Multiply Charged IonsQuantum Chemical ComputationsTrento, May 2002Lecture 2

Basis Sets - 1

basn

jjiji rcr

1

2

A

A

Rr

Rr

iqr

Ne

Ne

Ne

Usually atom-centered

N

iii rcr

Basis Sets - 2 2

,,, rnml ezyxrzyx l=m=n=o ‘s’

l or m or n = 1 ‘p’l + m + n = 2 ‘d’l + m + n = 3 ‘f’

63.typ

iiprimitiveicontracted c

Types of Basis Set• Minimal (one s, p (& d, f) function per shell.

STO-3G• Double- (two functions per shell or valence

shell). E.g. 3-21G, 6-31G, DZ, SV, …• Triple-, Quadruple-• Polarisation functions:

Poor Overlap

+ =

Better Overlap

Double- + polarisation (e.g. 6-31G*, 6-31G**, cc-pVDZ, SVP) usually adequate for HF, geometry.

Dynamic Correlation

CorrelationN

e e e

e

Nas likely as

N

e e

mixed with

He [1s2]

N

e

e

He [2p2]

Configuration Interaction

... ,

, , ,

,

, ,

,

,00CI

virtocc

virtoccvirtocc

nn

ai bj ck

abcijkijkabc

nn

ai bj

abijijab

nn

ai

aiia

c

ccc

kjiHF 0

kjaai

CI

• Only CISD practical (partly physically motivated!!)

• Slow convergence with basis set size (roughly 1/n3)

• Size inconsistent

H

H

H

H

al.et H ai0 H

MPn

nnnn

n

n

xxx

xxxxxx

nx

21

22221

11211

HF1

• Approximate solution to correct Schrödinger equation

• Exact solution to a different equation where Vee is replaced by <Vee>

•Apply perturbation theory (Møller-Plesset)•To second order (MP2), gives most of the

correlation energy in favourable cases

MP2

occ virtn

ji

n

ba

2

biai2

baji

ajbjMPE

• Poor performance of MP2 when small band gap

• Size consistent

Coupled Clusters

... ,

, , ,

,

, ,

,

,00exact

virtocc

virtoccvirtocc

nn

ai bj ck

abcijkijkabc

nn

ai bj

abijijab

nn

ai

aiia

c

ccc

0ˆ

exact Te

321

32ˆ

ˆˆˆ1 ˆ

ˆ21ˆ

21ˆ1

TTTT

TTTeT

virtocc nn

ai

aiiatT

,

,01

CC 32ˆ ˆ21ˆ

21ˆ1 TTTeT

41

212

22134

31123

2121

ˆ

ˆ241ˆˆ

21ˆ

21ˆˆˆ

ˆ21ˆˆˆˆ

21ˆˆ1

TTTTTTT

TTTTTTTeT

21ˆˆ1 ˆ TTTSD Restrict to T2 (CCSD):

0ˆ

CCSD SDTe

CCSD(T) et al.

41

212

22

3112

2121

ˆ

ˆ241ˆˆ

21ˆ

21

ˆ21ˆˆˆ

21ˆˆ1

TTTT

TTTTTTe SDT

• Only CCSD, CCSD(T) generally feasible• Size consistent• CCSD(T) very accurate where band gap is

large (“method of choice”)

Basis Sets and CorrelationSlow convergence with basis size!

N

e e

mixed with

He [1s2]

N

e

e

He [2p2]

3max

,,1l

EE exactcorrelcalccorrel

Correlation Consistent B.S.s

• cc-pVDZ: 3s, 2p, 1d (lmax = 2)• cc-pVTZ: 4s, 3p, 2d, 1f (lmax = 3)• cc-pVQZ: 5s, 4p, 3d, 2f, 1g (lmax = 4)• cc-pV5Z: 6s, 5p, 4d, 3f, 2g, 1h (lmax = 5)

T. H. Dunning et al.:

Extrapolation

Basis Set Dependence

Other ExtrapolationHF/STO-3G HF/DZ limite HF

MP2

CCSD

Full-CI exact result

Larger Basis Set

BetterDescription of

electronicCorrelation

G2, G3, etc.HF/STO-3G HF/DZ limite HF

MP2

CCSD

Full-CI exact result

Larger Basis Set

BetterDescription of

electronicCorrelation

• •

•

G2

EG2 = EQCISD(T)(6-311G**) + E(MP4, diffuse) + E(MP4, 3df) + …