Conventional and microwave-assisted synthesis of new indole-tethered benzimidazole-based 1,2,3-triazoles and evaluation of their antimycobacterial,antioxidant and antimicrobial activities

A new series of triheterocycles containing indole–benzimidazole-based 1,2,3-triazole hybrids have been synthesized in good yields via a microwave-assisted click reaction. All the compounds were characterized by IR, \(^{1}\hbox {H}\) NMR, \(^{13}\hbox {C}\) NMR and mass spectroscopy and were evaluated for their in vitro antitubercular activity against the Mycobacterium tuberculosis H37Rv strain. Compounds 4b, 4h and 4i displayed highly potent antitubercular activity with MIC 3.125–6.25 \(\upmu \hbox {g}/\hbox {mL}\). The antioxidant potential was evaluated using 2,2-diphenyl-1-picryl hydrazine and \(\hbox {H}_{2}\hbox {O}_{2}\) radical scavenging activity, and compounds 4e,4f and 4g showed excellent radical scavenging activity with \(\hbox {IC}_{50}\) values in the range of 08.50–10.05 \(\upmu \hbox {g}/\hbox {mL}\). Furthermore, the compounds were evaluated for antimicrobial activity against numerous bacterial and fungal strains, and compounds 4b, 4c and 4h were found to be the most promising potential antimicrobial molecules with MIC 3.125–6.25 \(\upmu \hbox {g}/\hbox {mL}\).     

Design,synthesis, in silico and biological evaluation of novel 2-(4-(4-substituted piperazin-1-yl)benzylidene)hydrazine carboxamides

A series of novel 2-(4-(4-substituted piperazin-1-yl)benzylidene)hydrazinecarboxamide derivatives has been successfully designed and synthesized to evaluate their potential as carbonic anhydrase (CA) inhibitors. The inhibitory potential of synthesized compounds against human CAI and CAII was evaluated. Compounds 3a–n exhibited \(\hbox {IC}_{50}\) values between \(1.89{-}415.1\,\upmu \hbox {M}\) against CAI and \(0.62{-}66.9\,\upmu \hbox {M}\) against CAII. Compound 3g was the most active inhibitor, with an \(\hbox {IC}_{50}\) value of \(0.62\,\upmu \hbox {M}\) against CAII. Molecular docking studies of compound 3g with CAII showed this compound fits nicely in the active site of CAII and it interacts with the zinc ion (\(\hbox {Zn}^{2+}\)) along with three histidine residues in the active site. Molecular dynamics simulation studies of compound 3g complexed with CAII also showed essential interactions which were maintained up to 40 ns of simulation. In vivo sub-acute toxicity study using 3g (300 mg/kg) was found non-toxic in adult Wistar rats.     

Synthesis,insecticidal activities and structure–activity relationship study of dual chiral sulfilimines

To investigate the “methyl” impact on bioactivity of sulfiliminyl dicarboxamides, a total of 16 novel N-cyano and N-trifluoroacetyl sulfiliminyl dicarboxamides containing m-heptafluoroisopropylated aromatic amino moiety were studied. Two series of sulfiliminyl substituents were designed, synthesized and evaluated against oriental armyworm (Pseudaletia separata Walker) for their insecticidal activities. Their chemical structures were established by corresponding \(^{1}\hbox {H}\) NMR, HRMS and optical polarimetry. Bioassay results revealed that some of the title compounds showed potent insecticidal activities against oriental armyworm. Notably, compounds IIa, IIIa, IVa exhibited 100% activity at \(1\, \hbox {mg}\,\hbox {L}^{-1}\), in particular, IIa showed a comparable control efficacy to that of the commercial product flubendiamide. The SAR of these N-cyano sulfiliminyl isomers can be summarized as follows (Sc, Ss) \(\ge \) (Sc, Rs), while the N-trifluoroacetyl sulfiliminyl isomers is (Sc, Rs) \(\ge \) (Sc, Ss). Comparative molecular field analysis indicated that an electropositive substituent, \(\hbox {CH}_{3}\) group in the benzene ring was very important for the improvement in biological activity. These results could hold promise for novel chiral sulfiliminyl RyR regulators.     

Synthesis and evaluation of bile acid amides of $$\alpha $$-cyanostilbenes as anticancer agents

A series of amino-substituted \(\alpha \)-cyanostilbene derivatives and their bile acid (cholic and deoxycholic acid) amides were designed and synthesized. A comparative study on the anticancer and antibacterial activity evaluation on the synthesized analogs was carried against the human osteosarcoma (HOS) cancer cell line, and two gram ?ve (E. coli and S. typhi) and two gram \(+\)ve (B. subtilis and S. aureus) bacterial strains. All the cholic acid \(\alpha \)-cyanostilbene amides showed an \(\hbox {IC}_{50}\) in the range 2–13 \(\upmu \hbox {M}\) against human osteosarcoma cells (HOS) with the most active analog (6g) possessing an \(\hbox {IC}_{50}\) of \(2\,\upmu \hbox {M}\). One of the amino-substituted \(\alpha \)-cyanostilbene, 4e, was found to possess an \(\hbox {IC}_{50}\) of \(3\,\upmu \hbox {M}\). An increase in the number of cells at the sub-\(\hbox {G}_{1}\) phase of the cell was observed in the in vitro cell cycle analysis of two most active compounds in the series (4e, 6g) suggesting a clear indication toward induction of apoptotic cascade. With respect to antibacterial screening, amino-substituted \(\alpha \)-cyanostilbenes were found to be more active than their corresponding bile acid amides. The synthesized compounds were also subjected to in silico study to predict their physiochemical properties and drug-likeness score.     

Synthesis of novel 7-aryl and 7-spiropyrazolo[4′,3′:5,6]pyrido[2,3-<Emphasis Type="BoldItalic">d</Emphasis>]pyrimidine derivatives and their study as AChE inhibitors

An efficient route for the synthesis of novel 7-aryl and 7-spiropyrazolo[4\(^{\prime }\),3\(^{\prime }\):5,6]pyrido[2,3-d]pyrimidine derivatives is described. These compounds were obtained by a cyclocondensation reaction between pyrazolopyridinediamines 4 and aldehydes 5 or cyclic ketones 6 in the presence of acetic acid as catalyst. This procedure provides the desired compounds in good yields under a simple two-step methodology. The obtained compounds were evaluated as AChE inhibitors and showed weak AChe inhibition with \(\hbox {IC}_{50} = 115{-}470 \, \upmu \hbox {M}\).     

Synthesis of first ever 4-quinolone-3-carboxylic acid-appended spirooxindole-pyrrolidine derivatives and their biological applications

A series of 4-quinolone-3-carboxylic acid-containing spirooxindole-pyrrolidine derivatives was synthesized via multicomponent 1,3-dipolar cycloaddition reactions of azomethine ylides with new (E)-4-oxo-6-(3-phenyl-acryloyl)-1,4-dihydroquinoline-3-carboxylic acids in good yields with high regioselectivity. The cycloadducts were characterized by analytical and spectral data including \(^{1}\hbox {H}\), \(^{13}\hbox {C}\), 2D NMR and mass spectroscopy. The structure of one of the compounds (8a) was investigated theoretically by computational techniques. DFT studies support the proposed mechanism for this cycloaddition reaction. Furthermore, antibacterial activities of the new compounds were evaluated against Gram-positive and Gram-negative bacterial strains. Compounds 8f, 8m and 8p showed potent inhibition activities against selected bacteria. The in vitro cytotoxicity of spirooxindole derivatives (8a–r) was evaluated against MCF-7 breast cancer cell line. Among the various compounds tested, compound 8f \((\hbox {IC}_{50} = 18.35~\upmu \hbox {M})\) showed significant cytotoxic activity compared to the standard drug doxorubicin \((\hbox {IC}_{50 }= 15.00~\upmu \hbox {M})\).     

Synthesis,in vitro $$\alpha $$-glucosidase inhibitory activity,and in silico study of ( E)-thiosemicarbazones and ( E)-2-(2-(arylmethylene)hydrazinyl)-4-arylthiazole derivatives

This study is focused on the identification of thiazole-based inhibitors for the \(\alpha \)-glucosidase enzyme. For that purpose, (E)-2-(2-(arylmethylene)hydrazinyl)-4-arylthiazole derivatives were synthesized in two steps and characterized by various spectroscopic techniques. All derivatives and intermediates were evaluated for their in vitro \(\alpha \)-glucosidase inhibitory activity. Thiosemicarbazones 20 and 35, and cyclized thiazole derivatives 2, 5–11, 13, 15, 21–24, 27–31, and 36–37 showed significant inhibitory potential in the range of \(\hbox {IC}_{50}=6.2\pm 0.19\)–\(43.6\pm 0.23~\upmu \hbox {M}\) as compared to standard acarbose (\(\hbox {IC}_{50}=37.7\pm 0.19~\upmu \hbox {M}\)). A molecular modeling study was carried out to understand the binding interactions of compounds with the active site of enzyme.     

Synthesis,characterization and molecular docking studies of thiouracil derivatives as potent thymidylate synthase inhibitors and potential anticancer agents

Thymidylate synthase (TS), one of folate-dependent enzymes, is a key and well-recognized target for anticancer agents. In this study, a series of 6-aryl-5-cyano thiouracil derivatives were designed and synthesized in accordance with essential pharmacophoric features of known TS inhibitors. Nineteen compounds were screened in vitro for their anti-proliferative activities toward HePG-2, MCF-7, HCT-116, and PC-3 cell lines. Compounds \(\mathbf{21}_{\mathbf{c}}\), \(\mathbf{21}_{\mathbf{d}}\), and 24 exhibited high anti-proliferative activity, comparable to that of 5-fluorouracil. Additionally, ten compounds with potent anti-proliferative activities were further evaluated for their ability to inhibit TS enzyme. Six compounds (\(\mathbf{21}_{\mathbf{b}}\), \(\mathbf{21}_{\mathbf{c}}\), \(\mathbf{21}_{\mathbf{d}}\), 22, 23 and 24) demonstrated potent dose-related TS inhibition with \(\hbox {IC}_{50}\) values ranging from 1.57 to \(3.89\,\upmu \hbox {M}\). The in vitro TS activity results were consistent with those of the cytotoxicity assay where the most potent anti-proliferative compounds of the series showed good TS inhibitory activity comparable to that of 5-fluorouracil. Furthermore, molecular docking studies were carried out to investigate the binding pattern of the designed compounds with the prospective target, TS (PDB-code: 1JU6).     

Substituted <Emphasis Type="Italic">N</Emphasis>-aminothioglycolurils containing thiosemicarbazone moiety and their cytotoxic activity in vitro

A library of hybrid molecules bearing thioglycoluril and (hetero)aromatic aldehyde thiosemicarbazone moieties was synthesized via a tandem hydrazone formation—ring contraction reaction of 5,7-dialkyl-3-thioxoperhydroimidazo[4,5-e]-1,2,4-triazin-6-ones with (hetero)aromatic aldehydes. All synthesized compounds were tested for their cytotoxic activity against rhabdomyosarcoma, A549, and MS human cancer cell lines by MTT-assay. Among the derivatives, (E)-4-benzylideneamino-1,3-dimethyl-5-thioxohexahydroimidazo[4,5-d]imidazol-2(1H)-one 1f was found to have the most marked antiproliferative activity toward the tested cell lines (1f: IC\(_{50}= 20.6,\) 23.7, and 6.4 \(\upmu \)M, respectively). The IC\(_{50}\) value of thioglycoluril 1f against normal human embryonic kidney cells HEK293 was 72.5 \(\upmu \)M, which appeared to be 3–11-fold higher than IC\(_{50}\) values of 1f against human cancer cells.     

Search of the neutrino-less double beta decay of $$^{}$$Se into the excited states of $$^{}$$Kr with CUPID-0

The CUPID-0 experiment searches for double beta decay using cryogenic calorimeters with double (heat and light) read-out. The detector, consisting of 24 ZnSe crystals 95\(\%\) enriched in \(^{82}\)Se and two natural ZnSe crystals, started data-taking in 2017 at Laboratori Nazionali del Gran Sasso. We present the search for the neutrino-less double beta decay of \(^{82}\)Se into the 0\(_1^+\), 2\(_1^+\) and 2\(_2^+\) excited states of \(^{82}\)Kr with an exposure of 5.74 kg\(\cdot \)yr (2.24\(\times \)10\(^{25}\) emitters\(\cdot \)yr). We found no evidence of the decays and set the most stringent limits on the widths of these processes: \(\varGamma \)(\(^{82}\)Se \(\rightarrow ^{82}\)Kr\(_{0_1^+}\))8.55\(\times \)10\(^{-24}\) yr\(^{-1}\), \(\varGamma \) (\(^{82}\) Se \(\rightarrow ^{82}\) Kr \(_{2_1^+}\))\(\,{<}\,6.25 \,{\times }\,10^{-24}\) yr\(^{-1}\), \(\varGamma \)(\(^{82}\)Se \(\rightarrow ^{82}\)Kr\(_{2_2^+}\))8.25\(\times \)10\(^{-24}\) yr\(^{-1}\) (90\(\%\) credible interval).     

Synthesis and preliminary biological evaluations of (+)-isocampholenic acid-derived amides

The synthesis of two novel (+)-isocampholenic acid-derived amines has been realized starting from commercially available (1S)-(+)-10-camphorsulfonic acid. The novel amines as well as (+)-isocampholenic acid have been used as building blocks in the construction of a library of amides using various aliphatic, aromatic, and amino acid-derived coupling partners using BPC and CDI as activating agents. Amide derivatives have been assayed against several enzymes that hold potential for the development of new drugs to battle bacterial infections and Alzheimer’s disease. Compounds 20c and 20e showed promising selective sub-micromolar inhibition of human butyrylcholinesterase \((h\hbox {BChE})\) (\(\hbox {IC}_{50}\) values \(0.80\,\pm 0.05\) and \(0.25\,\pm 0.02\,{\upmu }\hbox {M}\), respectively).     

First-principle calculations of structural,electronic, optical,elastic and thermal properties of $$\hbox {MgXAs}_{2}$$ ($$\hbox {X}=\hbox {Si}, \hbox {Ge}$$X=Si,Ge) compounds

First-principle calculations on the structural, electronic, optical, elastic and thermal properties of the chalcopyrite \(\hbox {MgXAs}_{2}\) (\(\hbox {X}=\hbox {Si}, \hbox {Ge}\)) have been performed within the density functional theory (DFT) using the full-potential linearized augmented plane wave (FP-LAPW) method. The obtained equilibrium structural parameters are in good agreement with the available experimental data and theoretical results. The calculated band structures reveal a direct energy band gap for the interested compounds. The predicted band gaps using the modified Becke–Johnson (mBJ) exchange approximation are in fairly good agreement with the experimental data. The optical constants such as the dielectric function, refractive index, and the extinction coefficient are calculated and analysed. The independent elastic parameters namely, \(C_{11}\), \(C_{12}\), \(C_{13}\), \(C_{33}\), \(C_{44}\) and \(C_{66 }\) are evaluated. The effects of temperature and pressure on some macroscopic properties of \(\hbox {MgSiAs}_{2}\) and \(\hbox {MgGeAs}_{2}\) are predicted using the quasiharmonic Debye model in which the lattice vibrations are taken into account.     

Cathepsin D inhibitors as potential therapeutics for breast cancer treatment: Molecular docking and bioevaluation against triple-negative and triple-positive breast cancers

The main aim of this study was to discover small molecule inhibitors against Cathepsin D (CatD) (EC.3.4.23.5), a clinically proven prognostic marker for breast cancer, and to explore the mechanisms by which CatD could be a useful therapeutic target for triple-positive and triple-negative breast cancers (TPBC & TNBC). The crystal structure of CatD at 2.5 Å resolution (PDB: 1LYB), which was complexed with Pepstatin A, was selected for computer-aided molecular modeling. The methods used in our study were pharmacophore modeling and molecular docking. Virtual screening was performed to identify small molecules from an in-house database and a large commercial chemical library. Cytotoxicity studies were performed on human normal cell line HEK293T and growth inhibition studies on breast adenocarcinoma cell lines, namely MCF-7, MDA-MB-231, SK-BR-3, and MDA-MB-468. Furthermore, RT-PCR analysis, in vitro enzyme assay, and cell cycle analysis ascertained the validity of the selected molecules. A set of 28 molecules was subjected to an in vitro fluorescence-based inhibitory activity assay, and among them six molecules exhibited \(>\)50 % inhibition at \(25\,\upmu \hbox {M}\). These molecules also exhibited good growth inhibition against TPBC and TNBC cancer types. Among them, molecules 1 and 17 showed single-digit micromolar \(\hbox {GI}_{50}\) values against MCF-7 and MDA-MB-231 cell lines.     

Spin-label Order Parameter Calibrations for Slow Motion

Calibrations are given to extract orientation order parameters from pseudo-powder electron paramagnetic resonance line shapes of ¹⁴N-nitroxide spin labels undergoing slow rotational diffusion. The nitroxide z-axis is assumed parallel to the long molecular axis. Stochastic-Liouville simulations of slow-motion 9.4-GHz spectra for molecular ordering with a Maier–Saupe orientation potential reveal a linear dependence of the splittings, \(2A_{\hbox{max} }\) and \(2A_{\hbox{min} }\), of the outer and inner peaks on order parameter \(S_{zz}\) that depends on the diffusion coefficient \(D_{{{\text{R}} \bot }}\) which characterizes fluctuations of the long molecular axis. This results in empirical expressions for order parameter and isotropic hyperfine coupling: \(S_{zz} = s_{1} \times \left( {A_{\hbox{max} } - A_{\hbox{min} } } \right) - s_{o}\) and \(a_{o}^{{}} = \tfrac{1}{3}\left( {f_{\hbox{max} } A_{\hbox{max} } + f_{\hbox{min} } A_{\hbox{min} } } \right) + \delta a_{o}\), respectively. Values of the calibration constants \(s_{1}\), \(s_{\text{o}}\), \(f_{\hbox{max} }\), \(f_{\hbox{min} }\) and \(\delta a_{o}\) are given for different values of \(D_{{{\text{R}} \bot }}\) in fast and slow motional regimes. The calibrations are relatively insensitive to anisotropy of rotational diffusion \((D_{{{\text{R}}//}} \ge D_{{{\text{R}} \bot }} )\), and corrections are less significant for the isotropic hyperfine coupling than for the order parameter.     

Pseudo-Hermitian Systems with $\mathcal {P}\mathcal {T}$-Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even \(\mathcal {P}\mathcal {T}\)-symmetry \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=1)\) admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd \(\mathcal {P}\mathcal {T}\)-symmetric systems \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=-1)\) which is appropriate to the fermions (Scolarici and Solombrino, Phys. Lett. A 303, 239 2002; Jones-Smith and Mathur, Phys. Rev. A 82, 042101 2010). We establish that the pseudo-Hermitian Hamiltonians with even \(\mathcal {P}\mathcal {T}\)-symmetry admit a degeneracy structure if the operator \(\mathcal {PT}\) anticommutes with the metric operator η _σ which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken \(\mathcal {P}\mathcal {T}\)-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even \(\mathcal {P}\mathcal {T}\) -symmetry.     

The envelope travelling wave solutions to the Gerdjikov–Ivanov model

Valence universal multireference coupled cluster (VUMRCC) method via eigenvalue independent partitioning has been applied to estimate the effect of three-body transformed Hamiltonian (\(\widetilde{{H}}_3\)) on ionisation potentials through full connected triple excitations \(S_{3}^{(1,0) }\). \(\widetilde{H}_3 \) is constructed using CCSDT1-A model of Bartlett et al for the ground-state calculation. Contribution of transformed Hamiltonian through full connected triples \(\overline{\widetilde{H}_3 S_3^{\left( {1,0} \right) }}\) involves huge amount of computational operations that is time-consuming. Investigation on \(\hbox {Cl}_{2}\) and \(\hbox {F}_{2}\) molecules using cc-pVDZ and cc-pVTZ basis sets shows that the above effect varies from 0.001 eV to around 0.5 eV, suggesting that inclusion of \(\overline{\widetilde{H} _3 S_3^{\left( {1,0} \right) } }\) is essential for highly accurate calculations.     

Ionic liquid-functionalized magnetic nanostructures as an efficient catalyst for the synthesis of 6H-chromeno[4,3-b]quinolin-6-ones

A novel imidazole ionic liquid (IL)-functionalized \(\hbox {silica}@\gamma \)-\(\hbox {Fe}_{2}\hbox {O}_{3}\) (\(\hbox {IL-SiO}_{2}@\hbox {MNP}\)) is prepared by the functionalization of \(\hbox {SiO}_{2}@\hbox {MNP}\) by 1-butyl-3-(3-trimethoxypropyl)-1H-imidazol-3-ium chloride as the IL moiety. The catalyst is characterized by transmission electron microscopy, scanning electron microscope, vibrating sample magnetometer, dynamic light scattering and Fourier transform infrared spectroscopy. \(\hbox {IL-SiO}_{2}@\hbox {MNP}\) showed good activity in the synthesis of 6H-chromeno[4,3-b]quinolin-6-one derivatives via multicomponent reaction of 4-hydroxycoumarin, anilines and benzaldehydes. The nanocatalyst is magnetically separable and easily recoverable and showed successful activity up to 10 runs.     

Facile synthesis and biological assays of novel 2,4-disubstituted hydrazinyl-thiazoles analogs

A convenient, one-pot multi-component synthesis of new 2,4-disubstituted hydrazinyl-thiazoles was accomplished using different aldehydes/ketones, thiosemicarbazide, and 4-methoxy phenacyl bromide in the presence of a catalytic amount of AcOH in EtOH. Products were obtained in reasonable yields and high purity. The in vitro antioxidant activity of hydrazinyl-thiazoles was evaluated by DPPH radical scavenging activity in comparison to ascorbic acid. Synthesized thiazoles 14c and 14g possessed the lowest \(\hbox {IC}_{50}\) values. Also, hydrazinyl-thiazoles were screened for their in vitro antibacterial activity against six strains of bacteria including S. aureus, M. luteus, E. coli, Ps. aeruginosa, B. subtilis, and A. hydrophila where some products showed good antibacterial activity. Moreover, compound 14a showed anticancer activity against melanoma cancerous cell lines A375 with \(\hbox {LC}_{50}= 0.55\hbox { mg}/\hbox {mL}\), slightly selective versus normal cell lines (Hu-2) with \(\hbox {LC}_{50}= 1.19\hbox { mg}/\hbox {mL}\).     

Synthesis of substituted 2<Emphasis Type="Italic">H</Emphasis>-chromenes by a three-component reaction as potential antioxidants

A series of alkoxy-substituted 2H-chromenes were synthesized by a one-pot three-component reaction using salicylaldehydes, acetyl acetone and alcohol as reactant and medium with tetra-n-butylammonium fluoride (TBAF) as catalyst. Simple reaction conditions, short reaction time and overall good yield of products make this synthetic strategy an efficient one to synthesize 2H-chromene molecules. All the synthesized compounds were evaluated for antioxidant activities. Among all the new compounds, 5j and 5k showed good inhibition \((70.41\, \pm \, 0.14\) and \(64.71\, \pm \, 1.02\%\)) at 100 \(\upmu \hbox {g/mL}\) concentrations.     

On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency \(\Omega \), amplitude \(F_{0}\) and phase \(\phi \), i.e. the system with the Hamiltonian of \(\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin \) \(\left( {\Omega t+\phi } \right) \). The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables \(\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }\) and \(\tau =\omega t\). The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading \(\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}\) which decreases first from quite macroscopic values of \(\sigma _{0} =2^{12,\ldots ,25}\) to minimal one of \(\sim \!(1/\sqrt{2})\) at times \(\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right) \) and then grows back unlimitedly. For certain phases \(\phi \) depending on the \(\Omega /\omega \) ratio and \(n=\log _2\!\sigma _0 \), the mass centre of the packet \(\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }\) delays approximately two natural ‘periods’ \(\sim \!(4\pi /\omega )\) in the area of the stationary point and then escapes to ‘\(+\)’ or ‘?’ infinity in a bifurcating way.  For ‘resonant’ \(\Omega =\omega \), the bifurcation phases \(\phi \) fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic \(\phi ( {\Omega ,n\rightarrow \infty } )\) obeying the classical formula \(\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega \) for initial energy \(E = 0\) in the wide range of \(\Omega =2^{-4},...,2^{7}\).