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Valence BondJoop van Lenthe

Theoretical Chemistry GroupUtrecht University

joop@chem.uu.nl

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Contents

• What is VB / Spinfunctions/Bonds • Orbital Optimisation / Brillouin Theorem• Energies and Gradients - Matrix elements• Applications/Illustrations

• Is the BSSE problem solved by using Non-orthogonal Orbitals ?• What is a π system in a bent benzene ?• Resonance and aromaticity• Local Resonance• Do VB structures mean something ?• What about O2

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What Universe are we in:• Time independent• Electronic Structure • Born Oppenheimer• Only scalar relativity• The Answer is 42• The Question ??

H = h(i)i

! +1

rij+ N .R.+ ~

i< j

!

1-electron 2-electron

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Starting from :AtomsBonding : Spin-pairingOrbitals : Non-Orthogonal

(historically) fixedSimplest : 1 structure, more determinantsIn general : Multi-structureExtend : Add Ionic.. structures

VB

MO

Starting from : MoleculesBonding : Sharing OrbitalsOrbitals : Orthogonal

OptimisedIn general : 1 configuration, 1 determinantExtend : Add configurations (CI)

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VB-atoms

H2O O3!g

"

1s22s

22pz

22px2py

1s22s

22px

22pz2py

1s22s

22py

22pz2px

#

$%%

&%%

Ha

2S 1s

a

Hb

2S 1s

b

H2O =O!H !H and

or

! = 1s2h2

2h3

2h1sah4sb

O

h1h2

h3 h4

sa

sb

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SPIN COUPLING

Perfect pairing - two orbitals are coupled to a singlet (bond)

! = N "1" 2# "

1"2( )

1 2

! = "1"2

( )

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1

3

2

213

Two Bonds

! = N "1" 2a

# " 1"2a( ) "

2b" 3# "

2b"3( ) = "

1"2a

( ) "2b"3

( )

! = "1"3

( ) "2 a"2b

( )

A

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VB-2

!(1,2) = c1sa(1)s

b(2)+ s

a(2)s

b(1)( )

Covalent

! "#### $####

+c2sa(1)s

a(2)( )

ionisch

! "# $#+ c

3sb(1)s

b(2)( )

ionisch

! "# $#

H2 vergeet spin

Covalent ionisch ionisch

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H2 VB possibilities!singlet =

1

21sA1sB +1sB1sA( ) "# $ #"( ) ms = 0, s = 0 (singlet %&)

! triplet =

1

21sA1sB $1sB1sA( ) "# + #"( ) ms = 0, s = 1 (triplet %%)

1

21sA1sB $1sB1sA( ) ""( ) ms = 1, s = 1 (triplet %%)

1

21sA1sB $1sB1sA( ) ##( ) ms = $1, s = 1 (triplet %%)

'

(

))))))

Kunt E =! H !

! ! berekenen; Grondtoestand is natuurlijk singlet

De Hamiltoniaan bevat (bij ons) enkel ruimte afhankelijketermen. Dus wordt de energie door het ruimte deel bepaald.

Volgorde elektronen - 1..2 1..2

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VB energy H2 - 3!singlet :

1

21sA1sB +1sB1sA( ) (singlet "#) E $ E1 + J + K

! triplet :

1

21sA1sB %1sB1sA( ) (triplet "")

1

21sA1sB %1sB1sA( ) (triplet "")

1

21sA1sB %1sB1sA( ) (triplet "")

&

'

((((((

E $ E1 + J % K

J en K : Bevoordelen triplet (denk He* / Regel van Hund)VB attractief door 1-elektron term (niet orthogonaal)But J,K,E are different

gives bonding

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Can we use orthogonal Orbitals in VB ??

H2(STO6G) ! = 1

2ab + ba [ ]

VB: Heitler-London : 0.70 Å 120.6 Kcal/molVB: Coulson-Fischer: (full CI) 0.73 Å 128.0 Kcal/molHartree-Fock : 0.71 Å 115.6 Kcal/molOrthogonal VB : >5.9 Å 0 Kcal/molExperimental (WWW): 0.74 Å 104 Kcal/mol

fixed orbitals optimised orbitals

NO

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Atomic Population Analysis (cf. McWeeny (1953))

H2 (STO6G/0.70 Å)

Orb-overlap qa qab EnergyVB: H-L 0.69 0.68 0.32 -1.134187VB: C-F 0.81 0.62 0.38 -1.144979HF : 0.0 0.60 0.40 -1.126100Orthog VB : 0.0 1.89 -0.89 -0.463516

VB:C-F (+p) 0.81 0.61 0.39 -1.148056

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VB-hierarchy

! = C

i"

i# L Wavefunction

! = s

d"d# L Structure

! = "1"2..."

nL Determinant

! = c

i"i# L Orbital

(H ! ES)C = 0 or fixed

Spin-projection

Optimised or fixed basis-function

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Spin-Functions(1)Rumer functions : Different orbitals spin-paired

Benzene (6-31G)

!1

= N "1" 2# "

1"2( ) "

3" 4# "

3"4( ) "

5" 6# "

5"6( )

-230.6614 -230.6614-230.6326-230.6326 -230.6326

0.44 0.44-0.07-0.07 -0.07CE = -230.6938

Spin-functions non-orthogonal!1!4 spin

= 0.25 !1!3 spin

= 0.5

!1!4

= 0.66 !1!3

= 0.82!1!2

= 0.52

C5

C4

C3

C2

C1

C6

C5

C4

C3

C2

C1

C6

C5

C4

C3

C2

C1

C6

C5

C4

C3

C2

C1

C6

C5

C4

C3

C2

C1

C6

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Spin-Functions(2)

Branching Diagrams : Orbitals in standard orderDifferent spin-functions

1 2

Spin-functions orthogonal;Obtain by e.g Schmidt orthogonalising Rumer functions

5 4

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# Atomic states Configuration spin Coefficient1 O(3P) H(2S) 1s22s2y2xzh -0.2352 O(1D) H(2S) 1s22s2y2xyz 0.1913 O+(2D0) H-(1S) 1s22s2(z2-y2)xh2 0.0534 O+(2P0) H-(1S) 1s22s2(z2+y2)xh2 -0.0285 O-(2P0)H+ 1s22s2z2y2x 0.327

Spin-Functions(3)

Branching Diagrams are often considered less informativeOH (2Π) (JHvL, G.G.Balint-Kurti (1980) DZP, Atomic Opt

!

• Oxygen 1D mixes in• O- ionic structure important due to atomic orbitals• O(z) => hybrid on bond formation

• Spin-function may be transformed from one basis to another

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VB=MCSCF

! = N c1"3" 3# c

2" 4"4( ) =

with

"1

=c1"3

+ c2"4( )c1

+ c2( )

"2

=c1"3# c

2"4( )c1

+ c2

( )

$

%

& &

'

& &

(

)

& &

*

& &

= N "1" 2# "

1"2( ) = "

1"2( )

"1"2

=c1# c

2

c1

+ c2

"3"4

= 0

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!

1!

2( ) !3!

4( ) = 2 " MCSCF( )* 2 " MCSCF( )

Only if (φ1φ2) orthogonal to (φ3φ4) :Strong Orthogonality

GVB : W.A. Goddard III et al (1972 - ..)Original version: generalPopular version : GVB-PP

strong orthogonalityperfect pairinguses Fock-operators (N4)size extensive

1999 : GVB-RP

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Spincoupled / Spincoupled VBJ. Gerratt†, M. Raimondi, D.L. Cooper - “The gang of Three”

Spincoupled : 1 configuration all spincouplings, optimised

! = cSk N!( )

12

A "1"

2"

3..." N#S ,M ;k

N{ }k

fSN

$

Fi

eff! i = "i! i

Each ψi satisfies her own

“stack” of virtual orbitals (ψi1,ψi

2,….)

Generate excited structures by applying excitations :

Ci!i1 orCi!i 2 ,... (singles),Ci!i1, j!ij1 (double)

Spincoupled-VB

This function can be obtained by restricting a CASSCF function : CASVB

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Orbital models

Fixed (VB)Optimised

AtomicPartial DelocalDelocalBreathing

F F

covalent ionic Bionic A

(overlaps to 99 %)

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+G. Balint-Kurti, G. Gallup, R. McWeeny, J. van Lenthe,M. Raimondi, Y. Mo, W.Wu, R. Harcourt, D. Cooper, J.Gerratt,R. Braam, , P-A Malmquist, R. Havenith,S. Shaik, P. Hiberty, S. Humbel, P. Karadov + .....and a lot of Physics (superconductivity, crystals) e.g.Masafumi Tamura, Akiko Nakao, Reizo Kato,“Frustration-induced valence - bond ordering in a new quantum triangular antiferromagnet based on [Pd(dmit)2]. “

XMVB, CASVB, TURTLE, VB2000, CRUNCH,...

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(Core)-hole states / Broken symmetry

eA AB B

Hit 1s

! =!A!

B

+

! =!A

1

2+

!B

1

2+

! =!A!

B

+

"!B!

A

+

!

A!

B

+

!A

+

!B

> 0.9 Ria Broer - GNOME

Glyoxaal+HC

O

CH

O+

Paul Ruttink

He2+

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NH (3Σ-) dimerG.Dhont, G. Groenenboom,A. vd Avoird

VB allows states to remain “themselves” as interaction (and overlap) kicks in

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Some wise (edited) remarks by J.N. Murrell, S.F.A. Kettle and J.M. Tedder

The Chemical Bond(Wiley, 1985)

Valence bond theory relies on chemical intuition for finding suitablevalence structures. In some respect, this may seem an advantage in that chemicalexperience can be directly introduced into a quantum-chemical calculation. Inanother respect, however, it is a weakness because it assumes some knowledge ofthe answer at the start. If chemical intuition is wrong, the VB calculation is wrong.The most valuable calculations are often those whose results refute intuition.In contrast, molecular orbital theory does not make any assumption about the wayorbitals are paired in a molecule.

Probably the most misunderstood concept of the theory is “resonance energy”.From resonance theory we deduce that a resonance hybrid will have a lower energythan any single canonical form. The theory tells us nothing about the absolutestability of one species relative to another.