Zos Chanukah = Shemini Atzeres
The Beis Aharon writes, "I think the eighth lamps of Chanukah are the same concept as עצרת שמיני. Tzaddikim say Zos Chanukah is the חתימה גמר when we are inscribed for a good year [because just as עצרת שמיני is a חתימה גמר, a day when we are inscribed for a good year, the same occurs on Zos Chanukah]."
Parashas Emor (Vayikra 23) lists the yomim tovim, concluding with Succos and Shemini Atzeres, and then the Torah discusses the menorah (see Vayikra 24:2). The Rokeiach says that the menorah hints at Chanukah. After the Torah discusses the yomim tovim, it hints at Chanukah. Chanukah is for eight days. It is because it is written next to Succos and Shemini Atzeres, which are eight days. Probably, the eighth day of Chanukah represents Shemini Atzeres. Rashi (Vayikra 23:36) explains the essence of Shemini Atzeres: "It can be compared to a king who invited his children for a meal for several days. When it was time for his children to leave, the king said, 'My children, please, remain with me one more day. פרידתכם עלי קשה, it’s hard for me when you leave.'"
Similarly, after seven days of Chanukah, Hashem says to the Jewish nation, "I enjoyed your mitzvos so much. Your Chanukah lecht, your Hallel, your Al HaNisim... Stay with me one more day. It’s hard for me when you leave."
About Shemini Atzeres, the Zohar (vol.3, 32.) states, "At this joyous time, the Jewish people are alone with the King. And when one is alone with the king, whatever he asks will be given to him." We can say that the same applies to Zos Chanukah. Whatever we ask Hashem for on this day will be granted.
The Beis Aharon zt'l teaches that we should say Tehillim on Chanukah. Certainly, one should say Tehillim on זאת חנוכה, because Hashem bestows His kindness on us on this day.
The Yismach Yisrael (Chanukah 53) says that when the Torah says זאת, it refers to something one can see with his eyes. זאת חנוכה means the kedushah is so great on this day, tzaddikim are literally able to see it. "But this revelation is only for the perfect tzaddikim. It states (Tehillim 118:23), היתה 'ה מאת זאת, we believe that Hashem gives this revelation on חנוכה זאת; however, נפלאת היא בעינינו, we can’t see it... Although we don’t see or feel this special revelation, we are happy for the tzaddikim, for we believe that they see it, and we rejoice in their joy." בי חפצת כי ידעתי בזאת (Tehillim 41:12). The Yismach Yisrael explains בזאת, since you gave us חנוכה זאת, I know that You desire me and that You want to grant me goodness and chasadim.
