PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom The Hydrogen Atom Z Y X + e-e- rere rprp r Coordinates Systems R z y x + e-e- r r spherical polar

PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom

The Hydrogen AtomThe Hydrogen AtomThe Hydrogen AtomThe Hydrogen Atom

Z

Y

X

+

e-

re

rp

r

Coordinates SystemsCoordinates SystemsCoordinates SystemsCoordinates Systems

R z

y

x+

e-

rr

spherical polar coordinatesspherical polar coordinatesspherical polar coordinatesspherical polar coordinates

z=r cosy=rsinsinx=rsincos

z=r cosy=rsinsinx=rsincos

center of mass or barycentercenter of mass or barycentercenter of mass or barycentercenter of mass or barycenter

pe

ppee

mm

rmrmR

pe

ppee

mm

rmrmR

reduced massreduced massreduced massreduced mass

pe

pe

mm

mm

pe

pe

mm

mm


RotationRotationRotationRotation

+ e-

If mp>>me: ≈me

r ≈ re

If mp>>me: ≈me

r ≈ re

moment of inertia:I=mr2

angular momentum:

kinetic energy:T=L2/(2I)

moment of inertia:I=mr2

angular momentum:

kinetic energy:T=L2/(2I)

tangential component:

radial component:

angular velocity

tangential component:

radial component:

angular velocity


angular momentum: vectorial representationangular momentum: vectorial representationangular momentum: vectorial representationangular momentum: vectorial representation

rr

pp L

L

meme

xyyxz

zxxzy

yzzyx

prprL

prprL

prprL

vmrprL

xyyxz

zxxzy

yzzyx

prprL

prprL

prprL

vmrprL

quantum mechanics:quantum mechanics:quantum mechanics:quantum mechanics:

-ˆ ˆ

x

yy

xiLx

ip zx -ˆ ˆ

x

yy

xiLx

ip zx

Analogies:Translation Rotation

m Iv

p=mv L=I

Analogies:Translation Rotation

m Iv

p=mv L=I


Coordinates TransformationCoordinates TransformationCoordinates TransformationCoordinates Transformation

spherical polar coordinatesspherical polar coordinatesspherical polar coordinatesspherical polar coordinatescartesian coordinatescartesian coordinatescartesian coordinatescartesian coordinates

),,(

),,(

:,

,,

2

2

2

2

2

22

zyx

zyxH

zyx

zyx

),,(

),,(

:,

,,

2

2

2

2

2

22

zyx

zyxH

zyx

zyx

),,(

),,(

)sin()sin(

1

)(sin

1

r

1

r

2 :,

,,

2

2

222

2

22

r

rH

rr

r

),,(

),,(

)sin()sin(

1

)(sin

1

r

1

r

2 :,

,,

2

2

222

2

22

r

rH

rr

r

z

y

x+

e-

rr


coulombic potential: coulombic potential: U=-1/rU=-1/rcoulombic potential: coulombic potential: U=-1/rU=-1/r

Z

Y

X

1-D1-D

r=|x|r=|x|

1-D1-D

r=|x|r=|x|

2-D2-D

r=|r=|√√xx22+ y+ y22||

2-D2-D

r=|r=|√√xx22+ y+ y22||

3-D3-D

r=|r=|√√xx22+ y+ y22+z+z22||

3-D3-D

r=|r=|√√xx22+ y+ y22+z+z22||


2

2

22

2

v

1

tx

2

2

22

2

v

1

tx

constant)(

)(v

1)(

)(

12

2

22

2

t

tT

tTx

xX

xXconstant

)(

)(v

1)(

)(

12

2

22

2

t

tT

tTx

xX

xX

constantv

)(

)(v

1)(

)(

12

2

2

2

22

2

dt

tTd

tTdx

xXd

xXconstant

v

)(

)(v

1)(

)(

12

2

2

2

22

2

dt

tTd

tTdx

xXd

xX

Equivalent to two ordinary (not partial) differential equations:

)()(

)(v

)(

22

2

2

2

2

2

tTdt

tTd

xXdx

xXd

)()(

)(v

)(

22

2

2

2

2

2

tTdt

tTd

xXdx

xXd

Space: f(x)Space: f(x) TIme: f(t)TIme: f(t)

)cos()sin(sin)()(),( tBtAL

xntTxXtx nnnn

)cos()sin(sin)()(),( tBtAL

xntTxXtx nnnn

Space: X(x)Space: X(x) Time: T(t)Time: T(t)


Schrödinger equation:Schrödinger equation:Schrödinger equation:Schrödinger equation:

),(),(2)(2

222

rRErRr

q

mmme

re

Rep

),(),(2)(2

222

rRErRr

q

mmme

re

Rep

)()()(2

2

RERmm RRep

)()()(2

2

RERmm RRep

)()(2

22

rErr

q

m rre

e

)()(

2

22

rErr

q

m rre

e

rR EEE

rRrR

)()(),(

rR EEE

rRrR

)()(),(

free particlefree particlefree particlefree particle

,,rr ,,rr

),,(),,(1

2

ˆ

2

ˆ2

22

rErrr

Lp

),,(),,(

1

2

ˆ

2

ˆ2

22

rErrr

Lp

kinetic, rotational, coulombic


Schrödinger equation:Schrödinger equation:Schrödinger equation:Schrödinger equation:

)()(2

22

rErr

q

m rre

e

)()(

2

22

rErr

q

m rre

e

),(),()sin()sin(

1

)(sin

1

r

1

)()(1

2r

1-

2

2

22

22

A

rArrdr

dr

rd

d

),(),()sin()sin(

1

)(sin

1

r

1

)()(1

2r

1-

2

2

22

22

A

rArrdr

dr

rd

d

)()()(),()(),,( rrr )()()(),()(),,( rrrradial, angular wavefunctions

),(),( r ),(),( r

)()(sin

)()sin()sin(

1

)()(

2

2

22

2

Am

d

d

d

d

md

d

l

l

)()(sin

)()sin()sin(

1

)()(

2

2

22

2

Am

d

d

d

d

md

d

l

l

)()(),( )()(),(


Wavefunctions Wavefunctions (solutions)(solutions)Wavefunctions Wavefunctions (solutions)(solutions)

)()()(),()(),,( rrr )()()(),()(),,( rrr

polynomialLaguerrerP

brrPrPerPrq

qrEln

:)(

)( ),( )()(0

2,

polynomialLaguerrerP

brrPrPerPrq

qrEln

:)(

)( ),( )()(0

2,

immlml eP )(cos()()(),(, immlml eP )(cos()()(),(,

functionsLegendreP

Pml

mll

ml

mlmlml

:

),(cos(|)!|(2

|)!|)(12()(),( ,,

functionsLegendreP

Pml

mll

ml

mlmlml

:

),(cos(|)!|(2

|)!|)(12()(),( ,,

im

lm e2

1)(,

im

lm e2

1)(,


Quantum numbers:Quantum numbers:Quantum numbers:Quantum numbers:

Principal: n: 1,2,3,……..Principal: n: 1,2,3,……..Angular: l: 0,1,2,…(n-1)Angular: l: 0,1,2,…(n-1)Magnetic: m: +l,(l-1)…0….-lMagnetic: m: +l,(l-1)…0….-l

Principal: n: 1,2,3,……..Principal: n: 1,2,3,……..Angular: l: 0,1,2,…(n-1)Angular: l: 0,1,2,…(n-1)Magnetic: m: +l,(l-1)…0….-lMagnetic: m: +l,(l-1)…0….-l

Energy:Energy:Energy:Energy:

22

1

nEn 22

1

nEn

Magnitude of the angular Magnitude of the angular momentum:momentum:Magnitude of the angular Magnitude of the angular momentum:momentum:

)1(|| llL )1(|| llL

z component of the z component of the angular momentum:angular momentum:z component of the z component of the angular momentum:angular momentum:

lmL || lmL ||


Effective potentialEffective potentialEffective potentialEffective potential

),,(),,(1

2

ˆ

2

ˆ2

22

rErrr

Lp

),,(),,(

1

2

ˆ

2

ˆ2

22

rErrr

Lp

),,(1

),,(1

),,()1(

),,(2

ˆ22

2

rr

rr

rr

llr

r

L

),,(1

),,(1

),,()1(

),,(2

ˆ22

2

rr

rr

rr

llr

r

L

eigenvalueseigenvalueseigenvalueseigenvalues

0 2 4 6 8 10 12 14 16 18 20-1,0

-0,5

0,0

0,5

Y A

xis

Titl

e

X Axis Title

The effective potential energy of an electron in The effective potential energy of an electron in the hydrogen atom. When the electron has the hydrogen atom. When the electron has zero orbital angular momentum, the effective zero orbital angular momentum, the effective potential energy is the Coulombic potential potential energy is the Coulombic potential energy. When the electron has nonzero orbital energy. When the electron has nonzero orbital angular momentum, the centrifugal effect gives angular momentum, the centrifugal effect gives rise to a positive contribution which is very rise to a positive contribution which is very large close to the nucleus. We can expect the large close to the nucleus. We can expect the ll = 0 and = 0 and l l 0 wavefunctions to be very 0 wavefunctions to be very different near the nucleus.different near the nucleus.

The effective potential energy of an electron in The effective potential energy of an electron in the hydrogen atom. When the electron has the hydrogen atom. When the electron has zero orbital angular momentum, the effective zero orbital angular momentum, the effective potential energy is the Coulombic potential potential energy is the Coulombic potential energy. When the electron has nonzero orbital energy. When the electron has nonzero orbital angular momentum, the centrifugal effect gives angular momentum, the centrifugal effect gives rise to a positive contribution which is very rise to a positive contribution which is very large close to the nucleus. We can expect the large close to the nucleus. We can expect the ll = 0 and = 0 and l l 0 wavefunctions to be very 0 wavefunctions to be very different near the nucleus.different near the nucleus.


Radial wavefunctionsRadial wavefunctionsRadial wavefunctionsRadial wavefunctions


Atomic orbitalsAtomic orbitalsAtomic orbitalsAtomic orbitals

What is an atomic orbital?What is an atomic orbital?Orbitals and orbitsOrbitals and orbits

When the a planet moves around the sun, you can plot a definite path for it When the a planet moves around the sun, you can plot a definite path for it which is called an orbit. A simple view of the atom looks similar and you may which is called an orbit. A simple view of the atom looks similar and you may

have pictured the electrons as orbiting around the nucleus. The truth is have pictured the electrons as orbiting around the nucleus. The truth is different, and electrons in fact inhabit regions of space known as different, and electrons in fact inhabit regions of space known as orbitals.orbitals.

Orbits and orbitals sound similar, but they have quite different meanings. It is Orbits and orbitals sound similar, but they have quite different meanings. It is essential that you understand the difference between them.essential that you understand the difference between them.

The impossibility of drawing orbits for electronsThe impossibility of drawing orbits for electronsTo plot a path for something you need to know exactly where the object is and To plot a path for something you need to know exactly where the object is and be able to work out exactly where it's going to be an instant later. You can't do be able to work out exactly where it's going to be an instant later. You can't do

this for electrons.this for electrons.The The Heisenberg Uncertainty PrincipleHeisenberg Uncertainty Principle says - loosely - that you can't know says - loosely - that you can't know with certainty both where an electron is and where it's going next. (What it with certainty both where an electron is and where it's going next. (What it actually says is that it is impossible to define with absolute precision, at the actually says is that it is impossible to define with absolute precision, at the

same time, both the position and the momentum of an electron.)same time, both the position and the momentum of an electron.)That makes it impossible to plot an orbit for an electron around a nucleus. Is That makes it impossible to plot an orbit for an electron around a nucleus. Is this a big problem? No. If something is impossible, you have to accept it and this a big problem? No. If something is impossible, you have to accept it and

find a way around it.find a way around it.

What is an atomic orbital?What is an atomic orbital?Orbitals and orbitsOrbitals and orbits

When the a planet moves around the sun, you can plot a definite path for it When the a planet moves around the sun, you can plot a definite path for it which is called an orbit. A simple view of the atom looks similar and you may which is called an orbit. A simple view of the atom looks similar and you may

have pictured the electrons as orbiting around the nucleus. The truth is have pictured the electrons as orbiting around the nucleus. The truth is different, and electrons in fact inhabit regions of space known as different, and electrons in fact inhabit regions of space known as orbitals.orbitals.

Orbits and orbitals sound similar, but they have quite different meanings. It is Orbits and orbitals sound similar, but they have quite different meanings. It is essential that you understand the difference between them.essential that you understand the difference between them.

The impossibility of drawing orbits for electronsThe impossibility of drawing orbits for electronsTo plot a path for something you need to know exactly where the object is and To plot a path for something you need to know exactly where the object is and be able to work out exactly where it's going to be an instant later. You can't do be able to work out exactly where it's going to be an instant later. You can't do

this for electrons.this for electrons.The The Heisenberg Uncertainty PrincipleHeisenberg Uncertainty Principle says - loosely - that you can't know says - loosely - that you can't know with certainty both where an electron is and where it's going next. (What it with certainty both where an electron is and where it's going next. (What it actually says is that it is impossible to define with absolute precision, at the actually says is that it is impossible to define with absolute precision, at the

same time, both the position and the momentum of an electron.)same time, both the position and the momentum of an electron.)That makes it impossible to plot an orbit for an electron around a nucleus. Is That makes it impossible to plot an orbit for an electron around a nucleus. Is this a big problem? No. If something is impossible, you have to accept it and this a big problem? No. If something is impossible, you have to accept it and

find a way around it.find a way around it.


It is not possible to determine the exact location of an electron in an atom. It is not possible to determine the exact location of an electron in an atom. However, the However, the probabilityprobability of finding an electron at a given position can be of finding an electron at a given position can be calculated. The higher the probability of finding an electron at a given calculated. The higher the probability of finding an electron at a given position, the larger the electron density at that position. position, the larger the electron density at that position.

It is not possible to determine the exact location of an electron in an atom. It is not possible to determine the exact location of an electron in an atom. However, the However, the probabilityprobability of finding an electron at a given position can be of finding an electron at a given position can be calculated. The higher the probability of finding an electron at a given calculated. The higher the probability of finding an electron at a given position, the larger the electron density at that position. position, the larger the electron density at that position.

An atomic orbital

is derived using the mathematical tools of quantum mechanics,

is a representation of the three-dimensional volume (i.e., the region in space) in which anelectron is most likely to be found, and

CANNOT be observed experimentally (electron density can, however, be observedexperimentally).


s (l=0) p (l=1) d (l=2) f (l=3)

n=1

n=2

n=3

n=4

n=5

. . .

n=6

. . . . . .


http://winter.group.shef.ac.uk/orbitron/

Some interesting Websites….Some interesting Websites….Some interesting Websites….Some interesting Websites….

http://www.orbitals.com/orb/

http://www.falstad.com/qmatom/

http://bouman.chem.georgetown.edu/atomorbs/aovis.html

http://electron6.phys.utk.edu/qm2/modules/m1-3/hydrogen.htm

http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/HydrogenAtom.htm

Documents

PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom The Hydrogen Atom Z Y X + e-e- rere rprp r Coordinates Systems R z y x + e-e- r r spherical polar