# VLD

## Definition

VLD ("Vive la Différence") is a phasing approach based on the properties of the Fourier transforms. Probabilistic techniques lead to the difference electron density with Fourier coefficients

$F_q\approx \bigg[(m|F|-D|F_p|)-|F_p|(1-D) \bigg( \frac{e-\sigma^2_A}{1-\sigma^2_A} \bigg) \bigg]\exp(\textrm{i}\varphi_p).~~~~~(1)$

$|F|$and $|F_p|$are the target and the model structure factor amplitudes, respectively, $\varphi$ and $\varphi_p$ the associated phases, $\textit{e}$ is related to the experimental error,

$\sigma_A=\Bigg(\frac{\sum_p}{\sum_N}\Bigg)^{1/2}D,$

$\sum$N and $\sum$p are the scattering powers of the target and of the model structure respectively. $D$ is related to the misfit between model and target structure via the relation $D=\langle\cos(2\pi \mathbf{h}\Delta \mathbf{r})\rangle$.

The difference electron density calculated via coefficients (1) is well correlated with the true difference density even when the model is uncorrelated with the target, and suggests the following cyclic algorithm:

(a) Assign random phases to the target structure factors and calculate the corresponding target electron density map.

(b) Select a model electron density and calculate the difference electron density via the coefficients (1). Once modified and Fourier inverted it will provide $F_q$ coefficients.

(c) A new electron density map may be obtained by summing model and difference electron densities via the tangent formula

$\tan\varphi=\frac{R_p\sin\varphi_p+w_qR_q\sin\varphi_q}{R_p\cos\varphi_p+w_qR_q\cos\varphi_q}~~~~~(2)$

where $w_q=\sqrt{2(1-\sigma_A)}$.

(d) Calculate the new electron density map for the target by using phases given by (2) and return to (b). Repeated cycles of the above algorithms lead from random phases to the correct values. The method may be generalized by replacing the difference electron density by hybrid electron densities. Furthermore, VLD may be used for extending and refining models provided by non ab initio approaches.