1. Time scale factors and functional forms
A CONSIDERATION OF MATHEMATICAL PROCEDURES and the MOLECULAR STRUCTURAL CRITERIA WHILE AVERAGING FOR FLUCTUATIONS
OP Book of Abstracts Page-36
UGC-SAP Seminar on
EMERGING TRENDS IN CHEMICAL SCIENCES-2015
November 05-06, 2015
Organized by
Department of Chemistry, Gauhati University
Guwahati
Time scale factors and functional forms
S. ARAVAMUDHAN
DEPARTMENT OF CHEMISTRY
NORTH EASTERN HILL UNIVERSITY
SHILLONG
4/18/2019
Dr.S.Aravamudhan

2. Intermolecular distances
Total energy ET
STATIC SYSTEM CONSIDERED
LARGE
compared to molecular sizes
2 molecules A and B A B
No interaction
ET = EA + EB
SMALL
comparable to molecular sizes
When the molecules are in a strong external magnetic field then the same electronic structure, will have additional electron circulations resulting in secondary magnetic fields.
WEAK INERTMOLECULAR
interaction
SHORT
Smaller than molecular sizes
Interactions arising due to these secondary fields can be effective even at much larger distances than at which electrostatic interactions are accountable.
STRONG ELECTROSTATIC
interaction
ET ≠ EA + EB
Repulsive or bonding attractive
4/18/2019
Dr.S.Aravamudhan

3. Intermolecular distance
DYNAMIC STSTEM; MOLEULES IN MOTION/ RANDOM FLUCTUATIONS A B
Intermolecular distance t
The intermolecular distances during the swing might go through large/small/short distance ranges at a single instance.
What is the time dependent functional form that is appropriate to describe the variation of oscillatory nature, triangular or sinusoidal?
The two points above have independent aspects to consider while the averaging over the time dependence is carried through.
Time axis
Intermolecular distance
4/18/2019
Dr.S.Aravamudhan

4. Intermolecular distance
Each one of the 8 structures will have equal weightage-equal dwell time for each
For structures near the center the weightage could be less (?)-unequal dwell time for each
triangular
Molecular structure might vary with distance
large
small
short
sinusoidal
Intermolecular distance 1 2 3 4 5 6 7 8
Causes for structural variations,
time scales for viability for changing structures,
electrostatic and magnetic interactions,
handling assignment of weight factors.
4/18/2019
Dr.S.Aravamudhan

5. The two lines intersecting at the point renders the point as a point where there can be two slopes simultaneously. At that point the slope changes from positive to negative with same magnitude
y1= + mx+c
y2= - mx+c
For such triangular waveforms, the equation for one oscillation may be written as equation of pair of lines. Sinusoidal form can be represented by much simpler function
Points of intersections have undefined slopes (indeterminates) and hence non differentiable. Inconvenient functions for mathematical modeling. Uniform weightage factor at all times is a convenience.
NO sharply defined points of intersections; have continuously varying slopes and hence differentiable. Convenient functions for mathematical modeling. Since Fourier transform can give rise to components of sine/cosine functions, linear combinations of sinusoidal functions are mostly applied for the descriptions. During a time period of oscillation dwell times are different requiring varying weightage factors.
4/18/2019
Dr.S.Aravamudhan

6. Sine and Cosine functions
Sinusoidal
Sine and Cosine functions
t = 0; or t0 t
The value of the function specified at any given point of time, enables the evaluation of the value of the function at any other point in time, provided the constants in the equation T, or ν is known. T is the time period of oscillation (characteristic time for one complete oscillation, may be in secs ) and ν is the frequency of oscillation – a number, ν per second.
2π is the angle (360º) in radians ( the total phase angle per oscillation) , therefore 2π ν is called the angular velocity, since 2π has dimension of angle for one oscillation, and for ν oscillations (per second) the corresponding angle would be 2π ν , which is the change in angle during 1 sec. The unit “per second” makes this a velocity in angles.
Y = sin ( 2π t /T )
Y = sin ( 2π ν t )
1/T= ν
Y = sin (ω t )
2π ν = ω
Yc = cos ( 2π t /T )
Yc = cos ( 2π ν t )
If the wave form at any stage does not correspond to one complete oscillation, then frequency is not definable since it has not completed the required 360º angle. Evolutions for less than 2π (less than one full oscillation are characterized by the change in angle from the reference time which is termed as the phase angle. At any time a phase angle is definable, and if this phase angle is divisible by 2π, the numerical coefficient of division is the frequency of oscillation.
Yc = cos (ω t )
4/18/2019
Dr.S.Aravamudhan

7. The function defined at one point of time remains defined for the entire time duration: Periodic function well defined.
Spurious / sporadic features increase the uncertainty of evaluating values of the function. (The sinusoidal trend in illustration of previous slide can be extended from -∞ to +∞ )
Trend towards random and spurious / sporadic fluctuations even for short time periods it appears to be periodic
A smooth function; appearing almost without any discontinuities is in actuality a sum of three different functions for evolution in the three time slots within a period of oscillation. From t0 to t1 : a sinusoidal evolution, from t1 to t2 : a exponential decay and from t2 to t3 : a sinusoidal form. Even if mathematically simple to look, evaluation would be more tedious than the simple sinusoidal form all through.
4/18/2019
Dr.S.Aravamudhan

8. Discontinuities at the specified points can be smoothened; experimentally may be filtering, or a set of numerical data can be subjected to a digital filtering; taking an average of points near the discontinuities and / or moving average techniques are numerical methods to smoothen the function; however after smoothening how to describe the resulting continuous function could be a matter of the several kind of fitting the points to a function (straight line/exponential/polynomial fittings by Least Squares or steepest descent procedures) the it could be a multi functional time dependence as described in previous slide. Smoothly varying functions can be subjected to mathematics of calculus without much complications, but could be tedious depending upon how many functions successively put together can describe the total time evolution.
Given a smoothly varying function of time over a certain interval, with respect to a given reference point of time, the value of the function can be calculated at any other point of time within the range. Numerical evaluations would typically consist of calculating in the value of the functions at equal intervals of time and tabulating them for processing digitally. Experimentally this is accomplished by Analogue to Digital Conversion using the A.D.C devices
4/18/2019
Dr.S.Aravamudhan

9. I1 =0.9*(0.2*E1+0.2*C1+0.2*B1+0.2*F1+0.1*D1+0.1*G1)+0.1*H1
E1=COS(2*PI()*A1/1.3)
C1=SIN(2*PI()*A1/3)
B1=COS(2*PI()*A1/3.5)
F1=COS(2*PI()*A1/2.3)
D1=SIN(2*PI()*A1/2.9)
G1=SIN(2*PI()*A1/1.9)
H1=COS(2*PI()*A1/0.9)
3.5 > 1.3 > 0.9
I1 =0.9*(0.2*E1+0.2*C1+0.2*B1+0.2*F1+0.1*D1+0.1*G1)+0.1*H1
A list of mathematical expressions for sinusoidal / cosinusoidal functions & the linear combinations of such functions.
From these mathematical equations numerical values at equal intervals of time can be obtained thus enabling numerical tables for each function.
These numerical values can be in turn plotted in a graphical form and the function can be visualized.
These aspects are illustrated in this slide.
J2=(EXP(-1*(0.5*2))*I1+EXP(-1*(0))*I2+EXP(-1*(-1*0.5)*2)*I3)/(EXP(-1*(0.5*2))+EXP(-1*(0))+EXP(-1*(-1*0.5)*2))
K1=I1+J1
L3=(0.1*K1+0.2*K2+0.4*K3+0.2*K4+0.1*K5)
M1=K1+L1
P1=COS(2*PI()*A1/16)
Q1=M1*0.2+P1
R1=M1*0.4+P1
S1=M1*0.8+P1
T1=M1*1.6+P1
4/18/2019
Dr.S.Aravamudhan

10. B to H; are all Sine OR Cosine Functions
I1 =0.9*(0.2*E1+0.2*C1+0.2*B1+0.2*F1+0.1*D1+0.1*G1)+0.1*H1
J2=(EXP(-1*(0.5*2))*I1+EXP(-1*(0))*I2+EXP(-1*(-1*0.5)*2)*I3)/(EXP(-1*(0.5*2))+EXP(-1*(0))+EXP(-1*(-1*0.5)*2))
K1=I1+J1
L3=(0.1*K1+0.2*K2+0.4*K3+0.2*K4+0.1*K5)
Consequence of adding increasing proportion of M to a COSINE FORM P can be seen in next slide
K --Linear Combinations of I & J ‘L’ addition of various proportions of K
M1=K1+L1
P1=COS(2*PI()*A1/16)
Q1=M1*0.2+P1
R1=M1*0.4+P1
S1=M1*0.8+P1
4/18/2019
Dr.S.Aravamudhan

11. P=COS ( 2*PI()*A1/16 )
Cosine form
Argument in the cosine form contains the frequency information
Starting with sine and cosine functions, a resulting FORM that appears to be a RANDOM function is obtained. Conclusion By Synthesis ! Given the resultant function to analyze for constituent component functions cannot be as easy ! Way out is to set up correlation function from the resultant and further subject the Correlation function to FOURIER TRANSFORMATION. A transformation in Time domain to be viewed in Frequency domain.
P+0.2M
P+0.4M
P+0.8M
P+1.6M
Since this is simulated by taking linear combination of well defined functional forms valid for infinite time, the amplitude at any point of time can be known provided a valid zero time is set for reference. The signal appears noisy, seemingly no useful information. It is supposed here that answer to the query "what is information ?” is well known.
Even at the outset it may be stated that this is not a random function of time, though it appears noisy, without any information to convey. To state so, that this function is a calculated function is not obvious from the visual inspection.
4/18/2019
Dr.S.Aravamudhan

12. <y(t)  y(t+)> = C() called  ”correltion function"
In a graphical plot establishing a mathematically expressed function that is valid for the entire range of values of independent variable is called a curve fitting. The method of east squares is a curve fitting procedure.
Seeking to establish a dependence one data (Y1 -value) point for a value of (X1) independent variable (conventionally X-axis) with for the entire range of X,Y pairs is called establishing correlation.
Thus for a fixed difference  of t-values on the entire X-axis range if the dependence is established, this factor is referred to as the correlation factor and by explicitly stating the dependence a ”correlation function” can be defined.
Difference t(2 – t1 )
= 
<y(t)  y(t+)> = C() called  ”correltion function" t2
y(t2) t1
y(t1)
4/18/2019
Dr.S.Aravamudhan