prof. dr. juan ignacio rodríguez...
TRANSCRIPT
Modelación Molecular
tiH
ˆ
Prof. Dr. Juan Ignacio Rodríguez Hernández Escuela Superior de Física y Matemáticas
Instituto Politécnico Nacional-México
Agosto 2018
I.1 The Schrödinger Equation
tiH
ˆ
VTH ˆˆˆ T is the kinetic energy operator
Vis the potential energy operator
Where H is the Hamiltonian operator:
Is the “famous” WAVE FUNCTION
The Schrödinger Equation
3
ALL INFORMATION of the
QUANTUM SYSTEM is obtained from
● single-valued
● continuous function and continuous n-th’s partial derivatives
● normalized (“quadratically integrable”= finite integral):
where the integral is over “all space”.
12
* dd
must be a “well behaved” function:
The Schrödinger Equation
4
),(),(),( *2txtxtx
If (x,t) is a one-particle (in 1D) wave
function where x is the spatial coordinate of
the particle then:
is the PROBABILITY DENSITY so that,
dxtx2
),(
is the PROBABILITY of finding the particle at
the position x at time t
The Schrödinger Equation
5
22)(),( xtx
If (potential energy operator)
is INDEPENDENT of time, it can be demonstrated*:
where 𝝍 is the time-independent wave
function which is a solution of the
STATIONARY Schrödinger equation:
)(ˆˆ xVV
The STATIONARY Schrödinger Equation
6
EH ˆ
VTH ˆˆˆ T is the kinetic energy operator
V is the potential energy operator
E is the ENERGY of the system
H is again the Hamiltonian operator:
The STATIONARY Schrödinger Equation
7
EH ˆ
● single-valued
● continuous function and continuous n-th’s partial derivatives
● normalized (“quadratically integrable”= finite integral):
where the integral is over “all space”.
12
* dd
must be a “well behaved” function:
● single-valued
● continuous function and continuous n-th’s partial derivatives
● normalized (“quadratically integrable”= finite integral):
where the integral is over “all space”.
12
* dd
The STATIONARY Schrödinger Equation
8
EH ˆ
If (potential energy operator)
is independent of time, ALL IFORMATION of
the system is obtain from ψ, which is obtained
In turn from the STATIONARY Schrödinger
equation:
)(ˆˆ xVV
The STATIONARY Schrödinger Equation
9
EH ˆ
«Operational» USE:
syssyssyssystem EH ˆQuantum
System systemH
sys ALL INFORMATION
OF THE SYSTEM!!!
???
The STATIONARY Schrödinger Equation
10
EH ˆ
Quantum
System
???
Molecules (water, aspirin,
oligomers, etc..)
Solids (gold, silicon, polymers,
galium arsenide, diamond,
etc.)
Nanosystems (clusters,
fullerens, rods, wires, etc. )
The Schrödinger Equation
11
EH ˆ
Highly predective:
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
The Schrödinger Equation
12
12
Experiment Theory (DFT PBE TZ)
C60 fullerene
RH= 1.453A (<1% error)
RH= 527 cm-1
Bond Length:
IP=7.61 eV IP=7.87 eV (<4% error)
RH= 526 cm-1
Rcc5= 1.458A
Rcc5= 1.401A RH= 1.399A (<1% error)
IR spectra:
(<1% error)
RH= 576 cm-1 RH= 576 cm-1 (<.1% error)
RH= 1183 cm-1 (<1% error) RH= 1179 cm-1
Ionization potential:
D. Sabirov RSC Adv. 3,194030(2013); J.Zhao Phys. Rev. B 65, 193401 (2002).
The Schrödinger Equation
13
EH ˆ
Highly predective:
Water
Isotropic Polarizability:
Molecule TZP/FC QZ4P/AE Exp.(a.u.)
CO 11.4 13.44 13.16a
C2H6 28.38 30.21 30.24b
p-nitroaniline 97.27 108.99 114.73c
The Schrödinger Equation
14
Highly predective:
Na6 Au20
Approximation
Approximation
TZP-FC 98.25 TZP-FC 134.72
TZP-AE 97.37 ZORA-TZP-FC 117.59
ETQZ3P 104.43 ZORA-TZP-AE 117.78
ETQZ3P1D 102.03 ZORA-QZ4P-AE 122.18
ETQZ3P2D 103.52 Idrobo et al.c 116.00
ETQZ3P3D 103.48
Fernández et al.d 104.32
Kummel et al.a 104.4 Zhao et al.e 129.27
Exp.b 111.8
The Schrödinger Equation
15
• Electronic properties
• optical properties
• Protein Folding
• X-ray spectra
• NMR spectra
• IR, VCD, CD
Typical property calculation::
The Schrödinger Equation
16
EH ˆ
§- Partial differential equation VERY VERY VERY difficult to solve.
§- Analytical solutions only for few and simple systems (harmonic osccilator,
hydrogen atom, rotor.)
§- Number of independent variables proportional to the size of the system
(number of atoms in the system).
Au20
Partial differential equation with
4800 independent variables!!!!!
The Schrödinger Equation
17
EH ˆ
An-initio/first principles
Methods (Computational
demanding) :
- Hartree-Fock
- Density Functional Theory
(DFT)
- MP2, MP3,…, CC, CI…
-Quantum Monte Carlo
Semiempirical Methods
(NON Computational
demanding) :
- Molecular Dynamics
- AM1, Huckel
- TB-DFT
Semiclasical statistical
methods (NON
Computational
demanding) :
-- Metropolis Monte Carlo
The Schrödinger Equation
18
• Electronic properties
• optical properties
• Protein Folding
• X-ray spectra
• NMR spectra
• IR, VCD, CD
Typical property calculation::
ONLY WITH COMPUTERS !!!
ATOMIC PHYSICS
I.2 The hydrogen atom
The hydrogen atom
20
Natural
abundance:
1H 99.985 %
2H 0.015 %
3H unstable
~75% of universe’s
mass is 1H !!
~90% of universe’s
atoms are 1H !!
Experimental ionization energy: IE= 1312 kJ/mol = 13.5eV
The hydrogen atom
21
The Schrodinger equation
EH ˆ
21
hydhydhydhydrogen EH ˆHydrogen
Atom hydrogenH
hyd ALL INFORMATION
OF THE HYDROGEN ATOM !!!
??
The Hamiltonian function (in SI):
epe
e
p
p
hydrogenrr
e
m
p
m
pH
2
0
22
4
1
2
)(
2
)(
),,(
),,(
eee
ee
ppp
pp
zyxiip
zyxiip
The Hamiltonian operator:
ep
e
e
p
p
hydrogenrr
e
mmH
2
0
22
22
4
1
22ˆ
Transformation of H to
Quantum Operator
pr
er
22
The Hamiltonian function (in SI):
The Hamiltonian operator (NO RELATIVISTIC):
ep
e
e
p
p
hydrogenrr
e
mmH
2
0
22
22
4
1
22ˆ
pr
er
23
22
22
22ˆ
e
e
p
p mmT
ep rr
eV
2
04
1ˆ
Time
independent!!
The hydrogen atom
24
The STATIONARY
Schrodinger equation
EH
),(),(}4
1
22{
2
0
22
22
epep
ep
e
e
p
p
rrErrrr
e
mm
pr
er
The hydrogen atom
25
The Schrodinger equation
),(),(}4
1
22{
2
0
22
22
epep
ep
e
e
p
p
rrErrrr
e
mm
Step 1: Solving SE
Step 2:Computing Properties
Solving SE: Transforming to CMS
26
),(),(}4
1
22{
2
0
22
22
epep
ep
e
e
p
p
rrErrrr
e
mm
ep
ep
eepp
rrr
mm
rmrmR
rmm
mRr
rmm
mRr
ep
p
e
ep
ep
?
Transforming to CMS
27
),,ˆ,ˆ(ˆepep rrppH
Chain rule
),,ˆ,ˆ('ˆ rRppH rR
),,,( epep rrppH
),,,(' rRppH rR
),,ˆ,ˆ('ˆ rRppH rR
It is better doing the transformation “before” (transformation of the
Hamiltonian FUNCTION:
Better transforming Hamiltonian function H
28
rmm
mRr
rmm
mRr
ep
p
e
ep
ep
r
ep
p
Re
r
ep
eRp
vmm
mvv
vmm
mvv
222222
2
1
2
1
2
1
2
1
2
)(
2
)(rReepp
e
e
p
pvMvvmvm
m
p
m
pT
Total mass: ep mmM
Reduced mass: ep
ep
mm
mm
The hydrogen atom SE
in the CMS system
),(),()}(22
{ 22
22
epepepe
e
p
p
rrErrrrVmm
),(),()}(22
{ 22
22
rRErRrVM
rR
29
Central Pontential
Separation of Schrodinger Equation
),(),()}(22
{ 22
22
rRErRrVM
rR
)()(2
22
RERM
RR
)()()}(
2{ 2
2
rErrV rr
rR EEE
rRrR
)()(),(
30
Separating the SE in the CMS
),(),()}(22
{ 22
22
rRErRrVM
rR
)()(2
22
RERM
RR
)()()}(
2{ 2
2
rErrV rr
31
The Schrodinger Equation for the
reduced mass particle
)()()}(
2{ 2
2
rErrV rr
Fictitious particle with mass μ,
e
e
ep
ep
m
mmm
mm
9994557.0
Note: Equation very similar to Born Oppenheimer approximation equation
Subjected to Coulomb potential
r
erV
1
4)(
0
2
kgm
kgm
e
p
31
27
10109.9
10673.1
ep mm 1837
32
33
Remember that this transformation can be
applied for any CENTRAL POTENTIAL
)()()}(2
{ 22
rErrV rr
Transforming to spherical coordinates
)()()}(
2{ 2
2
rErrV rr
)()()}(]sin
1cot
112[
2{
2
2
2222
2
22
22
rErrVrrrrrr
r
)()()}(ˆ2
1]
2[
2{ 2
22
22
rErrVLrrrr
r
}sin
1cot{ˆ
2
2
22
222
L
34
Angular Momentum Operator
}sin
1cot{ˆ
2
2
22
222
L
35
),,()(ˆˆzyx
riirprL r
prL
Angular momentum FUNCTION:
Angular momentum OPERATOR:
)(ˆ;)(ˆ;)(ˆx
yy
xiLz
xx
ziLy
zz
yiL zyx
2222ˆzyx LLLL
In spherical coordinates:
Transforming to spherical coordinates
)()()}(
2{ 2
2
rErrV rr
)()()}(]sin
1cot
112[
2{
2
2
2222
2
22
22
rErrVrrrrrr
r
)()()}(ˆ2
1]
2[
2{ 2
22
22
rErrVLrrrr
r
}sin
1cot{ˆ
2
2
22
222
L
36
Separating the SE in the CMS
),(),(ˆ2 ffL
)()()}(ˆ2
1]
2[
2{ 2
22
22
rErrVLrrrr
r
)()()})((2]2
[{ 2
2
222 rRrRErVr
rrrr r
),()(),,( frRr
37
The angular equation
)()( 2
2
2
m
d
d
)()(),( f
),(),(ˆ2 ffL
),(),(}sin
1cot{
2
2
22
22
ff
)()(}sin
cot{22
2
2
2
m
d
d
d
d
38
Solution of the angular equation:
Spherical Harmonics
imm
l
m
l ePml
mllYf )(cos
)!(4
)!)(12(),(),(
l
ml
mlm
l
m
l wdw
dw
lwP )1()1(
!2
1)( 222
Associated Lengendre Polynomials:
,...2,1,0l llllm ,1,...,2,1,0,...,1,
39
Solution of the angular equation:
Spherical Harmonics
,...2,1,0l llllm ,1,...,2,1,0,...,1,
),()1(),(ˆ 22 m
l
m
l YllYL
Conditions coming from the well behaved requirement on ψ
''
'
'
0
2
0
sin),(),(* mmll
m
l
m
l ddYY
40
The “radial” equation
)()1()()})((2]
2[{ 22
2
222 rRllrRErVr
rrrr r
1
0
21
12
1212
!)!12()!1(
})!{()1()()(
ln
k
kk
rl
ll
lnkklkln
lnL
d
dL
,...3,2,1n 1,...,2,1,0 nl
)2(})!{(2
)!1()
2()
2()( 12
3
3 naZrLrelnn
ln
na
Z
na
ZrR l
ln
lnaZrl
nl
2
2
04
ea
''
0
2
'' )()( llnnlnnl drrrRrR
41
The associated Laguerre polynomials:
Laguerre polynomials:
)()(
ed
deL r
r
r
r
Hydrogen Wave Functions and Energies
42
1,...,2,1,0 nl
llllm ,1,...,2,1,0,...,1,
,...3,2,1n
),()(),,( m
lnlnlm YrRr
222
0
42
222
0
42 1
8
1
)4(2 nh
eZ
n
eZEE rn
Principal quantum number
Azimuthal quantum number
magnetic quantum number
;1,...,2,1,0 nl llllm ,1,...,2,1,0,...,1, ;,...3,2,1n
Hydrogen eigenfunctions and eigenvalues
),,(),,(ˆ rErH nlmnnlm
r
eH r
1
42ˆ
0
22
2
),,( rnlm eigenfunctions
nE eigen values
43
Eigen -functions degeneracy
),,(),,(ˆ rErH nlmnnlm
),,( rnlm
nEThe degree of DEGENRACY is
equal to n2
44
;1,...,2,1,0 nl llllm ,1,...,2,1,0,...,1, ;,...3,2,1n
Once the problem (ES) is solved,
what else?
45
Hydrogen Properties!
Energy!
222
0
42
222
0
42 1
8
1
)4(2 nh
eZ
n
eZEE rn
eV
J
JsmNC
CkgZ
h
eZEn
598.13
1017868.2
)1062607.6)(/108541878.8(8
)6021765.1)(1010938.99994557.0(
8
18
2342212
4312
22
0
42
1
46
eVE 598.131 ???
Energy??
47
3
0
2
4 r
reF
Ley de Coulomb:
pr
er
r
r
dr
rdVVF
)(
This force produces a central potential:
2
0
2 1
4
)(
r
e
dr
rdV
Energy??
48
2
0
2 1
4
)(
r
e
dr
rdV
Cr
drerV 2
0
2
4)(
0)( rVr
erV
1
4)(
0
2
Energy??
49
0)( rV
r
0V
Hydrogen Properties!
Energy!
222
0
42
222
0
42 1
8
1
)4(2 nh
eZ
n
eZEE rn
eV
J
JsmNC
CkgZ
h
eZEn
598.13
1017868.2
)1062607.6)(/108541878.8(8
)6021765.1)(1010938.99994557.0(
8
18
2342212
4312
22
0
42
1
eVEn 598.131 eVIE 573.13exp !!
Theory Experiment 50
Ionization Energy:
eVEn 598.131 eVIE 573.13exp !!
Theory Experiment
51
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
52
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
epe
e
p
p
hydrogenrr
e
m
p
m
pH
2
0
22
4
1
2
)(
2
)(
pr
er
Hydrogen energies: quantum states
54
photon
Hydrogen energies: quantum states
55
photon
Series n 1
56
)1
1(8 222
0
42
11nh
eZEEE nn
)1
1(8
11232
0
42
1nch
eZE
chn
1710520973731568.1 mRtheory
32
0
42
8 ch
eZR
Lyman series
57
)1
1(1
2nR
17
exp 1009737315.1 mR eriment
1710520973731568.1 mRtheory
Hydrogen energies: quantum states
58
Spectral series n m
59
(ultraviolet)
(visible)
(infrared)
1,...,2,1,0 nl llm ,0,...,,...3,2,1n
Hydrogen eigenfunctions
),()(),,( m
lnlnlm YrRr
60
pr
er
222
0
42
222
0
42 1
8
1
)4(2 nh
eZ
n
eZEE rn
),,(),,(ˆ rErH nlmnnlm
1,...,2,1,0 nl llm ,0,...,,...3,2,1n
Hydrogen-like eigenfunctions
),()(),,( m
lnlnlm YrRr
61
pr
er
222
0
42 1
8 nh
eZEn
EH ˆ
N
),,(),,(ˆ rErH nlmnnlm
1Z ZZ
ep
ep
mm
mm
eN
eN
mm
mm
? He+
Li2+
Be3+
B4+
Hydrogen-like eigenfunctions
),()(),,( m
lnlnlm YrRr
62
)2(})!{(2
)!1()
2()
2()( 12
3
3 naZrLrelnn
ln
na
Z
na
ZrR l
ln
lnaZrl
nl
2
2
04
ea
222
0
42 1
8 nh
eZEn
Dependence on Z and :
Hydrogen-like eigenfunctions
),()(),,( m
lnlnlm YrRr
63
imm
l
m
l ePml
mllY )(cos
)!(4
)!)(12(),(
Complex
Hydrogen-like ORBITALS
),()(),,( m
lnlnlm YrRr
64
An ORBITAL function can be defined as an ONE-ELECTRON FUNCTION:
Hydrogen-like ORBITAL
Real Spherical Harmonics
65
imm
l
m
l
imm
l
m
l ePNePml
mllY )(cos)(cos
)!(4
)!)(12(),(
,...3,2,1;)(cos2)(2
1
,...3,2,1;cos)(cos2)(2
1
0
),(
0
msenmPNYYi
mmPNYY
mifY
Y
m
l
m
l
m
l
m
l
m
l
m
l
m
l
m
l
l
m
l
Notation:
,...,,,,,
,...5,4,3,2,1,0
hgfdps
l
Real hydrogen-like orbitals
66
A linear combination of eigenfunctions of the same
degenerate eigenvalor is eigenfunction.
),()(),,( m
lnlnlm YrRr
Are they eigen funtions of the hydrogen-like Hamiltonian?
Real hydrogen-like orbitals
67
n
l
m
Symbol for
Complex orbital
Symbol for real orbital
1 0 0 1s 1s 2 0 0 2s 2s 2 1 1 2p+1 2pX 2 1 0 2p0 2py 2 1 -1 2p-1 2pz
3 0 0 3s 3s 3 1 1 3p+1 3pX 3 1 0 3p0 3py 3 1 -1 3p-1 3pz 3 2 2 3d+2 3dz2 3 2 1 3d+1 3dxz 3 2 0 3d0 3dyz 3 2 -1 3d-1 3dx2-y2 3
2 -2 3d-2 3dxy
Real hydrogen-like orbitals
68
aZrea
Zs /2/3
2/1)(
11
aZrea
Zr
a
Zs 2/2/3
2/1)2()(
)2(4
12
xea
Zp aZr
x
2/2/5
2/1)(
)2(4
12
yea
Zp aZr
y
2/2/5
2/1)(
)2(4
12
zea
Zp aZr
z
2/2/5
2/1)(
)2(4
12
n=1
n=2
Real hydrogen-like orbitals
69
aZrea
rZ
a
Zr
a
Zs 3/
2
222/3
2/1)21827()(
)3(81
13
xea
Zr
a
Zp aZr
x
3/2/5
2/1
2/1
)6()()(81
23
n=3
yea
Zr
a
Zp aZr
y
3/2/5
2/1
2/1
)6()()(81
23
zea
Zr
a
Zp aZr
z
3/2/5
2/1
2/1
)6()()(81
23
Real hydrogen-like orbitals
70
)13()()6(81
13 23/2/7
2/12 zea
Zd aZr
z
xzea
Zd aZr
xz
3/2/7
2/1
2/1
)()(81
23
n=3
yzea
Zd aZr
yz
3/2/7
2/1
2/1
)()(81
23
)()()2(81
23 223/2/7
2/1
2/1
22 yxea
Zd aZr
yx
xyea
Zd aZr
xy
3/2/7
2/1
2/1
)()2(81
23
Hydrogen orbitals
),()(),,( m
lnlnlm YrRr
71
Fourth Postulate:
rdrnlm
2),,(
Probability of finding the electron in a
infinitesimal volumen around ),,(),,( zyxrr
v
What information????
Probability Density
72
???????),,(),,(2
'''
2rdrrdr mlnnlm
Probability of finding the
electron in a infinitesimal
volumen around ),,( rr
rdrrdr mlnnlm
2
'''
2),,(),,(
Probability of finding the
electron in a infinitesimal
volumen around ),,( rr
?????
Probability Density
73
Probability of finding the
electron IN THE STATE nlm
in a infinitesimal volumen
around ),,( rr
rdrrdr mlnnlm
2
'''
2),,(),,(
Probability of finding the
electron IN THE STATE n’l’m’
in a infinitesimal volumen
around ),,( rr
Probability Density
74
Probability of finding the
electron IN THE STATE 2px
in a infinitesimal volumen
around ),,( rr
rdrprdrp zx
22),,(3),,(2
Probability of finding the
electron IN THE STATE 3pz
in a infinitesimal volumen
around ),,( rr
Radial Distribution Function
75
r
ddrdrYYrRrD m
l
m
lnlnl sin),(),()()(
2
0 0
2*
22)]([)( rrRrD nlnl
1sin),(),(
2
0 0
*
ddYY m
l
m
l
Radial Distribution Function
76
r 22)]([)( rrRrD nlnl
Probability of finding the electron in
the state nl at a distance r from the
nucleus (proton)
Radial Distribution Function
77
Amme
a 529.010529.04 10
2
2
00
a0 is the Bohr radius:
1s
2s
2p { 2px , 2py , 2pz }
3s
3p { 3px , 3py , 3pz }
3d { 3dz2
, 3dxz , 3dyz , 3dx2 – y
2 , 3dxy }
Radial Distribution Function
78
Radial distribution functions for the 2s and 3s density distributions
drrRrdrrrDr nlnl
0
3
0
)()(
)})1(
1(2
11{
2
0
2
n
ll
Z
anr
Electron Density
79
)(1)( electronsofnumberNrdr
gives the probability of finding an electron at position )(r r
2)()()( rerer
Charge density:
0
2 )()(
rrV
2)()( rr
Experimental quantity !!
80
2
1 )()( rr s
Electron density
contour maps: 1s case
81
Electron density contour maps: 2s & 2p cases
Orbital Density
2s
2p { 2px , 2py , 2pz }
aZrea
Zr
a
Zs 2/2/3
2/1)2()(
)2(4
12
yea
Zp aZr
y
2/2/5
2/1)(
)2(4
12
z
y
z
y
82
Electron density contour maps: 2s & 2p cases
Orbital Density
2s
2p { 2px , 2py , 2pz }
aZrea
Zr
a
Zs 2/2/3
2/1)2()(
)2(4
12
yea
Zp aZr
y
2/2/5
2/1)(
)2(4
12
z
y
z
y
83
Electron density contour maps: 3d and 4f cases
Orbital (3dxz) Density
Orbital (4fxz2)
Average values: Properties
84
rdrPrPP nlmnlm
),,(ˆ),,(ˆ *
a the property operator P
Atomic Units (a.u.)
85
1 1em 02 e=
)()(}1
42{
0
22
2
rErr
er
)()(}1
2
1{ 2 rEr
rr
Atomic Units (a.u.)
86
5 1 15 31 27.2 2.20 10 6.58 10 2.63 10 / .)Hartree eV cm Hz kJ mol
)()(}1
2
1{ 2 rEr
rr
HartreesuaE 5.0..5.0
mABohr 111029.5529.01
Energy:
Lenth:
Mass: kgme
31101095.9
Charge:
Ce 19106022.1
Many Electron Atoms
87
),(),(}1
2
1
2{
11
2
1
222
ii
N
i
N
ij ji
N
i j
N
i
rR
n
rRErRrrrR
e
M i
EH
For gold N=79, so we have 3*79+3=240 independent variables !!!