化學數學（一）
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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
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