The proposal by Klier and colleagues (2002) was at odds with prevailing views of cervical dystonia, which focused on defects in the basal ganglia (Dauer et al., 1998) or some- times the cerebellum (Neychev et al., 2011) or propriocep- tive defects (Tempel and Perlmutter, 1990; Bove et al., 2004).

As a result, their idea was neglected until more recent studies resurrected the concept by characterizing the abnormalities of head movement in subjects with cer- vical dystonia, in relation to the neural integrator model (Shaikh et al., 2013).

Figure 3A-L depicts head positions for a subject with cervical dystonia, where the favoured head position (null) was 20 to the left of the centre position on the trunk (Fig. 3G-I).

Drifts of head position were minimal when the subject's head was voluntarily held at the null.

Coarse and jerky head movements were seen as the head turned 20 to the right (40 from the null) (Fig. 3A-C), or straight- ahead (20 from the null, Fig. 3D-F).


Typical of cervical dystonia, the jerky movements were present in all three rotational planes including the horizontal plane (torticollis), vertical plane (antero/retrocollis), and torsional plane (later- ocollis).

In such instances, the slow drift in head position was directed towards the null.

The rapid movements to- wards the desired head position corrected for the drifts toward the null.

Furthermore, in all three planes the drift velocity systematically changed with head position.

Downloaded from https://academic.oup.com/brain/article/139/10/2590/2196503 by McGill University Libraries - Serials Unit user on 15 May 2024 horizontal-torsional planes.

Qualitatively the drifts are dir- ected towards the null, minimal drifts are present at the null, and there is a further increase in drifts in the opposite direction as the head passes to the opposite side of the null.

These features of the drifts are quantitatively shown in Fig. 3O-Q. Such jerky, irregular, multidimensional head movements, composed of drifts and corrections in cervical dystonia, are often called 'dystonic tremor', a term that has generated considerable debate (Quinn et al., 2011; Elble, 2013).

It is interesting to note that the movements of dys- tonic tremor are analogous to the gaze-evoked nystagmus of the eyes.

Such a resemblance suggested an alternative label for the 'dystonic tremor', and that is 'head nystag- mus'.

Though 'nystagmus' is a term more commonly used in reference to various types of eye oscillations, it can also be applied more generally to any movement that has repeti- tive cycles of a slow drift followed by a rapid corrective movement in the opposite direction (i.e. a jerk nystagmus).


Interestingly, the term 'head nystagmus' was used more than three quarters of a century ago by Hyndman (1939) in his in descriptions of the jerky oscillations in cervical dystonia.

Regardless of the best terminology for these head movements, the results show that they can be inter- preted as deficits in a head neural integrator (Shaikh et al., 2013).


In addition to the coarse, jerky, oscillatory movements, which we have likened to gaze-evoked nystagmus of the eyes, some subjects with cervical dystonia also had smaller amplitude, sinusoidal, and more regular oscillations.

These oscillations, resembling 'essential tremor', were superim- posed on the coarse jerky movements (Fig. 3A and D).


It was subsequently shown that these sinusoidal head oscilla- tions had characteristics similar to the pendular eye oscil- lations that are thought to originate from instability in the ocular motor neural integrator (Das et al., 2000; Shaikh et al., 2015b).


Three characteristic features of these sinus- oidal head oscillations supported their origin in an abnor- mal neural integrator.

First, the amplitude of oscillation depended upon the head-on-trunk orientation; second there was no influence of head position on the frequency of oscillations; and finally there was a reset in phase of the oscillation after rapid head movements.


Thus, these quan- titative studies of head movements in cervical dystonia pro- vided strong support for a mixed disorder, combining at least two subtypes of oscillatory head movements, and both could be related to abnormal function of a head neural integrator (Shaikh et al., 2013, 2015b).


Proposals suggesting a relationship of the head neural inte- grators in the INC or its surrounding area to cervical dys- tonia do not imply that pathology in cervical dystonia must be intrinsic to these regions.

Major inputs to this area are from the cerebellum (Fig. 4), a region already implicated in cervical dystonia (Pelisson et al., 1998


These new concepts for the pathogenesis of cervical dys- tonia, based on the idea of an impaired neural integrator that receives multiple sources of feedback, are not mere academic exercises in anatomical localization or physi- ology.

They have direct implications for treatment.

Specifically, the new concept points to potentially novel stimulation targets for deep brain stimulation.

Traditionally, deep brain stimulation for cervical dystonia targets the internal segment of the globus pallidus or the subthalamic nucleus, but outcomes are not predictable (Vidailhet et al., 2005; Kiss et al., 2007).

The neural inte- grator concept suggests that stimulation of the INC or its cerebellar inputs may be a future strategy to treat cervical dystonia.


The concept of involvement of the midbrain in the con- trol of 3D head orientation is not new.

In the early 1950s Hassler and Hess applied monopolar electrical stimulation near the INC to evoke head movements in normal humans (Hassler and Hess, 1954).

Two decades later, during elec- trical stimulation of the INC, medio-superior red nucleus, and parts of medical longitudinal fasciculus posterior and inferior to the intercommisural line and lateral to the mid- plane, Sano noted vertical head movements accompanied by marked contraction and electromyographic discharges of posterior neck muscles bilaterally, such as the splenius capitis and trapezius in subjects with cervical dystonia (Sano et al., 1970).

Ablation of these regions reduced retro- collis (Sano et al., 1970).

Stimulation of an area superior, anterior and lateral to INC resulted in contraction of both sternocleidomastoids accompanied by anteroflexion of the neck (Sano et al., 1970).

Periaquaductal stimulation using a weaker electrical charge extinguished electromyographic discharges in bilateral posterior neck muscles followed by improvement in neck muscle tone (Sano et al., 1970).

Hassler later targeted the efferent projections of the INC, the ventro-oralis internus thalami, for the surgical treat- ment of torticollis (Hassler and Dieckmann, 1970) and the prestitial nucleus for retrocollis (Hassler et al., 1981).

These procedures were largely abandoned as a treatment for cervical dystonia because results were sometimes unpre- dictable.


This unpredictability is not surprising, because of the lack of a guiding conceptual model regarding how ma- nipulations of these regions might alter head positions.

However, in light of the neural integrator hypothesis, we can begin to understand the subtleties of these effects, such as the initial drift of head position after the inactivation of INC (Fig. 2A), and the holding of final head orientation after unilateral stimulation of this nucleus (Fig. 2B).

Moreover, these new hypotheses incorporating a neural in- tegrator in the pathogenesis of cervical dystonia, encourage us to revisit these target areas with more refined techniques that might offer therapeutic benefit to subjects with cervical dystonia.


Contemporary concepts emphasize the role of feedback in the pathogenesis of dystonias and also link to the most popular treatment of cervical dystonia, botulinum toxin in- jections.

Such therapy not only affects the neuromuscular junction, but it also modulates the afferent output of the cholinergic extra and intrafusal fibres (Filippi et al., 1993; Rosales et al., 1996, 2006).

In support of this idea about pathogenesis are the effects of botulinum toxin on the spinal and supra-spinal reflex pathways (Rosales et al., 1996).

Additional evidence comes from reduced central ex- citability with trans-cranial magnetic stimulation in humans after injection of botulinum toxin type A injection into the extensor digitorum brevis muscle (Kim et al., 2006).

It is also possible that the effects of botulinum toxin are com- bined with the secondary effects of muscle weakening on spindle activity and proprioception.

Clinical observations also support the idea that botulinum toxin affects the muscle spindle as there is a lack of a simple relationship between the required dose of the toxin and the observed clinical benefit.

It is often noted that the abnormal neck


In summary, we present a novel conceptual framework for the pathogenesis of cervical dystonia that emphasizes the role of abnormal feedback to the midbrain head neural integrator.

The idea that proprioception, the cerebel- lum, and basal ganglia are key sources of feedback to the head neural integrator is compatible with contemporary perspectives on cerebellar or proprioceptive abnormalities as contributors to dystonia, as well as conventionally sug- gested impairments in the function of basal ganglia.

This proposed framework for the pathophysiology of cervical dystonia departs from longstanding traditional concepts of dystonia that focused exclusively on the basal ganglia.

More importantly, this framework suggests novel therapies for cervical dystonia such as chronic electrical stimulation of the cerebellum or modulation of proprioception using vibration or electrocutaneous stimulation devices.


We read with great interest the article by Shaikh et al. ( 2016) presenting a pathophysiological explanation of cervical dystonia.

Cervical dystonia is a neurological disorder characterized by involuntary twisting and turning of the head in any of the three dimensions.

The authors proposed that the abnormalities of head movements seen in the disease stem from a malfunctioning head neural integrator caused by either an intrinsic dysfunction or as a result of impaired cerebellar, basal ganglia, or peripheral feedback.

Several studies hypothesized that the head neural integrator relies on feedback using visual information, neck proprioception, and input from the cerebellum [1][2][3].

Starting from the hypothesis that visual feedback may play a crucial role in the functionality of the head neural integrator and, subsequently, in the pathophysiology of dystonia, we decided to analyse the straight-ahead preference in healthy human subjects and patients with cervical dystonia.

Several studies had previously demonstrated a dysfunction of visuo-spatial perception in cervical dystonia [4][5].

In particular, it had been suggested that egocentric space representation was impaired in cervical dystonia and that visual straight-ahead perception was shifted towards the trunk compared to normal subjects whose visual straight-ahead were claimed to coincide with the head midsagittal plane [4][5].


Our study was based on the observation that healthy human subjects gazing to the side had been reported to respond faster to objects appearing in the peripheral visual field when the objects were displayed at a location in the subjects’ straight-ahead direction compared to objects displayed at an eccentric location to the subjects’ position [6].

This effect was independent of the subjects’ direction of gaze and could be found even though the objects formed similar images on the retina and had the same distance from the fovea.

Recently, the described behavioural observations of a preferential processing for visual information coming from the straight-ahead direction could be confirmed by a functional MRI study in healthy subjects on stimulus-evoked blood oxygen level-dependent responses in V1 and V2 [7].

The authors provided evidence that visual stimuli elicited an enhanced response when presented closer to the straight-ahead direction.

The straight-ahead preference for visual information coming from the peripheral visual field may be part of the head neural integrator system, however, the exact underlying mechanism for this effect remains unclear.

Possible explanations could be a top-down or attentional modulation of visual cortex processing that allows the subject to select behaviourally relevant information, or a bottom-up modulation by extra-retinal signals, such as proprioceptive information encoding the eye position.


In their study with human subjects, Durand et al. used an attention-demanding test that required their participants to not only disengage their gaze but also their attention from the straight-ahead position [6].

As the results of this particular task showed the same preference for the straight-ahead perception of visual objects, the authors concluded that while an attentional explanation could not be excluded, the observed effect does not seem to require full attentional resources, and that there might be an automatic early integration of visual and postural inputs, which facilitate straight-ahead perception.

This explanatory model would be in line with several studies reporting on the influence of gaze direction or proprioceptive eye position signals on visual cortex processing [8][9][10].


When adapting the paradigm of Durand et al. to an experimental setting (see Supplementary material for method) including healthy subjects and patients with cervical dystonia, we found that patients with cervical dystonia did not exhibit any preferential reaction for targets displayed closer to their head or their trunk axis at the population level (see Supplementary material for results).

Our study may reveal a subtle perturbation of the visual information processing in cervical dystonia.

It is unclear whether or not the described impairment of visual information processing in cervical dystonia is part of the cause or of the result of the disease’s pathophysiology.

However, our findings underline that other sensory inputs and cortical processes are affected in cervical dystonia and that their impairment may contribute to the hypothesized dysfunction of the head neural integrator.

Additionally, they may broaden the spectrum of possible novel therapies mentioned by Shaikh et al. in that physical therapy using visual feedback techniques could be helpful in treating the disease [11].


We appreciate the thoughtful comments from Amlang and colleagues regarding our recent viewpoint [1]. They raise the important point that the neural integrator responsible for controlling head position relies on visual feedback in addition to feedback from the cerebellum, proprioceptors, and the basal ganglia.

To address these ideas, Amlang and colleagues analyzed the ‘straight-ahead preference’ in subjects with cervical dystonia and compared it with that of healthy subjects.

The ‘straight-ahead preference’ is a classic physiological phenomenon in which the reaction time to a novel image appearing in the peripheral visual field is longer if the eyes and the gaze are straight ahead, aligned with the trunk than if the eyes are turned to an eccentric location and the target appears straight ahead, in line with head or trunk [2].

This privileged visual processing of the straight-ahead direction in humans is of fundamental functional significance.

As multi-tasking creatures, we scan the environment while in motion.

As a result, gaze often is not straight ahead, but it is beneficial to be able to refixate promptly to targets that appear directly in front to avoid potential obstacles or accidents.

Such behavior would require the brain to have internal knowledge of the relative alignment of the position of the eye (retina) in the orbit, the head on the body.

Disorders affecting head movements, such as cervical dystonia, could interfere with the coordinate transformations necessary for optimal visual function.

The elegant experiments by Amlang and colleagues [1] provide proof of such a principle.


They found an overall increase in reaction time in subjects with cervical dystonia.

The reaction time to the peripheral target location that was in line with the sagittal plane passing through the head was still less compared to a peripheral target that was eccentric compared to the head.

It was unclear whether impaired visuospatial processing in cervical dystonia is the cause or consequence of these phenomena.

The authors now speculate that altered visual input can affect the function of a presumed head neural integrator.

We agree with this possibility as an additional mechanism for an impaired head neural integrator.

Here we offer another insight on how impaired neural integration affects visuospatial processing and reaction times.


A fundamental function of our visual system is to generate a 3D map of our surroundings.

The brain uses this internal construct to launch appropriate reaching movements and locomotive behavior.

Such a seemingly simple goal requires a series of complex transformations from retinal to body-centered coordinates [3].

The parietal eye fields in primates represent space in a gaze-centered frame of reference [4][5][6].

However, this neural activity is modulated by eye-in-head or head-on-trunk orientation [7][8].

The goal of such modulation is to allow implicit transformations from a retinal to a head-centered and then to a body-centered frame of reference [4].

Eye, head, and body orientation signals are an important source of the signals required for these transformations [5][6].

The brainstem neural integrators are critical for generation of the motor commands for eye, head, and body position [9][10].

Thus, if these structures do not function correctly, one might expect errors to occur in coordinate transformations for gaze control, reaching, and other visuomotor behaviors.


These results can be explained given not only the head neural integrator hypothesis, but in general as a consequence of motor uncertainty or a mismatch between the efferent motor output and the efference copy being used for transformations.

There are two ways the brain receives head-on-trunk orientation.

One is via an efference copy from the neural integrator and the other is from proprioceptive afference.

Impairment in a head neural integrator could interfere with processing of both types of information.

As a result, the position signals necessary for coordinate transformations become less faithful, possibly more noisy and consequently increase the chances of inaccuracy.

To compensate for this potential for error the reaction time could be increased to give the brain more time to become more certain of the needed response.

It is also possible, however, that the head neural integrator has little role in such a phenomenon and that the increase in the reaction time is merely due to extraneous ‘noise’ in proprioceptive muscle output in cervical dystonia.

In such a situation, one might expect an indirect modulation of the output from the head neural integrator.


Subjects with cervical dystonia often present with a jerky head tremor [11][12][13].

These head oscillations and consequent eye oscillations (due to a tremor-induced but normal vestibulo-ocular reflex response) lead to a constantly changing head-on-trunk orientation and eye-in-orbit position.

As a result, there is a discrepancy as the target image on the retina remains stable, but the head-on-trunk and eye-in-head orientations are constantly in flux.

Such extraneous signals could also increase motor uncertainty and increase reaction time.

Several other studies suggested that visual-spatial orientation is disrupted in cervical dystonia [14][15][16] showing that the perception of visual-spatial orientation is referenced to the head in normal subjects, while in cervical dystonia it is shifted to the trunk.


In summary, we agree with the possibility that abnormal visuospatial processing could contribute to dysfunction of the head neural integrator.

However, it is also plausible that dysfunction of the head neural integrator itself can also affect visuospatial coordinate transformations.

This concept is consistent with the notion that cervical dystonia is not a disorder localized to a single brain region, but instead is a consequence of widespread impairment in a neural network responsible for sensorimotor integration.

With the considerable complexity of these neural networks controlling the position of the head, disentangling the core deficit causing cervical dystonia remains a challenge.

Nevertheless, the concept of a head neural integrator provides a conceptual model that may help to guide further studies of the pathophysiology of this disorder.


Employing advanced techniques in chemical synthesis consti- tutes an effective approach to realizing unconventional compounds, sometimes opening access even to new classes of materials featuring, e.g., novel bonding principles.

One such example concerns multinary silver oxides, which are thermally notoriously labile.

Using specially designed Bridgeman-type autoclaves made of distinctly scaling-resistant steel, 1 enduring conditions applied of up to 7 Â 10 8 Pa pressure of oxygen and up to 973 K temperature enables suppression of the thermal degradation of Ag 2 O and thus reaction of this oxide in all-solid state reactions with any other binary oxide in the periodic table.

In a preconceived view, one would expect the oxides attainable this way, Ag x M y O z (M ¼ nonmetal or metal), to represent analogues of respective alkali metal oxides.


However, systematic exploration of such systems has revealed that singular early observations of conspicuously short Ag + -Ag + separations in oxides are not strange exceptions, but manifestations of a general feature of the chemistry of silver.

2-4 Primarily in oxides with high silver contents, silver(I) ions tend to aggregate form- ing partial structures that are topologically reminiscent of elemental silver.

Furthermore, such structural motifs are asso- ciated with specic physical properties.

5, 12 for which precise dimensions have been determined here for the rst time, unaffected by disorder 13 or under-occupation of atomic positions.

14 The ndings shed light on the common effects of covalent bond length contractions caused by multiple bonding and/or superimposed polar contributions.

Comparison with isosteric P 4 O 10 reveals that the bond distances to the terminal oxygen atoms are signicantly less contracted in the borate, which does not feature any low lying orbitals that might mediate multiple bonding, while in accordance with the small difference in the electronegativities 15 of boron (2.051) and phosphorus (2.253), possible contractions due to polarities of the B-O and P-O terminal bonds are of comparable magnitudes.


The strange overall composition encountered is even more puzzling since such a kind of ternary oxide, a silver borate, is expected to follow heuristic rules of stoichiometry.

In the light of the electron counts given, which suggest the presence of itinerant excess electrons, it comes as a surprise that according to resistivity measurements the title compound is a small band gap semiconductor.

However, anal- yses of electron localizations using DFT based calculations comply with the experimental results.

Apparently, a new generalizable facet of the chemistry of silver becomes recog- nizable, which will complement text book knowledge of this element and bears potential for realizing interesting material properties due to the subtleties of electron localizations found.


Polycrystalline samples of Ag 16 B 4 O 10 were prepared by the solid state reaction of elemental Ag and H 3 BO 3 , or B 2 O 3 , in stainless- steel autoclaves at elevated oxygen pressures of 2 Â 10 7 to 5 Â 10 7 Pa and temperatures of 563-623 K. The starting materials were intimately mixed, pressed to pellets and placed in gold tubes, which were sealed at one end and crimped at the other.

Small amounts of H 2 O were added into the gold tubes to promote crystallization.

In a typical synthesis batch 0.03 mol of Ag and 0.01 mol of H 3 BO 3 were used, with the addition of 0.5 mL H 2 O.


Laboratory powder X-ray diffraction (PXRD) data (Fig. 1) were collected at room temperature on a Bruker D8 diffractometer with germanium monochromatized Cu-Ka1 radiation (l ¼ 1.5406Å), in steps of 0.01 over a 2q range of 10-90 degrees.

A Rietveld prole t was carried out with the TOPAS-4.2.0.2 (AXS) program.

16 The rened parameters were scale factor, sample displacement (mm), background as a Chebyshev polynomial of 5th degree, 1/x function, crystallite size, micro-strain (Stephens broaden model 17 ) and cell constants.

In addition, temperature dependent PXRD data were collected in the temperature range from room temperature to 773 K, in intervals of 50-100 K, as shown in the ESI, Fig. S1.†


A crystal suitable for single-crystal X-ray diffraction was selected in a drybox (M. Braun, Garching, Germany) under an argon atmosphere (<0.1 ppm O 2 , H 2 O) and mounted in a sealed glass capillary.

Diffraction data were collected at room temperature (298 K) with a SMART APEX-I CCD X-ray diffractometer (Bruker AXS, Karlsruhe, Germany), using Mo-Ka radiation.

The inten- sities of the Bragg reections were integrated with the SAINT subprogram in the Bruker Suite soware.

18 A multi-scan absorption correction was applied using SADABS.

19 The crystal structures were solved by direct methods and rened by full- matrix least-square tting with the SHELXTL soware package.

20,21 Experimental details and crystallographic data are given in Tables S1-S3.†


Magnetic properties were studied using a Quantum Design MPMS SQUID Magnetometer.

Zero-eld-cooled (ZFC) and eld- cooled (FC) magnetic susceptibility data were recorded in a 10 000 Oe eld while warming the sample from 5 to 300 K. Resistivities of polycrystalline bars (approximate dimensions 3 Â 3 Â 10 mm 3 ) of Ag 16 B 4 O 10 were recorded using a standard four-probe dc technique on a Quantum Design physical prop- erty measurement system.


Density functional (DFT) calculations, based on the experi- mental structure of Ag 16 B 4 O 10 , were performed using the CRYSTAL17 program package.

22 The bands in the semi-core and valence space, considering 19 valence electrons of each silver atom, 6 for O and 3 for B, are expanded in terms of local Gaussian basis functions.

The core electrons are represented by scalar relativistic pseudopotentials.

Details on the exponents and contractions are given in the ESI.† The integration in reciprocal space was based on 242 k-points in the irreducible part of the Brillouin zone.

The results presented here are ob- tained with the short-range-separated hybrid functional HSE- sol 23 for the exchange and correlation terms in the Kohn-Sham equations.


Atomic charges were evaluated by analyzing the electron density topologically according to the QTAIM approach.

24 A search of the atomic basins was performed with the critic2 (ref.

25 and 26) program on the basis of a 321 Â 321 Â 201 grid of data points of the valence electron density, augmented by core densities.

Within the basins, the valence density was integrated in order to get the net charges.

Data grids This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.


In more detail, all silver atoms are engaged in octahedral homoatomic building units, which are linked by sharing edges and vertices to form a 3D framework.

Every silver atom is coordinated by either one or two oxygen atoms at a distance typical of this pair of atoms d(Ag-O) ¼ 2.20-2.35Å (see Table S3†).


The rare complex anion B 4 O 10 8À has been observed here as an "isolated", i.e. unbridged, entity for the rst time.

12 Two previous reports present this building block as part of an extended framework.

Moreover, in both cases the structural analyses performed suffer from disorder, where occupation factors of 1/2 for the intra-cage bridging oxygen atoms even question the connectivity, i.e. the presence of such an integral anion, at all, 13 or from rening split atom positions for the anion in two orientations, impairing the accuracy of the data obtained.

14 The molecular anion, see Fig. 5, is isostructural and (valence) isoelectronic to P 4 O 10 .


The bond lengths are in the expected range, see Tables 1 and S3,† and the variations reect the position of the oxygen atoms, terminal or bridging within the adamantane type of cage.

Drawing the chemical bonds of the cage anion in terms of the Lewis concept requires placing a formal negative charge on both, the boron and the terminal oxygen atoms.

Although a formalism, this tells that the bonding situation is special and suggests looking for the respective structural distinguishing features by comparing with the This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.

View Article Online isosteric cage molecule P 4 O 10 .

Such an approach would imply comparison of the bonding of elements from the second and third row of the periodic table.


For the latter, contributions of d p -p p multiple bonding and superimposed polar interactions are factors of inuence, 34 while for the former such d p -p p , or p p -p p interactions are not relevant; consequently, the bonding within the borate anion, where all boron atoms are in tetrahe- dral coordination, has to be analyzed exclusively in terms of s bonding schemes.

Indeed, comparing the bond lengths in B 4 O 10 8À and P 4 O 10 (ref. 35) reveals signicant differences.

In the borate the terminal B-O bonds are only 4.4% shorter than the bridging ones, whereas the respective shrinkage amounts to 10.1% for the phosphorous oxide, see Table 1.

Obviously, for the P-O terminal bonds multiple bonding and polarity act in a syner- gistic fashion, while in the borate for the shrinkage of the B- O terminal bonds only polar inter-actions superimposing the s bond are relevant, resulting in a smaller net effect.


The B 4 O 10 8À cage like polyoxoanion is electron precise, and thus the excess electrons according to (Ag + ) 16 (B 3+ ) 4 (O 2À ) 10 Â 8e À have to be accommodated by the silver substructure.

As a consequence, one would expect itinerant electrons being present, giving rise to metallic conductivity.

Surprisingly, the compound is a small band gap semiconductor, and diamag- netic, as shown in Fig. 6.

By tting the resistivities in the high temperature range above 150 K to the Arrhenius equation (Fig. 6 inset), an experimental band gap of 4.71(1) meV was obtained.

This value is essentially identical to the calculated one, see below.

The diamagnetic susceptibilities are estimated to be À1147 Â 10 À6 cm 3 mol À1 , based on an analysis of the magnetic susceptibilities applying a Curie-Weiss, Pauli and diamagnetic term.

The upturn tail below 30 K is dominated by an unknown paramagnetic impurity (about 0.2% with S ¼ 1/2).

These results indicate the presence of localized and paired excess electrons.


However, no particularly shortened individual Ag-Ag bond- length can be identied that would provide evidence for normal 2e-2c bonds, although the interatomic silver separa- tions are on average smaller than those found for elemental silver.

As can be seen from Fig. 4 (red circle) the 'empty' octahedral silver units appear to cluster, forming blocks con- sisting of ve edge sharing octahedra.

Each such cluster entity features two tetrahedral voids, where three out of six edges formed by silver atoms are substantially shorter (2.80Å on average) than the separation in elemental silver (2.89Å).

This gives a rst qualitative hint to the fact that these are the regions in real space where the excess electrons might be accommo- dated.

Since these tetrahedral sites add up to four per formula unit, which in total may host four pairs of excess electrons, i.e. all of them, this view would comply with the experimentally found magnetic and transport properties.


We exclude the possible presence of hydride anions, lling the contracted voids, for chemical reasons.

Along any synthesis protocol yielding the title compound protons are essential constituents, and, moreover, applying an elevated oxygen pressure is an indispensable requirement.

The pronouncedly basic and strongly reducing hydride ion cannot exist under such conditions.


In order to back this explanation quantitatively, we per- formed a DFT based computational analysis on the density of states (DOS), electron densities and localizations.

DOS as well as the projections onto the atoms (PDOS) is shown in Fig. 7a.


Over the whole energy range, which is shied relative to the Fermi level (E F ¼ 0 eV), oxygen basis functions contribute to the DOS.

Below À7 eV, the silver PDOS is low.

Bands are at and can be associated with the B-O bonds in the B 4 O 10 8À unit.

The low contribution of boron to these bands points to the ionic char- acter of the B-O bonds.


The DOS between À7 eV and À3.2 eV is dominated by the 4d states of the silver atoms, however, it From À1.44 eV up to the Fermi level bands with higher dispersion are located.

The respective atomic contributions can be visualized by drawing isosurfaces of the electron density computed from these bands (Fig. 7b).


These bands are delo- calized over the whole crystal.

Note, for clarity reasons, only a cut-out of the isosurface is shown in Fig. 7b.

The bands are lled and the system is not metallic.

There is a distinct band gap in most parts of the Brillouin zone.


Only at k ¼ (1/2 1/2 1/6) it becomes as tiny as 15 meV (Fig. 7c).

The highest valence band and the lowest conduction band exhibit a parabolic dispersion.

Signicant effects of spin-orbit coupling are not expected, as the Ag 4d-shell is practically lled and the 5d-orbitals are only affected indirectly.


Moreover, the projected densities of states show the same character in both bands.

The band gap at k ¼ (1/ 2 1/2 1/6) can be seen only in calculations with hybrid func- tionals.

When using gradient corrected (GGA) functionals without fractional Hartree-Fock exchange the band gap is underestimated and vanishes, although still a minimum of the DOS can be seen at E F .


The projections of the DOS shown in Fig. 7a are based on a partitioning with respect to atomic basis functions.

In the linear expansion of the bands, a certain basis function may contribute to a state which is attributed to a neighboring atom, and an interpretation in terms of chemical bonding may not be unique.

Additional methods in position space give comple- mentary information.


The partitioning of space resulting from a topological analysis of the total electron density 24 gives atomic volumes and net charges, which are quite robust with respect to the choice of the basis set.

The values computed for Ag 16 B 4 O 10 are summarized in Table 2.

The volumes and charges at the various atomic positions are clearly related to the connectivity.

Ag3 and Ag4, which have a lower positive charge, have only one short distance <2.4Å to the neighboring oxygen, whereas Ag1 and Ag2 with a higher positive charge have two of them.


Like- wise, the negative charge of the oxygen atoms varies, depending on whether they occupy a bridging or terminal position in the B 4 O 10 8À unit.

The high positive charge of the boron atom points to a substantially ionic character of the B-O bond.


Further information on the bonding characteristics in posi- tion space can be obtained by the inspection of the electron localization function 27,28 (ELF), which can take values between 0 and 1.

High ELF values h are related to atomic shells, covalent bonds and lone pairs.

The ELF in Ag 16 B 4 O 10 comprises three distinct features.

At rst, there are nearly spherical domains around each silver atom, representing the lled 4d 10 shells.

In Fig. 8, these domains are omitted for clarity reasons.


The domains associated with the B 4 O 10 8À anion are repre- sented in Fig. 8a showing the lone pairs of the oxygen ions and one attractor respectively along each B-O bond.

The fact that the attractor is close to the oxygen atom is another hint to the polar character of this bond.

A special feature of the ELF, an attractor at h ¼ 0.23 appears in each tetrahedral void of the silver network.

Fig. 8b shows the domain with h ¼ 0.23 around one of them.


h ¼ 0.23 is a rather low value, though it represents a clear maximum in the ELF.

Attractors with low ELF values may originate from the inuence of close-by lled d-shells, 36 nevertheless pointing to covalent interactions in the valence shell.

Indeed, the face of the tetrahedron closest to the attractor contains two of the shortest Ag-Ag bonds (<2.80Å) within the range from 2.78Å to 3.23Å of all Ag-Ag contacts (cf.

metallic silver 2.89Å).


Putting


A new compound, Ag 16 B 4 O 10 , has been obtained as a coarse crystalline material via solid state synthesis.

The shiny black crystallites are insensitive to humid air and start decomposing thermally at $623 K with silver metal and amorphous B 2 O 3 resulting in the nal solid residues, see Fig. S1.† The title compound can be prepared from various starting materials in the required molar ratios, as there are boron(III) oxide, boronic acid, silver oxide and nally elemental silver as an essential component, while adding varying amounts of water as a mineralizing agent.

At rst glance it appears unintuitive that for the synthesis of such a considerably reduced material applying moderately elevated oxygen pressures of 2 Â 10 7 to 5 Â 10 7 Pa is an indispensable requirement.

However, running the experiments at ambient pressure results in the decomposition of silver oxide to the metal, while a too high oxygen pressure would end up in the formation of silver(I) borates.

So, the synthesis of the title compound is a delicate balancing act, and even for the optimized synthesis conditions, given above in detail, one or the other synthesis run may fail in yielding single phase products.

Fig. 1 displays a Rietveld prole t of an X-ray powder diffractogram of a sample obtained using the optimized synthesis procedure; a three-phase renement has revealed that only traces of Ag (2.1%) and Ag 2 O (2.3%) are present.

The atomic parameters obtained are given in Table S4.†


The constitution has been conrmed unambiguously by single crystal X-ray structure determination.

As illustrated in Fig. 2, the general structural organization of Ag 16 B 4 O 10 derives straight forwardly from a ccp pattern of elemental silver: out of every twenty silver atoms four adjacent ones are replaced by boron.

Since boron atoms are of considerably smaller size compared to silver, the substitution generates some empty space tolerating the insertion of oxygen atoms to form the polyoxoanion B 4 O 10 8À .


From this picture it is immediately obvious that the title compound ideally matches the notion of classifying crystal structures of extended inorganic solids rather based on the packing of the cations 31 than on that of anions.

Even beyond, it lends strong support to the further reaching concept according to which certain oxides may be regarded as alloys being stuffed with oxygen.

32,33 This particular structural interrelation is underpinned by Fig. 3 and 4, showing the crystal structure of the title compound along the view directions [001] and [121], respectively.


This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.


There are no conicts to declare.


A thorough systematic study of the Electron Localization Function (ELF) in fcc silver metal, the deficient vacant-type Ag16,4 structure, and the Ag16B4O10 title compound of the Chem.

Sci., 2020, 11, 962 edge article leads to a further understanding of the sub-valent characteristics of silver in the silver borate compound.

By visualizing the process in three consecutive steps, (fcc)eq-Ag / (fcc)ex-Ag / Ag16,4 / Ag16B4O10, the electron reduction of Ag atoms can be traced to be due to (i) the expansion (ex) of the host metallic array from its equilibrium (eq) geometry and (ii) the vacancy creation and subsequent insertion of guest borate clusters.

Our ELF analysis also allows us to identify to what extent metallic features remain in the title compound, providing an alternative explanation of why Ag16B4O10 is not a conductor whereas pure silver is.


As occurs with the chemical bond, the terms valence and oxidation state enclose elusive concepts that are very useful in everyday chemistry but it is difficult to establish a unique agreed definition for them [1].

These concepts lack both the possibility of being measured in the laboratory and a quantum-mechanical operator allowing their calculation.

In this context, sub-valent compounds tend to exhibit anomalous compositions and they are compounds where the bonding of one of the atoms involves fewer electrons than expected from the electron counting of its valence shell.

In the solid state, sub-valence is associated with oxidation numbers lower than nominal values for any of the atomic constituents of the compound [1].


According to a regular composition, one would expect silver atoms to participate in the electron counting of a chemical bonding network with one electron, that in its 5s valence shell.

In the Ag16B4O10 title compound of the edge article commented here [2], however, only 8 electrons are needed to account for the charge of the [B4O10]8− electron-precise supertetrahedral unit, which forces silver atoms in this compound to play a sub-valent role (only half electron from each Ag atom is required) to fulfill the observed chemical formula.


The sub-valent behavior of silver is not exclusive to this compound (see for example earlier discussions in [3-5]) and has been extensively investigated by the corresponding author of the article commented here who identified the existence of local clustering of Ag atoms hosting electron pairs as a general “new bonding pattern” (see [2] and [6] and references therein).

In this comment, we provide a consistent interpretation of the chemical bonding network of Ag16B4O10 that contains analogies but also qualitative and quantitative differences with respect to the discussion presented in [2].

These differences are essential for the understanding of the structure and bonding in Ag16B4O10 and are far-reaching since they can be extended to other silver oxide crystals such as Ag7Pt2O7 [6].


The idea that metallic behavior is partially maintained in Ag16B4O10 is in part suggested in the discussion of [2], where it is pointed out that silver sub-valent oxides present Ag(I) sub-arrays resembling the structure of the pure fcc-silver metal.

However, in our view, it was not explicitly applied to interpret the sub-valent nature of the title compound.

To investigate the residual metallic bonding properties of Ag16B4O10, we propose to follow the electron counting associated with the metallic subarray from the pure silver metal to the borate compound.


Bearing this idea in mind, we provide an alternative interpretation of the bonding network of Ag16B4O10 supported by a topological analysis of the Electron Localization Function (ELF) in (i) the Ag metal fcc structure, (ii) the Ag deficient fcc sub-array after removal of the [B4O10] units, and (iii) the Ag16B4O10 crystal structure.

By performing these topological analyses, we will be able to check whether metallic silver features are maintained or not in the title compound, and to what extent.


Our results clearly reveal that [B4O10]8− behaves as an independent anion in agreement with the analysis in [2].

The equivalence of [B4O10]8− with the well-known P4O10 molecule was also justified resorting to the extended Zintl–Klemm concept in [7].

However, we found differences with respect to the interpretation of [2] in the peculiarities of the silver subarray.

To solve the electron-excess, electron pairs confined in four approximate tetrahedral voids of the Ag16 silver fcc-distorted unit cell are proposed in [2].

Although this proposal successfully explains the observed semiconducting behavior and the diamagnetic ordering of this compound, the limited analysis of ELF values at the attractor positions carried out in [2] is not enough to suggest that one electron pair is localized in each of these voids.


By combining VASP electronic structure calculations [8] within the DFT-GGA-PBE approach [9] with critic2 topological searches of critical points and basin electron density integrations [10], we perform an extensive ELF analysis in the Ag-related structures detailed in the three subsections below (see computational details in the ESI†).

Although other methods such as ELI-D better describe the dynamic electronic correlation and could provide further insight into electron poor metal systems [11], ELF analysis reliably accounts for the valence description of inorganic materials and metals [12-14], and is simple and illustrative enough to analyze the bonding properties of the title compound.


There are three main results in the ELF analysis of pure silver metal that are relevant to understand the structure of Ag16B4O10.

The first one concerns the localization of the ELF attractors associated with the valence electrons of fcc-silver at its room conditions equilibrium geometry (a = 4.146 Å).


Two sets of valence attractors become apparent.

The first set corresponds to d-electrons located not further away than 1.0 Å from the Ag nuclear positions (small black balls in Fig. 1 (left)) with an ELF value of h = 0.27.

The large number of them is a typical consequence of the crystal environment.


More interesting are s-valence attractors easily identified at the eight tetrahedral interstices (8c positions) of the fcc unit cell (yellow balls in Fig. 1 (left)).

They appear around 1.84 Å away from the four neighbour Ag nuclei with h = 0.20.

Electron density integrations inside each of these tetrahedral voids result in 0.53e.


This yields a total of 4.24 electrons in the whole unit cell, a value that is consistent with the valence oxidation state (Nv_ox) of +1 for metallic silver, where Nv_ox is defined in this context considering the number of valence electrons hosted in the lattice interstices.

This definition is different from the standard Nox definition which obviously yields a zero value for a pure metal.


The second ELF feature is that the topology is qualitatively the same when we expand the fcc silver unit cell to the volume corresponding to that of the fcc distorted silver sub-lattice hosting the B4O10 cluster (a = 4.251 Å).

The relevant eight ELF attractors still appear at the tetrahedral positions.

However, and in concordance with the generalized stress–redox correlation [15], the electron population inside the basins of the tetrahedral voids decreases at this expanded sub-lattice.

This is a consequence of the lower overlap between the electron density clouds of neighbouring atoms when their internuclear distance increases.

In fact, the electron density integration within the ELF tetrahedral-attractor basins estimates now 0.38e in each void.

Thus, using these valence electrons as before to evaluate the valence oxidation state of silver, we obtain Nv_ox(Ag) = +0.76 for the expanded fcc-Ag lattice at the same volume of the Ag16B4O10 compound.

Under this perspective, the expanded fcc-Ag already behaves as sub-valent.


Finally, we find that all the ELF attractors at the tetrahedral voids of the fcc-Ag structure (regardless of the unit cell parameter) belong to the same ELF domain.

This means that the ELF values at the maxima are linked through ELF first-order saddle points or basin interconnecting points (bips) forming a three dimensional (3D) chemical entity usually called a superbasin.


This is the typical ELF picture of metals [12].

The small difference between the ELF value at the attractor (att) and at the bip reflects a highly delocalized nature of the electrons in the 3D-superbasins.

Specifically, hatt = 0.20 and 0.19, and hbip = 0.17 and 0.16 in the room conditions and in the expanded fcc-Ag structures, respectively.


The concept of ELF superbasins is not new and has been employed many times to describe the merging of ELF basins belonging to the same atom (see for example [13] and [14]) or belonging to different atoms [16, 17].

Being a key concept in our alternative interpretation, it is useful to realize that these superbasins constitute 3D channels in the real space that account for the electrical conductivity of metals, as first noted by Silvi and Gatti [12].


In the fcc-Ag structure, the electrical circuit originating from the connection of the tetrahedral ELF attractors belonging to the valence superbasin is easily visualized in Fig. S1.†


Let us now consider the Ag16,4 host structure with the same lattice parameters as in Ag16B4O10.

Ag–Ag nearest neighbour distances range from 2.76 Å to 3.23 Å, with an average value of 3.05 Å, pretty close to the Ag–Ag shortest distance of 3.01 Å in the expanded fcc-lattice.

Breaking the symmetry of the fcc structure by replacing with a vacancy 4 out of the 20 tetrahedral Ag4 units of the unit cell (Z = 4) makes tetrahedral attractors not equivalent any more.

Up to five different types of ELF attractors identified with different colors in Fig. 2 (left) are found.

Among then, four have a multiplicity of 16, whereas the other (cyan) has a multiplicity of 4.


Despite the symmetry breaking, there are particular ELF features in the defective silver-vacant structure that resemble the description obtained for the expanded fcc-lattice.

For example, the ELF at the tetrahedral maxima keeps similar values around hatt = 0.20 and the average basin population (0.40e) is only a little bit higher that the 0.38e value found in the expanded pure metal.

However, basin electron populations oscillate now between 0.50e and 0.30e (see Table S1†).

This range of values illustrates how relevant is the position of the attractor relative to the vacancy.

Vacancies not only break the symmetry of the structure but also interrupt the connectivity between attractors.


The existence of vacancies reduces the number of available tetrahedral voids which are the only positions where ELF attractors tend to be located in these silver compounds.

We observe that this reduction is not balanced by an overall increase of the basin electron populations (see Table S1†).

Thus, a lower valence oxidation state of the Ag atoms is produced.

Using the basin electron populations, the average valence oxidation state in the defective Ag-vacant structure was found to be Nv_ox(Ag) = +0.42.

These values evidence an increase of the sub-valent characteristics of the defective silver-vacant structure.


In the Ag16B4O10 title compound, ELF analysis allows us to differentiate an ELF domain for the [B4O10]8− unit with equivalent results to those in [2].

Attractors between B atoms and the bridge or terminal oxygens are detected, as well as lone pair superbasins containing up to three ELF maxima (see Fig. S2†), reinforcing the structural similarity to the (valence) isoelectronic P4O10 molecule previously emphasized in [2] and [7].


When the ELF map of the host silver subarray is examined, the same attractors at the tetrahedral voids as in [2] are found with ELF values around 0.25 (black balls of Fig. 2).

We found analogies only up to this point with the results of [2].

We understand that to be conclusive with respect to the chemical bonding and the structure of the new silver oxide compound, its bonding network should be compared with that of pure silver and the deficient silver structure.


As observed in Fig. 2 (right), only two types of attractors at tetrahedral voids (instead of one as in fcc Ag and five in Ag16,4) are identified.

The presence of borate clusters destroys and empties some of the basins at the tetrahedral voids close to the oxygen positions.

Oxygen atoms act as electron sinks of the otherwise populated basins in the defective silver-vacant structure.

Such an electron transfer to the oxygen atoms reflects an ionic interaction between the two fragments in the Ag16B4O10 compound.

In particular, pink, orange, and green attractors with distances shorter than 2.75 Å to oxygen atoms are not seen in the silver borate compound, whereas only black and cyan attractors beyond 3 Å of any oxygen atom still are present in this compound.


Electron density integrations within each of the basins of these two attractors lead to 0.71e and 0.36e, respectively.

We notice that it is the Ag4 tetrahedra associated with the basins that are more populated where some Ag–Ag distances are especially shorter than those in pure silver metal.

Nevertheless, these values show that the number of electrons hosted in these tetrahedral voids is definitely lower than 2, questioning the presence of pairwise electrons.

In this way, the sub-valent nature of Ag16B4O10 might be considered to be inherited from the Ag16,4 host lattice since we obtain Nv_ox(Ag) = +0.40, almost coincident with the value found in the defective silver-vacant structure.


Besides the quantitative picture provided by the electron density integration, the most interesting result of this third ELF analysis comes from the identification of different independent valence superbasins associated with the silver subarray of Ag16B4O10.

These superbasins are equivalent and result from the bips connecting black–black (hbip ≈ 0.16) and black–cyan (hbip ≈ 0.19) attractors of values hatt ≈ 0.25 (black) and ≈ 0.23 (cyan).

The ELF profile displaying all the possible connections between black and cyan silver attractors is shown in Fig. S3†. Focusing just on one of these equivalent superbasins, we observe that all black attractors are connected with one cyan and two black attractors forming a 1D infinite coil-like chain of helical symmetry with ramifications ending at the cyan attractors.

There are four of these electrical circuits per unit cell hosting the electrons delocalized among the tetrahedral voids and running along the c axis of the crystal as shown in Fig. 3.

The existence of these unconnected superbasins is used below to explain the different behavior of pure silver metal and the borate compound with respect to metallicity.


We have seen that, in the fcc-silver metal, s-type ELF attractors only appear inside the tetrahedral voids.

This fact is noted both in the defective and silver borate compounds, and points towards a general behaviour that could be used to explain other silver sub-valent compounds.

Tetrahedral attractors allow the formation of chemical entities called superbasins which identify regions where the metallic bonding properties are maintained.

Interestingly, in the pure fcc-silver metal a 3D superbasin is spread out along the whole crystal structure, whereas in the borate compound, it splits up into four independent 1D superbasins per unit cell resembling infinite circuits of helical symmetry along the c axis.

Since electron delocalization is an inherent property of superbasins reflected by the close ELF-values at the maxima and the bips, this striking result provides an alternative explanation of why the silver borate compound is a small band gap semiconductor whereas the silver metal is an electrical conductor.

Macroscopically, we can understand that borate clusters play the role of crystal defects interrupting the electric conductivity along the three directions of the crystalline space.


The chemical entities associated with these superbasins allow us to discuss the sub-valent characteristics of silver-based compounds too.

For the fcc unit cell of silver under room conditions, our ELF analysis and electron density integration lead to a single Ag4+ superbasin.

In contrast, four equivalent Ag16,6.4+ entities appear in the unit cell of Ag16B4O10.


From these compositions, it is evident that the sub-valent characteristics of the silver (sub)array originated during the transition from the fcc-metal to the silver borate compound.

If we ideally understand the transformation as a three step continuous process involving (i) the expansion of the fcc-structure, (ii) vacancy creation and (iii) borate cluster insertion, we can identify two mechanisms that are able to reduce Ag atoms.


The first one is associated with an increase of the lattice parameter.

Crystal expansions decrease the electron density inside the tetrahedral voids (and therefore in the superbasins) allowing transfer of the valence electrons to the atomic cores.

Firstly, we obtained Nv_ox(Ag) = +1 and Nv_ox(Ag) = +0.76 when a = 4.146 Å and a = 4.251 Å, respectively.


Secondly, vacancy creation or cluster (borate) insertion disrupts the 3D superbasin connectivity, reducing the number of tetrahedral positions where silver electrons prefer to be accumulated.

As there is no other mechanism to reduce Ag atoms, an almost identical Nv_ox(Ag) value is obtained for the defective (+0.42) and the borate (+0.40) silver compounds.


In conclusion, we believe that these findings may throw light on the interesting experimental observations reported in [2] and might help to interpret the new chemistry exhibited by the silver oxides highlighted in [2] and references therein.

Our results emphasize the decisive role that residual metallicity plays in the existence of sub-valent compounds and could represent a promising strategy to be applied to other sub-valent silver oxides showing singular behaviour like the recently published Ag7Pt2O7 compound [6].

Claimed ‘synergistic interactions among the filled 4d10-shells’ strengthening silver inter-cluster bonding [2] may be accounted for by means of the proposed metallic superbasins rather than localized electron pairs.

Suggested silver-oxide dispersion interactions need other scalar fields such as reduced gradient density [18] or chemical pressure formalism [19] to reveal so-called non-covalent interactions.

We are currently working in that direction.


Ag16B4O10 features an exotic scheme of chemical bonding and extends the growing family of subvalent silver oxides.

These findings constitute a new general and intrinsic facet of the chemistry of silver, which has not been fully understood, yet, and definitely deserves to be analysed from different perspectives.

Against this background, we distinctly appreciate the efforts made by A. Lobato, Miguel Á. Salvadó, and J. Manuel Recio (LSR) in studying these phenomena at the example of the title compound [1].

While the computational results presented in the Comment article well comply with those published in our original paper [2], the interpretations follow different routes.

Whereas LSR focus on the analogy of pattern of the Electron Localization Function (ELF) in position space in the title compound with those found in elemental silver, we interpreted the electronic structure of Ag16B4O10, both in position and reciprocal space, also considering the interactions between cationic and anionic partial structures.


In structuring our discussion, we qualitatively rank the contributions of the different types of chemical bonding to the total cohesion energy of the title compound.

The markedly largest share results from conventional covalent and ionic bonding, followed by much weaker unconventional d10–d10 interactions [3].

The latter contribution is a crucial ingredient in the formation of the agglomerates of equally charged Ag+, which represent excisions of the ccp structure of elemental silver, even with respect to interatomic distances.

These unique structural characteristics are evolving in silver-rich oxides already without presence of residual 5s electrons, i.e. in electron precise Ag+ compounds.

Still, in an early review [3], the expectation was expressed that these features may enable existence of subvalent species: “The substructures thereby formed have empty s and p conduction bands, which can easily accommodate further electrons on reduction”.

This assumption has become true, as meanwhile several such subvalent silver oxides have been discovered [4–14], in addition to previously confirmed Ag2F and Ag3O [15,16].


The most puristic manifestations of bonding interactions between presumably closed shell d10 species are found in the distorted hcp structures of elemental zinc and cadmium.

Here, the set of commonly equal distances to the 12 nearest neighbours of an atom in an ideal hcp structure is conspicuously split into subsets of six substantially shorter in-plane and six longer out-of-plane separations.

Density functional (DFT) calculations do not give an unequivocal picture of this anisotropy as the results strongly depend on the functionals used [17].

By applying wavefunction based methods in the framework of the method of increments [18,19], the potential energy surface (PES) with respect to the lattice parameters was analysed [20,21].

While the Hartree–Fock PES is overall repulsive, a structured PES consistent with the observed structures can only be obtained if the filled d-shells are included in the treatment of the dynamical electron correlation.


Such calculations, which are computationally highly demanding, have not yet been carried out for subvalent silver compounds, and a quantitative explanation for the d10–d10 bonding in silver-rich compounds, which on its turn would be a crucial prerequisite for rationalizing existence of the subvalent silver oxides under discussion, is still elusive.

However, this issue is not the central subject of the comment by LSR, which focuses on the localisation of the residual 5s electrons.


In their analysis, LSR follow a particular line of reasoning, spotting the evolution of the electron localisation function (ELF) starting from the fcc elemental silver, moving via the silver sub-array as excised from Ag16B4O10 to the integral compound.

The results on the two structures consisting of silver alone are illustrative as such, and underline the merits of this approach for visualising and interpreting the bonding in metals and intermetallics in real space [22].


However, except for confirming that the excess electrons preferably accumulate in the tetrahedral voids (the sites of highest positive potential), the worth of these findings in understanding the bonding in Ag16B4O10 is but limited.

At least from the Ag–O bond lengths, which indicate presence of strong, regular bonds between the silver and borate fragment structures, it is obvious that it is not possible to separate these structural parts without causing severe perturbations.

Thus, the fragment silver structure cannot serve as a reliable reference for the same unit containing the embedded borate anion.

For this reason, we focus in our reply on the electronic properties of the entire configuration of Ag16B4O10.


The computational results obtained in both studies comply satisfactorily.

LSR do not state if they encountered a gapped band structure.

In our PBEsol-GGA-calculations [23] the gap is closed, although showing a minimum of the density of states (DOS) at the Fermi level.

Only by applying a hybrid functional [24], a gap is opened.

According to our experience, however, this issue is not crucial for the evaluation of the ELF.

Importantly, in both studies [1,2] the most populated ELF basins within the silver substructure are found at the same sites.

Yet, the exegeses of the computational results are appreciably divergent.


One must keep in mind that the ELF is not an “observable” in terms of quantum chemistry, since there is no Hamiltonian which operating at an appropriate wave function would reproduce such features, and correspondingly there is no experimental tool available that would allow to directly validate such results.

ELF does not prove the existence of electron pairs, bonds, or lone pairs. But there is a strong and appealing analogy between ELF attractors and basins, and classical Lewis structures, enabling to interpret the 3D-ELF in terms of a conceptual view of bonding.

The analogy gets weaker in intermetallic compounds or in compounds including transition elements.

Especially with late transition elements (like Ag) the values of the valence-(s,p)-attractors are much lower than the values of the d-attractors, see analysis by Kohout, Wagner and Grin [25].

Clearly, interpretation of ELF features falls within the scope of chemical concepts [26].


In our attempt to rationalise the at first glance puzzling experimental observations of an electron imprecise extended oxide to show semiconducting and diamagnetic responses, we assume that the eight excess electrons per formula unit will localise pairwise with opposite spin orientation.

In consent with LSR, we regard the (B4O10)8− anion as electron precise, consequently the excess electrons would be hosted by the silver partial structure classifying this compound as subvalent with respect to silver.


For identifying possible localisations of such electron pairs, we inspected the silver part of the structure for short Ag–Ag separations, which might indicate presence of e.g. 2c–2e bonds, and performed band structure calculations along with an ELF analysis.

As a result, we identified contracted tetrahedral subunits where 3 out of 6 Ag–Ag contacts along the edges of the tetrahedron are significantly shortened and were the ELF shows the highest value not associated to conventional bonds or lone pairs.

Quite satisfactorily, the number of such building blocks per unit cell exactly corresponds to the number of electron pairs to be accommodated.


Being aware that a partition of space, be it for structuring the electron density distribution or the ELF of a chemical entity, always suffers from arbitrariness, even if performed by applying a formal algorithm, we did not claim that the ELF contour drawn within the tetrahedra would comprise a complete pair of electrons with anti-parallel spins.

With this respect LSR misinterpreted our statement that the excess electron pairs were related to the contracted Ag4 units.


LSR go beyond our interpretation by considering regions with lower ELF values, which indicate that the maxima are linked through ELF first-order saddle points or basin interconnecting points (bips) bringing about extended chemical entities, i.e. superbasins.

At this point, we are discussing the spatial extension of the localized electron pair.

Unfortunately, there is no means of reliably validating which of the two views, assuming extended, metallic super basins or more localised electron pairs, would rather be appropriate in interpreting the bonding situation encountered.


In fact, we do not consider the extension of the localized electron pair as being pivotal to the physical properties and the stability of Ag16B4O10.

The analysis of the electronic structure in reciprocal space shows that bands with notable Ag-s-character are found far apart from the Fermi level at −6 eV. Just below the Fermi level, the band structure as given in Fig 7 of ref.

2 shows a low DOS in a range of 1.4 eV. This is due to some bands with high dispersion.

The electron density of these bands is mapped in Fig. 7b of ref.

2, illustrating the linear combination of Ag-d-orbitals and oxygen lone-pairs.

This can be understood as a continuation of d10–d10 interactions in the silver partial structure to the lone-pairs of the B4O10 8− anion via dispersion interaction, stabilizing the whole structure and leading to the semiconducting behaviour of the compound.


From the chemistry perspective of trying to define a generalizable “concept” for this particular type of bonding, one would compare with analogous compounds.

In Fig. 1 we present the silver sub-structure of Ag5GeO4 (ref.

10 and 11) as an example, giving appreciable support to a situation of local bonding.

Here again, the number of contracted polyhedra (octahedral Ag6 units) corresponds to the number of electron pairs to be accommodated, further the silver clusters and are not aggregated which rules out significant delocalisation.


Manifestations of subvalent silver in extended solids can no longer be regarded as exotic singularities since in recent years a sizeable number of oxides displaying such characteristics have been reported [4–16].

From the empirical point of view there are two particularly noteworthy issues: (1) subvalent silver has been encountered in combination with quite diverse cations, e.g. B, Si, Ge, Pb, Ni, Mn, Os or Pt, apparently without any systematic showing up, which nourishes the expectation that many more such candidates will be accessible; (2) opposite to common presumption, these oxides are strikingly stable in humid air and in particular against oxidising conditions, they even form by solid–state reactions applying elevated oxygen pressure.

In a heuristic approach, one may separate the total of specific bonding interactions responsible for the phenomena featured in conventional covalent and ionic contributions, and dispersive d10–d10 forces, superimposed by additional bonding provided by the excess electrons populating bonding 5s bands or local 5s/5p skeleton orbitals.

While the first two contributions appear to be sufficient for forming the extended subarrays of Ag+, providing low lying, empty 5s states [3], suited to accommodate excess electrons, the third component is subordinate and just gently modulates the silver substructures as indicated by global or local contractions.

The low bonding energy of the latter explains the wide phenomenological spread with respect to structure modulations and properties encountered.

Thus, one finds extended silver subarrays e.g. of Ag0.5+ in Ag2NiO2 [6,13,14] or Ag2F [15], or locally shrank tetrahedral or octahedral units embedded in the silver substructure [2,9–12].

Correspondingly, metallic and semi-conducting conduction properties, respectively, have been found.


The novel bonding motif is reminiscent of charge density waves in solid materials as all intermediate stages between delocalized to localized excess electrons would be covered.

However, for the localized scenario there appears to be a closer analogy to the “Polyhedral Skeletal Electron Pair Theory” describing the bonding in the so-called Wade–Mingos molecular clusters [27,28].

In this sense the excess electron pair would occupy the lowest bonding skeleton MO of the embedded silver clusters.


Mac Giolla and Ly begin by "warning against an over-reliance on cut-offs" (p. 2).

A cut-off is the level of evidence that a researcher considers sufficient to conclude that their data supports either hypothesis, or that the data is inconclusive.


Although Mac Giolla and Ly suggest that cut-offs "encourage categorical rather than continuous thinking" (p. 2), we believe it is possible to interpret Bayes factors as a continuous measure while simultaneously acknowledging whether the Bayes factor meets a pre-specified threshold of evidence to enable a categorical interpretation; for example, to state whether an intervention was successful or not, or that the evidence was inconclusive.

This judgment will influence subsequent behaviours, such as whether they encourage or discourage uptake of the intervention when disseminating their results.

Thus, although the use of cut-offs should not prevent the interpretation of the Bayes factor as a continuous measure, cut-offs can nonetheless guide the researcher's decision-making process.


Our purpose is not to discuss where the minimum cut-off should be set, which is a matter of ongoing debate in psychology (e.g. (Aczel et al., 2018; Benjamin et al., 2018; for an argument against cut-offs, see Morey, 2015).

Instead, we urge researchers to clearly specify when they are using cut-offs, and what those cut-offs are.

We believe that Warmelink et al. ( 2019) provide a reasonable and not over-reliant model of how cut-offs can be used: (I) introduce the Bayes factor as a continuous measure, (II) specify their interpretation of the Bayes factor cut-offs, (III) use labels that indicate the increasing strength of evidence, and (IV) always report the exact Bayes factor.

Violating these guidelines might suggest an over- reliance on cut-offs.


The term "nominal support" in Mac Giolla and Ly's commentary is not part of the standard Bayesian vernacular (see Box 1 for a short glossary), but appears to be equivalent to interpreting posterior odds.

We would like to suggest three things that deception researchers should consider before reporting posterior odds.


1) If posterior odds are reported, they should be reported alongside Bayes factors.

Van Ravenzwaaij and Wagenmakers (2019) recently argued that, as the posterior odds are the product of the Bayes factor and the prior odds, both should be reported and justified alongside posterior odds.


2) Mac Giolla and Ly suggest that the use of cut-offs is problematic for Bayes factors, and they then offer posterior odds as an alternative.

Readers might assume that posterior odds therefore avoid the use of cut-offs in a manner that Bayes factors do not.

However, this will again not be the case for researchers who either desire to or are required to categorically interpret their results.

Again, the use of such thresholds to interpret either Bayes factors or posterior odds does not diminish their value as continuous measures.

Researchers can correctly interpret a Bayes factor as a continuous measure and acknowledge whether it met a pre-specified minimum threshold of evidence in order to draw certain categorical conclusions.

The use of such thresholds can help with identifying the required sample size and limit researcher flexibility when interpreting results.


3) Further guidance is required on how deception researchers can specify scientifically- informed prior odds.

Mac Giolla and Ly specify 8% "for the sake of argument", but not all experimental hypotheses are equally (im)plausible, and the degree of scepticism will differ between researchers.

Meanwhile, reporting the Bayes factor allows readers to calculate their own posterior odds based on their prior odds.


Mac Giolla & Ly encourage deception researchers to make their studies more informative by reporting the posterior distribution.

A distinction exists in statistical inference between estimation (e.g. using posterior distributions) and hypothesis testing (e.g. using Bayes factors) (see Dienes, 2020).

To calculate posterior distributions, researchers use vague "default" prior models, which allow the posterior distribution to be primarily determined by the data.

However, vague prior models are not very useful for testing theories, as it is unlikely that the vague prior model represents a scientific theory of interest.

In contrast, Warmelink et al. ( 2019) used prior models that represented their scientific theory (i.e., the prior model is a 'model of H1', see Dienes, 2019, box 4) to calculate informed Bayes factors.


Researchers should exercise caution before automatically calculating posterior distributions using the model of H1 used to calculate informed Bayes factors.

Consider the Bayes factor reported on p. 7 of Warmelink et al (2019): BH(0, 7.05) = 3.64.

The model of H1 was specified using a half-normal distribution (indicated by the subscript H in the notation and the solid black line in Figure 1A and 1B).

The half-normal distribution represents several aspects of the theory: (I) only effects in one direction are considered plausible, these are represented as positive in the analysis, (II) smaller effect sizes are considered more plausible than larger effect sizes, and (III) effect sizes up to twice the approximate scale-of-effect are considered plausible (the approximate scale-of-effect was 7.05, based on the results of Warmelink, 2012).


To see why caution should be exercised before calculating a posterior distribution, consider Figure 1.

The red dotted line in Panel A shows the actual posterior distribution for the data in this example.

There is nothing particularly disconcerting about Panel A - the most plausible effect sizes according to the posterior distribution lie midway between the most plausible effect sizes according to the model of H1 (i.e., those close to zero), and the most plausible effect sizes according to the data (i.e., those close to the observed mean difference of 9.41).


Now consider Panel B. The red dotted line in Panel B shows the posterior distribution assuming that Warmelink et al. ( 2019) had observed the same magnitude of effect, but in the opposite direction predicted by theory.

The posterior distribution is clearly not an accurate estimation of effect sizes in light of the data.

The model of H1 assigned zero plausibility to results in the opposite direction to the prediction and consequently so does the posterior distribution.


In sum, posterior distributions can be informative for estimation purposes when a vague prior model has been used, but researchers should exercise caution before automatically calculating posterior distributions when reporting informed Bayes factors.


In this section we provide two further suggestions that we believe will improve the reporting of Bayesian analyses.


In line with Jeffrey's platitude that "the answer should depend on the question", the result of the Bayes factor depends on the specified models of H1.

APA guidelines state that one should therefore report how sensitive the inference is to the specification of the model of Figure 1.


Both panels present the model of H1 used by Warmelink et al (solid black line), the model of the data (dotted black line) and the posterior distribution (dashed red line).

Panel A shows the actual results.

Panel B shows the posterior distribution for counterfactual data with the same effect size, but in the opposite direction to what was predicted.


H1 whenever reporting Bayes factors (Appelbaum et al., 2018, p. 20) 1 .

This can be done by reporting robustness regions alongside Bayes factors (for examples see Lakens et al., 2020; Lin, et al., in press).

Robustness regions communicate the extent to which the inference drawn from a Bayes factor is dependent on the model of H1 used to calculate the Bayes factor.

If all plausible effect sizes that could be used to model H1 fall within the robustness region, the inference is robust; if plausible effect sizes that could be used to model H1 fall outside the robustness region, the inference is not robust.


Robustness regions are reported as RR[LL, UL] where LL corresponds to the smallest scale-of-effect and UL corresponds to the largest scale-of-effect that draw the same conclusion as the model of H1 used to calculate the Bayes factor assuming the specified minimum cut-off for evidence is constant.

Consider again the Bayes factor discussed above: BH(0, 7.05) = 3.64.

Warmelink et al. ( 2019) concluded that they had obtained moderate evidence for an effect, when specifying their model of H1 with a scale-of-effect of 7.05 and a minimum cut-off of B = 3.


To create robustness regions, we calculated a series of Bayes factors, each time manipulating the scale-of-effect to identify the smallest and largest scale-of-effect that still produces B > 3 (see DeBruine and Dienes, 2020, for R code that calculates Robustness Regions non-iteratively).


In this case, the robustness region is RR[4.44, 15.74].

Whether this can be considered robust or not depends on the scientific context.

If effects outside the robustness region are scientifically plausible, the inference is not robust.

If the robustness regions are judged to cover all theoretically plausible effect sizes, the inference is robust.

In this example, we conclude that the results are not robust.


Many deception researchers may argue that an effect smaller than 4.44 details would be too small to be of interest.

However, an effect larger than 15.74 can be considered plausible and interesting (e.g. Vrij et al., 2018, report a mean difference of 29 details between two interviewing conditions) 2 .


As Mac Giolla and Ly highlight, 19 of the 22 Bayes factors reported by Warmelink et al. ( 2019) were inconclusive.

Here we provide a simple method (with corresponding R script available at https://osf.io/34nb6/, based on original by Palfi & Dienes, 2019) that researchers can adopt to estimate the number of participants required in order to obtain a pre-specified level of evidence (originally outlined by Dienes, 2015).


As sample size increases, and the standard error reduces towards zero, Bayes factors are better able to distinguish between evidence for two competing hypotheses.

The process requires one to (i) estimate an approximate scale-of-effect if H1 were true (for a tutorial on how to estimate what your theory predicts, see Dienes, 2019), (ii) estimate the standard error for a given sample size (e.g., the standard error from a past study), and (iii) specify a minimum level of evidence one wishes to obtain.

The R script then simulates the standard error in light of increasing sample size, and calculates Bayes factors for each sample size.

It does this assuming the evidence supports H1 and H0 separately.

The script also outputs a graph to show the Bayes factor for H0 and H1 at each tested sample size.


Figure 2 shows the graph created assuming a researcher wanted to replicate the result from Warmelink et al. ( 2019), BH(0, 7.05) = 3.64, with a minimum cut-off of 3.

In this case, a replication would require 130 participants to obtain a Bayes factor of 3 in favour of H1 assuming the study supported H1, or 328 participants to obtain a Bayes factor of 3 in favour of H0 assuming the study supported H0 3 .

It is common that more participants are required to obtain evidence for the 2 To interpret robustness regions, it can be useful to consider the smallest effect of interest.

Dienes (2020) provides guidelines on how to do so.

If one is uncertain whether a finding is robust, however, one can continue to collect data until the robustness region extends below any plausible smallest effect size of interest.

3 Assumptions include that the effect size and sample error obtained by Warmelink et al (2019) are accurate.

point-null hypothesis than for the alternative hypothesis.

In this example, to ensure enough participants are recruited that the study could obtain sufficient evidence for either hypothesis, a replication study should look to recruit at least 328 participants.


We whole-heartedly agree with Mac Giolla and Ly when they encourage researchers to design more informative studies and we are grateful for their commentary that sought to educate deception researchers on matters of reporting and interpreting Bayesian analyses.

Here, we have sought to add a number of clarifications, and also specified two additional ways that the Bayesian analyses in Warmelink et al. ( 2019) could have been more informative.

If researchers adopt these recommendations, we believe they will be better able to assess the evidence provided by their research.


There is potential for confusion between the different uses of the word prior and posterior in Bayesian analyses.

Here we provide an overview.


Prior odds specify the relative degree of conviction a researcher has in two theories.

If a person believes each hypothesis is equally plausible, that person has prior odds of P(H1) / P(H0) = .50/.50 = 1.

Prior odds are updated by the level of evidence in the data (i.e., the Bayes factor) to derive the posterior odds: Prior odds x Bayes factor = Posterior odds.

The posterior odds specify the relative degree of conviction a researcher has in two hypotheses following data collection.


A prior model is a probability distribution that shows the plausibility of different population effect sizes (x-axis = possible effect sizes, y-axis = plausibility, see Figure 1).

Warmelink et al (2019) used informed Bayes factors, meaning that the prior model was specified to represent the theory being tested (Dienes, 2019, Box 4).

Their prior models were created by specifying the approximate effect size (i.e., the "scale-of-effect") that is plausible if the theory is correct.

Prior models can be updated with the model of the data to create the posterior distribution.

The posterior distribution demonstrates the plausibility of population effect sizes following data collection assuming the hypothesis is true.

Note that "distribution" and "model" are used interchangeably.


In order to avoid confusion, we adopt Dienes's (2020) vernacular, referring to the prior model as the "model of H1", to reflect the fact that the prior model represents predictions made by the theory.


Bayes factors quantify the evidence in support of the null (absence of an effect) or the alternative hypothesis (presence of an effect).

Based on commonly used cut-offs, Bayes factors between 1/3 and 3 are interpreted as evidentially weak, and one typically concludes there is an absence of evidence.

In this commentary on Warmelink, Subramanian, Tkacheva, and McLatchie (2019) [1], we discuss how a Bayesian report can be made more informative.

Firstly, this implies a departure from the labels provided by commonly used cut-offs when reporting Bayes factors.

Instead, we encourage researchers to report the value of the Bayes factors, or to convert these values into nominal support for the hypotheses.

Secondly, researchers can provide recommendations to design follow-up studies by examining the posterior distribution of the magnitude of the effect size.

Lastly, we show how individual Bayes factors can be evaluated in the context of large-scale meta-analyses.


A crucial distinction when interpreting the results of a study is the difference between the absence of an effect from the absence of evidence.

This distinction is typically ignored within frequentist statistics, where ‘non-significant’ p values (p > .05) are incorrectly interpreted as an absence of an effect [2].

For this reason alone, Warmelink, Subramanian, Tkacheva, and McLatchie (2019) [1] should be commended for reporting Bayes factors, which can quantify the evidence for the alternative (presence of an effect) over the null hypothesis (absence of an effect), and vice versa.

For instance, a Bayes factor BF10 = 8 implies that the observations are eight times more likely under the hypothesis that the effect is present than under the hypothesis that the effect is absent.

Similarly, BF10 = 0.125 (equivalently, BF01 = 1/BF10 = 8) implies that the data are 8 times more likely under the null compared to the alternative.


To aid interpretation, cut-offs (e.g., Jeffreys, 1961) [3] for Bayes factors have been proposed.

These cut-offs suggest that Bayes factors between 1 and 3 (1/3 and 1) provide only weak evidence for the alternative hypothesis relative to the null hypothesis (or for the null hypothesis relative to the alternative; e.g., Jarosz & Wiley, 2014) [4].

Hence, when confronted with evidentially weak Bayes factors, the common conclusion is that there is an absence of evidence [5], and the results are perceived as inconclusive.


The reporting of Bayes factors is still quite novel within deception research, though it appears to be gaining popularity [6][7].

We believe this is a good thing, but it does bring with it new challenges for reviewers and editors.

Using Warmelink, Subramanian, Tkacheva, and McLatchie (2019) [1] as a concrete example, we elaborate on the continuous nature of Bayes factors and how to interpret them.

We begin by warning against an over-reliance on cut-offs.

Next, we give some suggestions on how to make a Bayesian report more informative.


Risk of cut-offs and reporting alternatives
As noted above, cut-offs have been suggested to aid the interpretation of Bayes factors.

These cut-offs, however, should be used with caution.

Let us take the results of Warmelink et al. ( 2019) [1] as a concrete example.

In total, they report 22 Bayes factors.

Nineteen of these are between 1/3 and 3, and, therefore, evidentially weak according to the cut-offs they themselves adopt.


The remaining three analyses resulted in Bayes factors of BF10 = 3.58, BF10 = 3.64, and in the opposite direction BF01 = 4.76, which were interpreted to provide ‘moderate’ evidence.

This interpretation highlights the main risk of cut-offs: They encourage categorical rather than continuous thinking.

Yes, a Bayes factor of 3.58 provides more evidence than a Bayes factor of 2.8, but to say that one provides moderate evidence, whereas the other provides weak or inconclusive evidence, seems somewhat misguided [8].


The boundaries between the categories further fade away, when we convert a Bayes factor into posterior model probabilities.

A Bayes factor only measures the strength of the relative evidence provided by the data, whereas the posterior model probabilities quantify the nominal support for the hypotheses after data observation.

To calculate the posterior model probabilities, we combine the prior plausibility of the hypotheses with the Bayes factor using the following formula:


P(H1|data) = (BF10 * P(H1)) / (BF10 * P(H1) + P(H0))
P(H0|data) = 1 - P(H1|data)


For instance, if the plausibility of the presence and the absence of an effect are equal before data observation, then we set P(H1) = 0.5 and P(H0) = 0.5.

Observations leading to a Bayes factor of BF10 = 3.58 then yield a nominal support of 78%, P(H1|data) = 3.58 * 0.5/(3.58 * 0.5 + 0.5), for the hypothesis that there is an effect, leaving a posterior probability of 22%, P(H0|data) = 1 - 0.78, for the hypothesis that the effect is absent.

These probabilities can in turn be visualized using a simple pie chart known as a pizza plot [8].


Now look at Figure 1.

The dark portion of the pie chart corresponds to the 78% in support of H1.

Imagine that this chart was on the table in front of you.

You close your eyes, spin the chart, and at random place your finger down.

Now open your eyes.

How surprised would you be to find your finger on the white portion of the pie chart? Your degree of surprise is an intuitive measure of the strength of the nominal support for H1 [8].


Our guess is you would not be that surprised – there was after all a 22% probability of your finger landing on the white portion.

Similar computations show that the ‘evidentially weak’ Bayes factor of 2.8 would lead to a posterior probability of 74% in support of H1 and 26% for H0.

In terms of nominal support for the hypothesis, the distinction between ‘evidentially weak’ and ‘moderate’ Bayes factors in this example is negligible.


Hence, the categorizations are not that informative, and we strongly encourage researchers to discuss their results using the value of the Bayes factors or the posterior probabilities for each hypothesis instead.


Again, we can use Warmelink et al. ( 2019) [1] as a concrete example.

In their discussion, they conclude that Hypothesis 2 was partly supported and that Hypothesis 3 was not supported.

We worry that such language exaggerates the strength of evidence of the data.

Ultimately, researchers should be fair to the uncertainty implied by the Bayes factors.

This uncertainty is lost when we focus on labels such as ‘weak’ and ‘moderate’, and when the relative evidence is not converted to nominal support.


However, simply reporting that results are uncertain will likely leave many reviewers and readers wanting.

We see at least two options to help researchers provide more informative Bayesian reports: first, by focusing on the posterior distribution of the effect size and second, by relating Bayes factors to the broader research field by discussing prior model probabilities.

We will discuss each approach in turn.


Beyond Bayes factors: Examining the posterior distribution of the effect sizes
A Bayesian report can be made more informative by studying the posterior distribution of the effect size parameter.

This can provide us with an estimate of the magnitude of the effect, given that it exists, or provide us with information on how to design a follow-up study.

For instance, say we conduct a large sampled between-group study (N = 1,000) and find a standardized between-groups difference (Cohen’s d) = 0.1.

With a default Bayesian t-test, this would produce a Bayes factor around 1.5 in support of the null hypothesis, that there is no difference between the two groups.

Although this result helps little in determining whether the null or alternative is true, by examining the posterior distribution of the parameter, and the associated credible interval (the Bayesian equivalent to the frequentist confidence interval), one can still learn something from the data.

Specifically, that if there is an experimental effect, it is likely to be very small indeed (for a pedagogical online tool to visual this point see https://rpsychologist.com/d3/bayes/).

Such a conclusion is considerably more valuable than what one can glean by solely focusing on the Bayes factor.

Furthermore, with the estimate of the effect in hand, one can then efficiently plan follow-up studies.


The potential role of prior model probabilities
Another way to move beyond a simple reporting of Bayes factors is to use more informed prior model probabilities when calculating the nominal support for hypotheses.

In order to calculate the nominal support for hypotheses, some form of prior model probabilities must be specified.

Above, for lack of better choice, we had equal prior model probabilities: P(H0) = 0.5 and P(H1) = 0.5.

That is, before looking at the data, H0 and H1 were seen as equally likely.

More informed prior model probabilities can be derived from large-scale field-wide meta-analyses, or better still can be agreed upon by the research community.

In particular, if one has reason to be sceptical for the presence of an effect, the prior model probabilities can be weighted in favour of H0, thereby correcting for false positives.


In the context of deception detection, we have strong reason to be highly sceptical to most cues to deceit.

It seems that the research on deception cues, prior to 2003, is compatible with a world in which there are no reliable cues to deceit at all [9].

Hence, in the best case, Luke’s findings suggest that only a small fraction of deception studies lead to the detection of an actual effect – a genuine cue to deception.


For the sake of argument, let us say that only 8% of the published articles report an actual deception cue, leaving 92% of the published articles reporting a false-positive finding.

Thus, P(H1) = 0.08 and P(H0) = 0.92.

This interpretation of Luke’s finding can now be used as a context to evaluate the moderate BF10 = 3.58 as reported in Warmelink et al. ( 2019) [1] using the Equation (1).


A direct calculation shows that the nominal support for H1 of 8% is then increased to 24%, P(H1|data) = 3.58 * 0.08/(0.92 + 3.58 * 0.08), which leaves a posterior probability of 76% in support for H0.

Hence, there is indeed relative evidence for H1, because the Bayes factor is larger than 1, but based on the posterior model probabilities we cannot conclude that there is more nominal support for H1 than H0.

In other words, the relative evidence of BF10 = 3.58 is not enough to overcome the initial scepticism brought about by Luke’s findings.


By changing P(H0) = 0.5 and P(H1) = 0.5 to the more stringent P(H0) = 0.92 and P(H1) = 0.08, we now require BF10 to be larger than 11.5 (=0.92/0.08) before the posterior model probability for H1 exceeds that of H0.

In other words, we changed the threshold of preference – the strength of the evidence needs to be at least 11.5 before we begin to consider a found deception cue feasible.

Moreover, we now need a Bayes factor of 40.77, which can be verified using Equation (1), before we get nominal support of P(H1|data) = 0.78.


Our aim here was to demonstrate how prior model probabilities can let us draw important conclusions about the hypotheses and thereby provide more informative results than simply reporting Bayes factors.

We wish to stress however that our suggested base rate of 8% for H1 is only meant for illustrative purposes.

This matter should be discussed at the level of the research community.


Concluding remarks
We provided some suggestions on how to make a Bayesian report more informative.

Firstly, we warn against an over-reliance on the standard Bayes factor cut-offs.

Instead, we encourage researchers to report the value of the Bayes factors or convert it to the nominal support for the hypotheses.

Secondly, we encourage researchers to go beyond a simple reporting of Bayes factors.

This can be achieved, for example, by studying the posterior distribution of the effect size or by evaluating Bayes factors in the context of a large-scale meta-analysis using Equation (1).


Ultimately, however, there is no quick fix for making studies more informative after the fact.

Greater care must be taken before data collection to increase the chances of producing compelling results (for guidelines see Schönbrodt & Wagenmakers, 2018 [10] and see Ly et al., 2019 [11], on how to quantify replicability).

Luke (2019) [9] brought the consequences of ignoring this advice to light.


It seems that the extant research on deception cues, prior to 2003, is compatible with a world in which there are no reliable cues to deceit at all.

Individual small-sample studies, whose effects may well be nothing more than sampling variation, were over-interpreted, published, and, in worst case scenarios, used for policy recommendations.

It took some 60 years for this issue to come to light in DePaulo et al. ’s (2003) meta-analysis [12]. And almost another 20 for us to understand the true consequences of this practice in Luke’s re-analysis of the data.

One can rightly wonder how much quicker we would have come to this conclusion if studies from the outset had been designed to provide more informative and compelling results.


Since DePaulo et al. ’s meta-analysis, deception researchers’ focus on traditional cues to deceit has waned.

Instead, there is a new focus on interviewers actively eliciting cues to deceit, (Vrij & Granhag, 2012) [13], of which Warmelink et al. ( 2019) [1] is one example.

Rather than passively observing truth tellers and liars, these methods consist of asking questions in a strategic manner in order to increase differences in the statements or behaviours between these groups.

This new approach may be a genuine path to a viable method of deception detection, or it may be another dead end. If we do not start designing our studies to produce more informative results, we may need to wait another 80 years to find out.


Acknowledgements
We would like to thank Timothy J. Luke and Eric-Jan Wagenmakers for their helpful comments on earlier versions of this paper.


Conflicts of interest
All authors declare no conflict of interest.