Resolution of Overapping Bands: Functions for Simulating Band Shapes R. D. B. Fraser and Eikichi Suzuki Division of Protein Chemistry, C.S.I.R.O.,343 Royal Parade, Parkvilk, Victoria 3052, Australia Functions suitable for representing band shapes when resolving overlapping bands by iterative least squares procedures are described. Expressions for symmetrical, skewed, and derivative bands are given and the types most suited to various analytical techniques are discussed.
treated as a parameter and included in the least squares optimization procedure. The area beneath the composite band is given by A
=
1/2YdXl/2Lf(*/ln2)'I2
y = l/(l
SYMMETRICAL BANDS
YOexp( -In 2+[2(X- X O ) / A X I / Z ] ~ I
(1)
and a Cauchy function,
Y
=
Yo/( 1
+ [2(X - Xo)/AX~/z121
In y
(2)
which are illustrated in Figure 1 ; YOis the peak height, XOthe peak position, and A X l I pthe band width at half height. One method of dealing with this situation ( I ) is to use a linear combination of the functions in Equations 1 and 2 in the ratiof : (1 -f),giving
Y
+ [2(X - Xo)/~xl/21p11,0 6 f 6 1
I
I
1
/
g
-~
+ ...
'~ / ~ U~' X '~
(6)
=
exp(-x2)
(7)
Thus as a varies from 0 to 1, the expression in Equation 5 varies smoothly from a Gaussian to a Cauchy function. Values of a > 1 produce band shapes with long tails (Figure 2). In terms of YO,XO,and AXI/Z,Equation 5 becomes
(3)
Y
(1) R. D. B. Fraser and E. Suzuki, ANAL.CHEM., 38, 1770 (1966). I
+ ~ 2 x 2 )= -x' +
a-0
The band shapes for f = 0,0.5,and 1.0 are shown in Figure 1. A suitable value off can be selected empirically or it may be
t
In (1
Lim ( y )
+
(1 - . f > / ( l
= -a+
whence
Yo[f.exp(-In 2.[2(X - XO)/AXIIPI~)
=
+ azx2)l'a2
which resembles a Cauchy function except that the denominator is raised to a variable power. For a = 1 , the expression reduces to a Cauchy function: for a = 0, Equation 5 is indeterminate but the limiting value'may be obtained by rewriting it as
In many applications, band shapes are intermediate between that of a Gaussian, =
(4)
Pitha and Jones (2, 3) investigated the effect of combining Gaussian and Cauchy functions with different half-band widths. However in most instances only a marginal improvement in fit is obtained compared with Expression 3 and the introduction of the additional parameter is not warranted. The shapes produced by taking the product of a Gaussian and a Cauchy function have also been investigated ( 2 4 , but these proved less satisfactory than those given by sum functions. An alternative approach is to use a function of the type
IN A PREVIOUS COMMUNICATION to this journal ( I ) a procedure for the resolution of overlapping absorption bands was described. The method has been applied to a variety of analytical problems with Gauss and Cauchy functions to represent band shapes, but its usefulness has recently been extended by the formulation of further functions for simulating band shapes. Details of these functions are given in the present communication.
Y
+ (1 - f ) x I
1
=
Yo/{ 1
+ [2a*- 1][2(X - Xo)/AXi/2]2]1'a'
(8)
(2) J. Pitha and R. N. Jones, Can.J. Chem., 44, 3031 (1966). (3) Zbid.,45,2347 (1967). (4) S. Abramowitz and R. P. Bauman, J. Chem. Phys., 39, 2757 (1965). . .
1
lX-X,)/AX;
(X-X,)
/AX;
Figure 1. Band shapes produced by using a linear combination of Gauss and Cauchy functions in the ratio f: (1 - f) respectively; f = 0 represents Expression 2 and f = 1.0 represents Expression 1
Figure 2. Band shapes generated by the general function for band shapes given in Equation 8. a = 0 corresponds to a Gaussian, a = 1 corresponds to a Cauchy function VOL. 41, NO. 1, JANUARY 1969
37
-Y v,
0.5
I
II
n -1.0
1.0
0
2.0
1-05
Q V ,1
-I0
I
I X-X,) /AX$
i
Figure 3. Skewed Gaussian band shapes generated by the function given in Equation 11; b = 0 gives a symmetrical Gaussian band
The area beneath the curve (5)is
I
IX-X,]
-IO
/AX,
IO
,
20
2
Figure 4. Comparison of the shapes of Gaussian band (Equation 1) and its first derivative function (Equation 15), illustrating the way in which Y,,, and AX,,, are d e fined
A = i / z Y d ~ l i 2 ~ 1(lid ' 2 r - 1/z)/[(2aP- I ) ~ W ( I / U ~ )(9) ]
O
