WASHINGTON » A Twitter post recently caught the eye of Bill McKibben, the environmental advocate and godfather of the Keystone XL pipeline protests. It included an image from "The Simpsons" showing Homer and his family basking in mountains of cash in their living room, followed by a report on Hillary Rodham Clinton’s appearance at a fundraiser with a lobbyist from the Keystone fight.

McKibben’s environmental organization, 350.org, has been trying to raise awareness about the ties it sees between lobbyists for the oil pipeline and former aides to Clinton. He promptly shared the post with his 150,000 Twitter followers, and the reaction was immediate.

"You expect different from a Clinton?" one person responded on Twitter. And from another: "Did you need another reason not to vote for Hillary Clinton?" Lost in the response was the source of the offending tweet. It was not another environmental organization or even a liberal challenger to Clinton. Instead, it was a conservative group called America Rising, which is trying, with laserlike focus, to weaken the woman who almost everyone believes will be the Democratic Party’s candidate for president in 2016.

For months now, America Rising has sent out a steady stream of posts on social media attacking Clinton, some of them specifically designed to be spotted – and shared – by liberals. The posts highlight critiques of her connections to Wall Street and the Clinton Foundation and feature images of Democrats like Sen. Elizabeth Warren of Massachusetts and Mayor Bill de Blasio of New York, interspersed with cartoon characters and pictures of Kevin Spacey, who plays the villain in "House of Cards." And as they are read and shared, an anti-Clinton narrative is reinforced.

America Rising is not the only conservative group attacking Clinton from the left. Another is American Crossroads, the group started by Karl Rove, which has been sending out its own digital content, including one ad using a speech Warren gave at the New Populism Conference in Washington last May

"Powerful interests have tried to capture Washington and rig the system in their favor," intones Warren, as images of Clinton with foreign leaders flash by.

The new-style digital campaign captures some basic facts about 21st-century communication: Information travels at warp speed on social media, it is sometimes difficult to know where that information comes from and most people like to read things with which they agree. The result, said Ken Goldstein, a professor of politics at the University of San Francisco who specializes in political advertising, is something more sophisticated.

"Politics is usually basic math," he said, "and this is a little bit of calculus, thinking a couple steps ahead."

The tactic is making for some awkward moments online. The AFL-CIO sent to its more than 60,000 followers an America Rising tweet praising its president, Richard L. Trumka, for a speech that was seen as challenging Clinton on economic issues, only to take it down a few hours later, saying it was a mistake.

Laura Hart Cole, of Verbank, New York, whose father, Philip A. Hart, was a senator from Michigan and a liberal icon, was shocked to learn that she had, like McKibben, shared the meme from America Rising on Twitter. Republican groups, she said, "have a history of sleazy tactics." But she added: "I guess it’s fair. If what they’re saying is factual, then I guess it’s fair play. It’s a dirty game."

Conservative strategists and operatives say they are simply filling a vacuum on the far left, as well as applying the lesson they learned in 2012, when they watched in frustration as Mitt Romney was forced to expend time and resources in a protracted primary fight. By the time he secured his party’s nomination, President Barack Obama hardly had to make the case that his opponent was a coldhearted plutocrat; Republicans like Newt Gingrich had already made the argument for him in the primaries.

Few Republicans are more familiar with that nightmare than Matt Rhoades, who was Romney’s campaign manager. He founded America Rising in response to a recommendation contained in an autopsy of Romney’s failed presidential run that was ordered by the Republican National Committee. The group’s original goal was to compete with American Bridge, the Democratic opposition research group, but its focus under Rhoades has been to subject Clinton to an ordeal similar to Romney’s.

"The idea is to make her life difficult in the primary and challenge her from the left," said Colin Reed, America Rising’s executive director. "We don’t want her to enter the general election not having been pushed from the left, so if we have opportunities – creative ways, especially online – to push her from the left, we’ll do it just to show those folks who she needs to turn out that she’s not in line with them."

No one thinks attacking Clinton from the left is likely to turn the most liberal Democrats into Republican voters. But Steven Law, president of American Crossroads, said the goal was simply to erode what should be her natural core of support.

"It can diminish enthusiasm for Hillary among the base over time," he said. "And if you diminish enthusiasm, lukewarm support can translate into lackluster fundraising and perhaps diminished turnout down the road."

This year, Zac Moffatt, a co-founder of Targeted Victory, a right-leaning political technology firm, who handled Romney’s digital operation and has worked with groups like America Rising and American Crossroads, laid out the strategy in a memo to several clients. "There was a hole to fill in the market," he said, and if Democrats were not willing to challenge Clinton, Republicans could do it themselves.

"We were seeing people on the left who were interested in content about Hillary Clinton, and that there would be opportunities for groups to share this information with Democrats on the left," Moffatt said.

To reach these groups, Moffatt had a plan: using micro-targeted advertising units on social media platforms like Facebook, Twitter and YouTube.

For example, as Clinton was traveling through Las Vegas this month on a campaign swing, "liberal Democrats" (as identified by Targeted Victory’s voter file) in the Las Vegas area saw a video pop into their Facebook news feeds, highlighting recent news reports about foreign government donations to the Clinton Foundation. The video was shared by America Rising, and received over 6,300 views, most from people who would never follow a group like America Rising on social media.

Other groups are also using micro-targeted advertising to inject their content into the Facebook and Twitter news feeds of "liberal Democrats," environmentalists and declared supporters of Warren, among others.

"You might start looking at union households. You might start looking at Bernie Sanders’s core of support," Moffatt said, referring to Sen. Bernie Sanders of Vermont, a candidate for the Democratic presidential nomination.

Law said members of his staff at American Crossroads had easily been able to inhabit the liberal role, despite being fervent Republicans. "We wear these little bracelets – WWEWD," Law joked, referring to "What would Elizabeth Warren do?"

In the face of Republican activity aimed at undermining its liberal support, the Clinton campaign has been publicly circumspect. Asked for a comment, it would only note that in a Quinnipiac University poll last month, Clinton led her closest opponent, Sanders, by 46 points among voters who consider themselves "very liberal."

Even some of those unhappy with Clinton like Joel Gombiner of Brooklyn – who posted the "Did you need another reason?" response to the Twitter message shared by McKibben – think the conservative groups may be outsmarting themselves.

"They view this as a means of weakening the Democratic Party and weakening the chance in a presidential election," said Gombiner, 26. But "that’s the whole point of a democracy, that the arguments make you stronger."

© 2015 The New York Times Company