Transcript Basis sets

Hartree-Fock Theory
1
The orbitalar approach
Slater
Mulliken
Y(1,….N) = Pi fi(i)
Pauli principle
spin
Combination
of Slater
determinants
Solutions for S2
2
MO approach
A MO is a wavefunction associated with a
single electron. The use of the term "orbital"
was first used by Mulliken in 1925.
We are looking for wavefunctions for a
system that contains several particules. We
assume that H can be written as a sum of
one-particle Hi operator acting on one
particle each.
H = Si Hi
Y(1,….N) = Pi fi(i)
E = Si Ei
Robert Sanderson
Mulliken
1996-1986
Nobel 1966
3
Approximation of
independent particles
Hartree
Fock
1927
Fi = Ti + Sk Zik/dik+ Sj Rij
Each one electron operator is the sum of one
electron terms + bielectronic repulsions
Rij is the average repulsion of the electron j upon i.
The self consistency consists in iterating up to
convergence to find agreement between the postulated
repulsion and that calculated from the electron density.
4
This ignores the real position of j vs. i at any given time.
Self-consistency
Given a set of orbitals Yi, we calculate the
electronic distribution of j and its repulsion
with i.
This allows expressing Fi = Ti + Sk Zik/dik+ Sj Rij
and solving the equation to find new Y1i
allowing to recalculate Rij. The process is
iterated up to convergence. Since we get
closer to a real solution, the energy
decreases.
5
Assuming Y= Y1Y2
Jij, Coulombic integral for 2 e
Let consider 2 electrons, one in orbital Y1, the other in orbital Y2, and
calculate the repulsion <1/r12>.
Assuming Y= Y1Y2
This may be written
Jij =
Dirac notation used in physics
= < Y1Y2IY1Y2>
= (Y1Y1IY2Y2)
Notation for 6
Quantum chemists
Assuming Y= Y1Y2
Jij, Coulombic integral for 2 e
Jij = < Y1Y2IY1Y2> = (Y1Y1IY2Y2) means the product of
two electronic density  Coulombic integral.
This integral is positive (it is a repulsion). It is large when dij
is small.
When Y1 are developed on atomic orbitals f1, bilectronic
integrals appear involving 4 AOs (pqIrs)
7
Assuming Y= Y1Y2
Jij, Coulombic integral involved in
two electron pairs
electron 1: Y1 or Y1 interacting with electron 2: Y2 or Y2
When electrons 1 have different spins (or electrons 2 have different spins)
the integral = 0. <aIb>=dij. Only 4 terms are  0 and equal to Jij.
The total repulsion between two electron pairs is equal to 4Jij.
8
Particles are electrons!
Pauli Principle
Wolfgang Ernst Pauli
Austrian 1900 1950
electrons are indistinguishable:
|Y(1,2,...)|2 does not depend on
the ordering of particles
1,2...:
| Y (1,2,...)|2 = | Y (2,1,...)|2
Thus either Y (1,2,...)= Y (2,1,...) S bosons
or Y (1,2,...)= -Y (2,1,...)
A fermions
The Pauli principle states that electrons are fermions.
9
Particles are fermions!
Pauli Principle
The antisymmetric function is:
YA = Y (1,2,3,...)-Y (2,1,3...)-Y (3,2,1,...)+Y (2,3,1,...)+...
which is the determinant
Such expression does not allow two electrons to be in the same state
the determinant is nil when two lines (or columns) are equal;
"No two electrons can have the same set of quantum numbers".
One electron per spinorbital; two electrons per orbital.
“Exclusion principle”
10
Two-particle case
The simplest way to approximate the wave function of a many-particle
system is to take the product of properly chosen wave functions of the
individual particles. For the two-particle case, we have
This expression is a Hartree product. This function is not antisymmetric.
An antisymmetric wave function can be mathematically described as
follows:
Therefore the Hartree product does not satisfy the Pauli principle. This
problem can be overcome by taking a linear combination of both Hartree
products
where the coefficient is the normalization factor. This wave function
is antisymmetric and no longer distinguishes between fermions.
Moreover, it also goes to zero if any two wave functions or two
fermions are the same. This is equivalent to satisfying the Pauli 11
exclusion principle.
Generalization: Slater determinant
.
The expression can be generalized to any number of fermions by writing it as a
determinant.
John Clarke Slater
1900-1976
12
Obeying Pauli principleY= 1/√ IY1Y2I Slater determinant
Kij, Exchange integral for 2 e
Let consider 2 electrons, one in orbital Y1, the other in orbital Y2, and
calculate the repulsion <1/r12>.
Assuming Y= 1/√ IY1Y2I
Kij is a direct consequence of the Pauli principle
Dirac notation used in physics
Kij =
= < Y1Y2IY2Y1A>
= (Y1Y2IY1Y2)
Notation for 13
Quantum chemists
Obeying Pauli principleY= 1/√ IY1Y2I Slater determinant
Kij, Exchange integral
It is also a positive integral (-K12 is negative and corresponds to a
decrease of repulsion overestimated when exchange is ignored).
two electron pairs
14
Obeying Pauli principleY= 1/√ IY1Y2I Slater determinant
Interactions of 2 electrons
15
Unpaired electrons: sgsu
Singlet and triplet states
bielectronic terms
│YiYj│+ │YiYj│
Jij+Kij
│YiYj│
Jij
Kij
│YiYj│
Jij-Kij
│YiYj│ │YiYj│
│YiYj│- │YiYj│
16
a(1)a() is solution of S2
a(1) (Sz2+Sz+S-S+) a(2)
[1/2]2 + 1/2 + 0
0.75
a(2) (Sz2+Sz+S-S+) a(1)
[1/2]2 + 1/2 + 0
0.75
2½½
0.5
S-a(1)S+a(2)
0
0
S+a(1)S-a(2)
0
0
2Sza(1)Sza(2)
total
S2 a(1) a() = 2 a(1) a()
2
17
a(1)b() is not a solution of S2
a(1) (Sz2+Sz+S-S+) b(2)
[1/2]2 - 1/2 + 1
0.75
b(2) (Sz2+Sz+S-S+) a(1)
[1/2]2 + 1/2 + 0
0.75
2 ½ (-½)
-.5
S-a(1)S+b(2)
b(1) a(2)
b(1) a(2)
S+a(1)S-b(2)
0
0
2Sza(1)Szb(2)
total
S2 a(1) b() = a(1) b() + b(1) a()
2
18
A combination of Slater
determinant then may be an
eigenfunction of S2
IabI± IbaI = a(1)b(2) a(1)b() - b(1)a(2) a()b(1)
± b(1)a(2) a(1)b() +- a(1)b(2) a()b(1)
IabI+ IbaI = [a(1)b(2) + b(1)a(2)] (a(1)b() - a()b(1))
IabI- IbaI = [a(1)b(2) - b(1)a(2)] (a(1)b() + a()b(1))
are a eigenfunction of S2
The separation of spatial and spin functions is possible
If the spatial function (a or b) is associated with opposite spins
19
UHF: Variational solutions are not
eigenfunctions of S2
Iab’I+ Iba’I = a(1)b’(2) a(1)b() - b’ (1)a (2) a()b(1)
+ b(1)a’(2) a(1)b() – a’(1)b(2) a()b(1)
=[a(1)b’(2) + b(1)a’(2)](a(1)b() - [a’(1)b(2) + b’(1)a(2)] a()b(1))
Not equal
The separation of spatial and spin functions is not possible
If 2 spatial functions (a  a’; b  b’) are associated with
opposite spins
20
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.
21
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
22
Energy (kcal/mol)
Importance of correlation effects on Energy
RHF in blue, Exact in red.
23
Valence Bond
24
Heitler-London
1927
Electrons are indiscernible:
If f1(1).f() is a valid solution
Walter Heinrich
Heitler
German
1904 –1981
f1().f(1) also is.
The polyelectronic function is therefore:
[f1(1).f() + f1().f(1)]
Fritz
Wolfgang
London
German
1900–1954
To satisfy Pauli principle this symmetric expression is associated
with an antisymmetric spin function: a (1).b() - a().b(1)
this represents a singlet state!
Each atomic orbital is occupied by one electron: for a bond, this
represents a covalent bonding.
25
The resonance,
ionic functions
-
H1 -H2
+
Other polyelectronic functions :
-
+
+
H1 -H2 ↔ H1 -H2
-
f1(1).f1()
[f1(1).f1() + f().f(1)]
f().f(1)
[f1(1).f1() -f().f(1)]
+
H1 -H2
-
symmetric
antisymmetric
According to Pauli principle, these functions necessarily
correspond to the singlet state:
a (1).b() - a().b(1)
26
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;
27
MO behavior of H2 dissociation
The cleavage is whether
homolytic, H2 → H• + H•
or heterolytic: H2 → H+ + H-
Energy
Energie
Yu
distance
A-B
internucdistance
léa ire
Yg
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) f28A(2)]
50% ionic
50% covalent
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.
29
Covalent function for electron pairs
Ordered on spins
a (1) b(1)
or
Y = IabI + IbaI
Ordered on electrons
IabI =
a (2) b(2)
Y = IabI - IabI
Y = [a(1).b(2) +a(2). b(1)].[a(1).b(2) - a(2).b(1)]
Y = [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)]
Y = IabI + IbaI
This is the Heitler-London expression
30
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).
31
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
32
IC: increasing the space of
configuration
4x4
2x2
Exact energy levels
3x3
33
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)
34
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
35
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
36
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.
37
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,
Y1 and Y2, that each yield r.
E1 < < Y2 IH 1I Y2 > = < Y2 IH 2I Y2 > + < Y2 IH1 -H 2I Y2 >
= E2 + ∫ r(r)(V1(r)- V2(r))dr
E2 < < Y1 IH 2I Y1 >
= 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.
38
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.
39
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
40
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.
41
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.
42
Kohn-Sham equations
Equation 3: the writing of an effective single-particle potential
Eeff(r) = <Y│Teff+Veff │ Y >
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
43
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
44
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↓
45
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
46
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
47
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
48
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!
49
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
50
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
51
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
52
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
53
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
54
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
55
Basis sets
There is no general solution for the Schrödinger except for
hydrogenoids. It is however natural to search for solutions
resembling them.
There is strictly no requirement to start by searching functions
close to hydrogenoids. We can use any function not necessarily
localized on atoms: for solids, plane waves are useful. We can
use functions localized on bonds, on vacancies…
56
Large basis sets
What is stronger than a turkishman? Turkishmen
What is better than a function? Several ones.
Minimizing parameters combing several functions is
generally an improvement (at the most, it is useless).
Basis set associated with hydrogenoids: minimum basis
set. more: extended basis set.
57
SCF limit
Increasing the number of (independent) functions leads to
improve the energy (variational principle). This improvement
saturates.
The limit is called SCF limit. This limit can be estimated by
interpolation.
58
SCF convergence for H+
H2+
d a0 (Ả)
E diss eV
exp
2.0
(1.06 )
2.791
1sH z=1
2.5
(1.32)
1.76
1sH z opt
2.0
(1.06)
2.25
+ 2p
2.0
(1.06)
2.71
+ more
2.0
(1.06)
2.791
59
H2
d a0 (Ả)
E diss
eV
1.40
(0.74)
4.746
1sH z=1
1.61
2.695
1sH z = 1.197
SCF Limit
VB 1sH z=1.166
1.38
3.488
1.40
3.636
1.41
3.782
VB + p (93%s 7%p)
A hundred of
functions
1.41
4.122
1.40
D = 0.5 10-4
4.746
D = 10-6
exp
60
Variation of the Slater exponent
• Hydrogen z=1.24 or z=1.30 smaller than z=1.0
• Diffuse orbitals: z small “soft”
• contracted orbitals: z large “hard”
z =Z/na0 < r> = a0 n(n+1/2)/Z = (n+1/2)/Z
61
Incomplete Basis sets
DZ: E(H2)=-1.128720 a.u.
TZ+0.31(TZ-DZ): E(H2)=-1.134308 a.u.
The term “ab initio” suggest no subjectivity. However the
choice of the functions is arbitrary; The variational
principle allows comparing several choices for
quantitative results (not always desirable for
understanding).
Some functions may be redundant. Therefore, it is better
to express the functions on a basis set of orthogonal
and normalized functions.
63
Basis set superposition error,
BSSE
Since basis sets are incomplete, there is an
error when calculating A + B → C.
Indeed, it seems fair to use the same basis
set for A, B and C. However in C, the
orbitals of B contribute to stabilize A if it the
basis set to describe A is incomplete. The
same is true the other way round. Each
monomer "borrows" functions from other
nearby components, effectively increasing
its basis set and improving the calculation
of derived properties such as energy
Thus A and B should be better described
using the orbitals centered on the other
fragment than alone. It follows that A+B is
underestimated relative to C.
64
Basis set superposition error,
BSSE (counterpoise method).
The energy gain for the reaction is
therefore overestimated.
• For a diatomic formation, the solution
is to calculate A and B using the full
basis set for A+B. (B being a dummy
atom when A is calculated). Ghost
orbitals are orbitals localized where
there is no nucleus (no potential).
• For an interaction between 2 large
fragments, there is a problem of
choosing the geometry for A: that of
lowest energy for A or that in the
fragment A-B. The method is
estimating the BSSE correction in the
fragment of A-B and assuming that it is
the correction for A.
65
Basis set superposition error, BSSE
(Chemical Hamiltonian approach)
The
(CHA) replaces the
conventional
Hamiltonian
with one designed to
prevent basis set mixing a
priori, by removing all the
projector-containing terms
which would allow basis set
extension.
Though conceptually different
from
the
counterpoise
method, it leads to similar
results.
66
Basis set transformation,
orthogonalization
Starting from scratch, a
set of functions is not
necessarily orthogonal.
Orthogonalization uses
matrix transformations.
There are 3 main
orthogonalization
processes:
depends on the ordering; this
may be useful
a' 2
a' 2
a2
a2
a1
a1
a' 1
Löwdin
S ch midt
a' 1
a' 2
a2
Insensitive to the
ordering.
a' 1
a1
C anon iqu e
67
hydrogenoids
Solving Schrödinger equation for hydrogenoids
leads to spatial functions:
Y(r,q,f)= Nn,l rl Pn,l(r) exp(-Zr/n) R(q,f)
• Nn,l is a normalization function (it contains the
dimension a0-3/2)
• rl is a power of r and Pn,l(r) a polynom of degree
n-l-1
• exp(-Zr/n) makes the summation in the universe
finite.
• R(q,f) is an spherical harmonic function.
69
Slater-type orbitals (STOs ) 1930
The radial part appears as a simplification of that of
the hydrogenoid.
Y(r)= Nn,l rl exp(- z r/n*)
Where
John Clarke Slater 1900-1976
Nn,l is a normalization constant,
z is a constant related to the effective charge of the nucleus,
the nuclear charge being partly shielded by electrons.
n* plays the role of principal quantum number, n = 1,2,...,
The normalization constant is computed from the integral
Hence
There is no node for the radial part!
70
Slater-type orbital
Why this simplification?
In LCAO, we make combinations of AOs.
It is therefore useless to start by imposing the polynoms.
What changes?
The hydrogenoids are orthogonal <1sI2s>=0.
(eigenfunctions associated with different quantum
numbers).
The Slater orbitals are not orthogonal. 1s+2s resembles
the hydrogenoid 1s (no node) and 2s-1s resembles
the 2s orbital (the combination makes the nodal
surface appear.
72
Double zeta
Using several Slater functions allows representing better
different oxidation states. When there is an electron
transfer, an atom could be A+, A° or A-. For a metal, often
several atomic configurations are close in energy: s2d8,
s1d9 or d10. This correspond to different exponents for the
s and d AOs.
Double and triple zeta functions 2 or 3 AOs adapted to one
oxidation state each and allowing variation in a linear
combination and flexibility.
As soon as the reference to oxidation states disappears
(Gaussian contractions) the terminology becomes less
justified and just qualify the number of independent
functions.
73
Gaussian functions
rn-1
2
-ar
e
with N= (2a/p)0.75 have expressions close to
Gaussians, N
Slater. They are decreasing exponentials (more rapidly than Slaters),
with different slope at r=0*. Close to r=req, differences are small.
Gaussians are not as appropriate than Slaters but not so different to
prevent considering them. The reason to use Gaussians is a computing
facility. In the H-F method, many integrals (N4) involve the product of 4
orbitals. The product of two Gaussian is easily calculated; it is another
Gaussian:
*
74
Gaussian functions fitting Slater
functions with z=1
A Gaussian orbital is a combination (contraction) of several individual
Gaussian functions called primitive. A STO-NG orbital is a linear
combination of N Gaussian fitting closely a Slater orbital.
Minimal basis set:
Sir John
Anthony Pople
1925-2004
Nobel 1998
STO-1G
e-0.27095 r2
STO-2G
.678914
e-0.151623 r2
STO- 3G
0.444635. 0.535328
e-0.109818 r2 e-0.405771 r2
0.430129
e0.9518195 r2
0.154329
e-2.22766 r2
75
Fit of a Slater-type orbital by STO-NG
0,6
Slater
Amplitude
Amplitude
0,5
STO3G
0,4
0,3
0,2
STO1G
0,1
-0,0
0
1
2
3
4
rayon
(bohr)
Radius
(a.u.)
76
density Radiale
probability
RadialDensité
de Probalibilté
Fit of a Slater-type orbital by STO-NG
0,05
ST O1G
0,04
ST O3G
0,03
SLATER
0,02
0,01
0,00
0
1
2
3
rayon (bohr)
Radius
(a.u.)
4
77
Correspondance for z≠1
There is scaling factor: r →z/z1 r
When the Slater exponent is multiplied by z/z1, the Gaussian
exponent is multiplied by (z/z1)2.
e-zr
=

-ar
e
=

-[√ar]
e
→ (z/z1) = (a/a1)0.5
For H, the Slater exponent is multiplied by 1.24; those of the
gaussian function are multiplied by 1.242= 1.5376
For C, the Slater exponent is multiplied by 1.625; those of the
gaussian function are multiplied by 1.6252= 2.640625
78
Minimal basis set and split valence
basis set.
STO-3G has been a long time used; with improvement of
computing facilities, this is not the case nowadays in
spite of the simplicity of using minimal basis.
One way to improve accuracy is taking more functions.
Releasing all contractions (N functions instead of 1 linear
combination) is expensive. We can split the Gaussian
into two sets. The partition may make groups or isolate
the outermost primitive. The first procedure perhaps
involves larger energy contribution but the second one is
more chemical. The flexibility in reaction is necessary for
the electron participating to the transformation (chemical
reaction). If only the outermost primitive is isolated, we
have the split-valence basis set named N-X1G by Pople.
79
Split valence basis set; N-X1G
Core orbitals are represented by a single
orbital with N primitives.
Valence orbitals are represented by 2
orbitals: one orbital with X primitives
and one diffuse orbital.
Sir John
Anthony Pople
1925-2004
Nobel 1998
basis
3-21G
4-31G
6-31G
core
3
4
6
valence valence
2
1
3
1
3
1 80
Forget about Slater :
Minimal basis set
Huzinaga and Dunning
One way to obtain contraction is making a full calculation with no
contraction and using the MO coefficients for contraction.
Basis set for O
81
Alternative partitioning for extended
basis sets;
Huzinaga and Dunning
The basis set for O [9s5p/3s2p] by Dunning means that there are 3 s
orbitals (using 1, 2 and 6 primitives) and 2 p orbitals (using 4 and 1
primitives). The “1s” orbital is represented by the first orbital (1
primitive).
The basis set for O2- [13s7p/5s3p] by Pacchioni and Bagus means that
there are 5 s orbitals (using 6, 2, 1, 2 and 1 primitives) and 2 p orbitals
(using 4, 2 and 1 primitives). The “1s” orbital is represented by the first
orbital (6 primitives).
DZHD Double-Zeta Huzinaga-Dunning DTZHD Double-triple-Zeta
Huzinaga-Dunning:
The contraction generates several orbitals and is qualified a N zeta
even if this is more mathematics than physics: the relation with Slater
orbitals and oxidation states desappears.
82
Polarized Basis sets
For H2, 7% of p improved the total energy from 3.782 eV to 4.122 eV.
The molecular potential is not spherical around each atom. The
directionality is provided by p-type orbitals.
Polarization functions consists to add “l+1” functions:
External valence
shell
s
p
d
polarization
p
d
f
Pople’s notation
6-31G**: first asterisk “heavy atoms”; second asterisk: p-type on H
84
6-21G(df) add d and f functions
and He
Basis sets for anions,
6-31++G**
Anions require diffuse orbitals and are more difficult to
calculate “in gas phase” than neutral or cationic species.
Usually one can use more diffuse (smaller) exponents.
The addition of diffuse functions, denoted in Pople-type
sets by a plus sign, +, and in Dunning-type sets by "aug"
(from "augmented"). Two plus signs indicate that diffuse
functions are also added to light atoms (hydrogen and
helium).
85
Pseudopotentials
Core orbitals do not participate to much to
chemistry and it is more useful and
simpler to calculate only the valence.
Then core electron interactions are
replaced by a pseudopotential.
The pseudopotential is an effective potential
constructed to replace the atomic allelectron potential such that core states
are eliminated and the valence electrons
are described by nodeless pseudowavefunctions.
Only the chemically active valence electrons
are dealt with explicitly, while the core
electrons are 'frozen'
Motivation:
Reduction of basis set size
Reduction of number of electrons
Inclusion of relativistic and other effects
87
88
Step calculations
To save calculation efforts, on can use
different accuracy for optimization of
geometry and calculation of properties on
the optimized geometry.
UHF/3-21G(d)
means that the geometry was optimized
using UHF/3-21G(d) and the final result
was calculated using UB3LYP/6-31G(d)
91
How many AOs? How many occupied
MOs? How many vacant MOs?
For C2H4
# MOs
# occ
# vac
EHT
STO-3G
3-21G
4-31G
6-31-G**
PS-31G
93
How many AOs? How many occupied
MOs? How many vacant MOs?
For C2H4
EHT
# MOs
12
# occ
6
# vac
6
STO-3G
14
8
6
3-21G
20
8
12
4-31G
20
8
12
6-31-G** 42
8
34
PS-31G
6
12
18
94