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So for about 90 minutes on Wednesday night, a dozen senators and the president gathered on neutral territory — a private dining room at one of this city's most elegant hotels — and tried to work out their frustrations over beef and wine.

Aside from the issue of how to handle the check, what was described by all as a convivial dinner raised difficult questions about how effective the new White House campaign to woo Republicans will be and whether Mr. Obama, even at his most contrite and conciliatory, can bridge the gap between two parties that remain deeply divided over fundamental questions of policy and the role of government.

Lawmakers in both parties say the president's efforts may make him a few new friends, but he is not going to change ideologies. Others privately complained that convening such a high-profile meeting seemed like an effort to distract from his failure to help forge a solution to avert the automatic budget cuts that went into effect last week.

Asked Thursday morning about the president's new social schedule, Speaker John A. Boehner chuckled before saying he hoped the talks would produce real compromise.

"After being in office four years, he's actually going to sit down and talk to members," Mr. Boehner said, his voice rising in feigned disbelief. Still, he added in a more serious tone, "I think it's a sign, a hopeful sign. And I'm hopeful that something will come out of it. But if the president continues to insist on tax hikes, I don't think we can get very far."

Those who have studied the relationship between presidents and Congress doubt seriously whether Mr. Obama's latest outreach will yield much.

"It's a rather shallow notion," said George Edwards, a political scientist at Texas A & M University and an expert who has written extensively on presidential power. "You're not going to get committed conservatives to change their long-held ideological commitments because you play a round of golf or invite them to the White House."

Or treat them to an expensive dinner, for that matter, experts said. The senators, as is their custom, did most of the talking on Wednesday night. They were grateful to Mr. Obama for the invitation, though they told him they wished he had reached out earlier, instead of first going on a campaign-style cross-country tour accusing Republicans of obstruction.

Mr. Obama's tone was amenable, and he kept his comments brief. "I don't think he came there to say, 'Here's the way it's going to be, and I need you to get in line,' " said Senator Mike Johanns of Nebraska, who estimated that Mr. Obama spoke only about 10 percent of the time. "I think he was saying the opposite."

They urged him to talk tougher on the need to bring down the cost of programs like Medicare. "You are in the bully pulpit," Senator Ron Johnson of Wisconsin said he told the president. "You can be honest with the American people and lay those facts on the table because that is what it's going to take."

Around 8:30 p.m., Senator John McCain of Arizona, the self-appointed timekeeper, interrupted to say that they had exhausted their allotted hour and a half and should let Mr. Obama be on his way. "The president's a busy guy," he said.

And with that, they went their separate ways into the cold, damp Washington night.

Next week Mr. Obama will take the extraordinary step of traveling to Capitol Hill to hold four separate meetings with members of Congress — one with Democrats and one with Republicans in each chamber. The last time he visited the Capitol to meet with the House Republican conference was January 2009; with Senate Republicans it was May 2010, though the president has met with them on occasion since. And on Thursday, Mr. Obama hosted a lunch at the White House that included Representative Paul D. Ryan of Wisconsin, the House Budget Committee chairman.

One popular theory that has gained currency in recent years is that Washington would be a much more civil and productive place if there were more bipartisan social gatherings like the one the president held on Wednesday. With members of Congress spending so little time in Washington — most flee back to their districts on Thursday after the week's legislative business is done — the kinds of bonds that foster cooperation and collaboration have failed to grow.

But political scholars say that bipartisan camaraderie is no substitute for the most crucial factor in advancing a president's agenda: large majorities in Congress.

"Stories about dramatic interaction between big personalities make for excellent reading, but they do tend to downplay the structural constraints against which that drama is playing out," said Andrew Rudalevige, a professor of government at Bowdoin College. "What does the Congress look like? Who holds seats? And what does public opinion look like?"

A closer look at the record of the president who is considered the most effective Congressional negotiator in modern times, Lyndon B. Johnson, shows that his success can be attributed as much to circumstance as to his famous powers of persuasion and intimidation.

Much of the Great Society legislation — laws that created the Medicare, Medicaid, the National Endowment for the Arts and financed public schools — passed Congress when Democrats commanded supermajorities in both houses in 1965 and 1966.

But after the Republicans picked up dozens of seats in the next Congress, weakening those majorities considerably by 1967, Johnson had a more difficult time. In battles that mirror today's big policy debates, he had to swallow considerable cuts in domestic spending before he could persuade Congress to pass a 10 percent income tax surcharge to help pay for the Vietnam War. And he failed to persuade Congress to create a national gun registry after the assassination of Robert F. Kennedy.

"Did Johnson just forget how to do it? No," Dr. Edwards said. "He didn't have the votes."

Members of both parties complain today that the president's outreach has been distant and dismissive. And they often cite Mr. Obama's immediate predecessor, George W. Bush, as being especially effective in his first term because of a productive relationship with a divided Congress. But political scientists note that two of Mr. Bush's signature achievements — the tax cuts and sweeping authority to combat terrorism — were won at a time when the president had soaring approval ratings and the political winds of Sept. 11 at his back.  

Still, many members of Congress acknowledge that there is nothing quite like taking a call from the president. And while he may not ultimately prevail on them, they always listen.

"Without these overtures, nothing would happen," said Senator Susan Collins of Maine, who took one of those phone calls on Monday. "It's not a sufficient step. But it's a necessary step."