Frederic Lanore

R Angus Silver
email: a.silver@ucl.ac.uk.

Extracting quantal properties of transmission at central synapses

Keywords: Synapse, Vesicle, Active zone, Release probability, Release site, Quantal analysis, MPFA, Variance-mean analysis

Chemical synapses enable neurons to communicate rapidly, process and filter signals and to store information. However, studying their functional properties is difficult because synaptic connections typically consist of multiple synaptic contacts that release vesicles stochastically and exhibit time-dependent behavior. Moreover, most central synapses are small and inaccessible to direct measurements. Estimation of synaptic properties from responses recorded at the soma is complicated by the presence of nonuniform release probability and nonuniform quantal properties. The presence of multivesicular release and postsynaptic receptor saturation at some synapses can also complicate the interpretation of quantal parameters. Multiple-probability fluctuation analysis (MPFA; also known as variance-mean analysis) is a method that has been developed for estimating synaptic parameters from the variance and mean amplitude of synaptic responses recorded at different release probabilities. This statistical approach, which incorporates nonuniform synaptic properties, has become widely used for studying synaptic transmission. In this chapter, we describe the statistical models used to extract quantal parameters and discuss their interpretation when applying MPFA.

Introduction

Fluctuations in the amplitude of evoked end-plate potentials recorded intracellularly from frog muscle fibers bathed in high Mg 2+ -containing solution (1, 2) together with their similarity to spontaneous end-plate potentials (which was first thought to be noise (3)) lead to the quantum hypothesis -the idea that neurotransmitter is released in discrete multimolecular packets or quanta. The probabilistic nature of this all-or-none process was rigorously tested using a number of different approaches [START_REF] Del Castillo | Quantal components of the end-plate potential[END_REF]. This work showed that fluctuations in the evoked end-plate potentials could be predicted with Poisson statistics under low probability conditions. Soon after, early electron microscopy studies revealed the presence of synaptic vesicles in motor nerve terminals [START_REF] De Robertis | Some Features of the Submicroscopic Morphology of Synapses in Frog and Earthworm[END_REF], providing a structural basis for the "quantum" [START_REF] Katz | The release of neural transmitter substances[END_REF].

Quantal analysis refers to a group of methods that use statistical models to extract the basic functional properties of synapses from postsynaptic responses, which are typically measured at the soma. Quantal analysis can provide insights into the function of synapses and identify the locus of changes in synaptic efficacy. Three quantal parameters are now commonly used to determine the properties of synaptic transmission: the first is the maximum number of vesicles that could be released at a synaptic connection by an action potential and is often referred to as the number of independent functional release sites (N), the second is the probability of vesicular release (P) and third, the amplitude of the postsynaptic response following the release of a single vesicle or quantum (Q). The size of the postsynaptic responses and its variability from trial-to-trial are determined by the values of these quantal parameters. Presynaptic modulation is associated with P, while postsynaptic changes are associated with Q. Formation of new contacts (or increasing P from zero at existing release sites) would be associated with a change in N.

Early attempts to apply quantal analysis to central synapses met with mixed success due to difficulties in resolving individual quantal events [START_REF] Kuno | Quantal components of excitatory synaptic potentials in spinal motoneurones[END_REF]. Nevertheless, it was recognized that binomial rather than Poisson statistical models were more appropriate given the relatively high release probability and few release sites present [START_REF] Kuno | Quantal components of excitatory synaptic potentials in spinal motoneurones[END_REF]. Considerable efforts were made to extract quantal parameters from amplitude distributions in subsequent studies, but this approach was challenging and it was often difficult to determine the reliability of the end results because the amplitude of quantal events often fell below the noise at central synapses [START_REF] Malinow | Presynaptic enhancement shown by whole-cell recordings of long-term potentiation in hippocampal slices[END_REF][START_REF] Bekkers | Presynaptic mechanism for long-term potentiation in the hippocampus[END_REF][START_REF] Larkman | Quantal analysis of excitatory synaptic action and depression in hippocampal slices[END_REF][START_REF] Edwards | LTP is a long term problem[END_REF][START_REF] Kullmann | Long-term potentiation is associated with increases in quantal content and quantal amplitude[END_REF]. Moreover, during this period, there was growing evidence of nonuniform quantal properties [START_REF] Walmsley | Nonuniform release probabilities underlie quantal synaptic transmission at a mammalian excitatory central synapse[END_REF][START_REF] Rosenmund | Nonuniform probability of glutamate release at a hippocampal synapse[END_REF]. The complications arising from such nonuniform quantal size and nonuniform release probability were highlighted in a simulation study that demonstrated the difficulties of using traditional quantal analysis approaches for studying central synaptic transmission [START_REF] Walmsley | Interpretation of "quantal" peaks in distributions of evoked synaptic transmission at central synapses[END_REF].

To overcome these problems a different statistical approach was developed to estimate quantal parameters at central synapses, which has its roots in non-stationary fluctuation analysis of ion channels [START_REF] Sigworth | The variance of sodium current fluctuations at the node of Ranvier[END_REF] and synaptic currents [START_REF] Traynelis | Estimated conductance of glutamate receptor channels activated during EPSCs at the cerebellar mossy fiber-granule cell synapse[END_REF][START_REF] Silver | Non-NMDA glutamate receptor occupancy and open probability at a rat cerebellar synapse with single and multiple release sites[END_REF]. Multiple-probability fluctuation analysis (MPFA) [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF] or alternatively variance-mean analysis [START_REF] Reid | Postsynaptic expression of long-term potentiation in the rat dentate gyrus demonstrated by variance-mean analysis[END_REF][START_REF] Clements | Unveiling synaptic plasticity: a new graphical and analytical approach[END_REF][START_REF] Clamann | Variance analysis of excitatory postsynaptic potentials in cat spinal motoneurons during posttetanic potentiation[END_REF] does not require that quantal events be distinguished from the background noise. Moreover, the statistical model underlying MPFA is multinomial and can therefore take into account nonuniformities in P and Q. MPFA has been used to determine the quantal parameters of transmitter release at low frequency [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF][START_REF] Lanore | Deficits in Morphofunctional Maturation of Hippocampal Mossy Fiber Synapses in a Mouse Model of Intellectual Disability[END_REF] and has been extended to short repetitive trains of synaptic responses [START_REF] Meyer | Estimation of quantal size and number of functional active zones at the calyx of held synapse by nonstationary EPSC variance analysis[END_REF][START_REF] Saviane | Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse[END_REF][START_REF] Hallermann | Bassoon Speeds Vesicle Reloading at a Central Excitatory Synapse[END_REF]. It has also been combined with analysis of covariance between successive stimuli within trains [START_REF] Scheuss | Estimating synaptic parameters from mean, variance, and covariance in trains of synaptic responses[END_REF][START_REF] Neher | Combining deconvolution and fluctuation analysis to determine quantal parameters and release rates[END_REF]. MPFA is a valuable tool to quantify the pre-and postsynaptic contributions to short-term plasticity changes [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Saviane | Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse[END_REF][START_REF] Hallermann | Bassoon Speeds Vesicle Reloading at a Central Excitatory Synapse[END_REF][START_REF] Oleskevich | Release probability modulates short-term plasticity at a rat giant terminal[END_REF][START_REF] Biró | Quantal size is independent of the release probability at hippocampal excitatory synapses[END_REF][START_REF] Biró | Release probability-dependent scaling of the postsynaptic responses at single hippocampal GABAergic synapses[END_REF][START_REF] Humeau | Fast changes in the functional status of release sites during short-term plasticity: involvement of a frequency-dependent bypass of Rac at Aplysia synapses[END_REF][START_REF] Valera | Adaptation of Granule Cell to Purkinje Cell Synapses to High-Frequency Transmission[END_REF]. This quantal analysis method has also been used to determine whether long-term plasticity changes are expressed pre-or postsynaptically [START_REF] Reid | Postsynaptic expression of long-term potentiation in the rat dentate gyrus demonstrated by variance-mean analysis[END_REF][START_REF] Sola | Increased neurotransmitter release during long-term potentiation at mossy fibre-granule cell synapses in rat cerebellum[END_REF][START_REF] Fourcaudot | cAMP/PKA signaling and RIM1alpha mediate presynaptic LTP in the lateral amygdala[END_REF].

In this chapter, we describe the statistical basis of MPFA, including the binomial and the multinomial models of synaptic transmission. We also describe how to apply MPFA at central synapses and discuss the interpretation of the quantal parameters with this method.

Statistical models of synaptic transmission 2.1 Binomial model

The binomial model assumes that each vesicle is released independently, that the release is synchronous, that P is uniform across vesicles and that Q is uniform both at the level of a single release site and across release sites. Under these assumptions the mean peak amplitude of the synaptic current response (I) can be expressed as a function of the number of functional release sites (N), the quantal amplitude (Q) and the probability of vesicular release (P) as follows:

I = NPQ (1)
and the associated variance can be expressed as:

σ 2 = NQ 2 P(1 -P) (2)
The relationship between the variance and the mean synaptic amplitude is therefore:

σ 2 = IQ - I 2 N ( 3
)
These results suggest that if a synaptic connection operates in a simple binomial manner, the relationship between the variance and the mean amplitude of synaptic responses is parabolic. Fitting a function of the form y = Ax -Bx 2 , (where y is the variance and x is the mean postsynaptic current) to the relationship between the variance and mean current recorded at different release probabilities can provide estimates of N and Q (Figure 1). P can be then calculated from equation (1).

Multinomial model

Several studies have shown that the assumptions of uniform release probability and quantal size required for a simple binomial model are not valid for most central synapses [START_REF] Walmsley | Nonuniform release probabilities underlie quantal synaptic transmission at a mammalian excitatory central synapse[END_REF][START_REF] Murthy | Heterogeneous release properties of visualized individual hippocampal synapses[END_REF][START_REF] Bekkers | Origin of variability in quantal size in cultured hippocampal neurons and hippocampal slices[END_REF][START_REF] Silver | Rapid-time-course miniature and evoked excitatory currents at cerebellar synapses in situ[END_REF]. These considerations lead to the application of a multinomial model for transmitter release [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Frerking | Effects of variance in mini amplitude on stimulus-evoked release: a comparison of two models[END_REF].

2.2.1

Nonuniform quantal size-In the multinomial model, quantal variability can be accommodated both at the single-site level (intrasite or type I variability; Figure 2a) [START_REF] Silver | Non-NMDA glutamate receptor occupancy and open probability at a rat cerebellar synapse with single and multiple release sites[END_REF][START_REF] Bekkers | Origin of variability in quantal size in cultured hippocampal neurons and hippocampal slices[END_REF] and across sites (intersite or type II variability; Figure 2b) [START_REF] Bekkers | Origin of variability in quantal size in cultured hippocampal neurons and hippocampal slices[END_REF][START_REF] Borst | Large amplitude variability of GABAergic IPSCs in melanotrophs from Xenopus laevis: evidence that quantal size differs between synapses[END_REF]. Intrasite quantal variability arises from fluctuations in the size of quantal events from an individual release site from trial-to-trial and from fluctuations in their latencies. These sources of type I variance can be defined in terms of their coefficient of variation (CV QS and CV QL respectively). Intersite variability arises from sampling quanta from different release sites, each of which has different mean quantal sizes. This type II variance can be defined in terms of the coefficient of variation CV QII .

Equation ( 1) can be extended to include nonuniform quantal behavior as follows:

I = NPQ P (4)
where Q P is the mean quantal size at the time of the peak of the mean synaptic current and is therefore affected by asynchronous release (Figure 2c, d) (for details see ref 41). The variance is then given by:

σ 2 = NQ P 2 P(1 -P)(1 + CV QII 2 ) + NQ P 2 PCV QI 2 (5)
Where CV QI combines all the variability observed at a single-site level:

CV QI = CV QL 2 + CV QS 2 (6)
The relationship between the variance and the mean I is therefore:

σ 2 = Q P I - I 2 N 1 + CV QII 2 + Q P ICV QI 2 (7)
The sum of the intra-and intersite variability can be defined in terms of the coefficient of variation as follows:

CV QT = CV QI 2 + CV QII 2 = CV QS 2 + CV QL 2 + CV QII 2 (8)
More details on how the different quantal variances can be estimated are given in section 3.3.

Nonuniform release probability-Studies

in the spinal cord [START_REF] Walmsley | Nonuniform release probabilities underlie quantal synaptic transmission at a mammalian excitatory central synapse[END_REF], hippocampal cultures [START_REF] Rosenmund | Nonuniform probability of glutamate release at a hippocampal synapse[END_REF][START_REF] Murthy | Heterogeneous release properties of visualized individual hippocampal synapses[END_REF][START_REF] Branco | Local dendritic activity sets release probability at hippocampal synapses[END_REF], and hippocampal slices [START_REF] Holderith | Release probability of hippocampal glutamatergic terminals scales with the size of the active zone[END_REF] have shown that the probability of release is nonuniform across individual synapses. The presence of nonuniform release probability tends to reduce the variance when compared to the uniform case. The impact on the variance of nonuniform P is largest at high P, leading to a wedge-shaped distortion of the variancemean relationship (Figure 2d) at high levels of nonuniformity [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Silver | Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations[END_REF]. This behavior can be captured in the multinomial model relatively neatly assuming that there is no correlation between release probability and quantal amplitude across sites [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Frerking | Effects of variance in mini amplitude on stimulus-evoked release: a comparison of two models[END_REF]. In this case the mean current and the variance are given by: I = NPQ P (9)

σ 2 = NQ P 2 P[1 -P(1 + CV P 2 )] 1 + CV QII 2 + NQ P 2 PCV QI 2 ( 10
)
where P is the mean release probability at the time of the peak of the current and CV P represents the coefficient of variation of release probability across sites. The relationship between the variance and the mean amplitude is then:

σ 2 = Q P I - I 2 N (1 + CV P 2 ) 1 + CV QII 2 + Q P ICV QI 2 (11)
Given that P is bounded by 0 and 1, CV P changes as a function of P. CV P and how it changes has been modeled using families of beta functions β(α,β) (41).These functions mimic the distribution of release probability across release sites and describe how the distribution might change when P varies. Furthermore, beta distributions approximate distributions that have been measured in hippocampal synapses in culture [START_REF] Murthy | Heterogeneous release properties of visualized individual hippocampal synapses[END_REF]. Using this approach CV P can be expressed as a function of the mean release probability and a family of beta distributions defined by a single parameter α:

CV P = 1 -P P + α ( 12
)
Substituting for CV P in equation [START_REF] Kullmann | Long-term potentiation is associated with increases in quantal content and quantal amplitude[END_REF] gives:

σ 2 = Q P I - Q P I 2 (1 + α) I + NQ P α (1 + CV QII 2 ) + Q P ICV QI 2 (13)
adding only one additional free parameter to the expression. Low α values (<2) indicate nonuniform release probability, while higher values indicate that the probability of release is uniform [START_REF] Silver | Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations[END_REF]. However, the beta function is an approximation and α is the least wellconstrained parameters in equation ( 13); thus estimates of CV P should be considered as a rough indicator of the level of nonuniformity in release probability.

MPFA during short-term plasticity

MPFA can be extended to examine how quantal parameters change during bursts of activity by determining the quantal parameters from first evoked postsynaptic current (PSC) in the burst and then analyzing the fluctuations in the subsequent events with CV analysis (Figure 3) [START_REF] Saviane | Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse[END_REF]. P can be estimated from the CV of evoked PSCs with the following equation:

CV = σ I = (1 -P) 1 + CV QII 2 + CV QI 2 NP (14)
Changes in Q P can be then calculated from equation ( 4) during the train, assuming N is constant. While simple to implement, this approach does assume that release probability is relatively uniform.

Interpretation of quantal parameters

The multinomial model provides a statistical description of release that incorporates both nonuniform presynaptic and postsynaptic properties. However, other synaptic properties, notably the occupancy of the postsynaptic receptors, affect the interpretation of the quantal parameters estimated with MPFA. This section describes how to interpret the synaptic parameters obtained with MPFA, discusses potential errors, and outlines some additional tests that can be performed to check that the interpretation is correct.

Interpretation of the number of release sites-When the postsynaptic

quantal responses sum linearly, the amplitude of the PSC reflects the number of vesicles released. Under these conditions N the maximum number of readily release vesicles that could be released by the synaptic connection following an action potential. N therefore corresponds to the number of anatomically distinct synaptic contacts only if a maximum of one vesicle is released per synaptic contact (univesicular release) [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF][START_REF] Biró | Quantal size is independent of the release probability at hippocampal excitatory synapses[END_REF][START_REF] Silver | Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations[END_REF].

In the case of multivesicular release, when multiple vesicles are released at each synaptic contact [START_REF] Biró | Release probability-dependent scaling of the postsynaptic responses at single hippocampal GABAergic synapses[END_REF][START_REF] Wadiche | Multivesicular release at climbing fiber-Purkinje cell synapses[END_REF][START_REF] Christie | Multivesicular release at Schaffer collateral-CA1 hippocampal synapses[END_REF], N will still reflect the total number of functional release sites or equivalently the maximum number of readily release vesicles that could be released by the synaptic connection following an action potential, if quanta sum linearly. But if the postsynaptic receptors become saturated [START_REF] Foster | Variance-mean analysis in the presence of a rapid antagonist indicates vesicle depletion underlies depression at the climbing fiber synapse[END_REF], quantal events no longer sum up because the postsynaptic membrane becomes insensitive to subsequent transmitter release. In the extreme case, where a single vesicle saturates the receptors at a synaptic contact, N would correspond to the number of synaptic contacts where release can occur rather than reflecting the maximum number of vesicles that can be released at those sites (which could be much larger) [START_REF] Silver | Non-NMDA glutamate receptor occupancy and open probability at a rat cerebellar synapse with single and multiple release sites[END_REF]. Thus, the interpretation of N depends on whether the postsynaptic receptors become saturated or not. Luckily, this can be tested by examining whether the postsynaptic receptor occupancy changes as a function of release probability using rapidly equilibrating low-affinity competitive antagonists (see section 3.4) [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF].

At some synapses, spillover of neurotransmitter from neighboring release sites gives rise to a slow current component [START_REF] Digregorio | Spillover of glutamate onto synaptic AMPA receptors enhances fast transmission at a cerebellar synapse[END_REF][START_REF] Barbour | Intersynaptic diffusion of neurotransmitter[END_REF][START_REF] Sakaba | Estimation of quantal parameters at the calyx of Held synapse[END_REF]. This spillover-mediated current can introduce a small low variance current component that can lead to an overestimation of N [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF]. At cerebellar mossy fiber-granule cell synapses, spillover current can be observed in isolation when direct release fails [START_REF] Digregorio | Spillover of glutamate onto synaptic AMPA receptors enhances fast transmission at a cerebellar synapse[END_REF]. If the spillover current and direct release summate linearly, the mean spillover current and its variance can simply be subtracted at the time of the peak of the postsynaptic response at each probability of release [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF].

Interpretation of CV P and P-As

for N, interpretation of P also depends on whether quantal responses sum linearly or not. Under conditions when quanta sum linearly, P represents the mean probability that a vesicle is released by the time of the mean peak postsynaptic response following an action potential, i.e., the integral of the release rate per functional release site up to that time (50). P can be described in terms of the product of the mean probability of a vesicle being in the release-ready state (P R ) and the mean probability that a release-ready vesicle will undergo fusion following an action potential (P F ) (51, 52):

P = P R P F ( 15
)
This interpretation is valid for both univesicular and multivesicular release, if the postsynaptic response is linear. On the other hand, if the postsynaptic receptors are saturated by a single vesicle, P indicates the mean probability that one or more vesicles have been released at each synaptic contact.

At many synaptic connections, release probability has been found to be reasonably uniform leading to a parabolic variance-mean relationship. When CV P is large, the variance-mean relationship takes on an increasingly wedge-like shape due to the loss of variance at high P (41). The distribution of release probability and how it changes with P can be approximated by a family of beta distributions with the same α value. This approximation is unlikely to give the fine details of P distribution but gives a rough estimate of nonuniformity in P under linear conditions. However, a combination of multivesicular release and receptor saturation could also reduce the variance at high P. This possibility can be tested for by examining whether receptor occupancy changes with release probability using low-affinity competitive antagonists (see section 3.4) [START_REF] Clements | Unveiling synaptic plasticity: a new graphical and analytical approach[END_REF][START_REF] Meyer | Estimation of quantal size and number of functional active zones at the calyx of held synapse by nonstationary EPSC variance analysis[END_REF][START_REF] Silver | Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations[END_REF]. Low-affinity antagonists can also help prevent the build up of desensitization during trains (53).

Interpretation of the quantal size-Estimation of the mean quantal amplitude

from the initial slope of the variance-mean relationship is robust to synaptic nonuniformities, because at low P the variance is little affected by nonuniform release probability. However, mean quantal amplitude at the time of the peak (Q P ), estimated from MPFA, does not correspond to the mean peak amplitude of the quantal waveforms unless the vesicular release time course (together with any temporal dispersion arising from axonal delays) is much shorter than the initial decay time course of the PSC. This is because when release is asynchronous, quantal events can occur and partially decay before the time of the peak of the evoked PSC. In cerebellar granule cells, Q P estimated from MPFA is about 20% lower than rise time-aligned quantal events [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF]. Estimation of Q P is also robust in the presence of multiquantal release and receptor saturation because these tend to occur at intermediate to high P. However, the presence of slow spillover-mediated currents can lead to an underestimate of Q P if it is not corrected for [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF].

Methods

In the following sections, we describe the experimental and analysis procedures to perform MPFA and explain how to estimate quantal variances and to discriminate between univesicular and multivesicular release at central synapses.

Experimental protocol

To perform MPFA, different release probability conditions should be imposed by changing the Ca 2+ concentration ([Ca 2+ ]) and [Mg 2+ ] in the extracellular medium [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Reid | Postsynaptic expression of long-term potentiation in the rat dentate gyrus demonstrated by variance-mean analysis[END_REF][START_REF] Saviane | Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse[END_REF]. The divalent cation concentration and thus membrane charge screening can be kept approximately constant by adjusting the ratio of these two ion species [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF]. The osmolarity should also be kept constant by adjusting the concentration of glucose and a phosphate-free solution can be used to avoid phosphate precipitation in high [Ca 2+ ]. The presence of NMDA receptor antagonists is useful for preventing Ca 2+ -induced plasticity and high [Ca 2+ ] conditions should be applied at the end of the recording to minimize the impact of plasticity induction.

Measuring synaptic responses under voltage clamp minimizes errors arising from changes in driving force and activation of voltage-gated channels in electrically compact cells such as cerebellar granule cells. However, voltage-clamp recordings from the soma are subject to errors in cells where synaptic inputs are located on electrically remote regions of the dendritic tree [START_REF] Spruston | Voltage-and space-clamp errors associated with the measurement of electrotonically remote synaptic events[END_REF][START_REF] Bar-Yehuda | Space-Clamp Problems When Voltage Clamping Neurons Expressing Voltage-Gated Conductances[END_REF]. Voltage-clamp error can be reduced by using capacitance and electrode series resistance compensation functions of the amplifier (but series resistance should be monitored throughout the experiment to ensure it remains constant). While this can improve voltage clamp of the soma and proximal dendrites, it does not compensate for poor voltage control of electrically remote regions of the cell. In cases where synapses are far from the recording site and the "space-clamp" is poor, postsynaptic potentials should be recorded and corrected for the deviation in driving force [START_REF] Silver | High-probability uniquantal transmission at excitatory synapses in barrel cortex[END_REF]. However, if synapses are located at different electrotonic locations, dendritic filtering will contribute to type II quantal variance and voltage-dependent dendritic conductances could complicate the variance-mean relationship by shaping postsynaptic potentials as they spread to the soma.

Data acquisition

Data acquisition and analysis can be performed with a variety of software applications and hardware configurations. We use NeuroMatic (http://www.neuromatic.thinkrandom.com/), a freely available acquisition and analysis package running under IgorPRO (Wavemetrics).

The sampling frequency of the recorded signal should be at least 2-fold higher than the highest-frequency component of the signal; otherwise the signal will be distorted by aliasing [START_REF] Nyquist | Certain Topics in Telegraph Transmission Theory[END_REF]. Aliasing can be avoided by filtering the data with a low-pass Bessel filter before digitizing and ensuring the cut-off frequency of filter is <1/3 of the digitization frequency. Moreover, low-pass filtering the data will reduce the background noise and increase the signal-to-noise ratio. In practice, reliable reconstruction of fast AMPAR-mediated currents can be achieved by low-pass filtering at 10kHz and digitized at 50-100kHz [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF][START_REF] Saviane | Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse[END_REF].

During MPFA experiments, the number of stimulated synapses should be constant. This can be achieved by using paired recordings [START_REF] Silver | High-probability uniquantal transmission at excitatory synapses in barrel cortex[END_REF] or minimal extracellular stimulation protocol [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Lanore | Deficits in Morphofunctional Maturation of Hippocampal Mossy Fiber Synapses in a Mouse Model of Intellectual Disability[END_REF]. In case of extracellular stimulation, changes in excitability or stimulation threshold could occur as a result of the different extracellular [Ca 2+ ] solutions or stimulation frequencies. These should be controlled for by establishing thresholds under the different conditions, monitoring failures and if possible measuring the afferent fiber volley. A minimum of 50 stimuli should be recorded for each release probability condition although more may be required to constrain the fit if few release probability conditions are used. In the case of uniform release probability, a minimum of three different release probability conditions is required and at least four in the nonuniform case.

The higher the frequency the more rapidly the data set can be acquired, but it should not be so high as to induce synaptic depression or facilitation. Time stability of the synaptic responses used to calculate the mean and the variance of the PSC amplitude is essential. Time stability of the amplitude of evoked PSCs can be tested with the Spearman rank order test (18), while amplitude stability across release probability conditions should be checked by returning to a specific condition (e.g., 2mM [Ca 2+ ]) and testing whether the mean PSC amplitude has changed.

Estimation of quantal variance

Unless a simple binomial model is used, quantal variance must be determined. Total quantal variance, CV QT , can be measured from PSCs evoked under conditions of low release probability, when failures of release reach 80-90%, because the proportion of multiquantal events is negligible regardless of the number of functional release sites [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF][START_REF] Silver | Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations[END_REF]. Recording under such low release probability conditions has several advantages: 1) quantal events can be measured from stimulus-aligned PSCs, minimizing the effect of spontaneous currents, allowing a precise measurement of the mean and variance; 2) it minimizes interactions between vesicles at synapses where multivesicular release occurs, and 3) it reduces postsynaptic interactions by reducing the build up of neurotransmitter via spillover.

In another variant of this approach, CV QT can be estimated by evoking PSCs while replacing Ca 2+ by Sr 2+ in the extracellular medium in order to induce asynchronous release of quanta [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Lanore | Deficits in Morphofunctional Maturation of Hippocampal Mossy Fiber Synapses in a Mouse Model of Intellectual Disability[END_REF][START_REF] Valera | Adaptation of Granule Cell to Purkinje Cell Synapses to High-Frequency Transmission[END_REF][START_REF] Foster | Variance-mean analysis in the presence of a rapid antagonist indicates vesicle depletion underlies depression at the climbing fiber synapse[END_REF]. CV QT can also be estimated from the distribution of miniature postsynaptic currents (mPSCs) recorded in TTX. However, this has the serious disadvantage that quantal events arise from all the synapses impinging onto the cell rather than being restricted to an individual synaptic connection.

Once the total quantal variability is determined, the contributions made by intra-and intersite variability can be estimated [START_REF] Silver | Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations[END_REF]. For rare cases where there is a single functional release site per synaptic connection, CV QI can be measured directly from the peak of stimulus-aligned PSCs [START_REF] Silver | Non-NMDA glutamate receptor occupancy and open probability at a rat cerebellar synapse with single and multiple release sites[END_REF]. However, for synaptic connections with multiple functional release sites, direct estimation of CV QI is not possible. In this case CV QI can be inferred from the PSC variance remaining when P is maximal. If quanta release is independent between release sites, then the variance remaining at the peak of PSCs should arise purely from intrasite quantal amplitude variability and quantal asynchrony (i.e., P 1; σ 2 ≈ IQ P CV QI

2 ). CV QI can be determined from the remaining variance by dividing it by IQ P and square rooting. CV QI can also be decomposed into CV QS and CV QL using equation ( 6), since CV QL can be determined from the difference in variability in the peak amplitude between stimulus-aligned and rise time-aligned events [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF]. Once CV QI is known, CV QII can be calculated from CV QT using equation ( 8). If estimation of CV QI is not possible, assuming that CV QI = CV QII , is likely to introduce relatively little error (41).

Distinguish between univesicular and multivesicular release

In order to distinguish between univesicular and multivesicular release, PSCs should be recorded in the presence of a rapidly equilibrating low-affinity competitive antagonist (such as γ-DGG or kynurenic acid for AMPA receptors), under low and high probabilities of release [START_REF] Clements | Unveiling synaptic plasticity: a new graphical and analytical approach[END_REF][START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF][START_REF] Wadiche | Multivesicular release at climbing fiber-Purkinje cell synapses[END_REF][START_REF] Christie | Multivesicular release at Schaffer collateral-CA1 hippocampal synapses[END_REF][START_REF] Silver | High-probability uniquantal transmission at excitatory synapses in barrel cortex[END_REF]. In the case of multivesicular release, the fractional block of the evoked PSC by the competitive antagonist will be less at high release probabilities than low release probabilities, because the postsynaptic AMPA receptors will be exposed to a higher concentration of neurotransmitter, which displaces the antagonist more effectively. Thus, if there is significant dependence of the agonist block on the release probability, this implies the presence of multivesicular release. Moreover, if the [Ca 2+ ]-dependence of release changes in the presence of antagonist, this implies that the postsynaptic receptors become saturated at high release probabilities. Under these conditions MPFA should be performed in the presence of a competitive antagonist [START_REF] Foster | Variance-mean analysis in the presence of a rapid antagonist indicates vesicle depletion underlies depression at the climbing fiber synapse[END_REF]. However, if the fractional block of the PSC by the antagonist is independent of release probability, this suggests that the postsynaptic receptor occupancy is constant across release probabilities and the interpretation of quantal parameters is straightforward. In this case MPFA should give the same number of functional release sites in the presence or the absence of a competitive antagonist.

Data analysis

The peak amplitude of PSCs should be measured from a window (Wp) centered around the maximum of the mean peak PSC and the baseline subtracted using a 1-2 ms window before the stimulus (Wb; Figure 4). Time stability of the PSC recorded for each condition should be assessed by using statistical tests such as linear regression or the Spearman rank order test. The variance arising from the background noise should be measured from each event using a window (Wn) that is similar to the one used to measure the amplitude at the time of the peak (e.g., 0.1 ms) (Figure 4). The two measurement windows (Wn and Wp) should be equidistant to ensure that both measurements pick up similar frequency components of the background noise. The baseline variance can then be calculated and subtracted from the variance of the synaptic amplitude.

Q P and N and their uncertainties can be estimated by performing a weighted fit of the variance-mean relationship to a binomial (equation 3), a multinomial model of release with nonuniform quantal size and uniform release probability (equation 7) or a model that also includes nonuniformity of release probability (equation 13). The initial slope is related to Q P , the degree of curvature to P and the size to N. P can be calculated for each release probability condition from the ratio of the mean current to NPQ P (the maximal possible response). Fits should be accepted when the χ 2 value gives p ≥ 0.05. P should be greater than 0.6 at high [Ca 2+ ] for an accurate estimate of N (41, 58).

Weighted least squares method

Conventional least squares fitting (equations ( 7) and ( 13)) assumes that the errors estimating the mean PSC amplitude are negligible, but a more sophisticated expression for the weights can be used when the errors in the current cannot be neglected [START_REF] Saviane | Errors in the estimation of the variance: implications for multipleprobability fluctuation analysis[END_REF]. Errors in the estimation of variance can be determined with unbiased estimators of the variance that can be obtained using sample moments [START_REF] Saviane | Errors in the estimation of the variance: implications for multipleprobability fluctuation analysis[END_REF]. A general expression of the variance of the sample variance s 2 is given by:

var(s 2 ) = n (n -2)(n -3) 3(3 -2n)(n -1) 2 -n(n -2)(n -3) 2 (n 2 -2n + 3)(n -1) 2 m 2 2 + m 4 ( 16
)
where n is the number of traces acquired and m 2 and m 4 are sample central moments about the mean and can be calculated as

m r = (1/n)∑ i = 1 n (X i -X) r ,
, where m 2 is the variance.

Equation ( 16) is valid even when small sample sizes are used. Maximizing the number of samples improves the estimators' reliability and is therefore anyway desirable. The use of weighted least squares improves the reliability in parameter estimation up to 30% (58).

Discussion

Summary of technique

MPFA [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF], which is also known as variance-mean analysis [START_REF] Reid | Postsynaptic expression of long-term potentiation in the rat dentate gyrus demonstrated by variance-mean analysis[END_REF][START_REF] Clements | Unveiling synaptic plasticity: a new graphical and analytical approach[END_REF], is a simple and robust method for estimating the quantal parameters of synaptic connections. MPFA overcomes the technical difficulties encountered with more traditional methods that involved identifying and interpreting peaks in amplitude histograms [START_REF] Walmsley | Interpretation of "quantal" peaks in distributions of evoked synaptic transmission at central synapses[END_REF][START_REF] Redman | Quantal analysis of synaptic potentials in neurons of the central nervous system[END_REF][START_REF] Stricker | Statistical analysis of amplitude fluctuations in EPSCs evoked in rat CA1 pyramidal neurones in vitro[END_REF], allowing estimation of quantal parameters at central synaptic connections, where quantal parameter are nonuniform, quantal variance is large, and where the signal-to-noise ratio is often low.

Limitations of MPFA

While MPFA is straightforward to apply, the accuracy of the results depends on the quality of the experimental data and the particular properties of the synapse under study. Time stable recordings of PSCs at multiple release probabilities are essential. Uncertainty in the values of quantal parameters can be reduced by increasing the number of recordings per release probability condition and the number of different release probabilities recorded, but clearly, for a finite length experiment, there is a trade-off between these two experimental parameters. At many central synapses the assumption that quantal events sum linearly has been borne out. Multiquantial release is not an issue for MPFA if postsynaptic receptor occupancy is in the low-to-intermediate range, but if postsynaptic receptors saturate following quantal release, the postsynaptic responses are no longer linear reporters of vesicular release. Under these conditions it is essential to lower receptor occupancy with rapidly equilibrating antagonists [START_REF] Clements | Unveiling synaptic plasticity: a new graphical and analytical approach[END_REF][START_REF] Meyer | Estimation of quantal size and number of functional active zones at the calyx of held synapse by nonstationary EPSC variance analysis[END_REF][START_REF] Silver | Estimation of nonuniform quantal parameters with multiple-probability fluctuation analysis: theory, application and limitations[END_REF][START_REF] Wong | Distinguishing between presynaptic and postsynaptic mechanisms of short-term depression during action potential trains[END_REF], to bring the postsynaptic response back into a linear regime. But perhaps the biggest limitation for this technique is that good quality voltage clamp is only feasible for synapses close to the soma. While this method can be applied to current clamp conditions, the impact of dendritic filtering and the impact of active dendritic conductances are difficult to control for.

As for all quantal analysis methods, MPFA relies on a statistical model of synaptic transmission. The results therefore depend on the accuracy of the model. Previous quanta analysis methods have been limited by their ability to account for nonuniform quantal parameters and the presence of high quantal variance. While MPFA can account for a wide range of nonuniformities including those arising from release probability and quantal size, it can only estimate the mean values of the quantal parameters and how they change during plasticity. While this is sufficient for many questions, a deeper understanding of the release process, including separation of mean probability of a vesicle being in the release-ready state and the mean probability that a release-ready vesicle will undergo fusion, requires more powerful descriptions of the release process (25-27, 52) that include short-term plasticity (61).

Comparison of MPFA to other methods

Several quantitative methods have been developed for studying synaptic function over the last decade that extends the basic MPFA approach. These include combining variance-mean approaches with covariance analysis [START_REF] Scheuss | Estimating synaptic parameters from mean, variance, and covariance in trains of synaptic responses[END_REF][START_REF] Scheuss | Separation of presynaptic and postsynaptic contributions to depression by covariance analysis of successive EPSCs at the calyx of held synapse[END_REF] and combining deconvolution and variance analysis enabling estimation of both the quantal properties and the time course of the release rate [START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF][START_REF] Sakaba | Quantitative relationship between transmitter release and calcium current at the calyx of held synapse[END_REF]. These approaches together with other more direct methods of measuring release (50) provide valuable additional information on the time course of release.

Combining variance-mean analysis and model of short-term plasticity is also effective for investigating the relationship between the quantal properties and short-term plasticity at central synapses [START_REF] Saviane | Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse[END_REF][START_REF] Hallermann | Bassoon Speeds Vesicle Reloading at a Central Excitatory Synapse[END_REF][START_REF] Valera | Adaptation of Granule Cell to Purkinje Cell Synapses to High-Frequency Transmission[END_REF][START_REF] Saviane | Estimation of quantal parameters with multiple-probability fluctuation analysis[END_REF].

Statistical models based on Bayesian inference have been used to extract information from the fluctuations of the postsynaptic response [START_REF] Turner | Bayesian analysis of mixtures applied to post-synaptic potential fluctuations[END_REF][START_REF] Bhumbra | Reliable evaluation of the quantal determinants of synaptic efficacy using Bayesian analysis[END_REF]. This technique assumes that the fluctuations in synaptic signals are described by mixtures of Gaussian distributions. Simulations show that a Bayesian Quantal Analysis (BQA) algorithm can accurately estimate the quantal parameters from a small data set with only two conditions of release probability (66) compared to MPFA which needs at least 3 release probability conditions (41). This could be advantageous if recording stability is a problem. The BQA algorithm does not make the distinction between intra-and intersite variability but characterizes their distributions using the "quantal likelihood function". Nevertheless, BQA performed comparably well to MPFA when compared using the same datasets (66).

Optical approaches and future directions

Most quantitative studies of synaptic connections have used electrophysiological recordings to assess synaptic function. However, optical approaches are potentially very powerful because they can also provide spatial information, thereby enabling the study of individual synaptic contacts [START_REF] Murthy | Heterogeneous release properties of visualized individual hippocampal synapses[END_REF][START_REF] Oertner | Facilitation at single synapses probed with optical quantal analysis[END_REF][START_REF] Yuste | Mechanisms of calcium influx into hippocampal spines: heterogeneity among spines, coincidence detection by NMDA receptors, and optical quantal analysis[END_REF][START_REF] Emptage | Optical quantal analysis reveals a presynaptic component of LTP at hippocampal Schaffer-associational synapses[END_REF][START_REF] Sylantyev | Cannabinoid-and lysophosphatidylinositol-sensitive receptor GPR55 boosts neurotransmitter release at central synapses[END_REF]. Optical approaches have shown, for example, that the probability of neurotransmitter release is nonuniform across release sites [START_REF] Murthy | Heterogeneous release properties of visualized individual hippocampal synapses[END_REF]. However, using postsynaptic Ca 2+ changes in spines as an assay of glutamate release is complicated by the presence of voltage-gated channels. The small number of NMDARs activated by a quantum [START_REF] Silver | Rapid-time-course miniature and evoked excitatory currents at cerebellar synapses in situ[END_REF] can also complicate interpretation of failures. Nevertheless, optical quantal analysis has been used to probe the locus of expression of short-term plasticity [START_REF] Oertner | Facilitation at single synapses probed with optical quantal analysis[END_REF][START_REF] Yuste | Mechanisms of calcium influx into hippocampal spines: heterogeneity among spines, coincidence detection by NMDA receptors, and optical quantal analysis[END_REF]. Presynaptic calcium imaging has also been used to assess changes of probability of release during long-term (69) and short-term plasticity [START_REF] Holderith | Release probability of hippocampal glutamatergic terminals scales with the size of the active zone[END_REF][START_REF] Sylantyev | Cannabinoid-and lysophosphatidylinositol-sensitive receptor GPR55 boosts neurotransmitter release at central synapses[END_REF]. However, these techniques provide only part of the quantal description of the synapse compared to electrophysiology-based MPFA. Moreover, unlike MPFA, which can be applied to both excitatory and inhibitory synapses, optical quantal analysis can only be applied to synapses that have Ca 2+ -permeable receptors. However, the development of new fluorescent reporters that sense transmitter directly, like glutamate-sensing fluorescent reporter (iGluSnFR) [START_REF] Marvin | An optimized fluorescent probe for visualizing glutamate neurotransmission[END_REF], and high speed 2photon imaging methods that can monitor synapses distributed in 3D space [START_REF] Kirkby | A compact Acousto-Optic Lens for 2D and 3D femtosecond based 2-photon microscopy[END_REF][START_REF] Fernández-Alfonso | Monitoring synaptic and neuronal activity in 3D with synthetic and genetic indicators using a compact acousto-optic lens two-photon microscope[END_REF] suggest that several of the limitations of optical quantal analysis could soon be overcome.

Conclusions

The quantum hypothesis was proposed to explain the stochastic nature of chemical synaptic transmission. However, the Poisson statistics that describe transmission at the neuromuscular junction under low release conditions are not applicable to most central synapses due to the low number of release sites. Another difficulty at central synapses is that the underlying release probability and quantal size are not uniform. MPFA is a simple approach for determining synaptic properties at synapses with nonuniform quantal parameters. It is robust to noise and large quantal variance, in contrast to more traditional methods that relied on identifying quantal peaks in amplitude histograms. MPFA has been widely used to determine quantal parameters at both glutamatergic [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Sargent | Rapid vesicular release, quantal variability, and spillover contribute to the precision and reliability of transmission at a glomerular synapse[END_REF][START_REF] Lanore | Deficits in Morphofunctional Maturation of Hippocampal Mossy Fiber Synapses in a Mouse Model of Intellectual Disability[END_REF][START_REF] Fourcaudot | cAMP/PKA signaling and RIM1alpha mediate presynaptic LTP in the lateral amygdala[END_REF] and GABAergic synapses [START_REF] Biró | Quantal size is independent of the release probability at hippocampal excitatory synapses[END_REF] and has been applied to trains of responses to determine which quantal parameters change during short-term plasticity [START_REF] Silver | Locus of frequency-dependent depression identified with multiple-probability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF][START_REF] Saviane | Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse[END_REF][START_REF] Oleskevich | Release probability modulates short-term plasticity at a rat giant terminal[END_REF][START_REF] Foster | Variance-mean analysis in the presence of a rapid antagonist indicates vesicle depletion underlies depression at the climbing fiber synapse[END_REF]. The quantal parameters estimated with MPFA can also be combined with models of short-term plasticity [START_REF] Saviane | Fast vesicle reloading and a large pool sustain high bandwidth transmission at a central synapse[END_REF][START_REF] Hallermann | Bassoon Speeds Vesicle Reloading at a Central Excitatory Synapse[END_REF][START_REF] Saviane | Estimation of quantal parameters with multiple-probability fluctuation analysis[END_REF], thereby providing a more complete picture of the quantal transmission at central synapses [START_REF] Rothman | Data-driven modeling of synaptic transmission and integration[END_REF].

Figure 1 .Figure 2 .

 12 Figure 1. Binomial model of synaptic transmission. (a) Postsynaptic currents for a synapse with N = 5, Q = -20 pA at P = 0.1, 0.5 and 0.9 were simulated stochastically (20 superimposed traces in grey and average trace of 200 events in black). (b) The dots represent the theoretical values of variance and mean amplitude at the 3 different release probabilities tested. The empty squares are values from a set of 200 simulated postsynaptic currents. The variancemean relationship of the theoretical values was fitted using equation (3). (modified from ref 58 with permission from Elsevier)

Figure 3 .

 3 Figure 3. (a) Mean EPSCs of 5 stimuli at 100 Hz at mossy fiber-granule cell synapse. Changes of P (b) and Q p (c) during the 100 Hz train of 9 different recordings. The graph in (d) shows the mean amplitude of P and Q p during the 100 Hz train relative to their inital value, with associated exponential fits. (adapted from ref 25)

Figure 4 .

 4 Figure 4. 200 postsynaptic currents were simulated with a multinomial model under a low and uniform probability of release (P = 0.05) which gives rise to 70% of failures. 50 stimulusaligned events are shown in grey. The average trace is displayed in black. Wb shows the window for the baseline correction and Wn and Wp the window for the measurement of the mean and variance of the baseline noise and the peak of the postsynaptic current, respectively. (modified from ref 64 with permission from Springer)

Neuromethods. Author manuscript; available in PMC 2018 September 20.

Acknowledgements

We thank Antoine Valera for comments on the manuscript. FL is supported by an IEF Marie Curie fellowship (FP7) and RAS holds a Wellcome Trust Principal Research Fellowship and an ERC Advanced Grant.


Craig A Bayse

Michael B Hall

Brett Pleune

Rinaldo Poli

Experimental and Theoretical Studies of Nonclassical d 0 Cyclopentadienyl Polyhydride Complexes of Molybdenum and Tungsten

Low-temperature protonation of compounds Cp*MH 5 (PMe 3 ) (M ) Mo, 1; W, 2) by HBF 4 'Et 2 O in CD 2 Cl 2 or CDFCl 2 affords the thermally unstable "hexahydride" derivatives [Cp*MH 6 (PMe 3 )] + (M ) Mo, 3; W, 4). The corresponding protonation of 1-and 2-d 5 affords 3-and 4-d 5 , respectively. The ∆δ on going from H 6 to HD 5 is small for both compounds, but positive for 3 and negative for 4, and no isotopic perturbation of resonance (IPR) is observed.

The T 1min at 400 MHz for [Cp*MH 6 (PMe 3 )] + apparently doubles on going from Mo to W (52 ms for 3 and approximately 100 ms for 4). Optimized geometries at the restricted Hartree-Fock (RHF) and second-order Møller-Plesset (MP2) levels and energy calculations at higher levels of theory show that these complexes are dihydrogen complexes [Cp*M(H 2 )(H) 4 (PR 3 )] + . The theoretical determination of a dihydrogen complex is consistent with the fact that the experimental T 1min values lie within the expected range for dihydrogen complexes. Examination of the potential energy surface at the MP2 level gives two mechanisms for hydride motion: exchange through a "trihydrogen anion" transition state and rotation of the dihydrogen ligand about its axis. The barrier for the hydride exchange (∼4 kcal/mol) is consistent with the inability to decoalesce the proton NMR signal.

Introduction

Fluxionality in transition-metal polyhydride complexes presents a challenging problem, especially in solution-phase proton NMR studies. At high temperatures, a transition-metal polyhydride complex can have a single peak due to ligands exchanging faster than the NMR time scale. This exchange-averaged signal often can be decoalesced by reducing the temperature of the experiment until multiple signals are observed corresponding to each nonequivalent proton in the system. A study of the low-temperature limit provides the best data for the determination of structural properties in the solution phase because relaxation times (T 1min ) and proton-deuteron coupling constants (J(H,D)) can be obtained to indicate the presence of a dihydrogen ligand and estimated M-H bond lengths and dihydrogen distances. In the worst case, the proton signal cannot be decoalesced above the freezing point of common lowtemperature solvents. For these complexes, J(H,D) and T 1min are exchange-averaged, making reasonable determination of structural parameters difficult. For these systems, theoretical studies can be used effectively to determine the low-temperature structure and to map out the mechanism of exchange. Various groups have demonstrated the utility of ab initio calculations in determining exchange mechanisms for transition-metal polyhydrides. 1 The protonation of polyhydride complexes has been shown, at least in some cases, to occur at one of the hydride metal bonds in some systems and to afford a coordinated dihydrogen (nonclassical) species, 2 which subsequently may or may not collapse to a polyhydride (classical) moiety depending on the electron richness of the metal. The latter is obviously possible only when there are two electrons in a metal-based orbital of suitable energy. Protonation of a d 0 complex, therefore, is limited to the formation of nonclassical products, and very few examples of this type of complex have been observed experimentally, for instance [ReH 6 (η 2 -H 2 )-(PR 3 ) 2 ] + , 2a [MoH 4 (η 2 -H 2 )(dppe)] 2+ . 3 The protonation of WH 6 (triphos) to form [WH 7-2x (η 2 -H 2 ) x (triphos)] + 4 is another possible entry into this category if x ) 1. These d 0 dihydrogen complexes are expected to be very labile toward the loss of dihydrogen since there are no d electrons to stabilize the interaction through d-σ* backbonding. We report here two new entries into the chemistry of d 0 dihydrogen complexes. These complexes have been synthesized through the protonation of the classical pentahydride d 0 complexes Cp*MH 5 (PMe 3 ) (M ) Mo, 1; W, 2) to form the dihydrogen products [Cp*MH 4 (η 2 -H 2 )(PMe 3 )] + (M ) Mo, 3; W, 4), which have been characterized by NMR and theoretical studies of the model systems [CpMH 6 (PH 3 )] + (M ) Mo, 3T; W, 4T).

Experimental Details

All manipulations were carried out under an inert atmosphere of nitrogen or argon by the use of vacuum-line, Schlenk, syringe, or drybox techniques. Solvents were dried by conventional methods and distilled under nitrogen prior to use. Deuterated solvents were dried over molecular sieves and degassed by three freeze-pump-thaw cycles prior to use. Methanol was degassed by three freeze-pump-thaw cycles prior to use. 1 H and 31 P{ 1 H} NMR measurements were made on Bruker AF200, WP200, or AM400 spectrometers; the peak positions are reported with positive shifts downfield of TMS as calculated from the residual solvent peaks ( 1 H) and downfield of external 85% H3PO4 ( 31 P). For each 31 P NMR spectrum a sealed capillary containing H3PO4 was immersed in the same NMR solvent used for the measurement, and this was used as the reference. The standard inversion-recovery-pulse (180-τ-90) sequence was used to determine T1. Values of T1 were obtained from the slopes of linear plots of ln(2Ieq/(Ieq -Iτ)) vs τ, where Ieq is the peak intensity at τ ) ∞. HBF4'Et2O, LiAlD4, LiAlH4, and PMe3 (Aldrich) were used without further purification. Compounds Cp*MH5(PMe3) (M ) Mo, W) and CDFCl2 were synthesized by previously described methods. 5 Compounds 1-and 2-d 5 were synthesized by adapting these methods from Cp*MCl4(PMe3) and LiAlD4 followed by CH3-OD quenching. Their NMR properties are identical to those of 1 and 2, except for the missing hydride resonances in the 1 H NMR and the P-D coupling in the 31 P{ 1 H} NMR spectra. For 1-d 5 : 1 H NMR (C6D6, δ): 2.19 (s, Cp*, 15H), 1.17 (d, Me, 9H, JPH ) 9.6 Hz). Residual hydride resonance: <4% (mol/ mol). 31 P{ 1 H} NMR (C6D6, δ): 12.6 (1:5:15:30:45:51:45:30:15: 5:1 undecet, JPD ) 8.9 Hz). For 2-d 5 : 1 H NMR (C6D6, δ): 2.31 (s, 15H, Cp*), 1.36 (d, Me, 9H, JPH ) 10.0 Hz). Residual hydride resonance: <5% (mol/mol). 31 P{ 1 H} NMR (C6D6, δ): -19.6 (1:5:15:30:45:51:45:30:15:5:1 undecet, JPD ) 6.5 Hz).

Protonation of Cp*MoH5(PMe3) (3). To a solution of 40 mg of Cp*MoH5(PMe3) (0.13 mmol) in 1 mL of CD2Cl2 at -80 °C in an NMR tube was added 20 µL of HBF4'Et2O (0.15 mmol) via microsyringe. The orange-brown solution darkened and began to slowly evolve H2. 1

Theoretical Details

The experimental complexes were modeled by replacing the methyl groups on the phosphine and Cp* ligands with hydrogens. The carbons of the Cp ring were constrained to be planar and equidistant from the metal, but the hydrogen atoms were allowed to bend in or out of the C5 plane. The phosphine ligand was constrained to local C3v symmetry.

Calculations were performed in two basis sets designated BSI and BSII. In both of these basis sets, the carbon 6a and phosphorus 6b atoms were treated with effective core potentials (ECPs) and double-ζ basis sets. The basis sets for the hydrogen atoms of the Cp and phosphine ligands were double-ζ quality. 7 In BSI, the metals were represented in a double-ζ basis set with a triple-ζ representation of the d orbitals (Mo (5s5p4d)/[3s3p3d]; W (5s6p4d)/[3s3p3d]) with the relativistic ECPs 8 of Ermler and Christiansen. The (n+1)s and (n+1)p orbitals of the metals have been included in accordance with recent studies on the importance of these basis functions. 9 The hydride ligands were represented by a triple-ζ basis set (5s)/ [3s]. 10 BSII is identical to BSI except that polarization functions (Mo (5s5p4d1f)/[3s3p3d1f]; W (5s6p4d1f)/[3s3p3d1f]; H (5s1p)/[3s1p]) have been added to the metals and hydride ligands (Mo, ζ ) 0.7, W, ζ ) 0.6; H, ζ ) 0.475). These exponents were optimized at the MP2 level on the RHF optimized geometry of the MoH6 and WH6 molecules. 11 All geometries were optimized at the restricted Hartree-Fock (RHF) 12 and second-order Møller-Plesset (MP2) 13 level of theory with full-gradient methods. For BSI, the geometries of the equilibrium and hydride-dihydrogen exchange transition states were calculated in C1 symmetry. The dihydrogen rotation transition state was optimized in Cs symmetry with two hydride ligands constrained to have the same dihedral angle with a carbon in the Cp ring. Since BSII contained both f functions and ECPs, geometry optimizations in this basis set were performed using a Fletcher-Powell 14 line-search algorithm in Cs symmetry. For BSI, energies were recalculated at the MP3, 12 configuration interaction truncated to single and double excitations (CISD), [START_REF] Saunders | [END_REF] and coupled cluster truncated to single and double excitations (CCSD). [START_REF] Lee | The TITAN Set of Electronic Structure Programs[END_REF] Relative energies were also calculated at the MP3 level in BSII. Numerical calcula-tions for the vibrational frequencies were performed at the RHF level in BSI. All calculations were performed using the GAMESS-UK package. 17

Experimental Results

The low-temperature (-80 °C) protonation of 1 and 2 with HBF 4 'OEt 2 in CD 2 Cl 2 or CDFCl 2 affords thermally unstable [Cp*MH 6 (PMe 3 )] + (M ) Mo, 3; W, 4). Complex 3 is sufficiently stable for NMR characterization only below 220 K (t 1/2 ) ca. 30 s at -60 °C), whereas 4 can be investigated up to ca. 250 K (t 1/2 ) ca. 2 min at -20 °C). Thermal decomposition leads to hydride-free (by NMR) uncharacterized products. The nature of the protonation products as [Cp*MH 6 (PMe 3 )] + is indicated by integration of the 1 H resonance, by the absence of immediate gas evolution during the protonation reaction, and by the binomial septet pattern of the 31 P-{selective-1 H} NMR resonance for 4.

Relative to the starting pentahydride compounds, the protonation products show deshielded 1 H NMR resonances with a reduced phosphorus coupling (∆δ ) +0.89, ∆J PH ) -18.4 Hz for Mo; ∆δ ) +1.25, ∆J PH ) -16.0 Hz for W) and shielded 31 P{ 1 H} NMR resonances (∆δ ) -8.1 for Mo, -0.2 for W). The thermal instability has hampered the growth of suitable crystals for a neutron or X-ray diffraction analysis. Given the pseudopentagonal bipyramidal structure of 1 and 2, 5b,18 a reasonable structure for complexes 3 and 4 is I (Scheme 1). However, protonation could induce formation of another dihydrogen ligand, 19 as in II. Such an occurrence was found for the protonation of IrH 5 (PCy 3 ) 2 to afford [IrH 2 (η 2 -H 2 ) 2 (PCy 3 )] + . 20 This structure would be isoelectronic with known half-sandwich 18-electron derivatives of Mo(IV) and W(IV) such as Cp*MoH 3 -(PMe 3 ) 2 21 and CpWH 3 (PMe 3 ) 2 . 22 In principle, even a tris-dihydrogen structure III could be adopted. This would be isoelectronic with Cp*MoH(PMe 3 ) 3 . 21 For this reason, we sought additional information on these systems via NMR spectroscopic and theoretical methods.

Structural information was sought experimentally by NMR through T 1 measurements, 23,[START_REF] Crabtree | The Organometallic Chemistry of the Transition Metals[END_REF] deuteration, and isotopic perturbation of resonance (IPR) studies. The use of the T 1min method for the determination of polyhydride structures (e.g., classical polyhydride vs nonclassical dihydrogen complexes) is now of widespread use and provides reliable results in many cases. 19,[START_REF] Crabtree | The Organometallic Chemistry of the Transition Metals[END_REF]25 The measured T 1min is 52 ms (400 MHz) at -90 °C for 3 (corresponding to 26 ms at 200 MHz; a minimum could not be reached at 200 MHz) and 51 ms (200 MHz) at -90 °C or approximately 100 ms (400 MHz) at -100 °C for 4. The results of the T 1min studies at 400 MHz are graphically shown in Figure 1. Stoichiometric protonation of 1-d 5 and 2-d 5 with HBF 4 'OEt 2 in CD 2 Cl 2 gave 3-and 4-d 5 , respectively. Unfortunately, the 31 P{ 1 H}, 1 H, and 2 H NMR resonances are too broad to establish the magnitude and multiplicity of the P-D and H-D couplings at all accessible temperatures. The chemical shifts of the hydride resonances of 3(4)-d 5 and 3(4) at different temperatures are shown in Figure 2. Given the limited temperature range of stability (especially for 3) and the large uncertainties in ∆δ ) δ(D 5 H)δ(H 6 ) (due to broadness), the presence of IPR is not conclusively demonstrated for either compound. The detection of IPR for nonclassical complexes is often difficult. 19 It is interesting and rather unexpected, however, that the sign of ∆δ is opposite for 3 and 4. The replacement of H with D usually provides an upfield shift for the resonance of the residual H ligands, while a downfield shift as observed for 3 usually indicates that an IPR effect is overriding the upfield isotopic shift. Thus, although the presence of an IPR effect is not conclusively established for either compound, the result for 3 is consistent with the presence of one or more H 2 ligands as expected. The negative ∆δ for 4 could be explained by a small IPR effect relative to the isotopic shift. A small IPR effect could result either from a small difference between the chemical shifts of the hydride and dihydrogen sites or from a small zero-point energy difference between the different isotopomers. It is interesting, however, to consider the additional possibility of a symmetric σ-delocalized structure such as IV. Although this structure is not precedented, it would lead to the expectation of no IPR effect and an upfield shift of the NMR resonance upon deuteration. Alternatively, structure IV could correspond to a transition state for the hydride-dihydrogen scrambling process. However, it is more likely that a dihydrogen complex would be favored due to the electron deficient nature of the hexahydride. Additional insight into the possible structure of 3 and 4, as well as on the mechanism of fluxionality, has been sought theoretically.

Theoretical Results

Geometry optimizations of the model complexes [CpMH 6 (PH 3 )] + (M ) Mo, 3T; W, 4T) at the RHF and MP2 levels in both BSI and BSII showed that the minimum energy structure of each of these systems is the η 2 -H 2 complex I (Figure 3). Frequency calculations in BSI at the RHF level established that these complexes are true minima on the potential energy surfaces. The geometric parameters for RHF and MP2 are very similar, and the MP2 values are listed in Table 1. Generally, both the addition of polarization functions (BSI to BSII) and electron correlation (RHF to MP2) shorten the metal-ligand distances and lengthen the intraligand bonds (H-H and C-C distances). These polyhydride complexes have distorted pentagonal bipyramidal geometries where the hydride ligands have large bond angles with the center of the Cp ligand (X Cp ) of 100-109°. This distortion of the hydride ligands away from the Cp is consistent with a pseudo-secondorder Jahn-Teller effect which results from the availability of low-lying unoccupied d orbitals. 26 Calculations (MP2, BSI) were also performed on the other possible geometries II-IV for the complex with M ) Mo. The bis-dihydrogen complexes reverted to I under full geometry optimization, but approximate structures were found by constraining the dihydrogen distances to a series of values between 0.78 and 1.14 Å. The lowest energy structures (d(H-H) ) 1.02 Å) were ∼14 (cis) and ∼22 kcal/mol (trans) less stable than I. It was not possible to calculate the tris-dihydrogen structure III, as this constrained pseudo-C 3v system reverted to the constrained pseudo-C 6v hexahydride structure IV, which lies 19.5 kcal/mol above I. The addition of the f polarization functions in BSII stabilized the energy of IV by ∼9 kcal/mol relative to I. Frequency calculations on the W hexahydride structure IV revealed that it is a second-order saddle point on the potential energy surface (two imaginary frequencies). If structure IV had six equivalent M-H bonds, they would transform as a 1 +b 1 +e 1 +e 2 in C 6v symmetry. As there is no metal orbital available of b 1 symmetry, the system is likely to be unstable. Since the available metal orbitals in IV (a 1 +e 1 +e 2 ) correspond to the bonding orbitals in C 5v , it is favorable for IV to rearrange to a geometry with five equatorial ligands: four hydrides and a dihydrogen occupying a single site.

A complex such as I can exchange by only one route without reduction of the metal, which is a fairly high energy process. This low-energy route (Scheme 2) involves stretching the dihydrogen toward an adjacent hydride and forming a "trihydrogen-anion ligand" at the transition state (TSE). Another transition state (TSR) for intramolecular motion involves the rotation of the dihydrogen ligand about the M-X H2 axis, where X H2 is the midpoint of the dihydrogen H-H bond. 1. To examine the exchange mechanism in these systems, the potential energy surface (MP2, BSI) for the molybdenum complex 3T was mapped out by constrained optimization, where the dihedral angles made by two hydride ligands, the metal, and the center of the Cp ring (X Cp ) to a carbon in the ring (∠(H-M-X Cp -C)) were fixed at various angles. Then, the surface was constructed by plotting the energy against the two smallest H-H distances from each optimization (Figure 4). An approximation to the transition state for exchange (TSE) was determined to lie at a geometry where two hydride ligands have dihedral angles (∠(H-M-X Cp -C)) of roughly 38°. For BSI, a Jorgensen optimization 27 was used to completely determine the exchange transition state (TSE). A similar process was used to obtain the TSE for 4T starting from the optimized transition state for 3T. The calculations in BSII were performed by constraining the system to C s symmetry and performing a Fletcher-Powell 14 geometry optimization. The rotational transition state (TSR) was obtained by optimizing the geometries of the systems in C s symmetry with equal dihedral angles (∠(H-M-X Cp -C)) for two hydride ligands, such that both hydrides are coplanar with a carbon in the Cp ring. Frequency calculations on TSE and TSR at the RHF level in BSI for both metal systems have one imaginary frequency, which demonstrates that these structures are true transition states. The relative energies of the transition states are shown in Table 2, and selected structural features (MP2) are shown in Figure 5 and listed in Tables 3 (TSE) and 4 (TSR).

At the MP2 level in BSI, the barrier to rotation is slightly higher than that for exchange, but at higher levels of electron correlation (see Table 2), the exchange barrier is 3-4 kcal/mol higher than that for rotation. In BSII, the difference in the MP2 barrier heights switches so that TSR is ∼5 kcal/mol higher than TSE. The values of the barrier height (TSE) from these calculations are consistent with the rapid exchange observed in the NMR experiment. Since the barrier heights at the MP3//MP2 and CCSD//MP2 levels give comparable results in BSI, we have calculated the barrier heights in BSII at the MP3 level. Overall, application of the larger basis set decreases the exchange barrier from ∼7 kcal/mol to ∼4 kcal/mol and increases the rotation barrier from 4 to 5 kcal/mol to 6-8 kcal/mol. The higher level calculations show that there is a slightly higher exchange barrier for 3T (Mo) 3 and4, respectively. a XCp is defined as the center of the Cp ring. b ∠(C-XCp-M-Ha) ) 0.0, ∠(C-XCp-M-Hd) ) 180.0. and a slightly higher rotation barrier for 4T (W), but these differences (∼1 kcal/mol) are not large enough to explain the differences observed experimentally. Since the rotational barrier is probably larger than the barrier for molecular tumbling in solution, 1b the dihydrogen ligand in the complex should not be considered to be rapidly rotating for the purpose of applying the T 1min criterion. Often rapidly rotating dihydrogen ligands are observed in d 4 or d 6 complexes where the occupied d orbitals provide a cylindrical area of electron density which facilitates the rapid rotation since two occupied quasi-degenerate (depending on the complex) metal orbitals can back-bond to the σ* orbital of the dihydrogen. 1b The d 0 complexes in this study lack this electron density (and thus lack any d f s* donor effects), a situation that results in a particularly strong cis-effect 28 in which the dihydrogen ligand is attracted to the cis hydride ligands. Thus, their rotational barriers are 2-4 times that observed for complexes with occupied d orbitals.

Discussion

The measurements of the dihydrogen distances by various methods are often inconsistent. For example, work performed on W(H 2 )(CO) 3 (P i Pr 3 ) 2 gives the following H-H distances: 0.75 Å (X-ray diffraction 29 ), 0.82 Å (uncorrected neutron diffraction 29 ), 0.89 Å (solid-state NMR 30 ), 0.96 Å (solution-phase NMR 31 ), 0.76 Å (solution-phase NMR with rapid rotation correction 31 ), 0.85 Å (solution phase J(H,D) 32 ), 0.87 Å (MP2-optimized geometry of W(H 2 )(CO) 3 (PH 3 ) 2 ) 33 ). X-ray diffraction is a poor method for the determination of any transitionmetal hydride complex, as the hydride ligands can easily be lost in the electron density around the metal center.

Neutron diffraction, on the other hand, can locate the hydrides easily, but the librational motion of a dihydrogen ligand may result in a shortened d(H-H). The solid-state NMR value is considered by some groups to be better than neutron diffraction values that have not been corrected for the librational motion of the dihydrogen ligand. 34 The librational correction lengthens the uncorrected neutron diffraction value by about 0.07 Å such that the corrected neutron diffraction value is in agreement with the solid-state NMR value. The MP2 theoretical value is in good agreement with the most accurate experimental methods.

However, the poor correlation of the solution-phase d(H-H) values calculated from T 1min should be noted. The inverse of T 1min (1/T 1min ) is the sum of the effect of the different relaxation rates upon the nucleus under question. [START_REF] Harris | Nuclear Magnetic Resonance Spectroscopy[END_REF] For this reason, Derosiers et al. 25d proposed including dipolar relaxations from all necessary nuclei as a correction to Crabtree's original formulation. However, while this correction gives overall good results for classical polyhydride (>3 H's) complexes, the agreement for nonclassical polyhydrides is not as good. In general, the T 1min method for determining dihydrogen distances is most reliable for cases where the only M-H interaction is a single dihydrogen ligand 34 (error ) (0.1 Å). In these cases, it is possible to obtain reasonable results if one accounts for librational motion and "hopping" between sites. 32 In fact, transition-metal polyhydride complexes with both nonclassical and classical hydrides in rapid exchange can be the most difficult to determine if the resonances of the nonequivalent hydride or dihydrogen ligands cannot be decoalesced. In these cases the T 1min value is exchange-averaged and a dihydrogen distance can only be estimated by using classical hydride structural parameters obtained from other methods. This procedure is also not necessarily reliable; for example, a recent study of [Cp*Os(H 2 )(H) 2 -(PPh 3 )] + gave a dihydrogen distance of 1.014 Å from neutron diffraction and 1.07-1.38 Å from T 1min . [START_REF] Gross | [END_REF] The problems in applying the T 1min method to several rhenium polyhydride complexes should also be noted. 25b,37 Another method for the determination of the dihydrogen bond distance in solution is the magnitude of the proton-deuteron coupling constant J(H,D). It was observed that classical hydrides had small values (J(H,D) < 5 Hz), while dihydrogen complexes had values of 20-30 Hz. A study of a large number of complexes resulted in the following relationship between J(H,D) and the dihydrogen bond distance: However, while this method is more reliable than T 1min , it can have the same problems if the hydride resonances cannot be decoalesced and cannot be determined if the resonances are broad. Note that in the case of W(H 2 )(CO) 3 (P i Pr 3 ) 2 the dihydrogen bond distance The systems discussed in this work are very problematic experimentally. They decompose almost immediately after formation and thus cannot be isolated. In the NMR experiments, the hydride signals did not decoalesce and were too broad to obtain J(H,D). The dihydrogen bond distances were estimated from Crabtree's isotopic formula 25a and the positions of the hydride ligands in the theoretical structures. This method results in dihydrogen bond lengths of 0.89-1.14 Å (3) and 1.00-1.32 Å (4). However, the theoretical calculations do not reflect the difference in bond lengths indicated by the solution-phase studies (Mo < W) and, in fact, predict the opposite results (Mo > W). In a study 38 of the elongated dihydrogen complex [CpRu(H 2 )-(dppm)] + , anharmonic vibrational corrections to the geometry were necessary to obtain the correct H-H distance. Without the correction, the H-H distances were calculated to be too short. The anharmonic corrections are critically important in that case because that d 6 system easily forms a dihydride. It is the low energy of the dihydride that results in the anharmonic potential energy. 38 Here, in the d 0 systems the potential is nearly harmonic because there is no low-lying allhydride structure.

In conclusion, while the experimental and theoretical data are in good agreement for the Mo complex, the W complex gives a T 1min that is too long relative to the value that can be calculated from the optimized model. It is most likely that the difference between theory and experiment here stems from the approximations involved in applying the isotopic T 1min formula to a fluxional polyhydride/dihydrogen complex.

Conclusions

The protonation of the half-sandwich pentahydride complexes [CpMH 5 PMe 3 ] (Mo, 1; W, 2) are shown by theoretical methods to give the dihydrogen complexes [CpMH 4 (H 2 )PMe 3 + ] (Mo, 3; W 4). The experimental T 1min studies of these rare, highly unstable complexes give dihydrogen bond distances of 0.89-1.14 Å (3) and 1.00-1.32 Å (4), while the theoretical results are 0.898 Å (3T) and 0.871 Å (4T). While there may be discrepancies between the theoretical dihydrogen distances and those measured by T 1min , we feel that the theoretical results in this study are more accurate. The exchange transition states for these complexes correspond to the formation of stretched trihydrogen-anion species and have a barrier of ∼4 kcal/mol, which is in agreement with the inability to decoalesce the hydride signals at -140 °C.

Scheme 1 Figure 1 .

 11 Scheme 1

( 26 )

 26 Bayse, C. A.; Hall, M. B. Inorg. Chim. Acta 1997, 259, 179, and references therein.

Figure 2 .

 2 Figure 2. Temperature dependence of the chemical shift for the hydride resonance in compounds 3, 3-d 5 (3D), 4, and 4-d 5 (4D).

Figure 3 .

 3 Figure 3. Geometry of [CpW(H 2 )H 4 (PH 3 )] + at the MP2 level in BSI. Structural parameters are listed in Table1.

(

  27) (a) Simons, J.; Jorgensen, P.; Taylor, H.; Ozment, J. J. Phys. Chem. 1983, 87, 2745. (b) Banerjee, A.; Adams, N.; Simons, J.; Shepard, R. J. Phys. Chem. 1985, 89, 52.

Figure 4 .

 4 Figure 4. Potential energy surface for the exchange of hydrides in [CpMoH 6 (PH 3 )] + in BSI at the MP2 level. Scheme 2

Figure 5 .

 5 Figure 5. Geometries of TSE and TSR for [CpW(H 2 )H 4 -(PH 3 )] + at the MP2 level in BSI. Structural parameters are listed in Tables3 and 4, respectively.

of Cp*MD5(PMe3) (M ) Mo, 3-d 5 ; W, 4-d 5 ).

  These experiments were carried out as detailed above for the corresponding pentahydride species. The results are shown in the Results and Discussion section.

	15H), 1.66 (d, Me, 9H, JPH ) 16.0 Hz), -2.19 (d, WH, 6H, JPH
	) 24.7 Hz). 31 P{ 1 H} NMR (CDFCl2, 193 K, δ): -21.1 (s). 31 P-{ 1 H-selective} NMR (CDFCl2, 193 K, δ): -21.1 (septet, JPH )
	24.0 Hz). Decomposition of 4 occurred rapidly at temperatures
	above -20 °C; 4 has a half-life of approximately 24 h at -80
	°C.
	Protonation
	H NMR (CD2Cl2, 193 K, δ): 2.23 (s, Cp*, 15H), 1.48 (d, Me, 9H, JPH ) 12.1 Hz), -3.03 (d, MoH,
	6H, JPH ) 35.5 Hz). 31 P{ 1 H} NMR (CDFCl2, 193 K, δ): 2.2
	(s). H2 evolution accelerated as the solution was warmed to 0
	°C. Decomposition of 3 occurs rapidly at temperatures above
	-60 °C; 3 has a half-life of approximately 2 h at -80 °C.
	Protonation of Cp*WH5(PMe3) (4). To a solution of 55
	mg of Cp*WH5(PMe3) (0.14 mmol) in 1 mL of CDFCl2 at -80
	°C in an NMR tube was added 25 µL of HBF4'Et2O (0.19 mmol)
	via microsyringe. The orange solution darkened and began
	to slowly evolve H2. 1 H NMR (CDFCl2, 193 K, δ): 2.39 (s, Cp*,

Table 1 . Theoretical Geometries of [CpM(H 2 )H 4 (PH 3 )] + in BSI and BSII at the MP2 Level

 1

		Mo, BSI	Mo, BSII	W, BSI	W, BSII
	parameter	MP2	MP2	MP2	MP2
	d(Ha-Ha), Å	0.826	0.898	0.825	0.871
	d(M-Ha), Å	1.873	1.830	1.911	1.814
	d(M-Hb), Å	1.732	1.709	1.735	1.706
	d(M-Hc), Å ∠(X Cp-M-Ha), a deg ∠(XCp-M-Hb), a deg ∠(XCp-M-Hc), a deg ∠(Ha-M-Ha), a deg ∠(C-XCp-M-Hb), a deg (75.36 (76.25 1.688 1.690 100.05 102.62 103.36 106.10 109.40 109.56 25.0 28.4 ∠(C-XCp-M-Hc), a deg (143.73 (144.48 (144.25 (143.89 1.709 1.680 100.73 102.80 104.37 105.78 108.22 108.69 24.9 27.8 (74.43 (75.21
	d(M-P), Å	2.553	2.506	2.550	2.502
	d(M-C), Å	2.363	2.346	2.372	2.345
	d(C-C), Å	1.474	1.476	1.473	1.475

a XCp is defined as the center of the Cp ring.

Table 2 . Energy Barriers for Hydride Exchange in [CpMH 6 (PH 3 )] + Relative to CpM(H 2 )H 4 (PH 3 ) +

 2

		Mo, TSE	Mo, TSR	W, TSE	W, TSR
	HF//HF a	11.42 (9.70) 3.09 (4.76) 10.50 (8.18) 3.96 (5.38)
	MP2//MP2 a	5.37 (2.80) 5.42 (8.45) 5.31 (2.80) 5.89 (9.32)
	MP3//HF	7.04	4.27	6.42	5.06
	MP3//MP2	7.39 (4.13) 4.20 (6.42) 6.95 (4.02) 5.24 (8.28)
	CISD//HF	8.40	4.18	7.82	4.89
	CISD//MP2	8.39	4.13	8.04	5.11
	CCSD//HF	6.96	4.44	4.35	3.07
	CCSD//MP2 7.35	4.19	6.95	5.23

a Value in parentheses in the energy barrier calculated in BSII.

Table 3 . Theoretical Geometries for the Transition State (TSE) for Hydride Exchange in BSI and BSII at the MP2 Level

 3

		Mo, BSI	Mo, BSII	W, BSI	W, BSII
	parameter	MP2	MP2	MP2	MP2
	d(Ha-Hb), Å	1.111	1.148	1.112	1.144
	d(M-Ha), Å	1.811	1.752	1.816	1.765
	d(M-Hb), Å	1.780	1.731	1.786	1.742
	d(M-Hc), Å	1.724	1.700	1.723	1.695
	d(M-Hd), Å ∠(XCp-M-Ha), a deg ∠(XCp-M-Hb), a deg ∠(XCp-M-Hc), a deg ∠(XCp-M-Hd), a deg ∠(C-XCp-M-Hb), a,b deg (38.60 1.712 113.45 108.97 103.93 99.37	1.693 114.52 110.56 105.41 100.48 (41.59	1.711 113.08 108.66 104.19 99.97 (38.31	1.686 113.93 109.59 105.36 100.70 (38.31
	d(M-P), Å	2.552	2.505	2.553	2.509
	d(M-C), Å	2.363	2.349	2.371	2.374
	d(C-C), Å	1.474	1.476	1.473	1.474

∠(C-XCp-M-Hc), a,b deg (109.18 (109.19 (109.48 (109.01

Table 4 . Theoretical Geometries for the Transition State (TSR) for Dihydrogen Rotation in BSI and BSII at the MP2 Level

 4 XCp is defined as the center of the Cp ring.

		Mo, BSI	Mo, BSII	W, BSI	W, BSII
	parameter	MP2	MP2	MP2	MP2
	d(Ha1-Ha2), Å	0.785	0.804	0.782	0.802
	d(M-Ha1), Å	1.956	1.907	1.990	1.910
	d(M-Ha2), Å	1.969	1.895	1.983	1.900
	d(M-Hb), Å	1.731	1.705	1.734	1.707
	d(M-Hc), Å ∠(XCp-M-Ha1), a deg ∠(XCp-M-Ha2), a deg ∠(XCp-M-Hb), a deg ∠(XCp-M-Hc), a deg ∠(Ha1-M-Ha2), deg ∠(C-XCp-M-Hb), a deg (78.77 1.710 99.55 122.62 106.53 98.59 23.0 ∠(C-XCp-M-Hc), a deg (145.23 (145.58 1.681 99.16 123.57 107.61 98.55 24.4 (80.53	1.706 97.86 120.55 107.34 98.57 22.7 (78.29 (145.79 (145.82 1.679 97.07 121.39 108.18 98.63 24.3 (79.65
	d(M-P), Å	2.550	2.508	2.547	2.501
	d(M-C), Å	2.362	2.340	2.375	2.340
	d(C-C), Å	1.475	1.477	1.474	1.476

a

(38) Gelabert, R.; Moreno, M.; Lluch, J. M.; Lledos, A. J. Am. Chem.Soc. 1997, 117, 9840.

Acknowledgment. Support for this work by the Department of Energy (Grant No. DEFG059ER14230 to R.P.) is gratefully acknowledged. We thank Prof. G. Parkin for providing the unpublished crystal coordinates of compound (η-C 5 Me 4 Et)MoH 5 (PMe 3 ). C.A.B. and M.B.H. thank the National Science Foundation (CHE-9423271) and the Robert A. Welch Foundation for support. The Texas A&M University Supercomputer Center is recognized for providing time on their machines. C.A.B. thanks the Robert A. Welch Foundation for support throughacompetitiveuniversity-widegraduatefellowship.