1

Hartree-Fock Theory

2

The orbitalar approach

Mulliken

(1,….N) = i i(i)

Slater

Combination of Slater determinants

Pauli principle Solutions for S2

spin

3

MO approachA MO is a wavefunction associated with a single electron. The use of the term "orbital" was first used by Mulliken in 1925.

Robert Sanderson Mulliken 1996-1986Nobel 1966

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 = i Hi (1,….N) = i i(i) E = i Ei

4

Approximation of independent particles

Fi = Ti + k Zik/dik+ j Rij

Each one electron operator is the sum of one electron terms + bielectronic repulsions

Hartree Fock

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.

This ignores the real position of j vs. i at any given time.

1927

5

Self-consistency

Given a set of orbitals i, we calculate the electronic distribution of j and its repulsion with i.

This allows expressing

and solving the equation to find new i

allowing to recalculate Rij. The process is iterated up to convergence. Since we get closer to a real solution, the energy decreases.

Fi = Ti + k Zik/dik+ j Rij

6

Jij, Coulombic integral for 2 e

Let consider 2 electrons, one in orbital 1, the other in orbital 2, and calculate the repulsion <1/r12>.

Assuming 12

This may be written

Jij = = < 12I12>

= (11I22)

Dirac notation used in physics

Notation for Quantum chemists

Assuming 12

7

Jij, Coulombic integral for 2 e

Jij = < 12I12> = (11I22) 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 1 are developed on atomic orbitals 1, bilectronic integrals appear involving 4 AOs (pqIrs)

Assuming 12

8

Jij, Coulombic integral involved in two electron pairs

electron 1: 1 or 1 interacting with electron 2: 2 or 2

When electrons 1 have different spins (or electrons 2 have different spins) the integral = 0. <>=ij. Only 4 terms are 0 and equal to Jij.

The total repulsion between two electron pairs is equal to 4Jij.

Assuming 12

9

Particles are electrons!Pauli Principle

electrons are indistinguishable:|(1,2,...)|2 does not depend on

the ordering of particles 1,2...: | (1,2,...)|2 = | (2,1,...)|2

Wolfgang Ernst Pauli Austrian 1900 1950

The Pauli principle states that electrons are fermions.

Thus either (1,2,...)= (2,1,...) S bosonsor (1,2,...)= - (2,1,...) A fermions

10

The antisymmetric function is: A = (1,2,3,...)- (2,1,3...)- (3,2,1,...)+ (2,3,1,...)+...

which is the determinant

Particles are fermions!Pauli Principle

Such expression does not allow two electrons to be in the same statethe 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”

11

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 exclusion principle.

Two-particle case

12

.The expression can be generalized to any number of fermions by writing it as a determinant.

Generalization: Slater determinant

John Clarke Slater 1900-1976

13

Kij, Exchange integral for 2 e

Let consider 2 electrons, one in orbital 1, the other in orbital 2, and calculate the repulsion <1/r12>.

Assuming √I12I

Kij is a direct consequence of the Pauli principle

Kij = = < 12I21A>

= (12I12)

Dirac notation used in physics

Notation for Quantum chemists

Obeying Pauli principle√I12I Slater determinant

14

It is also a positive integral (-K12 is negative and corresponds to a decrease of repulsion overestimated when exchange is ignored).

Kij, Exchange integral

two electron pairs

Obeying Pauli principle√I12I Slater determinant

15

Interactions of 2 electronsObeying Pauli principle√I12I Slater determinant

16

Jij-Kij

Unpaired electrons: gu

Singlet and triplet statesbielectronic terms

KijJij

│ij│ │ij│

│ij│ │ij│

│ij│- │ij│

│ij│+ │ij│Jij+Kij

17

is solution of S2

(1) (Sz2+Sz+S-S+) (2) [1/2]2 + 1/2 + 0 0.75

(2) (Sz2+Sz+S-S+) (1) [1/2]2 + 1/2 + 0 0.75

2Sz(1)Sz(2) 2 ½ ½ 0.5

S-(1)S+(2) 0 0

S+(1)S-(2) 0 0

total 2

S2 = 2

18

is not a solution of S2

(1) (Sz2+Sz+S-S+) (2) [1/2]2 - 1/2 + 1 0.75

(2) (Sz2+Sz+S-S+) (1) [1/2]2 + 1/2 + 0 0.75

2Sz(1)Sz(2) 2 ½ (-½) -.5

S-(1)S+(2) (1) (2) (1) (2)

S+(1)S-(2) 0 0

total 2

S2 =

19

A combination of Slater determinant then may be an

eigenfunction of S2

The separation of spatial and spin functions is possibleIf the spatial function (a or b) is associated with opposite spins

IabI± IbaI = a(1)b(2) - b(1)a(2)

are a eigenfunction of S2

± b(1)a(2) - a(1)b(2)

[a(1)b(2) + b(1)a(2)] (- IabI+ IbaI =

+

[a(1)b(2) - b(1)a(2)] (+ IabI- IbaI =

20

UHF: Variational solutions are not eigenfunctions of S2

The separation of spatial and spin functions is not possibleIf 2 spatial functions (a a’; b b’) are associated with opposite spins

Iab’I+ Iba’I = a(1)b’(2) - b’ (1)a (2)

+ b(1)a’(2) – a’(1)b(2)

=[a(1)b’(2) + b(1)a’(2)](- [a’(1)b(2) + b’(1)a(2)]

Not equal

21

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.

22

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

23

Importance of correlation effects on Energy

RHF in blue, Exact in red.

En

erg

y (k

cal/m

ol)

24

Valence Bond

25

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.

26

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

27

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;

28

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

29

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.

30

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

31

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).

32

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

33

IC: increasing the space of configuration

Exact e

nerg

y leve

ls

2x2

3x3

4x4

34

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()

35

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 

36

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.

37

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.

38

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.

39

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.

40

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

41

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.

42

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.

43

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

44

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

45

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 ↓

46

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

47

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

48

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

49

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!

50

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

51

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

52

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)

53

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)

54

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)

55

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

56

Basis setsThere 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…

57

Large basis setsWhat 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.

58

SCF limitIncreasing 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.

59

SCF convergence for H+

H2+ d a0 (Ả) E diss eV

exp 2.0

(1.06 )

2.791

1sH =1 2.5

(1.32)

1.76

1sH opt 2.0

(1.06)

2.25

+ 2p 2.0

(1.06)

2.71

+ more 2.0

(1.06)

2.791

60

H2 d a0 (Ả) E diss

eV

exp 1.40

(0.74)

4.746

1sH =1 1.61 2.695

1sH = 1.197 1.38 3.488

SCF Limit 1.40 3.636

VB 1sH =1.166 1.41 3.782

VB + p (93%s 7%p) 1.41 4.122

A hundred of functions

1.40

= 0.5 10-4

4.746

= 10-6

61

Variation of the Slater exponent

• Hydrogen =1.24 or =1.30 smaller than =1.0

• Diffuse orbitals: small “soft”

• contracted orbitals: large “hard”

=Z/na0 < r> = a0 n(n+1/2)/Z = (n+1/2)/Z

63

Some functions may be redundant. Therefore, it is better to express the functions on a basis set of orthogonal and normalized functions.

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).

Incomplete Basis sets

DZ: E(H2)=-1.128720 a.u. TZ+0.31(TZ-DZ): E(H2)=-1.134308 a.u.

64

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.

65

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.

66

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.

67

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:

Canonique

LöwdinSchmidt

a'2

a'2a'2

a2

a2a2

a'1

a'1

a1

a1

a'1

a1

depends on the ordering; this may be useful

Insensitive to the ordering.

69

hydrogenoidsSolving Schrödinger equation for hydrogenoids

leads to spatial functions:

(r,)= Nn,l rl Pn,l(r) exp(-Zr/n) R() • 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() is an spherical harmonic function.

70

Slater-type orbitals (STOs ) 1930The radial part appears as a simplification of that of the hydrogenoid.

                    (r)= Nn,l rl exp(- r/n*)    

WhereNn,l is a normalization constant,

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

                                                                 

John Clarke Slater 1900-1976

There is no node for the radial part!

72

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.

73

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.

74

Gaussian functionsGaussians, N rn-1 e-r2 with N= (2)0.75 have expressions close to 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:

*

75

Gaussian functions fitting Slater functions with =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.

Sir John Anthony Pople 1925-2004Nobel 1998

Minimal basis set:

STO-1G e-0.27095 r2

STO-2G .678914

e-0.151623 r2

0.430129

e0.9518195 r2

STO- 3G 0.444635.e-0.109818 r2

0.535328

e-0.405771 r2

0.154329

e-2.22766 r2

76

Fit of a Slater-type orbital by STO-NG

0 1 2 3 4-0,0

0,1

0,2

0,3

0,4

0,5

0,6

rayon (bohr)

Am

plit

ude

Slater

STO3G

STO1G

Radius (a.u.)

Am

plitu

de

77

Fit of a Slater-type orbital by STO-NG

0 1 2 3 40,00

0,01

0,02

0,03

0,04

0,05

rayon (bohr)

Den

sité

de

Pro

bali

bilt

é R

adia

le

STO1G

STO3G

SLATER

Rad

ial p

roba

bilit

y de

nsity

Radius (a.u.)

78

Correspondance for ≠1

There is scaling factor: r →r When the Slater exponent is multiplied by the Gaussian exponent is multiplied by ()2.

e-r = e-r = e-[√r] → () = ()0.5

For H, the Slater exponent is multiplied by 1.24; those of the gaussian function are multiplied by 1.242= 1.5376For C, the Slater exponent is multiplied by 1.625; those of the gaussian function are multiplied by 1.6252= 2.640625

79

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.

80

Split valence basis set; N-X1GCore 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 1925-2004Nobel 1998

basis core valence valence

3-21G 3 2 1

4-31G 4 3 1

6-31G 6 3 1

81

Forget about Slater :Minimal basis set

Huzinaga and DunningOne way to obtain contraction is making a full calculation with no contraction and using the MO coefficients for contraction.

Basis set for O

82

Alternative partitioning for extended basis sets;

Huzinaga and DunningThe 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.

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Polarized Basis setsFor 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

polarization

s p

p d

d f

Pople’s notation 6-31G**: first asterisk “heavy atoms”; second asterisk: p-type on H6-21G(df) add d and f functions and He

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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).

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PseudopotentialsCore 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 all-electron potential such that core states are eliminated and the valence electrons are described by nodeless pseudo-wavefunctions.

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

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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)

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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

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How many AOs? How many occupied MOs? How many vacant MOs?

For C2H4

# MOs # occ # vac

EHT 12 6 6

STO-3G 14 8 6

3-21G 20 8 12

4-31G 20 8 12

6-31-G** 42 8 34

PS-31G 18 6 12