Notes of literature[1]:JACS-2002-crownether by Chris Liu

Introduction:
1. Crown ethers have the remarkable property of recognizing and binding specific metal cations in complex mixtures.
2. By 2002, the one area of crown ether which remains unexplored is the conductance properties. Of particular interesting is the change in conductance as the crown binds and relinquishes a cation.
3. Such a chemical system could possibly be used as a molecular sensor or switch. These questions are, in fact, accessible to experimentalist. Recent synthetic advances enable researchers to form molecular wires(MW) by binding elongated organic molecules to metallic leads such as gold.
4. This has led to the development of various new prototypes of molecular scale electronic devices, among them, rectifiers, memory devices, switches, and transistors.
5. This communication combines molecular recognition and molecular electronic conductance to form a single electronic device. On a slightly larger scale, such an attempt was recently successful, where biotin molecules fictionalized a silicon nanowire, and conductance dropped by 12% upon exposure to m-antibiotin.
Here, they study a short molecular wire containing a crown-6 molecule connected via sulfur atoms to two long atomic gold wires(AGW) 

Methods
6. Using DFT method, one obtains an excellent zero-order Hamiltonian that describes the unbiased MW. The zero-bias conductance should now be calculated using time-dependent DFT. Yet, at present, this method is not well developed and is too time-consuming for this system. An alternative is to use the non-interacting particle model of Landauer, by which the current I associated with a voltage bias V is the equation (1) [see paper for more detail]
7. To calculate transmission probability T(E) ("transmittance"), they using the DFT method modifiedby themself. To the Kohn-Sham Fock-matrix Fks, they add two absorbing potentials stretched along the two gold wires. this results in a complex Fock matrix F=Fks-i(TL+TR). Here, TL and TR are positive diagonal matrices in the atomic orbital basis located on the left and right gold leads(线，绳索). These potential allow the computation of quantum transmittance using techniques adopted from reaction scattering theory.[see paper for more detail]

Results

8. Molecular wires containing a potassium cation show weakest conductance. The curve for sodium and lithium cations (not shown) is identical to that of potassium,but proton behaves differently. It is clear that the system with an empty crown conducts significantly better than a loaded one.
9. To better understand the conduction pattern, they also examine the transmittance and density of states(DOS) curve in the vicinity of the Fermi level.
10. All three systems exhibit a similar DOS at the Fermi level of about 10 states per eV (i.e. a HOMO-LUMO spacing of 0.1 eV).[ do not understand]
11.