1 electronic correlation vb method and polyelectronic functions ic dft the charge or spin...
TRANSCRIPT
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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.
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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
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Importance of correlation effects on Energy
RHF in blue, Exact in red.
En
erg
y (k
cal/m
ol)
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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/molNe−Ne: ~ 0.5 (kcal/mol however De is only ~ 0.3 kcal/mol)− ions are strongly correlated systemsF-: 246.5 kcal/molAl3+: 253.66 kcal/mol
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Importance of correlation effects on distances
RHF in blue, Exact in red.
Dis
tan
ces
pm
6
Relative errors in % Relative errors in ppm
Weak contribution: 0.1 Ả, 5%
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Fermi hole – Coulomb holeEach 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
HF account for the Fermi hole not the Coulomb hole!
This term vanishes for ↓
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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.
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types of correlation How electrons avoid each other?
Left-right radial (or In-out)
angular
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Valence Bond
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Heitler-London1927
Walter Heinrich Heitler German 1904 –1981
Fritz Wolfgang LondonGerman 1900–1954
Electrons are indiscernible:
If is a valid solution
also is.
The polyelectronic function is therefore:
To satisfy Pauli principle this symmetric expression is associated with an antisymmetric spin function: this represents a singlet state!Each atomic orbital is occupied by one electron: for a bond, this represents a covalent bonding.
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The total function has to be antisymmetric; for a triplet state since the spin is symmetric, the spatial function has to be antisymmetric.
Associated with one of the 3 spin functions
The triplet state
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The resonance, ionic functions
Other polyelectronic functions :
According to Pauli principle, these functions necessarily correspond to the singlet state:
H1--H2
+
H1+-H2
-
H1--H2
+ ↔ H1+-H2
-
symmetric
antisymmetric
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H+ + H- E = 2 √(1-)3/p = -0.472 a.u.
with =0.31
H2 dissociation The cleavage is whether homolytic, H2 → H• + H•
or heterolytic: H2 → H+ + H-
H + H E = -1 a.u;
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MO behavior of H2 dissociation
g
udistanceinternucléaire
EnergieEnergy
A-B distance
The cleavage is whetherhomolytic, H2 → H• + H•
or heterolytic: H2 → H+ + H-
g2 = (A +B)2 = [(A(1) A(2) + B(1) B(2)] + [(A(1) B(2) + B(1) A(2)]
50% ionic + 50% covalent The MO description fails to describe correctly the dissociation!
u
2 = (A -B)2 = [(A(1) A(2) + B(1) B(2)] - [(A(1) B(2) + B(1) A(2)] 50% ionic - 50% covalent
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H+ + H- E = 2 √(1-)3/p = -0.472 a.u.
with =0.31
H2 dissociation The cleavage is whether homolytic, H2 → H• + H•
or heterolytic: H2 → H+ + H-
H + H E = -1 a.u;
The MO approach, -0.769 a.u.*
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E= -0.769 a.u.
What are the solutions?
1. Uncouple ionic and covalent functions; this is the VB approach.
2. Let interact the g2 and u
2 states: these are of the same symmetry and interact. This is the IC approach.
g2 and u
2
E= -0.472 a.u.
E= -1 a.u.
ionic
covalent
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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.
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Covalent function for electron pairs
a (1) b(1)IabI = a (2) b(2)
= IabI + IbaI
= IabI - IabI
Ordered on electrons
Ordered on spins
or
= [a(1).b(2) +a(2). b(1)].[(1).(2) - (2).(1)] = [a(1).(1)b (2)(2) - b(1).(1). a(2).(2)] + [b(1).(1) a(2)(2) - a(1).(1).b (2).(2)]
This is the Heitler-London expression
= IabI + IbaI
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Triplet states
= IabI + IbaI
= IabI - IabI
Ordered on electrons
Ordered on spins
or
= [a(1).b(2) -a(2). b(1)].[(1).(2) +(2).(1)] = [a(1).(1)b (2)(2) - b(1).(1). a(2).(2)] - [b(1).(1) a(2)(2) - a(1).(1).b (2).(2)]
= IabI - IbaI
+
-= IabI
= IabI
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2 electrons – 2 orbitals
Jij-Kij
KijJij
│ ab │ │ ab│ │ ab │- │ ba │
│aa│- │ bb │Jij+Kij
│ aa │
│ bb │
Ground state
First excitedStates
Diexcited States
+│ bb │+
│ aa │+
│ ab │+│ ba │
│ ab │+│ ba │-
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Valence BondMake 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
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++
+
+
-
-
-
-
Resonance structures for the electrons of benzene:Neutral species have larger weights than charged structures,Kekulé more than Dewar.
Kekulé
It is easy to make approximation,
This structure has small weight
Kekulé Dewar
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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
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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
+- + -
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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
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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 - S2 - S2 + S4 = 1 -2S2 +S4 Using this rule, it is simpler to order the determinants on spins rather than on electrons
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covalent statesE
+2Sabhab+Kab
-2Sabhab-Kab
haa+hbb+Jab+1/R
distance H-H
triplet
singuletsinglet
AB distance
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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)
2Kab
│ ab │ │ ab│ triplets │ ab │- │ ab │
│aa│- │ bb │Antisymmetric
Ground state, mixed charactermore covalent than ionic
First excitedStates
Diexcited States , mixed characterMore covalent than ionic
│ aa │+ │ bb │Sym
│ab│+ │ ba │Sym2Kab/(1+S2)
2Sabhab+Jab
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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
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mixing of the singlet states Brillouin theorem 1934
Two functions are symmetric and one is antisymmetric: the ionic, antisymmetric state does not mix.
It is essentially a non-bonding state corresponding to gu (monoexcitation)
haa+hbb+Jab+2Sabhab-Kab
1+Sab2
Léon Nicolas Brillouin 1889 – 1969) was a French physicist. His father, Marcel Brillouin, grand-father, Éleuthère Mascart, and great-grand-father, Charles Briot, were physicists as well. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory.
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Interaction term between symmetric singlets
│ aa │+ │ bb │Sym
Ground state, mixed charactermore covalent (80%) than ionic
│ab│+ │ ba │Sym2Kab/(1+S2)
│ aa │+ │ bb │Sym
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)
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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
Singlet (covalent) Sym
haa+hbb+Jab+2Sabhab+Kab
1+Sab2
IaaI + IbbI
Singlet (ionic)
haa+hbb+Jab+2Sabhab-Kab
1+Sab2
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Resonance & covalence,
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:
The ionic VB structure: H+ H-
The covalent: H-H
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).
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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.
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Interaction term between symmetric singlets within Hückel approximation
│ aa │+ │ bb │Sym│ aa │+ │ bb │Sym
{(haa+hbb)S+2hab+ (aa│ab)+(bbba) } / (1+S2) becomes 2hab} and the two states are degenerate → forming 50% ionic 50% covalent statesThe bonding state is therefore the sum:this is 2 Eg the energy of the g
2 state!
haa+hbbE =
2hab
2 Eg
2hab
2 Eu
│ab│+ │ ba │Sym
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Interaction term between symmetric singlets Hückel approximation with spin
2 Eg = 2(haa+hab)/(1+S)
2 Eu= 2(haa-hab)/(1-S)
(haa+hbb)S+2hab - E
1+S2
(haa+hbb) +2habS - E S
1+S2 1+S2
(haa+hbb) +2habS - E S
1+S2 1+S2
(haa+hbb)S+2hab - E
1+S2
= 0
with S, the average energy is above the energy of pure VB structures
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Interaction term between symmetric singlets Hückel approximation with spin
│ aa │+ │ bb │
2 hbb +2habS 1+S2
E =
2 Eg = 2(haa+hab)/(1+S)
│ab│+ │ ba │
2 Eu= 2(haa-hab)/(1-S)
haa= hbbE =
2haa -2habS 1-S2
E =Mean energyAtomic energy
Pure VB structures
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Generalized Valence Bond, GVBIt starts by an orthogonalization
William A. Goddard III,
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Interaction Configuration
The OM calculated at the HF level are eigenfunctions of H°+<Repij> that is not H°+<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…)
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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).
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Interaction Configuration:mono, di, tri, tetra excitations…
Monexcitation: promotion of i to k
Diexcitation promotion of i and j to k and l
k
j
i
l
k
j
i
I >ki
I >k l I j
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Branching diagramHow many configurations for a spin state?
4 1
7/2 1
3 1 7
5/2 1 6
2 1 5 20
3/2 1 4 14
1 1 3 9 28
1/2 1 2 5 14
0 1 2 5 14
1 2 3 4 5 6 7 8
Number of open shells
SEach number isThe sum of the two previous ones.See circles!
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Matrix elements within a state and an excited state: Slater rules
<GSI > = [(ppIqr)-(pqIpr)] - [(ttIqr)-(tqItr)] pq p q
All the matrix elements are bielectronic terms (since H-HHF concerns the bielectronic repulsion).Many of them are equal to zero.
For a single excitation, the terms are the off-diagonal 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).
<GSI > = (ppIqr)-(pqIpr) - (ttIqr)-(tqItr)pq
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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.
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Full symmetry, truncated symmetrySince 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.
excitations % of excitations
single 0.6 % (Brillouin)
diexcitations 94 %
tri 0.8 % (similar to Brillouin)
tetra 4.4 %
All the others 0.17 %
Example of H2O
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For other properties (Dipole moment) the monoexcitations count!
Energy (D)
SCF -112.788 -0.108
SCF+di -112.016 -.068
SCF+mono+di -113.018 +0.030
Exp. +.044
:C≡O: or :C=O ¨
¨
Cδ- Oδ+
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perturbations• Moller-Plesset
• MP2 perturbation for di-excitations
• MP3 perturbation up to tri-excitations
• MP4 perturbation up to tetra-excitations
Coherence in size
Correlation should depend on NEcorr~ -(N-1) eV Considering mono and diexcitations (SDIC) on gets a size dependence in N1/2. This requires correction as proposed by Davidson.
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IC: increasing the space of configuration
Exact e
nerg
y leve
ls
2x2
3x3
4x4
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Which linear combination of g2 and
u2 is 80% covalent and 20% ionic?
g2 and u
2 are 50% covalent and 50% ionic:
g2 = 1/√2 (aa+bb) + 1/√2 (ab+ba)
u2 = 1/√2 (aa+bb) - 1/√2 (ab+ba)
= g2+u
2 =(√(aa+bb)+(√(ab+ba) 2= 0.2 2= 0.82= 0.4 2= 1.6= 0.9467 = -0.3162The diexcitation allows flexibility between covalency and ionicity.
The square of the coefficents is the weight
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VB vs. ICBoth 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
approximationAdvantage of IC: • Many efficient softwares• Less thinking
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Density Functional TheoryWhat 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)]
E() = T() + VN-e() + Ve-e()
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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 pFermi
3 . Equating #of electrons in coordinate space to that in phase space gives:
n(r) = 8/(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
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DFT
Two Hohenberg and Kohn theorems : Walter Kohn
1923, Austrian-born American nobel 1998
Pierre C Hohenberg Kohn 1923, german-bornamerican
The existence of a unique functional.
The variational principle.
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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 time-dependent 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.
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First theorem on Existence : demonstrationThe 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 . The associated Hamiltonians,H1 and H2, will therefore have different ground state wavefunctions, 1 and 2, that each yield .
E1 < < 2 IH 1I 2 > = < 2 IH 2I 2 > + < 2 IH1 -H 2I 2 > = E2 + ∫ rV1r V2r))dr
E2 < < 1 IH 2I 1 > = E1 + ∫ rV2r V1r))dr
E1 + E2 < E1 + E2
Therefore ∫ rV1r V2r))dr=0 and V1r V2r)The electronic energy of a system is function of a single electronic density only.
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Second theorem: Variational principleThe 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) is the exact density, E[(r)] is minimum and we search for by minimizing E[(r)] with ∫ (r)dr = N
(r) is a priori unknown
As for HF, the bielectronic terms should depend on two densities i(r) and j(r) : the approximation 2e(ri,rj) = i(ri) j(rj) assumes no coupling.
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Kohn-Sham equations3 equations in their canonical form:
Lu Jeu Sham San DiegoBorn in Hong-Kongmember of the National Academy of Sciences and of the Academia Sinica of the Republic of China
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Kohn-Sham equationsEquation 1:
This reintroduces orbitals: the density is defined from the square of the amplitudes. This is needed to calculate the kinetic energy.
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Kohn-Sham equationsEquation 2:
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.
Using an effective potential, one has a one-body expression.
63
Kohn-Sham equationsEquation 3: the writing of an effective single-particle potential
Eeff() = <│Teff+Veff │ > Teff+Veff = T + Vmono + RepBi Veff = Vmono + RepBi + T – Teff
Veff(r) = V(r) + ∫e2(r’)/(r-r’) dr’ + VXC[(r)]as in HF
Aslo a mono electronic expressionUnknown except for free electron gas
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Exchange correlation functionalsVXC[(r)]
This term is not known except for free electron gas: LDA
EXC[(r)] = ∫(r) EXC(r) dr EXC[(r)] = VXC[(r)]/ (r) = EXC( ) + (r) EXC( )/
EXC( ) = EX( ) + EC( ) = -3/4(3/)1/3 (r) + EC( ) determined from Monte-Carlo
and approximated by analytic expressions
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Exchange correlation functionalsSCF-X
The introduction of an approximate term for the exchange part of the potential is known as the X method.
VX() -6(3/4 )1/3
is the local density of spin up electronsand is a variable parameter.with a similar expression for ↓
66
Exchange correlation functionalsVXC[(r)]
EXC = EXC( ) LDA or LSDA (spin polarization)
EXC = EXC( ) GGA or GGSDA
- Perdew-Wang- PBE: J. P. Perdew, K. Burke, and M. Ernzerhof
EXC = EXC( ) metaGGA
67
Hybrid methods:B3-LYP (Becke, three-parameters,
Lee-Yang-Parr)
Axel D. Becke german 1953
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
68
Weitao YangDuke university USABorn 1961 in Chaozhou, China got his undergraduate degree at the University of Peking
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.
Robert G. ParrChicago 1921
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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 ab-initio 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!
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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
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DFT good for Bond Energies
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
Bond energies (eV); dissociation are better than in HF
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DFT good for distances
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
Distances (Å)
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DFT polarisabilities (H2O)
Exp. HF LDA
0.728 0.787 0.721
xx 9.26 7.83 9.40
xx 10.01 9.10 10.15
xx 9.62 8.36 9.75
Distances (Å)
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DFT dipole moments (D)
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
Distances (Å)
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Jacobs’scale, increasing progress according Perdew
Paradise = exactitude
Steps Method Example
5th step Fully non local -
4th stepHybrid Meta GGA B1B95
Hybrid GGA B3LYP
3rd step Meta GGA BB95
2nd step GGA BLYP
1st step LDA SPWL
Earth = Hartree-Fock Theory