There is no law of political physics that predicts what happens when the most unconventional type of presidential campaign collides with the most timeworn kind of Washington scandal.

So in deciding to discard just about every rule in the playbook as he grapples with accusations of sexual harassment, Herman Cain is making a big bet that the quirky, anti-orthodox instincts that brought him to his unlikely perch at the top of opinion polls for the 2012 Republican nomination will carry him through.

There is little evidence that he is wrong about that. Even as Cain stumbles — his recollections shifting and his explanations defying consistency and coherence — his campaign says $1.2 million has poured in since the first allegations appeared in Politico on Sunday.

In coming days, polls will show how well Cain’s popularity is holding up against the storm.

“We think it will pass,” his spokesman J.D. Gordon said of what he called “a continued appalling smear campaign that we’re trying to put behind us so we can focus on the economy.”

Yet some of those who have cast their lot with Cain are beginning to worry that the candidate and his tight-knit team are not up to dealing with what has hit them, particularly as the number of allegations stemming from his years in the 1990s as head of the National Restaurant Association has grown. At least three women who were employees — none of them publicly identified — have suggested that he behaved improperly; two of them reportedly received five-figure settlements from the organization.

What is most inexplicable to many old political hands is this: Cain’s campaign caught wind of Politico’s reporting more than a week before the story broke. So why didn’t it use that time to get its facts straight?

Twice while the story was in the works, Gordon and Mark Block, the campaign’s chief of staff, asked Cain to recall everything he could, said one campaign official, who agreed to talk about internal deliberations on the condition of anonymity.

The candidate insisted that his memory was vague and suggested that they ask the restaurant association’s chief counsel, Peter Kilgore — who was there during Cain’s tenure — for details. According to the campaign official, the association declined to provide them.

Cain’s closest advisers assumed the allegations were not credible, the official said, because they were based on unnamed sources and provided no documentation, no dates and no locations.

Meanwhile, Cain’s team was focused elsewhere as the campaign attempted to capi­tal­ize on the candidate’s soaring poll numbers to build an organization and a list of high-profile endorsements.

“The macro view is, we increased in popularity so quickly, we are still trying to catch up in a lot of areas,” the campaign adviser said.

The shape and scope of the controversy may continue to evolve, given that one of the accusers has asked the association to release her from their nondisclosure agreement so that she may discuss her allegations.

Cain has denied any wrongdoing. But where he initially claimed not to remember any accusations, he now says he was aware of at least one. Other details of his story have changed as well.

“I think he’s getting bad advice, probably because they didn’t know what was coming their way,” said a top Iowa supporter, Pottawattamie County Republican Chairman Jeff Jorgensen. “They’re behind the eight ball, and maybe not as organized as they should be.”

But Jorgensen allowed that what looks like a disaster to political veterans such as himself could prove to be an affirmation of why so many Republicans liked Cain in the first place.

“Herman is new to this game, is not a polished politician, which really is what draws a lot of people to his campaign,” he said. “If you had a polished response and had your Ps and Qs and ducks lined up in a row and deniability and all of that — a lot of politicians follow that routine.”

The political climate may have altered the old rules as well.

“People expect presidential politics to be rough-and-tumble, but times are different, and so are the circumstances of this campaign,” former senator Fred D. Thompson (R-Tenn.) wrote in the conservative National Review Online. “Republicans are looking for someone to believe in. They won’t all vote for Herman Cain, but right now, he symbolizes something they want to protect. Try to take him down at your peril.”

Indeed, so intense is the backdraft from the revelations that no one wants to be caught with fire-starter on his hands. After Cain accused operatives of his rival Rick Perry of leaking the story, the Texas governor’s campaign denied any connection — and pointed to possible links between the National Restaurant Association and the presidential campaign of former Massachusetts governor Mitt Romney. That campaign also insisted it had nothing to do with it.

Some of Cain’s more seasoned supporters are urging him to stay clear of the superheated Beltway and start making more appearances nationwide, perhaps at tea party rallies, to stoke his supporters.

But — typically, they are learning — Cain is insisting on doing things, including managing his schedule, his own way.

On Thursday, he met privately in New York with former secretary of state Henry A. Kissinger. But on Friday, he is scheduled to be back in Washington, appearing at the Defending the American Dream Summit, sponsored by the conservative organization Americans for Prosperity. And on Monday, Cain is planning to appear on ABC’s “Jimmy Kimmel Live.”

Cain’s campaign is also pondering a counterattack on the news organization that broke the story. “This is likely not over with Politico from a legal perspective,” an official said.

In one nod to conventionality, Cain has hired a lawyer to advise him on crisis management; the campaign, however, declined to identify who that is.