LTL to Büchi automaton, deterministic?
There are languages that are representable by an LTL formula but by no deterministic Büchi automaton (DBA).
Thus a translation from LTL to DBA cannot exist.
An example is the formula $\mathbf{F}\mathbf{G} a$, which states that from some moment on $a$ has to hold forever.
A nondeterministic Büchi automaton can guess when this moment occurs.

No deterministic Büchi automaton can accept the corresponding language. To see this, assume that there is a DBA for the language and note that when reading only $a$ an accepting state must be visited at some point.
After visiting the accepting state, let the next symbol of the input word be $b$, and then $a$ again until another accepting state is reached.
Such a state must be reached again eventually, as $a^* b a^{\omega}$ is part of the language.
By iterating this idea a word can be constructed that is not in the language but nevertheless is accepted by the automaton, which contradicts the assumption.