Tao Zhang

Dongxin He

Junyu Wei

Hongshun Liu

Qingquan Li

Gilbert Teyssedre

CHARGE VIBRATION BEHAVIOUR IN POLYIMIDE UNDER THE PULSE VOLTAGE WITH DIFFERENT RISE AND FALL TIMES

High-voltage electronic equipment is widely used in power systems, which normally operate under the pulse voltage condition. However, compared to DC or AC stress, the electrical insulation suffers a more severe ageing process under a pulse voltage with steep edges and so may cause electrical equipments to fail early. The charge behaviour in polyimide under the pulse voltage with different edge times was investigated in this study. The experimental results show that charges vibrate at the pulse edge, while the amplitude of charge vibration increases with decreasing edge times. Furthermore, the detected vibration signal is strong within the nanosecond edge times, while being negligible in the microsecond range. Notably, the amplitude of charge vibration at the falling edge is larger than that at the rising edge in the presence of the column-plate electrode. It is believed that the resultant force acting on the charge increases drastically at the pulse edges, leading to the charge movement and the generation of ultrasonic waves, with the resultant force becoming larger when the edge times decrease. Charge vibration behaviour is harmful to the microstructure of insulation materials, and so it may be concluded that the rise and fall times of the pulse voltage play an important role in insulation failure.

Introduction

With the development of high-voltage direct current transmission projects and the large-scale application of new energy power generation and grid connection, the high-voltage (HV) electronic technology plays an increasingly important role in the transmission and distribution network. HV power electronic equipment can realise flexible control and free conversion of electric energy. It can also reduce the volume of power transmission and transformation equipment, which improves the transmission capacity and stability of the system [START_REF] Yang | Power electronic technology application in power system reactive power compensation study[END_REF][START_REF] Sun | On the application and research of electronic and electrical technology in power system[END_REF].

Power electronic equipment is generally operated under high-frequency pulse voltages for long periods. On the one hand, the power electronic technology, such as pulse width modulation (PWM), is applied to realise flexible control of electronic equipment by subjecting to a high-frequency pulse voltage [START_REF] Wang | Influence of repetitive square voltage duty cycle on the electrical tree characteristics of epoxy resin[END_REF][START_REF] Kheraluwala | Coaxially wound transformers for high-power high-frequency applications[END_REF]. For example, the power electronic transformer is a new type of transformer that combines the power electronic technology with a high frequency transformer. Its core component, the high frequency transformer, works under the high-frequency pulse voltage because of rectifying and inverter circuits. On the other hand, the power electronic devices generate the pulse voltage and current with steep rise time at the time of turn-on and turn-off [START_REF] Bilbao | Development and testing of an active high voltage saturation probe for characterization of ultra-high voltage silicon carbide semiconductor devices[END_REF]. The polyimide (PI) film is widely used for insulation in different kinds of HV power electronic equipments, such as the inverter-ted motor, power electronic transformer and HV insulated gate bipolar transistor (IGBT) [START_REF] Akram | Multilayer polyimide nanocomposite films synthesis process optimization impact on nanoparticles dispersion and their dielectric performance[END_REF][START_REF] Wang | Design of Archimedes spiral antenna to optimize for partial discharge detection of inverter fed motor insulation[END_REF]. Thus, the stability and electrical strength of PI under the pulsed voltage is important for the safe operation of such HV power electronic equipments.

However, electrical insulation is more prone to failure under the pulse voltage than under DC and AC voltages. It is a very important feature that the pulse voltage at the rising and falling edges has a very rapid change in the amplitude compared with AC or DC voltages, and the rise and fall times can reflect this characteristic clearly. Here, the rise generally means the time interval that a pulse rises from 10% to 90% of its peak amplitude while the fall time is inverse. P. Facinelli compared the initiation voltage of an electric tree under sinusoidal and pulse voltages, and the result showed that the initiation voltage of the electrical tree is lower under the pulse voltage [START_REF] Manciinelli | Electrical treeing in silicone gel under square voltage: Frequency, rise time and crosslinking influence[END_REF]. Peng Wang investigated the partial discharge (PD) characteristics under a pulse voltage. The results showed that the PD magnitude and insulation endurance under repetitive square wave voltages are around 10 times and one-third times of those under sinusoidal voltages (the frequency of both is 50 Hz) [START_REF] Wang | Comparison of PD characteristics for inverter-fed motor insulation under sinusoidal and repetitive square wave voltage conditions[END_REF]. Furthermore, the PD magnitude and high frequency energy of PD pulses increases with increasing slew rates (from 200 to 400 ns), which leads to a decrease in the average lifetime of insulation [START_REF] Wang | Partial discharge phenomenology and induced aging behavior in rotating machines controlled by power electronics[END_REF][START_REF] Wang | Effect of rise time on PD pulse features under repetitive square wave voltages[END_REF]. Shin Nakamura studied the electrical treeing in silicone gel under the pulse voltage with the rise time from 10 ns to 10 μs and the results show that the treeing length is longer when the rise time is shorter [START_REF] Du | Effect of direct fluorination on surface charge of polyimide films using repetitive pulsed power[END_REF]. It indicates that the pulse voltage causes more damage to insulation and that the rise time of the pulse voltage has an important effect on the insulation lifetime, which puts forward higher requirements for PI films [START_REF] Bhutta | Steady-state conduction current performance for multilayer polyimide/SiO2 films[END_REF][START_REF] Nakamura | Electrical treeing in silicone gel under repetitive voltage impulses[END_REF][START_REF] Yang | Surface morphology and Raman analysis of the polyimide film aged under bipolar pulse voltage[END_REF]. Thus, it is necessary to clarify the mechanism of insulation failure under the pulse voltage with different rise and fall times, in order to improve the reliability of power electronic equipment.

Charge accumulation is a very important factor in the acceleration of insulation deterioration [START_REF] Akram | Charge transport and trapping of surface modified stator coil insulation of motors[END_REF][START_REF] Wu | Effect of surface modification of electrodes on charge injection and dielectric characteristics of propylene carbonate[END_REF][START_REF] Zhao | Interfacial space charge characteristic of PPLP insulation for HVDC cables[END_REF], which distorts the electric field, causing flashover [START_REF] Akram | Impact of impulse voltage frequency on the partial discharge characteristic of electric vehicles motor insulation[END_REF][START_REF] Akram | Degradation mechanism of Al2O3 nano filled polyimide film due to surface discharge under square impulse voltage[END_REF][START_REF] Yuan | A review on factors that affect surface charge accumulation and charge-induced surface flashover[END_REF][START_REF] Akram | Cavity and interface effect of PI-film on charge accumulation and PD activity under bipolar pulse voltage[END_REF] and promoting the initiation and growth of the electrical tree [START_REF] Zhu | Electrical treeing initiation and breakdown phenomenon in polypropylene under DC and pulse combined voltages[END_REF]. Previous research has illustrated that the charges may accelerate under the pulsed voltage (100-300 Hz) and collide with the molecular chain, breaking the microstructure of the insulation material and resulting in the increase of the fractal dimension of the electrical tree [START_REF] Du | Electrical tree characteristics in silicone rubber under repetitive pulse voltage[END_REF]. Ying Liu studied the characteristics of electrical tree initiation under the DC and pulse voltage (with a maximum amplitude of 30 kV and the edge time adjustable from ns to μs). The results showed that the electrical tree can hardly be generated and that the tree initiation voltage is very high under the DC voltage, while the electrical tree grows faster and has a lower initiation voltage under the pulse voltage [START_REF] Liu | Electrical tree growth characteristics in XLPE cable insulation under DC voltage conditions[END_REF]. It is believed that the steep pulse rising edge can stimulate charge detrapping, releasing a large quantity of electro-mechanical energy quickly, which will break the molecular chains and induce the electrical tree [START_REF] Liu | Electrical tree initiation in XLPE cable insulation by application of DC and impulse voltage[END_REF]. Kai Zhou investigated the relationship between the space charge and PD characteristics under the pulse voltage and the results illustrated that the decrease in pulse rise time (from 10 to 1 μs) enhances the memory effect of the space charge leading to more intense PD activities [START_REF] Zhou | Space charge effects on partial discharge activities under pulse voltage[END_REF]. M. Fukuma reported on an increase of the space charge density with the decrease in the rise time (|dV/dt| from 3040 to 434 kV/s) and it is believed that a larger slope may enhance the efficiency of charge provision or leave less time for charge recombination [START_REF] Fukuma | Millisecond time-range analysis of space-charge distribution and electroluminescence in insulating polymers under transient electric stress[END_REF].

It can be inferred from the above studies that the rising and falling edges of the pulse voltage have a triggering effect on insulation failure, and this impact is related to special charge behaviours occurring at the pulse edges, due to the rapid change in the voltage amplitude. The rise and fall times depend on the performance of switching devices and the external circuit structure. The inductance and capacitance of the external circuit make the rising and falling edges slow and its degree of influence depends on the time constant of the circuit. With the development and application of the wide-bandgap semiconductor silicon carbon (SiC) and gallium nitride (GaN), the power and switching frequency become much higher than in conventional Si semiconductor devices [START_REF] Ma | Review on driving circuits for wide-bandgap semiconductor switching devices for mid-to high-power applications[END_REF] and can generate a pulse voltage with shorter rising and falling times [START_REF] Wang | A simulation study of high voltage 4H-SiC IGBTs[END_REF][START_REF] Mantooth | Wide bandgap technologies and their implications on miniaturizing power electronic systems[END_REF]. The shorter rise time helps to increase the frequency of the switching devices and thus improves the efficiency of the power electronic equipment. But this requires higher insulation strength. Thus, charge behaviours at the nanosecond rising and falling edges need to be investigated for further understanding of the mechanism of insulation degradation under the pulse voltage. However, the current research mainly focusses on the accumulation and decay characteristics of the charge under the pulse voltage, while the dynamic charge characteristics at the rising and falling edges are still not clear. Thus, it is important to clarify the motion mechanism of the space charge at the pulse edge.

In this study, an experimental set up has been built to probe the electro-mechanical response under the pulsed voltage with different rise and fall times. The experimental results show that there exists a special charge behaviour at the pulse edges, which has a close relationship with the rise and fall time. The mechanism of charge behaviour at the pulse edges is analysed from the perspective of the resultant force acting on the charge.

Experiment Setup

In this study, a method to investigate the electro-acoustic response at the pulse edge is proposed. The structure of the charge measurement unit is shown in Figure 1. The experimental system is composed of a high-voltage pulse source, a DC power supply, a coupling circuit, a trigger control circuit, a charge measurement unit, an oscilloscope and a computer. The high-voltage pulse source can generate a pulse voltage with rise and fall times from the nanosecond to the microsecond level. The trigger control circuit can achieve a space charge measurement at the rising and falling edges and the DC and pulse voltages are superimposed of the coupling circuit. The measurement unit is mainly composed of an upper electrode (applying a high voltage), a semi-conducting layer, PI insulation, a grounding electrode, a PVDF sensor, and an amplifier, in which the upper electrode is a copper cylindrical electrode with a diameter of 15 mm and the grounding electrode is a flat aluminium electrode with a diameter of 100 mm. In addition, the whole electrode system is put in a confined coaxial space to avoid interference.

Figure 1. The illustration of experimental system

If the pulse rising and falling edges have an effect on charge behaviour, it may lead to space charge vibration and generate an ultrasonic waveform. The ultrasonic waveform passes through a grounding electrode and acts on the polyvinylidene fluoride (PVDF) piezoelectric sensor, converting acoustic signals to electrical signals. In this way, the charge dynamic behaviours at the rising and falling edges can be studied.

In this experiment, a square-wave pulse voltage with a frequency of 500 Hz, an amplitude of ±1 kV and a duty cycle of 50% was applied to the PI film (Kapton 500 HN, a standard series without nano-particles and coating) with a size of 5×5 cm and thickness of 125µm. The waveform of the negative pulse voltage is shown in Fig. 2. In this study, the 'rising edge' denotes the increase in the voltage amplitude and the 'falling edge' represents the decreasing amplitude of the pulse voltage. For example, the rising edge of negative pulse voltage is the voltage from 0 kV to -1 kV and the falling edge is the voltage from -1 kV to 0 kV. The rise and fall times were set to 100 ns, 200 ns, 400 ns, 1 s, 2 s, 4 s, 10 s, 20 s, 50 s, 80 s, 100 s, 150 s and 200 s.

Experiment Results

Acoustic response at pulse edges

Figure 3 shows the experimental results under the square-wave pulse voltage. The red curve in Figure 3 is the electroacoustic signal measured from the PVDF sensor that reflects charge vibration. It can be seen from the picture that there exist vibration signals at the pulse edge. However, no signal is detected during the constant voltage stage. Figure 4 shows the electroacoustic waveform measured by the PVDF sensor at the rising and the falling edge, shown inside the rectangular box in Figure 3. As shown in Figure 4, the electro-acoustic waveform can be divided into three parts. This phenomenon indicates that the rising and falling edges of the pulse voltage indeed have a special effect on the space charge. Part 1 of the waveform corresponds to the electromagnetic noise caused by the step of the electric field at the pulse edge. The starting position of the electromagnetic oscillation is at the same position as the inflection point of the rising edge because the speed of the electromagnetic wave is very fast, and is close to the speed of light.

Part 2 may be the ultrasonic signal generated by the charge because some delay time is needed for the PVDF sensor to measure the ultrasonic signal. The theoretical delay time can be calculated and compared with the delay time measured by the oscilloscope to determine whether it is generated by the charge. The signal generated by the charge near the grounding electrode first propagates to the sensor and the schematic of the propagation path is shown in Figure 5.

In Figure 5, x is the propagation distance of the ultrasonic wave, which is the thickness of the grounding aluminium electrode (15 mm). The speed v of the ultrasonic wave in aluminium is approximately 6300 m/s. The delay time Tdelay can be calculated using the following equation:

delay x T v  (1)
The theoretical delay time is approximately 2.4 μs, which is consistent with the actual value of the signal measured using the oscilloscope. Thus, it can be determined that the second part of the waveform is generated by charge vibration. The shape of the third part is the same as that of the second part and the delay time between parts 2 and three is about 4.8 μs, that is twice the propagation time of sound waves in the aluminium plate. Therefore, it can be deduced that part 3 is the catadioptric waveform of part 2 in the aluminium electrode.

Since there is an electro-acoustic response that reflects the charge vibration at the rising and falling edges, it can be used to probe the changes in the charge behaviour with the variation of rising and falling edges of the pulsed voltage, which can help clarify the mechanism of insulation failure under the pulsed electric field.

Charge vibration laws with different rise and fall times

The electro-acoustic signal amplitude measured from the piezoelectric sensor reflects the vibration level of the space charge. Under the same condition of the voltage amplitude, the higher the voltage measured, the stronger will be the force charge generated and the more will be the damage caused to the microstructure of insulation materials.

Figure 6 shows the electro-acoustic waveform under the -1 kV pulse voltage with the rise times of 100 ns, 400 ns, 1 μs and 10 μs. The first peak is the space charge near the grounding electrode (closest to the PVDF sensor) which demonstrates that its attenuation of amplitude is the smallest. It can therefore be used to reflect the change of the charge characteristics. It is apparent from the figure that there is a dramatic decline of the charge vibration amplitude: the vibration amplitude is 1.9 V at the 100 ns rising edge while it decreases to 0.067 V at the 10 μs rising edge. Different rise and fall times were applied to clarify the decreasing trend of the charge vibration amplitude and the results are shown in Figure 7.

Figure 7 shows the variation of the electro-acoustic signal amplitude with different rise and fall times under the negative and positive pulse voltage, respectively. A logarithmic X-scale is adopted to show the effect of changes in rise and fall times over several orders of magnitude. It can be seen from the graph that the amplitude of charge vibration decreases with the increase in rise and fall times. In particular, the vibration amplitude is much higher at the nanosecond rise and fall times. Then, the response drops sharply when the rise and fall times change from the nanosecond to the microsecond range, and the vibration signal is very weak at the microsecond level, meaning that it is almost undetectable. The transition zone is approximately between 1 and 5 μs, and the vibration amplitude is nearly zero when the pulse edge time is longer than 50 μs.

The characteristic of pulse edges is sudden changes in the voltage amplitude and this can be accounted for by the absolute of the slope, α = |dV/dt|. Thus, it is necessary to study the change law of the slope under different pulse edge times. Figure 8 shows the 200 ns falling edge of the negative pulse. The voltage variation is not linear: the slope first increases, arrives at a maximum point, then decreases gradually. The place where α is the highest may have the greatest impact on charge behaviour.

To clarify the relationship between the pulse edge time and charge vibration, the maximum value of the absolute slope under different rise and fall times was calculated and the results are shown in Figure 9. It is apparent that the change law of the slope with the pulse edge time is similar to that of the charge vibration amplitude. The maximum slope is very large at the nanosecond rise and fall time, while it decreases rapidly with the increase in pulse edge time. When the rise and fall times increase to microsecond levels, the slope is small and the rate of decline slows down. This reveals that the vibration amplitude of the charge may be determined by the maximum value of the absolute slope of the pulse edge. It can be seen from Figure 7 that the amplitude of the acoustic signal at the falling edge is larger than that at the rising edge, regardless of whether the polarity of the pulse voltage is negative or positive. However, the maximum value of the absolute slope is the same at the rising and falling edges. These features tend to show that the response is not only that due to capacitive charges.

There exists a period of high level to reach the falling edge and more charges are injected to the PI film during the high-voltage level stage. The charge density at the falling edge is larger than that at the rising edge, because the constant stage before the falling edge is in a high-voltage level stage while in a low-voltage level stage before the rising edge. Thus, the reason leading to the difference in the acoustic signal amplitude at the rising and falling edges may be that more charge accumulation enhances the overall effect of charge vibration. It can be concluded that the falling edge of the pulse voltage may cause more damage to the insulation material than the rising edge.

Charge vibration laws under the pulse superimposed dc voltage

Figure 10 shows the variation of the electro-acoustic signal amplitude with rise and fall times under the condition of the -1 kV pulsed superimposed DC voltage to -1 kV DC voltage and the 1 kV pulsed superimposed DC voltage to 1 kV DC voltage, respectively. With the superimposed DC voltage, the charge vibration amplitude increases compared with that under the merely pulsed voltage condition in Figure 7. The higher voltage injects more charge to the PI film and increases the total vibration effect.

Although the charge vibration amplitude increases with the superimposed DC voltage, the transition zone does not change showing the same change law as when the vibration amplitude decreases significantly due to change in the pulse edge time from nanosecond to microsecond. The vibration amplitude decreases close to zero when the pulse edge time is longer than 50 μs.

Analysis of Charge Behaviour at Pulse Edges

Mechanism of charge vibration

The signal measured from the PVDF sensor is caused by charge displacement and the movement of all objects occurs under the drive of the force. Thus, the resultant force acting on the charge can be analysed to clarify the mechanism of charge vibration at the rising and falling edges.

How the resultant force changes with the variations in rise and fall times can be roughly calculated by the kinetic energy theorem. The function of the kinetic energy theorem is described as follows:

0 F t mv mv    ( 2
)
where F is the resultant electrostatic force acting on the charge, Δt is the change of time, m is the quality of charge, v is the velocity of charge in the terminal state, and v0 is the velocity of charge in the initial state. The charge velocity can also be reflected from the voltage amplitude measured by the PVDF sensor: a charge with a larger velocity will generate a stronger ultrasonic wave and the higher voltage is measured by the PVDF sensor. In other words, the higher voltage signal represents the larger charge velocity.

Before the rising and falling edges (during the constant voltage stage of pulse), v0 and v are all approximately zero because no charge signal was measured. This means that the resultant force acting on the charge is small and the charge is in the equilibrium state. At the rising and falling edges, the charge vibrates and has a certain velocity. With the decreasing rise and fall times, the amplitude of charge vibration increases significantly, which represents that the charge has a larger velocity and the change in kinetic energy increases; while, Δt decreases. Thus, according to Equation ( 2), it can be calculated that the resultant force is larger with the rise and fall times decreasing.

The reason the resultant force acting on the charges increases with the decrease in pulse edge time is not clear. An analysis method based on the balance of forces acting on the charge has been proposed in a previous study [START_REF] He | Space charge behaviors in cable insulation under a direct current-superimposed pulsed electric field[END_REF]. This theory can also be applied to explain this phenomenon and the forces acting on the charge need to be further analysed to explore the variation of the resultant force with pulse edge time. Figure 11 shows two kinds of forces acting on the charge: the electrical field force and the other forces. The other forces can be summarised as 'material stress' and include the bondage of the nucleus and the electrons to the charge, the attraction and repulsion of other charges around etc. These forces depend on the relative distance between charges. Therefore, the magnitude of the material stress is related to the position of the charge: the material stress would remain constant if the charge position does not change.

The change of forces acting on the charge at the rising and falling edges is shown in Figure 12. Under the constant voltage stage of the pulse, the electric field force is balanced with the material stress. The resultant force is close to zero and the charges remain stationary. When the rising or falling edge is met, the electric field force can increase or decrease rapidly. The charge cannot migrate immediately when the rise or fall time is less than the relaxation time of the charge; thus, the material stress cannot change at once. Therefore, the equilibrium of the forces acting on the charge is broken and the resultant force suddenly increases at both the rising and falling edges, causing the charge to gain a large acceleration, move and generate the ultrasonic wave measured by the PVDF sensor.

With the rise and fall times increasing, the change of the electric field force becomes slower and the rise and fall times become closer to the relaxation time of the charge. The position of charges will make an adjustive shift, and change the relative distance from other charges. This allows the material stress to adjust, attempting to balance with the electric field force. Therefore, the resultant force acting on the charge will decrease with the increase in pulse edge time; as a result, the amplitude of charge vibration decreases.

Relationship Between Polarity of Charge Waveform and Resultant Direction

Figure 13 shows the electro-acoustic waveform at the 100 ns rising and falling edges of +1 and -1 kV pulse voltages, respectively. Comparing Figure 13a,b (or Figure 13c,d), it can be found that although the polarity of the pulse voltage does not change, the polarity of the voltage measured from the PVDF sensor is opposite to that at the rising and falling edges. Moreover, comparing Figure 13a,c (or Figure 13b,d), the polarity of the charge waveform is the same as that at the rising or falling edge despite the change in the polarity of the pulse voltage. This may be related to the movement direction of the charge. This phenomenon can also be explained using the force analysis theory. Figure 14 shows the movement direction of the charge. At the rising edge of the negative pulse voltage, the electric field force increases suddenly but the material stress changes slowly. This leads to the electric field force being greater than the material stress and the direction of the resultant force is the same as that of the electric field force, which in turn will lead to the movement of the charge in the direction of the electric field force, as shown in Figure 14a. The directions of the electric field force acting on the negative charge near the cathode and the positive charge near the anode are reverse. Therefore, two charge peaks with opposite polarity are observed. On the contrary, the material stress is greater than the electric field force at the falling edge of the negative pulse voltage. Thus, the charge will move in the direction of the material stress, as shown in Figure 14b. This can also be used to analyse the charge movement behaviour at the rising and falling edges of the positive pulse. It can be seen that charges have an opposite movement direction at the rising and falling edges under both negative and positive pulses. This is consistent with the charge waveform having opposite polarities at the rising and falling edges (the comparison between Figure 13a,b or 13c,d).

In addition, although the polarity of the charge in the PI film is opposite under the negative and positive pulses, the movement direction of the charge with different polarities is the same at the rising or falling edge, as is shown in Figure 14a,c or 14b,d. This explains why the charge vibration waveform has the same polarity at the rising or falling edge of both negative and positive pulses.

Through the above discussion, it can be concluded that the polarity of the charge waveform obtained from the force analysis is consistent with that measured by the experiment. This also confirms that it is reasonable to analyse the charge vibration behaviour at the rising and falling edges from the perspective of the resultant force change.

Discussion

Under the condition of the DC or constant voltage stage of the pulse, space charges injected from the electrode or produced by the ionisation of internal impurities are captured by the trap and the trapped charges are in the equilibrium state. When the rising and falling edge of the pulsed voltage arrives, they can deviate from the steady state due to the disturbing force and return to the equilibrium position after the force is removed [START_REF] Bernstein | Analysis of the electrically stimulated acoustic-wave method for observing space charge in semi-insulating films[END_REF]. The suddenly increased resultant force at the rising and falling edges acts as the driving force, which compels the charge to move and vibrate. However, if the resultant force is strong enough or the force acts enough times, the charge may not be able to return to the equilibrium position. This could result in the charge being detrapped, followed by a large amount of energy being released in this process. As a result, the microstructure of the insulation may be broken by the released energy. Then, the detrapped charge will move back and forth under the action of the pulses edge, colliding with the molecular chain. The charges will migrate to another position after being detrapped, which leads to a more uneven electric field. Distortion of the local electric field will change the force distribution inside the insulation materials. The chemical structure of the molecular chain will be deformed under the action of the non-uniform force, resulting in the reduction of mechanical and electrical properties of the insulation material. The influence mechanism of the pulse edge on insulation degradation is shown in Figure 15. The behaviour of charge vibration at the rising and falling edges is harmful to the insulation material, which will accelerate the destruction of the insulating material microstructure. In the process of charge vibration, the charge will drive the molecular chain to vibrate in the same way. The molecular structure of the PI film will be damaged and cracked, resulting in a tiny crevice forming inside the insulation. This will drop the electrical strength and eventually lead to a breakdown. With the rise and fall times decreasing, charge vibration becomes more violent, which will cause more damage to the insulation material. The phenomenon that the charge vibration only exists at the condition of rising and falling edges instead of being at the constant voltage stage of the pulse voltage could be used to explain why the electrical tree has a lower initiation voltage and grows faster under the pulse voltage than that under the DC voltage, and the insulation has a shorter lifetime under the pulse voltage.

The analysis of the resultant force acting on the charges in this study is a preliminary explanation and is not yet mature. In further studies, the intrinsic mechanism of material stress needs to be investigated in a quantitative manner. Additionally, the quantitative descriptive relationship between the charge vibration amplitude and the degree of insulation degradation needs to be clarified, which could help formulate insulation standards under pulse electric fields. Based on the charge behaviour theory, an effective strategy to suppress charge vibration under the pulse voltage can be explored and proposed, in order to enhance the electrical properties and stability of the PI film and other insulation materials operated in high-voltage power electronic equipment.

Conclusion

Through the measurement of ultrasonic waves generated from the PI material by the piezoelectric transducer, the experimental results reveal that there exists the charge vibration behaviour at the rising and falling edges, and that the rise and fall times have an important effect on charge vibration. The main conclusions are as follows:

(1) Space charge vibrates under the motivation of the pulse voltage at the rising and falling edges from the nanosecond to microsecond level. The amplitude of charge vibration increases with the decrease in the rise and fall times; and the degree of charge vibration at the nanosecond level of the rise and fall times is more severe than that at the microsecond level. It is believed that the pulsed voltage edge acts as a disturbance source on trapped charges, and breaks the equilibrium of the functional forces on charges. The sudden increased resultant force due to the rapid change of the electric field force results in charge movement and vibration. The resultant force decreases with the increase in rise and fall time, because of the adjustment of material stress. Furthermore, the vibration amplitude at the falling edge is higher than that at the rising edge, due to more charge accumulation.

(2) The charge vibration waveform at the rising edge has an opposite polarity to that at the falling edge and the polarity of the charge waveform is the same at the rising or falling edge despite the change in the polarity of the pulse voltage. The direction of the resultant force determines the movement direction of the charge and the latter determines the waveform polarity of charge vibration. Under the same polarity of the pulse voltage, the movement direction at the rising edge is opposite to that at the falling edge; and the movement direction of the charge is the same at the rising or falling edges with different polarities in the pulse voltage.

(3) Charge vibration is a harmful behaviour for insulation, which will destroy the microstructure of materials and induce the electrical tree initiation. In future studies, the quantitative descriptive relationship between the charge vibration amplitude and the degree of insulation degradation needs to be explored. Lastly, it is important to propose a strategy suppressing charge vibration, which can improve the insulation performance under pulse electric fields.

Figure 2 .

 2 Figure 2. Waveform of pulse voltage applied to the polyimide

Figure 3 .

 3 Figure 3. Electro-acoustic signals measured by the polyvinylidene fluoride sensor

Figure 4 .

 4 Figure 4. Waveform measured from the polyvinylidene fluoride sensor under -1 kV pulse voltage with 100 ns rise and fall time.

Figure 5 .

 5 Figure 5. Schematic of the propagation path of the ultrasonic waveform generated by the charge

Figure 6 .

 6 Figure 6. Electro-acoustic waveform under -1 kV pulse voltage with different rise times.

Figure 7 .

 7 Figure 7. Electro-Acoustic signal amplitude at different rise and fall times under pulsed voltage. (a); (b).

Figure 8 .Figure 9 .

 89 Figure 8. Change law of the absolute of slope at 100 ns negative pulse falling edge

  -1kV pulsed voltage superimposed -1 kV DC voltage (b) +1 kV pulsed voltage superimposed +1 kV DC voltage.

Figure 10 .

 10 Figure 10. Electro-acoustic signal amplitude under the pulse superimposed DC voltage.

Figure 11 .Figure 12 .

 1112 Figure 11. Forces acting on the charge.

Figure 13 .

 13 Figure 13. Electro-acoustic waveform measured by the polyvinylidene fluoride sensor at ± 1 kV pulse voltage at the 100 ns edge.

( a )Figure 14 .

 a14 Figure 14. Movement direction of charge.

Figure 15 .

 15 Figure 15. Influence mechanism of pulse edge on insulation degradation

Acknowledgments

This research was financed by the National Natural Science Foundation of China (Grant No.51907105 and No. U1966209) and the Shandong Provincial Natural Science Foundation (Grant No. ZR2019QEE013).


Holger S P Mueller

Ananya M Brahmi

Jean-Claude Guillemin

Frank Lewen

Stephan Schlemmer

Holger S P Müller

Rotational spectroscopy of isotopic cyclopropenone, c-H2C3O, and determination of its equilibrium structure

Keywords: Molecular data -Methods, laboratory, molecular -Techniques, spectroscopic -Radio lines, ISM -ISM, molecules -Astrochemistry

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Introduction

Cyclopropenone is one of the smallest aromatic molecules (Anonymous 1983;Wang et al. 2011), after cyclopropenylidene (c-C 3 H 2 ), and the first cyclic molecule with a ketone functional group which was detected in space. It was first found in the cold and less dense envelope of the giant star-forming region Sagittarius B2(N) (Hollis et al. 2006). More recently, c-H 2 C 3 O was detected in four dark clouds or prestellar cores, TMC-1, B1-b, L483, and Lupus-1A (Loison et al. 2016). It was found in addition in the low-mass star-forming region L1527 (Araki et al. 2017). Benson et al. (1973) presented the first account on the rotational spectrum of cyclopropenone. They reported between 6 and 14 rotational transitions for five isotopic species recorded between 9 GHz and 40 GHz. The main isotopolog and the two isotopomers containing one 13 C atom were studied in a sample in natural isotopic composition. They employed isotopic en-Transition frequencies from this work as well as related data from earlier work are given for each isotopic species as supplementary material. Given are also quantum numbers, uncertainties, and residuals between measured frequencies and those calculated from the final sets of spectroscopic parameters. The data are available at CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.ustrasbg.fr/cgi-bin/qcat?J/A+A/ riched samples for the investigations of the isotopologs containing 18 O or two D. Additionally, they derived structural parameters of c-H 2 C 3 O and determined its dipole moment and rotational g values through Stark and Zeeman measurements, respectively. Later, Guillemin et al. (1990) extended the data set of the main isotopic species to 247 GHz. These data are sufficient to identify c-H 2 C 3 O in cold astronomical sources up to the lower submillimeter region, possibly up to ∼500 GHz, but are not enough for searches in warmer astronomical environments. Although cyclopropenone was identified to date only in colder sources, it may well be detected in the warmer and denser parts of star-forming regions one day. Propanal (Lykke et al. 2017), propene, and propenal (Manigand et al. 2021) are recent examples of molecules initially found only in cold astronomical sources later found in the hot corino, the warmer and denser parts of a low-mass star-forming region, of IRAS 16293-2422 B in these cases. The isotopic data may be suitable for astronomical searches in cold environments in the microwave and possibly lower millimeter regions if good estimates of the lowest order centrifugal distortion parameters are available, but are certainly too limited for searches throughout the entire millimeter region. Moreover, no data are available for c-HDC 3 O, which is the most promising isotopolog to be found in cold dark molecular clouds. Therefore, we investigated the rotational spectra of ACCEPTED MANUSCRIPT -CLEAN COPY A&A proofs: manuscript no. c-H2C3O_05 five isotopologs of cyclopropenone in the millimeter region to facilitate their detection in space and extended measurements of the main isotopic species to the lower submillimeter region to allow more secure searches in warmer astronomical sources.

Experimental details

Sample preparation

The synthesis of normal cyclopropenone was the same as in Guillemin et al. (1990) and followed the procedure described by Breslow et al. (1977). The synthesis of the cyclopropenone sample highly enriched in deuterium followed largely Breslow & Oda (1972) with n-Bu 3 SnH replaced by n-Bu 3 SnD. Please note that cyclopropenone decomposes at room temperature. It is very stable in a sealed container at temperatures below ∼240 K. We kept the sample at dry ice or liquid nitrogen temperature in our experiments.

Spectroscopic measurements

The investigation of the rotational spectra of cyclopropenone isotopologs was carried out with two different spectrometers. We employed two 7 m long coupled glass cells with 10 cm inner diameter in a double path arrangement for measurements in the 70-126 GHz region, yielding an optical path length of 28 m. We used a 5 m long double path cell with 10 cm inner diameter for the 160-245 GHz and the 342-505 GHz ranges. Both spectrometers use Virginia Diode, Inc. (VDI), frequency multipliers driven by Rohde & Schwarz SMF 100A synthesizers as sources and Schottky diode detectors. Frequency modulation was employed to reduce baseline effects with demodulation at twice the modulation frequency. This causes absorption lines to appear approximately as second derivatives of a Gaussian. Additional information on the spectrometers are available in Ordu et al. (2012) and Martin-Drumel et al. (2015), respectively.

We recorded individual transition frequencies in all three frequency windows covering 10 MHz for well-predicted lines up to 100 MHz in the search for first lines of the singly deuterated isotopolog. The pressure was around 1.0 Pa in the 3 mm region as test measurements showed that the peak intensity was best between ∼0.75 and ∼1.5 Pa. The pressure was between 1.0 and 2.0 Pa at shorter wavelengths. We refilled the cells after roughly one hour because the molecule is only moderately stable at room temperature; its half-life was close to one hour in our cells. Uncertainties were evaluated mostly based on the symmetry of the line shape and were as small as 5 kHz for isolated and very symmetric lines. Such small uncertainties were achieved earlier, for example, in the case of 2-cyanobutane with a much richer rotational spectrum (Müller et al. 2017). Uncertainties for good lines were 10-20 kHz, and larger ones, up to ∼100 kHz, were used for example for weaker lines and lines close to other lines.

Initial measurements using the highly deuterated cyclopropenone sample yielded no clear signs of rotational transitions of c-D 2 C 3 O, but instead strong signals of c-H 2 C 3 O. We suspected rapid D-to-H exchange on the cell walls, even though this may appear unusual for a molecule with substantial aromatic character. On the other hand, at least some unsaturated molecules are known to exchange D and H quite readily; for example, the reaction between HC 3 N and D 2 O was used to generate DC 3 N [START_REF] Spahn | [END_REF]

Quantum-chemical calculations

We carried out quantum-chemical calculations at the Regionales Rechenzentrum der Universität zu Köln (RRZK) using the commercially available program Gaussian 09 [START_REF] Frisch | Gaussian 09[END_REF]. We performed B3LYP hybrid density functional (Becke 1993;Lee et al. 1988), Møller-Plesset second (MP2) and third order perturbation theory (MP3) calculations (Møller & Plesset 1934), along with coupled cluster calculations with singles and doubles excitations augmented by a perturbative correction for triple excitations, CCSD(T) (Raghavachari et al. 1989). We employed correlation consistent basis sets which were augmented with diffuse basis functions aug-cc-pVXZ (X = T, Q, 5) (Dunning, Jr. 1989), which we abbreviate here as 3a, 4a, and 5a, respectively. These basis sets were further augmented with core-correlating basis functions in some cases, yielding the aug-cc-pwCVXZ basis sets (Peterson & Dunning, Jr. 2002), which we denote as 3aC, 4aC, and 5aC, respectively.

Equilibrium geometries were determined by analytic gradient techniques, harmonic force fields by analytic second derivatives, and anharmonic force fields by numerical differentiation of the analytically evaluated second derivatives of the energy. The main goals of these anharmonic force field calculations were to evaluate initial spectroscopic parameters for the minor isotopic species of cyclopropenone and first order vibration-rotation parameters (Mills 1972), see also Sect. 6. Core electrons were kept frozen in MP2, MP3, and CCSD(T) calculations unless "ae" indicates that all electrons were correlated.

Spectroscopic properties of cyclopropenone

Cyclopropenone is a planar asymmetric rotor with κ = (2B -A -C)/(A -C) = -0.8801, somewhat close to the prolate symmetric limit of -1. Its dipole moment of 4.39 D (Benson et al. 1973) is along the a-inertial axis which is aligned with the CO bond, see The K c = J -1 transitions are approaching oblate pairing, that is, the two lines coalesce at somewhat higher J. Also demonstrated is the 1:3 para to ortho ratio for transitions with K a being even and odd, respectively; see Sect. 4 for further details.

-1 and transitions with ∆K a = ±2 are much weaker, but can be observed quite readily under favorable circumstances.

Carbon has two stable isotopes with mass numbers 12 and 13 and with terrestrial abundances of 98.89% and 1.11%, respectively (Berglund & Wieser 2011). The respective abundances are 99.76%, 0.04%, and 0.20% for 16 O, 17 O, and 18 O, and 99.98% and ∼0.015% for H and D.

The main isotopolog and those with 18 O or 13 C in the keto group (C1) have C 2v symmetry. Spin-statistics caused by the two equivalent H lead to ortho and para states with a 3:1 intensity ratio, see Fig. 2. The ortho states are described by K a being odd. The doubly deuterated isotopolog has also C 2v symmetry, but the ortho to para ratio is 2:1, and the ortho states are described by K a being even. The effect of substituting 13 C at C2 or C3 is the same because the two carbon atoms are structurally equivalent. The corresponding species is thus referred to as 13 C2 isotopolog. Note that we do not consider isotopologs with more than one 13 C. The abundance of the 13 C2 isotopolog is ∼2.2% in a sample of natural isotopic composition. Its symmetry is C S , as is that of the HD isotopolog. No non-trivial spin-statistics exist in these two isotopologs. The HD isotopolog has a very small bdipole moment component of ∼0.21 D as derived from the structure and from quantum-chemical calculations; the corresponding value for the 13 C2 isotopolog is ∼0.06 D.

Spectroscopic results

Pickett's programs SPCAT and SPFIT (Pickett 1991) were used to calculate and fit the rotational spectra of the cyclopropenone isotopologs. The results of Guillemin et al. (1990) were used for the main species. Too few transition frequencies have been published for the minor isotopic species to determine quartic centrifugal distortion parameters reliably. We evaluated ground state rotational and equilibrium quartic centrifugal distortion parameters for six isotopic species of cyclopropenone at the B3LYP/3a quantum-chemical level; vibrational corrections to centrifugal distortion parameters are, to our knowledge, not available in publicly available quantum-chemical programs. We approximated the experimental value X iso exp of an isotopic spectroscopic parameters by scaling the calculated value X iso calc with the ratio of the experimental and calculated value X main exp /X main calc of the main iso-topolog. These values were used as starting parameters to reproduce the limited sets of transition frequencies of the minor isotopic species. One important aspect in this procedure, and in all of our fitting, was to determine or to float as few parameters as possible to improve the weighted rms of the fit as a useful measure of the quality of the fit. This implies that we searched in each fitting round for the spectroscopic parameter that reduced the weighted rms most and that was useful in the context of parameters already determined or floated in the fit. It was sufficient to float B and C in the cases of the 13 C1, 13 C2, and 18 O isotopologs of cyclopropenone. One transition of the 18 O species, 4 1,3 -4 1,4 , and two transitions of the 13 C2 species, 10 2,8 -10 2,9 and 15 3,12 -15 3,13 , had large residuals between the reported experimental frequency and the calculated one that we weighted out these transitions; eventually, they were omitted from the final fits. The reported frequencies were larger than in our final calculations by 0.46 MHz for the line attributed to the 18 O species and by 43.51 and 92.56 MHz, respectively, for the two lines assigned to the 13 C2 isotopolog. We had to float also A to reproduce the transition frequencies of the doubly deuterated isotopolog well.

The adjusted rotational parameters of c-D 2 C 3 O differed somewhat more from the initial parameters than the adjusted B and C values of the other minor isotopic species. We assumed the deviations in the case of c-HDC 3 O will be roughly half as large as those of c-D 2 C 3 O. We corrected the c-HDC 3 O accordingly to improve the calculations of this isotopolog for the first searches of rotational transitions.

The main isotopic species

Our investigations started with measurements of the main isotopic species in the 342-505 GHz region. We targeted the strongest R-branch transitions with K a ≤ 5 in a first step, subsequently extending to K a = 29 in few steps. Transitions with K a ≤ 14 were found within 0.3 MHz of the initial calculations.

The deviations increased rapidly with K a and exceeded 10 MHz for the highest K a transitions. We searched then for Q-branch transitions with ∆K a = 0 and for various types of ∆K a = 2 transitions, most of them had also ∆J = 0. Later, very limited measurements were carried out in the 160-245 GHz region and then again more extensive ones in the 70-126 GHz region. The ∆K a = ∆J = 0 transitions reached K a = 8, and the ∆K a = 2 transitions extended to K a = 5 ↔ 7. Whereas the rotational spectrum of cyclopropenone is quite sparse on the level of the strongest lines, it is much richer on the level of the weakest transitions recorded with the consequence that the desired line has an increased chance of being blended with or being close to a usually unassigned line.

In the end, we had recorded 398 different transitions newly which correspond to 311 different frequencies because of 87 unresolved asymmetry doublets. The majority of these doublets are prolate paired transitions, which have the same J and K a for each pair, as may be expected for an asymmetric rotor somewhat close to the prolate symmetric limit; seven are oblate paired transitions with K c = J in each case. In addition, we remeasured three transitions already reported by Guillemin et al. (1990). These data were combined with 14 transition frequencies from Benson et al. (1973) and 45 from Guillemin et al. (1990), which corresponded to 51 transitions. Uncertainties of 30 kHz were used for these lines as stated by Guillemin et al. (1990) for their data and commensurate for the data by Benson et al. (1973).

We subjected the transition frequencies initially to fits employing Watson's A reduction, as done previously (Guillemin et al. 1990). We tried out also Watson's S reduction because cy- Notes. Watson's S and A reductions were used in the representation I r . Numbers in parentheses are one standard deviation in units of the least significant figures. Empty fields indicate parameters not used in the fit. (a) Ground state rotational and equilibrium centrifugal distortion parameters.

(b) Guillemin et al. (1990). (c) The labels (new), (B), and (G) refer to our new data, those from Benson et al. (1973), and from Guillemin et al. (1990). (d) Weighted rms, unit-less.

clopropenone is quite close to the prolate symmetric limit, in which case the S reduction is usually preferable. As can be seen in Table 1, we required an almost complete parameter set up to sixth order in the A reduction, only Φ K was not used in the fit. Two fewer parameters were used in the S reduction, albeit with a slightly larger wrms of 0.972 compared to 0.921. Nevertheless, we decided to view the S reduced fit as the preferred one. Trial fits with Φ K or H K , H J , and h 1 from B3LYP/3a calculations added to the fit as fixed parameters did not improve the quality of the fits and resulted in changes of the remaining spectroscopic parameters roughly corresponding to their uncertainties. Moreover, we were uncertain concerning the reliability of the quantum-chemically calculated sextic distortion parameters, see also Sect. 7.1. Therefore, these parameters were omitted from the final fits. The quantum-chemically calculated spectroscopic parameters are given in Table 1 for comparison purpose along with the parameters from Guillemin et al. (1990). We also provide rms values for the whole data set and for the individual sources for completeness reason. The rms is meaningful if only one uncertainty was used in all instances or if the uncertainties differ only by a factor of a few. In the present case, the uncertainties differ by around a factor of 10. In this and similar cases, the rms is dominated by the lines with large residuals.

5.2. The 13 C1, 13 C2, and 18 O isotopologs

We began our investigations of the minor isotopologs of cyclopropenone containing one 13 C or 18 O in the 160-245 GHz region searching for the stronger R-branch transitions first with K a ≤ 2 and then up to K a = 5. Additional measurements were carried out in the 70-126 GHz region, in which we recorded R-branch transitions up to K a = 7 and several ∆K a = ∆J = 0 transitions for the two isotopomers with one 13 C. The extent of measured lines was more limited for the 18 O because of its lesser abundance.

The final line lists of the 13 C1, 13 C2, and 18 O isotopologs consist of 104, 95, and 48 different and new transition frequencies, respectively, in addition to the small number of previously reported lines (Benson et al. 1973), for which uncertainties of 70 kHz were assigned. The resulting sets of spectroscopic parameters, determined as described at the beginning of this sec-Table 2. Spectroscopic parameters (MHz) from a minimum parameter set, a maximum parameter set, and initial parameters along with details on the fits of the 13 C1, 13 C2, and 18 O isotopologs of cyclopropenone. (Benson et al. 1973). Parameters without uncertainties were estimated and kept fixed in the analyses, see Sect. 5. (a) With omitted lines after the comma, see Sect. 5. (b) Weighted rms, unit-less. The parameter values, their uncertainties, and additional details on the fits are presented in Table 2 as minimum parameter sets which are the preferred parameter sets. We also tested how many parameters can be determined with sufficient significance. These were all quartic centrifugal distortion parameters for the isotopomers with one 13 C and all but ∆D K for the 18 O isotopolog.

The resulting values are given as maximum parameter sets in the same table along with the initial parameters whose derivation was also described at the beginning of this section.

The singly and doubly deuterated isotopic species

Our procedure for studying the rotational spectrum of the D 2 isotopolog was very similar to that for the minor isotopic species described in the previous subsection. The use of a sample highly enriched in deuterium made it possible to record considerably more transitions extending to higher quantum numbers. The spectral recordings were line richer than those of the sample in natural isotopic composition because of the presence of lines caused by the HD species and even some of the main species. In addition, we identified lines of traces of CH 2 Cl 2 which was used as a solvent in the course of the preparation of the deuterated sample and which was difficult to remove completely.

The R-branch transitions with ∆K a = 0 were recorded up to K a = 9, transitions with ∆K a = ∆J = 0 up to 2, and those with ∆K a = 2 up to K a = 5 ← 3.

We employed a different strategy to search for transitions of the HD isotopolog because no previous data existed. A pattern of two or more transitions is obviously more decisive than a single line. The best candidates are nearly prolate or nearly oblate paired R-branch transitions because the small asymmetry splitting changes rapidly with J. The asymmetry splitting of the K c = J nearly oblate paired transitions were deemed to be too large in the 160-245 GHz region. As can be seen in Fig. 3, the K a = 6 nearly prolate paired transitions were suited very well, even though they were considerably weaker. The measured asymmetry splittings of about 1.8, 4.4, and 9.5 MHz for J" = 12 to 14 compared very well with the calculated splittings. We searched for R-branch transitions with equal or lower K a with an improved calculation of the rotational spectrum. Subsequently, we sought Q-branch transitions with ∆K a = 0 and 2, respectively. Our line list reached K a = 3 in the first case and K a = 3 ← 1 in the second case of these weaker transitions.

We did not target any b-type transitions of c-HDC 3 O as these were calculated to be even weaker than many of the ∆K a = 2 transitions. Some b-type transitions were covered by accident; even much longer integration times may not have helped because the amount of lines with similar or larger intensities were so large that it would be difficult to identify unblended lines unambiguously. In addition, it is important to note that ∆K a = 1 transitions carry less information on the purely K dependent parameters than do those with ∆K a = 2 at similar values of K a .

The minimum parameter sets for both deuterated isotopologs were determined as described above. Interestingly, we obtained a satisfactory fit for 172 different frequencies of c-D 2 C 3 O after floating only two of the five quartic centrifugal distortion parameters whereas we required four for 126 different frequencies of c-HDC 3 O; this is probably a consequence of the choice of the covered transitions. In the fit of c-D 2 C 3 O with the full parameter set, ∆D J and ∆D JK improved the wrms by similar amounts. The parameter values, their uncertainties, and additional details on the fits are presented in Table 3.

Structural parameters of cyclopropenone

Several methods exist to derive structural parameters of a molecule from its moments of inertia, which are inversely proportional to the rotational parameters. The results depend on the model to a varying degree, and some methods do not even yield the same result for different data sets in theory. Data of different isotopologs are required for molecules with three or more symmetry-inequivalent atoms; it is usually desirable to have each symmetry-inequivalent atom substituted once to be able to obtain reliable structural parameters.

The ground state effective (or r 0 ) structure is the most straightforward model in which the structural parameters are fit to the ground state moments of inertia. The ground state values are usually the ones determined first, and often they are the only ones. The ground state moments of inertia contain vibrational contributions from the zero-point vibrations, causing the r 0 structure to be one of the less meaningful structure models with sometimes relatively large changes with isotopic data set.

The differences in the moments of inertia caused by the substitution of one or more symmetry-inequivalent atoms can be used to determine the Cartesian coordinates of these atoms (Kraitchman 1953;Costain 1958). The resulting substitution (or r s ) structure reduces the effects of vibrational contributions to Notes. Watson's S reduction was used in the representation I r . Numbers in parentheses of parameters are one standard deviation in units of the least significant figures. Numbers in parentheses associated with numbers of quantum numbers, transitions, and lines, respectively, refer to data from the previous study (Benson et al. 1973). Parameters without uncertainties were estimated and kept fixed in the analyses, see Sect. 5. (a) Weighted rms, unit-less.

the ground state moments of inertia, albeit to a varying degree. The r s structure is equivalent to r ∆I , the structure that is obtained by fitting structural parameters to the differences of the moments of inertia (Rudolph 1991). This structure is, in turn, equivalent to the r I, structure, in which vibrational contributions i to the ground state moments of inertia, that is I ii,0 = I ii,e + i , with i = a, b, c, are assumed to be equal for different isotopologs of a given molecule. The advantage of taking i into account explicitly is that rotational parameters of isotopic species to be studied can be estimated much better, in particular for atoms with more than two isotopes or for multiply substituted isotopologs (Epple & Rudolph 1992;Müller & Gerry 1994;Müller et al. 2019) 2 CS to H 13 2 C 34 S. The equilibrium (or r e ) structure of a molecule in the potential minimum is the most meaningful structure. It is, however, also the most elusive structure because not only data of more than one isotopolog are needed for molecules with three or more symmetry-inequivalent atoms, but for each isotopolog we require the knowledge of several vibration-rotation parameters according to

B e = B 0 + 1 2 j α B j - 1 4 j≤k γ B jk -... ( 1
)
in order to evaluate the equilibrium parameter B e from the ground state value B 0 . Here, the α B j are first order vibrational corrections, the γ B jk are second order vibrational corrections, and so on. Equivalent formulations hold for A e and C e . An attractive and lately very common approach is to calculate j α B j /2 by quantum-chemical means to derive semi-empirical equilibrium rotational parameters B i,e from the experimental ground state values (Stanton et al. 1998). Second and higher order vibrational contributions are neglected. Numerous quantum-chemical programs are available to carry out such calculations; examples have been mentioned in Sect. 3. We used B3LYP, MP2, and ae-MP2 calculations with a triple zeta basis set to evaluate the first order vibrational corrections for isotopologs of cyclopropenone which are summarized in Table 4 together with ground state values and the resulting semiempirical equilibrium values. The inertia defect ∆ = I cc -I bb -I aa is also given for each set of rotational parameters.

We employed the RU111J program [START_REF] Rudolph | Advances in Molecular Structure Research[END_REF] to derive semi-empirical equilibrium structural parameters r SE e for each of the three quantum-chemical calculations. The results are given in Table 5 together with structural parameters purely from quantum-chemical calculations and with the r s parameters from Benson et al. (1973). The differences between the r s parameters and our r SE e parameters turned out to be larger than expected and will be discussed in detail in Sect. 7.2. We calculated subsequently r I, structures, which are equivalent to r s structures as mentioned above, in order to evaluate the dependence of the resulting parameters on diverse input data. The first set of input data were our present rotational data from six isotopic species as summarized in Table 4. The second set employed our starting values as described in Sect. 5. The third set were the rotational parameters from Benson et al. (1973), but without A( 13 C2), as it deviated by 61 MHz from the value of our preferred fit. The results of these structure fits are also given (in reverse order) in Table 5.

Article number, page 7 of 11 Notes. All values from this work unless indicated otherwise. Numbers in parentheses are one standard deviation in units of the least significant figures. (a) Quantum-chemical calculations as detailed in Sect. 3; structural parameter determinations as described in Sect. 6. (b) Derived parameter in present calculations. (c) Benson et al. (1973); structural parameters derived from substitution coordinates; ∠(C1C2H) calculated from ∠(C2C3H) and ∠(C3C1C2). (d) This work; rotational parameters from Benson et al. (1973), but A( 13 C2) omitted; see also Sects. 6. (e) This work; rotational parameter derived as described in Sect. 5 using data from Benson et al. (1973) and Guillemin et al. (1990). ( f ) This work; rotational parameters from this work. (g) Quantum-chemical method in parentheses as detailed in Sect. 6; see also Table 4.

Discussion

Spectroscopic parameters

As can be seen in Table 1, the J and K a quantum number ranges and also the number of different transition frequencies of the main isotopolog of cyclopropenone have been greatly increased in the course of the present investigation with respect to the previous study (Guillemin et al. 1990). The previous spectroscopic parameters agree well with ours in the A reduction taking into account the uncertainties in that work. The B3LYP/3a quantumchemically calculated ground state rotational and equilibrium quartic centrifugal distortion parameters agree quite well with our experimental values in both reductions. The comparison of the sextic centrifugal distortion parameters is more mixed as far as experimental values are available; trial fits showed that the experimental values are only slightly affected by the absence of three and one sextic parameter in the S and A reduction, respectively. This shows that the deviations seen in particular in H K J and the corresponding Φ K J are not mainly caused by the omission of the three or one parameter. Moreover, a trial fit with h 1 in the fit yielded a value of 19 ± 10 µHz, not compatible with the calculated 96 µHz.

Our final fit in the S reduction has two parameters less than the one in the A reduction at the expense of a slightly larger wrms. We consider this improvement to be sufficient to favor the S reduction. Additional reasons for preferring the S reduction are that it is more versatile than the A reduction as the latter can exhibit convergence problems for molecules close to the prolate or oblate symmetric limit. Moreover, [START_REF] Watson | Vibrational spectra and structure[END_REF] recommended to use the S reduction because such fits yield smaller correlation coefficients. This conclusion was recently reemphasized in an extensive study of the rotational spectra of SO 18 O and S 18 O 2 in their lowest two and three vibrational states, respectively, with detailed analyses of the fits (Margulès et al. 2020).

Our spectroscopic parameters for the 13 C1, 13 C2, and 18 O isotopologs in Table 2 agree very well with our initial values both for the minimum and maximum parameter sets. The agreement is good in the cases of the deuterated isotopologs, as can be seen in Table 3. The parameter D K from the full parameter sets of both deuterated isotopologs agrees very well with the initial values, which may indicate that the small changes in D K seen for the isotopomers containing one 13 C may, nevertheless, be too large.

The initial spectroscopic parameters of all minor isotopic species were derived from quantum-chemically calculated ones scaled with the respective ratios between experimental and calculated value of the main species as described in slightly more detail at the beginning of Sect. 5. Such scaling was used, for example, to evaluate some sextic centrifugal distortion parameters of two minor isotopologs of TiO 2 (Kania et al. 2011). Cazzoli et al. (2014) demonstrated in a study of the rotational spectrum of H 36 2 S that such scaling works very well even for heavy atom substitutions of relatively light molecules such as H 2 S. Higher level quantum-chemical calculations, as employed by Cazzoli et al. (2014), may lead to better starting values than the lower level calculations used here. However, higher level calculations are computationally more demanding, in particular for somewhat larger molecules. Moreover, Morgan et al. (2018) have shown that considerable computational effort is necessary to obtain systematically converged very good results even for the fairly small molecule formaldehyde, H 2 CO. The quality of the scaling of centrifugal distortion parameters may be limited in addition by the lack of vibrational corrections. Quartic and sextic equilibrium centrifugal distortion parameters can be calculated with several quantum-chemical programs, but the derivation of vibrational corrections is, to the best of our knowledge, not possible with any publicly available program.

Scaling of quantum-chemically derived distortion parameters should be better than scaling by appropriate powers of the ratios of 2A -B -C, B + C, and B -C, as done, for example, in the recent case of isotopic H 2 CS (Müller et al. 2019). This type of scaling works usually quite well for heavy atom substitutions, but less so for H to D substitutions, in particular for molecules with few atoms. Fixing parameters of minor isotopologs, which cannot be determined sufficiently well, to values from the main isotopic species work usually less well, but is in most cases better than fixing such parameters to zero.

We may ask ourselves what are the advantages or disadvantages of our minimum parameter fits compared to our maximum parameter fits. Increasing the parameter set beyond the optimum increases the uncertainties, usually the correlation among the parameters, and affects the parameter values frequently outside the initial uncertainties. Therefore, transition frequencies calculated with fewer parameters are often better in cases of interpolation or modest extrapolation compared to values from a larger parameter set. On the other hand, calculated uncertainties become too optimistic sooner upon extrapolation for a smaller parameter set. Extrapolation in J from our present data should be reasonable to two times the upper frequency limit for the strong R-branch transition which is ∼1.0 THz for the main isotopolog and 450 to 500 GHz for the minor isotopologs. Extrapolation in K a is much more limited.

The ground state inertia defects in Table 4 are relatively small and positive, as is common for planar molecules with small out-of-plane vibrational amplitudes. The semi-empirical equilibrium inertia defects, obtained from the ground state rotational constants after subtracting off the first order vibrational corrections, are much smaller in magnitude and negative, which is also very common.

Structural parameters

Our semi-empirical equilibrium structural parameters in Table 5 are largely very similar among the three different sets. The CO bond length has fairly large uncertainties, but displays almost no scatter; the C=C bond shows, in contrast, more pronounced, though still modest variations, which are only a few times the combined uncertainties for the parameters employing B3LYP and ae-MP2 corrections. The CH bond varies about equally much, albeit has larger uncertainties than the C=C bond.

The previously published r s structure (Benson et al. 1973) agrees, at best, reasonably well with our r SE e structures; the C=C bond length difference is, with more than 4 pm, particularly large. We calculated r I, structures which are equivalent to the r s structure (Rudolph 1991). Using our ground state rotational data of six isotopologs, as summarized in Table 4, we obtained structural parameters close to our r SE e values, the largest difference of 0.4 pm occurs in the CO bond length and is even within the combined uncertainties. The initial parameters of five isotopic species, as derived in Sect. 5, yielded essentially the same parameters, the main differences are larger uncertainties in some parameters, most likely caused by the absence of data of the HD isotopolog. And even the fit with values from Benson et al. (1973) yields results quite similar to the other two r I, fits. The deviation of the r s structure parameters appears thus to be caused by the difference of 61 MHz in A( 13 C2) which is mainly caused by two lines falsely attributed to the 13 C2 isotopolog, see also Sect. 5.

The quantum-chemical calculations in Table 5 show very small changes with basis set and also rather small changes between the different methods. The bond lengths display the usual shortening upon increasing basis set size and upon correlation of all electrons as far as applicable. The bond lengths differ substantially among the different methods. The B3LYP parameters show little change with basis set size and agree quite well with our r SE e values. The largest deviation is seen for the C=C bond length, which is too short by about 0.5 pm. The C-C bond length is also too short, but by smaller amounts, whereas the CO and CH bonds were calculated slightly too long. The MP2/5a and ae-MP2/5aC structures agree between well and very well with our r SE e structure, the MP2/5a values are slightly closer for the C-C and CH bonds, whereas the ae-MP2/5aC are slightly closer for the other two bond lengths. Calculations using MP3 are sometimes better than those with MP2, but usually worse than those with CCSD(T). In the case of cyclopropenone, already some bond lengths are too short at the MP3/3a level, even more so at ae-MP3/3aC level. Bond lengths at the CCSD(T)/3a level are rather long. The effects of correlating all electrons compared to those with frozen core are similar to those of MP2 calculations employing basis sets of triple zeta quality. If we assume the relative differences are the same for basis sets of quadruple zeta quality, we can estimate the following ae-CCSD(T)/4aC bond lengths: r(CO) ≈ 120.10 pm, r(C-C) ≈ 142.95 pm, r(C=C) ≈ 134.53 pm, and r(CH) ≈ 108.00 pm, all very close to our r SE e values; this behavior is common for ae-CCSD(T)/4aC calculations.

Structural parameters of cyclopropenone are compared with those of other molecules in Table 6. Cyclopropenylidene is suited best for a comparison as it is basically cyclopropenone without the O atom. The O atom withdraws electron density from the cyclopropenylidene rest, causing an elongation of all bond lengths; the effect is smallest for the distant CH bond. Formaldehyde is probably the prototypical molecule with a CO double bond. Its CO bond is only slightly longer than that of c-H 2 C 3 O. The C=C bond lengths of c-H 2 C 3 O and c-C 3 H 2 are essentially equal to the one in C 2 H 4 , and thus typical CC double bonds. The C-C bonds of both cyclic molecules, however, are much shorter than that of ethane, whose C-C bond can probably be viewed as a prototypical CC single bond. In fact, these bonds are only slightly longer than the aromatic CC bonds in benzene. Finally, the CH bond lengths in c-H 2 C 3 O and c-C 3 H 2 are substantially shorter than in C 2 H 4 , but much longer than ∼106.2 pm in acetylene (Liévin et al. 2011;Tamassia et al. 2016). All this reflects the aromatic character of cyclopropenone (Anonymous 1983;Wang et al. 2011) and cyclopropenylidene from a structural point of view.

Conclusion and outlook

Our data for five minor isotopic species of cyclopropenone are sufficient for astronomical searches in cold molecular clouds even without any extrapolation as the Boltzmann peak at 10 K is at ∼95 GHz. Extrapolation of the data in J should be reasonable up to 450 or 500 GHz. Such extrapolations should be possible up to 1 THz for the main isotopic species, which is sufficient even Lohilahti (2006); experimental r e values. (d) Harmony (1990); r ρ m values according to the second set of data; see Harmony & Taylor (1986) for the definition of this structure model. (e) Gauss & Stanton (2000); r SE e at the CCSD(T)/4 level. ( f ) Martin & Taylor (1996); r e values from CCSD(T) calculations extrapolated to infinitely large basis set with ae corrections and empirical adjustments.

for rotational temperatures around 300 K. We assume, however, that rotational temperatures of c-H 2 C 3 O in warmer sources will be closer to 100 K, based of findings for propanal, propenal and propene in IRAS 16293-2422 B (Lykke et al. 2017;Manigand et al. 2021).

Dedicated searches will probably be necessary to detect minor isotopologs of cyclopropenone in space. c-HDC 3 O is the most promising one to be found in cold molecular clouds. One of the most sensitive molecular line surveys of such sources is the one by Cernicharo et al. (2020) between 31.4 and 50.3 GHz carried out with the Yebes 40 m radio telescope with additional data from the 3 mm region taken earlier with the IRAM 30 m dish. Cernicharo et al. (2020) report the detection of HC 3 O + and investigated the chemistry of oxygen-containing molecules in TMC-1. They derived a column density of (4.0±0.2)×10 11 cm -2 for c-H 2 C 3 O. The most constraining transition of c-HDC 3 O in that survey is the 3 13 -2 12 transition at 37833.721 MHz. The 3σ upper limit to its intensity in brightness temperature is ∼2.6 mK (with corrections for beam dilution and beam efficiency); the corresponding intensity of the 3 13 -2 12 ortho transition of c-H 2 C 3 O is 42 mK, yielding a 3σ column density ratio of > 16 for c-H 2 C 3 O to c-HDC 3 O (J. Cernicharo, private communication to HSPM, Nov./Dec. 2020). This is less constraining than the corresponding ratios of, for example, ∼34 and ∼25 found for c-C 3 H 2 to c-C 3 HD (Bell et al. 1986) and l-C 3 H 2 to l-C 3 HD (Spezzano et al. 2016), respectively, in TMC-1. There are, however, sources with higher degrees of deuteration that may be considered if their column densities of c-H 2 C 3 O are sufficiently large. A search for the cyclopropenone isotopomers with one 13 C may be more promising in the envelope of Sagittarius B2(N) because the 12 C to 13 C ratio in the Galactic center is as low as 20 to 30 (Müller et al. 2008(Müller et al. , 2016;;Halfen et al. 2017;Jacob et al. 2020).

We determined structural parameters of cyclopropenone which compare very favorably with those of cyclopropenylidene as far as comparison is possible. The CO bond length agrees almost exactly with that of formaldehyde. Relatively large deviations between our structural parameters and the earlier substitution structure were traced to misassignments of two transitions to the 13 C2 isotopolog which led to a too large A rotational parameter.

  Fig. 1. Model of the cyclopropenone molecule. Carbon atoms are symbolized by gray spheres which are numbered. Hydrogen atoms are indicated by small, light gray spheres and the oxygen atom by a red sphere.

Fig. 1 .

 1 Fig.2. Section of the submillimeter spectrum of c-H 2 C 3 O. The K c = J -1 transitions are approaching oblate pairing, that is, the two lines coalesce at somewhat higher J. Also demonstrated is the 1:3 para to ortho ratio for transitions with K a being even and odd, respectively; see Sect. 4 for further details.

Fig. 3 .

 3 Fig. 3. Section of the millimeter spectrum of cyclopropenone highly enriched in deuterium. Transitions with small, but rapidly increasing asymmetry splitting are shown for K a = 6 of c-HDC 3 O. These transitions were the first which we assigned because of their easily recognizable patterns.

  Numbers in parentheses are one standard deviation in units of the least significant figures. If no uncertainties are given, none were published. (a) This work, r SE e with ae-MP2 corrections. (b) Spezzano et al. (2012); r SE e , ae-CCSD(T)/4C values; r(C=C) calculated from r(C-C) and ∠(C3C1C2).

  (c)

Table 1 .

 1 Present and previous experimental spectroscopic parameters (MHz) of the main isotopolog of cyclopropenone and details on the fits in comparison to values calculated at the B3LYP/3a level.

	S reduction

Table 3 .

 3 Spectroscopic parameters (MHz) from a minimum parameter set, a full parameter set, and initial parameters along with details on the fits of the mono and doubly deuterated isotopologs of cyclopropenone.

	Parameter	Minimum	Full	Initial
	HD isotopolog
	A	28049.092 (3)	28049.090 (4)	28035.946
	B	7510.3156 (1)	7510.3156 (1)	7509.008
	C	5916.1823 (1)	5916.1823 (1)	5915.416
	D K × 10 3	46.16	45.95 (33)	46.16
	D JK × 10 3	29.541 (3)	29.541 (3)	29.509
	D J × 10 6	1347.93 (15)	1347.99 (18)	1349.3
	d 1 × 10 6	-412.68 (8)	-412.71 (9)	-414.44
	d 2 × 10 6	-180.56 (68)	-180.54 (72)	-181.29
	J max	20 (0)	20 (0)
	K a,max	7 (0)	7 (0)
	No. trans.	130 (0)	130 (0)
	No. freqs.	126 (0)	126 (0)
	rms	0.0126	0.0122
	wrms a	0.817	0.815
	D 2 isotopolog
	A	24535.524 (1)	24535.521 (3)	24514.520
	B	7254.9417 (1)	7254.9422 (1)	7252.557
	C	5591.7853 (1)	5591.7855 (1)	5590.549
	D K × 10 3	30.37	30.20 (11)	30.37
	D JK × 10 3	27.185	27.189 (1)	27.185
	D J × 10 6	1179.79	1180.35 (13)	1179.8
	d 1 × 10 6	-410.13 (10)	-410.31 (11)	-412.33
	d 2 × 10 6	-199.06 (4)	-199.10 (5)	-200.75
	J max	24 (10)	24 (10)
	K a,max	9 (2)	9 (2)
	No. trans.	189 (10)	189 (10)
	No. freqs.	172 (10)	172 (10)
	rms	0.0222	0.0214
	wrms a	0.917	0.810

Table 4 .

 4 Holger S. P.Müller et al.: Rotational spectroscopy of isotopic c-H 2 C 3 O Ground state rotational parameters B i,0 of cyclopropenone isotopic species, vibrational corrections ∆B i,v a , resulting semi-empirical equilibrium rotational parameters B SE i,e , and inertia defects ∆. bNotes. (a) ∆B i,v = j α B i j /2 calculated by different quantum-chemical means as detailed in Sect. 3.(b) All numbers in units of MHz, except ∆ in units of amu Å 2 . extrapolated the residuals of H 2 C 32 S, H 2 C 33 S, and H 2 C 34 S to H 2 C 36 S, and those of H 2 C 32 S, H 2 C 34 S, and H 13

					B3LYP		MP2		ae-MP2
	Species B i /∆	B i,0	∆B i,v	B SE i,e	∆B i,v	B SE i,e	∆B i,v	B SE i,e
	main	A	32040.686	167.049 32207.735	179.586 32220.272	183.294 32223.980
		B	7825.001	33.674	7858.675	34.377	7859.378	34.577	7859.578
		C	6280.728	36.969	6317.697	37.689	6318.417	37.999	6318.727
		∆	0.1068		-0.00547		-0.00273		-0.00321
	13 C1	A	32040.141	167.513 32207.654	180.110 32220.251	183.804 32223.945
		B	7816.553	33.050	7849.603	33.778	7850.331	33.969	7850.522
		C	6275.289	36.543	6311.832	37.281	6312.570	37.584	6312.873
		∆	0.1065		-0.00550		-0.00276		-0.00324
	13 C2	A	31174.389	161.272 31335.661	173.586 31347.975	177.251 31351.640
		B	7709.476	32.969	7742.445	33.660	7743.136	33.863	7743.339
		C	6172.746	36.126	6208.872	36.842	6209.588	37.155	6209.901
		∆	0.1083		-0.00548		-0.00270		-0.00321
	18 O	A	32040.339	167.399 32207.738	180.001 32220.340	183.743 32224.082
		B	7344.877	31.126	7376.003	31.751	7376.628	31.941	7376.818
		C	5967.532	34.415	6001.947	35.072	6002.604	35.363	6002.895
		∆	0.1079		-0.00537		-0.00264		-0.00314
	HD	A	28049.092	130.214 28179.306	137.620 28186.712	140.330 28189.422
		B	7510.316	31.968	7542.284	32.521	7542.837	32.719	7543.035
		C	5916.182	34.032	5950.214	34.480	5950.662	34.764	5950.946
		∆	0.1142		-0.00590		-0.00267		-0.00325
	D 2	A	24535.524	98.171 24633.695	102.770 24638.294	105.086 24640.610
		B	7254.942	31.074	7286.016	31.600	7286.542	31.729	7286.671
		C	5591.785	31.521	5623.306	31.843	5623.628	32.081	5623.866
		∆	0.1210		-0.00641		-0.00272		-0.00337

, if residuals of known isotopologs can be transfered to the values of the desired isotopolog. In the case of H 2 CS,

Müller et al. (2019)

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Table 5 .

 5 Quantum-chemical and experimental bond lengths (pm) and bond angles (deg) of cyclopropenone.

	Method a	r(CO)	r(C-C)	r(C=C) b	r(CH)	∠(OCC)	∠(C3C1C2) b ∠(C1C2H)
	B3LYP/3a	120.157	142.672	133.964	108.054	151.999	56.001	153.841
	B3LYP/4a	120.000	142.626	133.894	108.005	152.005	55.989	153.852
	B3LYP/4aC	119.989	142.613	133.887	108.018	152.004	55.991	153.856
	MP2/3a	120.681	143.399	135.270	107.970	151.858	56.284	154.166
	MP2/4a	120.388	143.046	134.907	107.850	151.865	56.270	154.150
	MP2/5a	120.313	142.962	134.815	107.817	151.868	56.264	154.142
	ae-MP2/3aC	120.395	142.885	134.752	107.822	151.866	56.268	154.148
	ae-MP2/4aC	120.140	142.612	134.496	107.689	151.865	56.270	154.139
	ae-MP2/5aC	120.078	142.531	134.408	107.655	151.868	56.264	154.132
	MP3/3a	119.628	142.810	134.383	107.698	151.934	56.133	153.478
	ae-MP3/3aC	119.355	142.333	133.908	107.556	151.940	56.121	153.460
	CCSD(T)/3a	120.684	143.735	135.308	108.155	151.921	56.158	153.739
	CCSD(T)/4a	120.346	143.347	134.912	108.030	151.928	56.144	153.731
	ae-CCSD(T)/3aC 120.404	143.267	134.832	108.010	151.929	56.142	153.724
	r s	c	121.2 (2)	141.2 (3)	130.2 (3)	109.7 (3)		54.9 (7)	152.5 (5)
	r I, (old) d	120.29 (36) 143.18 (46) 134.90 (85) 107.82 (31) 151.89 (22) 56.21 (45)	153.91 (41)
	r I, (old) e	120.47 (30) 142.88 (38) 134.84 (38) 107.88 (16) 151.84 (12) 56.31 (23)	154.02 (22)
	r I, (new) f	120.44 (28) 142.91 (35) 134.73 (14) 107.90 (9)	151.88 (8)	56.24 (16)	153.95 (14)
	r SE e (B3LYP) g r SE e (MP2) g r SE e (ae-MP2) g	120.04 (21) 142.91 (17) 134.532 (6) 107.77 (4) 120.05 (10) 142.88 (8) 134.480 (4) 107.84 (2) 120.05 (12) 142.87 (10) 134.468 (4) 107.85 (2)	151.92 (4) 151.93 (2) 151.93 (2)	56.16 (7) 56.15 (3) 56.14 (4)	153.74 (6) 153.73 (3) 153.73 (3)

Table 6 .

 6 Equilibrium bond lengths (pm) and selected bond angles (deg) of cyclopropenone in comparison to related molecules.

	Molecule	r(CO)	r(C-C)	r(C=C)	r(CH)	∠(C3C1C2) ∠(C1C2H)
	c-H 2 C 3 O a 120.05 (12)	142.87 (10) 134.468 (4) 107.85 (2)	56.14 (4)	153.73 (3)
	c-C 3 H 2	b		141.72 (1)	132.19	107.51 (2)	55.60 (1)	147.78 (3)
	H 2 CO c		120.461 (19)			110.046 (16)
	C 2 H 6	d			152.2 (2)		108.9 (1)
	c-C 6 H 6	e		139.14 (10)	108.02 (20)
	C 2 H 4

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Acknowledgements. We are grateful to José Cernicharo for communication of unpublished results of a search for c-HDC 3 O in TMC-1. We acknowledge support by the Deutsche Forschungsgemeinschaft via the collaborative research center SFB 956 (project ID 184018867) project B3 as well as the Gerätezentrum SCHL 341/15-1 ("Cologne Center for Terahertz Spectroscopy"). We thank the Regionales Rechenzentrum der Universität zu Köln (RRZK) for provid-ing computing time on the DFG funded High Performance Computing System CHEOPS. J.-C. G. is grateful for financial support by the Centre National d'Etudes Spatiales (CNES; grant number 4500065585) and by the Programme National Physique et Chimie du Milieu Interstellaire (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES. Our research benefited from NASA's Astrophysics Data System (ADS).