IIIBV, AIIBVI, and potential alternatives to formerly mentioned—“third wave” superlattices (SL) and two-dimensional (2D) materials infrared (IR) APDs. In the beginning, the APDs fundamental operating principle is demonstrated together with progress in architecture. It is shown that the APDs evolution has moved the device’s performance towards higher bandwidths, lower noise, and higher gain-bandwidth products. The material properties to reach both high gain and low excess noise for devices operating in different wavelength ranges were also considered showing the future progress and the research direction. More attention was paid to advances in AIIIBV APDs, such as AlInAsSb, which may be used in future optical communications, type-II superlattice , and 2D materials-based IR APDs. The latter—atomically thin 2D materials exhibit huge potential in APDs and could be considered as an alternative material to the well-known, sophisticated, and developed AIIIBV APD technologies to include single-photon detection mode. That is related to the fact that conventional bulk materials APDs’ performance is restricted by reasonably high dark currents. One approach to resolve that problem seems to be implementing low-dimensional materials and structures as the APDs’ active regions. The Schottky barrier and atomic level thicknesses lead to the 2D APD dark current significant suppression. What is more, APDs can operate within visible (VIS), near-infrared (NIR)/mid-wavelength infrared range (MWIR), with a responsivity ~80 A/W, external quantum efficiency ~24.8%, gain ~105 for MWIR . It is believed that the 2D APD could prove themselves to be an alternative providing a viable method for device fabrication with simultaneous high-performance—sensitivity and low excess noise.Avalanche photodiodes (APDs) have drawn huge interest in recent years and have been extensively used in a range of fields including the most important one—optical communication systems due to their time responses and high sensitivities. This article shows the evolution and the recent development of A That output signal suppression requires an extra amplifier along with a system to correctly detect the signal at the detector stage. The devices based on avalanche photodiodes (APDs) exhibiting high bandwidth (BW) and gain (M)—high gain-bandwidth product (GBW) and low excess noise [F(M)] at the same time are well matched to detect suppressed optical signals, e.g., in the long-distance applications such as free-space optical communications (FSO), night vision, light detection, and ranging (LIDAR/LADAR), time of flight (ToF), intelligent robotic and finally in battlefield conditions (military applications). Therefore, improvement in GBW and F(M) reduction has been a key goal for APD’s progress. The methods to suppress the F(M) may be divided into three tactics. The initial approach could be to choose a material  with advantageous multiplication properties. Next, the F(M) may be substantially limited by the reduction of the avalanche layer to use the non-local effect of the multiplication phenomena. The final method may be widely categorized as impact ionization engineering (I2E) exploiting properly constructed heterojunctions.The avalanche multiplication effect can be used to detect low-power optical signals and even single-photons due to the amplification mechanism within all main: near- (NIR), short- (SWIR), mid- (MWIR), and long wavelength infrared radiation (LWIR) ranges. An advanced laser radar and weapons systems implemented in long-range army and space applications must detect, recognize and track various targets under a diversity of atmospheric conditions including absorption by CO, CO2. In the field of fiber optic communication (FOC), InGaAs ternary alloy is much more expensive in terms of fabrication than Ge, but provides lower noise and higher time response. The Ge APDs are advised for detection systems where noise generated by the amplifier is high. The development of device technology with active regions built of the narrow gap semiconductors, such as HgCdTe and T2SLs  has contributed to the development of the new passive/active detection applications and capabilities. In active imagery systems, a laser source is implemented to the observed region, and reflected radiation is temporally examined. The output signal may be amplified in the APD itself, before going to the readout integrated circuit (ROIC). In addition, the grouping of dual-band capability and multiplication gain is another technology enabling dual-band detection for a wide temperature selection.The APD materials’ selection is conditioned by the potential applications to include the most common: high-speed receivers, single-photon counters, and laser range finders3. The SPAD is more sensitive than a photomultiplier, however, when the avalanche process is initiated at infinite gain, additional photons detected during the pulse and circuit regeneration are discounted which makes SPAD more like a Geiger counter than a photomultiplier. SPADs build a variety of approaches to reach single-photon detection (SPD) mode and compete with superconducting nanowire single-photon detectors (SNSPDs). The main reason for this tendency is unquestionably the move to optical quantum information applications—quantum key distribution (QKD) putting severe requirements on detector parameters that move away from the performance of the well-developed typical APDs. Effective single-photon numeration, with a single-photon detection efficiency (SPDE) > 50%, was reached just for wavelengths shorter than <2 μm4. SNSPDs exhibit outstanding performance on a wide wavelength range, but their applications is restricted by cryogenic cooling requirements. In contrast, SPADs circumvent the SNSPDs’ fundamental restrictions by offering a reasonable option at ≤300 K mainly by AIIIBV material leader—InGaAs. Reaching the high performance in the MWIR exhibits potential in astronomy, LIDAR, dark matter research applications, and examination of chemistry and molecular dynamics, to include many absorption fingerprints for molecules: H2O, CO2, O2, O3, CH4, and N2O35. Figure 8.The APD could operate under the conditions where applied bias is higher than the infinite gain voltage meaning that the appearance of the single-photon causes an avalanche breakdown producing a high signal marking the presence of another photon (passive or active imaging). This mode of operation is referred to as a counting or single-photon avalanche detector (SPAD)—called a Geiger mode avalanche detector by Cova et al. pioneering paperM and low F(M), to detect low radiation flux from far located stars. The avalanche ionization in LWIR can be more simply reached in comparison with SWIR and MWIR devices. Even though a higher M may be reached under a given bias, the large dark current is an issue for LWIR APDs significantly impacting the device performance. Derelle et al. presented the n+/n−/p planar APD deposited by molecular beam epitaxy (MBE), exhibiting M = 16 at −2.7 V and cut-off wavelength, λc ~ 9 μm at 80 K. Authors showed that the F(M) assessment in HgCdTe LWIR APDs is restricted to low M [F(M) = 1–1.25 at M = 6] caused by tunneling currents influence9. Since the photocurrent to dark current ratio is low at high reverse voltage in LWIR it is difficult to estimate F(M) with high M11.The focal plane array (FPA) intended to operate in LWIR is advantageous because the number of photons in the 8–12 μm atmospheric transparency window is significant for reaching high detectivity and response time. In addition, astronomy applications need FPAs exhibiting high third wave” group—two-dimensional (2D) layered materials and van der Waals (vdW) heterostructures can be also used to fabricate avalanche multiplication to include single-photon-counting technologies. Recently, the spectacular growth in the quantity of research papers related to the promising 2D photodetectors has been observed, however, those materials exhibit low absorption caused by their thin atomistical nature. To use those unique 2D materials properties for device design, the considerable latest attempts have been directed at combining with photonic structures (dielectric waveguides), plasmonic structures, or photonic crystals. That combination with photonic structures allowed to demonstrate single graphene layer with high absorption, modulators, detectors, and lasers12. Impact ionization leading to the carrier avalanche is a favorable approach to fabricating 2D photodetectors exhibiting high detection efficiency.The member of the “14. Those materials are characterized by different impact ionization coefficients versus carrier transport direction. The electric field needed for avalanche multiplication in out-of-plane transport is hundreds of kV/cm, while for in-plane close to tens of kV/cm is confirmed by measured results15. The 2D layered gapless graphene can detect radiation from ultraviolet (UV) to microwave making it an alternative for numerous photodetector designs operating in wide spectral ranges, however, its zero-bandgap characteristics limit the fabrication of photodetectors with high detectivity. Alternatively, 2D transition metal dichalcogenides (TMDs), thickness-dependent energy bandgap MoS2 and WSe2, exhibit promising photodetection capabilities mainly in the visible (VIS) to NIR ranges to include impact ionization effect. In comparison to the graphene and TMDs, 2D black phosphorus (BP) exhibiting a direct energy bandgap within the range from 0.3 eV (bulk) to 2 eV (monolayer) proved to be a proper material candidate for APD technology8. The multiplication was also observed in 2D InSe for the VIS range. In addition, not only the conventional impact ionization effect but also the ballistic avalanche mechanism was observed in the 2D materials family. The effectiveness of the multiplication mechanism varies versus the material’s intrinsic capabilities. Consequently, the research of innovative materials characterized by the low electric field for avalanche multiplication is significant in reaching energy-effective devices. The avalanche multiplication mechanism in conventional materials is restricted by high bias which could be circumvented by 2D materials-based APDs15.In comparison to standard bulk, the 2D materials exhibit numerous exceptional capabilities, such as mechanical flexibility, strong light-matter coupling, self-passivated surfaces, and gate-tunable Fermi-level providing flexibility in heterostructure designIIIBVbased material systems including well-known “third wave” material family member—superlattices (SLs). In addition, the current progress in the new materials and architectures for high-performance IR APDs is presented to include innovative “third wave” 2D materials. In addition, the strategies to reach high-performance APDs are presented. The field related to the APD advances in telecommunications is generally omitted due to the excellent review papers published recently17.Taking the above into consideration this paper shows the current status and future development of IR-based APDs. It encompasses both bulk HgCdTe and AF(M) deterioration limiting GBW. This is because both carriers (electron and hole) may be multiplicated. The carriers are photogenerated in the low electric field active layer and, ideally, only the carriers exhibiting the utmost impact ionization probability are moved toward the high-electric field multiplication area.Figure αe (electrons) and αh (holes) ionization coefficients. Those parameters describe the multiplication probability per unit length meaning that ~1/αe and ~1/αh represent an average distance carrier moves before impact ionization occurs ;excess noise factor , operating temperature (T), manufacturability, and limitations with technology readiness level (TRL)34.The current bulk and type-II superlattice T2SLs) materials suitable for high-performance APDs’ fabrication and their spectral ranges are gathered in Table SLs mater36. The InAlAs or InP submicron multiplication areas with InGaAs absorption layers (InGaAs is reported to be lattice matched to InAlAs and InP) could be used to reach lower F(M) because of “dead space” effect. The InAlAs k = αh/αe is reported to be much higher than the InP k reached for low E. The InAlAs F(M) is much lower than in InP at a given gain due to the high InAlAs αh/αe ratio and the favorable InP “dead space” effect. Moreover, light with a wavelength, λ > 1.4 μm called “eye-safe”, goes to the eye anterior potions eye (primarily the cornea) consequently not reaching the retina. Since Si does not absorb beyond > 1 μm, AIIIBV semiconductors offer the potential for longer wavelengths of LIDARs.The semiconductor’s selection for APDs fabrication is conditioned by applications where the most common are fiber optic communications, high-speed receivers, single-photon counters, and laser range finders. Even though the IV-group semiconductor materials such as Si and Ge exhibit superior performance among APDs, Si and Ga-based APDs cannot operate in a 1.55 μm optical communication band due to their cut-off wavelength limitations. For this reason, the research efforts have been directed at InGaAs/InP APDs. Much current research on the APDs has been focused on the development of the new architecture and the materials substitutions/alternatives to lower dark current, to reach higher speed and lower excess noise maintaining optimal gain levels at the same time. Recently GeSn APDs have been introduced to circumvent the longer cut-off wavelength limitationsGBW are required. Time response and GBW are mainly restricted by the profile of the heterojunction between the active and avalanche regions and the doping distribution within the detector. Attempts to reach improvement in avalanche gain for InGaAs by rising the electric field are not feasible which is related to the tunneling effects resulting in high leakage currents. The low value of the electron effective mass causes a sharp increase in tunneling current for electric fields, E > 150 kV/cm38. That drawback was circumvented by combining an InGaAs absorber layer operating with low E and a lattice-matched InP multiplication layer with a wider bandgap responsible for impact ionization. That architecture is an example of the previously mentioned SAM-APD design (isolated absorption and multiplication layers). The InGaAs/InP SAM-APD device structure, with a double-diffused floating guard ring, is presented in Fig. F(M)] are transported to InP heterojunction, where the impact ionization occurs. That design allows for low surface current caused by the junction being located in the wide energy gap InP providing responsivity in the longer wavelength range by the low energy gap InGaAs active layer.For high-speed telecommunication receivers, the APDs exhibiting short response time and high 16 cm−3. The p+-layer is thin and doped up to the level of 1017–1018 cm−3. The junction is typically produced by Zn p+-type diffusion into the InP avalanche region and Cd diffusion (or implantation) for the guard ring into the top InP layer through the SiO2 mask.InGaAs/InP heterojunction APDs are usually built of the 1–2 μm thick undoped active layer. The 0.1–0.3 μm thick InGaAsP grading and 1–2 μm thick multiplication regions are doped up to the level 1 × 10It must be stressed that the valence band offset at the InGaAs/InP junction accumulates holes in the valence band which deteriorates the device’s response time. That valence band discontinuity is reduced by grading the bandgap of the quaternary InGaAsP layer grown between the InP and InGaAs regions. That improved architecture is referred to as the separate, absorption, graded, multiplication avalanche photodiode (SAGM).xInx1–AsySby1– separate absorption, charge, and multiplication (SACM) APD is presented in Fig. 0.7In0.3As0.3Sb0.7  blocking region, AlxInx1–AsySby1– within the region x = 0.4–0.7 grading region, p−-type Al0.4In0.6As0.4Sb0.4 1000 nm thick active region, 150 nm thick, p+-type Al0.7In0.3As0.3Sb0.7 charge region (1.25 × 1017 cm−3), 1000 nm thick p−-type Al0.4In0.6As0.4Sb0.4 multiplication region, and finally n-type GaSb contact region [(1–9) × 1017 cm−3]. Lastly, N2/Cl2 inductive coupled plasma (ICP) and typical photolithography with bromine methanol and SU-8 treatment to reduce leakage current were used to define circular mesas.The AlE within the multiplication layer enables the avalanche effect, and photogenerated electrons drift is realized by a small electric field suppressed by the charge region in the absorber layer. Figure xInx1–AsySby1– SACM APD performance. The dark current at 95% breakdown voltage assumes ~120 nA, being roughly ~100× lower than the current for APDs based on Ge on Si and like AlInAs/InGaAs APDs39. The experimental M ~ 50 values were confirmed by the Monte Carlo simulations.Under the strong reverse bias, the high 40. Their high potential is conditioned by the low production expenses related to the easily available AIIIBV fabrication foundries and the relatively low price of the 6” native substrates, as well as the operation using thermoelectric cooling. The historical problem with the surface leakage of InAs photodiodes is gradually reduced by elaborating wet chemical etching recipes like the solutions of phosphoric and sulfuric acid-based etchants41.In the last decade, a new breakthrough in the development of InAs APDs has been reachedM normalized dark current density (JDark) at the level of ~5 × 10−6 A/cm2 for LN2 temperature (77 K) has been published42. The onset of BTB tunneling at moderately low E requires a thicker multiplication region to achieve high multiplication gain exacerbating passivation difficulties. To resolve that issue, planar structures as shown in Fig. 40. To form a p–n junction, Beryllium (Be) implantation at low energy was used.Both mesa and planar InAs APDs were fabricated. For mesa p-i-n structures, the Table 44. These include the introduction of the continuous or grading bandgap for absorption/avalanche layers to limit carrier trapping and insertion of the electric-field control layers. Advanced APDs structures require the multiplication region thickness to be shrinked to reach fast response times. The local McIntyre theory does not properly justify the excess noise characteristics for the devices with thin multiplication regions. The InP APD with the 0.25 μm-thick multiplication region reaches the excess noise performance scaling with 1/k ~ 0.25 for the hole-initiated avalanche process (h-APD), however, according to the McIntyre model, the multiplication factor is reported at the level of ~0.7. The significant reduction in excess noise can be reached by the “dead space” effect in the thin multiplication layers. Additional improvements in the detector’s excess noise may be reached by the implementation of the InAlAs/InAlAsSb (instead of InP) being lattice-matched to InGaAs and InP as the multiplying layer. The InAlAs/InAlAsSb αh/αe was estimated to be much higher than the InP αh/αe ratio at low E. The InAlAs F(M) at a given M is much smaller than in InP being related to the high InAlAs αh/αe ratio and the favorable “dead space” effect in InP.Progress in materials’ properties and advanced detector structures have increased the APDs performance for fiber optic communication systems over the past decadeF(M) versus M for selected material systems. The solid lines present the F(M) for k = 0–1 values simulated by the local field theory24. In general, F(M) should increase versus k. Typical excess noises are shown by shaded regions37. The k = αh/αe values for the best commercially available Si APDs stay within the range 0.01–0.06. InP and InAlAs commonly implemented as avalanche regions of the APDs for telecommunication applications assume higher k-values:k = 0.4–0.5;InP within the range k = 0.2–0.344.InAlAs within the range Figure IIIBV bulk compound semiconductors, AlxInx1–AsySby1– to GaSb and AlxGax1–AsySby1– lattice-matched to InP were reported to reach excess noise comparable to Si17. The AlxGax1–AsySby1– APDs within the range of Al chemical composition, x = 0.5–0.7 exhibit k-values at the level of 0.01. This behavior is explained by the significant domination of electron impact multiplication in comparison to the holes, which is related to the Sb contribution/content. It is suggested that Sb-content increases photon scattering rates and increases effective hole mass causing a significant suppression of hole ionization coefficient, αh. Both AlxGax1–AsySby1– and AlxInx1–AsySby1– quaternary compounds are considered to have a potential for ≥2 μm wavelength optical communication band. Figure F(M) versus M for InAs APDs compared with data for selected materials: HgCdTe and InAlAs. The F(M) values estimated for InAs p-i-n avalanche photodiodes do not follow McIntyre theory falling below the local field model assessed for k = 0 being similar to the reported for SWIR HgCdTe and slightly higher than published for MWIR HgCdTe electron-initiated, e-APDs. This F(M) dependence on gain falling under the lower limit of the local field theory is related to “dead space” effect. As marked in the section “Background”, if the avalanche layer is thick, “dead space” may be ignored, and McIntyre local field theory correctly justifies the APD’s performance. The InAlAs APD F(M) dependence on M is comparable to the standard APDs where both electrons and holes undergo multiplication.Lately, two quaternary AIIIBV materials system, enabling broader applications that were previously only possible with the less accessible HgCdTe system. The InAs APDs properties make them an attractive approach for a wide range of NIR and MWIR purposes, including active/passive imaging, LIDAR, and remote gas sensing.The progress in InAs electron-initiated, e-APDs’ fabrication allowed the ideal properties of avalanche multiplication and excess noise to transfer to the readily available Ax1–CdxTe, x Cd chemical compositions where k = αh/αe is either much higher k » 1 or much lower than k «1 what was presented in Fig. x1–CdxTe capability dependence on the bandgap energy46. For a cut-off wavelength shorter than about λc < 1.9 μm (x = 0.65 for 300 K), authors estimated αe « αh due to the resonant enhancement of the hole multiplication coefficient when bandgap energy corresponds to the Eg ≅ ESO = 0.938 eV  and their heterojunctions /p-type Si (Eg = 1.1 eV)) exhibit promising avalanche performance in VIS to NIR ranges8.Different types of 2D photodetectors with the flexibility in forming heterostructures have already been widely studied with the advantages of weak vdW interactions. The most popular are photoconductive, photovoltaic, phototransistor (hybrid detectors), and photothermoelectric2 layer79. At a bias voltage above 12 V, the E in InSe is large enough to speed up photogenerated electrons and generate electron-hole pairs by carrier multiplication. Further increase of the voltage (>50 V) causes the metal/semiconductor junction breakdowns leading to the dramatic rise of both photocurrent and dark current and lowering signal-to-noise ratio. Also, Atalla et al. have observed increasing in photocurrent versus bias voltage in the Ti/BP Schottky barrier due to the avalanche effect80. Comparable results were reported by Gao et al. on the avalanche effect in the graphite/InSe Schottky detector  related to the multiplication nature has been constant because the excess noise restricts the detector’s sensitivity, detectivity and reduces the operating BW. The solution is a higher and fully controlled—deterministic impact ionization mechanism which can be achieved either by the proper multiplication material selection, by device design , or by material engineering. It was demonstrated that the non-local effect of impact multiplication allows to limit of the noise in many materials covering the wide radiation range. In addition, the “third wave” materials and related technologies have opened the prospect of I2E to design and fabricate heterojunctions to further suppress noise and reach higher GBW.The detectors for optical telecommunication applications and quantum information technologies have mainly pushed the APDs to progress with high GBW being strictly related to the F(M). The conditions and variables allowing to reduce noise contribute also to high GBW. Therefore, the GBW increase and F(M) suppression have been an effort for the progress and investigation. The following methods to improve APDs performance must be implemented:M, F(M), and GBW products. The local-field multiplication model explains that the APDs F(M) and GBW are conditioned by the material’s carriers multiplication coefficients in the avalanche layer. Higher detection parameters are reached if one of the multiplication coefficients is substantially higher than the other, i.e., the k = αh/αe differs significantly from unity. Attempts to increase APD’s parameters have moved to electric field profile optimization and research on the new compounds to include bulk AIIIBV, AIIBVI, “third wave” materials and technologies—T2SLs InAs/GaSb, “Ga free”—InAs/InAsSb and 2D materials. Flexibility in bandgap energy tuning and the energy band profile optimization in 2D materials makes the impact ionization be monitored by the number of layers modification. It is feasible to adjust the k = αh/αe level by varying the number of 2D materials constituting layers. For bulk and “third wave” avalanche layers, the minimal F(M) has been reached with materials such as Si, HgCdTe, InAs, AlxGax1–AsySby1–, AlxInx1–AsySby1–, T2SLs InAs/GaSb, and MoS2 exhibiting k « 1);1) choosing a material with advantageous carrier multiplication coefficients. The APD’s avalanche layer contributes to the F(M)]—proved for many bulk materials used for avalanche regions: InP, GaAs, Inx1–AlxAs, Si, AlxGax1–As, SiC, GaP, GaInP). As 2D materials-based APDs are inherently reduced to the submicrometer level, the absorbers based on those materials are under large lateral electric fields leading to the breakdown;2) thickness reduction  of the avalanche layer to utilize the non-local nature of the multiplication process .Further optimization of the APD performance is possible allowing to design and fabrication of devices with supreme parameters over conventional avalanche devices. For instance, by choosing 2D materials with promising band alignments and structures, it is feasible to implement appropriate Schottky junctions to suppress the dark currents and widen operating wavelengths. Moreover, improvement in processing allows us to reduce the response time and current noise. The 2D APD has been reported to be operating within VIS, NIR and MWIR ranges with a third wave” APDs. The progress in the development of the APD operating in the IR range was presented covering materials based on HgCdTe as well as AIIIBV alloys including “Ga-based” and “Ga-free” T2SLs. The non-local characteristic approach and technological achievements have opened up the option of multiplication engineering incorporating different materials and heterojunctions to reach better performance: suppressed noise with higher GBW in broader spectral regions. It is believed that the 2D/vdW APD could prove itself to be an alternative to the bulk multiplication devices providing a possible method for developing devices exhibiting high sensitivity and low excess noise.That paper has reviewed the multiplication effect generated by the avalanche process and sketchily reviewed the latest research on bulk and “


Mendelian randomization (MR) is an instrumental variable approach used to infer causal relationships between exposures and outcomes and can apply to summary data from genome-wide association studies (GWAS). Since GWAS summary statistics are subject to estimation errors, most existing MR approaches suffer from measurement error bias, whose scale and direction are influenced by weak instrumental variables and GWAS sample overlap, respectively. We introduce MRBEE (MR using Bias-corrected Estimating Equation), a novel multivariable MR method capable of simultaneously removing measurement error bias and identifying horizontal pleiotropy. In simulations, we showed that MRBEE is capable of effectively removing measurement error bias in the presence of weak instrumental variables and sample overlap. In two independent real data analyses, we discovered that the causal effect of BMI on coronary artery disease risk is entirely mediated by blood pressure, and that existing MR methods may underestimate the causal effect of cannabis use disorder on schizophrenia risk compared to MRBEE. MRBEE possesses significant potential for advancing genetic research by providing a valuable tool to study causality between multiple risk factors and disease outcomes, particularly as a large number of GWAS summary statistics become publicly available. OriginaFrom a statistical standpoint, both UHP and CHP in an MR model exhibit characteristics similar to outliers in traditional regression analysis, and hence can be addressed by applying robust tools. In the literature, MR pleiotropy residual sum and outlier (MRPRESSO)  and iterWhile the aforementioned single-exposure MR methods allow for some IVs to exhibit horizontally pleiotropic effects, they typically assume that the overwhelming majority of IVs influence the outcome solely through the mediation of the exposure. However, considerable evidence suggests that common human traits share a significant amount of causal variants, such as systolic blood pressure (SBP) and diastolic blood pressure (DBP) , making However, multivariable MR is often subject to substantial weak instrument bias because the instruments only need to be associated with one exposure in a set for them to be considered to satisfy assumption (i) in practice. In other words, the set of IVs used in multivariable MR is the union set of exposure-specific IV sets used in univariable MR. As GWAS sample sizes become larger, increasing numbers of causal variants with moderate or small effects are being identified, making weak instrument bias – the violation of assumption (i) – more significant and difficult to disregard. For instance, Measurement error bias occurs when explanatory variables are measured with random error, which generally exists in all statistical models including linear and generalized linear regression models, and leads to biased estimates of model parameters . Since cIn this paper, we propose a multivariable MR method, MR using Bias-corrected Estimating Equations (MRBEE), to eliminate measurement error bias while simultaneously accounting for horizontal pleiotropy in the presence of many weak IVs. In contrast to existing methods that only address measurement error bias in specific cases such as no sample overlap ⊤ be a vector of m independent genetic variants where each variant is standardized with mean zero and variance one, xi = ⊤ be a vector of p exposures, and yi be an outcome. Consider the following linear structural model:B = ⊤ is an (m × p) matrix of genetic effects on exposures with βj = ⊤ being a vector of length p, θ = ⊤ is a vector of length p representing the causal effects of the p exposures on the outcome, γ = ⊤ is a vector of length m representing horizontal pleiotropy, which may violate the (IV2) or (IV3) conditions, and ui and vi are noise terms. Substituting for xi in (2), we obtain the reduced-form model:Let ui and vi are usually correlated, and hence traditional linear regression between xi and yi cannot obtain a consistent estimate of θ. In contrast, the genetic variant vector gi is generally independent of the noise terms ui and vi because the genotypes of individuals are randomly inherited from their parents and do not change during their lifetime  is the score function of (7) and SIVW(θIVW) = 0, and HIVW is the 2nd derivative matrix of (7). Since That is, ximately :(11)Eβj) can be regarded as the average information carried by each IV, while βj) is not substantially larger than Here, cov. This is the reason why in some empirical studies  ≠ 0, there is additional pleiotropy bias due to the horizontal pleiotropy that violates the InSIDE assumption. In univariable MR, it is challenging to guarantee γj = 0 or cov = 0 for all 1 ≤ j ≤ m, resulting in a potentially biased IVW estimate. Traditional solutions to horizontal pleiotropy bias require that only a small proportion of IVs exhibit horizontally pleiotropic effects, and robust tools or Gaussian mixture models can be employed to identify these IVs  = 0 after accounting for multiple exposures, which can be then detected and removed using the robust tools such as a pleiotropy hypothesis test.If cov sub-matrix of p − 1 elements of the last column of Here, FBEE is the Fisher information matrix, i.e., the expectation of the Hessian matrix of SBEE(θ), and the inner matrix VBEE(θ) is the covariance matrix of SBEE(θ). A consistent estimate of ΣBEE(θ) isThe covariance matrix of Where 2.4Spleio which tests the same null hypothesis for each SNP as MR-PRESSO  distribution under Hj0. The only assumption here is that Spleio for all candidate IVs and remove IVs with large Spleio values in an iterative manner, which is summarized in The statistic otically . In pracHj0 using Spleio becomes more powerful and more UHP/CHP IVs can be detected. Specifically, the variance of Spleio vanishes with a rate O(1/nmin) where nmin is the minimum sample size of exposures and outcome GWAS, while the effect size of γj under the alternative hypothesis is of O(nmin/m). Panel (A) in Spleio will be removed from causal estimation using our algorithm in practice.It should be pointed out that as GWAS sample sizes increase, the test of Spleio tests a very general null hypothesis, we can also calculate Spleio for all SNPs across the genome after estimating the causal effects of p exposures on the outcome used in MR. Results from these tests can be used to (i) find novel loci associated with the MR outcome and (ii) draw inferences about pathways of genetic association with the MR outcome. Specifically, when a SNP has a negative effect on the exposure βj and a positive pleiotropic effect on the outcome γj, and simultaneously the causal effect θ is positive, then the total effect of this variant on the outcome αj is canceled and hence cannot be detected in the outcome GWAS. In contrast, the pleiotropy test directly tests the effect γj and therefore is able to detect novel loci. For example, Spleio are displayed in Since 2.5m = 50,100, and 250 genetic variants G for 30k individuals from a binomial distribution with minor allele frequency (MAF) τ that followed a Uniform distribution. One true exposure x with variance 1 was generated. The effect sizes β of the m genotypes on the exposure followed a Uniform distribution and were scaled to explain 5% of exposure variation. Thus, increasing m was equivalent to introducing more weak IV bias. In the true MR model α = βθ + γU + γC, the term γU representing UHP was random noise and the term γC representing CHP was negatively correlated with β. UHP and CHP effects were either generated for 0% or 10% of IVs depending on the simulation scenario, and were scaled to match the patterns of horizontal pleiotropy that we observed in Real Data Analysis I  with variance ϵx was generated from a normal distribution x and y, we performed linear regression of x and y on each Gs separately to produce the respective GWAS estimate pairs https://github.com/noahlorinczcomi).For the univariable MR results presented in were IVW , MR-Eggewere IVW , dIVW  explained by the m = 50, 100, and 250 SNPs of 5% for each exposure. Effect sizes  of G on x = ⊤ were generated fromη1, η2) are scaling factors to ensure 5% heritability in  explained by the m SNPs. We then generated x asU) = (1−h2) × (0.15/2)2, var(ϵx1) = var(ϵx2) = 1−h2 −var(U), and h2 = 0.05. CHP in univariable MR methods is automatically introduced by generating two genetically correlated exposures. Additional UHP βs1θ1 + βs2θ2 using the same procedure described above in the univariable setting described above. We then simulated the outcome y asϵY) = 1 − var(x⊤θ + U). We then performed association testing of  and y for all SNPs and phenotypes separately using randomly drawn values for the quantities above and linear regression on Gs to produce the estimates For the multivariable MR results presented in 2.6We performed two real data analyses, the first of which is described here and the second in −8) with the exposure in a within-phenotype and between-ancestry fixed-effects meta-analysis of EAS and EUR GWAS, had the same sign in the EAS and EUR GWAS, and had at least P<0.05 in both GWAS. We then selected only independent SNPs from this set using ancestry-specific linkage disequilibrium (LD) reference panels from 1000 Genomes Phase 3 . Only ancestry-specific GWAS estimates were used in ancestry-specific MR. For multivariable MR, we filtered the full set of all IVs used in univariable MR to only independent SNPs that had linkage disequilibrium r2<0.01 in a 1Mb window using ancestry-specific LD reference panels from 1000 Genomes. This resulted in 3,097 IVs used in EAS and 2,821 in EUR. Results from alternative selections of the IVs are available in the We generally followed the methods of  Phase 3  and the  Phase 3 : r2<0.01Spleio with causal estimates from multivariable MRBEE. These tests were used for inferences of direct, exposure-mediated, novel, and pleiotropic genetic associations with CAD as described in For all available SNPs genome-wide, we performed horizontal pleiotropy testing using the statistic 2.7In Real Data Analysis II, we estimated direct causal effects of seven exposures on risk of schizophrenia (SCZ) and bipolar disorder  with GWAS data from European populations using multivariable MRBEE and existing alternatives.We estimated causal effects of the following risk factors: Cannabis use disorder (CUD), left handedness (LH), Attention-Deficit/Hyperactivity Disorder (ADHD), sleep duration, education, intelligence, and neuroticism (SESA). All GWAS data were from studies in strictly EUR individuals. Exposure GWAS sample sizes ranged from 55k for ADHD  to 1.7M −5 in multivariable MR analysis. We then restriced this set of IVs to only those with P<5×10−8 in a 7-degree of freedom chi-square joint test of association with any of the 7 exposures. This test accounting for sample overlap among the exposure GWAS. We then excluded 3 IVs whose minor allele frequencies differed by more than 0.10 from all other exposures. This resulted in 1,227 IVs that were used in multivariable MR which were standardized by their GWAS standard error.Because some exposure GWAS did not detect many genome-wide significant signals , we initially considered all independent SNPs with exposure GWAS P<5×10Spleio using all MR exposures with a causal effect P-value less than 0.05 for either SCZ or BP. Including nonsignificant exposures in genome-wide pleiotropy testing would have only increased the variance term used in Spleio and not otherwise affected the inferences we could make. We performed a sensitivity analysis in which non-significant MR exposures were included, the results of which are presented in Spleio was performed separately for SCZ and BP.We performed genome-wide horizontal pleiotropy testing with 33.1Univariable simulation results in Multivariable simulation results in 3.23.2.1−11) using IMRP but only 1.43 (P=0.140) using MR-Egger. Full univariable MR results are presented in the Univariable MR results suggested nonzero causal effects of all exposures on CAD in either EAS or EUR populations. However, there was widespread evidence of unbalanced horizontal pleiotropy as indicated by large differences in causal estimates between estimators that differ only in how UHP/CHP is addressed. For example, the odds ratio of causal effect of DBP on CAD in EAS was estimated to be 2.03 (P=2.8×10−100) and 1.76 in EUR (P<1×10−20), the latter of which was undetected in −100). The direct causal effect of SBP on CAD in EAS was similarly underestimated by MR-Median compared to MRBEE, where MRBEE produced an odds ratio estimate of 1.94 (P=1.3×10−5) and MR-Median 1.49 (P=1.3×10−15).−25) was completely mediated by SBP  and in EUR was 0.65 (P=0.058)  see . This su058) see , a simil3.2.2Spleio test to all SNPs genome-wide using causal estimates from MRBEE to search for SNPs with pleiotropic effects. The original CAD GWAS in EAS and EUR respectively identified 65 (λGC = 1.16) and 39 (λGC = 1.00) loci, defined as 1 megabase (Mb) windows with r2<0.01 between lead SNPs (P<5×10−8). Genome-wide horizontal pleiotropy testing with Spleio correspondingly identified 27 (λGC = 1.08) and 41 (λGC = 1.01) loci in EAS and EUR. In EUR, nine loci that were detected in horizontal pleiotropy testing were not detected in the original CAD GWAS, as http://www.nealelab.is/), all of which could only be detected in a recent larger CAD GWAS (We then applied the CAD GWAS . In EUR 3.33.3.1−34), which is of opposite sign for SCZ . It is consistent with Univariable MR results suggested nonzero total/unmediated causal effects of CUD, ADHD, left handedness, neuroticism, sleep duration, intelligence, and education on either BP or SCZ. We found a strong protective causal effect of left handedness on BP risk  and 2.71 (P=5.7×10−8), the latter of which is more consistent with the literature. That is, the odds ratio for association between CUD and schizophrenia is 3.90, 95% CI: 2.84–5.34 in Full multivariable MR results are presented in Spleio removed – to the bias we expected in the multivariable IVW estimator using −4) and r=0.97 for SCZ (P=2.5×10−4). Only 3 IVs (<0.25%) had significant Spleio values in MR, and they had no impact on causal estimates.As before, we compared differences between MRBEE and IVW* – the multivariable IVW estimator with pleiotropic IVs identified using Spleio but had P>5×10−8 in the original GWAS  coronary artery disease (CAD) in East Asian and European populations and (ii) schizophrenia and bipolar disorder. Causal risk factors were generally consistent for CAD between EAS and EUR and between SCZ and BP in EUR, where there was evidence that any differences between MRBEE estimates and those made by alternative methods were the results of uncontrolled bias in other methods. For example, the IVW causal estimate of LDL on CAD in EAS was expected to have 55.3% downward bias from Spleio that, when applied genome-wide, identified the pathways through which many genomic loci were associated with CAD, SCZ, and BP. Spleio testing revealed that many genetic associations with disease endpoints were non-direct, suggesting that a large portion of the heritability of these complex traits may be conferred indirectly through their causal risk factors. This test also identified 9 novel loci for CAD in EUR – seven of which were replicated in UKBB – 11 for SCZ and 3 for BP, for which no adequate independent replication data exists. This method of pleiotropy testing using Spleio is therefore a valuable tool both for gaining better insight into how genetic risk of disease is conferred and in detecting new risk loci.We finally introduced a multivariable horizontal pleiotropy test using the statistic MRBEE has the following limitations. As with all MR methods, the reliability of causal estimates produced by MRBEE depends on the quality of GWAS data used in MR. For example, biases in GWAS from assortative mating or dynastic effects may propagate through to MR and bias causal estimation . Second,In conclusion, univariable MR analysis is inherently limited in its ability to reduce bias, but univariable MR methods and their applications have so far dominated the literature compared to multivariable analyses. We developed multivariable MRBEE to reduce known biases in MR and estimate direct causal effects of multiple exposures in robust way. MRBEE can be a useful tool in studying causality between risk factors and disease outcomes as more large GWAS summary statistics are made publicly available.https://github.com/noahlorinczcomi/MRBEE. R code used in simulations and real data analyses are available at https://github.com/noahlorinczcomi/MRBEE. The MRBEE software, written in the R language, is available at https://github.com/noahlorinczcomi/MRBEE.All GWAS data used for the analyses were retrieved from publicly available repositories whose online locations are presented in Supplement 1


For the periodic tableof elements as well as small molecules, our model accurately reproducesatomic  polarization potentials and multipolar dispersioncoefficients, elucidating the high promise of the presented modelin the development of next-generation quantum-mechanical force fieldsfor (bio)molecular simulations.The quantum Drude oscillator (QDO) is an efficient yetaccuratecoarse-grained approach that has been widely used to model electronicand optical response properties of atoms and molecules as well aspolarization and dispersion interactions between them. Three effectiveparameters  fully characterize the QDOHamiltonian and are adjusted to reproduce response properties. However,the soaring success of  The many-bodyextension of the QDO model (the coupled QDO model)has been widely employed to study both molecules and materials, includingtheir electronic12 and optical13 properties, polarization,15 dispersion,24 and exchange27 interactions, as well as a wealth of nonadditivefield effects in quantum mechanics28 and quantumelectrodynamics.29 Coupled QDOs are also extensivelyused in the development of van der Waals (vdW) density functionals,31 quantum mechanical6 and polarizable force fields,36 as well as recent machine learning force fields.38 Despite such a wide applicability of the coupled QDO model, itssuccess in describing real atoms remains fundamentally unexplained,and the optimal mapping between atoms and oscillators has not beenestablished. In this Letter, we develop an optimized parametrization(OQDO) in which the parameters are fixed by using only the well-knownatomic dipolar properties. Remarkably, the OQDO reproduces spatialatomic polarization potentials and atomic multipolar dispersion coefficients.Our OQDO model for atoms and small molecules also paves the way tothe development of next-generation quantum-mechanical force fieldsfor (bio)molecular simulations.The development of predictivemodel Hamiltonians that can describe various properties of realisticmolecules and materials is a cornerstone of modern physicsq, μ, ω} fullydefine the QDO, and three atomic response properties could be chosento fix them, meaning that the choice of QDO parameters is not unique.In addition, all QDO response properties, multipolar polarizabilitiesand dispersion coefficients, are uniquely fixed by the three parameters via closed-form relations.6 Thestatic dipole polarizability of a QDO, α1 = q2/μω2, conveniently combines all three parameters, and it is natural toset this expression to the reference atomic α1. TheQDO expression for the dipole–dipole dispersion coefficient C6 and α1 are given. Since α1 and C6 are accurately known forall elements in the periodic table,41 they form a baselinefor the QDO parametrization. However, one more condition is requiredto obtain {q, μ, ω}, for which differentconstraints can be imposed. A reasonable idea is to fix q = 1 au since a QDO should reproduce the response of electrons. Thisresults in the fixed-charge QDO (FQDO):q and usingQDO recursion relations for high-order response usually yields largeerrors in the multipolar response properties  parametrization scheme:44 However, the C8 dispersion coefficient is not directly measurable, and accurate ab initio calculations of quadrupole (α2) and octupole (α3) polarizabilities and C8–C10 dispersioncoefficients are currently technically feasible only for closed-shellspecies  or alkali and alkaline-earthatoms with s valence shells.48 For other open-shell atoms ,convergence of quantum-chemical response calculations becomes a technicalhurdle49 . Thus, usinghigher-order atomic response properties does not lead to a parametrizationthat would be universally applicable across the periodic table aswell as for small molecules.The three parameters {ties see  and 43).1, C6} to the oscillator parameters. The third parameteris fixed by using the force balance equation for vdW-bonded dimersderived recently.27 Two equations for q and ωfollow the JQDO scheme, whereas the third one is replaced with a transcendentalequation for a product μω to be solvednumerically (vide infra)fsc = e2/4πε0ℏc is the fine-structure constant and a0 is Bohr’s radius. The vdW radius (RvdW) is calculated via the universalformula connecting it with the dipole polarizability26 delivers μω.Here, we introduce an optimizedQDO parametrization (OQDO), wherewe effectively map dipolar atomic quantities {αi and j and usinginteratomic perturbation theory,51 the interactionenergy can be written as the integrated product of the electron densityof moiety i with the electric potential generatedby moiety j:5252 and it was shown that exchange53 anddispersion56 interactions can be represented using the form of V being effective quantities differentfrom free-atom counterparts. The argument that dispersion interactionscan be written using 54 which was further elaborated by Hunt55 with a focus on dispersion forces and finallyextended to dispersion energies with a demonstration of its validityfor real molecules and materials.58The QDO Hamiltonianeffectively captures the integrated atomicresponse. However, when modeling molecules or solids, coupled QDOsmust properly describe noncovalent interactions between atoms. Consideringtwo fragments Eint asE, which can also model the effect of environment, δρ(r) = ρE(r) – ρ(r), where ρE(r) is the electron density under the externalfield. Then the dominant contribution to δVj(r) is generated by the corresponding δρj(r′) via the polarization potential5960 In the Supporting Information, we present VpolQDO(r) in comparison to Vpol(r) calculated for 21 atoms (between H–Ca andKr) within hybrid density functional theory DFT-PBE065 shown to yield a highly accurate description of electronic response66 comparable to coupled-cluster calculations (seethe Supporting Information). Here, we remarkthat the strength of the electric field was chosen individually foreach element depending on its reference static dipole polarizability41 so that the field-induced dipole moment is set as d = α1E = 0.01 au for all atoms.67Responseproperties are given by variations of Vpol(r) for real atoms with different QDO flavors,it is instructive toconsider which atomic properties can be faithfully captured by a QDO.First, the QDO does not aim to describe static properties of the atomicelectron density but rather its response under applied static andfluctuating fields, as demonstrated also by the insets in q determinesits magnitude. This explains why VelFQDO yields good agreement with VelDFT for hydrogen. However, the QDO model does not describe Vel for many-electron atoms because q ≈1 au, while the ESP of atoms scales nonlinearly with Z (see the example of carbon in the inset of VpolQDO(r) in Vpol curves of real atoms and VpolQDO(r) employingthe three QDO models discussed above. We used the accurate ab initio reference data on α1 and C6(C8(Vpol(r) curves obtained from DFT calculationswithout any adjustments. In fact, the performance of the OQDO is sensitiveto variations in the QDO parameters (solutions A or B in Vpol(r) underline the importance of optimal mapping between atomicresponse properties and QDO parameters.Before comparing α1 and C6,41,68 toand C6(C8−47 to paSupporting Information) and their connection to real atoms.The starting point is 26μω in RvdW as from 1 = α1(RvdW). For simplicity, we rewrite x asa and b given byRvdW in termsof α1. For all elements in the periodic table, wefound that 1 ≈400 au) of Cs.41 The existence of two solutionsextends beyond the employed QDO model. We obtained an analogous resultby using the Tang-Toennies potential69 withthe repulsive interaction treated by the Born-Mayer form (see the Supporting Information).We discuss now the technical aspects of derivingthe two solutionsof the OQDO model ( labeled as OQDO(A) and OQDO(B) in μω differ both in mass and charge, yielding quite different results.First, VpolQDO(r) constructed from solutionB does not resemble DFT potentials, while A is in good agreement withthem  and 12 small molecules. Overall, our results showthat the OQDO parametrization improves the dispersion coefficientscompared to the FQDO one, reducing the MARE from 31% to 25% for C8 and from 68% to 33% for C10 when averaged over all 28 (26 for C10) systems. The OQDO model consistently surpasses theFQDO in accuracy for all systems except for alkaline-earth metalswhere the FQDO gives more accurate results. Moreover, C8 and C10, but roughly for one-third of them, the dispersioncoefficients are overestimated. The maximal errors of FQDO are observedfor Xe in the case of both C8 (66%) and C10 (190%). In contrast, the OQDO consistentlyunderestimates both dispersion coefficients for all systems, exceptfor C8 of Li as well as C10 of Li and Cs. The maximum errors of the OQDO are observedfor CO2 (46%) in the case of C8 and CO (58%) in the case of C10, whichare significantly smaller than maximum errors of the FQDO. The consistencyof the OQDO errors allows a straightforward rescaling of dispersioncoefficients: with our best rescaling factors, 1.3 for C8 and 1.5 for C10, one candecrease the MARE of the OQDO to 15% and 22%, respectively, whichis commensurate with the uncertainty of the reference molecular C8 and C10 values.A more detailed analysis of the dispersion coefficients  can be foundin the Supporting Information, where wealso discuss static polarizabilities α2 and α3. The latter becomes less important in the QDO approach wherethe dispersion coefficients, determining the dispersion energy, aredirectly expressed in terms of QDO parameters. It is important tomention that the effects of three-body interactions are captured bythe OQDO scheme on an equal footing with the JQDO scheme. The accuracyin determination of C6 and C9 coefficients is known to be comparable.70 In ref molecular simulations.We presented the OQDO model based on robust parametrizationthatsolely employs dipolar α


Specifically, we observed the mode at around 1850 cm–1 and assigned it to H–O–D bend + rockinglibration. Second, we found that the H–O–D bend overtoneband and the OD stretch + rocking libration combination band contributeto the band located between 2850 and 3050 cm–1.Furthermore, we assigned the broad band located between 4000 and 4200cm–1 to be composed of combinational modes of high-frequencyOH stretching modes with predominantly twisting and rocking librations.These results should help in a proper interpretation of Raman spectraof aqueous systems as well as in the identification of vibrationalrelaxation pathways in isotopically diluted water.Overtones and combinational modes frequently play essentialrolesin ultrafast vibrational energy relaxation in liquid water. However,these modes are very weak and often overlap with fundamental modes,particularly in isotopologues mixtures. We measured VV and HV Ramanspectra of H This isotope shift is instrumental in infrared(IR) and two-dimensional infrared (2D IR) spectroscopy of proteins,where it allows for measurements of protein’s amide I vibrations(1620–1680 cm–1) free of solvent (H2O) interference.118The isotopic dilutionof liquid water (the substitution of H forD atom) results in significant changes in the vibrational spectrumof the system. The most striking change is the redshift of the stretchingvibrations frequency from around 3400 cm2O, and H2O isotopologuesin a mixture, all having their vibrational manifestation in a spectrum.The fundamental modes of these components are well-known and frequentlystudied. In addition to already mentioned OD and OH stretching modes and H–O–H and D–O–D bending modes,there is also the H–O–D bending mode at around 1460cm–1.11 Moreover, inthe low frequency region, there are strongly overlapped librations,rotations of water molecules that are essentially hindered (restrained)by hydrogen bonds. There are three rotations that a “free”water molecule would have in the gas phase: around its C2 symmetry axis, in-plane, and out-of-plane. Correspondinglytwisting librations around 450 cm–1 (vL1) are associated with hindered rotations of water moleculesaround the C2 symmetry axis, rocking librationsaround 550 cm–1 are hindered in-plane rotations(vL2), and wagging librations around 725cm–115 are restrained out-of-plane vibrations of water molecules (vL3). These bands are red-shifted by a factorof ≈√2 for D2O to, respectively, around 330,425, and 565 cm–1.15 Theso-called translational region 20–300 cm–1 contains two broad bands, signatures of low-frequency modes: hydrogen-bondbending at 62 cm–1 (vT2)17 and hydrogen-bondstretching at 175 cm–1 (vT1).20 These bands were also interpreted in terms of restricted translationsof water molecules.22The isotopic substitution, let aloneits usefulness, leads to thepresence of the three HDO, D2O/HDO/D2O mixture, one shouldexpect overtones and combinational modes originating from all threecompounds. Walrafen and co-workers predicted the spectral positionsof most of the combination bands and observed some.23 However, the then-existing state of technology allowed for determinationof the position of these “weak bands” just from the“negative/positive concavities” of a baseline.15 Recently, Verma et al. pointed at the intermolecularnature of the combinational band bend + libration of H2O at 2130 cm–1, designating it as an excellentprobe of water-hydration behavior.24 Therefore,this band is also called the “association band”.26 Also recently, the discussion on the nature of the band around 4100cm–1 has been reignited. While Walrafen and Pughassigned the band to the combination of low frequency part of theOH stretching mode with twisting libration,23 Morawietz et al. correlated the band with the combination of thehigh frequency part of the OH stretch and water wagging librations.27Overtones and combinationsof fundamental modes, due to very lowintensity, are typically difficult to observe. Nevertheless, in thespectrum of the H2O/HDO/D2O mixtureswith different concentrations of isotopologues with polarization-resolvedfemtosecond stimulated Raman scattering spectroscopy (FSRS).29 This method offers numerous advantages over spontaneous Raman scatteringalso in stationary (not time-resolved) studies. While only one Ramanphoton out of a million pump photons is produced in spontaneous methods,in stimulated Raman, that efficiency may be up to 10%.30 Moreover, the fluorescence background is suppressedin FSRS, and the transmission geometry of the measurement with thewhole SRS signal contained in the coherent Raman probe beam allowsfor very high reproducibility of the experimental condition from sampleto sample. Moreover, measurements for two relative polarizations  between the Raman pump and probe offer additional informationon the symmetry of observed modes. In perpendicular polarization spectra,the intense symmetric components are not present, which facilitatesobservation of very weak asymmetric modes. Our highly sensitive FSRSstudies revealed, for the first time, some combinational modes andprovided additional features of the known modes.In this work, we studied H47. In this work two computational approaches were employed. Mixedquantum-classical line shape theory based on electrostatic spectroscopicmaps32 was used to calculate Raman spectra of H2O/D2O mixtures. Within this approach, low-frequencyvibrational modes are treated classically via molecular dynamics (MD)simulations, whereas the high-frequency vibrations are treated quantum-mechanically.Specifically, the OH(OD) fundamental stretching modes and H–O–H(D–O–D/H–O–D)bending overtone modes are treated based on the exciton Hamiltonianexpressed in the local mode basis. This approach has been used tocalculate IR, 2D IR, and Raman spectra of liquid water and ice andis known to provide a good agreement with experiments.48Simulationshave been integral in interpreting experimental spectraof liquid water.50 The anharmonictreatment of vibrations is necessary because, as shown previously,2551 the descriptionof combination bands of water in terms of harmonic normal modes isinadequate. Calculated combination bands allowed us to identify thedominant modes forming combination bands in terms of low- and high-frequencymodes.To interpret the origin of combination bands formed betweenhigh-and low-frequency vibrations, Raman spectra of the three isotopologuesof the book isomer of the water hexamer were studied using vibrationalperturbation theory (VPT2).22.12O/HDO/D2O mixtures of different compositions were preparedby mixing ultrapure H2O (from Millipore Direct Q3 UV system)with D2O  with the following initial (not equilibrated) concentrationsof H2O: 0, 1, 5, 15, 30, 50, 70, 85, 95, and 100 mol %.The solutions were subsequently left in sealed glass flasks for equilibrationfor at least 1 day.The H2.2–1) and is around 3 ps long. The Raman pump energywas around 0.5 μJ. The beam waists of the pumps in the focalpoints in a sample were about 25 μm (1/e2). The femtosecond Raman probe beam (supercontinuum) was generatedby focusing a small portion of the laser pulse on a sapphire plate.The measurements were performed with both parallel (VV) and perpendicularpolarization (HV) between the Raman pump and Raman probe in forward(transmission) geometry. Samples were measured in a 2 mm sample-pathfused silica cuvette (Hellma QS quartz) at 20 °C. The cuvettewas placed in a cuvette holder with a fixed position, and only thecuvette content was changed, which assured a fixed geometry  between measurements.The Raman pump and the probe were temporarily overlapped with theuse of a manual optical delay line. The Raman probe spectrum withand without the presence of the Raman beam  in the sample was recorded by the spectrometer(spectrograph Andor Shamrock SR 500i with CCD camera Andor NewtonU971N).The stimulatedRaman scattering studies were performed with the setup for pump–probefemtosecond stimulated Raman scattering, which was described in refs  and 53, 2.32O/D2O mixture,the equilibrium concentrations for components  were calculated with the use of the followingequation:K ≈ 3.9.56 The spectra referredto as the “HDO spectra” below were extracted from thespectra of H2O/D2O mixtures by subtracting spectraof pure components  multiplied bytheir respected equilibrium fractions  from the spectra of the mixturescalculated by solving The baseline was subtracted from each spectrum with the nod points of the polynomialattached at the same wavenumbers of a spectrum. Spectra were not normalized.For each composition of the H32O (D2O), aswell as H2O/D2O/HDO mixtures. For a detaileddescription of the approach, the reader is referred to refs . The main idea is to describe low-frequencyvibrations using classical MD simulations and then treat the chosenhigh-frequency vibrations quantum-mechanically. This approach accountsfor the inhomogeneous broadening and motional narrowing effects. Inthe quantum-mechanical treatment, the system of coupled vibrationalchromophores is described by the excitonic Hamiltonian expressed inthe local mode basis. The excitonic Hamiltonian is constructed byincluding the vibrations of interest. In this work, OH and OD stretchingbands were modeled by the excitonic Hamiltonian with the diagonalelements being the OH/OD hydroxyl stretching and H–O–H,H–O–D, and D–O–D bending overtone local-modefrequencies as done previously.48 By including the bendingovertone chromophores, we account for the stretch–bend Fermiresonance, which is believed to contribute to the OH/OD stretchingband.65 The off-diagonal elements of the excitonic Hamiltonian are the vibrationalcouplings. There are two types of couplings between hydroxyl stretchingchromophores: intramolecular and intermolecular. The former arisesbetween two hydroxyl stretching modes of the same water molecule,while the latter corresponds to two hydroxyl stretching modes thatbelong to two different water molecules. Bending overtone chromophoresare intramolecularly coupled to hydroxyl stretch chromophores viaFermi coupling.35Mixed quantum-classicalline shape theory was used to calculateVV and HV Raman spectra of pure H to refs , and 57 67 that correlatethe local mode frequency with the electric field created by the pointcharges of all atoms comprising the local atomistic environment ofthe water molecule. The charges were taken directly from the watermodel used in MD simulations described below. Intermolecular couplingsbetween OH(OD) stretch chromophores were calculated within the transitiondipole approximation.68 In the absence of the electrostaticmap for the H–O–D bending overtone, the correspondinglocal-mode frequency was assumed to be independent of the environmentand set at its experimental value of 2950 cm–1.59 How strongly the H–O–D bendingovertone frequency depends on the atomistic environment of the HDOmolecule is still an open question. The approximation used in thiswork is expected to work well, because the frequency of the H–O–Dbending overtone remains essentially unchanged from that of its valuefixed in the simulations due to a frequency mismatch with other high-frequencyvibrations that were treated quantum-mechanically in this work. Similarly,the local mode frequency of D–O–D bending overtone wasassumed independent of the environment and set to its experimentalvalue of 2380 cm–1.69 This assumption is more questionable, although the dependence ofD–O–D bending overtone frequency on the atomistic environmentof the D2O molecule is unclear. It is known that includingan electric-field-dependent H–O–H bending overtone mapin the calculation of the OH stretching band does improve the agreementwith the experiment.35 However, the effectsof stretch–bend Fermi resonance on the OD stretching bandshave not been computationally studied. In the future, it would behighly desirable to parametrize H–O–D and D–O–Dfundamental and overtone bending maps as it has been done for theH–O–H bending fundamental and overtone vibrations.3570The diagonal frequenciesand intramolecular OH(OD) couplings weremodeled by the electrostatic maps71 Theratio between longitudinal and transverse bond polarizability derivativesis taken to be 5.6.72 More details canbe found in refs  chromophoresvia the intramolecular Fermi coupling.Another necessaryquantity for calculating Raman spectra, transitionpolarizability of the OH(OD) stretch chromophores is approximatedusing the bond polarizability model. in refs  and 48 a73 The intrinsic lifetime broadening is accountedfor approximately based on the experimentally measured lifetime ofthe vibrational excited states.Diagonal frequencies,excitonic couplings, and transition polarizabilitieschange in time because a fluctuating local environment gives riseto changing electric fields. These effects are captured by spectroscopicmaps. The calculation of Raman spectra amounts to generating the excitonicHamiltonian and the transition polarizability tensor for each MD configurationand computing the corresponding VV and HV response functions.74 Raman spectra of pureliquid water calculated using the mixed quantum-classical approachdescribed above with E3B2 water model have shown to be in very goodagreement with experiment.35 As comparedin ref 6, (H2O)5D2O, and (H2O)5(HDO) were calculated using vibrational perturbation theory (VPT2),50 which is a widely used method for the anharmonicanalysis of molecular vibrations of water.95 VPT2 calculations were performed using density functional theorywith the B3LYP97 functional and 6-311++G98 basis set. All calculations were performed usingthe Gaussian 1699 software package. Beforecalculating anharmonic frequencies, the geometry of the book hexamerwas optimized at the same level of theory and basis set. All calculationswere performed using “verytight” geometry convergencecriteria. No negative harmonic or anharmonic frequencies were observedfor all three water hexamer isotopologues indicated above.As shown by McCoy, the harmonic analysis basedon normal modescannot be used to study combination bands.44.12O/D2O/HDO mixtures with differentconcentrations are shown in –1 (dark green) and around 2500 cm–1 (light green)in the HV spectra arise from OH and OD stretching vibrations of diluteH2O in D2O and dilute D2O in H2O, respectively. We will discuss stretching bands in 2O/D2O mixtures.StimulatedRaman spectra of H–1 region that are believed to contain spectroscopicsignatures of bending vibrations and combination bands between bendingvibrations and librations. We will then discuss low-intensity bandsin the 2800–3200 cm–1 region that are thoughtto contain H–O–D bending overtone and OD stretching+ librations combination bands. Finally, we will analyze combinationbands between OH stretching vibrations and librations, giving riseto low-intensity features in the 3900–4500 cm–1 frequency range.In what follows, we will discuss low-intensitybands in the 1100–2400cm4.22O/D2O mixtures andcomprise symmetric ν1 and antisymmetric ν3 stretching motions. These bands arise from a complex interplaybetween hydrogen bonding100 and intra- and intermolecularresonance couplings.102 They are also influenced bysolute–solvent interactions.106 VV and HV spectra of H2O/D2O mixtures calculated using the mixed quantum-classicalapproach described above are compared to the experiment in 2O. This hasalso been correctly reproduced in our simulations. However, in theH2O/D2O mixtures with concentrations of H2O greater than 15%, the relative intensities of the two peakswere not captured. This might be due to an overestimation of the effectsof the Fermi resonance on the OD stretching band. Such effects havenot been investigated before. It should be noted that the low-frequencyfeature of the OH stretching region seen in Raman VV spectra appearsto be underestimated in the simulated spectra compared to the experiment.The agreement between theory and experiment was better in our previouswork which was based on experimental data available at that time.35 We experimentally observed that the intensityof the low-frequency feature of the OH band nonlinearly depends onthe Raman pulse energy when studied by FSRS. That nonlinearity wasobserved only in VV polarization, so it is likely related to the polarized“collective” mode of water. We determined the onsetof the nonlinear behavior between 0.5 and 0.6 μJ. Therefore,to avoid introducing nonlinear effects into the spectra and to providethe best signal-to-noise ratio of the experiment, the Raman pump wasset to 0.5 μJ. Perhaps, that energy still introduced some smallnonlinearities causing additional discrepancies between experimentaland calculated spectra. Since the presented study was performed withthe 515 nm Raman pump, the Raman resonance of the low-frequency partof the OH stretch with the water overtones absorption in the visiblered range should not occur.107We startwith a brief discussion of the fundamental OH/OD stretching bands,which dominate spectra of H48 In light of the new FSRS measurements presented here, further worktoward improving transition polarizability maps is in order, and thenonlinear pump dependence of the red side of the OH stretch band willbe furthermore examined.The discrepancy between the blue sides of the simulatedand experimentalpolarized OH and OD bands has been observed previously and is attributedto the inaccuracy of transition polarizability maps.As seen in 2O and H2O, they may cover some low intensityspectral features. One way to better discern low-intensity featuresis to use isotope diluted water, e.g., HDO in D2O. In thiscase, the OH band is much narrower compared to pure H2O.2 Moreover, it is known that the low-frequencyshoulders of the stretching modes  are relatedto highly polarized collective vibrations of water108 and thus are inactive in the HV polarization configuration.Hence, both the isotopic dilution and HV polarization measurementsare tools to access low-intensity modes that might be hidden behindthe broad OD/OH stretching bands.Since this work is explicitlydevoted to the overtones and combinationalbands of water, the OD and OH stretching bands will be analyzed inmore detail in our future work. Here, it is essential to point outthat due to the inhomogeneous broadening of the stretching bands inneat D4.3Supporting Information.In what follows, we attempt to interpretcombination bands observed in the experimental Raman spectra in termsof the corresponding modes of the water hexamer. We caution againstthe direct interpretation of combination bands of liquid water byanalogy with those of a finite-size water cluster, but such a comparisoncan still be very helpful. The optimized geometry of the book hexamerstudied in this work can be found in the –1. In this range, thecalculated Raman VPT2 spectra of the water hexamer are clearly dominatedby hydrogen bond bending and stretching. The former involves O–O–Obending with the corresponding feature in the experimental Raman ofliquid water found at 62 cm–120 The latter involveshydrogen-bond stretching modes, namely, the O–O stretchingalong the hydrogen bond direction. In the VPT2 spectra of the waterhexamer these modes are found to be in the 94–197 cm–1 frequency range. We connect these modes to the broad feature inthe experimental Raman spectrum of liquid water with the peak at 175cm–1,20 which is in good agreementwith the calculated frequencies of the book hexamer.The translational frequency range correspondsto frequencies below 300 cmC2 molecular symmetry axis. We label these modes of thehexamer as vc2. Note that we purposefullyuse notation that is distinct from vL1, which was used to label the corresponding band of the liquid water,to emphasize that these modes belong to a cluster. The correspondingfrequencies are found to be higher than 210 cm–1. We also found that C2 rotations ofsome water molecules contribute to librations with frequencies ashigh as 439 cm–1. The corresponding feature in theexperimental spectra of liquid water, vL1, is located at around 450 cm–1.13 In-plane librations (vip) ofthe water hexamer are found to have frequencies in the 334–754cm–1 range. The corresponding experimental signature, vL2, is found at around 550 cm–1. Finally, out-of-plane wagging librations of the water hexamer,which we denote as voop, are found tohave frequencies in the range 666–909 cm–1, with the corresponding experimental feature, vL3, centered at around 730 cm–1.The lowestfrequency Raman active librations are associated witha hindered rotation of the water molecule around its We stress that the assignment described above sometimes becomesambiguous because some modes of the water hexamer involve differentmotions of different water molecules and, expectedly, have frequenciesin between the frequencies of the respective pure modes. In such cases,our assignment is based on the dominating motion. Overall, we finda reasonable correspondence between the librations of the book hexamerand those of liquid water. We will use this to interpret experimentallyobserved combination bands that involve librations.4.4–1 spectral region of VVand HV Raman spectra of H2O/D2O mixtures. First,we note that a weak band around 1060–1070 cm–1 is associated with silica glass (cuvette). The bending modes, δ,of D2O, HDO, and H2O are easily identifiableand are located at around 1210, 1460, and 1650 cm–1, respectively. 2O/D2O mixtures according to the proceduredescribed in –1. This band has been observed in IR and Ramanspectra of liquid water and identified as H–O–D bendingbefore.111 Moreover, for samples containing up to 15mol % of H2O . Yet, it is not observed in the raw spectraof the mixtures. To uncover this band, we analyze the HDO extractedspectra shown in –1 is clearly seen for bothVV and HV polarizations and particularly well for samples containing49.6 mol % of HDO and 42 mol % of HDO in H2O and 25.5 mol% of HDO in H2O. We noticed that the peak is most pronouncedin the samples in which HDO is “dissolved” in H2O, yet presently we could only speculate on the physical backgroundof this fact.One should also expect the presence of the bandrelated to HODbending + libration to be located between these two, particularlyin the sample containing around 50 mol % of HDO (50/50 H2O)6, (H2O)5(HDO), and (H2O)5(D2O) isotopologuesof the book hexamer. The calculations were performed at 0 K. For presentationclarity, all peaks were convoluted with a 5 cm–1 full-width at half-maximum Lorentzian line shape. Raman spectraof (H2O)5(HDO), and (H2O)5(D2O) clusters were calculated for the water hexamer withthe water molecule highlighted in 2O molecules,respectively. We choose to alter the isotopic composition of the moststrongly hydrogen-bonded water molecule because its bending mode isthe most blue-shifted (1686 cm–1) compared to theother five anharmonic bending modes at 1645, 1621, 1612, 1606, and1592 cm–1, which makes it easier to identify andtrack as isotopes are introduced into the structure.2O)6 with those of (H2O)5(HDO) and(H2O)5(D2O), one can follow the spectralchangescaused by isotope substitution. First, we note that the combinationband designated as bend + libration in the Raman spectra of liquidH2O and experimentally observed at around 2140 cm–1  stretching bands in thesample containing 15 mol % of H2O. However, it is seenthere just as a small “bump” between intense OD andOH stretching peaks, which causes problems with proper baseline subtraction.As a result, its intensity for the VV polarization is smaller in thatrange compared to HV polarization. The band at around 2950 cm–1 is much better seen in the HV polarized spectra asa distinct band for H2O concentrations in the range 0–30mol %, but it is still discernible in the sample containing 50 mol% of H2O. The band’s presence is also evident inHDO “extracted spectra”, containing over 25 mol % ofHDO, as illustrated in 2O, this band is centered around 2990cm–1 and is somewhat broader compared to the bandsin the spectra of H2O/D2O mixtures, where itis centered around 2975 cm–1. A band in this spectralrange was observed by Walrafen and Blatz in neat D2O (at2960 cm–1) and assigned to the combination bandof OD stretch + vL2.15 However, since this mode is also present in the spectraof H2O/D2O mixtures containing mostly HDO isotopologue,it was assigned by other authors to the overtone of H–O–Dbending mode (2δHOD) at 2950 cm–1.113 Since we observe a band in that region inexperimental data in neat D2O also, we claim, based onour experimental results, that the band around 2970 cm–1 contains a contribution from both HDO overtone and a combinationband of OD stretch with, most likely, rocking libration vL2 centered at 425 cm–1. The fit ofthat band with two Gaussian peaks, at 2930 ± 19 and 3020 ±70 cm–1, in the HV spectrum of 50 mol % HDO is shownin Supporting Information, Figure S1.In HOD and 1νOD + vL2) overlapwith the red shoulder of the broad OH stretching mode. This overlapis a likely reason for resonant vibrational energy transfer from theOH stretch to the OD stretch in H2O/D2O/HDOmixtures, possibly through both 2δHOD and 1νOD + vL2, which was observed byde Marco et al. in 2D IR studies59 andby Pastorczak et al. in their femtosecond IR pump–SRS probemeasurements.52It is clear from the VV spectra that both modes6 in the 2800–3000 cm–1 rangein agreement with the experiment. The sharply rising feature at higherfrequencies in (H2O)6 and (H2O)5(HDO) spectra belongs to the red shoulder of the most red-shiftedOH stretching frequency band of the book hexamer located at 3031 cm–1.To interpret spectral features inthe 2900–3000 cm2O)5(HDO) shows a set of combination modes formedby OD stretch fundamentalof the HDO molecule (vOD = 2611 cm–1) and libration modes. There is also H–O–Dbending overtone vibration at 2914 cm–1 in reasonableagreement with mixed quantum-classical simulations and previous experiments.More combination bands extend beyond 3000 cm–1 wherethey overlap with OH stretching bands. By analyzing the vibrationalmodes of the book hexamer, we found that lower frequency bands around2800–2900 cm–1 are combination bands involvingOD stretching and vc2 librations. The two bands in the spectrum of (H2O)5(HDO) around 2990 cm–1 correspond to ODstretch and in-plane hindered rotations of water molecules vip.Calculated anharmonic Raman spectrum of (H2O)5(D2O) has more featuresin the 2900–3000 cm–1 frequency range becausethe cluster has two OD stretches to formcombination bands with librations. The high-frequency bands that wereobserved near 2990 cm–1 in (H2O)5(HDO) spectrum and assigned to OD stretch + vip are present in the (H2O)5(D2O) spectrum at lower frequencies 2850–2870 cm–1 with the most intense band having the same character: OD stretch+ vip. Low-frequency features are shiftedto around 2700–2850 cm–1 and correspond tolower frequency OD stretch, since there are two OH stretches, mixedwith vc2 (lower range) and vip (higher range) and 2855–3000 cm–1 corresponding to higher frequency OD stretch mixed with vc2 (lower range) and vip (higher range) librations.VPT2 Raman spectrumof (H–1 has contributions from H–O–D bending overtone as wellas from OD stretch + vL2 (rocking librations)combination band with some contribution from OD stretch + vL1 (twisting librations) combination band. Thisassignment fits our experimental results and agrees with the combinedassignment of Walrafen and Blatz15 andothers113.Based on our theoreticalanalysis, we conclude that the band seenin experimental Raman spectra between 2950 and 3000 cm4.62O/H2O mixtures in the 3200–3900 cm–1 frequency range are dominated by the fundamental OH stretching band.OH stretch–H–O–H bend Fermi resonance is believedto be important as well and gives rise to a feature in the VV Ramanspectrum at 3250 cm–1.65 It is known that the low-frequency feature of the OH stretch decaysfaster with isotopic dilution than the high-frequency one and thehigh-frequency feature shifts to higher frequencies. Interestingly,the VV spectrum of pure D2O in this region contains a broadpeak centered at 3520 cm–1. Similarly, the HV spectrumcontains a broader feature peaking at roughly the same frequency.This feature is likely due to the contamination of D2Owith H2O. We tried to estimate the “true”level of contamination of D2O with H2O by lookingat the intensity of the H–O–H bending mode. That wouldeither confirm the presence of around 0.4 mol % H2O contaminationor allow us to extract the spectral residuals unrelated to the contamination.Unfortunately, the H–O–H bending band was weak, barelydistinguishable from noise.Both VV and HV Raman spectra of D2O contaminationis not the only contribution to the broad feature observed in theRaman spectrum of D2O. The comparison of this band in theD2O sample with the OH stretch band in the sample containing1 mol % of H2O shows a clear difference in the centralfrequency (around 3520 cm–1 vs around 3460 cm–1) and shapes of respective bands. To illustrate this,we scaled the experimental Raman VV spectra of 1 mol % H2O in D2O such that its intensity at 3450 cm–1 matches that of the spectrum of “pure” D2O. The difference between the “pure” D2Ospectrum and the scaled spectrum of 1 mol % H2O is shownin –1. We assignthis difference to a combination band between OD stretching mode (∼2500cm–1) and D–O–D bending mode (∼1210cm–1).115However, H–1. The former is in agreement with the experimentalresult. However, we emphasize that these frequencies are for the particular(H2O)5(D2O) cluster with the D2O molecule chosen, as explained above  + vL2 observed at3990 ± 25 cm–1 and OH asymmetric stretch (ν3) + vL3 observed at 4170 ±50 cm–1. Moreover, the authors reported but didnot observe experimentally ν1 + vL3 combination mode at 4130 cm–1. Walrafenand Puigh, in the later work,23 mentionedonly the band at 4000 ± 10 cm–1 and assignedit to ν1 + vL3 combinationmode. Recently, Morawietz et al.27 presentedmeasured and ab initio MD Raman spectra of liquidwater polemizing with the Walrafen’s and Puigh’s interpretation.Their 2D correlation spectra pointed at the correlation of the bandobserved around 4100 cm–1 with the low-frequencylibration (vL1) and high-frequencypart of the OH stretching band.27Next, we address the weak band located between4000 and 4100 cm2O/HDO/D2O ratios are shown in Supporting Information, Figure S2), yet at least bimodal structure of the bands isclear. The more intense feature is centered around 4080 cm–1, in agreement with the band observed by Morawietz et al., and aweak feature spreads from around 4200 cm–1 to about4460 cm–1. A comparison of the intensities of thebands located at 4080 cm–1 for both polarizationshows that this mode is depolarized.Our SRS spectra in this frequency range for chosen H–1 frequency range, we turn to Raman spectra ofwaterhexamer isotopologues calculated using VPT2 and shown in 2O)6. Theonly significant difference between isotopologues spectra is a distinctpeak near 4000 cm–1 that is present only in the(H2O)6 spectrum. We find that this peak is acombination band between a vc2 librationand 3732 cm–1 OH stretch whose hydrogen is replacedby D in HDO and D2O.To understand spectroscopic features in the 3900–4500cm–1 frequency range of the VPT2 spectrum of (H2O)6. We found that the most intense peaks in thelowerfrequency band 3950–4100 cm–1 were clearlydominated by high-frequency OH stretches around 3700 cm–1 and vc2 librations. The higher frequencyband 4100–4200 cm–1 is found to be dominatedby combination modes between high-frequency OH stretches and vip librations.We start our analysis by focusing on the 3950–4200cm–1 and below, and librations. We found that combinationbands formed by these modes and voo and vip librations appear in the Raman spectrum in4120–4190 cm–1 range but their intensityis much smaller compared to the intensities of combination bands formedby high-frequency OH stretches. Overall, in agreement with Morawietzet al.,27 we conclude that combinationsbands in this spectral region are dominated by high-frequency OH stretches,but in line with Walrafen and Blatz,15 thestretching vibrations are combined with both vL1 and vL2 librations.We also examined VPT2 spectraof the book hexamer for the combinationbands formed by lower frequency OH stretching vibrations, 3450 cm2O, HDO, and D2O, together with their assignmentssupported by our simulations, are summarized in 15 for neat H2O and D2O andlater by Walrafen and Pugh23 for H2O, D2O, and HDO in D2O (with unspecifiedconcentration). In the former work, the authors postulated many combinationalbands involving sums and differences of bending mode with librations:δ + vL1 – vL3, δ + vL1 – vL2, δ + vL3 – vL2 basing on the concavityof a baseline. However, they could not observe ternary sum combinationsbands , which should generally be more intense than the difference bands.Therefore, they called the presence of ternary bands into question.In this work, with the stimulated Raman technique, we were also unableto single out ternary combination bands of water. Nevertheless, weclearly observed combinations of librations with D–O–Dbending at 1590 cm–1, H–O–H with bendingat 2140 cm–1, and (for the first time) H–O–Dbending at 1850 cm–1. For all the isotopologues,the position of the combinational band indicates that the combinationcannot involve wagging librations, vL3, which has the highest energy (565 cm–1 for D2O and 725 cm–1 for H2O), so themixing involves twisting, vL1 and/or rocking vL2 librations only. Calculated anharmonic Ramanspectra of the water hexamer fully support these conclusions.The wavenumbers of the experimentallyobserved Raman bands forH2O/HDO/D2O mixtures of different isotopologueratios withfemtosecond stimulated Raman and compared the results with calculatedspectra. Thanks to the superior sensitivity of FSRS and careful measurementprocedure, we observed a number of weak low-intensity peaks in themixture spectra and with the help of computational methods, we assignedthem to the combinational modes and overtones of particular isotopologuespresent in the mixture. Specifically, we observed, for the first timeto our knowledge, the mode at around 1850 cm–1 andassigned it to H–O–D bend + rocking libration. Second,we found that both H–O–D bend overtone band and OD stretch+ rocking libration combination band contribute to the band locatedbetween 2850 and 3050 cm–1. That mode is most likelyresponsible for the reported earlier59 resonant vibrationalenergy transfer from the OH stretch to OD stretch in the mixture oflight and heavy water. Furthermore, we assigned the broad band locatedbetween 4000 and 4200 cm–1 to be composed of combinationalmodes of high-frequency OH stretching modes with, predominantly, twistingand rocking librations.To conclude, we measured VV and HV Raman spectra ofH