Use the subjunctive mood in that-clauses that are the complements of verbs expressing an obligation or a demand (for instance, advise, ask, command, demand, desire, insist, propose, recommend, request, or suggest) and to express conditions contrary to fact.
The consultant recommended that the staff be recertified in laboratory safety procedures.[that-clause expressing obligation or demand]
If the Earth were the size of a basketball, its surface
would be smoother than a basketball's. [condition contrary to fact]
Do not use the auxiliary do when you negate the that-clause with not.
When you are performing this procedure, it is crucial that the temperature does not rise significantly.
When you are performing this procedure, it is crucial that the temperature
not rise significantly.
The conditional, often considered a variety of the subjunctive, is used to express states or actions contrary to fact, but whose truth or possible implications you would like to consider. In particular, use the conditional in if clauses whose conditions are not true. To form the conditional for the present tense or the future tense, use the past tense form of the verb.
Researchers assume that a nuclear war would raise an enormous pall of thick, sooty smoke from massive fires that would burn for days, even weeks, following an attack.
--C. Donald Ahrens, Meteorology Today
If we could travel far beyond the protective blanket of
air that shelters our planet, beyond the ancient craters of the moon, and beyond
the orbits of the nine known planets, we would reach the realm
of interstellar space, where our solar system leaves off and the rest of the
--Andrew Fraknoi, "The Universe: An Introduction"
To express the past tense of the conditional, add the auxiliary have to your verb phrase.
Scientists should have recognized the possibility of black holes in 1916, when the German astronomer Karl Schwarzchild succeeded in solving Einstein's equations for the gravity surrounding a point mass.[The combination of should (conditional) and have (past) signals that this was an unreal past action; the recognizing did not take place.]
--William J. Kaufmann, "The Black Hole"