application of swe to h like system

7/23/2019 Application of SWE to H Like System

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Application of S.W.E. to Hydrogen

& Hydrogen Like Systems

1Dr. D. Ilangeswaran, M.Sc., M.Phil., Ph.D.,


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Dr. D. Ilangeswaran, M. Sc., M. Phil., Ph. D.,

Assistant Professor of Chemistry

Rajah Serfoji Govt. College(Autonomous)

Thanjavur - 613005


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Hydrogen Like Atom

A system with a central nucleus and an

electron is known as H like atom.

In such systems the nucleus & the electron

are held together by means of electrostatic

attraction.

The wave function for either the H or H

like atoms (i.e. one e- systems) like He+, Li2+

can be calculated accurately.



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The SWE for H atom

1. Potential energy

In H atom, the +vely charged proton is at thecentre whereas thevely charged e- is at adistance, r from the nucleus.

The potential energy of attraction betweenthe nucleus & the e- is given in equation (1).Where 40 is the permitivity, in atomic unit = 1.SI unit of it is 1.1126 10-10 C2 m-1 J-1.

and Z = 1 for H atom.

V =-Z e2

4 r

(1)



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2. Reduced Mass

Generally the mass of an e- (9.1 10-28 g)

is negligible when compared to the mass of

proton (1.76 10-24 g).

Due to this reason we may assume that the

mass of an electron is roughly equal to the

reduced mass of H atom.

mp. m

e

mp+ m

e

mp. m

e

mp

me= =



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The SWE for H & H like atom is

Using the values of m & V in this equation we get

Transforming the above equation from Cartesiancoordinate to polar coordinates (r, , ), we get



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Equation (4) involves three variables, r, , and .Where is the zenith angle and is azimuthal

angle.

Separation of variables

The dependence of r, , and occur indifferent terms of the above equation. Since the r

term involves the potential energy, it is possible

to separate the partial differential equations, onein each of the variables.



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The position of a

particle on the surface

of a sphere of radius r

is more convenientlydetermined in terms of

two angular variables

(coordinates) -,called the azimuthal

angle and , called thezenith angle.

The angle is the angle measured in the xy plane between the x axis

and the projection of the line r joining the particle, P with the centreof the sphere. It varies from 0 to 2.

The angle is the angle measured between the line r and the z axis. Itvaries from 0 to .



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The above mentioned variables can be separated by inserting into

equation (3) a solution of the form

(r, , ) = R(r), Y(), Z()

Where R, Y and Z are functions of only three variables r, and respectively.The function R(r) is referred to as the radial function since it

describes how the wave function varies with the radial distance, r.

The functions Y() and Z() combined together as sayY(,) represents angular part of the wave function.

The following necessary derivatives can be obtained by

proper differentiation.



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Since the two sides of equation (7) are functions of different

variables, this equation can be correct, only if each side of this

equation is equal to the same constant, say m2.

Dividing equation (9) by sin2



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Thus equation (4) is now separated into three ordinary

differential equations (8), (13) and (14) each of which involves

only one variable. These equations are called as , r and

equations respectively.



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Atomic Orbitals

For an atom, use Schrdingers equation

Find permissible energy levels for electrons aroundnucleus.

For each energy level, the wave function defines anorbital, a region where the probability of finding anelectron is high

The orbital properties of greatest interest are size,

shape (described by wave function) and energy. Solution for multi-electron atoms is a very difficult

problem, and approximations are typically used



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The hydrogen atom

The electron of the hydrogen atom moves in threedimensions and has potential energy (attraction to

positively charged nucleus)

The Schrodinger equation can be solved to find

the wave functions associated with the hydrogenatom

In 1-D particle in a box, the wave function is afunction of one quantum number; the 3-D

hydrogen atom is a function of three quantumnumbers



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Wave functions of hydrogen

The solution of the Schrodinger equation

for the hydrogen atom is:

Rnl describes how wave function varies withdistance of electron from nucleus

Ylm describes the angular dependence of the wavefunction

Subscripts n, l and m are quantum numbers ofhydrogen

;,,, ,,, lmlnmln rRr



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Principal quantum number, nHas integral values of 1,2,3 and is relatedto size and energy of the orbital

As n increases, the orbital becomes larger andthe electron is farther from the nucleus

As n increases, the orbital has higher energy(less negative) and is less tightly bound to thenucleus



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Angular quantum number, l

Can have values of 0 to n-1 for each

value of n and relates to the angular

momentum of the electron in an orbital

The dependence of the wave function

on l, determines the shape of the orbitals

The value of l, for a particular

orbital is commonly assigned a

letter:

0s

1p

2d

3f

s orbital

p orbital d orbital



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Magnetic quantum number, ml

Can have integral values between l and - l, including

zero and relates to the orientation in space of theangular momentum.

s orbital:

l=0, m=0

p orbital:

l=1, m=-1,0,1

d orbital:l=2, m=-2,-1,0,1,2



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Quantum

Numbers

Name Allowed Values Allowed

States

n principalquantum number

1,2,3.. Anynumber

l Angular quantum

number

0,1,2,(n-1) n

ml

magnetic

quantum number

-l ,- l+1,0,( l-1),

l

2 l + 1

Calculation of quantum numbers



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Shells and subshells

All states with the same principal quantum number are

said to form a shell; the states having specific values ofboth n and l are said to form a subshell

Shell (n) l Subshell symbol

1 0 1s

2 0 2s

2 1 2p3 0 3s

3 1 3p

3 2 3d

0s1p2d3 - f



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Y1,0,0

Wave

Function

1s

Subshell

symbol

3

2

001

mlln

Example



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Orbital shapes

Solution of the

Schrodinger wave equation

for a one electron atom :

o

2/1

3o

0,0,1

a

rexp

a

1,,r

2/1

0,04

1,

Cbxe

CbJmxk

kgxm

Jsx

h

mxmke

ao

19

29

31

34

10

2

2

10602.1chargeelectron

/10988.8constantsCoulomb'

10109.9electronofmass

10055.12

10529.0



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Y1,0,0

Wave

Function

1s

Subshell

symbol

Y2,1,1

Y2,1,0

Y2,1,-1

Y2,0,0

2p

2p

2p

2s

012

112

-112

002

001

mlln

Other orbitals



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Allowed energies of hydrogen

The energy En of the wave function Ynlm depends only on

n:

m - mass of electron

e - electron charge

hPlanck constant

permittivity of free space

Because n is restricted to integer values, energy levels are

quantized

222

4

8 nh

meE

o

n



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Atomic Orbitals: Multi-electron atoms

Electron spin quantum number, msThis quantum number only has two values: and.

This means that the electron has two spin states, thusproducing two oppositely directed magnetic moments

This quantum number doubles the number of allowed statesfor each electron.

Pair of electrons in a given orbital must have opposite spins



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n l ml Wave

Function

Subshell

symbol

ms(1/2), (-1/2)

1 0 0 Y1,0,0 1s

2 0 0 Y2,0,0 2s

2 1 -1 Y2,1,-1 2p

2 1 0 Y2,1,0 2p

2 1 1 Y2,1,1 2p

Example



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Pauli exclusion principleNo two electrons can have the same set of quantum

numbers: n, l, ml

andms

Aufbau principle

Electrons fill in the orbitals of successively increasingenergy, starting with the lowest energy orbital

Hunds rule

For a given shell (example, n=2), the electron occupies

each subshell one at a time before pairing up



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Orbital energies: multi-electron atoms

Energy depends on both n and l



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n l ml Wave

Function

Subshell

symbol

ms(1/2), (-1/2)

1 0 0 Y1,0,0 1s2 0 0 Y2,0,0 2s

2 1 -1 Y2,1,-1 2p

2 1 0 Y2,1,0 2p

2 1 1 Y2,1,1 2p

Example: Nitrogen (1s22s22p3)



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n l ml Wave

Function

Subshell

symbol

ms(1/2), (-1/2)

1 0 0 Y1,0,0 1s2 0 0 Y2,0,0 2s

2 1 -1 Y2,1,-1 2p

2 1 0 Y2,1,0 2p

2 1 1 Y2,1,1 2p

Example: Carbon



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Atomic Orbitals: Summary

In the quantum mechanical model, the electron is

described as a wave. This leads to a series of wave

functions (orbitals) that describe the possible energiesand spatial distribution available to the electron

The orbitals can be thought of in terms of probability

distributions (square of the wave function)



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Sizes, Shapes, and orientations of orbitals

n determines size; l determines shape

mldetermines orientation



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Nodes in orbitals: s orbitals: 1s no nodes, 2s one node,3s two nodes



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Nodes in orbitals: 2porbitals:

angular node that passesthrough the nucleus

Orbital is dumb bell shaped

Important: the + and - thatis shown for a p orbital

refers to the mathematical

sign of the wavefunction, notelectric charge!



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Nodes in orbitals: 3dorbitals:

two angular nodes that

passes through thenucleus

Orbital is four leafclover shaped

d orbitals are importantfor metals


The fourth quantum number: Electron Spin


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The fourth quantum number: Electron Spin

ms = +1/2 (spin up) or -1/2 (spin down)

Spin is a fundamental property of electrons, like itscharge and mass.

(spin up)

(spin down)


El b l h d ff l f


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Electrons in an orbital must have different values ofms

This statement demands that if there are twoelectrons in an orbital one must have ms = +1/2 (spinup) and the other must have ms = -1/2 (spin down)

This is the Pauli Exclusion Principle

An empty orbital is fully described by the threequantum numbers: n, land ml

An electron in an orbital is fully described by thefour quantum numbers: n, l, ml and ms



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Born-Oppenheimer Approximation

the nuclei are much heavier than the electrons andmove more slowly than the electrons

in the Born-Oppenheimer approximation, we freezethe nuclear positions, Rnuc, and calculate the

electronic wavefunction, Yel(rel;Rnuc) and energyE(Rnuc) E(Rnuc) is the potential energy surface of the

molecule (i.e. the energy as a function of thegeometry)

on this potential energy surface, we can treat themotion of the nuclei classically or quantummechanically



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Born-Oppenheimer Approximation

freeze the nuclear positions (nuclear kinetic energy is zero inthe electronic Hamiltonian)

calculate the electronic wavefunction and energy

E depends on the nuclear positions through the nuclear-electron attraction and nuclear-nuclear repulsion terms

E = 0 corresponds to all particles at infinite separation

nuclei

BA AB

BAelectrons

ji ij

nuclei

A iA

Aelectrons

i

i

electrons

i e

el

r

ZZe

r

e

r

Ze

m

2222

2

2

H

YY

YYYY

d

dEE

elel

elelel

elelel *

*

, H

H



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Nuclear motion on the

Born-Oppenheimer surface

Classical treatment of the nuclei (e,g. classical

trajectories)

Quantum treatment of the nuclei (e.g. molecular

vibrations)

22 /,/, tEnucnuc

RaRFmaF

)(2

,

22

nuc

nuclei

A A

nuc

nucnucnucnuceltotal

Em

RH

H

YY



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Hartree Approximation

assume that a many electron wavefunction can bewritten as a product of one electron functions

if we use the variational energy, solving the manyelectron Schrdinger equation is reduced tosolving a series of one electron Schrdingerequations

each electron interacts with the averagedistribution of the other electrons

)()()(),,,( 321321 rrrrrr Y



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Hartree-Fock Approximation

the Pauli principle requires that a wavefunction for electronsmust change sign when any two electrons are permuted since |Y(1,2)|2=|Y(2,1)|2, Y(1,2)=Y(2,1) (minus sign for fermions)

the Hartree-product wavefunction must be antisymmetrized

can be done by writing the wavefunction as a determinant determinants change sign when any two columns are switched

n

nnn n

n

n

n

21222

111

)()1()1(

)()2()1(

)()2()1(

1 Y



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Spin Orbitals

each spin orbital I describes the distribution of one electron in a Hartree-Fock wavefunction, each electron must be in a

different spin orbital (or else the determinant is zero)

an electron has both space and spin coordinates

an electron can be alpha spin (, , spin up) or beta spin (, ,spin down)

each spatial orbital can be combined with an alpha or betaspin component to form a spin orbital

thus, at most two electrons can be in each spatial orbital



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Fock Equation

take the Hartree-Fock wavefunction

put it into the variational energy expression

minimize the energy with respect to changes in the orbitals whilekeeping the orbitals orthonormal

yields the Fock equation

n 21Y

YYYY

d

dE*

*

var

H

iii F

*

var/ 0,i i j ijE d



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Fock Equation

the Fock operator is an effective one electron

Hamiltonian for an orbital

is the orbital energy each orbital sees the average distribution of all the

other electrons

finding a many electron wavefunction is reduced tofinding a series of one electron orbitals

iii F



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Fock Operator

kinetic energy operator

nuclear-electron attraction operator

22

2

em

T

nuclei

A iA

A

ner

Ze2V

KJVTF NE



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Fock Operator

Coulomb operator (electron-electron repulsion)

exchange operator (purely quantum mechanical -

arises from the fact that the wavefunction must

switch sign when you exchange to electrons)

ijij

j

electrons

ji dr

e

}{

2

J

ji

ij

j

electrons

j

i dr

e }{

2

K

KJVTF NE



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Solving the Fock Equations

1. obtain an initial guess for all the orbitals i

2. use the current Ito construct a new Fock operator

3. solve the Fock equations for a new set of I

4. if the new Iare different from the old

I, go back to

step 2.

iii F



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Hartree-Fock Orbitals

for atoms, the Hartree-Fock orbitals can be computed

numerically

the s resemble the shapes of the hydrogen orbitals

s, p, d orbitals

radial part somewhat different, because of interaction withthe other electrons (e.g. electrostatic repulsion and

exchange interaction with other electrons)



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Hartree-Fock Orbitals

for homonuclear diatomic molecules, the

Hartree-Fock orbitals can also be computed

numerically (but with much more difficulty)

the s resemble the shapes of the H2+

orbitals , , bonding and anti-bonding orbitals



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LCAO Approximation

numerical solutions for the Hartree-Fock orbitalsonly practical for atoms and diatomics

diatomic orbitals resemble linear combinations ofatomic orbitals

e.g. sigma bond in H2 1sA + 1sB

for polyatomics, approximate the molecular orbital

by a linear combination of atomic orbitals (LCAO)

c



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Basis Functions

s are called basis functions

usually centered on atoms can be more general and more flexible than atomic

orbitals

larger number of well chosen basis functions yields

more accurate approximations to the molecularorbitals

c



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Roothaan-Hall Equations

choose a suitable set of basis functions

plug into the variational expression for the energy

find the coefficients for each orbital that minimizesthe variational energy

c

YY

YY

d

dE

*

*

var

H



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Roothaan-Hall Equations

basis set expansion leads to a matrix form of the

Fock equations

F Ci = i S Ci F Fock matrix Ci column vector of the molecular orbital

coefficients

I orbital energy S overlap matrix



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Fock matrix and Overlap matrix

Fock matrix

overlap matrix

dF F

dS



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Intergrals for the Fock matrix

Fock matrix involves one electron integrals of kinetic andnuclear-electron attraction operators and two electronintegrals of 1/r

one electron integrals are fairly easy and few in number(only N2)

two electron integrals are much harder and much morenumerous (N4)

dh ne )( VT

21

12

)2()2(1

)1()1()|( dd

r



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Solving the Roothaan-Hall Equations

1. choose a basis set

2. calculate all the one and two electron integrals

3. obtain an initial guess for all the molecular orbital

coefficients Ci

4. use the current Ci to construct a new Fock matrix

5. solve F Ci = i S Ci for a new set of Ci

6. if the new Ci are different from the old Ci, go back tostep 4.



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Solving the Roothaan-Hall Equations

also known as the self consistent field (SCF) equations, since

each orbital depends on all the other orbitals, and they are

adjusted until they are all converged

calculating all two electron integrals is a major bottleneck,

because they are difficult (6 dimensional integrals) and verynumerous (formally N4)

iterative solution may be difficult to converge

formation of the Fock matrix in each cycle is costly, since it

involves all N4 two electron integrals



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Summary

start with the Schrdinger equation

use the variational energy

Born-Oppenheimer approximation

Hartree-Fock approximation

LCAO approximation


Th P li i i l d Sl t d t i t


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The Pauli principle and Slater determinants



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application of swe to h like system

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