| Summary: | Vibrational overtones in deeply-bound molecules are sensitive probes for variation of the proton-to-electron mass ratio <inline-formula> <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> </inline-formula>. In nonpolar molecules, these overtones may be driven as two-photon transitions. Here, we present procedures for experiments with <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow></mrow> <mn>16</mn> </msup> <msubsup> <mi mathvariant="normal">O</mi> <mn>2</mn> <mo>+</mo> </msubsup> </mrow> </semantics> </math> </inline-formula>, including state-preparation through photoionization, a two-photon probe, and detection. We calculate transition dipole moments between all <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi> <msup> <mspace width="0.166667em"></mspace> <mn>2</mn> </msup> <msub> <mo>Π</mo> <mi>g</mi> </msub> </mrow> </semantics> </math> </inline-formula> vibrational levels and those of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <msup> <mspace width="0.166667em"></mspace> <mn>2</mn> </msup> <msub> <mo>Π</mo> <mi>u</mi> </msub> </mrow> </semantics> </math> </inline-formula> excited electronic state. Using these dipole moments, we calculate two-photon transition rates and AC-Stark-shift systematics for the overtones. We estimate other systematic effects and statistical precision. Two-photon vibrational transitions in <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow></mrow> <mn>16</mn> </msup> <msubsup> <mi mathvariant="normal">O</mi> <mn>2</mn> <mo>+</mo> </msubsup> </mrow> </semantics> </math> </inline-formula> provide multiple routes to improved searches for <inline-formula> <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> </inline-formula> variation.
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