S. Aime et al. / Journal of Magnetic Resonance 178 (2006) 184–192
187
daj
dt
2.3. Polarization transfers
¼ ꢀiTrð½H; rꢂGjÞ ¼ iTrð½H; GjꢂrÞ.
ð9Þ
In other situations, where G1 + G2 + G3 is still present in
the Hamiltonian without commuting with it, we can expect
transfers toward other quantities through equations derived
from ½H; rstꢂ ¼ 0. These transfers will be assessed by the
2.2. Invariants
2.2.1. The A2X and AA0X spin systems
The immediate consequence of (9) is that the quantity Gj
does not evolve if the associated operator commutes with
the Hamiltonian of the spin system corresponding to the
hydrogenated molecule (in that case,
asjt ¼ ajð0Þ). In particular, if G1 + G2 + G3 commutes with
the Hamiltonian, then rst = r(0) and no transfer can occur
towards any other quantity. This is the case whenever the
two hydrogen nuclei remain magnetically equivalent after
hydrogenation since the part of the Hamiltonian concerning
coefficients asjt involved in the expansion in the steady state
P
density operator (see (6)) rst ¼ jasjtGj whereas the effective
P
part of the Hamiltonian can be expanded as H ¼ iJiGi
where Ji is the J coupling between two spins and Gi the
two-spin product operator corresponding to this coupling.
daj
dt
¼ 0 and
Resorting to ½H; rstꢂ ¼ 0 leads to equations of the type
X
Ji½Gi; Gjꢂasjt ¼ 0.
ð10Þ
i;j
A
A0
The commutator in (10) is non-zero provided that the two-
spin product operators Gi and Gj share one spin and that
[Gi,Gj] = Fk, where Fk is a three-spin product operator
involving the three different spins which appear in Gi and
Gj. An equation is obtained for all pairs (Gi,Gj) leading to
0
J
c0oupling0 between these two nuclei, JAA ðIx Ix þ
IAy IyA þ IAz IzA Þ (which is, except the factor JAA , nothing else
than G1 + G2 + G3) is known to commute with the whole
Hamiltonian [13]. Thus, no hyperpolarization transfer oc-
curs if the hydrogenated molecule proceeds from an A2X
spin system while transfer toward X becomes possible for
an AA0X spin system (the subject of the present study).
0
the same Fk. This means that, if we start from an hyperpolar-
0
ized state (the source, say G1 ¼ IAx IxA ), there will be a cou-
pling (thus transfer) with another state (target) represented
by Gj, provided that the commutator [G1,Gj] on the one hand,
and the coupling constants J1 and Jj on the other hand, are non-
zero. This leads at least to the following equation:
2.2.2. The A2A02X spin system
A related situation is when the two hydrogens in the
para state are embedded, upon hydrogenation, in two
different sets of magnetically equivalent nuclei. This is
precisely the case for the hydrogenation of dimethylmal-
eate which results in dimethylsuccinate (see above). For
molecules involving a carbon-13 within the initial double
ꢃJ1ajst ꢄ Jja1st ¼ 0.
ð11Þ
Additional terms may be involved if other commutators
lead to the same Fk as [G1,Gj].
It can be noticed that the right hand-side member of all
equations derived from relation (10) is zero. These equa-
tions need therefore to be appended by an equation, the
right hand-side member of which is non-zero. This can be
provided by a conservation equation based on the norm
of the density operator. In a very general way, the norm
of an hermitian operator (as the density operator) is equal
to the trace of its square. The norm of the density operator
must be kept constant provided that it describes a steady
state (a state devoid of time evolution, i.e., a state for which
no precession occurs). This is because the square of the
density operator represents the ‘‘populations’’ (contribu-
tions) of the various spin operators upon which it is
expanded. This is true for r(0) and rst. One has thus
bond, we have to deal with a A2A0 X spin system, one
2
of the two para hydrogens is A, the other A0. As shown
in the next section, hyperpolarization transfer, if it oc-
curs, originates necessarily from G1 + G2 + G3 and is
0
mediated
by
JAA
.
In
other
words,
only
0
0
0
0
ðIAx IAx þ IAy IyA þ IAz IzA Þ must be multiplied by JAA in
the Hamiltonian of the final spin system. It turns out
that such a term no longer exists in the Hamiltonian
of a A2A0 X spin system. Rather one has (A1 and A01
2
being the two additional protons of the spin system)
0
0
0
A
A0
A0
A10
JAA ðIx Ix þ IxAIxA þ IxA Ix þ IxA Ix þ IyAIyA þ IyAIAy
1
1
1
1
0
A0
A10
A0
A10
0
0
þ IAy Iy þ IAy Iy þ IAz IzA þ IzAIzA þ IAz Iz þ IAz Iz Þ.
1
1
1
1
1
Clearly G1 + G2 + G3 vanishes by losing its identity
through admixture with the three other terms and by the
fact that (A,A1) on one hand, and (A0, A10 ) on the other
hand, are indistinguishable. This feature thus precludes
Trðr2stÞ ¼ Trðr2ð0ÞÞ.
This can be written as
ð12Þ
X
2
2
2
2
2
jasjtj ¼ ja1ð0Þj þ ja2ð0Þj þ ja3ð0Þj ¼ 3ja1ð0Þj .
ð13Þ
any polarization transfer toward X in an A2A0 X spin system
2
j
and explains the above experimental results. It can, howev-
er, be noted that the hyperpolarization originating from p-
H2 cannot be lost and remains amid the proton spin system
(observed experimental results not shown). Albeit not
transferred to X, it actually appears as enhanced 13C satel-
lites in the proton spectrum since the presence of a carbon-
13 is necessary for breaking the symmetry, thus unraveling
the existence of hyperpolarization.
It can be noticed that the rightmost member of (13) is not
of paramount importance as it represents merely a scaling
factor associated with the rate of hyperpolarization (which
may depend on a lot of experimental factors).
Additional relations can possibly be obtained by looking
at linear combinations of Gj which commute with the Ham-
iltonian. In that case, the linear combination is invariant