CHM695Feb. 9

UHF vs RHF: H2 dissociation

UHF

RHF (H + H+)

(H + H)E R

S=0

ROHF (Restricted openshell HF): paired electrons have the constraint of same

spatial orbitals. S2 ROHF = S(S + 1) ROHF

Electronic Correlation

iWhile solving one-electron wfns.

only mean field is taken!

electron i electron j missing!

Electronic motion is correlated: they tend to avoid each other

(more than HF)

HF energy is higher than

actual energy

Ecorr = EHF Eexact> 0

Configuration Interaction

linear combination of n-electron wfns.

= d0 0 + d1 1 +

coefficients excited state wfn.ground state

wfn.

=Xi=0

di i

exact if the basis set is complete

1s2 1s1 2s1, 2s2 ,

Number of configurations for a full CI:

(2K)!

n! (2K n)!very large number even for a small system!

Many determinants will not contribute much => CI on selected electrons only

and/or selected excitations only

CIS: CI Singles (only single excitations) CID: CI Doubles (only double excitations) CISD: CI singles & doubles Valence only CI (only valence electrons) HOMO+LUMO only CI

=Xi=0

di i

Usual way: do independent calculations to get i

following by variationally obtaining di

MCSSF: Multiconfiguration SCF ,{cj} {di} obtained by SCF

CASSCF: Complete Acitve Space SCFMCSSF is done on selected MOs only

Mller-Plesset Perturbation Theory (MP){di} from perturbation theory

Xi

Fi

H = H0 + V

Xi,j

1

rijXj

Jj Kj

= (0) + (1) + 2 (2) + =Xr=0

r (r)

E = E(0) + E(1) + 2E(2) + =Xr=0

rE(r)

E(0) =D (0)|H0| (0)

EE(1) =

D (0)|V | (0)

EE(2) =

D (0)|V | (1)

EE(3) =

D (0)|V | (2)

E..

MP2

MP3

HF

(1) =Xj

c(1)j (0)j

single, double excitations to

virtual orbitals of the ground state

wfn.

Coupled Cluster Approaches:CCSD => singles & doubles CCSD(T) => CCSD+Triples

RHF

MP2CCSD

CCSD(T)Full CI

STO-3G 6-31G(d,p) 6-31G++(d,p)| | |

exact

Computational Time:HF ~ K4

MP2 ~ K5 CCSD ~ K6

CCSD(T) ~ K7 only for ~10 atoms

a few tens of atoms

Performance: Energetics

Errors are due to missing correlationAtomization energy of small molecules: error ~60 kcal/molHeat of reactions: error ~5-10 kcal/mol

Hartree Fock:

Energy difference between conformers: error ~1-2 kcal/mol(due to cancellation

of errors)Dispersion is not captured by HF (Note: dispersion is due to e-

correlation)MP2

Compared to HF, error decreases by 25-50%Dispersion is well captured in MP2

Performance: StructureHartree Fock:

Usually good. Bond distances are underestimated

Typical errors: Bond distance~0.02

Bond angles~1.5 For transition state structures:

Bond distance~0.2 correlation!

MP2Usually very good Errors decrease ~50%

Distance between non-bonded atoms is better predicted than HF

(as dispersion is accounted)

Performance: Charge Distribution

Reproduced very well in HF as well as MP2

HF dipole moments are over estimated (10-25%) HF predicts a molecule more polar than it is.

Electrostatic potential is very good.

Density functional theory

Density: (r) =X

i

(r) (r)

.

r

number of electrons in unit volume at r

(r1) =

Zdr2 dr3 drn d!1 d!2, , d!n

(1, 2, , n) (1, 2, , n)integrate over all the coordinates except 1

Density is function of only 3 coordinates

Many electron wave function is a function of 4n coordinates

f(x) = x2 function of x1 12 4

number numberfunction

What does a functional mean?

function numberfunctional

f(x) = x2

Z 10

f(x)dx Z 10x2dx =

x3

3

10

=1

3functional

Another example: (x) E

Z (x)H (x)

E[ (x)]

Our interest:

(r) EF [(r)]

density functionalinput density ??

Does it make sense?

number of electrons+ position of nuclei+ nuclear charges

H energy

Density has the following properties:Z(r) = n

(r) has maxima at the positions of nuclei

density at the position of nuclei has information regarding nuclear charge

(r) H energy

unique

unique

Formally this is shown by Hohenberg and Kohn (1964)

H = T + Vee + Vne

E[(r)] = T [(r)] + Vee[(r)] + Vne[(r)]

Vne , (r)unique

This implies, (r) Hunique

or, (r) F [(r)]

E

E[(r)] = T [(r)] + Vee[(r)] + Vne[(r)]

universal functional

external potential

Functional obeys variational theorem

E[(r)] E[0(r)]exact density

variation of

density

Only valid for the exact density functional