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The question isn't if it can run them, it's whether it can run them well. It varies from game to game. It also varies with emulators and what options you selected.
With that said, I emulated PSX games successfully with a shitter computer than what you have, several years ago. Mostly games like Xenogears and Super Robot Wars. I don't know what the current state of PS2 emulation is, but it's likely also acceptable.