The recent growth of large language models (LLMs) has enabled more authentic, human-centered interactions through multi-agent systems. However, investigation into how conversations affect the psychological states of LLMs is limited, despite the impact of these states on the usability of LLM-based systems. In this study, we explored whether psychological states change during multi-agent interactions, focusing on the effects of conversation depth, topic, and speaker. We experimentally investigated the behavior of 10 LLMs in open-domain conversations. We employed 14 questionnaires and a topic-analysis method to examine the behavior of LLMs across four aspects: personality, interpersonal relationships, motivation, and emotion. The results revealed distinct psychological trends influenced by conversation depth and topic, with significant variations observed between different LLM families and parameter sizes.