“When the chemistry works, it communicates enthusiasm and a team and the likelihood of success and progress,” said Michael Feldman, who was an aide to former Vice President Al Gore. “But it can’t be an arranged marriage with someone she meshes well with on paper,” he added. “It very much has to be real.”  Mrs. Clinton’s aides began collecting information last week on as many as 10 candidates. James Hamilton, a Washington lawyer who is overseeing the vetting, will begin meeting with candidates as early as this week.  Senior Democrats, thinking about both the fall election and a potential re-election in four years, said the right choice would stir up enthusiasm on the campaign trail, enhancing Mrs. Clinton’s strengths while not outshining her or overtaking events.  Mrs. Clinton is warm and personable one on one, the Democrats said, which creates an easy camaraderie when she teams up with people she likes.  In San Antonio, Mrs. Clinton seemed to light up in the presence of Julián Castro, the city’s former mayor and now the secretary of housing in the Obama administration. She grabbed his hand and thrust it skyward at an outdoor rally, and the two glided naturally on the rope line, snapping selfies.  The photos from the event, with Mrs. Clinton, 68, beaming next to Mr. Castro, 41, turned out so well that cable networks often show the campaign’s images of the two together on screen when Mrs. Clinton calls in for phone interviews.  On a Saturday night in Youngstown, Ohio, Representative Tim Ryan and Mrs. Clinton made a surprise visit to O’Donold’s Irish Pub and Grill for a couple pints of Guinness. Mrs. Clinton leaned comfortably on the bar next to Mr. Ryan and smiled widely as “Born in the U.S.A.” blared from the jukebox.  Mr. Ryan, holding his pint in the air, as the packed bar followed suit, said, “To Hillary! The next president of the United States.” Mrs. Clinton threw her head back and laughed, then took a long swig of beer.  Last month, at a stop at the Court Street Diner in Athens, Ohio, Mrs. Clinton warmly embraced Senator Sherrod Brown, whose policy expertise Mrs. Clinton has said she admires.  They made their way, practically arm in arm, through the narrow 1950s-themed diner, talking to voters and meeting a developmentally disabled man, Noah, whom Mr. Brown introduced to Mrs. Clinton as “my friend.”  “How can you not love Athens?” Mrs. Clinton said. Mr. Brown nodded.  Dr. Lillian Glass, a body language expert, was struck by Mrs. Clinton’s ease with Mr. Brown. “Her body is leaning into his, and she never leans into anybody. She adores that guy,” she said in a telephone interview. “And you can tell that he knows his place. You don’t see him trying to take over.”  In many ways, Mrs. Clinton’s search mirrors that of George Bush, who already had deep relationships with major players in Washington by the time he captured the Republican nomination in 1988, Dan Quayle, his running mate, said in an interview.  The contenders most frequently mentioned by her advisers and senior Democrats close to the campaign include Senator Michael Bennet, from the key state of Colorado; Thomas E. Perez, President Obama’s secretary of labor and a Hispanic civil rights lawyer; Representative Xavier Becerra of California; and Senators Tim Kaine and Mark Warner, both former governors from Virginia. Senator Elizabeth Warren of Massachusetts is a favorite of liberal Democrats, though an all-female ticket is unlikely. All of these candidates have relationships with Mrs. Clinton, and several have appeared with her at campaign events.  “This isn’t a consensus-type thing,” Mr. Quayle said in a phone interview. “It’s much more the feeling of the two principals, and someone she’s going to feel comfortable with every day.”  Still, a connection can take hold between two candidates on a presidential ticket who are just getting to know each other.  In 1992, Bill Clinton selected Mr. Gore, another youthful Southern Democrat, to reinforce his message that a new kind of Democratic Party was emerging. The two men and their young families proved an electric combination on the campaign trail.  In a telephone interview, Walter Mondale recalled when he and his wife, Joan, first met with Jimmy and Rosalynn Carter in Georgia in 1976. “Joan said, ‘He’s going to pick you,’” Mr. Mondale said. “I said, ‘Why?’ and she said, ‘Because we all get along perfectly.’ And we still do today.”  Barack Obama and Joseph R. Biden Jr. did not know each other well when the Obama campaign decided the first African-American nominee of a major party needed a scrappy and experienced running mate who could appeal to white working-class voters.  Mrs. Clinton has made clear she wants a partner who shares her granular knowledge of policy and who she will enjoy working closely with in the West Wing.  Get the latest national and state polls on the presidential election between Hillary Clinton and Donald J. Trump.  Although she is at ease with younger hopefuls like Mr. Castro, Mrs. Clinton has also relished policy conversations with Mr. Perez, a favorite of labor unions, her advisers said. (Senator Harry Reid, Democrat of Nevada, has expressed concern that selecting Mr. Brown would leave a vacancy in the Senate that a Republican would probably be appointed to fill.)  Mrs. Clinton may be forced to ignore such factors, Democrats said, and think about whom she most prefers to spend the next four to eight years working with. “It’s like getting married without any divorce proceedings possible,” Mr. Mondale said.  The partnership extends to the running mates’ spouses, who will spend ample time together, and in Mrs. Clinton’s case includes an opinionated former president.  Nancy Reagan famously did not like Mr. Bush and his wife, Barbara, but Mr. Bush was determined to “show his skills in figuring out how to make Reagan his friend,” said Joel K. Goldstein, a professor and vice-presidential expert at Saint Louis University School of Law.  Michael S. Dukakis said that he had not spent much time with Senator Lloyd M. Bentsen Jr. of Texas before he picked him as his running mate in 1988, but that they ended up getting along well, adding, “Our wives had a great relationship.”  Of course, Mr. Dukakis ran years before social media and a hyperkinetic press corps dissected every aspect of the most critical partnership in politics. That means the comfort level has to be real, or the internet will pick up on any hint of tension and turn it viral.  In fact, interest in how the first female presidential nominee at the top of a major ticket would interact with a vice-presidential nominee is such a source of speculation and intrigue, it has inspired Hollywood writers.  In HBO’s comedy “Veep,” Selina Meyer, the foulmouthed title character portrayed by Julia Louis-Dreyfus who stumbled her way into the presidency, is hilariously overshadowed by her charming and more competent running mate.  “You never want a vice president that makes people start to think, ‘Boy, it’s a shame he or she isn’t at the top of the ticket,’” said David H. Mandel, an executive producer of the show. “It’s that magical balance.”
The Clinton campaign is searching for a charismatic Democrat who can click with, but not outshine, its candidate. Auditions have already started.