It is theoretically possible

The structures of ATP and ADP are shown above, with the phosphate groups designated α, β and γ. The value of the Gibbs standard free energy of hydrolysis of the α–β phosphoanhydride bond of ADP is similar to that of the β–γ phosphoanhydride bond of ATP at approx. 30 kJ per mol. Hence it is theoretically possible to couple the hydrolysis of the α–β phosphoanhydride bond of ADP to an energy-requiring process and produce an overall negative ΔG.
(The reader may wish to consult an external source — e.g. this previous SE Biology answer — if unfamiliar with a chemical account of the popular but imprecise idea of ATP as a “source of energy”.)
The hydrolysis of the analogous bond of ATP is so used
Most reactions and processes for which ATP provides the driving free energy change involve the hydrolysis of the β–γ phosphoanhydride bond in the reaction:
ATP → ADP + Pi
However in several synthetic processes the energy is provided from the hydrolysis of the α–β phosphoanhydride bond in the reaction:
ATP → AMP + PPi
These include RNA synthesis (to form a phosphodiester bond) and amino-acyl tRNA synthesis (to form the aminoacyl bond, which itself provides the energy to drive peptide bond formation).
Examples of ADP hydrolysis driving energy-requiring reactions?
Although this may merely reflect my ignorance, I know of no reactions of the sort:
ADP → AMP + Pi
where the energy of hydrolysis is used to form a chemical bond etc.
So, although I may be proved wrong, my answer is NO. But…
…a possible exception — not producing inorganic phosphate — is the adenylate kinase reaction, for which the overall standard free energy change is near zero:
2 ADP ⇔ ATP + AMP
This could be considered as using the energy of hydrolysis of the α–β phosphoanhydride of one of the molecules of ADP to form the β–γ phosphoanhydride bond of ATP. This is thought to be important in the regulation of glycolysis, the AMP produced serving as a signal for the requirement of energy and acting as an activator of the enzyme phosphofructokinase.
Why — or why not?
“Why” questions are dangerous in biology, as there is a tendency to argue that the way things are are the way they must be; and there are generally no tests of hypotheses in support of such a view. Nor did the original poster ask why. But with the strict understanding that what follows is just speculation, I offer these thoughts in support of my generally negative answer:
- I have argued previously that an NTP (ATP — perhaps by chance)
developed as the “universal energy currency” because it was already
involved in RNA synthesis in the hypothetical RNA world. Clearly this
requires hydrolysis of the α–β phosphoanhydride bond of a nucleoside
phosphate molecule. The argument for ATP over ADP would be that the
pyrophosphate produced in the former process is further hydrolysed to
inorganic phosphate by pyrophosphates, and the irreversibility of the
reaction (heat is produced) is important to prevent RNA synthesis
reversing. As NTPs had to be synthesized anyway, they could also be
used for other purposes, but it was energetically more efficient to
hydrolyse the β–γ phosphoanhydride bond if irreversibility was not
crucial (e.g. the product removed).
- Once ATP became fixed as the energy currency, metabolism was tuned
for its production and, later regulation. Thus the development of
systems for anaerobic and aerobic synthesis of ADP would not have
conferred any advantage on an organism. When regulatory systems
developed in involving the relative concentrations of ATP, ADP and
AMP (as mentioned above), using ADP to drive reactions would have
been disruptive.