The Mathematics for Chemists (I)

(Fall Term, 2004)(Fall Term, 2005)(Fall Term, 2006)

Department of ChemistryNational Sun Yat-sen University

化學數學（一）

Review: General

• Vector• Determinant • Matrix• 1st Order LODE• 2nd Order LODE• PDE• Orthogonal

Expansion

Dot, cross, orthonormal basis, gradient, convergence, curl (rot)

Order-lowering, row and column operations,Triangular form, linear inhomogeneous equations

Linear homogeneous equationsTranspose, cofactor, adjoint, inverseSymmetric,orthogonal,Hermitian,unitaryTrace, eigenvalue, eigenvector, diagonalization,Canonical form of quadratic forms

Constant coefficient inhomogeneousVariable coefficient inhomogeneous

Constant coefficient + special inhomogeneousVariable coefficient + homogeneous:(Associated) Legendre, Hermite,(Associated) Laguerr

Separation of variables: PDE ODE

General expansion, Fourier series, FT

Base Vectors

)0,0,1(i )0,1,0(j )1,0,0(k

),,(

),0,0()0,,0()0,0,(

)1,0,0()0,1,0()0,0,1(

zyx

zyx

zyx

zyx

aaa

aaa

aaa

aaa

kjia

axi

azk

ayj

Orthogonal basis:

Nonorthogonal basis:

a b c v a b c

a

bc

x1

x2

x3

1 1 2 2 3 3c c c v x x x

Vector Spaces

,1 ,2 ,3 , ,1 ,2 ,3 ,

,1 ,2 ,3 , ,1 ,2 ,3 ,

, ,1

( , , , , )

( , , , , )i i i i i n i i i i n

j j j j j n j j j j n

n

i j ij i k j k ijk

x x x x x x x x

x x x x x x x x

x x

1 2 3 n

1 2 3 n

x e e e e

x e e e e

x x

)0,,0,0,1( 1e)0,,0,1,0( 2e

)1,,0,0,0( ne

n321 eeeea nn aaaaaaaa 321321 ),,,,(

…

j i

j iijji if 0

if 1ee

n321 eeeeb nn bbbbbbbb 321321 ),,,,(

Orthonormal basis

General orthogonal basis:

1 1 2 2

,1 ,2 ,3 ,1 1

...

( )

n n

n n

i i i i i i i ni i

v v v

v v x x x x

1 2 3 n

v x x x

x e e e e

Vector Algebra

cosab ba zzyyxx bababa ba

baυ sinabυbbaυ

abaυ

c)a(bc)b(acba )(zyx

zyx

bbb

aaa

kji

ba

c)a(bc)bacba

b)c(ac)bacba

()(

()(

Vector Calculus

kjizyx

kji

kji

z

f

y

f

x

f

fzyx

ffgrad

zyxdiv zyx

υυ

2 2 22

2 2 2

f f ff f

x y z

x y z

curl rotx y z

i j k

υ υ υ 0)( f

0)( A

Major Theorems in Integration

( )S L

d d B S B l

( )V S

dV d A A S

LS

V

S

adF(x)

dxb

dx=F(a)-F(b) b a

( )

( ) ( ) ( )

y yx xz z

S

x y z x y z

L L

B BB BB Bdydz dxdz dxdy

y z z x x y

B B B dx dy dz B dx B dy B dz

i j k i j k

( )

( )

yx z

V

x y z

S

AA Adxdydz

x y z

A dydz A dxdz A dxdy

The Solutions of Linear Equations

nnnnnnn

nn

nn

nn

bxaxaxaxa

bxaxaxaxa

bxaxaxaxa

bxaxaxaxa

332211

33333232131

22323222121

11313212111

nnnnn

n

n

n

aaaa

aaaa

aaaa

aaaa

D

321

3333231

2232221

1131211

nnnnn

n

n

n

aaba

aaba

aaba

aaba

DD

Dx

31

333331

223221

113111

22

1

D

Dx 1

1

D

Dx 2

2

D

Dx n

n

…

Cramer’s Rule:

Properties of Determinants

321

321

321

333

222

111

ccc

bbb

aaa

cba

cba

cba

1. Transpose:

321

321

321

333

222

111

ccc

bbb

aaa

cba

cba

cba

2. Multiplication by a scalar:

0

000

333

222 cba

cba

333

222

111

333

222

111

3333

2222

1111

cbd

cbd

cbd

cba

cba

cba

cbda

cbda

cbda

333

111

222

333

222

111

cba

cba

cba

cba

cba

cba

333

222

111

333

222

111

bca

bca

bca

cba

cba

cba

3. Zero row or column:

0

0

0

0

33

22

31

ca

ca

aa 4. Addition rule:

5. Interchange of rows/columns: antisymmetry:

0

333

111

111

cba

cba

cba6. Two equal rows/columns:

0

333

111

111

333

222

222

cba

cba

cba

cba

cba

cba

7. Proportional rows/columns:

1 1 1 1 1 1 1

2 2 2 2 2 2 2

3 3 3 3 3 3 3

a b b c a b c

a b b c a b c

a b b c a b c

8. Additions of rows/columns:

dx

dc

dx

dc

dx

dcbbb

aaa

cccdx

db

dx

db

dx

dbaaa

ccc

bbbdx

da

dx

da

dx

da

ccc

bbb

aaa

dx

d

321

321

321

321

321

321

321

321

321

321

321

321

9. Differentiation:

Reduction to Triangular Form

nn

nn

n

n

n

n

aaaa

a

aa

aaa

aaaa

aaaaa

332211444

33433

2242322

114131211

0000

000

00

0

Square Matrix: Transpose and Inverse

TTTTT AB...CXABC...X

IABBA detA

AA 1

ˆ

( )C T^

A A

111 AB...CXABC...X 11

AA detdet TTij jia a

symmetric matrixAAT If

Orthogonal Matrices: General Cases

)...,,( 321

321

3333231

2232221

1131211

n

nnnnn

n

n

n

aaaa

aaaa

aaaa

aaaa

aaaaA

nn

n

n

n

n

nnn a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

.

.

.

,...,

.

.

.

,

.

.

.

,

.

.

.3

2

1

3

33

23

13

3

2

32

22

12

2

1

31

21

11

1 aaaa

ijji

ji ji

ji

δaa

1

0aa

)(

)(

Again, Kronecker symbol.

T1 AA

Orthogonal Matrices: General Cases

nnnnnn

n

n

n

aaaa

aaaa

aaaa

aaaa

a

a

a

a

A

...3

2

1

321

3333231

2232221

1131211

ijji

ji ji

ji

δaa

1

0aa

)(

)(

nnnnnn

n

n

n

aaaa

aaaa

aaaa

aaaa

...

,...,

,...

,...

,...

321

33332313

22322212

11312111

a

a

a

a

Matrix Multiplication

kj

n

kiknjinjijijiij bababababac

1

332211

11 12 13 111 12 1 1

21 22 23 221 22 2 2

31 32 3 3

1 2 3

1 2

1 2 3

11 12 1 1

21 22 2 2

nj p

nj p

j p

i i i in

n n nj np

m m m mn

j p

j p

i

a a a ab b b b

a a a ab b b b

b b b ba a a a

b b b ba a a a

c c c c

c c c c

c

C AB

1 2

1 2

i ip

m m mj

j

mp

ic c

c c c c

c

Other Useful Matrices in Chemistry

1. Hermitian matrices:

2. Unitary matrices:

AAA T *)(

* +( ) 1 TA A A A A = I

01

10

2

1xS

0

0

2

1

i

iyS

10

01

2

1zS

A. For real matrices, Hermitian means symmetric.B. All physical observables are Hermitian matrices.

zyx icibia eee SSS ,,For real matrices, unitary means orthogonal.

(Hermitian matrix is the complex extension of symmetric matrix)

(Unitary matrix is the complex extension of orthogonal matrix)

Matrix Algebra

ABCCABBCA

ACABCBA

BAAB

The associative law:

The distributive law:

The (non-)commutative law:

A,B = AB - BACommutator:

The Determinant and Trace of a (Square ) Matrix Product

( ) ( ) ( ) ( ) det AB det A det B det BA

n

kkiikii bac

1

BAtrABtrC tr1 1

n

i

n

kkiik ba

ik

n

ikikk abd

1

ik

n

k

n

ikiab

1 1

trBAtrD

tr trtrAB B A

The Matrix Eigenvalue Problem and Secular Equations

nnnnnnn

nn

nn

nn

xxaxaxaxa

xxaxaxaxa

xxaxaxaxa

xxaxaxaxa

332211

33333232131

22323222121

11313212111

0)(

0)(

0)(

0)(

332211

3333232131

2323222121

1313212111

nnnnnn

nn

nn

nn

xaxaxaxa

xaxaxaxa

xaxaxaxa

xaxaxaxa

0

)(

)(

)(

)(

321

3333231

2232221

1131211

nnnnn

n

n

n

aaaa

aaaa

aaaa

aaaa

The condition for the existence of nontrivial solution:

(secular determinant)

Ax x

1,2,3,...k nkkk xAx

Matrix Diagonalization , k=1,2,3, ,nkk kAx x

11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

n

n

n

n n n nn

x x x x

x x x x

x x x x

x x x x

1 2 3 nX x x x x

1 2 3

=

n

1 2 3 n

1 2 3 n

AX Ax Ax Ax Ax

x x x x

XD

1

2

3

0 0 0

0 00= 00 0

0 0 0 n

D

- 1D=X AX

Diagonalization of a square matrix is essentially the same as finding the eigenvaluesand their respective eigenvectors.

Application: Quadratic Forms

2 22 2 2 2 2

1 1 2 2 3 2 3, 3x 4 2 , x 2 3 3x y xy y x x x x x x

2 2 2 22( , )Q x y ax xy cy ax xy yb xb cyb

( )

a xQ x

b

by

c yTx x Ax

2 2

2 2

3 13 2

1 1

3 1 3

1 1

3(3 ) ( ) 3 2

xQ x xy y x y

y

x x y

y x y

x yQ x y x x y y x y x xy y

x y

General Quadratic Forms

1

1 2 3

211 1 12 2 13 1 3 1 1

221 2 1 22 2 23 2 3 2 2

( , , , , )

=a +a

+a x x +a

+

+

n n

n ij i ji j

n n

n n

Q x x x x a x x

x x x a x x a x x

x a x x a x x

2n1 1 n2 2 3 3a x +an n n n nn nx x x a x x a x

1 11 12 13 1

2 21 22 23 2

3 31 32 33 3

1 2 3

, =

n

n

n

n n n n nn

Q

x a a a a

x a a a a

x a a a a

x a a a a

T(x) =x Ax

x A

Solving First Order ODE

Separable Equations:

First-order linear equations:

),( yxFdx

dy

)(

)(

yg

xf

dx

dy

cdxxfdyyg )()(

)()( xfdx

dyyg

cdxxfdxdx

dyyg )()(

+ initial conditions

)()( xryxpdx

dy

Reduction to Separable Form: Homogeneous Equations

For n=0: ),(),( yxFyxF 22 2

( , ) 1x y y y y

F x y fx xxy x

),(),( yxFx

yf

x

yfyxF

x

yfyxF

dx

dy),( ( )dy d x d

x fdx dx dx

x

dx

f

d

cxf

d

ln

),,(),,( yxfyxf n

Example:

y

x

First-Order Linear Equations:The inhomogeneous Case

( )( )

p x dxF ex

)()( xpdxxp

dx

d

)()()(

xpxFdx

xdF

)()()()()( xrxFyxpxFdx

dyxF

yxFdx

dy

dx

xdF

dx

dyxFyxpxF

dx

dyxF

)()()()()(

xrxFyxFdx

d r xF x y d cF x x

( ( ))p xdy

xy r x

d

Three Cases

2

0 two real roots

4 0 real double root

0 two complex roots

a b

1 2

1 2

1 2

and are linearly independent

, a new solution is needed!

and are linearly independent

y y

y y

y y

xx ececxycxycy 21212211 )()(

21

1)( axx eexy

22 1( ) ( ) axxy x x xy e

2 21 2

21 2

( )

( )

ax ax

ax

y x c e c xe

c c x e

xixiax ececexycxycxy 212

2211 )()()(

xdxdexy ax sincos)( 212

211 ccd 2 1 2( )d i c c

2 / 4i a b

2

20

d y dya by

dx dx 2 0a b

term in

( 0,1,2,...)

cos (or sin )

cos (or si

(

n )

)ax

n

ax ax

ce

cx n

c x c x

ce x ce x

r x

20 1 2

Choice of

...

cos sin

( cos sin )

ax

nn

ax ax

p

ke

a a x a x a x

k x l x

e k x le x

y

The determination of the coefficient(s) in yp is obtained by substituting it back to theinhomogeneous equation. However, if yp is already in yh then the general solution should be: ( ) () () )( phy cx xx yy x

where the choice of c(x): If the characteristic equation of the corresponding homogeneous equation has two(real or complex) roots, then c(x) =x, or else, c(x)=x2 .

If r(x) is the sum of terms given in above table, the total yp(x) is the sum of respective yp of all terms. [This leads to a method of series expansion for general r(x) ]

2

2( )

d y dya by

dx dxr x

2

2 2 2

2

(1- ) 2 ( 1) 0

(1- ) 2

Legendre:

Associated Legendre:

Hermite:

Laguerre:

Associated Laguerre:

Bessel:

[ ( 1) /(1 )] 0

2 2 0

(1 ) 0

( 1 ) ( ) 0

x x l l

x x l l m x

x n

x x n

x m x

y y y

y y

n m

y

y y y

y y y

y y y

yx x

2 2( ) 0x yny

Second-Order ODE: Special Cases of Variable Coefficients

It’s hard or impossible to obtain the solution of a general second-order ODE

2

2( ) ( ) ( ) ( )

d y do x p x q x

yy

dx dxr x

Inhomogeneous, linear, variable coefficients:

The Legendre Equation2 ( 1)(1 ) 2 0x y xy yl l

2 4

1 3 5 (2 1)( ) ( )

!

( 1) ( 1)( 2)( 3)

2(2 1) 2 4(2 1)(2 3)

l

l l l

ly x P x

l

l l l l l lx x x

l l l

0 ( ) 1P x 1( )P x x

1 1( 1) ( ) (2 1) ( ) ( ) 0l l ll P x l xP x lP x

1

1

0 if '

( ) ( ) (2 1) if '

2

l l

l l

P x P x dx ll l

The Associated Legendre Functions2

2

2

( 1(1 ) 2 0(1 )

)m

x y xy yx

l l

Under conditions: ,3,2,1,0l lm ,,2,1,0

22( ) (1 ) ( )m

m ml lm

dP x x P x

dx

The particular solutions are associated Legendre functions:

1

1( ) ( ) 0 ( ')m m

l lP x P x dx l l

21

1

( )!2( )

2 1 ( )!m

l

l mP x dx

l l m

The Hermite Equation

022 nyyxy

42 )2(!2

)3)(2)(1()2(

!1

)1(2)( nnn

n xnnnn

xnn

xxH

1)(0 xH xxH 2)(1

0)(2)(2)( 11 xnHxxHxH nnn

)()( 22

xHexy nx

n

2

,( ) ( ) ( ) ( ) 2 !x nm n m n m ny x y x dx e H x H x dx n

The Laguerre Equation

01 nyyxyx

!)1(

!2

)1(

!1)1()( 2

221

2

nxnn

xn

xxL nnnnnn

1)(0 xL xxL 1)(12

2 42)( xxxL

323 9186)( xxxxL

0)()()21()( 12

1 xLnxLxnxL nnn

n: real number

Laguerre polynomials:

Recurrence relation:

Associated Laguerre Functions

0)(1 ymnyxmyx

)()( xLdx

dxL nm

mmn

)()( 122, xLxexf l

lnlx

ln

04

1)1(2

f

x

ll

x

nf

xf

2 2 2 1 2 1 2, ,0 0

3

,

( ) ( ) ( ) ( )

2 ( 1)!

( 1)!

x l l ln l n l n l n l

n n

f x f x x dx e x L x L x x dx

n n

n l

The associated Laguerre equation

It’s solution is associated Laguerre polynomials:

they arise in the radial part of the wavefunctions of hydrogen atom in the form ofassociated Laguerre functions:

which satisfy:

and are orthogonal with respect to the weight function x2 in the interval [0,∞]:

Hydrogen-Like Atoms )1(V ][

2

2

0

2

2

22

24eff2

2

llERRV

rrZe

effrR

rrR

2

20

0

40

2

2/,,.

;

)()()(

emaZr

nln

lnlnln

e

a

eLNrR

Normalization factor

Laguerre polynomails

,...3,2,1;222

02

42

32, nE

n

eZln

),()(),()(),,( ,2

,,,,,,

lll mln

lnl

lnmllnmln YeLn

NYrRr

2

2 2

2 1 1sinsin

sin

)]1([

221

2

22 llconstY

Y

Separation of Variables: Turn PDE into ODE

0

y

f

x

f )()(),( yYxXyxf

dx

dXY

x

XY

x

f

dy

dYX

y

f

0)(

)()(

)( dy

ydYxX

dx

xdXyY

1 ( ) 1 ( )0

( ) ( )

dX x dY y

X x dx Y y dy

Cdx

dX

X

1 Cdy

dY

Y

1

CXdx

dX CY

dy

dY

CxAexX )( CyBeyY )( ( )

( , ) ( ) ( )Cx Cy C x y

f x y X x Y y

Ae Be De

1 ( ) 1 ( )

( ) ( )

dX x dY y

X x dx Y y dy

Separation of Variables: Turn PDE into ODE

yYxXy,x

Eyxm2 2

2

2

22

XEdx

Xd

m2 x2

22

2 2

22 y

d YE Y

m dy

yx EEE

)()()()()()(

2 2

2

2

22

yYxEXy

yYxX

x

yYxX

m

)()()()()()(

2 2

2

2

22

yYxEXy

yYxX

x

yYxX

m

22

2

2

)(1

2

2

)(1 )()(

mE

yYxX y

yY

x

xX

A 2D problem reduced to two 1D problems!

Orthonormal Expansion

)(1

)( xgg

xg nn

n

*( ) ( )b

n nac g x f x dx

*,( ) ( )

b

m n m nag x g x dx

0

)()(n

nn xgcxf

Fourier Series

0

1

( ) ( cos sin )2 n n

n

a n x n xf x a b

l l

1( )cos

l

n l

n xa f x dx

l l

1( )sin

l

n l

n xb f x dx

l l

f(x)

l-l 0x

+A

-A

Example: Fourier Series

2( )

22

( )2 2

2( )

2

A ll x for l x

lAx l l

f x for xlA l

l x for x ll

Since ( ) is odd,

0 if ( ) is odd( ) ( )

0 if ( ) is even

l

l

f x

g xf x g x dx

g x

1

( ) sinnn

n xf x b

l

0

2( )sin

l

n

n xb f x dx

l l

2

0 2

2

2 0 2

2( )sin ( )sin

4sin (1 )sin

l l

n l

l l

l

n x n xb f x dx f x dx

l l l

A n x n xx dx x dx

l l l

2 2

2 2

0

81,5,9,13,

83,7,11,15,

n

if n even

Ab if n

nA

if nn

2 2 2 2

8 1 1 3 1 5( ) sin sin sin

1 3 5

A x x xf x

l l l

f(x)

l-l 0x

+A

-A

Fourier Transform Pairs

1( ) ( )

2ixyf x g y e dy

1( ) ( )

2ixyg y f x e dx

( )f x dx

if exists.

Example( ) , 0 0axf x e for x and a

0

( )

0

1 1( ) ( )

2 21

2

ixy ax ixy

a iy x

g y f x e dx e e dx

e dx

( )

0

1 1 1( )

2 2

a iy xeg y

a iy a iy

2 2

1 1( )

2 2

a iy a iyg y

a iy a iy a y

2 2

1Re ( )

2

ag y

a y

y

a/π

0

FT