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Correlation

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Electronic correlation
• VB method and
polyelectronic
functions
• IC
• DFT
The charge or spin interaction between 2
electrons is sensitive to the real relative
position of the electrons that is not
described using an average distribution. A
large part of the correlation is then not
available at the HF level. One has to use
polyelectronic functions (VB method), post
Hartree-Fock methods (CI) or estimation
of the correlation contribution, DFT.
1
Electronic correlation
A of the correlation refers to
HF: it is the “missing energy”
for SCF convergence:
Ecorr= E – ESCF
Ecorr< 0 ( variational principle)
Ecorr~ -(N-1) eV
2
Energy (kcal/mol)
Importance of correlation effects on Energy
RHF in blue, Exact in red.
3
Importance of correlation effects on reactions
− Bond cleavage: H2O → HO + H : ~ 130 kcal/mol
− same number of electron pairs :
H2O + H+ → H3O+: ~ 7 kcal/mol
− intermediate case:
2 BH3 → B2H : ~ 70 kcal/mol
− weak interactions (H-bonds, van der Waals)
H2O - H2O: ~ 3 kcal/mol
Ne−Ne: ~ 0.5 (kcal/mol however De is only ~ 0.3
kcal/mol)
− ions are strongly correlated systems
F-: 246.5 kcal/mol
Al3+: 253.66 kcal/mol
4
Distances pm
Importance of correlation effects on distances
RHF in blue, Exact in red.
5
Relative errors in %
Relative errors in ppm
Weak contribution: 0.1 Ả, 5%
6
Fermi hole – Coulomb hole
Each electron should be surrounded by an “empty
volume” excluding the presence of another electron
– of the same spin (Fermi hole)
– of opposite spins (Coulomb hole)
In HF, the Probability of finding an electron with the same spin
at the same place is 0; that of finding an electron with  spin
is not!
P(↑↑)2=dV1dV2/2 [a2(r1)b2(r2)+a2(r2)b2(r1)-2a(r1)b(r2)a(r2)b(r1)]
P(↑↑)2 = 0 for r1 = r2
P(↑↓)2=dV1dV2/2 [a2(r1)b2(r2)+a2(r2)b2(r1)]
P(↑↓²)2  0 for r1 = r2
This term vanishes for ↓
HF account for the Fermi hole not the Coulomb hole!
7
Correlation Left-Right
The probability of finding 1 electron in
the left region and one in the right is 1.
In HF it is only ½;
Taking into account correlation reduces
the probability of finding the electrons
together; it decreases the weight of the
ionic contribution.
8
types of correlation
How electrons avoid each other?
radial (or In-out)
Left-right
angular
9
Valence Bond
10
Heitler-London
1927
Electrons are indiscernible:
If f1(1).f2(2) is a valid solution
Walter Heinrich
Heitler
German
1904 –1981
f1(2).f2(1) also is.
The polyelectronic function is therefore:
[f1(1).f2(2) + f1(2).f2(1)]
Fritz
Wolfgang
London
German
1900–1954
To satisfy Pauli principle this symmetric expression is associated
with an antisymmetric spin function: a (1).b(2) - a(2).b(1)
this represents a singlet state!
Each atomic orbital is occupied by one electron: for a bond, this
represents a covalent bonding.
11
The triplet state
The total function has to be antisymmetric; for a triplet state since
the spin is symmetric, the spatial function has to be antisymmetric.
[f1(1).f2(2) - f1(2).f2(1)]
Associated with one of the 3 spin functions
a(1).a(2)
b(1).b(2)
a (1).b(2) + a(2).b(1)
12
The resonance,
ionic functions
-
H1 -H2
+
Other polyelectronic functions :
-
+
+
H1 -H2 ↔ H1 -H2
-
f1(1).f1(2)
[f1(1).f1(2) + f2(2).f2(1)]
f2(2).f2(1)
[f1(1).f1(2) -f2(2).f2(1)]
+
H1 -H2
-
symmetric
antisymmetric
According to Pauli principle, these functions necessarily
correspond to the singlet state:
a (1).b(2) - a(2).b(1)
13
H2 dissociation
The cleavage is whether homolytic, H2 → H• + H•
or heterolytic: H2 → H+ + H-
H+ + H- E = 2 √(1-s)3/p = -0.472 a.u.
with s=0.31
H+H
E = -1 a.u;
14
MO behavior of H2 dissociation
The cleavage is whether
homolytic, H2 → H• + H•
or heterolytic: H2 → H+ + H-
Energy
Energie
u
distance
A-B
distance
internucléaire
g
sg2 = (fA +fB)2 = [(fA(1) fA(2) + fB(1) fB(2)] + [(fA(1) fB(2) + fB(1) fA(2)]
50% ionic
+
50% covalent
The MO description fails to describe correctly the dissociation!
su2 = (fA -fB)2 = [(fA(1) fA(2) + fB(1) fB(2)] - [(fA(1) fB(2) + fB(1) f15A(2)]
50% ionic
50% covalent
H2 dissociation
The cleavage is whether homolytic, H2 → H• + H•
or heterolytic: H2 → H+ + H-
H+ + H- E = 2 √(1-s)3/p = -0.472 a.u.
with s=0.31
*
H+H
The MO approach,
-0.769 a.u.
E = -1 a.u;
16
What are the solutions?
1. Uncouple ionic and covalent functions;
this is the VB approach.
2. Let interact the sg2 and su2 states: these
are of the same symmetry and interact.
This is the IC approach.
E= -0.472 a.u.
E= -0.769 a.u.
sg2 and su2
ionic
E= -1 a.u.
covalent
17
The Valence-Bond method
It consists in describing electronic states of a
molecule from AOs by eigenfunctions of
S2, Sz and symmetry operators. There
behavior for dissociation is then correct.
These functions are polyelectronic. To
satisfy the Pauli principle, functions are
determinants or linear combinations of
determinants build from spinorbitals.
18
Covalent function for electron pairs
Ordered on spins
a (1) b(1)
or
 = IabI + IbaI
Ordered on electrons
IabI =
a (2) b(2)
 = IabI - IabI
 = [a(1).b(2) +a(2). b(1)].[a(1).b(2) - a(2).b(1)]
 = [a(1).a(1)b (2)b(2) - b(1).b(1). a(2).a(2)]
+ [b(1).a(1) a(2)b(2) - a(1).b(1).b (2).a(2)]
 = IabI + IbaI
This is the Heitler-London expression
19
Triplet states
Ordered on spins
 = IabI
 = IabI
or
 = IabI -+ IbaI
Ordered on electrons
 = IabI +- IabI
 = [a(1).b(2) -a(2). b(1)].[a(1).b(2) +a(2).b(1)]
 = [a(1).a(1)b (2)b(2) - b(1).b(1). a(2).a(2)]
- [b(1).a(1) a(2)b(2) - a(1).b(1).b (2).a(2)]
 = IabI - IbaI
20
2 electrons – 2 orbitals
│ ab │+│ ba │- l │ aa │+│ bb │
Diexcited States
│aa│- │ bb │
Jij+Kij
Jij
Kij
First excited
States
Jij-Kij
│ ab │ │ ab│
│ ab │- │ ba │
Ground state
│ aa │ +│ bb │+ l │ ab │+│ ba │
21
Valence Bond
Make the list of all the resonance structures. (a complete
treatment necessitates considering all of them).
To each one is associated a VB expression (and a Lewis
structure).
• Rule 1: electrons belonging to a pair have opposite spins
• Rule 2: bonds are represented by covalent singlet functions
• Rule 3: The total wavefunction is antisymmetric relative to
exchange of spins (symmetric relative to exchange of
electrons).
• The VB functions interact to give linear combinations
(coefficients and energies are obtained through a variational
principle and a secular determinant)
• The square of the coefficients are the weights of the VB
structure
22
Resonance structures for the p electrons of benzene:
Neutral species have larger weights than charged structures,
Kekulé more than Dewar.
Dewar
Kekulé
+
+
Kekulé
+
+
-
-
It is easy to
make
approximation,
This structure
has small weight
23
Example of Butadiene
a b c d
C=C-C=C
• A determinant: IabcdI
• ab is a bond:IabcdI+IbacdI = IabcdI-IabcdI
• cd is a bond:IabcdI+IbacdI + IbadcI+IabdcI
or IabcdI-IabcdI - IabcdI+IabcdI
24
excited Butadiene (zwiterion)
a b c d
+
+
C-C=C-C  C-C=C-C
• A determinant: IaabcI (2 electrons in a, none in d)
• aa is a pair: IaabcI
• cd is a bond: IaabcI+IaacbI = IaabcI-IaabcI
• Resonance: IaabcI+IaacbI + IbcddI+IcbddI
= IaabcI-IaabcI + IbcddI-IbcddI
25
Calculation of matrix elements
Q = <IabIIHIIabI>
H = Hmono + Hbi
<abIHmonoIab> = <aIa> <bIHmonoIb> + <aIHmonoIa> <bIb> = haa+hbb
<abIHbiIab> = Jab
Q = <abIHIab> = haa+ hbb+ Jab
K = <IabIIHIIbaI>
<abIHmonoIba> = <aIb> <bIHmonoIa> + <aIHmonoIb> <bIa> = 2habSbb
<abIHbiIba> = Kab
K = <abIHIba> = 2Sabhab+ Kab
26
Calculation of matrix elements
<IaabbIIIaabbI>
General Method: 1/n! <all permutationsIHIall permutations>
• Remove 1/n! and retain only one permutation
<single permutationIHIall permutations>
• Keep only permutations involving the same spin.
1/24<IaabbIIIaabbI>
= <aabbIIaabbI>
= <aabbIaabb> - <aabbIbaab> - <aabbIabba> + <aabbIbbaa>
=
1
-
=
1 -2S2 +S4
S2
-
S2
+
S4
Using this rule, it is simpler to order the
determinants on spins rather than on electrons27
E
triplet
covalent states
-2S abhab-Kab
haa+h bb+J ab+1/R
+2S abhab+K ab
singulet
singlet
AB distance
distance H-H
28
H2 singlet states: the symmetric ionic and covalent functions mix to
generate the ground state and the diexcited state;
the antisymmetric covalent state does not mix (Brillouin theorem)
Diexcited States , mixed character
More covalent than ionic
│aa│- │ bb │Antisymmetric
2Kab
│ ab │ │ ab│ triplets │ ab │- │ ab │
2Sabhab+Jab
First excited
States
│ab│+ │ ba │Sym
2Kab/(1+S2)
│ aa │+ │ bb │Sym
Ground state, mixed character 29
more covalent than ionic
Energies of pure VB structures
IabI + IbaI
Singlet (covalent) Sym
haa+hbb+Jab+2Sabhab+Kab
1+Sab2
IaaI + IbbI
Singlet (ionic)
haa+hbb+Jab+2Sabhab-Kab
1+Sab2
IabI, IabI, IabI - IbaI
triplet
haa+hbb+Jab-2Sabhab-Kab
1-Sab2
IaaI - IbbI
Singlet (ionic) Antisym
haa+hbb-Jab-2Sabhab+Kab
1-Sab2
30
mixing of the singlet states
Brillouin theorem 1934
Two functions are symmetric
and one is antisymmetric: the
ionic, antisymmetric state
does not mix.
Léon Nicolas Brillouin 1889 –
It is essentially a non-bonding 1969) was a French physicist. His
state corresponding to sgsu father, Marcel Brillouin, grandfather, Éleuthère Mascart, and
(monoexcitation)
great-grand-father, Charles Briot,
were physicists as well. He made
haa+hbb+Jab+2Sabhab-Kab
contributions
to
quantum
mechanics,
radio
wave
1+Sab2
propagation in the atmosphere,
solid
state
physics,
and
31
information theory.
Interaction term between symmetric singlets
1/√(1+S2){│ aa │+ bb │} │H│ 1/√(1+S2){│ ab │+ │ ba │}
= {(haa+hbb)S+2hab+<aa│1/r12│ab> +<bb│1/r12│ba> } / (1+S2)
= {(haa+hbb)S+2hab+ (aa│ab)+(bbba) } / (1+S2)
│ab│+ │ ba │Sym
2Kab/(1+S2)
│ aa │+ │ bb │Sym
Ground state, mixed character 32
more covalent (80%) than ionic
Sign of K
Earlier we have define K>0 from a single determinant; it
was a consequence of the Pauli principle <IabIIIabI>
Here we have the determinant with permutation;
<IabIIIbaI>
This changes the sign: K<0.
The covalent energy is the lowest.
IabI + IbaI
haa+hbb+Jab+2Sabhab+Kab
Singlet (covalent) Sym
1+Sab2
IaaI + IbbI
Singlet (ionic)
haa+hbb+Jab+2Sabhab-Kab
1+Sab2
33
34
Resonance & covalence,
The ionic VB structure: H+  HThe covalent: H-H
Both structures contribute to the
bonding, equally for MOs or
when K is neglected:
(1+S) (aa+bb)+(1-S) (ab+ba)
There is 1 electron in each orbital
so that the density is the same:
In the middle plane, the amplitude is not zero for the two VB
structures. When they combine equally, they double (bonding
state) or vanish (antibonding state).
35
Resonance & covalence,
electron pair & AF diradical
The ionic VB structure: H+  H(aa+bb) matches better the description of electron
pairs, the two electrons being located at the same
place
The covalent:
H-H
(ab+ba) corresponds better to an antiferromagnetic
state with one electron on each side and a diradical.
36
Interaction term between symmetric singlets
within Hückel approximation
{(haa+hbb)S+2hab+ (aa│ab)+(bbba) } / (1+S2) becomes
2hab} and the two states are degenerate → forming 50%
ionic 50% covalent states
The bonding state is therefore the sum:
this is 2 Esg the energy of the sg2 state!
2 Esu
2hab
│ aa │+ │ bb │Sym
2hab
E = haa+hbb
│ab│+ │ ba │Sym
2 Esg
37
Interaction term between symmetric singlets
Hückel approximation with spin
(haa+hbb)S+2hab - E
(haa+hbb) +2habS - E S
1+S2
1+S2
1+S2
(haa+hbb) +2habS - E S
(haa+hbb)S+2hab - E
1+S2
1+S2
1+S2
=0
2 Esu= 2(haa-hab)/(1-S)
2 Esg = 2(haa+hab)/(1+S)
with S,
the average energy is above the energy of pure VB structures
38
Interaction term between symmetric singlets
Hückel approximation with spin
2 Esu= 2(haa-hab)/(1-S)
2haa -2habS
Mean energy
E=
1-S2
Atomic energy
2 hbb +2habS
Pure VB structures E =
1+S2
│ aa │+ │ bb │
E = haa= hbb
│ab│+ │ ba │
2 Esg = 2(haa+hab)/(1+S)
39
Generalized Valence Bond, GVB
It starts by an orthogonalization
William A. Goddard40III,
Interaction Configuration
The OM calculated at the HF level are
eigenfunctions of H°+S<Repij> that is not
H°+S<1/rij>. We can form linear
combinations of the determinants using
variational theory. This is the IC. If the
basis set is “complete”, a full IC should
lead to experimental results (excepting
relativistic effects, Born-Oppenheimer…)
41
Interaction Configuration
OM/IC: In general the OM are those
calculated in an initial HF calculation.
Usually they are those for the ground
state.
MCSCF: The OM are optimized
simultaneously with the IC (each one
adapted to the state).
42
Interaction Configuration:
mono, di, tri, tetra excitations…
Monexcitation: promotion of i to k
k
I ki>
j
i
Diexcitation promotion of i and j to k and l
l
kl
Ij
I >
k
j
i
43
Branching diagram
How many configurations for a spin state?
S
4
1
7/2
1
3
1
5/2
1
2
1
3/2
1
1
1/2
1
1
0
1
2
5
3
1
14
5
4
20
9
2
3
6
4
2
7
28
14
5
5
Each number is
The sum of the
two previous
ones.
See circles!
6
14
7
Number of open shells
8
44
Matrix elements within a state and
an excited state: Slater rules
<GSI pq> = S [(ppIqr)-(pqIpr)] - S [(ttIqr)-(tqItr)]
p
q
All the matrix elements are bielectronic terms
(since H-HHF concerns the bielectronic
repulsion).
Many of them are equal to zero.
<GSI pq> = (ppIqr)-(pqIpr) - (ttIqr)-(tqItr)
For a single excitation, the terms are the offdiagonal terms of the Fock-matrix, which are 0
for HF eigenfunctions.
There is no mixing between the GS and the
mono-excited states. Brillouin theorem
(already seen using symmetry).
45
Mind the spatial symmetry
Only determinants with the same symmetry
interact.
To perform an IC:
Generate all the configurations
corresponding to an electronic state
(solutions of Sz and S2) and eliminate
those that are of different symmetry.
46
Full symmetry, truncated symmetry
Since it is expensive, instead of making a full IC, one
restricts the “space of configuration” to a small space.
Note that VB includes more physics in truncations.
Example of H2O
excitations
single
diexcitations
tri
tetra
All the others
% of excitations
0.6 % (Brillouin)
94 %
0.8 % (similar to Brillouin)
4.4 %
0.17 %
47
For other properties (Dipole moment) the
monoexcitations count!
:C≡O:
or
Energy
¨
:C=O
¨
m(D)
SCF
-112.788
-0.108
SCF+di
-112.016
-.068
SCF+mono+di
-113.018
+0.030
Exp.
Cδ- Oδ+
+.044
48
Coherence in size
Correlation should depend on N
Ecorr~ -(N-1) eV
Considering mono and diexcitations (SDIC)
on gets a size dependence in N1/2. This
requires correction as proposed by Davidson.
perturbations
•
•
•
•
Moller-Plesset
MP2 perturbation for di-excitations
MP3 perturbation up to tri-excitations
MP4 perturbation up to tetra-excitations
49
IC: increasing the space of
configuration
4x4
2x2
Exact energy levels
3x3
50
Which linear combination of sg2 and
su2 is 80% covalent and 20% ionic?
sg2 and su2 are 50% covalent and 50% ionic:
sg = 1/√2 (aa+bb) + 1/√2 (ab+ba)
2
The square of
the coefficents
is the weight
su2 = 1/√2 (aa+bb) - 1/√2 (ab+ba)
= lsg2+msu2 =(l+m)/√2(aa+bb)+(l-m)/ √ 2(ab+ba)
(l+m)2/2) = 0.2
(l-m)2/2 = 0.8
(l+m)2 = 0.4
(l-m)2 = 1.6
l = 0.9467
m = -0.3162
The diexcitation allows flexibility between covalency and
ionicity.
51
VB vs. IC
Both methods are equivalent provided that
• The same basis set are used
• there is no simplification (truncation of the # of VB
structures or limited IC)
They both take into account correlation and allow mixing
covalent and ionic contributions in variable amounts.
Advantage of VB:
• Close to Lewis structures and chemical language
• Easier to visualize and then easier for making
approximation
Advantage of IC:
• Many efficient softwares
• Less thinking
52
53
Density Functional Theory
What is a functional? A function of another function:
In mathematics, a functional is traditionally a map from a
vector space to the field underlying the vector space,
which is usually the real numbers. In other words, it is a
function that takes a vector as its argument or input and
returns a scalar. Its use goes back to the calculus of
variations where one searches for a function which
minimizes a certain functional.
E = E[r(r)]
E(r) = T(r) + VN-e(r) + Ve-e(r)
54
Thomas-Fermi model (1927): The kinetic energy
for an electron gas may be represented as a
functional of the density.
It is postulated that electrons are uniformely distributed in space.
We fill out a sphere of momentum space up to the Fermi value,
4/3 p pFermi3 . Equating #of electrons in coordinate space to that
in phase space gives:
n(r) = 8p/(3h3) pFermi3
and
T(n)=c ∫ n(r)5/3 dr
T is a functional of n(r).
Llewellen Hilleth
Thomas
1903-1992
Enrico Fermi
1901-1954 Italian
nobel 1938
55
DFT
Pierre C
Hohenberg
Kohn 1923,
german-born
american
Two Hohenberg and
Kohn theorems :
The existence of a unique functional.
The variational principle.
Walter Kohn
1923,
Austrian-born
American
nobel 1998
56
First theorem: on existence
The first H-K theorem demonstrates that the ground state properties of
a many-electron system are uniquely determined by an electron
density.
It lays the groundwork for reducing the many-body problem of N electrons with
3N spatial coordinates to only 3 spatial coordinates, through the use of
functional of the electron density.
This theorem can be extended to the time-dependent domain to develop timedependent density functional theory (TDDFT), which can be used to describe
excited states.
The external potential, and hence the total energy, is a unique
functional of the electron density.
57
First theorem on Existence : demonstration
The external potential, and hence the total energy, is a unique
functional of the electron density.
The proof of the first theorem is remarkably simple and proceeds by reductio
ad absurdum.
Let there be two different external potentials, V1 and V2 , that
give rise to the same density r. The associated Hamiltonians,H1
and H2, will therefore have different ground state wavefunctions,
1 and 2, that each yield r.
E1 < < 2 IH 1I 2 > = < 2 IH 2I 2 > + < 2 IH1 -H 2I 2 >
= E2 + ∫ r(r)(V1(r)- V2(r))dr
E2 < < 1 IH 2I 1 >
= E1 + ∫ r(r)(V2(r)- V1(r))dr
E1 + E2 < E1 + E2
Therefore ∫ r(r)(V1(r)- V2(r))dr=0 and V1(r) = V2(r)
The electronic energy of a system is function of a single
electronic density only.
58
Second theorem: Variational principle
The second H-K theorem defines an energy functional for the
system and proves that the correct ground state electron density
minimizes this energy functional :
If r(r) is the exact density, E[r(r)] is minimum and we
search for r by minimizing E[r(r)] with ∫ r(r)dr = N
r(r) is a priori unknown
As for HF, the bielectronic terms should depend on two
densities ri(r) and rj(r) : the approximation r2e(ri,rj) =
ri(ri) rj(rj) assumes no coupling.
59
Kohn-Sham equations
3 equations in their canonical form:
Lu Jeu Sham
San Diego
Born in Hong-Kong
member of the National Academy of
Sciences and of the Academia
Sinica of the Republic of China
60
Kohn-Sham equations
Equation 1:
This reintroduces orbitals: the density is defined
from the square of the amplitudes. This is needed to
calculate the kinetic energy.
61
Kohn-Sham equations
Equation 2: Using an effective potential, one has a
one-body expression.
The intractable many-body problem of interacting electrons
in a static external potential is reduced to a tractable problem
of non-interacting electrons moving in an effective potential.
The effective potential includes the external potential and the
effects of the Coulomb interactions between the electrons,
e.g., the exchange and correlation interactions.
62
Kohn-Sham equations
Equation 3: the writing of an effective single-particle potential
Eeff(r) = <│Teff+Veff │  >
Teff+Veff = T + Vmono + RepBi
Veff = Vmono + RepBi + T – Teff
Aslo a mono electronic
expression
Unknown except for free
electron gas
Veff(r) = V(r) + ∫e2r(r’)/(r-r’) dr’ + VXC[r(r)]
as in HF
63
Exchange correlation functionals
VXC[r(r)]
This term is not known except for free electron gas: LDA
EXC[r(r)] = ∫r(r) EXC(r) dr
EXC[r(r)] = VXC[r(r)]/ r(r)
= EXC( r) + r(r)  EXC( r)/ r
EXC( r) = EX( r) + EC( r)
= -3/4(3/p)1/3 r(r) + EC( r)
determined from Monte-Carlo
64
and approximated by analytic expressions
Exchange correlation functionals
SCF-Xa
The introduction of an approximate term for the
exchange part of the potential is known as the Xa
method.
VXa () = -6a(3/4 pr  )1/3
r  is the local density of spin up
electrons
and a is a variable parameter.
with a similar expression for r↓
65
Exchange correlation functionals
VXC[r(r)]
EXC = EXC( r) LDA or LSDA (spin polarization)
EXC = EXC( r,r) GGA or GGSDA
- Perdew-Wang
- PBE: J. P. Perdew, K. Burke, and M. Ernzerhof
EXC = EXC( r,r,Dr)
metaGGA
66
Hybrid methods:
B3-LYP (Becke, three-parameters,
Lee-Yang-Parr)
Incorporating a portion of exact exchange from HF theory with
exchange and correlation from other sources :
a0=0.20 ax=0.72
aC=0.80
List of hybrid methods: B1B95 B1LYP
MPW1PW91 B97 B98 B971 B972 PBE1PBE
O3LYP BH&H BH&HLYP BMK
67
Axel D. Becke german 1953
Robert G. Parr
Chicago 1921
Chengteh Lee received his
Ph.D. from Carolina in
1987 for his work on DFT
and is now a senior
scientist at the
supercomputer company
Cray, Inc.
Weitao Yang
Duke university USA
Born 1961 in
Chaozhou, China
got his undergraduate
degree at the
University of Peking
68
DFT
Advantages : much less expensive than IC or VB.
adapted to solides, metal-metal bonds.
Disadvantages: less reliable than IC or VB.
One can not compare results using different
functionals*. In a strict sense, semi-empical,not abinitio since an approximate (fitted) term is
introduced in the hamiltonian.
* The variational priciples applies within a given functional and not
to compare them. The only test for validity is comparison with
experiment, not a global energy minimum!
69
DFT good for IPs
IP for Au (eV)
Without f
With f functions
SCF
7.44
7.44
SCF+MP2
8.00
8.91
B3LYP
9.08
9.08
Experiment
9.22
Electron
affinity H
Exp.
SCF
IC
B3LYP
Same
basis set
0.735
-0.528
0.382
0.635
70
DFT good for Bond Energies
Bond energies (eV); dissociation are better than in HF
Exp.
HF
LDA
GGA
B2
3.1
0.9
3.9
3.2
C2
6.3
0.8
7.3
6.0
N2
9.9
5.7
11.6
10.3
O2
5.2
1.3
7.6
6.1
F2
1.7
-1.4
3.4
2.2
71
DFT good for distances
Distances (Å)
Exp.
HF
LDA
GGA
B2
1.59
1.53
1.60
1.62
C2
1.24
0.8
7.3
6.0
N2
1.10
1.06
1.09
1.10
O2
1.21
1.15
1.20
1.22
F2
1.41
1.32
1.38
1.41
72
DFT polarisabilities (H2O)
Distances (Å)
Exp.
HF
LDA
m
0.728
0.787
0.721
axx
9.26
7.83
9.40
axx
10.01
9.10
10.15
axx
9.62
8.36
9.75
73
DFT dipole moments (D)
Distances (Å)
Exp.
HF
LDA
GGA
CO
-0.11
0.33
-0.17
-0.15
CS
1.98
1.26
2.11
2.01
LiH
5.83
5.55
5.65
5.74
HF
1.82
1.98
1.86
1.80
74
Jacobs’scale, increasing progress according Perdew
Steps
Paradise = exactitude
Method
Example
5th step
Fully non local
-
Hybrid Meta GGA
B1B95
Hybrid GGA
B3LYP
3rd step
Meta GGA
BB95
2nd step
GGA
BLYP
1st step
LDA
SPWL
th
4 step
Earth = Hartree-Fock Theory
75
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