Theoretical Chemistry

Theoretical Chemistry

Marek Kręglewski

Wydział Chemii UAM

Course content

I. Experimental background of quantum mechanics II. Observables and operators

III. Models in quantum chemistry IV. Hydrogen atom and orbitals

V. Multi-electron atom, spin and Pauli Exclusion Principle VI. Molecule H₂+

VII. Two-atomic molecules VIII. Ab initio calculation IX. Vibrations of molecules X. Rotations of molecules

Literature

1) Peter Atkins & Julio de Paula, Atkin’s Physical Chemistry (Part2),

Oxford University Press 2014

2) Andrew R.Leach, Molecular Modelling – Principles and

Applications, Pearson Education Limited 2001

3) Lucjan Piela, Idee chemii kwantowej, PWN, Warszawa 2001

4)

Włodzimierz Kołos, Chemia kwantowa, PWN, Warszawa 1991.

Black body emission spectrum

0 100 200 300 400 500 0 500 1000 1500 2000 2500 3000 3500 u(20) u(100)

 

1

8

,

₃ 2







kT h

e

h

c

T

u





_



Spectral density:

Planck hypothesis (1900): ΔE=hν (quantum of energy)

h = 6,62∙10-34 J s c = 2,99792458∙108 _{m s}-1 k = 1,380662∙10-23 _{J K}-1

T

b



max



max



Photoelectric effect

(-) (+)

spark

hν The Lenard laws (1899)

1) The number of emitted electrons is proportional to the intensity of light 2) The maximum velocity of electrons

depends on wavelength not on intensity of light

Einstein formula (1905, Nobel prize in 1921):

hν = ½ m

_e

v

2

_{+ W}

Zn

„In fact, it seems to me that the observations on "black-body radiation", photoluminescence, the

production of cathode rays by ultraviolet light and other phenomena involving the emission or conversion of light can be better understood on the assumption that the energy of light is distributed discontinuously in space. According to the assumption considered here, when a light ray starting from a point is

propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localised in space, which move without being divided and which can be absorbed or emitted only as a whole.”

Photoelectric effect

Es scheint mir nun in der Tat,

daß die Beobachtungen über die

„schwarze Strahlung‟, Photolumineszenz, die Erzeugung von

Kathodenstrahlen durch ultraviolettes Licht und andere die

Erzeugung

bez.

Verwandlung

des

Lichtes

betreffende

Erscheinungsgruppen besser

verständlich erscheinen unter der

Annahme,

daß die Energie des Lichtes diskontinuierlich im Raume

verteilt sei. Nach der hier ins Auge zu fassenden Annahme ist bei

Ausbreitung eines von einem Punkte ausgehenden Lichtstrahles die

Energie nicht kontinuierlich auf

größer und größer werden der

Räume verteilt, sondern es besteht dieselbe aus einer endlichen

Zahl von in Raumpunkten lokalisierten Energiequanten, welche sich

bewegen, ohne sich zu teilen und nur als Ganze absorbiert und

erzeugt werden können.

"Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt”. Albert Einstein, Annalen der Physik. Vol. 322 No. 6 (1905): 132–148.

Compton effect / scattering (1923)

θ φ λ λ’ m_e v 

A

m

c

m

h

e

02426

,

0

10

426

,

2

2

sin

2

12 2

































λ’ > λ

p

_e

= m

_e

v

p

_f

= h/λ

p

_e

= p

_f

m

_e

v = h/λ

Spectrum of the hydrogen atom

ΔE(i←j) = T

_i

– T

_j

λ = hc / ΔE

de Broglie hypothesis (1923)



h

p



Why is the duality not observed in classical mechanics? m = 2g v = 1000 m/s p = 2 kg m /s h = 6,62∙10-34 J s c = 2,99792458∙108 _{m /s} λ = (6,62 ∙10-34 / 2) m = 3,31∙10-25 _nm

p

h





Wave-particle duality: a key concept of

quantum mechanics

Uncertainty principle

2











p

_x

x











E

t

1923 - Werner Heisenberg Exact formula:

4

2 2 2











p

_x

x

Wavefunction

Postulate I

All information about a system are included in its wavefunction.

The square of the wavefunction│ Ψ│2 _{describes the probability density}







  





x

,

y

,

z

2

dxdydz

1

Operators in Quantum Mechanics

Postulate II

Operators of position and momentum

x

i

p

x

x

x















ˆ

ˆ

The operator is defined through its action on a function.

In order to build an operator of a complex physical quantity the positions and momenta in classical Newtonian expression are replaced by corresponding operators.

Hamiltonian – the operator of total mechanical energy







ˆ ˆ ˆ



ˆ( , , ) 2 1 ˆ ˆ ˆ ) , , ( 2 1 ) , , ( 2 ) , , ( 2 2 2 2 2 2 2 2 2 z y x V p p p m V T H z y x V p p p m z y x V m p z y x V v m E E E z y x z y x pot kin total                  

Time evolution of a wavefunction

Postulate III

t

i

H













ˆ

If potential energy does not change with time the Schrödinger equation converts to:







E

H

ˆ

Time evolution

Values of observables

Postulate IV

If a wavefunction Ψ is the eigenfunction of the operator Â





a

A

ˆ



The experimental measurement can give the value a, only

.

Expectation (mean) value of the operator





       



dV

dV

A

A









ˆ

ˆ

_A

_ˆ

_

_a

_{when Ψ is an eigenfunction} of the operator Â

a

A

ˆ



when Ψ is not an eigenfunction

Model 1: Free particle in space

x 2 2 2 2 2 2 2

2

2

1

0

2

ˆ

ˆ

ˆ

ˆ

0

2

0

2

dx

d

m

dx

d

i

m

m

p

V

T

H

m

p

mv

E

E

E

x x x pot kin total











































 

 

 

E

 

x

dx

x

d

m

x

E

x

H











₂





2 2

2

ˆ



 

 

iax iax

Ne

x

Ne

x







2 1





Test of solutions:

 

 

 

 





m

a

E

ENe

e

Na

m

e

Na

e

ia

N

x

dx

d

Niae

x

dx

d

Ne

x

iax iax x i iax iax iax

2

2

2 2 2 2 2 2 1 2 2 1 1































Model 2: Particle in a box

0 a x I V= II V=0 III V=

 

 

 

0 sin 2 0 ˆ           x x a n a x x E H III II I       2 2 2 2 2ma n E   

The motion of a particle limited to the well <0,a>.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 n2

Particle in a box

2 2 2 2 0 2 0 2 2 2 0 2 0 2 2 2 0 0 2 2 2 2 sin 2 sin 2 2 sin 2 sin sin 2 2 sin 2 sin 2 sin 2 2 sin 2 n ma dx x a n a dx x a n a n m a dx x a n a dx x a n dx d x a n m a dx x a n a x a n a dx x a n a dx d m x a n a E _a a a a a a                                                                                                                 

  sin cos 1  sin 2 0

2 sin 2 sin 2 0 0 0                                             x dx a n a n i a dx x a n x a n a n i a dx x a n a dx d i x a n a p a a a x          

 

 

2 2 2 2 0 2 2 0 2 2 2 0 2 2 2 2 2 cos 1 2 1 2 sin 2 sin 2 sin 2 n a dx x a n a n a dx x a n a n a dx x a n a dx d x a n a p a a a x            _                                                     _{  }

Expectation values of energy, momentum, momentum squared and position

2 2 2 cos 2 2 1 2 sin 2 2 1 2 2 sin 2 2 1 2 sin 2 sin 2 sin 2 0 2 2 0 0 0 2 0 a a a x a n n a x a a dx x a n n a x a x a n n a x x a dx x a n x a dx x a n a x x a n a x a a a a a                                                                           _{  }         

Particle in a box

                                                                                   2 2 2 0 2 2 2 2 2 2 0 0 3 0 0 2 0 3 0 2 0 2 0 2 2 0 2 2 2 1 3 1 2 sin 2 2 2 3 2 cos 2 1 2 cos 2 1 3 2 sin 2 2 1 2 sin 2 1 3 1 2 cos 2 1 2 2 1 2 sin 2 sin 2 sin 2                    n a x a n n a n a n a a dx x a n n a n x a n n a x n a dx x a n x n a a x a n n a x a x a dx x a n x a dx x a dx x a n x a dx x a n a x x a n a x a a a a a a a a a a



 





 

 

 









4 2 3 1 4 2 1 12 1 2 1 4 1 3 1 0 2 2 1 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2             _         _        _{ }        _                                               n n n n n a n a n a p p x x p p x x p p p p x x x x p p x x p x p p p x x x x x x x x x x x x x x x x x       

Test of the uncertainty principle for the particle in a box: Expectation values of position squared

Particle in a box

                2 2 4 2 3 2 2 1 sin 2 x a Nx x x a Nx x x a Nx x x a Nx x x a n a x                      Eigenfunction (exact)

Trial functions (approximate):

        xa x a x a N a N a N a a a N x x a x a N dx x ax x a N dx x a x N a a a             _{ }        _{ }      _  5 1 5 5 2 5 2 5 5 5 2 0 5 4 3 2 2 0 4 3 2 2 2 0 2 2 30 30 30 1 30 5 4 2 3 5 4 2 3 2          x a x a x a N a N a N a a a N x x a x a N dx x ax x a N dx x a x N a a a                             _  2 7 2 7 7 2 7 2 7 7 7 2 0 7 6 5 2 2 0 6 5 4 2 2 0 2 2 2 105 105 105 1 105 7 3 5 7 6 2 5 2   





     2 7 3 7 7 2 7 2 7 7 7 7 7 2 0 7 6 5 2 4 3 3 4 2 0 6 5 4 2 3 3 2 4 2 0 2 2 2 105 105 105 1 105 7 3 2 5 6 3 7 6 4 5 6 4 4 3 4 6 4 x a x a x a N a N a N a a a a a N x x a x a x a x a N dx x ax x a x a x a N dx x a x N a a a                                   _    





    1 630 5 1 3 2 7 6 2 1 9 1 5 6 4 7 6 8 4 9 4 6 4 4 2 4 4 9 2 9 2 0 5 4 6 3 7 2 8 9 2 0 4 4 5 3 6 2 7 8 2 0 7 6 2 5 3 8 6 2 4 4 2 0 2 2 2 2         _{   }                             a N a N x a x a x a x a x N dx x a x a x a ax x N dx ax x a x a x x a x a N dx x a x N a a a a   2 2 9 4 9 9 2 630 630 630 x a x a x a N a N     

Particle in a box

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 sin f1 f2 f3 f4 The graph of the eigenfunction and trial functions

Particle in a box

2 2 2 2 2 0 2 2 0 2 2 2 0 2 2 2 2 87 , 9 2 2 cos 1 2 1 2 2 sin 2 2 sin 2 2 sin 2 ma ma dx x a a m a dx x a a m a dx x a a dx d m x a a a a a      _{ }                                                                            2 2 2 1 2ma E                    2 2 9 4 2 7 3 2 7 2 5 1 630 105 105 30 sin 2 x a x a x x a x a x x a x a x x a x a x x a a x                            2 2 0 3 2 2 5 0 2 5 5 0 2 2 2 5 2 10 3 2 2 30 2 30 30 2 30 ma x x a m a dx x a x m a dx x a x a dx d m x a x a a a a     _                                             2 2 0 5 4 3 2 2 7 0 4 3 2 2 2 7 0 2 2 7 2 7 0 2 2 2 2 7 2 14 5 6 4 8 3 2 2 105 6 8 2 2 105 6 2 2 105 105 2 105 ma x x a x a m a dx x ax x a m a dx x a x a x m a dx x a x a dx d m x a x a a a a a                                         _        _                 2 2 0 5 4 3 2 2 3 2 7 0 4 3 2 2 3 2 7 0 2 2 7 2 7 0 2 2 2 2 7 2 14 5 6 4 16 3 14 2 4 2 105 6 16 14 4 2 105 6 4 2 105 105 2 105 ma x x a x a x a m a dx x ax x a x a m a dx x a x a x m a dx x a x a dx d m x a x a a a a a            _{   }      _           _           _        _                 2 2 2 2 0 7 6 5 2 4 3 3 4 2 9 0 6 5 4 2 3 3 2 4 2 9 2 2 0 2 2 2 9 2 2 9 0 2 2 2 2 2 9 2 12 7 12 6 36 5 38 4 16 3 2 2 630 7 12 6 36 5 38 4 16 3 2 2 630 12 36 38 16 2 2 630 12 12 2 2 630 630 2 630 ma m a x x a x a x a x a m a dx x ax x a x a x a m a dx x ax a x a x m a dx x a x a dx d m x a x a a a a a             _{    }             _{    }                                              

Two-dimensional box

x y a b 0 V=0 V=  V=  V=  V= 

 

 

 

   

   

   

   

   

   

 

 

 

 

   

   

 

 

 

 

 

 

 

 

 

 

 

y E

 

y y m x E x x m E E E E y y y m E x x x m E y y y m x x x m y x y x E y y x m x x y m y x E y x y m y x x m y x E y x y m x m y x y x y x E y x y m x m y y y x x x y x y y y x x x y y x x y x y x y x x y y x y x y x y x y x y x                                                                                                      2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 / 2 2 2 2 2 2 , , , 2 2              

 

 

         x a n a x a n m n E_x  _n 2 sin  2 2 2 2 2 

Two-dimensional box

x y a b 0 V=0 V= V= V= V=

 

                      y b k b x a n a y x b k a n m E_nk  _nk , 2 sin  2sin  2 2 , 2 2 2 2 2 ,  Degenarate states: If a=b then E_1,2= E_2,1

 

                      y a a x a a y x a a m E    2 sin 2 sin 2 , 2 1 2 2 , 1 2 2 2 2 2 2 2 , 1 

 

                      y a a x a a y x a a m E    sin 2 2 sin 2 , 1 2 2 1 , 2 2 2 2 2 2 2 1 , 2 

Hydrogen atom – classical picture

e c

F

r

e

r

m

F





₂



2 0 2

4

1

v



1 15

10

.

6

1

_





s

T







1/2 6 1 0

10

.

2

4

v





ms



mr

e



eV

V

T

E







13

.

6

Reduced mass



motion of the reduced mass around a center of mass

mr

e



Mr

N

r



r

e



r

N



n

r

M

r

m



_e2





_N2



 n r M m mM   2









M

m

mM



e

m

/



0.99945 (H)

0.99972 (D)

Hydrogen atom

2 2 2 2 2 2 2 2 2 2 2 e

z

y

x

r

r

e

z

y

x

2

H

ˆ













































Schrödinger equation:

)

z

,

y

,

x

(

E

)

z

,

y

,

x

(

H

ˆ

_e







Spherical coordinates:

x = r sinθ cosφ

y = r sinθ sinφ z = r cosθ z P x y r θ φ

Hydrogen atom 1



 

2

 

2



2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2

2

ˆ

e j e j e j e e e e j j j j

z

z

y

y

x

x

r

r

e

z

y

x

m

z

y

x

M

H





















































































Center of mass coordinates: e j e e j j e j e e j j e j e e j j m M z m z M Z m M y m y M Y m M x m x M X          Relative coordinates: 2 2 2 z y x r z z z y y y x x x j e j e j e          x z y j e x_e x_j z_j y_j z_e y_e r



x

j

y

j

z

j

x

e

y

e

z

e



E

c



x

j

y

j

z

j

x

e

y

e

z

e



H

ˆ



,

,

,

,

,





,

,

,

,

,

Hydrogen atom 2

        m M X x m M x X M m x M X M m M x X M m x m X M m m x M x m x X M m M x X M m M x X M m M x x X M m m x X M m m x X M m m x x X M m M x x x X x X x x X M m m x x x X x X x j e j e j j j e e e 1 1 1 2 2 2 1 2 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2                                                                                                                                                          

Transformation of the Hamiltonian to the center of mass and relative coordinates

Similarly for Y, Z, y, z:





r e z y x Z Y X m M H e j 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ˆ _                                  

Hydrogen atom 3





r

e

z

y

x

Z

Y

X

m

M

H

e j 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2

2

ˆ

_

_





































































Schrödinger equation after the separation of coordinates:









)

,

,

(

)

,

,

(

ˆ

,

,

,

,

ˆ

z

y

x

E

z

y

x

H

Z

Y

X

E

Z

Y

X

H

e trans trans













Translation of atom relative motion of nucleus and electron

E

_total

=E

_tr

+E



x

_j

,

y

_j

,

z

_j

,

x

_e

,

y

_e

,

z

_e









X

,

Y

,

Z

,

x

,

y

,

z









X

,

Y

,

Z

 



x

,

y

,

z





Hydrogen atom 4

Schrödinger equation:

)

z

,

y

,

x

(

E

)

z

,

y

,

x

(

H

ˆ

_e







Współrzędne sferyczne: x = r sinθ cosφ y = r sinθ sinφ z = r cosθ

r

e

z

y

x

H

_e 2 2 2 2 2 2 2 2

2

ˆ

_

_



































_{2 2 2}

z

y

x

r







0≤r<, 0≤θ≤π, 0≤φ<2π z P x y r θ φ

Hydrogen atom 5

r

e

sin

1

sin

sin

1

r

r

r

r

2

H

ˆ

2 2 2 2 2 2 2 e



















































































)

,

,

r

(

E

)

,

,

r

(

H

ˆ

_e















   















(

r

,

,

)



R

(

r

)

After the separation a set of 3 equations:

 

 

  

  





R

 

r

E

R

 

r

r

e

l

l

r

r

r

r

l

l

m

m

i

nlm n nlm lm lm m m





























_

_















































































2 2 2 2 2 2 2

1

2

1

sin

sin

sin

1





























The equation in spherical coordinates equations: azimuthal horizontal radial

Hydrogen atom 6

The boundary conditions generate quantum numbers

Asymuthal equation:

 













2







m=0,±1, ±2, ±3, … Horizontal equation:

 





Square-integrable l=0, 1, 2, 3, … m=-l,-l+1,…,0,…,+l Radial equation: R(r) Square-integrable n=1, 2, 3, … l=0,1,…,n-1 2 2 2 4

1

2

n

R

n

e

E









_H





Energy of the hydrogen atom ₂ 1

4

109677

2







e

cm

R

_H





Rydberg constant





2 2 1 0 4

3

.

109737

2

4

 





cm

e

m

R

e





Hydrogen atom 7

The wavefunctions of the H atom



_nlm

(

r

,



,



)



R

_nl

(

r

)

Y

_lm

 



,



Radial functions: 0 2 5 0 2 3 0 2 3 2 0 21 2 0 0 20 0 10 6 2 1 ) ( 2 2 2 1 ) ( 2 ) ( a Zr a Zr a Zr re a Z r R e a Zr a Z r R e a Z r R                                a₀ = 0,529 Ǻ = 0,529 · 10-10 _{m Bohr radius} -0.5 0 0.5 1 1.5 2 2.5 0 5 10 15 R10 R20 R21 a₀

Hydrogen atom 8

Radial probability density: R2_{(r) r}2

Volume element: dV = dx dy dz = r2 sinθ dr dθ dφ

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 12 14 (R10*r)^2 (R20*r)^2 (R21*r)^2 Notice: For l=n-1 there is a single maximum at r=n2_*a 0 Normalization integral:





_

 

_{ }

 

 

 





0 0 2 0 2 2 2 2 0 0 2 0 2

1

sin

,

sin

,

,

   























_nlm

r

r

dr

d

d

R

r

r

dr

Y

_lm

d

d

[a₀]

Hydrogen atom 9

Qualitative graphs of orbitals of type: s, p, d

s

p_z p_y p_x

d_x2-y2 d_xy d_xz

Hydrogen atom 10

Linear combinations of atomic orbitals Atomic orbitals              i a Zr p i a Zr p s a Zr p s a Zr s s a Zr s e re N e re N a Z N re N a Z N e a Zr N a Z N e N                                         sin 2 1 sin 2 1 4 1 cos 2 4 1 2 1 0 0 2 5 0 2 3 0 2 3 0 2 2 1 21 2 2 211 0 2 2 2 210 0 2 2 0 2 200 0 1 1 100

















0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 1 21 211 1 21 211 sin sin 2 sin 2 cos sin 2 sin 2 2 2 1 a Zr p a Zr p i i a Zr p y a Zr p a Zr p i i a Zr p x ye N re N i e e re N p xe N re N e e re N p i                                   

Spin

The Stern-Gerlach experiment

The beam of silver atoms passing through the magnetic field Electron configuration of silver:

Ag: 1s2_/2s2_2p6_/3s2_3p6_3d10_/4s2_4p6_4d10_/5s1 m_s = +½ m_s = -½ magnet Spinorbital s s nlm m nlmm









State of an electron α β

Indistinguishable particles

a,b – particles 1,2 - detectors a b 2 1

Probability of detecting different particles P₁=|φ_a(1) φ_b(2)|2

P₂=|φ_a(2) φ_b(1)|2

If particles identical P₁ = P₂ , then: φ_a(1) φ_b(2) = ± φ_a(2) φ_b(1) The particles can interfere with each other.

Indistinguishable particles

The amplitude of interference of identical particles:

Bosons φ_a(1) φ_b(2) + φ_a(2) φ_b(1) integer spin

Fermions φ_a(1) φ_b(2) - φ_a(2) φ_b(1) half-integer spin

The wavefunction for fermions is antisymmetrical with respect to permutation: Φ(1,2,3,…) = - Φ(2,1,3,…)

If two fermions occur in the same state1=2, thus φ_a(1) φ_b(1) - φ_a(1) φ_b(1) ≡ 0 It is the content of the Pauli exclusion principle.

The wavefunction for bosons is symmetrical.

Multielectron atoms

7s

7p

6s

6p

6d

5s

5p

5d

5f

4s

4p

4d

4f

3s

3p

3d

2s

2p

1s

Electron shells: n = 1,2,3,… → K,L,M,… l = 0,1,2,…→ s,p,d,… Hund’s rule:

For a given electron configuration, the lowest energy term is the one with the greatest value of spin multiplicity

Multielectron atoms

The electronic term 2S+1_L J

2S+1 means multiplicity, where S is the total spin.

How to determine the values of L, J, S ? J = L+S, L+S-1, … , |L-S| The carbon atom electronic configuration 1s2 _2s2 _2p2

Closed shells give the total spin S = 0

l₁ l₂ s₁ s₂ m₁ m₂ M_L L M_S S 1 1 +½ -½ 1 1 2 2 0 0 ½ ½ 1 0 1 1 +1,0,-1, 0 1 ½ ½ 1 -1 0 0 +1,0,-1, 0 0 +½ -½ 0 0 0 0 ½ ½ 0 -1 -1 +1,0,-1, 0 +½ -½ -1 -1 -2 0 Terms: 3_P 2, 3P1, 3P0, 1D2, 1S0

Helium atom1

12 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ˆ r e r e r e z y x m z y x m z y x M H e e j j j j                                                       12 2 2 2 1 2 2 2 1 2 2 2 2 2 2 2 ˆ r e r e r e m m M H e e j j              12 2 2 1 2 2 2 2 ˆ r e r e m H i i i e e          





 The electronic Hamiltonian in the

approximation of the infinitely heavy nucleus

 

1,2 ₁

   

1₂ 2

 The one-electron approximation

 

1 ₁

   

1 ₁ 1

1  

  Spinorbital = orbital * spin_function

 

   

   

 

   

   

 

1,2

 

2,1 1 2 1 2 1 , 2 2 1 2 1 2 , 1 1 2 2 1 1 2 2 1                 

 Antisymmetrized multi-electron function

 

H

   

i i E

 

i r e m i H _{n n n} i i e e     ˆ    2 2

Helium atom 2

 

1,2  

   

1,2  1,2 

 



   

   



 

 



   

1 2

   

1 2



 

2,1 2 1 2 , 1 1 , 2 2 1 2 1 2 1 2 , 1 1 2 2 1 1 2 2 1 a a s s                   

Spinorbital function = orbital function * spin function

Symmetry of the orbital function

 



       



 

 

   

 

 

   

 

 



       

1 2 1 2



 

2,1 2 1 2 , 1 1 , 2 2 1 2 , 1 1 , 2 2 1 2 , 1 1 , 2 2 1 2 1 2 1 2 , 1 s s s s s s a a                                  

Symmetry of the spin function

 



   

   





       



 



   

   



   

   

       





               2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 , 1 2 1 2 1 2 1 2 1 2 1 2 1 2 , 1 1 2 2 1 triplet 1 2 2 1 singlet                     and The singlet (S=0) and triplet (S=1) functions

Multielectron atoms

 



               1 1 1 2 1 2 2 2 ˆ n i n i j ij n i i i e e r e r ne m H 

 



   

   



 

_{ }

 

_{ }

2 1 2 1 2 1 2 1 2 , 1 2 2 1 1 2 1 1 2 2 1 2 1            

Determinant form of the electron function for the helium atom





 

 

 

 

 

 

 

 

 

n n n n n n n n          ... 2 1 ... ... ... ... ... 2 1 ... 2 1 ,..., 2 , 1 2 2 2 1 1 1 ! 1  

The antisymmetrized function for the n electron system fulfills the Pauli exclusion principle

E_HF : the Hartree-Fock energy – the lowest energy in the frame of one-electron approximation

Variational method

How to solve a Schrödinger equation when the exact solution is not possible? We search for a trail function Φ which gives the lowest ground state energy.

































dV

S

dV

H

H

c

dV

dV

H

E

H

j i ij j i ij N i i i

















* * 1 * *

ˆ

ˆ

ˆ

If Φ is the same as ψ, then ε is equal to E0.

If Φ is an approximation of ψ, then ε is higher than E₀. Linear combinations method:

The best trial function Φ is searched as a linear

combination of functions φ_i, which form a set of basis functions. ε is minimized with respect to coefficients c_i:



H



c



H

S



dla

i

N

c

i j ij ij j ii i











0



1

,...,







N

i

dla

c

_i



0



1

,...,







The variational method for a particle in a box (1)

 





 





 

x c

 

x c

 

x x a x a x x a x a x 2 2 1 1 2 2 9 2 7 1 630 30          

 The basis functions φ1 i φ2 are normalized, thus

S₁₁=1 i S₂₂=1.

The set of secular equations:

















0

0

22 2 21 21 1 12 12 2 11 1

























H

c

S

H

c

S

H

c

H

c

The necessary conditions for the existence of non-trivial solutions:

0 0 22 12 12 11 21 12 21 12 22 21 21 12 12 11                      H S H S H H S S S H H H S H S H H

The normalization of the function Φ(x):

   

 



 

 



1 2 1 2 2 1 2 2 2 1 2 1 2 1 2 2 2 2 2 1 2 1 2 2 2 1 1 2 *             













S c c c c dx c c dx c dx c dx x c x c dx x dx x x     

The variational method for a particle in a box (2)

















0

0

22 2 12 1 12 2 11 1

























H

c

S

H

c

S

H

c

H

c

0

22 12 12 11



















H

S

H

S

H

H





S

H

H

c

c

S

c

c

c

c















12 11 1 2 2 1 2 2 2 1

2

1





 

























































































































S

H

H

H

S

S

H

H

H

S

H

H

H

H

S

H

H

S

H

H

H

H

H

H

S

S

H

H

H

H

H

H

S

H

H

H

S

S

H

H

H

12 22 11 2 2 12 22 11 2 1 12 22 12 11 22 11 2 12 2 22 11 2 12 22 11 2 2 12 22 11 2 12 22 11 12 22 11 2 2 2 12 22 11

2

1

2

1

2

1

2

1

4

4

1

4

2

0

2

1

0















For every calculated energy ε₁ or ε₂ the set of equations on coefficients c₁ i c₂ is solved.

The variational method for a particle in a box (3)

The calculation of integrals :

   





14 21 3 140 20 70 84 35 21 30 7 1 2 1 5 3 4 1 21 30 7 6 3 5 3 4 21 30 3 3 630 * 30 630 30 0 7 6 5 2 4 3 7 0 3 2 2 3 3 0 7 2 2 9 5 21 12          _{  }                     





a a a x x a x a x a a dx x ax x a a x a dx x a x a x a x a S S S    











2 2 2 2 0 5 4 3 2 2 3 7 2 0 4 3 2 2 3 2 2 3 7 2 0 2 2 2 0 7 2 2 2 9 2 2 2 5 12 21 2 2 5 12 6 3 14 1 21 30 2 5 12 4 24 3 14 2 2 21 30 2 12 12 2 12 12 2 21 30 2 12 12 2 630 * 30 2 630 2 30 a m a m x x a x a x a a m dx x ax x a ax x a x a a m dx x ax a x ax a m dx x a x a dx d m x a x a H a a a a           _{   }       _{   }                         







   





2 2 2 2 0 5 4 3 2 7 2 0 4 3 2 2 0 7 2 5 2 2 2 2 2 9 21 21 2 2 5 2 1 3 2 21 30 2 5 2 4 4 3 2 21 30 2 2 2 21 30 2 30 2 630 a m a m x x a x a a m dx x ax x a a m dx x a x a dx d m x a x a H a a a           _{ }                        





The variational method for a particle in a box (4)

The calculation of integrals :

14 21 3  S 2 2 21 12 21 2 2m a H H            2 ₂ 0 2 2 9 2 2 2 2 2 9 22 2 2 0 5 2 2 2 5 11 12 2 630 2 630 10 2 30 2 30 a m dx x a x a dx d m x a x a H a m dx x a x a dx d m x a x a H a a                          





The set of secular equations:





0 12 21 2 21 2 10 0 ) 12 ( 14 21 3 21 2 0 ) 14 21 3 21 2 ( 10 14 21 3 14 21 3 2 1 2 1                            c c c c















4 3,295017884



*(28/2) 102,13025038 869719621 , 9 ) 2 / 28 ( * 295017884 , 3 4 295017884 , 3 7 / 76 36 * 28 / 1 * 4 16 0 36 4 28 1 0 21 * 4 12 * 10 14 / 21 * 3 * 21 * 2 * 2 12 10 196 21 * 9 1 2 1 2 2                                   A solution:

The integrals for energy

in units of 2

2

2ma 

The variational method for a particle in a box (5)

Calculation of the linear combination coefficients:

2 1 1 2 1 2 1 1 2 1 2 1

19882025

,

0

80405626

,

0

19882025

,

0

80405626

,

0

14

/

21

*

2

*

247271562

,

0

*

2

247271562

,

0

1

1

247271562

,

0

14

/

21

*

3

*

869719621

,

9

21

*

2

869719621

,

9

10

869719621

,

9



































c

c

c

c

c

c

c





S

H

H

c

c

S

c

c

c

c















12 11 1 2 2 1 2 2 2 1

2

1

The exact value of energy E₁ for a particle in a box [in units of ]

1 1 2 1 9,869604401 E E   





2 2 2ma 

The variational method for a particle in a box (6)

A graph of the functions:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.5 1 1.5 sin f1 f2 f1+f2

The function sin(x) and the linear combination of functions φ₁ and φ₂ overlap in a scale of the graph

The Hartree-Fock method for an atom

 



                1 1 1 2 1 2 2 2 ˆ n i n i j ij n i i i e e r e r Ze m H 





 

 

 

 

 

 

 

 

 

n n n n n n n n          ... 2 1 ... ... ... ... ... 2 1 ... 2 1 ,..., 2 , 1 2 2 2 1 1 1 ! 1  

   

i

i

 

i

p

n

F

ˆ



_p





_p



_p



1

,

2

,

3

,...,

 



   











_p i  _p i _p i _p  lub

 

 

 

 

 

   

 

 

 

   

1 1

 

2

 

2

 

1 ˆ 1 2 2 1 1 ˆ 2 1 ˆ 1 ˆ 1 ˆ 1 ˆ 1 ˆ 2 12 2 * 2 12 2 * 1 2 1 2 j i j i i j j i A A A e dV r e K dV r e J r e Z m h K J h F                             







 _{One-electron operator} Two-electron operators: Coulomb Exchange The orbital energy

Unrestricted Hartree-Fock method (UHF)

 



   











_p

i



_p

i

_p

i

_p



or

where: φ_p is real, and numbers of electrons with spin α and β are not equal Used for the open-shell systems (atoms or molecules)

(different orbital energies for spins α and β)

A form of the graph

Restricted Hartree-Fock (RHF)

Even number of electrons, equal number of electrons of spin α and β. The number of spinorbitals is twice as many as that of occupied orbitals.

 

   

 

i

   

i

i

i

i

i

p p p p

















 2 1

2 The number of spinorbitals is equal to

the number of electrons, wheras each orbital is occupied by two electrons.

LUMO (Lowest unoccupied molecular orbital)