content_image/441179.jpg content_image/441176.jpg content_image/441177.jpg content_image/441178.jpg content_image/441180.jpg content_image/441181.jpg content_image/441182.jpg content_image/441186.jpg content_image/441187.jpg content_image/441161.jpg content_image/441183.jpg content_image/441184.jpg content_image/441185.jpg content_image/441188.jpg
if { ϕ$_{k}$ } is. The inverse relation is 

Similarly we define the electron creation operators in the new states ˜ c † k σ in terms of the Bloch creation operators c † k σ by 

It is clear that our new basis functions do not satisfy Bloch’s theorem, since 

<img src='content_image/441179.jpg'>

Nevertheless, I will argue that the new basis has very interesting properties and is relevant to the description of the superconducting state. 

## II. NATURE OF THE WAVE FUNCTIONS 

Consider the case where the Bloch functions are well approximated by plane wave states 

Then 

describing standing waves. The values of r for which | ˜ ϕ$_{k}$ ( r ) | 2 is maximum satisfy 

with n integer, and for those r values 

so that | ˜ ϕ$_{−}$$_{k}$ ( r ) | 2 = 0. Consequently this will cause a strong reduction of the direct Coulomb repulsion between an electron in state ˜ ϕ$_{k}$ and another electron in state ˜ ϕ$_{−}$$_{k}$ compared to the plane wave case where the density is uni- form, as illustrated in Fig. 1. Conversely, the in-phase 

<img src='content_image/441176.jpg'>

<img src='content_image/441177.jpg'>

<img src='content_image/441178.jpg'>

<img src='content_image/441180.jpg'>

<img src='content_image/441181.jpg'>

<img src='content_image/441182.jpg'>

<img src='content_image/441186.jpg'>

<img src='content_image/441187.jpg'>

modulation of the wavefunction amplitude for two elec- trons in the same state ˜ ϕ$_{k}$ will give a strong enhancement of the direct Coulomb repulsion in that case compared to the plane wave case. 

Similarly, consider a band described by the tight bind- ing approximation with a single orbital φ$_{i}$ ( r ) per site. The Bloch functions are 

and the new orbitals are given by 

It is clear that for small k the situation is the same as for the plane-wave case: the values of R$_{i}$ for which the amplitude is maximum in Eq. (12a) coincide (or nearly coincide) with those where the amplitude in Eq. (12b) is zero. However, this changes when the wavevector k becomes an appreciable fraction of a reciprocal lattice vector G . We have 

<img src='content_image/441161.jpg'>

<img src='content_image/441183.jpg'>

<img src='content_image/441184.jpg'>

<img src='content_image/441185.jpg'>

<img src='content_image/441188.jpg'>

