i i V (1.16) P= ε χo eE (1.17) D= εE (1.18

Chapter 1

N . .

s = σ ρ

(1.1.a)

Φ σ . n .

s =

(1.1.b)

p h

= cν

p mv= (1.2) E h= ν E mv

k = ²

2 (1.3)

( )

[

_{1 2 1}

]

2 2 1

1 E ε E hν ν hν

ε + = + − << (1.3)

ε₁+E₁=E^*₂ (1.4) hν₁+E₁^*=2hν₁+E₂ (1.5)

ε₁+E₁^*=ε₂+E₂ (ε₂ >ε₁) (1.6a) ε₁+E₁ =ε₂+E₂^* (ε₂ <ε₁) (1.6b) ε_1a+ε_1b+E₁=2ε_1b+E₂^* (ε_1a >ε_1b) (1.7)

E₁^*=E₂+ ε₂ (1.8)

ε_1a+E₁=ε_2b+E^*₂ (1.9) h I m v

I E E

a e e

ν₁ ² _{2 1}

= + 2 = ^* − (1.10) U Q_i

i i

=_4πε¹

∑

_{R r}− ^(1.11) p=

∑

^{Qi i}r

i

(1.12) F=Q E (1.13) p p= _o+αE+1βE +

2

2 ... (1.14)

T p E= × (1.15)

P

p

=

∑

ⁱ

i

V (1.16)

P= ε χ_{o e}E (1.17)

D= εE (1.18)

(

_e

)

o

o + =ε 1+χ

ε

= E P

D (1.19)

ε ε

ε χ

r ≡ o = +1 e (1.20)

P M

M ≡ r − + ε

ε ρ

1

2 (1.21)



 



α+

= ε ρ + ε

−

≡ ε

kT 3 3

M N 2 P 1

2

o A r

r M

p (1.22) F=Qv B× (1.23)

T m B= × (1.24)

e

m_es= −m^s (1.24)

s

Bs mˆ

ˆ=−2h

µ (1.25)

e

B 2m

eh

=

µ (1.26)

m l 2 m

e

e

−

= (1.27)

m

l ˆ

ˆ=−h

µ_B (1.28)

M Lˆ ˆ

gµ_B =−h (1.29)

I N

N ˆ ˆ

g µ I=hM (1.30)

p

N 2m

eh

=

µ (1.31)

∆m e r m B

e

= ^{2 2}

4 (1.32) ω µ

γ

=g B Bh = eB (1.33)

ω=g_Nµ^B B=γ_NB

h (1.34)

^M=

∫

ψ^*i∆^mψ^jdτ ^(1.35)

∑

=

i i

V

M m (1.36)

M= µ χ_{o m}H (1.37)

B= µH (1.38)

µ

_r

= + 1 χ

_m (1.39) χ_m A

T B

= + (1.40) ω

≡ ν

=

−

≡

∆E E_i E_j h h (1.41)

( ) ( )

ij ² t

0 2 ij j

i 1 K t i t dt

a

p

ω

=

∫

← exp

h (1.42)

τ ψ ψ

=

∫

^{K d}

K_ij ^*_i ˆ _j (1.43)

∆pE

=

V (1.44)

τ ψ

∆ ψ

=

∫

^d

K_ij E ^*_i p _j (1.45) τ

ψ

∆ ψ

=

∫

^d

P ^*_i p _j (1.46)



 



− ν

=



 



 −

−

= kT

h kT

E E N

N i j

j

i exp exp (1.47) 1

t a

r+ + = (1.48) δ δE. t≥ hπ

2 (1.49)

δν≥ 1πτ

2 (1.50)

c ν v ν ν

∆ν= − ₀ = _o (1.51) T I

I_o

≡ (1.52)

( )

T I lg

lg I

A ^o =−



 



≡  (1.53)

( )

r I lg

lg I R

r o=−

 



≡  (1.54)

c

~≡ν

ν (1.55)

Chapter 2

ψ

=

ψ E

Hˆ (2.1)

(

ϕ θ

)

=

( ) ( )

ϕ θ

ψ r, , R r Y , (2.2) V

T

Hˆ =ˆ +ˆ (2.3)

V e

o r

= − 1 4

2

πε (2.3)

( )

p p 2 r

e 2

m T T 2

m T

Tˆ 2 ˆ ˆ ˆ

, h

h + −

−

= _ϕθ (2.4)

( )



( )

−

( ) [ ( )

ϕθ

]

=

( ) ( )

ϕ θ



 



− +

θ

ϕ, ˆ ˆ R r Tˆ_ϕ,θ Y , ER r Y , m

r 2 R V m T

Y 2

e 2 r

e

2 h

h (2.5)

( ) ( ) ( )

T

[

Y

( ) ]

E

Y 1 m r 2 R V m T

2 r R

1

e 2 r

e

2 ϕ θ −

θ

= ϕ



 



− + ˆ_ϕθ ,

, ˆ

ˆ h ,

h (2.6)

E_n e m^e hc

o

= −32πε⁴ h_{2 2}n = − RH₂

n (2.7)

( )



( )

ϕ θ



 



= 

ψ , , _l^m ,

o l n m l,

n, Y

a n

r R 2 r N 1 m l

n (2.8)

a_o m e^o pm

e

=4πε₂h² =52 9. (2.9) 1

dτ= ψ

∫

ψ^{* (2.10)}

( )

² _l^m

m l

2 1

ˆl Y =l l+ h Y (2.11)

( )

l 1_h ^*_h

l l

l = + = (2.12)

m l m

l m

ˆY Y

l_z = h (2.13) mh

z =

l (2.14)

l*

m=−µB (2.15)

Bm

mz =−µ (2.16)

( ) (

σ = +

) ( )

ϕσ ϕ ss 1h ˆ²

s (2.17)

(

s 1

)

h s^*h s + =

=

s (2.18)

( )

σ m

( )

σ

sˆ_zϕ = _shϕ (2.19) h

ms z =

s (2.20)

s*

2 _B

ms =− µ (2.21)

s B z

ms_, =−2µ m (2.22)

j l s= + (2.23)

( )

j 1_h j^*_h j + =

=

j ( j=l+s) (2.24) h

j

jz =m

(

^−j≤mj ^≤+j

)

(2.25)

( )

ψ

=

ψ ²

2 I I+1

ˆ h

I (2.26)

(

I 1

)

_h I^*_h I + =

=

I for hydrogen 2

I=1 (2.27) h

I

Iz =M

(

-I≤M_I ≤+I

)

(2.28) I*

N p

I g

M = µ (2.29)

I N p z

I g

M _, = µ M (2.30) ν

=

−

=

∆E E_i E_j h (2.31) 0

jd

i∆ ψ τ≠

ψ

=

∫

^p

P ^* (2.32)

∆p= −e∆r (2.33)

∆p_x = −e r∆ sin cosθ ϕ (2.34)

∆p_y = −e r∆ sin sinθ ϕ (2.35) θ

∆

−

=

∆p_z e rcos (2.36) dr

d d r

dτ= ²sinθ ϕ θ (2.37)

(

r

) (

r

)

r r d d dr e

P _j ^{2 2}

0 0 2

0 i

x =−

∫ ∫ ∫

^{∞ π π}ψ^{* ,}ϕ^,θψ ^,ϕ^,θ ∆ ^sin θ^cosϕ ϕ θ ^(2.38)

(

r

) (

r

)

r r d d dr e

P _j ^{2 2}

0 0 2

0 i

y =−

∫ ∫ ∫

^{∞ π π}ψ^{* ,}ϕ^,θψ ^,ϕ^,θ ∆ ^sin θ^sinϕ ϕ θ ^(2.39)

(

r

) (

r

)

r r d d dr e

P _j ²

0 0 2

0 i

x =−

∫ ∫ ∫

^{∞ π π}ψ^{* ,}ϕ^,θψ ^,ϕ^,θ ∆ ^sinθ^cosθ ϕ θ ^(2.40)

j 2 i

i 2 j

H n n

n 1 n R 1 hc

E  >









 −

∆ =

=

ν~ (2.41)

lim 2

n

~ R

j H j

it =T =

ν (2.42)

E z hc

H hc z

n R

n

R

= − ² ₂ = − n₂ (2.43)

( ) ( )













− −

= −

ν ₂

i i 2 j j

z n a

1 a

n R 1

~ (2.44)

∑ ∑ ∑∑

>

+

−

∇

−

=

i i i j i ij

2

i 2 2

e 2

r e r

ze m

Hˆ 2h (2.45)

5 ...

4 3 4 3 3 2 2

1s E s E p E s E p E s E d E p E s

E < < < < < < < < (2.46)

∑

=

i

li

L (2.47)

(

L 1

)

_h L^*_h

L + =

=

L (2.48)

∑

=

i

si

S 2.49)

(

S 1

)

_h S^*_h S + =

=

S (2.50)

S L

Jˆ = ˆ + ˆ (2.51)

(

J 1

)

J J=L+S,L+S-1,...,L-S J + h= ^*h

=

J (2.52)

S L J J

J

Jz =M h −J≤M ≤+J M =M +M (2.53)

( )

(

L S,veryrarely

)

1 L 2

usually S,

L 1

S 2

<

+

= ι

≥ +

=

ι (2.54)

2 1 2

1 2

1 j ,j j 1,..., j j j

J ˆ :

ˆ =

∑

^{for two electrons} = + + − −

i

ji

J (2.55)

( )

^{J g M B}

2A E 1

E= _o + ^{* 2} + µ_{B J} (2.56)

( ) ( ) ( ) ( )

^{* 2}

* 2

* 2

* 2

J 2

L S

1 J

g = + + − (2.57)

(

M 2M

)

B g

M M A E

E= _o + _{L S} + _L + _S (2.58)

E Eo gN N IB= − µ M (2.59)

( )

_



 



 



 



 −

+ +

= _o ^{2 2}_J J ²

3 M 1 b 2 a 2E E 1

E ^* (2.60)

A e− ⁻ =A⁺ (2.61) A e A

A e A

+ =

+ =

− −

− − 2− (2.62)

( )

O A

( )

O

I ⁻ =− (2.63)

A e+ ⁻ =A⁺ +2e⁻ (2.64) M e+ ⁻ =M⁺+2e⁻ (2.65)

ABC e+ ⁻ =AB⁻ +C (2.66)

A h+ ν=A⁺+e⁻ (2.67)

A^*+ = +B A B⁺+e⁻ (2.68)

A^*+BC AC= ⁺+ +B e⁻ (2.69)

A^*+ =B A⁺+ +B e⁻ (2.70) A^*+ =B A⁺+B⁻ (2.71) A^*+ =B AB⁺+e⁻ (2.72)

X⁺+M X M= + ⁺ (2.73)

X⁺+M XM= ⁺ (2.74)

XH⁺+M X MH= + ⁺ (2.75)

F=Q (2.76) E

F=Qv B× (2.77)

Chapter 3 X

E X X

E× = × = (3.1) E

Y

X× = (3.2) Y X= ⁻¹ and X Y= ⁻¹ (3.3)

(

A×B

)

×C=A×

(

B×C

)

(3.4)

1

1 .ie. X Z Y Z

Z X Z

Y= × × ⁻ = × × ⁻ (3.5)

(3.6)

(3.7) 1

n 2 1 n p

2 p 2

j 1  = −



 

 π +

±

=

χ cos , ,..., (3.8)

∑∑

∑∑

∑∑

∑

α= β>α αβ β α

= α= α

α

= >

=

+

− +

∇

−

= ^N

1

N 2

n

1 i

N

1 i

n 2

1 i

n

i

j ij

n 2

1 i

2 e

2

r e Z Z r

e Z r

e m

Hˆ 2h _(3.9)

τ ψ ψ

=

∫

^d

S_{12 1}^* ₂ (3.10)

(

I A

)

2

X=1 + (3.11)

species molecular orbital Σ_g⁺

( ) (

A B

)

g 1s 1s

S 1 2

1 +

= +

σ _(3.12)

Σ_u⁺

( ) (

_{A B}

)

u 1s 1s

S 1 2

1 −

= −

σ^* _(3.13)

( ) ( ) ( ) ( )

[

x y z

]

1 2s 2p 2p 2p

2

1 χ +χ +χ +χ

=

ψ (3.14)

( ) ( ) ( ) ^{( )}

[

2s 2px 2py 2pz

]

2

2=1 χ +χ −χ −χ

ψ (3.15)

( ) ( ) ( ) ( )

[

2s 2px 2py 2pz

]

2

3=1 χ −χ +χ −χ

ψ (3.16)

( ) ( ) ( ) ^{( )}

[

x y z

]

4 2s 2p 2p 2p

2

1 χ −χ −χ +χ

=

ψ (3.17)

S_ij=0 i≠ j (3.18) S_ij=1 i= j (3.19)

τ ψ ψ

=

∫

^{H d}

H_{ij i}^* ˆ _j (3.20) H_ij= α if i and j belongs to the same atom (Coulomb integral) (3.21)

H_ij= β if i and j belong to neighbour atoms (resonance integral) (3.22)

H_ij=0 in all other cases (3.23)

H−ES =0 (3.24) α− E β

β α− E = 0

(3.25)

α_X = +α h_Xβ (3.26) β_XY =k_XYβ (3.27) T I L

r I

=E = 1 = 2 1

2 2 2

ω (3.28)

(

^{J 1}

)

^{J J 0,1,2,3,...}

J + = ^* =

= h h

L (3.29)

J M J

M_J − ≤ _J ≤

= h

Lz (3.30)

(

J 1

)

B J

(

J 1

)

I J E 2

2

r =h + = ' + (3.31) B I

2

' =h2 (3.32) I m r_i

i N

= i

∑

= 1

2 (3.33)

I= µr_o² (3.34) µ = +

m m

m m

1 2

1 2

(3.35)

∆J = ±1 and M_J = ±1 (3.36)

( )( ) ( )

[

J 1 J 2 J J 1

]

2B

(

J 1

)

hc B E - E_{ri r,j}

+

= +

− + +

=

=

ν ^,

~ (3.37)

cI h hc

B B ₂

=8π

= ′ (3.38)

( ) ( ) ( )



 +

− +

=



 



− +

= kT

J J J B

kT N N

N ^r ' 1)

exp 1 E 2

exp 1 J

2 ^,J ₀

0

J (3.39)

∆J = ±2 for identical atoms (3.41)

∆J = ± ±1 2, for different atoms (3.42)

( ) ( )

[

²

]

r hcBJ J 1 A B K

E = + + − (3.43)

( ) ( )

[

²

]

r hcBJ J 1 C BK

E = + + − (3.44)

( ) ( )

[

^{J J 1 K2}

]

E_r =hcB + + A−B (3.43)

( ) ( )

[

^{J J 1 K2}

]

E_r =hcB + + C−B (3.44)

∆J = ±1 ∆K=0 (IR) (3.45)

∆J = ± ±1 2, ∆K=0 (RA) (3.46)

∆J = ±1 (IR) (3.47)

∆J = ±2 (RA) (3.48)

2 2

2 2

2kq 1 dq

d H 2 +

− µ

= h

ˆ (3.49)

,...

2 , 1 , 2 v

v 1 h

E_v  =



 

 + ν

= (3.50)

∆v= ±1 (+: absorption, -: emission) (3.51)











 



 

 +

−



 

 + ν

=

2

v 2

v 1 2 x

v 1 h

E (3.52)

∑

∑

⁻

=

−

=

= ν

π

= ^{3N 6}

1 i

2 i 6

N 3

1 i

2 i 2 i 2

2c Q 2T Q

4 V

2 ~ & _(3.53)

q g q fq

q 2T & ¹&

V

2 = ' = ' ⁻ (3.54)

S G S FS

S 2T & ¹&

V

2 = ' = ' ⁻ (3.55)

GF−λE =0 (3.56)

6 3

,..., 2 ,

~ 1

4 π

^{2 2}

ν

²

= −

=

λ

_i

c i N

_(3.57)

Q L S= ⁻¹ (3.58)

j 0 i

2

ij S S

F E 











∂

∂

= ∂ or

j 0 i

2

ij q q

f E 











∂

∂

= ∂ (3.59)

ρ=

Ι

Ι (3.60)

1 n ,..., 2 , 1 n p

2 p cos 2

j 1  = −



 

 π +

±

=

χ (3.61)

( )

_{ij i}

j j

j

i n R r

h

m =1

∑

χ χ − ^(3.62)

2 nc m 2

h hc

kT hc

E

= λ

=

=

ν~ (3.63)

c

A=αl (3.63)

∫

^α

( )

^{ν ν}

=

band

d

A (3.64)

( )

^{19 3 1 1 1}

2 A

o

e 10 144 10 A dm cm s mol

e N

c m A4

f ε ≈ × ⁻ × ^{− − −}

= ln . (3.65)

f m e^e

=4 3 ²

π ν

h P² (3.66)

( )

<−

( )

<− ⁻ < ⁻

−

<

−

<

−CH₃ C₂H₅ CHCH_{3 2} C CH_{3 3} S O (3.67)

− > −F NO₂> −OH> −Cl> −NH₂−Br> − > − =I C O> −COOH> −CN> −SH> −R N₃ ⁺ (3.68)

− < −F Cl< −Br< −OH< −OCH₃< −NH₂< −O⁻ (3.69)

−NO₂ > −CHO> −COCH₃> −COOH COO> ⁻ ≈ −CN> −SO NH_{2 2} (3.70) I

v 2m

hν= 1 _e ² + (3.71) hν = 1m v_e + +I E_v+ E_r

2 ² ∆ ∆ (3.72)

v

n= c (3.73)

νλ

=

v (3.74)

r r

n= εrµ ≈ ε (3.75)



 

 − πν

= nz

t 2 i E

E exp (3.76)



 



 



 

 −

 πν



 



− π

= c

t nz 2 c i

z 2 n E

E _o exp ^k exp (3.77)

in

k

n −

=

n

(3.78)

ν α

= lnπ~ 4

10

n_k c (3.79)

2 2 o o

p a

ν

−

=ν

∆ (3.80)

R n

n

M ≡ − M +

2 2 1

2 ρ (3.81)

n n

N Ci i

o i i 2

2 2 2

1 2 3

− + =

∑

− ρ

ν Aν

,

(3.82)

λ π −

= ϑ

∆ n^L n^R

2 l (3.83)

[ ]

lc ϑ

= ∆

α (3.84)

[ ]

M =10⁻³M

[ ]

α (3.85)

L

R −α

α

= α

∆ (3.86)

zvB mv

= r² (3.87)

zU= mv²

2 (3.88)

m z B r / = ^{2 2}U

2 (3.89)

m z/ = 5 7. V

4π ν^{2 2} (3.90)

m z U

s t / = 2

2 2 (3.91)

m m

*≈ m²²

1 (3.92)

(

µ B

)

2µ B B

µ

∆E=+ _B − − _B = _B (3.93)

hν=2µ_BB (3.94)

∆m_s= ±1 (3.95)

B h

= 2µ νB (3.96)

µ ν

=

−

B

I 2

aM h

B (3.97)

( )

B

B'= 1−σ (3.98)

I 1 I 1 I - -I, M M

g

M

_I,z

=

_a

µ

_N

= + ,..., − ,

(3.99)

B g

E = − M

_I

B = −

_a

µ

_N

M

(3.100)

B g

E =

_a

µ

_N

∆ m

^(3.101)

2 B

a

0

π

= γ

ν

_(3.102)

( )

^a

( )

_r

a B 1

1 2

2 B = −σν

π σ γ

− π =

= γ

ν _l (3.103)

( )

^a

( )

_o

a B 1

1 2

2 B = −σν

π σ γ

− π =

= γ

ν _l (3.104)

δ

−

=

τ 10

(3.105)

( )

∑

^{−σ −}

∑∑

µ

−

=

i i i i j ij i j

B a

o J M M

2 M h 1

B g E

E (3.106)

n ,..., 2 , 1 i M

1

M

_A

= ± ∆

_Bi

= ± =

∆

_(3.107)

1 M

i

i =±

∆

∑

^(3.108)

( )

∑

∑

≠ ≠

+ ν

= σ +

−

= µ ν

i j

j ij i

j

0 i j ij i

N a

i J M J M

h 1 B

g (3.109)

2

2 J

q qJ 1 2

I= ± + (3.110)

(

+η

)

=



 





γ γ + + + −

= I1

P 2 P P

P 1 P

I I

C H o 41 32

41 32

NOE (3.111)

(

−

)

= ₊ − ₋

=2P n n n N N dt

dn

e (3.112)

( )

 ^t  = ¹

−

−

=

−

π

= γ

ϑ 2

t B_{i r}

(3.114)

(

B B_l

)

2 T

1 T

1 _a

2 2

π − + γ

* = (3.115)

1 2

*

* 2 2 2

/

1

T T T

T

1 ≤ ≤

= π ν

∆

_(3.116)

o

o hk

I = (3.120)

k

I=h (3.121)

=λ ν

=~ 1

k (3.122)

sin 2 2

s= k_o ϑ (3.123)

( ) ( )

j

oR sr

k f s expi R

) i

Aexp( _j

j =

ψ (3.124)

∑

=

ψ

=

Ψ ^N

1 j

j (3.125)

( ) ( ) ( ) ( )

jk jk k

N

1 j

N

1 k

j sf s f

K s

I sr

sr

∑∑

sin

= =

= (3.126)

( ) ( ) ( ) ( )

k j s

f s f K

s I

jk jk k

N

1 j

N

1 k

j

m =

∑∑

≠

= = sr

sr

sin (3.127)

( )

s

( ) (

' exp isr'

)

dr'

f_j^ϕ =

∫

ρ_j ^r − ^(3.128)

( ) [

^{Z f}

( )

^s

]

s s C

f_j^e = _{2 j} − _j^ϕ (3.129)

( ) ( ) ( )

s I

s s I

M

g

= m (3.130)

( )

_dr

r r rp r

0

g ^=∞

∫

^(3.131)

( ) ( )

_dr

r r p

r dr r rp r

0 0 a

∫

∫

∞

∞

= (3.132)

Chapter 4

∑

=

−

=

^N

1 i

ia

E E

E

_{i (4.1)}

∑ ∑ ∑ ∑ ∑

= =+ = =+ = +

+ + +

=

^N

i N

i j

N i

N i j

N j k

1 1 ij 1 1 1 ijk i,j,k,...,N

ia

E E ... E

E

_(4.2)

∑ ∑

= =+

=

^N

i N

i j

1 1

ij

ia

E

E

_(4.3)

Vˆ Hˆ Vˆ Hˆ Hˆ

Hˆ

_ij

=

_i

+

_j

+ =

⁰

+

_(4.4)

τ ϕ ϕ ϕ ϕ

=

∫

^{Vˆ d}

E_ij ^*_j ^*_{i i j} (4.5)

2 ij 1 ij

ij E E

E = + (4.6)

( ) ( ) ( ) _



 



 ϑ ϕ − ρ ϑ ϕ

+

= π ∑

^{α α}

∫

=

α α α i i j j j j j j

N

1 i

i

Z r Y , r r Y , dr

1 2

Q

^lm

4

^{j l} _m^l _m^l

l

^(4.7)

10 10 8 8 6 6 4 2 4

R C R C R C R E C

ij ij ij ij

ij =− − − − (4.8) dx Eψ

ψ d 2m ²

2 e

2 =

− h

(4.9)

( )

x exp

( )

ikx

ψ_k = (4.10)

e 2 2

k

2m

h

E = k

_(4.11)

= λ

= κ

= h

d h hk N

I (4.12)



 

 +  −

=

kT E 1 E

N 2 N

F F

exp

(4.13)

2dsinα=nλ n= ± ±0 1 2, , ,... (4.14)

( )

s F E

E= _o (4.15)

( ) ( )





 



 

 



 + +

=

=

∑ ∑

=

= j

3 j 2 j 1 N

1 j

j j

N

1 j

j z

d y l d x k d i h exp f isr

exp f s

F _(4.16)

ϑ

=

∆x dsin (4.17) λ ϑ

= π λ π∆

= ϕ

∆ 2 sin

2 x d

(4.18) ,...

3 , 2 , 1 2

4 sin

4 ϑ= π = λ

= π λ π∆

= ϕ

∆ x d n n

(4.19)

( )

N j

i d n

d_{i ij}

ij 4 sin 2 1,2,3 1,2,...,

=

= π

= λ ϑ

±

= π ϕ

∆ (4.20)