The macroscopic evaluation and the microscopic study, including the degree of adhesion and inflammation, were investigated after three and five weeks. The resultant electrospun TPU nanofibers had diameters ranging from about 200 to 1000 nm. The diameters and morphology of the nanofibers were significantly affected by the concentration of polymer. Uniform TPU nanofibers without beads could be prepared by electrospinning through reasonable control of the process concentration. These nanofibers' biodegradability and antibacterial properties were investigated by weight loss measurement and microdilution methods, respectively. The purpose of this study was to provide electrospun nanofibers having biodegradability and antibacterial properties that prevent any adhesions or inflammation after pelvic and abdominal surgeries. The in vivo experiments revealed that electrospun TPU nanofibers reduced the degree of abdominal adhesions. The histopathological study confirmed only a small extent of inflammatory cell infiltration in the 8% and 10% TPU. Conclusively, nanofibers containing 8% TPU significantly decreased the incidence and severity of postsurgical adhesions, and it is expected to be used in clinical applications in the future.Intra-abdominal adhesions following surgery are a challenging problem in surgical practice. This study fabricated different thermoplastic polyurethane (TPU) nanofibers with different average diameters using the electrospinning method. The conditions were evaluated by scanning electron microscopy (SEM), atomic force microscope (AFM), and Fourier transform infrared spectrometer (FTIR) analysis. A static tensile test was applied using a strength testing device to assess the mechanical properties of the electrospun scaffolds. By changing the effective electrospinning parameters, the best quality of nanofibers could be achieved with the lowest bead numbers. The electrospun nanofibers were evaluated  Surgical intra-abdominal adhesions are a common cause of postoperative complications, which is found in most cases of abdominal and gynecological surgeries . AbdominSome common prior strategies are applying antiadhesion materials and fibrinolytic agents and improving surgical procedures . A methoNanofibers are a large category of nanosized fibers with various physicochemical properties made from different polymers, which were successfully used in many biomedical sciences such as tissue engineering, wound coatings, physical barriers, and drug delivery , 8. TherE. coli and Staphylococcus aureus, play an essential role in the progression and consequences of surgery, especially in this age when engineered bacteria are almost everywhere [Nanofiber formation through electrospinning is a versatile and straightforward technique using biopolymers and nanocomposites that have gained much attraction to provide a proper environment , 11. Furerywhere –23. Antierywhere . Nanofiberywhere , 26.in vivo antiadhesion efficacy of nanofibers compared to routine surgical mesh was evaluated in a rat cecal abrasion model to demonstrate the possible application for the antiadhesion property.Although new materials have been selected to satisfy the standards for adhesion prevention, none of them have met all the desirable features. So, there is still a need for an advanced and inexpensive membrane to be effectively used in surgeries for a wide range of patients to reduce postoperative adhesions. In this work, some nanostructured barriers made from TPU were fabricated by the electrospinning technique in different polymer solution concentrations. The nanofibers' morphology, size, and structure in different concentrations were analyzed by various characterization techniques. The mechanical properties, biodegradability, and antibacterial feature of the samples were also studied by strength testing device, weight loss measurement, and microdilution method, respectively . Then, tCommercial medical-grade thermoplastic polyurethane  was used as a raw material. Tetrahydrofuran  and N,N-dimethylformamide  were purchased solvents to prepare the electrospun nanofibers. Commercial soft polypropylene nonabsorbable synthetic surgical mesh  and phosphate-buffered saline (PBS) were purchased from the Sigma-Aldrich Company, St. Louis, US.W/V) were prepared in such a way that 0.6, 0.8, 1, and 1.2 g of the TPU was dissolved in 4 mL of DMF and 6 mL of THF and stirred to obtain a homogenous solution at 25°C. Solutions of TPU were embedded in a 10 mL syringe with an internal radius of 23 gauges and were fed through the needle to create stable fibers. Nanofibers were electrospun at a rotation speed of 500 rpm, 19 kV voltage, and 1.5 mL/h flow rate, whereas the distance between syringe tip and collector was 140 mm at room temperature  consists of a voltage regulating system up to 30 kV, which could adjust the flow rate of the solution through the uniaxial needle. The rotation speed of the collector (in rpm) and the possibility to set the appropriate distance between collectors and needle were under control. Different concentrations of TPU solutions  at an accelerating voltage of 15 kV. A total of 100 fibers were counted for each sample, and diameters and distributions of the electrospun nanofibers were measured by the SEM images using the Digimazer image analysis software version 4.3.5.Atomic force microscope  measurements were performed to determine the morphology and topographic properties of TPU nanofibers.Fourier Transform Infrared spectrometer  was used to determine the chemical properties of nanofibers.To assess the mechanical properties of the electrospun scaffolds, a static tensile test was applied using a strength testing device . All samples were cut into strips with the gauge 6 mm width dimension by 0.03 mm length. 0.2 N measuring cell was employed under tensile loading at 37°C in a temperature chamber and 70% relative humidity with an elongation speed of 10 mm/min.in vivo study to assess the antiadhesion efficacy of electrospun TPU nanofibers in the abdominal cavity was carried out using 96 healthy female Sprague Dawley rats (divided into 12 groups (n = 8)) aged at 8-10 weeks with an average body weight of 200–250 g. The rats were purchased from the Laboratory Animal Centre of Shiraz University of Medical Sciences. The rats' observation was done upon coming to the laboratory for a week to adapt to the new place. Throughout the experiment, rats were housed in standard plastic cages at 23-25°C and 40–60% relative humidity with a controlled 12-hour light/dark cycle for at least two weeks before the experiment. Standard pellet diet and water were freely available for the animals throughout the investigation, and there was no difference between the experimental groups.The antiadhesion characteristics of the TPU nanofibers were investigated using a rat cecal abrasion model. The procedures and handling of the animals were performed according to the guidelines approved by the Ethics Committee (IR.SUMS.REC.1398.072) of Shiraz University of Medical Sciences, Shiraz, Iran. The An aseptic condition was provided for the surgical procedure. Anesthesia was performed intramuscularly in all groups using a combination of two drugs, ketamine (80 mg/kg) and xylazine (10 mg/kg). After anesthesia, the rats were placed on the surgical table in the supine position, and the surgical field was prepared with 1% of antiseptic povidone-iodine . The hain = 8, For each group of rats, one type of electrospun TPU nanofibers was implanted. The rats were divided into two categories; each category included five groups n = 8, . In cateThe scaffolds of TPU nanofibers in different concentrations were implanted into the body subcutaneously compared to a control group. At the end of each course (three and five weeks), the rats were sacrificed, and the adhesion to surrounding tissue and its shape was evaluated macroscopically. The formation of adhesion was assessed according to a semi-quantitative scoring system as follows : no adhen is the number of rats with the same grade score in each group, G is the macroscopic score, and N is the total number of rats in each group. Finally, the summary of all scores obtained in each group is reported in Abdominal adhesions were assessed by two independent persons blinded to experiments. Then, the grade scores were measured using the following equation:μm sagittal slices and stained with H&E. The stained sections were examined by an independent and blinded investigator with a CKX3 Olympus microscope (Japan), and images were captured [After an animal sacrifice and macroscopic evaluations, the surgery site was excised and collected for histological evaluation. The tissues were fixed in 4% formalin for 24 hours and embedded in formalin. Samples were cut into 5 captured . Histolocaptured . The tisW1) and immersed in a PBS buffer (pH: 7.4) containing penicillin-streptomycin (100 U/mL) and incubated for 1, 2, 4, 8, 16, 32, and 64 days at 37°C. All samples were then washed with double distilled water and dried in a vacuum oven. At the end of each period, the weight of the samples was calculated in milligram (W2), and the percentage of weight loss in the respected time was determined using the following formula:The synthesized electrospun TPU nanofibers in different concentrations and synthetic surgical mesh as a standard in dimensions of 1 × 20 × 80 mm pieces were weighed , and 10 μL of inoculated bacteria was applied for each microorganism. The first and last rows of the microplate were left empty to achieve a better optical contrast after plate reading. The prepared microplates were incubated for 24 hours at 37°C; then, the optical density was measured at 600 nm by a microplate reader . This procedure was repeated three times. A blank 96-well microplate consisting of 45 μL of culture media and 45 μL of the sample (as explained) was prepared. In the end, 10 μL of culture media was added to each well. The turbidity of each well in a sample microplate (a microplate with the concerned microorganism) was compared to an equivalent well in a blank microplate. Microorganism viability was calculated as follows:n et al. . The orgAll antimicrobial studies were evaluated using IBM SPSS software.U-tests using statistical software SPSS IBM.; p value <0.05 was considered significant. The number of animals in each group was 8, and the data were expressed as means ± SD.The data obtained from macroscopic and microscopic evaluations and histological analysis were analyzed by Kruskal–Wallis one-way analysis of variance and Mann–Whitney p value ≤0.05 was considered to be statistically significant. The experiment was repeated three times.To determine the differences between the means of degradation and antimicrobial assay results, the one-way ANOVA procedure and the post hoc Tukey test were performed. Several processing parameters are involved in the fabrication of nanofibers by the electrospinning method. These parameters, including applied voltage, the flow rate of the solution, and needle tip to collector distance, should be optimized to obtain the desired nanofibers. In this study, the optimum parameters for nanofiber preparation are as follows in W/V) on the chemomechanical properties and biological activity of fabricated nanofibers were investigated. The SEM images and morphologies of the resulting nanofibers in two different magnifications are shown in W/V) had smooth surfaces.Then, the effects of changing the concentration of polymer solutions , C-H stretching, C=O stretching (amid and ester), C=C stretching (aromatic), CH2 bending, C-C stretching, C-O stretching, and long-chain bendings, respectively. A nearly similar absorption band appeared at the 8% concentration of TPU at 3375, 2918, 1666, 1571, 1403, 1186, 1115, and 916 cm−1. At 10% concentration of TPU, the absorption bands are 3333, 3001, 1698, 1603, 1455, 1074, 1015, and 885 cm−1. At 12% concentration of TPU, the absorption bands are 3329, 2958, 1700, 1596, 1413, 1220, 1018, and 770 cm−1. Previous studies confirmed the presence of several amid groups as a characteristic feature of polyurethan [−1, bending vibration of amide bond N-H at around 1500 cm−1, and C–N stretching vibration amide bond at around 1220 cm−1 [The FT-IR results as depicted in 220 cm−1 . These rOne of the most critical properties needed in tissue engineering is good mechanical properties. Mechanical stability and compatibility of the scaffolds play an essential role in the biomedical approach. Electrospun nanofibers could potentially supply useful mechanical features such as tensile strength, the breakage mechanism, and fiber morphology. The 6% TPU nanofibers were illustrated low tensile strength compared to the other nanofibers. On the other hand, 8% and 10% TPU nanofibers were found to have increased tensile strength and elongation at break value than the 12% TPU nanofibers. Therefore, it was seen that electrospinning of 8% and 10% concentrations of TPU nanofibers affected the elastic and plastic deformation under the same experimental condition. The mechanical properties results showed no significant difference in tensile strength values between the 8% and 10% concentrations of nanofibers. Lower tensile strength values in low concentrations of electrospun TPU nanofibers are due to the presence of beads in the nanofiber mats, which possess an unpleasant effect on a fiber . TherefoAt 6% TPU, it is expected that low concentration and the presence of beads lead to lower elongation at break. Also, at 12% TPU, the higher concentration would diminish tensile properties due to the increasing diameter, causing reduced fiber surface area and decreased fiber interaction points. Both higher diameter and lower beads at 8% and 10% of TPU compared to the others account for high elongation at break. Therefore, an optimum point of average diameter and bead extent could ultimately be reached at 8% TPU. Recently, there is a report that showed a consistent result. Li et al. showed that the 8% TPU concentration in two different solvent systems resulted in smooth and uniform nanofibers .The antiadhesion potency of TPU made from different concentrations compared to synthetic surgical mesh was evaluated by a rat cecum abrasion model. The TPU nanofibers were appropriately flexible, well-handled during surgery, and effectively protected the injured area. Untreated rats  and rat groups treated with synthetic surgical mesh were considered as control and standard groups, respectively.The tissue antiadhesion potential of the nanofibers was evaluated for 3 and 5 weeks after surgical operation macroscopically and histopathologically.p < 0.05) and received a score equivalent to 2.50 and 1.625 after 3 and 5 weeks. The scores were not significantly different compared to the synthetic surgical mesh group (p < 0.05). The nanofibers containing 8% and 10% TPU significantly reduced peritoneal adhesions to different extents than those of the control and synthetic surgical mesh group (p < 0.05) both after 3 and 5 weeks. The nanofibers containing 8% TPU remarkably inhibited adhesion formation. Adhesion scores for this group were significantly lower than those for the control group (p < 0.001) as well as the synthetic surgical mesh group .A digital camera was used to take photographs for macroscopic evaluation of nanofibers' effects on postsurgical adhesion in the rat's peritoneal cavity shown in Interestingly, the nanofibers containing 8% and 10% TPU also inhibited inter-visceral adhesions. According to the best of our knowledge, there is no study using different TPU nanofiber concentrations to evaluate postsurgical adhesion. However, several animal studies are using other nanofibers to evaluate postsurgical adhesion. Yamaoka et al. applied a copolymer of polylactic acid- (PLGA-) polyethylene glycol (PEG) for inhibiting postsurgical adhesion . AlthougHistological evaluation was performed on the specimen as previously described. A series of images of histological sections taken at magnification ×100 to represent the overall picture of the implantation site are given in As the concentration was increased from 6% to 12%, the scores were slightly decreased. This trend was detected in specimens that were sacrificed at the 3rd and 5th weeks of implantation. As a general tendency, the scores gradually reduced with time, which means that the tissue reaction was gradually subsided. The histological pictures have represented these changes during time. Overall, regarding histopathological findings, a few points are mentioned as follows.A severe inflammatory reaction could be seen in the control group. In a scaffold with 12% TPU, a relatively dense capsule was formed on both sides of the scaffold, gradually degraded as the macrophages and giant cells were easily seen in the membrane's pores. The capsule was always rich in capillary-like blood vessels and spindle-shape fibroblasts which surrounded the membranes. There was no difference in capsular thickness regarding the membrane's side, near the abdominal wall, or the peritoneal side. The type of infiltrated inflammatory cells was gradually changed from polymorphonuclear to lymphoblast cells, macrophages, and giant cells. Neither necrosis was observed in any of the samples at any time point.As a result, the microscopic evaluations showed that postsurgical usage of TPU in 8, 10%, and 12% concentrations relieved abdominal adhesion, and their effects were a little better than the commercial synthetic surgical mesh. Besides, macroscopic evaluations revealed that TPU nanofibers, especially in 8% and 10% concentrations, significantly prevent abdominal adhesion compared to commercial synthetic surgical mesh. It seems that these nanofibers exerted strong fibrinolytic activity due to chemical occlusion to inhibit adhesion formation in addition to being an excellent physical barrier. Some antiadhesion agents using drug-impregnated physical barriers, paclitaxel-loaded hyaluronic acid films, and sirolimus-eluting polypropylene meshes also showed effectively prevent postsurgical adhesions , 47.in vivo behavior of these scaffolds. This assay was performed over a span of 64 days , removal of molar mass, and weight loss to metabolize into oligomeric components , 49. A m 64 days . ObtaineOur animal model created defects in the abdominal cavity after three and five weeks of postimplantation to monitor the implants. S. aureus and E. coli, were used as the target bacteria to screen the in vitro antibacterial susceptibility of the different concentrations of electrospun TPU nanofibers by the microdilution method. The compounds reveal a dose-dependent bacterial growth inhibition . The purpose of this study was to provide electrospun nanofibers having biodegradability and antibacterial properties that prevent any adhesions or inflammation after pelvic and abdominal surgeries. As a result, the morphology observed from SEM and AFM confirmed that the nanofibers had a uniform structure and fine morphology in higher concentrations. According to this study, the nanofiber containing 8% TPU meets the intended applications for postsurgical antiadhesive treatment. An in vivo study using a rat cecal abrasion model exhibited good antiadhesive properties, with biodegradation in the proper time for the nanofibers mentioned above. These electrospun TPU nanofibers had a healing effect on postsurgical adhesion compared to those of the control group, and even the standard mesh judging from macroscopic observation further confirmed the histological analysis. It seems that this valuable feature is due to the enhanced interaction between the nanofiber and body fluids, which in turn is a result of the high surface-to-area ratio of TPU nanofibers. This study also revealed that the rats did not show symptoms of fever, severe inflammatory reactions, or death in any of the experimental groups. It seems that electrospun TPU nanofibers create a physical barrier to prevent exogenous fibroblasts from entering the inflammatory area and invoking inflammatory mediators. It also helps the wound healing process by preventing the spread of the inflammatory process to nonsurgical areas [The electrospun TPU nanofibers were fabricated in different concentrations (6%, 8%, 10%, and 12% al areas , 51. Theal areas . These eIt should be noted that these nanofibers are currently produced inexpensively and economically. However, it is essential to pay attention to the long-term clinical and inflammatory effects and unpredictable reactions, including allergic reactions, before entering the clinical arena. We will probably see more studies on the clinical applications of electrospun TPU nanofibers in the near future.


World Health Organization (WHO) classification of central nervous system tumors. To address this question, a meeting of neuropathologists with expertise in molecular diagnosis was held in Haarlem, the Netherlands, under the sponsorship of the International Society of Neuropathology (ISN). Prior to the meeting, participants solicited input from clinical colleagues in diverse neuro‐oncological specialties. The present “white paper” catalogs the recommendations of the meeting, at which a consensus was reached that incorporation of molecular information into the next WHO classification should follow a set of provided “ISN‐Haarlem” guidelines. Salient recommendations include that (i) diagnostic entities should be defined as narrowly as possible to optimize interobserver reproducibility, clinicopathological predictions and therapeutic planning; (ii) diagnoses should be “layered” with histologic classification, WHO grade and molecular information listed below an “integrated diagnosis”; (iii) determinations should be made for each tumor entity as to whether molecular information is required, suggested or not needed for its definition; (iv) some pediatric entities should be separated from their adult counterparts; (v) input for guiding decisions regarding tumor classification should be solicited from experts in complementary disciplines of neuro‐oncology; and (iv) entity‐specific molecular testing and reporting formats should be followed in diagnostic reports. It is hoped that these guidelines will facilitate the forthcoming update of the fourth edition of the WHO classification of central nervous system tumors.Major discoveries in the biology of nervous system tumors have raised the question of how non‐histological data such as molecular information can be incorporated into the next  The accurate classification of human neoplasms not only has implications for the care of individual patients (in estimating prognosis and guiding therapy) and for the conduct and interpretation of clinical studies of new diagnostic and therapeutic approaches, but also for the analysis and understanding of basic scientific experiments, for the elucidation of population‐based disease trends that may implicate environmental or other etiologies, and for the allocation of resources by governments and health insurers to support health care. Periodic revisions of tumor classifications therefore have diverse and important effects on many aspects of individual and population health. In recognition of this level of importance, over the past approximately half century, the World Health Organization (WHO) has sponsored tumor classifications by international experts. The last classification of tumors of the central nervous system emerged from a consensus meeting in 2006 Over the past decade, insights into the molecular basis of human tumors have significantly improved both our biological understanding of neoplasms as well as our abilities to diagnose tumors and estimate their prognosis and likelihood of response to specific therapies. Brain tumors have shared in this molecular revolution, and in some areas have been at the forefront. Such molecular information can have diagnostic, prognostic and/or predictive value, with diagnostic and prognostic data often tightly linked with classification and grading, and predictive data also linked with the efficacy of particular therapeutic approaches. A critical question with major practical consequences has therefore arisen: how should clinically relevant molecular information be incorporated into nervous system tumor classification?The 2007 WHO classification of tumors of the central nervous system included well over 100 distinct entities. Many of these tumors do not have molecular characteristics that at present require their implementation in the clinical diagnostic work‐up. However, other tumors are known to have clinically relevant molecular characteristics, and the basic question on how to incorporate this information optimally in their classification is similar across the entities. Before attempting to answer this question separately for each tumor entity, the major conceptual issues that underlie the question need to be addressed. Once the major conceptual issues are resolved, it would then be possible to use this information to guide the next WHO classification of individual tumor entities.A proper answer to the above question also involves a balance between incorporating the latest molecular findings and the practical issues of clinical diagnosis and current patient management. One needs to identify the best balance between the most accurate, cutting‐edge molecular approaches and the everyday necessities of brain tumor patients and diagnostic laboratory resources, all of which are rapidly evolving. To do so requires the input of experts proficient in both the latest molecular advances and their clinical relevance as well as the most practical diagnostic procedures. In order to address the above question, we undertook a meeting of expert, molecularly oriented oncologic neuropathologists who had solicited input on this question from their neuro‐oncological colleagues  prior to the meeting. The meeting addressed the critical question of how molecular information could be used for tumor classification as well as a series of related subquestions. The current publication outlines the consensus recommendations of that meeting.The meeting, entitled “WHO's Next?: A Colloquium to Guide Next Steps in Brain Tumor Classification and Grading,” was held from May 1 through May 3, 2014 in Haarlem, the Netherlands. Twenty‐eight neuropathologists from 10 different countries were invited, with only one unable to attend. In advance of the meeting, the participants were asked to canvass their colleagues involved in brain tumor diagnosis and therapy, both from their own institutions and from other institutions and clinical trial groups in which they participate. The group obtained input from over 150 neuro‐oncological specialists . In this manner, the participants provided their own views and those of colleagues from other disciplines at the meeting.The meeting focused on the overall question and four subquestions shown in Table BCR‐ABL1”), thereby excluding contaminating outliers that potentially confuse clinicopathological correlations. It was recognized that, with this approach, some tumors may not fit neatly into a single diagnostic category , but that this approach minimizes the risk of “wastebasket” diagnoses. In order to achieve this goal, entity definitions must thus be as precise and evidence‐based as possible.The meeting began with an invited lecture by Dr. Daphne de Jong, who provided critical perspective on how hematopathologists have handled similar questions in their field. One of the key operational principles that the group unanimously agreed to adopt from the hematopathology experience is that clinically relevant disease entities should be defined as narrowly as possible in order to establish highly biologically uniform groups ; The meeting was sponsored by the International Society of Neuropathology and made possible through generous support from the STOPbraintumors Foundation, the Netherlands. Neither of these organizations had a role in determining the recommendations made by the participants, but the executive committee of the International Society of Neuropathology reviewed and approved the plans for the meeting as well as the recommendations prior to publication.The meeting reached consensus regarding a broad set of conclusions that pertain to future classification systems for nervous system tumors, including the widely utilized WHO classification scheme. The group did not make comprehensive recommendations about individual tumor entities but, as shown below, utilized examples to illustrate how such decisions could be made in the future.(1) Disease entities should be defined as precisely and objectively as possible in order to establish highly biologically and clinically uniform groups . With this approach, some tumors may not fit into a diagnostic category and may require a descriptive diagnosis ; such “gray zone” tumors require further study before their exact position in the classification could be established.(2) (a) (i) For some entities, molecular information is required to provide an “integrated” diagnosis (see below) and only a descriptive histological diagnosis is acceptable if no molecular diagnostic testing is available. (ii) For other entities, molecular information will be necessary to provide an “integrated” diagnosis but a formal “NOS” (not otherwise specified) can be used if no molecular testing is performed. (iii) To do the above, the definitions of some disease entities need to be refined, while others need to be added.Molecular information should be incorporated into the definitions of some diagnostic entities.(b) For some diagnostic entities, histology alone will remain the basis for definition and diagnosis.Regarding the use of molecular information in diagnosis:(3) (a) The “integrated diagnosis” will be the top line in order to emphasize its primacy over the other lines even though it will be the last portion of the diagnostic format completed, as it is dependent on all diagnostic information being present. It is anticipated therefore that the integrated diagnosis will be “pending” for a period of time between histological examination and the availability of molecular information.(b) The “histological classification” is the standard microscopic diagnosis that is based on hematoxylin and eosin staining and additional histological techniques such as histochemistry, immunohistochemistry and electron microscopy.(c) IDH‐mutant glioblastoma or WNT‐subgroup medulloblastoma in a child), stressing the WHO grade in the diagnosis may be more confusing than helpful, and that such situations may require a comment stating that the prognosis is better in such a molecular subtype than suggested by the grade.The “WHO grade” is the standard histological grade. As in the past, WHO grade reflects natural history after surgery alone, rather than expected patient prognosis following current adjuvant therapies. For example, despite substantially improved control rates with current therapy, a medulloblastoma is still considered WHO grade IV as, if left without adequate postoperative treatment, it will follow a rapidly progressive, typically fatal course. Thus, the question arose as to whether a separate, additional grade reflecting expected behavior following therapy should be considered. However, because both current therapies and responses are subject to changes and having two different grades on a single report is confusing, the group opined that only a WHO grade based on natural history should be reported. This discussion raised the additional challenge that in some tumor types  The “molecular information” is a synoptic account of the results of the molecular tests recommended for that particular tumor entity. Notably, the specific molecular tests recommended vary among tumor entities and will likely change over time. Moreover, the reporting of such molecular information should follow a set of guidelines, which are outlined in the following section.A key concept was that diagnoses should be “layered” in order to provide a format for displaying multiple types of information Tables , 3, 4, 5(4) (a) Whether particular tests are required or recommended for diagnosis will depend on the biological properties of individual tumor types and whether the reported biomarkers are diagnostic, prognostic and/or predictive.(b) Future decisions to incorporate such testing into diagnostic definitions will be based on conclusive published evidence from multiple independent studies.(c) IDH1/2 sequencing to exclude rare mutations if IDH1 R132H immunohistochemistry is negative).For some genetic tests, some general methodological approaches may be recommended over others . In some situations, second‐level tests should follow first‐level tests  MGMT promoter methylation status”).In settings in which molecular testing is required or recommended, a report should state if it was not done (“unknown”) or if ordered (“pending”), along with a reason if not performed  The methodological and results parameters of the assays performed should be included in reports in order to provide testing details and interpretive significance. This was felt important as, in some institutions, molecular reports are separate from surgical pathology reports and the results of molecular testing are either left out of the pathology report entirely or only abstracted in addenda. The group felt that by incorporating the pertinent details into the original surgical pathology report, this would facilitate comparability of data and multi‐institutional patient care, given that the pathology report alone is often forwarded to outside centers.(f) Molecular testing must be based on histologically representative tissue. While this practice is routine in most academic centers, it remains possible that fragments of tissue are sent directly by neurosurgeons to molecular testing laboratories without histological confirmation; this practice risks false‐negative results and must be avoided.Regarding molecular testing and reporting:(5) The grading of adult type diffuse gliomas will follow standard, current WHO criteria for astrocytomas and oligodendrogliomas with the caveat that in some circumstances, assigning a precise grade is not possible. The latter is most relevant for the category of diffuse glioma that is not clearly of pure astrocytic or pure oligodendroglial subtype, either in the setting of a small biopsy in which selective sampling may be a concern or because of lack of molecular studies being performed, a discordance between morphology and molecular studies , or a molecular pattern that does not fit neatly into a single tumor type. In such circumstances, the WHO grade may either be left off altogether (preferably with an explanatory comment) or may appear as “high grade” or “at least WHO grade …”. For example, a phenotypically ambiguous diffuse glioma with atypia, mitoses, microvascular proliferation, and necrosis could initially be diagnosed as being “at least WHO grade III” given that it would qualify as grade III if oligodendroglial  or grade IV if astrocytic . In compliance with WHO terminology, the term “anaplastic” will precede any astrocytic or oligodendroglial tumor qualifying for a grade III designation. The term glioblastoma will be utilized for astrocytic neoplasms qualifying as grade IV.(6) H3.3 K27M mutations in diffuse pediatric high‐grade gliomas/intrinsic pontine gliomas and a rarity of 1p/19q co‐deletion in pediatric oligodendrogliomas).Some pediatric tumor types will require separation from their adult histological “look‐alikes.” Separating these pediatric entities becomes critical now that there is clear evidence that the underlying molecular basis is different  The inclusion of non‐tissue‐based information  is not required in the layered final diagnosis, but can be of clear utility in reaching a final diagnosis or determining sampling adequacy in individual cases. As such, non‐tissue‐based data can be included in the clinical history or comments section, as is already common practice.(8) Input from a broad constituency of clinical and scientific colleagues involved in neuro‐oncology is important in guiding future decisions regarding brain tumor classification.The conclusions reached can be summarized as follows:(1) For each tumor that involves a wide age spectrum, WHO working groups should decide whether the available molecular and clinicopathological data justify separating it into distinct pediatric and adult subtypes. Decisions on suggested clinical‐radiological phenotypes and defining molecular features would be necessary. In some situations, temporary “gray zone” designations may be needed.(2) (a) For those entities for which molecular information is required for diagnosis, terminology should be considered for otherwise histologically compatible tumors that have discordant molecular profiles, lack sufficient tissue for molecular analyses or are diagnosed at centers that cannot perform molecular testing. In most of the latter situations, it is expected that only a descriptive diagnosis will be possible and that such descriptive diagnoses may not have distinct International Classification of Diseases for Oncology (ICD‐O) codes .(b) For those entities for which molecular diagnostic information is suggested (but not required) for diagnosis, terminology should be considered for otherwise histologically compatible tumors that either have discordant molecular profiles or are diagnosed at centers that cannot perform the molecular tests. In such situations, it is likely that “NOS” (not otherwise specified) categories will need to be created and that such “NOS” diagnoses will need to have distinct ICD‐O codes .For each tumor entity, the working groups should decide whether molecular testing is required or suggested to make the diagnosis, or whether clinically meaningful diagnostic or prognostic molecular information is lacking, such that the diagnosis remains entirely histological in nature.(3) IDH‐mutant glioblastoma may be somewhat misleading in terms of expected behavior. In such situations, it can be helpful to add a comment stating that the prognosis may be better than suggested by the grade.For each tumor entity, the working groups should make recommendations for situations in which clarification of the WHO grade may be helpful. For example, as WHO grade reflects natural history following surgery alone, a “grade IV” designation in an (4) For each tumor entity with an altered name or definition, the working groups should consider the addition of a section entitled “synonyms” in order to list the corresponding prior names of entities, thus allowing for ready connection between the updated and prior classification systems for clinical care as well as clinical, experimental and epidemiological study purposes. For instance, if the diagnosis of “oligoastrocytoma” is no longer rendered, the synonym “oligoastrocytoma” would be listed for histologically ambiguous diffuse gliomas, whether or not molecular testing has resolved tumor diagnosis.(5) IDH1 and ATRX analysis may be carried out initially, but would 1p/19q analysis still need to be performed in gliomas with ATRX mutation or loss of protein expression ? When a diffuse glioma has whole‐arm 1p/19q co‐deletion , is it still necessary to assess the IDH and ATRX status as well? Another major question would be whether sequencing analysis for IDH mutation is needed when IDH1 R132H immunohistochemistry is negative. One might argue in such a situation that additional sequence analysis would be recommended in the setting of a low‐grade tumor in a young adult patient in which the likelihood of IDH mutation is high, but not in the setting of an elderly patient with a glioblastoma that had no prior history or histological evidence of a lower grade precursor . It is also entirely possible that genetic tests not discussed at this meeting  will be incorporated into diagnostic definitions at the time of the eventual WHO classification revisions. Lastly, it may be that certain important “negative” findings should be included, for example, that amplification of the chromosome 19 miRNA cluster was not found in the evaluation of an embryonal tumor.For each group of tumors , considerations should be given to defining recommended tests and the order of carrying out such tests if done sequentially. For example, in a diffuse glioma without morphological evidence of oligodendroglioma, (6) It is expected that a combined histological and molecular approach can classify the vast majority of cases and all of the common combinatorial scenarios. Nonetheless, there will likely be unusual combinations that will necessitate descriptive diagnoses and the working groups should address recommendations on how to list these rare situations.(7) The WHO working groups should avail themselves of the broad input that will be solicited from the clinical and scientific neuro‐oncology communities.The meeting did not address the above issues across the spectrum of tumor entities in the 2007 WHO classification, but instead addressed these issues in “test” cases, which included the diffuse gliomas, medulloblastoma and atypical teratoid/rhabdoid tumor (AT/RT). Nonetheless, the group felt that the principles derived from analyzing these tumor types could be applied across the spectrum of WHO entities in the forthcoming update of the fourth edition. As such, the group proposes the following guidelines with a decision flowchart included as Figure sine qua non in order to use that data and information to generate knowledge. The ISN‐Haarlem guidelines provide logical formats to convey tissue‐based data and information. As such, it is hoped that these templates be used by the various WHO working groups as they begin defining and redefining nervous system tumor entities and that these definitions form the basis for the knowledge that will improve the treatment of brain tumor patients in the near future.As data has become more plentiful and complex in the modern world, there has been a need to distinguish among data, information, knowledge and wisdom (“DIKW”) and to develop systems that facilitate the conversion of data to information to knowledge and eventually to wisdom. Similarly, in medicine, as data have become more plentiful and more complex, there is an increasing challenge to convert that data into clinically meaningful information that can be used to treat patients. Presenting that data in logical and accepted formats is a


Molecular Biomedicine is a peer-reviewed and open-access journal launched by Springer Nature in 2020, publishing the pioneer works in molecular medicine. We appreciate for the sincere support of many talented scientists and academic institutions worldwide that inspired us to start this journal.Molecular medicine has a wide academic scope across basic, translational and clinical medicine, as well as physics, chemistry and biology from a molecular point of view to manage human diseases. The landmark event in the origin of molecular medicine was the achievement in 1949 published by Linus Carl Pauling, the two-times Nobel Prize laureate, who pointed out that sick cell anemia was a molecular disease caused by the chemical modification of a single well-identified protein. In 1990s, the launch of Human Genome Project guided molecular medicine into a new era, leading to the progress of epigenetics, genomics, transcriptomics, proteomics and metabolomics, as well as the invention of novel biotechnologies such as CRISPR/Cas9, single-cell sequencing and machine learning. Driven by these advances, many novel pathogenic factors including DNA methylations in cell development, non-coding RNAs in cancer, and protein post-translational modifications in neurodegenerative disease, were identified, promoting the rational understanding of the molecular mechanism in pathogenesis. Under the efforts of translational medicine research in recent years, many cutting-edge theories and technologies have been transformed into products that improve human health, such as cobas EGFR Mutation Test  for the liquid biopsy of non-small cell lung cancer, Keytruda  for cancer therapy, and Praluent  for hypercholesterolemia treatment. The emergence of these products in turn triggers the progress of molecular medicine.Molecular Biomedicine is established to meet the current and future demand.Indeed, molecular medicine is a rapidly developing realm of research. The number of papers published each year in this field has dramatically increased over the past years. Many governments and foundations around the world are increasingly providing financial support to the research in molecular biomedicine, aiming to improve the diagnosis, prevention and treatment of human diseases. Undoubtedly, numerous important discoveries will be accomplished in the near future, but the limited number of existed high-impact journals in this field cannot ensure the timely publication of relevant works that will emerge at an increasing rate. On this basis, Molecular Biomedicine is to publish the significant findings across the full spectrum of medicine, such as clinical medicine, genetics, bioinformatics, molecular biology, cell biology, pharmacology, medicinal chemistry and pharmacy. This journal covers major human diseases including cancer, cardiovascular diseases, diabetes, neurodegenerative diseases, bacterial and virus infection, and chronic bronchitis, particularly emphasizing on the molecular process to disclose innate cause of diseases and suggest new techniques for diagnosis and therapy. The scope of Molecular Biomedicine includes but is not limited to epigenetics, genomics, proteomics, gene editing, RNA biology, signal transduction, structural biology, early diagnosis and biomarkers of disease, gene therapy, cell therapy and immunotherapy, targeted therapy by small molecules, molecular imaging, artificial intelligence in medicine and regenerative medicine.The goal of Molecular Biomedicine, suggesting a new opportunity to share information across the specialists and the public.We aim to establish a leading platform for medical research and offer the latest advances to meet the needs of academic readers. A range of content types such as original research articles, reviews, letters-to-editor, and research highlights related to the original discoveries or critical thinking in biomedicine are interested by Molecular Biomedicine has a globally renowned Editorial Board members composing of high-impact scientists from all over the world. We welcome researchers to submit valuable works to Molecular Biomedicine and promise to give a fair and prompt first decision within 3 weeks after submission. Our editorial staffs will also provide high-quality support to ensure your article to publish online within 2 weeks after formal acceptance.Molecular Biomedicine gives a forum for students, scientists, and clinicians interested in biomedicine to share their discoveries and viewpoints. We would complete our peer-review process as quick as possible, publish high-quality papers, and would like to invite you to submit your manuscripts to Molecular Biomedicine.In summary, with advances in new technologies including high-throughput DNA/RNA sequencing, gene editing, artificial intelligence and nano-biomaterials, we have entered an unprecedented era, in which the molecular process of life is increasingly understood, and precise biomarkers and targeted therapies are timely emerged.


Accuratelydescribing excited states within Kohn–Sham (KS)density functional theory (DFT), particularly those which induce ionizationand charge transfer, remains a great challenge. Common exchange-correlation(xc) approximations are unreliable for excited states owing, in part,to the absence of a derivative discontinuity in the xc energy (Δ),which relates a many-electron energy difference to the correspondingKS energy difference. We demonstrate, analytically and numerically,how the relationship between KS and many-electron energies leads tothe step structures observed in the exact xc potential in four scenarios:electron addition, molecular dissociation, excitation of a finitesystem, and charge transfer. We further show that steps in the potentialcan be obtained also with common xc approximations, as simple as theLDA, when addressed from the ensemble perspective. The article thereforehighlights how capturing the relationship between KS and many-electronenergies with advanced xc approximations is crucial for accuratelycalculating excitations, as well as the ground-state density and energyof systems which consist of distinct subsystems. I1 In principle,this is possible within density functional theory (DFT)8 as the ground-state density, n(r), contains all the information about the many-electron system’sground and excited states according to the first Hohenberg–Kohn(HK) theorem.9 However, in practice, sucha description is extremely challenging. The excitation spectrum, thefundamental gap , I, and the electron affinity (EA), A),and charge-transfer energies  are of particular importance.28 The unreliable performance of standard exchange-correlation (xc)approximations for these quantities is in contrast to the remarkablesuccess of Kohn–Sham (KS) DFT for various applications to ground-stateproperties of materials.37 In this article, we explore the exact relationship between KS excitationenergies and the corresponding many-electron quantities with standardand ensemble DFT. We study the consequences of this relationship onthe exact KS potential and its importance for the advancement of approximatexc density functionals.Describingmany-electron excited states at an affordable computationalcost remains an important goal within solid-state physics, quantumchemistry, and materials science.40 within KS-DFT the relationshipbetween the KS energy levels, {εi}, and the many-electron energies, {Ei}, is not generally straightforward.For example, while for the exact KS potential the highest occupied(ho) KS energy level, εho, equalsminus the IP,47 −I, the fundamental gap, Eg = I – A, doesnot simply equal the KS gap, EgKS = εlu –εho  and the ho KS energies), even for the exact KS potential. Instead,the KS gap differs from the fundamental gap by Δ, known as thederivative discontinuity:61Eg = I – A = εlu – εho + Δ. Δ manifests in the exact xc potential asa uniform shift when the number of electrons within the system infinitesimallysurpasses an integer. It occurs because the xc energy of the systemhas discontinuities in its derivative as a function of electron number, N, at integer values of N.Unlike other commonly used methods forelectronic structure calculations,e.g., many-body perturbation theory,62 that the charge-transferenergy in stretched systems differs fromthe corresponding KS energy difference by the charge-transfer derivativediscontinuity (CTDD), ΔCT, which occurs when a fractionof charge is transferred from one subsystem to another within thewhole system. The CTDD proved to be an important concept for accuratelymodeling charge transfer within KS theory in practice.63Similarly,it has been shown recentlyℏωog), is relatedto the corresponding KS gap (εlu – εho = ℏωogKS) via a derivative discontinuity,64 as such ℏωog = ℏωogKS + Δog.In 1995, Levy proposed that the optical(uncharged) gap, i.e.,the energy to excite an electron from the ground to its first excitedstate  method;78 however, this approach has known numerical difficulties which arisefrom the use of a finite basis set.81 Solutions have been proposedto overcome these numerical issues;83 however, the OEP methodis yet to be adopted as a mainstream approximation within DFT owingto the numerically challenging nature of the approach.In general, sharp spatial steps may occur in the g., refs  and 68).68 which stems from the decay rateof a given subsystem. Moving from one subsystem to another leads toa change in the LEIP, which causes a sharp spatial step in the xcpotential. The height of the step is analytically derived below fromthe density decay rate, following refs ,ing refs  and 68.89 and a molecule between two metallic contacts in a transport experiment.96 Therefore, understanding the steps in the exact KS potential iscrucial, as it allows one to accurately describe various scenariosin real materials of high practical importance with KS DFT.A simple and instructiveexample of a system with a step in thexc potential is a stretched diatomic molecule L···Rsketched in d = |d|, AtomL is located at x being the interatomicaxis , as suchNL0 electrons on AtomL and NR0 electrons on R; see NL···R0 = NL0 + NR0. d → ∞; those are the systems on whichwe focus below. In the diatomic molecule L···R with interatomicdistance axis see 1.In theexact KS potential ofthe whole molecule, vL···RKS(r), takesfor large d,97 and howit relates to the atomic potentials, vLKS(r) and vRKS(r). Isit that, similarly to the molecular density, vL···RKS(r) has to reproduce the atomic density 9 we know that this potential is unique, up toa constant,and equals 98 The sameis, of course, also true for Atom R. However, the simple superpositionof the atomic KS potentials can create the following problem , εLho. Then, from the perspectiveof the KS system, the electron which should localize on L will spuriouslydo so on R, resulting in the wrong number of electrons on each atom.99 In the case in which the atoms within the moleculeare bonded, the molecular ho levels of Atoms L and R ought to be aligned;this does not always happen if the two atomic potentials are simplysuperimposed.Now we ask what formthe plateau, which results in a spatiallyabrupt step in the KS potential between the atoms(and a complementary step far to the right of Atom R).65 In the vicinity of Atom R, the molecular potential equals What must the exact KS potential do to maintainthe correct atomicdensities in the vicinity of each atom while yielding the correctdistribution of charge within the molecule? The answer is to raisethe level of the potential around one of the atoms, in our case AtomR, forming a IL and IR, and the molecular orbitalenergies of the system as a whole , as wellas the molecular energy levels are known; see molecular KS energy levels by {ηi} to clearly distinguish them from the atomic KS energy levels, {εi}. We alsoexplicitly indicate whether the molecular orbitals localize on oneof the atoms by the subscripts L and R. Generally, in the vicinityof Atom L, the molecular KS potential vL···RKS(r) is identical to the atomic potential, v′,and in the vicinity of R, vL···RKS(r) isidentical to v″. The difference v″ – v′ is therefore the interatomicstep height, S.100 Furthermore,in the vicinityof Atom L, the molecular density is nL···R(r), which equals the (shifted) atomicdensity, Lho(r)|2. This orbital decays as106v′ = ηLho + IL. Similar analysis for Atom R yields v″ = ηRho + IR. Combining these two results and recalling that S = v″ – v′, we arrive at an expression for the interatomic step:68S inthe potential.107 The step is generallynonzero, because the KS energy differences do not equal the many-electronenergy differences, as mentioned in the IR – IL, doesnot equal the KS energy difference, ηLho – ηRho. The step forms at the point in the electron density wherethe decay from the left meets the decay from the right, and the LEIPabruptly changes.Followingref , we now also ref  and refety,  see 2, and desee refs −105). Frmolecular energy levels, {ηi}, and not the atomic levels, {εi}. Therefore, in general, S, relying only on atomic calculations. This equation rather showsthe relationship between S, the molecular KS energies,and the many-electron energies, IL and IR, associated with each atom.We wish to emphasize that the right-hand sideof ηLho and ηRho need not be assumedequal. The latter is true when L and/or R are closed-shell atoms.In the particular case that L and R are bonded, the ho KS orbitalstretches over both atoms, and therefore, it follows that, in thenotation adopted here, ηRho = ηLho. As a result, S = IR – IL by Almbladh and von Barth.108IL or IR is the overall IP of the molecule; in the case depicted in IL. Thus,the overall highest occupied molecular orbital (HOMO) energy is ηLho and is equal to the atomic orbital εLho when v′= 0. Furthermore, due to the IP theorem in DFT,110 which we discuss indetail below, εLho = −IL. It then followsthat S = IR + ηRho. It doesnot necessarily follow, however, that S vanishes.A generally nonzero S stems from the inclusion ofthe molecular energy, ηRho, opposed to the atomic energy,εRho,in Rho equals−IR, whereas the molecular energy ηRho does not, as it is elevated relative to the atomic energyby the step height S: ηRho = εRho + S.Depending on the atoms L and R, either 111 in which theexact KS potential is separated into the KS potential for each individualsubsystem plus the “partition potential”. In the limitthat the subsystems are completely separated–in our case thetwo atoms–the partition potential consists of the interatomicstep described above.112 The partitionpotential is a functional of the density of each fragment of the system113 and hence is nonlocal in character.114 In addition, the exact partition potentialis known to contain derivative discontinuities.115 The perspective allowed by PDFT offers an approach to developingapproximations which capture these discontinuous features, yieldingaccurate binding energies of disassociated diatomics117 or a reliable description of charge transfer.119 The partition potential has also been shown to be a chemically significantreactivity potential.121Our decomposition of this molecule into fragments is reminiscentof Partition DFT (PDFT)II.BN, varies continuously and infinitesimally surpasses aninteger value. A fractional number of electrons in our systems ofinterest may be considered as a time average of the number of electronsin an open system, namely in a system which is freeto exchange electrons with its surroundings  states.41The uniformjump Δ occurs in the KS potential when the number of electrons, .g., ref  (§14)). N to change continuously. Second, in In the following, we consider three typesof many-electron systems. First, in this section, we describe in detaila finite system that is connected to an electron reservoir, whichallows N = N0 + α electrons,where N0 is an integer number, and 0 ⩽α ⩽ 1. As mentioned above, the ground state of such asystem is an ensemble, which combines states each with a differentinteger number of electrons. For systems with Coulomb interactionat zero temperature, this ensemble consists only of states for N0 and N0 + 1 electrons,|ΨN0⟩ and |ΨN0+1⟩:125 As a direct consequenceof Ô in the ensemble stateis O = Tr{Λ̂Ô} = (1−α)⟨ΨN0|Ô|ΨN0⟩ + α⟨ΨN0+1|Ô|ΨN0+1⟩.41 In particular, the average density of a system with N electrons isn is the ground-state densityfor the N0-electron system, and n is the ground-state density for the (N0+1)-electron system. Furthermore, the total energy asa function of N equalsE(N) is piecewise-linear in N: for any fractional N, the energy is linear, but it can change its slope when N passes an integer. Consequently, the chemical potential,μ = ∂E/∂N, isa stair-step function of N. For example, in the groundstateI = E(N0–1) – E(N0) is the IP, and A = E(N0) – E(N0+1) is the EA of the system.Clearly, the chemical potential is generally discontinuous at integer N; the height of this discontinuity equals the fundamentalgap of the system, Eg = I – A.We start with a finite system, like an atom or a molecule, with 126 whichstates that the ith KS eigenenergy εi = ∂E/∂fi–the derivativeof the total energy with respect to the occupation of the ith level, fi–we find that the ho KS energy level, εho(N), equals the chemical potential,μ(N), and is also discontinuous at integer N  = −I and infinitesimallyabove εho(N0+) = −A. The IP theorem in KS DFT is an exact result, for the exact xc potential.Furthermore, from a combination of the piecewise-linearityof theenergy and Janak’s theorem,vxc(r). From the perspective of the KS system, increasing N above an integer means occupying the next KS level, εlu(N0–). As εlu(N0–)does not necessarily equal −A, evenfor the exact KS potential , can only come from vxc(r), becausethe Hartree potential is continuous and the external potential is N-independent. Therefore,ho(N0+) ought to have, namely −A, and the value it has in the absence of discontinuity, εlu(N0–): Δ = −A – εlu(N0–). Together with εho(N0–) + I = 0, and dropping here the argument N0– for brevity, we arrive at the following familiar form for Δ:Eg = I – A, and theKS gap, EgKS = εlu – εho. The derivative discontinuity is a topic of great importanceand has received much attention over the years .II.CNL0 electrons on Atom L and NR0 electronson Atom R, so the total number of electrons equals NL···R0 = NL0 + NR0. Next, we allow the total number ofelectrons to vary continuously: NL···R = NL···R0 + α (−1 ⩽ α⩽ 1). We consider the specific case for which any additionalcharge localizes on Atom R, whereas any charge deficiency resultsin a decrease of charge around Atom L. As we show by a direct charge-transfercalculation in Let us now consider a stretched diatomic molecule L···R,where the separation between the atoms is large enough for the energyand density of the molecule to satisfy NL···R, or equivalentlyto α, is a stair-step function discontinuous at integers, qualitativelysimilar to the chemical potential depicted in L···R isthe left-to-right charge-transfer energy, EL→RCT = IL – AR, namelythe energy required to remove one electron from Atom L minus the energygained by adding an electron to an infinitely distant Atom R. As forthe finite system discussed above, the stretched molecule L···Ralso obeys the IP theorem. Namely, the overall HOMO energy, ηho(NL···R), hasto equal μL···R(NL···R). For NL···R slightly below NL···R0, the overall ho energy equals ηLho(NL···R0), which in our case, as explained in IL. As the overall number of electrons increasesabove NL···R0, the overall ho level is localized aroundAtom R and has to equal −AR. Asa result, the molecular potential vL···RKS(r) jumps by the constantcharge-transfer derivativediscontinuity. ΔL→RCT is thedifference between the charge-transfer energy, EL→RCT = IL – AR, andthe corresponding quantity in the KS system (ηRlu – ηLho)  cf. 10.NL···R0, is constant and aninteger. When the molecule is excited, a fraction of q electrons is transferred from Atom L to Atom R. We define an ensembleconsisting of the ground state, |Ψ0⟩, of themolecule and the first excited state |Ψ1⟩where the latter has charge-transfer character, i.e., the nature of|Ψ1⟩ is such that compared to the ground state,one electron is transferred from Atom L to Atom R. The statisticaloperator describing this ensemble is given by0⟩and |Ψ1⟩, have a fixed (integer) particlenumber, N0. The ensemble expectation valueof any operator Ô, by virtue of O = Tr{Γ̂Ô} = (1–q)⟨Ψ0|Ô|Ψ0⟩ + q⟨Ψ1|Ô|Ψ1⟩. In particular, the ensemble density is given byn0(r) and n1(r) are the densities ofthe ground stateand the first excited state, respectively. Likewise the total ensembleenergy as a function of q equalsE1 = E0 + ECT; for this system with alarge but finite atomic separation, ECT = ĨL – ÃR for q > 0 and ECT = ĨR – ÃL for q < 0, where ĨL is the ionization energy of the wholemolecule which corresponds to an electron localized to the left atomwhile ÃR is the molecule’saffinity and corresponds to the addition of an electron to the rightatom once the electron on the left atom has been ionized–this is the nature of a charge-transfer excitation. Consequently,both ĨL and ÃR are influenced by the Coulomb interaction between theleft and right atoms; this effect has previously been omitted becausethe atoms were assumed to be infinitely separated: limd→∞ĨL – ÃR = IL–AR (as defined above).By modeling the system with a finite separation we more closely modela real donor–acceptor pair for short- to medium-range chargetransfer. The difference between ĨL – ÃR and IL – AR is the electron–holeelectrostatic interaction. For a large separation between the donorand acceptor, it is usually approximated as −1/d.154Finally, we consider a stretched but finite diatomic molecule inwhich the atomic separation is large enough to define individual atomswithin the molecule but in which the electrons localized on the leftatom experience the Coulomb repulsion of the electron localized onthe right atom and vice versa. The total number of electrons withinthe molecule, ECT inthis system into q (see m(q) = ∂EL···R/∂q, whichis the change in energy as a result of transfer of charge, is a stair-stepfunction:157 we can express the charge-transfer energyas suchηiq is the ith KS energy of the ensemblesystem. As q→ 0+, ηN0+1q – ηN0q = ηRlu – ηLho. Therefore,recalling that in the limit of infinite atomic separation q-dependentxc functional.160 In ref  equals the differencebetween the chemical potentials of the constituent atomsm(q) around 0, denoted here m(−q), equalsD = Eg,L + Eg,R. Using D canbe also expressed in terms of the KS quantitiesD mayalso be expressed solely in terms of the KS gaps andΔ’s of the constituent atoms using D, can equally be expressed in theKS system in terms of the derivative discontinuities of the individual atoms and also in terms of the charge-transferderivative discontinuities of the system as a whole. We shall see below in S, derivedin Note that in the limit that Atom L and Atom R becomeinfinitelyseparated, Finally, we emphasize two additional results.From III162 in which the exact, fully correlated many-electron wavefunction may be calculated for an arbitrary external potential. Inaddition to the ground state, the many-electron excited states arecalculated by solving the many-electron Schrödinger equation.163 As a result, we have access to the exact many-electronground-state and excited-state electron densities, from which theexact corresponding KS potentials can be calculated by a numericalinversion of the KS equations. Our inversion algorithm to calculatethe KS potential is that of ref (ntar(r) and an initialguess for the KS potential, vKS(0)(r), the followingiterative procedure is performed: For the kth iteration,a DFT calculation with vKSk)((r) ismade, and the density nk)((r) is obtained. Then, the KS potential for the nextiteration is updated, as follows: vKSk+1)((r) = vKSk)((r) + λ[(nk)((r))p – (ntar(r))p], where λ and p areparameters .The procedure continues up to numerical convergence, which in ourcase happens when the mean absolute error between the many-electronand KS densities is <10–9 Bohr–3. More details for this algorithm can be found in ref /ntarget(r)) < 10–4, enforced for r∈, with L′ = 30 Bohr. Finally,the parameters a and b requiredfor the alignment of the KS potentials, which show the asymptoticbehavior of ∼a/r + b . Fromthe KS perspective, the density decay is governed by the ho orbitalsquared, ho(N0–) = −I. If the number of electrons is now slightly increased above N0 by a small fraction of an electron, α,the density becomes a linear combination of n and n, as in n decays ∝ A-decay), which is slowerthan the decay of n, because I > A for all known systems (known as the convexity conjecture123). Therefore, the A-decay asymptotically dominatesthe density decay. From the KS perspective, the decay of the densityis dominated by the now highest, partially occupied orbital . The problem arises when taking ho(N0+), relative tothe overall potential shift, Δ (cf. ho(N0+) = εlu(N0–) + Δ, one may further inferthat the density decays ∝lu(N0–) = −A and hence Δ = 0. In other words, if the jump Δ is uniform,its height is zero.If the number of electronsin the system equals an integer t, Δ cf. 3. Recall+. Although n decaysslower and is thus the asymptotically dominant term, it is multipliedby the small coefficient, α. As a result, we have a competitionbetween the two decay rates: when we reduce α to 0, while lookingat a fixed and large r, the region in which the A-decay is dominant moves away from the nucleus as the term αn vanishes and the term (1−α)nprevails. The process is illustrated in vext(x) = −2.0/(0.4·|x|+1), with 1 + α same-spin electrons.168 It is useful to look at the natural logarithmof the density in order to clearly see the decay rates, as such aregion of an exponential decay appears as a straight line of negativeslope. Indeed, in I- and A-regions of exponential decay. As α decreases, the A-decay region appears further away from the nucleus. Next,recalling our conclusion from not uniform. Thisresolves our paradox: the correct decay rate of the density in theregion of A-decay is ∝ ho(N0+) = εlu(N0–)+ Δ = −A, as required; whereas in theregion of I-decay the potential is elevated by Δ.As a result, steps form in the potential as shown in N, the quantities Δand S have the following relationship: lim+α→0S = Δ. For the system presentedin To resolve this paradox, we look more closelyat order-of-limits problem, namely limr|→∞| lim+α→0vKS = Δ, whereas lim+α→0 limr|→∞|vKS = 0. In other words, if we examinethe value of the KS potential at some finite point |r| while continuously decreasing α to zero,for a certain α the plateau will be wide enough to reach |r| and elevate the potential there. Takingthen |r| to infinity will result withthe height Δ for the KS potential. Conversely, taking |r| to infinity first, while keeping αfinite, ensures that for any finite α, no matter how small,we will reach the edge of the plateau, and the potential value willdrop to 0.Finally, we wish to add several comments on plateausin finitesystems. First, the shape of the steps observed includes characteristicdips clearly seen in IV.BParadox 2 – The transfer of charge in a diatomicmoleculeresults in a plateau, Δ, around the acceptor atom. Yet, theoverall interatomic step height must remain S. To furtherexplore the relationship between Δ and S, westudy the stretched diatomic molecule presented in R to emerge around the acceptor atom, in our case Atom R (withno significant change around L), but this is contrary to the resultsof S around Atom R, irrespectiveof any infinitesimal transfer of charge, to ensurethe correct distribution of charge in the ground-state KS system.173 As S ≠ ΔR, and (thinking of the complementary scenario of right-to-leftcharge transfer) S ≠ ΔL either,there appears to be a contradiction.three regions of exponential decaybetween the atoms: going from right to left, the density decay isfirst governed by IR and then by AR (changing at point (2); cf. AR-decay meets the IL-decay at point (1), simply due to the factthat the two atoms form one molecule. As a result, we expect not onebut two steps in the KS potential between the atomsin this diatomic molecule (62 similarly to the derivation of S(2), which depends solely on quantities related to Atom R,equals ΔR, whereas the step S(1) equals −ΔL→RCT. Importantly, the steps S(1) and S(2) combine to yieldthe overall step S of R is expected to formon the receiving Atom R upon charge transfer or addition, but in conjunction,in the region of Atom L, the KS potential shifts when the “localelectron number” decreases below an integer.The combination of these two plateaus yields an overall interatomicstep of height S.To resolve this paradox,we refer again to the density of the system. For both Cases (i) and(ii), the natural logarithm of the density in between the two atomsis sketched in S in Case (i)has been illustrated and extensively discussed in ref (S(1) and S(2), havebeen identified both in a 1D model of a stretched diatomic moleculeand in a 3D (Li ··· Be)3+ ion. Case (ii)is numerically illustrated in The internal structureof the step d in ref . The twoVexact KS potential that describes the process,is by no means a trivial task.174 To thisend, it is necessary to exactly obtain not only the ground state ofthe system but also its first excited state that corresponds to acharge transfer.Simulation of a charge-transferprocess, and particularly obtainingthe integer number of same-spin electrons, in this case NL···R0 = 2. vext(x) =−4/(0.6·|x–7.5|+1) – 2/(0.4·|x+7.5|+1), is asymmetric, chosen such that the ground-stateelectron density corresponds to a system with one electron localizedon Atom L and one electron localized on Atom R, whereas in the firstexcited state both electrons are localized on Atom R. Hence, by excitingthis system we can initiate a transfer of charge from L to R. We firstfind the exact many-electron ground-state density n0(x) and the first excited-state density n1(x). Then, we construct anensemble electron density, which corresponds to a transfer of a fractionof q electrons from left to right by a linear combinationof the ground-state and excited-state densities, given by q ⩽ 1.174 We emphasizethat all the densities present in In this section, we present a prototypical1D stretched diatomicmolecule L···R, which we excite in order to transfercharge from Atom L to Atom R. Our system consists of an q ⩽ 0.5. Hence, there exists a KS system, which exactlyreproduces the electron density of nby numerical inversion  = |ϕ0(x)|2 + (1–q) · |ϕ1(x)|2 + q · |ϕ2(x)|2. When the system is excited, a fraction of the electron (q) initially occupying the first excited KS orbital is transferredinto the second excited KS orbital localized in our case on Atom R,while the overall number of electrons stays constant and an integer;in this sense, this type of excitation is uncharged , but in the vicinityof each atom, this excitation corresponds to a charged excitation. This observation may explainwhy approximate KS theories, such as linear response time-dependentDFT (TDDFT), struggle to accurately describe charge transfer.177The GOK theorem ensures a one-to-one mapping between the densityand the local potential for this excited system, provided that 0 ⩽ nversion III. In o178 The authorsmodeled a spin singlet which in its ground state consisted of twoelectrons localized to one potential well; the first excited statecorresponded to an electron localized each to a distinct potentialwell. The authors found an interatomic step in the exact xc potentialupon charge transfer, the overall height of which acted to align thechemical potentials of the two wells.178 However, Pribram-Jones et al. did not observe a plateau which correspondsto Δ localized to the acceptor as in their ground state bothelectrons were localized to the donor, and hence initially no electronswere localized to the acceptor. Our model charge-transfer system consistsof one electron localized to the donor and another (same-spin) electronlocalized to the acceptor in the ground state. Therefore, our donor–acceptoris more general in character, and hence, upon excitation we expectto observe the double step structure, one which corresponds to Δfor the acceptor atom and one to the CTDD, as described in The exact ensemble xcpotential for charge transfer was first studiedby Pribram-Jones et al.ground-state electrondensity, ln(n0(x)), forour diatomic molecule: each electron occupies its own potential well,and far from the well the density decays exponentially. There aretwo regions of decay between the atoms–the IL- and the IR-decay–andhence one step at the point where the decay of the density changesyielding a change in the LEIP; see S isexpected far to the right of Atom R, when the IL-decay will prevail over the IR-decay (not shown in the figure). Both steps together form a plateauof height S around Atom R. n), the natural logarithm of the exact excitedmany-electron density, given by q = 5 × 10–4. For reference, ln(n0(x)) is also shown. There are now three regions of exponential decayin the density n: the IL-, AR-, and IR-decay, as we expected  and S(2) (arrows). The right (acceptor) atom experiences thejump in the KS potential characteristic of the derivative discontinuity,i.e., S(2) = ΔR, owingto the local number of electrons of Atom R surpassing an integer bya small amount (q). Simultaneously a plateau formsin the vicinity of the left donor atom. The height of the plateauis ΔL→RCT, i.e., the CTDD associated with transferring an electronfrom the left to the right atom (S(1) is therefore equalto −ΔL→RCT (The minus sign describes the fact that S(1) is a step down betweenthe atoms, whereas ΔR is a step up.). The sum ofthe two steps equals the overall step of cted cf. IV.B. Theght atom 13. S(1) fractional number of electrons NL···R = 2.0005 in the ground state. This means that the approach chosen in ref  system, upon excitationfrom its ground to first excited state, not necessarily related toa transfer of charge? Shall we expect steps in the potential alsoin this case? To explore this question, we model a single atom withan integer N in its ground and excited states, tofind whether its KS potential forms any plateaus upon excitation.This concept was first proposed by Levy64 and was analyzed numerically by Yang et al.179 Below we study how the change in the exact KS potentialof the excited ensemble state varies with the ensemble weight, β,which allows us to compare this scenario with those studied above.In vext(x) = −2.0/(0.4|x|+1) with N0 = 2 . We calculate the exact ground-state and the first excited-statedensity. We then find the ensemble electron density employing the1D version of q = β in this case, for β = 10–4, 10–3, and 10–2, and invert the KS equations to find the corresponding exact KSpotential associated with each density.We model a single atom in 1D with the external potential –4, 10–3, and 10–2. The exciteddensity has two regions of decay in each case: closer to the originthe I-decay region is present, but then the decayrate changes and the density decays slower. The rate of decay of thisexcited density is determined by I – ℏω01, where ℏω01 is the energy required to excite the many-electronsystem from the ground to the first excited state. Due to this changein the density decay rate, we expect steps in the potential of thecorresponding KS system.vxc–the difference betweenthe xc potential of theexcited system and the ground-state xc potential. In the central regionof the system, the excited KS potential is elevated by a plateau ofheight Δ01. The height of the plateau can be analyticallydeduced, as before: Δ01 = I –(I–ℏω01) – (εN0+1β–εN0β), where εiβ is the ith KS energy of the ensemble system. As β → 0+, εN0+1β – εN0β = εlu – ε ho = ℏωogKS is the energy required to excite a KS electronfrom the ho to the lu KS orbital. ℏω01 = ℏωog isthe many-electron optical gap. Thus, Δ01 = Δog and64 and Yang et al.179 It is a time-independent way of calculating exact excitationenergies,158 similar to the calculationof the fundamental gap (discussed above).180The steps are clearly seen in og is always relatively small,below0.03 hartree (<1 eV), for different 1D atoms with a slightly lessor more confining external potential, e.g., vext(x) = −8.0/(|x|+1),with N = 2–this implies that exciting oneKS electron for this system is indeed a good model for the many-electronexcitation of the two-electron system. Hence, for this system, ℏωogKS ≈ ℏωog which implies that as long as Δog issmall, the ground-state KS energy levels are reasonably good approximationsto the many-electron excitation energies in their own right, i.e.,neglecting the contribution of the Hartree-xc (Hxc) kernel withinTDDFT, which has been observed by others.182 For more strongly correlated systems, or indeed the charge-transfersystem above, this is not the case, and the role of the Hxc kernelor the corresponding Δ becomes crucial.185We find that Δdecreasing below aninteger. This discontinuity emerges because a truly isolated systemwith a fractional number of electrons cannot exist in reality; theremust be a source of electrons, e.g., an electron reservoir (the donor),with which a finite system, like an atom or molecule, can exchangeelectrons (the acceptor). Imagine that the chemical potential of thereservoir is adjusted such that an infinitesimal amount of chargeis transferred to the finite system. The xc potential of the systemas a whole (reservoir plus the finite system) experiences a uniformshift of height ΔCT, which is the CTDD associatedwith transferring an electron from the reservoir to the finite system;see From the analysis in the sections above, we conclude thatany electrondonor experiences a discontinuous shift in its xc potential despitethe local number of electrons L→RCT, manifests as a uniformshift in the xc potential of the donor–acceptor when the transferred(excited) charge is infinitesimal. As the amount of charge is increased,a plateau of height ΔR localizes to the acceptoratom which in the vicinity of just the acceptor looks to be uniform–|S(2)| = ΔR in L→RCT becausethe donor and acceptor form one system–|S(1)| = ΔL→RCT. Consequently, the shift to the xcpotential associated with the derivative discontinuity of the finitesystem when the local number of electron increases above an integer,Δ, can never be truly uniform.As the amount of charge transferred fromthe reservoir is steadilyincreased, a plateau localizes in the vicinity of the acceptor whichis associated with the derivative discontinuity of that finite system,Δ. In conjunction, the shift in the xc potential associatedwith the CTDD localizes to the donor. This occurs for the diatomicmolecule of VIIexact many-electron densitiesand the corresponding exact KS potentials obtained from the densitiesby means of numerical inversion: but what happens when working withinone of the common approximations to the xc functional,like the local density approximation (LDA) or a generalized gradientapproximation (GGA)? Does the resultant approximate KS potential possessany steps or form any plateaus in the various scenarios discussedabove?So far we have addressed N = N0 + α, with, e.g.,the LDA in its standard implementation , one obtains a gradually changing xc potential,without any plateau of the sort presented in The immediate answer to this question is negative. Itis well-knownthat if one addresses a finite system with a varying number of electrons, N, relying on LDAdensities, also in a different way: First, one solves the system self-consistentlyfor N0 and separately for N0 + 1 electrons, within a given xc approximation. Second,one creates the ensemble density, n, using However, in the spirit of thepresent work, it is possible to obtainthe KS potential for fractional N0 = 2) for varying α.Remarkably, the potentials show a clear asymptotic behavior of ∼a/r + b far from the nucleus, rather than the exponential decay of the standard LDA.This allows us to align each potential such that it decays to 0  and subsequentlysubtract from it the KS potential for N = 2. Theresultant differences are shown in does form a plateau of height SLDA = 0.134 hartree in the vicinity of the nucleus. Asα → 0+, the height of the plateau converges,and its width logarithmically approaches infinity (cf. refs (We obtained this“inverted LDA” (invLDA) potentialfor the Li ion (cf. refs  and 186)A qualitative understanding ofthe emergence of plateaus in theinvLDA can be gained by looking at the density decay rates, presentedin SLDA found with invLDA and ΔLDA for Li+ obtained from KS-LDA quantities. ForLi+ with LDA, ILDA = 2.8712hartree, ALDA = 0.1924 hartree , εLDAho = −2.1899hartree, and εLDAlu = −0.2399 hartree. Hence, according to LDA = 0.7288hartree. Alternatively, ΔLDA′ = −ALDA – εLDAlu = 0.0475 hartree. For the exact xc functional, Δ′= Δ, but for an approximate one, like the LDA, the above equalityis not necessarily true, because the IP theorem is not obeyed. Inany case, neither ΔLDA nor Δ′LDA seem to equal SLDA.Next, we establish the quantitative relationshipbetween realignment ofthe KS potentials to satisfy the IP theorem.187 This means that for each α the KS potential is shifted bythe amount required for the ho level to equal the IP. For N0 = 2, this shift is v0 = −ILDA – εLDAho(N0–)= −0.6812 hartree. Notably, for all α > 0, the same shift of v1 = −ALDA – εLDAho(N0–+α) = −0.0868hartree is required. We denote Δv = v1 – v0 =−0.5945 hartree and recall that εho(N0+) = εlu + lim+α→0S, to find thatv = 0, and we return to the basic relationshipbetween Δ and S derived in We resolvethe above conundrum by N there are two, equallylegitimate approaches to obtain the KS potential. They lead to twoqualitatively different results: the standard approach yields a smoothlyvarying potential, without steps, which exponentially decays at infinity.The invLDA approach yields steps in the KS potential, and the asymptoticdecay is ∼a/r. We relatethese improvements to the piecewise-linearity in the density, whichis enforced in the invLDA approach. This internal inconsistency withinsemilocal xc approximations closely relates to another inconsistency:the IP of finite systems, like atoms and small molecules, can be obtainedwith common xc approximations from total-energy differences with highaccuracy of a few percent, whereas obtaining the same quantity directlyfrom the ho energy level results in discrepancies of ∼50%  potential.Stepscan occur in the exact potential in different scenarios:(i) a finite system (an atom) in the ground state with a varying numberof electrons ; (ii) a 68 This rule is true irrespectively of the specificphysical or chemical process the system undergoes, be it adding asmall fraction of an electron to the system, exciting the system,inducing transfer of charge, or even bringing two subsystems together.In a sense, the complex step structure of the potential is the priceone pays for the decision to describe an interacting many-electronsystem via a noninteracting system with a multiplicative potential.107 An expression for the height of the step inthe exact KS potential can be derived from the changes in the LEIP.As a general rule, steps in the potential occur at points wherethe exponential decay rate of the density changes and hence changesthe ‘local effective ionization potential’ (LEIP).By analyzing the exact KS potential, we show the general relationshipbetween the step structures in the potential and derivative discontinuitiesin the xc energy: in the cases discussed here, the many-electron energydifference equals the corresponding KS energy difference plus theassociated derivative discontinuity.Eg = EgKS + Δ.This relationship manifests in the potential as a uniform shift asthe system’s electron number infinitesimally surpasses an integervalue. For a small finite fraction of an additional electron, spatialstep structures form in the exact xc potential on the periphery ofthe system in order to elevate the level of the potential in the centerby Δ; as this additional amount of electron tends to zero, theplateau created by the steps becomes the uniform shift.The well-known derivativediscontinuity of the xc energy (Δ)of a system with a varying number of electrons relates the fundamentalgap and the KS gap: S, which forms ina stretched diatomic molecule in order to correctly distribute theelectron density throughout the system has usually been regarded asunrelated to the derivative discontinuity because the system typicallyconsists of a fixed number of electrons and the height of the stepis seemingly unrelated to the Δ’s of any of the constituentatoms. We demonstrate that upon the transfer of charge from one atomto another within the diatomic molecule, the acceptor atom experiencesa shift which corresponds to Δ of that atom owing to the “localnumber of electrons” on that atom surpasses an integer, Δa. Simultaneously the donor atom experiences a shift whichcorresponds to the charge-transfer derivative discontinuity (CTDD),62 Δd→aCT.Therelationship between a particular step structure in the xcpotential and derivative discontinuities is not always straightforward.The infamous interatomic step, S = Δa – Δd→aCT, and hence the interatomic step iscomprised of two derivative discontinuities, which are revealed whencharge transfer occurs. In addition, this derivative discontinuityoccurs when a fraction of an electron is added to the overall system,while the additional charge localizes on one of the atoms. In bothcases, ΔCT is related to the discontinuity of thederivative of the xc energy of the stretched molecule.We demonstrate that this discontinuityoccurs within the exactKS potential within ensemble DFT of a system which undergoes chargetransfer when excited. Analysis of this potential can offer valuableinsight for the development of advanced approximations to the xc energywithin ensemble DFT. In this case, we show that 64 We demonstrate this numerically for a single atom and show thatthis excitation is well approximated by the ground-state KS energydifferences for this system alone, i.e., in this case the Δis small. This implies that the Hartree-xc kernel plays a small rolein yielding accurate spectra for our single atom. This is not thecase for the charge-transfer system, however, as we typically findthe CTDD to be large. Hence, in this case, the Hxc kernel must haveimportant features which, at least in part, correspond to the CTDDin the potential. Capturing these features in approximations to theground-state and excited xc potential of DFT and ensemble DFT, respectively,as well as the xc kernel of time-dependent DFT, is crucial for accuratelyobtaining many-electron excitation energies from KS theory.We alsoshow that the many-electron excitation energy from theground to the first excited state is related to the KS energy differenceplus the associated derivative discontinuity.Finally, we demonstrate that step structures are obtainable alsofrom approximate xc functionals, as simple as the LDA. With the “invertedLDA” (invLDA) approach introduced here, we construct an ensembleof LDA densities with an integer number of electrons for each. Upon“reverse-engineering” these densities we find that thecorresponding potential possesses step structures, which resemblethose present in the exact potential. Ensuring that our invLDA potentialsobey the IP theorem, we establish the relationship between the stepheight and the derivative discontinuity in approximate xc functionals.