MolecularThermodynamics Quantum Part Atkins9e

8/9/2019 MolecularThermodynamics Quantum Part Atkins9e

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7Quantum theory: introduction and principle

Postulates of quantum mechanics:

The wavefunction

ψ has all the information.

ψ can be obtained from the Schrodinger equation, ^ H ψ = Eψ .

The Born interpretation

⌈ ψ ⌉2 is the probability density for finding a particle.

Acceptable wavefunctions

ψ must be continuous, a continuous first derivative, be single-valued, and be square-

integrable.


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Observables

Ω → Ω̂ built from the position and momentum operators of the form

^ x= x × ̂p x=ℏid

dx

The eisenberg uncertainty principle

[Ω̂1 , Ω̂2 ]=Ω̂1 Ω̂2−Ω̂2 Ω̂1≠0 ! ∆ Ω1 ∆ Ω2 ≥1

2|⟨ [Ω̂1 , Ω̂2 ] ⟩|


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8Quantum theory: techniques and applications

Translational motion

Free motion in one dimension

−ℏ2

2m

d2

ψ

d x2= Eψ

The solutions of this equation have the form

ψ = A eikx +B e−ikx E=k

2ℏ

2

2 m where A and B are constants.

! all values of " are possible

! the translational energy of a free particle is not quanti#ed

$n either case of eikx or e−i kx , |ψ |2 is independent of %

! the position of the particle is completely unpredictable


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A particle in a box


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(a) The acceptable solutions

The Schrodinger equation for the region where & ' ( is the same as for a free

particle, so the general solutions are given as)

ψ k = A eikx +B e−ikx= A (cos kx+isin kx )+B (coskx+isin kx )=( A+B ) coskx+ ( A−B ) isin kx

Then, with new coefficients

ψ k =C coskx+ D sin kx E k =k

2ℏ

2

2 m


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oundary conditions

The wavefunction must be #ero where & is infinite.

ψ k ( x )=0 for the regions of x ≤ 0 and x ≥ L

! by the requirement of wavefunction continuity

ψ k (0)=0 and ψ k ( L)=0

! C =0 andkL=¿0

ψ k ( L )= Dsin¿

!kL=nπ n=1,2,…

! ψ n ( x )= D sin(nπx/ L)n=1,2, …

! En=n2h2/8 m L2


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!onclusion:

The need to satisfy boundary conditions implies that only certain

"a#efunctions are acceptable$ and hence restricts obser#ables to discrete

#alues%


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(b) &ormali'ation

The normali#ation condition)

∫0

L

ψ ¿ψ d x=C 2∫0

L

sin2 nπx L

d x=C 2 × L2=1 , so C =(

2 L )

1 /2

The complete solution is

En= n

2h

2

8m L2 n=1,2,…

ψ n ( x )=(2

L )1/2

sin( nπx L )

The energies and wavefunctions are labeled with the *quantum number+ n .

A quantum number is an integer in some cases, as we shall see, a half-integer

that labels the state of the system.


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c The properties of the solutions

n ↑ ! the number of nodes ↑

ψ n has n - nodes

the number of nodes ↑ ! the average curvature of the wavefunction

! "inetic energy of the particle ↑

/ach wavefunction is a superposition of momentum eigenfunctions)

ψ n ( x )=( 2 L )1/2

sin(nπx

L )=( 2 L )

1

2 ( eikx−e−i kx ) k =nπ L

! equal probability of travel to both directions


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Because n cannot be #ero, the possible lowest energy is

E1= h

2

8 m L2

This lowest, irremovable energy is called the 'eropoint enery.

The physical origin of the #ero-point energy can be e%plained)

. The uncertainty principle

! a confined particle some information for the position

! the momentum cannot be #ero precise value

0. Acceptable wavefunctions

! must be #ero at both ends and need to be survived

! must have curvature

! non#ero "inetic energy


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The separation between ad1acent energy levels

En+1− En=(n+1)2 h2

8 m L2 −

n2

h2

8 m L2=(2 n+1)

h2

8 m L2

Separation ↓ as L↑

Separation → 0 as L→ ∞ ! not quanti#ed li"e a free particle

Separation ↓ as m↑


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!orrespondin principle

2lassical mechanics emerges from quantum mechanics as high quantum

numbers are reached.

(d) *rthoonality


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3avefunctions corresponding to different energies are orthogonal)

Suppose for two different energies of En and Em

^

H ψ n= En ψ n ^

H ψ m= Em ψ m

Then ∫ψ m¿ ^ H ψ n d x= En∫ψ m

¿ψ n d x ∫ψ n

¿ ^ H ψ m d x= Em∫ψ n¿

ψ m d x

4ow consider noting that energies are real

∫ψ m¿ ^ H ψ n d x−{∫ψ n¿ ^ H ψ md x }¿= En∫ψ m¿ ψ nd x− Em∫ψ nψ m¿ d x

5sing hermiticity, the sum on the left is #ero, E

(¿¿n− Em)∫ψ m¿ ψ n d x0=¿

owever, the two energies are different6 therefore the integral on the right

must be #ero)

∫ψ m¿ ψ n d x=0


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+otion in t"o and more dimensions

Schrodinger equation for a particle in a bo% in two dimensions)

−ℏ2

2 m (∂2ψ

∂ x2 +

∂2ψ

∂ 2 )= Eψ


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(a) ,eparation of #ariables

,eparation of #ariables technique

7ivides the equation into two or more ordinary differential equations, one for

each variable

3rite the wavefunction as a product of functions)ψ ( x , )= ! ( x )" ( )

3ith this substitution to the equation)

−ℏ2

2m

d2 !

d x2 = E ! !

−ℏ2

2 m

d2

"

d 2 = E" " E= E ! + E"

ψ n1 ,n2 ( x , )=2

( L1 L2 )1 /2 sin(

n1 πx

L1)sin(

n2 π

L2)

En1

,n2=( n1

2

L12+

n22

L22 ) h

2

8 m


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$n same way, the solutions for a particle in a bo% in three dimensions)

ψ n1 ,n2 ,n3 ( x , , # )=(8

L1 L2 L3 )1/2

sin(n1 πx

L1)sin(

n2 π

L2)sin (

n3 π#

L3)

En1

,n2

, n3=( n1

2

L12+

n22

L22+

n32

L32 ) h

2

8 m


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(b) -eeneracy

3hen L1= L2= L in two-dimensional case6

ψ n1 ,n2 ( x )=2 L sin (n

1πx

L )sin (n

2π

L )

En1 ,n2=(n12+n2

2) h2

8 m L2

2onsider the cases n1=1,n2=2 and n1=2,n2=1 )

ψ 1,2 ( x )=2

Lsin(

πx

L )sin(

2 π

L ) E1,2=

5h2

8m L2

ψ 2,1 ( x )= 2 Lsin (2 πx

L )sin( π

L ) E2,1= 5h

2

8m L2

! Although the wavefunctions are different, they are deenerate.

! The occurrence of degeneracy is related to the symmetry of the system.


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.ibrational motion

armonic oscillation if the potential function is

$ =1

2 kx2

with restoring force % =−kx from % =−d$ /dx

! The Schrodinger equation becomes

−ℏ2

2 m

d2

ψ

d x2+

1

2kx

2ψ = Eψ


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The enery le#els

The permitted energy levels are

E&=(&+ 12 )ℏ' '=( k m )1 /2

&=0,1,2, …


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armonic oscillator has #ero-point energy6

E0=1

2ℏ '

! Because the particle is confined, its position is not completely uncertain.

! Therefore, its momentum cannot be e%actly #ero.

! The particle fluctuates incessantly around its equilibrium position6 though

classical mechanics would allow the particle to be perfectly still.


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The "a#efunctions

(a) The form of the "a#efunctions

The wavefunctions for a harmonic oscillator,ψ ( x )= ( × (polynomial in x ) ×(bell-shaped Gaussian function )

. ψ → 0as x →) ∞

0. *heexp+nen 2 is proportional to x2× (mk )1 /2 , so it decays more rapidly for large masses

and stiff springs.

8.$t spread over wider range as& increases .

! 2orresponding principle wor"s at high & .

ψ & ( x )= ( & H & ( )e− 2 /2

= x-

- =( ℏ2

mk )1/4


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/otational motion

/otation in three dimensions: the particle on a sphere

A particle of mass m that is free to move anywhere on the surface of a sphere of

radius r .


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a The Schrodinger equation

The Schrodinger equation is

−ℏ

2

2 m ∇2 ψ = Eψ

9rom :ustification ;.< on p8(

1

.2 /

2ψ =

−2mEℏ

2 ψ /

2= 1

sin20

∂2

∂ϕ2+

1

sin0

∂

∂0 sin 0

∂

∂ 0

/2

ψ =− ψ ϵ ϵ=2 1E

ℏ2

2e3456e 1 =m. 2

*7+854n5mn5m2e.6,9∧m9

9=0,1,2, … m9=9 , 9-1,…, - 9 +1,- 9

Angular momentum quantum number 9

=agnetic quantum number m9


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The normali#ed wavefunctions and energies are

" 9, m9 (0 ,ϕ ) :spherical harmonics !able "#3 on p302$

E=9 (9+1 ) :2

2 1 9=0,1,2, …

! 854ni#ed∧independen +; m9

! (29+1 ) -foldde%enerate


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(b) Anular momentum

2omparing with the classical rotational energy, E=ni5de +; 4n>594. m+men5m={9 (9+1 ) }1/2 : 9=0,1,2,…

&-co mp+nen +; 4n>594.m+men5m=m9 : m9=9 ,9 -1,… , - 9

The number of nodes in " 9, m9 (0 ,ϕ ) increases with 9

! higher angular momentum implies higher "inetic energy

! a high "inetic energy arises from the motion parallel to the equator

because curvature is greatest in that direction


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(c) ,pace quanti'ation

9or each 9 , there are only 29+1 possible m9 values)

The orientation of a rotating body is quanti#ed) space quanti'ation

The plane of rotation of a particle can ta"e only a discrete range of

orientations.

*tto ,tern and 0alther 1elach (in 2342)

5xperiment: Shot a beam of silver atoms through an inhomogeneous

magnetic field

6dea: A rotating, charged body behaves li"e a magnet and interacts with the

applied field


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5xpectation:

2lassical mechanics)

Angular momentum can have all orientations)

! the associated magnet can ta"e any orientation

! a broad band of atoms is e%pected

>uantum mechanics)

The angular momentum is quanti#ed)

! the associated magnet lies in a number of discrete orientations

! several sharp bands of atoms are e%pected


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/esults:

*bser#ed discrete band$ and so confirmed the quantum mechanics%


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(d) The #ector model

9 x ,9 , and 9 # are complementary observables)

[ ̂9 x ,

^9 ]=i :

^9 # [

^9 ,

^9 # ]=i :

^9 x [

^9 # ,

^9 x ]=i :

^9

And

[ 9̂2 ,9̂8 ]=0 8= x , , and #

! Although we can specify the magnitude of the angular momentum and any

of its component, if 9 # is "nown, then it is impossible to ascribe values to

the other two components.


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8%2 ,pin

Stern and ?elach observed two bands of Ag atoms in e%periment)2onflict with the fact that 9 must be an integer.

! from spin , not due to orbital angular momentum

,pin:

,pin quantum number: 6

+anitude: { 6 (6+1 ) }1/2ℏ

,pin manetic quantum number: m6=6 , 6 -1,… , -6


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5lectrons ha#e spin:

6=1

2

+anitude: {12 (

12+1)}1 /2

: =(34 )

1/2

:

m6=+12

(denoted as - or ↑ )

m6=−12

(denoted as ? or ↓ )


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The ground state configuration of Ag atom)

[ @. ] 4 d10 5 61 , a single unpaired electron outside a closed shell


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Particles:

Fermions: with half-integer spin

/lectron, proton, neutron,@osons: with integer spin including (

photon

+atter is an assembly of fermions held toether by forces con#eyed by bosons%


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3Atomic structure and atomic spectra

The structure of hydroenic atoms

amiltonian for hydrogenic atom

^ H =−ℏ2

2 me∇e

2− ℏ

2

2m ( ∇ (

2 − e2

4 π 0.

(a) The separation of #ariables

7ivide Schrodinger equation into two parts)

.The motion of the atom as a whole

0.The motion of electron relative to the nucleus


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Schrodinger equation becomes)

−ℏ2

2 C ∇2 ψ −

A e2

4 π 0 . ψ = E ψ

1

C=

1

me+ 1

m ( C :.ed53edm466

Because potential energy is centrosymmetric independent of angle,

we can write

ψ (. , 0 ,ϕ )= D (. ) " (0 ,ϕ)


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The equation does separate, we have to solve

/2

" =−9 (9+1 ) "

spherical harmonics: 'uantum number 9 , m9

−ℏ2

2 C

d2

5

d .2+$ e;; 5= E55=.D

7he.e $ e;; = − e2

4 π 0. +

9 (9+1 ) ℏ2

2 .2

! a particle of mass in one-dimensional region where the potential

energy is &eff ) radial "a#e equation


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b The radial solutions

$ e;; =− e2

4 π B0 .+

9 (9+1 )ℏ2

2 .2

The st term)

2oulomb potential energy

The 0nd term)

stems from centrifugal force from the angular momentum of electron


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$ e;; = − e2

4 π 0 .+

9 (9+1 ) ℏ2

2 .2

0hen 9=0

/lectron has no angular momentum

ure 2oulombic and attractive at all radii

0hen 9 ≠ 0

The centrifugal term is positive repulsive

As . →0 , this repulsive term dominates the 2oulombic term.

! $ e;; is repulsive


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The allowed energies

En= − 2 e4

32 π 2

B 02ℏ

2n

2 n=1,2,…

adial wavefunctions have the form D ( . )=( polynomial in . ) ×(decayin% e(ponential in . )

adial wavefunctions depend on both n and 9

Dn, 9 (. )= ( n, 9 9 Ln+1

2 9+1 ( ) e− /2

7he.e =2 A.

n 4040=

4 π 0ℏ2

me e2 =52#" pm(B+h. .4di56)

L ( ):466+3i4ed L4>5e..e p+9n+mi49


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6nterpretation of the radial "a#efunction$ D (. ) :

./%ponential factor ensures that D (. )→0 as . →∞ .

0.The factor

9

ensures that provided9>0

D ( . )=0

at the nucleus.8. L ( ) oscillates and accounts for the presence of radial nodes.


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3%2 Atomic orbitals and their eneries

Atomic orbital:A one-electron wavefunction for an electron in an atom

7efined by three quantum numbers) n , 9 , m9

ψ n , 9 , m9 F|n , 9 , m9 ⟩


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>uantum numbers)

rincipal quantum number) n

n=1,2,3, …

Dee.minehe ene.> +; 4 hd.+>eni34+m

En= − 2 e4

32π 2

02

ℏ2

n2

Angular momentum quantum number) 9

9=0,1,2,… , n−1

=4>ni5de +; 4n>594. m+men5m={9 (9+1 ) }1/2ℏ

=agnetic quantum number) m9

m9=−9 ,−9+1,… ,9−1, 9

&-co mp+nen +; 4n>594. m+men5m=m9 ℏ


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The value of the principal quantum number, n ,

controls the ma%imum value of 9

9

controls the range of values ofm

9

/lectron has its own spin) 6=1

2m6=)

1

2

! 4eed four quantum numbers (n , 9 , m9 , m6) to specify

the state of an electron in a hydro)enic atom


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(a) The enery le#els

/nergy levels of hydrogenic atoms) En=

− 2 e4

32 π 2

B 02ℏ

2n

2

2+5nd 64e6 : E0 → n+854ni#ed )

E ∝ 2

ydberg constant)

h3 D H =

H e4

32 π 2 02 ℏ2 H i6 he .ed53ed m466 ;+. hd.+>enG

D H =− H

me D D=

me e4

8 π 2

02

h3

3

! The only discrepancies arise from the neglect of relativistic corrections

the increase of mass with speed.


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(b) Atomic orbitals

The ground state) ψ 1,0,0 F|1,0,0 ⟩

ψ =1

(π 403 )1 /2 e−. /40(;+. =1)

1 6 +.2i49 i6 6phe.i3499 6mme.i349G

m4xim5m495e+; ψ =1

(π 403 )1/2

(4 .=0)∧exp+neni499 de346

2ontributions of the potential and "inetic energy

otential energy prefers to sit on the top of the nucleus.

Cinetic energy li"es to be spread out uniformly.

! A compromise between the two e%tremes)

! The wavefunction spread away from the nucleus and has a reasonably

low average curvature.


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3%4 ,pectroscopic transitions and selection rules

Transition

ψ i → ψ I ∆ E=h J

,election rule) a statement about which transitions are allowed

*allowed+ or *forbidden+

!onser#ation of anular momentum:

A photon has an intrinsic spin angular momentum) 6=1 .

! ∆ 9=) 1 ;+.he .4n6ii+n +; ψ i → ψ I


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Quantum mechanical #ie": (ustification 2%9 on p;)

#, ;i=∫ψ ; ¿ # ψ i d x=⟨ ; | #|i ⟩;+. +ne - e9e3.+n4+m

=−e . )ith components: x=−e x , =−e , #=−e#

After calculating ;i for all three components

! ;i ≠ 0 )hen∆ 9=) 1∧∆ m9=0,) 1

! 4o restrictions for n , because it does not relate directly to angular

momentum


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1rotrian diaram:

Summari#es the energies of the states and the transitions between them


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The structures of manyelectron atoms

The Schrodinger equation for a many-electron atom is highly complicated and no

analytical e%pression for the orbitals and energies can be given.

! 3e are forced to ma"e appro%imations.


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Pauli principle:

3hen the labels of any two identical fermions are e%changed, the total

wavefunction changes sign6 when the labels of any two identical bosons are

e%changed, the total wavefunction retains the same sign.

auli principle for two-electron total wavefunction) K (1,2 )

K (2,1 )=−K (1,2 )

9or two electrons, there are four possible spin states)

- (1 ) - (2)- (1) ?(2) ? (1)- (2) ? (1) ? (2)


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Because we cannot tell which electron is which or there must be symmetry

−¿(1,2) ? (1) ? (2)+¿(1,2) ¿

- (1 )- (2) ¿

3here ) (1,2)=(1/2)1 /2 {- (1 ) ? (2)) ?(1)- (2)}

−¿(1,2)−¿ (2,1)=− ¿+¿ (1,2 ) ¿+¿ (2,1 )= ¿

¿

9our possible total wavefunctions for the same special function ψ )

ψ (1 ) ψ (2)- (1 ) - (2)

ψ (1 ) ψ (2) ? (1) ? (2)

+¿ψ (1 ) ψ (2) ¿

−¿ψ (1 ) ψ (2) ¿

! Only the last one satisfies the auli principle.


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Any acceptable wavefunction for closedshell species)

,later determinant form:

K (1,2,… , ( )= 1( ( M )1/2

[ψ 4 (1) - (1 ) ⋯ ψ 4 ( ( ) - ( ( )

⋮ ⋱ ⋮

ψ # (1) - (1 ) ⋯ ψ # ( ( ) - ( ( ) ]! Antisymmetric under any e%change of any pair of electrons


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(a) Penetration and shieldin


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5ffecti#e nuclear chare:

e;; = −

7he.e i6he 6hie9din>3+n64n

The shielding constant is different for 6 and p electrons.6 electron has a greater penetration than a p electron.9ig (.0( on

p8D(

6 electron has less shielding than a p electron.

! The energies of subshells depend on 9 , but not on m9

6


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.alence electrons:

/lectrons in the outermost shell of an atom in its ground state

! esponsible for the chemical bonds that the atom forms

(b) The buildinup principle


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Building-up principle or Aufbau principle )

9or an atom with atom number )

9eed into the orbitals the electrons in succession.

The order of occupation is)

1 62 6 2 p 3 63 p 4 63 d 4 p 5 6 4 d * 6


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und+s ma%imum multiplicity rule)

An atom in its ground state adopts a configuration with the greatest number

of unpaired electrons

Spin correlation)

/lectrons with parallel spins behave as if they have a tendency to stay well

apart, and hence repel each other less.


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2onsider to satisfy the auli principle

+¿(1,2) 4(1,2)−¿(1,2) p(1,2)K 2(1,2)=ψ ¿

K 1(1,2)=ψ ¿

7he.e ψ ) (1,2 )=(1/2 )1/2 {ψ 4 (1) ψ 2(2))ψ 2 (1 ) ψ 4(2) }

p (1,2 ) is parallel spin state

4 (1,2) is paired spin state

3hen two electrons occupy at the same position, .1=.2 :

−¿ψ ¿

vanishes while +¿ψ ¿ survives

−¿ (1,2)=(1/2 )1 /2

{ψ 4 (1 )ψ 2 (2 )−ψ 2 (1 )ψ 4(2)}ψ ¿

! spin correlation


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5lectron confiurations:

On average, 3 d electrons are closer to nucleus than 4 6 electrons.

! 3 d electrons repel each other more strongly than the two 4 6

electrons.

! Sc has [ A. ] 3 d1 4 62

2r and 2u have [ A. ] 3 d5 4 61 and [ A. ] 3 d10 4 61 , respectively.

The subtle shades of energy differences and electron-electron repulsion

! The rich-comple%ity of inorganic d- me49 chemistry


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2alculations show that the energy difference between [ A. ] 3 d3 and

[ A. ] 3 d1 4 62 depends on e;;

! As A e;; increases, transfer of a 4 6 electron to a 3 d orbital becomes

more favorable because electron-electron repulsions are compensated

by attractive interactions.

!2+¿ : [ A. ] 3d3

$ ¿ , the observed [ A. ] 4 60 3 dn configurations for

2+¿ =

¿ cations of

Sc through En


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The spectra of complex atoms

The actual energy levels are not given solely by the energies of the orbitals,

because the electrons interact with one another in various ways.


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3% ,inlet and triplet states

,inlet:

−¿ (1,2)=(1/2 )1 /2 {- (1 ) ? (2 )− ?(1)- (2)} ¿

Triplet:

- (1 ) - (2 )

+¿ (1,2)=(1

2 )1

2 {- (1 ) ? (2 )+ ? (1 )- (2 ) } ¿

? (1) ? (2)


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9or states arising from the same configuration, the triplet state is generally lies

lower than the singlet state. The two states of 1 61 2 61 He differ by FD0 cm-

corresponding to (.G e&.

T"o simple features in the spectrum of atomic


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3%9 ,pinorbit couplin

,pinorbit couplin:

$nteraction of the spin magnet moment with the magnetic field arising from the

orbital angular momentum


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(a) The total anular momentum

The energies of the levels with quantum numbers 6 , 9 , I )

E9 , 6 , I=1

2

h3A { I ( I+1 )−9 (9+1 )−6 ( 6+1 ) }

The strength of the spin-orbit coupling depends on the nuclear charge.

! The greater the nuclear charge, the stronger the spin-orbit interaction.


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(b) Fine structure

Fine structure:

The structure in a spectrum due to spin-orbit coupling

$n 4a, the spin-orbit coupling affects the energies by about < cm-

3%;Term symbols and selection rules


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a The total orbital angular momentum

b The multiplicity

c The total angular momentum

Term symbol:

L❑2 N+1

594. m+men5m854n5mn5m2e. , L

m4>ni5de: { L ( L+1 ) }1 /2ℏ

495e : L=91 +92 ,9 1 +92−1,… ,∨91−92∨¿ L=012345* …

L=N OD %P H 1 …

4 39+6ed 6he99h46 4 #e.+ +.2i49 4n>594. m+men5mG

+496pin 854n5m n5m2e. , N

495e : N=61+62, 61+62−1,… ,∨61−62∨¿

m59ip9i3i=2 N+1

+494n>594. m+men5m 854n5mn5m2e. , 63heme :

f the spin-orbit couplin% is )ea, then it is effectie

only )hen all the orbital momenta are operatin% cooperatiely


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495e :


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== couplin

ussell-Saunders coupling fails when the spin-orbit coupling is large in

heavy atoms. $n that case, the individual spin and orbital momenta of the

electrons are coupled into individual 1 values6 then these momenta are

combined into a grand total : .

!orrelation diaram )

The labels derived by using the ussell-Saunders scheme can be used to

label the states of the 11 -coupling scheme.


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(d) ,election rules

,election rules:

∆ N=0 ∆ L=0,) 1 ∆9=)1 ∆ 2e3+me6 m+.e4pp.+p.i4e

! .4n6ii+n62e7een 6in>9e ∧.ip9e 64e6 (;+. 7hi3h∆ N=)1 ) ,

7hi9e;+.2idden∈9i>h 4+m6, 4.e499+7ed∈he4 4+m6G


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MolecularThermodynamics Quantum Part Atkins9e

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