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…
1
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.
2
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.
3
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
4
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
5
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
6
Correlation
The SCF energy is always above the
exact energy. The difference is called
the correlation energy, a term coined
by Löwdin.
A certain amount of electron correlation
(Fermi
correlation)
is
already
considered
within
the
HF
approximation, found in the electron
exchange term describing the
correlation between electrons with
parallel spin.
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)
or post Hartree-Fock methods (CI).
7
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.
8
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.
9
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.
10
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.
11
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'
a
2
a
2
a
a'
1
2
a
1
Löwdin
Schmidt
a'
a'
2
a
1
Insensitive to the
ordering.
2
a'
1
a
Canonique
1
1
12
Orthogonal basis sets
Advantage:
- Leads to easier calculations
(no rectangle terms).
- objectivity (canonic or Löwdin)
Inconvenient:
- Interpretation becomes difficult;
it is not possible to talk of AO
occupancy if they are
delocalized.
Per-Olov Lowdin
Uppsala 1916-2000
Orthogonalization:131950
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.
14
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!
15
Slater-type orbital
They are solution for a spherical potential V’ different from V allowing
the same eigenfunctions and the same eigenvalues than the
Schrödinger equation for the atom.
V' = -Z/r + n* (n*-1)/2r2
or
2r2[E-V']-2r2[E-V’] = n*(n*-1)
Let verify for Y2s= N r e-Zr/2:
16
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.
17
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.
18
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:
*
19
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
20
Fit of a Slater-type orbital by STO-NG
0, 6
Slater
Amplitude
A m plitu de
0, 5
STO3G
0, 4
0, 3
0, 2
STO1G
0, 1
-0,0
0
1
2
3
4
rayon
(bohr)
Radius
(a.u.)
21
densityR adiale
probability
RadialD ensité
de P robalibilté
Fit of a Slater-type orbital by STO-NG
0, 05
STO1G
0, 04
STO3G
0, 03
SLATER
0, 02
0, 01
0, 00
0
1
2
3
rayon (bohr)
Radius
(a.u.)
4
22
Correspondance for z≠1
There is scaling factor: r →z2/z1 r
When the Slater exponent is multiplied by z2/z1, the Gaussian
exponent is multiplied by (z2/z1)2.
e-zr
=
2
-ar
e
=
2
-[√ar]
e
→ (z2/z1) = (a2/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
23
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.
24
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 25
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
26
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.
27
cc-pVDZ and others
These basis are designed to converge systematically to the complete basis set
(CBS) limit using extrapolation techniques. The 'cc-p', stands for 'correlation
consistent polarized' and the 'V' indicates they are valence only basis sets.
Examples of these are: cc-pVDZ - Double-zeta, cc-pVTZ - Triple-zeta, cc-pVQZ
- Quadruple-zeta, cc-pV5Z - Quintuple-zeta, aug-cc-pVDZ, ( Augmented
versions of the preceding basis sets with added diffuse functions)...
H-He
B-Ne
Al-Ar
cc-pVDZ [2s1p] → 5 func.
[3s2p1d] → 14 func.
[4s3p1d] → 18 func.
cc-pVTZ [3s2p1d] → 14 func.
[4s3p2d1f] → 30 func.
[5s4p2d1f] → 34 func.
cc-pVQZ [4s3p2d1f] → 30 func. [5s4p3d2f1g] → 55 func. [6s5p3d2f1g] → 59 func.
28
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
29
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).
30
Rydberg functions
These orbitals are much more diffuse than
the others associated with the valence.
They are associated with loosely tight
electrons occupying atomic orbitals with a
quantum number n+1
31
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
32
33
Pseudopotentials
Small core – Large core: More or less electrons can be frozen. In a small core,
not only the valence electrons are treated explicitly but also the outer shell
beneath.
Norm-conserving pseudopotentials are such that the pseudo- and all-electron
valence eigenstates have the same energies and amplitude (and thus density)
outside a chosen core cutoff radius rc.
Pseudopotentials with different cutoff radius are said to be harder or softer.
Softers are more rapidly convergent, but at the same time less transferable,
that is less accurate to reproduce realistic features in different environments.
Ab initio? The expression of the pseudopotential is a parameterized expression
fitted on experiment. It needs some estimated formula and not the only use of
Schrödinger equation. As for DFT, it is ab-initio in the sense of not introducing
empirical data to evaluate integrals, but it does through the fitting of a potential.
History: First introduced by Hellmann in the 1930s. By construction of this
pseudopotential, the valence wavefunction is guaranteed to be orthogonal to all
34
the core states.
Used with plane waves to avoid introducing an excessive number of
PW to represent the oscillations close to the nucleus.
35
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)
36
Counterintuitive effect
Hij-ESij is usually negative since Hij is larger than -ESij.
For high S values and low lying orbitals this can be not
true.
Then since interacting terms change sign, the out-ofphase combination become lower in energy than the
in-phase combination.
The change of sign also imposes some negative
population of atomic orbitals and some values
exceeding 2.
Mulliken population are less reliable when the basis set
is extended, since there are large S values between
two functions that have nearly the same localization.
37
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
38
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
39