Quantum chaos is reviewed from the viewpoint of “what is molecule?”, particularly placing emphasis on their dynamics. Molecules are composed of heavy nuclei and light electrons, and thereby the very basic molecular theory due to Born and Oppenheimer gives a view that quantum electronic states provide potential functions working on nuclei, which in turn are often treated classically or semiclassically. Therefore, the classic study of chaos in molecular science began with those nuclear dynamics particularly about the vibrational energy randomization within a molecule. Statistical laws in probabilities and rates of chemical reactions even for small molecules of several atoms are among the chemical phenomena requiring the notion of chaos. Particularly the dynamics behind unimolecular decomposition are referred to as Intra-molecular Vibrational energy Redistribution (IVR). Semiclassical mechanics is also one of the main research fields of quantum chaos. We herein demonstrate chaos that appears only in semiclassical and full quantum dynamics. A fundamental phenomenon possibly giving birth to quantum chaos is “bifurcation and merging” of quantum wavepackets, rather than “stretching and folding” of the baker’s transformation and the horseshoe map as a geometrical foundation of classical chaos. Such wavepacket bifurcation and merging are indeed experimentally measurable as we showed before in the series of studies on real-time probing of nonadiabatic chemical reactions. After tracking these aspects of molecular chaos, we will explore quantum chaos found in nonadiabatic electron wavepacket dynamics, which emerges in the realm far beyond the Born-Oppenheimer paradigm. In this class of chaos, we propose a notion of Intra-molecular Nonadiabatic Electronic Energy Redistribution (INEER), which is a consequence of the chaotic fluxes of electrons and energy within a molecule. The origin of such statistical properties arising even from small molecular systems are widely ascribed to classical [Molecules consist of heavy nuclei and light electrons, which give rise to quite interesting and characteristic properties. All of the living bodies and biological systems on the globe are made up with molecules and are subject to the beautiful chemical laws. In the biological activities most of those molecular dynamics and properties should appear to be lassical  and quanlassical ,10,11,12Light electrons in a molecule move very fast and yet are accompanied with long wavelength in their matter waves, while the nuclear motion is much slower, but the relevant wavelengths are relatively far shorter due to the heavy masses. Thus, the two intrinsic hierarchical structures emerge in time-scale and length-scale within the dynamics of a molecule. Consequently the general picture of a molecule are: (i) Nuclei move on mean-field potential functions that are generate by the electrons. (ii) The nuclear motion can bear classical and semiclassical nature due to the short wavelength, while electrons are utterly quantum. Therefore, in contrast to classical chaos of the nuclear motion, it is often claimed that there is no chaos in electronic states because of discreteness in the energy levels with large gaps among them. Physics of the mapping of classical chaos onto quantum systems is often referred to as quantum chaology . As a maHowever, the theoretical separation of the electronic and nuclear motions due to the fast and slow modes, which is known as the Born-Oppenheimer separation, can sometimes break down through large kinematic interaction between them, which cause a mixing of the relevant electronic states. This interaction is generally and vaguely referred to as nonadiabatic interaction ,18,19,20Chaos theory for Hamilton mechanics was initiated (resumed from the view point of Poincaré) around 1950s . Many beWe begin this review with the very basic structure of molecular science in In Multidimensional semiclassical mechanics is indispensable in the treatment of the dynamics of heavy nuclei in molecules. We review our developed semiclassics ,30 in SeThen we study the characteristic molecular science, in which the nuclear dynamics kinematically couples with the electronic states, where the electronic-nuclear separation is broken down. In The final part of this review is devoted to the nonadiabatic electron dynamics in This review concludes with some remarks in We first outline briefly the very basic frame of quantum description of molecules suggesting how chaos studies of molecules proceed.jth and kth component of ath electron and Ath nuclei, respectively, and Ath nuclei.The nonrelativistic molecular Hamiltonian is given ast. This in turn requires that Due to the great difference in the time scales between electrons and nuclei, their motion is assumed to be separated at each time, as the so-called the Born-Oppenheimer (BO) approximation represents the total wavefunction Notice that Equation  is thereThe time independent figure terature .The lowest-order correction to the Born–Oppenheimer energy is found to be in the order ofThis analysis  clearly nuclear wavepackets. This is referred to as the Born-Huang expansion [To approach the exact solution of Equation  from thexpansion . The eleThe nuclear wavefunctions are subject to the following coupled equationsSince f motion  are due adiabatic wavefunctions, while others are to diabatic wavefunctions collectively. There are many ways to determine the diabatic functions [Note that the above Born-Huang expansion can be performed with use of other basis functions that do not satisfy Equation . In suchEquation , have siEquation  are refeunctions ,17,18,20k from initial molecules to products in the formIt was Arrhenius who established the well-known thermodynamical reaction rate rate or speed has been derived based on the Maxwell-Boltzmann distribution for equilibrium states. The physical insight of this rate expression was uncovered by Eyring [k  [40,41,While being subject to statistical laws on one hand, the dynamics of molecular motions are generally deterministic on the other hand. Let us see such a rather intricate example. See see ref.  for the Another example of statistical reaction rate theory was given by Marcus , which aesonance ,50. In celectronic states is less than 1.0 eVelectronic state manifold of densely quasi-degenerate states, ending up with finding chaos.The key quantity that brings about stochastic and statistical nature into small molecular dynamical systems is the density of state (DOS), not the number of particles (atoms and molecules). In case where the DOS is low, the quantum dynamics should be subject to the Schrödinger equation and naturally deterministic. The higher is DOS, the harder become the relevant coupled equations to solve, and statistical treatments turn out to be more feasible. Due to the heavy masses of nuclei, the vibrational and rotational degrees of freedom of a molecule tend to have very high DOS, which can readily amount to the order 10Given the statistical assumption of molecular vibrational energy distribution, the RRKM theory actually estimates the rate of unimolecular dissociation based on statistical inference at the bottle neck position on the way to products in phase space , using t∠HCC in due time ordering. More complicated energy redistribution can follow such as a very large amplitude motion of the hydrogen atoms, but we do not pursue them here. Since the spectra of the longer time-width can track the long physical process of energy redistribution. The related experiments and precise analyses have vividly shown the real-time and actual energy flow within a molecule. The notion of chaos has thus physical reality in chemical dynamics.Such experimental studies were made possible by the advent of short pulse laser, which has unified two then separated subfields in molecular science, namely, spectroscopy of molecular structures and chemical reaction dynamics. As such the so-called stimulated emission pumping (SEP) spectroscopy applied to acetylene molecule H-C≡C-H (linear in shape having a triple bond) has given a huge impact ,52,53,54The energy transfer among the different vibrational modes are induced by high anharmonicity of the relevant potential functions, leading to nonlinear canonical equations, and are also due to the overlap among the Fermi resonances  or the FIncidentally, energy transfer, relaxation, and redistribution of intra- and inter-molecular systems in condensed molecular systems ,59 and bBeck and Steve Berry are the first, to the best of out knowledge, who found the critical roles of classical chaos in the statistical behavior of “melting and phase transition” appearing in structural isomerization dynamics of small atomic clusters . Berry asolid, while at some high energy the isomerization begins to take place by surmounting the potential barriers, and further in considerably high energy frequent isomerization keeps taking place, which can be viewed as extensive large scale vibrational motions over the entire molecule or as a liquid-phase analog [Chaos not only makes the relevant phase space ergodic and mixing but also gives interesting qualitative features to the dynamics of molecules. Such properties can be typically seen in the structural isomerization dynamics of atomic clusters. Take an example from Are analog . Even hie analog ,66. Thise analog ,63,67.In the liquid-like phase, chaotic dynamics manifest themselves in the following phenomena. (1) Each classical path wanders among the potential basins for individual isomers in a random fashion see . The timch basin ,69. (2) ch basin ,69. (3) ch basin . (4) Micch basin .branching and remerging of the reaction tubes exist behind the memory loss.Behind the above chaotic dynamics we can imagine geometrical structures in phase space, referred to as the reaction tubes. Suppose that a trajectory enters a PBP basin from the neighboring COCT by passing through a divided surface . Then there should exist a large bundle of similar trajectories flowing from the COCT to this PBP that includes this particular trajectory. See exp−βt ,72. Thisexp−βt ,72. By cThe original idea of reaction tube is due to De Leon and their colleague, which they called the cylindrical manifold in the study of two dimensional two potential basin dynamics ,74. “BifIn the early 1980s the studies of quantum chaos in terms of quantum wavepacket propagation were made rather extensively in the community of chemical physics ,77,78,. We herein study pure quantum chaos in terms of wavepackets and eigenfunctions confined in the separatrix on the modified Hénon-Heiles potential. This potential function is regarded as a two dimensional potential energy surface (PES) on which for molecular vibrational motion to develop.weak chaos.The Hamiltonian we adopt isx-coordinate such thatWe first track a wavepacket propagating in time within the major quasi-separatrix. To do so, we extended the Poincaré surface of section to a quantum version as follows ,80. GiveThen Fourier back it using only the positive We then plot An example of the quantum Poincaré section is exhibited in y Heller  .The next question is whether the eigenfunctions can be built in the quasi-separatrix without decaying to the stable tori. We next explore this aspect.Prior to the numerical presentation we need to show how only (or at least mostly) the separatrix eigenfunctions are extracted from the very many eigenfunctions. This is a difficult task if we resort to the straightforward method of diagonalization of the Hamiltonian matrix, which may be expanded in arbitrary basis functions. Fortunately, those wavepackets we have shown above in E from a wave packet, one may want to calculate the time-energy Fourier transform [As a standard way to obtain eigenfunctions at any chosen energy ransform ,84(21)ψE. Moreover one may need to know in advance whether E should indeed belong to one of the quasi-separatrix eigenfunctions.However, this approach needs to pick extremely accurate see ref.  to extrat in the denominator.An efficient alternative to calculate eigenfunctions directly without the prior knowledge on the energy spectrum is the screening method . Supposet at all ,86. Incirated asΦϵ1,ϵ2=∫ϵrated asΦϵ1,ϵ2=∫ϵquasi-tunneling.Davis and Heller found a quantum mechanical tunneling between two (or more) tori, which are separated by separatrix(es). This phenomenon is referred to as the dynamical tunneling through dynamical barrier ,88,89. Iℏ as demonstrated in terms of the Poincaré section. The above two panels in We are curious about the possible effects of the magnitude of the Planck constant on quantum chaos. To save the space, we refer this aspect to our original paper . Yet we chematic . As a chThe classical trajectories within the thin quasi-separatrix are generally not periodic nor quasi-periodic by definition (meaning a zero measure for the exception), and moreover they wander around circulating near one tori to another one from time to time, each torus of which represents different mode of vibration see . TherefoThe mechanism of energy quantization is a landmark lying in between classical and quantum mechanics. It is therefore quite natural for semiclassical mechanics to play a major role there. Besides, the chemical dynamics quite often demands semiclassical treatment for the nuclear motion apart from chaos. There are many beautiful studies in the literature on the EBK semiclassical quantization for regular systems  semiclassical approximation to the Feynman kernel or its energy representation (the energy Green function) to the density of states, thereby extracting the essential roles of phase-space periodic orbits . The sucsee ref.  for instDE≃12πℏN∫ℏ is obviously a canonical candidate to improve the trace formula [It seems that there are two major opposite directions to cope with the fundamental difficulty of the trace formula. One is an attempt to recount the periodic orbit contributions in Equation  in such  formula . Deep maThe other one is to take an easier, simpler and more tractable approach, mainly based on the semiclassical propagation of wavepackets. Our approach below is along this line. Besides, semiclassical wavepacket theory is one of the main methodologies in chemical reaction dynamics. As such, we here present our own approach, which is intended to step further beyond semiclassical approximation to the exact one. We then attain some insights about the mechanism of energy quantization, that is, the explicit roles of not only constructive but destructive interferences in forming discrete energy levels. We also see that the energy levels in chaos can be determined without the tedious pre-exponential factor (or the Monodromy matrix) of the semiclassical kernel.Many semiclassical theories have been proposed in the literature ,103,104,q in configuration space, where m being the mass. The classical factors are thus factored out for this short interval and thereby the purely quantum factors are thus supposed to be included only in the function p is a momentum at We represent a wavefunction lov type , such thon F,33. F ,30. Thenm) are scaled to unity p is set to be numerically equivalent to the corresponding velocity v. In transforming the Euler picture to the Lagrange by definingWe use the mass weighted coordinates so that all the masses FG energy peaks are under focus without paying much attention to to-be-nullified part of N-dimensional system, where the pre-spectral density is defined ast. We implicitly assume that the wavepacket adopted here has been taken from the above ADF framework, Equation  in suchThen this function should appears to be an exactly periodic function. Furthermore, since the preexponential factor L and rewrite it with the help of Equation PThe correlation function It is now clear from Equation  that onlEquation  gives riprior quantization condition [E but destructive interference upon integration over the trajectory space, which will be discussed below.This is a quantization (resonance) condition for ondition ,115. NotYet, the true peaks should be contained in the set of peaks fulfilling Equation . Note alEquation , therebyThe off-resonant trajectories should suppress the values of rs, ref. ,112, fora pair of very high spurious peaks” that contain the peak of Suppose there is a peak satisfyingles, see .)n and the energy There is yet another mechanism to generate spurious peaks. A trajectory with Equation  with somEquation  may be s of n,Ex . These hHaving presented the specific roles of off-resonant trajectories to suppress the spurious peaks, we are concerned whether the trace formula implements such a function in it.One of the theoretical and numerical difficulties of the trace formula manifests itself explicitly in the denominator of Equation  of the ss has been introduced. The value To explicitly monitor the sensitivity of the amplitude factor to the variation of the preexponential factor, we first make the following preliminary calculations , in whic spectra . In partℏ, the quality of semiclassical quantization is accordingly deteriorated and should not be used.)After somewhat extensive analysis and numerical studies, we have confirmed that the spectrum positions are accurately reproduced through an amplitude correlation function ,118, of ition q0 ,116. We The partial success of the amplitude-free quantization without paying full attention to the semiclassical amplitude factors, except for the Maslov phase, suggests that one does not necessarily have to faithfully track the every fine complexity of the Lagrange manifold behind classical chaos in order to perform semiclassical mechanics, partly due to the quantum smoothing.Tunneling effect is one of the key phenomena to characterize chemical reactions not only in very low temperature circumstances but even in excited states. It is now widely accepted that complex semiclassical theory is very effective in the study of quantum tunneling . HoweverWe first need to formulate a tunneling theory in a semiclassical context ,123. Letintegral . It is sernativeK→Insertion of Equation  into EquN is the number of dimensions and the additional phase The tunneling paths induce additional phases when they are incorporated into the semiclassical Feynman kernel such that (89)K to the direction of zero momentum.The stationary phase condition in the semiclassical approximation to the Feynman kernel requires that the quantum phase Equation  should bsolution . TherefoTunneling phenomena are quite important in many chemical reaction dynamics, and some applications of the present treatment are found in Refs. ,127,128.In view of the nature of this review article, we briefly show an example of chaos that is induced by tunneling dynamics, just for curiosity. The Hamiltonian isnonadiabatic interactions as one of the key factors to modulate the structure of the energy levels. As such, the theory of level dynamics [nuclear kinematic coupling on the occasion of the variation of nuclear configuration (molecular geometry) and are one of the most critical interactions in chemical dynamics [One of the characteristic features of quantum chaos is reflected on the energy level statistics such as the patterns in the nearest neighbor level spacing distribution (NNLSD) ,129. Thedynamics ,131,132 dynamics ,134 expldynamics ,19,,20. bifurcate and coherently merge (remix) many times at the molecular geometry of significant nonadiabatic interactions. It is quite natural therefore to suspect that the wavepacket bifurcation and merging should be the dynamical origin of purely quantum chaos in place of “stretching and folding” in the baker’s transformation for classical chaos. We hence survey the nuclear wavepacket dynamics of bifurcation and remixing in this section first and then proceed to the nonadiabatic electron wavepacket dynamics in the next section.Besides, we are quite often concerned with the wavepacket dynamics of electrons rather than the electronic energy levels at a given nuclear configuration. In other words, the essential characteristics of molecular nonadiabatic dynamics is seen when the relevant quantum wavepackets We here show that “bifurcation and merging” of quantum wavepackets can be observed with the modern real-time experiments. Among the cutting-age experimental methodologies to track the femtosecond-scale nuclear motions in chemical dynamics, the time- and angle-resolved photoelectron spectroscopy (TRPES) has been very well established since 1999 ,136,137,For readers who might be interested in the recent experimental progress since then, which are related to observation nonadiabatic dynamics, we refer to the following works (we apologize for not a complete list). T. Suzuki is the first who used photoelectron imaging technique to observe the dynamics passing through an intersystem crossing of pyradine ,144. Wörtroscopy  and timetroscopy . Furthertroscopy , time-retroscopy , ultrafatroscopy , and so troscopy .The basic experimental setting for the time-resolved photoelectron spectroscopy is as follows: (1) First excite a ground state wavepacket to an electronically excited state by a pump pulse-laser. This laser is of a sufficiently short width so that the nuclear wavepacket thus excited begins to run spontaneously, which eventually passes across the nonadiabatic region(s). Then, (2) shine another laser (probe laser) to ionize the excited state. (3) Measure the energy and angular distributions of photoelectrons as a function of the time delay (the time lag between the probe and pump), and thereby retrieve the information of the dynamics of the nuclear  dynamics. This TRPES technique is quite successful in that there is no closed channel to be probed.The theoretical framework of TRPES is outlined below ,139,140.ondition  with k lechanics ,139,140.T is a nuclear kinetic energy (nuclear masses are assumed to be unity), i, J the nonadiabatic coupling constant between the electronic states induces an interaction between the two otherwise uncoupled harmonic potentials. a and x and y. Unlike the original Heller model, we chose the frame for We next survey how the infinite repetition of bifurcation and merge of the quantum wavepackets results in the specific signature in eigenfunctions. Fujisaki systematically investigated this aspect by placing a focus on the transition from regular to chaotic states of the wavefunctions as functions of energy and the magnitude of nonadiabatic coupling elements ,152. TheJ and A) are shown. The definition of the amplitude distributions isSince the Heller model is simply an uncoupled pair of harmonic potentials at nd so on . Here innd so on , on one butions) ,152. Thuenergy levels has been extensively studied R(t)  definedor  .I, which gives the theoretical foundation for the so-called ab initio dynamics, molecular dynamics, the first principle dynamics, and so on.The non-zero off-diagonal elements of ths R(t) . If, on To save the work needed for the path branching calculations , we ofteH˜  is givenHence only the atom We here define the Lorentz-like nonadiabatic force with the second term of the right hand side of this equation as (117)fIJThe expression in Equation  can be rIn terms of the vector m A only .left-handed rule in classical electromagnetic theory. The Lorentz force distinguishes the front and back sides of a plane made up by an electron current and a magnetic field. See It is immediately noticed that this expression is essentially of the same mathematical form as that behind the Fleming We thus recognize that the nonadiabatic force can distinguish right and left with respect to the plane of the vector field of nonadiabatic interactions, which can serve as one of the fundamental mechanisms of the origin of molecular chirality. Yet, further study is needed to quantify the actual magnitude of anisotropic force.As a candidate among molecules possibly having electronic-state chaos, we study some of the highly excited states of a cluster consisting of 12 boron atoms, Bc states ,164,165,In contrast to classical chaos, there is no crisp criterion with which to define or identify chaos in quantum dynamics, to the best of our knowledge. Below we only quantify the extent of chaoticity of the relevant electron dynamics represented as in Equation . A speciAt any selected time Equation . The enesee ref.  and its It is experimentally hard to pick and prepare a pure adiabatic excited state out of the densely quasi-degenerate electronic state. Even if one could successfully pump the ground state to one of the highly adiabatic states, the resultant state would quickly begin to undergo continual nonadiabatic mixing with other states, each of which in turn further mixes with other states. These enduring state-mixings look like a diffusion as a whole in the electronic Hilbert space. Equation  in a colt starting from this initial To quantify the extent of the diffusive motion of the wavepackets in the electronic state-space explicitly, we survey the scaling property of the dynamics in terms of the fractional Brownian motion (fBm) of Mandelbrot . Consideexamined . Further∑J=1 displays two exponents Equation  is shownEquation  for the To observe one of the physical consequences from the present stochastic electron dynamics, we next survey the internal electron current driven by the nonadiabatic interactions. The probability current density of electrons articles . For manrticles [j→=ℏnly  is pumped to the higher levels by borrowing the kinetic energy of an incident particle A. The Feshbach resonances and their overlapping set the foundation of Intramolecular Vibrational energy Redistribution (IVR) in the unimolecular dissociation dynamics, as discussed in The above diffusive and chaotic phenomena due to the very frequent nonadiabatic interactions prevent molecular dissociation and bring about a notion of quantum resonance we have not seen before. As resonance phenomena the Pauling resonance in the theory of chemical electronic structure  and the hyper-resonance (resonance among the resonant states), since each adiabatic state is already in a Feshbach and/or Pauling resonance state. We refer to such a very long life-time tentative chemical bonding as dynamical bonding [During the continual state-mixing in the diffusive motion in B bonding .Despite the very high energy of those excited-state boron clusters, they do not readily break apart, since the dissociation channels of them tend to be closed by very frequent nonadiabatic transitions. This is a natural consequence of hyper-resonance. We then survey a possible mathematical ground of the long life-time in terms of a drastically simplified statistical theory based on the phase space theory, originally proposed by J. C. Light . This thN-body cluster  Thus, the critical quantity for the statistical rate of the dissociation probability turns out to be the ratioN-body cluster. Then the related entropy We further suppose that the phase space of a system under study has the mixing property and is completely statistical due to chaos. Then we may regard the quantityfor simplicity that the transition among the electronic states occur freely. Then the entropy confined in the individual state Although the state mixing is caused by the interactionEquation , we hereIn such an extreme case the connected phase-space volume within On the other hand, the phase space volume directly connected to the dissociation channel should be represented as the simple sum of each asN-body cluster, and the probability for the cluster to come across the dissociation channel is extremely small. Consequently, the life-time of the clusters is expected to be far longer than in case of no nonadiabatic transitions.Hence the the dissociation probability can be dramatically lowered from the order of Equation  to(147)Nonadiabatic electronic-state chaos thus inflates the volume of phase-space of nuclear motion of clusters to a very large extent. Due to this inflation, the molecular states lose chances to reach one of their dissociation channels. That is, a dissociating molecule can be brought back to the bound state by nonadiabatic interactions, which would dissociate otherwise. Here is an example of such an event . In Figuesonance . This maTo review “quantum chaos” in the scope of chemical dynamics, we herein have tracked the following subjects. We began with full quantum dynamics, wavepackets and eigenfunctions localized in the quasi-separatrix in two-dimensional potential function. This study was made in search for a novel molecular vibrational mode. Indeed, in this small dimensional system lie several vibrational modes, which give birth to very long-time wandering motions in the quasi-separatrix. A pair of the eigenstates of the dynamical tunneling of the second kind have been identified.Semiclassical theory and energy quantization thereof, which is indispensable in the study of molecular vibration were next reviewed. We took a wavepacket approach in studying semiclassical dynamics rather than the direct semiclassical reduction of the Feynman kernel. With this approach, we discussed the mechanism of energy quantization for not only making the peaks but suppressing the off-resonant components. One of the most annoying parts in semiclassical kernel and related quantities is the divergence at caustics and/or turning points, which casts an inherent difficulty to the Gutzwiller trace formula. To avoid the difficulty, we have shown that the amplitude-free energy quantization actually works quite well. Chaotic paths that appear only through quantum tunneling have been shown along with a semiclassical tunneling formalism based on the notion of the parity of motion.One of the main aims of the present review has been to guide the readers to chaos in molecular nonadiabatic dynamics, which is induced by the kinematic couplings between nuclei and electrons. The entanglement between electrons and nuclei is one of the most important subjects in general chemical science. We have emphasized the critical roles of “bifurcation and merging” of quantum wavefunctions. The most characteristic molecular chaos is found in nonadiabatic electron dynamics in the densely quasi-degenerate electronic-state manifold, every state of which mixes together with other members of the manifold. We have shown that the dynamical properties of electronic-state chaos can be measured in terms of the quantities related to diffusion and memory loss in the Hilbert space, turbulent flow of electron current, and so on. The chaos of the electronic excited state should become more and more vital in future molecular science, which will proceed far beyond the Born-Oppenheimer paradigm of the current theoretical framework of chemistry.Some of the characteristics found for molecular quantum chaos are summarized below:The basic mechanism of chaos in nonadiabatic wavepacket dynamics is “bifurcation and merging”, which is a quantum mechanical synonym of “stretching and folding” in the classical baker’s  transformation. Indeed repeated occurrences of “bifurcation and merging” of quantum wavepackets result in genuine chaos in quantum mechanics not only in energy spectra but more importantly in the dynamics of wavepackets. We would like to particularly emphasize again that the bifurcation and merging of wavepackets are indeed physically observable. Interestingly, “bifurcation (branching) and merging of the reaction tubes” sets also a geometrical foundation of classical chaos in structural isomerization dynamics. Extension of the tube geometry to quantum mechanics has not yet been explored.“Loss of memory” is one of the central concepts to characterize chaos in general. The positive Lyapunov exponent in classical systems indicates an exponential separation of nearby orbits, leading to the loss of the memory of their initial stage. In our quantum chaos, the exponential decay of the memory of an electronic state in the time-propagation against the nonadiabatic (kinematic) interactions can take place in the Hilbert space. Again, we have observed that classical chaos in the structural isomerization dynamics of liquid-like clusters is associated with a characteristic feature of memory loss, which leads to a random yet statistical appearance of the isomers in the time series.The advent of the ultrafast pulse laser has revolutionized molecular science since the 1970s. The femtosecond laser chemistry  to date A final remark before closing. The essential nature of classical and quantum chaos may lie in mathematical beauty and intricate geometry behind the complicated-looking phenomena. However, the complexity of molecules often makes such elegant mathematical structures obscure and even less mighty. Moreover, We are afraid that sticking to the too rigorous definitions of quantum chaos would isolate itself from other natural science. Therefore, I would like to conclude this review by stressing that more studies on quantum chaos made mathematically easy and conceptually useful are demanded than ever, since there are so many unknowns left (not only) in the molecular phenomena, notions, and laws that need an understanding of quantum chaos.


Voice conversion (VC) consists of digitally altering the voice of an individual to manipulate part of its content, primarily its identity, while maintaining the rest unchanged. Research in neural VC has accomplished considerable breakthroughs with the capacity to falsify a voice identity using a small amount of data with a highly realistic rendering. This paper goes beyond voice identity manipulation and presents an original neural architecture that allows the manipulation of voice attributes . The proposed architecture is inspired by the fader network, transferring the same ideas to voice manipulation. The information conveyed by the speech signal is disentangled into interpretative voice attributes by means of minimizing adversarial loss to make the encoded information mutually independent while preserving the capacity to generate a speech signal from the disentangled codes. During inference for voice conversion, the disentangled voice attributes can be manipulated and the speech signal can be generated accordingly. For experimental evaluation, the proposed method is applied to the task of voice gender conversion using the freely available VCTK dataset. Quantitative measurements of mutual information between the variables of speaker identity and speaker gender show that the proposed architecture can learn gender-independent representation of speakers. Additional measurements of speaker recognition indicate that speaker identity can be recognized accurately from the gender-independent representation. Finally, a subjective experiment conducted on the task of voice gender manipulation shows that the proposed architecture can convert voice gender with very high efficiency and good naturalness. Voice conversion (VC) consists of digitally altering the voice of an individual, e.g., its identity, accent, or emotion, while maintaining its linguistic content unchanged. Primarily applied to identity conversion ,2, VC haNeural VC, i.e., VC based on neural networks, was first introduced in , followiThrough the multiple and various contributions in neural VC that have been presented over the recent years, an important progressive change in the VC paradigm can be distinguished from the initially agnostic learning of one-to-one VC using parallel datasets to today’s structured and informed learning of many-to-many VC from non-parallel datasets. Following the historical paradigm of one-to-one parallel VC, cycle-GAN and S2S with attention mechanism, VC models have been proposed to learn the acoustic mapping from pairs of sentences from source and target speakers. Inspired by , cycle-GTo overcome the shortcomings of this paradigm, research efforts have gradually moved towards many-to-many and non-parallel datasets, allowing the scalability of neural VC with large and multiple speakers datasets, with the assumption that the increase of data can induce a substantial increase in the quality and naturalness of the VC. Among the first attempts, starGAN VC ,23 was pIn another line of research, VC based on comprehensively structured speech representations has been investigated. According to the fundamental model of speech communication, speech conveys verbal and non-verbal information: linguistic , para-linguistic , and extra-linguistic . VC architectures have started to integrate explicitly linguistic content and speaker identity  informat-An extension of the VC architecture presented in  that all-An implementation of a network for voice attribute disentanglement based on a fader network , an adve-The application of the proposed neural architecture to voice gender manipulation. While this study only focuses on voice gender manipulation, we foresee extending it in the future to manipulate other identity-related attributes such as age, accent, or speaking style.This paper proposes a structured neural VC architecture for manipulating voice attributes by means of disentangling the attributes in the latent representation. The main contributions of the paper can be listed as follows:The remainder of this paper is organized as follows: T time frames, and the sequence of T phonemes The neural VC architecture used in this paper is based on the architecture presented in , which wThe speaker encoder ion task . This inThe content encoder oss , meaning that the time information is fully preserved during encoding.Learning a shared encoding among both audio and text modalities can be related to cross-modal domain adaptation, in which one wants to learn a code that is independent of the input modality or distribution. As a result, the content encoder is trained to transcribe the phonetic content from the speech signal. In this paper, the linguistic embedding has the same length nth time frame:The content recognition loss u denotes a uniform distribution in which all speakers have the same probability S being the total number of speakers in the dataset.In order to disentangle content and identity information, an adversarial strategy is added to remove identity information from the linguistic embedding A decoder essed in  During cIn the previous section, we presented the disentanglement of speech content and speaker identity which is processed adversarially between parallel encoding. In the present section, we introduce further disentanglement of voice attributes by proposing cascade disentanglement using a fader network , as illuThe speaker embedding First, the reconstruction loss of the auto-encoder Second, a discriminator loss Third, an adversarial loss Finally, the total adversarial loss of the fader network can be written asThe model configuration parameters are the same as those described in , with thas Da in ), which Following , our sys and Lim  algorith and Lim . This deAll training runs were performed on a single GPU (GForce GFX 1080Ti). The inference and the mel inversion were run on the CPU (Xeon(R) CPU E5-2630 v4 @ 2.20 GHz), while a number of the figures have been generated using the GPU. The duration of the VC model training is 20 min per epoch with 80 epochs (roughly 27 h), and the training of the gender autoencoder model lasts 1 min and 30 s per epoch with 400 epochs . The inference of one sentence of 1.5 s takes around 2 s for computing the mel-spectrogram plus two seconds for mel inversion when using our CPU.With respect to the training parameters, the VC system makes use of the Adam optimizer with a learning rate equal to The English multi-speaker VCTK corpus  is used Ask her to bring these things with her from the store” is uttered by a male speaker (left) and by a female speaker (right). The top figures show the original signals and the bottom figures correspond to the conversion conditioned on the opposite gender. The gender conversion algorithm clearly transposes the average F0 in line with what we would have done to convert between male and female speakers using traditional vocoders (±1 octave) [ octave) . HoweverTo assess whether the proposed framework is successful in disentangling speaker identity and gender representation, a set of objective evaluations were conducted: a gender classification task (including a short ablation study on the fader structure), a speaker classification task, the mutual information between the embeddings and the true gender, and a 2D visualization of the embeddings.original) or the reconstructed speaker embedding w: with the estimated gender est. gender), by swapping to the opposite gender inv. gender), or by neutralizing the gender de-gender). With the adversarial setting, the original speaker embedding and the reconstructed speaker embedding with the estimated gender have very high accuracies. When swapping the gender by conditioning the reconstructed embedding on the opposite of the estimated gender the accuracy becomes zero, which is expected because the gender is inverted. With reconstruction conditioned on w degrades the EER in all cases; however, these rates remain relatively low, at around 6.8% for the gender estimated from the pre-trained classifier and A Receiver Operating Characteristic curve, or ROC curve, is a graphical plot that illustrates the diagnostic ability of a binary classifier as its discrimination threshold is varied. The ROC curve is created by plotting the true positive rate (TPR) against the false positive rate (FPR) at various threshold settings. The Equal Error Rate (EER) is the error rate of a binary classifier when the operating threshold for the accept/reject decision is adjusted such that the probability of false acceptance and that of false rejection become equal. On the ROC curve, it corresponds to the intersection with the anti-diagonal line. To assess whether the proposed architecture is efficient at converting the gender of the voice, a subjective evaluation was conducted.To the best of our knowledge, there are no neural gender conversion algorithms available in the literature; therefore, we used a traditional signal processing approach as our baseline for perceptual tests. Classic voice transformation algorithms perform gender manipulation by means of modifying the average of the fundamental frequency (F0) and the positions of the vocal tract resonances (known as formants). Due to physiological differences between the female and male voice organs, notably the size of the vocal folds and vocal tract, these two parameters have average values which generally differ for male and female voices. These differences have been measured and documented in the literature ,49. Cons(1)feminine, rather feminine, uncertain, rather masculine, or masculine;whether the voice is typically perceived as: (2)bad) to 5 (perfect), which is commonly used for experimental evaluation of Text-To-Speech and Voice Conversion systems.the sound quality on a standard Mean Opinion Score (MOS) 5-degree scale from 1 (The task consisted of listening to one speech sample (converted or not) and judging the following:(1)True) and converted audio signal with:the original audio signal ((2)VC);the original VC system ((3)phase voc.; see supplementary for details) with two cases: female-to-male conversion (f2m) and male-to-female conversion (m2f);a phase vocoder ((4)base) with five conditioning values of the parameter the VC system with the proposed gender autoencoder ((5)nofader), with the five values of the the VC system with the gender autoencoder but trained without the fader loss ((6)adapt) with the five values of the the VC system with the gender autoencoder with the VC decoder re-trained  and two females (p253 and p300), with five randomly chosen sentences per speaker in the validation set. Six configurations were compared , This paper presents a structured neural VC architecture that allows the manipulation of voice attributes  based on adversarial learning of a hierarchically structured speech and speaker encoding. The proposed VC architecture employs multiple auto-encoders to encode speech as a set of idealistically independent linguistic and extra-linguistic representations, which are learned adversarially and can be manipulated during VC. Moreover, the proposed architecture is time-synchronized, meaning that the original voice timing is preserved during conversion; this enables its use in lip-syncing applications. A set of objective and subjective evaluations conducted on the VCTK dataset shows the efficiency of the proposed framework on the task of voice gender manipulation. Our further work will investigate the generalization of the proposed framework to other voice attributes, such as age, attitude, and emotion.


N-linked is the most common type of glycosylation which plays a significant role in identifying various diseases such as type I diabetes and cancer and helps in drug development. Most of the proteins cannot perform their biological and psychological functionalities without undergoing such modification. Therefore, it is essential to identify such sites by computational techniques because of experimental limitations. This study aims to analyze and synthesize the progress to discover N-linked places using machine learning methods. It also explores the performance of currently available tools to predict such sites. Almost seventy research articles published in recognized journals of the N-linked glycosylation field have shortlisted after the rigorous filtering process. The findings of the studies have been reported based on multiple aspects: publication channel, feature set construction method, training algorithm, and performance evaluation. Moreover, a literature survey has developed a taxonomy of N-linked sequence identification. Our study focuses on the performance evaluation criteria, and the importance of N-linked glycosylation motivates us to discover resources that use computational methods instead of the experimental method due to its limitations. The process of glycosylation is considered to be one of the most complex type of post translation modification (PTM) in eukaryotes cells . The posThere are various techniques to identify such sites, broadly it can be classified into experimental and computational method . The expThe N-linked glycosylation is the primary glycosylation type, as 90% of glycosylated sites belong to the N-linked glycosylation . UsuallyThe glycosylated region of N-linked sites appears at the specific location within the protein sequence, as protein sequence consists of the chain of amino acid and each amino acid out of known 20 is represented by specific alphabetic character . In compThe existing reviews are compared on various perspectives such as quality assessment scores, availability of N-linked model, feature set construction method, training model algorithm, specie type, performance metric and target repositories as shown in The rational of our work is to provide the comprehensive systematic literature review on the identification of N-linked sites to bring out the detail of exiting computational models. The researchers have performed numerous efforts to identify such sites computationally in the recent past. The work presented by these researchers has been reviewed by few authors to ensure the effectiveness of the proposed prediction model to identify the N-linked sites . The autThis SLR will help the medical scientists in the targeted identification of cancer, type I diabetic cell for treating the patients, and help the pharmacists in effective drug development by opting the accurate predictor of N-Linked sites. Furthermore, it will facilitate the researchers to develop more accurate and efficient predictive model by analyzing the techniques used by existing researchers.The proposed article is presented in the following sequence: the methodology adopted to conduct survey along with objectives and research questions is presented in “Survey methodology”. The analysis of the research question is described in “Assessment and discussion”. The “Discussion and future direction” presents synthesis of reviewed literature. Finally, the article has been concluded in “Conclusion”.The survey methodology consists of three phases: plan, conduct of review and conclusion as shown in The process involved to conduct the review is shown in The steps involved to conduct the review were: (a) Search of relevant primary study from different search venues. (b) Selection of relevant research articles from searched articles obtained in previous step through predefined inclusion/exclusion criteria. (c) The selected articles were then assigned score based on their defined quality parameters. (d) Backward snowballing to include the important articles.http://scholar.google.com/)Google Scholar (https://ieeexplore.ieee.org/search/advanced)IEEE Xplore (https://link.springer.com/)Springer Link (https://academic.oup.com/bioinformatics)Bioinformatics PLOS ONE  is the major step to conducting any systematic review. In this study, questionnaire has been designed to measure the quality of selected articles. The score is computed on the following criteria:The study has awarded score (1) if N-linked predictive tool has developed, otherwise scored (0).The study has awarded score (2) if the method developed to extract feature from data based on computational approach, score (1) for hybrid approach and score (0) in-case of experimental approach.The study has awarded score (1) if the computation method for training has provided, otherwise scored (0).The score (1) has been awarded if the data set used is available otherwise scored (0).The score (1) has been awarded if the organism type is available otherwise scored (0).The studies were rated by taking conference and journal rating list into account. The possible score for publication is shown in The resultant score has been calculated for each study by aggregating the points of all question. Article achieving minimum score (5) has been included in the review.After performing the quality assessment, back-word snowballing to extract the relevant articles from the references of the selected articles. The articles by The glycosylation sites especially N-Linked identification is very important domain, therefore in this review, systematic and empirical method is adopted to extract the relevant article from the digital libraries mentioned in The shortlisted articles are then filtered based on title, abstract, introduction and examined the full article if required for each search result. The article contains less than four pages and irrelevant articles were eliminated. The results of primary search, filtering and inspection phase, covering five digital libraries, are presented in After the preprocessing of articles, inclusion/exclusion test has been performed and after that quality assessment score has been computed. The article having at least five score have included in this study and it is total of 70 in count as given in In this section, the research questions have been analyzed based on 70 primary studies.To find the relevant publishing channel, channel type and geographical aspects for the N-linked glycosylation sites requires the meta information. To achieve this purpose, channel type, publishing year and demographical distribution is presented for the analysis of selected studies.The importance of selected topic can be evaluated from the yearly publication on the relevant domain. The 28 out of 70 articles has been published in 2021 which also of 40% of selected article as shown in It is clear from It is observed, 42 out of 70 studies have been published in the different regions of the Europe as shown in Quality assessment score for each finalized study awarded according to defined criteria in quality assessment score section, shown in The overall classification result and QA studies have presented in of Furthermore, the sources of finalized studies, and total number/percentage of studies per publication source mentioned in The available tool to identify the N-Linked glycosylation sites and for which kind species it can identify the relevant site is the parameter of this study. There is hierarchy of N-Linked Glycan to PTM. Where PTM is classified into various type and Glycosylation in one of them and glycosylation is further classified into five group and N-linked is one of them.The summarized detail of eight is represented in It is important to specify for which kind of species these tools will be operating, therefore to achieve this purpose the information is also extracted from the selected studies. Some authors  did not The data is the major component to develop any machine learning model . In bioiMost of the authors used the computational feature extraction approach while few used the experimental data obtained from mass spectrometry, human plasma and psycho-chemical method as mentioned in The choice of algorithm to train any predictive model is most important factor which impact the performance of any model . TherefoIt is observed from the selected articles that most of the authors  used theThe result comparisons are used to present the performance to various based on which conclusion can be drawn with respect specific dimension. In this systematic review, the performance comparison of N-linked model in the selected articles has performed. The parameter used for the performance consists of (a) availability of data set. (b) accuracy metric (c) sensitivity metric. (d) specificity (e) availability of developed tool (f) comparison on independent data and the type of glycosylation as mentioned in This section summarizes and discuss the detail of this systematic literature review regarding the identification of N-linked sites.The objective of this study was to analyze the current progress to identify the N-linked glycosylation sites. To achieve this objective, a taxonomy has built based on the coding scheme as mentioned in Feature set construction method The performance of computational model deeply depends on the quality of feature set extracted from the data set which later used for training the machine learning model ((a) ng model . The disng model  only useng model  applied Machine training algorithm The most significant part of computational model after the feature extraction method is to develop the method to train the machine model ((b) ne model . The perne model  widely ane model  method. ne model  in whichne model  who propPerformance evaluation Once the model has trained, it then validated on the independent data to evaluate the performance. There are various techniques to measure the validity of model, the most significant metrics to evaluate the performance are Accuracy metric, Sensitivity and Specificity metric. The sensitivity test measures the true positive accuracy of a model while specificity measures the true negative accuracy of the model. In this study, the performance has evaluated on aforementioned metrics. Around 40% of the authors have not validated their model on any of above mentioned performance metrics. Only 20% of the authors have performed each of the defined performance metrics. The predictive models in which PTM type is specialized to N-linked have better accuracy as compared to those in which PTM type is not specified or are the generalized ones. The highest accuracy of −99% was achieved by author (c) Several possible observations can be made in the finding of this SLR based on the taxonomy as shown in (a) Identify the O-linked glycosylation sites for threonine and serine using ANN.(b) How the performance of C-linked glycosylation can be enhanced through exiting neural network classifiers.(c) Develop a comprehensive predictive model to classify the type of glycosylation.(d) How effective are the exiting classifier to predict the other PTM sites?Bioinformatics is an emerging filed, there are lot of problems that needs the computational solution over the experimental. As it was mentioned earlier, the researchers have identified almost ∼ 200 types of PTM which plays key role in various biological functions. Apart from N-linked glycosylation, the other types of glycosylation such as O-linked and C-linked also play vital role in protein functioning and various drug discovery techniques. Therefore, it is the opportunity for the researchers, pharmaceutical and academia to develop the efficient computational model to solve the problem that needs better computational solution. Few of the existing problem that needs to be addressed are given belowThe significance of N-linked glycosylation promotes the discovery of such sites using computational methods instead of experimental method due to its limitations. In this systematic study, existing information to identify such sites was studied which covered the possible challenges and their solutions through systematic method. The research articles, related to the keywords associated with N-linked glycosylation were evaluated through five major digital libraries. In the result of search query applied to digital libraries, more than 800 articles have found and after filtering process 70 article were remained for further analysis. The results show that approximately 75% of the articles were published in recognized journals and rest belong to top conferences. It was observed that more than 40% of articles were published in the American journal followed by the Middle East with 20%. Most of the selected studies focused on the feature construction method and training algorithm, but less focused on the performance evaluation criteria and development of tool or web server.The major shortcomings of any SLR primarily are related to search strategy, poor classification, and inaccurate data extraction. In this SLR, these deficiencies were overcome by applying the search query on five major digital libraries to reduce biasness. The results of search queries were then filtered through well-defined inclusion/exclusion criteria.


A randomized crossover trial was designed to investigate the influence of muscle activation and strength on functional stability/control of the knee joint, to determine whether bilateral imbalances still occur six months after successful anterior cruciate ligament reconstruction (ACLR), and to analyze whether the use of orthotic devices changes the activity onset of these muscles. Furthermore, conclusions on the feedforward and feedback mechanisms are highlighted. Therefore, twenty-eight patients will take part in a modified Back in Action (BIA) test battery at an average of six months after a primary unilateral ACLR, which used an autologous ipsilateral semitendinosus tendon graft. This includes double-leg and single-leg stability tests, double-leg and single-leg countermovement jumps, double-leg and single-leg drop jumps, a speedy jump test, and a quick feet test. During the tests, gluteus medius and semitendinosus muscle activity are analyzed using surface electromyography (sEMG). Motion analysis is conducted using Microsoft Azure DK and 3D force plates. The tests are performed while wearing knee rigid orthosis, soft brace, and with no aid, in random order. Additionally, the range of hip and knee motion and hip abductor muscle strength under isometric conditions are measured. Furthermore, patient-rated outcomes will be assessed. The incidence of anterior cruciate ligament (ACL) tears has been reported as 68.6 per 100,000 cases per year . Long-teMultimodal programs are already established and focus on strength, flexibility, proprioception, and balance training. In addition, feedback training, understanding of biomechanics, and correct landing patterns are becoming increasingly important in the rehabilitation process ,12. NeveLess than ten years ago, the Back in Action (BIA) test battery, a standardized tool for deciding whether a patient is ready for RTS after an ACLR was developed and published . The tesKnee orthotic devices are often recommended and used after ACLR. However, it is neither proven if there is a mechanical advantage of using them nor which orthotic device has the most effective influence on body balance and coordination and, thus, the function of the knee joint . In a reSoft or rigid orthotic devices may support feedback strategies through skin contact and thus increase the subjective feeling of safety, resulting in a willingness to perform. In addition, braces and orthoses provide patients with security  but are not mechanically proven. The immediate clinical relevance is that orthotic devices either positively or negatively influence the tested performance on RTS or not.For sporting performance and benefits to be taken from orthotic devices, the feedforward and feedback strategies model is becoming increasingly important. Feedforward and feedback control are generally explained as mechanisms to restore homeostasis in an organism . TherefoSurface electromyography (sEMG), is an objective method to analyze muscular activation and coordination patterns during the execution of movements . Cross eFurthermore, the role of gluteus muscles on DKV is not fully clarified . Due to Motion tracking systems are a common instrument to track postural control . The MicFurthermore, device-based strength measurements are considered the gold standard for assessing muscle function . TherefoThe influence of rigid orthoses and soft braces on the performance of the knee jointThe activation of the gluteus medius and semitendinosus muscles and the associated influence on the cumulative DKV during landing maneuversConclusions on feedforward and feedback mechanisms during the test batteryTo the best of our knowledge, there have been no studies conducted to investigate the following aspects using the specifically designed BIA test battery, yet:® company, Medi Bayreuth, Germany) or no aid.In this study, we will perform the functional and biomechanical assessment of patients six months (+/−1 month) after unilateral primary ACLR using an autologous hamstring tendon graft with a test battery, using either a knee rigid orthosis, knee soft brace  concerns that individual use of supports/orthoses will affect athletic performance and RTS test results, which may help athletes who like to continue wearing aids for psychological reasons to decide on their usefulness. Based on the research questions, it can be hypothesized:Hypothesis 1:Soft braces or rigid orthoses improve functional performance during the modified BIA test battery.Hypothesis 2:Muscle activity of the gluteus medius and semitendinosus muscles is impaired in the ACL-reconstructed leg during the modified BIA test battery.Hypothesis 3:Impairment of gluteus medius and semitendinosus muscle function is related to lower performance in the operated limb in the modified BIA test battery.Hypothesis 4:An impairment of gluteus medius strength and semitendinosus muscle activation correlates with DKV and pelvic drop emergence during the modified BIA test battery.Ethical approval is currently under consideration in the county medical chamber as an extension to a previously approved ethical application. The processing number is 22-TEMP897788-BO.The assessment will be conducted in the Rehathleticum, a functional diagnostic physiotherapy center in Berlin (Germany).A signed informed consent form and the confirmation of personal data protection will be obtained. Participants can withdraw from the study at any time without giving a reason. All data is anonymized by assigning a patient number so that a later assignment of personal data is only possible via the Sporthopaedicum in Berlin and the University Hospital of Brandenburg, Center for Orthopaedics and Traumatology. All measures are taken to protect the privacy of the subjects and the confidentiality of personal data.This study is a randomized, prospective, single-blind clinical trial with a three-group cross-over design. Due to the design of the study, complete blinding is not possible. Statistical analysts are blinded on the groups to be compared. The randomization procedure and allocation are done through a person who was not further involved. SJ and CC are responsible for the test setup and finally carry out the tests.Patients are only informed about the performance but not about the target character of the tests. After receiving written informed consent, three patients are randomly assigned to either group A, B, or C  and therThe bioelectrical activity of the gluteus medius and semitendinosus muscles are recorded using sEMG according to Surface ElectroMyoGraphy for the Non-Invasive Assessment of Muscles (SENIAM) guidelines . SurfaceParticipants will be recruited at the Sporthopaedicum, Center for Orthopaedics, Sports Medicine and Orthopaedic Surgery in Berlin (Germany) and the University Hospital of Brandenburg, Center for Orthopaedics and Traumatology (Germany). Suitable patients are selected via a file review and invited for a surveillance test. The test battery is performed six months (+/−1 month) postoperatively to observe the greatest possible difference while reducing the risk of reinjury to a minimum.The studied cohort of 28 patients consists of female and male patients. Only a large impact on performance is internally valid for estimating whether braces or orthoses should be worn as they are used to reduce adverse events. For this reason, a potential cut off for wearing braces or orthoses was set at a large estimated effect. A large effect for the investigated three degrees of freedom, with an acceptable alpha error of 0.05 and a 1-beta error of 0.9, results in a required total sample size of 27 when using an ANOVA: fixed effects, special, main effects, and interactions model. To catch up one possible drop out, which is not expected, a sample size of 28 is included.six months (+/−1 month) after post-traumatic primary unilateral arthroscopically assisted ACLR, with the use of autologous ipsilateral semitendinosus graft;no additional procedures including meniscus sutures during the arthroscopically assisted ACLR;no osteoarthritis surgery during the arthroscopically assisted ACLR other than shaving;no history of injury or disease in the ACL-reconstructed limb prior to the ACL injury within the last 6 months;no history of injury or disease in the adjacent joints, contralateral limb, or spine within the last 6 months;age 18–40 years;2;Body Mass Index (BMI) smaller or equal to 24.9 kg/mmoderate activity with regard to the rehabilitation plan, physiotherapy, and training therapy;frequent recreational sporting activity prior to the ACL injury.secondary ACLR;ACLR with the use of a method other than autologous ipsilateral semitendinosus tendon graft;diagnosed additional systematic diseases;current impaired performance due to previous injuries apart from the ACL injury;current permanent knee pain or swelling.The examination is performed in a single session, including three runs of the modified BIA test battery.The patients are asked to abstain from unaccustomed strenuous exercise for at least 24 h before the testing and avoid eating a heavy breakfast in the morning before the test and no rich meals within two hours of the test. They are advised to wear comfortable sports outfits and indoor sports shoes. Only the double-leg and single-leg stability tests are carried out barefoot.SEMG electrodes are applied to all participants on the ACL-reconstructed and unaffected legs on gluteus medius and semitendinosus muscles to measure muscle activation throughout the whole test battery.After block randomization to determine the group, participants complete the test battery as presented in Patients perform a double-leg stability test and single-leg stability tests on the Zebris Balance Force Plate, starting with the non-injured leg. While maintaining stability for 10 s on one leg with arms crossed in front of the chest, the contralateral leg needs to be flexed and is not allowed to touch the ground. If balance is lost, the test has to be repeated . The cenOne practice trial with submaximal power  is given, followed by three subsequent test trialsHands are placed on the hips during the jumpsStarting position in the center of the plate, with feet parallel and shoulder-width apartKnee and hip joints are extended during the flight phaseSoft landing in the center of the 3D force plate with knee over toe position, stable leg axisSingle-leg landing: the contralateral leg is not allowed to touch the ground, otherwise the test is stopped immediatelyAll tests are explained in detail and demonstrated by the tester. The subject is given time to ask questions about the performance of the jumps. All tests are guided by the same examiner.The patient bents the knees until a semi-squat (knee ～90°), hips flexed, and then jumps explosively upward, trying to reach maximum height in the shortest possible time . The subject stands with one leg in the center of the plate and performs the jump as described above. The uninjured leg is tested first followed by the injured leg .Starting from an upright position on the top of a vaulting box, the subject drops down from the box onto the center of the force plate considering the shortest possible ground contact time (<250 ms). This is followed by an immediately performed “vertical push-off action as quickly and as explosively as possible in order to perform the highest possible jump in the shortest possible time; thereby using an eccentric-concentric muscle action” . The secStarting from an upright position on the top of a vaulting box the patient takes a step forward and drops down onto the center of the 3D force plate. The landing is performed softly and with a stable leg axis on one leg. The patient must hold the position for five seconds without losing the balance before stepping off the force plate. The uninjured leg is tested first followed by the injured leg.The jump coordination path is performed with the Speedy Basic Jump Set .The patient performs a coordination parkour as quick as possible without breaks. Then, 16 single-leg jumps through a course of red (forward-backward-forward jumps) and blue (sideways jumps) hurdles has to be completed. The hips have to be parallel during the performance. The contralateral leg is not allowed to touch the ground and the limbs are not allowed to touch the hurdle . Time isThe patient completes 15 steps in and out with one foot at a time, while being allowed to use the arms to keep the balance. A repetition is defined when the starting leg returns to the initial position. In case of a mistake or a step on the speedy pole, the test is stopped immediately . Time isThe patient lies supine with the pelvis fixed straight on an examination table, arms lay along the upper body, and the extremity to be tested is abducted by 30°. Starting with the uninvolved leg, the examiner fixes the expander two centimeters proximal to the lateral malleolus. No support from the hands or the opposite limb is allowed during the test. On command the patient pulls the leg to the side evenly and without momentum (4–6 s) while the tester fixes the expander with the help of a weight. It is important that only the muscle group to be tested, the abductor muscles of the hip, is activated. The sEMG records the muscle activation.The patients complete the International Knee Documentation Committee (IKDC), the ACL-Return to Sport Injury Scale (ACL-RSI) and the Tegner Activity Scale (TAS). The patient-reported results on symptoms, recovery and functionality of the knee are crucial and of great importance as they represent the patient’s subjective assessment in addition to the objective biomechanical analysis.Prior to electrode placement the skin is cleaned and degreased with alcohol. Surface electrodes with a two cm center-to-center distance are attached along the direction of the muscle fibers on the evaluated muscle bellies. The signals are registered with 16-bit accuracy at a sampling rate of 1500 Hz and stored for subsequent analysis using Noraxon DTS unit (Noraxon USA). The EMG signals are filtered with a Butterworth high-pass filter (cutoff frequency 10 Hz) and a low-pass filter (cutoff frequency 500 Hz). The sEMG signal from the evaluated muscles are measured continuously during the modified BIA test battery.The collected data is normalized to a reference value. Maximum voluntary contraction (MVC) is performed separately for each muscle investigated before the test trials. The MVC contractions is performed against a static resistance. The parameters of each muscle are expressed as a percentage of its MVC (%MVC).®, OH, USA). The piezoelectric force plate provides data on forces and moments in three dimensions . The Bertec assesses the forces and moments acting at the contact point where the force is applied.Kinematic data is recorded with the 3D force plate Bertec FP4060-08 , fixed on tripods 90 cm above the ground and placed approximately three meters from the circle center. The motion capture system is synchronized to the sEMG system and force plate using an illuminated ball controlled by the sEMG software. This ball is placed to be seen by one camera. From the color stream of this camera, we infer the ball’s on and off phases. Subsequently, the synchronization shift is calculated.The Azure Kinect Body Tracking SDK delivers 3D joint positions and orientations for each joint segment. From this, 3D knee angles are obtained by regarding the left and right ankle, knee, and hip joint positions by calculating the angles between the hip-knee and knee-ankle vectors. Additionally, the local 3D joint segment orientations are used to calculate flexion and extension, abduction and adduction, and internal and external rotation. The resulting kinematic and angular data is filtered using a 4-th-order Butterworth filter with a cutoff frequency of 10 Hz.Studies show good reliability when using the Azure Kinect for Motion analysis purposes . So far,All data are presented in a descriptive manner, including anthropometric data and absolute measured values. Side-to-side differences are calculated for all SL tests, using the limb symmetry index (LSI). Therefore, absolute values of the ACL-reconstructed leg are divided by the values of the unaffected leg and multiplied by 100. For the stability, quick feet and speedy tests LSI is calculated by dividing the measured value of the unaffected leg by the value of the ACL-reconstructed leg and multiplied by 100 according to the calculation done in the study by Herbst et al. .Power is quantified in the sagittal plane and defined through force and velocity. This parameter is directly extracted from the force plate data and reported as extracted. It is assessed for jumping during loading phase and total jump height. Power is based on peak power calculations for the lower extremity, which is defined in newton for maximal applied force during contraction. Additionally, it will be interpreted in the context of recent LSI-papers.Descriptive quantitative data are calculated as the mean ± standard deviation. The Kolmogorov-Smirnov test is used to test the normality assumption. To compare the variances across the means of the three groups we use ANOVA repeated measures. Wilcoxon and Friedman-Test are used as well to calculate between-group differences.All statistical analyses are performed using SPSS . The significance level for all statistical tests is set a priori to 0.05. A post-hoc power analysis is conducted when significant effects are found.The purpose of the present study was to investigate the impact of orthotic devices and to analyze muscle activation and strength on functional stability of the knee during the modified test battery six months after ACLR.It might be possible that soft braces or rigid orthoses improve functional performance during the modified BIA test battery. However, it is also possible that patients may be hindered in performing the test by wearing these aids, resulting in a decrease in functional performance.As already mentioned, feedback and feedforward strategies play a major role in the completion of this test battery as well as in the general practice of sports ,24.We assumed that an ACL injury as well as ACLR negatively influence the mentioned feedback and feedforward strategies. Therefore, athletes may benefit from consistent and long-term use of soft braces or rigid orthoses at this time until they reach their initial performance level.On the other hand, if the performance deteriorates due to the wearing of soft braces or rigid orthoses, the fundamental use of these aids in the rehabilitation process would have to be discussed and investigated more closely.In addition to functional performance, the psychological aspect of wearing soft braces or rigid orthoses should also be considered.Through to semitendinosus stripping with only partially regeneration of the tendon and postoperative immobility it might be possible that the sEMG results of the medial knee flexor show a strong imbalance of muscle activation patterns between the affected and unaffected limb and thus show worse results in the BIA test battery. In this case one could discuss the following aspect. Since the ischiocrural muscles act as agonist of the ACL function and as a medial stabilizer one could focus more on targeted neuromuscular training in order to strengthen these muscles. On the other hand, one could also discuss whether the use of other ACL grafts such as peroneus longus split could be an alternative to the semitendinosus graft as it has no primary influence on the biomechanics of the knee.Ueno et al. found that increased knee adduction is associated with pelvic tilt and reduced gluteus medius strength and could therefore increase the risk of ACL injury . MoreoveIn summary, the obtained results will provide further information on the rehabilitation process. Accordingly, conclusions can be drawn, and training adapted to avoid reinjuries, e.g., due to muscular imbalances. However, if the tests provide excellent results, this would be a reason to establish jump tests earlier in the rehabilitation process in order to be able to release people earlier into their regular everyday life. For personal and economic reasons, this could be especially important for professional athletes so that they can hopefully return to their preinjury level as quickly as possible.