A realistic cap on itemized deductions that protects charities and the middle class would raise only about $450 billion over the next decade, according to new White House estimates — too little to make a serious dent in the soaring national debt.

The estimates are contained in a document, obtained by The Washington Post, that is being shared with congressional Democrats and other White House allies as President Obama and top Republican leaders negotiate for an agreement on how to avert the year-end “fiscal cliff.” It offers a rebuttal to Republican arguments that eliminating loopholes and deductions could raise just as much money for deficit reduction as raising the income tax rates on top earners.

While that may be true in theory, the document argues, it would be difficult in practice — particularly if policymakers adhered to White House demands to keep the middle class from harm and preserve tax breaks for charitable giving.

“Some have suggested that limits on high-income tax expenditures could substitute for rate increases and that it would be possible to raise $1 trillion or more while keeping the top income tax rate at 35 percent,” the document says. “But a careful look at the math of these types of caps and limits shows that, once one takes into account the reality of their impact on middle-class families and on charitable donations, plausible limits raise only a fraction of the $1 trillion or more some have suggested.”

For example, the document looks at a $25,000 cap on itemized deductions, recently offered as a potential compromise by the bipartisan Committee for a Responsible Federal Budget. Such a cap, if applied to people all income levels, would raise taxes by an average of $2,400 on about 17 million households with income under $250,000 a year — the White House definition of middle class.

Limiting the cap to those with incomes over $250,000 would raise about $800 billion over the next decade — similar to the $850 billion the White House estimates would be generated by letting the George W. Bush-era tax rates expire for high earners.

But implementing the cap at $251,000 would create an unacceptable “cliff,” the document argues. So policymakers would have to draft a “realistic phase-in” that would implement the cap gradually on higher earnings. That would reduce the potential revenue to about $650 billion, the document argues.

Exempting charitable deductions would knock revenues back even further, to around $450 billion over the next decade, the estimates suggest — barely one-fourth of the $1.6 trillion in revenue Obama has said would be needed as part of a far-reaching “grand bargain.”

“Plausible tax expenditure limitations that protect middle-class families and incentives to give to charity would raise far less revenue from the well-off than is needed for a major budget agreement,” the document concludes. “A budget framework that raises only these amounts from high-income tax deductions while committing to no rate increases on high-income Americans would inevitably force any tax reform designed to further reduce the deficit to raise taxes on middle-class families simply to preserve lower rates for the most fortunate.”