Download Valence Bond Concepts Applied to MM Force Field Development

Valence Bond Concepts Applied to Molecular Mechanics Force
Field Development
I. Hybridization and Molecular Shapes
II. Resonance in MM Computations
III. Valence Bond Theory and Shapes of Covalent Transition
Metal Complexes
IV. Modelling the Splitting of N 2 by Simple
Mo(Amide) 3 Complexes.
Funding Provided by the National Science Foundation
and Molecular Simulations Inc.
Mr. Dan Root
Mr. Tom Cleveland
Mr. Tim Firman
Collaborators
Prof. Tony Rappè (CSU)
Prof. Notker Rösch (TU-Muenchen)
Special thanks to Schrödinger, Inc.
Molecular Mechanics Computations
A Classical Mechanical Approach (Ball and Spring)
O
Bond Spring
(length = r 0; force constant = k r )
H
H
Bond Angle Spring
E = k r (r-r 0)2 + k θ(θ-θ0)2 + k φ(1+cos(n φ+δ)
bond stretch
bond angle bend
torsional motion
+ van der Waals + electrostatic terms
Issues Impacting Rational Design
of Homogeneous Catalysts
Mechanistic
What step(s) control the reaction rate and selectivity?
Structural
What are the important steric interactions that guide selectivity?
Can the structures of new catalyst designs be predicted?
Synthetic
Can promising designs of new catalysts be synthesized?
What Makes Transition Modeling
So Difficult?
• Transition Metal Complexes Have Complicated and Varied Shapes
O
C
OC
Fe
N
CO
N
CO
Fe
N
PR 3
N
Rh +
R3P
N
C
O
OC
Ni
CO
R3P
CO
R3P
+
Rh
PR 3
PR 3
PR 3
Trigonal Bipyramid
Square Pyramid
T-Shape
Square Plane
Trigonal Plane
• Transition Metal Complexes Often Have Indistinct Topologies
Cl
+
Zr
R
PR 3
Pt
Cl
Rh
Cl
PR 3
Method Development and the
Pauling Point
Empirical
Too Good To
Be True
Ab Initio
Too True To Be Good
Increasing Reliability
Pauling Point
Increasing Effort
The VALBOND/UFF Force Field
Goals
• Description of Inorganic Molecular Shapes Including Conformational
Dynamics
• Theory-based Derivation of New Potential Energy Functions
• Application to Full Periodic Table
• Minimal Parametrization
• Accuracy in Structures and Vibrational Frequencies Similar to MM3
for Organics
• Bond Making and Bond Breaking
Premise
Because molecular mechanics bonded terms are based on
a localized bond topology, Valence Bond Theory is the
natural viewpoint for the derivation of new potential
energy functions.
Valence Bond Theory and Molecular Shapes
Principles of the Directed Covalent Bond
• Covalent bonds are formed by the interaction of singly-occupied orbitals of the
central atom and the ligands.
• Hybridization of these orbitals provides a mechanism for maximizing bond
strength by concentrating electron density in the bonding region.
• Hybrid orbitals located on the same atom must be orthogonal and normalized.
• Two sp3 hybrid orbitals have maxima in their eigenfunctions at tetrahedral
angles, sp2 hybrid orbitals have maxima at trigonal planar angles, sp hybrid
orbitals have maxima at linear angles, and pure p orbitals have maxima
at right angles.
Construction of Hybrid Orbitals
“The dependence on r of s and p hydrogen-like eigenfunctions is not greatly
different ... the problem of determining the best bond eigenfunctions reduces
to a discussion of the θ, ψ eigenfunctions.”
3
1
px
ψ1 = s +
2
2
“... the best bond eigenfunction will be
that which has the largest value in the
1
1
px +
ψ2 = s bond direction ... along the x-axis the
2 3
2
best eigenfunction is ψ1with a maximum
For Methane
value of 2, considerably larger than 1.732
for a p eigenfunction.”
2
pz
3
Pauling, L. J. Am. Chem. Soc. 1931, 53, 1367
“A second eigenfunction can be introduced in the xz plane... This eigenfunction
is equivalent and orthogonal to ψ1, and has its maximum at an angle of 109o28’.”
Hybrid Orbitals for Other Geometries
Geometry
Hybridization
Strength
Linear
sp1
sp6d5
sp2
1.91
2.96
1.991
Tetrahedron
sp3
sp1. 125d1. 875
2.00
2.950
Square Plane
sp2d1
2.694
Octahedron
sp3d2
2.923
Trigonal Plane
“I have not succeeded in determining whether or not these octahedral
eigenfunctions are the strongest ...”
Pauling, L. J. Am. Chem. Soc. 1931, 53, 1367
Forty Years Later Pauling Returned to
the Problem of Constructing Hybrids
“ I have now found a simple relation between
the strength (the bond-forming power) of a
hybrid spd bond orbital and the angles that it
makes with other similar orbitals...”
2
Ssp
(
α
)
=
0.5
+
1.5
cos
(α 2)
0
+ 0.5 − 1.5cos2 (α 2)
2
4
Sspd
(
α
)
=
3
−
6
cos
(
α
2)
+
7.5
cos
(α 2)
0
+ 1.5 + 6 cos2 (α 2) − 7.5cos4 (α 2)
Pauling, L. Proc. Nat. Acad. Sci. 1975, 72, 4200.
Generalized Hybrid Orbitals
For any pair of hybrid orbitals with hybridization spmdn making the bond
angle α, the strength functions are given by
2
1.95
1.9
S(α ) = Smax 1−
1−
1.85
sp
sp2
sp3
p
1.8
where ∆ = overlap inte
1.75
(
1.7
and Smax =
1
1
1+ m + n
1.65
Bond Angle (degrees)
Root, D. M.; Landis, C. R.; Cleveland, T. J. Am. Chem. Soc. 1993, 115, 4201.
180
165
150
135
120
105
90
75
60
45
30
15
0
1.6
Pair Defect Sum Approximation
For two ligands forming electron pair bonds with two spn
orbitals, the energy of bond angle distortions is
approximated by the strength defects in each of the bonds.
Pair Defect = Smax - S(α)
For H2O, the total energy as a function of bond angle is given b
O
H
α
H'
E(α) = kO-H(Smax - S(α)) + kO-H'(Smax - S(α))
where kO-H is a scaling parameter (VALBOND parameter)
Strength Functions Model Potential
Energy Surfaces
Assuming that
the VALBOND force field
models high quality ab initio
energies over large variations
in bond angles
Water
60
Energy (kcal/mol)
• hybridizations are known
• potential energies scale
linearly with pair-defects
80
valbond
ab initio
harmonic
40
20
0
40
60
80
100
120
140
Angle (degrees)
Root, D. M.; Landis, C. R.; Cleveland, T. J. Am. Chem. Soc. 1993, 115, 4201.
160
180
Assignment of Hybridizations
Given hybridizations for each bond orbital, the hybrid orbital strength
functions accurately simulate bending potential energy surfaces.
But how are hybridizations determined?
H
•
•
•
B
•
F
••
F
Lewis Structure
gross
3 sp2 hybrids
hybridization
quantitative
Bent's Rule
sp
1.81
H
sp2.09
120.9
B
F
A quantitative expression of Bent's rule is used to
distribute p-character among each ligand, lone pair,
and singly occupied orbital.
F
118.1
O rganic Rad icals and Carb enes
M ole cu le
An gle
VALBO N D
Exp
1
CH 2
H -C-H
103.0
102.4
3
CH 2
H -C-H
131.7
136
CH 3
H -C-H
120
120
CF 2
F-C-F
103.8
104.8
CH F
H -C-F
103.6
101.8
Cl-C-Cl
103.9
100(9)
1
1
1
CCl2
A m ines, Pho sp hines, and A rsines
M ole cu le
BF 2N H 2
N Cl3
N H Cl2
N O2
N ClO
PH 3
PCl3
CH 3PH 2
AsH 3
AsF 3
AsCl3
AsBr 3
AsI 3
An gle
VALBO N D
F-B-F
119.9
H -N -H
115.0
Cl-N -Cl
106.5
H -N -Cl
106.7
Cl-N -Cl
106.1
O -N -O
149.2
Cl-N -O
120.0
H -P-H
93.8
Cl-P-Cl
100.1
C-P-H
97.1
H -P-H
91.3
H -As-H
91.7
F-As-F
96.0
Cl-As-Cl
98.7
Br-As-Br
99.6
I-As-I
100.2
Exp .
117.9
116.9
107.1
102
106
134.1
113.3
93.3
100.1
96.5
93.4
92.1
96.0
98.6
99.7
100.2
Ino rganic Flo tsam and Jetsam
M ole cu le
Sn 6(Ph 2)6
Ph2
Sn SnPh2
Sn
Ph SnPh2
Ph 2Sn
Ph 2Sn
An gle
VALBO N D
<Sn -Sn -Sn >
112.4
C-Sn -C
105.5
Exp .
112.5
106.7
2
B 3Ph 3O 3
Ph2
B
O
O
Ph 2B
O1
O2
P1 O
4
P2
O
O
P
P3
O
Ga 2Pyr 2Br 4
pyridine
Br
Br
Br
Ga Ga Br
121.7
118.0
O 1-P 1-O 2
O 2-P 1-O 4
O 2-P 2-O 3
P 1-O 2-P 2
P 2-O 3-P 2
114.1
104.5
99.3
122.9
127.3
115
103
99
124
128
Br-Ga -Br
Ga -Ga -Br
107.0
115.1
105.8
116.3
C-In -C
C-As-C
101.9
124.7
99
126
pyridine
As 3(CH 3)6In 3(CH 3)6
As
In
In
As
121.8
118.2
BPh2
O
PO 4P 3O 3
O3
<B-O -B>
<O -B-O >
As
In
A Comparison of Parametrization: VALBOND & MM3
H C C CH3
H H
HC
H
O
H
CH3
H
C
H
C
H
O
O
C
H
C
H H
C H
H
H3C
MM3 Requires:
CH3
C
H3C
O
CH3
20 equilibrium bond angles
20 bending force constants
VALBOND Requires:
4 valbond parameters (scaling factors)
4 hybridization weighting factors
Simple Valence Bond Hybrids (Localized Bonds)
Are Poor Descriptors of Hypervalent Molecules
• Use of d2sp3 (e.g. SF6) and dsp3 (e.g. PF5) hybridization
schemes is incompatible with ab initio computations.
• If d-orbitals are excluded, it is not possible to generate
enough hybrid orbitals to accommodate all bonds and lone
pairs.
• Ionic resonance helps explain molecular stabilities but does
not lead to simple justification of molecular shapes.
F
-
+
Xe
F
F
+
Xe
-
F
F
Xe
F
Origins of the Angular Distortion Potential
of XeF 2: Orbital or 1,3 Repulsion?
120
MP2
1-3-van der Waals
100
1-3-Coulombic
80
MP2
60
VDW
40
Coulombic
20
0
60
90
120
Angle (degrees)
150
180
Hypervalent VALBOND Uses Both 2c-2e and 3c-4e
Bonds as Fundamental Bonding Units
Consider ClF3,
• VALBOND uses three resonance structures, each with two
lone pairs, one 2c-2e bond, and one 3c-4e bond
I
F
Cl
F
III
II
F
F
Cl
F
F
F
Cl
F
3c-4e bond
• Resonance structures with linear 3c-4e bonding
arrangements are preferred.
F
Resonance Structure Populations Are Geometry Dependent
3c-4e angles
Population=
Σ cos2θ
Res. Structures
Σ
3c-4e bond
2c-2e bond
3c-4e angles
Σ cos2θ
SHAPE
Resonance Structures and Populations
T-shape
F
F
F
F
F
F
F
F
100%
0%
0%
F
Trigonal Planar
F
F
F
F
F
F
33%
33%
F
F
F
33%
Results for Hypervalent VALBOND
All structures use one set of generic VALBOND
parameters
180o (180o )
178o (182o )
F
F
Xe
F
F
Cl
F
F
F
S
F
o
174 o
(186o )
o
F
92 (88 )
90o
(90o )
F
F
P
180o (180o )
F
F
F
F
Xe
F
F
F
F
F
90o (90o )
S
120o
90o (90o )
F
o
F
(120 )
Inclusion of Lone Pairs Brings Computed Structures
Into Agreement with Experiment!
F
F
Dynamic Motions of PF
5
Bending Frequencies
VALBOND
151 174 300 340
Experiment
500 175 520 533
Axial - Equatorial Exchange
Transition States
VALBOND
F
F
F
P
Ab Initio
99o
F
F
∆E† = 3.0 kcal/mol
F
F
F
P
101o
F
F
2 - 5 kcal/mol
Dynamic Motions of ClF
3
Bending Frequencies
VALBOND
418 381 300
Experiment
442 328 328
Axial - Equatorial Exchange
Pathways - Possible TS's
F
Cl
F
F
Cl
F
F
C 3v
F
F
Cl
F
FF
F
Cs
F
Cl
Cl
F
C 2v
D 3h
F
F
Axial - Equatorial Exchange in ClF
3
MP2
VALBOND
130o
o
134
F
Cl
F
91o
F
∆E†= 40 kcal/mol
F
The Transition State is C
2v
!
Cl
F
100o
F
∆E†= 37 kcal/mol
Simple (?) Geometries of Hydrides
BeH 2
H 2O
Cu(Me) 2- PtH 2
NH 3
BH 3
ZrH 3+
PdH 3-
CH 4
RhH 4-
SF6
WH 6
Hybridization and Metal Complexes:
the Intriguing Case of WH 6
• 1989 Girolami reports that ZrMe
6
2-
is not octahedral
Morse, P. M.; Girolami, G. S. J. Am. Chem. Soc. 1989, 111, 4547
• 1990 Haaland demonstrates that WMe
6
is not octahedral (either C
3v
or trigonal prismatic)
Haaland, A. et al.
J. Am. Chem. Soc. 1990, 112, 4547
• 1992/1993 Albright and Schaefer independently report that WH
• is not octahedral
• exhibits four minima of nearly equivalent energy
• two minima are C 3v and two are C 5v
• distortion to octahedral geometry requires ca. 130 kcal
Kang, S. K.; Tang, H.; Albright, T.
Shen, M.; Schaefer, H. F.; Partridge, H.
J. Am. Chem. Soc. 1993, 115, 1971.
J. Chem. Phys. 1992, 98, 508.
6
:
Transition Metal Bonds Have Little p-Orbital Character
Schilling, Goddard, Beauchamp J. Am. Chem. Soc. 1987, 109, 5565.
Hybridizations of M-H+ Bonds
100
90
%s
80
%p
*d
%d
* NBO values
70
60
50
**ds
40
30
20
*p
Rh
Ru
Tc
Mo
Nb
Zr
Y
Sr
0
Cd
*sp
*
Pd
10
Hybridization Rules for
Transition Metal Complexes
• Use only s and d orbitals in forming hybrid orbitals.
• To form n covalent electron-pair M-H bonds use sdn-1 hybridization.
• Lone pairs prefer high d-orbital character.
• When the metal valency exceeds 12 electrons, delocalized bonding units
are used (e.g. hypervalent, linear 3-center 4-electron bonds).
Landis, C. R.; Cleveland, T.; Firman, T. K. J. Am. Chem. Soc. 1995, 117, 1859.
0.25
0.20
sd
sd
4
0.15
0.10
66Þ
90Þ
0.05
114Þ
0.00
Energy
0.20
sd
2
sd5
0.15
0.10
63Þ
90Þ
0.05
117Þ
0.00
0.20
sd3
d
0.15
0.10
71Þ
0.05
109Þ
55Þ
125Þ
0.00
30
60
90
120
150/
30
60
Angle (degrees)
90
120
150
+
ZrH3
Electron count:
6 eBonding orbitals:
3 localized pairs
sd2
Bonding hybrids:
Expected bond angles: 90Þ
93Þ
MP2 Geometry Optimization
RuH4
Electron count:
Bonding orbitals:
Bonding hybrids:
Nonbonding orbitals:
Nonbonding hybrids:
Expected bond angles:
12 e4 localized pairs
sd3
2 lone pairs
d
109.5Þ
71Þ and/or 109Þ
Shapes of 6 and 12 Electron MHn
Landis, C. R.; Cleveland, T.; Firman, T. K. J. Am. Chem. Soc. 1995, 117, 1859.
The Beguiling Case of WH6
Shen, Schaefer, Partridge J. Chem. Phys. 1993, 98, 508.
Four Local Minima Were Proposed
63o
63o
C3v
116
116 o
o
116o
C5v
63o
63o
"To inorganic chemists comfortable with the idea that WMe6 is
effectively octahedral, the present theoretical results for WH6 will b
unsettling."
Shapes of 12 Electron MH6
TcH6+ at MP2
PdH3
190Þ
-
85Þ
Electron count:
14 e
Bonding orbitals:
2 localized pairs
MP2 Optimized Geometry
1 3-center 4-electron bond
sd
Bonding hybrids:
Nonbonding orbitals:
Nonbonding hybrids:
4 lone pairs
d
Expected bond angles: 90Þ
180Þ delocalized bond
The Structures of Homoleptic Pt-Aryls
Pt(C 6Cl 5)4
Pt(C 6F 5)42(Usón, R., Forniés, J., et al., J. Chem. Soc.,
( Forniés, J., et al., J. Am. Chem. Soc. 1995, 117, 4295
Dalton Trans. 1980, 2, 1386)
Electron Counting
Electron Counting
16 e - - 12 e - >> 2 3center- 4e - interactions
14 e - - 12 e - >> 1 3center- 4e - interaction
>> sd hybridization (90
>> 4 pure d lone pairs
C 6F5
Pt
)
>> sd 2 hybridization (90
>> 3 pure d lone pairs
See-Saw Geometry
Square Planar
C 6F5
o
C 6F5
C 6F5
C 6Cl 5
C 6Cl 5
C 6Cl 5
Pt
C 6Cl 5
o
)
Seam-Searching:
Approximation of Transition States
Rappè,
Landis
Example: Diels-Alder Reaction
Reactant
1.42Å
(1.39)
Product
Energy
Seam
2.11Å
(2.24)
C
C
C
True TS
Reaction Coordinate
C
C
1.44Å
(1.40)
C
102o
(102o)
1.42Å
(1.38)
Bond Breaking/ Making: Homolytic Cleavage of CH4
H
H
H
C
+
H
H
sp3
H
sp2
Rappè, Landis
H
H
p
VALBOND/UFF Models this by:
• Extended Rydberg Function for Bond Stretch Energy
• VALBOND for Angle Energy
• Bond Order Dependent Hybridization
• For 90o<θ<120o, 1.0Å< RC-H< 5.0Å
the Maximum Energy Deviation < 3 kcal/mol !!
Simple Mo(NRR') 3 Complexes Effect
N 2 Cleavage
Laplaza, C.E.; Cummins, C. C.
N
Science 1995,268,861-863
N
Mo
Mo
NRR'
'RRN
N
NRR'
red-orange, paramagnetic
N2
N
R
R
N
N
'R
Mo
R
N R'
R'
N
purple, paramagnetic
'RRN
N
N
N
R
NRR'
NRR'
Mo
N
Mo
'R
N
R'
N R'
N
'RRN
R
R
Mo
NRR'
NRR'
N
2
R'
Mo
'R
N
R
N
R
N R'
R
N
gold, diamagnetic
Mo
'RRN
NRR'
NRR'
Why Does the Cummins Complex Split N
2?
" It is thought that the M-N triple bond is one of the
strongest metal-ligand bonds, and its formation
clearly provides the thermodynamic driving force
for the N 2 cleavage reaction elucidated here."
"Monomeric Mo(NRAr) 3 is formally related to the
well-known dimeric Mo(III) complexes X 3Mo-MoX 3
(X=alkyl, amide, alkoxide), which have unbridged
metal-metal triple bonds. Severe steric constraints
apparently render Mo(NRAr) 3 immune to
dimerization, endowing the complex with the stored
energy required for the observed reactivity toward
N 2."
Why Does the Cummins Complex Split N
Closely related complexes form
N
N
SiR3
Mo
µ-N 2 bridged dimers ...
R3Si
N
2?
N
Mo
N
N N
N
N N
SiR3SiR3
R3SiR Si
3
Shih, K.Y.; Schrock, R. R.; Kempe, R.
8804-8805.
J. Am. Chem. Soc. 1994, 116,
2-
V
N
N
V
Ferguson, R.; Solari, E.; Floriani, C.; Chiesi-Villa, A., Rizzoli, C.
Angew. Chem. Int. Ed. Engl. 1993, 32, 396-397.
... but do not cleave N
2 to
yield metal nitrides.
Is N 2 Cleavage Thermodynamically
Favorable for Simple Mo(NR 2)3 Complexes?
Results of DFT Computations
A collaboration with the research group of Prof. Notker
Rösch, TU-Muenchen
1.99Å (2.00)
distances: DFT (Schrock Structure)
Energy (kcal/mol)
NH 2
H 2N
NH 2
Mo
1.20Å(1.20)
N
1.90Å
(1.91)
H 2N
2.00Å
1.67Å
N
Mo
NH 2
NH 2
N
2 H N Mo NH
2
2
NH 2
6 kcal/mol
Reaction Coordinate
UFF/VALBOND Evaluation of Ligand Effect
2.05Å(2.03)
'RRN
distances:
R,R'=t-Bu; 3,5-Me 2Ph (R,R'=H)
NRR'
NRR'
Mo
1.39Å(1.35)
N
2.02Å (1.99)
1.81Å
(1.80)
N
Mo
'RRN
Energy (kcal/mol)
'RRN
NRR'
NRR'
Mo
NRR'
NRR'
7.0 kcal/mol
1.21Å(1.20)
N
1.94Å
(1.90)
'RRN
N
Mo
NRR'
NRR'
39.7 kcal/mol
2.00(2.00)Å
1.69 (1.69)Å
N
2
*-10 kcal/mol
* With DFT-based
exothermicity correction
Reaction Coordinate
Mo
'RRN
NRR'
NRR'
5.0 kcal/mol
Developments in Progress
• New Valence Bond Consistent Improper, π-Bond, and
Torsional Terms
• Improved Hypervalent Descriptions for Transition
Metal Complexes
• Explicit Application of Resonance for
• Conjugated Aromatics
• Ionic-Covalent Resonance
• Hypervalency (esp. Metal Complexes)
• Donor Bonding
• Reactant-Product Mixing
UFF2/VALBOND Transition State
Searching
(NH 2)3Mo-N 2-Mo(NH 2)3
2 NMo(NH 2)3
2.03Å
distances: MM (DFT)
NH 2
H 2N
NH 2
Mo
1.35Å
N
1.99Å (2.00)
1.80Å
N
Energy (kcal/mol)
Mo
H 2N
NH 2
H 2N
NH 2
Mo
1.20Å(1.20)
NH 2
NH 2
N
1.90Å
(1.90)
H 2N
N
Mo
NH 2
NH 2
46.6 kcal/mol
2.00(2.00)Å
1.69 (1.67)Å
N
2 H N Mo NH
2
2
*6 kcal/mol
* Adjusted to match
DFT exothermicity
Reaction Coordinate
NH 2
Localized Hybrid Orbitals Are Good
Descriptors of Molecular Electron Densities
Natural Bond Orbital (NBO) Analyses
Fraction of e- density
Molecule
Hybridization
in Localized Hybrids
BH3
sp2
99.98%
CH4
sp3
99.98%
NH3
sp3.37 (N-H)
99.99%
sp2.18 (lone pair)
H2O
sp4.49 (O-H)
sp0.57 (lone pair)
pure p (lone pair)
99.99%
The Remarkable Robustness of
the Pauling Legacy
Explorations of the Directed Covalent Bond
• Mathematical Formulation of Hybrid Orbitals
• Molecular Mechanics and Valence Bond Concepts
• Bent’s Rule and Molecular Shapes
• Hypervalent Molecules and Resonance
• Simple Transition Metal Complexes
• New Rules for Hybridization in Simple Metal Complexes
Is the Pauling Legacy More
Harmful than Helpful?
" Pauling's enormous influence has entrenched VB theory
to a degree that it still receives consideration and deference,
which some believe excessive. In the final analysis, only the
MO theory (at the first approximation level) provides a unified,
self-consistent view of bonding that is equally applicable
across the periodic table."
Butler, I. S.; Harrod, J. F. Inorganic Chemistry. Principles and
Applications Benjamin/Cummings: Redwood City, CA, 1989;
page 75.
Principles of the Directed Covalent
Bond: Lewis’ Rules Updated
• The electron-pair bond is formed through
the interaction of an unpaired electron on
each of the two atoms.
• The spins of the electrons are opposed
when the bond is formed, so that cannot
contribute to the paramagnetic susceptibility
of the substance.
• Two electrons which form a shared pair
cannot take part in forming additional pairs.
Pauling, L. J. Am. Chem. Soc. 1931, 53, 1367
Principles of the Directed Covalent Bond:
Qualitative Interpretation of Wave Equations
• The main resonance terms for a single electronpair bond are those involving only one eigenfunction from each atom.
• Of two eigenfunctions with the same dependence on r, the one with larger value in the bond
direction will give rise to the stronger bond, and
for a given eigenfunction the bond will tend to be
formed in the direction with largest value of the
eigenfunction.
• Of two eigenfunctions with the same dependence on θ and φ, the one with the smaller mean
value of r will give rise to the stronger bond.
Pauling, L. J. Am. Chem. Soc. 1931, 53, 1367
The Pair-Defect Approximation
“ The approximate bond strength (Sapprox) of an orbital i at angles αij with the
other orbitals j is given by...”
Sapprox = Smax − ∑ [Smax − S0 (α i )]
i
“ We now have subjected it to an extensive test ... it is seen that the pair-defectsum approximation to the bond strength seems to be an excellent one.”
Pauling, L.; Herman, Z.; Kamb, B. J. Proc. Natl. Acad. Sci., USA 1982, 79, 1361.
Are Hybrid Orbitals Good
Descriptors of Electron Density?
Natural Bond Orbitals analysis
provides a method for extracting localized bond descriptions
from high quality electronic
structure computations.
For non-hypervalent molecules
localized bond descriptions
account for >99.98% of the
density matrices.
Reed, A. E.; Curtiss, L. A.; Weinhold, F.
Chem. Rev. 1988, 899.
Molecule
Hybridization
Fraction of e- density
in Localized Hybrids
BH3
sp2
99.98%
CH4
sp3
99.98%
NH3
sp3.37 (N-H)
99.99%
sp2.18 (lone pair)
H2O
sp4.49 (O-H)
sp0.57 (lone pair)
pure p (lone pair)
99.99%
ClF3
sp9.6d2.0 (axial Cl-F)
sp11.3d6.9 (eq. Cl-F)
99.15%
How are Hybridizations Determined
in VALBOND?
Based on Pauling’s rules and a simple, parametrized algorithm based on
Bent’s rule†, hybridizations for simple non-hypervalent molecules of the
p-block are determined readily.
H
•
•
•
B
F
•
••
gross
3 hybrids with ~sp2
hybridization
F
Lewis Structure
quantitative
Bent's Rule
H
sp1.81
sp2.09
† Bent,
H. Chem. Rev. 1961, 275.
120.9
B
F
F
118.1
Hypervalency Challenges VBBased Bonding Descriptions
• spmdn hybridization schemes are incompatible with high level electronic
structure computations.
Magnusson, E. J. Am. Chem. Soc. 1990, 112, 1434.
• therefore simple hybridization schemes cannot be used to create one
electron-pair bond between the central atom and each ligand.
• Resonance is important
F
Cl+
F-
F
but why is ClF3 T-shaped?
F-
Cl+
F
F
F
Cl+
F-
F
Ionic-Covalent Resonance
Maximizes at Linear Arrangements
θ
NBO analyses indicate that the
3-center 4-electron bond is modeled
well as donation of a lone pair from
F- into a localized σ* orbital of the
ClF2+ fragment.
F-
F
Maximum stabilization
at θ = 180o
F
According to Natural Resonance Theory† analysis, two resonance structures
account for 99.95% of the total MP2 electron density.
F
Cl+
F
50%
†
F-
F-
Cl+
F
50%
F
F
Cl+
F
F-
<1%
Glendening, E. D.; Weinhold, F. “Natural Resonance Theory” University of Wisconsin
Theoretical Chemical Institute, 1994
Can You Predict the Structures of
These Molecules?
WH6
TcH6+
ZrH3+
RhH3
PtH42-
FeH64-
RhH4-
RuH4
PtH2
How Does Site Isomerization in
ClF3 Occur?
Structures of Simple Metal Hydrides
Challenge All Bonding Models
• On the basis of ab initio computations Albright et al.1 and Schaefer et al.2
suggest that WH6 is not octahedral. Instead they propose that four lower
symmetry minima (2 C3v and 2 C5v) exist at nearly equal energies.
• Schaefer estimates that the octahedral structure lies ~140 kcal/mol above
global minimum!
• Gas phase diffraction data3 for WMe6 and the crystallographic structure4 of
ZrMe6+ demonstrate non-octahedral structures.
1
2
3
4
Albright, T. A.; Kang, S. K.; Tang, H. J. Am. Chem. Soc. 1993, 115, 1971.
Shen, M.; Schaefer, H. F.; Partridge, H. J. Chem. Phys. 1992, 194, 109.
Haaland, A. et al. J. Am. Chem. Soc. 1990, 112, 4547.
Morse, P. M.; Girolami, G. S. J. Am. Chem. Soc. 1989, 111, 4114.
Shape of 16 Electron MH4
y Functions for sdn
d Orbitals