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Application of S.W.E. to Hydrogen
& Hydrogen Like Systems
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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
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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)
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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
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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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