Why was the Persian ceremonial capital of Persepolis left unguarded for Alexander the Great to storm and loot in 330 BC? With how vast and powerful the Persian Empire was at the time, wouldn't they have had a standing army ready to defend it?

Persepolis was actually guarded by a Persian Army. I'll start with the aftermath of the Persian defeat at Gaugamela. Following his rather decisive loss Darius retreated to Ecbatana where he began to build another army with which to oppose Alexander. Persepolis itself was held by a force under the control Ariobarzanes although the exact number of men he had is disputed, contemporary accounts credit him with around 4,000 infantry and about 700 cavalry. Alexander had around 17,000 men at his disposal. What made the Persian defense plausible was the terrain over which Alexander would have to traverse in order to seize Persepolis. Following Alexander's conquest of Susa he divided his forces in two and sent Parmenion and half his army down the Royal Road while he took the direct route to Persepolis proper with around 17,000 men. The problem was that the direct route required him to pass through the Persian Gates, a narrow and treacherous mountain pass well suited to ambush. Making it even more treacherous and difficult was the fact that Alexander made his push during the winter of 330 BC. After initially encountering no resistance in his initial approach Alexander made a key and rare tactical error by failing to send forward scouts for a possible ambush. Thus Ariobarzanes was able to catch Alexander completely by surprise and drive him back with heavy casualties. The Macedonians and Persians were at an impasse for around a month with neither side able to gain a decisive advantage. Eventually, either through a prisoners of war or a local guide, he flanked them and caught Ariobarzanes in a pincer. After routing them the reports conflict as to what happened to the Persians who escaped. Some claim that Ariobarzanes and his remnants escaped north and eventually surrendered to Alexander. Others, that he and his remaining forces died to a man rather than surrender. The final one is that his forces escaped to Persepolis, but were betrayed by the Persian noble in charge of Persepolis who refused to open the gates leaving Ariobarzanes and his men to be slaughtered before surrendering the city. The city itself was not stormed, but surrendered peacefully to Alexander along with most of the Persian treasury. That is why the looting and pillaging of Persepolis is so unique in that Alexander allowed it in spite of its rather peaceful surrender which is different than how he normally treated cities that peacefully surrendered like Susa which was completely spared. Most historians agree that some form of revenge or spite played a role in the city wide destruction Alexander allowed. Whether retaliation for the destruction of Athens or because Alexander was not recognized as the legitimate successor to Darius while Darius still lived is unknown. In short the Persians did have an army (albeit a small one) guarding the capital. Its size was probably due to the misguided belief that the Persian Gates, the winter, and the actual siege of Persepolis (which never happened) would take long enough for Darius to assemble an army and come to its aid. Thus Alexander's audaciousness and relentlessness was underestimated by Darius and the difficulties involved either proved less difficult than believed or never materialized at all. In any case Persepolis was guarded by an army and following that armies defeat surrendered in the hope that a peaceful surrender would spare them looting and pillaging. Further, while the Persian Empire remained formidable in its own right much of its power and prestige had taken a severe beating from the repeated losses to Alexander so they were not the once mighty power they had been by the time of the fall of Persepolis.