New pairing scheme to overhaul BCS theory

•Two intrinsic difficulties of BCS theory identified.•A new pairing scheme introduced.•High accuracy of the new pairing scheme demonstrated.•Matthias׳ rule explained for the first time.•High Tc of metallic hydrogen questioned.

In a superconductor pair occupancy probabilities are doubly defined with conflicting values when normal and umklapp scattering coexist with the same destination momentum. To resolve this issue a new pairing scheme is introduced to assert normal–umklapp frustration under such circumstances. Superconductivity then arises solely from residual umklapp scattering to destination momenta not reached by normal scattering. Consequent Tc calculations from first principles for niobium, tantalum, lead and aluminum turn out to be accurate within a few percent. A new perspective is revealed to support Matthias׳ rule. New light is also shed relevant to the future study of metallic hydrogen.