1

Computational chemistry

I. part: Basics of signal processing with Octave

notes for lecture and practice

BSc in Chemistry, computational lab in 6th semester

Gergely Tóth

Loránd Eötvös University, Institute of Chemistry

6 hours of practice

Octave introduction, functions and basic programming 2-6

8 hours of lecture and practice

Solution of a set of linear algebraic equations 7-9

Numerical integration 10-12

Interpolation 12-13

Smoothing 13-14

Numerical differentials 14-15

Data storage in matrices – conventions, statistical applications 15-16

Principal component analysis 17-19

Fourier-transformation 20-29

References 29

version 2016.02.01

"Supported by the Higher Education Restructuring Fund allocated to ELTE"

2

Octave introduction, functions and basic programming

Set working directory, logbook

> pwd

> cd mydir (Linux) cd c:\a\teaching\compchem (Windows)

> diary logbook.dat # write a logbook

> ls

> quit

Initial steps

> x=2

> y=3;

> z=x+y**3

> x=cos(z)

> disp(" order of commands ")

> y=exp(x)*log(x)-log10(x)+sqrt(x)-ceil(x)+floor(x)-

fmod(x,2)+tan(x)

> function y=f(x,z) y=x+z*3-sqrt(x);endfunction; #preparation

of a function

> f(3,4)

> x=2

> z=6

> f(x,z)

> round(2.4)

> round(2.6)

> sign(3.2)

> sign(-3.2)

> factor(12312)

> lookfor factor

> factorial(12)

> s=1

> n=5

> for i=1:n s=s*i; endfor; #writing loops

> s

> help primes

3

> primes(100)

Row and column vectors

> v=[2;34;2]

> v=[2,34,2]

> u=rand(1,3)

> size(u)

> v=resize(v,1,4)

> u=resize(u,1,4)

> v*u'

> dot(v,u)

> sum(v)

> function s=sumvtg(v,n) s=0;for i=1:n s=s+v(i); endfor;

endfunction;

> sumvtg(v,4)

> sumvtg(u,4)

> function [su,sv]=sumuvtg(u,v,n) su=0;sv=0;for i=1:n

sv=sv+v(i);su=su+u(i); endfor; endfunction;

> sumuvtg(u,v,4)

> [s1,s2]=sumuvtg(u,v,4)

s1 = 0.35839

s2 = 38

> function [su,sv]=sumuvtg(u,v,n) su=0;sv=0;for i=1:2:n

sv=sv+v(i);su=su+u(i); endfor; endfunction;

> [s1,s2]=sumuvtg(u,v,4)

> prod(v)

> sumsq(v)

> sumsq(v.*u) # . = multiply one by one element

> v

> sort(v) # sorting of vectors

Exercise:

Write a function to calculate sum of squares of vector

elements

4

Commands for programming

< > =< == & | ! !=

while( ) body endwhile

if( ) body elseif() body else body endif

do body until( ) – stops if it is true

while( ) body endwhile – runs if it is true

break – jump out from for or while

continue – jump to next in for or while, the rest is skipped

from the commands

Exercises:

Write a function to calculate sum of squares of vector

elements, as long as there is a 0 element.

Write a function to calculate the number of vector elements

smaller than a given number.

Write a function to find the roots of second order

polynomials.

Write a function to calculate cross and triple product of

vectors.

Complex numbers

> z=0.1+2i

> real(z)

> imag(z)

> abs(z)

> i=sqrt(-1) #define complex unit

Exercises:

Write a function to calculate the conjugate of a complex

number.

Write a function to calculate the conjugate of a complex

number, if the imaginary part was negative.

Coordinate transformation

> [theta,r]=cart2pol(1,0)

5

theta = 0

r = 1

> cart2pol(1,0)

> [x,y]=pol2cart(0,1)

> [x,y,z]=sph2cart(0,0,1)

Exercise:

Write a function to calculate degree to radian.

Write a function to calculate the highest common factor or

lowest common multiple. Look for Euclidean algorithm and

relation of the two numbers.

Graphics

> plot(u,v,"4+") #4 colour, + sign

> print -djpg proba.jpg

Matrices

> A=[1,2;3,4]

> A(1,2) # given element

> A(:,1) # all elements of column 1

> B=randn(2,2)

> A*B

> C=randn(2,3)

> C*A

error: operator *: nonconformant arguments (op1 is 2x3, op2 is

2x2)

> A*C

> C'*A

Exercise:

Write a function to multiply matrices (multiple loops)

> det(A)

> inv(A)

> A*inv(A)

> eig(A)

6

> [vA,eA]=eig(A)

> A==B

> v=vec(A)

> u=v'

> B=eye(2) #Diagonal Matrix

> trace(A)

Exercise:

Write a function to substitute trace command

> sortrows(A,2)

Exercise:

Create an algorithm and write a function to sort vector

elements.

> save amatrix.dat A

> A=A'

> load amatrix.dat

7

Solution of a set of linear algebraic equations

Theory

General form of the equations:

a11x1 + a12x2 + …….. + a1nxn = b1

a21x1 + a22x2 + …….. + a2nxn = b2

…………………………………………………….

an1x1 + an2x2 + …….. + annxn = bn

n equations n unknowns (xi) n×n coefficients (aii)

using matrix and column vector notation

Ax=b

Unique solution exists, if ∃ bi ≠ 0 and det(A) ≠ 0

Cramer-method

)Adet(

)Adet( k=kx , where

=

+−

+−

nknnknn

nkk

aabaa

aabaa

......

.....................

......

A

1,1,1

11,111,111

k

Inverse matrix method

Ax=b

A-1Ax=A-1b see A-1A = I (identity matrix)

x=A-1b

Gauss-elimination

Extended matrix formalism

The size of the extended matrix is n×(n+1)

8

Schematic view of the two steps:

elimination step back substitution step

Example for the elimination process

−−→

−−

−−→

−− 105200

13500

21320

15123

42760

13500

21320

15123

42760

35140

21320

15123

)) ba

a) Subtract from line III 2 times line II

b) Subtract from line IV -3 times line II

Pivoting to avoid division failure:

if aii=0 (or very small), change lines as long as to eliminate aii ≠ 0

Practice

> A=[2,4,6;2,3,1;-1,0,5]

> b=[8;7;-2]

> inv(A)*b

> B=A #for Cramer-method

> B=A

> B(:,1)=b

> det(B)/det(A)

> B=A

> B(:,2)=b

> det(B)/det(A)

Theory

Overdetermined case

n number of lines, m number of unknowns, n>m

Anxmxmx1≈bnx1

9

Derivation of the analytical form:

Ax≈b

ATAx≈ATb where AT is the transpose of A

(ATA)-1(ATA)x=(ATA)-1ATb where (ATA)-1(ATA)=I

x=(ATA)-1ATb

Overdetermined case ≈ least square regression of lines, planes and hyper-planes

Practice

> A=resize(A,4,3)

> A(4,:)=[2,3,2]

> b=resize(b,4,1)

> b(4)=4

> inv(A'*A)*A'*b

Theory

Addition of a constant term - intercept

Extend A with a column vector of 1-s

1

1

1

A

Practice

> A=resize(A,5,4)

> A(:,4)=1

> A(5,:)=[-1,5,-1,1]

> b=resize(b,5,1)

> b(5)=0

> inv(A'*A)*A'*b

10

Numerical integration

Theory

n known xi-yi pairs (e.g. from experiment); equidistant x -h=xi+1-xi

task: numerical determination of the area

Newton-Cotes equations

trapezoidal rule – approximation of the area with trapezoids

( )∫+

++≈ +

1

)(2

)( 3

1

i

i

x

x

ii hyyh

dxxf ο local form for one trapezoid

( )∫ +++++++≈ −

b

a

nni hyyyyyh

dxxf )(2.....22

)( 2

121 ο global form for the total

interval

)( 2hο ordo means in math that the error is directly proportional to h2

Simpson 1/3 rule

( )∫+

+++≈ ++

2

)(43

)( 5

21

i

i

x

x

iii hyyyh

dxxf ο local form

( )∫ ++++++++≈ −−

b

a

nnn hyyyyyyyyh

dxxf )(42...24243

)( 4

1254321 ο global form

11

n should be odd number

Practice

> u=rand(1,20);

> for i=1:20 v(i)=i*0.1; endfor;

> u

> v

> function s=trint(x,y,n) s=0; for i=1:n-1 s=s+(x(i+1)-

x(i))/2*(y(i)+y(i+1));endfor;endfunction

> trint(v,u,20)

ans = 0.87450

> trapz(v,u)

ans = 0.87450

> function s=simpint(x,y,n) s=0; for i=1:2:n-2 s=s+(x(i+1)-

x(i))/3*(y(i)+4*y(i+1)+y(i+2));endfor;endfunction

> simpint(v,u,20)

ans = 0.73852

> quad('cos',0,pi()/2)

ans = 1.0000

> quadl('cos',0,pi()/2)

ans = 1.0000

Polynomials in Octave

> c=[2,3,4,5,1]; #coefficients (last one is the

constant)

> a=[1,2,3,4];

> conv(a,c) #convolution

> polyder(a) #derivation

> q=polyder(a,c) #derivate a convolutions

> [q,r]=polyder(a,c) #rational functions

> roots(a) #roots

> for i=1:20 u(i)=v(i)+cos(v(i)*0.3); endfor;

> p=polyfit(v,u,5) #fit of polynomials

> pint=polyint(p) #integration

12

> polyval(pint,pi()/2)-polyval(pint,0) #evaluation of

polynomials

Interpolation

Theory:

n known xi-yi pairs

task: find y for a given x

basic idea fit polynomial max Pn-1(x)

interpolation xЄ[xmin ; xmax]

extrapolation x<xmin or xmax<x (dangerous, if no e.g. physical law behind it)

Lagrange-interpolation

Cubic polynomial using the 4 next neighbouring points

4

342414

321

3

432313

4212

423212

4311

413121

4323

))()((

))()((

))()((

))()((

))()((

))()((

))()((

))()(()(

yxxxxxx

xxxxxx

yxxxxxx

xxxxxxy

xxxxxx

xxxxxxy

xxxxxx

xxxxxxxp

−−−

−−−+

−−−

−−−+

−−−

−−−+

−−−

−−−=

Practice

> interp1(v,u,1.05)

> interp1(v,u,1.05,'linear')

> interp1(v,u,1.05,'cubic')

Theory

Spline interpolation

simple polynomial interpolation is not smooth at the measured points

for equidistant x -h=xi+1-xi

13

local, e.g. cubic polynomials, altogether n-1 polynomials

gi(x)=ai(x-xi)3+ bi(x-xi)

2+ ci(x-xi)+ di

gi(xi)=yi

gi(xi+1)=yi+1

gn-1(xn)=yn

g’i(xi+1)= g’i+1(xi+1)

g’’i(xi+1)= g’’i+1(xi+1)

= set of linear equations (2 extra equations are necessary, e.g. ‘natural’ spline)

spline can be differentiated (e.g. use in mechanics dr

rdUrF

)()( −= )

in real applications:

a) calculation of all coefficients (ai, bi, ci, di)

b) use for different x-s to calculate y(x)-s

Practice

> interp1(v,u,1.05,'spline')

Smoothing

Theory

Golay-Savitzky filter

n known xi-yi pairs; equidistant x -h=xi+1-xi

Fit, e.g. cubic polynomials on 5 points (k=2 case)

Replace yi with iy~ (xi and h are not used in this form)

∑+=

−=

−=kij

kij

jiji ycF

y1~

Table of Golay-Savitzky coefficients for general points and endpoints

yi c-2 c-1 c0 c1 c2 F

-3 12 17 12 -3 35

y1 c1 c2 c3 c4 c5 F

14

yn cn cn-1 cn-2 cn-3 cn-4 F

69 4 -6 4 -1 70

y2 c1 c2 c3 c4 c5 F

yn-1 cn cn-1 cn-2 cn-3 cn-4 F

2 27 12 -8 2 35

Practice

> function [w]=golay(v,n) for i=3:n-2 w(i)=1/35*(-3*v(i-

2)+12*v(i-1)+17*v(i)+12*v(i+1)-

v(i+2));endfor;w(1)=v(1);w(2)=v(2);w(n-1)=v(n-

1);w(n)=v(n);endfunction;

> golay(v,20)

Numerical differentials

Theory

Simple forms

n known xi-yi pairs; equidistant x h=xi+1-xi

)(2

)(' 211 hh

yyxf ii

i ο+−

≈ −+

)(2

)('' 2

2

11 hh

yyyxf iii

i ο++−

≈ −+

)(12

163016)('' 4

2

2112 hh

yyyyyxf iiiii

i ο+−+−+−

≈ −−++

symmetric choice of points has better performance

Differentials with Golay-Savitzky filter

∑+=

−=

−=kij

kij

jiji ydFh

y1

'~

Table of Golay-Savitzky differential coefficients

yi d-2 d-1 d0 d1 d2 F

1 -8 0 8 -1 12

Practice

> function [w]=golayder(u,v,n) h=u(2)-u(1);for i=3:n-2

w(i)=1/12*(v(i-2)-8*v(i-1)+8*v(i+1)-v(i+2));endfor;w(1)=(v(2)-

15

v(1))/h;w(2)=(v(3)-v(1))/2/h;w(n-1)=(v(n)-v(n-

2))/2/h;w(n)=(v(n)-v(n-1))/h;endfunction;

> golayder(u,v,20)

> quit

Data storage in matrices – conventions, statistical applications

Theory

n rows - n objects m columns – m variables

=

333231

232221

131211

ddd

ddd

ddd

Dnxm

mean: n

d

d

n

i

ij

j

∑== 1

variance:

( )

1

1

2

2

−

−

=∑

=

n

dd

s

n

i

jij

j

standard deviation

( )

1

1

2

−

−

=∑

=

n

dd

s

n

i

jij

j

centering jij

n

i

ij

ijij ddn

d

dc −=−=∑

=1

scaling

( )

1

1

2

−

−

==

∑=

n

dd

d

s

dc

n

i

jij

ij

j

ij

ij

standardization (z-scoring, studentization)j

jij

ijs

ddc

−=

covariance:

( )( )

1

1

−

−−

=∑

=

n

dddd

s

n

i

kikjij

jk

correlation coefficient kj

jk

jkss

sr = range [-1;1]

around -1 close to directly proportional j and k variables with negative slope

16

around 1 close to directly proportional j and k variables with positive slope

Practice

> D=rand(10,5)

> D(:,2)=cos(D(:,1))

> D(:,3)=D(:,1)*2+rand()

> D(:,4)=exp(D(:,1))

> D(:,5)=D(:,2)+D(:,4)+rand()*0.1

> mean(D)

> median(D)

> meansq(D)

> std(D)

> var(D)

> sortrows(D,2)

> statistics(D)

> help statistics

> center(D)

> A=studentize(D)

> mean(A) #mean of standardized equals 0

> std(A) #standard dev. of standardized equals 1

> cov(D) # in a matrix

> cor(D)

> cov(A) # covariance of standardized equals correlation

> anova(D)

One-way ANOVA Table:

Source of Variation Sum of Squares df Empirical Var

*********************************************************

Between Groups 142.7943 4 35.6986

Within Groups 30.2420 45 0.6720

---------------------------------------------------------

Total 173.0363 49

Test Statistic f 53.1194

p-value 0.0000

17

> p=var_test(D(:,1),D(:,2))

> p=t_test(D(:,1),0.5)

> p=t_test(D(:,1),1)

> p=t_test(D(:,1),2,”<>”)

> p=t_test(D(:,1),0.78,”<”)

> p=t_test_2(D(:,1),D(:,2),”<>”)

Principal component analysis

Theory

to reduce the number of variables

to find relationship among variables

to cluster objects or variables

to perform principal component regression

Factoring D Dnxm=TPT

=mxaP loading matrix: coefficients of old variables in the new principal

components (super variables) a<=min(n,m)

=nxaT score matrix: object coordinates in the new variables;

PPT=I and TTT=I (orthogonal matrices)

T,P first vectors contain the majority of the total variance in the data matrix. Total variance:

sum of column variances, 2

js -s.

Outer product representation of the principal component factorization:

18

Numerical methods to determine principal components

Eigenvalue determination

Cmxn=DcTDc C covariance matrix, Dc centred data matrix

C=ZΛZ-1= ZΛZT

P=Z, where Z eigenvectors of C

T= DZ Λ, where Λ eigenvalues of C (diagonal matrix, λi-s are in the diagonal)

Singular value decomposition

Dnxm=UnxmWmxm(Vmxn)T

T=UnxmWmxm

PT=(Vmxn)T

UTU=I VTV=I (U and V are orthogonal matrices)

λi=wii2

How many principal components are necessary?

explained variance of the i-th principal component ∑

=i

i

λ

λ

a) sum up to K components as long as 95% is explained

b) scree plot

19

Graphical representation of loading and score matrices

Loading plot Score plot

for variables for objects

For example x-axis = principal component 1, y-axis = principal component 2

Practice

> A=studentize(D)

> [Ve,E]=eig(A'*A)

> [U,W,V]=svd(A)

> size(U)

> size(W)

> size(V)

> trace(E)

> for i=1:5 E(i,i)/trace(E) endfor;

> s=0;

> for i=1:5 s=s+W(i,i)**2; endfor;

> for i=1:5 W(i,i)**2/s endfor;

> UW=U*W

> plot(UW(:,1),UW(:,2),"4+")

> plot(V(:,1),V(:,2),"4+")

20

Theory

Fourier transformation

Jean-Baptiste Joseph Fourier (1768—1830) French mathematician

Transformation: change, replace

In mathematics function and variable changes:

e.g., Legendre-transform in thermodynamics

U(S,V) S→T variable replacing, but keep the whole information content of U

A(T,V)=U-TS

V→p replacement: G(T,P)=U-TS+pV

Integral-transforms: Laplace, Fourier...

Why it is important for chemists?

• mathematics, physics: solution of partial differential equations

• one function is measured or observed, its transform has expressive meaning

(spectroscopy, diffraction)

Eustach

density changes of air → frequency, amplitude

(20Hz-20kHz) low and high voice at different place in the spiral

← cilia

← liquid

21

Fourier-transformational and canonical infrared spectroscopy (FTIR and IR)

Same is measured by using different variables: FTIR-in time, IR-in frequency. (non-

mathematically one function in two representation)

22

FTIR: less optic, better resolution, better signal/noise ratio, shorter measurement time,

because the whole frequency domain is measured at once, scanning in time domain is fast.

Michelson-interferometer: Replace the magnitude of the signal to a measurable time domain

vmτ = λ/2

f = 1/τ = 2νm/λ

λ = c/ν

f = (2νm/c) ν

2νm/c ≈ 10-10

vm moving speed of mirror, τ equals the time for λ/2 length, f frequency, c speed of light, v

optical frequency

23

• mathematics is different in the transformed space (theoretical chemistry / physical

chemistry calculations, medical diagnostics, image processing, image compression,

data evaluation, noise filtering)

Sum of electrostatic interactions in periodic systems:

Coulomb interaction decays 1/r, infinite amount of periodic images should be calculated.

Ewald: transform the problem to the reciprocal space (1/length unit) better convergence of the

sum

Medical diagnostics: 3D methods

The Radon transformed representation of the density of the human body is measured in layers

Image compression, noise filtering: After transformation we can select between noise and

signal, satellite images

Convolution: enhance of contrast in images

Deconvolution: e.g., kinetics

Mathematical background of Fourier series

Periodic f(t) function as sum of sines and cosines, if:

T periodic

T finite number min., max, and discontinuity

and ∞<∫T

dttf0

)(

∑∑∞

=

∞

=

++=11

0 2sin

2cos

2)(

n

n

n

nT

ntb

T

nta

atf

ππ

∫+

=

Tt

t

n dtT

nttf

T

a0

0

2cos)(

1

2

π ∫

+

=

Tt

t

n dtT

nttf

T

b0

0

2sin)(

1

2

π

∫+

=

Tt

t

dttfT

a0

0

)(1

0 usually small n is enough

24

Periodic function in exponential form:

)sin()cos( titeit +=

∑∞

−∞=

=n

ti

nectfω)(

t

nπω

2=

t

n=ν

2

nnnn

ibacc

−== n≠0

Fourier-transformation – continuous, aperiodic function

Fourier-transform: inverse Fourier-transform:

∫∞

∞−

−= dtetfF ti πνν 2)()( ∫∞

∞−

= νν πνdeFtf

ti2)()(

Attributes:

)()()()( 22112211 νν FaFatfatfa +⇔+ linear

⇔

aF

aatf

ν1)( time scaling

( )νbFb

tf

b⇔)(

1 frequency scaling

( ) 02

0)(ti

eFttfνπν⇔− time shift

( )0

2 0)( νννπ −⇔−Fetf

ti frequency shift

25

Convolution: )()()()()( ννττ HGdtthtghg ⇔−≡Ψ≡∗ ∫∞

∞−

1) g(t) (on scheme v(t)) input impulse causes h(t) response

2) many impulses – sum of many h(t)-s

3) if g(t) is a function, then response is a convolution

The Fourier transform of the convolution of two functions is the product of the individual

Fourier transforms

Image processing: f(t) is convolved with other functions, e.g. contrast

(Femto)kinetics: Convolved function is measured (kinetics + instrument), but if we know the

instrument function, we can get the kinetic function (deconvolution)

Correlation )()()()()(),( ννττ ∗

∞

∞−

⇔+≡Ω≡ ∫ HGdtthtghgcorr

autocorrelation, if )()( thtg =

Velocity autocorrelation function: memory of the particle velocity in a liquid

0 1 2 3

vacf(t)

t/ps

sebesség autokorrelációs függvény

0 50 100 150 200 250

F(v)

v/(1/cm)

spektrális sűrűség

Many measurements provide information on correlations, e.g., IR on a liquid =

autocorrelation of the total dipole moment of the system

26

Some transform

To notice:

rectangular signal:

transform ≈ sin(x)/x shape

broader signal → narrower transformed signal

Gaussian-curve:

transform is Gaussian, as well

broader Gaussian signal → narrower transformed signal

cosines:

transform: Dirac-delta pair

comb function:

transform: comb, as well

Discrete Fourier-transform

N samples ∆t frequency

notations: tN∆

=∆1

ν N

nkt

ππν

22 =

kFkFF =∆= )()( νν nftnftf =∆= )()(

Transforming equation:

∑−

=

−

≡1

0

21 N

n

N

nki

nk efN

F

π

Inverse:

∑−

=

≡1

0

2N

k

N

nki

kn eFf

π

all points are necessary for one point in the other space

28

Sampling theory: Nyquist-frequency:

tc

∆=

2

1ν

c

tν2

1=∆ the frequency depends on the frequencies in the system

advantage:

if F(ν)=0 for all ν >νc-re, then all F(ν) can be determined for all ν < νc

disadvantage:

if F(ν)≠0 for ν >νc-re, it biases F(ν) in νЄ[-νc,νc]

broadening effect:

continuous cosine ↔ Dirac-delta pair

discrete cosines ↔ new lines close to the original ones

determination of νc:

- trial and error refining of the sampling frequency

- filtering with low transmission filters

- windowing: multiplication before the transformation with a window function

Even and odd functions: cosines or sine solely

Structure of liquids: g(r) (pair-correlation function) can be calculated (relative density at r

distance from a particle), S(q) (structure function) measured in diffraction experiments, not

expressive. S(q) and g(r) related with sine transform.

∫ −+= drrgqr

qrrqS )1)((

)sin(41)( 2πρ

0

1

2

3

4

0 5 10 15 20

g(r)

r/Angstrom

párkorrelációs függvény

-1

0

1

2

3

4

0 5 10 15

S(q)

q / (1/Angstrom)

szerkezeti függvény

29

Fast Fourier-transformation (FFT)

Fk calculation scales N2

Danielson-Lánczos theory:

Fk can be calculated as special sum of (Fkeven) and (Fk

odd)

The factorization is recursive down to 1-1-elements. It can be used to build an algorithm,

where all multiplication or summation is performed only one time:

final scaling Nlog2N

N must be here power of 2 (…128, 256, 512, 1024, 2048 ,5096…). Fill with zeroes the rest up

to power number of 2, if we have less data.

Fast cosine transform, e.g., JPEG image compression

References:

Further reading:

S. Dowdy and S. Wearden: Statistics for research, Wiley 1982, New York

W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery: Numerical recipes in FORTRAN, Cabridge Univ.

Press. 1982, Cambridge

Valkó P., Vajda S.: Műszaki-tudományos feladatok megoldása személyi számítógéppel, Műszaki könyvkiadó,

1987, Budapest.

Source and original idea of some exercises:

Deutsch T., Vajda S., Valkó P., Riedel M: Kémiai számítástechnika füzetek, ELTE TTK Kémiai Kibernetikai

Laboratórium, 1976-1987