ARTICLE pubs.acs.org/JPCA
Two-Dimensional Chirped-Pulse Fourier Transform Microwave Spectroscopy David S. Wilcox, Kelly M. Hotopp, and Brian C. Dian* Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, Indiana, 47907-2084
bS Supporting Information ABSTRACT: Two-dimensional (2D) correlation techniques are developed for chirped-pulse Fourier transform microwave (CP-FTMW) spectroscopy. The broadband nature of the spectrometer coupled with fast digital electronics permits the generation of arbitrary pulse sequences and simultaneous detection of the 818 GHz region of the microwave spectrum. This significantly increases the number of rotation transitions that can be simultaneously probed, as well as the bandwidth in both frequency dimensions. We theoretically and experimentally evaluate coherence transfer of three- and four-level systems to relate the method with previous studies. We then extend the principles of single-quantum and autocorrelation to incorporate broadband excitation and detection. Global connectivity of the rotational energy level structure is demonstrated through the transfer of multiple coherences in a single 2D experiment. Additionally, open-system effects are observed from irradiating many-level systems. Quadrature detection in the indirectly measured frequency dimension and phase cycling are also adapted for 2D CP-FTMW spectroscopy.
’ INTRODUCTION The traditional utility of microwave spectroscopy has been its ability to precisely determine the structure and shape of gas phase molecules, as well as associated molecular parameters, including centrifugal distortion, quadrupole coupling, and barriers to inversion or internal rotation that perturb rotational energy levels.1 High sensitivity and high resolution are signatures of Fourier transform microwave (FTMW) spectrometers employing molecular beam techniques. Efficient cooling through the molecular beam free expansion eliminates the contribution of large amplitude vibrations to the overall rotation. Although rigidity and symmetry often limit the number of observed rotational transitions, the “pure” rotational spectra of small- to medium-sized molecules (310 heavy atoms) can nevertheless be dense and difficult to interpret. Measuring and structurally assigning the rotational spectra of larger molecules, such as those of astronomical or biological relevance, pose significantly greater challenges. Here we report the development and application of a twodimensional (2D) broadband microwave strategy, which facilitates the collection and structural assignment of the rotational spectra of flexible polyatomic molecules. Some flexible, and relatively small, biomimetic molecules including N-acetyl-alanine N0 -methylamide,2 N-methylacetamide,3 ethylacetamidoacetate,4 and N-acetyl alanine methyl ester5 have recently been investigated with rotational spectroscopy. The results have been used to understand conformational preferences of protein backbone folding and experimentally test the accuracy of quantum chemical calculations. Such studies are complicated by multiple equilibrium conformations, lack of symmetry, tunneling-splitting of methyl rotations, and 14N quadrupole coupling. r 2011 American Chemical Society
While it is possible to eliminate quadrupole angular momenta by isotopic substitution,2 a high spectral density originating from multiple stable asymmetric configurations is expected. The presence of one or more internal rotors further congests the rotational spectrum, particularly when low-barrier toptop interaction becomes non-negligible (the implications of which are an active field of research6). 2D double resonance techniques simplify multicomponent spectra by associating transitions sharing a common energy level. Using a chirped-pulse Fourier transform microwave (CP-FTMW) spectrometer, we devise global coherence transfer strategies by exploiting the broad bandwidth, fast electronics, and arbitrary waveform generation (AWG) of the CP-FTMW technology. Pulse sequences are designed to transfer energy level structure information to multiple coherence channels in a single 2D experiment. The transitions of different rotational components are efficiently isolated, thereby assisting in the assignment of complex spectra. Developed originally for NMR spectroscopy,7 2D spectroscopy has been adapted for microwave,810 infrared,11 and electronic radiation,12,13 finding rich applications in the study of modern chemical problems in all but rotational spectroscopy. Aside from revealing the energy level structure of a molecule, 2D techniques decouple the temporal and frequency resolution into separate dimensions, thus, permitting time-resolved tracking of coherent states in an evolving system.14 Protein structure and Received: May 9, 2011 Revised: June 30, 2011 Published: July 05, 2011 8895
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The Journal of Physical Chemistry A kinetics have been studied with 2D NMR15 and 2D IR16 spectroscopy. Energy transport pathways of the FennaMathews Olson light harvesting complex have also been illuminated with 2D electronic spectroscopy.17 In the microwave region, the majority of work developing 2D spectroscopy culminated in the late 1980s due largely to technological limitations. For example, multiple frequency synthesizers were required to correlate rotational transitions outside the bandwidth of transform-limited pulses, placing a practical restriction on the number of energy levels involved in coherence transfer. Transform-limited pulses and the low resolution in the indirectly measured frequency dimension (ω1) restricted the information obtained in a 2D plot to a narrow region around a signal frequency. Global rotational level structural changes caused by a dynamically evolving molecule, for example, would not be observable in a single narrowband experiment. Despite these limitations, a significant body of work was accomplished detailing coherence transfer of three- and four-level systems. Aue et al.7 first suggested applying 2D NMR techniques at microwave frequencies. The basic principles of microwave 2D correlation were subsequently realized8,9 followed by the first demonstration of a 2D microwave spectrum.10 A considerable connection with 2D NMR methods was established by Bauder and co-workers who adapted many of the acronymic NMR techniques for microwave radiation: COSY (autocorrelation),18 hetero-COSY (single-quantum correlation),19 INADEQUATE (double-quantum correlation),19 n-quantum filtering (n = zero or double),19 and novel pulse sequences20 were shown to correlate both progressive (ladder-type) and regressive (v- or w-type) connected rotational energy levels of three-18,19 and four-level20 systems and to access dipole-forbidden multiple quantum coherences. NOESY21 was used to investigate collision-induced population transfer and relaxation of a four-level system. The phasecycling procedure of Stahl and Dreizler22 was extended to select rotational coherence transfer pathways and eliminate unwanted signals for diverse pulse sequences. J€ager et al.23 applied RF radiation in one-dimension, obtaining low energy l-type doublet information with microwave detection. Extending the Bloch vector model of two-level spin systems, the earlier theoretical descriptions of 2D microwave spectroscopy used linear combinations of density matrix elements to form “pseudo-spin” Bloch vector components.21,23 The solutions of these equations described coherence pathways as a function of the applied radiation and were later extended to include relaxation24 and phase cycling.19,22 The qualitative picture of the Bloch vector model is helpful in understanding the coherence transfer mechanism in analogy with NMR, but the physical interpretation is not always clear. Consequently, Vogelsanger and Bauder’s formalism19 explained 2D FTMW spectroscopy entirely by the evolution of the density matrix, successfully handling relaxation and phase cycling of progressive and regressive three-level systems. The 2D CP-FTMW spectroscopy is first illustrated with spectra arising from approximately isolated three- and four-level systems. Broadband detection affords 10 GHz of signal frequency bandwidth in a single acquisition. The distinction between “pump” and “signal” coherences, strictly speaking, is no longer required, although the terminology will be used for clarity in description. AWG eliminates the need for multiple frequency synthesizers without sacrificing multiple coherence excitations and also increases the bandwidth in the indirectly measured dimension. Accordingly, the single-quantum and autocorrelation pulse sequences are then extended to incorporate broadband excitation
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and detection in both frequency dimensions. Finally, phase cycling and quadrature detection in ω1 with chirped excitation pulses is discussed.
’ EXPERIMENTAL METHODS A detailed description of the CP-FTMW spectrometer has been previously described25 and only a brief overview will be given here. The microwave polarizing pulses were produced by an arbitrary waveform generator (Tektronix AWG 7101) in the DC to 5 GHz range at a sampling rate of 10 GS/s. Pulses were filtered with a 5 GHz low pass filter (Lorch 10LP-5000-S) and amplified with a preamplifier (Mini-circuits ZX60-6013E-S +6000 MHz, +10 dB gain). The signal was then mixed with a 13.0 GHz phase-locked dielectric resonator oscillator (PLDRO), which was held in a phase-locked loop with a 100 MHz quartz oscillator plate assembly (Wenzel Associates 501-10137B) driven by a 10 MHz rubidium frequency standard (Stanford Research Systems FS725). The upper and lower sidebands were then passed through a 13 GHz cavity notch filter (Lorch 6BR6 13s000/100-S) and a manual step attenuator (Weinschel AF117A69-11) before amplification via a 200 W traveling wave tube amplifier (Amplifier Research 200T8G18A). The microwave signal was then broadcast into the molecular interaction chamber by a gain enhanced microwave horn antenna (Amplifier Research AT4004). The sample molecules were introduced into the chamber, which was evacuated to 1 106 Torr, through a 2 mm orifice pulsed valve (General Valve Series 9) with a backing pressure of 2 psi. The sample molecules were mixed with a helium/ neon (30/70%) buffer gas and cooled in the supersonic expansion to approximately 5 K. The molecular free induction decay (FID) was detected with a receiving horn antenna. The signal was then transmitted through a PIN diode limiter (Advanced Control Components ACLS 4619F-C36-1K) and a reflective single pole single throw switch (Advanced Technical Materials S1517D). After mixing down with an 18.9 GHz PLDRO (Microwave Dynamics PLO-2000-18.90), a 4 μs FID was digitized at a rate of 40 GS/s by a 12 GHz oscilloscope (Tektronix TDS6124C). A typical 2D CP-FTMW spectroscopy experiment utilized two polarizing microwave pulses separated by a variable time delay designated t1. Pulses would excite single or multiple rotational transitions; in the latter case, the AWG was used to simultaneously generate multiple pulses. This was achieved by coadding two or more chirp-pulse envelopes with different frequency sweeps over the same duration of time. The t1 delay between the pulses was automatically incremented in 2048 or 4096 steps with a dwell time of 1 ns or 500 ps, respectively, using National Instrument’s LabVIEW software. At each measurement step, the FID was signal averaged and recorded. The time domain data of all t1 steps were processed offline with a digital Kaiser-Bessel filter and Fourier transformed, yielding spectra with 125 kHz resolution in the ω2 domain. Modulations in the magnitude intensity of the irradiated rotational transitions with respect to t1 were then filtered with a Kaiser-Bessel function and Fourier transformed once more, revealing coherence peaks in the ω1 magnitude spectrum with approximately 400 kHz resolution. All experiments were processed by this method unless stated otherwise. The resolution of the ω1 and ω2 transition frequencies reflect the sampling time in each dimension (and not the highest resolution achievable with the CP-FTMW spectrometer). 8896
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The Journal of Physical Chemistry A Coherence transfer pathways in 2D experiments are dependent on the pulse angles of the excitation radiation. In the Bloch vector model, a π/2 pulse maximizes the coherent superposition between two dipole-allowed states, whereas a π pulse induces a population inversion. Experimentally, pulse excitation angles are a function of the integrated electric field strength and the rotational transition moment. The π/2 pulse for a specific rotational transition was determined by varying the pulse length and the manual step attenuator to maximize the signal intensity in the ground state rotational spectrum. It was found that, at full power (i.e., zero attenuation), pulses with durations from 100 to 125 ns and bandwidths from 5 to 25 MHz satisfactorily approximated the π/2 pulse for molecules with dipole moments on the order of a few Debye. The π/2 pulse was then fine-tuned with the manual step attenuator. Doubling the π/2 pulse length at a fixed attenuation then provided an approximate π pulse. Further adjustment of the pulse duration to minimize the spectral intensity gave the most precise π pulse for a given transition. In practice, the pulse angle parameters were often not optimal. This particularly included cases where the transition moments differed between dipole-allowed transitions or when chirpedpulses were coadded to excite multiple transitions simultaneously, therefore, diluting the total power over multiple frequency ranges. However, because the π/2 and π pulses were well-separated within the parameter space, it was not necessary to correct for these small deviations from optimal conditions. Approximate pulse angles affect the intensities in the ω1 dimension, but as will be demonstrated, these measurements are fairly robust and coherence information is resolved under the experimental conditions. Finally, it should be noted that after mixing the excitation pulse generated by the AWG with the 13.0 GHz PLDRO, only one of the sidebands was resonant with a molecular transition in the systems investigated in this study. For more dense spectra, it would be advisable to upconvert the signal with a frequency multiplier rather than mixing with a PLDRO if the sidebands unintentionally excite unwanted coherences. The details of each pulse are given in the text and figures when necessary. A full description of each pulse sequence is available in the Supporting Information
’ RESULTS AND DISCUSSION General Description and Modeling. We first consider the single-quantum correlation (SQC) of a progressive three-level system to explain the origin of signals in a 2D experiment (Figure 1). Two π/2 pulses separated by a delay (t1) are used to decode the molecular coherences induced by the light fields along two frequency axes: ω1 and ω2. The first polarizing microwave pulse prepares the Ea r Eb superposition, σab, out of thermal equilibrium. Population cycles between the coupled states at the angular Rabi frequency for the duration of the pulse. Though not directly excited, changes in the number density of level Eb necessarily create a population difference, Nbc, between the Eb and Ec levels. Indirect population cycling does not contribute to the final signal in one-photon experiments, but in multiple dimensions it will be demonstrated that indirect population cycling of Nbc can give rise to coherence signal in ω2. When the light field is turned off, the σab superposition evolves coherently with a frequency of ωab and intensity proportional to the pulse angle and the population difference at thermal equilibrium (N0ab). Following the t1 free precession period, a second π/2 pulse simultaneously irradiates both Ea r Eb and Eb r Ec transitions, and the FID is measured in t2. The σab and σbc t2 coherence
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Figure 1. Schematic of the single-quantum correlation pulses sequence of a three-level progressive system. The initial π/2 pulse prepares the σab coherence out of thermal equilibrium. Following a variable time delay, t1, a second nonselective pulse irradiates both the Ea r Eb and Eb r Ec transitions, mixing the σab and σbc coherences. A free induction decay is then measured with respect to t2 and Fourier transformed into the ω2 spectrum. Stepping the t1 time delay changes the magnitude of the freely precessing σab coherence, thereby modulating the intensities of the σab and σbc coherences in the ω2 spectrum. Fourier transform with respect to t1 reveals coherence peaks sharing a common energy level that are coupled by the mixing pulse.
intensities are not only a function of the population differences induced by the first pulse but are also parametrically coupled to the magnitude of the σab coherence at the end of t1. Stepping the time delay between the pulses changes the instantaneous magnitude of the σab coherence which oscillates with a frequency of ωab. Diagonal and off-diagonal peaks in a ω1 by ω2 2D plot originate from this mechanism. The σab coherence intensity in ω2 parametrically oscillates at a frequency of ωab with respect to t1, appearing as a diagonal peak in a 2D plot. No energy level connectivity is obtained as this is a result of irradiating the same transition multiple times. The σbc coherence intensity in ω2 also oscillates at a frequency ωab with respect to t1 because the two coherences are coupled (mixed) by the second pulse. This oscillation appears as an off-diagonal cross peak in the 2D plot and directly reflects the connectivity between energy levels. The SQC sequence, which has been demonstrated experimentally,19 is relatively simple in terms of the coherence transfer pathway, but the general concepts are applicable to more complex pulse sequences. The quantitative multiphoton response of a statistical ensemble of molecules can be described by the time evolution of the density matrix : ipFðtÞ ¼ ½HðtÞ, FðtÞ ð1Þ where the Hamiltonian is given by μ 3 EBðtÞ HðtÞ ¼ H0 ~
ð2Þ
The eigenvalues of the diagonal field-free rotational Hamiltonian, H0, are coupled by the off-diagonal projection of the electric field onto the transition moment of the molecule. Constructing the Hamiltonian matrix for a progressive (ladder-type) or regressive (v- or w-type) energy level structure is a straightforward application of one-photon selection rules. The most general Hamiltonian describes a combination of progressive and regressive energy level configurations. Following the derivation of Vogelsanger and Bauder,19 eq 1 was transformed into a rotating frame. When irradiated by a classical sinusoidal field, the density matrix elements in this interaction picture oscillate as a function of the difference between 8897
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The Journal of Physical Chemistry A
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Figure 2. Real portion of the coherence oscillation in the interaction picture subject to an arbitrary linear frequency sweep covering 10 MHz. The resonant frequency was set to zero. As the field passes through the center frequency, the coherence terms oscillate slowly, corresponding to an adiabatic following of the Bloch vector with the field.
the resonant and excitation frequencies. In the rotating wave approximation (RWA), counter-rotating terms are dropped, leaving the near resonant, slowly oscillating functions that are principally responsible for describing coherence information of the system. The complex, time-independent amplitudes of the coherence terms include contributions from the phase of the excitation radiation. As a result, phase cycling schemes can be designed to effectively select coherence transfer pathways and eliminate unwanted signals. In our lab, the solutions to these equations were solved numerically according to XðtÞ ¼ Pdiagðeλt ÞP1 Xðt0 Þ
ð3Þ
where P and λ are the eigenvectors and eigenvalues, respectively, of the coefficient matrix of the coupled linear differential equations and X is the tetradic column vector representation of the density matrix. With the CP-FTMW spectrometer, coherences in the rotating frame are more complex owing to the time-dependent phase, ϕ(t), of the chirp waveform: xðtÞ ¼ sin½2πðf0 t þ ϕðtÞÞ
ð4aÞ
Defining ϕðtÞ ¼
ðf1 f0 Þ 2 t 2ttotal
ðf1 f0 Þ xðtÞ ¼ sin 2π f0 þ t t 2ttotal
ð4bÞ
where f0 and f1 are the initial and final frequencies of the chirp and ttotal is the pulse length. The dynamics of the Bloch vector subject to the linear frequency sweep of eq 4 are best analyzed in the interaction picture. Figure 2 shows the real portion of the coherence response as the field sweeps through resonance. When the light field is far from resonance, rapid oscillations integrate to zero and can be neglected in accordance with the RWA. As the incident frequency approaches the transition (center) frequency, the Bloch vector adiabatically follows the field through resonance as reflected by the slow variation in coherence amplitude. The interaction between the density matrix and Hamiltonian is then approximately sinusoidal near resonance leading to a simplification in modeling coherence transfer: the incident field is treated sinusoidally and the time-dependence is solved with eq 3. Given that the T(2) dephasing time is much longer than the duration of a typical pulse, irradiating transitions over different periods of the frequency sweep does not affect the frequency information of coherence transfer. Intensities of low J transitions, for this reason, are only approximately modeled with thermally
equilibrated population differences (given optimal experimental conditions). Nonequilibrium conditions induced at the start of the frequency sweep produce small changes in coherence intensity. At higher J transitions, intensities deviate more profoundly, as we have neglected to include the M-degeneracy contribution to intensity into the model, although the general trends are predicted. In multiphoton sequences, nonzero initial conditions of the density matrix coherence elements after subsequent pulses give rise to off-diagonal peaks in a 2D plot. The final density matrix is tailored using multiple pulses associated with different Rabi angles. The angular Rabi frequency, R = εμgp1, is a function of the transition moment, μg, (g = ac in the principal axis frame) and integrated electric field envelope, ε, and represents the tipping angle of the Bloch vector off of the ground state axis in the pseudospin analogy. The power under the time integrated chirped field envelope decreases as a function of one over the square root of the bandwidth, increasing the tipping angle range and efficiency with respect to transform-limited excitation. These concepts will be demonstrated experimentally in the following sections with three- and four-level systems. Regressive Narrowband 2D Autocorrelation. The microwave spectra of molecules containing atoms with nonvanishing quadrupole moments exhibit hyperfine splitting from coupling of the nuclear field gradient with the rotational angular momentum.26 Selection rules for hyperfine multiplets permit transitions from a single lower (upper) state to multiple upper (lower) levels. These regressively connected systems have been previously studied with a 2D autocorrelation technique that probes connections between close-lying transitions (
