physical chemistry - advanced materials h - atom the hydrogen atom z y x + e-e- rere rprp r...
TRANSCRIPT
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
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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||
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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)
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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
)()(),( )()(),(
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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)(,
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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 ||
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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.
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
Radial wavefunctionsRadial wavefunctionsRadial wavefunctionsRadial wavefunctions
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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.
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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).
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
s (l=0) p (l=1) d (l=2) f (l=3)
n=1
n=2
n=3
n=4
n=5
. . .
n=6
. . . . . .
PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS H - AtomH - AtomH - AtomH - Atom
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