symmetry, the N01 bands were not detectable, but we inferred their positions by adding the experimentally known ground-state fundamental to the N11 band position measured in the PE spectrum. For these modes, experimental uncertainty in the T1 fundamental is on the order of 5 cm–1, because the required ground-state frequencies are available (30) only for condensed-phase rather than dilute gas samples.
Table 3. Vibrational Frequencies (cm–1) in the T1(n,π*) and S0 Electronic States of 4PN
  ν18 (b1) oo-plane ring bendν17 (b1) oo-plane C═O wagν16 (b1) ring in-versionν15 (b1) oo-plane C–H wagν13 (a2) ring twistν27 (b2) in-plane C═O wagν26 (b2) in-plane ring bendν10 (a1) ring breatheν9 (a1) ring stretchν8 (a1) ring breathe
T1(n,π*)TDPBE0/def2-TZVP137270584676422353671476821933
 EOM/cc-pVTZa70131548720521347667471803928
 EOM/ANO173137553721515349668471803926
 EOM/6-311G(2pd,2df)106155560728─b347669472805932
 Expt. fundamentalc123.0206.2531.6602.9407331663463.3786.3894.8
            
S0Expt. fundamental148.76d423.92d720e847e395f453f641f504f822f920.e
a

EOM-EE-CCSD calculations employed the frozen-core approximation, a preferred choice when using the cc-pVTZ or ANO1 basis set. Computed EOM-EE-CCSD results differ from those reported previously (4) because the earlier calculations correlated all electrons.

b

Frequencies of a2 modes are not available because of a computed conical intersection that causes repulsion between the 3(n,π*) and 3(π,π*) states at a C2 (twisted) geometry.

c

From jet-cooled phosphorescence excitation spectrum (this work).

d

From gas-phase infrared spectrum. (31)

e

From gas-phase infrared spectrum. (30)

f

From Raman spectrum of molten sample. (30)

For comparison with experimental fundamentals in the T1(n,π*) state, Table 3 contains harmonic-frequency predictions obtained using TDDFT and EOM-EE-CCSD computational methods. For the present TDDFT calculation, we chose the PBE0 XC functional because of its documented successes (32−34) in predicting excited-state properties of carbonyl-containing molecules. The PBE0 functional also slightly outperformed the widely used B3LYP functional in our earlier examination (4) of T1(n,π*) and S1(n,π*) vibrational frequencies of 4PN. The present investigation is similar to our previous (4) computational study of 4PN, but we now employ basis sets better suited to the computational methods. For example, we previously carried out the TDDFT calculations using the cc-pVTZ basis set. However, the Dunning cc basis sets were designed for use with the frozen-core approximation, whereas DFT methods correlate all electrons. Thus, for the TDDFT calculations in the present work, we employed def2-TVZP, which is an all-electron basis set and contains polarization functions optimized (35) for DFT calculations.
Unlike TDDFT calculations, the EOM-EE-CCSD (ab initio) frequency calculations reported in Table 3 do incorporate the frozen-core approximation, a conventional choice for post-Hartree–Fock methods. For these calculations, we used several different triple-ζ basis sets, all of the same size, that are compatible with a frozen core. In the Discussion section, we assess the relative performance of TDDFT (PBE0) vs EOM-EE-CCSD in predicting T1(n,π*) vibrational frequencies of 4PN.

Rotational Analysis of the 000 Band Contour

Figure 7 shows the 000 band of the T1(n,π*) ← S0 transition of 4PN, recorded using a helium backing pressure of 2 atm. The wavelength increment between data points in the scan is 0.0005 nm, corresponding to about 0.04 cm–1 in this region of the spectrum. The wavenumber scale on the spectrum is relative to the band origin (ν0), determined to be 27 290.6 cm–1 using a simulation procedure described later in this section. The simulation also provided rotational branch labeling shown in the figure.

Figure 7

Figure 7. Jet-cooled T1(n,π*) ← S0 phosphorescence excitation spectrum of 4PN, recorded using a helium backing pressure of 2 atm. The region of the 000 band is shown. The band origin, listed in Table 4, was determined using a simulation procedure outlined in the text. Branch labels are given in the format ΔNΔJ, where the O-form branch corresponds to ΔN = −2; the P-form branch corresponds to ΔN = −1, etc.

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Our analysis of the band contour employs an angular-momentum coupling scheme analogous to Hund’s case (b) for diatomic molecules. (10,29) This limiting case applies to triplet states of most organic molecules lacking heavy atoms. For these species, spin–orbit interactions with nearby electronic states are weak, and the electronic spin S is not coupled electrostatically to the nuclear framework. Thus, S is a good quantum number, but its projection along a molecular axis is not. Another good quantum number in the case (b) limit is N, which represents the rotation of the nuclear framework in space. The projection of N on the a-axis of the 4PN molecule is represented by Ka, which has the same meaning as for a near-prolate asymmetric top molecule lacking spin. The total angular momentum J = N + S is conserved, so that for a triplet species (S = 1), the quantum number J can have the values N – 1, N, and N + 1. The corresponding rotational sublevels have the labels F3, F2, and F1, respectively. The labeled features in Figure 7 specify ΔNΔJ branches that are relatively intense according to selection rules. (29,36) Each branch is the superposition of ΔKa = 0, K″ = 0, 1, 2... sub-branches.
We employed the STROTA computer program, written by Judge et al., (29) to ascertain the branch labels in Figure 7. STROTA is capable of fitting or predicting the rotational structure of singlet–triplet spectra, irrespective of the chosen angular-momentum coupling scheme.
The dye-laser system in the present experiment does not permit resolution of individual rotational lines, and hence it is not possible to obtain a quantitative fit to the spectrum. However, we have been able to extract reasonable values of T1(n,π*) spin constants from the spectrum by using the simulation feature of the STROTA program. We fixed the T1(n,π*) inertial constants (A', B', and C') at values obtained in the TDDFT (PBE0) calculation outlined in the previous section. We used ground-state inertial constants known from microwave experiments. (37) We varied other molecular parameters manually, including spin constants (10) a, a0, α, and β, and ran the STROTA program in simulation mode to check the quality of the fit. We also varied the band origin ν0, effective rotational temperatures characterizing the jet expansion, and transition dipole moment components. The inertial constants were kept fixed at the values referred to above.
The band contour was simulated by convoluting the predicted rotational lines with a Gaussian function corresponding to the bandwidth of the doubled dye-laser output. We changed the variable parameters interactively until we achieved optimal qualitative agreement between the simulated and observed band contour. The final simulation, shown in Figure 8, was produced using molecular parameters listed in Table 4. The ratio of transition dipole moment components, μx:μz, was chosen to be 0.2:1, which reflects contributions to oscillator strength from 1(n,π*) B1 and 1(π,π*) A1 states, respectively.

Figure 8

Figure 8. Experimental spectrum of Figure 7 (black trace), along with simulation (red trace) produced using the molecular constants listed in Table 4. The simulation used a two-temperature model (see text) to describe the distribution of rotational levels in the ground state. The two temperatures were chosen to be 4 and 16 K, with weighting factors of 0.4 and 0.6, respectively.

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Table 4. Molecular Constants (cm–1) Used To Simulate the T1(n,π*) ← S0 000 Contour of 4PN
Ground-state inertial constantsa
A″0.19544
B″0.090566
C″0.061895
Excited-state inertial constantsb
A′0.19789
B′0.089883
C′0.061809
Spin constants
a0.1468
a00.1247
α–1.245
β–0.56
Band origin
ν027290.6
a

From the microwave spectrum. (37)

b

From the TDDFT (PBE0)/def2-TZVP calculation reported in this work.

The simulation reproduces the locations of the band maxima well, though intensities are slightly inaccurate, particularly in the wavenumber extremes of the branches. This inaccuracy is due, in part, to the model we used to describe the rotational temperature of 4PN in the free-jet expansion. This experiment does not incorporate a skimmer, and therefore, the laser excites a collection of molecules located over a relatively large range of angle (θ) with respect to the centerline of the expansion. We estimate that the light-collection system in this experiment images phosphorescence from excited-state molecules spanning an angular range of θ = ±25°. The density of buffer gas is dependent on cos2(f·θ), where f is a numerical factor that depends on the heat-capacity ratio of the buffer gas. At |θ| = 25°, the density of the helium buffer gas is reduced by a factor of 0.77 compared to the centerline of the expansion. (38) Thus, the equilibrium rotational temperature of the 4PN molecules could vary considerably over the sampled range.
The simulation feature of the STROTA program incorporates a single Boltzmann factor corresponding to rotational temperature. To account for temperature variation across the sampled region of the jet, we simply used a weighted sum of two spectra simulated at different rotational temperatures. We generated the optimized simulation in Figure 8 by choosing the two temperatures to be 4 and 16 K, with weighting factors of 0.4 and 0.6, respectively.
The two-temperature approach is known to be successful for a skimmed molecular beam, (39) though for the present free expansion, the model is likely oversimplified, and this contributes to intensity deviations in the overall simulation. Nonetheless, these deviations are relatively small and are not significantly worse for slightly colder or warmer expansions. The Supporting Information contains band contours measured using 3-atm and 1-atm expansions of helium. Simulations using the two-temperature model (with the same molecular parameters as in Table 4) are also included. At the highest backing pressure used in this investigation (5 atm, inset of Figure 2), the two-temperature model is unable to reproduce the observed intensity profile suitably.
A caveat of the overall rotational analysis presented above is that the spin-rotation constants (a and a0) and the inertial constants both depend on the molecule’s moments of inertia, and hence the two sets of constants could be correlated with each other. We investigated this premise by using T1(n,π*) inertial constants from an EOM-EE-CCSD calculation discussed earlier, rather than the TDDFT (PBE0) calculation, in the band-contour simulation. The inertial constants from the EOM-EE-CCSD/cc-pVTZ calculation are (in cm–1) A′ = 0.19502; B′ = 0.089349; and C′ = 0.061276. We used the EOM-EE-CCSD inertial constants to simulate the band contour observed in the 2-atm helium expansion. The result is shown in Figure 9.

Figure 9

Figure 9. Experimental spectrum of Figure 7 (black trace), along with simulation (red trace) produced using T1(n,π*) inertial constants from an EOM-EE-CCSD calculation rather than TDDFT (PBE0).

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In generating this simulation, we kept the spin–spin constants (α and β) and the band origin (ν0) the same as those optimized previously in conjunction with the TDDFT calculation of inertial constants. However, because of the correlation between inertial and spin-rotation constants, we needed to adjust the latter in order to achieve a qualitatively reasonable fit when using EOM-EE-CCSD inertial constants. The spin-rotation constants used in the simulation of Figure 9 are a = 0.1559 cm–1 and a0 = 0.1392 cm–1. These values are about 10% higher than those used with the TDDFT inertial constants, an outcome that places a rough uncertainty range on the spin-rotation constants available from the observed band contour.
In contrast to the spin–rotation constants, the spin–spin constants, α and β, do not involve the molecular moments of inertia directly. The spin–spin constants are related to the average positions of the unpaired electrons with respect to the molecular framework and can also include spin–orbit contributions. (10) In the triplet state of a near-symmetric rotor, the value of α controls the splitting among F1, F2, and F3 sublevels for a given (N,K). In the case of 4PN, the magnitude of α is large enough that three separate T1(n,π*) ← S0 subbands, corresponding to the three F sublevels, are resolvable within the 000 band contour. The intensity within a given F subband depends sensitively on the ΔN branch, (29) but the most intense branches are readily differentiated from each other, as seen in Figure 7.
We confirmed the role of α described above by generating simulations of the 000 band using a range of α values that depart from its optimum (−1.245 cm–1) by ±20%. These simulations are shown, along with the observed band contour for the 2-atm expansion, in Figure 10. In the simulations, the Q-, R-, and S-form branches (ΔN = 0, +1, and +2, respectively) shift together as α is varied, because these intense branches belong to the same F3 subband (see Figure 7 for branch labels). The P-form branch shifts minimally because it is part of the F2 subband, whose position depends less sensitively (10) on α.

Figure 10

Figure 10. Origin band of the 4PN T1(n,π*) ← S0 transition, simulated using a range of α values departing from its optimum (red trace) by +20% to −20%. Other molecular constants are those in Table 4. Simulations employing positive and negative deviations in α are shown in blue and green, respectively. Black trace is the experimental spectrum.

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The optimized simulation in Figure 10 reproduces the separation between F2 and F3 subbands very well, indicating that the parameter value α = −1.245 cm–1 is accurate. When α is varied by ±20% of its optimum, the shift in the F3 subband results in an extreme departure from the observed spectrum and suggests that the uncertainty in α is less than about 10%.
The spin–spin constant β is an asymmetry parameter that does not affect the triplet rotational energies unless Ka = 1. (10) In the observed jet-cooled spectrum of 4PN, triplet levels up to Ka = 5 make significant contributions, and therefore the optimal value of β cannot be determined precisely at the resolution of this experiment. We find that values within ±50% of β = −0.56 cm–1 produce acceptable simulations.

Discussion

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This investigation provides new details about the structure and dynamics of 4PN in its T1(n,π*) state. The jet-cooled T1(n,π*) ← S0 spectrum precisely and unambiguously gives fundamental frequencies for the lowest-energy modes of the molecule, ν15 through ν18. We determined these frequencies directly, through measurement of N01 bands. The very weak (Franck–Condon-forbidden) N01 bands for out-of-plane modes had not been detectable in a previous room-temperature investigation (4) because of spectral congestion.
Experimental fundamentals (Table 3) for the T1(n,π*) ring modes and higher-frequency vibrations allow a comprehensive assessment of TDDFT (PBE0) vs EOM-EE-CCSD performance, presented below. This analysis also reflects basis-set choices that are better suited to the computational methods than in our previous study. (4)
Success of these excited-state frequency calculations hinges on the method’s ability to characterize the π*← n chromophore in 4PN. Figure 11 shows the canonical (Hartree–Fock) HOMO–LUMO pair. Occupancy of the π* LUMO not only weakens the carbonyl bond but also affects the bonds within the ring, in accordance with this orbital’s nodal structure. The most direct consequence of this delocalization is frequency lowering of most of the ring vibrational modes upon T1(n,π*) ← S0 excitation.

Figure 11

Figure 11. Isosurfaces of the canonical molecular orbitals for the lowest-energy electronic transition of 4PN. The value of |ψ| on each isosurface is 0.050 Å–3/2.

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For the in-plane modes (a1 and b2 symmetry) listed in Table 3, the TDDFT and EOM calculations of the T1(n,π*) state have about the same high accuracy, producing harmonic-frequency values that are greater than the experimental fundamentals by a just a few percent or less. Overestimates of this magnitude can be attributed to neglect of anharmonicity.
For out-of-plane modes (a2 and b1 symmetry), computed frequencies are generally much less accurate, with errors in both directions that vary considerably among computational method, basis set, and mode number. We can understand these errors on a case-by-case basis.
The ν18 (b1) mode is the lowest-frequency vibration in the 4PN molecule. This normal mode is an out-of-plane ring-bending motion in which the a axis acts as a hinge. The ν18 mode also involves significant torsion about each of the C–C single bonds flanking the carbonyl group. The torsional motion becomes less stiff upon T1(n,π*) ← S0 excitation because electron promotion to the LUMO (Figure 11) disrupts O═C–C═C conjugation. The observed ν18 fundamental frequency drops from 149 cm–1 in the ground state to 123 cm–1 upon excitation to T1(n,π*). Table 3 shows that the EOM calculation significantly overestimates this drop, and the TDDFT calculation modestly underestimates it.
We investigated these errors with the aid of the Natural Bond Orbital (NBO) (40) program. We carried out an NBO analysis of the T1(n,π*) electron density obtained in the EOM and TDDFT calculations. According to the TDDFT calculation, T1(n,π*) ← S0 excitation promotes effectively 0.999 electron to the LUMO of Figure 11. The EOM calculation predicts an occupancy of 0.972 electron for this LUMO. In the EOM calculation, 0.011 electron is associated with the HOMO →LUMO+1 promotion. Compared to the LUMO, the electron density in the LUMO+1 is more localized to the region of the carbonyl moiety, and the LUMO+1 has nodes at the C–C bonds adjacent to the carbonyl. These characteristics lower the resistance to torsion about these bonds and help to explain why the EOM calculation produces a ν18 frequency significantly lower than the observed fundamental.
The ν17 (b1) mode is a ring-bending vibration similar to ν18, except ν17 involves out-of-plane carbonyl wagging (i.e., pyramidalization) in addition to ring bending. As shown in Table 3, the EOM calculation of T1(n,π*) seriously underpredicts the ν17 harmonic frequency. In this case, as with ν18, the EOM calculation appears to overestimate the contribution of the LUMO+1 (Figure 11). The nodal structure of this orbital diminishes repulsion between the C–C bonds adjacent to the carbonyl. The consequence for ν17 is that the carbonyl carbon is subjected to unrealistically low resistance to pyrimidalization in the EOM calculation. On the other hand, it is possible that the TDDFT calculation lacks sufficient LUMO+1 character, leading to a significant overestimate of the ν17 frequency.
Other observed modes are more delocalized to include atoms distant from the carbonyl group, and it is not possible to analyze their calculated frequencies in terms of the simple HOMO–LUMO picture presented above. For example, the calculation of ν13 (a2) is influenced by configuration interaction in a subtle way. The ν13 vibration takes the molecule from its C2v equilibrium structure to a ring-twisted geometry of C2 symmetry. For this mode, the TDDFT harmonic-frequency prediction is very accurate, producing a ν13 value that is just 4% greater than the observed T1(n,π*) fundamental of 407 cm–1. However, the EOM calculation of this mode is pathological, with results sensitive to the basis set. The correlation consistent ANO1 and cc-pVTZ basis sets lead to a 28% overestimate of ν13, whereas with the 6-311G(2pd,2df) basis, the EOM calculation produces an imaginary frequency.
We have traced (4) these problems to a computed conical intersection between the T(n,π*) and T(π,π*) surfaces. The intersection occurs at a C2v geometry, where the two electronic states are classified as 3A2 and 3A1, respectively. The T(n,π*) state has a nearby equilibrium geometry, also C2v, and at this point in coordinate space, the 3A2 and 3A1 states remain close to each other energetically, but they do not interact because they still have different electronic symmetry. However, at ring-twisted (C2) geometries associated the with ν13 coordinate, the T(n,π*) and T(π,π*) states have the same symmetry and can interact with each other. The EOM calculation finds the two states to be nearly isoenergetic at the T(n,π*) minimum, and thus they mix significantly along the ν13 coordinate. With the ANO1 and cc-pVTZ basis sets, the T(π,π*) state is located slightly below T(n,π*), and the mixing of the two states pushes T(n,π*) upward along the ν13 coordinate. This leads to overestimation of the ν13 frequency. However, the 6-311G(2pd,2df) basis set locates T(π,π*) slightly above T(n,π*), so that the latter is repelled downward, giving the surface a downward curvature and an imaginary frequency.
By contrast to EOM, the TDDFT calculation locates the T(π,π*) state significantly higher than T(n,π*) in this potential-energy region. The TDDFT calculation predicts a T2(π,π*) – T1(n,π*) energy difference of about +0.16 eV, or 1300 cm–1, at the T1(n,π*) minimum. The larger state separation allows the TDDFT calculation to produce a highly accurate ν13 frequency for T1(n,π*); i.e., 422 cm–1, compared to the measured value of 407 cm–1.
Aside from these unique findings for ν13, our conclusion is that the EOM calculation tends to underestimate the frequencies of out-of-plane modes in the T1(n,π*) state of 4PN, whereas the TDDFT calculation tends to overestimate them slightly. The EOM description appears to contain unduly large contributions from molecular orbitals like the LUMO+1 shown in Figure 11. Such orbitals have more extensive antibonding character at the carbonyl than does the LUMO.
These considerations help to interpret the T1(n,π*) equilibrium geometries computed by TDDFT and EOM, as well as the quality of 000 band contour simulations. As seen in the contour analysis (Results section), the TDDFT and EOM calculations do not differ significantly in their prediction of the B′ or C′ inertial constant for the T1(n,π*) state. However, the predicted A′ constant from the TDDFT calculation is slightly greater than that of EOM, with values of 0.1978 and 0.1950 cm–1, respectively. For comparison, the experimentally measured A″ constant for the ground state is 0.1954 cm–1. (37) Thus, the TDDFT calculation, but not EOM, predicts a contraction of the ring structure toward the a axis upon T1(n,π*) ← S0 excitation. The contraction predicted by TDDFT can be traced, in part, to a pronounced shortening of the two C–C bonds flanking the carbonyl group. The TDDFT calculation predicts a length of 1.418 Å for these bonds, whereas EOM predicts 1.436 Å.
Irrespective of the particular computational method, the π* ← n promotion should cause shortening of these C–C bonds, in accord with the LUMO isosurface shown in Figure 11. The TDDFT description of the T1(n,π*) ← S0 excitation does not include contributions from the LUMO+1 or higher molecular orbitals, according to the NBO analysis discussed above. This helps explain why the TDDFT calculation predicts a more extreme shortening of the C–C bonds adjacent to the carbonyl, compared to the EOM calculation. The latter explicitly includes contributions from orbitals such as LUMO+1, which have nodes at these C–C bonds.
As seen in Figure 8, the simulated 000 band contour arising from the TDDFT (PBE0) inertial constants agrees well with the observed spectrum. The agreement is slightly better for the TDDFT structure than for EOM, as seen by comparison to Figure 9. The larger A′ constant from the TDDFT calculation subtly affects the branch structure (labeled in Figure 7) and leads to the better agreement. Hence, it appears that the TDDFT calculation is more accurate than EOM in predicting the ring bond lengths of the T1(n,π*) excited state.
We conclude this section by noting a surprising outcome of the 000 band contour analysis. The spin constants determined for the T1(n,π*) state of 4PN (Table 4) are an order of magnitude greater than is expected for monocyclic organic molecules. (7,9,10,41) For example, the value of α in the T1(n,π*) state of 2-cyclopenten-1-one (2CP) is −0.25 cm–1, (7) in contrast to −1.25 cm–1 determined for 4PN in the present study. In the triplet states of molecules without significant spin–orbit coupling, the α constant is proportional to ⟨(1–3 cos θ)/r3⟩, (41) where r is the distance between the unpaired electrons, and θ describes the orientation of the spin–spin vector in the molecular frame. (The negative sign of α in 4PN indicates that the distribution of spin density is prolate. (41)) Given the similarity of 2CP and 4PN, in terms of their size, shape, and π* ← n chromophore, one expects similar α constants for their T1(n,π*) states. But the experimental α values are significantly different in the two molecules, and the magnitude of α for 4PN is closer to that of O2 (1.32 cm–1 (42)) in its triplet ground state.
In triplet states of O2 and isoelectronic diatomics, spin–orbit interaction makes a dominant contribution to α, (43) and the angular momentum coupling is close to the Hund’s case (a) limit. In the T1(n,π*) state of 4PN, the spin–orbit energy should be negligible because the orbital angular momentum averages out to zero in nonlinear polyatomic molecules. Nonetheless, the value of α in the T1(n,π*) state of 4PN is large enough to split the F spin-rotation components by an amount greater than the rotational intervals─indicating that the angular momentum coupling has departed from the Hund’s case (b) limit and begins to resemble that of case (a).
As the case (a) limit is approached, the |z⟩ triplet spin substate (body-fixed axis) becomes relatively uncontaminated by |x⟩ and |y⟩, and the projection of electron spin on the molecular symmetry axis becomes a good quantum number. At the same time, the N quantum number becomes less useful for describing the rotational level pattern and is replaced by J. In the singlet–triplet spectra of case (a) molecules having C2v symmetry, each triplet spin state (|x⟩, |y⟩, and |z⟩) is represented as a separate band having the characteristic P, Q, R branch structure (referring to ΔJ) of a singlet–singlet transition. (29) The branch labels in Figures 7 and 10 show that this behavior is manifested in the F3 subband of the 4PN T1(n,π*) ← S0 transition. This appears to be an example of Hund’s case (ab) triplet-state coupling, (44) in which one of the three F components of the case (b) representation is sufficiently distant from the other two that case (a) labeling becomes meaningful for the unique component. In the T1(n,π*) ← S0 transition of 4PN, the F3 component is aptly described as a |z⟩ subband. (29)
To be sure, it is unusual for triplet states of an asymmetric-top molecule to fall into the case (a) or (ab) category, because the first-order contribution to spin–orbit coupling is zero. However, second-order spin–orbit interaction, which couples two electronic states, can be highly significant─particularly if heavy atoms are involved. In deuterated selenoformaldehyde (D2C80Se), for example, the T2(π,π*) state interacts particularly strongly with the |x⟩ and |y⟩ components of T1(n,π*). (45) In the 401 band of the selenoformaldehyde-d2 T1(n,π*) ← S0 system, this spin–orbit interaction leads to a |z⟩ subband that is separated from the other two by more than 100 cm–1─a definitive example (46) of case (ab) coupling.
In the T1(n,π*) state of 4PN, the allowed spin–orbit interactions with other electronic states are isomorphic with those of selenoformaldehyde, as both the 4PN and selenoformaldehyde T1(n,π*) states have A2 orbital symmetry. In 4PN, the magnitude of second-order spin–orbit matrix elements must be considerably smaller than in selenoformaldehyde, because the latter has the “heavy-atom” effect. However, the energy gap between T2(π,π*) and T1(n,π*) is likely much smaller in 4PN than in selenoformaldehyde. Our TDDFT calculation of 4PN predicts a T2 – T1 energy difference of 1300 cm–1 near the T1(n,π*) minimum, whereas this difference is calculated (45) to be about 5000 cm–1 in selenoformaldehyde. Because of the relatively small T2 – T1 energy gap in 4PN, it is entirely plausible that second-order spin–orbit perturbations are responsible for the splittings, on the order of a few cm–1, observed among the T1(n,π*) spin states in the T1(n,π*) ← S0 spectrum of 4PN.

Conclusions

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The 4PN molecule provides an excellent testbed for understanding structural effects of electronic excitation. The canonical HOMO–LUMO picture of π*← n excitation is straightforward and leads to qualitatively predictable changes in vibrational frequencies. It should be possible to predict these changes with quantitative accuracy by using modern computational approaches for treating excited states. To test these expectations, we have measured the vibronically resolved T1(n,π*) ← S0 band system of 4PN under jet-cooled conditions. We observe significant frequency decreases for out-of-plane ring modes upon excitation. These changes stem from the nodal structure of the π* LUMO. However, configuration interaction within the triplet manifold can lead to subtleties that are difficult to model accurately, even with high-level computational methods. The computationally intensive EOM-EE-CCSD technique tends to overestimate frequency drops for the out-of-plane modes. The much more economical TDDFT method modestly underestimates the drops but generally performs impressively with regard to frequency predictions. Also, the TDDFT (PBE0) calculation of T1(n,π*) inertial constants leads to very good agreement between simulated and observed rotational contours for the T1(n,π*) ← S0 000 origin band.
Both computational methods reveal that the T2(π,π*) state is nearly isoenergetic with T1(n,π*) at molecular geometries close to the T1(n,π*) potential-energy minimum. The EOM calculation places the two states too close together, leading to unphysical predictions of the lowest ring-twisting frequency. However, the TDDFT (PBE0) calculation locates the T2(π,π*) at an energy that leads to a very accurate ring-twist frequency. Moreover, the TDDFT energies support the hypothesis that the T1(n,π*) and T2(π,π*) are coupled via spin–orbit interaction. A spin–orbit perturbation of this sort could explain the signficant departure from the expected Hund’s case (b) branch structure we observe in the T1(n,π*) ← S0 000 band. We hope this hypothesis promotes expansion of computational tools for investigating triplet states.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.3c01059.

  • This document shows origin-band contours for the T1(n,π*) ← S0 transition of 4PN, measured using 3-atm and 1-atm expansions of helium. Simulations using the two-temperature model (with the same molecular parameters as in Table 4) are also included. (PDF)

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Author Information

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  • Corresponding Author
  • Authors
    • Sean W. Parsons - Department of Chemistry and Biochemistry, University of Wisconsin-Eau Claire, 105 Garfield Avenue, Eau Claire, Wisconsin 54701, United States;  Present Address: Department of Chemistry, University of Southern California
    • Devon G. Hucek - Department of Chemistry and Biochemistry, University of Wisconsin-Eau Claire, 105 Garfield Avenue, Eau Claire, Wisconsin 54701, United States;  Present Address: College of Pharmacy, University of Michigan
    • Piyush Mishra - Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, Indiana 47907, United States;  Present Address: Department of Chemistry, Massachusetts Institute of Technology
    • David F. Plusquellic - Applied Physics Division, National Institute of Standards and Technology, 325 Broadway Avenue, Boulder, Colorado 80305, United States
    • Timothy S. Zwier - Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, Indiana 47907, United States;  Present Address: Current address: Combustion Research Facility, Sandia National Laboratories;  Orcidhttps://orcid.org/0000-0002-4468-5748
  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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This work was supported by grants from the National Science Foundation (CHE-1362897 and CHE-1955137), under the Research in Undergraduate Institutions program. We also acknowledge support from the Office of Research and Sponsored Projects and the Blugold Center for High-Performance Computing at the University of Wisconsin-Eau Claire. We are grateful to Prof. Dennis Clouthier for very helpful discussions.

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    Journal of Computational Chemistry (2019), 40 (2), 387-399CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    Ab initio and d. functional CCSD(T)-F12/cc-pVQZ-f12//B2PLYPD3/6-311G** calcns. have been performed to unravel the reaction mechanism of triplet and singlet methylene CH2 with ketene CH2CO. The computed potential energy diagrams and mol. properties have been then utilized in Rice-Ramsperger-Kassel-Marcus-Master Equation (RRKM-ME) calcns. of the reaction rate consts. and product branching ratios combined with the use of nonadiabatic transition state theory for spin-forbidden triplet-singlet isomerization. The results indicate that the most important channels of the reaction of ketene with triplet methylene lead to the formation of the HCCO + CH3 and C2H4 + CO products, where the former channel is preferable at higher temps. from 1000 K and above. In the C2H4 + CO product pair, the ethylene mol. can be formed either adiabatically in the triplet electronic state or via triplet-singlet intersystem crossing in the singlet electronic state occurring in the vicinity of the CH2COCH2 intermediate or along the pathway of CO elimination from the initial CH2CH2CO complex. The predominant products of the reaction of ketene with singlet methylene have been shown to be C2H4 + CO. The formation of these products mostly proceeds via a well-skipping mechanism but at high pressures may to some extent involve collisional stabilization of the CH3CHCO and cyclic CH2COCH2 intermediates followed by their thermal unimol. decompn. The calcd. rate consts. at different pressures from 0.01 to 100 atm have been fitted by the modified Arrhenius expressions in the temp. range of 300-3000 K, which are proposed for kinetic modeling of ketene reactions in combustion.
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  3. 3
    Naskar, S.; Das, M. Singlet and Triplet excited State Energy Ordering in Cyclopenta-fused Polycyclic Aromatic Hydrocarbons (CP-PAHs) Suitable for Energy Harvesting: An Exact Model and TDDFT Study. ACS Omega 2017, 2, 1795,  DOI: 10.1021/acsomega.7b00278
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    3
    Singlet and Triplet Excited State Energy Ordering in Cyclopenta-Fused Polycyclic Aromatic Hydrocarbons (CP-PAHs) Suitable for Energy Harvesting: An Exact Model and TDDFT Study
    Naskar, Sumit; Das, Mousumi
    ACS Omega (2017), 2 (5), 1795-1803CODEN: ACSODF; ISSN:2470-1343. (American Chemical Society)
    We calcd. the ground and low-lying excited states of cyclopenta-fused polycyclic arom. hydrocarbons (CP-PAHs) using exact diagonalization in full CI (CI) within the model PPP Hamiltonian as well as a time-dependent d. functional theory technique. The CP-PAHs include acenaphthylene, isomers of pyracylene, cycloocta-pentalene, and three isomers of dicyclo-pentacyclo-octenes (DCPCO). We used the inherent symmetries of these systems to calc. the energy ordering of the lowest singlet (S1) and lowest triplet excited (T1) states with respect to the ground state (S0). The calcn. shows that the lowest dipole allowed singlet absorption varies from 0.43 to 1.42 eV for most of these systems. Such an optical absorption range is very promising in harvesting solar radiation ranging from the visible to near-IR region improving the efficiency of photovoltaic device application. The calcd. optical gaps for pyracylene, acenaphthylene, and two isomers of DCPCO are in very good agreement with exptl. results reported in the literature. The calcd. S1-T1 energy gaps (ΔST) in cycloocta-pentalene and in the DCPCO isomers are very small ranging from 0.01 to 0.2 eV, which is highly desirable in improving their electroluminescence efficiency in light-emitting device applications.
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  4. 4
    Sessions, A.; McDonnell, M.; Christianson, D.; Drucker, S. Triplet and Singlet (n, π*) Excited States of 4H-Pyran-4-one Characterized by Cavity Ringdown Spectroscopy and Quantum-Chemical Calculations. J. Phys. Chem. A 2019, 123, 6269,  DOI: 10.1021/acs.jpca.9b04238
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    4
    Triplet and Singlet (n,π*) Excited States of 4H-Pyran-4-one Characterized by Cavity Ringdown Spectroscopy and Quantum-Chemical Calculations
    Sessions, Anna G.; McDonnell, Michael P.; Christianson, Drew A.; Drucker, Stephen
    Journal of Physical Chemistry A (2019), 123 (29), 6269-6280CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    The 4H-pyran-4-one (4PN) mol. serves as a model for investigating structural changes following π* ← n electronic excitation. We have recorded the cavity ringdown (CRD) absorption spectrum of 4PN vapor at room temp., over the wavelength region from 350 to 370 nm. This spectral region includes the T1(n,π*) ← S0 band system as well as the low-energy portion of the S1(n,π*) ← S0 system. Aided by predictions from ab initio (equation-of-motion excitation energies with dynamical correlation incorporated at the level of coupled cluster singles doubles, EOM-EE-CCSD) and d. functional theory (time-dependent d. functional theory with PBE0 functional, TDPBE0) calcns., we have made vibronic assignments for about 30 features in the CRD spectrum, mostly T1(n,π*) ← S0 transitions. We have used these results to correct certain vibronic assignments appearing in the previous literature for both T1(n,π*) ← S0 and S1(n,π*) ← S0 band systems. We conclude that the lowest-energy carbonyl wagging fundamentals (ν27, in-plane and ν17, out-of-plane) undergo significant frequency drops (28 and 50%, resp.) upon T1(n,π*) ← S0 excitation and similar drops (29 and 39%, resp.) for S1(n,π*) ← S0 excitation. We find that vibrational modes involving the conjugated ring atoms undergo relatively small frequency changes upon π* ← n excitation, for both T1 and S1 states. We have used the present spectroscopic results and vibronic assignments to test the accuracy of computed excited-state frequencies for 4PN. This benchmarking process shows that the economical time-dependent d. functional theory method is impressively accurate for certain (but not all) vibrational modes. The highly correlated EOM-EE-CCSD ab initio method is capable of making accurate frequency predictions, but the results, unexpectedly, depend sensitively on basis set family. This anomaly is traceable to a computed conical intersection between the T1(n,π*) and T2(π,π*) surfaces near the T1(n,π*) potential min. Relatively small errors in the location of the conical intersection lead to enhanced mixing of the two electronic states and incorrect T1(n,π*) vibrational frequencies when certain triple-ζ quality basis sets are used.
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  5. 5
    McAnally, M. O.; Zabronsky, K. L.; Stupca, D. J.; Pillsbury, N. R.; Phillipson, K.; Drucker, S. Lowest Triplet (n, π*) State of 2-Cyclohexen-1-one: Characterization by Cavity Ringdown Spectroscopy and Quantum-Chemical Calculations. J. Chem. Phys. 2013, 139, 214311– 1,  DOI: 10.1063/1.4834655
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    5
    Lowest triplet (n,π*) state of 2-cyclohexen-1-one: Characterization by cavity ringdown spectroscopy and quantum-chemical calculations
    McAnally, Michael O.; Zabronsky, Katherine L.; Stupca, Daniel J.; Phillipson, Kaitlyn; Pillsbury, Nathan R.; Drucker, Stephen
    Journal of Chemical Physics (2013), 139 (21), 214311/1-214311/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The cavity ringdown (CRD) absorption spectrum of 2-cyclohexen-1-one (2CHO) was recorded over the range 401.5-410.5 nm in a room-temp. gas cell. The very weak band system (ε ≤ 0.1 M-1 cm-1) in this spectral region is due to the T1(n, π*) ← S0 electronic transition. The 000 origin band was assigned to the feature obsd. at 24 558.8 ± 0.3 cm-1. We have assigned 46 vibronic transitions in a region extending from -200 to +350 cm-1 relative to the origin band. For the majority of these transitions, we have made corresponding assignments in the spectrum of the deuterated deriv. 2CHO-2,6,6-d3. From the assignments, we detd. fundamental frequencies for several vibrational modes in the T1(n, π*) excited state of 2CHO, including the lowest ring-twisting (99.6 cm-1) and ring-bending (262.2 cm-1) modes. These values compare to fundamentals of 122.2 cm-1 and 251.9 cm-1, resp., detd. previously for the isoconfigurational S1(n, π*) excited state of 2CHO and 99 cm-1 and 248 cm-1, resp., for the S0 ground state. With the aid of quantum-mech. calcns., we have also ascertained descriptions for these two modes, thereby resolving ambiguities appearing in the previous literature. The ring-twisting mode (ν39) contains a significant contribution from O=C-C=C torsion, whereas the ring-bending mode (ν38 in the ground state) involves mainly the motion of C-5 with respect to the plane contg. the other heavy atoms. The CRD spectroscopic data for the T1(n, π*) state have allowed us to benchmark several computational methods for treating excited states, including time-dependent d. functional theory and an equation-of-motion coupled cluster method. In turn, the computational results provide an explanation for obsd. differences in the T1(n, π*) vs. S1(n, π*) ring frequencies. (c) 2013 American Institute of Physics.
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  6. 6
    O’Keefe, A.; Deacon, D. A. G. Cavity Ring-Down Optical Spectrometer for Absorption Measurements Using Pulsed Laser Sources. Rev. Sci. Instrum. 1988, 59, 2544,  DOI: 10.1063/1.1139895
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    Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources
    O'Keefe, Anthony; Deacon, David A. G.
    Review of Scientific Instruments (1988), 59 (12), 2544-51CODEN: RSINAK; ISSN:0034-6748.
    A technique was developed which allows optical absorption measurements to be made using a pulsed light source and offers a sensitivity significantly greater than that attained using stabilized continuous light sources. The technique is based upon the measurement of the rate of absorption rather than the magnitude of absorption of a light pulse confined within a closed optical cavity. The decay of the light intensity within the cavity is a simple exponential with loss components due to mirror loss, broadband scatter (Rayleigh, Mie), and mol. absorption. Narrowband absorption spectra are recorded by scanning the output of a pulsed laser (which is injected into the optical cavity) through an absorption resonance. The sensitivity of this technique was demonstrated by measuring several bands in the very weak forbidden b 1Σg - χ3Σg transition in gaseous mol. O. Absorption signals of <1 part in 106 can be detected.
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  7. 7
    Pillsbury, N. R.; Zwier, T. S.; Judge, R. H.; Drucker, S. Jet-cooled phosphorescence excitation spectrum of the T1(n, π*) ← S0 transition of 2-cyclopenten-1-one. J. Phys. Chem. A 2007, 111, 8357,  DOI: 10.1021/jp072353r
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    7
    Jet-Cooled Phosphorescence Excitation Spectrum of the T1(n,π) ← S0 Transition of 2-Cyclopenten-1-one
    Pillsbury, Nathan R.; Zwier, Timothy S.; Judge, Richard H.; Drucker, Stephen
    Journal of Physical Chemistry A (2007), 111 (34), 8357-8366CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    The T1(n,π*) ← S0 transition of 2-cyclopenten-1-one (2CP) was investigated by using phosphorescence excitation (PE) spectroscopy in a free-jet expansion. The origin band, near 385 nm, is the most intense feature in the T1(n,π*) ← S0 PE spectrum. A short progression in the ring-bending mode (ν'30) is also obsd. The effective vibrational temp. in the jet is estd. at 50 K. The spectral simplification arising from jet cooling helps confirm assignments made previously in the room-temp. cavity ring-down (CRD) absorption spectrum, which is congested by vibrational hot bands. In addn. to the origin and ν'30 assignments, the jet-cooled PE spectrum also confirms the 2801 (C:O out-of-plane wag), 2901 (C:C twist), and 1901 (C:O in-plane wag) band assignments that were made in the T1(n,π*) ← S0 room-temp. CRD spectrum. The temporal decay of the T1 state of 2CP was investigated as a function of vibronic excitation. Phosphorescence from the v' = 0 level persists the entire time the mols. traverse the emission detection zone. Thus the phosphorescence lifetime of the v' = 0 level is significantly longer than the 2 μs transit time through the viewing zone. Higher vibrational levels in the T1 state have shorter phosphorescence lifetimes, on the order of 2 μs or less. The concomitant redn. in emission quantum yield causes the higher vibronic bands (above 200 cm-1) in the PE spectrum to be weak. It is proposed that intersystem crossing to highly vibrationally excited levels of the ground state is responsible for the faster decay and diminished quantum yield. The jet cooling affords partial rotational resoln. in the T1(n,π*) ← S0 spectrum of 2CP. The rotational structure of the origin band was simulated by using inertial consts. available from a previously reported d. functional (DFT) calcn. of the T1(n,π*) state, along with spin consts. obtained via a fitting procedure. Intensity parameters were also systematically varied. The optimized intensity factors support a model that identifies the S2(π,π*) ← S0 transition in 2CP as the sole source of oscillator strength for the T1(n,π*) ← S0 transition.
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  8. 8
    Spangler, L.; Pratt, D. Laser-Induced Phosphorescence Spectroscopy in Supersonic Jets - the Lowest Triplet-States of Glyoxal, Methylglyoxal, and Biacetyl. J. Chem. Phys. 1986, 84, 4789,  DOI: 10.1063/1.449965
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    8
    Laser-induced phosphorescence spectroscopy in supersonic jets. The lowest triplet states of glyoxal, methylglyoxal, and biacetyl
    Spangler, Lee. H.; Pratt, David W.
    Journal of Chemical Physics (1986), 84 (9), 4789-96CODEN: JCPSA6; ISSN:0021-9606.
    The lowest triplet states of 3 α-dicarbonyls (glyoxal, methylglyoxal, and biacetyl) are reported using laser-induced phosphorescence spectroscopy in supersonic jets. At the level of vibrational resoln., 3Au glyoxal apparently has a geometry similar to that of the ground state. The T1 ← S0 transitions of methylglyoxal and biacetyl each exhibit strong progressions in the torsional vibrations of the Me groups, showing that these mols. undergo a conformational change on excitation to the lowest triplet state. A Franck-Condon anal. of the methylglyoxal spectrum, with proper consideration for nuclear spin statistics, yields a Me barrier of V3 = 115 ± 5 cm-1 in this state. This value was confirmed by a direct measurement of the tunneling splitting of A and E torsional levels. The hindering potential in the lowest triplet state of methylglyoxal is different from those in the ground (V3 = 269 cm-1) and 1st excited singlet (V3 = 190 cm-1) states. Possible reasons for these differences are discussed.
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  9. 9
    Tomer, J. L.; Holtzclaw, K. W.; Pratt, D. W.; Spangler, L. H. Phosphorescence Excitation Spectroscopy in Supersonic Jets – The Lowest Triplet-State of Pyrazine. J. Chem. Phys. 1988, 88, 1528,  DOI: 10.1063/1.454132
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    Phosphorescence excitation spectroscopy in supersonic jets. The lowest triplet state of pyrazine
    Tomer, J. L.; Holtzclaw, K. W.; Pratt, D. W.; Spangler, L. H.
    Journal of Chemical Physics (1988), 88 (3), 1528-38CODEN: JCPSA6; ISSN:0021-9606.
    The laser-induced phosphorescence excitation spectrum of pyrazine was examd. in the collision-free environment of a supersonic jet. The origin of the lowest triplet state (T1) lies at 26,820 cm-1 and exhibits a sym. parallel-type rotational contour, confirming that this state is 3B3u (nπ*) with an equil. geometry that is similar to those of the S0 (1Ag) and S1 (1B3u,nπ*) states. Thirty vibrational bands were also obsd. in the ∼4000 cm-1 interval between the T1 and S1 origins. Of these, the 13 lower energy bands all exhibit parallel-type contours and may be assigned as T1 ← S0 transitions, principally involving totally sym. modes. The 17 higher energy bands exhibit both parallel and perpendicular contours and may be assigned as S1 ← S0 hot band transitions, some involving nontoally sym. modes. No evidence for a 2nd, ππ* triplet state lying below the S1 origin was found, nor is there any evidence for rapid relaxation of any of the zero-order T1 levels at a resoln. of ∼1 cm-1. The intersystem crossing dynamics of S1 pyrazine is governed by the interaction of the 2 largely nested potential surfaces, S1 and T1, zero-order nπ* states that appear to differ primarily in the extent to which they interact vibronically with other zero-order states in manifolds of the corresponding multiplicity.
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    Raynes, W. Spin Splittings and Rotational Structure of Nonlinear Molecules in Triplet Electronic States. J. Chem. Phys. 1964, 41, 3020,  DOI: 10.1063/1.1725668
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    10
    Spin splittings and rotational structure of nonlinear molecules in doublet and triplet electronic states
    Raynes, W. T.
    Journal of Chemical Physics (1964), 41 (10), 3020-32CODEN: JCPSA6; ISSN:0021-9606.
    Matrix elements are presented for the Hamiltonian of a nonlinear, nonrigid polyat. mol. in a multiplet electronic state. Their use is appropriate only for electronic and vibrational spectra, since hyperfine interactions involving nuclear spins and nuclear quadrupole moments are not considered. For the most general case, 9 parameters are required to take full account of spin-rotation interactions, and 5 are required for spinspin interactions. For mols. of orthorhombic symmetry only 3 spin-rotation parameters and 2 spin-spin parameters are nonzero. For nonlinear mols. in doublet and triplet electronic states, explicit formulas are presented for (a) the rotational term values of sym. rotors and (b) spin splittings of asym. rotors possessing orthorhombic symmetry. All these formulas reduce to well-known expressions for diat. mols. in 2Σ and 3Σ states when K-dependent terms are ignored. Application of these formulas to the results of Dressler and Ramsay (CA 54, 19155f) on the 2B1 ground states of NH2 and ND2 permits the detn. of the spin-rotation parameters of these mols. All 5 spin parameters of HCHO in its lowest 3A2 state are given, together with curves of spin splittings in the lower K levels. The spin parameters of HCHO, NH2, and ND2 are compared with those of NO2 and ClO2 found by recent microwave studies. For a triplet state of an orthorhombic mol., the spin-spin consts. detd. by band spectroscopy are simply related to the spin consts. D and E detd. from zero field splittings in ESR spectroscopy. The surprisingly small value of D = 0.42 cm.-1 for the lowest triplet state of HCHO is discussed briefly in terms of a breakdown of the orbital approxn. for this prototype n-π* state.
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  11. 11
    Stanton, J.; Bartlett, R. The Equation of Motion Coupled-Cluster Method ─ a Systematic Biorthogonal Approach to Molecular-Excitation Energies, Transition-Probabilities, and Excited-State Properties. J. Chem. Phys. 1993, 98, 7029,  DOI: 10.1063/1.464746
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    The equation-of-motion, coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited-state properties
    Stanton, John F.; Bartlett, Rodney J.
    Journal of Chemical Physics (1993), 98 (9), 7029-39CODEN: JCPSA6; ISSN:0021-9606.
    A comprehensive overview of the equation of motion coupled-cluster (EOM-CC) method and its application to mol. systems is presented. By exploiting the biorthogonal nature of the theory, it is shown that excited-state properties and transition strengths can be evaluated via a generalized expectation-value approach that incorporates both the bra and ket state wave functions. Reduced d. matrixes defined by this procedure are given by closed form expressions. For the root of the EOM-CC effective Hamiltonian that corresponds to the ground state, the resulting equations are equiv. to the usual expressions for normal single-ref. CC d. matrixes. Thus, the method described in this paper provides a universal definition of coupled-cluster d. matrixes, providing a link between EOM-CC and traditional ground state CC theory. Excitation energy, oscillator strength, and property calcns. are illustrated by means of several numerical examples, including comparisons with full CI calcns. and a detailed study of the 10 lowest electronically excited states of the cyclic isomer of C4.
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  12. 12
    Krylov, A. Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitchhikeras Guide to Fock Space. Annu. Rev. Phys. Chem. 2008, 59, 433,  DOI: 10.1146/annurev.physchem.59.032607.093602
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    Equation-of-motion coupled-cluster methods for open-shell and electronically excited species; the Hitchhiker's guide to Fock space
    Krylov, Anna I.
    Annual Review of Physical Chemistry (2008), 59 (), 433-462CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)
    A review. The equation-of-motion coupled-cluster (EOM-CC) approach is a versatile electronic-structure tool that allows one to describe a variety of multiconfigurational wave functions within single-ref. formalism. This review provides a guide to established EOM methods illustrated by examples that demonstrate the types of target states currently accessible by EOM. It focuses on applications of EOM-CC to electronically excited and open-shell species. The examples emphasize EOM's advantages for selected situations often perceived as multireference cases [e.g., interacting states of different nature, Jahn-Teller (JT) and pseudo-JT states, dense manifolds of ionized states, diradicals, and triradicals]. I also discuss limitations and caveats and offer practical solns. to some problematic situations. The review also touches on some formal aspects of the theory and important current developments.
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    Furche, F.; Ahlrichs, R. Adiabatic Time-Dependent Density Functional Methods for Excited State Properties. J. Chem. Phys. 2002, 117, 7433,  DOI: 10.1063/1.1508368
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    Adiabatic time-dependent density functional methods for excited state properties
    Furche, Filipp; Ahlrichs, Reinhart
    Journal of Chemical Physics (2002), 117 (16), 7433-7447CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    This work presents theory, implementation, and validation of excited state properties obtained from time-dependent d. functional theory (TDDFT). Based on a fully variational expression for the excited state energy, a compact derivation of first order properties is given. We report an implementation of analytic excited state gradients and charge moments for local, gradient cor., and hybrid functionals, as well as for the CI singles (CIS) and time-dependent Hartree-Fock (TDHF) methods. By exploiting analogies to ground state energy and gradient calcns., efficient techniques can be transferred to excited state methods. Benchmark results demonstrate that, for low-lying excited states, geometry optimizations are not substantially more expensive than for the ground state, independent of the mol. size. We assess the quality of calcd. adiabatic excitation energies, structures, dipole moments, and vibrational frequencies by comparison with accurate exptl. data for a variety of excited states and mols. Similar trends are obsd. for adiabatic excitation energies as for vertical ones. TDDFT is more robust than CIS and TDHF, in particular, for geometries differing significantly from the ground state min. The TDDFT excited state structures, dipole moments, and vibrational frequencies are of a remarkably high quality, which is comparable to that obtained in ground state d. functional calcns. Thus, yielding considerably more accurate results at similar computational cost, TDDFT rivals CIS as a std. method for calcg. excited state properties in larger mols.
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  14. 14
    Jacquemin, D.; Duchemin, I.; Blase, X. Is the Bethe-Salpeter Formalism Accurate for Excitation Energies? Comparisons with TD-DFT, CASPT2, and EOM-CCSD. J. Phys. Chem. Lett. 2017, 8, 1524,  DOI: 10.1021/acs.jpclett.7b00381
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    14
    Is the Bethe-Salpeter Formalism Accurate for Excitation Energies? Comparisons with TD-DFT, CASPT2, and EOM-CCSD
    Jacquemin, Denis; Duchemin, Ivan; Blase, Xavier
    Journal of Physical Chemistry Letters (2017), 8 (7), 1524-1529CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
    Developing ab initio approaches able to provide accurate excited-state energies at a reasonable computational cost is one of the biggest challenges in theor. chem. In that framework, the Bethe-Salpeter equation approach, combined with the GW exchange-correlation self-energy, which maintains the same scaling with system size as TD-DFT, has recently been the focus of a rapidly increasing no. of applications in mol. chem. Using a recently proposed set encompassing excitation energies of many kinds [J. Phys. Chem. Lett. 2016, 7, 586-591], we investigate here the performances of BSE/GW. We compare these results to CASPT2, EOM-CCSD, and TD-DFT data and show that BSE/GW provides an accuracy comparable to the two wave function methods. It is particularly remarkable that the BSE/GW is equally efficient for valence, Rydberg, and charge-transfer excitations. In contrast, it provides a poor description of triplet excited states, for which EOM-CCSD and CASPT2 clearly outperform BSE/GW. This contribution therefore supports the use of the Bethe-Salpeter approach for spin-conserving transitions.
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  15. 15
    Tajti, A.; Stanton, J.; Matthews, D.; Szalay, P. Accuracy of Coupled Cluster Excited State Potential Energy Surfaces. J. Chem. Theory Comput. 2018, 14, 5859,  DOI: 10.1021/acs.jctc.8b00681
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    Accuracy of Coupled Cluster Excited State Potential Energy Surfaces
    Tajti, Attila; Stanton, John F.; Matthews, Devin A.; Szalay, Peter G.
    Journal of Chemical Theory and Computation (2018), 14 (11), 5859-5869CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    The validation of the quality of the description of excited electronic states is of special importance in quantum chem. as the general reliability of ab initio methods shows a much larger variation for these states than for the ground state. In this study, we investigate the quality of excited state energy gradients and potential energy surfaces on selected systems, as provided by the single ref. coupled cluster variants CC2, CCSD, CCSD(T)(a)*, and CC3. Gradients and surface plots that follow the Franck-Condon forces are compared to the resp. CCSDT ref. values, thereby establishing a useful strategy for judging each variant's accuracy. The results reveal serious flaws of lower order methods - in particular, CC2 - in several situations where they otherwise give accurate vertical excitation energies, as well as excellent accuracy and consistency of the recently proposed CCSD(T)(a)* method.
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  16. 16
    Loos, P.-F.; Lipparini, F.; Boggio-Pasqua, M.; Scemama, A.; Jacquemin, D. A Mountaineering Strategy to Excited States: Highly Accurate Energies and Benchmarks for Medium Sized Molecules. J. Chem. Theory Comput. 2020, 16, 1711,  DOI: 10.1021/acs.jctc.9b01216
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    16
    A Mountaineering Strategy to Excited States: Highly Accurate Energies and Benchmarks for Medium Sized Molecules
    Loos, Pierre-Francois; Lipparini, Filippo; Boggio-Pasqua, Martial; Scemama, Anthony; Jacquemin, Denis
    Journal of Chemical Theory and Computation (2020), 16 (3), 1711-1741CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Following our previous work focusing on compds. contg. up to 3 non-hydrogen atoms [J. Chem. Theory Comput.2018, 14, 4360-4379], we present here highly accurate vertical transition energies obtained for 27 mols. encompassing 4, 5, and 6 non-hydrogen atoms: acetone, acrolein, benzene, butadiene, cyanoacetylene, cyanoformaldehyde, cyanogen, cyclopentadiene, cyclopropenone, cyclopropenethione, diacetylene, furan, glyoxal, imidazole, isobutene, methylenecyclopropene, propynal, pyrazine, pyridazine, pyridine, pyrimidine, pyrrole, tetrazine, thioacetone, thiophene, thiopropynal, and triazine. To obtain these energies, we use equation-of-motion/linear-response coupled cluster theory up to the highest tech. possible excitation order for these systems (CC3, EOM-CCSDT, and EOM-CCSDTQ) and selected CI (SCI) calcns. (with tens of millions of determinants in the ref. space), as well as the multiconfigurational n-electron valence state perturbation theory (NEVPT2) method. All these approaches are applied in combination with diffuse-contg. at. basis sets. For all transitions, we report at least CC3/aug-cc-pVQZ vertical excitation energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3 almost systematically delivers transition energies in agreement with higher-level methods with a typical deviation of ±0.04 eV, except for transitions with a dominant double excitation character where the error is much larger. The present contribution gathers a large, diverse, and accurate set of more than 200 highly accurate transition energies for states of various natures (valence, Rydberg, singlet, triplet, n → π*, π → π*, ...). We use this series of theor. best ests. to benchmark a series of popular methods for excited state calcns.: CIS(D), ADC(2), CC2, STEOM-CCSD, EOM-CCSD, CCSDR(3), CCSDT-3, CC3, and NEVPT2. The results of these benchmarks are compared to the available literature data.
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  17. 17
    Andersson, K.; Malmqvist, P.; Roos, B.; Sadlej, A.; Wolinski, K. Second-Order Perturbation-theory with a CASSCF Reference Function. J. Phys. Chem. 1990, 94, 5483,  DOI: 10.1021/j100377a012
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    17
    Second-order perturbation theory with a CASSCF reference function
    Andersson, Kerstin; Malmqvist, Per Aake; Roos, Bjoern O.; Sadlej, Andrzej J.; Wolinski, Krzysztof
    Journal of Physical Chemistry (1990), 94 (14), 5483-8CODEN: JPCHAX; ISSN:0022-3654.
    Second-order perturbation theory based on a CASSCF ref. state is derived and implemented. The first-order wave function includes the full space of interacting states. Expressions for the contributions to the second-order energy are obtained in terms of up to four-particle d. matrixes for the CASSCF ref. state. The zeroth-order Hamiltonian reduces to the Moeller-Plesset Hamiltonian for a closed-shell ref. state. The limit of the implementation is given by the no. of active orbitals, which dets. the size of the d. matrixes. It is presently around 13 orbitals. The method is illustrated in a series of calcns. on H2, H2O, CH2, and F-, and the results are compared with corresponding full CI results.
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  18. 18
    Christiansen, O.; Koch, H.; Jorgensen, P. Response Functions in the CC3 Iterative Triplet Excitation Model. J. Chem. Phys. 1995, 103, 7429,  DOI: 10.1063/1.470315
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    18
    Response functions in the CC3 iterative triple excitation model
    Christiansen, Ove; Koch, Henrik; Joergensen, Poul
    Journal of Chemical Physics (1995), 103 (17), 7429-41CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The derivation of response functions for coupled cluster models is discussed in a context where approxns. can be introduced in the coupled cluster equations. The linear response function is derived for the approx. coupled cluster singles, doubles, and triples model CC3. The linear response functions for the approx. triples models, CCSDT-1a and CCSDT-1b, are obtained as simplifications to the CC3 linear response function. The consequences of these simplifications are discussed for the evaluation of mol. properties, in particular, for excitation energies. Excitation energies obtained from the linear response eigenvalue equation are analyzed in orders of the fluctuation potential. Double replacement dominated excitations are correct through second order in all the triples models mentioned, whereas they are only correct to first order in the coupled cluster singles and doubles model (CCSD). Single replacement dominated excitation energies are correct through third order in CC3, while in CCSDT-1a, CCSDT-1b, and CCSD they are only correct through second order. Calcns. of electronic excitation energies are reported for CH+, N2, and C2H4 to illustrate the accuracy that can be obtained in the various triples models. The CH+ results are compared to full CI results, the C2H4 results are compared with complete active space second order perturbation theory (CASPT2) and expt., and the N2 results are compared to expt. Double replacement dominated excitations are improved significantly relative to CCSD in all the triples models mentioned, and is of the same quality in CC3 and CCSDT-1a. The single replacement dominated excitation are close to full CI results for the CC3 model and significantly improved relative to CCSD. The CCSDT-1 results for the single replacement dominated excitations are not improved compared to CCSD.
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  19. 19
    Mooneyham, A.; McDonnell, M.; Drucker, S. Cavity Ringdown Spectrum of 2-Cyclohexen-1-one in the CO/Alkenyl CC Stretch Region of the S1(n, π*) – S0 Vibronic Band System. J. Phys. Chem. A 2017, 121, 2343,  DOI: 10.1021/acs.jpca.7b00826
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    19
    Cavity Ringdown Spectrum of 2-Cyclohexen-1-one in the CO/Alkenyl CC Stretch Region of the S1(n, π*)-S0 Vibronic Band System
    Mooneyham, Ashley E.; McDonnell, Michael P.; Drucker, Stephen
    Journal of Physical Chemistry A (2017), 121 (12), 2343-2352CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    The cavity ringdown (CRD) absorption spectrum of 2-cyclohexen-1-one (2CHO) vapor at room temp. was recorded over λ = 360-380 nm. This portion of the spectrum encompasses the S1(n,π*) ← S0 vibronic band system in the region of the C=C and C=O stretch fundamentals. Assignments were made for ∼40 vibronically resolved features in the spectrum, affording fundamental frequencies for 7 different vibrational modes in the S1(n,π*) state, including the C=C (1554 cm-1) and OC-CH (1449 cm-1) stretch modes. The C=O stretch character is spread over at least 4 different vibrational modes in the S1(n,π*) state, with fundamentals spanning the 1340-1430 cm-1 interval. This finding stems from a significant redn. in C=O bond order upon excitation, which leads to near-coincidence of the C=O stretch and several CH2 wag frequencies. Such complexities make 2CHO an ideal candidate for testing excited-state computational methods. The present spectroscopic results were used to test EOM-EE-CCSD harmonic-frequency predictions for the S1(n,π*) state. The performance was benchmarked the of less costly computational methods, including CIS(D) and TDDFT. For certain d. functionals (e.g., B3LYP and PBE0), the accuracy of TDDFT frequency predictions can approach but not meet that of EOM-EE-CCSD.
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  20. 20
    Jacquemin, D.; Adamo, C. In Density-Functional Methods for Excited States; Ferre, N., Filatov, M., HuixRotllant, M., Eds.; Topics in Current Chemistry-Series; 2016; Vol. 368; pp 347– 375.
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    Gordon, R. D.; Park, W. K. C. The 353 nm 1nπ* Transition of 4H-Pyran-4-one and a Deuterated Derivative. Can. J. Chem. 1993, 71, 1672,  DOI: 10.1139/v93-208
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    The 353-nm 1nπ* transition of 4H-pyran-4-one and a deuterated derivative
    Gordon, Robert D.; Park, William K. C.
    Canadian Journal of Chemistry (1993), 71 (10), 1672-5CODEN: CJCHAG; ISSN:0008-4042.
    The 353-nm vapor absorption of 4H-pyran-4-one and its 3,5-d2 deriv. has been measured and assigned to a forbidden, 1A2 (nπ*)-1A1 electronic transition, with electronic origin near 28,360 cm-1, made allowed by vibrations involving out-of-plane ring and C-H motions. Although the mol. remains planar upon excitation, other effects of conjugation on the nature of the excited state are less marked than in 2-cyclopenten-1-one.
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  22. 22
    Kaliman, I. A.; Krylov, A. I. New Algorithm for Tensor Contractions on Multi-Core CPUs, GPUs, and Accelerators Enables CCSD and EOM-CCSD Calculations with over 1000 Basis Functions on a Single Compute Node. J. Comp. Chem.J. Comp. Chem. 2017, 38, 842,  DOI: 10.1002/jcc.24713
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    New algorithm for tensor contractions on multi-core CPUs, GPUs, and accelerators enables CCSD and EOM-CCSD calculations with over 1000 basis functions on a single compute node
    Kaliman, Ilya A.; Krylov, Anna I.
    Journal of Computational Chemistry (2017), 38 (11), 842-853CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    A new hardware-agnostic contraction algorithm for tensors of arbitrary symmetry and sparsity is presented. The algorithm is implemented as a stand-alone open-source code libxm. This code is also integrated with general tensor library libtensor and with the Q-Chem quantum-chem. package. An overview of the algorithm, its implementation, and benchmarks are presented. Similarly to other tensor software, the algorithm exploits efficient matrix multiplication libraries and assumes that tensors are stored in a block-tensor form. The distinguishing features of the algorithm are: (i) efficient repackaging of the individual blocks into large matrixes and back, which affords efficient graphics processing unit (GPU)-enabled calcns. without modifications of higher-level codes; (ii) fully asynchronous data transfer between disk storage and fast memory. The algorithm enables canonical all-electron coupled-cluster and equation-of-motion coupled-cluster calcns. with single and double substitutions (CCSD and EOM-CCSD) with over 1000 basis functions on a single quad-GPU machine. We show that the algorithm exhibits predicted theor. scaling for canonical CCSD calcns., O(N6), irresp. of the data size on disk. © 2017 Wiley Periodicals, Inc.
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    Medvedev, E.; Pratt, D. Hund’s Case (a)-case (b) Transition in the Singlet-Triplet Absorption Spectrum of Pyrazine in a Supersonic Jet. J. Exp. Theor. Phys. 1998, 87, 35,  DOI: 10.1134/1.558642
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    Certain equipment, instruments, software, or materials, commercial or noncommercial, are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement of any product or service by the authors’ institutions (including NIST), nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

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  25. 25
    Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T. B.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X. Advances in Molecular Quantum Chemistry Contained in the Q-Chem 4 Program Package. Mol. Phys. 2015, 113, 184,  DOI: 10.1080/00268976.2014.952696
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    Advances in molecular quantum chemistry contained in the Q-Chem 4 program package
    Shao, Yihan; Gan, Zhengting; Epifanovsky, Evgeny; Gilbert, Andrew T. B.; Wormit, Michael; Kussmann, Joerg; Lange, Adrian W.; Behn, Andrew; Deng, Jia; Feng, Xintian; Ghosh, Debashree; Goldey, Matthew; Horn, Paul R.; Jacobson, Leif D.; Kaliman, Ilya; Khaliullin, Rustam Z.; Kus, Tomasz; Landau, Arie; Liu, Jie; Proynov, Emil I.; Rhee, Young Min; Richard, Ryan M.; Rohrdanz, Mary A.; Steele, Ryan P.; Sundstrom, Eric J.; Woodcock, H. Lee, III; Zimmerman, Paul M.; Zuev, Dmitry; Albrecht, Ben; Alguire, Ethan; Austin, Brian; Beran, Gregory J. O.; Bernard, Yves A.; Berquist, Eric; Brandhorst, Kai; Bravaya, Ksenia B.; Brown, Shawn T.; Casanova, David; Chang, Chung-Min; Chen, Yunquing; Chien, Siu Hung; Closser, Kristina D.; Crittenden, Deborah L.; Diedenhofen, Michael; DiStasio, Robert A., Jr.; Do, Hainam; Dutoi, Anthony D.; Edgar, Richard G.; Fatehi, Shervin; Fusti-Molnar, Laszlo; Ghysels, An; Golubeva-Zadorozhnaya, Anna; Gomes, Joseph; Hanson-Heine, Magnus W. D.; Harbach, Philipp H. P.; Hauser, Andreas W.; Hohenstein, Edward G.; Holden, Zachary C.; Jagau, Thomas-C.; Ji, Hyunjun; Kaduk, Ben; Khistyaev, Kirill; Kim, Jaehoon; Kim, Jihan; King, Rollin A.; Klunzinger, Phil; Kosenkov, Dmytro; Kowalczyk, Tim; Krauter, Caroline M.; Lao, Ka Un; Laurent, Adele; Lawler, Keith V.; Levchenko, Sergey V.; Lin, Ching Yeh; Liu, Fenglai; Livshits, Ester; Lochan, Rohini C.; Luenser, Arne; Manohar, Prashant; Manzer, Samuel F.; Mao, Shan-Ping; Mardirossian, Narbe; Marenich, Aleksandr V.; Maurer, Simon A.; Mayhall, Nicholas J.; Neuscamman, Eric; Oana, C. Melania; Olivares-Amaya, Roberto; O'Neill, Darragh P.; Parkhill, John A.; Perrine, Trilisa M.; Peverati, Roberto; Prociuk, Alexander; Rehn, Dirk R.; Rosta, Edina; Russ, Nicholas J.; Sharada, Shaama M.; Sharma, Sandeep; Small, David W.; Sodt, Alexander; Stein, Tamar; Stuck, David; Su, Yu-Chuan; Thom, Alex J. W.; Tsuchimochi, Takashi; Vanovschi, Vitalii; Vogt, Leslie; Vydrov, Oleg; Wang, Tao; Watson, Mark A.; Wenzel, Jan; White, Alec; Williams, Christopher F.; Yang, Jun; Yeganeh, Sina; Yost, Shane R.; You, Zhi-Qiang; Zhang, Igor Ying; Zhang, Xing; Zhao, Yan; Brooks, Bernard R.; Chan, Garnet K. L.; Chipman, Daniel M.; Cramer, Christopher J.; Goddard, William A., III; Gordon, Mark S.; Hehre, Warren J.; Klamt, Andreas; Schaefer, Henry F., III; Schmidt, Michael W.; Sherrill, C. David; Truhlar, Donald G.; Warshel, Arieh; Xu, Xin; Aspuru-Guzik, Alan; Baer, Roi; Bell, Alexis T.; Besley, Nicholas A.; Chai, Jeng-Da; Dreuw, Andreas; Dunietz, Barry D.; Furlani, Thomas R.; Gwaltney, Steven R.; Hsu, Chao-Ping; Jung, Yousung; Kong, Jing; Lambrecht, Daniel S.; Liang, WanZhen; Ochsenfeld, Christian; Rassolov, Vitaly A.; Slipchenko, Lyudmila V.; Subotnik, Joseph E.; Van Voorhis, Troy; Herbert, John M.; Krylov, Anna I.; Gill, Peter M. W.; Head-Gordon, Martin
    Molecular Physics (2015), 113 (2), 184-215CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
    A review. A summary of the tech. advances that are incorporated in the fourth major release of the Q-Chem quantum chem. program is provided, covering approx. the last seven years. These include developments in d. functional theory methods and algorithms, NMR (NMR) property evaluation, coupled cluster and perturbation theories, methods for electronically excited and open-shell species, tools for treating extended environments, algorithms for walking on potential surfaces, anal. tools, energy and electron transfer modeling, parallel computing capabilities, and graphical user interfaces. In addn., a selection of example case studies that illustrate these capabilities is given. These include extensive benchmarks of the comparative accuracy of modern d. functionals for bonded and non-bonded interactions, tests of attenuated second order Moller-Plesset (MP2) methods for intermol. interactions, a variety of parallel performance benchmarks, and tests of the accuracy of implicit solvation models. Some specific chem. examples include calcns. on the strongly correlated Cr2 dimer, exploring zeolite-catalyzed ethane dehydrogenation, energy decompn. anal. of a charged ter-mol. complex arising from glycerol photoionisation, and natural transition orbitals for a Frenkel exciton state in a nine-unit model of a self-assembling nanotube.
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    Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0Model. J. Chem. Phys. 1999, 110, 6158,  DOI: 10.1063/1.478522
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    26
    Toward reliable density functional methods without adjustable parameters: the PBE0 model
    Adamo, Carlo; Barone, Vincenzo
    Journal of Chemical Physics (1999), 110 (13), 6158-6170CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We present an anal. of the performances of a parameter free d. functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amt. of exact exchange. The results obtained for structural, thermodn., kinetic and spectroscopic (magnetic, IR and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chem. and condensed matter physics.
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    CFOUR (version 2.1), a Quantum Chemical Program Package written by J. F. Stanton, J. Gauss, M. E. Harding, P. G. Szalay with contributions from A. A. Auer, R. J. Bartlett, U. Benedikt, C. Berger, D. E. Bernholdt, Y. J. Bomble, L. Cheng, O. Christiansen, M. Heckert, O. Heun, C. Huber, et al. CFOUR uses the integral packages MOLECULE (J. Almlöf and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van Wüllen.

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  28. 28
    Polik, W. F.; Schmidt, J. R. WebMO: Web-Based Computational Chemistry Calculations in Education and Research. WIREs Computational Molecular Science 2022, 12, e1554,  DOI: 10.1002/wcms.1554
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    Judge, R. H.; Korale, A. A.; York, J. J.; Joo, D. L.; Clouthier, D. J.; Moule, D. C. Computerized Simulation and Fitting of Singlet-Triplet Spectra of Orthorhombic Asymmetric Tops – Theory and Extensions to Molecules with Large Multiplet Splittings. J. Chem. Phys. 1995, 103, 5343,  DOI: 10.1063/1.470569
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    Computerized simulation and fitting of singlet-triplet spectra of orthorhombic asymmetric tops: theory and extensions to mols. with large multiplet splittings
    Judge, R. H.; Korale, A. A.; York, J. J.; Joo, Duck-Lae; Clouthier, Dennis J.; Moule, D. C.
    Journal of Chemical Physics (1995), 103 (13), 5343-56CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Motivated by our recent finding that the singlet-triplet bands of selenoformaldehyde involve an upper state with large zero field splittings, we have extended the theory and written a program for predicting and fitting such rotationally resolved spectra. Triplet state matrix elements for a case (A) basis have been developed, including corrections for centrifugal and spin-centrifugal distortion. The full Hamiltonian matrix has been symmetry adapted, simplifying the problem to four individual matrixes of approx. equal size for mols. of orthorhombic symmetry. Diagonalization of these matrixes yields triplet state energies that are in agreement with previous treatments using a basis in which the spin splittings are small relative to the rotational intervals. Methods have been developed for sorting the eigenvalues and assigning quantum labels regardless of the magnitude of the spin splittings. The calcn. of the relative intensities of the rotational lines within a band has been programmed using transition moment matrix elements from the literature. The selection rules for various upper state symmetries have been developed in a form useful for the anal. of spectra. Band contour predictions of spectra for various coupling cases have been presented.
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    Csaszar, P.; Csaszar, A.; Somogyi, A.; Dinya, Z.; Holly, S.; Gal, M.; Boggs, J. E. Vibrational Spectra, Scaled Quantum-Mechanical (SQM) Force Field and Assignments for 4H-Pyran-4-one. Spectrochim. Acta 1986, 42A, 473,  DOI: 10.1016/0584-8539(86)80043-5
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    Vibrational spectra, scaled quantum-mechanical (SQM) force field and assignments for 4H-pyran-4-one
    Csaszar, Pal; Csaszar, Attila; Somogyi, Arpad; Dinya, Zoltan; Holly, Sandor; Gal, Miklos; Boggs, James E.
    Spectrochimica Acta, Part A: Molecular and Biomolecular Spectroscopy (1986), 42A (4), 473-86CODEN: SAMCAS; ISSN:0584-8539.
    The gas-phase IR spectrum of 4H-pyran-4-one (γ-pyrone) was recorded in the 4000-400 cm-1 region by a Nicolet 7199 FTIR spectrometer and interpreted using a general valence force field calcd. quantum mech. at the ab initio level with a split-valence 4-21 basis. Assignment of certain fundamentals was facilitated by information gained from the IR and Raman spectra of the melt and from the IR spectrum of the satd. soln. in CCl4. To account for systematic computational errors, the theor. ab initio force field was scaled using a set of consts. derived by the empirical fitting of force fields computed for related mols. to their obsd. spectra. Either the scale factors derived for a family of open-chain mols. or, better, for C6H6 could be used to yield a scaled force field which gave unequivocal assignments for γ-pyrone. The method promises to be of general applicability for mols. of this complexity.
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    Smithson, T. L.; Ibrahim, N.; Wieser, H. Cyclohexanones: Evidence of Chair Inversion and Estimate for Barriers to Planarity from the Far-Infrared Spectra. Can. J. Chem. 1983, 61, 1924,  DOI: 10.1139/v83-330
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    Cyclohexanones: evidence of chair inversion and estimate for barriers to planarity from the far-infrared spectra
    Smithson, Tracy L.; Ibrahim, Nan; Wieser, Hal
    Canadian Journal of Chemistry (1983), 61 (8), 1924-32CODEN: CJCHAG; ISSN:0008-4042.
    The far-IR, at 50-450 cm-1 are reported for the vapors of cyclohexanone and the structurally related compds. tetrahydro-4H-pyran-4-one, tetrahydro-4H-thiopyran-4-one, tetrahydro-4H-pyran-3-one, 1,3-dioxan-5-one, and 4H-pyran-4-one. Except for the last mentioned, the IR are characterized by 2 or in some cases 3 sequences of Q branches which are assigned to the out-of-plane deformations of the cyclohexanone ring. The lowest wavenumber sequence in each compd. arises from a vibration which, if completely executed, would take the chair conformation to its equiv. via an inversion through the planar form. Each sequence is amenable to soln. by a one-dimensional Hamiltonian incorporating a quadratic-quartic potential function, providing ests. for the magnitudes of the barriers to planarity.
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    Wathelet, V.; Preat, J.; Bouhy, M.; Fontaine, M.; Perpete, E.; Andre, J.; Jacquemin, D. Assessment of PBE0 for Evaluating the Absorption Spectra of Carbonyl Molecules. Int. J. Quantum Chem. 2006, 106, 1853,  DOI: 10.1002/qua.20982
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    Assessment of PBE0 for evaluating the absorption spectra of carbonyl molecules
    Wathelet, Valerie; Preat, Julien; Bouhy, Michael; Fontaine, Michele; Perpete, Eric A.; Andre, Jean-Marie; Jacquemin, Denis
    International Journal of Quantum Chemistry (2006), 106 (8), 1853-1859CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
    Using the parameter-free PBE0 hybrid functional, the authors compute the UV/visible spectra of solvated compds. presenting a carbonyl chromophoric unit linked to a C-C double bond. PBE0 is extremely efficient for accurately reproducing exptl. values, with a mean unsigned error of 7 nm for an extended set of compds., although no fitting or statistical treatments are performed. PBE0 has a predictive efficiency comparable to the known Woodward-Fieser empirical formula, and can therefore be used to extend these rules without requiring addnl. exptl. results. Consequently, the UV/visible spectra of several compds. that have not yet been synthesized are predicted.
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    Benchmarking the performance of time-dependent density functional methods
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    The performance of 24 d. functionals, including 14 meta-generalized gradient approxn. (mGGA) functionals, is assessed for the calcn. of vertical excitation energies against an exptl. benchmark set comprising 14 small- to medium-sized compds. with 101 total excited states. The exptl. benchmark set consists of singlet, triplet, valence, and Rydberg excited states. The global-hybrid (GH) version of the Perdew-Burke-Ernzerhoff GGA d. functional (PBE0) is found to offer the best overall performance with a mean abs. error (MAE) of 0.28 eV. The GH-mGGA Minnesota 2006 d. functional with 54% Hartree-Fock exchange (M06-2X) gives a lower MAE of 0.26 eV, but this functional encounters some convergence problems in the ground state. The local d. approxn. functional consisting of the Slater exchange and Volk-Wilk-Nusair correlation functional (SVWN) outperformed all non-GH GGAs tested. The best pure d. functional performance is obtained with the local version of the Minnesota 2006 mGGA d. functional (M06-L) with an MAE of 0.41 eV. (c) 2012 American Institute of Physics.
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    Accuracy of TD-DFT Geometries: A Fresh Look
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    We benchmark a panel of 48 DFT exchange-correlation functionals in the framework of TD-DFT optimizations of the geometry of valence singlet excited states. To this end, we use a set of 41 small- and medium-sized org. mols. for which ref. geometries were obtained at high level of theory, typically, CC3 or CCSDR(3), with the aug-cc-pVTZ at. basis set. For the ground-state parameters, the tested functionals provide av. deviations that are small (0.010 Å and 0.5° for bond lengths and valence angles) and not very sensitive to the selected (hybrid) functional, but the errors are larger for the most polarized bonds (CO, CN, and so on). Nevertheless, DFT has a tendency to provide too compact distances, a trend slightly enhanced for functionals including a large amt. of exact exchange. The av. errors largely increase when going to the excited-state for most bond types, i.e., TD-DFT delivers less accurate excited-state distances than DFT for ground state. In particular TD-DFT combined with hybrid functionals provides significantly too short CO and CS/CSe bonds with resp. av. errors in the -0.026/-0.052 Å and -0.015/-0.082 Å ranges, depending on the selected hybrid functional. For the carbonyl bonds, the sizes of the TD-DFT deviations obtained when selecting std. hybrid functionals are of the same order of magnitude as the EOM-CCSD ones.
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    A review. The choice of basis set in quantum chem. calcns. can have a huge impact on the quality of the results, esp. for correlated ab initio methods. This article provides an overview of the development of Gaussian basis sets for mol. calcns., with a focus on four popular families of modern atom-centered, energy-optimized bases: at. natural orbital, correlation consistent, polarization consistent, and def2. The terminol. used for describing basis sets is briefly covered, along with an overview of the auxiliary basis sets used in a no. of integral approxn. techniques and an outlook on possible future directions of basis set design. © 2012 Wiley Periodicals, Inc.
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    Microwave spectra were detd. of the planar mols. 4H-pyran-4-one (I), 4H-thiapyran-4-thione (II), 4-H-thiapyran-4-one, and 4H-thiapyran-4-thione (III). Extensive isotopic results for I, II, and III enabled precise, complete structures to be calcd. The structures gave little evidence for enhanced aromaticity in these mols. The high dipole moment values (∼4 D) obtained for I, II, and III in soln. (Rolla, M., et al., 1952) were confirmed and the values more closely defined. Vibrational satellite spectra indicate that the lowest frequency fundamentals of these mols. are out-of-plane motions near 100 cm-1.
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    The authors present a 1st practical demonstration of Raman spectroscopy to obtain high definition maps of rotational temps. and abs. densities in mol. supersonic jets.
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    The low and high resoln. fluorescence excitation spectra of d3-tryptamine were obsd. in the environment of a cold, supersonic mol. beam. As in the case of unlabeled tryptamine, 6 bands due to the origins of different conformers were found in the low-resoln. spectrum of deuterated tryptamine. A previous paper reported that conformers labeled A and B and conformers labeled D and E of tryptamine have identical rotational structures. However, for deuterated tryptamine those conformers have distinguishable rotational structures. Anal. of the rotational structure in the high-resoln. electronic spectra of 5 of the bands was used to det. the geometries of the different conformers. Conformers A, B, and F have a gauche conformation with respect to the rotation about the Cα-Cβ bond while conformers D and E have an eclipsed conformation. The geometries of conformers A and B and conformers D and E differ only in the orientation of the amino group. In these structures the angles of the internal rotation of the amino group are 180°, 60°, 180°, 60°, and -60° for conformers A,B, and D-F, resp. Feature C consists of 2 overlapped conformers; these conformers may have the amino group trans to the indole ring.
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    We describe principal features of the newly released version, NBO 6.0, of the natural bond orbital anal. program, that provides novel "link-free" interactivity with host electronic structure systems, improved search algorithms and labeling conventions for a broader range of chem. species, and new anal. options that significantly extend the range of chem. applications. We sketch the motivation and implementation of program changes and describe newer anal. options with illustrative applications. © 2013 Wiley Periodicals, Inc.
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    The dominant interaction of O with a magnetic field is through the electronic spin magnetic moment. However, a precise comparison with expt. of the results of calcg. the microwave paramagnetic spectrum, assuming only this interaction, shows a systematic discrepancy, which is removed by introducing 2 corrections. The larger (∼0.1%, or 7 gausses) is a correction for the second-order electronic orbital moment coupled in by the spin-orbit energy. Its magnitude is proportional to the second-order term μ'' in the spin-rotation coupling const. The smaller (∼1 gauss) is a correction for the rotation-induced magnetic moment of the mol. Since the dependence of this contribution on quantum nos. is unique, this coeff. can also be detd. by fitting the magnetic spectrum. A total of 120 X-band and 78 S-band lines were observed. The complete corrections were made on 26 lines with a mean residual error of ∼0.5 Mc./sec. This excellent agreement confirms the anomalous electronic moment to 60 p.p.m. and also confirms the validity of the Zeeman-effect theory. A new result is the rotational magnetic moment of -0.25 ± 0.05 nuclear magnetons per quantum of rotation. Knowledge of this moment allows the electronic contribution to the effective moment of inertia to be detd. This correction of 65 p.p.m. is made, the latest fitting of the universal at. consts. is used, and the equil. internuclear distance is recalcd. to be Rc = 1.20741 ± 0.00002 A. λ'', the second-order spin-orbit contribution to the coupling of the spin to the figure axis, is 465 ± 50 Mc./sec., less than 1% of the total coupling const. λ. Theoretical intensities of a no. of the microwave transitions are calcd. and successfully compared with expt. over a range of 100 to 1 in magnitude. The ΔM = 0 transitions are less than 0.01 of the strength of the δM = ± 1 transitions, and are too weak to observe. J breaks down as a quantum no. in the presence of a magnetic field. This allows ΔJ = ±2 transitions to comprise roughly half of all lines observed.
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    We present calcns. of electron spin-spin (SS) coupling strengths evaluated as expectation values over multi-configuration and restricted high-spin SCF wave functions. Together with the spin-orbit (SO) CI methodol., this enables us to analyze the full spin Hamiltonian including the zero-field splitting (ZFS) parameters of the triplet state. The calcd. ZFS parameters include both the SS coupling to first order and SO coupling to second order of perturbation theory. The relative importance of these two contributions is strongly system dependent. In the lowest triplet state of the benzene mol., the main ZFS parameter - the D parameter - is detd. entirely by the SS coupling, with D calcd. to be 0.1583 cm-1. In contrast, the calcd. D parameter in the X3Σg- ground state of the oxygen mol. (3.77 cm-1) includes a large one-center SS contribution (DSS = 1.455 cm-1) but an even larger SO coupling contribution (DSO = 2.315 cm-1). ZFS parameters for mol. oxygen excited states, A3Σu+ and B3Σu-, which belong to Hund's case (b), are also calcd. A large neg. D value (-10.2 cm-1) for the A3Σu+ state is to 90% detd. by the DSO contribution, while the Schumann-Runge state spin splitting is mainly detd. by SS coupling. The calcd. values for benzene and oxygen are in good agreement with data from EPR and rotational fine-structure spectra. The applicability of response theory is with this contribution expanded to include the calcn. of the SS coupling of the spin Hamiltonian, complementing the previous implementations of hyperfine A- and g-tensors.
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    Moule, D.; Chantranupong, L.; Judge, R.; Clouthier, D. Ab-Initio Predictions of the Zero-Field Splittings and tthe Singlet-Triplet Transition Strengths for the ã3A2 (T1) ← x̃ 1A1 (S0), n → π* Transition of Selenoformaldehyde. Can. J. Chem. 1993, 71, 1706,  DOI: 10.1139/v93-212
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    Experimental proof of the case (ab) coupling hypothesis in the first excited triplet state of selenoformaldehyde (H2C:Se)
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    A high-resoln. laser-induced phosphorescence spectrum of the vibronically induced 401 band of the a~ 3A2-X~ 1A1 system of D2C80Se was recorded with Doppler-limited resoln. Rotational anal. revealed that transitions to all three spin components occur within the 12,265-12,525 cm-1 region. The rotational structure of the band was fitted to the exptl. precision of measurement using a triplet state Hamiltonian that allowed for slight variations in the geometries of the individual spin components. The triplet state of selenoformaldehyde involves case (ab) coupling, in which two of the spin components are close together and the 3rd is ∼125 cm-1 higher in energy. This unusual occurrence of large zero-field splittings in the triplet state of an asym. top arises because of the close proximity of the 3A1(π,π*) state which substantially displaces two of the spin components to lower energy.
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  • Abstract

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    Figure 1

    Figure 1. Structural formula of 4H-pyran-4-one (4PN), along with the molecule-fixed coordinate system used here. The molecule has a C2v planar equilibrium geometry in its ground and T1(n,π*) states.

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    Figure 2

    Figure 2. Overview of the T1(n,π*) ← S0 phosphorescence excitation spectrum of 4PN, recorded using a supersonic free-jet expansion. The buffer gas was 5 atm of helium. The maximum of the 000 origin band is located at 27293.2 cm–1. Inset shows the origin band on an expanded horizontal scale.

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    Figure 3

    Figure 3. Low-wavenumber region of the spectrum in Figure 2. Blue tie lines show assignments of vibronic cold bands (v″ = 0), and red tie lies show assignments of hot bands (v″ > 0). Franck–Condon factors (FCF) were computed using a procedure described in the text. Assignment labels without tie lines, shown above the spectrum, refer to FC-forbidden bands.

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    Figure 4

    Figure 4. Continuation of the spectrum in Figure 3.

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    Figure 5

    Figure 5. Continuation of the spectrum in Figure 4. The black trace is plotted using the same vertical scale as in Figures 3 and 4. The green trace is plotted on a contracted (× 1/4) vertical scale in order to show peaks near 800 and 900 cm–1, which have maximum intensity.

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    Figure 6

    Figure 6. Region of T1(n,π*) ← S0 phosphorescence excitation spectrum of 4PN, about 700 cm–1 above the origin band. These traces show the suppression of S1(n,π*) ← S0 hot bands under colder helium expansion conditions.

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    Figure 7

    Figure 7. Jet-cooled T1(n,π*) ← S0 phosphorescence excitation spectrum of 4PN, recorded using a helium backing pressure of 2 atm. The region of the 000 band is shown. The band origin, listed in Table 4, was determined using a simulation procedure outlined in the text. Branch labels are given in the format ΔNΔJ, where the O-form branch corresponds to ΔN = −2; the P-form branch corresponds to ΔN = −1, etc.

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    Figure 8

    Figure 8. Experimental spectrum of Figure 7 (black trace), along with simulation (red trace) produced using the molecular constants listed in Table 4. The simulation used a two-temperature model (see text) to describe the distribution of rotational levels in the ground state. The two temperatures were chosen to be 4 and 16 K, with weighting factors of 0.4 and 0.6, respectively.

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    Figure 9

    Figure 9. Experimental spectrum of Figure 7 (black trace), along with simulation (red trace) produced using T1(n,π*) inertial constants from an EOM-EE-CCSD calculation rather than TDDFT (PBE0).

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    Figure 10

    Figure 10. Origin band of the 4PN T1(n,π*) ← S0 transition, simulated using a range of α values departing from its optimum (red trace) by +20% to −20%. Other molecular constants are those in Table 4. Simulations employing positive and negative deviations in α are shown in blue and green, respectively. Black trace is the experimental spectrum.

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    Figure 11

    Figure 11. Isosurfaces of the canonical molecular orbitals for the lowest-energy electronic transition of 4PN. The value of |ψ| on each isosurface is 0.050 Å–3/2.

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  • References

    This article references 46 other publications.

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      Merz, T.; Bierhance, G.; Flach, E.-C.; Kats, D.; Usvyat, D.; Schutz, M. Description of Excited states in Photochemistry with Theoretical Methods. Physical Sciences Reviews 2021, 6, Art. No. 20170178. DOI: 10.1515/psr-2017-0178
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      Mechanism and rate constants of the CH2 + CH2CO reactions in triplet and singlet states: A theoretical study
      Savchenkova, Anna S.; Semenikhin, Alexander S.; Chechet, Ivan V.; Matveev, Sergey G.; Konnov, Alexander A.; Mebel, Alexander M.
      Journal of Computational Chemistry (2019), 40 (2), 387-399CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      Ab initio and d. functional CCSD(T)-F12/cc-pVQZ-f12//B2PLYPD3/6-311G** calcns. have been performed to unravel the reaction mechanism of triplet and singlet methylene CH2 with ketene CH2CO. The computed potential energy diagrams and mol. properties have been then utilized in Rice-Ramsperger-Kassel-Marcus-Master Equation (RRKM-ME) calcns. of the reaction rate consts. and product branching ratios combined with the use of nonadiabatic transition state theory for spin-forbidden triplet-singlet isomerization. The results indicate that the most important channels of the reaction of ketene with triplet methylene lead to the formation of the HCCO + CH3 and C2H4 + CO products, where the former channel is preferable at higher temps. from 1000 K and above. In the C2H4 + CO product pair, the ethylene mol. can be formed either adiabatically in the triplet electronic state or via triplet-singlet intersystem crossing in the singlet electronic state occurring in the vicinity of the CH2COCH2 intermediate or along the pathway of CO elimination from the initial CH2CH2CO complex. The predominant products of the reaction of ketene with singlet methylene have been shown to be C2H4 + CO. The formation of these products mostly proceeds via a well-skipping mechanism but at high pressures may to some extent involve collisional stabilization of the CH3CHCO and cyclic CH2COCH2 intermediates followed by their thermal unimol. decompn. The calcd. rate consts. at different pressures from 0.01 to 100 atm have been fitted by the modified Arrhenius expressions in the temp. range of 300-3000 K, which are proposed for kinetic modeling of ketene reactions in combustion.
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      Naskar, S.; Das, M. Singlet and Triplet excited State Energy Ordering in Cyclopenta-fused Polycyclic Aromatic Hydrocarbons (CP-PAHs) Suitable for Energy Harvesting: An Exact Model and TDDFT Study. ACS Omega 2017, 2, 1795,  DOI: 10.1021/acsomega.7b00278
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      Singlet and Triplet Excited State Energy Ordering in Cyclopenta-Fused Polycyclic Aromatic Hydrocarbons (CP-PAHs) Suitable for Energy Harvesting: An Exact Model and TDDFT Study
      Naskar, Sumit; Das, Mousumi
      ACS Omega (2017), 2 (5), 1795-1803CODEN: ACSODF; ISSN:2470-1343. (American Chemical Society)
      We calcd. the ground and low-lying excited states of cyclopenta-fused polycyclic arom. hydrocarbons (CP-PAHs) using exact diagonalization in full CI (CI) within the model PPP Hamiltonian as well as a time-dependent d. functional theory technique. The CP-PAHs include acenaphthylene, isomers of pyracylene, cycloocta-pentalene, and three isomers of dicyclo-pentacyclo-octenes (DCPCO). We used the inherent symmetries of these systems to calc. the energy ordering of the lowest singlet (S1) and lowest triplet excited (T1) states with respect to the ground state (S0). The calcn. shows that the lowest dipole allowed singlet absorption varies from 0.43 to 1.42 eV for most of these systems. Such an optical absorption range is very promising in harvesting solar radiation ranging from the visible to near-IR region improving the efficiency of photovoltaic device application. The calcd. optical gaps for pyracylene, acenaphthylene, and two isomers of DCPCO are in very good agreement with exptl. results reported in the literature. The calcd. S1-T1 energy gaps (ΔST) in cycloocta-pentalene and in the DCPCO isomers are very small ranging from 0.01 to 0.2 eV, which is highly desirable in improving their electroluminescence efficiency in light-emitting device applications.
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    4. 4
      Sessions, A.; McDonnell, M.; Christianson, D.; Drucker, S. Triplet and Singlet (n, π*) Excited States of 4H-Pyran-4-one Characterized by Cavity Ringdown Spectroscopy and Quantum-Chemical Calculations. J. Phys. Chem. A 2019, 123, 6269,  DOI: 10.1021/acs.jpca.9b04238
      4
      Triplet and Singlet (n,π*) Excited States of 4H-Pyran-4-one Characterized by Cavity Ringdown Spectroscopy and Quantum-Chemical Calculations
      Sessions, Anna G.; McDonnell, Michael P.; Christianson, Drew A.; Drucker, Stephen
      Journal of Physical Chemistry A (2019), 123 (29), 6269-6280CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      The 4H-pyran-4-one (4PN) mol. serves as a model for investigating structural changes following π* ← n electronic excitation. We have recorded the cavity ringdown (CRD) absorption spectrum of 4PN vapor at room temp., over the wavelength region from 350 to 370 nm. This spectral region includes the T1(n,π*) ← S0 band system as well as the low-energy portion of the S1(n,π*) ← S0 system. Aided by predictions from ab initio (equation-of-motion excitation energies with dynamical correlation incorporated at the level of coupled cluster singles doubles, EOM-EE-CCSD) and d. functional theory (time-dependent d. functional theory with PBE0 functional, TDPBE0) calcns., we have made vibronic assignments for about 30 features in the CRD spectrum, mostly T1(n,π*) ← S0 transitions. We have used these results to correct certain vibronic assignments appearing in the previous literature for both T1(n,π*) ← S0 and S1(n,π*) ← S0 band systems. We conclude that the lowest-energy carbonyl wagging fundamentals (ν27, in-plane and ν17, out-of-plane) undergo significant frequency drops (28 and 50%, resp.) upon T1(n,π*) ← S0 excitation and similar drops (29 and 39%, resp.) for S1(n,π*) ← S0 excitation. We find that vibrational modes involving the conjugated ring atoms undergo relatively small frequency changes upon π* ← n excitation, for both T1 and S1 states. We have used the present spectroscopic results and vibronic assignments to test the accuracy of computed excited-state frequencies for 4PN. This benchmarking process shows that the economical time-dependent d. functional theory method is impressively accurate for certain (but not all) vibrational modes. The highly correlated EOM-EE-CCSD ab initio method is capable of making accurate frequency predictions, but the results, unexpectedly, depend sensitively on basis set family. This anomaly is traceable to a computed conical intersection between the T1(n,π*) and T2(π,π*) surfaces near the T1(n,π*) potential min. Relatively small errors in the location of the conical intersection lead to enhanced mixing of the two electronic states and incorrect T1(n,π*) vibrational frequencies when certain triple-ζ quality basis sets are used.
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    5. 5
      McAnally, M. O.; Zabronsky, K. L.; Stupca, D. J.; Pillsbury, N. R.; Phillipson, K.; Drucker, S. Lowest Triplet (n, π*) State of 2-Cyclohexen-1-one: Characterization by Cavity Ringdown Spectroscopy and Quantum-Chemical Calculations. J. Chem. Phys. 2013, 139, 214311– 1,  DOI: 10.1063/1.4834655
      5
      Lowest triplet (n,π*) state of 2-cyclohexen-1-one: Characterization by cavity ringdown spectroscopy and quantum-chemical calculations
      McAnally, Michael O.; Zabronsky, Katherine L.; Stupca, Daniel J.; Phillipson, Kaitlyn; Pillsbury, Nathan R.; Drucker, Stephen
      Journal of Chemical Physics (2013), 139 (21), 214311/1-214311/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The cavity ringdown (CRD) absorption spectrum of 2-cyclohexen-1-one (2CHO) was recorded over the range 401.5-410.5 nm in a room-temp. gas cell. The very weak band system (ε ≤ 0.1 M-1 cm-1) in this spectral region is due to the T1(n, π*) ← S0 electronic transition. The 000 origin band was assigned to the feature obsd. at 24 558.8 ± 0.3 cm-1. We have assigned 46 vibronic transitions in a region extending from -200 to +350 cm-1 relative to the origin band. For the majority of these transitions, we have made corresponding assignments in the spectrum of the deuterated deriv. 2CHO-2,6,6-d3. From the assignments, we detd. fundamental frequencies for several vibrational modes in the T1(n, π*) excited state of 2CHO, including the lowest ring-twisting (99.6 cm-1) and ring-bending (262.2 cm-1) modes. These values compare to fundamentals of 122.2 cm-1 and 251.9 cm-1, resp., detd. previously for the isoconfigurational S1(n, π*) excited state of 2CHO and 99 cm-1 and 248 cm-1, resp., for the S0 ground state. With the aid of quantum-mech. calcns., we have also ascertained descriptions for these two modes, thereby resolving ambiguities appearing in the previous literature. The ring-twisting mode (ν39) contains a significant contribution from O=C-C=C torsion, whereas the ring-bending mode (ν38 in the ground state) involves mainly the motion of C-5 with respect to the plane contg. the other heavy atoms. The CRD spectroscopic data for the T1(n, π*) state have allowed us to benchmark several computational methods for treating excited states, including time-dependent d. functional theory and an equation-of-motion coupled cluster method. In turn, the computational results provide an explanation for obsd. differences in the T1(n, π*) vs. S1(n, π*) ring frequencies. (c) 2013 American Institute of Physics.
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    6. 6
      O’Keefe, A.; Deacon, D. A. G. Cavity Ring-Down Optical Spectrometer for Absorption Measurements Using Pulsed Laser Sources. Rev. Sci. Instrum. 1988, 59, 2544,  DOI: 10.1063/1.1139895
      6
      Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources
      O'Keefe, Anthony; Deacon, David A. G.
      Review of Scientific Instruments (1988), 59 (12), 2544-51CODEN: RSINAK; ISSN:0034-6748.
      A technique was developed which allows optical absorption measurements to be made using a pulsed light source and offers a sensitivity significantly greater than that attained using stabilized continuous light sources. The technique is based upon the measurement of the rate of absorption rather than the magnitude of absorption of a light pulse confined within a closed optical cavity. The decay of the light intensity within the cavity is a simple exponential with loss components due to mirror loss, broadband scatter (Rayleigh, Mie), and mol. absorption. Narrowband absorption spectra are recorded by scanning the output of a pulsed laser (which is injected into the optical cavity) through an absorption resonance. The sensitivity of this technique was demonstrated by measuring several bands in the very weak forbidden b 1Σg - χ3Σg transition in gaseous mol. O. Absorption signals of <1 part in 106 can be detected.
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    7. 7
      Pillsbury, N. R.; Zwier, T. S.; Judge, R. H.; Drucker, S. Jet-cooled phosphorescence excitation spectrum of the T1(n, π*) ← S0 transition of 2-cyclopenten-1-one. J. Phys. Chem. A 2007, 111, 8357,  DOI: 10.1021/jp072353r
      7
      Jet-Cooled Phosphorescence Excitation Spectrum of the T1(n,π) ← S0 Transition of 2-Cyclopenten-1-one
      Pillsbury, Nathan R.; Zwier, Timothy S.; Judge, Richard H.; Drucker, Stephen
      Journal of Physical Chemistry A (2007), 111 (34), 8357-8366CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      The T1(n,π*) ← S0 transition of 2-cyclopenten-1-one (2CP) was investigated by using phosphorescence excitation (PE) spectroscopy in a free-jet expansion. The origin band, near 385 nm, is the most intense feature in the T1(n,π*) ← S0 PE spectrum. A short progression in the ring-bending mode (ν'30) is also obsd. The effective vibrational temp. in the jet is estd. at 50 K. The spectral simplification arising from jet cooling helps confirm assignments made previously in the room-temp. cavity ring-down (CRD) absorption spectrum, which is congested by vibrational hot bands. In addn. to the origin and ν'30 assignments, the jet-cooled PE spectrum also confirms the 2801 (C:O out-of-plane wag), 2901 (C:C twist), and 1901 (C:O in-plane wag) band assignments that were made in the T1(n,π*) ← S0 room-temp. CRD spectrum. The temporal decay of the T1 state of 2CP was investigated as a function of vibronic excitation. Phosphorescence from the v' = 0 level persists the entire time the mols. traverse the emission detection zone. Thus the phosphorescence lifetime of the v' = 0 level is significantly longer than the 2 μs transit time through the viewing zone. Higher vibrational levels in the T1 state have shorter phosphorescence lifetimes, on the order of 2 μs or less. The concomitant redn. in emission quantum yield causes the higher vibronic bands (above 200 cm-1) in the PE spectrum to be weak. It is proposed that intersystem crossing to highly vibrationally excited levels of the ground state is responsible for the faster decay and diminished quantum yield. The jet cooling affords partial rotational resoln. in the T1(n,π*) ← S0 spectrum of 2CP. The rotational structure of the origin band was simulated by using inertial consts. available from a previously reported d. functional (DFT) calcn. of the T1(n,π*) state, along with spin consts. obtained via a fitting procedure. Intensity parameters were also systematically varied. The optimized intensity factors support a model that identifies the S2(π,π*) ← S0 transition in 2CP as the sole source of oscillator strength for the T1(n,π*) ← S0 transition.
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    8. 8
      Spangler, L.; Pratt, D. Laser-Induced Phosphorescence Spectroscopy in Supersonic Jets - the Lowest Triplet-States of Glyoxal, Methylglyoxal, and Biacetyl. J. Chem. Phys. 1986, 84, 4789,  DOI: 10.1063/1.449965
      8
      Laser-induced phosphorescence spectroscopy in supersonic jets. The lowest triplet states of glyoxal, methylglyoxal, and biacetyl
      Spangler, Lee. H.; Pratt, David W.
      Journal of Chemical Physics (1986), 84 (9), 4789-96CODEN: JCPSA6; ISSN:0021-9606.
      The lowest triplet states of 3 α-dicarbonyls (glyoxal, methylglyoxal, and biacetyl) are reported using laser-induced phosphorescence spectroscopy in supersonic jets. At the level of vibrational resoln., 3Au glyoxal apparently has a geometry similar to that of the ground state. The T1 ← S0 transitions of methylglyoxal and biacetyl each exhibit strong progressions in the torsional vibrations of the Me groups, showing that these mols. undergo a conformational change on excitation to the lowest triplet state. A Franck-Condon anal. of the methylglyoxal spectrum, with proper consideration for nuclear spin statistics, yields a Me barrier of V3 = 115 ± 5 cm-1 in this state. This value was confirmed by a direct measurement of the tunneling splitting of A and E torsional levels. The hindering potential in the lowest triplet state of methylglyoxal is different from those in the ground (V3 = 269 cm-1) and 1st excited singlet (V3 = 190 cm-1) states. Possible reasons for these differences are discussed.
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    9. 9
      Tomer, J. L.; Holtzclaw, K. W.; Pratt, D. W.; Spangler, L. H. Phosphorescence Excitation Spectroscopy in Supersonic Jets – The Lowest Triplet-State of Pyrazine. J. Chem. Phys. 1988, 88, 1528,  DOI: 10.1063/1.454132
      9
      Phosphorescence excitation spectroscopy in supersonic jets. The lowest triplet state of pyrazine
      Tomer, J. L.; Holtzclaw, K. W.; Pratt, D. W.; Spangler, L. H.
      Journal of Chemical Physics (1988), 88 (3), 1528-38CODEN: JCPSA6; ISSN:0021-9606.
      The laser-induced phosphorescence excitation spectrum of pyrazine was examd. in the collision-free environment of a supersonic jet. The origin of the lowest triplet state (T1) lies at 26,820 cm-1 and exhibits a sym. parallel-type rotational contour, confirming that this state is 3B3u (nπ*) with an equil. geometry that is similar to those of the S0 (1Ag) and S1 (1B3u,nπ*) states. Thirty vibrational bands were also obsd. in the ∼4000 cm-1 interval between the T1 and S1 origins. Of these, the 13 lower energy bands all exhibit parallel-type contours and may be assigned as T1 ← S0 transitions, principally involving totally sym. modes. The 17 higher energy bands exhibit both parallel and perpendicular contours and may be assigned as S1 ← S0 hot band transitions, some involving nontoally sym. modes. No evidence for a 2nd, ππ* triplet state lying below the S1 origin was found, nor is there any evidence for rapid relaxation of any of the zero-order T1 levels at a resoln. of ∼1 cm-1. The intersystem crossing dynamics of S1 pyrazine is governed by the interaction of the 2 largely nested potential surfaces, S1 and T1, zero-order nπ* states that appear to differ primarily in the extent to which they interact vibronically with other zero-order states in manifolds of the corresponding multiplicity.
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    10. 10
      Raynes, W. Spin Splittings and Rotational Structure of Nonlinear Molecules in Triplet Electronic States. J. Chem. Phys. 1964, 41, 3020,  DOI: 10.1063/1.1725668
      10
      Spin splittings and rotational structure of nonlinear molecules in doublet and triplet electronic states
      Raynes, W. T.
      Journal of Chemical Physics (1964), 41 (10), 3020-32CODEN: JCPSA6; ISSN:0021-9606.
      Matrix elements are presented for the Hamiltonian of a nonlinear, nonrigid polyat. mol. in a multiplet electronic state. Their use is appropriate only for electronic and vibrational spectra, since hyperfine interactions involving nuclear spins and nuclear quadrupole moments are not considered. For the most general case, 9 parameters are required to take full account of spin-rotation interactions, and 5 are required for spinspin interactions. For mols. of orthorhombic symmetry only 3 spin-rotation parameters and 2 spin-spin parameters are nonzero. For nonlinear mols. in doublet and triplet electronic states, explicit formulas are presented for (a) the rotational term values of sym. rotors and (b) spin splittings of asym. rotors possessing orthorhombic symmetry. All these formulas reduce to well-known expressions for diat. mols. in 2Σ and 3Σ states when K-dependent terms are ignored. Application of these formulas to the results of Dressler and Ramsay (CA 54, 19155f) on the 2B1 ground states of NH2 and ND2 permits the detn. of the spin-rotation parameters of these mols. All 5 spin parameters of HCHO in its lowest 3A2 state are given, together with curves of spin splittings in the lower K levels. The spin parameters of HCHO, NH2, and ND2 are compared with those of NO2 and ClO2 found by recent microwave studies. For a triplet state of an orthorhombic mol., the spin-spin consts. detd. by band spectroscopy are simply related to the spin consts. D and E detd. from zero field splittings in ESR spectroscopy. The surprisingly small value of D = 0.42 cm.-1 for the lowest triplet state of HCHO is discussed briefly in terms of a breakdown of the orbital approxn. for this prototype n-π* state.
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    11. 11
      Stanton, J.; Bartlett, R. The Equation of Motion Coupled-Cluster Method ─ a Systematic Biorthogonal Approach to Molecular-Excitation Energies, Transition-Probabilities, and Excited-State Properties. J. Chem. Phys. 1993, 98, 7029,  DOI: 10.1063/1.464746
      11
      The equation-of-motion, coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited-state properties
      Stanton, John F.; Bartlett, Rodney J.
      Journal of Chemical Physics (1993), 98 (9), 7029-39CODEN: JCPSA6; ISSN:0021-9606.
      A comprehensive overview of the equation of motion coupled-cluster (EOM-CC) method and its application to mol. systems is presented. By exploiting the biorthogonal nature of the theory, it is shown that excited-state properties and transition strengths can be evaluated via a generalized expectation-value approach that incorporates both the bra and ket state wave functions. Reduced d. matrixes defined by this procedure are given by closed form expressions. For the root of the EOM-CC effective Hamiltonian that corresponds to the ground state, the resulting equations are equiv. to the usual expressions for normal single-ref. CC d. matrixes. Thus, the method described in this paper provides a universal definition of coupled-cluster d. matrixes, providing a link between EOM-CC and traditional ground state CC theory. Excitation energy, oscillator strength, and property calcns. are illustrated by means of several numerical examples, including comparisons with full CI calcns. and a detailed study of the 10 lowest electronically excited states of the cyclic isomer of C4.
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    12. 12
      Krylov, A. Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitchhikeras Guide to Fock Space. Annu. Rev. Phys. Chem. 2008, 59, 433,  DOI: 10.1146/annurev.physchem.59.032607.093602
      12
      Equation-of-motion coupled-cluster methods for open-shell and electronically excited species; the Hitchhiker's guide to Fock space
      Krylov, Anna I.
      Annual Review of Physical Chemistry (2008), 59 (), 433-462CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)
      A review. The equation-of-motion coupled-cluster (EOM-CC) approach is a versatile electronic-structure tool that allows one to describe a variety of multiconfigurational wave functions within single-ref. formalism. This review provides a guide to established EOM methods illustrated by examples that demonstrate the types of target states currently accessible by EOM. It focuses on applications of EOM-CC to electronically excited and open-shell species. The examples emphasize EOM's advantages for selected situations often perceived as multireference cases [e.g., interacting states of different nature, Jahn-Teller (JT) and pseudo-JT states, dense manifolds of ionized states, diradicals, and triradicals]. I also discuss limitations and caveats and offer practical solns. to some problematic situations. The review also touches on some formal aspects of the theory and important current developments.
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    13. 13
      Furche, F.; Ahlrichs, R. Adiabatic Time-Dependent Density Functional Methods for Excited State Properties. J. Chem. Phys. 2002, 117, 7433,  DOI: 10.1063/1.1508368
      13
      Adiabatic time-dependent density functional methods for excited state properties
      Furche, Filipp; Ahlrichs, Reinhart
      Journal of Chemical Physics (2002), 117 (16), 7433-7447CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      This work presents theory, implementation, and validation of excited state properties obtained from time-dependent d. functional theory (TDDFT). Based on a fully variational expression for the excited state energy, a compact derivation of first order properties is given. We report an implementation of analytic excited state gradients and charge moments for local, gradient cor., and hybrid functionals, as well as for the CI singles (CIS) and time-dependent Hartree-Fock (TDHF) methods. By exploiting analogies to ground state energy and gradient calcns., efficient techniques can be transferred to excited state methods. Benchmark results demonstrate that, for low-lying excited states, geometry optimizations are not substantially more expensive than for the ground state, independent of the mol. size. We assess the quality of calcd. adiabatic excitation energies, structures, dipole moments, and vibrational frequencies by comparison with accurate exptl. data for a variety of excited states and mols. Similar trends are obsd. for adiabatic excitation energies as for vertical ones. TDDFT is more robust than CIS and TDHF, in particular, for geometries differing significantly from the ground state min. The TDDFT excited state structures, dipole moments, and vibrational frequencies are of a remarkably high quality, which is comparable to that obtained in ground state d. functional calcns. Thus, yielding considerably more accurate results at similar computational cost, TDDFT rivals CIS as a std. method for calcg. excited state properties in larger mols.
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    14. 14
      Jacquemin, D.; Duchemin, I.; Blase, X. Is the Bethe-Salpeter Formalism Accurate for Excitation Energies? Comparisons with TD-DFT, CASPT2, and EOM-CCSD. J. Phys. Chem. Lett. 2017, 8, 1524,  DOI: 10.1021/acs.jpclett.7b00381
      14
      Is the Bethe-Salpeter Formalism Accurate for Excitation Energies? Comparisons with TD-DFT, CASPT2, and EOM-CCSD
      Jacquemin, Denis; Duchemin, Ivan; Blase, Xavier
      Journal of Physical Chemistry Letters (2017), 8 (7), 1524-1529CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      Developing ab initio approaches able to provide accurate excited-state energies at a reasonable computational cost is one of the biggest challenges in theor. chem. In that framework, the Bethe-Salpeter equation approach, combined with the GW exchange-correlation self-energy, which maintains the same scaling with system size as TD-DFT, has recently been the focus of a rapidly increasing no. of applications in mol. chem. Using a recently proposed set encompassing excitation energies of many kinds [J. Phys. Chem. Lett. 2016, 7, 586-591], we investigate here the performances of BSE/GW. We compare these results to CASPT2, EOM-CCSD, and TD-DFT data and show that BSE/GW provides an accuracy comparable to the two wave function methods. It is particularly remarkable that the BSE/GW is equally efficient for valence, Rydberg, and charge-transfer excitations. In contrast, it provides a poor description of triplet excited states, for which EOM-CCSD and CASPT2 clearly outperform BSE/GW. This contribution therefore supports the use of the Bethe-Salpeter approach for spin-conserving transitions.
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    15. 15
      Tajti, A.; Stanton, J.; Matthews, D.; Szalay, P. Accuracy of Coupled Cluster Excited State Potential Energy Surfaces. J. Chem. Theory Comput. 2018, 14, 5859,  DOI: 10.1021/acs.jctc.8b00681
      15
      Accuracy of Coupled Cluster Excited State Potential Energy Surfaces
      Tajti, Attila; Stanton, John F.; Matthews, Devin A.; Szalay, Peter G.
      Journal of Chemical Theory and Computation (2018), 14 (11), 5859-5869CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      The validation of the quality of the description of excited electronic states is of special importance in quantum chem. as the general reliability of ab initio methods shows a much larger variation for these states than for the ground state. In this study, we investigate the quality of excited state energy gradients and potential energy surfaces on selected systems, as provided by the single ref. coupled cluster variants CC2, CCSD, CCSD(T)(a)*, and CC3. Gradients and surface plots that follow the Franck-Condon forces are compared to the resp. CCSDT ref. values, thereby establishing a useful strategy for judging each variant's accuracy. The results reveal serious flaws of lower order methods - in particular, CC2 - in several situations where they otherwise give accurate vertical excitation energies, as well as excellent accuracy and consistency of the recently proposed CCSD(T)(a)* method.
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    16. 16
      Loos, P.-F.; Lipparini, F.; Boggio-Pasqua, M.; Scemama, A.; Jacquemin, D. A Mountaineering Strategy to Excited States: Highly Accurate Energies and Benchmarks for Medium Sized Molecules. J. Chem. Theory Comput. 2020, 16, 1711,  DOI: 10.1021/acs.jctc.9b01216
      16
      A Mountaineering Strategy to Excited States: Highly Accurate Energies and Benchmarks for Medium Sized Molecules
      Loos, Pierre-Francois; Lipparini, Filippo; Boggio-Pasqua, Martial; Scemama, Anthony; Jacquemin, Denis
      Journal of Chemical Theory and Computation (2020), 16 (3), 1711-1741CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Following our previous work focusing on compds. contg. up to 3 non-hydrogen atoms [J. Chem. Theory Comput.2018, 14, 4360-4379], we present here highly accurate vertical transition energies obtained for 27 mols. encompassing 4, 5, and 6 non-hydrogen atoms: acetone, acrolein, benzene, butadiene, cyanoacetylene, cyanoformaldehyde, cyanogen, cyclopentadiene, cyclopropenone, cyclopropenethione, diacetylene, furan, glyoxal, imidazole, isobutene, methylenecyclopropene, propynal, pyrazine, pyridazine, pyridine, pyrimidine, pyrrole, tetrazine, thioacetone, thiophene, thiopropynal, and triazine. To obtain these energies, we use equation-of-motion/linear-response coupled cluster theory up to the highest tech. possible excitation order for these systems (CC3, EOM-CCSDT, and EOM-CCSDTQ) and selected CI (SCI) calcns. (with tens of millions of determinants in the ref. space), as well as the multiconfigurational n-electron valence state perturbation theory (NEVPT2) method. All these approaches are applied in combination with diffuse-contg. at. basis sets. For all transitions, we report at least CC3/aug-cc-pVQZ vertical excitation energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3 almost systematically delivers transition energies in agreement with higher-level methods with a typical deviation of ±0.04 eV, except for transitions with a dominant double excitation character where the error is much larger. The present contribution gathers a large, diverse, and accurate set of more than 200 highly accurate transition energies for states of various natures (valence, Rydberg, singlet, triplet, n → π*, π → π*, ...). We use this series of theor. best ests. to benchmark a series of popular methods for excited state calcns.: CIS(D), ADC(2), CC2, STEOM-CCSD, EOM-CCSD, CCSDR(3), CCSDT-3, CC3, and NEVPT2. The results of these benchmarks are compared to the available literature data.
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    17. 17
      Andersson, K.; Malmqvist, P.; Roos, B.; Sadlej, A.; Wolinski, K. Second-Order Perturbation-theory with a CASSCF Reference Function. J. Phys. Chem. 1990, 94, 5483,  DOI: 10.1021/j100377a012
      17
      Second-order perturbation theory with a CASSCF reference function
      Andersson, Kerstin; Malmqvist, Per Aake; Roos, Bjoern O.; Sadlej, Andrzej J.; Wolinski, Krzysztof
      Journal of Physical Chemistry (1990), 94 (14), 5483-8CODEN: JPCHAX; ISSN:0022-3654.
      Second-order perturbation theory based on a CASSCF ref. state is derived and implemented. The first-order wave function includes the full space of interacting states. Expressions for the contributions to the second-order energy are obtained in terms of up to four-particle d. matrixes for the CASSCF ref. state. The zeroth-order Hamiltonian reduces to the Moeller-Plesset Hamiltonian for a closed-shell ref. state. The limit of the implementation is given by the no. of active orbitals, which dets. the size of the d. matrixes. It is presently around 13 orbitals. The method is illustrated in a series of calcns. on H2, H2O, CH2, and F-, and the results are compared with corresponding full CI results.
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    18. 18
      Christiansen, O.; Koch, H.; Jorgensen, P. Response Functions in the CC3 Iterative Triplet Excitation Model. J. Chem. Phys. 1995, 103, 7429,  DOI: 10.1063/1.470315
      18
      Response functions in the CC3 iterative triple excitation model
      Christiansen, Ove; Koch, Henrik; Joergensen, Poul
      Journal of Chemical Physics (1995), 103 (17), 7429-41CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The derivation of response functions for coupled cluster models is discussed in a context where approxns. can be introduced in the coupled cluster equations. The linear response function is derived for the approx. coupled cluster singles, doubles, and triples model CC3. The linear response functions for the approx. triples models, CCSDT-1a and CCSDT-1b, are obtained as simplifications to the CC3 linear response function. The consequences of these simplifications are discussed for the evaluation of mol. properties, in particular, for excitation energies. Excitation energies obtained from the linear response eigenvalue equation are analyzed in orders of the fluctuation potential. Double replacement dominated excitations are correct through second order in all the triples models mentioned, whereas they are only correct to first order in the coupled cluster singles and doubles model (CCSD). Single replacement dominated excitation energies are correct through third order in CC3, while in CCSDT-1a, CCSDT-1b, and CCSD they are only correct through second order. Calcns. of electronic excitation energies are reported for CH+, N2, and C2H4 to illustrate the accuracy that can be obtained in the various triples models. The CH+ results are compared to full CI results, the C2H4 results are compared with complete active space second order perturbation theory (CASPT2) and expt., and the N2 results are compared to expt. Double replacement dominated excitations are improved significantly relative to CCSD in all the triples models mentioned, and is of the same quality in CC3 and CCSDT-1a. The single replacement dominated excitation are close to full CI results for the CC3 model and significantly improved relative to CCSD. The CCSDT-1 results for the single replacement dominated excitations are not improved compared to CCSD.
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    19. 19
      Mooneyham, A.; McDonnell, M.; Drucker, S. Cavity Ringdown Spectrum of 2-Cyclohexen-1-one in the CO/Alkenyl CC Stretch Region of the S1(n, π*) – S0 Vibronic Band System. J. Phys. Chem. A 2017, 121, 2343,  DOI: 10.1021/acs.jpca.7b00826
      19
      Cavity Ringdown Spectrum of 2-Cyclohexen-1-one in the CO/Alkenyl CC Stretch Region of the S1(n, π*)-S0 Vibronic Band System
      Mooneyham, Ashley E.; McDonnell, Michael P.; Drucker, Stephen
      Journal of Physical Chemistry A (2017), 121 (12), 2343-2352CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      The cavity ringdown (CRD) absorption spectrum of 2-cyclohexen-1-one (2CHO) vapor at room temp. was recorded over λ = 360-380 nm. This portion of the spectrum encompasses the S1(n,π*) ← S0 vibronic band system in the region of the C=C and C=O stretch fundamentals. Assignments were made for ∼40 vibronically resolved features in the spectrum, affording fundamental frequencies for 7 different vibrational modes in the S1(n,π*) state, including the C=C (1554 cm-1) and OC-CH (1449 cm-1) stretch modes. The C=O stretch character is spread over at least 4 different vibrational modes in the S1(n,π*) state, with fundamentals spanning the 1340-1430 cm-1 interval. This finding stems from a significant redn. in C=O bond order upon excitation, which leads to near-coincidence of the C=O stretch and several CH2 wag frequencies. Such complexities make 2CHO an ideal candidate for testing excited-state computational methods. The present spectroscopic results were used to test EOM-EE-CCSD harmonic-frequency predictions for the S1(n,π*) state. The performance was benchmarked the of less costly computational methods, including CIS(D) and TDDFT. For certain d. functionals (e.g., B3LYP and PBE0), the accuracy of TDDFT frequency predictions can approach but not meet that of EOM-EE-CCSD.
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    20. 20
      Jacquemin, D.; Adamo, C. In Density-Functional Methods for Excited States; Ferre, N., Filatov, M., HuixRotllant, M., Eds.; Topics in Current Chemistry-Series; 2016; Vol. 368; pp 347– 375.
      There is no corresponding record for this reference.
    21. 21
      Gordon, R. D.; Park, W. K. C. The 353 nm 1nπ* Transition of 4H-Pyran-4-one and a Deuterated Derivative. Can. J. Chem. 1993, 71, 1672,  DOI: 10.1139/v93-208
      21
      The 353-nm 1nπ* transition of 4H-pyran-4-one and a deuterated derivative
      Gordon, Robert D.; Park, William K. C.
      Canadian Journal of Chemistry (1993), 71 (10), 1672-5CODEN: CJCHAG; ISSN:0008-4042.
      The 353-nm vapor absorption of 4H-pyran-4-one and its 3,5-d2 deriv. has been measured and assigned to a forbidden, 1A2 (nπ*)-1A1 electronic transition, with electronic origin near 28,360 cm-1, made allowed by vibrations involving out-of-plane ring and C-H motions. Although the mol. remains planar upon excitation, other effects of conjugation on the nature of the excited state are less marked than in 2-cyclopenten-1-one.
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    22. 22
      Kaliman, I. A.; Krylov, A. I. New Algorithm for Tensor Contractions on Multi-Core CPUs, GPUs, and Accelerators Enables CCSD and EOM-CCSD Calculations with over 1000 Basis Functions on a Single Compute Node. J. Comp. Chem.J. Comp. Chem. 2017, 38, 842,  DOI: 10.1002/jcc.24713
      22
      New algorithm for tensor contractions on multi-core CPUs, GPUs, and accelerators enables CCSD and EOM-CCSD calculations with over 1000 basis functions on a single compute node
      Kaliman, Ilya A.; Krylov, Anna I.
      Journal of Computational Chemistry (2017), 38 (11), 842-853CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      A new hardware-agnostic contraction algorithm for tensors of arbitrary symmetry and sparsity is presented. The algorithm is implemented as a stand-alone open-source code libxm. This code is also integrated with general tensor library libtensor and with the Q-Chem quantum-chem. package. An overview of the algorithm, its implementation, and benchmarks are presented. Similarly to other tensor software, the algorithm exploits efficient matrix multiplication libraries and assumes that tensors are stored in a block-tensor form. The distinguishing features of the algorithm are: (i) efficient repackaging of the individual blocks into large matrixes and back, which affords efficient graphics processing unit (GPU)-enabled calcns. without modifications of higher-level codes; (ii) fully asynchronous data transfer between disk storage and fast memory. The algorithm enables canonical all-electron coupled-cluster and equation-of-motion coupled-cluster calcns. with single and double substitutions (CCSD and EOM-CCSD) with over 1000 basis functions on a single quad-GPU machine. We show that the algorithm exhibits predicted theor. scaling for canonical CCSD calcns., O(N6), irresp. of the data size on disk. © 2017 Wiley Periodicals, Inc.
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    23. 23
      Medvedev, E.; Pratt, D. Hund’s Case (a)-case (b) Transition in the Singlet-Triplet Absorption Spectrum of Pyrazine in a Supersonic Jet. J. Exp. Theor. Phys. 1998, 87, 35,  DOI: 10.1134/1.558642
      There is no corresponding record for this reference.
    24. 24

      Certain equipment, instruments, software, or materials, commercial or noncommercial, are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement of any product or service by the authors’ institutions (including NIST), nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

      There is no corresponding record for this reference.
    25. 25
      Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T. B.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X. Advances in Molecular Quantum Chemistry Contained in the Q-Chem 4 Program Package. Mol. Phys. 2015, 113, 184,  DOI: 10.1080/00268976.2014.952696
      25
      Advances in molecular quantum chemistry contained in the Q-Chem 4 program package
      Shao, Yihan; Gan, Zhengting; Epifanovsky, Evgeny; Gilbert, Andrew T. B.; Wormit, Michael; Kussmann, Joerg; Lange, Adrian W.; Behn, Andrew; Deng, Jia; Feng, Xintian; Ghosh, Debashree; Goldey, Matthew; Horn, Paul R.; Jacobson, Leif D.; Kaliman, Ilya; Khaliullin, Rustam Z.; Kus, Tomasz; Landau, Arie; Liu, Jie; Proynov, Emil I.; Rhee, Young Min; Richard, Ryan M.; Rohrdanz, Mary A.; Steele, Ryan P.; Sundstrom, Eric J.; Woodcock, H. Lee, III; Zimmerman, Paul M.; Zuev, Dmitry; Albrecht, Ben; Alguire, Ethan; Austin, Brian; Beran, Gregory J. O.; Bernard, Yves A.; Berquist, Eric; Brandhorst, Kai; Bravaya, Ksenia B.; Brown, Shawn T.; Casanova, David; Chang, Chung-Min; Chen, Yunquing; Chien, Siu Hung; Closser, Kristina D.; Crittenden, Deborah L.; Diedenhofen, Michael; DiStasio, Robert A., Jr.; Do, Hainam; Dutoi, Anthony D.; Edgar, Richard G.; Fatehi, Shervin; Fusti-Molnar, Laszlo; Ghysels, An; Golubeva-Zadorozhnaya, Anna; Gomes, Joseph; Hanson-Heine, Magnus W. D.; Harbach, Philipp H. P.; Hauser, Andreas W.; Hohenstein, Edward G.; Holden, Zachary C.; Jagau, Thomas-C.; Ji, Hyunjun; Kaduk, Ben; Khistyaev, Kirill; Kim, Jaehoon; Kim, Jihan; King, Rollin A.; Klunzinger, Phil; Kosenkov, Dmytro; Kowalczyk, Tim; Krauter, Caroline M.; Lao, Ka Un; Laurent, Adele; Lawler, Keith V.; Levchenko, Sergey V.; Lin, Ching Yeh; Liu, Fenglai; Livshits, Ester; Lochan, Rohini C.; Luenser, Arne; Manohar, Prashant; Manzer, Samuel F.; Mao, Shan-Ping; Mardirossian, Narbe; Marenich, Aleksandr V.; Maurer, Simon A.; Mayhall, Nicholas J.; Neuscamman, Eric; Oana, C. Melania; Olivares-Amaya, Roberto; O'Neill, Darragh P.; Parkhill, John A.; Perrine, Trilisa M.; Peverati, Roberto; Prociuk, Alexander; Rehn, Dirk R.; Rosta, Edina; Russ, Nicholas J.; Sharada, Shaama M.; Sharma, Sandeep; Small, David W.; Sodt, Alexander; Stein, Tamar; Stuck, David; Su, Yu-Chuan; Thom, Alex J. W.; Tsuchimochi, Takashi; Vanovschi, Vitalii; Vogt, Leslie; Vydrov, Oleg; Wang, Tao; Watson, Mark A.; Wenzel, Jan; White, Alec; Williams, Christopher F.; Yang, Jun; Yeganeh, Sina; Yost, Shane R.; You, Zhi-Qiang; Zhang, Igor Ying; Zhang, Xing; Zhao, Yan; Brooks, Bernard R.; Chan, Garnet K. L.; Chipman, Daniel M.; Cramer, Christopher J.; Goddard, William A., III; Gordon, Mark S.; Hehre, Warren J.; Klamt, Andreas; Schaefer, Henry F., III; Schmidt, Michael W.; Sherrill, C. David; Truhlar, Donald G.; Warshel, Arieh; Xu, Xin; Aspuru-Guzik, Alan; Baer, Roi; Bell, Alexis T.; Besley, Nicholas A.; Chai, Jeng-Da; Dreuw, Andreas; Dunietz, Barry D.; Furlani, Thomas R.; Gwaltney, Steven R.; Hsu, Chao-Ping; Jung, Yousung; Kong, Jing; Lambrecht, Daniel S.; Liang, WanZhen; Ochsenfeld, Christian; Rassolov, Vitaly A.; Slipchenko, Lyudmila V.; Subotnik, Joseph E.; Van Voorhis, Troy; Herbert, John M.; Krylov, Anna I.; Gill, Peter M. W.; Head-Gordon, Martin
      Molecular Physics (2015), 113 (2), 184-215CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
      A review. A summary of the tech. advances that are incorporated in the fourth major release of the Q-Chem quantum chem. program is provided, covering approx. the last seven years. These include developments in d. functional theory methods and algorithms, NMR (NMR) property evaluation, coupled cluster and perturbation theories, methods for electronically excited and open-shell species, tools for treating extended environments, algorithms for walking on potential surfaces, anal. tools, energy and electron transfer modeling, parallel computing capabilities, and graphical user interfaces. In addn., a selection of example case studies that illustrate these capabilities is given. These include extensive benchmarks of the comparative accuracy of modern d. functionals for bonded and non-bonded interactions, tests of attenuated second order Moller-Plesset (MP2) methods for intermol. interactions, a variety of parallel performance benchmarks, and tests of the accuracy of implicit solvation models. Some specific chem. examples include calcns. on the strongly correlated Cr2 dimer, exploring zeolite-catalyzed ethane dehydrogenation, energy decompn. anal. of a charged ter-mol. complex arising from glycerol photoionisation, and natural transition orbitals for a Frenkel exciton state in a nine-unit model of a self-assembling nanotube.
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    26. 26
      Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0Model. J. Chem. Phys. 1999, 110, 6158,  DOI: 10.1063/1.478522
      26
      Toward reliable density functional methods without adjustable parameters: the PBE0 model
      Adamo, Carlo; Barone, Vincenzo
      Journal of Chemical Physics (1999), 110 (13), 6158-6170CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present an anal. of the performances of a parameter free d. functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amt. of exact exchange. The results obtained for structural, thermodn., kinetic and spectroscopic (magnetic, IR and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chem. and condensed matter physics.
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    27. 27

      CFOUR (version 2.1), a Quantum Chemical Program Package written by J. F. Stanton, J. Gauss, M. E. Harding, P. G. Szalay with contributions from A. A. Auer, R. J. Bartlett, U. Benedikt, C. Berger, D. E. Bernholdt, Y. J. Bomble, L. Cheng, O. Christiansen, M. Heckert, O. Heun, C. Huber, et al. CFOUR uses the integral packages MOLECULE (J. Almlöf and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van Wüllen.

      There is no corresponding record for this reference.
    28. 28
      Polik, W. F.; Schmidt, J. R. WebMO: Web-Based Computational Chemistry Calculations in Education and Research. WIREs Computational Molecular Science 2022, 12, e1554,  DOI: 10.1002/wcms.1554
      There is no corresponding record for this reference.
    29. 29
      Judge, R. H.; Korale, A. A.; York, J. J.; Joo, D. L.; Clouthier, D. J.; Moule, D. C. Computerized Simulation and Fitting of Singlet-Triplet Spectra of Orthorhombic Asymmetric Tops – Theory and Extensions to Molecules with Large Multiplet Splittings. J. Chem. Phys. 1995, 103, 5343,  DOI: 10.1063/1.470569
      29
      Computerized simulation and fitting of singlet-triplet spectra of orthorhombic asymmetric tops: theory and extensions to mols. with large multiplet splittings
      Judge, R. H.; Korale, A. A.; York, J. J.; Joo, Duck-Lae; Clouthier, Dennis J.; Moule, D. C.
      Journal of Chemical Physics (1995), 103 (13), 5343-56CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Motivated by our recent finding that the singlet-triplet bands of selenoformaldehyde involve an upper state with large zero field splittings, we have extended the theory and written a program for predicting and fitting such rotationally resolved spectra. Triplet state matrix elements for a case (A) basis have been developed, including corrections for centrifugal and spin-centrifugal distortion. The full Hamiltonian matrix has been symmetry adapted, simplifying the problem to four individual matrixes of approx. equal size for mols. of orthorhombic symmetry. Diagonalization of these matrixes yields triplet state energies that are in agreement with previous treatments using a basis in which the spin splittings are small relative to the rotational intervals. Methods have been developed for sorting the eigenvalues and assigning quantum labels regardless of the magnitude of the spin splittings. The calcn. of the relative intensities of the rotational lines within a band has been programmed using transition moment matrix elements from the literature. The selection rules for various upper state symmetries have been developed in a form useful for the anal. of spectra. Band contour predictions of spectra for various coupling cases have been presented.
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    30. 30
      Csaszar, P.; Csaszar, A.; Somogyi, A.; Dinya, Z.; Holly, S.; Gal, M.; Boggs, J. E. Vibrational Spectra, Scaled Quantum-Mechanical (SQM) Force Field and Assignments for 4H-Pyran-4-one. Spectrochim. Acta 1986, 42A, 473,  DOI: 10.1016/0584-8539(86)80043-5
      30
      Vibrational spectra, scaled quantum-mechanical (SQM) force field and assignments for 4H-pyran-4-one
      Csaszar, Pal; Csaszar, Attila; Somogyi, Arpad; Dinya, Zoltan; Holly, Sandor; Gal, Miklos; Boggs, James E.
      Spectrochimica Acta, Part A: Molecular and Biomolecular Spectroscopy (1986), 42A (4), 473-86CODEN: SAMCAS; ISSN:0584-8539.
      The gas-phase IR spectrum of 4H-pyran-4-one (γ-pyrone) was recorded in the 4000-400 cm-1 region by a Nicolet 7199 FTIR spectrometer and interpreted using a general valence force field calcd. quantum mech. at the ab initio level with a split-valence 4-21 basis. Assignment of certain fundamentals was facilitated by information gained from the IR and Raman spectra of the melt and from the IR spectrum of the satd. soln. in CCl4. To account for systematic computational errors, the theor. ab initio force field was scaled using a set of consts. derived by the empirical fitting of force fields computed for related mols. to their obsd. spectra. Either the scale factors derived for a family of open-chain mols. or, better, for C6H6 could be used to yield a scaled force field which gave unequivocal assignments for γ-pyrone. The method promises to be of general applicability for mols. of this complexity.
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    31. 31
      Smithson, T. L.; Ibrahim, N.; Wieser, H. Cyclohexanones: Evidence of Chair Inversion and Estimate for Barriers to Planarity from the Far-Infrared Spectra. Can. J. Chem. 1983, 61, 1924,  DOI: 10.1139/v83-330
      31
      Cyclohexanones: evidence of chair inversion and estimate for barriers to planarity from the far-infrared spectra
      Smithson, Tracy L.; Ibrahim, Nan; Wieser, Hal
      Canadian Journal of Chemistry (1983), 61 (8), 1924-32CODEN: CJCHAG; ISSN:0008-4042.
      The far-IR, at 50-450 cm-1 are reported for the vapors of cyclohexanone and the structurally related compds. tetrahydro-4H-pyran-4-one, tetrahydro-4H-thiopyran-4-one, tetrahydro-4H-pyran-3-one, 1,3-dioxan-5-one, and 4H-pyran-4-one. Except for the last mentioned, the IR are characterized by 2 or in some cases 3 sequences of Q branches which are assigned to the out-of-plane deformations of the cyclohexanone ring. The lowest wavenumber sequence in each compd. arises from a vibration which, if completely executed, would take the chair conformation to its equiv. via an inversion through the planar form. Each sequence is amenable to soln. by a one-dimensional Hamiltonian incorporating a quadratic-quartic potential function, providing ests. for the magnitudes of the barriers to planarity.
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    32. 32
      Wathelet, V.; Preat, J.; Bouhy, M.; Fontaine, M.; Perpete, E.; Andre, J.; Jacquemin, D. Assessment of PBE0 for Evaluating the Absorption Spectra of Carbonyl Molecules. Int. J. Quantum Chem. 2006, 106, 1853,  DOI: 10.1002/qua.20982
      32
      Assessment of PBE0 for evaluating the absorption spectra of carbonyl molecules
      Wathelet, Valerie; Preat, Julien; Bouhy, Michael; Fontaine, Michele; Perpete, Eric A.; Andre, Jean-Marie; Jacquemin, Denis
      International Journal of Quantum Chemistry (2006), 106 (8), 1853-1859CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
      Using the parameter-free PBE0 hybrid functional, the authors compute the UV/visible spectra of solvated compds. presenting a carbonyl chromophoric unit linked to a C-C double bond. PBE0 is extremely efficient for accurately reproducing exptl. values, with a mean unsigned error of 7 nm for an extended set of compds., although no fitting or statistical treatments are performed. PBE0 has a predictive efficiency comparable to the known Woodward-Fieser empirical formula, and can therefore be used to extend these rules without requiring addnl. exptl. results. Consequently, the UV/visible spectra of several compds. that have not yet been synthesized are predicted.
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    33. 33
      Leang, S. S.; Zahariev, F.; Gordon, M. S. Benchmarking the Performance of Time-Dependent Density Functional Methods. J. Chem. Phys. 2012, 136, 104101,  DOI: 10.1063/1.3689445
      33
      Benchmarking the performance of time-dependent density functional methods
      Leang, Sarom S.; Zahariev, Federico; Gordon, Mark S.
      Journal of Chemical Physics (2012), 136 (10), 104101/1-104101/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The performance of 24 d. functionals, including 14 meta-generalized gradient approxn. (mGGA) functionals, is assessed for the calcn. of vertical excitation energies against an exptl. benchmark set comprising 14 small- to medium-sized compds. with 101 total excited states. The exptl. benchmark set consists of singlet, triplet, valence, and Rydberg excited states. The global-hybrid (GH) version of the Perdew-Burke-Ernzerhoff GGA d. functional (PBE0) is found to offer the best overall performance with a mean abs. error (MAE) of 0.28 eV. The GH-mGGA Minnesota 2006 d. functional with 54% Hartree-Fock exchange (M06-2X) gives a lower MAE of 0.26 eV, but this functional encounters some convergence problems in the ground state. The local d. approxn. functional consisting of the Slater exchange and Volk-Wilk-Nusair correlation functional (SVWN) outperformed all non-GH GGAs tested. The best pure d. functional performance is obtained with the local version of the Minnesota 2006 mGGA d. functional (M06-L) with an MAE of 0.41 eV. (c) 2012 American Institute of Physics.
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    34. 34
      Bremond, E.; Savarese, M.; Adamo, C.; Jacquemin, D. Accuracy of TD-DFT Geometries: A Fresh Look. J. Chem. Theory Comput. 2018, 14, 3715,  DOI: 10.1021/acs.jctc.8b00311
      34
      Accuracy of TD-DFT Geometries: A Fresh Look
      Bremond, Eric; Savarese, Marika; Adamo, Carlo; Jacquemin, Denis
      Journal of Chemical Theory and Computation (2018), 14 (7), 3715-3727CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We benchmark a panel of 48 DFT exchange-correlation functionals in the framework of TD-DFT optimizations of the geometry of valence singlet excited states. To this end, we use a set of 41 small- and medium-sized org. mols. for which ref. geometries were obtained at high level of theory, typically, CC3 or CCSDR(3), with the aug-cc-pVTZ at. basis set. For the ground-state parameters, the tested functionals provide av. deviations that are small (0.010 Å and 0.5° for bond lengths and valence angles) and not very sensitive to the selected (hybrid) functional, but the errors are larger for the most polarized bonds (CO, CN, and so on). Nevertheless, DFT has a tendency to provide too compact distances, a trend slightly enhanced for functionals including a large amt. of exact exchange. The av. errors largely increase when going to the excited-state for most bond types, i.e., TD-DFT delivers less accurate excited-state distances than DFT for ground state. In particular TD-DFT combined with hybrid functionals provides significantly too short CO and CS/CSe bonds with resp. av. errors in the -0.026/-0.052 Å and -0.015/-0.082 Å ranges, depending on the selected hybrid functional. For the carbonyl bonds, the sizes of the TD-DFT deviations obtained when selecting std. hybrid functionals are of the same order of magnitude as the EOM-CCSD ones.
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    35. 35
      Hill, J. G. Gaussian Basis Sets for Molecular Applications. Int. J. Quantum Chem. 2013, 113, 21,  DOI: 10.1002/qua.24355
      35
      Gaussian basis sets for molecular applications
      Hill, J. Grant
      International Journal of Quantum Chemistry (2013), 113 (1), 21-34CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
      A review. The choice of basis set in quantum chem. calcns. can have a huge impact on the quality of the results, esp. for correlated ab initio methods. This article provides an overview of the development of Gaussian basis sets for mol. calcns., with a focus on four popular families of modern atom-centered, energy-optimized bases: at. natural orbital, correlation consistent, polarization consistent, and def2. The terminol. used for describing basis sets is briefly covered, along with an overview of the auxiliary basis sets used in a no. of integral approxn. techniques and an outlook on possible future directions of basis set design. © 2012 Wiley Periodicals, Inc.
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    36. 36
      Hougen, J. Rotational Structure of Singlet-Triplet Transitions in Near Symmetric Tops. Can. J. Phys. 1964, 42, 433,  DOI: 10.1139/p64-039
      36
      Rotational structure of singlet-triplet transitions in near-symmetric tops
      Hougen, Jon T.
      Canadian Journal of Physics (1964), 42 (3), 433-51CODEN: CJPHAD; ISSN:0008-4204.
      Expressions are presented for the intensities of the rotational lines in singlet-triplet transitions in mols. which belong to asymmetric-top point groups, but which are nevertheless near symmetric tops. The general selection rules for the rotational branches in such singlet-triplet transitions are: ΔN = 0, ±1, ±2, and ΔK = 0, ±1, ±2. They differ from the selection rules for singlet-singlet transitions in near symmetric tops by the occurrence of branches with ΔN = ±2 and ΔK = ±2. For certain asymmetric-top mols., it is possible to divide transitions into 2 mutually exclusive types: those characterized by the selection rules ΔK = 0, ±2 and those characterized by ΔK = ± 1.
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    37. 37
      MacDonald, J. N.; Mackay, S. A.; Tyler, J. K.; Cox, A. P.; Ewart, I. C. Microwave Spectra, Structures, and Dipole Moments of 4H-Pyran-4-one and Its Sulfur Analogs. J. Chem. Soc. Faraday Trans. 2 1981, 77, 79,  DOI: 10.1039/f29817700079
      37
      Microwave spectra, structures, and dipole moments of 4H-pyran-4-one and its sulfur analogs
      MacDonald, John N.; Mackay, Susan A.; Tyler, J. Kelvin; Cox, A. Peter; Ewart, Ian C.
      Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics (1981), 77 (1), 79-99CODEN: JCFTBS; ISSN:0300-9238.
      Microwave spectra were detd. of the planar mols. 4H-pyran-4-one (I), 4H-thiapyran-4-thione (II), 4-H-thiapyran-4-one, and 4H-thiapyran-4-thione (III). Extensive isotopic results for I, II, and III enabled precise, complete structures to be calcd. The structures gave little evidence for enhanced aromaticity in these mols. The high dipole moment values (∼4 D) obtained for I, II, and III in soln. (Rolla, M., et al., 1952) were confirmed and the values more closely defined. Vibrational satellite spectra indicate that the lowest frequency fundamentals of these mols. are out-of-plane motions near 100 cm-1.
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    38. 38
      Tejeda, G.; Mate, B.; FernandezSanchez, J.; Montero, S. Temperature and Density Mapping of Supersonic Jet Expansions Using Linear Raman Spectroscopy. Phys. Rev. Lett. 1996, 76, 34,  DOI: 10.1103/PhysRevLett.76.34
      38
      Temperature and density mapping of supersonic jet expansions using linear Raman spectroscopy
      Tejeda, G.; Mate, B.; Fernandez-Sanchez, J. M.; Montero, S.
      Physical Review Letters (1996), 76 (1), 34-7CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The authors present a 1st practical demonstration of Raman spectroscopy to obtain high definition maps of rotational temps. and abs. densities in mol. supersonic jets.
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    39. 39
      Wu, Y. R.; Levy, D. H. Determination of the Geometry of Deuterated Tryptamine by Rotationally Resolved Electronic Spectroscopy. J. Chem. Phys. 1989, 91, 5278,  DOI: 10.1063/1.457573
      39
      Determination of the geometry of deuterated tryptamine by rotationally resolved electronic spectroscopy
      Wu, Yenchune R.; Levy, Donald H.
      Journal of Chemical Physics (1989), 91 (9), 5278-84CODEN: JCPSA6; ISSN:0021-9606.
      The low and high resoln. fluorescence excitation spectra of d3-tryptamine were obsd. in the environment of a cold, supersonic mol. beam. As in the case of unlabeled tryptamine, 6 bands due to the origins of different conformers were found in the low-resoln. spectrum of deuterated tryptamine. A previous paper reported that conformers labeled A and B and conformers labeled D and E of tryptamine have identical rotational structures. However, for deuterated tryptamine those conformers have distinguishable rotational structures. Anal. of the rotational structure in the high-resoln. electronic spectra of 5 of the bands was used to det. the geometries of the different conformers. Conformers A, B, and F have a gauche conformation with respect to the rotation about the Cα-Cβ bond while conformers D and E have an eclipsed conformation. The geometries of conformers A and B and conformers D and E differ only in the orientation of the amino group. In these structures the angles of the internal rotation of the amino group are 180°, 60°, 180°, 60°, and -60° for conformers A,B, and D-F, resp. Feature C consists of 2 overlapped conformers; these conformers may have the amino group trans to the indole ring.
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    40. 40
      Glendening, E. D.; Landis, C. R.; Weinhold, F. NBO 6.0: Natural Bond Orbital Analysis Program. J. Comput. Chem. 2013, 34, 1429,  DOI: 10.1002/jcc.23266
      40
      NBO 6.0: Natural bond orbital analysis program
      Glendening, Eric D.; Landis, Clark R.; Weinhold, Frank
      Journal of Computational Chemistry (2013), 34 (16), 1429-1437CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      We describe principal features of the newly released version, NBO 6.0, of the natural bond orbital anal. program, that provides novel "link-free" interactivity with host electronic structure systems, improved search algorithms and labeling conventions for a broader range of chem. species, and new anal. options that significantly extend the range of chem. applications. We sketch the motivation and implementation of program changes and describe newer anal. options with illustrative applications. © 2013 Wiley Periodicals, Inc.
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    41. 41
      Richert, S.; Tait, C.; Timmel, C. Delocalisation of Photoexcited Triplet States Probed by Transient EPR and Hyperfine Spectroscopy. J. Magn. Reson. 2017, 280, 103,  DOI: 10.1016/j.jmr.2017.01.005
      41
      Delocalisation of photoexcited triplet states probed by transient EPR and hyperfine spectroscopy
      Richert, Sabine; Tait, Claudia E.; Timmel, Christiane R.
      Journal of Magnetic Resonance (2017), 280 (), 103-116CODEN: JMARF3; ISSN:1090-7807. (Elsevier B.V.)
      Photoexcited triplet states play a crucial role in photochem. mechanisms: long known to be of paramount importance in the study of photosynthetic reaction centers, they have more recently also been shown to play a major role in a no. of applications in the field of mol. electronics. Their characterization is crucial for an improved understanding of these processes with a particular focus on the detn. of the spatial distribution of the triplet state wavefunction providing information on charge and energy transfer efficiencies. Currently, active research in this field is mostly focussed on the investigation of materials for org. photovoltaics (OPVs) and org. light emitting diodes (OLEDs). As the properties of triplet states and their spatial extent are known to have a major impact on device performance, a detailed understanding of the factors governing triplet state delocalisation is at the basis of the further development and improvement of these devices. ESR (EPR) has proven a valuable tool in the study of triplet state properties and both exptl. methods as well as data anal. and interpretation techniques have continuously improved over the last few decades. In this review, we discuss the theor. and practical aspects of the investigation of triplet states and triplet state delocalisation by transient continuous wave and pulse EPR and highlight the advantages and limitations of the presently available techniques and the current trends in the field. Application of EPR in the study of triplet state delocalisation is illustrated on the example of linear multi-porphyrin chains designed as mol. wires.
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    42. 42
      Tinkham, M.; Strandberg, M. W. P. Interaction of Molecular Oxygen with a Magnetic Field. Phys. Rev. 1955, 97, 951,  DOI: 10.1103/PhysRev.97.951
      42
      Interaction of molecular oxygen with a magnetic field
      Tinkham, M.; Strandberg, M. W. P.
      Physical Review (1955), 97 (), 951-66CODEN: PHRVAO; ISSN:0031-899X.
      The dominant interaction of O with a magnetic field is through the electronic spin magnetic moment. However, a precise comparison with expt. of the results of calcg. the microwave paramagnetic spectrum, assuming only this interaction, shows a systematic discrepancy, which is removed by introducing 2 corrections. The larger (∼0.1%, or 7 gausses) is a correction for the second-order electronic orbital moment coupled in by the spin-orbit energy. Its magnitude is proportional to the second-order term μ'' in the spin-rotation coupling const. The smaller (∼1 gauss) is a correction for the rotation-induced magnetic moment of the mol. Since the dependence of this contribution on quantum nos. is unique, this coeff. can also be detd. by fitting the magnetic spectrum. A total of 120 X-band and 78 S-band lines were observed. The complete corrections were made on 26 lines with a mean residual error of ∼0.5 Mc./sec. This excellent agreement confirms the anomalous electronic moment to 60 p.p.m. and also confirms the validity of the Zeeman-effect theory. A new result is the rotational magnetic moment of -0.25 ± 0.05 nuclear magnetons per quantum of rotation. Knowledge of this moment allows the electronic contribution to the effective moment of inertia to be detd. This correction of 65 p.p.m. is made, the latest fitting of the universal at. consts. is used, and the equil. internuclear distance is recalcd. to be Rc = 1.20741 ± 0.00002 A. λ'', the second-order spin-orbit contribution to the coupling of the spin to the figure axis, is 465 ± 50 Mc./sec., less than 1% of the total coupling const. λ. Theoretical intensities of a no. of the microwave transitions are calcd. and successfully compared with expt. over a range of 100 to 1 in magnitude. The ΔM = 0 transitions are less than 0.01 of the strength of the δM = ± 1 transitions, and are too weak to observe. J breaks down as a quantum no. in the presence of a magnetic field. This allows ΔJ = ±2 transitions to comprise roughly half of all lines observed.
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    43. 43
      Vahtras, O.; Loboda, O.; Minaev, B.; Agren, H.; Ruud, K. Ab initio Calculations of Zero-field Splitting Parameters. Chem. Phys. 2002, 279, 133,  DOI: 10.1016/S0301-0104(02)00451-2
      43
      Ab initio calculations of zero-field splitting parameters
      Vahtras, O.; Loboda, O.; Minaev, B.; Agren, H.; Ruud, K.
      Chemical Physics (2002), 279 (2-3), 133-142CODEN: CMPHC2; ISSN:0301-0104. (Elsevier Science B.V.)
      We present calcns. of electron spin-spin (SS) coupling strengths evaluated as expectation values over multi-configuration and restricted high-spin SCF wave functions. Together with the spin-orbit (SO) CI methodol., this enables us to analyze the full spin Hamiltonian including the zero-field splitting (ZFS) parameters of the triplet state. The calcd. ZFS parameters include both the SS coupling to first order and SO coupling to second order of perturbation theory. The relative importance of these two contributions is strongly system dependent. In the lowest triplet state of the benzene mol., the main ZFS parameter - the D parameter - is detd. entirely by the SS coupling, with D calcd. to be 0.1583 cm-1. In contrast, the calcd. D parameter in the X3Σg- ground state of the oxygen mol. (3.77 cm-1) includes a large one-center SS contribution (DSS = 1.455 cm-1) but an even larger SO coupling contribution (DSO = 2.315 cm-1). ZFS parameters for mol. oxygen excited states, A3Σu+ and B3Σu-, which belong to Hund's case (b), are also calcd. A large neg. D value (-10.2 cm-1) for the A3Σu+ state is to 90% detd. by the DSO contribution, while the Schumann-Runge state spin splitting is mainly detd. by SS coupling. The calcd. values for benzene and oxygen are in good agreement with data from EPR and rotational fine-structure spectra. The applicability of response theory is with this contribution expanded to include the calcn. of the SS coupling of the spin Hamiltonian, complementing the previous implementations of hyperfine A- and g-tensors.
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    44. 44
      Creutzberg, F.; Hougen, J. T. Triplet-State Rotational Energy Levels for Near-Symmetric Rotor Molecules of Symmetry C2v, D2, and D2h. Can. J. Phys. 1967, 45, 1363,  DOI: 10.1139/p67-102
      44
      Triplet-state rotational energy levels for near-symmetric rotor molecules of symmetry, C2v, D2, and D2h
      Creutzberg, F.; Hougen, Jon T.
      Canadian Journal of Physics (1967), 45 (3), 1363-87CODEN: CJPHAD; ISSN:0008-4204.
      The ideas involved in Hund's coupling cases for diat. mols. are extended to allow the definition of three coupling cases, (a), (ab), and (b), for the triplet states of orthorhombic polyat. molecules. Case (a) is dealt with by second-order perturbation theory. Case (b) has recently been discussed by Raynes. Case (ab), which can be subdivided into types I, II, and III, is studied by examg. the results of numerical calcns. for selected values of the rotational consts., rotational quantum nos., and spin-splitting parameters for a near-prolate rotor. The assignment of symmetry species to the various spinrotation functions under consideration is described.
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    45. 45
      Moule, D.; Chantranupong, L.; Judge, R.; Clouthier, D. Ab-Initio Predictions of the Zero-Field Splittings and tthe Singlet-Triplet Transition Strengths for the ã3A2 (T1) ← x̃ 1A1 (S0), n → π* Transition of Selenoformaldehyde. Can. J. Chem. 1993, 71, 1706,  DOI: 10.1139/v93-212
      45
      Ab initio predictions of the zero-field splittings and the singlet-triplet transition strengths for the ~a3A2(T1) ← ~x1A1(S0), n → π* transition of selenoformaldehyde
      Moule, D. C.; Chantranupong, L.; Judge, R. H.; Clouthier, D. J.
      Canadian Journal of Chemistry (1993), 71 (10), 1706-12CODEN: CJCHAG; ISSN:0008-4042.
      The energy levels of the lower valence and Rydberg states of CH2Se have been calcd. by the SCF/CI method. Wavefunctions for the ROHF (restricted open-shell Hartree-Fock) states were obtained with the Binnings-Curtis double-ξ basis set augmented with Ryberg and polarization functions. CI was applied to the parent configurations, PCMO (parent configuration MO). Oscillator strengths were evaluated for the allowed elec. dipole transitions by the RPA and SOPPA (second-order polarization propagator approxn.) methods. The spin-orbit contribution to the zero-field splitting of the first triplet state, 3A2(n,π*) as well as the oscillator strengths to the 3 spin components were calcd. by perturbation theory. These calcns. predict that the Sx, Sy, and Sz components are shifted by -96.091, -96.707, and +29.167 cm-1, resp., from their unperturbed position. The oscillator strengths for the 3 components fx,fy, and fz of the 3A2(n,π*) ← 1A1(g.s.) transition were calcd. to be 3.45 × 10-7, 1.15 × 10-7, and 173.0 × 10-7.
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    46. 46
      Joo, D.; Clouthier, D.; Judge, R. Experimental Proof of the Case (ab) Coupling Hypothesis in the First Excited Triplet state of Selenoformaldehyde (H2C═Se). J. Chem. Phys. 2000, 112, 2285,  DOI: 10.1063/1.480827
      46
      Experimental proof of the case (ab) coupling hypothesis in the first excited triplet state of selenoformaldehyde (H2C:Se)
      Joo, Duck-Lae; Clouthier, Dennis J.; Judge, R. H.
      Journal of Chemical Physics (2000), 112 (5), 2285-2291CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A high-resoln. laser-induced phosphorescence spectrum of the vibronically induced 401 band of the a~ 3A2-X~ 1A1 system of D2C80Se was recorded with Doppler-limited resoln. Rotational anal. revealed that transitions to all three spin components occur within the 12,265-12,525 cm-1 region. The rotational structure of the band was fitted to the exptl. precision of measurement using a triplet state Hamiltonian that allowed for slight variations in the geometries of the individual spin components. The triplet state of selenoformaldehyde involves case (ab) coupling, in which two of the spin components are close together and the 3rd is ∼125 cm-1 higher in energy. This unusual occurrence of large zero-field splittings in the triplet state of an asym. top arises because of the close proximity of the 3A1(π,π*) state which substantially displaces two of the spin components to lower energy.
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  • Supporting Information

    Supporting Information

    The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.3c01059.

    • This document shows origin-band contours for the T1(n,π*) ← S0 transition of 4PN, measured using 3-atm and 1-atm expansions of helium. Simulations using the two-temperature model (with the same molecular parameters as in Table 4) are also included. (PDF)


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