In our study, we explored the relationship between race and shared decision-making (SDM), while recognizing that other social factors affect such communication and may interact with race.

As we note in our paper, an important next step will be to explore how perceptions of race interact with other social variables to influence SDM.


Is there evidence/theoretical model(s) that link institutional racism to population-level health disparities?


Several well-known models broaden our understanding of how institutional racism may lead to population-level health disparities.

The chronic stress induced by personal experiences with discrimination is one mechanism by which institutional racism may affect health [19, 2].

For example, McEwen’s model (1998) of allostatic load emphasizes interactions between cognitive processes (i.e. responses to perceived stress) and physiological responses (e.g. cardiovascular, immunological effects) to explain how environmental stressors, major life events and trauma (e.g. racism) result in physiological changes [20].

In Massey’s biosocial model of racial stratification (2004), concentrations of poverty and violence (due to socioeconomic inequalities and residential segregation) result in high allostatic loads that have downstream health effects such as coronary artery disease, inflammatory disorders and cognitive impairment [21].

Among African-Americans, perceptions of discrimination are independently associated with C-reactive protein, a marker of systemic inflammation that correlates to cardiovascular disease and other health outcomes and precursors to cardiovascular disease (e.g. coronary artery calcification) [22, 23].

Institutional racism may also affect health through negative health behaviors such as delays in healthcare utilization and treatment non-adherence [24], and cigarette and alcohol use [25].

Institutional racism, as perceived discrimination within society, is associated with poor health measures, including depression, high blood pressure, cardiovascular disease, and pulmonary disease [26, 27].


In the above, we have highlighted some of the evidence linking institutional racism to health outcomes, and illustrated models proposing pathways between racism and population-level health disparities.


Is there evidence and/or theoretical model(s) that link the patient/provider relationship and communication disparities to population-level health disparities?


Evidence-based models have shown how health providers may contribute to racial/ethnic disparities in health [28, 29].

These models focus on patient/provider relationships and how bias, health and interpersonal behaviors, cognitive and affective factors, perceptions, and professional decision-making influence healthcare delivery and health outcomes.


Based on our research, we have developed a conceptual model for exploring relationships between race, shared decision-making and health outcome.

It is important to note that our model is nested within the broader context of the healthcare system and macro-level factors which also affect health outcomes and health disparities.


The model demonstrates how shared decision-making is a joint endeavor, and its successful execution depends on the preference of both patients and providers to engage in this process, and on the patient/provider relationship, which creates the situational context for SDM.

Race may potentially affect SDM through several mechanisms.

First, racial differences may exist in patient preferences for SDM or in how SDM is conceptualized.

A study of a multi-ethnic population found no differences between white and African-American patients in preferences for SDM [30], suggesting that patient preference is unlikely to drive SDM disparities.

In a prior analysis, we found that African-Americans with diabetes defined SDM in ways that are different than how it is conceptualized in the literature [31], which may influence SDM behaviors and contribute to differential experiences of SDM among African-Americans.


Race may also impact SDM through its influence on the patient/provider relationship.

In a prior analysis involving African-American diabetes patients, one of the most powerful SDM facilitators was physicians’ interpersonal skills, which were described as essential to establishing a meaningful patient/provider relationship and creating an environment for patients to express concerns and play active roles [32].

Interpersonal skills and relationship-building may be an important way to address the disproportionate physician mistrust among African-Americans [33].


Physician mistrust is associated with lower preferences for shared decision-making [34], and in our conceptual model, trust and normative beliefs (e.g. biases) are two mechanisms by which race may influence SDM, either directly or through the patient/provider relationship [35].

For example, providers may have preconceived ideas about who is more likely to prefer an active role, which may influence how engaging they are of different racial/ethnic groups in decision-making.

In general, African-Americans have less participatory physician visits with more physician verbal dominance, less information delivery, and less patient-centered communication than whites [36, 37].


Our study explored patient perceptions about the influence of race on patient/provider communication and specific domains of shared decision-making, an area that has received little attention to date.

Consistent with our conceptual model, participants described patient trust, bias/stereotypes and the patient/provider relationship as mediating factors between race and SDM.

We also found that race has the potential to negatively influence SDM within each of its domains—information-sharing, deliberation/physician recommendation and decision-making—through cultural discordance, patient beliefs arising from internalized racism, and unconscious provider bias (personally-mediated racism).


While SDM and patient-centered care are associated with health outcomes, the mechanisms are not fully understood.

However, we do know that information-sharing and joint goal-setting (important SDM components) are associated with self-efficacy and diabetes self-management, and research suggests that patient understanding, self-efficacy, patient satisfaction and trust predict adherence and self-management, and may be the mechanisms through which shared decision-making impacts health [38, 39, 40].


Are there potentially effective solutions to address institutional racism, particularly unconscious provider bias?


The question of how to effectively address U.S. racial/ethnic health disparities is an important one, and for which there is no simple or single solution.

Rather, the answers must address the range of causes of disparities (e.g. inequalities in education, housing, and health insurance) and empower multiple levers of change (e.g. patients, providers, health systems, policymakers, communities).

For example, researchers at the University of Chicago are working to reduce diabetes disparities on the city’s South Side, a predominantly working class African-American community, through a multi-site intervention that combines patient education/activation, provider training, health systems redesign and community engagement [41].


Our paper focuses on patient-physician communication, and as such, our proposed solutions target patients and physicians.

The issue that Bradby (2010) raises about how to address unconscious provider bias is an important one.

The training of health providers involved in the Tuskegee syphilis experiment in racial eugenics showed that medical education can create racial bias and exacerbate health disparities [42].

To date, however, no research has shown that physician education through cultural competency training can reduce disparities in health outcomes [43, 44].

However, such training can improve provider knowledge, attitudes and skills, which may be an important precursor to addressing unconscious provider bias.


There is evidence that with sufficient motivation, cognitive resources and effort, people can inhibit stereotypes and focus on individuals rather than the sociodemographic groups they represent [45, 46].

Drawing upon evidence in social cognitive psychology, Burgess et al (2007) have outlined strategies and skills for healthcare providers to prevent unconscious racial biases from influencing the clinical encounter.

Their framework includes: 1) Enhancing internal motivation and avoiding external pressure to reduce bias, 2) Enhancing understanding of the psychosocial basis of bias, 3) Enhancing providers’ confidence in their ability to successfully interact with socially dissimilar patients, 4) Enhancing emotional regulation skills specific to promoting positive emotions, 5) Increasing perspective taking and affective empathy, and 6) Improving the ability to build partnerships with patients.

Teal et al. ( in press) developed a medical student elective designed to help manage patient bias that incorporates many of Burgess’ principles; they reported significant improvements in students’ strategies to identify and address their racial biases.


Thus, medical science is currently on the cutting edge of identifying and implementing strategies to reduce unconscious provider bias in healthcare, but much work remains to be done.

Addressing such bias is only one of many strategies needed to make strides in eliminating racial/ethnic health disparities.


In summary, Bradby (2010) raises important issues about racism within healthcare and its potential effects on patient/provider communication and health disparities.

Yet the evidence is clear: race and racism affect the U.S. healthcare system and the patients and providers that interact within it.

Our current study builds upon a large body of evidence and gives voice to the perceptions and experiences of African-Americans with diabetes.

Only through continued work to understand racism can we make strides in addressing its effects on healthcare delivery and health disparities.


Pre-oxygenation prolongs the time to oxygen desaturation during apnoea and is a critical component in providing safe general anaesthesia [1].

Currently, an end-tidal oxygen concentration (EtO 2 ) ≥ 90% via a tight-fitting facemask is recommended for obstetric patients before rapid sequence induction and tracheal intubation [2].

However, despite the importance of pre-oxygenation, it is often performed poorly in obstetric patients due to air entrainment even around a tight-fitting facemask [3].

Furthermore, the requirement for rapid delivery of the fetus, along with well-recognised time pressures and human factor issues can all make pre- oxygenation suboptimal [4].

The accepted standard of practice for pre-oxygenation is that parturients undergoing 450 © 2019 Association of Anaesthetists Anaesthesia 2019, 74, 450-456 doi:10.1111/anae.14567 general anaesthesia either complete 3 min of tidal volume breathing or eight vital capacity breaths via a facemask, with both methods being considered to be equally effective [5, 6].


High-flow nasal oxygen (HFNO) has gained increasing popularity in non-obstetric patients, increasing time to desaturation [7, 8], including in paediatric patients [9].


Although it would seem rational to apply HFNO to obstetric patients [10, 11], its use appears described only in case reports and journal correspondence, and a direct comparison with facemask has not been performed [12- 14].

We wished to fill this gap in the literature, and hypothesised that HFNO pre-oxygenation would be as effective as standard flow rate facemask pre-oxygenation in the obstetric population.


The study received ethics committee approval from the University of British Columbia Children's and Women's Research Ethics Board and was performed between November 2017 and January 2018.

Following written informed consent, healthy, term (≥ 36 weeks gestation), non-labouring parturients admitted for elective caesarean delivery or induction of labour were recruited.

Exclusion criteria were: ASA physical status 3 or above; any medical condition that could affect gas exchange; BMI ≥ 40 kg/m 2 ; known airway pathology; obstructed nasal airway; or unable to tolerate a tight-fitting facemask.


Microsoft Excel 2010 was used to generate a randomisation sequence with 1:1 allocation to either HFNO or control groups.

This sequence, contained in sealed envelopes, was not known to the study investigators and was opened after enrolment.

Further blinding of the study was not feasible due to the nature of the intervention.


All patients were first placed in a 30-degree head up position using the Troop TM (Goal Medical, LLC, Eugene, OR, USA) elevation pillow and with a right lateral pelvic wedge to minimise aortocaval compression.

A nose clip was applied and each patient was asked to breathe 21% oxygen at 12 l.min À1 through a standardised mouthpiece using tidal volume breathing.

The mouthpiece was connected to an anaesthetic circle system with a two-litre reservoir bag, heat moisture exchange filter and an oxygen analyser.

At the end of 30 s, baseline values for oxygen saturation, respiratory rate, heart rate and EtO 2 were recorded.


Patients in the standard oxygen flow rate facemask group (control) were then pre-oxygenated with 100% oxygen, using tidal volume breathing and a tight-fitting facemask attached to a circle system, with an oxygen flow rate of 15 l.min À1 .

End-tidal carbon dioxide waveform analysis was monitored to ensure a tight seal was achieved between the patient and the facemask.

After 3 min, patients were asked to hold their breath for a few seconds and the nose clip was re-attached.

The patients then exhaled fully and continued to breathe for several breaths via the mouthpiece (delivering 21% oxygen) to ensure the highest value for EtO 2 was measured.


Patients in the HFNO group were pre-oxygenated with 100% oxygen using tidal volume breathing and the Optiflow TM HFNO nasal high-flow cannula system (Fisher and Paykel Healthcare Ltd, Auckland, New Zealand).

The oxygen flow rate was initially started at 30 l.min À1 and gradually increased to a maximum of 70 l.min À1 over the first 30 s. If a patient did not tolerate the flow rate, it was reduced to a level the patient could tolerate, typically 50- 60 l.min À1 .

Patients were specifically instructed to breathe via their nose with a closed mouth and the percentage of time this was achieved was recorded (0%, 25%, 50% or 100%).

After 3 min, the patients were asked to hold their breath for a few seconds while the HFNO was quickly removed and the nose clip re-attached.


Patients in both groups then underwent a washout period for up to 5 min until the EtO 2 returned to within 10% of their baseline value.

Once washout was completed, both groups repeated 30 s of tidal volume breathing followed by eight vital capacity breaths using their respective pre- oxygenation methods.

During the initial 30 s of tidal volume breathing, the oxygen flow rate in the HFNO group was increased from 30 to 70 l.min À1 and EtO 2 was measured at the conclusion of eight vital capacity breaths.

The time taken to complete 30 s of tidal volume breathing and eight vital capacity breaths was also recorded.


Immediately after the study, patient satisfaction was assessed using a 5-category Likert scale for acceptability (acceptable, slightly acceptable, neutral, slightly unaccept- able or unacceptable) and comfort (completely comfortable, slightly comfortable, neutral, slightly uncomfortable or completely uncomfortable).

The study flow diagram is depicted in Fig. 1.


The primary outcome was EtO 2 after pre-oxygenation, with factorial comparisons also being made between HFNO and standard flow rate facemask, and also between 3 min of tidal volume breathing and eight vital capacity breaths.

Secondary outcomes were the proportion of patients achieving an EtO 2 ≥ 90%, percentage of time patients had their mouths closed during HFNO pre-oxygenation and tolerability (comfort and acceptability) of both pre- oxygenation techniques.


The study sample size was determined based on a prior study demonstrating that the mean (SD) EtO 2 achieved following 3 min of pre-oxygenation in healthy, non-pregnant subjects with HFNO and facemask was 85.6% (6.4%) and 88.5% (6.2%), respectively [15].

We initially powered the study for a non-inferiority comparison.


The criterion for non-inferiority with respect to EtO 2 concentration achieved after 3 min was considered to have been met if the upper limit of a two-sided 95%CI for the absolute difference of EtO 2 concentration between groups was less than 10%.

Using a non-inferiority design, assuming a non-inferiority margin of 10 and a standard deviation of 6.3, and that the HFNO group would have a mean approximately 3% lower than the control group, 26 patients (13 per group) were required to achieve a power of 90% with a type-1 error of 0.05.

To account for attrition and study dropouts due to equipment errors (i.e. lack of a tight seal for waveform analysis), the sample size was increased to 40 (20 per group).

However, after patient recruitment we realised that as a physiological and not clinical study, a standard superiority design was more appropriate.

Using a superiority analysis with 20 patients per group there would have been approximately 31% power to detect a difference in means of 3% with a standard deviation of 6.3.


Consequently, data for primary outcomes were analysed using mixed-effects linear regression.

Mixed- effects models take into account the repeated-measures aspect of the data and control for correlations among measurements from the same patient.

To determine if there was a difference in how the treatment groups responded, we tested for an interaction term between group and time- point.

If a significant interaction was found, we followed it up with pairwise tests of the estimated marginal means between the groups at each time-point (e.g. HFNO baseline vs. control baseline) using the Kenward-Roger method to estimate confidence intervals, and controlling for multiple comparisons using Tukey's method [16].


The proportion of patients achieving an EtO 2 ≥ 90%, comfort and acceptability was compared using Fisher's exact test.

A p value < 0.05 was considered statistically significant.

Statistical analysis was performed using RStudio version 1.0.153 (Integrated Development for R. RStudio, Inc., USA; http://www.rstudio.com) and R version 3.4.1, release date June 30, 2017 (The R Foundation for Statistical Computing, Austria).


We recruited a total of 40 patients between November 2017 and January 2018; 68 patients were assessed for eligibility and 40 patients were randomly allocated to HFNO (n = 20) or control (n = 20) groups (Fig. 2).

All patients received the randomised pre-oxygenation method, but three patients in the HFNO group were not analysed due to EtO 2 measurement errors; two patients were unable to make a tight seal around the mouth piece when exhaling and one patient did not tolerate the oxygen flow rate of 70 l.min À1 and discontinued the intervention.

Patient characteristics between the groups were similar (Table 1).


There was a significant interaction between group and time in the mixed-effects regression model (p = 0.001).

Table 2 shows the estimated marginal means and confidence intervals.

Post-hoc pairwise Tukey tests comparing the means between HFNO vs. control groups at each time-point suggested that there was no significant difference at baseline (16.1% vs. 15.9%, p = 0.99) or after the washout period (17.2% vs. 17.3%, p = 1.0).

However, HFNO had a significantly lower estimated marginal mean EtO 2 compared with the control group after 3 min of tidal volume breathing (87.4% vs. 91.0%, p = 0.02), and also after eight vital capacity breaths (85.9% vs. 91.8%, p < 0.0001).

Figure 3 shows the range of data for each group at each time-point.


After 3 min of tidal volume breathing, there were significantly fewer number of patients in the HFNO group compared with the standard flow rate facemask group who were able to achieve an EtO 2 ≥ 90% (8/17 (47%) vs. 17/20 (85%), p = 0.032).

The maximum oxygen flow rate of 70 l.min À1 was tolerated by 82% (14/17) of the HFNO patients.

One patient required reduction in the flow rate to 60 l.min À1 and two patients required the rate to be further reduced to 50 l.min À1 .

There was no significant difference in comfort or acceptability between the two groups (Table 3).


We found only 24% (4/17) of patients in the HFNO group were able to breathe with their mouths closed for the duration of the pre-oxygenation period.

The mean (SD) EtO 2 for patients who were able to breathe with their mouths closed for at least 50% of the pre-oxygenation period compared with breathing with their mouths open for more than 50% of the time, was 90.3% (3.4%) and 82.0% (8.9%), respectively, p = 0.07.


Our main result that HFNO pre-oxygenation performs worse than standard flow rate facemask pre-oxygenation in healthy term parturients is surprising given its efficacy in other settings [7, 8, 10].

Yet, some aspects of our results are consistent with earlier studies.

In a small pilot pre-oxygenation study of four parturients, only one patient (25%) was able to achieve an EtO 2 ≥ 90% after 3 min of HFNO pre-oxygenation [14].

In another study of 20 healthy non-obstetric volunteers, only 10 (50%) participants were able to achieve an EtO 2 ≥ 90% within 3 min of tidal volume breathing [17].

In contrast, in a facemask pre-oxygenation study of 20 parturients, 75% achieved an EtO 2 ≥ 90% after 3 min of tidal volume breathing [5].

Similarly, in a physiological study of five healthy male volunteers, 100% of the participants achieved an EtO 2 ≥ 90% after 3 min of tidal volume breathing with a mean (SD) of 92% (1%) [18].

Together, these findings suggest that standard facemask pre- oxygenation is quite effective and HFNO may not always be fully effective.


One reason why HFNO did not achieve similar or higher EtO 2 compared with the standard flow rate facemask may be related to air entrainment.

A small non-obstetric Another possible reason for the poorer performance observed in the HFNO group may simply be because HFNO does not actually work by increasing the patient's oxygen reserve but instead by maintaining oxygenation when the patient is apnoeic.

We did not assess this important phase of induction.

Although measuring EtO 2 has clear advantages, pre- oxygenation method may be revealing [18].

Recently, Lodeniuset al. found that when HFNO pre-oxygenation was used in non-pregnant patients requiring rapid sequence induction for emergency surgery, a similar situation to obstetric general anaesthesia, there was a lower incidence of oxygen desaturation below 93% compared with facemask pre-oxygenation, with no evidence of gastric regurgitation [19].

We did not investigate time to oxygen desaturation as an end-point since we could not justify subjecting healthy parturients to general anaesthesia and prolonged apnoea in a randomised study.


We chose to use the maximum flow rate of 70 l.min À1 with HFNO compared with 40-50 l.min À1 recommended by the manufacturer because this was previously effective [7, 12].

Importantly, we found HFNO was well tolerated and acceptable to most patients, despite using maximum oxygen flow rates.

We did not compare HFNO flow-rate directly to 70 l.min À1 using a tight-fitting facemask as this may have resulted in lung barotrauma by forming a semi- closed circuit.


There are several limitations to our study.

First, it was not possible to blind the study due to the nature of the intervention.

Second, we used EtO 2 as a surrogate measure of effectiveness for pre-oxygenation.

Previous studies have investigated declines in oxygen saturations and arterial blood gases during apnoea after induction [7, 8, 10].

We did not examine patients who may benefit the most from HFNO pre-oxygenation, such as the obese or those with a difficult airway.

We also do not know how the two methods might compare in a real-world emergency scenario [3].

Finally, we originally powered the study as non-inferiority and future study designs that use a different de novo power analysis and overcome these limitations would be desirable to ascertain the place of HFNO in obstetric practice.


We read with interest the study by Shippam et al. [ 1] regarding high-flow humidified nasal oxygen (HFNO) for pre-oxygenation of pregnant women in preparation for general anaesthesia.

The findings from this study do not support the use of HFNO to pre-oxygenate term pregnant women.

The primary end-point used was an end-tidal oxygen fraction (FETO2) ≥ 0.9 which is a recommended target for pre-oxygenation by the Obstetric Anaesthetists’ Association [2] and the Difficult Airway Society.

However, with the advancement in the use of HFNO as an adjunct in both pre-oxygenation and apnoeic oxygenation, should this target be revised?


The use of HFNO maximises the physiological phenomenon of aventilatory mass flow, driving oxygen into the lungs and flushing out other dead space gases present [3].

This, coupled with the splinting of upper airways and reduction in associated shunt, can improve oxygenation [3].

If this is the case, then is it a prerequisite to pre-oxygenate to a value of FETO2 ≥ 0.9 in a situation where we can continually replenish our oxygen stores via continuous HFNO? If the patency of the airway can be maintained can HNFO redefine our end-point for pre-oxygenation?


We thank Tanna and Saha [1] for their interest in our study and for raising a thought-provoking discussion about the potential lowering end-tidal oxygen fraction endpoint in obstetric patients when using high-flow nasal oxygenation.

The current recommended pre-oxygenation target of end-tidal oxygen fraction ≥ 0.9 pre-dates the advancement of high-flow nasal oxygenation [2].

This specific value originated from the idea that it is desirable to reduce the risk of hypoxia by providing a reservoir of 95% oxygen assuming an obligatory 5% alveolar carbon dioxide, which corresponds to an end-tidal oxygen fraction of 0.9 according to an earlier study by Berry et al. [ 3].

In our study, after 3 min of tidal volume breathing, there were significantly fewer numbers of parturients who were able to achieve the pre-oxygenation target of an end-tidal oxygen fraction ≥ 0.9 in the high-flow nasal oxygenation group compared with the standard flow rate facemask group (47% vs. 85%, respectively).

If we had used a lower target of end-tidal oxygen fraction ≥ 0.8 as a primary endpoint, the high-flow nasal oxygenation group would have achieved a substantially greater proportion above the threshold (88% high-flow nasal oxygenation vs. 95% facemask), altering the conclusion of the study [4].

Our findings, along with a number of associated studies, have consistently found that not all pregnant women can achieve end-tidal oxygen fraction ≥ 0.9 after 3 min of pre-oxygenation [5, 6].


As such, we agree with Tanna and Saha that the threshold of end-tidal oxygen fraction ≥ 0.9 should be re-examined.

Indeed, reducing the endpoint of pre-oxygenation would mean a potentially shortened time to delivery of the fetus.

Also, using a simple theoretical model, the additional loss in safe apnoea time by lowering the threshold of end-tidal oxygen fraction from 0.9 to 0.8 may not be clinically significant, especially when arterial desaturation rate can be slowed by apnoeic oxygenation (Table 1).

However, just as it is difficult to justify the strict need for end-tidal oxygen fraction ≥ 0.9, the paucity of compelling evidence demonstrating reliability of apnoeic oxygenation in obstetric patients makes it just as difficult to challenge the status quo and adopt a lower endpoint.


Using a lower target of end-tidal oxygen fraction may be reasoned if apnoeic oxygenation can consistently maintain the oxygen reservoir following induction of general anaesthesia for obstetric patients; however, the success of this technique relies on a number of conditions to be met.

The benefit of apnoeic diffusion oxygenation is highly dependent on airway patency; yet, even with the most careful of pre-assessment, it is impossible to predict which patient may not have the patent airway necessary to support apnoeic diffusion [7].

Additionally, an early non-obstetric study by Fraioli et al. [ 8] demonstrated that apnoeic oxygenation is less effective in patients with a low predicted functional residual capacity to body weight ratio.

Although the results are difficult to generalise to obstetric patients, this study highlights the possibility that apnoeic oxygenation may not be as effective in individuals with altered respiratory physiology and thus the need to further affirm its role in pregnancy.


We need to better understand how we could identify parturients who would not benefit from apnoeic oxygenation.

Studying apnoeic oxygenation in pregnancy is ethically and practically difficult.

Most parturients who receive general anaesthesia are higher risk parturients who have lower physiological reserves and are often not included in research studies.

However, we believe time should be dedicated to investigate how to conduct apnoeic oxygenation effectively in the parturient and factors associated with its ineffectiveness, as difficult as that may be.

Results from ongoing high-flow nasal oxygenation studies may offer further useful insights and information to guide this discussion, but until then, many obstetric anaesthetists would likely practice with the devil they know [9].

Much more convincing data would be necessary before advocating the lowering of a safety margin that has persisted for decades.


The scope of the concept of oxidation states in main group compounds has remarkably expanded in the last two decades.

1-3 Stable singlet carbenes featuring non-oxidative electron-pair donation such as N-heterocyclic carbenes (NHCs) 4 and cycli- c(alkyl)(amino) carbenes (cAACs) 5,6 have been crucial to preparatively access unique low oxidation states.

For the description of chemical bonding, donor-acceptor interactions have been invoked, traditionally connected to transition metals.

7 Thus, the formal electron deciency of the central atom is alleviated by s-donation from the ligand, which is counter- balanced by a somewhat weaker p-backdonation.

7,8 While the chemistry of low-valent p-block compounds has substantially benetted from this coordinative bonding concept, the s-block chemistry has lagged behind.

1 The Group 2 chemistry has been long dominated by the +2 oxidation state, given the strong propensity of these elements to lose the valence electrons.

Signicant progress in the obtention of species with different oxidation states has been reported by Jones and co-workers on the Mg(+1) dimer compound I (Scheme 1), containing Mg I -Mg I bonds stabilized by guanidinate or b-dike- timinate (NacNac) ligands.

9 Over the years, the Jones compound has evolved from curiosity into a highly selective reducing agent.

9,10 Although species with a Be I -Be I single bond have been computationally predicted, stable molecules featuring this bonding motif remain unrealized so far.

11-14 Zero-valent Be(0) or Mg(0) compounds II were also elusive until recently.

In 2016, Braunschweig and co-workers reported the seminal isolation of dicoordinated neutral Be(0)(cAAC) 2 complexes III.

15 The unusual bonding situation has been rationalized in terms of donor- acceptor interactions between cAAC ligands acting as s-donors to empty s-type orbitals of a Be(0) atom, which would have available p-type electrons to furnish a signicantly strong p-backdonation towards the ligands, cAAC ) Be / cAAC (Scheme 1B).


This bonding scheme provides access to a stable beryllium radical cation V and a neutral species VI, with beryllium in the formal oxidation state of +1. 16,17 Attempts to prepare the Mg(0) congener were unsuccessful, leading instead to the ligand activation product IV.

18 The reaction outcome has been ascribed to the formation of the highly reactive Mg(0)(cAAC) 2 species, followed by ligand rearrangement.

Only recently, large charge transfer has been recognized from Mg to the cAAC ligands.

19 The quest for Mg(0) compound has been recently fullled by Harder and co-workers, using an extraordinarily bulky ligand (BDI* ¼ HC{C(tBu)N[2,6-(3-pentyl)-phenyl]} 2 ) to stabilize the Mg(+2) precursor.

20 The reduction with sodium powder fur- nished a Mg(0) compound {[(BDI*)Mg À ][Na + ]} 2 , which upon heating yielded a three-magnesium atom cluster VII (recall Scheme 1).

Notably, the bonding situation of the latter differs from that of the previous species, as the stability is driven by two † Electronic supplementary information (ESI) available.

Mg-Mg electron-sharing bonds rather than a donor-acceptor interaction.

21 The oxidation state assessment of these species is connected to the molecular orbital theory picture.

22 The donor-acceptor interaction in a closed-shell singlet conguration, similar to that in the Dewar-Chatt-Duncanson (DCD) model, assumes an electronic structure preorganization of E(0) from the ground state 1 S (ns 2 np 0 ) into the doubly excited singlet state 1 D (ns 0 np 2 ) to interact with the s-donor/p-acceptor ligands (Scheme 1B).

Within this description, applying the ionic approximation to the s- and p-type bonds could indeed lead to the zero oxidation state picture of the alkali earth metal.

Note, however, that Be (Χ P ¼ 1.57) and Mg (Χ P ¼ 1.31) are much less electronegative than C (Χ P ¼ 2.55).

23 Moreover, while the energy for such electron promotion is accessible for transition metals, the experimental gas-phase values for Be and Mg are as high as 178.3 and 399.9 kcal mol À1 , respectively.

24 A plausible alternative scenario can be postulated, whereupon bonding of the ligands, the metal centre oxidizes, and its electron pair ends up at the ligands, forming a diradical(oid) species (Scheme 1C).

The interaction between the unpaired electrons would be signicant in a closed- shell description, leading to the three-centre two-electron system (3c-2e).

However, a broken-symmetry solution would suggest a diradical(oid) species, where the paring between the electrons is lower than optimal.

A relatively small singlet-triplet gap (DE S-T ) is a key indicator for observing the diradical character.

25 However, distinguishing between these two pictures using single-reference methods is not straightforward, if not impos- sible, as the incomplete description of the spin polarization can mislead the wavefunction analysis.

26 Previous investigations have pointed out the multi-reference character of related systems such as germanium Ge(cAAC) 2 and zinc Zn(cAAC) 2 counterparts.

27,28 Herein, we pinpoint the subtle features of prominent low-valent Be- and Mg-based compounds through quantum chemistry calculations.


As an outset, we included NHC and cAAC ligands where the anking groups have different stereoelectronic properties, i.e. methyl (Me) and 2,6-diisopropyl-phenyl (Dip).

Both the singlet and triplet states of the systems were optimized at the B3LYP level of theory.

Broken symmetry (BS) solutions, such as the open-shell singlet (OSS), lower in energy were found for the systems stabilized by cAAC ligands.

In fact, the closed-shell B3LYP solution for Be-cAAC Dip is not stable.

Thus, the ground state is singlet in all cases, either closed-shell or open-shell.

The relative energies of the singlet and triplet states at the B3LYP- D3(BJ)/def2-TZVPP level are relatively close, ranging from 7.9 to 2.6 kcal mol À1 for Be-NHC Dip and Be-cAAC Me , respectively and from 13.3 to 2.4 kcal mol À1 for Mg-NHC Dip and Mg-cAAC Me , respectively.

The triplet state in Mg-cAAC Dip lies 9.1 kcal mol À1 below the closed-shell singlet, but the ground state is of OSS nature.

Similar observations have been obtained with different functionals (Tables S1-S3†).


Fig. 1 displays the ground-state geometries of the studied compounds, together with their dissociation energies (D 0 ) and singlet/triplet energy differences, and Table 1 shows the numerical values of their main geometrical parameters.

The Be- C bond lengths vary slightly with the nature of the ligand, i.e. from 1.634 to 1.648Å, in good agreement with previous studies.

11-15,19 These values fall in the expected bond lengths of single and double bonds (1.77 and 1.57Å, respectively).

29 Moreover, the bond angles are almost collinear for all cases 167.4-179.9 , favouring a strong delocalization on the C-Be-C p-system.

On the other hand, the Mg-C bond lengths are shorter than those reported by Couchman et al. for Mg 2 (NHC) 2 and Mg 2 (NHC) 4 systems.

12 Note, however, that the computed values are, in fact, longer than expected for a single bond Mg-C (2.14Å).

Only in the case of the Mg-cAAC Dip compound, the This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.


Å , when the C-Be-C angle bends so that the Be atom dissociates perpendicularly to the interatomic C-C axis.

From this point on, the occupation of Be 2s-type EFO rapidly increases and the change of formal OS from Be(+2) to Be(0) occurs at a Be-C distance of ca.


In Be-cAAC Me , the BS solution is lower in energy at the equilibrium geometry.

The spin polarization is mostly noto- rious on the ligands' frontier EFOs, which have a and b occu- pations much larger than those of the Be centre.

The dissociation starts in a collinear fashion.

The diradical char- acter increases with the Be-C distance, reaching its maximum (<S 2 > ¼ 0.60) at ca.

1.95Å, thus resulting in a much marked Be(+2) character.

Beyond this point, the C-Be-C angle bends so that once again the Be atom dissociates perpendicularly to the interatomic C-C axis.

In the process the diradical character rapidly decreases, inducing a charge transfer from the ligands to Be's 2s-type EFO until the formal OS changes at around 2.1Å, where the open-shell solution merges into the closed-shell one.

This again shows that a stable angular geometry is a key to achieving genuine low valent species.


In the case of the Mg-based species, the equilibrium struc- ture already points towards a Mg(0) species, so that the increase of the Mg-C distance further increases monotonically the gap in the EFO occupations in favour of the Mg moiety in closed-shell dissociation proles (see Fig. 5).

Coincidentally, in Mg-cAAC Me , the ground state is still of OSS nature aer spin-contamination correction, and the occupations of Mg and cAAC frontier EFOs are almost equal.

When using the CASSCF wavefunction at this geometry, the situation is more clear, resulting in an Mg(0) description with R(%) ¼ 80 as shown in Table 1.


Since the OS of the central E atom is clearly inuenced by the C-E-C angle, we also analysed the EOS performance along the C-E-C bond angle for Be-cAAC Me and Mg-cAAC Me species at the B3LYP level of theory (Fig. 6).

The occupation of the frontier EFO on E monotonically increases as the C-E-C angle deviates from collinearity.

In the case of Be, the EFO occupation remains always below 0.3 and that of the ligand remains always larger, even for closed C-Be-C angles (up to ca. 130 ) where the CS solution prevails.

However, in the case of Mg-cAAC Me one can observe a crossing point at around 110 where the occupation of the Mg EFO becomes large enough to yield a Mg(0) picture.

This occurs even before the closed-shell regime is reached, and in line with the CASSCF results.


At this point we can safely state that the +2 oxidation state is a consequence of the strong electron p-accepting properties of the ligands.

It is interesting to contrast this analysis with the strong s-donor and weak p-acceptor monoanionic b-diketimi- nate magnesium ligands.

Optimizing the experimentally accomplished Mg-BDI* leads to an almost collinear structure with an Mg-Mg-Mg angle of 175.4 .

Structures resulting from reducing the steric encumber hold the same structural features with a bond angle of 180 .

However, the bond lengths are sharply increased from 2.800Å (Mg-BDI*) to 2.917Å (Mg-Nac- Nac).

No open-shell singlet solutions were found for these compounds, and large vertical singlet-triplet gaps were ob- tained at both DFT and CASSCF levels (>25 kcal mol À1 ).

As ex- pected, the ligand interaction with the central element is explained by the s-type natural orbital (NO) with an occupancy close to 1.90 (see Fig. S17 and S19†).

The p- and p*-type natural orbitals present occupancies lower than 0.1, a ngerprint of dynamic correlation.

In the triplet state, one electron from the s-type NO needs to be transferred to a p-type NO from the central element.

Besides, the bond dissociation energies 49.0 and 63.9 kcal mol À1 suggest stable compounds for NacNac and BDI* derivatives, respectively, in agreement with the large Mg- Mg BOs (0.76 and 0.52).

Note that Mg-BDI* has a lower BO despite the higher dissociation energy, due to the dispersion interaction between the ligands.


In line with these ndings, EOS analysis also yields a relatively straightforward Mg(0) assignation (see Table S4 and Fig. S38†).


This concept can be used to take beryllium to even lower oxidation states.

Be-NacNac and Be-BDI* are predicted to be stable towards the dissociation, with 70.0 and 83.9 kcal mol À1 , respectively.

The description of the electronic structures shows no appreciable diradical character, with singlet-triplet gaps of 30.2 and 35.7 kcal mol À1 .

Given the higher electronegativity of Be with respect to Mg, the partial charges at Be are strongly negative À0.90 and À0.42 au.

Formally, these molecules bear a beryllium atom with an oxidation state of À2, which is further corroborated by EOS analysis (see Table S4 and Fig. S39†).


In summary, we re-examine the features of the structure, chemical bonding, and stability of the low-valent group 2 compounds.

In contrast to the accepted understanding, beryl- lium still remains in the +2 oxidation state territory.

The strong s-donor stabilized approach produces an internal electronic rearrangement furnishing diradical(oid) species with two unpaired electrons on the ligands.

Magnesium analogues might present oxidation state zero when the ligands are not too p- acidic, but the chemical bond is too weak to consider these molecules thermally stable.

Nonetheless, the effective oxidation state analysis suggests that the strongly Mg-based ligands are key to accessing genuine low-valent compounds.

Our study does not only give more insight into the peculiar features of the molecules considered, but also suggest a promising novel type of beryllium À2 oxidation state.

The presented results indicate that many more are yet to come to the fore from these combinations.


We challenge the assignment of the oxidation state +2 for beryllium and magnesium in the complexes Be(cAACDip)2 and Mg(cAACDip)2 as suggested by Gimferrer et al., Chem.

Sci. 2022, 13, 6583 in a recent study.

A careful review of the data in the ESI contradicts their own statement and shows that the results support the earlier suggestion that the metals are in the zero oxidation state.

The authors reported wrong data for the excitation energies of Be and Mg to the 1D (np2) state.

We also correct some misleading statements about the EDA method.

In a recent study by Gimferrer et al. [ 1], the authors report quantum chemical calculations of di-coordinated beryllium and magnesium compounds ML2 (M = Be, Mg) with various ligands L, which have been synthesized by different groups in recent years.

The focus of the work lies on the assignment of the oxidation state of the metal atoms in the compounds Be(cAACDip)2 and Mg(cAACDip)2 with the ligands cAAC (cyclic Alkyl Amino Carbene) that carry bulky Dip (2,6-diisopropylphenyl) substituents.

The beryllium complex Be(cAACDip)2 was isolated by Braunschweig and co-workers who also analyzed the bonding situation and suggested that it is the first example of a stable beryllium compound in the zero oxidation state Be(0) [2].

The magnesium homologue Mg(cAACDip)2 was postulated as an intermediate that quickly rearranges to a more stable species [3].


Using a variety of methods, Gimferrer et al. [ 1] question the assignment of the zero oxidation state and propose that Be(cAACDip)2 and Mg(cAACDip)2 are actually M(+2) (M = Be, Mg) compounds where the ligands bind as anionic diradicals with a doubly charged metal ion.

We have carefully reviewed the work, and found erroneous data and contradictions between the numerical results and the conclusions in the paper.

There is also a confusing mixture of statements about the electronic structure and bonding situation in low-valent compounds and the importance of the diradicaloid character in the molecules.

In addition, we encountered some misleading statements about the EDA method, which we want to clarify.

Finally, we noticed that the authors did not consider the possibility that the metal atoms could be in the formal oxidation state +1.

We will focus in our comment on the beryllium compound Be(cAACDip)2.

Fig. 1 shows the bonding scenarios for the compounds M(cAACDip)2 (M = Be, Mg) that were considered by Gimferrer et al. [ 1].

The authors say in the introduction that the model of dative interactions shown in Fig. 1 is unlikely, because the electronegativities of Be (1.57) and Mg (1.31) are much lower than for C (2.55).


This argument does not consider the electronic state of the metal atom, which may significantly change the effective electronegativity of the atom.

This was already reported a long time ago by Hinze [4], who showed that electronegativity strongly depends on the atomic orbitals.

He derived electronegativity values for the atomic valence orbitals responsible for atomic electronegativities, which may differ significantly from the average value that was used by the authors.

A striking example is the related boron complex (BH)(cAACDip)2, which was isolated and structurally characterized by Bertrand and co-workers in 2011 [5].

The bonding analysis clearly showed that it is a boron(I) complex that is best described by dative bonds (cAACDip)/(BH))(cAACDip) where the carbon atoms of the ligands are donors and boron is an acceptor.

The authors then state that the model of dative bonding shown in Fig. 1a would require excitation energies of 178.3 kcal mol−1 and 399.9 kcal mol−1 for the Be and Mg atoms to acquire the atomic valence state 1D with doubly occupied (n)p2 AOs, respectively, citing a study by Moore and Russell (their ref.

24) [6].

The values are wrong!

The cited work gives the binding energies in atomic valence orbitals of the atoms but not the double excitation energies.

A value of 399.9 kcal mol−1 for the excitation energy of the magnesium atom 1S(3s2) / 1D(3p2) would be much higher than the ionization energy of Mg (176.3 kcal mol−1) which is physically unreasonable.

We checked the literature and found that the experimental excitation energy to the 1D (2p2) state of Be is 162.5 kcal mol−1 [7] and that the excitation energy to the 1D (3p2) state of Mg is unknown, because it strongly interacts with the 3s and 1D series [8].


Since the experimentally known promotion energies to the excited states of the third-row atom Mg are lower than for the second-row atom Be [7], it can be concluded that the excitation energy to the 1D (3p2) state of Mg is <162.5 kcal mol−1.

More importantly, the alternative binding model proposed by Gimferrer et al. [ 1] requires a dramatically higher extraction energy of 634.9 kcal mol−1 for the double ionization of Be and 523.0 kcal mol−1 for Mg [7], which is only slightly reduced by the electron affinity of the cAACDip ligands (2 × 18.4 kcal mol−1) [9].

The data indicate that the bonding model of dative interactions shown in Fig. 1a needs a significantly lower promotion energy than the model of diradical binding displayed in Fig. 1b.

Gimferrer et al. [ 1] analyzed the actual interactions in Be(cAACDip)2 and Mg(cAACDip)2 with the EDA-NOCV (energy decomposition analysis with natural orbitals for chemical valence) [10,11] method.

Table 1 shows the numerical results, which were given in their ESI (Table S8) [1].

Crucial information about the most appropriate interacting fragments is provided by the orbital term, which indicates the energy change during the bond formation.

Fragments which give the lowest energy change for the orbital interactions are the most suitable species to describe the bonding interaction.

This has been shown to be very useful for elucidating the nature of the chemical bonds in a large variety of main-group molecules, transition metal complexes as well as lanthanide and actinide compounds [12].


The authors divided the orbital term without explanations in contributions as DEorb and DEorb-HF giving a total value DEorb-corr, which is irrelevant for the assignment of the most suitable fragments and which are misleading nominations, because the data come from DFT calculations.

The orbital interactions come from the Kohn–Sham orbitals and not from the Hartree–Fock orbitals.

Table 1 shows that the neutral fragments M + (cAAC)2 give a significantly lower orbital interaction than the doubly charged fragments M2+ + (cAAC)2−. The smaller energy contribution of DEorb is the reason why Braunschweig and co-workers assigned the oxidation state zero to Be in Be(cAACDip)2 [2].

Gimferrer et al. [ 1] suggested that one should use the density change rather than the energy change as a criterion for the choice of the oxidation state, saying that the electron flow to the empty pz AO of Be2+ is only 0.42e when doubly charged fragments are used, while 1.49e is transferred from the occupied pz AO of Be(0) when one uses neutral fragments.

But a careful inspection of the data gives a different result.

Table 1 shows that the largest orbital term for the interaction of the doubly charged fragments comes from DEorb-p, which amounts to −211.6 kcal mol−1 and contributes 47.7% to the total orbital interactions.

Fig. 2 shows the associated deformation density, which is provided as Fig. S41 in the ESI provided by the authors.

It clearly shows that the orbital interaction describes the formation of the C–Be–C p bond, which is depicted by the blue region of charge accumulation.

The calculated charge migration amounts to 1.8e, which is much higher than the charge migration between the neutral fragments (1.5e).


Gimferrer et al. [ 1] mentioned the charge migration of 1.8e stating that “75% of the electron flow is associated with the internal reorganization of the fragment density, which certainly has an important energy impact, but has no influence of the oxidation state”. But the deformation density clearly reveals that the electron flow of 1.8e is associated with the charge flow from the occupied ligand orbitals to the vacant C–Be–C p orbital but not with an internal reorganization of the fragment density!

The cited statement of Gimferrer et al. [ 1] is in conflict with their own results and contradicts their suggestion to use the density change and not the energy change as a criterion for choosing the oxidation state.

The deformation density in Fig. 2 clearly shows that DEorb-p is associated with the formation of C–Be–C p. It follows that even the size of the charge rearrangement suggests that the neutral fragments Be + (cAAC)2 are the best choice for describing the bonding in Be(cAACDip)2 and that the oxidation state of beryllium is Be(0) and not Be(+2).

The EDA-NOCV method is a very sophisticated approach for analyzing the nature of the chemical bond [13].

It has been the subject of critical comments by some workers [14], which has been clarified in a recent study [15].

The authors state that “EDA cannot distinguish an electron-sharing interaction from a spin-polarized one (diradicaloid)” citing a previous study by them [16].

The goal of EDA is to analyze the electronic structure in the form of energy components that can be interpreted in a physically meaningful manner.

This information is very useful to understand chemical bonds in terms of bonding models based on covalent (orbital) interactions, electrostatic (Coulomb) interactions and Pauli repulsion.


Subtle differences, such as between electron-sharing interactions and spin-polarized interactions, fall outside the scope of EDA, and it is improper to criticize the method for not providing information about a particular property that is not the target of EDA.

Clarifying notes about the EDA have recently been given in the literature [15].

EDA uses the electronic structure of the respective method for an energy analysis.

Information about the open-shell and closed-shell nature of the wavefunction, which may have a diradicaloid character due to a small HOMO–LUMO gap, comes from the electronic structure method which is used but not from EDA.

EDA is as good as the quantum theoretical method on which it is based.

But this issue is irrelevant to the assignment of the oxidation state, which is a model that is based on well-defined rules.

The same holds true for EDA-NOCV.

Models are not right or wrong but more or less useful.

It is odd when the authors question the reliability of the KS-DFT and the energy-based EDA criterion for assigning the valence state of the metal atoms in Be(cAACDip)2 and Mg(cAACDip)2, when they are using arguments that are in conflict with their own data.

The comments by Gimferrer et al. [ 1] could lead to the impression that the EDA-NOCV method using DFT calculations may not be used for molecules with multi-reference characters or open-shell singlet states.

This is not necessarily the case.


Such systems must be carefully tested for the reliability of the DFT approach, which may or may not be valid.

A pertinent example is the bonding situation in the calcium complex Ca–CO2, which was recently reported [17].

The complex has an open-shell singlet ground state, where the bonding interactions take place between a Ca+ (4s1) cation in the doublet ground state and a doublet ground state CO2− anion.

The EDA-NOCV calculations using standard DFT calculations provided deep insights into the nature of the chemical bonds.

Since the paper by Gimferrer et al. [ 1] focusses on the topic of oxidation state, it is useful to cite the revised definition given by the IUPAC which states “The OS of a bonded atom equals its charge after ionic approximation.

In the ionic approximation, the atom that contributes more to the bonding molecular orbital (MO) becomes negative.” [ 18] Inspection of the MOs of Be(cAACDip)2 shows that the largest coefficient of the cAACDip) Be/cAACDip p bonding MO is at beryllium (see Fig. S11, occupation 1.80, in the ESI), which suggests that Be(0) is the appropriate formal oxidation state of the metal atom.

It is important to realize that the formal oxidation state of an atom is not identical to the atomic charges and the real electronic state in a molecule.

This is sometimes confused.

A famous example is the debate in the 90s, when Snyder questioned the assignment of the oxidation state Cu(III) in [Cu(CF3)4]− [19].


A more recent example concerns the oxidation state in the coordinatively saturated metal carbonyls M(CO)8 of group 2 (M = Ca, Sr, Ba) which were recently synthesized and theoretically analyzed with the EDA-NOCV method [20a].

Following the Dewar–Chatt–Duncanson model of chemical bonding, all carbonyls are M(0) complexes, but the EDA-NOCV method showed that the best description for the metal–CO interactions comes from M+ + (CO)8− [20b–d].

Gimferrer et al. [ 1] report a large number of numerical results using a variety of methods which suggest that the electronic structures of some of the investigated complexes have a diradicaloid character due to a low singlet–triplet gap which are assigned as open-shell singlets.

They also report a global reliability index R(%) measuring to which extent the formal OS model matches the actual electron distribution, which varies between 59 and 100% (Table 1 in their paper).

But the reported bond dissociation energies (BDEs) given in Tables S5–S7 of the ESI of their work show very similar BDE values at different levels of theory, which question the value of the R(%) data.

The authors also report the results of the EOS (Effective Orbital State) [21] method, which is based on atomic fragments that are defined in different ways and can lead to different results.

For example, the EOS analysis using the Mulliken or Löwdin approaches gives Be(0) for Be(cAACMe) whereas the NAO and QTAIM procedures suggest Be(+2) [1].

Another concern is the EOS result that the magnesium atom in Mg(cAACMe) is Mg(0) while in Mg(cAACDip) it is Mg(+2).

A drastic alteration in the nature of the Mg-cAACR bond from Mg(0) to Mg(+2) by changing the rather remote substituent R is questionable.

Another puzzling feature is the complete omission of the possibility that the metal atoms have the oxidation state +1.

In the paper by Braunschweig and co-workers it was shown that the EDA-NOCV results of Be(cAACDip) using singly charged fragments give slightly higher values for the orbital term DEorb than using neutral fragments [2].


We repeated the calculations at the same level of theory as the authors but with the inclusion of dispersion interactions, which had not been considered.

We found that the DEorb values using singly charged and neutral fragments become nearly equal, which means that Be(cAACDip) may be described with the oxidation states Be(0) or Be(+1) but clearly not with Be(+2).

In summary, we note that the assignment of the oxidation state +2 for beryllium and magnesium in the complexes Be(cAACDip)2 and Mg(cAACDip)2 as suggested by Gimferrer et al. is not valid and that the conclusion is not supported by their own data.

A critical inspection of their data corroborates the earlier assignment of the oxidation state zero.

The authors reported wrong data for the excitation energies of Be and Mg to the 1D (np2) state, which are the electronic reference state of the metal atoms for the dative interactions with the cAACDip ligands.


A recent article by Pan and Frenking challenges our assignment of the oxidation state of low valent group 2 compounds.

With this reply, we show that our assignment of Be(+2) and Mg(+2) oxidation states in Be(cAACDip)2 and Mg(cAACDip)2 is fully consistent with our data.

Some of the arguments exposed by Pan and Frenking were based on visual inspection of our figures, rather than a thorough numerical analysis.

We discuss with numerical proof that some of the statements made by the authors concerning our reported data are erroneous.

In addition, we provide further evidence that the criterion of the lowest orbital interaction energy in the energy decomposition analysis (EDA) method is unsuitable as a general tool to assess the valence state of the fragments.

Other indicators based on natural orbitals for chemical valence (NOCV) deliver a more reliable bonding picture.

We also emphasize the importance of using stable wavefunctions for any kind of analysis, including EDA.


Pan and Frenking [1] have challenged our recent assignment of the formal oxidation state of the beryllium and magnesium family of di-coordinated compounds ML2 (M = Be, Mg) with ligands of cyclic (alkyl) (amino) carbene (cAAC) and N-heterocyclic carbene (NHC) types [2].

In a landmark study, the group of Braunschweig isolated and characterized the Be(cAACDip)2 complex [3].

The authors described the system as the first example of a stable beryllium compound in the zero oxidation state Be(0), where the bonding situation consists of two donor–acceptor interactions between cAAC ligands and the central atom cAAC(0) $ Be(0) % cAAC(0).

The preparation of the heavier analogue Mg(cAACDip)2 has been attempted by Turner, but under the reaction conditions a ligand activation is observed, giving evidence for a highly reactive transient species [4].

In our original work [2], we provided an alternative picture where the strong p-acidity of the ligand causes its own reduction giving rise to the singlet diradicaloid L(−1) / E(+2) ) L(−1) situation [2].

The non-innocent behavior of cAAC ligands in main group and transition metal complexes has been discussed in a number of computational and experimental studies [5-9].


One of the most relevant findings of our work [2] is that the closed-shell single determinant wavefunction is either not stable (B3LYP) or higher in energy than a broken-symmetry KS-DFT solution.

This indication of significant static correlation prompted us to rely on multireference wavefunctions (CASSCF in particular) to study the electronic structure of these species.

Besides the usual electronic structure indicators, we applied the effective oxidation state (EOS) analysis to assign the oxidation state of the central atom and that of the ligands.

This approach, introduced in 2015 by some of us [10], has already been tested on a wide variety of challenging systems, exposing some limitations of algorithms preferred by IUPAC [11].

In contrast to energy decomposition analysis (EDA) [12-14], EOS was specifically devised solely for the purpose of assigning oxidation states.

It should be noted that other approaches specifically designed for OS assignment such as Head-Gordon's Localized Orbital Bonding Analysis (LOBA) [15] or the recently introduced Oxidation State Localized Orbitals (OSLO) [16] can only be applied for single-determinant wavefunctions.

EOS is the only scheme of that kind that can be applied on equal footing for single-determinant and multiconfigurational wavefunctions, and hence it was the natural choice for our study.


Remarkably, only a small part of the criticism of Pan and Frenking actually refers to our preferred methodological framework, with arguments that are rather bizarre or simply faulty.

For instance, they state that the EOS scheme “[.] is based on atomic fragments that are defined in different ways and can lead to different results” [1].

While this is a pretty accurate description of the EDA approach, it does not apply to EOS analysis.

In EOS, one needs to define the composing fragments, in this case the central atom and each ligand, but not their reference state.

As such, the EOS approach is free of bias (vide infra) [6].

Then, the occupation of the fragment's domain natural orbitals (i.e. Mayer's effective fragment orbitals, henceforth EFOs) is used to assign electron pairs (or individual electrons in open-shell systems) to the fragments.

Rather than rounding the occupation numbers to the closest integer, the EFOs of all fragments are sorted in descending order.

The most plausible OS assignment (for all fragments) is the one in which the electron pairs are assigned to the highest occupied EFOs.

A reliability index R (%) quantifies the extent to which the discrete OS values match the actual electronic structure.

It is simply derived from the occupation numbers of the frontier EFOs in the EOS procedure [10].

In fact, it is not the way the fragments are defined, but rather the identification of the atom in the molecule (AIM) that influences the numerical results to some extent.

In Tables S9 and S10 of our original work, we already provided numerical evidence of the effect of using one or another AIM definition to obtain the EFOs [2].


Pan and Frenking pinpoint the case of Be(cAACMe)2, for which using classical Löwdin analysis results in a Be(0) OS (with a very low R (%) value, a close-call situation, in contrast with a clear Be(+2) OS when using more reliable AIM approaches like the Quantum Theory of Atoms in Molecules (QTAIM) [18] or the Natural Atomic Orbital (NAO) basis [19].

Pan and Frenking failed to notice that Löwdin's partial atomic charge on Be is −0.47 e, in sharp contrast with the other AIM approaches tested (well over +1.0 e).

They also missed the case of Mg(cAACMe)2.

Here, the Löwdin partial charge on Mg is as low as −0.75 e, while it is ca. +0.4 e when using NAO or QTAIM.

Remarkably, in all three cases the EOS analysis still yields the same undisputable OS assignment of Mg(0).

A less biased perception of Tables S9 and S10 indicates that (i) EOS analysis is significantly more robust than partial atomic charges and (ii) partial atomic charges neither correspond to nor correlate with oxidation states, as repeatedly observed, and in line with IUPAC's definition of OS and the winner-takes-all principle [20].


At this point it should become clear that we did not “omit the possibility that the metal atoms has the oxidation state +1”: such a faddy result (which necessarily implies a formal OS of −1 for each of the ligands) simply cannot stem from the wavefunction analysis.

Artificially forcing the EOS scheme to produce a particular set of OS on the fragments, by definition, results in a lower R (%) value (in fact, even below the worst-case scenario of R (%) = 50).

Pan and Frenking also cast doubts on the fact that for Mg(cAACMe)2 we obtain Mg(0) while for Mg(cAACDip)2 it is Mg(+2).

Accordingly, “a drastic alteration in the nature of the Mg-cAACR bond from Mg(0) to Mg(2+) by changing the rather remote substituent R is questionable”. They regrettably failed to notice in Table 1 and Fig. 1 of our manuscript that the C–Mg–C bond angle goes from 107.8° to almost collinear (179°) by the steric effect of the remote substituent.

Finally, we must also refer to Pan and Frenking most unjustified criticism: “[.] the reported bond dissociation energies (BDEs) given in Tables S5–S7 of the ESI of their work show very similar BDE values at different levels of theory, which question the value of the R (%) data”.

In fact, the R (%) values are obtained from the EOS analysis of the multireference CASSCF wavefunctions.

There is no relationship between the BDEs and the R (%).


Pan and Frenking also confound some statements and data we provide in the introductory part of our original manuscript, which merely served as hints to illustrate that the original Be(0) assignment might not be appropriate.

This background information was not implying that Be(0) assignment must be incorrect, but was rather highlighting the uncertainty in previous assessment that prompted the more comprehensive analysis we performed in our work.

From their comments it appears that we actually share the view on most of the fundamental aspects of the oxidation state concept, but again, they distorted our statements.

First of all, we agree with Pan and Frenking that merely considering the atomic electronegativities of the contact atoms to decide the fate of the electrons of a bond is troublesome.

Some of us have already discussed this issue in detail [10], and that is precisely why we advocate for alternative approaches to simplistic IUPAC algorithms, such as EOS.

Now, we did not question the existence of dative bonds between the ligands and Be/Mg on the basis of their respective electronegativities as Pan and Frenking insinuated.

This is clearly not the point.

The point is, in the context of IUPAC's ionic approximation (IA), any bond (dative or covalent) will have the corresponding electron pair assigned to either Be/Mg or C, and in the most straightforward application of IA, it should be assigned to the most electronegative atom [20], which is the carbon atom.

This is what stems from EOS analysis in the cases in question such as Be(cAACDip)2, as illustrated in Fig. S37 of our original manuscript.

The occupation of the s-type EFO on the ligands (0.85) is much higher than that of the s-type EFO on Be (0.18), which results in the electron pair of the s M–L bond unambiguously assigned to the ligands.

What is really relevant from the OS point of view is the assignment of the p electrons of the system.

In the Be(0) picture, the electrons of the 3c–2e p-bond are assigned to the less electronegative atom, at odds with IUPAC's simple rules [20].

Since the latter are not foolproof we conducted EOS analysis, which in this case is crystal clear: as shown in Fig. S37 of ref.

[ 2], the occupation of the p-type EFO on Be (0.12) is much smaller that of the p-type EFO of the ligands (0.44), which keep the electrons, resulting in a Be(+2) picture.

In our original manuscript, we also explicitly stated that Ponec and Cooper had also questioned the fate of the p-electrons using domain-averaged Fermi-hole (DAFH) analysis for a Be(cAACH)2 model system [21].

According to Ponec and Cooper, the electron pair formally originally occupying the Be(2pp) orbital contributes symmetrically to such an extent into the originally empty p orbitals on the carbene-like moieties that there is only a very small residual population on Be.

In fact, the DAFH analysis reveals a contribution to the p bonding of 0.14 from Be and 0.95 from each of the ligands (see Fig. 1 of ref.

[ 21]).

This is in full agreement with our results and with a Be(+2) formal valence state.


Pan and Frenking also appear to estimate the contribution of an AO to a given MO just by looking at the orbital plot.

They focus on a p-type natural orbital with a natural occupation of 1.80.

They claim without any proof that the largest coefficient of that orbital is at the Be atom.

Then, according to IUPAC's IA, the electron pair should be assigned to Be.

They quote a statement from ref.

[ 20] that in our view represents a naïve and ambiguous point on the relationship between AO coefficients and OS.

This issue has already been discussed elsewhere by some of us [22], so we will only provide a quick answer to Pan and Frenking.

First of all, the orbital they refer is a natural orbital corresponding to our CASSCF description of Be(NHCDip)2, not Be(cAACDip)2.

For the latter, the relevant orbital is given in Fig. S15 of our original manuscript [2], with a natural occupation of 1.64.

It is not so obvious which is the contribution of a given fragment holding a set of non-orthogonal AOs to a molecular or natural orbital.

One may consider an orthogonal basis instead, such as a Löwdin's variant implemented in pySCF [23].

The orbital coefficients of the 2pz and 3pz AOs of Be are 0.399 and −0.096 for Be(NHCDip)2, and 0.291 and −0.095 for Be(cAACDip)2.

For comparison, just the coefficient of the 2pz AOs of each of the two contact carbon atoms of the ligands is 2 × 0.495 and 0.541 and 0.512 (non-symmetric orbital), respectively.

The population of these natural orbitals on Be (i.e. sum of the square of the coefficients on an orthogonal basis) is merely 0.18 and 0.11, respectively, not too different from the DAFH and EFO occupations discussed above.

Looks can be deceiving.


We do concede to Pan and Frenking that the excitation energies of Mg that we discussed in the Introduction of ref.

[ 2] are plainly wrong.

That was a regrettable mistake and we are grateful to them for pointing it out, despite being irrelevant to the discussion.

Pan and Frenking used the ionization potential and electron affinity of the fragments.

They took the electron affinity of cAACDip from a PBE0/def2-TZVP calculation without Zero Point Vibration Energy (ZPVE) correction [24], which does not meet the definition of molecular electron affinity [25] (it is not 18.4 kcal mol−1 but 14.5 kcal mol−1).

Surely, the ionization energies required to achieve a real (not formal!) Be2+ species are much higher than those needed to reorganize the valence electron in a Be(0) picture, but this does not preclude the possibility of a formal Be(+2) OS.

Otherwise, how to justify the O(−2) formal OS in H2O, where the formation of O2− (which is unstable, by the way) from O(0) is endothermic by +167.8 kcal mol−1 (refs.

[ 26] and [27]) and the ionization potential of H is +313.3 kcal mol−1 [28]?

Surely, Pan and Frenking are not challenging the formal OS assignments in the water molecule.


Another set of criticism from Pan and Frenking points towards our illustrative EDA calculations and EDA-NOCV analysis of these Be and Mg complexes, reported in the ESI of our original work [2].

Pan and Frenking stated: “we repeated the calculations at the same level of theory as the authors but with inclusion of dispersion interactions, which had not been considered.

We found that the DEorb values using singly charged and neutral fragments become nearly equal, which means that Be(cAACDip) may be described with the oxidation states Be(0) or Be(+1) but clearly not with Be(+2).” Unfortunately, these calculations were neither reported nor used in their comment, and instead they reused our original Table S8 as Table 1 in their comment, which in fact always included dispersion corrections.

First of all, assuming they did not reoptimize the geometry, the inclusion of dispersion will only provide a new stabilizing term DEdisp, with exactly the same DEorb values.

Secondly, they complain about our notation, where the total orbital interaction term DEorb-corr is presented as a sum of DEorb and DEorb-HF contributions, and they erroneously assume that we used them for our interpretation.

Taking into account that the analysis is performed at the B3LYP-D3(BJ)/TZ2P level of theory, the latter corresponds to the contribution from 20% of exact-exchange of the density functional approximation.

The reason the two terms are given explicitly is connected to the fact that the current implementation of EDA-NOCV does not decompose the non-local HF-like exchange contribution to DEorb [29].

Such information has been described in dedicated technical literature and also could be easily checked in the outputs of their own calculations, or even in former tables reported by Pan and Frenking [30,31].


Finally, and more importantly, Pan and Frenking claim that our analysis of the electron flow of the NOCVs is wrong just by how the deformation density looks like in our Fig. S41.

They ignore the actual contributions of the symmetry-adapted fragment orbital (SFO) basis to the deformation density, which are also provided in our Fig. 1.

Let us repeat here our point concerning the p channel from the NOCV analysis.

For the sake of simplicity, we will consider the total a and b contributions together.

For each channel k, the deformation density is written as Drk(r) = |nk| Σm ck m ck* v c* m(r)cv(r) - Σm ck* m ck v c* m(r)cv(r), where nk is the eigenvalue (in absolute value) and the pair of NOCVs are expressed in the (orthogonal) SFO basis {c} by the coefficients ck.

Integrating the deformation density leads to 0 = ∫ Drk(r)dr = Σm |ck m|^2 - Σm |ck* m|^2, which readily affords a decomposition in terms of the SFO basis.

The most significant contributions of the SFOs can be found in the output of a typical NOCV run.

When using a Be0 reference, the corresponding eigenvalue is ca.

|1.5|, meaning that an electron flow of 1.5 e is associated with this particular channel of the deformation density.

In this case, it can be seen that the 2pz-type SFO on Be exhibits a very large and negative contribution of −1.49 e, meaning that electrons move from this orbital to those with positive contribution (ligands, in this case).

This SFO contribution essentially fully accounts for the whole charge transfer of this channel.

When using a Be2+ reference, the eigenvalue of the deformation density is |1.8|, but the contribution of the 2pz-type SFO on Be is merely +0.42 e. The positive contributions from other SFOs of the Be moiety are negligible, so ca. necessarily +1.4 e move into the ligand (and from the ligands, as indicated by the ligand-centered larger negative SFO contributions).

Regrettably, Pan and Frenking strongly relied on our selected isocontour value (our Fig. S41 of the original manuscript and Fig. 2 in their comment) although the orbital interaction shows a Be contribution only at low isocontour values, while for bigger isocontour values it shows only the participation of the carbene carbon of the cAAC ligands (Fig. 2).

This intrafragment charge-transfer process can be even more clearly seen with the NOCV orbitals, where the major contribution comes from the p-orbital of the cAAC ligands (Fig. 1).

A similar analysis can now be carried out not in the orbital space but in the real space, using a Hirshfeld-type partitioning for the entire fragments (A) [32], ∫ wA(r)Drk(r)dr = ΣA Dqk A, which affords the same picture albeit with a much smaller charge transfer (−0.78 e on Be for the Be0 reference and +0.04 e on Be for the Be2+ reference) (Fig. 1).

Hence, the actual numbers, and not a mere picture, evidence that there is much less electron flow between fragments using the Be2+ reference, thus invalidating the Pan and Frenking claim.

For completeness, we have performed the EDA-NOCV analysis with the correct wavefunction also for the rather odd Be1+ fragmentation (see Table 1).

The NOCVs of the p-bonding channel using a Be1+ fragmentation indicate an intermediate situation between using Be2+ and Be0, as might be anticipated.

The eigenvalue of the deformation density is |1.54|, while the contribution of the 2pz-type SFO on Be is −0.51 e, slightly larger than the value using Be2+ discussed above (+0.42 e).

In the Be1+ case, however, it originates from a partial cancelation of a (−0.75 e) and b (+0.24 e) contributions, thus indicating a much significant charge-reorganization than when using Be2+ as reference.

Yet, from an energy perspective, Be0 and Be1+ show lower DEorb values than the Be2+ fragmentation scheme.

Does this necessarily mean that the oxidation state of Be cannot be Be(+2), while all other evidence clearly points toward this situation? Has the criterion of the lowest orbital interaction term been carefully tested in the past?

In this sense, some of us considered the simplest case of H2O when evaluating an OS assignment method based on the position of the centroids of localized orbitals with respect to the atoms [33].

We found that the method failed to describe H2O as formally H(+1) and O(−2).

As will become clear later on, this kind of proof of concept calculations were regrettably skipped when putting forward EDA's orbital interaction energy as a general criterion to establish appropriate valence states and OS (vide infra).


After having addressed Pan and Frenking's inaccurate statements, let us go back to square one.

As mentioned above, the closed-shell KS-DFT wavefunction of many of these systems are either not stable or a lower in energy broken-symmetry solution is found.

In the case of Be(cAACDip)2, the corresponding ⟨S2⟩ value is 0.57, which cannot be ignored at all.

Indeed, this is supported by the computed D1 diagnostic [34], which exceeds in all the cases the reference value.

Consistently, the weight of the closed-shell configuration is for all the complexes below 0.85.

Local spin analysis [35] thus reveals a significant diradicaloid character, with local spins centered at both cAACDip ligands.

That is why we claim that these systems have diradicaloid character, not because they exhibit a small singlet–triplet gap, as Pan and Frenking stated.

The CASSCF description also confirms this picture, both from the significant occupation of the antibonding natural orbital of the Be(cAACDip)2 p bond (0.36) and from the local spin analysis values reported in Table 1 of our original paper [2].


This situation is quite similar to that of the largely discussed bonding in the [NaBH3]− cluster [36].

In our earlier work on that subject [36], we already showed that “EDA cannot distinguish an electron-sharing interaction from a spin-polarized one (diradicaloid)”, at least by only looking at the energy components.

Pan and Frenking do not confront the statement and clearly underestimate the implications of this fact, by stating that these are subtle differences and fall outside the scope of EDA.

We could not agree more with their own statement: “the EDA is as good as the quantum theoretical method on which it is based” [1].

We in fact extend it to any wavefunction analysis method.

That is precisely why one should make sure that an appropriate wavefunction is considered.

At least one can check whether it has internal instabilities or not. But how can this not be relevant for OS assignment from wavefunction analysis? If a given bond between A and B is “electron-sharing”, the electron pair will be allocated to either A or B depending on the direction of the ionic approximation.

If there is enough spin polarization of the bond, a more reasonable result might be the homolytic distribution of electrons [37], since in the extreme case (a pure diradical) there is no “sharing” of electrons.

The structure of the single-determinant closed-shell wavefunction and that of a broken-symmetry one can be very different, and this can even be taken into consideration for the EDA analysis (Fig. 3).

In contrast, they justify their systematic misuse of EDA with “[.] it is odd to criticize the method for not providing information about a particular property that is not the target of EDA.” Should we stop caring about using the correct wavefunction?

Notably, Pan and Frenking brought the case of the Ca–CO2 complex in their comment [39], as an example where EDA-NOCV works for a system with open-shell singlet ground state character.

Indeed, this is not their best example since one can clearly notice in their original paper that they have in fact used once more a closed-shell unstable wavefunction in their EDA-NOCV analysis, ignoring their own described diradical character [39].

We have carried out the correct calculations, and we also present herein the obtained results.


Table 2 gathers the outcome of the EDA for the closed-shell (unstable) and open-shell wavefunctions for the situation where the fragments are a Ca+(4s1) cation in the doublet ground state and a doublet CO2− anion.

The results for both calculations are very similar: the only change in the energy components is in the DEorb, (i.e. 3.2 kcal mol−1 in the DEint), as illustrated in Fig. 3.

This is because using closed-shell (unstable) or open-shell (stable) wavefunctions only affects the last step of the EDA process.

Interestingly, the associated NOCVs exhibit significant differences in the charge transfer, as shown in Fig. 4.

In the wrongly computed example by Pan and Frenking, the first deformation density channel has an eigenvalue of 1.26, with main contributions from Ca+(4s1) to the CO2− SOMO (Fig. 4 CSS wavefunction).

Note that their original article does not include this information [39].

Applying EDA to the correct OSS wavefunction leads to a decrease of the eigenvalue to only 0.60, indicating a significant change in the electron density.

Indeed, applying EOS to the closed-shell wavefunction indicates a Ca(+2) with R (%) = 94.4, while with the OSS wavefunction, EOS unambiguously indicates now an oxidation state Ca(+1) with R (%) = 93.6, which is the assignation Frenking reported by analyzing the wrong wavefunction.


It is evident that Pan and Frenking are only stacked in the criterion of the lowest orbital interaction term, which looks more like a “credo” than a carefully tested indication of the best electronic reference situation.

It is a good exercise to track back where their recited creed “those fragments which give the lowest energy change for the orbital interactions are the most suitable species to describe the bonding interaction” comes from.

According to them, this is a proven fact; however, their citations only go to articles with the same phrase and no proof.

To the best of our knowledge, the first mention of this was given in 2007 [40,41], but there is no sign of a thorough systematic study of this matter, only a belief [42-46].

One should not forget the memorable Feynman's quote: one must not verify an idea using the same data that suggested the idea in the first place.

It is interesting to point out that some authors have already questioned its validity [6,47].

Also, as repeatedly stated by the original developers of the EDA, the orbital interaction term includes both charge-transfer and polarization contributions, also within the fragments as shown here, which can certainly blur the analysis [48-51].

Some of us have been diagnosing the problems associated with such an energetic criterion when it comes to OS assignment in general (notice that it is never considered in any of the IUPAC technical reports), and found that EDA-NOCV can still be a valuable tool if the interfragment electron flow is considered instead.

It is out of the scope of the present response to discuss the details, which will be given elsewhere soon [52], but we can already disclose a quite surprising result.

There is no need to use the exotic Be(cAACDip)2 system to observe that the orbital interaction is not a good indicator of the best electronic reference.

As shown on Table 3, BeH2 would be in fact best described as Be(0) and H(0) by the lowest orbital interaction energy criterion.

Are Pan and Frenking also challenging the oxidation state of BeH2?


Conclusions Our assignment of Be(+2) and Mg(+2) oxidation states in Be(cAACDip)2 and Mg(cAACDip)2 is fully consistent with our data.

Although Pan and Frenking have claimed a critical inspection of our data, their arguments are based on superficial visual inspection of our figures rather than a thorough numerical analysis.

Additionally, we showcase their recent flawed application of EDA to diradicaloid species Ca–CO2.

We provide evidence that the criterion of the lowest orbital interaction energy using EDA to determine the valence state of fragments is intrinsically unreliable, since it already fails for a simple system such as BeH2.

As the oxidation state is a density based property, NOCV analysis can be a more reliable indicator of the best reference state.

EDA-NOCV is without doubt a very useful method for bonding analysis, yet it should be used with appropriate care.


Schizophrenia is a chronic mental disorder characterized by acute phases of positive symptoms (e.g. hallucinations, delusions, and behavioral disturbances) as well as negative symptoms (e.g. lack of emotion, reduced interest in day- to-day activities, and social withdrawal).

These symptoms are often present even in stable phases, although they often worsen during acute exacerbations or relapses.

Repeated relapses may cause the functioning of patients to deteriorate further [3].

Relapse prevention is therefore a fundamental element in the management of schizophrenia [4].


Maintenance treatment with regular long-term use of antipsychotic medication [4, 5] and continuity of care is essential in the prevention of relapses [6-8].

Poor adher- ence and its associated fluctuations in blood plasma levels of treatment medications is a critical factor in the poor out- comes experienced by some patients [9, 10] and has been


The application of pharmacokinetic and pharmacody- namic evidence provided a flexible modeling framework as well as robust results, as shown by the successful validation.

The framework has potential for further appli- cation in other therapies and disease areas with similar data restrictions, which is relevant for model-informed drug development programs.


The findings may help physicians and US payers make informed decisions considering both the clinical and the economic consequences of the different available long- acting injectable dose regimens in patient care.

from different aripiprazole LAI dose regimens, (2) a phar- macodynamic model linking these blood plasma levels with relapse rates over time, and (3) a pharmacoeconomic model outlining the health and economic outcomes of the LAI dose regimens, based on the modeled relapse rates.

This paper presents an application of a pharmacokinetic-pharmacody- namic-pharmacoeconomic (PK-PD-PE) [17], or pharma- cometric-pharmacoeconomic (PMPE) [16], model for the pharmacoeconomic assessment of different aripiprazole LAI dose regimens for the treatment of schizophrenia to inform clinical and payer decisions in the USA in a postmarketing setting.


The modeled population consisted of adults with schizo- phrenia, in accordance with the indications of AM and AL [12, 13].

A patient cohort was simulated by bootstrapping the pivotal trial data of AM [18].

The cohort had a mean age of 39 years, a mean height of 170 cm, and mean weight of 81 kg.

In total, 36% of the cohort was female, and 5% had a poor cytochrome P450-2D6 metabolizer status.


The analysis compared eight LAI dose regimens: two AM dose regimens (400 or 300 mg every 4 weeks [q4wk]) and six AL dose regimens (441 or 662 or 882 mg q4wk, 882 or 1064 mg every 6 weeks [q6wk], 1064 mg every 8 weeks [q8wk]).

Please note that according to expert opinion, in clinical practice, AL 441 mg and AM 300 mg are generally used only when patients do not tolerate higher doses [6].

The model assumed that AM and AL were administered as single intramuscular injections according to the package insert [12, 13].

For the first 14 days of AM treatment and for the first 21 days of AL treatment, oral aripiprazole monohy- drate 15 mg daily was administered concomitantly [12, 13].

The model assumed complete adherence to medication.

The analysis assumed treatment after discontinuation of LAI was standard of care (SoC), consisting of the oral medications olanzapine, risperidone, quetiapine, and ziprasidone.


The analysis took a US healthcare payer perspective and considered only direct healthcare costs (cost year of 2021).

The time horizon was 1 year starting at LAI initiation, a scenario typically relevant for US payers.

A scenario analy- sis evaluated a 2-year time horizon.

In line with guidelines, costs were discounted by 3% per year in this scenario [19].

linked with symptom exacerbation, disease relapse, and increased need for hospitalization [11].


Long-acting injectable (LAI) antipsychotics avoid the need to take oral medication every day and may, as such, be more effective than oral antipsychotics in improving medica- tion adherence and persistence [6, 10].

Aripiprazole mono- hydrate (AM) [12] and aripiprazole lauroxil (AL), a prodrug that is converted to aripiprazole [13], are LAI formulations of aripiprazole with different pharmacokinetic profiles.

Dif- ferent dose regimens of AM and AL are currently used for the maintenance treatment of adults with schizophrenia [12, 13], and clinical data have shown that these result in vary- ing steady-state average aripiprazole plasma concentrations (C avg ) [14].


The aim of this study was to compare the costs and effects, specifically relapses, of the available aripiprazole LAI dose regimens and, in particular, to assess the conse- quences of different frequencies of administration to inform clinical and US payer decisions.

Pharmacoeconomic mod- els comparing the costs and effects of different treatment options typically rely on comparative long-term outcome data from phase III randomized controlled trials (RCTs).

This type of evidence is not, and likely will not be, avail- able for all the AL doses in the current comparison, as two dose regimens (662 and 1064 mg) were granted US FDA approval based on combining phase I pharmacokinetic data and simulations.

The simulated steady-state exposures of these doses were compared with those dose regimens with phase III outcome data supporting their approval using a so-called "bridging approach" [15].


In the absence of RCT data, pharmacometric models can be used to simulate clinical inputs for the pharmaco- economic analysis [16, 17].

This analysis consists of three sequential components: (1) a pharmacokinetic model charac- terizing the blood plasma concentrations over time resulting 123 Integrated Pharmacokinetic-Pharmacodynamic-Pharmacoeconomic Modeling of Treatment for Schizophrenia


A targeted literature review was conducted to identify pub- lished PK-PD-PE models of adults with schizophrenia to inform model structure and model inputs.

Figure 1 provides an overview of the model, consisting of the pharmacoki- netic, pharmacodynamic, and pharmacoeconomic com- ponents, and shows how these are linked together.


First, the pharmacokinetic component of the model was used to simulate the aripiprazole plasma concentration over time and the minimum concentration per dosing interval (C min ) for each LAI dose regimen.

Second, the pharmacodynamic component was used to derive the probability of relapse conditional on the simulated aripiprazole C min .

This served as effectiveness data in the pharmacoeconomic model.


The pharmacoeconomic model itself was a Markov patient-level simulation with five health states representing remission on LAI, relapse on LAI, remission on SoC, relapse on SoC, and death.

Patients entered the model in the health state "remission on LAI," where they were treated with an LAI dose regimen.

Patients experiencing a relapse moved to the health state "relapse on LAI.

" Patients who discontinued LAI moved to "remission on SoC" or "relapse on SoC" if they also experienced a relapse.

Patients who recovered from their relapse moved to the "remission" health state.


From all health states, patients could move to the absorbing health state "death." Adverse events were not modeled because evi- dence regarding adverse events at different C min was unavail- able and evidence also suggested that the safety profiles of AM and AL were similar [20, 21].

The model had a cycle length of 2 weeks, which was the highest common denomi- nator of the 4-, 6-, and 8-week regimens of the evaluated LAIs, was built in R version 4.0.2 [1], and made use of the RxODE package [2].


The following (interim) outcomes were generated.


• In the pharmacokinetic model:


o the minimum aripiprazole plasma concentration per dosing interval, i.e. C min


• In the pharmacodynamic model:


o the probability of relapse per patient over time based on C min over time, and o the average number of relapses per treatment regi- men within the time horizon.


• In the pharmacoeconomic model:


Two pharmacokinetic models, one for each LAI, were selected based on methodological robustness and similarity in model structures [18, 22].

Both pharmacokinetic models were published by the respective manufacturers and based on clinical trials.


The pharmacokinetic model for AM was a three-compartment model with one central and two periph- eral compartments [18].

The pharmacokinetic model for AL was a two-compartment model with one central and one peripheral compartment [22].


In both models, the absorp- tion of aripiprazole from the oral depot during the initiation phase was described by a first-order process [18, 22].

In the AM pharmacokinetic model, the absorption of aripipra- zole from the intramuscular depot was modeled by a first- order process to reflect the bolus injection [18].

In the AL pharmacokinetic model, the enzymatic conversion of AL to aripiprazole was described by a zero-order process with lag time, and the absorption of aripiprazole was modeled by a first-order process [22].


Details of the equations used can be found in electronic supplementary material (ESM)- 1.

Both models were built in NONMEM software and were replicated in R for seamless integration with the pharmaco- dynamic and pharmacoeconomic elements.


The pharmacokinetic parameters were dependent on a set of covariates that were randomly bootstrapped for each simulated patient and subject to uncertainty.

The C min of each simulated patient during each dosing interval follow- ing different LAI regimens was simulated based on the patients' baseline characteristics and the administered LAI dose regimen.


Based on the estimated Cmin values from the aforemen- tioned pharmacokinetic models, a pharmacodynamic model characterizing the relationship between aripiprazole C min and relapse was used to derive the probability of relapse for each simulated patient during each dosing interval.

The pharma- codynamic model was developed using SAS software [23] by the sponsor of this study using data from 315 patients receiving either placebo or 300/400 mg AM.

It modeled relapse probability as a function of aripiprazole C min using a survival model with an exponential hazard function [24].

The proportional hazard assumption did not hold for a con- tinuous hazard function.

A dichotomous hazard function with a cut-off value of C min = 95 ng/mL was used in line with previous analyses [14].

Different models were fitted, and the exponential hazard function was chosen based on goodness-of-fit statistics.

As an alternative scenario, a con- tinuous hazard rate as a function of C min was fitted.

The haz- ard rates generated were transformed into a 14-day relapse probability to match with the model's cycle length.

The probability of transition from remission to relapse with LAI treatment could therefore be calculated conditional on the estimated C min value of each simulated patient.


The pharmacoeconomic model calculated the costs of treat- ment and relapse associated with each LAI dose regimen.

Table 1 shows an overview of the transition probabilities, including the C min -dependent relapse probability for LAI estimated in the pharmacodynamic model.

The transition probability from remission to relapse with SoC treatment was informed by the weighted average of probabilities of olanzapine, risperidone, quetiapine and ziprasidone [25].

The probability of transitioning from relapse to remission was derived from Medicaid data indicating a duration of first relapse of 4 weeks and was equal for all LAIs and SoC [26].


The discontinuation rate was informed by a medication dis- continuation study using Truven MarketScan administrative claims data, which reported an annual all-cause discontinu- ation probability of 75.2% for patients with schizophre- nia treated with AM [27].

The rate of 5.2% per cycle was assumed to also apply to patients treated with AL.


Mortality among people with schizophrenia is known to be higher than in the general population [28].

The age- and sex-dependent background mortality [29] was therefore adjusted with a standardized schizophrenia mortality ratio of 3.7 [30].

The mortality risk was assumed equal in all alive health states.


Wholesale average drug acquisition costs were sourced from the IBM Micromedex RED BOOK, and an overview of the costs is presented in Table 2 [31].

SoC treatment was assumed to consist of equal proportions of oral olanzapine, risperidone, quetiapine, and ziprasidone, in line with previ- ous analyses [25].

Additional costs for the IM administration of AM and AL of $US14.31 per injection applied [32].


Disease management costs of patients in remission were included as a monthly monitoring visit relating to routine psychiatric and nonpsychiatric care, at a price of $US103.93 per visit [25].

The same source informed the costs associated with a relapse, reporting that 77.3% of patients experiencing relapse required hospitalization (Table 3) [25].


To inform the patient-level simulation, the population was bootstrap-simulated until convergence of C min was reached (N = 2000), and distributions of C min and C avg in steady state were generated as well as a pharmacokinetic profile over time for each LAI dose regimen.

The number of relapses and the costs of LAIs, relapses, and SoC were presented per dose regimen as well as incremental results comparing dose regimens and the incremental cost per relapse avoided, using the dose regimen with the lowest mean number of relapses as the reference treatment.