Ab Initio Calculation of Isotopic Fractionation in B(OH)
3
(aq) and BOH
4
-
(aq)
James R. Rustad*
,²
and Eric J. Bylaska
‡
Department of Geology, UniVersity of California, DaVis, One Shields AVenue, DaVis, California 95616, and
Pacific Northwest National Laboratory, MS K9-96, Richland, Washington 99352
Received November 21, 2006; E-mail: jrrustad@ucdavis.edu
Computational methods are becoming increasingly central to
understanding the fractionation of stable isotopes in geochemical
systems. A prominent application is the fractionation of
11
B and
10
B between boric acid B(OH)
3
and borate ion B(OH)
4
-
in seawater.
Boron isotopes are key indicators of the pH and CO
2
content of
oceans in the geologic past.
1
In seawater solutions at 25 °C and
[B]
total
) 4.5 ppm, B(OH)
3
and B(OH)
4
-
exist in equal concentra-
tions at pH ∼8.8. Since the
11
B/
10
B ratio differs between B(OH)
3
and B(OH)
4
-
, the isotopic composition of each species is a function
of pH over the range where both species coexist in appreciable
concentrations. If it is assumed that only B(OH)
4
-
is incorporated
into minerals, then the pH of seawater in equilibrium with the min-
erals at the time of deposition can be obtained from the
11
B/
10
B
ratio in the mineral phases. The calculation of pH requires know-
ledge of the isotope fractionation factor R
34
. In the harmonic limit,
R
34
)
B(OH)3
/
B(OH)4
-
(
3
/
4
) where is the reduced partition
function ratio:
where u
(h,l)i
) pc2πω
(h,l)i
/kT, h and l refer to the heavy isotope
and light isotopes, respectively, and the product runs over all
frequencies.
2
Paleo-ocean pH estimates have until recently relied on semiem-
pirical estimates of R
34
,
3
in part because of the difficulty of
measurements at [B]
total
) 4.5 ppm. Several problems have lately
been identified in the original work, and improvements have been
sought through electronic structure calculations.
4,5
Recent measure-
ments of the isotope-induced shift in the pK
a
of B(OH)
3
have
reported R
34
values close to 1.028, within 1 per mil of the Hartree-
Fock calculations.
5c-e
This level of agreement is surprising as the
Hartree-Fock calculations involve extensive frequency scaling. The
approach combines the effects of solvation, inadequacies in the
wave function, and anharmonicity into a single factor, making
systematic improvement difficult.
Here, we take a new approach to estimating R
34
for aqueous
species, using ab initio molecular dynamics (AIMD).
6
The AIMD
approach is useful because the fractionating species are embedded
in real solvent at configurations typical of 300 K. Moreover, the
vibrational frequencies obtained through AIMD are not restricted
to be harmonic. This study is the first time AIMD has been applied
to the calculation of isotopic fractionation. First, we establish
whether the AIMD calculations are capable of reproducing the
observed frequencies for B(OH)
4
-
(aq) and B(OH)
3
(aq). Second,
we explore the utility of eq 1 in reproducing R
34
. Work on gas-
phase systems, for example, indicates that it is better to use
harmonic frequencies in eq 1 rather than anharmonic frequencies
derived from experiment.
7
It is not clear whether this would remain
true in aqueous systems.
B(OH)
4
-
and B(OH)
3
are inserted in a small periodic cell of
water molecules, and forces are calculated using density functional
theory. The forces are used to perform a molecular dynamics
simulation. The classical vibrational density of states responsible
for boron fractionation can then be obtained by Fourier transforma-
tion of the velocity autocorrelation function for the boron atom.
All calculations were carried out with NWChem,
8
employing a
plane-wave basis (energy cutoff ) 90 au) with Troullier-Martins
pseudopotentials
9
and the PBE96 exchange-correlation functional.
10
We used Car-Parrinello dynamics with the fictitious mass set at
100 au. Electronic and nuclear degrees of freedom were attached
²
University of California, Davis.
‡
Pacific Northwest National Laboratory.
)
(
Q
h
Q
l
29
)
∏
i
u
hi
u
li
e
-u
hi
/2
1 - e
-u
hi
1 - e
-u
li
e
-u
li
/2
(1)
Figure 1. Vibrational density of states for the boron atom in (a) B(OH)3-
(aq), (b) B(OH)4
-
(aq), and (c) teepleite (Na2ClB(OH)4). Green:
11
B; Blue:
10
B. Multiple lines represent spectra calculated from replicas of the VAF
within the standard error.
Published on Web 02/01/2007
2222 9 J. AM. CHEM. SOC. 2007, 129, 2222-2223 10.1021/ja0683335 CCC: $37.00 © 2007 American Chemical Society