This is an argument that's still debated -- reasonable people can and do disagree with the extent to which the bombs themselves or the Soviet invasion of Manchuria or other factors pushed Japan into surrender. You can find more about this in this section of our FAQ; the linked answers include book recommendations. Our resident nuclear expert, u/restricteddata, has on his blog this piece that reviews some of the debates historians have over use of the bombs.

I will take a few stabs. Part 1/2

Gar Alperovitz and Nitze's chief argument is that Japan is a 'defeated' nation and that even without the usage of nuclear bombs they would have surrendered by Nov or Dec of the year.

See their writings in The United States Strategic Bombing Survey Japan's Struggle to End the War and Atomic Diplomacy.

The key faults at it were their belief of 'defeat' and surrender are the same thing. The idea when the US military says Japan has been defeated is pretty common but what you don't hear the next phrase is 'how do we convince her of that.'

We have to discuss several aspects of the war in order to counter the idea that Japan wouldn't require some kind of nudge to surrender.

We should make a note of the people we will mention their titles and why they are important.

Henry L. Stimson, SecDef for Roosevelt & Truman, SecState for Hoover

Paul H. Nitze, director and Vice Chairman of the Strategic Bombing Survey

Supreme War Council of Japan on Aug, from here on as SWC

Admiral Kantarō Suzuki, PM from here on as Suzuki

Shigenori Tōgō, Minister of State here on as Togo

General Korechika Anami, Minister of War here on as Anami

Admiral Mitsumasa Yonai, Minister of Navy here on as Yonai

General Yoshijirō Umezu, Chief of Army General Staff here on as Umezu

Admiral Soemu Toyoda, Chief of Navy General Staff here on as Toyoda

Kōichi Kido, Lord Keeper of the Privy Seal of Japan (chief advisor of the Emperor), here on as Kido

Fujita Hisanori, Grand Chamberlain

Code name MAGIC, ULTRA, Decryption process OLYMPIC, US invasion of Kyushu

The chief thing to counter any attacks that the atomic bombs DID NOT force the unconditional surrender of Japan was the Supreme War Council Meeting that was well described by all sources both pro and against the dropping of the atomic bomb whether it was Nitze, or Kido, or Togo.

When Togo realize on Aug 7about the bomb the Americans boasted might actually be real despite the fact that the Army claims it's just a bigger bomb on Hiroshima, Togo proposed to accept a surrender base on the conditions set forth by the Postdam Declaration (with the condition of the retention of the Emperor as head of [symoblic?] government). This request was rejected in the SWC, the idea that the US could deploy another weapon of this scale is simply unrealistic, and whether or not the US has another of this weapon is also in question.

The same day, Kido receives a report of the devastating report of the atomic weapon unleashes upon Hiroshima requested an immediate audience with the emperor at 13:30, and he recorded how worried the emperor was given the devastating effect that was given to him. It was recorded that he was very much unhappy that the information he obtain from the SWC is limited and without much detail. On the night of Aug 8, Suzuki stated that given now they are aware of the weapon known as the atomic bomb he like his aide to prepare for him his views to terminate the war at the SWC.

The following day, the SWC received notification that the USSR has declare war on Japan. While Suzuki has thought the Japanese could hold Manchuria for two month, he was told by Lt Gn Ikeda Sumihsa that Chaochou will fall into Russian hands in 2 wks. Chaochu is the capital of Manchuria. It's capture would signal any formal resistance of the Kwantung Army.

The Emperor summoned Kido to his chamber in 9:55 on the 8th after receiving words that USSR has declare war on Japan. He asked Kido to have a 'heart to heart' talk with Suzuki to accept the Postdam Declaration and end the war at once. Since Suzuki had business in palace that day Kido discussed this with Suzuki who assured Kido that his own determination was also to end the war as quickly as he can. At 10:55, Kido refer Suzuki's reply to the emperor. Much of this information came from Kido and Hisanori's memoir and writing.

At the same time, after Suzuki confirm with Kido the FIRST SWC meeting took place for the 9th. While the SWC recognize that it's impossible to continue the war much longer they disagree on how to end the war. Due to the nature of SWC the cabinet must achieve unanimity of views. They were unable to arrive at that conclusion. Suzuki and Togo discussed the impossibility of continuing the war given atomic weapon are now available, Toyoda recalled years later on the US capability to deploy them in rapid succession and Anami doubted the US possession of more than one bombs. At that moment, 13:00, news reaches the SWC that a second atomic bomb was deployed in Nagasaki. Suzuki feared that unlike the bolstering of Anami of the readiness to defend the Japanese coast, the Americans will simply continue dropping atomic weapons on Japan until the destruction of the Japanese state and people.

Now we hit this point, it's obvious normal people would be like OK uncle. You got us. We can't stop your airplanes we can't stop your bombs we are out of boats. It's time to quit. But no. No, no, no, no, no.

At this point, Yonai supported Togo and Suzuki, agreeing with them that they should accept the Postdam with the condition of keeping the emperor. Anami, Umezu, and Toyoda decline to support. While they all acknowledge the seeming impossibilities of the war they insist on 3 additional conditions

1. there would be no military occupation of the Japanese homeland

2. that the Japanese military would conduct self voluntary disarmenement.

3. the Japanese government would prosecute their own war crimes

Suzuki cannot find himself to support these as that would fly in the face of the Postdam and it would suggest to the US that Japan would fight to the bitter end. The meeting ends at 14:30 without any conclusion.

The second SWC began at 18:00, Togo began by saying it is impossible to obtain the 3 conditions Anami wanted, and Anami said that the Decisive Homeland Battle would bring the Americans to the negotiation table. [we will discuss this battle plan later]

Yonai statd that Japan has suffered defeat after defeat from Saipan to Luzon to Okinawa. It would be irrational to suggest the Japanese home defense would carry through. Anami stated that the defense [of Kyushu] would sent so many body bags to the US homeland that the US would have no choice but to accept Japanese conditions.

The Second SWC end in an impasse, with neither side able to convince each other. This is after BOTH atomic weapons were deployed and the USSR beginning their offensive against Japan.

Suzuki at this point with Kido involved the Emperor at midnight, Hirohito was in his military uniform, and in attendance with him the entire SWC. Everyone basically repeats their points, and the Emperor said nothing, and after 2 hrs, the cabinet was at an impasse again.

It was then Suzuki went to the throne and begged for a Divine Intervention. Likely realizing that the state is at the verge of destruction, Hirohito did the unprecedented action in modern Japanese history, he stated he agree with Togo, and asked that the surrender be according to Postdam with the condition of keeping the emperor as an institution.

The decision solidified the SWC and at 2:30 on the 10th the Japanese government was finally able to come to the conclusion of a conditional surrender after the Divine Intervention.

But, wait for it, the US rejected.

The Japanese cabinet sent the decision to the Swiss government and relay to the US government, and the reply was "the Authority of the Japanese Emperor shall be subject to the Supreme Commander of the Allied powers."

Togo ask to accept the American demand, whereas Anami rejects and demand one final battle. It took ANOTHER Divine Intervention from the Emperor that led to the Japanese acceptance to the unconditional surrender.

So after all of this typing, you must ask, but why? Why not accept the Japanese' conditional surrender? It was simple. Japan was defeated militarily in all aspects, Japan has no real way to hold out against the American forces. They are defeated in the sense that the ice caps are melting [according to current trajectory]. But Japan refuses to acknowledge of its defeat. The US has no reason to accept any demand from the Japanese. Japan had no position to make demands and the US was in no mood to accept any demand.

Then you may ask, as some have, WHY then did the US drop the atomic bomb?

Part 2/2

We now refer back to Anami's claim that he wanted one more battle to 'win' the war for Japan. OLYMPIC was to operate in the southern Kyushu region under MacArthur. The original estimation was that perhaps 200,000 troops would be defending Kyushu. ULTRA estimated about 230,000 in April and the troop number continue to rise from May to July. By June, 280k troops were in Kyushu, and by July 21 the number reached 455k. End of July saw the number further increased to 525k. The increase of the troop deployment is staggering as were the defense preparation. Two army headquarters now station in southern Kyushu, as the Japanese now determined that southern Kyushu is where the US invasion will begin. Instead of spreading their forces as the US military planners hoped, all these troops are now concentrated in southern Kyushu.

Further more, planes were readied to attack transports in sucid mission from across EA and Japan. About 1/4 of all Japanese sucide triaining planes were located in Kyushu. More than 2000 biplanes, fighters, and floatplanes were converted into suicide attacks at night. From Shanghai to Korea to Honshu, planes were ready for their one way trip to Kyushu. Their targets? US transports. As Japanse forces dug in to the beaches, ULTRA was telling the American planners 2 things, Japan is digging in, reinforcing Kyushu with everything they got, and they are going to take as Americans down with them as possible.

This tells us that no, at no point in the prior month of Aug did Japan think about surrendering to the Postdam Declaration. The idea that Japan MAY HAVE surrender after BOTH atomic bombs and the Soviet invasion short of a Divine Intervention is also unrealistic. Anami was clearly pinning for a last ditch battle at Kyushu, moving 1/3 of ALL his forces into Kyushu. He is going to make Iwo Jima like a school day parade with his last battle. That is why even when they told him US can drop more atomic bombs, he was unwilling to compromise, because he was already dug in.

Did the BOMB directly cause Japan to surrender? No. Because as we seen, the SWC rejected unconditional surrender. But the bomb did move the Emperor to surrender.

Sources:

RICHARD B. FRANK, Ending the Pacific War: The New History

RICHARD B. FRANK, Downfall: The End of the Imperial Japanese Empire

Japan's Struggle to End the War: The United States Strategic Bombing Survey

東郷茂徳と太平洋戦争, Shigenori Tōgō and the Pacific War, 松本繁一 Matsumoto Shigekazu (I am guessing)

「終戦工作」と「国体」に関する一試論, The end of war and the idea of state (I am doing a poor translation) 和彦君島 Kimijima, Kazuhiko

終戦と無決定の本質, The end of war and the essence of indecision by 李炯 Hyong Cheol LEE

昭和天皇実録 Record of Emperor Showa

終戦史録 Record on the End of War

Hiroshima in History: The Myth of Revisionism