Least squares method Let adjustable parameters for structure refinement be uj Then if R = S w(hkl) (|Fobs| – |Fcalc|)2 = S w D2 Must get ∂R/∂ui = 0 one eqn/parameter hkl hkl hkl hkl

Least squares method Let adjustable parameters for structure refinement be uj Then if R = S w(hkl) (|Fobs| – |Fcalc|)2 = S w D2 Must get ∂R/∂ui = 0 one eqn/parameter Then S w D ∂|Fc|/∂ui = 0 hkl hkl hkl hkl

Least squares Simple example – again To solve simultaneous linear eqns: a11x1 + a12x2 + … = y1 a21x1 + a22x2 + … = y2 If: Then simultaneous eqns given by A x = y

Least squares Suppose: a11x1 + a12x2 + … ≈ y1 a21x1 + a22x2 + … ≈ y2 Then: a11x1 + a12x2 + … – y1 = e1 a21x1 + a22x2 + … – y2 = e2 No exact solution as before – but can get best solution by minimizing S ei 2 i

Least squares a11x1 + a12x2 + … – y1 = e1 a21x1 + a22x2 + … – y2 = e2 No exact solution as before – but can get best solution by minimizing S ei Also – note that no. observations > no. of variable parameters (n > m) Minimize: 2 i

Least squares Minimize:

Least squares To illustrate calcn, let n, m = 2 (a11x1 + a12x2 – y1)2 = e12 (a21x1 + a22x2 – y2)2 = e22 Take partial derivative wrt x1, set = 0: (a11x1 + a12x2 – y1) a11 = 0 (a21x1 + a22x2 – y2) a21 = 0

Least squares To illustrate calcn, let n, m = 2 (a11x1 + a12x2 – y1)2 = e12 (a21x1 + a22x2 – y2)2 = e22 Take partial derivative wrt x1, set = 0: (a11x1 + a12x2 – y1) a11 = 0 (a21x1 + a22x2 – y2) a21 = 0 (a11 a11) x1 + (a11 a12) x2 = (a11) y1 (a21 a21) x1 + (a21 a22) x2 = (a21) y2 (a11 a11 + a21 a21) x1 + (a11 a12 + a21 a22) x2 = (a11 y1 + a21 y2 )

Least squares (a11 a11 + a21 a21) x1 + (a11 a12 + a21 a22) x2 = (a11 y1 + a21 y2 ) x1 S ai1 + x2 S ai1 ai2 = S ai1 yi 2 2 2 2 i=1 i=1 i=1

Least squares (a11 a11 + a21 a21) x1 + (a11 a12 + a21 a22) x2 = (a11 y1 + a21 y2 ) x1 S ai1 + x2 S ai1 ai2 = S ai1 yi Now consider: 2 2 2 2 i=1 i=1 i=1

Least squares (a11 a11 + a21 a21) x1 + (a11 a12 + a21 a22) x2 = (a11 y1 + a21 y2 ) x1 S ai1 + x2 S ai1 ai2 = S ai1 yi Now consider: AT A 2 2 2 2 i=1 i=1 i=1

Least squares (a11 a11 + a21 a21) x1 + (a11 a12 + a21 a22) x2 = (a11 y1 + a21 y2 ) x1 S ai1 + x2 S ai1 ai2 = S ai1 yi Now consider: AT A And: (AT A) x = (AT y ) 2 2 2 2 i=1 i=1 i=1

Least squares In general:

Least squares In general: And: (AT A) x = (AT y )

Least squares In general: (AT A) x = (AT y ) x = (AT A)-1 (AT y )

Least squares Again: ƒs are not linear in xi

Least squares Again: ƒs are not linear in xi Expand ƒs in Taylor series

Least squares Again: ƒs are not linear in xi Expand ƒs in Taylor series

Least squares Solve, as before:

Least squares Solve, as before:

Least squares Solve, as before:

Least squares Weighting factors matrix:

Least squares So: Need set of initial parameters xjo Problem solution gives shifts ∆xj, not xj

Least squares So: Need set of initial parameters xjo Problem solution gives shifts ∆xj, not xj Eqns not exact, so refinement process requires no. of cycles to complete the refinement Add shifts ∆xj to xjo for each new refinement cycle

Least squares How good are final parameters? Use usual procedure to calculate standard deviations, s(xj) no. observations no. parameters

Least squares Warning: Frequently, all parameters cannot be “let go” at the same time How to tell which parameters can be refined simultaneously?

Least squares Warning: Frequently, all parameters cannot be “let go” at the same time How to tell which parameters can be refined simultaneously? Use correlation matrix: Calc correlation matrix for each refinement cycle Look for strong interactions (rij > + 0.5 or < – 0.5, roughly) If 2 parameters interact, hold one constant