extending our previous approach for bcc [14]. As a first step, we match the peak liquid structure factor properties of the two-mode PFC model to the standard expression from classical DFT. The expression for the PFC liquid structure factor is obtained by varying ψ around its liquid value, ψ = ¯ ψ$_{l}$ + δψ , and evaluating the corresponding variation Δ F of the dimensional free-energy difference between solid and liquid using Eqs. (2) and (4), and the relation (15) between φ and ψ . Dropping terms of δψ higher than quadratic order, we obtain Equating Δ F PFC = Δ F DFT and using the expression for the liquid structure factor S ( k ) = 1 / (1 − C ( k )), we obtain By evaluating the above expression at the peak of the liquid structure factor, we obtain or, using Eq. (11) and the relationship ¯ ψ$_{l}$ = ψ$_{c}$ϵ 1 / $^{2}$, Substituting the Fourier transform, we obtain A second relation is now needed to determine ϵ and λ independently. To obtain it, we substitute Eq. (58) into the relation C ( k ) = ( S ( k ) − 1) /S ( k ) and compute the second derivative of C ( k ) evaluated at the peak of the liquid structure factor to obtain Eqs. (60) and (61) combined now give and In addition, the relation (55) between the real and di- mensionless densities expresses A second expression for the free-energy of a spatially in- homogeneous liquid is obtained from classic DFT in terms of the solid amplitude A$_{s}$ of the first q$_{0}$ -mode. The two solid amplitudes A$_{s}$ and B$_{s}$ can be computed for a given R by using the scaling relations A$_{s}$ = ϵ 1 / $^{2}$A and B$_{s}$ = ϵ 1 / $^{2}$B where A and B are the equilibrium values of the scaled amplitudes in solid. The latter are obtained, together with ψ$_{c}$ (Fig. 9), by using the conditions (46) and (47) with Δ F AE defined by Eq. (45). For a given R , Eqs. (62), (63), and (64) fix the three parameters ϵ , λ , and g of the PFC model uniquely in terms of peak liquid structure factor properties, S ( q$_{0}$ ) and C ′′ ( q$_{0}$ ), where q$_{0}$ = | ⃗ K$_{111}$ | here, and the solid density wave amplitude u$_{s}$ = u$_{111}$ . This still leaves the freedom to vary R within the range where fcc is stable with re- spect to bcc (Fig. 9). Varying R changes the shape of the liquid structure factor as shown in Fig. 12 and de- creasing R below some threshold produces a second peak at q$_{1}$ = | ⃗ K$_{200}$ | , and generally increases the contribution of the second mode. Thus decreasing R increases the am- plitude of the second mode u$_{200}$ as shown in Fig. 11. For where and is the Fourier transform of the direct correlation function. Fourier transforming again, we obtain Δ F PFC = λq 8 0 g ∫ d⃗ r$^{[}$ δψ 2 [ a + 3 ¯ ψ 2 l λq 8 0 + λ ( ∇ 2 + q 2 $_{0}$)$^{2}$(( ∇ 2 + q 2 $_{1}$) 2 + r$_{1}$ )] δψ$^{]}$. (51) δψ = ∫ d ⃗ k (2 π ) 3 / $_{2}$δψ$_{k}$ e i ⃗ k · ⃗ $^{r}$, (52) Δ F PFC = λq 8 0 g ∫ ∫ d ⃗ kd ⃗ k ′ (2 π ) 3 δψ$_{k}$δψ$_{k}$ ′ 2 { [ a + 3 ¯ ψ 2 l λq 8 0 + λ ( − k 2 + q 2 $_{0}$)$^{2}$(( − k 2 + q 2 $_{1}$) 2 + r$_{1}$ )] ∫ d⃗ re i $_{(}$⃗ k $_{+}$⃗ k $^{′}$) · ⃗ r } = λq 8 0 g ∫ d ⃗ k δψ$_{k}$$^{δψ}$− k 2 [ a + 3 ¯ ψ 2 l λq 8 0 + λ ( − k 2 + q 2 $_{0}$)$^{2}$(( − k 2 + q 2 $_{1}$) 2 + r$_{1}$ )] . (53) λ = − 9 k$_{B}$T C ′′ ( q$_{0}$ ) 8 n$_{0}$q 6 0 + 9 k$_{B}$T R n$_{0}$S ( q$_{0}$ )(1 − 3 ψ 2 $_{c}$) q 8 0 . (63) ϵ = 8 9( q 2 $_{0}$S ( q$_{0}$ ) C ′′ ( q$_{0}$ )(1 − 3 ψ 2 $_{c}$) − 8 R ) , (62) λ = − k$_{B}$T C ′′ ( q$_{0}$ ) 8 n$_{0}$q 6 $_{0}$( 1 9 + ϵR ) . (61) ϵ = − k$_{B}$T n$_{0}$S ( q$_{0}$ ) λq 8 $_{0}$(1 − 3 ψ 2 $_{c}$) . (60) a + 3 λq 8 0 ¯ ψ 2 l = k$_{B}$T n$_{0}$S ( q$_{0}$ ) , (59) S ( k ) = k$_{B}$T n$_{0}$ ( a + 3 λq 8 0 ¯ ψ 2 l + λ ( − k 2 + q 2 $_{0}$)$^{2}$(( − k 2 + q 2 $_{1}$) 2 + r$_{1}$ )) . (58) g = λq 8 $_{0}$A 2 s n 2 $_{0}$u 2 s , (64) Δ F DFT = k$_{B}$T 2 ∫ ∫ d⃗ rd⃗ r ′ δn ( ⃗ r )$^{[}$ δ ( ⃗ r − ⃗ r ′ ) n$_{0}$ − C ( | ⃗ r − ⃗ r $^{′}$| )$^{]}$ δn ( ⃗ r ′ ) , (54) δn ( ⃗ r ) = n ( ⃗ r ) − n$_{0}$ = δφ ( ⃗ r ) = $^{√}$λq 8 0 g δψ ( ⃗ r ) , (55) C ( k ) = n$_{0}$ ∫ d⃗ rC ( | ⃗ r | ) e − i ⃗ k · ⃗ $^{r}$, (56) Δ F DFT = λq 8 0 g k$_{B}$T 2 n$_{0}$ ∫ d ⃗ kδψ$_{k}$$^{δψ}$− k [1 − C ( k )] . (57) 11