1. 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

2. 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

3. 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

4. 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

5. 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

6. Least squares
Minimize:

7. 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

8. 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 )

9. 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

10. 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

11. 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

12. 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

13. Least squares
In general:

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

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

16. Least squares
Again:
ƒs are not linear in xi

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

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

19. Least squares
Solve, as before:

20. Least squares
Solve, as before:

21. Least squares
Solve, as before:

22. Least squares
Weighting factors matrix:

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

24. 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

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

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

27. 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 > or < – 0.5, roughly)
If 2 parameters interact, hold one constant