Robot-assisted pedicle screw placement is usually performed under general anesthesia to keep the body still. The aim of this study was to compare the accuracy of the robot-assisted technique under regional anesthesia with that of conventional fluoroscopy-guided percutaneous pedicle screw placement under general anesthesia in minimally invasive lumbar fusion surgery.  This study recruited patients who underwent robot-assisted percutaneous endoscopic lumbar interbody fusion (PELIF) or fluoroscopy-guided minimally invasive transforaminal lumbar interbody fusion (MIS-TLIF) between December 2017 and February 2020 at a single center. Based on the method of percutaneous pedicle screw placement used, patients were divided into the robot-assisted under regional anesthesia (group RE-RO) and fluoroscopy-guided under general anesthesia (group GE-FLU) groups. The primary outcome measures were screw accuracy and the incidence of facet joint violation (FJV). Secondary outcome measures included X-ray and visual analogue scale (VAS) scores which were used to evaluate the degree of the postoperative pain at 4 hours and on postoperative days 1, 2, and 3. Intraoperative adverse events were also recorded. p=0.44) or overall Gertzbein and Robbins screw accuracy grades (p=0.35). Only the top screws were included in the analysis of FJVs. The percentages of FJV  were 5.6% and 28.3%, respectively. This difference was statistically significant (p=0.01). Overall, the FJV grades in group RE-RO were significantly better than those in group GE-FLU (p=0.009). The mean fluoroscopy time for each screw in group RE-RO was significantly shorter than that in group GE-FLU . The postoperative pain between the RE-RO and GE-FLU groups was not statistically significant. The intraoperative adverse events included 1 case of registration failure and 1 case of guide-wire dislodgment in group RE-RO, as well as 2 cases of screw misplacement in group GE-FLU. No complications related to anesthesia were observed.  Eighteen patients were included in group RE-RO, and 23 patients were included in group GE-FLU. The percentages of clinically acceptable screws (Gertzbein and Robbins grades A and B) were 94.4% and 91.5%, respectively. There was no significant difference in the percentages of clinically acceptable screws ( Robot-assisted pedicle screw placement under regional anesthesia can be performed effectively and safely. The accuracy is comparable to the conventional technique. Moreover, this technique has the advantage of fewer FJVs and a lower radiation time. Pedicle screw fixation, a rigid surgical technique, has been widely used in spine surgery since the 1970s –4. In adRobot-assisted pedicle screw placement is usually performed under general anesthesia to keep the body still and improve screw placement accuracy. However, general anesthesia may be associated with high percentages of perioperative complications and medical costs, especially for elderly patients , 11. In Our medical team found that patients could remain motionless and painless during fluoroscopy-guided percutaneous pedicle screw placement under regional anesthesia in percutaneous endoscopic lumbar interbody fusion (PELIF) surgery. We predict that accurate, robot-assisted placement of pedicle screws in this patient state can be achieved. Therefore, we attempted to use a spine robot instead of fluoroscopy to guide pedicle screw placement. To the best of our knowledge, no previous study has been reported focusing on robot-assisted pedicle screw accuracy under regional anesthesia. This study, therefore, aimed to evaluate the accuracy and safety of robot-assisted pedicle screw placement under regional anesthesia in lumbar fusion surgery.This study was approved by the Ethics Committee of the Hebei General Hospital before data collection and analysis. This retrospective study recruited patients with lumbar degenerative disease who underwent robot-assisted PELIF or fluoroscopy-guided minimally invasive transforaminal lumbar interbody fusion (MIS-TLIF) between December 2017 and February 2020. The diagnoses included lumbar spinal stenosis, lumbar disc herniation, and lumbar spondylolisthesis. The patients were divided into two groups according to the pedicle screw implantation method: robot-assisted under regional anesthesia (group RE-RO) and fluoroscopy-guided under general anesthesia (group GE-FLU).The inclusion criteria were as follows: (1) scheduled 1- or 2-level PELIF or MIS-TILF surgery with either robot-assisted or fluoroscopy-guided percutaneous pedicle screw placement as the internal fixation technique; and (2) postoperative computed tomography (CT) scans taken before discharge with images meeting the measurement requirements. The exclusion criteria were as follows: (1) patients with a degree of lumbar spondylolisthesis or lumbar spondylolysis of II or higher; (2) patients with infection, tumors, or scoliosis of the spine; and (3) history of previous spinal surgery.μg/ml) was pumped at a rate of 3–8 ml/h. The administration of epidural anesthesia was performed using the loss-of-resistance technique through the interlaminar space of the operating segments  were copied from the inspection equipment and input into the robotic surgical plan workstation  for preoperative planning. Before surgery, the patient was told how to cooperate with the surgery, including keeping still and breathing evenly for some special period. During surgery, the patients were placed in a comfortable prone position on the operating table, with oxygen inhalation and ECG and vital sign monitoring. Dexmedetomidine , three needles were inserted into the spinous process of the upper lumbar spine and bilateral posterior superior iliac spines to fix the frame . After iThe pedicle screw placement procedures were completed by two senior spine surgeons who had each performed more than 50 cases of fluoroscopy-guided pedicle screw insertion. After general anesthesia, the patient was placed in a prone position. A C-arm was used to locate the targeted vertebral pedicles and plan the screw route. A puncture needle was inserted through a 1.5 cm incision with fluoroscopy guidance. After a final fluoroscopy check on the AP and lateral views, the puncture needle was replaced with a spacer. Screw  placement was performed after decompression and interbody fusion .The primary outcome measures were screw accuracy and the incidence of FJV. All patients underwent thin-slice CT scans (≤1.2-mm slices) of the lumbar spine postoperatively. Screw accuracy was evaluated using the Gertzbein and Robbins criteria : grade AAs secondary outcome measures, we compared the X-ray exposure and intraoperative adverse events related to the screw placement procedure as well as to anesthesia. X-ray exposure measurements were determined by the fluoroscopy time for each screw including the robot registration and intraoperative screw evaluation (sum of exposure times of the whole screw implantation and rod connecting procedures/number of screws inserted). A visual analogue scale (VAS) score was used to evaluate the degree of the postoperative pain at 4 hours and on postoperative days 1, 2, and 3.t tests were used for group comparisons of age, body mass index (BMI), the superior facet joint angle, fluoroscopy time for each screw, and the VAS score at 4 hours and on postoperative days 1, 2, and 3. The Mann–Whitney U tests were used for group comparisons of accuracy and FJV grades. The statistical significance of these parameters was set at p < 0.05.Statistical analyses were performed using IBM SPSS Statistics for Windows, version 23.0 . Fisher's exact test and Pearson's chi-squared test were used for group comparisons of sex, the distribution of diagnosis and screw location, as well as the percentages of clinically acceptable screws and facet violation screws. Two-sample Ninety-four consecutive patients were initially included in this study. Because of the requirement of postoperative CT results and other disease-related criteria, only 41 patients (22 women and 19 men) met the inclusion criteria. Eighteen patients (10 women and 8 men) who underwent PELIF were included in group RE-RO. Twenty-three patients (12 women and 11 men) were included in group GE-FLU; 20 patients (9 women and 11 men) underwent MIS-TLIF and 3 women underwent PELIF. The baseline characteristics of age, sex distribution, BMI, the superior facet joint angle, distribution of diagnosis, and screw location did not differ between the groups RE-RO and GE-FLU .p=0.44). The difference in the Gertzbein and Robbins screw accuracy grades was also not statistically significant (p=0.35). Considering the relationship between FJV and ASD as well as the surgeon's level of concern during insertion in different segments, only the 82 top screws were included in the analysis. In group RE-RO, 5.6% of the 36 screws analyzed violated the facet joint . In group GE-FLU, the incidence of FJV was 28.3%. This difference between these two groups was statistically significant (p=0.01). The FJV grades in group RE-RO were significantly better than those in group GE-FLU (p=0.009). A detailed list of the pedicle screw accuracy grades is presented in A total of 168 screws were inserted into patients' 4 vertebrae. Among them, 74 were implanted using the robot-assisted technique under regional anesthesia (group RE-RO), and 94 were implanted using the fluoroscopy-guided technique under general anesthesia (group GE-FLU). The incidence of pedicle breach  in the two groups was 10.8% (8/74) and 20.2% (19/94), respectively. There was no significant difference in the incidence of clinically acceptable screws (grades A and B), with percentages of 94.4% and 91.5% for groups RE-RO and GE-FLU, respectively (p=0.03). The VAS scores at 4 hours and on 1, 2, 3 days after surgery in group RE-RO were 4.7 ± 2.5, 4.7 ± 1.9, 3.9 ± 1.1, and 2.7 ± 1.0, which were lower than those in group GE-FLU at each time point . However, this difference was not statistically significant . No patients suffered from neurovascular complications postoperatively or underwent revision surgery due to screw misplacement. No cases of cerebrospinal fluid (CSF) leakage or surgical site infection were observed. Other adverse events of the screw placement procedure were as follows: 1 case of registration failure in group RE-RO. The screws were placed by the fluoroscopy-guided technique instead of the robot-assisted technique.1 case of guide wire dislodgment in group RE-RO. It was found during the operation, and the fragment was removed under the guidance of fluoroscopy.2 screws in 2 cases of screw misplacement in group GE-FLU. They were revised intraoperatively by the fluoroscopy-guided technique after an X-ray check. There were no complications related to anesthesia in either group.The mean fluoroscopy time for each screw in group RE-RO was significantly shorter than that in group GE-FLU (group RE-RO, 5.4 ± 1.9 seconds; group GE-FLU, 6.8 ± 2.0 seconds; In this study, we showed that robot-assisted percutaneous pedicle screw placement under regional anesthesia has a high accuracy of 94.6%. The accuracy reported under general anesthesia is 85%–99% –23. AlthIn terms of the baseline, there were some differences in the sequence of screw placement and decompression between the two groups. In the RE-RO group, to avoid the accuracy reduction in robot image acquisition and registration procedure after decompression, screw placement was carried out before decompression. In order to prevent the screw tail from affecting the decompression operation, screw placement was carried out after decompression and interbody fusion in the 20 patients who underwent MIS-TLIF. However, the screw path was established and marked with a guide wire before the decompression step. Both groups completed the screw path preparation before decompression. Therefore, we think that different decompression methods have little effect on the accuracy of screws.In the robot-assisted cohort, the robotic platform was a Hover-T frame, which is designed for minimally invasive surgery. The frame is fixed on the spine and pelvis during the whole insertion procedure. Relative resting of the body and platform can reduce the influence of accidental body motion on screw accuracy. Ringel et al.  reportedIn terms of FJV events, the robot-assisted technique was better than the fluoroscopy-guided technique. This finding is consistent with previous studies , 25, 26.Because of the different decompression methods, we only compared the radiation exposure time during the screw placement procedure. The results showed that the robot performed significantly better in minimizing this time. The advantage of the short radiation exposure time is more remarkable as the number of screws increases, especially in some spinal deformity surgeries. This is because most of the radiation exposure in robotic surgery occurs during the registration and platform process, which only needs to be performed once per operation. Fan et al.  comparedIntraoperative adverse events were equal in the two techniques. However, it seems that the complications in the robot group are less likely to cause serious consequences. Keric et al.  also fouThe findings of our study provide a new anesthesia method for the clinical application of spine robots. It proves that the spinal robot can be used under regional anesthesia. The accuracy is clinically acceptable. Considering that regional anesthesia has many advantages over general anesthesia , we expeRobot-assisted pedicle screw placement under regional anesthesia can be performed effectively and safely. The accuracy is comparable to that of the fluoroscopy-guided technique. Moreover, this technique has the advantage of fewer FJVs and a lower radiation time.


Inula species, is a pseudoguaianolide-type SL equipped with a typical and highly reactive α-methylene-γ-lactone moiety. The bioproperties of BRT and related pseudoguaianolide SLs, including helenalin, gaillardin, bigelovin and others, have been reviewed. Marked anticancer activities of BRT have been evidenced in vitro and in vivo with different tumor models. Three main mechanisms are implicated: (i) interference with the NFκB/ROS pathway, a mechanism common to many other SL monomers and dimers; (ii) blockade of the Keap1-Nrf2 pathway, with a covalent binding to a cysteine residue of Keap1 via the reactive α-methylene unit of BRT; (iii) a modulation of the c-Myc/HIF-1α signaling axis leading to a downregulation of the PD-1/PD-L1 immune checkpoint and activation of cytotoxic T lymphocytes. The non-specific reactivity of the α-methylene-γ-lactone moiety with the sulfhydryl groups of proteins is discussed. Options to reduce or abolish this reactivity have been proposed. Emphasis is placed on the capacity of BRT to modulate the tumor microenvironment and the immune-modulatory action of the natural product. The present review recapitulates the anticancer effects of BRT, some central concerns with SLs and discusses the implication of the PD1/PD-L1 checkpoint in its antitumor action.Sesquiterpene lactones (SLs) are abundant in plants and display a large spectrum of bioactivities. The compound britannin (BRT), found in different  Sesquiterpene lactones (SLs) represent a large class of natural products found in many plant species. The compounds are characterized by the presence of an α-methylene-γ-lactone moiety appended to a mono- or bi-cyclic system . The natpseudo-guaiacanes) has been less investigated than other SLs. This subgroup includes the lead compound helenalin being potently active but causing allergic reactions . Th. Th+ andThe mode of action of BRT is multifactorial. One of the prime pathways regulated by BRT is the Keap1-Nrf2 pathway which is the main protective response to oxidative and electrophilic stresses. BRT is a potent inducer of the transcription factor Nrf2 (nuclear factor erythroid 2-related factor 2) and does so via a direct binding to a conserved cysteine residue (Cys-151) of Keap1 (Kelch-like ECH-associated protein-1) . Keap1 iIn the presence of BRT, NRF2 can escape ubiquitination. Then, it can accumulate within the cell and translocate to the nucleus, where it promotes its antioxidant transcription program. The Keap1–Nrf2 pathway is implicated in multiple human diseases, as a regulator of oxidative stress . Nrf2 inBeyond its antioxidant effect, BRT also functions via the regulation of cell cycle progression in cancer cells. In MOLT-4 cells, BRT was found to prevent the S-phase cell cycle transition through the up-regulation of proteins p21 (cip1) and p27 (kip1) which are the two main cyclin-dependent kinase (CDK) inhibitors . In MCF-In cells, energy metabolism is regulated by the activity of several transcription factors, among which is the triad of c-Myc, HIF-1 and p53, essential to control glycolysis . The int+) T cells.BRT also negatively regulates downstream targets, such as the immune checkpoint protein PD-L1 (programmed cell death-ligand 1) that is frequently overexpressed in tumor cells. PD-L1 is largely implicated in the escape of tumor cells to T-cell killing, and contributes to promote tumor cell survival, migration and proliferation. BRT has been found to markedly inhibit the expression and protein synthesis of PD-L1 in various types of cancer cells, via HIF-1α/c-Myc . Both PDThis is an important discovery, because PD-L1 is a co-inhibitory molecule often expressed on tumor cells and considered a prime target in oncology. PD-L1-targeted monoclonal antibodies are largely used to treat solid tumors. The use of small molecules regulating PD-L1 expression can be extremely useful to design chemo-immunotherapy combinations . Zhang aBRT operates via multi-target and multi-pathway mechanisms. One of the signaling routes implicated in the mechanism of action of BRT is the AMP-activated protein kinase (AMPK) pathway. BRT has been found to activate AMPK in liver cancer cells and to induce apoptosis and autophagy. The process is associated with the production of reactive oxygen species (ROS) promoted by BRT in hepatic cancer cells . IncreasThese different mechanisms or pathways modulated by BRT are interconnected. ROS-modulating drugs can effectively regulate PD-L1 expression on cancer cells  and NFκBCarpesium faberi and C. abrotanoides, also presents a reactive group and functions as an anticancer STAT3 inhibitor . Th. Th67].


Transcranial magnetic stimulation has emerged as a noninvasive brain stimulation tool that can be utilized to investigate cortical excitability with the aim of probing neural plasticity mechanisms. In particular, in pathological disorders, such as schizophrenia, cortical dysfunction, such as an aberrant excitatory-inhibitory balance in cortical networks, altered cortical connectivity, and impairment of critical period timing are very important to be studied using different TMS paradigms. Studying such neurophysiological characteristics and plastic changes would help in elucidating different aspects of the pathophysiological mechanisms underlying schizophrenia. This review attempts to summarize the findings of available TMS studies with diagnostic and characterization aims, but not with therapeutic purposes, in schizophrenia. Findings provide further evidence of aberrant excitatory-inhibitory balance in cortical networks, mediated by neurotransmitter pathways such as the glutamate and GABA systems. Future studies with combining techniques, for instance, TMS with brain imaging or molecular genetic typing, would shed light on the characteristics and predictors of schizophrenia.Neural plasticity refers to the capability of the brain to modify its structure and/or function and organization in response to a changing environment. Evidence shows that disruption of  Schizophrenia is a complex chronic mental health disorder characterized by a positive and negative array of symptoms . DiagnosApplication of local stimulation of the neocortex based on a neurophysiological can enhance a greater understanding of the complex architecture of cortical anatomy and its correlation with brain functions. Such methods were applied some time ago to explore a wide range of cerebral functions such as sensory, motor, emotional, and subcortical vascular cognitive impairment .Transcranial magnetic stimulation (TMS) is a “non-invasive method for brain stimulation using the magnetic field to stimulate nerve cells in the brain meninges, subsequently inducing an electric current in the brain tissues that produces neuronal depolarization and generate action potentials” . IntroduTMS therapeutic effects are applied either through single-pulse TMS, repetitive pulse (rTMS), or theta-burst stimulation (TBS). A single pulse measures the motor threshold in an experimental setting . HoweverTMS is investigated for the treatment of positive schizophrenic symptoms . NeuronaThese disorders are associated with an increased subcortical dopaminergic activity leading to decreased activation of inhibitory cortical afferents, ultimately increasing cortical excitability results. In morphological studies of neuropathological changes, the cortical inhibition is mediated by cortical GABAergic interneurons, closing the final inhibitory dysfunction , 19. WhiThe present study is aimed at reviewing and analyzing the current literature available on the efficacy and assessment of brain plasticity index in schizophrenia through TMS protocols.We searched for studies published in the following electronic databases including the Cochrane Library, PubMed, EMBASE, and Web of Science, using the search terms as follows: e.g., “Schizophrenia” AND “transcranial magnetic stimulation “or “psychiatric Disorders” AND “TMS” or “schizophrenia” AND “brain plasticity Index” or paired pulse stimulation, from inception to September, 2021. In addition, combinations of Medical Subject Headings (MeSH) and text were used. One hundred and twenty-nine articles were selected. One author performed the initial selection of studies based on title and abstract. Then, each potentially eligible selected study was screened against the inclusion criteria. Finally, for 85 studies, three authors independently reviewed and evaluated the full-text articles to decide and determine whether these articles meet the predefined inclusion criteria. Any related disagreement was resolved by discussion. All selected studies collected from the search were critically examined. The studies for the review were grouped according to the paradigms and parameters examined in schizophrenia patients .μV in at least 50% of pursued trials  [The motor threshold (MT) is defined as the lowest intensity of TMS, required to provoke a predefined motor-evoked potential (MEP) with a peak-to-peak amplitude of 50  trials) . MinimumMoreover, medications such as carbamazepine and phenytoin increase the MT activities by changing the sodium channel properties which reflect the transsynaptic excitability of corticospinal response neurons, modulated by altering the presynaptic sodium/calcium channel conductivity . Other sMerton and Morton  note thaPatients with schizophrenia who are medicine-free demonstrate a higher degree of enhanced I-wave inhibition than those taking medication thus implying that their cortical inhibitory GABAergic activity is impaired , 43. WorCorticocortical excitability can be more directly examined by using paired-pulse TMS in which two consecutive TMS stimuli are applied to the brain at different interstimulus intervals (ISIs). The conditioning stimulus (CS) with approximately 80-95% of the RMT is the first, where the cortex of the actual test stimulus (TS) is set at 110-120% of RMT. Thus, the neuromodulatory effect of the CS on the amplitude of TMS-evoked MEPs depends mostly on two factors: the intensity of CS and TS and the interstimulus interval (ISIs) between both types of stimuli .An ISI of 1-5 ms is “short” and produces inhibitory effects with a lower MEP amplitude. An ISI in the interval of 8-20 ms demonstrates facilitatory effects with the MEP amplitude over 1 mV (amplitude height of the TS without previous CS). When using the bilateral stimulation , a significant reduction of the contralateral MEP amplitude was remarked in the untreated schizophrenic patients, revealing that the antipsychotic treatment may repeal the increased transcallosal inhibition (TCI) in schizophrenia . First-dIntracortical inhibition can identify changes in variable conditions. Administering GABA-A receptor agonists such as lorazepam primarily caused cortical inhibition  and mighn = 30 schizophrenic patients) served as subjects and half were untreated [n = 40 schizophrenic patients), it demonstrates primarily less cortical inhibition in schizophrenic patients [A study explored important aspects of the pathophysiological mechanisms using the paired-pulse method, in which , which included 24 schizophrenia-spectrum patients and 30 healthy controls. The study indicated that the left middle prefrontal gyrus-motor cortex rsFC was linked to both cortical inhibition and schizophrenia symptoms as stronger the rsFC indicates stronger the motor cortical inhibition . MultiplIf the interstimulus interval between the 2 pulses is long, i.e., 9-25 ms, a facilitatory effect is observed, referred to as intracortical facilitation (ICF) , 45–56. This is opposite to two studies that found there is a difference in ICF, as one study involved 13 schizophrenia patients and 13 controls of ICF measured after changing their antipsychotic and were followed up for 8-weeks. ICF measurements were done at baseline and 8 weeks posttreatment. The study reported significantly lower ICF at 8 weeks compared to baseline; this could be attributed to the interstimulus interval which was between 7 and 30 ms, and previous studies have been cross-sectional in design, and this might explain the lack of change in ICF between healthy controls and schizophrenia patients . AnotherLong-interval intracortical inhibition (LICI) refers to suppression of cortical neuronal firing, following a paired-pulse TMS, mediated in part by the primary inhibitory neurotransmitter of the central nervous system; gamma-aminobutyric acid (GABA) receptor B. LICI can be measured either from motor-evoked potentials (MEPs) in the interossei muscles of the hand or directly recorded from the cortex using a concomitant electroencephalography (EEG) method.Several studies examined the prefrontal cortex, motor cortex TMS-EMG LICI in patients with schizophrenia (SCZ). In a study including 18 SCZ patients (9 were medicated with antipsychotics) and 18 HC, Fitzgerald et al., in a study including 18 patients with schizophrenia (9 were medicated with antipsychotics) and 18 healthy controls, showed that there was no significant difference across both groups related to LICI, suggesting that no influence of neuroleptic medication . SimilarIt is noteworthy to mention that the studies investigating LICI in the motor cortex of patients with schizophrenia failed to exhibit any significant correlation between LICI changes and neural correlates of schizophrenia symptoms. However, studies examining LICI in the dorsolateral prefrontal cortex (DLPFC) of patients with schizophrenia revealed significant findings. This suggests that altered inhibitory process related to DLPFC, and not to motor cortex, might be underlying the deficits of cognitive function that exist in patients with schizophrenia –66.The application of TMS can assist in studying the transcallosal connection between primary motor cortices (M1) in both human hemispheres, which include transcallosal inhibition (TCI) and facilitation (TCF) with onset latency, typically (30-40 ms). Moreover, the latency and duration of the TCI are known as the transcortical conduction time  . The resA previously early study, on the CSP in schizophrenia, revealed no significant differences in the duration of CSP between schizophrenic patients and controls . HoweverTMS is a noninvasive procedure that uses a magnetic field to stimulate nerve cells of the brain. For the past few decades, TMS has been established as a potential treatment for schizophrenia. TSM enhances casual relationships between the brain and behavior. Furthermore, rTMS is known for its positive effects on neuroplasticity. Given its neuroprotective effects, rTMS can benefit patients with various neuropsychiatric disorders in a short period , 72–76, Investigating cortical excitability  can measure cortical inhibition. In patients with schizophrenia, TMS is an auspicious approach to record changes in cortical excitability and to elucidate the underlying pathophysiological mechanisms of the schizophrenia –79.Methodological issues can cause measurement inaccuracies . In addiIncreased MEP is a major observation among patients with schizophrenia due to interference in the cortical inhibitory mechanism –39. MEP In several lines of evidence, cortical inhibition was observed in schizophrenic patients compared to controls , 26–28. Confirmed through imaging and TMS technique, schizophrenia patients are suspected to have other neurophysiological and neuropsychological considerations, impaired by interhemispheric transfer . DespiteTMS is an auspicious noninvasive method that can be used to study detailed aspects of cortico-motor-neural excitability in schizophrenia on a pathophysiological basis, with a focus on detecting inhibitory dysfunction . Combining it with other methods, like molecular genetic typing, neuropsychological examination, structural and functional imaging including magnetic resonance-volumetric, event-related potentials, and functional magnetic resonance imaging techniques, might contribute to an understanding of future neurobiological vulnerability about schizophrenia population.


Data availability and temporal resolution make it challenging to unravel the anatomy  of the Last Glacial abrupt climate changes. Here, we address these limitations by investigating the anatomy of abrupt changes using sub-decadal-scale records from Greenland ice cores. We highlight the absence of a systematic pattern in the anatomy of abrupt changes as recorded in different ice parameters. This diversity in the sequence of changes seen in ice-core data is also observed in climate parameters derived from numerical simulations which exhibit self-sustained abrupt variability arising from internal atmosphere-ice-ocean interactions. Our analysis of two ice cores shows that the diversity of abrupt warming transitions represents variability inherent to the climate system and not archive-specific noise. Our results hint that during these abrupt events, it may not be possible to infer statistically-robust leads and lags between the different components of the climate system because of their tight coupling. Palaeodata resolution and dating limit the study of the sequence of changes across Earth during past abrupt warmings. Here, the authors show tight decadal-scale coupling between Greenland climate, North Atlantic sea ice and atmospheric circulation during these past events using two highly resolved ice-core records. Oxygen isotope (δ18O) profiles from Greenland ice cores provide master records of this climate variability5, illustrating fluctuations between Greenland Stadial (GS) phases with full glacial conditions and milder Greenland Interstadial (GI) phases .Paleoclimatic records of the Last Glacial reveal a series of abrupt warming events occurring in the North Atlantic region, known as Dansgaard-Oeschger (D-O) events, with counterparts in lower latitudes18O value of Greenland ice is mainly affected by local surface temperature changes, past changes in precipitation seasonality, the temperature at the moisture source regions, and elevation changes18. Hence, although δ18O is not a direct temperature proxy, it can be used as a qualitative tracer of local Greenland surface temperature changes. The second-order parameter d-excess (d-excess = δD−8·δ18O) is commonly interpreted as a record of past changes in evaporation conditions or shifts in mid-latitude moisture sources20, whereas Ca2+ concentrations ([Ca2+]) in Greenland ice cores reflect both source strength and transport conditions from terrestrial sources, which are mainly the mid-latitude Asian deserts22. Finally, changes in Na+ concentrations ([Na+]) can be interpreted as qualitative indicators of the sea-ice cover extent in the North Atlantic at the stadial-interstadial scale23, whereas relative site accumulation rate changes can be estimated from the annual-layer thickness (denoted λ)24. Hence, ice-core multi-tracer studies are well suited to evaluate the precise phasing and duration of changes between different regions without relative dating uncertainty as all records come from the same core.The mechanisms proposed to explain D-O event dynamics can be confronted with annual-to-decadal-scale observations of climatic changes across the globe over the GS–GI transitions. Indeed, such data sets provide a basis to map out the sequence of events, infer possible causal relations and evaluate hypothetical sets of governing mechanisms by comparing model output with the spatial expression and relative phasing of the observed changes, hereafter referred to as the “anatomy” of the changes. However, looking at the anatomy of abrupt events in paleoclimate data is challenging because it requires a high temporal resolution not attainable in most climatic archives, and because of relative dating uncertainties between paleoclimate records from different archives. Records of annual or close-to-annual resolution from Greenland ice cores overcome this challenge since they contain tracers recording conditions in different parts of the Earth System with each year’s precipitation, all in one archive. The δ26. For each of those transitions, a lead of a few years in changes in terrestrial aerosol concentrations, accumulation rate, and mid-latitude moisture sources relative to the changes in marine aerosols and the isotopic temperature was found. Such results suggest that the Greenland surface warming was preceded by changes in the conditions at the dust sources or changes to the transport to Greenland . In parallel, the phasing between the high- and lower-latitude climate responses was investigated using ice-core gas-phase measurements: the δ15N of N2 as a tracer for Greenland surface temperature changes28 and the methane concentration (CH4) as a proxy for tropical climate change30. Although the first studies31 estimated a lag of a few decades of tropical CH4 emissions behind δ15N at the onset of the abrupt warmings, a more recent study32, focusing on the Bølling transition and using 5-yr-resolution δ15N and CH4 records, estimated that high- and low-latitude climate changes occurred essentially synchronously at that time, with Greenland surface temperature leading atmospheric CH4 emissions by 25.This approach was initially applied to characterize the sequence of events at the onsets of the Holocene, GI-1e (Bølling), and GI-8c25 and 26 to all transitions back to 60 ka b2k (thousand years before 2000 C.E.) and derived an average sequence of changes characteristic of the GI onsets by combining the estimated leads and lags for all studied transitions23. Based on the assumption that the relative timing differences between different tracers at all GI onsets are the result of the same underlying process, it was found that changes in both local precipitation and terrestrial dust aerosol concentrations led the change in sea-salt aerosol concentrations and δ18O of the ice by about a decade. Event-stacking-based approaches are often applied to extract the common signal from highly variable climatic records36. Although this is useful, it is also worth looking into the details of the sequence of changes over each event, especially considering the high diversity observed in the amplitude of the warming8, the shape and duration of GS and GI38  and NGRIP  ice cores. Having so many highly resolved ice-core records from two different locations over numerous D-O events provides the most comprehensive opportunity so far to assess the geographical representativeness of single ice-core records. Second, the anatomy of D-O warmings inferred from Greenland ice-core data is compared with new simulations from the coupled Community Climate System Model Version 4 (CCSM4) as the basis for discussing the processes involved in D-O warmings.18O, d-excess) at high resolution (5 cm) from the NGRIP ice core5  to each data series within a prescribed search interval across each GS–GI transition , previously identified in Greenland ice cores20. Its simplest explanation is that atmospheric warming (δ18O) over the ice sheet takes place roughly in parallel to a moisture source shift to a cooler region (d-excess), although other factors contribute to d-excess changes. Indeed, the values of d-excess in Greenland precipitation are thought to be primarily an indicator of conditions over the subtropical North Atlantic19, with possible small contributions from other source areas43. Although d-excess bears a signature of vapor source characteristics (sea-surface temperature (SST) and relative humidity), it is also affected by changes in condensation temperature, the temperature difference between the source and condensation regions44 and possibly different changes in seasonality at the two sites45 and by North Atlantic sea-ice removal46. A simple visual observation of the two records suggests that the variability is slightly smaller in NEEM compared with NGRIP. This is likely related to the stronger thinning rate in the NEEM glacial section, leading to the fact that each individual sample is the average isotopic values over a longer period at NEEM than at NGRIP. The difference is related to the influence on flow-induced thinning of the presence of bottom melting at NGRIP38, which results in NGRIP annual layers being thicker by a factor of 1.5–2 compared with NEEM during most of the Glacial38.To start, we describe the general characteristics of NGRIP and NEEM water isotopic records. Fig. 18O, d-excess, λ, [Na+] and [Ca2+] relative to the onset of the δ18O transition, together with the uncertainty intervals taken as the marginal posterior 5–95% credible intervals. The uncertainty intervals associated with the timing of the transition onset have a width varying from 1 to up to 205 yr depending on the considered event and ice tracer, with an average of 71 yr. First, we observe that for most studied transitions, the onsets in the four tracers occur within a few decades. In 15 out of the 21 transitions with NEEM δ18O, d-excess, [Na+] and [Ca2+] data, all onsets fall within ≤40 yr of each other. In NGRIP, five out of the seven transitions with δ18O, d-excess, [Na+], [Ca2+] and λ data start within ≤40 yr of each other. Second, we observe large variability of the relative timing between the onsets of different tracers between events, but the observed leads and lags are consistent with zero lag considering the uncertainty intervals. Still, we observe that for eight out of 13 transitions, the lead/lag relationship between δ18O and d-excess is consistent between the two cores. Also, for all events except GI-10 and GI-12c, an agreement within the uncertainty intervals is observed in the transition onsets for δ18O, d-excess, [Na+] and [Ca2+] between NEEM and NGRIP. In summary, we do not identify a clear dominant pattern in the sequence of changes of the different tracers within their uncertainty intervals when looking at the transitions individually, in line with the ramp-fitting analysis of ref. 23. Finally, our results show that the transition for any given tracer never reaches the post-transition stable level before the other two (or three) tracers have started to change, suggesting a very tight coupling between the different climatic components reflected by the proxies.Then, we present the inferred timing of changes in ice tracers at the D-O warming onsets. Figs. 18O, [Ca2+] and [Na+] , such as previously highlighted across the Last Deglaciation abrupt warmings25 . In particular, the d-excess and δ18O and transitions into GI-18 stand out as they both take ≥150 yr. The consistency in duration patterns between the two ice cores breaks down at the bottom of the cores, i.e., in the time period earlier than GI-21.1e. We interpret this as an effect of the decreasing temporal resolution of the records through the flow-induced thinning of the ice at those depths. Because of these possible limitations, we do not consider those transitions in the following discussion.Finally, we focus on the inferred duration of the transitions in ice tracers. The transitions in the five parameters vary from ≤10 yr to almost 200 yr, with 19 of the 24 transitions in NEEM and 20 of the 25 transitions in NGRIP being faster than a century  compensating changes in the other parameters49.First, we discuss our results over the Last Deglaciation abrupt warmings, i.e., across the onsets of the Holocene and GI-1e (Bølling), and we compare them with results from refs. 25 Fig. . The seq45. However, the patterns of the transition durations in NGRIP and in NEEM are overall similar despite large event-to-event variability  δ18O or [Na+] records individually, (2) to the combined water isotopic tracers, (3) to the combined impurity tracers, and (4) to all tracers together  4 is characterized by a local ice-sheet maximum and GI-18 follows one of the longest stadials, GS-19.1, which lasts 5300 yr. Interestingly, the low-latitude counterpart of GI-18 as recorded in the Hulu cave isotopic record is not as clear as for the other abrupt events50 and neither is it in Antarctica where the classic bipolar seesaw pattern cannot be identified unambiguously37. In this context, we investigate first the potential role of changes in the climatic background state , also come into play and complicate the response further, e.g., the relative importance of these controlling factors on the water isotopic records may differ from one event to the other leading to a variety of transitions recorded in these proxies. Owing to their design, those simulations only illustrate that transitions can be expressed differently when a single forcing , modulated by regional feedbacks mechanisms, is involved. However, recent modeling studies suggest that D-O events could result from unforced climate oscillations linked to dynamics internal to the climate system such as atmosphere-ice-ocean interactions altering poleward energy transport13.Next, we provide a model perspective on the anatomy of D-O warmings. Early simulations from a coupled global ocean–atmosphere–sea-ice model forced with different freshwater amounts in the North Atlantic have shown that multiple stable or quasi-stable states of the AMOC strength could exist and that rapid transitions between these states do not have identical climate expressions as modeled temperature and precipitation changes are not linearly related to the AMOC strength55. We consider three simulations run under slightly different prescribed atmospheric CO2 concentrations , each containing two spontaneous D-O-like transitions  that could be different from one event to the next. Although the precise forcing and sequence of events vary between these examples, the end-result of abrupt climate change characterized by large sea-ice, atmospheric circulation, and temperature anomalies, is common to all.The absence of a systematic pattern in the phasing or duration of transitions in different parameters, as seen in both model- and proxy-based climate parameters, could be the result of internal climate variability superimposed on a common set of mechanisms. In such a context, the ref. 23. Finally, we support the push for (1) higher spatial coverage of well-resolved and well-dated paleoclimate records to study the range of variability in stadials, interstadials, and the transitions between them, (2) additional Earth System Model simulations, and (3) of particular relevance to the future, more work on understanding if the glacial climate state is particularly prone to the strong sea-ice, ocean, atmosphere feedbacks that characterize the abrupt D-O transitions or if that is a general feature of the Earth’s climate.The emerging picture of the D-O warmings is one in which the components of the climate system are so tightly coupled that it may not be possible to resolve significant leads and lags between them, and consequently, it may be elusive to search for a single sequence of events in proxy data that can adequately describe all Last Glacial abrupt climate transitions. The challenge is likely to be even larger when studying the phase relationship between different types of climate archives where inter-archive dating uncertainties must also be considered. Our results underline that drawing general conclusions on the mechanisms of D-O-scale climate dynamics from phasing studies based on a single event should be avoided, as the results might not necessarily be transferred to other events. Combining our new data-based results with climate model simulations, we show that the diversity of abrupt warming transitions observed in ice-core tracers is reproduced between records and thus, not related to archive noise. Although the observed differences could be interpreted as consequences of internal climate variability superimposed onto a common set of mechanisms, the lack of a unique sequence of changes representing D-O warmings could alternatively illustrate that there is not a unique trigger per se to these abrupt changes. Such interpretation would challenge the common assumption that the rapid climatic events within the Last Glacial are the results of the same set of underlying mechanisms. Hence, although stacking of multiple abrupt transitions is useful for examining the average sequence of events and increasing the signal-to-internal-variability ratio, such an approach should be employed with caution, and the underlying assumptions in terms of physics or statistics and associated limitations should be spelled out explicitly as, for example, in ref. 39. Analyses have been performed at the Niels Bohr Institute at the University of Copenhagen using IR Cavity Ring Down Spectrometry on discrete samples with a resolution of 5 cm. The combined uncertainty of the measurements (1σ) is 0.05‰ and 0.4 ‰ for δ18O and δD, respectively. In the present study, we use and show 600 yr-long sections covering 25 abrupt transitions  Stable Isotope Lab (SIL) (University of Colorado)  have a temporal resolution of better than 1 yr at 10 ka b2k, ~3 yr at 45 ka b2k, ~4 yr at 80 ka b2k, and ~5 yr at 105 ka b2k for NGRIP and of ~1 yr at 10 ka b2k, ~4 yr at 45 ka b2k, ~7 yr at 80 ka b2k, and ~18 yr at 105 ka b2k for NEEM.The NEEM water isotope sections are part of the continuous high-resolution water isotope record covering 8–130 ky b2k2+] and [Na+] records over the past 60 ka are published in ref. 23, and in our study, we present the records extended back to ~108 ka b2k (Supplementary Data 62). The effective resolution of the CFA records is between 1 and 2 cm and the relative concentration uncertainty is typically 10%. Here, we use the [Ca2+] and [Na+] records averaged to annual resolution. For the depth range corresponding to each year in the GICC05 time scale, average impurity concentrations are calculated directly for NGRIP. For NEEM, the corresponding NEEM depth range is first calculated by interpolation between the time scale transfer match points of ref. 40, and then the annual average impurity concentration for that depth range is calculated. The interpolation scheme thus assumes similar accumulation variability patterns between NGRIP and NEEM.The NGRIP and NEEM high-resolution Ca+ and  only, (4) [Na+] only, then, (5) combined water isotopic tracers (δ18O and d-excess), and (6) combined impurity tracers ([Ca2+] and [Na+]), and finally, (7) all tracers together. Scatter plots are presented in Supplementary Figure R2 values are low  because of the large scatter of the durations around the 1:1 line. We estimate the significance of the correlations for the seven different transition duration data series in the following way: we randomly permute the NEEM durations 105 times  and calculate the resulting slope s of the  plot , the significance is extremely convincing , demonstrating that despite the large scatter in durations between events and sometimes even between species, the results from the two cores show a highly consistent picture. Finally, note that comparing the results of the ramp-fitting procedure on NEEM δD and δ18O records, we observe that the timing of the onsets and the durations of the transitions are in agreement within ≤6 yr for 15 out of 19 events when considering the transition durations  δ+] only,  [Na+] on+] only,  [Na+] on8O only,  d-excessy, then,  combined67. The low-resolution version of CCSM4 is a coupled climate model of ~3.75° × 3.75° resolution in the atmosphere with 26 levels in the vertical, including a spectral dynamical core (T31). The ocean model resolution is ~3° with 60 vertical levels. The model uses POP2 and CAM4 as the ocean and atmosphere components, respectively70. The land-surface and sea-ice components are CLM4 and CICE4, respectively72. The land and atmosphere share the same Cartesian latitude and longitude grid, whereas the ocean and sea ice share the same curvilinear grid with poles centered on Greenland and Antarctica. The four earth system components are coupled using a climate coupler that conserves momentum, energy, and freshwater fluxes between earth system components (CPL7). The climate in the model has been validated against modern observations and is known to have some significant deviations from the observed climatology. Northern Hemisphere Arctic sea-ice thickness and extent is rather excessive in the model. Atlantic mass and heat transport are somewhat weak compared with the high-resolution version of the model and observations. The low heat transport results in excessive sea-ice extent and a bias in globally averaged SSTs of ~1 °C. The climate validation of the low-resolution model is fully described in ref. 55.We present a new set of glacial climate simulations obtained with a low-resolution version of the CCSM4 model74. The set-up of the boundary and initial conditions for a high-resolution glacial simulation has been discussed in detail previously76. The same format for applying the boundary and initial conditions in the high-resolution version of CCSM4 has been applied to its low-resolution version. Continental land ice is represented by Laurentide, Fennoscandian, Greenland, and Antarctic distributions representative of approximately the Last Glacial Maximum (LGM). Insolation is set according to LGM orbital parameters and an initial background atmospheric CO2 concentration of 185 ppmv was used in the initial 2100 years of simulation. Two other simulations were also run to produce a set of three runs with different fixed background atmospheric CO2 forcing  to investigate the effect of background atmospheric CO2 on D-O variability. The two additional new low-resolution glacial simulations were spun off the initial 185 ppmv run at year 2100 and each of the three simulations was run for another ~5000 years. Each of the three runs exhibits two unforced climate oscillations that are approximately the same magnitude, shape and length as the D-O events observed during MIS 3, like the high-resolution simulations reported previously12, but with periods and background variability in better agreement with observations. The branch point for each of the three sensitivity runs was chosen during the first stadial of the regular unforced D-O stadial at 185 ppmv, which was close to surface statistical equilibrium in the stadial phase of the simulation. In this study, we use the two D-O-like oscillations from each of the three simulations, totaling six different abrupt warming transitions to be analyzed using our ramp-fitting approach  model for palaeotopography, bathymetry, and land ice cover18O profile is mainly controlled by the condensation temperature77, whereas the annual-layer thickness inferred from the NGRIP GICC05 chronology24 is determined by the annual precipitation rate and subsequent flow-induced strain, the first two modeled time series selected are the annual average surface temperature and annual precipitation changes at the model grid point closest to the NGRIP drilling site. Model studies have also highlighted that the variations in sea-ice area surrounding Greenland between GSs and GIs are also of primary importance for the temperature and precipitation response recorded in Greenland ice cores, e.g. 79. In addition, sea-ice extent in the North Atlantic has been suggested to impact d-excess values as sea-ice removal during abrupt warmings could cause a change in the original moisture source location associated with the retreating polar front in the North Atlantic46. In subsequent work, it has been proposed that abrupt changes in d-excess may be recording abrupt atmospheric circulation change and an associated change in moisture source location17. Sea ice has also a direct and significant influence on the formation and transport of sea-salt aerosol as parametrized by [Na+]. On the one hand, sea-ice formation acts as efficient, alternative source of sea-salt aerosol in addition to the open ocean. On the other hand, sea-ice extension increases the transport distance of sea-salt aerosol from the open ocean to the ice-core drill sites. [Ca2+] changes across GS–GI transitions are also attributed to hemispheric scale atmospheric circulation patterns in addition to changes in aridity and dust storm activity in central Asian deserts62. As a result, we also look into two additional model-derived measures of the influence of the state of the atmosphere, ocean and cryosphere in the surrounding regions in the North Atlantic, i.e., a measure of annually averaged sea-ice area changes and one of the atmospheric circulation variations in the North Atlantic. In our simulations, the sea-ice extent in the North Atlantic, including the Greenland, Irminger and Norwegian (GIN) seas varies to a large degree on both seasonal and millennial timescales. One of the determining factors that drives the large variations in sea ice on longer-than-annual timescales in these simulations is the sub-sea-ice thermohaline instability that occurs prior to each D-O warming58. Therefore, we extract a measure of annual average sea-ice area (or concentration) in the Irminger Seas as this is the region in the model that shows the largest loss of sea ice during the warming transition  changes in sea-level pressure in the region between 30 and 90°N and −90°W and 25°E. From this we can obtain both the Empirical Orthogonal Functions (EOFs) and PC time series associated with the dominant modes of variability. The EOFs in this region of the North Atlantic are well characterized by the sea-level pressure dipole pattern associated with the NAO in the modern climate  index84. Therefore, the results obtained for the first set of D-O-like transitions (modeled events 1, 3, and 5, Fig. Looking at the ramp-fitting results, the durations of the transitions in the different modeled physical parameters seem to be slightly longer during the second transition compared with the first one Figure . An explSupplementary InformationPeer Review FileDescription of Additional Supplementary FilesSupplementary Data 1Supplementary Data 2Supplementary Data 3Supplementary Data 4Supplementary Code 1


MalrieuJ. Chem. Phys.1973, 58, ( (12), ), 5745−5759). We describe the main parametersthat enter in this algorithm and perform benchmark calculations ona set of 21 small molecules and compare ground state energies withfull configuration interaction (FCI) results (FCI21 test set). Thefocus is the comparison of the performance of three different typesof many-particle basis functions (MPBFs): (1) individual Slater determinants(DETS), (2) individual spin-adapted configuration state functions(CSFs), and (3) all CSFs of a given total spin that can be generatedfrom spatial configurations (CFGs). An analysis of the cost of thecalculation in terms of the number of wavefunction parameters andthe energy error is evaluated for the DET-, CFG-, and CSF-based ICE.The main differences for the three many-particle basis representationsshow up in the number of wavefunction parameters and the rate of convergencetoward the FCI limit with the thresholds of the ICE. Next, we analyzethe best way to extrapolate the ICE energies toward the FCI resultsas a function of the thresholds. The efficiency of the extrapolationis investigated relative to the FCI21 test set as well as near FCIcalculations on three moderately sized hydrocarbon molecules CH4, C2H4, and C4H6. Finally, we comment on the size-inconsistency error for the threemany-particle representations and compare it with the error in thetotal energy. The implication for selected CI implementations withany of the three many-particle representations is discussed.The present workis the second part in our three-part series onthe comparison of many-particle representations for the selected configurationinteraction (CI) method. In this work, we present benchmark calculationsbased on our selected CI program called the iterative configurationexpansion (ICE) that is inspired by the CIPSI method of Malrieu andco-workers ( The absence of comparable experimentaldata on the one hand and infeasibility of full configuration interaction(FCI) calculations for large molecules on the other renders a rigorousbenchmarking of sCI calculations on realistic molecules a rather difficultendeavor. There have been attempts toward a thorough benchmark ofsCI methods by various groups recently, such as the benchmarking ofthe Gaussian-2 set using semistochastic heat-bath configuration interaction12 (SHCI) by Yao et al.13 Stochastic methods such as the full configuration interaction quantumMonte Carlo (FCIQMC) method pioneered by Alavi and co-workers alsohave their own standardized algorithms for benchmarking and extrapolation,which depends on the type of FCIQMC algorithm used.17 Another such effort is illustratedby the recent work of the adaptive sampling CI by Tubman et al.18 Benchmarking efforts have also been make for the calculation ofvertical excitation energies employing sCI methods by Loos et al.19 Along with benchmarking efforts, recent collaborativeinitiatives on comparing various approaches together with sCI havealso appeared.21The selected configuration interaction (sCI) method has recentlyseen a resurgence and has established itself as a powerful tool forquantum chemistry as evidenced by various recent studies.dN)as the ordinate for convergence and parallel scaling analysis.23 These criteria make transferability of thresholds difficult forsCI methods that are based on CFGs and CSFs. In the present work,we examine a general approach for the analysis of the characteristicsof a sCI method by comparing the thresholds, extrapolation techniques,and error bars for three different types of MPBFs including individualDETs, CFGs, and CSFs. The similarities and differences between thethree MPBFs are presented and discussed. We present the advantagesand drawbacks of each type of MPBF in terms of the types of problemsadapted for each case.A subject that, to thebest of our knowledge, has hardly ever beenstudied before, is the question of which many particle basis is bestsuited for sCI calculations? Here, we address this question by exploringthree different types of many-particle basis functions (MPBFs). Themost straightforward choice is to expand the many particle wavefunctionin terms of individual Slater determinants (DETs). Alternatively,one can construct individual spin-adapted configuration state functions(CSFs). Lastly, selection can be performed on individual spatial configurations(CFGs) that are then allowed to bring in all CSFs arising from anygiven configuration. Most existing sCI methods are based on the DETbasis and use the total DET count  is devised tosystematize future benchmarking and comparisons such as those thatexist for density functional theory methods.26 In the present paper (Part II), we shall describethe numerical performance of the associated thresholds and investigatea simple but effective extrapolation schemes. In Part III, we willpresent systematic case studies on inorganic and organic moleculesfor accessing the strengths and weaknesses of the three MPBFs andexplore the limits of the ICE method.This paper is the second in a series ofpapers on our sCI methodcalled the “iterative configuration interaction” (ICE)method. In Part I of this series, we have presented an in-depth descriptionof the algorithm and implementation details of the ICE in terms oftree data structures and recursive matrix element algorithms.The outline of the paperis as follows: First, we briefly recallthe algorithm of the ICE method followed by a description of the thresholdsentering the protocol. We also describe the methodology for obtainingthe statistics from benchmark data. This is followed by the results,which consist of three parts: First, we present the benchmark resultson the FCI21 set for the three types of MPBF. Second, we present ageneral extrapolation scheme for obtaining near FCI energies, andfinally, we test the size-inconsistency error as a function of thethresholds of the ICE method. In the conclusions, the main strategiesand guidelines for performing calculations using the ICE are outlined.2The details of the algorithm and the associatedimplementationshave been explained in Part I of this series. Therefore, we will onlysketch the main steps of the algorithm and provide details about theparts necessary for the benchmarking and extrapolation.2.127 whichappeared in 1973. The original CIPSI algorithm was later modifiedto a “three class” CIPSI algorithm by Evangelisti etal.28 Our ICE algorithm closely resemblesthe three class CIPSI but differs in some important aspects. The maindifference of the ICE with the three class CIPSI algorithm is thatthe ICE is designed as an approximate full CI method rather than asa multi-reference perturbation theory (MRPT) method. Consequently,emphasis is placed on convergence of the variational energy wherepossible. Only the selection part relies on MRPT. However, the perturbativeenergy calculated can be used as a measure of the quality of the initialreferences during the iterations and for extrapolating to the FCIlimit as will be explored below. Second, the ability to work withthree different types of MPBFs is a significant difference from bothalgorithms and has important implications for practical applications.However, the fundamental intellectual basis of this work unambiguouslyis the pioneering work for Malrieu and co-workers.The ICE algorithmis inspired by the groundbreaking and originalconfiguration interaction by perturbation with multiconfigurationalzeroth-order wavefunction selected by iterative process (CIPSI) paperby Huron, Malrieu, and Rancurel,The algorithmis summarized in 31 are as follows:a)th order set of MPBFs|ΦI(0)⟩ ∈|Ψ0⟩,which are expected to represent a dominant part of the state of interest.However, this is not strictly required as the algorithm will alsofind states that have not been anticipated. Other than a manuallyinput set of initial MPBFs, |Ψ0⟩ can be alsobe constructed automatically by performing an initial complete activespace self-consistent field (CASSCF) calculation with a smaller CASspace. Here, the |ΦI(0)⟩′s ∈ΨCAS will correspond to the CASSCFroot of interest.The ICE procedure is “seeded”by a rationally chosen |0⟩b)33 (PT2) relative to all initial MPBFs individually. Excited MPBFswith a perturbation contribution of larger than the first thresholdTVar are included in the (“selected”) variational space.JI–1 is the energy difference and is given by Aninitial selection is performed bygenerating the possible single and double excitations relative tothe initial configurations. The selection is performed by evaluatingthe second-order Epstein–Nesbet perturbation energyc)The many-particle Hamiltonian is diagonalizedover the set of the set of presently selected MPBFs, thus definingthe initial many particle states.d)The eigenfunctions arising from thediagonalization are analyzed with respect to the leading contributionsto the roots found. MPBFs with a weight of larger than the secondthreshold TGen are considered as “generator” MPBFs.This is also illustrated in the flowchart of the ICE algorithm in e)I ∈ generatorsIC|ΦI(0)⟩) is an eigenfunction of the Hamiltonian.35GP is the projector in the generator space and Ĥgen is the projected Hamiltonian. Carehas to be takento ensure that the energetic ordering of the states is maintainedin the projected Hamiltonian Ĥgen.The generator part of the present wavefunctionis renormalized thus defining the “contracted generator wavefunction”.The renormalized wavefunction and energy are obtained by diagonalizingthe Hamiltonian in the generator space as given by f)Jn–1 represents the energy denominatorfor the contracted selection forthe nth root and is given by eq C̃In and Ẽnk)k to k + 1 iteration can be calculatedas ΔnE = Enk – Enk + 1. Once the energy difference at the k + 1th level is smaller than a small, predefined value, the calculation is considered tobe converged.Convergence of the wavefunction andenergy is checked. If convergence has not yet been achieved, the algorithmproceeds by going back to step (d). The energy change from h)EJPT2, k + 1 for all the newly generated MPBFs has been calculated,the selection of important new MPBFs can be made. Here again, TVardictates the value of EJPT2, k + 1 beyond which a newly generated MPBF is to beincluded in the variational space, i.e., EJPT2, k + 1≥ TVar. Note that a vast majorityof the generated MPBFs will be rejected and only a small number willsatisfy the criteria listed before as will be shown later. Importantly,the total PT2 energy ErestPT2, (k + 1) of those MPBFs that have been rejected canthen be estimated by summing over their contribution over all states.For amulti-state ICE, we follow thestrategy of state-averaging. Hence, before selection begins, the PT2contribution of each new MPBF needs to be summed for all roots asshown in The main steps of the algorithm which has been implementedin theORCA program packageThe rest of the steps are similarto the single-state ICE.EPT2 =∑JEJPT2, also referred toas the “rest” energy, which will be used during theextrapolation.The three parameters that will be requiredin the following benchmarkinganalysis are the two thresholds TGen and TVar and PT2 energy of thediscarded MPBFs A, τ) (A = –Log10(TGen)). For example, a calculation with TGen =10–4 and TVar =10–11 will be labeledas ICE.In the following analysis, we shall use the combinedparameter 2.2ICEE vs the FCI energy FCIE as a functionof the two thresholds TGen and TVar. In order to test the largestspread of the values of TGen and TVar, we performed calculations insteps of factors of 10. The range for TGen is chosen to be from 10–2 to 10–8 and for TVar from 10–6 to 10–14, thus leading to a totalof 64 calculations for each of the 21 molecules and a total of 1344data points. The average error was estimated by taking the mean ofthe 21 molecules for each pair of TGen and TVar values. Note thatsince the ICE is a variational method, the errors in the energy withrespect to FCI are always positive. The error bars were estimatedby calculating the variance σ2 of the error in the21 molecules for each pair of TGen and TVar parameters.35Thenatural reference for the ICE methods is provided by actual FCI calculations.Hence, the FCI21 benchmark set is chosen such that FCI calculationsare possible on all the molecules included. Benchmarking mainly concernsthe study of the error in the ICE energy 336 wasused for all the molecules except N2, O2, F2, and CH4 for which the Ahlrichs split valence(SV)37 basis (without polarization functions)was kept on the heavy atoms in order to make the FCI calculationsfeasible. The geometry was optimized at the FCI level for the groundstate with the given basis set, the converged distances, and anglesare given in 38 The converged FCI energies along with the coupled-clusterenergies for the FCI21 set are given in The accuracy of theICE algorithm for DETs, CFGs, and CSFs is performedby comparison of the molecules in the FCI21 set with the FCI energy.The cc-pVDZ double-ζ basis set3.1In order to clearly present all the data and the various aspectsof the error analysis with TGen and TVar thresholds, two types ofplots are chosen. First, the error vs TVar is plotted in order toshow the rate of convergence with the parameters along with the variancein the error for the FCI21 set. In the second type of plot, the variationof the average energy error is shown in a contour plot with TGen andTVar together to facilitate the comparison of DET, CFG, and CSF ICE.In this way, all the different aspects of the dependence of errorand error bars with TGen and TVar can be studied. Each point in a)k⟩ (with respect to |Ψ0⟩) is controlled by TGen, whereas TVar has no effect on theexcitation order. Therefore, for a sufficiently small TVar, an improvementin energy is achieved by adding functions, which have a higher excitationorder compared to those present in |Ψk – 1⟩. This precisely is what is achievedby decreasing TGen.Decreasing TVar witha fixed valueof TGen leads to a plateau behavior, and hence decreasing TVar beyondthe plateau does not improve the error. This implies that in orderfor there to be substantial change in the wavefunction, TGen has tobe decreased. Actually, the excitation order of MPBFs in |Ψb)The convergence with TGen for a givensufficiently small TVar is exponential for all three variants of theICE. This is very satisfying since this implies that one does nothave to tighten both TGen and TVar for approaching the FCI limit.It is enough to fix a sufficiently large TGen/TVar ratio (as doneby introducing the parameter τ) and then decrease TGen to systematicallyapproach the FCI energy. This strategy will be used in the extrapolationscheme described below.c)The plateau behavior with TGen fora fixed large TVar is less pronounced with CFG-ICE and CSF-ICE. However,the DET-ICE variant shows a plateau with TGen for a fixed large TVaras can be seen in d)The errorbars giving the varianceof the energy have a constant value with decreasing TVar for a fixedTGen. Therefore, it appears that the variance decreases mainly witha decrease in TGen and is unaffected by decreasing TVar.e)k⟩ willbe closer to the FCI one for the CFG and CSF MPBF than the DET MPBFirrespective of the type of molecule.The variance in the error of the FCI21set slightly decreases upon going from DET, CFG to CSF-ICE, as expected.First, this is due to the generally larger absolute error for DET-ICEcompared to CFG-ICE and CSF-ICE with a given TGen and TVar. The secondreason is due to the normalization of the wavefunction. The CFG andCSF MPBFs are more compact compared to the DET MPBF, and hence thewavefunction expansion in CFG and CSF basis is shorter than that inthe DET basis. This implies that for a given threshold TGen and TVar,the wavefunction |ΨFrom inspection of A comparison of variations with both TGen and TVar together canbe made using a contour plot of the error versus the two parametersas shown in 3.2In this section, we present a comparison of the variationand perturbative energy for the FCI21 benchmark set. Before presentingthe data for the three types of MPBFs, we first present an illustrativeexample to highlight the difference between DET, CFG, and CSF MPBFsduring the calculation of the perturbative energy contributions.3.2.14H6) molecule  has 62 orbitals and a total of 22 electrons. The closed shell restricted Hartree–Fock (RHF) configuration is chosen as the |0⟩thorder wavefunction |Ψ0⟩ is large enough for a somewhat general comparisonof the behavior of perturbative energy contribution in the DET, CFG,and CSF MPBFs. The main conclusions from a)EIPT2 ≥ 10–7 mEh as shown in The CFG and CSF many-particlerepresentationsshow a similar spread of the perturbative contribution. Both MPBFscontain about a maximum of 50,000 MPBFs having a perturbative energycontribution of b)sM = 0 basis and not a spin eigenbasis.The DET many-particle representationon the other hand has a total of 912,186 single and doubly excitedDETs, which is about 1 order of magnitude larger than the CFG andCSF MPBFs. This is not unexpected as DETs are expanded in an c)2H4.The spread of the perturbativeenergycontribution for the DET basis is much larger than CFG and CSF basis.As shown in d)EIPT2 contribution.A direct consequence of this largerspread in the DET basis is that the perturbative contribution broughtin by a single DET is about 1 order of magnitude smaller than thatof a single CSF. This is more clearly seen in e)EIPT2 brought in by a single MPBF. This valueis about 10–7 mEh for CFG and CSF MPBF and about10–9 mEh for DET MPBF. Therefore, the energy broughtby most of the DETs is more than two orders of magnitude smaller thanthat for the CFG and CSF MPBFs.From the comparison of the PT2 energydensity for DET, CFG, and CSF MPBFs given in f)EIPT2 can be used in orderto estimate the minimum values of thethresholds TGen and TVar adequate for a reliable FCI energy approximation.This average value of The butadiene molecule was chosen for a demonstrativepurpose heresince the space of singly and doubly excited MPBFs from |Ψ3.2.2Now, we shallpresent the analysis on the FCI21 benchmark data. Once the new MPBFsare chosen as described in EPT2) that need to be emphasized:a)38 The extent of thisoverestimation needs to be benchmarked as a function of TGen and TVar.It is expected that the CFG-ICE will show a relatively larger overestimationdue to the fact that all the CSFs of a given singly or doubly excitedCFG are included while calculating its PT2 contribution.As the perturbativeestimate is calculatedusing the Epstein–Nesbet zeroth-order Hamiltonian, it willtend to be overestimated.b)Note that since only the generatorMPBFs are used to calculate the perturbative contribution, part ofthe PT2 correction due to the “spectator” MPBFs is lostand a bias is introduced. On the other hand, this PT2 contributiondue to generators comes essentially free of cost, whereas a full PT2contribution would require additional non-trivial computational effort.This bias can be systematically removed upon decreasing the TGen threshold.The following aretwo points about the “rest” PT2 energy (The error in the energies with the “rest”PT2 correctionof the FCI21 set can be compared to the FCI results for the threevariants as given in a)The PT2correction for the DET-ICEbehaves very differently compared to the CFG-ICE and CSF-ICE. Themain reason for this fundamental difference is due to the fact thatthe PT2 contribution due to a single DET is in general much smallerthan the PT2 contribution due to a single CSF belonging to the sameconfiguration as shown in the previous section. Consequently, thethresholds cannot be directly compared for DET-ICE and CSF/CFG-ICE.b)–6 and large TVarvalue of 10–6, the perturbative contribution isnegligible compared to the variational correlation energy. However,for the CFG-ICE and CSF-ICE, the perturbative contribution for TGen10–6 and TVar 10–6 is so largethat the total energy error increases from 1 to about 10 mH as canbe seen in –6 mH.The most significant observationisthat the perturbative energy estimate is larger by at least an orderof magnitude for the CFG-ICE and CSF-ICE compared to the DET-ICE,thus corroborating the finding of the previous section. For the DET-ICE,even with a small TGen value of 10c)Evar + PT2 energy errorincreases with decreasing TGen ( and a fixed TVar) for the CFG-ICEand CSF-ICE variant due to the fact that upon increasing TGen The variance of the errors shows asimilar behavior as compared to the variational energies.There is a range of interestingobservations and differences betweenthe DET, CFG and CSF-ICE numbers. The major points are summarizedbelow:EPT2 contribution, in EPT2  for the DET-ICE, CFG-ICE, and CSF-ICE with TGen and TVarthresholds. As expected, there is a large PT2 contribution for largervalues of TVar as shown by the horizontal yellow block in 40 However, the perturbative contribution decreases exponentially withdecreasing TVar. Note that, as remarked earlier, the PT2 contributionis largest for the CFG-ICE as shown by the yellow region in In order to better understand the Therefore, not surprisingly, small values of TGen and TVar seemto be ideal for an accurate prediction of the FCI energy. However,as we shall analyze in detail in the next section, decreasing thethresholds implies an increase in the total number of variationalparameters in the wavefunction.In summary, adding the PT2 energyto the variational ICE energyis a mixed blessing. On one hand, it can reduce the error of the calculationrelative to the FCI results. On the other hand, the convergence tothe FCI limit is far less smooth due to the overshooting of the Epstein–Nesbetsecond-order energy and one can also undershoot the FCI energy. Hence,extrapolation appears to be a more promising strategy to improve thevariational ICE results as will be discussed below.3.3As the solutionof a large eigenvalue problem is the rate-limitingstep in a sCI procedure, the number of variational parameters in DET,CFG, and CSF representation plays a major role in determining theefficiency of the method. Clearly, the number of variational parametersdepends on the values of the two thresholds TGen and TVar in additionto the choice of the MPBF used. As the thresholds approach zero, thenumber of variational parameters  approachestheir respective FCI dimensions, which can be prohibitive. Analysisof the number of variational parameters with respect to the two thresholdsTGen and TVar can provide important information about the compactnessof the wavefunction for the respective threshold regimes. In the presentsection, we study the increase in the number of wavefunction parametersfor the three variants of the ICE as a function of the two parametersTGen and TVar.–6 for CSF/CFG and 10–9 for DET) addingadditional MPBFs has a negligible effect on the total energy.The number of wavefunction parameters also increases upon a decreasein TGen. A comparison of the logarithm number of wavefunction parameterswith both thresholds TGen and TVar together is given in The increase in the number of wavefunction parameterswith decreasingthresholds is accompanied by a decrease in the error in the energieswith respect to the FCI values. Notice that, as one would expect,the smallest energy error occurs at the top right corner of 3.4k + 1⟩, which is expanded in thebasis of the “selected” MPBFs. Therefore, the size ofthe selected space of MPBFs at each ICE step is directly related tothe cost of the total calculation. As shown in the previous section,the total number of MPBFs increases with decreasing thresholds TGenand TVar; consequently, the cost of the calculation increases withdecreasing TGen and TVar as illustrated for the NH3 moleculein double-ζ basis , respectively. Herethe number of MPBFs in the calculation has been used as a proxy ofthe total computational time as described in A was set to 1 and B to 0.7 in order to obtain comparablemagnitudes for CBindex.In order to beable to extrapolate energies obtained with progressively tighter thresholds,one needs to identify the optimal way to reduce the thresholds. Thischoice can be made by looking at the efficiency of an ICE calculation,which is dependent on the following two factors: First, the cost ofthe calculation, which is proportional to the number of wavefunctionparameters needed to optimize. Second, the accuracy of the calculationobtained at a given TGen and TVar. A tighter choice of thresholdswill permit a more accurate result while at the same time incurringa large cost of calculation. In order to obtain a “cost-benefitindex” (CBindex), the function shown in a)–4 and TVar = 10–9, itis more advantageous to decrease the TGen to 10–6 while keeping TVar constant (=10–9) than keepingTGen = 10–4 and decreasing TVar to 10–11. This manner of optimal change of parameters is shown visually withthe two lines on the upper section of In order to decrease the error, onewould benefit from decreasing the TGen while keeping TVar fixed ratherthan the other way around. This is more important for smaller TGenthresholds. For example, in order to improve the error obtained atTGen = 10b)In c)–4 to achieve CCSDT(Q) quality resultsas will be described below in more detail.For small molecules, it is convenientto use a single parameter τ see 2.1 such d)3) or τ =4  and decrease TGen alone. Thisstrategy leads to the smallest computational cost while at the sametime obtaining the best possible total energy. This will be discussedin more detail in In practice, for larger molecules,the optimal thresholds seems to be to fix τ to a small valuesuch as τ = 3 , which leads to resultsthat are better than CCSDT(Q) quality. In these calculations due tothe small value of τ, the rest energy is so small that addingit to the variational energy leads to insignificant changes. Noticethat the setting with ICE provides similar results to the muchtighter ICE setting at a much lower computational cost  the extrapolationwith weight of the generator coefficients  as done by the original papers by Buenker and Peyerimhoffand by Angeli and Persico et al. and others46 and (2) extrapolation with respect to the PT2 energy correctiondue to the rejected configurations as done, e.g., by Holmes et al.48 Inspired by the later studies, we have followed a slightly modifiedapproach here.As the objectiveof a sCI calculation is to approximate the FCI energy as closely aspossible and any sCI method must introduce truncation thresholds,it is tempting to devise extrapolation schemes that allow one to estimatethe FCI energy obtained at zero threshold. Such an extrapolation schemehas been studied in great detail by Buenker and Peyerimhoff in theearly 1970s10(TGen), τ) as explained in vide infra), the convergence of the correlation energy tends to be exponentialwith respect to the threshold TGen (keeping τ fixed) shown 40 and others.In our case, there are two thresholds TGen andTVar. However, as pointed out above below, one can devise a compositethreshold by fixing the ratio of TGen/TVar represented by τand varying only TGen. Once a composite parameter (τ) is chosen,the energies can be extrapolated against TGen (this scheme is representedas ICE(−LogEPT2) known asthe “rest” energy. This is in agreement with the findingthat the energies can be extrapolated against the “rest”PT2 energy (EPT2) withthe extrapolated value (EPT2∞) obtained as EPT2 → 0. Moreover, ithas been realized that an extrapolation of the energy with the “rest”PT2 contribution tends to be more linear than the extrapolation relativeto the thresholds.47 Therefore, we usea simple (two-point) linear function to estimate the extrapolatedenergy as a function of EPT2 as shown in Moreover, by design of the ICE algorithm, the error in the energydecreases with a decrease in the PT2 contribution of the rejectedMPBFs  or more, one can also obtain the confidence interval (ΔEICE∞) of the extrapolated energy as shown in The parameters Epoint)ICE(2∞) and three-point extrapolatedenergy (Epoint)ICE(3∞) will be very close andhence ΔEICE∞ will be close to 0. Thus, a confidenceinterval of ΔEICE∞ = 0 will imply a high confidenceof the extrapolation for small enough TGen and τ values. Notethat this extrapolation scheme is general and transferable for anyMPBF and essentially comes without additional overhead or changesto the ICE algorithm.Upon convergence or close to convergence, a two-point extrapolatedenergy   . As shown in EPT2 toward the FCI energy for all four schemes ICE, ICE, ICE,and ICE.As an illustration of our extrapolation scheme,we plot the energyconvergence of the FCI21 set with a series of calculations ICE – ICE. A three-point extrapolation would be denoted as EP– ICE.A linear fit to obtain the extrapolated energy EICE∞ with a two-point fit,we have performed an analysis of a seriesof systematic choices EP – ICE for the FCI21benchmark set.In order to access the accuracyof the extrapolated energy m = 4to m = 7 show a monotonically increase in accuracyfor the FCI21 set as shown in EP(m/n). However, the accuracy of the extrapolatedenergy increases with decreasing TGen, as expected. Note that thecalculations with τ = 3 are at least an order of magnitude cheaperthan that with τ = 7. However, the quality of the extrapolatedenergies with τ = 7 is better than that with τ = 3.The series of extrapolated energies with As seen from It is now tempting to test this for larger systems for which FCIvalues cannot be obtained and compare the extrapolated FCI energiesto coupled-cluster ones (CCSDT(Q)) or DMRG energies as will be donein the next section.3.7.3In order to test the extrapolation scheme proposedin the previoussection, we carry out ICE calculations on three small polyenes ina double ζ-basis set as shown in 2H4 andC4H6. The cc-pVDZ basis was used for all atomsfor the case of methane (CH4). The geometries of the threemolecules are given in Section 1.4.1 of the Supporting Information.The cc-pVDZ basis set is usedfor hydrogen atoms, and the SV basisset is chosen for the carbon atoms for C3.7.3.1a)The scheme ICE has beenused for all the three molecules with TGen =3,4,5,6,7, and 8 wherepossible.b)EICE∞) by fitting to EPT2 with a straight line as we have shown for the benchmark set. Thisgives a unique value of the extrapolated energy EICE∞, which can be obtained by setting EPT2 → 0 in Once the seriesof energies has beenobtained, an extrapolated FCI energy is chosen  is shownin 3.7.3.2–8 for CH4 and 10–7 for C2H4 and C4H6, which we used to obtain thebest variational energies, respectively. As can be seen from Now we can proceed tothe results for the three molecules together.These three molecules are too large for an exact FCI calculation inthe given basis set and hence comparison cannot be directly made withFCI results. Nevertheless, one can obtain near-FCI quality energyfor such closed shell molecules from CCSDT(Q) calculations. First,we compare the convergence of the energies as a function of the thresholdTGen keeping τ = 3, i.e., a scheme with ICE for decreasingvalues of TGen as shown in Following the convergence of the pure variational energy, one canproceed to the extrapolation. The variational and extrapolated ICE,CCSDT(Q), and DMRG energies are shown in Figure 11 of the SupportingInformation. The difference between the EICE∞ obtainedby progressively adding more accurate ICEE, and EPT2 values cangive reliable confidence intervals for the extrapolated FCI energy EICE∞.The quality of the extrapolatedenergies given in 4. In all cases, the variational energiesare lower than the DMRG estimates and even the CCSD(T) energies forCH4 and C2H4 as shown in The main point to emphasize is that the variational and extrapolatedenergies are very close to the CCSD(T) energies for all moleculesand CCSDT(Q) for CH3.7.4m/n, τ) can be usedto systematically reduce the error by at least an order of magnitudewith respect to the variational ICE value. We have observed that,for small molecules, i.e., about 14 electrons, a tighter scheme withτ = 4 or τ = 5 can give an accurate approximation (lessthan 0.1 mEh error) to FCI as shown in m/n, τ) schemewith τ = 3 can be a viable alternative to get at least resultsof about CCSD(T) quality. However, unlike the latter, the ICE is notrestricted to single reference systems. Moreover, we clearly acknowledgethat the energies obtained here suffer from size-inconsistency errors,which is the primary reason behind the lack of accuracy with respectto CCSDT(Q) values. This size-inconsistency error will be studiedin the next section.In conclusion, a judiciouschoice of the extrapolation scheme EP associated with the ICE method arises due to the truncatedCI expansion and contributes to the deviation from the FCI energy.For a pedagogical and clear description of the SIE for approximateCI methods, see the description by Malrieu et al.3.8.1The SIE in the presentformulation of the DET, CFG, and CSF-ICE can be demonstrated usingthe neon dimer. In order to calculate the size-inconsistency error,we proceed as follows: the FCI energy of the Ne atom in the cc-pVDZbasis with  (with a frozen core) is calculated followed bythe energy of the Ne dimer  at a 10 Å distance. TheSIE can then be calculated by subtracting twice the neon energy fromthat of the neon dimer as shown in q 14:For a size-consistentmethod, this difference should vanish since the distance between theNeon atoms (10 Å) makes it an essentially a non-interacting pairof atoms. A deviation from zero therefore results from the size-inconsistencyerror introduced by the truncation of the CI space. The error in asa function of TGen and TVar is shown in 3.8.2A more rigoroustest of the SIE can be performed using a larger system consistingof the ethene molecule with a neon atom separated by 10 Å, thusensuring that the two molecules are essentially non-interacting 18. The cA size-consistent method will give total energies, which can beexactly written as a sum of the energies of the ethene and Ne moleculesseparately. Therefore, the SIE can be obtained by subtracting theFCI energy (or extrapolated FCI energy in the case of ethene) of etheneand Ne molecules calculated separately from the ICE calculating involvingthe full system as given in 2 is almost equal to the magnitude ofthe total deviation from the FCI energy of the benchmark set as shownin The similarity of 4a)Due to the spin-adapted formalism ofthe CFG-ICE and CSF-ICE, these two many-particle bases lead to a significantlymore compact many particle wavefunction than the DET-ICE many-particlebasis representation. This allows for a better accuracy with a similarnumber of wavefunction parameters. This will be capitalized on inmuch greater detail in concrete chemical applications in Part IIIof the series.b)The behaviorof the perturbative energycontribution for the CFG and CSF-ICE is quite different from the DET-basedICE. This has been demonstrated by the comparison of the spread ofthe PT2 energy contribution for the three MPBFs for the butadienemolecule. The main conclusion from this analysis is that the PT2 energyis spread out over about 1 order of magnitude more DETs compared toCSFs. Due to this larger spread of the PT2 energy contribution, theamount of PT2 energy brought in by one CSFs is about four times largerthan that brought in by one DET. Therefore, even for the perturbativeenergy calculation, the CSF many-particle basis is more compact thanthe DET basis.c)–4 reproduces99.8% of the correlation energy, which is better than the CCSD(T)result with only 2% of the total time of a FCI calculation.Fromour benchmark FCI21 set, the defaultsetting of τ = 7 and TGen =10d)–4, which leads to excellentresults and an optimal cost/benefit ratio. Relaxing these thresholdsfurther comes at the expense of additional penalties in accuracy.However, depending on the application, slightly reduced accuracy maystill be acceptable.Based on the cost-benefitanalysis,an extrapolation scheme was devised, which was shown to be effectivefor the benchmark set. We recommend different settings for high-accuracystudies on small molecules and studies on larger systems. For smallsystems, we find that ICE provides results that surpass CCSDT(Q)quality, which is considered to be a converged level of theory. Forlarger molecules, compromises have to be made and such large τvalues are not feasible. Here, we recommend τ = 3 together witha decreased TGen value of 10e)An open-ended linear extrapolationscheme was used to reduce the residual error of the calculation whilenot increasing the computational cost. Based on our results, a two-pointextrapolation EP leads to a reduction in the error relativeto ICE of one order of magnitude (FCI21 set) while stillbeing no more expensive than the ICE calculation itself.Again, for larger number of correlated electrons, some compromiseshave to be made.f)Forlarger systems, we have observedsignificant errors in the ICE final energies that we mainly attributeto the size-inconsistency inherent in the CI procedure. We have quantifiedthese errors using the neon dimer and the ethene/neon molecule.In this second part ofour three series of papers, which aims acomparison of the three many-particle representations, we have performeda benchmark test of comparison of the three methods with FCI for aset of 21 diatomic molecules. There are some interesting conclusionsthat emerge from the present analysis:In the third part of the present seriesof papers, we shall performcase studies on different types of molecules in order to shed lighton the strengths and weaknesses of the three types of many-particlebasis representations.