= 0 associated Legendre dierential equation

~

²

2 m

1

r

²

^

²

Y

_`;m`

(#; ') = EY

_`;m`

(#; ');

E =`(` + 1) ~

²

2 I ` = 0; 1; 2; : : : m

_`

= `; (` 1); : : : ; 0; : : : ; ` 1; ` every energy level is (2`+1)-fold degenerate

` - orbital angular momentum quantum number m

_`

- magnetic quantum number

~

²

^

²

Y

_`;m`

(#; ') = ~

²

`(` + 1) Y

_`;m`

(#; ')

119

L = r p ,^L = ~ i ^r r L ^

_z

= ^ x p ^

_y

y ^ p ^

_x

= ~

i

@

@ L ^

z

() = ~

i

@

@ = ~ m () ) () = e

^im

From the periodic boundary condition:

( + 2) =(); m = 0; 1 2; 3; : : :

^L

²

= ^ L

²_x

+ ^ L

²_y

+ ^ L

²_z

= ~

²

1

sin #

@

@# (sin # @

@# ) + 1 sin

²

#

@

²

@'

²

= ~

²

^

²

` = 0; 1; 2; : : :

wavefunctions

` m

_`

Y

_`;m`

(#; ')

0 0

₄¹

¹₂

1 0

₄³

¹

2

cos #

1

₈³

¹₂

sin # e

^i'

2 0

₁₆⁵

¹

2

(3 cos

²

# 1)

1

₈¹⁵

¹₂

cos # sin # e

^i'

2

₃₂¹⁵

¹

2

sin

²

# e

^2i'

real combinations: p

x

= Y

_{1; 1}

Y

_1;1

p 2 ; p

y

= Y

_{1; 1}

+ Y

_1;1

i p

2

121

~

²

2 r

²

e

²

4

0

r

( r ; #; ') = E ( r ; #; ') r

²

= 1

r

²

@

@ r ( r

²

@

@ r ) + 1 r

²

sin #

@

@# (sin # @

@# ) + 1 r

²

sin

²

#

@

²

@'

²

r

²

= 1

r

²

@

@r ( r

²

@

@r ) +

²

r

²

123

Separation of variables: (r; #; ') =R(r)Y_{`;</sub>m<sub>`}(#; ')

~² 2r²

Y_`;m` @

@r(r²@R

@r ) +R(r)²Y_`;m`

e²

40rRY_`;m` =ERY_`;m`

~² 2r²

Y_{`;</sub>m<sub>`} @

@r(r²@R

@r ) +R(r)`(` + 1)Y_{`;</sub>m<sub>`}

e²

4₀rRY_{`;</sub>m<sub>`} =ERY_{`;</sub>m<sub>`}

~² 2r²

@

@r(r²@R

@r ) +R(r)`(` + 1)

e²

40rR=ER

The solution can be obtained using the Sommerfeld's polynomial method.

The results:

R

_n^`

( r ) = 1

r e

^r=r0

P

_n^`

( 2 r

r

₀

); where r

₀

= n 4

0

~

²

e

²

, 0 l < n ; n = 1; 2; : : : and P

_n^l

(x) are the so-called Laguerre polynomials.

Schrödinger: E

n

= m

e

e

⁴

8

²₀

h

²

n

²

Bohr: E

n

= m

e

e

⁴

8

²₀

h

²

n

²

125

radial wavefunctions, Laguerre polynomials

(r; #; ') =Rn;`(r)Y<sub>`;</sub>m_`(#; ')

n ` Rn;`

1 0 2

Z a

³₂ e ^%=2

2 0 ^p1₈

Z a

³₂

(2 %)e ^%=2

2 1 ^p1₂₄

Z a

³₂

%e ^%=2

3 0 ^p₂₄₃¹

Z a

³₂

(6 6% + %²)e ^%=2

3 1 ^p₄₈₆¹

Z a

³₂

(4% %²)e ^%=2

3 2 ^p₂₄₃₀¹

Z a

³₂

%²e ^%=2

% =_4"^2Ze_~²2^rn=²_na^Zr; wherea=^4"_e⁰2^~2 is the Bohr radius

atomic units, xed nucleus

~

²

2 m

_e

r

²e

e

²

4

₀

r

= E x ! x

⁰

, y ! y

⁰

, z ! z

⁰

@

²

@ x

²

= 1

²

@

²

@ x

⁰²

r = p

x

²

+ y

²

+ z

²

) r ! q

( x

⁰

)

²

+ ( y

⁰

)

²

+ ( z

⁰

)

²

= r

⁰

127

atomic units, xed nucleus ~²

2me

r²e

e² 40r

=E +

~²

2me²(r⁰e)² e² 40r⁰

0=E ⁰ select to fulll

~²

me² = e² 40 =Ea

+ Ea

1

2(r⁰e)² 1 r⁰

0= E ⁰

Ea

1

2(r⁰e)² 1 r⁰

0= E ⁰ +

1

2(r⁰e)² 1 r⁰

0= E^{0 0} E⁰= E

Ea

atomic units for distance and energy:

= 4⁰~²

mee² =a₀; Bohr radius Ea= ~²

mea²₀; hartree

hydrogenlike atomic wavefunctions: n;`;m<sub>`</sub>(r; #; ') =Rn;`(r)Y<sub>`;</sub>m_`(#; ')

n ` m<sub>`</sub> Rn;` Y<sub>`;</sub>m_`

1 0 0 2

Z a

³₂

e ^%=2 ₄¹¹₂

2 0 0 ^p1₈

Z a

³₂

(2 %)e ^%=2 ₄¹¹₂

2 1 0 ^p1₂₄

Z a

³₂

%e ^%=2 ₄³¹₂ cos #

2 1 +1 ^p1₂₄

Z a

³₂

%e ^%=2 ₈³¹₂ sin #e^i'

2 1 1 ^p1₂₄

Z a

³₂

%e ^%=2 ₈³¹₂

sin #e ^i'

3 0 0 ^p1₂₄₃

Z a

³₂

(6 6% + %²)e ^%=2 ₄¹¹₂

% =_4"^2Z₀^e_~²2^rn

129

shells and subshells

atomic orbital (AO) - one electron wavefunction (

n;`;m<sub>`</sub>

) quantum numbers:

n - principal

` - azimuthal (orbital angular momentum) m

_`

- magnetic

131

shells and subshells

a shell consists of AOs with the same principal quantum number n (K, L, M, N, . . . )

subshell same n dierent ` (s, p, d, f, g, . . . subshells)

for example: n=1,2, and 3

s orbitals, ` = 0,m<sub>`</sub>= 0

s=c

Z a

³₂

Pn(%)e ^%=2 ₄¹¹₂

the angular wavefunction is constant,Y_0;0(#; ') = ₄^{1 1}₂ spherical symmetry

thePn(%)s are Laguerre polynomials, and their roots give the number of nodal surfaces

133

p orbitals, ` = 1,m<sub>`</sub>= 0; 1

p₀= 1p 24

Z a

³₂

%e ^%=2 3

4 ¹₂

cos # = % cos #f(%) =zf(%) = pz

p₊₁= 1p 24

Z a

3

2%e ^%=2 3

8 1

2sin #e^i'

p ₁= 1p 24

Z a

³

2%e ^%=2 3

8 ¹

2sin #e ^i'

p orbitals, ` = 1,m<sub>`</sub>= 0; 1

to get rid of the complex variable we take linear combinations of ^p+1 and ^p 1

px= p₊₁+ p ₁=p1 24

Z a

³₂

%e ^%=2 3

8 ¹₂

sin # (e^i'+e ^i')

py= p₊₁ p ₁=p1 24

Z a

³

2%e ^%=2 3

8 ¹

2sin # (e^i' e ^i')

% sin # (e^i'+e ^i') = % sin # 2 cos ' ) ^px=xf(%)

% sin # (e^i' e ^i') = % sin # 2isin ' ) ^py =yf(%)

135

p orbitals, ` = 1

d orbitals, ` = 2

similarly to p orbitals we make linear combinations of complex WFs to get real functions

d

xy

= xyf ( r ) d

yz

= yzf ( r ) d

zx

= zxf ( r )

d

_x2 y²

= 1

2 ( x

²

y

²

) f ( r ) d

_z2

=

p 3

2 (3 z

²

r

²

) f ( r )

137

d orbitals, ` = 2

shells and subshells

atomic orbital (AO) - one electron wavefunction (

n;`;m<sub>`</sub>

) quantum numbers:

n - principal

` - azimuthal (orbital angular momentum) m

_`

- magnetic

m

s

- spin

139

an intrinsic angular momentum of a particle

Stern - Gerlach experiment(1922)

N

S

Inhomogeneous magnetic eld!

Magnetic dipole moment, relation to the angular momentum

Classical description

m = SI n, where I is the current in an electric current loop, S is the surface of the loop, and vector n perpendicular to the loop.

If the current is produced by a single charged particle I = e = T , where T is the periodic time of the motion.

I =

_2m^2m_e^e_rT^re

=

_2m^pe_er

=

_2m^erp_er2

=

_2m^e_e^`_r2

m =

₂^r_m²_e^e_r^`2

=

₂^e_m^`_e

Force on a moment : F = r (mB)

141

an intrinsic angular momentum of a particle

Stern - Gerlach experiment

to conrm the Bohr-Sommerfeld theory

Ag atoms are in ` = 0 state ) no splitting

the spatial orientation is quantized

an intrinsic angular momentum of a particle

Stern - Gerlach experiment(1922)

Uhlenbeck and Goudsmit - spin(1925): An internal angular momentum of the electron ( ^ S ) produces on additional

magnetic moment: ^ m

z

=

^g_~^B

^S

z

, where g is the g-factor, and

B

=

₂^e_m^~

is the Bohr magneton, e is the positive unit charge.

no spin in non-relativistic quantum mechanics ad hoc introduction by Pauli

it occurs naturally in Dirac's relativistic QM(1928) ( g = 2) correction from quantum electrodynamics (1948):

g = 2:002319

143

an intrinsic angular momentum of a particle

The intrinsic angular momentum (S) can be characterized by the eigenvalues of the ^Sz and ^S²operators, where ^S²= ^Sx²+ ^Sy²+ ^Sz².

S^z = ~sz S^² = ~²s(s+ 1)

The possible values ofs are 0;¹₂; 1;³₂; 2; : : : , whilems= s; s+ 1; : : : ;s fermions like electron, proton, neutron (half-integer

spin,s=¹₂;³₂;⁵₂; : : : )

bosons like photon, W bosons,⁴He (integer spin, s= 0; 1; 2; : : : )

The eigenvalues of spin for an electron:

Wave function of the particle and the spin

The wave function of the electron must be extended by the spin: E.g., the wave function of the electron in the H atom: n;`;m<sub>`</sub>;m_s=¹₂(r; ; ) = n;`;m<sub>`</sub>(r; ; ) Wave functions with dierent spins are orthogonal to each other.

Vector representation: n;`;m<sub>`</sub> = 0

@ ^n;`;m` 0

1

A, n;`;m<sub>`</sub> = 0

@ 0

n;`;m<sub>`</sub>

1 A

145

Total angular momentum of an electron

The x , y , z component of the total angular momentum of a particle is the sum of the orbital and spin angular momentums:

J ^

_i

= ^ L

_i

+ ^ S

_i

; i = x ; y ; z

In the non-relativistic case (the speed of the particles are negligible with respect to the speed of light) the ^ J

²

; ^ J

_z

; ^ S

²

; ^ S

_z

; ^ L

²

; ^ L

_z

operators commute with each other and with the Hamilton

operator, i.e., we can nd a common set of eigenfunctions for all

these operators. These operators belong to the compatible

measurable physical quantities.

Magnetic dipole moment in QM

In general ^mz= g₂^e_m^Jz

for the orbital angular momentum of the electron: ^m = _2m^e_e^L = _~^B^L, i.e.,gL= 1

for an electron without orbital angular momentum: ^m = ²_~^B^S, i.e., gS = 2

in general the LandégJ factor should be used: ^m = ^gJ_~^B^J, where gJ =gLj(j+1) s(s+1)+`(`+1)

2j(j+1) +gSj(j+1)+s(s+1) `(`+1) 2j(j+1)

abs. value of magnetic moment: M=gJ

pj(j+ 1)B

147

Zeeman eect

In magnetic eld the Hamiltonian contains an additional term:

V ^

_mag

= ^ mB, where B is the magnetic induction vector.

Supposing that the magnetic eld is oriented along the z axis, V ^

mag

= ^ m

_z

B

_z

=

^gJ_~^B

J ^

z

B

z

Due to this term the energy levels depend on the j

z

quantum

numbers.

total angular momentum quantum number: j = j` s j, e.g.,

` = 0, s orbital, j =

¹₂

` = 1, p orbital, j =

¹₂

;

³₂

` = 2, d orbital, j =

³₂

;

⁵₂

The energy is slightly j-dependent (ne structure of the H atom: splitting of the spectral lines of atoms due to electron spin)

E

_{j n}

c

²

²

2 n

²

"

1 +

²

n

²

n j +

¹₂

3 4

!#

,

where =

_4"^e2_0~c

=

₁₃₇¹

is the ne-structure constant

149

j-dependent relativistic correction: spin-orbit splitting With respect the resting frame of the electron the proton is orbiting around the electron and producing a magnetic eld B, B =

_c¹2

v E

From a brief derivation the magnetic eld is:

B =

_me¹_ec21 r

@U(r)

@r

L

As the energy shift is E

_mag

= m

_z

B

_z

and ^ m

_z

=

²_~^B

S ^

_z

than ^ H

_mag

=

¹_2~m²_eec^B21

r

@U(r)

@r

^L ^S, where the 'Thomas-half'

is also included (Llewellyn Thomas, 1926).

Lyman alpha transition in hydrogen

B = 0 B 6= 0 The slitting of energies according to the j values is a relativistic eect.

The Zeeman eect splits the energy levels of the H atom. As the value of g

_J

depends on the j ; `; s values the extent of the splitting is dierent for the energy levels.

151

cyclic permutations h ^`

z

; ^`

x

i = i ~^`

y

h ^`

y

; ^`

z

i = i ~^`

x

h ^`

²

; ^`

z

i = 0; h

^`

x

; ^`

y

i = i ~^`

z

The angular momentum can be visualized as a vector with length

~ p

`(` + 1) rotating around the z

cyclic permutations [^ s

_x

; ^ s

_y

] = i ~^ s

_z

[^ s

_z

; ^ s

_x

] = i ~^ s

_y

[^ s

_y

; ^ s

_z

] = i ~^ s

_x

s ^

²

; ^ s

_z

= 0

153

singlet combination:

p1

2((1)(2) (2)(1)) multiplicity: 1

triplet combinations:

(1)(2)

p1

2((1)(2) + (2)(1))

(1)(2)

multiplicity: 3

In general, if ^J = ^J

₁

+ ^J

₂

!

j = j j

₁

j

₂

j; j j

₁

j

₂

j + 1; : : : ; j j

₁

+ j

₂

j

155

Time dependent perturbation

Let's suppose that the stationary system is eected by a small time-dependent external force (perturbation, ^K(t)):

~ i @

@t +

H^₀+ ^K(t) = 0

The eigenfunctions of the unperturbed Hamiltonian are r, H^₀ r =Er r. Att= 0 the system is in state i.

Due to the perturbation attthe wave function is the lin. comb. of the eigenstates of ^H₀: =P

rcr(t) re ^~iErt, wherecr(t= 0) = ir, i.e., ci(t= 0) = 1 andcr(t= 0) = 0 ifr6=i.

Time dependent perturbation

One can easily show that ^dc_dt^k = _~ⁱ P

rKkrcre^i!krt, where !kr =^Ek_~^Er andKkr =R

kK(t) ^ rd.

As a "rst order" approximation at the rhs of the

dc_k

dt = _~ⁱ P

rKkrcre^i!krt equationcr is set to zero exceptci which is one.

Integrating the ^dc_dt^k = _~ⁱKkie^i!kit equations with respect to time, the newc_k⁽¹⁾(t) = ki i

~

Rt

0Kki()e^i!kid denes the transition probability:

W(i!k) = jck(t)j²= _~¹2R^t

0Kki()e^i!kid², ifi 6=k.

157

Electric dipole transition

H atom in visible light. E eld is homogeneous in the scale of the H atom.

Potential energy in the electric eld:Epot=e =R

(r)(r)d³r, where is the density of electric chargeEpot =R

(r)(r)d³r= R(r)((0) + rjr=0 r +¹₂Pi;j=x;y;z

i;j @²

@x_i@x_jjr=0xixj+ : : : )d³r. As the total charge is zero and derivatives of E is supposed to be small,

R

Electric dipole transition

Transitions induced by a light beam, perturbation operator:

K^=eExx sin^ (!t) !Kkr=eExxkrsin(!t)

W(i!k) =^e2_~^E2^x2jxkij²R₀^tsin(!t)e^i!kid², wheresin(!t) can be replaced by ₂¹_i e^i!t e ^i!t

W(i!k) =^e_4~^2E2^x2jxkij²Rt

0e^i((!ki+!)d Rt

0e^{i((!ki !)}d²

The above transition probability large if ! !ki or ! !ki: absorption and induced emission of a photon.

The transition probability is proportional to the square of the transition dipole moment: exki=R

kex rd

159

Electric dipole transition

if axki is zero the k ) i transition is called forbidden.

As an example, investigate the n=1;`=0;m<sub>`</sub>=0;m_s=¹₂ ) n=2;`=0;m<sub>`</sub>=0;m_s=¹₂

transition! x_1;0;0;1

2)2;0;0;¹₂ =R

2;0;0;¹₂x _1;0;0;1

2d. The value of this integral is zero because of the symmetry. _2;0;0;1

2 and _1;0;0;1 2 are symmetric functions, e.g., _2;0;0;1

2(r) = _2;0;0;1

2( r), on the other hand x is anti-symmetric.

Similarly, s ) s, p ) p, d ) d, : : : transitions are all forbidden.

The selection rules for the hydrogen atom:

`<sup>0</sup> = ` 1,m⁰_{`</sub>=m<sub>`};m_` 1, andm⁰_s=ms

Pauli exclusion principle

Pauli exclusion principle (postulate VI of quantum mechanics):

No more than two electrons may occupy any given orbital, and if they do so, their spins must be paired

There cannot exist two electrons having the same set of quantum numbers

The total wavefunction must be antisymmetric with respect to the interchange of all coordinates of two electrons (fermions)

161

Pauli exclusion principle

(x

₁

; x

₂

; : : : ; x

i

; : : : ; x

j

; : : : ) = (x

₁

; x

₂

; : : : ; x

j

; : : : ; x

i

; : : : ),

where x

_i

is a composite notation for the spatial coordinates and the

spin, x

i

= (r

i

; ).

He ground state: 1s² (xed nucleus, independent particle approximation)

H^H= 1 2r² 1

r H^He= 1

2r²₁ 1 2r²₂ 2

r₁ 2 r₂+ 1

r₁₂ = ^h₁+ ^h₂+ 1 r₁₂ For the sake of simplicity thee e interac. is neglected:

H^He^approx= ^h₁+ ^h₂

(1; 2) = (r₁;r₂) = ₁(r₁) ₂(r₂) = ₁(1)₂(2);

these are H atom-like wavefunctions (see page 126)

^hⁱi =Eii

E^approx=E₁+E₂; hereE₁andE₂are the H atom-like energies (Z=2)

163

He ground state: 1s² (xed nucleus, independent particle approximation)

let's label the electrons ^a(1) = 1s(1)(1) and b(2) = 1s(2)(2)

ground(1; 2) = 1s(1)(1) 1s(2)(2) It is not anti-symmetric!

1ground(1; 2) =^p1₂(1s(1)(1) 1s(2)(2) 1s(2)(2) 1s(1)(1)) = 1s(1)1s(2)^p1₂((1)(2) (2)(1)).

It is the only possible anti-symmetric wave function. ¹_groundis the eigenfunction of the ^Sz= ^Sz(1) + ^Sz(2) and ^S² spin operators withms= 0 ands= 1 quantum numbers.

He excited states

aand bare the occupied atomic orbitals

Degenerate product states (e-e interaction is not considered):

1(1; 2) = a(r1)b(r2), 2(1; 2) = a(r2)b(r1) These are orthogonal to each other,R

dr₁³R

dr₂³1(1; 2)2(1; 2) = 0, and degenerate withE^approx:=Ea+Eb energy:

(^h₁+ ^h₂)1= (^h₁a(r1)b(r2) + a(r1)^h₂b(r2)) = Eaa(r1)b(r2) + a(r1)Ebb(r2) = (Ea+Eb)1

To include the e-e interaction the wave function can be approximated by a linear combination: =b₁1+b₂2

(^h₁+ ^h₂) = (Ea+Eb) =) (^h₁+ ^h₂+ ^V) = (Ea+Eb+ ^V)

165

He excited states

Introducing some shorthand notations:

V^=_r12¹ C=D

1j ^Vj1

E=D 2j ^Vj2

E=R d³r1R

d³r2ja(r1)j²jb(r2)j²

r₁₂ ,

K=D

1j ^Vj2

E=D 2j ^Vj1

E

=R d³r1R

d³r2_a(r1)_b(r2)b(r1)a(r2)

r₁₂ ,

E = (E Ea Eb)

Ea+Eb+ ^V

=E ) ( E+ ^V) = 0

b₁

E+ ^V

1+b₂

E+ ^V

2= 0

He excited states

h1j = )b₁

E+ ^V

1+b₂

E+ ^V

2= 0 h2j = )b₁

E+ ^V

1+b₂

E+ ^V

2= 0 b₁

D1j ^Vj1

E E

+b₂

D1j ^Vj2

E= 0 b₁

D2j ^Vj1

E+b₂

D2j ^Vj2

E E

= 0 The result is a simple homogeneous linear equation:

b₁(C E) +b₂K= 0 b₁K+b₂(C E) = 0

To have a non-trivial solution the determinant of the coecient matrix should be zero: (C E)² jKj²= 0

167

He excited states

We obtained two solutions for the energy: E =C jKj or E=Ea+Eb+C jKj

If E=C+ jKj thanb₁=b₂=^p1₂ ) singlet state.

If E=C jKj thanb₁= b₂=^p1₂ ) triplet state.

Pauli exclusion principle )

1= ^p1₂(a(r1)b(r2) + a(r2)b(r1)) ((1)(2) (2)(1))

3= ^p1₂(a(r1)b(r2) a(r2)b(r1)) ((1)(2) + (2)(1))

He excited states

What are the meaning of theC andKcoecients?

C=R d³r1R

d³r2ja(r1)j²jb(r2)j²

r₁₂ is the classical coulomb interaction of two charged particle. It is always a positive quantity.

K=R d³r1R

d³r2a(r1)_b(r2)b(r1)a(r2)

r₁₂ is the so-called exchange interaction, no classical analog.

In the ground state, a= b= n=0;`=0;m<sub>`</sub>=0, only the singlet combination, ¹ can appear.

In the rst excited state a= n=0;`=0;m<sub>`</sub>=0and b= n=1;`=0;m<sub>`</sub>=0.

169

He excited states

For states arising from the same conguration, the triplet state generally lies lower than the singlet state (see Hund's rule). Qualitative explanation:

3(r1; r1) = 0, i.e., the two electrons can not be at the same place. $

parahelium, orthohelium

Excitation of both of the electrons requires an energy larger than the ionization energy: only 1s¹nl¹ excitations appear in the spectra No radiative transitions between singlet and triplet states

Spectroscopically, He behaves like two distinct species, parahelium and orthohelium

171

the easy way to build antisymmetric wavefunctions

ground= 1s1s[(1)(2) (1)(2)]

1s(1)(1) 1s(1)(1) 1s(2)(2) 1s(2)(2)

= 1s(1)(1)1s(2)(2) 1s(1)(1)1s(2)(2)

= 1s(1)1s(2)[(1)(2) (1)(2)]

rows ! electrons columns ! spinorbitals

Determinant

A homogeneous system of linear equations:

c₁₁x₁+c₁₂x₂+c₁₃x₃+ : : : c_1nxn= 0 c₂₁x₁+c₂₂x₂+c₂₃x₃+ : : : c₂nxn= 0

... ...

cn1x₁+cn2x₂+cn3x₃+ : : : cnnxn= 0 Matrix notation: C x = 0, where

C = 0 BB BB BB

@

c₁₁ c₁₂ : : : c₁n

c₂₁ c₂₂ : : : c₂n

... ... ... cn1 cn2 : : : cnn

1 CC CC CC A

; x =

0 BB BB BB

@ x₁ x₂ ... xn

1 CC CC CC A

173

Determinant

Formal solution of a inhomogeneous system of linear equation, C x = b, needs the inverse of matrix C: x = C ¹ b

C ¹= adj(C)=det(C) (see wikipedia page: Invertible matrix) To have a non-trivial solution of the homogeneous system of linear equation, the matrix C ¹ should not exist. ! det(C) = 0

det(C) = X

fp1;p2;:::;png

( 1)^pc₁p₁c₂p₂c₃p₃: : :cnpn, where the sum runs on the whole set of permutations of numbers 1; 2; 3; : : : ;nandp is the parity (number of exchange of indices requiered to obtain the given

permutation) of the given permutation.

Determinant

Some properties of determinants:

det(AB) = det(A)det(B)

det(A

^T

) = det(A), where A

^T

denotes the transpose of A.

If matrix A is composed from column vectors,

A = ([a

₁

] ; [a

₂

] ; [a

₃

] ; : : : ; [a

n

]), and vectors [a

i

] are linearly dependent than det(A) = 0.

det([a

1

] ; [a

2

] ; : : : ; [a

i

] ; : : : ; [a

j

] ; : : : ; [a

n

]) = det([a

₁

] ; [a

₂

] ; : : : ; [a

j

] ; : : : ; [a

i

] ; : : : ; [a

n

]).

175

Determinant

Expansion of a determinant along a column (or a row):

a₁₁ a₁₂ a₁₃ : : : a_1n a₂₁ a₂₂ a₂₃ : : : a_2n a₃₁ a₃₂ a₃₃ : : : a_3n

.. . ...

a_n1 a_n2 a_n3 : : : a_nn

= ( 1)¹⁺²a₁₂

a₂₁ a₂₃ : : : a_2n a₃₁ a₃₃ : : : a_3n

.. .

a_n1 a_n3 : : : a_nn

+( 1)²⁺²a₂₂

a₁₁ a₁₃ : : : a_1n a₃₁ a₃₃ : : : a_3n

.. .

a_n1 a_n3 : : : ann

+ ( 1)³⁺²a₃₂

a₁₁ a₁₃ : : : a_1n a₂₁ a₂₃ : : : a_2n

.. . ...

a_n1 a_n3 : : : ann

+ ( 1)⁴⁺²a₄₂ : : :

Li atom

Li= 1 p3!

1s(1)(1) 1s(1)(1) 2s(1)(1) 1s(2)(2) 1s(2)(2) 2s(2)(2) 1s(3)(3) 1s(3)(3) 2s(3)(3) rows ! electrons

columns ! spinorbitals

177

Li atom

if two columns are equal - three electrons are on one spatial orbital the Pauli exclusion principle is not fullled

Li= 1 p3!

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

= 1s(1)(1)

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

1s(2)(2) 1s(2)(2) 1s(3)(3) 1s(3)(3) + 1s(1)(1)

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

= 0

Li atom

if two rows are interchanged - the determinant changes sign antisymmetric wavefunction

1st row expansion

_Li=

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

= 1s(1)(1)

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

1s(2)(2) 2s(2)(2) 1s(3)(3) 2s(3)(3) + 2s(1)(1)

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

2nd row expansion

^1!2_Li =

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

= 1s(1)(1)

1s(2)(2) 2s(2)(2) 1s(3)(3) 2s(3)(3) + 1s(1)(1)

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

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

179

General properties

The electrons are indistinguishable...

The individual one-particle orbitals have no physical meaning:

the Slater determinant is invariant with respect to any

orthogonality and scalar product keeping linear combination of

the original orbitals.

Hamiltonian

^H = X^N

i

1 2r²i

XN i

ZA

RiA+ XN

i

XN j>i

1 rij + Hso

Energy of atoms is basically n dependent, moderate dependents on L, S values and slightly dependents on J value (light atoms).

Spherical symmetry =) ^J²and ^Jz commute with the Hamiltonian: Jand MJ are good quantum numbers.

Without Hso theL,ML,S,MS are also good quantum numbers

181

Aufbau/building-up principle, diagonal rule

orbitals with a lowern+ ` value are lled before those with highern+ ` values

in the case of equaln+ ` values, the orbital with a lowernvalue is lled rst

Examples: He, Li, C, N, O

In most of the cases the hr₃di < hr₄si =) on the 3d orbitals the e-e repulsion is stronger: Sc, [Ar] 3d¹4s²

There are exceptions too: Cu, 1s²2s²2p⁶3s²3p⁶4s²3d⁹is predicted instead of 1s²2s²2p⁶3s²3p⁶4s¹3d¹⁰

Due to the e-e interaction the shell-, sub-shell conguration can not describe the atomic spectra (see He atom)

Atomic term symbols, vector and scalar,z-projection, additions

total orbital angular quantum number

^L = P ^`i orM_L=P

m_`i,M_L= 0; 1; 2; : : : ; L

L= 0 1 2 3 4

S P D F G

total spin angular momentum quantum number

^S = P ^si orMS =P

ms_i,MS = 0; 1; 2; : : : ; S total angular quantum number

^J = ^L + ^S,M_J = 0; 1; 2; : : : ; J

maxfM_Lg =LandmaxfM_Sg =S; (2L+ 1)(2S+ 1) =P

J2J+ 1

183

Atomic term symbols, Clebsch-Gordan series

total orbital angular quantum number L = `

₁

+ `

₂

; `

₁

+ `

₂

1; :::; j`

₁

`

₂

j

total spin angular momentum quantum number S = s

₁

+ s

₂

; s

₁

+ s

₂

1; :::; j s

₁

s

₂

j

total angular quantum number

J = L + S ; L + S 1; :::; j L S j

Atomic term symbols

atomic term symbol:

^2S+1

L

_J

term:

^2S+1

L

microstate: a unique conguration of quantum numbers n = num of spin orbitals; k = num. of electrons

number of microstates:

_kⁿ

multiplicity: 2 S + 1

S= 0 1/2 1 3/2

2S+1= 1 2 3 4

singlet doublet triplet quartet

185

H electronic transitions,²S₁₌₂,²P₁₌₂,²P₃₌₂,²D₅₌₂,²D₃₌₂, etc.

Atomic term symbols, helium atom

187

Atomic term symbols,^2S+1LJ

1s²:¹S₀ 2p⁶: ¹S₀ 3d¹⁰: ¹S₀ 1s¹:²S₁₌₂

1s²2s²2p¹, i.e. [Ne]2p¹: ²P₃₌₂,

2P₁₌₂

atoms with closed subshells are in the¹S₀ state

atoms with onee in an open subshell n` are in the²Lstate In general, the open subshells dene the atomic term

Atomic term symbols, non-relativistic case, LS / Russel-Saunders coupling

In the non-relativistic case^2S+1Ldenes the energy.

The relativistic eects (e.g., spin-orbit coupling) are small perturbations.

The spin-orbit coupling for the individual electrons is small. An average can be calculated using the total ^Land ^Soperators: ^Hso=A(L;S)^L ^S.

The energy leveles are splitted according to the various values ofJ:

Eso =¹₂A(L;S)(J(J+ 1) L(L+ 1) S(S+ 1))

As in a given term theLandSare constant (and J= 1) the observable splitting isE(J) E(J 1) =A(L;S)J.

=) Fine or multiplett structure of the spectra

189

Atomic term symbols,^2S+1LJ, spin-orbit coupling

Relativistic case, jj-coupling

In the relativistic case (Z1) the spin-orbit eect dominates over the e -e repulsion, thus P

i<j 1

rij

can be considered as a perturbation.

H ^

_so

= P

i

i

^`

i

^ s

_i

= P

i i

2

(^ j

_i²

`

²_i

s

_i²

).

Spin and orbital momenta of the electrons coupled into ^ j

_i

eigenfunctions. The anti-symmetrized products of these functions are the eigenfunctions of the zero-order Hamiltionian ( ^ H without the e -e repulsion).

E

_so

= P

i i

2

( j

_i

( j

_i

+ 1) `

i

(`

i

+ 1) s

_i

( s

_i

+ 1)).

The good quantum numbers are J , j

₁

, j

₂

, etc.

191

LS- and jj-coupling

Coupling of `=1 and s=1/2 results in either a j=1/2 or a j=3/2 state.

Possible J values:

j₁ j₂ J

1=2 1=2 0; 1

1=2 3=2 1; 2

3=2 3=2 0; 1; 2; 3

Hund's rules

an atom in its ground state adopts a conguration with the greatest number of unpaired electrons

193

Hund's rules

Rules to determine the lowest state for a given electron conguration the term of highestS(maximum multiplicity, 2S+ 1) will lie lowest in energy

if more than one term exist with maximum multiplicity then the term having the highestLwill lie lowest in energy

for terms having a spin-orbit splitting, if the outermost subshell is half-full or less than half-full the states will be ordered with the lowestJvalues lying lowest; if the outermost subshell is more than half-lled, the level with the highest value ofJ, is lowest in energy

Selection rules for electronic transitions

transition dipole moment:

^

= e X

electrons

^r

fi

=

Z

f

^

i

d one electron

s = 0

` = 1; m

_`

= 0; 1

multi electron S = 0 L = 0; 1

J = 1; 0, J = 0 = J = 0

195

2S+1

L

_J

any atomic state can be specied

any spectroscopic transition can be described

Purpose: analysis of the elementary composition.

Sample preparation: heating to high temperature.

Atomic absorption spectroscopy and atomic emission spectroscopy Concentration of atoms can be measured (BeerLambert law[see later]/intensities)

197

Composition of stars

Relative speed and temperature of stars and galaxies.

argument: elephant herd and the ies

electrons

light particles fast

nuclei

heavy particles slow

199

In document Physical Chemistry and Structural Chemistry (Pldal 118-200)