Estimating pKa values for pentaoxyphosphoranes.

Article abstract of DOI:10.1021/ja025779m

pKa values are estimated independently, by two entirely different methods, for the ionizations of the apical and equatorial OH groups of two representative hydroxyphosphoranes. A bond length-pKa correlation based on crystal structures of cyclohexanol deri

Full text of DOI:10.1021/ja025779m

Published on Web 05/15/2002
Estimating pKa Values for Pentaoxyphosphoranes
J. E. Davies, N. L. Doltsinis, A. J. Kirby,* C. D. Roussev, and M. Sprik
Contribution from the UniVersity Chemical Laboratory, Cambridge UiVersity,
Cambridge CB2 1EW, United Kingdom
Received February 1, 2002
Abstract: pKa values are estimated independently, by two entirely different methods, for the ionizations of
the apical and equatorial OH groups of two representative hydroxyphosphoranes. A bond length-pKa
correlation based on crystal structures of cyclohexanol derivatives gives values of 13.5 ( 1.5 and 8.62 (
1.87, respectively, for the apical and equatorial OH groups of tetracyclohexyloxyhydroxyphosphorane, and
an ab initio molecular dynamics calculation gives values of 14.2 and 9.8 for the corresponding first ionizations
of pentahydroxyphosphorane.
Methods and Results
Phosphate transfer reactions involving diesters typically go
by associative mechanisms,1 involving pentacovalent transition
states and, in at least some cases, intermediates. Such pentaoxy-
phosphorane intermediates 1 are of particular interest in the
hydrolysis of RNA, and their lifetimes and protonation states
are of paramount importance in understanding the mechanisms
of action of ribonucleases.2 So the pKa values of the monoanion,
as a base and as an acid, are a key to understanding the
mechanisms of these reactions. They are not, however, directly
accessible, because lifetimes of pentaoxyphosporanes are too
short. Pentaalkoxyphosphoranes are isolable but rapidly hydro-
lyzed, but no pentaoxyphosphorane with a free OH group has
been isolated.3 The derived anions are less stable still: the
current consensus is that a phosphorane monoanion in solution
has a short but significant lifetime, but a phosphorane dianion
only a borderline existence.2
CrystalStructureCorrelation: pKaEstimatesfor(Cyclohexyloxy)4POH.
There is a simple linear relationship between the length of a
C-OX bond and the pKa of the acid HOX:4,5 the stronger the
acid the longer the bond. This correlation can in principle be
used to estimate pKa values of compounds HOX too unstable
to exist as the free acid or anion.4,5 Such structure-reactiVity6
correlations are very general: they have been demonstrated for
14 of 15 classes of compounds tested, and have been extended
to N-O and P-O bonds. The single exception, which may be
said to prove the rule, is the system ArCHMe-OX, where the
torsion angle between the plane of the aromatic ring and the
C-OX bond is a second significant variable.7 The (stabilizing)
π-σ*C-OX interaction responsible for the effects on bond length
in this benzylic system is strongest for the best leaving groups
XO-, and for the (perpendicular) geometry where the π and
σ*C-OX orbitals are parallel. Since this preference is in opposi-
tion to the usual steric preference (for the largest, methyl, group
to be perpendicular to the plane of the ring) the conformation
about the Ar-C(OX) bond changes for the most electronegative
groups OX, so that the π-σ*C-OX interaction makes an
increased contribution to the lengthening of the C-OX bond.7
This effect on conformation is limited to those systems where
a strong π-type interaction affects the length of the C-OX bond
directly. There is little or no dependence on torsion angle for
σ-σ*C-OX interactions.4
Previous estimates (summarized by Perrault and Anslyn2)
range from 6.5 to 11 for pKa(1) and 11.3-15 for pKa(2). We
report two new, independent estimatessin remarkably close
agreementsfor pKa(1) for both apical and equatorial OH groups
of simple hydroxyphosphoranes, based on two completely
different approaches: (a) a crystal structure correlation and (b)
ab initio molecular dynamics calculations.
The sensitivity of the length of a C-O bond to pKa depends
on the substituents on C.4 Good correlations have been
established for acetals, and for derivatives of cyclohexanol and
(4) Kirby, A. J. AdV. Phys. Org. Chem. 1994, 29, 87-183.
(5) The method was recently used by White to estimate the pK of 2,4-
dinitrobenzenesulfenic acid. Green, A. J.; Giordano, J.; White, J. M. Aust.
J. Chem. 2000, 53, 285-292.
(6) Structure-reactiVity correlations in general: because reactivity (the rates
and equilibria of reactions of compounds R-OX) is typically correlated
(by linear free energy relationships) with the pKa values of the conjugate
acids HOX of the leaving groups.
* Address correspondence to this author. E-mail: ajk1@cam.ac.uk.
Phone: +44 1223 336370. Fax: +44 1223 336913.
(1) Thatcher, G. R. J.; Kluger, R. AdV. Phys. Org. Chem. 1996, 25, 99-265.
(2) Perreault, D. M.; Anslyn, E. V. Angew. Chem., Int. Ed. Engl. 1997, 36,
432-450.
(3) Hanes, R. E.; Lynch, V. M.; Anslyn, E. V.; Dalby, K. N. Org. Lett. 2002,
4, 201-203.
(7) Edwards, M. R.; Jones, P. G.; Kirby, A. J. J. Am. Chem. Soc. 1986, 108,
7067-7073.
9
6594
J. AM. CHEM. SOC. 2002, 124, 6594-6599
10.1021/ja025779m CCC: $22.00 © 2002 American Chemical Society

Estimating pK Values for Pentaoxyphosphoranes
A R T I C L E S
a
of primary alcohols,8 with the sensitivity to the pKa falling in
this order. Higher sensitivity promises more accurate estimates
of pKa, because the standard deviation of the relevant bond
lengths in accurate crystal structures, of the order of 0.003 Å,
is of the same order of magnitude as the sensitivity coefficient
of the correlation (i.e. (0.003 Å corresponds to (1 pKa unit).
A few attempts to make methoxymethoxyphosphoranes (con-
taining MeOCH2O-P groups) convinced us that such systems
were likely to be inconveniently unstable, so we chose to work
with phosphoranes derived from secondary alcohols, and
specifically cyclohexanol, for which good data and a good
correlation (eq i) are available.8
Scheme 1. General Procedure for Preparation of
Pentaoxyphosphoranes
Table 1. Observed Cyclohexyl-O Bond Lengths for
Cyclohexyloxyphosphoranes 3 and 4
bond lengths (Å)
moleculea
apical
1.443(4)
equatorial
1.449(4)
3(i)
1.435
1.459
1.460
1.451
1.452
1.444
1.447
1.445
1.443
1.449
bond length (cyclohexyl-OX) )
3(ii)
3(iii)
3(iv)
1.438
1.429
1.475 - (2.90 × 10-3)pKa(HOX) (i)
1.440
1.435
There are no relevant crystal structures in the Cambridge
Structural Database of phosphoranes derived only from simple
primary or secondary alcohols.
1.443
1.443
1.452
To estimate the pKa of the P-OH group of systems of interest
to the bioorganic chemist we have prepared and tried to
crystallize a series of pentaalkoxyphosphoranes from tricyclo-
hexyl phosphite. Of most interest are structures close to 1, and
we have prepared various tricyclohexyloxyphosphoranes 2
derived from uridine (U ) 3-N-methyl-uridyl), as well as simpler
cyclic derivatives of ethan-1,2-diol and cis-cyclopentane-1,2-
diol. Crystallization proved impossible for all these compounds,
using either conventional methods or the Cambridge freeze-
grow technique.9 We eventually obtained crystals of the simplest
system, pentacyclohexyloxyphosphorane [(CyO)5P, 3]. We
report its structure and derived estimates of pKa values for the
apical and equatorial OH groups of (CyO)4POH.10
1.454
1.45042 ( 0.0054
mean
4
1.4383 ( 0.0050
1.433(3)*
1.461(3)
1.466(4)
aIndependent data for four molecules of 3 in the asymmetric unit.10
solid (mp 53-54 °C) is extremely hygroscopic. Its spectroscopic
properties and 31P NMR chemical shift (δ 139.2 ppm) agree with
previously reported values. Cyclohexyl benzenesulfenate was prepared
in 60-75% yield, according to Chang et al.13 (1,2-diol bis-benzene-
sulfenates can be prepared by using the same procedure). These
compounds were not isolated but redissolved immediately after workup
(estimated yields 57-68%) in dry pentane or dry DCM, and used
immediately. To a cooled (-78 °C) solution of alkyl benzenesulfenate
(10 mmol) in dry pentane or DCM (150 mL) was added, over a period
of 1 h, a solution of tricyclohexyl phosphite (2.5 mmol) in 30 mL of
dry pentane. After standing for a further 1 h the mixture was allowed
to warm to room temperature and stirred for a further 2 h. Cooling to
-78 °C precipitated diphenyl disulfide, which was removed by
filtration. The filtrate was concentrated to 50 mL and the process
repeated. The residues were washed with anhydrous propylene carbonate
(3 × 50 mL) and acetonitrile (3 × 50 mL), and the combined extracts
were evaporated to dryness. As the primary criterion of purity we used
the peak in the 31P NMR with the characteristic chemical shift of
pentacovalent phosphorus. After purification the peak assigned to the
desired phosphorane accounted for over 95% of phosphorus compounds
present, and the 1H NMR spectrum indicated that less than 5% of
diphenyl disulfide remained.
We have a crystal structure for only one compound with a
five-membered ring spanning apical and equatorial positions,
the adduct 4 of tricyclohexyl phosphite and phenanthraquinone:
11
results for this compound are discussed briefly below.
Experimental Section
Pentacyclohexyloxyphosphorane (3) was eventually obtained as
colorless crystals, mp <15 °C, by slow cooling from pentane. The
literature mp13 of 3 is 90-92 °C: we made no attempt to measure the
melting point of the crystals we obtained at low temperatures. 31P NMR:
1H and 31P NMR spectra were recorded on a BRUKER Avance-
DRX 400 spectrometer at 300 K, using tetramethylsilane and 85% H3-
PO4 as internal and external standards, respectively. Chemical shifts
are reported in δ (ppm). All solvents used were very carefully dried,
and reactions carried out in anhydrous conditions under argon.
Tricyclohexyl phosphite was prepared as previously described12 and
recrystallized from acetonitrile. The carefully dried white crystalline
1
8.49 Hz. H and 13C NMR agreed with
31
1
-73.68 ppm; sextet, J
P- H
reported values.13 (Other tricyclohexyloxyphosphoranes failed to crys-
tallize, and were not fully characterized.)
3 crystallized in space group P1, with four molecules in the unit
cell, giving 20 relevant C-O bond lengths from the one experiment
(Table 1).10 Of these eight involve apical and twelve equatorial oxygens,
allowing independent estimates of the pKa values of apical and
equatorial P-OH groups. The mean (cyclohexyl)C-OP bond lengths
(Table 1) are 1.436 ( 0.0044 and 1.450 ( 0.005 Å for bonds to apical
(8) Amos, R. D.; Handy, N. C.; Jones, P. G.; Kirby, A. J.; Parker, J. K.; Percy,
J. M.; Su, M. D. J. Chem. Soc., Perkin Trans. 2 1992, 549-558. Equation
i is taken from this paper.
(9) Bond; A. D.; Davies, J. E.; Kirby, A. J. Acta Crystallogr. E 2001, 57,
o1242-o1244.
(10) For full structural information see: Davies, J. E.; Kirby, A. J.; Roussev,
C. D. Acta Crystallogr. E 2001, 57, o994-o995.
(11) Jones, P. G.; Kirby, A. J.; Pilkington, M. Acta Crystallogr. E 2002, 58,
o268-o269.
(13) Chang, L. L.; Denney, D. B.; Denney, D. Z.; Kazior, R. J. J. Am. Chem.
Soc. 1977, 99, 2293-2297.
(12) Saunders, B. C.; Stark, B. P. Tetrahedron 1958, 4, 169.
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J. AM. CHEM. SOC. VOL. 124, NO. 23, 2002 6595

A R T I C L E S
Davies et al.
Figure 1. Molecular conformations of 3 (molecule 1 of 4 in the asymmetric unit)10 and 4,11 showing displacement ellipsoids at the 30% probability level.
and equatorial O, respectively, corresponding (eq i) to pKa values of
13.5 ( 1.5 and 8.62 ( 1.87 for the apical and equatorial OH groups of
the tetracyclohexyloxyhydroxyphosphorane.
the 10 ps time-window accessible to current ab initio molecular
dynamics (MD) simulation technology. Dissociation of weak
acids, however, can only be studied by using special sampling
methods. Previous work on water autodissociation21,22 has shown
that the free energy for breaking an OH bond can be computed
by thermodynamic integration. In this approach an explicit order
parameter q, such as the distance r between the leaving proton
H+ and the residual basic anion A-, is fixed by mechanical
constraint techniques. A series of constrained runs is performed
at consecutive values of q bridging two states of interest. The
mean force f(q) is evaluated for each of the intermediate values
of q. Numerical integration of these data gives an estimate of
the potential of mean force w(q) relative to its value w(q0) at
some reference state q0.
Tricyclohexyloxyphenanthren-9,10-dioxyphosphorane (4) was pre-
pared by the addition of triisopropyl phosphite to phenthraquinone. The
crystal structure (Figure 1)11 shows that the CsO bond of the apical
cyclohexyloxy group in the crystal is axial on the cyclohexane ring.
The equatorial preference of an alkoxy group is relatively small, even
for tert-butoxy (A-value 0.7514), so this inversion is reasonably explained
as a crystal packing effect. The axial geometry does not vitiate the
bond length-pKa correlation, because our best information is that axial
and equatorial cyclohexyl-OX bonds to the same OX leaving group
do not differ significantly in length,15 but phosphorane 4 also differs
from 3 in two more important respects. Two of the oxygen atoms
attached to P are phenolic, and thus substantially more electronegative
than alkoxyl groups; and they are constrained apical-equatorial in a
five-membered ring. We have no simple way of estimating how far
either difference would affect the pKa values of phosphorane P-OH
groups. It may be presumed that more electronegative atoms will
increase the acidity of geminal P-OH groups, by the generalized
anomeric effect, and the relevant C-OP bonds are indeed significantly
shorter than the corresponding bonds of 3. But in the absence of more
data the pKa estimates (formally about 11 and 4, respectively16) we
can derive for the apical and equatorial P-OH groups of structures,
based on a single compound, can only be indicative.17
∆w(q) ) w(q) - w(q ) ) - qdq′ f(q′)
(ii)
∫
0
q0
The order parameter q can be a general function of particle
coordinates. This feature is the basis for the constraint method
used in the present calculation: as an alternative to imposing
an AH distance constraint, proton abstraction is enforced by
controlling the hydrogen coordination of nH of anion A-.22 This
has the advantage that all protons are treated on a strictly
equivalent footing.
Density Function Based ab Initio Molecular Dynamics
Calculations: P(OH)5
The “Car-Parrinello” approach18 has proved particularly
suitable for simulating aqueous acid-base reactions, dominated
as they are by solvent effects. An important element in this
success is that the time scales for transfer of excess protons in
water19 and the dissociation of strong acids20 are well within
To maximize the duration of a run in real time, the spatial
dimensions of the model were kept to a minimum, with one
P(OH)5 molecule solvated in an cubic cell of side L ) 9.86 Å.
Filled with pure liquid water this box contains 32 H2O
molecules. A single P(OH)5 solute molecule must displace 3
water molecules to maintain ambient pressure, leaving 29 solvent
molecules in the cell. This system (see Figure 2) was used for
all protonation states, without further adjustment of solvent
density. The temperature was kept at 300 K.23 We verified that
under these conditions, P(OH)5 and monanion are stable on the
(14) Eliel, E. L.; Wilen, S. H. Stereochemistry of Organic Compounds; John
Wiley & Sons: New York, 1994.
(15) Amos, R. D.; Handy, N. C.; Jones, P. G.; Kirby, A. J.; Parker, J. K.; Percy,
J. M.; Su, M. D. J. Chem. Soc., Perkin Trans. 2 1992, 549-558.
(16) Nominally 11.03 ( 1.03, 4.82 ( 1.04, and 3.10 ( 1.38, based on standard
deviations of bond length from the crystal structure solution.
(17) Published structures for the phenanthraquinone-triisopropyl phosphite adduct
are not accurate enough to be useful in this context (Hamilton, W. C.;
LaPlaca, S. J.; Ramirez, F.; Smith, C. P. J. Am. Chem. Soc. 1967, 89, 2268-
2272. Spatley, R. D.; Hamilton, W. C.; Ladell, J. J. Am. Chem. Soc. 1967,
89, 2272-2278).
(21) Trout, B. L.; Parrinello, M. Chem. Phys. Lett. 1998, 288, 343. Trout, B.
L.; Parrinello, M. J. Phys. Chem. B 1999, 103, 7340.
(22) Sprik, M. Chem. Phys. 2000, 258, 139.
(23) All simulations were carried out with the CPMD package developed by J.
Hutter, A. Alavi, T. Deutsch, M. Bernasconi, S. Goedecker, D. Marx, M.
Tuckerman, and M. Parrinello at the MPI solid-state research institute in
Stuttgart and the IBM Research Laboratory Zurich.
(18) Car, R.; Parrinello, M. Phys. ReV. Lett. 1985, 55, 2471.
(19) Tuckerman, M.; Laasonen, K.; Sprik, M.; Parrinello, M. J. Phys. Chem.
1995, 99, 5749.
(20) Laasonen, K.; Klein, M. L. J. Phys. Chem. A 1997, 101, 98.
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6596 J. AM. CHEM. SOC. VOL. 124, NO. 23, 2002

Estimating pK Values for Pentaoxyphosphoranes
A R T I C L E S
a
Figure 2. The aqueous P(OH)5 model system used, with a snapshot of a
coordination-enforced proton transfer. P(OH)5 and the H2O involved are
drawn as balls and sticks: the OH under coordination control is black.
Periodic boundary conditions are applied.
Figure 3. (a) Mean force as a function of proton coordination number for
apical (squares) and equatorial (circles) P(OH)5 oxygens and (b) the free
energy curves obtained by thermodynamic integration (eq ii), using the
equilibrium at nH ≈ 1 as the reference point.25
time scale of the simulation (5 ps). The apical monoanion
relaxed by a pseudorotation to the equatorial form, consistent
with the expected conformational stability. The dianion was
found to be strongly basic, converting to a monoanion by
deprotonation of a coordinated water molecule.24
a strong acid, spontaneously releases a further proton within
femtoseconds). The resulting potentials of mean force (PMF,
see eq ii) for the two deprotonation routes are compared in
Figure 3b. The equatorial curve terminates at a free energy
difference of 10.2 kcal/mol for nH ) 0.2. This is well below
the ∆w ) 14.4 kcal/mol at the final point of the apical curve.
In two separate series of calculations, the method of depro-
tonation by coordination control was applied to an apical and
an equatorial hydroxyl group. Starting from the equilibrium
value of nH ) 0.98, which was chosen as the reference state,25
the proton coordination number nH was decreased toward zero,
thus transferring a proton from the donor hydroxyl group to an
acceptor solvent molecule (Figure 2). For each fixed value of
nH a trajectory was computed, roughly 2 to 3 ps in length. In
Figure 3a, the “blue-moon” corrected average constraint (i.e.
mean) force22 is plotted as a function of nH. The steep rise in
restoring force for small perturbation of nH, reaching a maximum
at nH ) 0.925, is a reflection of the initial stretching of the OH
bond. With further reduction of coordination number the mean
force is seen to fall off toward zero due to increased binding to
the solvent. The P-O bond donating the proton shrinks by 0.1
Å in the process.26
Computation of pKa: Methodology
Computation of pKa values has been applied in organic
chemistry on a number of occasions,27 and has been imple-
mented in widely available quantum chemistry packages such
as Gaussian. These calculations follow a thermodynamic
cycle: the acid is taken out of solution and dissociated in a
vacuum and the fragments are finally reinserted into solution.
Whereas the vacuum proton affinities can be computed with
high accuracy by using ab initio electronic structure methods
(SCF or DFT), the calculation of the solvation energies is carried
out in a reaction field depending on a set of empirical
parameters.27b (For a very recent example of the application of
this approach applied to hydroxyphosphoranes see the Note
Added in Proof.) In the approach adopted here, by contrast, the
acid remains solvated throughout the dissociation process. The
pKa is then determined from the free energy profiles for
deprotonation. The calculation is “in principle” parameter free
(but see below).
At nH ≈ 0.2 transfer is complete and a contact ion-pair
+ -
OH3 | OP(OH)4 is formed. At this point the proton, now
effectively attached to the water, can initiate a Grotthus-type
diffusion process. In our small periodic cell this rapidly brings
the excess proton back into contact with the newly created
phosphorane monoanion. The four remaining, unconstrained OH
groups are vulnerable to attack by the acid solvent, converting
them to PO+H2. Our attempts at enforced deprotonation
invariably ended in such solvent-mediated tautomerization,
followed by rapid dehydration to phosphoric acid (which being
The basis of our estimation of pKa is the statistical mechanical
method described by Chandler.28 This purely classical approach
is valid for dissociation reactions AB f A + B occurring in
(24) The ab initio MD approach applied here is similar to previous density
functional plane wave based studies of aqueous systems (see refs 19-22).
An important difference, however, is that we use the HCTH instead of
BLYP gradient-corrected density functional, because of the much improved
treatment by HCTH of phosphorus and sulfur compounds (Boese, A. D.;
Doltsinis, N. L.; Handy, N. C.; Sprik, M. J. Chem. Phys. 2000, 112, 1670-
1671).
(27) (a) Lyne, P. D.; Karplus, M. J. Am. Chem. Soc. 2000, 122, 166-167. (b)
Schuurman, G.; Cossi, M.; Barone, V.; Tomasi, J. J. Chem. Phys. A 1998,
102, 6706-6712. (c) Shapley, R. A.; Bacskay, G. B.; Warr, G. G. J. Chem.
Phys. A 1998, 102, 1938-1944. (d) Major, D. T.; Laxer, A.; Fischer, B. J.
Org. Chem. 2002, 67, 790-802.
(28) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford
University Press: Oxford, 1987.
(25) For an explanation of the small deviation from unity see ref 22.
(26) For details see: Doltsinis, N. L.; Sprik, M. To be submitted for publication.
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A R T I C L E S
Davies et al.
the gas phase or solution. The key equation is the reversible
work theorem relating the AB radial distribution function (RDF)
to the PMF according to
g(r) ) exp[-w(r)/kBT]
(iii)
The RDF in eq iii can be interpreted as a probability
distribution for distance r between atoms A and B. Equation
iii, supplemented by a suitable maximum bonding radius r )
Rc distinguishing between reactant AB and product A + B,
provides us with all the information needed to determine the
reaction quotient at equilibrium and, hence, to derive the
classical expression for the (inverse) dissociation constant
Rc
Kc-1 ) c
dr4πr2 exp[-w(r)/kBT]
(iv)
∫
0
0
with c0 the standard concentration. To apply eq iv to a PMF
for proton coordination number, the dependence on nH must be
transformed to a function of r. This was done numerically by
inversion of the relationship between the expectation value of
AH distance and nH calculated by averaging over the MD
trajectories at fixed proton coordination number. Equation iv
holds subject to the condition that w(r) is the total work needed
for bringing reactants together from infinite separation. What
is available from simulations, however, are relative PMF values
(eq ii) referred to a finite maximum separation r0 ) Rmax e L/2
. This introduces an uncertainty of w(Rmax)/(2.3 kBT) in the pKa
that for the small system sizes used in ab initio MD can easily
amount to underestimation of several pK units. To reduce this
systematic error we have adapted the procedure of Chandler28
for small systems, exploiting the fact that in controlled dis-
sociation of a weak acid there is only one reactive group AH
present, namely the group to which the constraint is applied.
This enables us to fix the reference free energy by normalizing
the corresponding RDF to unity:
Figure 4. Dependence of the pK for the ionization of liquid water on the
bonding criterion Rc. The circles give the result applying eqs vi and viii to
the potential of mean force (PMF) of ref 21 generated by using a distance
constraint. The squares are the pK values obtained when the PMF for proton
coordination from ref 22 is used. The crossing dashed lines locate the value
of Rc that reproduces the experimental ionization constant of water.
replacing eq vii. The activity of the undissociated reactant (H2O)
is set to unity giving
2
R(Rc)N
Kw )
(viii)
(
)
c0V
and instead of N ) 1 the total number of solvent molecules in
the model cell is used (N ) 32).21,22
Computation of pKa: Results
As a first crucial test, the modified scheme for computing
pKa was applied to the free energy profiles for autodissociation
of water obtained previously.21,22,29 The result is shown in Figure
4, plotted as a function of the bonding radius Rc.
1
V
dr exp[-∆w(r)/k T] ) 1
(v)
∫
B
V
Considering that the two sets of PMF data were generated
with different deprotonation constraints, the good agreement can
be seen as a validation of our methodology. Moreover, when
the parameter Rc is adjusted to reproduce the experimental
ionization constant (pKw ) 14), a value of Rc ≈ 1.2 Å is found,
consistent with accepted criteria for maximum extension of OH
bond lengths. Repeating the same procedure for the PMF values
for acid dissociation of P(OH)5 we find the result displayed in
Figure 5. For distances where the OH bond can be considered
broken, the curves for the equatorial and apical hydroxyl groups
differ by at least 4 pK units.
As is also evident from Figures 4 and 5, the computed
equilibrium constants show significant dependence on Rc,
making specification of this parameter a critical step in the
calculation. This dilemma was resolved by treating Rc as an
adjustable parameter. Fitting the pKw obtained by the coordina-
tion constraint method to the experimental value of 14 gives Rc
) 1.22 Å. Applying this value to the phosphorane PMF of
Figure 3b we obtain pKa values of 9.8 and 14.2 for the equatorial
where V ()L3) is the volume of the model cell. By using the
normalized RDF of eq v the dissociation fraction R can now be
calculated directly from the expectation value of the number of
protons within radius Rc from the reactive O atom
Rc
dr 4π2 exp[-∆w(r)/kBT]
∫
0
R(Rc) ) 1 -
(vi)
Rmax
dr 4πr2 exp[-∆w(r)/kBT]
∫
0
where the volume integral in eq v has been approximated by
the integral over a sphere with radius Rmax. To obtain our
estimate of the equilibrium constant, we express Ka in terms of
the dissociation fraction and insert eq vi
R(Rc)2
N
Ka )
(vii)
c V
1 - R(Rc)
0
setting N ) 1. A similar approach can be employed for the
estimation of the ionization constant of liquid water from free
energy profiles for autodissociation.21,22 Since dissociation is
enforced for only one selected H2O molecule, eq vi remains
valid. The equivalence of the water molecules is accounted for
by the relationship between Kw and the dissociation fraction R
(29) The agreement between the work for deprotonation of a water molecule
and the pKw must be considered fortuitous and the claim made in ref 22
that this number can be used as an estimate of the equilibrium constant is
unjustified. for a proper treatment of the kinetics of water autodissociation
see: Geissler, P.; Dellago, C.; Chandler, D.; Parrinello, M. Science 2001,
291, 2121.
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Estimating pK Values for Pentaoxyphosphoranes
A R T I C L E S
a
esters. The thermodynamically stable form at pH 7 is clearly
identified as the neutral phosphorane, so any anionic intermedi-
ate that lives long enough to reach proton transfer equilibrium
will be converted to, and thus be present primarily as the neutral,
OH form. (Phosphorane dianions, understood, as discussed
above, to have at best a borderline existence toward P-O
cleavage, also appear from our calculational results to be basic
enough to be protonated by solvent water.) Of course the
predominant neutral species is unlikely to be the reactiVe form
of a hydroxyphosphorane near pH 7, and the familiar problems
of kinetic equivalence in systems involving a mobile proton or
protons still have to be resolved.
The crystal structure correlation method depends crucially
on the quality and quantity of the data available. It offers a
simple route to a rough estimate of almost any pKa of interest
that cannot be measured directly. But extensive, high-quality
data are needed to allow precise estimates, for example for the
cyclic phosphoranes of interest in ribonucleotide reactions; and
it seems clear that more such data will be hard won. In this
situation theory may turn out to be the more promising way
forward. So it is important and encouraging that our first
calculations at this level give estimates closely similar to those
based on experiment.
Figure 5. pKa values for equatorial (circles) and apical (squares) acid
dissociations of P(OH)5 as a function of maximum bonding radius Rc,
corresponding to the potentials of mean force in Figure 3. The vertical
dashed line indicates the optimum value of Rc reproducing the experimental
pKw of liquid water (see Figure 4).
and apical dissociations, respectively, in good agreement with
our experimental estimates for a related system.
Note Added in Proof. In a very recent investigation of acidity
constants of phosphoranes with structures related to 1, using
partly empirical continuum dielectric methods, Lopez et al. (J.
Am. Chem. Soc. 2002, 124, 5010-5018) report a value of pKa
) 7.9 for the first ionization of an equatorial OH group of
ethylene phosphorane. This value is in good agreement with
our experimental estimate of 8.6 for the corresponding OH group
of (CyO)4POH, though lower than our calculated value of pKa
) 9.8 for P(OH)5. Applying the statistical correction described
by Lopez et al. gives a macroscopic value of 9.3. Thus the two
calculations bracket the experimental estimate, and the result
of Lopez et al. lends further support to a best estimate for the
pKa of an equatorial OH group of a hydroxyphosphorane close
to our experimental value of 8.6.
Discussion
The two methods used in this work allow independent
estimates of the pKa values of both apical and equatorial P-OH
groups of phosphoranes: in systems (CyO)4P-OH and (HO)4P-
OH, respectively. The two systems are different, and not
expected to have identical acidities, so small differences in the
estimates are to be expected. The relative values of the pKa
estimates also make good sense: the pentahydroxyphosphorane
is expected to be the weaker acid because hydrogen-bonding
solvation will reduce electron withdrawal by the remaining four
OH groups and thus raise the pKa of a given P-OH.30 The
results confirm, as is well-known, that equatorial P-OH groups
are more strongly acidic than apical, and put an order of
magnitude on the difference (expected to be more accurate than
the absolute values of the estimates) of some 5 pKa units. They
also fall reassuringly near the middle of the range of previous
estimates of pKa(1).2
Acknowledgment. We thank Christoph Dellago for helpful
discussions. C.D.R. was a NATO/Royal Society Postdoctoral
Fellow, further supported within the TMR Programme of the
European Commission by the European Network ENDEVAN.
N.L.D. is grateful to EPSRC for financial support. Computa-
tional resources were provided by the High Performance
Computing Facility of the University of Cambridge.
The estimated value of pKa(1) of ca. 9 for a hydroxyphos-
phorane bearing only OH or O-alkyl groups provides a firm
basis for discussing phosphate transfer mechanisms of phosphate
(30) Pentahydroxyphosphorane is used here as a model for the sorts of
intermediates described in the Introduction. It is not a viable target for
direct measurements in aqueous solution, so statistical corrections to the
estimated pKa are not appropriate.
JA025779M
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J. AM. CHEM. SOC. VOL. 124, NO. 23, 2002 6599