According to the gaon Rabbi Meshulam Roth, since Yom HaAtzmaut is a holiday, commemorating the great salvation that Israel experienced on that day, it is appropriate to recite the SheHechiyanu blessing on the day, as we do on all the holidays, including Purim and Chanukah. He believes, however, that the obligation to say SheHechiyanu depends on one’s joy. Therefore, one who is not particularly happy may recite the blessing but is not obligated to do so, while one who is joyous about the establishment of the State is obligated to recite it on Yom HaAtzmaut.
Many others hold that one should not say SheHechiyanu on Yom HaAtzmaut, because the Sages instituted the recitation of this blessing only for holidays on which one is forbidden to do work, like the three pilgrimage festivals, Rosh HaShanah, and Yom Kippur. The SheHechiyanu blessing that we say on Chanukah and Purim, which do not entail a prohibition against work, pertains to the special mitzvot we perform on these days – the Megillah reading on Purim and the candle lighting on Chanukah – not to the very essence of the day. For, as we said, one does not recite SheHechiyanu over the essence of a holiday unless it is forbidden to do work on that day.
One who wants to be meticulous and satisfy all opinions should wear a new garment and recite SheHechiyanu on it, having in mind that the blessing relates to the holiday as well. If he is the chazzan (prayer leader), it is best to say the blessing on the garment right before Hallel. This way, the listeners can also discharge their obligation8.
Some say that the salvation of Yom HaAtzmaut is similar to the exodus from Egypt, requiring us to say Hallel at night. This was Rabbi Goren’s custom, and some communities follow this practice. However, many authorities hold that the halachah of saying Hallel on the night of Pesach is unique, and we cannot deduce other holidays from it, the proof being that we say Hallel on the other holidays during the daytime alone. Therefore, one should not say Hallel on the night of Yom HaAtzmaut. This is the practice of most of Rav Tzvi Yehuda HaKohen Kook’s students9.
 Rabbi Meshulam Roth’s words are found in Responsa Kol Mevaser 1:21. The author of the Mishna Berura concurs in his Bi’ur Halachah (692, s.v. she’hechiyanu), writing that one recites SheHechiyanu over the essence of a holiday, even if it does not entail a prohibition against work. Rabbi Roth adds that it is preferable to say the blessing before the recitation of Hallel, for then it might be considered a blessing over the mitzvah, like the SheHechiyanu we say before lighting the candles on Chanukah. He also relies on the opinion of the Bach and his adherents, who hold that the rule of, “A doubtful case, involving blessings, is decided leniently” [i.e., one omits the blessing] does not apply to the blessing of SheHechiyanu. The Chatam Sofer (Orach Chaim 55) adds that in any doubtful case involving SheHechiyanu, one is obligated to recite the blessing if he knows that he is happy. Rabbi Goren concurs in Torat HaMo’adim, as does Sefardic Chief Rabbi Ben-Zion Meir Chai Uziel in Mishpatei Uziel, third edition, O.C. 23. On the other hand, the author of Yaskil Avdi (6:10) writes that one should not recite SheHechiyanu on Yom HaAtzmaut, for the reasons stated above. He also doubts whether the miracle took place specifically on Yom HaAtzmaut. Rabbi Ovadyah Yosef writes the same in Yabi’a Omer 6:42, quoting many poskim who hold that the rule of, “A doubtful case, involving blessings, is decided leniently,” applies to SheHechiyanu, as well. This is also the Beit Yosef’s opinion, and, according to the Rambam, reciting a blessing in vain is a biblical transgression. Rabbi Moshe Tzvi Neryah agrees that one should not recite the SheHechiyanu blessing on Yom HaAtzmaut.
 Rabbi Goren’s words are found in Torat HaShabbat VeHaMo’ed. See Rabbi Shmuel Katz’s article HaRabbanut HaRashit VeYom HaAtzmaut (4, notes 7, 8, 17, 18). A book entitled KeLavi Shachein, in memory of Gad Ezra Hy”d, contains articles on this issue – one by Rabbi Sharki, who supports the saying of Hallel at night, and one by R. Yaakov Ariel, who opposes it. Rabbi Neryah expressed this view before Rabbi Ariel did (see Kovetz Hilchot Yom HaAtzmaut VeYom Yerushalayim), quoting several reasons why Hallel is recited specifically on Pesach night, none of which are relevant to Yom HaAtzmaut. Rabbi Ariel Edri concurs in a booklet called Shachar Ahalileka. According to Rav Hai Gaon, it seems that we recite Hallel on the first night of Pesach because a person is obligated to view himself as if he is actually leaving Egypt that night. Consequently, he must sing praise as the miracle occurs. This does not apply to any other holiday. Most Rabbis accept this viewpoint. After all, until Rabbi Goren’s tenure as Chief Rabbi, everyone agreed that Hallel pertains to the day alone. And when he publicized his ruling to say Hallel at night, it stirred a dispute, and it is unclear whether the Rabbinic Council agreed. Rav Tzvi Yehuda Kook instructed Yeshivat Mercaz HaRav to recite Hallel at night, in accordance with Rabbi Goren’s ruling. However, it seems that he personally disagreed. Later on, when Rabbi Avraham Shapiro became Chief Rabbi (and Rosh Yeshiva of Mercaz HaRav), the Yeshiva stopped saying Hallel at night. Most of Rav Tzvi Yehuda’s students follow this practice.