so that the amplitudes in Eq. (12) become the same when k → G / 2, since the second term in Eq. (13) vanishes. Thus, we expect the direct Coulomb repulsion between an electron in state ˜ ϕ$_{k}$ and one in state ˜ ϕ$_{−}$$_{k}$ to be reduced, and that between two electrons in the same state ˜ ϕ$_{k}$ to be enhanced, when k is near the bottom of the band but not when k is near the top of the band. The Coulomb repulsion between an electron in Bloch state k and one in Bloch state k ′ is independent of k and k $^{′}$. Instead, in the new basis The bare Coulomb interaction between opposite spin electrons in a plane wave basis is given by showing in particular that the interaction is most reduced for k ′ = − k , and most enhanced for k ′ = k . However, for k and k ′ denoting states near the top of the band, ∼ G / 2, with G a reciprocal lattice vector, the effect disappears, in agreement with the discussion in the previous section. However, Eq. (19) appears to suggest that the Coulomb interactions are the same also for states around the half-filled band, e.g. k = G / 4. This is an artifact caused by keeping only the on-site Coulomb matrix el- ement U . When considering other Coulomb matrix ele- ments also it is found that the Coulomb interaction be- tween states in the new basis and in the conventional basis become the same only when the states approach the top of the band and the wavevectors are connected by reciprocal lattice vectors. Next we consider the Coulomb interaction in the triplet and singlet ( k ↑ , − k ↓ ) pairs. We have with Ω the volume of the system. The Coulomb interac- tion between an electron of spin ↑ in the plane wave state ϕ$_{k}$ and an electron of spin ↓ in ϕ ′ k is simply independent of k and k $^{′}$. The Coulomb interaction be- tween an electron of spin ↑ in state ˜ ϕ$_{k}$ and an electron of spin ↓ in state ˜ ϕ $^{′}$$_{k}$: is given by so that the triplet state in the new basis, is identical to the triplet state in the conventional basis, | t > . The Coulomb interaction is The singlet states are different however, since and we have showing indeed a reduction for ( k ↑ , − k ↓ ) in the new basis as well as an increased repulsion for ( k ↑ , k ↓ ), since the Coulomb matrix element V (2 k ) is always positive. Note also that the effect is largest for small k . Similarly consider the Coulomb interactions in the tight binding model. We will assume for simplicity that the only non-zero matrix element of the Coulomb inter- action in the tight binding model is the onsite Coulomb repulsion U: so the Coulomb repulsion is smaller in the singlet state of the new basis than in the singlet state of the conventional basis. The average of Eqs. (22) and (24a) or (24b) yields Eq. (16b) or Eq. (15b) for k ′ = − k , since the state | k ↑ , − k ↓ > is a linear combination of singlet and triplet states with equal weight. U = ∫ d $^{3}$rd$^{3}$r $^{′}$| ϕ$_{i}$ ( r ) | 2 e 2 | r − r $^{′}$| | ϕ$_{i}$ ( r $^{′}$) | 2 (17) < s | V$_{c}$ | s > = 1 Ω ( V (0) + V (2 k )) (24b) < ˜ s | V$_{c}$ | ˜ s > = V (0) Ω (24a) ˜ c † k $_{↑}$˜ c † − k ↓ + ˜ c † − k $_{↑}$˜ c † k ↓ = i ( c † k $_{↑}$c † k ↓ + c † − k $_{↑}$c † − k $_{↓}$) (23) < ˜ t | V$_{c}$ | ˜ t > = < t | V$_{c}$ | t > = 1 Ω ( V (0) − V (2 k )) . (22) < ˜ k ↑ , ˜ k ′ ↓ | V$_{c}$ | ˜ k ↑ , ˜ k ′ ↓ > = 1 Ω V (0) + 1 2Ω V (2 k )( − δ$_{k}$ ′ , − k + δ$_{k}$ ′ , $_{k}$) (16b) | ˜ k ↑ , ˜ k ′ ↓ > ≡ ˜ c † k $_{↑}$˜ c † k ′ ↓ | 0 > (16a) | ˜ t > = ˜ c † k $_{↑}$˜ c † − k ↓ − ˜ c † − k $_{↑}$˜ c † k ↓ 2 | 0 > (21) ˜ c † k $_{↑}$˜ c † − k ↓ − ˜ c † − k $_{↑}$˜ c † k ↓ = c † k $_{↑}$c † − k ↓ − c † − k $_{↑}$c † k ↓ (20) < k ↑ , k ′ ↓ | V$_{c}$ | k ↑ , k ′ ↓ > = ∫ d $^{3}$rd$^{3}$r $^{′}$| ϕ$_{k}$ ( r ) | 2 e 2 | r − r $^{′}$| | ϕ$_{k}$ ′ ( r ) | 2 = 1 Ω V (0)(15b) | k ↑ , k ′ ↓ > ≡ c † k $_{↑}$c † k ′ ↓ | 0 > (15a) V ( q ) = 4 πe 2 q 2 (14b) V$_{c}$ = 1 Ω ∑ kk ′ q V ( q ) c † k + q $_{↑}$c † k ′ − q $_{↓}$c$_{k}$ ′ $_{↓}$c$_{k}$$_{↑}$ (14a) < ˜ k ↑ , ˜ k ′ ↓ | V$_{c}$ | ˜ k ↑ , ˜ k ′ ↓ > = U N 2 ∑ i (1 + sin (2 k · R$_{i}$ ))(1 + sin (2 k ′ · R$_{i}$ ))(19) < k ↑ , k ′ ↓ | V$_{c}$ | k ↑ , k ′ ↓ > = U N (18) III. COULOMB REPULSION IN THE NEW BASIS 3