From skimming the FAQ it seems that the Soviet invasion had a greater (though not necessarily insignificant) psychological impact on the Japanese leadership than the atomic bombs (though there seems to be some contention).
Did the American leadership believe the Soviet invasion would be a more effective surrender incentive?
If yes, why did they 'waste' the debut of their secret weapon on an enemy already on the verge of surrender? Were the Americans trying to speed surrender up even further? Trying to cut the Soviets out of the surrender negotiations? Making a 'show of force' towards the Soviets in anticipation of the imminent Cold War?
If no, what caused the Americans to underestimate the psychological impact of the Soviet invasion? What role did they expect the atomic bombs to play in bringing about surrender?
In short, they hoped to force Japan to surrender before the Soviets could help us, if at all possible, to avoiding having to clash with them over post-war Japan, in addition to post-war Europe.
At the Yalta conference, the Soviets agreed to join us in subduing Japan, "two or three months" after the Germans surrendered. This happened on May 8/9, 1945. However, after the Trinity test in July 1945, Truman realized that if we could build and use nuclear bombs quickly enough, we could force Japan to surrender before the Soviet Union could join us. This would mean that that US would have unilateral control over post-war Japan, and the Soviet Union wouldn't have any standing to ask for concessions. Part of these concessions, as specified at the Yalta conference, were the Sakhalin and Kuril islands.
Also, the Japanese were not quite "on the verge of surrender." They were losing, and were pretty much assured to lose at that point in the war, but the Americans were expecting heavy military casualties, and a prolonging of an already long war, if they were forced to invade the Japanese home islands. They had plans drawn up, in the form of Operation Downfall, and while estimates vary, we were probably looking at an additional 18 months of war and somewhere between 500,000 and 1,000,000 dead, and another 500,000-1,500,000 injured. And that's just the US military casualties. That doesn't take into account the Japanese casualties, both civilian and military, or the Soviet casualties.
I don't have any hard sources at my fingertips, as I'm recalling this from memory, but I know that I read a lot about this in "The Most Controversial Decision: Truman, the Atomic Bombs, and the Defeat of Japan" by Wilson D. Miscamble.