[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-177850-105":53,"doc-detail-177850-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","weight-evaluation-supplementary-information","Weight evaluation - Supplementary information","","Weight evaluation supplementary information focuses on computationally efficient schemes for line-position evaluation and damping weights used in a numerical integral-transform approximation. It explains how damping weights are tabulated across rotational quanta, while position weights require per-line complex evaluation split into real and imaginary parts via Euler’s identity. It further derives an ultrafast pump–probe delay scan formulation, introduces time-domain approximation through inverse Fourier transform reasoning, and analyzes approximation error in the frequency domain, providing upper bounds linked to probe-field envelope and grid parameters.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/weight-evaluation-supplementary-information/177850/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/weight-evaluation-supplementary-information/177850.png","ImageObject",442,249,{"name":88,"@type":89},"WPS_1786070896","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/vnd.openxmlformats-officedocument.wordprocessingml.document","2026-10-19","2026-09-02",true,{"@type":98,"interactionType":99,"userInteractionCount":8},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How are damping weights computed and reused across lines?","Question",{"text":108,"@type":109},"Damping weights are computed from an algebraic formula and tabulated for all rotational quanta present in the line list. For each line, the appropriate weight is obtained by indexing the table using that line’s rotational quantum number.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Why is the position weight split into real and imaginary parts?",{"text":113,"@type":109},"The position-weight formula involves complex exponential terms, producing complex values. Splitting into real and imaginary components allows complex exponentials to be evaluated as sines and cosines using Euler’s identity, facilitating numerical computation.",{"name":115,"@type":106,"acceptedAnswer":116},"How is approximation error derived and bounded?",{"text":117,"@type":109},"The error is analyzed by discretizing the polarization and separating it into the exact polarization plus an error term. The signal-intensity error is then expressed via dimensionless variables, showing that the error magnitude is dominated by low-order terms and is upper-bounded using probe-field envelope and grid parameters.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},177850,1791722126,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":8,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":79},549768072016,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Supplementary information\nWeight evaluation\nThe numerical implementation of the integral transform approximation requires the adoption of computationally-efficient schemes for the evaluation of the line position and damping weights introduced in Eq. 15 in the main text.\nThe damping weight \u001a𝑎\u001bΓ,𝑖\u001b is computed according to the algebraic formula in Eq. 8 in the main text. As the damping changes for each line only as a result of different rotational quanta, the associated weight is tabulated for all rotational quanta present in the line list. For each line the weight is then obtained from the table by indexing it with the rotational quantum number of that line.\nThe position weight \u001a𝑎\u001b𝜔,𝑖\u001b depends on the exact line position, so it must be computed for each line. Since its formula involves complex exponential terms, its value is also complex. To facilitate computation, it is convenient to split up the weight in real and imaginary parts and evaluate the complex exponentials as sines and cosines making use of Euler’s identity. The expression to be numerically evaluated is then:\nwith \u001a𝜑\u001b𝑖\u001b=\u001a2\u001a𝜆\u001b𝜔,𝑖\u001b−1\u001b\u001a\u001a𝜃\u001bΔ𝜔\u001b\u001b, \u001a𝜆\u001b𝜔,𝑖\u001b=\u001a\u001a𝜔\u001b𝑖\u001b−𝜔[\u001a𝑘\u001b𝑖\u001b]\u001bΔ𝜔\u001b, and \u001a\u001a𝜃\u001bΔ𝜔\u001b\u001b=𝜏𝛥𝜔/2 as defined in the main text. Compared to other equivalent expressions, this expression minimizes the per-line evaluations of sine/cosine functions. While faster than complex exponentials, sine and cosine functions are still fairly expensive to compute, especially considering they are evaluated for each line. A polynomial approximation can be used when \u001a𝜃\u001bΔ𝜔\u001b≪1. The real and imaginary parts are separately parametrized by the polynomial expressions:\nThe real part \u001a𝑎\u001b𝜔,ℜ\u001b is linear with 𝜆, in addition to a third and a fifth order correction term. The value for \u001a𝐴\u001b0\u001b is determined by matching the value of \u001a𝑎\u001b𝜔,ℜ\u001b(𝜆) to Re[\u001a𝑎\u001b𝜔,𝑖\u001b] at the polynomial peaks, for \u001a𝜆\u001b±\u001b=\u001a1\u001b2\u001b±\u001a1\u001b2\u001a\u001b3\u001b\u001b. The value of \u001a𝐴\u001b1\u001b is found by matching the slope at 𝜆=\u001a1\u001b2\u001b. This results in:\nThe imaginary part \u001a𝑎\u001b𝜔,ℑ\u001b(𝜆) is zero at 𝜆=0, 1 and has a peak at 𝜆=\u001a1\u001b2\u001b. \u001a𝐵\u001b0\u001b and \u001a𝐵\u001b1\u001b are obtained by setting the value at 𝜆=\u001a1\u001b2\u001b and slope at 𝜆=0 equal to the respective values of  Im[\u001a𝑎\u001b𝜔\u001b𝑖\u001b], resulting in:\nIt is worth noting that, while the expressions for the A and B constants are rather involved, they are only evaluated once for each iteration, i.e. \u001a𝜃\u001bΔ𝜔\u001b is not a line property.\nUltrafast probe delay scan\nThe evolution of the molecular coherence following the impulsive excitation by a fs pulse can be measured in the time domain by employing a pump-probe scheme. The resulting probe delay scan is performed by integrating the CRS signal over 𝜔 and scanning the probe delay time 𝜏:\nUsing Eq. 2 from the main text we find that:\nwhich is a cross-correlation that can be expressed as a convolution in time:\nAn ultrafast approximation for \u001a𝜒\u001b(3)\u001b\u001a𝜏\u001b is then found by using the definition of \u001a\u001a𝜒\u001b𝑖\u001b\u001b(3)\u001b\u001a𝜏\u001b and the integral transform approach to express \u001a𝜒\u001b(3)\u001b\u001a𝜏\u001b:\nHere the factor exp\u001a𝑗𝜔𝜏\u001b, combined with the integral over 𝜔, is recognized as an inverse Fourier transform (from 𝜔 to 𝜏) to arrive at the expression for evaluating the time-domain signal:\nWe note that the reasoning above requires the integral to be taken over the entire frequency domain. This domain can be effectively limited by excluding from the calculation of the line strength distribution function \u001a𝑊\u001b\u001a𝜔,Γ\u001b those lines that lie outside a certain frequency range. We further note that the weights in Eq 8 in the main text are not quite as good in reducing the approximation error in \u001a𝜒\u001b(3)\u001b\u001a𝜏\u001b, because they only target a single value of 𝜏. However, this shortcoming can be offset by using a higher resolution for the Γ grid, as briefly outlined in the main text and discussed in more detail in the section “Approximation error – Time domain approximation”.\nApproximation error\nFrequency-domain approximation\nIn the following we provide an analytical derivation of the approximation error introduced in Eq. 9 in the main text, with the choice of weights in Eq. 8 (ibid.). We furtherm","cbCairDM1F9wgRY9","https://ap.wps.com/l/cbCairDM1F9wgRY9","docx",723141,10,"English","# Weight evaluation\n## Damping weights and position weights\n## Ultrafast probe delay scan\n## Approximation error\n### Frequency-domain approximation","[{\"question\":\"How are damping weights computed and reused across lines?\",\"answer\":\"Damping weights are computed from an algebraic formula and tabulated for all rotational quanta present in the line list. For each line, the appropriate weight is obtained by indexing the table using that line’s rotational quantum number.\"},{\"question\":\"Why is the position weight split into real and imaginary parts?\",\"answer\":\"The position-weight formula involves complex exponential terms, producing complex values. Splitting into real and imaginary components allows complex exponentials to be evaluated as sines and cosines using Euler’s identity, facilitating numerical computation.\"},{\"question\":\"How is approximation error derived and bounded?\",\"answer\":\"The error is analyzed by discretizing the polarization and separating it into the exact polarization plus an error term. The signal-intensity error is then expressed via dimensionless variables, showing that the error magnitude is dominated by low-order terms and is upper-bounded using probe-field envelope and grid parameters.\"}]","Weight evaluation - Supplementary information | DOCX",1788329391]