{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:26:51Z","timestamp":1760146011339,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,25]],"date-time":"2024-09-25T00:00:00Z","timestamp":1727222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study the nonlinear absorptive and dispersive optical properties of molecular systems immersed in a thermal reservoir interacting with a four-wave mixing (FWM) signal. Residual spin-orbit Hamiltonians are considered in order to take into account the internal structure of the molecule. As system parameters in the dissipation processes, transverse and longitudinal relaxation times are considered for stochastic solute\u2013solvent interaction processes. The intramolecular coupling effects on the optical responses are studied using a molecule model consisting of two coupled harmonic curves of electronic energies with displaced minima in nuclear energies and positions. In this study, the complete frequency space is considered through the pump\u2013probe detuning, without restricting the derivations to only maximums of population oscillations. This approach opens the possibility of studying the behavior of optical responses, which is very useful in experimental design. Our results indicate the sensitivity of the optical responses to parameters of the molecular structure as well as to those derived from the photonic process of FWM signal generation.<\/jats:p>","DOI":"10.3390\/sym16101263","type":"journal-article","created":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T04:05:46Z","timestamp":1727323546000},"page":"1263","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Parametric Characterization of Nonlinear Optical Susceptibilities in Four-Wave Mixing: Solvent and Molecular Structure Effects"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6177-7919","authenticated-orcid":false,"given":"Jos\u00e9 L.","family":"Paz","sequence":"first","affiliation":[{"name":"Departamento Acad\u00e9mico de Qu\u00edmica Inorg\u00e1nica, Facultad de Qu\u00edmica e Ingenier\u00eda Qu\u00edmica, Universidad Nacional Mayor de San Marcos, Lima 15081, Peru"}]},{"given":"Alberto","family":"Garrido-Schaeffer","sequence":"additional","affiliation":[{"name":"Departamento de Operaciones Unitarias, Facultad de Qu\u00edmica e Ingenier\u00eda Qu\u00edmica, Universidad Nacional Mayor de San Marcos, Lima 15081, Peru"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0458-4691","authenticated-orcid":false,"given":"Marcos A.","family":"Loro\u00f1o","sequence":"additional","affiliation":[{"name":"Departamento Acad\u00e9mico de Qu\u00edmica Fisicoqu\u00edmica, Facultad de Qu\u00edmica e Ingenier\u00eda Qu\u00edmica, Universidad Nacional Mayor de San Marcos, Lima 15081, Peru"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0023-7342","authenticated-orcid":false,"given":"Lenin","family":"Gonz\u00e1lez-Paz","sequence":"additional","affiliation":[{"name":"Departamento de Biolog\u00eda, Laboratorio de Gen\u00e9tica y Biolog\u00eda Molecular (LGBM), Facultad Experimental de Ciencias (FEC), Universidad del Zulia (LUZ), Maracaibo 4001, Venezuela"},{"name":"Laboratorio de Biocomputaci\u00f3n (LB), Centro de Biomedicina Molecular (CBM), Instituto Venezolano de Investigaciones Cient\u00edficas (IVIC), Maracaibo 4001, Venezuela"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7503-1528","authenticated-orcid":false,"given":"Edgar","family":"M\u00e1rquez","sequence":"additional","affiliation":[{"name":"Grupo de Investigaciones en Qu\u00edmica y Biolog\u00eda, Departamento de Qu\u00edmica y Biolog\u00eda, Facultad de Ciencias Exactas, Universidad del Norte, Carrera 51B, Km 5, V\u00eda Puerto Colombia, Barranquilla 081007, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6128-9504","authenticated-orcid":false,"given":"Jos\u00e9 R.","family":"Mora","sequence":"additional","affiliation":[{"name":"Grupo de Qu\u00edmica Computacional y Te\u00f3rica (QCT-USFQ), Instituto de Simulaci\u00f3n Computacional (ISC-USFQ), Departamento de Ingenier\u00eda Qu\u00edmica, Universidad San Francisco de Quito (USFQ), Quito 170157, Ecuador"}]},{"given":"Ysaias J.","family":"Alvarado","sequence":"additional","affiliation":[{"name":"Centro de Biomedicina Molecular (CBM), Laboratorio de Qu\u00edmica Biof\u00edsica Te\u00f3rica y Experimental (LQBTE), Instituto Venezolano de Investigaciones Cient\u00edficas (IVIC), Maracaibo 4001, Venezuela"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"15380","DOI":"10.1039\/C9CP02662B","article-title":"Partition of optical properties into orbital contributions","volume":"21","author":"Sitkiewicz","year":"2019","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1021\/acsphotonics.7b00728","article-title":"Theory of Nanoscale Organic Cavities: The Essential Role of Vibration-Photon Dressed States","volume":"5","author":"Herrera","year":"2018","journal-title":"ACS Photonics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1800219","DOI":"10.1002\/lpor.201800219","article-title":"Strong Coupling in Microcavity Structures: Principle, Design, and Practical Application","volume":"13","author":"Yu","year":"2019","journal-title":"Laser Photonics Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1850038","DOI":"10.1142\/S0218863518500388","article-title":"Effects of spectral diffusion on the nonlinear optical properties in two- and three-state quantum systems in a four-wave mixing signal","volume":"27","author":"Moncada","year":"2019","journal-title":"J. Nonlinear Opt. Phys. Mater."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.optcom.2019.01.018","article-title":"Non-Markovian evolution of a two-level system interacting with a fluctuating classical field via dipole interaction","volume":"437","author":"Hesabi","year":"2019","journal-title":"Opt. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1650209","DOI":"10.1142\/S021797921650209X","article-title":"Linear and nonlinear optical absorption coefficients and refractive index changes in Morse quantum wells under electric field","volume":"30","author":"Sakiroglu","year":"2016","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2073","DOI":"10.1021\/cr020681b","article-title":"Many-Body Approaches for Simulating Coherent Nonlinear Spectroscopies of Electronic and Vibrational Excitons","volume":"104","author":"Mukamel","year":"2004","journal-title":"Chem. Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1016\/j.jcp.2018.06.040","article-title":"Application of the differential evolution for simulation of the linear optical of photosynthetic pigments","volume":"372","author":"Pishchalnikov","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1080\/09500340.2020.1810341","article-title":"Study of the nonlinear optical responses in the Four-wave mixing signal in saturation regimen of a two-level system with intramolecular coupling","volume":"67","author":"Paz","year":"2020","journal-title":"J. Mod. Opt."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"138885","DOI":"10.1016\/j.cplett.2021.138885","article-title":"Absorptive and dispersive responses in a two-level molecule with vibronic coupling: Permanent dipole moments effects","volume":"779","author":"Paz","year":"2021","journal-title":"Chem. Phys. Lett."},{"key":"ref_11","first-page":"215001","article-title":"Nonlinear optical responses of molecular systems with vibronic in fluctuating environments","volume":"31","author":"Paz","year":"2021","journal-title":"J. Nonlinear Opt. Phys. Mater."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1080\/09500340.2021.1976856","article-title":"Solvent randomness and intramolecular considerations of optical responses in four-wave mixing","volume":"68","author":"Paz","year":"2021","journal-title":"J. Mod. Opt."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Paz, J.L., Espinoza-Montero, P., Loro\u00f1o, M., Torres, F.J., Gonz\u00e1lez-Paz, L., M\u00e1rquez, E., Vera-Villalobos, J., Moncada, F., and Alvarado, Y. (2022). Propagation and Parametric Amplification in Four-Wave Mixing processes: Intramolecular Coupling and High-Order Effects. Symmetry, 14.","DOI":"10.3390\/sym14020301"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"065322","DOI":"10.1063\/5.0086769","article-title":"A kinetic model for the equilibrium dynamics of absorption and scattering processes in four-wave mixing spectroscopy","volume":"12","author":"Paz","year":"2022","journal-title":"AIP Adv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.nima.2018.03.051","article-title":"Advances in instrumentation for FEL-based four-wave-mixing experiments","volume":"907","author":"Mincigrucci","year":"2018","journal-title":"Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zhang, G., Lefkidi, G., Murakami, M., H\u00fcbner, W., and George, T. (2021). Introduction to Ultrafast Phenomena: From Femtosecond Magnetism to High-Harmonic Generation, CRC Press. [1st ed.].","DOI":"10.1201\/9780429194832"},{"key":"ref_17","unstructured":"Boyd, R.W. (2008). Nonlinear Optics, Academic Press."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Noudem, P., Fouejio, D., Mveme, C.D., Nya, F.T., and Zekeng, S.S. (2023). Electronic, nonlinear optical, UV\u2013vis and NBO analysis of methyl methacrylate for optoelectronic and optical applications: DFT study and impact of conformation. Spectrochim. Acta Part A Mol. Biomol. Spectrosc., 303.","DOI":"10.1016\/j.saa.2023.123267"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"128036","DOI":"10.1016\/j.molstruc.2020.128036","article-title":"Design and growth of novel organic molecular Quinoline 4-nitrophenol (QNP) single crystals: For Nonlinear optical (NLO) applications","volume":"1210","author":"Karuppasamy","year":"2020","journal-title":"J. Mol. Struct."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Binish, B., and Mani, R.K. (2023). Synergic effects of Sn4+ doping on the nonlinear optical limiting properties of SnxCd1-xMoO4 nanostructures for optoelectronic applications. J. Photochem. Photobiol. A Chem., 439.","DOI":"10.1016\/j.jphotochem.2023.114614"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"171331","DOI":"10.1016\/j.ijleo.2023.171331","article-title":"Application of slow-light engineered chalcogenide and silicon photonic crystal waveguides for double-dimensional microwave frequency identification","volume":"290","author":"Emami","year":"2023","journal-title":"Optik"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"107874","DOI":"10.1016\/j.optlastec.2022.107874","article-title":"Enhanced frequency up-conversion based on four-wave mixing assisted by a Bessel-Gaussian beam in 85Rb atoms","volume":"149","author":"Wang","year":"2022","journal-title":"Opt. Laser Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"128960","DOI":"10.1016\/j.optcom.2022.128960","article-title":"Competition of collimated ultraviolet lights generated by four-wave mixing process in Cs vapor","volume":"527","author":"Hu","year":"2023","journal-title":"Opt. Commun."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"127629","DOI":"10.1016\/j.optcom.2021.127629","article-title":"High-speed modulating retro-reflectors with optical phase conjugation compensation","volume":"507","author":"Chen","year":"2022","journal-title":"Opt. Commun."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"170740","DOI":"10.1016\/j.ijleo.2023.170740","article-title":"Applications of nonlinear four-wave mixing in optical communication","volume":"279","author":"Andrekson","year":"2023","journal-title":"Optik"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"100802","DOI":"10.1016\/j.isci.2019.100802","article-title":"Chalcogenide Taper and Its Nonlinear Effects and Sensing Applications","volume":"23","author":"Gao","year":"2020","journal-title":"iScience"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.optlastec.2016.03.013","article-title":"Nonlinear optical properties and optical power limiting effect of Giemsa dye","volume":"82","author":"Abdulkareem","year":"2016","journal-title":"Opt. Laser Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"053014","DOI":"10.1088\/1367-2630\/16\/5\/053014","article-title":"Third order optical nonlinearity of graphene","volume":"16","author":"Cheng","year":"2014","journal-title":"New J. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.optmat.2014.05.003","article-title":"Study of the nonlinear optical properties and behavior in phenoxy-phthalocyanines liquid at nanosecond laser pulses","volume":"37","author":"Yao","year":"2014","journal-title":"Opt. Mater."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.molstruc.2019.04.091","article-title":"Synthesis, structural, vibrational, thermal, dielectric and optical properties of third order nonlinear optical single crystal for optical power limiting applications","volume":"1191","author":"Siva","year":"2019","journal-title":"J. Mol. Struct."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"128901","DOI":"10.1016\/j.molstruc.2020.128901","article-title":"Molecular structure, linear optical, second and third-order nonlinear optical properties of two non-centrosymmetric thiophene-chalcone derivatives","volume":"1222","author":"Naik","year":"2020","journal-title":"J. Mol. Struct."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"126371","DOI":"10.1016\/j.physleta.2020.126371","article-title":"Atomic nonclassicality in the Jaynes-Cummings model","volume":"384","author":"Dai","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Blum, K. (2010). Density Matrix Theory and Applications, Springer.","DOI":"10.1007\/978-3-642-20561-3_2"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2292","DOI":"10.1002\/jrs.6223","article-title":"Ultraviolet resonance Raman spectroscopy of anthracene: Experiment and theory","volume":"52","author":"Holtum","year":"2021","journal-title":"J. Raman Spectrosc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4318","DOI":"10.1039\/C7SC00586E","article-title":"Quantitative first principles calculations of protein circular dichroism in the near ultraviolet","volume":"8","author":"Li","year":"2017","journal-title":"Chem. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2367","DOI":"10.1021\/acs.jpclett.8b00827","article-title":"2018 On-the-Fly Ab Initio Semiclassical Evaluation of Absorption Spectra of Polyatomic Molecules beyond the Condon Approximation","volume":"9","author":"Patoz","year":"2018","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2751","DOI":"10.1016\/j.chempr.2019.07.008","article-title":"Electronic Circular Dichroism Spectroscopy of Proteins","volume":"5","author":"Rogers","year":"2019","journal-title":"Chem"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1006\/jtbi.1993.1049","article-title":"Role of a Stochastic Friction Coefficient in Open Channel Noise","volume":"161","author":"Colmenares","year":"1993","journal-title":"J. Theor. Biol."},{"key":"ref_39","unstructured":"Mukamel, S. (1995). Principles of Nonlinear Optical Spectroscopy, Oxford University Press. [1st ed.]."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1080\/09500340.2014.986235","article-title":"Rotating wave approximation effects on the nonlinear optical responses of complex molecular systems using a four-wave mixing","volume":"62","author":"Paz","year":"2015","journal-title":"J. Mod. Opt."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1103\/PhysRevA.26.1589","article-title":"Analytical and numerical studies of multiplicative noise","volume":"26","author":"Sancho","year":"1982","journal-title":"Phys. Rev. A"},{"key":"ref_42","first-page":"107765","article-title":"Analyzing coherent and incoherent optical susceptibilities in four-wave mixing: Stochasticity, symmetry, and intramolecular coupling","volume":"61","author":"Paz","year":"2024","journal-title":"Res. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1242","DOI":"10.1063\/1.448444","article-title":"Power effects in polarization spectroscopy on a water solution of malachite green","volume":"82","author":"Marcano","year":"1985","journal-title":"J. Chem. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1103\/PhysRevA.17.309","article-title":"Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing I. Theory","volume":"17","author":"Yajima","year":"1978","journal-title":"Phys. Rev. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1103\/PhysRevA.17.324","article-title":"Study of ultra-fast relaxation processes by resonant Rayleigh type optical mixing. II. Experiment on dye solutions","volume":"17","author":"Yajima","year":"1978","journal-title":"Phys. Rev. A"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1263\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:03:04Z","timestamp":1760112184000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1263"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,25]]},"references-count":45,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["sym16101263"],"URL":"https:\/\/doi.org\/10.3390\/sym16101263","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,9,25]]}}}