WASHINGTON Dc, United States — The jobs crisis has brought an unwelcome discovery for many unemployed Americans: Job openings in their old fields exist, yet they no longer qualify for them.

They’re running into a trend that took root during the recession. Companies became more productive by doing more with fewer workers. Some asked staffers to take on a broader array of duties — duties that used to be spread among multiple jobs. Now, someone who hopes to get those jobs must meet the new requirements.

As a result, some database administrators now have to manage network security.

Accountants must do financial analysis to find ways to cut costs.

Factory assembly workers need to program computers to run machinery.

The broader responsibilities mean it’s harder to fill many of the jobs that are open these days. It helps explain why many companies complain they can’t find qualified people for certain jobs, even with 4.6 unemployed Americans, on average, competing for each opening. By contrast, only 1.8 people, on average, were vying for each job opening before the recession.

The total number of job openings does remain historically low: 3.2 million, down from 4.4 million before the recession. But the number of openings has surged 37 percent in the past year. And yet the unemployment rate has actually risen during that time. Companies still aren’t finding it easy to fill job vacancies.

Take Bayer MaterialScience, a unit of Bayer. When the company sought earlier this year to hire a new health, safety and environment director for one of its plants, it wanted candidates with a wider range of abilities than before. In particular, it needed someone skilled not just in managing health and safety but also in guiding employees to adapt to workplace changes.

Joe Bozada, chief of staff for Bayer’s CEO, said the company initially interviewed 30 candidates. Then it did final interviews with seven. But none had the additional experience the company now wanted. Ultimately, Bozada said, the company chose one of its own employees it had already trained.

That shift, across multiple industries, has caught the eye of David Altig, research director at the Federal Reserve Bank of Atlanta. Workers aren’t just being asked to increase their output, Altig says. They’re being asked to broaden it, too.

A company might have had three back-office jobs before the recession, Altig said. Only one of those jobs might have required computer skills. Now, he said, “one person is doing all three of those jobs — and every job you fill has to have computer skills.”

The trend is magnifying the obstacles facing the unemployed. Economists have long worried that millions of people who have lost jobs in depressed areas like construction don’t qualify for work in growing sectors like health care. But it turns out that some of the jobless no longer even qualify for their old positions.

Frustrated in their efforts to find qualified applicants among the jobless, employers are turning to those who are already employed.

“They’re hiring a known quantity that already has this specific experience on their resume,” said Cathy Farley, a managing director at Accenture. “It is slowing some of the re-hiring from the ranks of the unemployed.”

Only 49 percent of people laid off from 2007 through 2009 were re-employed by January 2010, according to a Labor Department survey. It’s the lowest such proportion since the survey began in 1984.

And more than 40 percent of the nearly 15 million unemployed Americans have been out of work for six months or longer. That’s near the record high set during the recession.

Some of the unfortunate ones are information technology workers. One reason is that tech companies are increasingly combining business analyst and systems analyst positions.

Suppose a company wants a new software application. A business analyst would seek the least expensive approach and then propose the technical requirements. Separately, a systems analyst would build the technology.

But now, employers want “those two skill sets in one human being,” said Harry Griendling, chief executive of DoubleStar Inc., a staffing firm outside Philadelphia.

The trend reflects the push that companies made during the recession to control costs, squeeze more output from their staffs and become more productive. Productivity measures output per hour worked. Economy-wide, it soared 3.5 percent last year. It was the best performance in six years.

And it means workers are bearing heavier burdens. In manufacturing, employees increasingly must be able to run the computerized machinery that dominates most assembly lines. They also have to carry out additional tasks, such as inspecting finished products, notes Mark Tomlinson, executive director of the Society of Manufacturing Engineers.

Manufacturers advertised nearly 200,000 jobs at the end of August, a jump of about 40 percent from a year ago, according to government data. Yet hiring by manufacturers has risen less than 6 percent over that time — evidence that they are having a hard time finding qualified workers.

“There are jobs available, but the worker just has to have more skills than before,” Tomlinson said.

Bob Brown, 49, of Dayton, Ohio, has felt the demand for broader skills firsthand. After working for 30 years in manufacturing, including 20 as a plant supervisor, Brown was laid off in July 2009.

He spent a year looking for a new job. His efforts yielded only three calls from employers in the first four months.

But once things began to pick up, Brown noticed something else: The plant manager jobs he used to have, and that he was aiming for again, all required certifications in productivity-boosting management practices.

So Brown paid for courses at a community college to learn a management strategy known as “six sigma.” It’s an approach to cutting waste and raising efficiency popularized by General Electric. The courses allowed him to obtain his certification. In August, he was hired by an electrical product assembly plant near Williamsport, Pennsylvania.

“That’s the way the industry’s going,” Brown said. “Everybody wanted certifications.”

Human resource specialists say employers who increasingly need multi-skilled employees aren’t willing to settle for less. They’d rather wait and hold jobs vacant.

HR specialists even have a nickname for the highly sought but elusive job candidate whose skills and experiences precisely match an employer’s needs: the “purple squirrel.”

“There are lots of requests for purple squirrels nowadays,” said Joe Yesulaitis, chief executive of Aavalar Consulting, an IT staffing firm.