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We perform first-principles calculation to investigate the dynamic conductance of atomic wires of the benzenedithiol (BDT) as well as carbon chains with different length in contact with two Al(100) electrodes (Al-C n -Al). Our calculation is based on the combination of the non-equilibrium Green's function and the density functional theory. For ac conductance, there are two theories that ensures the current conservation: (1). the global formula which is a phenomenological theory that partitions the total displacement current into each leads so that the current is conserved.(2). the local formula which is a microscopic theory that includes Coulomb interaction explicitly so that the current is conserved automatically. In this work, we use the local formula to calculate the dynamic conductance, especially the emittance. We give a detailed comparison and analysis for the results obtained from two theories. Our numerical results show that the global formula overestimates the emittance by two orders of magnitude. We also obtain an inequality showing that the emittance from global formula is greater than that from local formula for real atomic structures. For Al-C n -Al structures, the oscillatory behavior as the number of carbon chain N varies from even to odd remains unchanged when local formula is used. However, the prediction of local formula gives rise to opposite response when N is odd (inductive-like) as compared with that of global formula. Therefore, one should use the local formula for an accurate description of ac transport in nanoscale structures. In addition, the ‘size effect’ of the ac emittance is analyzed and can be understood by the kinetic inductance. Since numerical calculation using the global formula can be performed in orbital space while the local formula can only be used in real space, our numerical results indicate that the calculation using the local formula is extremely computational demanding.


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