Table of contents

1. Intro

2. Scales

3. Visualisation

4. Mechanics

5. MO Reactants

6. MO TS

7. Stereo

8. Problems

Molecular Modelling for Organic Chemistry

http://www.ch.ic.ac.uk/local/organic/mod/

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A course of seven lectures and lecture-demonstrations given by

Henry Rzepa,

at the Department of Chemistry, Imperial College.

Importance of Molecular Modelling

The 1998 Nobel Chemistry Prize was awarded to Pople and Kohn for their work in Computational Chemistry and Molecular Modelling.

The 1999 Nobel Chemistry Prize was awarded to Ahmed Zewail for his work in developing spectroscopic methods for studying reactions and in particular transition states, an essential aspect of molecular modelling.

Modelling is an essential component of the Grand challenge of Dial-a-Molecule.

Course Aims and synopsis

Molecular modelling is a collection of (computer based) techniques for deriving, representing and manipulating the structures and reactions of molecules, and those properties that are dependent on these three dimensional structures. This lecture course aims to introduce in a simple way the hierarchy of computational modelling methods used nowadays as standard tools by organic chemists for searching for, rationalising and predicting structure and reactivity of organic, bio-organic and organometallic molecules. The emphasis will be on helping to develop a feel for the correct "tool" to use in the context of a typical problem in structure, activity or reactivity, by describing the limitations and strengths of each method. The techniques described include:

1. Molecular visualisation

2. Equilibrium and transition state geometry location

3. Molecular mechanics methods

4. Semi-empirical, ab initio and density functional molecular orbital methods

5. Methods for topological analysis of wavefunctions

The course does NOT deal with: excited states and photochemistry, hybrid (layer) models and inorganics.

The use of these tools is illustrated via the following case studies:

- Molecular visualisation: Using known nuclear positions or templates to construct a model.

1. The helical structure of DNA

2. Environmentally Friendly Herbicides.

3. Chiral Resolution using the Pirkle Reagent

4. Nanotechnology and Molecular cavities: Molecular self-assembly of Nanobiotics, Dihydrogen bonding

- Molecular Mechanics and Conformational analysis: Calculating the Nuclear positions

- 3 revisited: Where IS the H-bond in the Pirkle Reagent?

5. The mechanism and stereochemistry of Carbocation rearrangements

6. Design of organic crystals: A molecular anvil

- Reactivity using Quantum Mechanics: Calculating where the electrons are

- 3 revisited: Where ARE the electrons in the Pirkle Reagent?

- 2 revisited: How do orbitals interact within a molecule?

- 4 revisited: The structure of DNA

- 5 revisited: Non-classical Carbocations

- Mechanism determined using Quantum Mechanics

7. Regiospecificity: The mechanism of electrophilic aromatic substitutions

- Nitration of Pyridine

- Aryl C-C bond formation using a metal catalyst

- 6 revisited: The Diels Alder reaction

8. Stereospecificity: Asymmetric C-C bond formation via Enol Borinates.

9. Stereospecificity in the Strecker amino acid synthesis.

10. Stereospecificity in Catalytic polymerisation: The Ziegler-Natta mechanism and the role of agostic interactions

2011 Exam Paper and outline answer

Background texts and workbooks:

1. Wikipedia entry

2. Computational Organic Chemistry, Steven Bachrach, Wiley, 2007, ISBN 978-0-471-71342-5.

- This book also has a very interesting Blog, where the latest news and interesting publications in the area are discussed.

- Another blog on Computational Chemistry

3. Computational Chemistry Workbook: Learning Through Examples Thomas Heine, Jan-Ole Joswig, Achim Gelessus ISBN: 978-3-527-32442-2, 2009.

4. Essentials of Computational Chemistry, Chris Cramer, Wiley, 2002.

5. Chemical Applications of Molecular Modelling, Jonathan Goodman, RSC, 1999. (ISBN 0-85404-579-1)

6. Molecular Modelling Workbook for Organic Chemistry, W. J. Hehre, A. J. Shusterman and J. E. Nelson, 1998.

7. Brief Guide to Molecular Mechanics and Quantum Chemical Calculations, W. J. Hehre, J. Yu, P. E. Klunzinger and L. Lou, 1998.

8. Laboratory Book of Computational Organic Chemistry, W. J. Hehre, A. J. Shusterman, W. Wayne Huang, 1996.

9. Laboratory Exercises using HyperChem from Hypercube Educational Publications, M. L. Caffery, Caffery, Dobosh and Richardson. (ISBN 1-896164-30-7)

10. Molecular Mechanics across Chemistry. Anthony Rappe and Carla Casewit (ISBN: 0-935702-77-6).

11. Molecular Modelling. Principles and Applications, A. R. Leach, Longman, 1996. ISBN 0-582-23933-8

12. Introduction to Computational Chemistry, by Frank Jensen, Wiley, 1999 (ISBN 0-471-98425-6).

13. Introduction to Theoretical Organic Chemistry and Molecular Modelling by William B. Smith. ISBN: 978-0-471-18643-4

14. Encyclopaedia of Computational Chemistry, the definitive reference work for Computational Chemistry.

15. WATOC Congress, 1999, 2002, 2005, 2008, 2011.

Return to overview| Forward to scales|Forward to visualisation|Forward to Mechanics| Forward to MO Reactant|Forward to MO TS| Forward to MO Advanced|

(c) H. S. Rzepa 1998-2012. No reproduction rights granted to this material without permission.

Techniques of Molecular Modelling: Molecular Visualisation

Historical Preamble

Molecular Visualisation originated from the physical models of molecules often used in conjunction with lectures, the grand daddy of which was constructed by August Wilhelm von Hofmann as early as 1860 whilst lecturing at what was the precursor to the modern Chemistry department at Imperial College (and who also introduced the still familiar colour scheme). Note the planar carbon (tetrahedral carbon was some years away!).

Josef Loschmidt introduced the spacefill form of representation still in use today to visualize molecules, and also of bond types representations (single, double, triple) in a booklet published in 1861. We can judge how accurate his visualisation was by comparing his diagram (of an aromatic triazine) with a modern computer generated model. Note that the cyclohexatriene structure for the benzene nucleus itself (as proposed by Kekulé) was still some four years into the future at that point! (there is still controvery and debate about whether the representation shown here fully anticipates Kekulé or not. It certainly does for the 1,3,5,-triazine ring). Loschmidt's diagrams are key because he introduces (implicitly) the idea that different atoms have different sizes. It was not until 1873 that van der Waals formalised the idea of atomic radii (inferred from his equations of state). These radii are now the basis of all molecular visualisation (see blog here).

Following the recognition of the tetrahedral nature of saturated carbon in 1874, Hermann Sachse in 1890 using only trigonometry, derived the non-planar chair and boat conformations of cyclohexane and the axial and equatorial positions of the hydrogens. He even encouraged readers of his articles to construct a paper model to help them visualise this! Molecular visualization was well and truly born (except that no-one recognised the significance of Sachse's work, and he died in obscurity in 1893!)

Nowadays, molecular visualization is almost entirely accomplished using computers. What you will see in the lecture comprises "state-of-the-art" computer technology. In this demo, we will switch on the stereoscopic lecture theatre projection system. The computer generates two separate images of the molecule in question, and via the dual-channel graphics card, sends each to a different projector. The projection uses circularly polarized light reflected off an alumina-coated screen which preserves this polarization. The direction of polarization is different for the two projectors. The polarized glasses you will have been issued with also filter only one sense of polarization each; in effect the left eye sees only one projected image, the right the other. The brain then combines this information to generate its own perceived 3D image. An example of this technique as applied to one of the Laboratory experiments can be viewed here.

A second technique is also demonstrated here, using a program called Jmol to script annotations to the structure, in this case highlighting particular bonds, lengths and angles. The scripts are invoked using in-lined buttons in this case. Jmol is not (yet) Stereo-enabled, so a different program is used to enable this (DS Viewer Pro). Later in the course, animations (in stereoscopic display) will be used to enhance the visualisation process.

Case Study 1. The structure of DNA

This model relies exclusively on effective visualisation. It is said to be the most famous scientific diagram of the 20th century! Even so, it took 3-4 years after its publication in 1953 before comprehension dawned more widely. Perhaps the lack of a readily available model which people could see in their own office did not help! Even now, its a challenge to build this from commercially available plastic models! And the coordinates provided do not help much unless you are a very experienced crystallographer.

We will return to this later in the course, but I ask one question at this stage. How did Watson and Crick know that the helix was right handed?. The answer (in part) lies in van der Waals radii!

Case Study 2: Environmentally Friendly Herbicides.

The environmental impact of chemicals is of major contemporary concern. When a new herbicide is being developed, one of the very first properties that must be understood is the likely persistence of the substance in soil. As part of a screening program, two closely related potential herbicides were found to have unexpectedly different stabilities in water, one surviving hydrolysis for hours, the other only for a minute or so. There was clearly an important need to understand this behaviour so that some design control could be exercised over this property.

The mechanistic chemists had already established that the critical center was the O-C-O group, in which cleavage of one C-O bond was the important step. In modelling this system, some important early strategic decisions have to be taken. Is the reactivity related to some intermolecular property involving say interaction with water, or is it some structural feature of the molecule itself? Some knowledge of chemistry (the "anomeric" effect) suggests that intramolecular interaction of the C-O bond with the adjacent oxygen electron lone pairs is likely to be important, which may in turn affect say the C-O bond strength and hence its length. This in turn rules out simple molecular mechanics (MM) as the first line of attack, since this is principally a method that does not deal easily with electrons and their wavefunctions. A quantum mechanical (QM) approach should work, but the relative size of the molecule, and its potential conformational complexity implies a major task. Background literature searching and reading shows moreover that QM methods have had a checkered history in dealing with anomeric effects! Its time for a exploratory approach by inspecting the crystal structure.

Using molecular visualisation soon reveals that the solvolytically unstable isomer (1, lhs) has two quite different C-O bond lengths, whereas the relatively stable form (2, rhs) shows almost no difference in these lengths. Thus (1, lhs) is well on the way to breaking a C-O bond, even with no water getting in on the act. Thus this really does appear to be an intramolecular and not an intermolecular problem, where analysing a single structural feature can give great insight into the structure-activity relationships.

Unstable Isomer (twist boat)

Stable Isomer (chair)

Firstly, you notice that the conformations of the two 7-membered rings are quite different, but one common aspect is that in both cases the aryl group is "equatorial", an orientation generally preferred on steric grounds. This "locking" of the 7-ring then induces a particular angular relationship between each C-O bond and the lone pairs of the other oxygen atom. We know that this property should be important from our reading about the "anomeric" effect. Although we cannot easily directly measure angles involving lone pairs, we can "idealize" these with two dummy atoms arranged tetrahedrally around each oxygen atom. By doing this, we discover that for the stable compound, these idealised lone pairs are never quite "antiperiplanar" to the opposite C-O bond (164°) With the unstable isomer however, one particular lone pair is found to be more closely "antiperiplanar" to the elongated C-O bond (174°), a classical manifestation of the anomeric effect.

Armed with this highly focussed conformational interpretation of our reactivity, we could now go on to design new variants, hopefully with the particular properties we want.

Literature Citations. Isoxazolinyldioxepines. Part 1. Structure-Reactivity Studies of the Hydrolysis of Oxazolinyl-dioxepin Derivatives, P. Camilleri, D. Munro, K. Weaver, D. J. Williams, H. S. Rzepa and M. Z. Slawin, J. Chem. Soc. Perkin Trans. 2, 1989, 1929. DOI: 10.1039/P29890001929

Case Study 3: Chiral Resolution using the Pirkle Reagent

The so called Pirkle compound is much used as a chiral resolving and nmr shift reagent. In its optically active form, it will help to resolve racemic mixtures of chiral molecules by binding slightly differently with each of the two enantiomers of say an amino acid. Such chiral resolution is of paramount importance in the development of a new drug; the Thalidomide pregnancy pill case arose because of incomplete separation of two such enantiomers, one of which turned out to be toxic to the foetus.

But what sort of intermolecular interactions are involved in the binding process? The simple truth is that for the Pirkle reagent, no-one really knew until recently. Intermolecular interactions can be quite difficult to calculate accurately, especially for a molecule that size and the quickest and most direct way to probe this question is to look at the crystal packing structure if it can be measured. A search of the Cambridge crystal structure data base revealed that although no structures between the Pirkle reagent and chiral molecules had been reported, the structure of the racemic reagent itself had already been determined as part of a study on triboluminescence (the crystals emit purple light when you grind them!). The authors conclusion was that a highly unusual hydrogen bond was formed between the -OH group of one molecule and the F_3C- group of another. But a small alarm bell sounds at this point. The crystallographers had not actually located the position of the hydrogen atom, they had assumed it could only sit between the O and the F. Could they be mistaken? Visualization enables this assumption to be easily tested.

S,S

S,R

The effect of the crystal lattice diagram can initially be overwhelming, the trees being well and truly hidden by the wood. The tools offered by the molecule editor make a very good wood-cutter, and soon the unique interactions involved can be displayed on the screen. Immediately apparent is that an alternative explanation involving an interaction between the OH group and the π-face of an adjacent molecule is feasible. This hypothesis was rapidly confirmed by explicitly locating the vital hydrogen atom, both in the original racemic crystal and in the optically active form (above). Many profound implications follow from this discovery. It has helped focus on the role of π-facial hydrogen bonding in protein interactions, it has now been discovered as a common feature of certain types of silanols, phenols and amines and it occurs in inclusion compounds known as calixarenes. The implications for modelling are also significant. By their very nature, most molecular mechanics methods would not reproduce the phenomenon. Moreover, the unexpected orientation of the OH bond resulting from π-facial interaction sets up specific "stereoelectronic" interactions in the molecule which profoundly affect calculated quantum mechanical properties such as electrostatic potentials, often used to predict intermolecular interactions. Thus much new insight and direction is given to a project, the germ of which was started by 3D visualization of a structure which revealed detail that others without such a tool had missed.

Literature Citations. A. M. Sweeting and A. L. Rheingold, J. Chem. Phys, 1988, 93, 5648; π-Facial Hydrogen Bonding in the Chiral Resolving agent (S) 2,2,2-Trifluoromethyl-1-(9-Anthryl)ethanol and its Racemic Modification, H.S. Rzepa, M. L. Webb, A. M. Z. Slawin and D. J. Williams, J. Chem. Soc., Chem. Commun., 1991, 765. DOI: 10.1039/C39910000765

Fast forward to next installment of this story

Case Study 4. Nanotechnology and Molecular Cavities/Lattices

Molecular self-assembly of Nanobiotics: Some of the most convincing "nanoscale" molecular engineering has been carried out by Reza Ghadiri's group. They have created cyclic peptides which act as bacteriostatic agents by inserting into the cell wall, and allowing all the water inside to escape. The following cyclic peptide is known as cyclic -PHE-ALA*-PHE-ALA*-PHE-ALA*-PHE-ALA*- (* indicates the un-natural amino acid configuration).

The key breakthrough was "engineering" the nanoscale device to stack correctly so that water channels form in the lattice structure. If amino acids with identical chirality are used to form the cyclic peptide (isotactic), the stacking does not exhibit this feature. If however, the peptide polymer is "heterotactic", ie the peptide comprises alternating R and S forms of the amino acid constituents, then the stacking changes completely, allowing the water channels to form. This in fact is very similar to the Pirkle reagent phenomenon noted above.

The same authors also note the effect of an apparently tiny change to two of the four phenylalanine residues. This involved changing the side chain from Ph-CH_2 into Ph-CH_2-CH_2 (atoms with halo). The packing is now changed from an infinite polymer with long water channels, to discrete dimers, where the water channels are interrupted. The substance is no longer as effective as a nanobiotic!

Literature Citations. M. R. Ghadiri, K. Kobayashi, J. R. Granja, R. K. Chadha, D. E. McRee, Angew. Chem. Int. Ed., 1995, 34, 93. DOI: 10.1002/anie.199500931; D. T. Bong, M. R. Ghadiri, "Self-Assembling Cyclic Peptide Cylinders as Nuclei for Crystal Engineering", Angew. Chem. Int. Ed., 2001, 40, 2163. DOI.

Molecular self-assembly- Dihydrogen bonding and Hydrogen storage: Ethane (C_2H_6) is a gas at room temperature, with a melting point of -181°C. Ammoniaborane (BH_3NH_3) is iso-electronic with ethane, i.e. it has the same molecular weight, but it has a melting point of +104°C, an unparalled 285° higher. What is the origin of this effect? The neutron diffraction structure is shown below. Visualisation reveals that each of the B-H hydrogens is more or less equidistant to two N-H from separate molecular units, making a N-H...H(B)...H-N T-shaped unit. Conversely, each of the N-H hydrogens is more or less equidistant to two B-H from separate molecular units, again revealing a T-shaped motif. Thus any single BH_3NH_3 interacts via twelve hydrogen bonds to other molecules in the crystal lattice. The effect is to produce a remarkably rigid lattice. Why does one H interact with another? Well, the N-H hydrogens are +ve charged, and the B-H hydrogens are -ve. The two are drawn together by electrostatic interaction to form a dihydrogen bond. Such substances can extrude gaseous hydrogen gas easily, and are the focus of interest for hydrogen storage devices.

Literature cited. W. T. Klooster, T. F. Koetzle, P. E. M. Siegbahn, T. B. Richardson, and R. H. Crabtree, "Study of the N-H...H-B Dihydrogen Bond Including the Crystal Structure of BH_3NH_3 by Neutron Diffraction", J. Am. Chem. Soc., 1999, 121, 6337 - 6343; DOI: 10.1021/ja9825332

Case Study 4a. 2012 Bonus extra: The nanocar

The smallest driven/steered car was recently constructed and driven 6nm in a straight line.

The coordinates were in fact quite difficult to acquire (from the journal), but are presented her for your enjoyment, together with thoughts of how the motor might work!

Preparing the Coordinates: Crystallographic coordinates relate to what is called a unique unit cell, and not necessarily to a discrete entity we call a "molecule". If one is lucky, the unit cell will contain at least one "molecule", but it may also contain fragments of molecules which span the cell. In creating a set of coordinates suitable for visualisation, one may have to multiply one or more of the x, y or z boundaries of the unit cell by e.g. a factor of 2 (or even more) to recover particular effects, such as intermolecular hydrogen bonding, molecular stacking, etc. A good program for doing this sort of operation is Mercury, from the CCDC centre. If you want to go one step further and actually change the structure and re-optimise the coordinates, you will have to worry about atom hybridisation (next section).

Pros and Cons of Molecular Visualisation

Pros: Needs no (expensive) software, data often available from databases, interpretation uses simple "arrow pushing" concepts, fast.

Cons: Recording structure can be slow (2 days per structure), need to "normalise" result, not quantitative, ie can be difficult to relate geometric features to measurable quantities such as the relationship between a bond length and the rate of hydrolysis, no modifications can be made to the original structure.

Back to scales|Forward to Mechanics| Forward to MO Reactant|Forward to MO TS| Forward to MO Advanced|

(c) H. S. Rzepa 1998-2013. No reproduction rights granted to this material without permission.

Techniques of Molecular Modelling: Molecular Mechanics

As noted before, the origins of the molecular mechanics method can be traced to work done by eg Barton (see reprints on van der Waals, electrostatic or the joined up analysis), Bright-Wilson (Molecular Vibrations, 1938. DOI: 10.1063/1.1750232) and others in the late 1940s and early 1950s and e.g. Allinger (starting in 1959, DOI: 10.1021/ja01530a049) and the MM2/3/4 force fields and Kollman/Amber in the 1970s-80s.

1. The Mechanics Force Field

- Steric energy normalised relative to zero rather than any thermodynamic quantity.

1. The parameters in the above equations require definition of the atom types (hybridisation of each atom; carbon has perhaps six different useful hybridisations) and an explicit definition of where one thinks the bonds are and of what type (single, double, delocalised etc). For bond lengths (1st term), each parameter (force constant K) is specific to the particular atom-type pair. A great many different atom-type pairwise combinations are possible from the elements of the periodic table; few force fields have even a fraction of these permutations.

2. A force constant K is also needed for each three-atom bend combination.

3. For the torsional 3rd term, parameters are inserted only for the common single and double bonds.

1. ψ is the dihedral angle

2. γ is a phase factor depending on whether ψ=0 is a minimum or maximum

3. n is a` symmetry number

4. The fourth term relates to non-bonded interactions (it implies we know where there are no bonds!). It requires knowledge of

1. van der Waals radii (for a recent table, see 10.1021/jp8111556)

2. The second part of this term accounts for the dispersion energy.

3. The electrostatic last part of this term requires charges on each atom pair (approximated with various relatively crude charge schemes, often obtained from quantum mechanical calculations on tiny molecules representing typical functional groups)

5. Many (often proprietary) force fields include a fifth additional term in the sum above, relating to:

- hydrogen bonding, or

- π-conjugation, or

- anomeric effects, etc

- There are also a rather simpler so-called universal or full periodic table force field (UFF, DOI: 10.1021/ja00051a040) where an attempt is made to interpolate atom pair or atom triple values from the basic hybridization specified for each atom, and specific values for each atom. These are not very good for energies, but suffice for cleaning up a crudely drawn structure. They can also be applied to many more elements of the periodic table than the MM2/3/4 style force fields.

- Some mechanics implementations insert cross-terms, ie stretch-bend. This implies that say a bond far from its equilibrium length may bend differently from one of normal length. One could also envisage a need for e.g changing the atom charges on say a polar carbonyl group as a function of its stretched bond length (this is never in practice done). There are an enormous number of potential cross terms, the vast majority of which are in fact neglected. For large molecules (say DNA) this may in fact be a significant error.

- As an approximate rule of thumb, the first three terms in the above equation tend to zero for unstrained and unhindered molecules. A high positive value of the steric energy is therefore taken as an indication of instability. The fourth term can take on negative values, particularly if the molecule is globular (lots of non-bonded attractive terms) or if the molecule has many ionic interactions where qi and qj have opposite signs.

- Force field parameters can be of widely differing quality. Some are derived from accurate experimental measurements, others may be pure guesses, or interpolations. Some force fields are good for energies, others for geometries.

- A good summary of the more widely used force fields can be found here.

- Energies should only ever be compared for isomers with the same total number of atom/bond types, in other words using the same parameters.

- Energies should not be compared across different force fields, across different programs implementing ostensibly the same force field, or even across different versions of the same program!

2. Using MM to optimise a molecular geometry

- The steric energy is used in conjunction with a "geometry optimiser" to find local minima in the energy. The process starts with a Taylor series expansion of the potential energy as a function of a the coordinates (show in the diagram below for just one coordinate, but generalized later to 3N-6). The expansion is truncated to correspond to a harmonic or quadratic potential surface:

There are four ways of populating the inverse Hessian matrix:

- steepest descent, which sets the diagonal terms to 1, and all the others to zero. Very fast, particularly initially, and suitable for the largest molecules since no matrix inversion need be performed.

- conjugate gradient, which evaluates the value of only the diagonal terms, setting all the others to zero. Slower, but meanders less when the gradient is no longer "steep".

- BFGS, which evaluates the diagonal terms, and approximates the off-diagonal terms from the history of the optimisation. Much slower, and less suitable for large molecules, since inverting the Hessian is now increasingly time consuming.

- Newton-Raphson, which evaluates all the terms. Extremely slow, but can succeed when all other methods fail (opt(calcall) keyword in Gaussian!).

The value of Δx is obtained from a one-dimensional line search to obtain a scaling factor (normally around 0.25), and an iterative process started. Convergence is achieved when either the value of Δx reaches a small predetermined value, or the change in energy between successive cycles becomes small.

3. Summary of Pros and Cons of Molecular Mechanics

Pros:

1. Simple energy functions are quick to solve,

2. can cope with large molecules,

3. accurate for systems which are close to the models used to reproduce the force field (interpolative),

4. The minima found are restricted to the "atom-connectivity" specified in the input file; no bonds can make/break during the optimisation process (and hence no matter how bad the initial geometry guess is, that connectivity will be preserved). Hence very useful for instantly converting sketches of molecules into tidied coordinates, even in an on-the-fly manner (elastic band optimisers)

5. deals reasonably accurately with van der Waals interactions at non-bonded distances for larger molecules (which increase vastly in number as the molecule gets bigger and which are notoriously difficult to model using purely quantum mechanics)

6. Mechanics, when used with high quality parameter sets, can be very reliable for calculating vibrational properties (IR frequencies, but not intensities which are an electronic property). See also the cons

7. Mechanics is good for conformational exploration (Monte-Carlo, Genetic algorithms, etc) where the number of possible conformations can number millions. It is used in this context for NMR predictions of molecules where many possible conformations may exist.

8. Mechanics is highly applicable to molecular dynamics, and hence derived statistical thermodynamic quantities such as free energies etc. In this context it is frequently used for analysing proteins, and other large biomolecules.

Cons:

1. No unique functional form for the energy terms (= proprietary?)

2. Number of parameters rises rapidly for non-CHNO elements,

3. Need to know atom/bond type for all the molecule (non-classical bonding)

4. Does not include majority of cross terms in force field

5. Most mechanics implementations do not compute 2nd derivatives (entropy)

6. Cannot cope with unusual (= unpredicted) bonding or interactions

7. Mechanics is prone to "false minima", allowed by the equations, but physically unrealistic (e.g. hemispherical carbon)

8. Cannot be used for bond-breaking transition states and hence reactions.

Case Study 5: The Relative Stability of Carbonium Ions

Organic chemists are taught that carbonium ion stability is predominantly due to electronic factors such as whether they are primary, secondary, tertiary, allylic, benzylic etc. What is mentioned much less is that their relative stability is also very sensitive to their geometries, and in particular the angles of the three substituents at the carbon. Geometries of carbonium ions are particularly inaccessible; it is not easy at all to get crystal structures! So how can Molecular Mechanics model the following carbonium ions? They are arranged in the expected order of increasing stability. Notice that the MM2 method gradually decreases the predicted angle energies. Notice also however how it predicts that the tertiary carbonium ion is actually LESS stable than the preceeding primary ion, which is surprising to say the least. Clearly, electronic factors must be playing an important role as well. Suffice to say that even without ANY consideration of electronic factors, the MM2 method does not do too badly in its prediction of relative stabilities. This re-enforces the conclusion that bond angles in carbonium ions are just as important as substitution! See also DOI: 10.1021/ed800058c for more detail.

stretch       =   0.868  angle         = 19.173

stretch bend  =  -0.672  dihedral      =  6.497

improp torsion=   0.155  van der Waals =  2.663

electrostatics=  -1.341  hydrogen bond =  0.000

Energy of the final structure is 27.342 kcal/mol.

(QM B3LYP/cc-pVTZ → rearranges to final

product directly by breaking/forming bonds)

stretch       =   0.447 angle         =  18.758

stretch bend  =  -0.078  dihedral      =   6.916

improp torsion=   1.124  van der Waals =   1.622

electrostatics=   0.000  hydrogen bond =   0.000

Energy of the final structure is 28.788 kcal/mol.

(Relative QM B3LYP/cc-pVTZ free energy

= 25.7 Kcal/mol)

stretch       =   0.571  angle         =  5.694

stretch bend  =  -0.027  dihedral      =  7.179

improp torsion=   0.009  van der Waals =  5.678

electrostatics=  -0.545  hydrogen bond =  0.000

Energy of the final structure is 18.559 kcal/mol.

(Relative QM B3LYP/cc-pVTZ free energy

= 23.3 Kcal/mol)

stretch       =   0.512 angle         =  3.439

stretch bend  =  -0.153 dihedral      =  6.662

improp torsion=   0.002 van der Waals =  3.872

electrostatics=   0.683 hydrogen bond =  0.000

Energy of the final structure is 15.017 kcal/mol.

(Relative QM B3LYP/cc-pVTZ free energy

= 0 Kcal/mol)

Case Study 3 revisited: π-facial Hydrogen Bonding

The H-bond energy functions in the Mechanics method are normally specifically parameterised against standard models, which involve three atoms, ie X...H...Y. The model of hydrogen bonding in this molecule is unusual to say the least, since it involves a whole ring and not a single atom as the H acceptor, and hence the Mechanics method (in theory) has no functional form which can reproduce it. In particular, aromatic rings are attractive to electrophiles by virtue of their quadrupole moment, which is a higher order property of the charge distribution (thus the so-called multipole expansion goes as monopole charge, dipole, quadrupole, octapole and hexadecapole. The equations above in effect only include the first and possibly second term of this expansion).

The optimised geometry does predict what looks like H-bonding, but its directionality is not quite correct. The steric energy is -49 compared with -18 if the two components are infinitely separated. This makes the binding energy = -31 kcal/mol. In this case, it is predicted to arise almost entirely from the non-bonded van der Waals term rather than electrostatic or hydrogen bonding (dipole-dipole) terms.

One monomer

Dimer

Stretch: 1.3257

Bend: 2.8090

Stretch-Bend: 0.2180

Torsion: -24.8762

Non-1,4 VDW: -3.1495

1,4 VDW: 10.7361

Dipole/Dipole: 4.1782

Total Energy: -8.7587 kcal/mol

Stretch:          2.5669

Bend:             6.6539

Stretch-Bend:     0.3319

Torsion:        -54.0415

Non-1,4 VDW:    -29.6591

1,4 VDW:         15.6989

Dipole/Dipole:    9.4549

Total Energy:   -48.9942 kcal/mol

There is one final aspect which must be considered, namely the entropy of the interaction. The equation

ΔG = ΔH - T.ΔS

implies that if entropy decreases (i.e. ΔS is -ve) as a result of a reaction (or in this case a binding, in which two free molecules are reduced to one bound complex), then ΔG must increase (two -ve signs = a positive result) as a result by the amount T.ΔS. The Mechanics method could in principle provide an estimate of this term via calculation of the normal modes of vibration (and solution of the appropriate partition function equations), but few Mechanics programs actually perform this calculation (which requires the second derivatives to be evaluated, something only available using MM4, accurate to between 25-50 cm-1). An approximate estimate of T.ΔS is about -(12-15) kcal/mol at 298K (i.e. approx 2 kcal/mol per degree of lost freedom), which has to be added (-/- = +) to the binding energy obtained above, to obtain a final value of -(15) kcal/mol. This value is certainly in the correct ballpark.

Another property of this molecule can also be studied using molecular mechanics, namely the rotation and preferred orientation of the aryl-C bond, for which mechanics is very well suited. The following "dihedral driving" calculation shows the MM2 energy as a function of the dihedral angle between the trifluoromethyl group and the anthracene ring. The barrier (about 10-11 kcal/mol) compares reasonably well with the measured value of ΔG (from NMR lineshape analysis) of about 14 kcal/mol. The discrepancy may well be due to ignoring solvation of the OH group, which increases its size, and hence the barrier.

Reference: M. L. Webb and H. S. Rzepa, Chirality, 1994, 6, 245-250. DOI: 10.1002/chir.530060406

Fast forward to next installment of this story

Case Study 6: Molecular Anvils and Tennis Balls

Now that we have a quantitative method for estimating energies of molecules, we can tackle a problem such as shown below. An application of "shape design" is the recent area of designing catalysts to promote reactions by ensnaring them in a cavity, or binding pocket. Whilst designing enzymes to do this is somewhat beyond the scope of the current course, a simple example of small molecule design can be shown here. A resorcinol-anthracene based aromatic molecule was designed in the form of spacers (the aromatic rings) and links (in the form of hydrogen-bonds) to create in the solid state a system with well defined cavities, which are large enough to accommodate small organic molecules, such as ethyl methacrylate and cyclohexadiene. The former is held within the cage by further hydrogen bonds to the cavity lattice. In particular, various changes to the structure can be made, to increase or decrease steric bulk, etc and the effects of these changes can be modelled, ie predicted.

The previous systems were modelled by drawing/creating the model using the built-in editor (ChemBio3D, etc). Here we will be importing the X-ray derived coordinates. There are some important considerations to doing so.

- Applying the Mechanics model requires explicit assignment of force constants for ALL atom types, which in turn requires the hybridisation (sp, sp2, sp3) of all centres to be fully known. No intermediate hybridisations are allowed.

- Conversion from X-ray coordinates to a format where all atom and bond types are fully defined (e.g. the MDL Molfile) is not 100% reliable. In particular, the hybridisation must be deduced from a combination of the bond lengths of all substituents at a given atom, the total coordination, and probably the angles (ie sp2 assumes the central atom is coplanar with three surrounding atoms).

- The Molfile we are loading here was very carefully edited by hand to ensure that:

1. the aromatic rings all had correct atom types, to allow aromaticity to be detected

2. the small organic molecules trapped in the cavity were actually "disordered" in the X-ray structure. This means that even the H count on any given carbon atom has to be carefully checked.

- In this example, all H-bonds were in fact "conventional" and hence the H-bond term in the Mechanics force field can be automatically applied.

- We will see in the Quantum Mechanics section how all these limitations can be removed.

When a standard MM2 mechanics force field is applied to the resorcinol-anthracene anvil, the following results are obtained:

Component

MM2, kcal/mol

Supermolecule

-157.2

Supermolecule without substrates (Anvil)

-140.1

Anvil computed as four components

-92.8

Three Substrates together, without anvil

+14.2

Three separated Substrates

+23.4

Anvil + Substrates separately

-140.1+14.2=-125.9

Binding Energy of substrates to anvil

-157.2-(-125.9)=-31.3

These energies are NOT corrected for entropy (i.e. they are closer (but not quite) enthalpic, and not free energies) a significant limitation. Very likely this sort of catalyst functions by converting enthalpic energies (the binding of the catalyst components) into entropic energies (i.e. pre-organising the reactants in the cavity, which of course consumes entropy. The free energy cost of assembling the three substrates is likely to be around +24-30 kcal/mol, but this is still likely to result in a win!).

What IS useful with the mechanics approach is the ability now to make minor variations to the environment, such as changing the nature of the substrates included in the centre of the anvil to see how the binding energy responds. Its a tuning procedure!

Literature Citation. Catalysis by Organic Solids. Stereoselective Diels-Alder Reactions Promoted by Microporous Molecular Crystals Having an Extensive Hydrogen-Bonded Network, K. Endo, T. Koike, T. Sawaki, O. Hayashida, H. Masuda, and Y. Aoyama, J. Am. Chem. Soc., 1997, 4117 - 4122; DOI: 10.1021/ja964198s

An interesting alternative to the above anvil is the molecular "tennis ball" (e.g. DOI) elaborated by Julius Rebek. Quite large molecules can be enclosed in these balls, and once in, they organize themselves very specifically, a phenomenon Rebek has termed molecular "social isomerism" (DOI: 10.1002/anie.200462839)

Just as with the anvil, the tennis ball is held together with hydrogen bonds, and this can be modelled using the Mechanics method, as can the organisation of the ligands inside the cavity.

One monomer

Dimer

Stretch:          5.1147

Bend:            49.5243

Stretch-Bend:     0.4564

Torsion:         -3.2591

Non-1,4 VDW:    -20.0160

1,4 VDW:         21.6595

Dipole/Dipole:   25.2888

Total Energy:    78.7685 kcal/mol

Stretch:          9.8872

Bend:           103.0995

Stretch-Bend:     1.2134

Torsion:         -3.8372

Non-1,4 VDW:    -87.3993

1,4 VDW:         43.0785

Dipole/Dipole:   29.6762

Total Energy:    95.7182 kcal/mol

Here, the energy liberated upon dimerisation is 62 kcal/mol, of which 41 is due to the non-bonded dispersion terms and 21 to hydrogen bonds. This is more than enough to overcome the entropic terms which acrue when substrates are trapped inside.

Back to scales|Back to Visualisation| Forward to MO Reactant|Forward to MO TS| Forward to MO Advanced|

(c) H. S. Rzepa 1998-2013. No reproduction rights granted to this material without permission.

Techniques of Molecular Modelling: Quantum Mechanical Methods

Molecular orbital methods date back to the origins of quantum mechanics and the Schrödinger equation.

1. The first practical chemical solution to this equation was formulated by Erich Hückel in 1931 and dealt exclusively with planar conjugated systems.

2. This molecular orbital theory was thereafter extended by Pauling, Hartree and Fock, Mulliken and many others,

3. and first effectively implemented in the 1960s by Hoffmann and Lipscomb to all valence electron systems (EHT),

4. by Pople in the 1970s who defined a family of semi-empirical methods

5. which were first effectively paramterised by Dewar (MINDO/3, MNDO, AM1), Stewart PM3 and 6)

6. In parallel, effective implementations of ab initio and density functional methods emerged in the 1980s (Gaussian, GAMESS, Turbomole, Orca, etc)

In essence, QM methods differ from the MM methods in providing explicit solutions for the electronic component of a molecule rather than just the nuclear terms.

These methods all adopt the Born-Oppenheimer approximation. Stated approximately, we assume that the motion and (point) position of the nuclei is described by classical equations, whereas the electrons is described by non-classical wavefunctions. This means the nuclear-nuclear energy contributions are classical mechanical (electrostatic point charge), but the nuclear-electron and electron-electron terms are quantum mechanical and the total energy is the sum of these nuclear and electronic components. The starting point for solving the electronic part is the Schrödinger equation:

H.Ψ = E.Ψ

A Quick Summary of the Principle Quantum Mechanical Methods used today

Methods which integrate Hamiltonian (H) and Basis set (Ψ)

- AM1 (Austin Model 1, based on a parametrised Hartree-Fock model), a Semiempirical method parametrised for ~42 elements. The method uses only valence shell atomic orbitals, representing them with Slater functions (see below), whilst the inner shells (e.g. 1s for C) and other terms in the Hartree-Fock equations are modelled with parametric functions (often referred to as the NDDO set of approximations) in a manner similar to molecular mechanics.

- PM3, a reparametrisation of AM1, introduced in 1989 and a bit like the curate's egg (much better for e.g. H-bonding). Parameters for 42 elements in all combinations.

- PM6, a reparametrisation of PM3 introduced in 2007 and about twice as accurate. Includes parameters for 69 elements (but not necessarily all possible combinations) DOI: 10.1007/s00894-007-0233-4. In 2009, the method was extended with explicit terms for dispersion energies (van der Waals) and delocalized hydrogen bonds, PM6-DH2+.

- RM1 is the latest (2008) method in this stable, offering high accuracy parameters for all combinations of the 10 most common elements (H, C, N, O, P, S, F, Cl, Br, I).

- Oniom: A method first introduced in 1996 which partitions a molecule into layers, each treated using a different approximation. The most common is a two layer model, the outer layer treated using molecular mechanics (or even a simpler unified field model), the inner one treated using MO theories. Three layer models apply one of mechanics, one of semi-empirical and one of ab initio MO theories (DOI: 10.1021/jp962071j)

Η: List of Ab initio/Density functional Hamiltonians

Ψ: List of Basis sets

- RHF, or (Spin) Restricted Hartree Fock. Applied to singlet electronic states (all electrons paired) and used for relatively rapid prototyping of a calculation. No consideration for electron correlation.

- UHF, or (Spin) Unrestricted Hartree Fock, for use with open shell systems (doublets, triplets, etc).

- ROHF. Restricted openshell Hartree Fock, for open shell systems. More limited than UHF for some systems (ie 2nd derivatives, etc) but free of spin contamination.

- B3LYP (UB3LYP): A popular density functional method (not strictly an ab initio one), offering good reliability and short range electron correlation treatment at the penalty of about half the speed of a RHF calculation. There are many other functionals that have been proposed, but none tested so thoroughly as this one.

- wB97XD (Density functional theory with empirical long-range correlation correction; DOI: 10.1039/b615319b). These dispersion corrections have been incorporated into new generations of double-hybrid methods, such as BML and B2-PLYP/B2GP-PLYP which address many deficiencies of B3LYP (DOI: 10.1021/jp710179r) and which occupy the so-called fifth rung of Jacob's ladder of approximations (DOI: 10.1063/1.1390175)

- MP2 (MP4), Moller-Plesset methods based on perturbation theory, an older method than the density functionals for incorporating electron correlation. Perhaps 10-100 times slower than HF, but in some cases more reliable, especially for so-called "long range correlation effects". New double-hybrid DFT methods rely on MP2 for the correlation part of the DFT functional.

- CISD; Configuration Interaction based on a Hartree-fock reference state, for systems where multiple electronic configurations and electron correlation are important, e.g. excited states etc.

- CASSCF(n,n); a keyword for invoking multi-configurational SCF where the Hartree-Fock reference state is simply not good enough (i.e. transition metal series, excited states, etc).

- CCSD(T), CCSDTQ; the so-called coupled-cluster approach to the correlation problem. Very very expensive, and can only be properly applied to about 10-12 (non-hydrogen) atoms.

- STO-3G. A minimal (single-ζ) basis set. Use for very rapid prototyping of a calculation, or initial geometry optimisation from a poor starting guess. The 3 means using three Gaussian-type functions sharing a common exponent (ζ) to represent a shell (modelled on the Slater-type-function or STO, this being adapted from the Schrödinger solutions for a hydrogen atom).

- 3-21G*. a small double-ζ basis set due to Pople, and available for most of the periodic table. The 3 is used for the core electrons (1s for eg carbon), the "2" and "1" are used for the valence electrons, and are represented by respectively two and one Gaussian functions, each with its own ζ exponent. The "*" means polarization (d) functions on e.g. S, P, Cl etc. Used for relatively rapid prototyping of a calculation.

- SDD, an alternative basis set for the entire period table using effective core potentials (Pseudopotentials) to reduce the number of basis functions (for the core electrons) and to include relativistic effects (for heavy elements).

- 6-31+(d,p). A reasonably high quality double-ζ basis set with anionic diffuse (+) and "polarisation functions" on the atoms ("p" orbital for Hydrogen, "d and f" orbital for carbon, "f and g" orbital for transition metals, etc).

- cc-pVTZ. A triple-ζ correlation-consistent basis set with valence polarization functions, now regarded as appropriate for reasonably high quality on small to medium sized molecules.

- aug-cc-pV5Z and aug-cc-pV5Z-pp. An augmented and polarized "pentuple ζ", correlation-consistent basis set for best quality calculations; available significantly for the first transition series. The higher elements are treated using pseudopotentials (pp). Also included in this type of basis is the aug-pcx (x=1-4) Kohn-Sham consistent basis for use on the fifth rung of Jacob's ladder.

- Gen: A general input for basis sets. Get them from here and mix-n-match your basis sets for the problem at hand.

Methods which extrapolate Hamiltonian and Basis set

- G2/G3 (Gaussian-3): Effectively an extrapolation of both the basis set, and the correlation treatment, the combination of which heads off towards exact solutions of the Schrödinger equation. Hugely expensive in computer time.

- W1-W4 Theories, currently considered the most accurate solutions to the Schrödinger equations available. This level of theory is now routinely correcting "faulty" experimental data! See DOI: 10.1063/1.2348881, 10.1021/jp071690x, 10.1021/jp900056w and 10.1063/1.3489113

How the various methods scale with size: Let us define N as ~ the number of atoms (slightly more accurately, it represents the number of basis functions associated with all the atoms in the molecule under consideration, and so it also goes up with the quality of the basis set)

- Methods which do not require 2-electron integral evaluations scale as N^2 (due entirely to diagonalising the Hamiltonian (Fock) matrix, but this only kicks in for very large matrices)

- Semi-empirical methods ~ N^2 (but can be made to scale as N^1 for larger molecules, say > 500 atoms)

- DFT methods ~N^3-N^4

- MP2 methods ~N^5-N^6

- Coupled cluster methods, ~N^6-N^8

One also has to consider how the 1st and 2nd derivatives of the energy with respect to the nuclear coordinates behave. The most recent implementations of density functional theory have low scaling 1st and 2nd analytical derivatives, which now routinely enable good geometry optimisations using BFGS-like algorithms.

Case Study 6: Directing effects in the Electrophilic substitution of Aromatic rings

There are two ways of approaching this problem. The simplest is to look at either the properties of the reactant, or the properties of the initial product, the Wheland intermediate. We will look at the first of these initially.

1. The Properties of the Reactant: A substituted Benzene.
Henry Armstrong, working at (the precursor to) Imperial College around 1887, was one of the first chemists to categorise substituents (R, or Radicles as he calls them) on a benzene ring in terms of the effect they had towards electrophilic substitution reactions (of X) of the ring. With the electron yet to be discovered, he attributed the two observed modes of behaviour (i.e. o/p in Table I vs m in Table II of 10.1039/CT8875100258) as being due to an electrically polarizable entity he called an "affinity", of which he suggested a benzene ring had six. He also suggested these affinities acted at a distance over the whole ring, but which differed in their directionality (or Resultant as he puts it, the modern term for which is Vector) according to the nature of R (a difference we nowadays categorize as being the electron donating or withdrawing properties of R). Armstrong's description of the properties of his affinity (See DOI: 10.1039/CT8875100258 where he writes "the introduction of a radical/substituent [onto the benzene ring] doubtless involves an altered distribution of the affinity, much as the distribution of the electric charge in a body is altered by bringing it near to another body". In 10.1039/PL8900600095, p102, he also clearly describes what we now know as a Wheland intermediate) may well constitute the first glimmering of the wave-like properties of the electron!

Erich Hückel some 44 years later was able to derive from Schrödinger's wave equation a simple expression which predicted how these "affinities", now named electrons of course, could be polarized. The resulting (π) molecular orbital functions (eigenvectors) can be used to illustrate graphically how both the directing effects (i.e. the probability of finding electrons in any particular position, i.e. the MOs) and the activating/deactivating effects (i.e. the probability of interacting with the electrons, i.e. the orbital energies) operate.

To maximise the visual effect, we are going to use two substituents R that were not in fact in Armstrong's original list; CH_2+ for an electron withdrawing group and CH_2- for an electron donating group. Its worth considering just for a moment why we are not using more conventional neutral groups (i.e. NO_2 and NH_2 respectively) for this illustration. Being neutral, the latter groups can only polarize the molecule by separating charge to produce a dipolar ionic species. Such ionic charge separation always takes a fair bit of energy to achieve compared to neutral covalent bonding, and actually requires proper solvation to treat quantitatively. It is much easier if the R group is already ionic, because then moving the charge from one part of the molecule to another takes little energy and no change in ionicity, and hence eliminates the need for any solvation treatment.

o/p Directing

m-Directing

HOMO, R=CH_2-

HOMO, R=CH_2+

HOMO-1, R=CH_2+

The Highest occupied π canonical Molecular orbital (HOMO) for R=CH_2- corresponds to the electron-pair least bound to the molecule and is hence the most available to react with an Electrophile. It has an energy of -1.7eV, which shows how relatively weakly it is bound (most π electrons have energies around -10ev or even less). This weak binding is also associated with it being a good nucleophile. Note how this wavefunction has density on the ortho and the para positions of the ring, but none at all on the meta position. This of course matches the resonance bond formulation familiar to all organic chemists, and hence the recognition of o/p direction and activation for such a substituent.

Contrast this with R=CH_2+. The HOMO in this case has an energy of -15.0eV, very much more tightly bound, with hence with vastly reduced nucleophilic properties. It has a node in the para position. The next highest orbital is almost degenerate with the HOMO (-15.6 eV) and has a node in the ortho positions. Because these two orbitals are essentially the same in energy, they have to be considered together; the only position where both of them do have density is in the meta position. This substituent therefore is meta directing, but strongly deactivating (due to the very negative orbital energies).

If we now assume that the reactant has similar properties to the transition state for this reaction (the Hammond principle, in part) we can infer that the transition state will inherit the same properties, and that the electrophilic reactivity of such substituted benzene derivatives can be derived purely from the properties of the reactant (if its an early transition state)

2. The Properties of a Heterocycle: Pyridine.

What about a real system? The differing behaviour of Pyridine and its N-Oxide towards electrophilic substitution is well known. In particular, these two, apparently almost identical species, have quite different properties when nitrated. The former nitrates (albeit slowly) in the 3-position, whilst the latter nitrates (rather faster) in the 4-position. Does the HOMO for each reflect this? The answer is that the corresponding orbitals look very similar to the ones shown above! Inded, the size of the orbital at the 4-position even looks larger than that at the 2-position.

HOMO, Pyridine, m-directing

HOMO, Pyridine N-Oxide, o/p directing

3. Properties of the Wheland/Armstrong Intermediate.

An prequel to looking at reactivity of aromatics is to consider the relative energy of the possible Wheland intermediates as being better indicators of the outcome of the reaction than simply the wavefunction of the reactant. How can one calculate the relative energies? The mechanics method, since in general the force field constants do not cancel out for the various pararmeters, is going to give a very unreliable relative energy. It will also not reflect on the polarization of the π-electrons as illustrated above. Only proper treatment of the electrons via quantum mechanics can treat this sort of problem. The energies shown below have been obtained from the PM6 semi-empirical method (which calculates the energies and optimizes the geometries of the protonated Wheland intermediate in just a few seconds). The outcome shows clearly the differentiation between Pyridine and its N-oxide, and also reproduces the preference for 4- rather than 2-substitution in the latter (But see DOI: 10.1039/P29750000277 where an argument is presented for 4-nitration resulting not by reaction of the free N-oxide but the protonated species).

Substrate, X

ortho (Relative to para)

meta (Relative to para)

para

Pyridine, X=N

-3.9 (234.6)

-15.1 (223.4)

0.0 (238.5)

Pyridine N-Oxide, X=NO

4.8 (208.9)

25.6 (231.2)

0.0 (205.6)

X=NS

3.3 (232.4)

23.5 (251.1)

0.0 (227.6)

Phosphabenzene X=P

0.1 (210.7)

0.0 (210.6)

0.0 (210.6)

Phosphabenzene P-oxide, X=PO

1.7 (145.4)

45.1 (188.8)

0.0 (143.7)

X=PS

2.2 (170.7)

45.1 (213.6)

0.0 (168.5)

Arsabenzene X=As

3.6 (229.6)

19.6 (245.6)

0.0 (226.0)

Iridiabenzene X=Ir(PH_3)3

3.4

39.4

0.0

What about other substituents? An almost entirely unstudied problem is the electrophilic substitution of phosphabenzene. It's a great surprise to find that in modelling terms at least, it behaves quite differently to pyridine! Using this approach, one can even predict reactivity for something as unconventional as metallabenzenes (for a review of the chemistry of metallabenzenes, s ee 10.1039/b517928a). For examples of such metallabenzenes with or

. For details of the bonding in the Ru system, see this blog post.

Case Study 2 Revisited: Where ARE the electrons in the Pirkle Reagent?

As a visualisation problem, viewing the Crystal structure of the Pirkle reagent provided a rationalisation of how the intermolecular interactions might proceed via a novel form of hydrogen bonding involving an entire face of one aromatic ring. We saw how Molecular mechanics is not parameterised to deal with such an unusual interaction, and so might not handle it properly. Clearly, the location of the electrons is critical, since the H+ binds to regions of high electron density. There are three aromatic rings to choose from in this molecule, and moreover, because of the chiral centre present, two faces to each ring, ie six possibilities in all. Can calculating the wavefunction of this molecule cast light on this?

Part 1: The π-electrons. One solution of the Schrödinger equation provides a canonical set of energy levels for electron (pairs) in the molecule, that having the highest energy being referred to as the HOMO (Highest occupied molecular orbital). As we saw before, the HOMO corresponds to the electron (pair) which is least strongly bound to the molecule, and hence can be regarded as describing where the most basic (= proton seeking) or nucleophilic (= nucleus seeking) electrons in the molecule are. So a good first start to understand where a hydrogen bond might occur is to plot the HOMO for the Pirkle reagent. Unfortunately, the HOMO is a π type orbital and shows almost no discrimination for binding to a proton!

Part 2: The integrated σ- and π-electrons. In fact the effect we are seeking is actually transmitted via the σ framework (originating in the inductive effect of the C-F bonds) and not the π system. The next level of approximation is to recognise that the basicity of a molecule derives from not just a single MO, but a suitably weighted sum of the contributions of all the electrons, and particularly of the σ bonds. This function is called the molecular electrostatic potential or MEP. This is a workfunction, and represents the amount of energy needed to remove a bare proton from any specified position close to a molecule out to an infinite distance away. If this energy is positive, the proton was clearly bound to the molecule, if the energy is negative, it was clearly repelled by the molecule. The function needs to be computed for all points in space around the molecule, and it is conventional to calculate an "iso-value surface", or actually two iso-surfaces, one representing a positive value of the MEP, the other a negative value. This is then rendered using computer graphics. In this display the negative potential which in effect attracts a proton (i.e. give rise to a positive work function) is rendered in purple.

Notice how the two faces of the anthracene ring are quite different, an effect induced by the electron withdrawing effect of the C-CF_3 group which withdraws electrons anti-periplanar to the bond, i.e. on the face opposite to it. But in turn the lone pair of the oxygen atom re-focuses density on to one ring of this opposite face, and in fact the hydrogen bond forms at almost exactly the centre of this electron density. Thus this hydrogen bond is created by a complex stereo-electronic effect caused by the shaping of the electron density by one withdrawing group and one lone pair.

Part 3: The electron Topology (AIM or Atoms in Molecules). What happens when one places two of these molecules into the close proximity found in the crystal structure? The π-MOs are of little help, and the electrostatic potentials of the two components, as used above, interfere to the point of not really revealing anything. Yet another way of analyzing where the electrons are is needed. This is found in a technique known as Quantum Chemical Topology. It is the study of the topology of electron density in molecules, and uses the language of dynamical systems (critical points, manifolds, gradient vector fields etc, first introduced by the mathematician Henri Poincare) to identify key points in the electron densities.

There are precisely four types of points, differing in the characteristics of the curvature of the electron density ρ(r) in the region of the critical point. This curvature is obtained from the second derivative of the electron density with respect to cartesian space (the density Hessian). The eigenvalues of this 3 by 3 matrix can have exactly four conditions, which are summarised by the signature of this matrix, being the sum of the signs of these eigenvalues:

1. NCP: A critical point (all three eigenvalues -ve) which identifies where an attractor (i.e. a nucleus) is in a molecule

2. BCP: A critical point (two -ve, one +ve) which identifies bonds in a molecule. Normally found between a pair of attractors, but can also involve three attractors (a 3-centre bond).

3. RCP: A critical point (one -ve, two +ve) which identifies rings in a molecule

4. CCP: A critical point (all three eigenvalues -+ve) which identifies cages in molecules

5. The total number of these various points must satisfy a topological relationship known as the Poincaré-Hopf condition, which states that num(NCP-BCP+RCP-CCP) = 1.

6. One property is known as the Laplacian of the electron density, ∇2ρ(r) (being the sum of the diagonal elements of the density Hessian) at a BCP is used to characterise the type of bond in that region

- Covalent, ∇2ρ(r) negative, ρ(r) large (> 0.2)

- Ionic, ρ(r) small

- Charge shift, ∇2ρ(r) positive, ρ(r) large (> 0.2)

The most relevant of the critical points is the second type; a BCP. It tells us in effect if a bond exists between any pair of nuclei. What it does not tell is is what the bond order of that bond is; it could easily for example be less than 1. That is a chemical rather than a topological interpretation.

What happens when such an analysis is conducted on the dimer of the Pirkle reagent? In the resulting analysis (left), the light blue points are the positions of some of the critical bond points identified in the electron density topology (there are many more than shown here). These reveal some fascinating insights into how the Pirkle reagent binds with itself.

The QTAIM Critical point analysis

1. There is indeed a bond critical point between the H of the OH group, and the π-face of one ring. It shows it binding to approximately three atoms of that ring, in exactly the manner first revealed by visualisation, and then by the MEP plot above. The electron density at this point (0.013 - 0.015 au) corresponds very approximately to an interaction energy of about 2.5 - 3.0 kcal/mol.

2. A set of four bond critical points occur between four pairs of atoms in the π-π- stacked anthracene rings, with densities of around 0.004 - 0.005 au. This reveals that such stacking is indeed (weakly) attractive and of significance (totalling around 3 kcal/mol).

3. A further pair of bond points is shown between the oxygen lone pair and the adjacent ring C-H bond; the density (0.018) indicates each to be worth about 3 kcal/mol.

4. Thus the OH group actually participates in two concurrent types of hydrogen bonding, one to a π-face via the H, and one to a C-H bond via the lone pair. This last interaction was not spotted in this molecule until this topological analysis was done! It also re-inforces the concept that these interactions are co-operative rather than independent.

The NCI Method: Surface based on (reduced) density gradient ∇ρ(r); surface colour based on ∇2ρ(r)

- red =strongly repulsive

- yellow = weakly repulsive

- green = weakly attractive

- blue= strongly attractive.

- For more explanation, see this summary.

The very recent NCI (non covalent interactions) method takes this analysis one step further by developing the concept of surfaces of interaction rather than just critical points. These surfaces are derived from the density gradient ∇ρ(r) (the deviation from a homogenous electron distribution), coloured by the value of the λ_2 eigenvalue of the Laplacian of the density ∇2ρ(r). The NCI method is showing great promise for explaing e.g. stereoselectivity in chemical reactions.

Part 4: Energies again. Our analysis of the Pirkle reagent concludes here with a revisitation of the binding energy. We saw previously how this could be estimated using Molecular Mechanics. This had the advantage of including the so called dispersion, or long range correlation effects, but the disadvantage that there was no ready access to the entropy of the process. A very new procedure which incorporates both advantages is the dispersion-corrected density functional method, one very recent implementation of which is ωB97XD (DOI: 10.1039/b810189b), and this gives the following values:

ΔH dimerisation -25.1; ΔS -49.2, TΔS = -14.7; ΔG dimerisation -10.4 kcal/mol.

Case Study 1 Revisited: How do orbitals interact within a molecule?

We (think we) know where covalent σ-bonds are in molecules, they lie along the straight line connecting the nuclei of two atoms. We can idealize lone pairs on atoms such as oxygen, and in case study 1, all we needed was an (X-ray) structure to spot a connection between the (antiperiplanar) orientation of (certain) such idealized lone pairs and the C-O bonds in the molecule. But are the lone pairs really where we have idealized them? And in dealing the orientation of the lone pair with a σ-bond, we become aware that we are actually dealing with a concept derived from the Klopman-Salem equation, which uses not so much the two-electron σ-bond, as a vacant orbital occupying the (assumed) same orientation known as the σ*-bond. This was the basis of the earlier lecture course on stereo-electronics and also conformational analysis. It is time to take some of these concepts and show how they can be properly quantified. Let us proceed as follows:

1. We will solve the Schrödinger equation for a molecule (to any required degree of accuracy). This gives us functions (the canonical molecular orbitals) which tell us the electron density probability over the molecule as a whole.

2. We now need to convert this (global) function to some sort of local property, since the concept of a bond is local, and not descriptive of the molecule as a whole. One such procedure is known as the NBO (Natural bond orbitals). A more formal description of this type of orbital is: Natural Bond Orbitals (NBOs) are localized few-center orbitals ("few" meaning typically 1 or 2, but occasionally more) that describe the Lewis-like molecular bonding pattern of electron pairs in optimally compact form. This can be precisely expressed in mathematical form (but we shall not do so here!); it is sufficient to say that the electron density probability predicted by summing all the occupied canonical molecular orbitals and alternatively all the local NBOs comes out the same! (i.e. there is more than one way of constructing a wavefunction for a molecule that predicts its overall electron density distribution, which in fact is the only properly measurable property).

3. Just as with canonical MOs, where we had a (doubly) occupied set and a virtual (unoccupied) set, so the NBOs turn out to geometrically occupy the bonds (BD), the lower energy non-valence cores (CR), and the valence lone pairs (LP) of the molecule, with the unoccupied orbitals being represented by BD* and RY* (don't worry about this last one, it stands for Rydberg).

4. In the NBO formalism, one can calculate an interaction energy between any doubly occupied BD or LP NBO, and any unoccupied BD* NBO. This can be represented in very simple terms by the diagram:

5. This is exactly the form of the Klopman-Salem diagram you may well recollect from 2nd year conformational analysis lectures. But we have now made progress. By using the NBO expressions, we can obtain an exact value for the value of the quantity E(2).

6. When this is done for the two dioxepins discussed in case study 1, the following results (at the B3LYP/6-31G(d) level) are obtained (with the keyword pop(nbo)) for the stable (green) and unstable (pink) forms;

7. The only two large E(2) terms (both ~15.5 kcal/mol) are between an oxygen LP and a C-O BD* involving the adjacent oxygen. They correspond exactly to the two antiperiplanar interactions we first noticed by visualisation, but now we can precisely quantify these interactions.

8. For the unstable isomer, only one large E(2) term is found (13.4); the other (7.5) is now significantly reduced because the orientation is no longer exactly antiperiplanar. The molecule is less stabilised and the two C-O bonds are no longer equal in length.

NBO analysis is very useful in a variety of situations, and is increasingly found in the literature. For example, it can be used to rationalise the gauche preference of 1,2-difluoroethane, and many others. Such E(2) NBO terms are thought to indicate useful chemistry down to energies of ~3 kcal/mol, and ones as high as 130 kcal/mol have been found (~15 is however typical of the anomeric effect). See this blog for another example where the unexpected was highlighted by such techniques.

Case Study 4 Revisited: The Helical structure of DNA

I left this case study dangling, by posing the question: how was it determined that the helix is right handed? Watson and Crick actually answered this by stating (very briefly with no details) that a left handed helix could only be constructed by violating the permissible van der Waals contacts. The vdW contacts are actually very well modelled using molecular mechanics (the 4th term in the force field described here) and so a reasonable model could have been constructed using this technique to test this assertion. It suffers one slight deficiency. Because it uses force constants to define the terms, these are assumed to be the same everywhere throughout the molecule. Thus the rungs of the DNA ladder in the middle are treated the same as at the end of the helix. Is this justified? It is only very recently that vdW effects have been incorporated into quantum mechanical models in a manner which can be applied to large molecules (ωB97XD, see above). So we will kill two birds with one stone by jumping straight to this technique. We will also overcome another defect of mechanics, and that is the general lack of implementation for incorporating entropic effects on the energies. We will pose the new question: what are the relative free energies of and handed DNA helices (which, by the way are diastereomers, not enantiomers!). It turns out the answer depends on the nature of the base pairs.

1. If they are exclusively C-G, as in d(CGCG)2, ΔG_298 favours the left handed Z-helix by ~12 kcal/mol.

2. If they are A-T, as in d(ATAT)2, the right handed B form is favoured by ~4.3 kcal/mol. As it happens, this is exactly what is known for small (tetramers) of DNA!

3. Dispersion terms alone (van der Waals) make Z-d(CGCG)2 5.1 kcal/mol LESS stable than B-d(CGCG)2,

4. Dispersion terms alone (van der Waals) make Z-d(ATAT)2 12.5 kcal/mol LESS stable than B-d(ATAT)2.

System

ΔΔG

ωB97XD/6-31G(d)/water.

0.0

1. Furanose oxygen within 2.85Å of a guanine ring

2. Furanose OC-H hydrogen within 2.48Å of a second furanose oxygen

3. Facilitated by C-Hσ/C-Oσ* anti-periplanar acidification (E2 5.8, magenta bonds)

4. Anomeric interaction between iguanine and the ribose; NLp/C-Oσ*; E2 11.6 (violet bond).

5. Cytosine-furanose anomeric (E2 6.8, indigo bond).

6. Gauche-like conformation of the ethane-1,2-diol fragment (gold bond).

0.0

10.0

We conclude that Watson and Crick's model building, based on van der Waals analysis, did indeed favour the B-DNA helical form, but that there are other aspects that can overcome this for CG-rich DNA strands. See for example this blog for more details.

Bsck to scales|| Back to Visualisation| Back to Mechanics|On to Transition states| On to Advanced|

(c) H. S. Rzepa 1998-2012. No reproduction rights granted to this material without permission.

Erich Hückel

1896 - 1980

preliminary - sorry, untranslated -

and will still be modified in some points.

Erich Hückel (in Engl. often spelled Huckel or Hueckel) was born in 1896 in Berlin and died in Marburg (Germany) in 1980 where he had been professor of physics for many years. Hückel had studied physics and developed with Debye a theory of strong electrolytes which became known as the Debye-Huckel theory. He then worked for a while on capillary effects and wrote a book about it. Later on he introduced a theory of the C=C bond based on quantum mechanics which was to become the famous Huckel theory. He left this field of interest after a short while and published thereafter only occasionally on various other topics of physics.

Erich Hückel - one view of many

by U. Anders, 1992

In den Jahren 1931 - 1932 erschien eine Artikelserie von Hückel, die die Grundlage für die später nach ihm benannte Theorie bildete (4a-4d). Diese Theorie war, wie er in einem 7 Jahre später von ihm selber geschriebenen Übersichtsartikel feststellte "... von den Zweifeln zahlreicher Fachgenossen begleitet ...". Und gleichzeitig stellte er bei dieser Gelegenheit etwas betrübt fest

"Jene eben genannten Arbeiten {d.h die Hückels zur Hückel-Theorie A.d.A.} haben bisher bei den Chemikern - wenigstens in Deutschland - wenig Beachtung gefunden ... "(5a; 752)

Die Anstrengungen Hückels waren dabei, abgesehen von der kompetenten Hilfe durch seinen Bruder Walter, ebenso gewaltig wie riskant gewesen. Denn seit Schrödingers grundlegenden Arbeiten waren nur ein paar Jahre vergangen und nur einige wenige Theoretiker wagten sich überhaupt an größere Atome und Moleküle. Die mathematischen Schwierigkeiten der uneingeschränkten Schrödinger Gleichung jenseits von Wasserstoff und Helium waren unübersehbar.

Die ersten atomaren und molekularen Anwendungen der neuen Quantenmechanik auf dem Gebiet der Chemie erfolgten in dieser Zeit meistens durch Forscher, die den Weg über die Spektroskopie gegangen waren, wie zum Beispiel Herzberg, Hund, Mulliken, Slater. Bei Mulliken ist der langsame, tastende Weg von der Spektroskopie über kleine zweiatomige Moleküle wie N2, CO etc. zu organischen Molekülen hin über Jahrzehnte besonders gut zu verfolgen. Hückel wagte den Übersprung: er ging, sagen wir einmal furchtlos, direkt auf die interessanteren ´normalen´ Verbindungen der organischen Chemie los, insbesondere auf das Benzol. Das war im Jahre 1931, und wenn man sich unter der damaligen Literatur umsieht, ein absolut revolutionärer Versuch, ein unerhörtes Novum. Er war damit in seiner Zeit den quantentheoretischen Ansätzen der theoretischen organischen Chemie weit voraus und wurde dementsprechend mißachtet, wie eingangs dargestellt.

Schauen wir uns diesen Hückel einmal genauer an - seine Autobiografie (67) läßt einige Aussagen zu.

1921 hatte Hückel in Göttingen, wo er auch zur Schule gegangen war und studiert hatte, seine Doktorarbeit Zerstreuung von Röntgenstrahlen (68) unter Debye fertiggestellt. Bedingt durch die Wirren nach dem ersten Weltkrieg wurden die Semester zu Trimestern umgebaut. Hückel sagt dazu:

"In einem Trimester sollte soviel erarbeitet werden wie früher in einem Semester. Daß die Rechnung nicht aufgehen konnte ist eigentlich trivial. Man kann ein Schwein auch nicht in der halben Zeit mästen, indem man ihm doppelt soviel zu fressen gibt...."(67; 77)

Die Folge davon waren Lücken in seinem Wissen, die Hückel klar sah:

"So blieben speziell meine mathematischen Kenntnisse mager und ich mußte mir sie erst als Assistent und später als Professor selber aus Büchern aneignen, und zwar - immer lückenhaft - das was ich gerade brauchte... "(67; 77)

Einge der Lücken konnte er füllen: Er war 1921 Hilfsassistent bei Hilbert, 1922 Assistent bei Born, mit dem er 1923 und noch einmal später, 1951, veröffentlichte; in der Veröffentlichung von 1923 von Hückel und Born ging es um die Quantentheorie mehratomiger Moleküle.

Anschließend wurde Hückel bei Debye, der 1921 einen Ruf an die ETH erhalten hatte, Vorlesungsassistent, nachdem das Arbeitsverbot für Ausländer in der Schweiz aufgehoben worden war. Die bekannte Debye-Hückel Theorie wurde in dieser Zeit teilweise entwickelt - sie erschien 1923 (75). Aber in charakteristischer Weise schreibt Hückel

"Die Debyesche Theorie fand zunächst wenig Anklang und Verständnis bei den Chemikern ... " (67; 92)

Haben wir das eingangs nicht schon einmal gehört, als es um die Hückel Theorie ging? Diese zwei Aussagen, das kann kein Zufall sein. Es spiegelt Einstellungsgrundlagen der Persönlichkeit Hückels wieder, die man nicht übersehen sollte. Dies waren keine literarischen, zu unterschiedlichen Zeiten an unterschiedlichen Stellen abgegebene Ausrutscher mehr, das waren grundlegende Einstellungen.

1925 habilitierte sich Hückel (76) zum ersten Mal. Es war wirklich keine schlechte Arbeit - die Schlußfassung der Debye-Hückel Theorie ist auch heute noch Standard-Lehrinhalt physikalisch-chemischer und elektrochemischer Bücher. Danach kam die Heirat:

" Mein Schwiegervater {Zsigmondy, A.d.A.} hatte meine Habilitation als Vorbedingung für den Konsens zur Heirat mit meiner Braut gestellt. Er wurde erteilt und eine neue Lebensart stand mir bevor." (67; 102)

1928 erhielt Debye einen Ruf nach Leipzig - aber ´den Hückel´ wollte er als Assistent nicht mitnehmen, wie dieser selber zugab:

"Ich wußte selbst sehr genau, daß das für mich niemals möglich war. Meine theoretischen Kenntnisse und Fähigkeiten waren wahrhaftig viel zu beschränkt und dürftig, als daß ich daran auch nur gedacht hätte. (67; 122)

Im Frühjahr 1928 ging Hückel mit einem auf ein Jahr befristeten Rockefeller-Stipendium nach London. Er beschäftigte sich anfangs theoretisch mit der Adsorption von Gasen. Das gewählte Arbeitsgebiet erwies sich ihm bei genauerem Hinsehen als unergiebig für den Beginn einer akademischen Laufbahn:

"Ich sah ein, daß ich auf ein falsches Pferd gesetzt hatte, und mußte mich also in meinen Plänen ganz umstellen." (67; 125)

Was tun war nun die große Frage.

Hier in London studierte E. Hückel daher in der Folge die Arbeiten von E. Schrödinger, den er mit seiner Frau übrigens gut aus der Züricher Zeit persönlich kannte. Hückel studierte jetzt auch die Bücher: Courant-Hilberts Methoden der mathematischen Physik, Weyls Gruppentheorie und Diracs Principles of Quantum Mechanics, später Heisenbergs Die physikalischen Prinzipien der Quantentheorie. Und er hielt Vorträge darüber:

" ... hielt ich in London auch eine Reihe von Vorträgen hierüber. Das ist die beste Methode, ein Gebiet kennenzulernen, daß einem neu ist. (67; 126)

Debye, den er zufällig in London traf, " ... fragte ich um Rat, was ich weiterhin tun könnte" (67; 126):

"Er riet mir, die Quantentheorie auf chemische Probleme anzuwenden. Ich war also auf dem richtigen Wege" (67; 126)

Aus familiären Gründen unterbrach er seinen Aufenthalt in England und setzte ihn, gemäß seinen neuen wissenschaftlichen Vorstellungen, im Frühjahr 1929 in Kopenhagen bei Niels Bohr fort. Dort traf er auf Oskar Klein, der ihm mathematisch weiter half, auf Casimir, der ihn in die Permutationsgruppen einweihte, auf Gamow, auf Ehrenfest.

Und er suchte und suchte - nach einer Idee, nach einem eigenen Arbeitsgebiet.. Wahrscheinlich fand er nichts rechtes. Anstöße mußten für Hückel offenbar von Außen kommen. Denn er sagt:

"Bohr machte mich darauf aufmerksam, daß es vor kurzem Heitler und London mit Hilfe der Schrödingerschen Wellenmechanik gelungen war, für die chemische Bindung zwischen zwei Wasserstoffatomen im Wasserstoffmolekül eine Erklärung zu finden und die Bindungsenergie angenähert zu berechnen...(67; 131)

"Bohr meinte nun, man könne versuchen, auch die sogenannte Doppelbindung, wie sie insbesondere in vielen Fällen zwischen zwei benachbarten Kohlenstoffatomen in der Organischen Chemie angenommen wird, - oder,wenn man lieber will, vorkommt - zu erklären...(67; 131)

"Dies wurde dann tatsachlich das Problem, das ich anzugreifen versuchte und das mich durch Erweiterung auf daran anschließende Probleme und Fragen mehrere Jahre beschäftigte. So entstand schließlich daraus mein wissenschaftliches Hauptwerk. Das konnte man aber damals natürlich noch nicht voraussehen. (67; 131)

Anschließend besorgten ihm Debye und v. Laue (Motto: "für Hückel muß etwas getan werden") für 1930 ein Notgemeinschaftsstipendium (67; 133).

Schließlich trug Hückels geistiges Ringen erste Früchte. 1930 meldete er sich aus dem Institut für Theoretische Physik an der Universität Leipzig mit dem Artikel Zur Quantenchemie der Doppelbindung zu Wort (26). Diese Arbeit war, wie er am Ende des Artikels ehrlich schreibt, auf Anregung von N. Bohr in Kopenhagen begonnen worden. Wertvolle quantentheoretische Diskussionen, so fährt er weiter fort, habe er dann in Leipzig Eduard Teller zu verdanken, jenem Wissenschaftler nämlich, der Anfang der fünfziger Jahre unter dem Namen Edward Teller als ´Vater der Wasserstoffbombe´ bekannt wurde. (Literatur hier etwa (34), (35)).

Die Art und der Stil des ersten Hückelschen Artikels über die Doppelbindung sollte für seine Schreib- und Denkweise kennzeichnend bleiben: sie bestand in dem Versuch, die seit Van´t Hoff (...), Lewis (12), Kossel (13a-13c) übliche Schreibweise der chemischen Bindung unter Verwendung von Valenzstrichformeln, insbesondere bei Vorliegen der Doppelbindung, mit den neuen Ideen der eben erst enstandenen Quantenchemie zu integrieren, unter gleichzeitiger Berücksichtigung der Erklärungsmöglichkeiten bei chemischen Reaktionen wie Substitution, Addition etc. Und genau diese letzgenannte Einbindung der Reaktionsmechanismen brachte ihn, wie noch gezeigt werden wird, später in Schwierigkeiten mit den organischen Chemikern. Dabei störte Hückel die pragmatische Valenzschreibweise mit Strichen, doch mußte er konstatieren, daß

"... auch wohl die meisten organischen Chemiker trotz verschiedentlicher, auch gerade in letzter Zeit erhobener Einwände {an dieser Auffassung A.d.A.} festhalten ... "(26; 423)

Was er damit eigentlich genau sagen wollte, mag der Spekulation des Lesers überlassen bleiben.

Ein weiteres Problem, das die physikalischen Chemiker in dieser Zeitspanne beschäftigte war die die Frage, ob die Doppelbindung gegenüber einer Verdrehung starr sei oder nicht und wie dies begründbar wäre.

Für die Starrheit gab es die ersten optischen experimentellen Aussagen. Sein Doktorvater Debye hatte zu dieser Zeit außerdem gerade mittels der ganz neuen Möglichkeiten der Röntgenuntersuchungen experimentell nachgewiesen, daß eine Rotation um die - Achse nicht stattfindet (28).

Hückel klinkte sich also in ein aktuelles Thema ein, als er sich sich mit der "... von den Chemikern sogenannten ´Starrheit´ der Doppelbindung ..." befaßte. Dabei ging er folgendermaßen vor: Zunächst katalogisierte er die bis dahin bekannten chemischen und physikalischen Eigenschaften dieser Stoffklassen nach den Methoden der damaligen theoretischen organischen Chemie. Schützenhilfe dazu erhielt er von seinem Bruder Walter Hückel, der zu dieser Zeit am Institut für Organische Chemie an der Universität Freiburg im Breisgau tätig war und der auch schon ein Buch zu seinem Fachgebiet veröffentlicht hatte (27). Der uns hier mehr interessierende Erich Hückel benötigte nämlich diese Indizien für seine Theorie.

Dieser ´I. Teil´ der Veröffentlichung, 14 Seiten lang, liest sich wie eine Abhandlung aus der physikalischen oder theoretischen organischen Chemie. Man fragt sich beim Lesen sofort, wer hier eigentlich der Adressat sein sollte: sprach er die Physiker an oder die Chemiker? Für die Physiker war dieser Anwendungsbereich erst einmal sicher äußerst uninteressant, und der praktische organische Chemiker las sicher nicht diese Art von Zeitschrift, denn der folgende ´II. Teil´ war dazu für ihn weder in seiner Notwendigkeit noch in seinem Inhalt einsehbar. Eine Tatsache, die seine Kritiker natürlich nicht übersehen sollten, wir werden gleich darauf zurückkommen.

Im anschließenden ´II. Teil´ analogisiert E.Hückel zuerst die -Bindung in Ethen mit der des Sauerstoffs, aufbauend auf Arbeiten, die der Brite Lennard-Jones 1929 in einem diesbezüglichen Artikel veröffentlicht hatte (19). Vergleicht man, nebenbei bemerkt, die Nomenklatur der beiden Artikel, so sticht sofort ins Auge, daß es zu diesem Zeitpunkt noch eine Bezeichnungsproblematik der Valenzorbitale zwischen den verschiedenen Schulen gab (Mulliken/Hund versus hier Lennard-Jones). Zusätzlich dazu mögen dem damaligen Chemiker, falls er die Zeitschrift für Physik je in die Hand nahm, Hückelsche Sätze wie

" ... s und p bedeuten in üblicher Weise das Impulsnormal 0, bzw. 1 (in Einheiten h/2p ), welches dem Zustand nach Aufhebung der Wechselwirkung zwischen den Elektronen um die Kernverbindungslinie zukommt ... "(26; 438)

mit einiger Sicherheit etwas mystisch angemutet haben. Schon hier ein Beispiel für ein Theoretisieren ohne praktisch anwendbare Modellvorstellungen, ganz im Gegensatz z.B. zu dem vorher erwähnten Artikel von Lewis aus dem Jahre 1916, in dem dieser in seiner amerikanisch-zugreifenden Art und Weise auch für die Praxis anwendbare Modellvorstellungen entwickelt hatte. Übrigens sollte diese Art des amerikanischen Pragmatismus auch stets die Arbeiten von Mulliken bestimmen, über die in einem späteren Kapitel noch zu reden sein wird.

Im ´Teil II, B´ kommt E.Hückel dann schließlich zu seinem eigentlichen Thema, wobei er zuerst einmal als Modellsubstanz vorstellt. Die Idee dahinter war, von über zu zu kommen. Der Absatz beginnt folgendermaßen:

"Um nun vom Sauerstoff zum Äthylen zu kommen, denken wir uns zunächst von einem der Sauerstoffkerne zwei positive Elementarladungen (H-Kerne) aus der Verbindungslinie herausgeführt, sodaß wir zunächst das Formaldehyd erhalten ... "(26; 443)

Es mag dahingestellt sein, ob es sich hier um einen Druckfehler oder einen Gedankensprung handelt, aber diese Stelle ist kennzeichnend für viele ähnliche. Es ist eine schwerfällige, umständliche, fast beamtische Sprache und Argumentationsweise, deren sich E. Hückel bedient.

Hückel interessiert sich dabei, im wesentlichen Lennard-Jones folgend, fast wie ein Spektroskopiker für die Zustände der s und p Elektronen. Er geht von den s- und p - Funktionen aus, deren Symmetrie-Verhalten er eingehend untersucht: symmetrisch/antisymmetrisch, gerade/ungerade sind die Begriffe, mit denen er argumentiert. Einige der Integrale werden zwar definiert aber nicht berechnet. Über den Hamilton-Operator selbst schweigt er. Er spekuliert geschickt über die relativen Größen der einzelnen Energieterme unter Ausnutzung bereits bekannten Tatsachen der Molekülspektroskopie (etwa der Arbeiten von Hund, Mulliken und anderer) und der eingangs angeführten Erkenntnisse der theoretischen organischen Chemie. Neu war das Zusammenfügen der Indizien symmetrisch und antisymmetrisch, gerade und ungerade für den von ihm betrachteten Fall der Doppelbindung.

Für das organische Molekül Ethen konnte er zeigen:

1.) s - und p -Elektronen können unabhängig voneinander betrachtet werden.

2.) der p -Term spaltet sich in 2 Zustände p u und pg.

3.) obwohl nicht berechenbar, folgert Hückel aus den chemischen Indizien, daß Eu < Eg sein muß.

4.) Beim Verdrehen der Doppelbindung nimmt die Energie des symmetrischen p -Zustandes ab und die des antisymmetrischen zu. Ein Verdrehung um die - Achse würde also einer Vergrößerung der Energie bedeuten. Da dies nicht geschehen kann, verhält sich die C==C- Bindung starr.

Mit Hückels eigenen Worten:

"Dabei wird, wie man leicht einsieht, einer Verdrehung von 90° ein Energieaufwand von der Größenordnung Eg - Eu entsprechen. Der Zustand p u2liefert in der Tat also ´die Stabilität der Doppelbindung´ {Hervorhebung durch Hückel} gegen Verdrehung ... "(26; 453)

Kaum ein Jahr nach dem Artikel Zur Quantenchemie der Doppelbindung legt E. Hückel seine Hauptschrift vor: Quantentheoretische Beiträge zum Benzolproblem, bestehend aus drei Teilen von insgesamt 132 Seiten; zum Vergleich: Schrödingers Artikelserie, die die Quantenchemie eingeläutet hatte, war 140 Seiten lang gewesen (1a-1d).

Hückels Arbeit kommt jetzt aus dem Institut für Theoretische Physik der Technischen Hochschule Stuttgart. In seinen 1975 erschienenen Erinnerungen ´Ein Gelehrtenleben - Ernst und Satire´ (67) stellt Hückel seine neue Anstellung als Dozent für Chemische Physik sehr plastisch dar. Seine Frau sollte diese Jahre später als "die sieben Jahre unserer Schmach" bezeichnen (67; 137). Debye hatte Hückel diese Stelle verschafft. Es war keine planmäßige Stelle, vielmehr eine, die aus verschiedenen Stiftungen bezahlt wurde, die jedes Semester wieder erneuert werden mußte. Außerdem enthielt sein Arbeitsvertrag die Klausel, daß sich mindestens 3 Hörer einschreiben mußten - was gelegentlich nur mit der Hilfe der Kinder von Kollegen erreicht werden konnte. Hückel meinte später daß der Dr. rer. nat. e.h., den er Ende der 60-er Jahre von der TH Stuttgart erhielt, "wohl eine Kompensation für die ´sieben Jahre der Schmach´ darstellten."(67; 167). Die Stuttgarter ließen ihn damals aber erst einmal eine zweite Habilitation erstellen. Die oben erwähnte Veröffentlichung (4a) war gleichzeitig die geforderte zweite Habilitationsschrift.

Diesmal bedankt sich Hückel im Anhang der ersten Veröffentlichung (4a), d.h. der Habilitationsschrift, bei F. Hund und W. Heisenberg für "... wertvolle Diskussionen". Wie vorher assistiert auch hier, ebenfalls namentlich genannt, Bruder Walter mit seinem Wissen aus der theoretischen organischen Chemie.

.angenommen wird, - oder, wenn man lieber will, vorkommt - zu erklären." (67; 131)(?????)

Ohne zu psychologisieren, muß man sich einmal die gewaltigen persönlichen Anstrengungen vorstellen, die Hückel unternommen hatte: Er geht zurück nach Leipzig, schreibt den Artikel, tritt 1930 die neue Stelle in Stuttgart an, zieht um und schreibt in der ganzen Zeit an seinem Artikel. Sicherlich hat er daneben an der Technischen Hochschule Stuttgart auch noch akademische Aufgaben zu erfüllen. Ferner besucht er Hund und Heisenberg, die er schließlich erst einmal kennenlernen mußte. Ohne allgemeine gesellschaftlichen Kontakte war dies wahrscheinlich nicht möglich, auch wenn der Weg dorthin über Kopenhagen sicherlich etwas geebnet war. Hückels Karriereprogramm schien wahrlich scharf eingestellt, sollte man meinen. Dem war aber nicht ganz so, wie wir gesehen haben, es war wohl mehr seine schiefe Lage in Stuttgart, aus der er heraus wollte. Und seine Frau war schließlich die Tochter des deutschen Nobelpreisträgers von 1926, Zsigmondy (Kolloide), wenngleich auch sein Schwiegervater trotz seiner hohen Auszeichnung offenbar nichts für Hückel im Bereich Karriere tun konnte.

Wie immer nimmt Hückel zu Beginn des Artikels Quantentheoretische Beiträge zum Benzolproblem einen breiten, neun-seitigen Anlauf durch die theoretische organische Chemie, um dann zu seinem Thema zu kommen. Dies beginnt er mit der Formel der Schrödingergleichung, so, wie sie für das Benzol aussehen würde. Anschließend verfolgt er zwei Wege. Als ´erste Methode´ bezeichnet er den Weg über die Slater-Determinante, " ... welche unter Vermeidung der Gruppentheorie die Behandlung der Wechselwirkung zwischen einer großen Anzahl von Atomen ermöglicht ... "(4a; 219). Dies war eine Absage an den langsamen Weg, an das tastende Sich-bewegen über 2,3,4 ...n - atomige Moleküle, wie ihn viele anderen Forscher unternahmen. Darin sah Hückel offenbar nicht zu Unrecht einen mühsamen Weg.

Auch seine ´zweite Methode´ war von diesem Zuschnitt (*). Er wandte ein Verfahren an, das sich von vornherein an einer großen Anzahl von Atomen und Elektronen orientierte: F. Bloch hatte schon 1928 ein solches für Elektronen im Kristallgitter ausgearbeitet (36). Übrigens hatte F. Bloch auch über eine, wenngleich andere Anwendung, der Slater-Determinante 1930 berichtet. Beide Artikel waren in der Zeitschrift für Physik erschienen (37), einer Zeitschrift, die damals ganz offensichtlich zu den ersten zählte und die das Ohr am Puls der Zeit hatte. Auch Hückels Artikel sollten ja darin erscheinen.

Was dann in Hückels Veröffentlichung folgt, ist ein Lehrstück im Vermeiden von Schwierigkeiten. Hückel hatte es eilig, ohne Zweifel. Wie im vorigen Absatz schon angedeutet, übernahm er Methoden, die seiner Fragestellung - Erklärungsmechanismen der Reaktionen am Benzol - am nächsten kamen. Jetzt weicht er den neuralgischen Punkten der Berechnung der Integrale unter Anwendung von Symmetrie- und Termbetrachtungen äußerst geschickt aus. Wie sagt doch F.A. Cotton in seinem Buch ´Chemical Applications of Group Theory´ 30 Jahre später:

"Symmetry can tell us that, ´in principle´ {Hervorhebung durch Cotton A.d.A.}, two states must differ in their energy, but only by computation or measurement can we determine how great the difference will be ... "(38; 4)

Der Hamilton-Operator wird nicht genau definiert und es sollte Jahrzehnte dauern, um herauszufinden, wie dieser durch Hückel mehrfach vereinfachte und mehrfach genäherte Operator nun wirklich zu verstehen sei. Es wurde eine gute Übung für Autoren von Lehrbüchern und deren Studenten, herauszuarbeiten, wie er nun wirklich auszusehen habe. Die entscheidende Analyse war die von Xxxx im Jahr 19xx (xx).(*)

Interessant schon an dieser Stelle zu vermerken, daß die Ergebnisse Hückels auch auf graphentheoretischem Hintergrund erhalten werden können. Eine gute Darstellung diesbezüglich ist z.B. die von Trinajstic (47).

Doch wie immer die Ableitungen einzuschätzen sind: die Methode Hückels bot damit, nur 6 Jahre nach dem Beginn der Quantenmechanik, bereits die Möglichkeit, Reaktionsmechanismen über Elektronendichten an aromatischen und ungesättigten Verbindungen gedanklich zu verfolgen!

Im Jahre 1957 meldete sich E. Hückel nochmals mit einer weiteren Zusammenfassung zum anstehenden Thema (45). Gezwungenermaßen - der Großmeister der deutschen Quantenchemie hatte ihn dazu aufgefordert. Doch Hückel hatte so seine Schwierigkeit mit der Literatur im allgemeinen und der Hückel-Theorie im besonderen:

"Ich brauchte etwa ein halbes Jahr zur Vorbereitung hierfür, um mich zu informieren, wie sich das Gebiet entwickelt hatte. (67; 167)

Diese Veröffentlichung ist leichter lesbar als die von 1937, berücksichtigt auch andere semiempirischen Techniken, und - was besonders interessant ist - bringt keine Anwendungen mehr aus dem Bereich der physikalischen organischen Chemie. Den Kernteil beginnt er:

"Die - mehr quantitativen - Überlegungen, die ich seinerzeit anstellte, um zu einer einigermaßen befriedigenden theoretischen Behandlung zu kommen, waren die folgenden ... "(45; 869)

Diese haben wir weiter oben erwähnt. Ferner aber rückt er dann im Folgenden unübersehbar von der Paulingschen VB-Methode ab:

"Neuerdings hat McWeeny es unternommen, die VB- Methode auf eine begründetere Basis zu stellen, um die Unzulänglichkeiten, die dieser ... anhaften, zu beheben ... "(45; 875)

Insbesondere verurteilt er in diesem Zusammenhang scharf die Paulingsche Verwendung des Begriffes Resonanz (anstelle von Mesomerie), da dieser Begriff nach seiner Meinung zu Verwechslungen beitragen konnte.

Aber auch zu seiner eigenen Methode hat er Kritik bereit :

"Den beiden {d.h. der VB- und der MO-Methode A.d.A} hier bisher besprochenen Näherungsverfahren, die ich bereits in meinen Arbeiten um 1930 anwandte, haften grundsätzliche Mängel an, auf die ich zum Teil schon hingewiesen habe ... "(45; 883)

Dabei bezieht er sich dann auf die Unmöglichkeit, innerhalb dieser Theorie angeregte Molekülzustände zu berücksichtigen, d.h. Spektren quantitativ zu analysieren, Ionisierungsenergien oder Doppelbindungsenergien zu berechnen. Roothaan, McWeeny, Parr - in den Literaturzitaten des Meisters selber zeichnet sich die Änderungen ab. Damit soll nicht gesagt werden, daß die Hückeltheorie damit abgelegt war - selbst beim ´Fußballmolekül´, dem Buckminsterfulleren leistete die Hückelmethode Ende der 80-ger Jahre noch gute Dienste (..).

Und damit ist die wissenschaftliche Geschichte der Hückeltheorie eigentlich schon erzählt. Die eben zitierte Veröffentlichung aus dem Jahre 1957 war, nach der von 1937, die letzte zum Thema Hückeltheorie. 1975, fünf Jahre vor seinem Tode, schrieb Erich Hückel:

"Meine frühere Arbeitskraft aus den dreißiger Jahren erlangte ich nicht wieder. 1946 wurde ich fünfzig, und mein Gedächtnis hatte durch die lange Hungerzeit gelitten." (67; 166)

Und er begründet gleich noch weiter:

"Es hatte auch keinen Zweck, zu versuchen, in dem Wissenschaftsgebiet, das sich aus meinen Arbeiten über das Benzol und meine damit verbundenen Untersuchungen entwickelt hatte, weiter zu arbeiten. Es war nicht möglich gewesen {ob er das jetzt gesundheitlich bedingt oder kriegsbedingt meint, bleibt offen A.d.A.}, die Erweiterungen, die dieses Gebiet insbesondere in den USA und in England erfahren hatte, zu verfolgen oder nachträglich zu studieren." (67; 166)

Jeder Diplomand, Doktorand, angehender Professor hätte sich solche Aussagen nicht zu schreiben gewagt, vielleicht nicht einmal zu denken gewagt, um seine Karriere nicht zu unterminieren. Natürlich stellt sich, besonders im Rahmen dieses Buches da natürlich die Frage nach dem warum, weshalb. Schauen wir uns daher diesen eigenwilligen E. Hückel menschlich einmal etwas genauer an.

Erich Hückel war der mittlere von drei Brüdern. Sein Bruder Walter wurde ein Jahr vorher, sein Bruder Rudi drei Jahre nach ihm geboren. Sein Vater hatte Medizin studiert und liebäugelte immer wieder mit einer akademischen Karriere, die aber an der Berliner Universität, an der er es versuchte, nicht vorankam. Tatsache war, daß die Universität Berlin seine Zulassung zum Privatdozenten mit allerlei fadenscheinigen Gründen abgelehnt hatte. Sicher hätte es einige Entbehrungen für die Familie gekostet, wenn Vater Hückel die geforderte zweite Habilitation nachgeholt hätte. Aber vielleicht hätte er diese, thematisch basierend auf seiner ersten Habilitation, der in Tübingen angefertigten, erstellen können? Nun, dies wollte oder konnte er nicht - Erich Hückel schreibt dazu:

"Mein Vater gab also sein bisheriges Lebensziel, Hochschullehrer zu werden, resignierend auf. Er hatte seiner Frau aus Liebe seine Wissenschaft geopfert." (67; 29).

Und nach Tübingen wollte er nicht zurück,

" ... nicht zum wenigsten, weil er den Klatsch der Kollegen über die ´niedrige Herkunft´ seiner Braut und späteren Frau fürchtete." (67; 26)

Der Vater Erich Hückels ließ sich in der Folge als Arzt in Göttingen nieder. Eine Universitätsstadt sollte es schon sein, auch der Kinder wegen, meinte er. Er war sehr an den damaligen Entdeckungen der Physik und Chemie interessiert und hatte dazu in seinem Haus, ein Labor und eine Werkstatt eingerichtet. Hier erhielten die Kinder ihren ersten Zugang zu den Wissenschaften: Erich zur Physik, Walter zur Chemie; Bruder Rudi dagegen wurde Arzt. Hückel hatte im Alter von 3-6 Jahren einen Vorschullehrer, der ihm die Dinge beibrachte die

"... ein Kind von 6-9 Jahren können muß: richtiges Lesen, Schreiben und Rechnen - und keinen Mumpitz ... " (67; 41)

In seiner Schulzeit konnte sich Erich Hückel aber noch nicht entscheiden, was er werden wollte. Erst beim Abitur entschied sich Hückel schließlich für die Physik. Interessant die folgende Sequenz:

"Von der Medizin riet mir mein Vater ab, dieser so verantwortungsvolle Beruf würde mich bei meiner sensiblen Natur und meiner Gewissenhaftigkeit aufreiben." (67; 48)

Und diese ´sensible Natur´ Erich Hückels durchzieht, liest man seine Autobiografie, offenbar tatsächlich sein ganzes Leben. In dieser Autobiografie (67) stellt sich uns Hückel eigentlich nicht wie ein Naturwissenschaftler reinen Wassers dar. Er liebt es, über kleine, nebensächliche Details, die aber offenbar für ihn das gelebte Leben ausmachten, zu reden, über Dinge wie z.B. Stilblüten seiner Gymnasiallehrer, eine ganze Seite lang. Oder er schimpft über die in den 70-er Jahren üblichen Mengenlehre, die er seiner Enkeltochter, für die er 8 Jahre Ersatzvater war (seine Tochter war geschieden), beibringen wollte oder mußte. Das seine vorgelegte Biografie keine normale war, war ihm völlig klar und ebenso egal:

"Dieses Buch ist wohl in einem etwas ungewöhnlichen Stil, im Wechsel zwischen Ernst und Satire, geschrieben. Der eine Leser mag das amüsant finden, ein anderer mag sich darüber ärgern. Mir ist beides recht." (67; 7)

Manche, die ihn kannten fanden sein Buch in der Tat ´gräßlich´. Andere besuchten ihn und fanden sein Ausweichen bei Fragen nach seiner Theorie bedauerlich. Heilbronner z.B. erzählte, daß er einmal Hückel auf seine Theorie hin angesprochen habe. Hückel habe mit einem nichtssagenden, abschließenden Satz darauf reagiert, weiter nichts. Eine sensible Natur, zwei Kriege, die er allerdings in der Heimat hinter sich bringen konnte, demütigende Behandlungen durch die deutsche Beamtenriege, dazu noch eine nazistische, im Zuge seiner Anstellung als Professor in Marburg (nach dem Motto: take it or leave it) - all das mag zu seiner Unlust beigetragen haben. Wie bei seinem Vater wurde auch beim ihm die erste Habilitation nicht anerkannt.

Psychologisch könnte zur Deutung seiner Persönlichkeit vielleicht die Tatsache reflektiert werden daß Hückel an Migräne litt. An einer Stelle berichtet er von einem sehr schweren Migräneanfall:

"Wiederholte heftige Migräne-Anfälle kamen hinzu, so daß ich für ein paar Wochen in ein Sanatorium mußte. (67; 147)

Vielleicht, das wäre näher zu untersuchen, war Hückel Migräniker? Denn eine Migräne kommt selten allein. Oft sind Migräniker sehr auf süßes Essen eingestellt. In seiner Biografie spricht Hückel öfters davon, daß er sich mittags mit Kuchen versorgt hätte. Kampf und Auseinandersetzungen jeder Art schien er zu hassen - er war ein weicher Mensch, der, wenn er über seine Krankheiten berichtet (67), schon richtiggehend wehleidig werden konnte (67; 165). War er ein Hypochonder? Heilbronner erzählte, daß Hückel in Marburg seine Vorlesungen prinzipiell zeitlich so legte, daß sie möglichst wenig Studenten besuchen konnten. Seine Klagen über zuwenig Studenten kennen wir aus Stuttgart her, wo es noch begründet gewesen sein mag. War es das? Wer weiß.

Und wie meinte Hückel doch:

"Ein glückliches Zusammentreffen der Gene der Eheleute {seiner Großeltern A.d.A.}, die beide vorzügliche Anlagen mitbrachten, dürfte zu dem ungewöhnlichen Intellekt meines Vaters und damit seiner 3 Söhne Walter Erich und Rudi beigetragen haben. (67; 16)

Eigenartig. Viel kritisiert wurde die simple Art seines Schreibstils....(*)

Aber trotz allem ist es eine menschlich-angenehme Biografie, durch die wir immerhin einiges über die Entstehung seiner Theorie und den Menschen Hückel erfahren. Man erkennt, daß hinter den Theorien menschliche Schicksale stehen. Daß diese sogenannten Genies die Theorien nicht ohne Kampf, ohne Opfer, ohne Eigenheiten zustande bringen. Immer wieder wird dieser Mythos besonders von den Medien gepflegt: der Wissenschaftler hat in der Badewanne eine herrliche Idee, rennt mehr oder weniger bekleidet ins Labor, um mit wenigen Griffen - der Zuschauer wartet - die bahnbrechende Arbeit für die nächsten tausend Jahre zu realisieren. Diejenigen der Leser, die in den Wissenschaften arbeiten wissen, daß diese Geschichten ausnahmslos in den Bereich der Fiktion gehören. Es sei hier - als ein Beispiel von unzähligen - nur an den gut dokumentierten Kampf Plancks mit der Strahlung schwarzer Körper erinnert. Es war ein mühsamer Weg bis zur Planck´schen Konstanten, vor allem für Planck selber. Und andere mogelten, halfen etwas nach, rundeten in die richtige Richtung: Newton, Mendel, Galilei, Coulomb... Und nur ganz, ganz selten gibt es diese kurze effektiven Gedankenblitze - Einstein sei hier genannt. Vergessen wir aber nicht, daß nach einigen Blitzen von Einstein nichts mehr gehört wurde. Und auch da gab es menschliche Aspekte. Wir alle kennen Einsteins Geschichte - die von Milva Einstein, seiner Ehefrau damals in Zürich, deren Geschichte kennen wir nicht, nicht wirklich. Und warum gab Einstein ihr die Gelder des Nobelpreises? In welcher menschlichen Schuld stand er ihr gegenüber? Darüber schweigt man immer - es gelten nur die Lehrsätze. Wirklich? Wissenschaft gibt es durch den Menschen und sie wird mit dem letzten Menschen sterben. Nein, man sollte Autobiografien wie die Hückels vielleicht doch nicht zu sehr verurteilen, auf keinen Fall verurteilen, um dieses fragwürdige, menschenlose Bild, das die Naturwissenschaften gerne von sich gezeichnet haben wollen, zu stützen. Wissenschaft und Mensch gehören zusammen, Wissenschaft per se kann es nicht geben.

Theorie kommen und gehen so wie Menschen kommen und gehen. Und so schließt Erich Hückel sein Buch mit dem berühmten Satz

Sic transit gloria mundis.

Literatur

1 E. Schödinger : "Quantisierung als Eigenwertproblem."

a) Annalen Physik 79, 361-376 (1926).

b) Annalen Physik 79, 489-527 (1926).

c) Annalen Physik 80, 437-490 (1926).

d) Annalen Physik 81, 109-139 (1926).

2 M. Jammer: Conceptual Developments of Quantum Mechanics. McGraw-Hill,

New York, 1966.

3 W. Heitler,F. London: "Wechselwirkungen neutraler Atome und homöopolare

Bindung in der Quantenmechanik." Z. für Physik 44, 455-472 (1927).

4 E. Hückel: "Quantentheoretische Beiträge zum Benzolproblem."

a) Z. für Physik 70, 204-286 (1931).

b) Z. für Physik 72, 310-337 (1931).

c) Z. für Physik 76, 628-648 (1932).

d) Z. für Physik 83, 632-... (1933)

5 E. Hückel: "Grundzüge der Theorie ungesättigter und aromatischer Verbindungen."

a) Z. Elektrochem. 43, 752-788 (1937).

b) Z. Elektrochem. 43, 827-849 (1937).

10 Berzelius

11 Kekule

12 G.N. Lewis : "The atom and the molecule."

J. Amer. Chem. Soc. 38, 762-785 (1916).

13 W. Kossel : "

a) Annalen Physik 49, 229-... (1916).

b) Naturwissenschaften 7, 339-... (1919).

c) Naturwissenschaften 7, 360-... (1919).

21 R.S. Mulliken: Life of a Scientist. Springer-Verlag, Berlin, 1989.

26 E. Hückel: "Zur Quantenchemie der Doppelbindung."

Z. für Physik 60, 423-456 (1930).

27 W. Hückel: Theoretische Grundlagen der organischen Chemie. Leipzig 1931.

Verlag Chemie, Weinheim, 1955.

28 P.Debye: "?????

Physik. Zeitschrift 30, 524-... (1929).

29 E. Wigner, E.E. Witmer : "?????

Z. für Physik 51, 827-... (1928).

30 A. Lapworth, R. Robinson: "Distribution of electrons in the aromatic nucleus and the

early stages of aromatic substitutions."

Nature 129, 278 (1932).

31 E. Hückel, W. Hückel: "Theory of induced polarities in benzene."

Nature 129, 937 (1932).

32 A. Lapworth, R. Robinson: "Theory of induced polarities in benzene."

Nature 130, 273 (1932).

45 E. Hückel, "Zur modernen Theorie ungesättigter und aromatischer Verbindungen."

Z. Elektrochem. 61, 866-890 (1957).

46 C.A. Coulson, B. O'Leary, R.B. Mallion: Hückel Theory for Organic Chemists.

Academic Press. London, 1978.

47 N. Trinajstic: Chemical Graph Theory." Vol. 1 und 2. CRC Press, Inc.

Boca Raton (Flor.,USA), 1983.

67 E. Hückel: Ein Gelehrtenleben - Ernst und Satire. Verlag Chemie. Weinheim, 1975.

68 E. Hückel, ´Zerstreuung von Röntgenstrahlen´. Dissertation Göttingen 1921.

69 G.W. Wheland, D.E. Mann: Dipole Moments of Fulvenes and Azulenes.

J. Chem. Phys. 17, 264-268 (49).

74 [67] E. Hückel: Grundzüge der Theorie ungesättigter und aromatischer Verbindungen.

Verlag Chemie. Weinheim, 1938 und 1940.

75 P. Debye und E. Hückel: Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung

und verwandte Erscheinungen. Physik. Z. 24, 185 (1923).

76 E. Hückel: Zur Theorie konzentrierter wässeriger Lösungen starker Elektrolyte.

Physik. Z. 26, 93 (25). (Habilitationsschrift)

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Copyright © May 9, 2002 by U. Anders, Ph.D.

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Last updated : May 9, 2002 - 10:05 CET

Overview of Molecular Modelling

Molecular modelling is a very diverse subject, ranging from the acquisition and subsequent display of molecular coordinates through to highly accurate (i.e. better than experiment) numerical simulation using theoretically derived functions. Depending on the context and the rigour, the subject itself is also often referred to as "molecular graphics", "molecular visualisation", "computational chemistry", "computational quantum chemistry" or "theoretical chemistry". A related area known as "molecular simulation" relates the use of molecular modelling techniques to describing and understanding the statistical behaviour and properties of collections of molecules on a "macroscopic" scale. "Molecular dynamics" deals with those time-dependent properties of collections of molecules, and uses many of the techniques of molecule modelling and statistical mechanics. Both these last two methods are beyond the scope of this lecture course.

Six Characteristic Features and Classifications

1. Molecular Scales. Molecular modelling spans an enormous range of molecule size.

- Solutions of the Quantum mechanical Schroedinger equation which predict properties at least as accurate as can be measured by experiment are only available for molecules with up to ~8 atoms (rather more with symmetry)

- See this article for how such theory can validate experiment.

- Acceptably accurate "QM" solutions are nowadays available for molecules with perhaps 250 atoms if large-scale computational resources are available (see DNA).

- The "macromolecular" region of up to 10,000 atoms can be treated using semi-empirical QM methods, giving ~ chemical accuracy.

- Beyond this, in the region up to about 50,000 atoms, non-quantum mechanical solutions based on "Molecular Mechanics" (MM) methods can be used.

- For larger systems (the "Mesoscopic" region), the "atom" is replaced by "unified groups" of several atoms, or by simple descriptions (spheres, ellipsoids, etc) of entire molecules and one can treat collections of perhaps up to a million molecules.

2. Molecular Coordinates. Molecular modellers were amongst the first to take full advantage of on-line databases and information sources, and were the first chemists to adapt to the modern Internet. 3D atom coordinates are an essential feature of many modelling methods.

- Sources of Coordinates: Accurate or approximate 3D coordinates can be obtained from several experimental sources, in order of preference:

- Expt: The Cambridge Crystal database centre

- Expt: Microwave, electron diffraction and other specialised spectroscopies (small molecules only)

- Expt: NMR data (NOE experiments which determine how close eg to H atoms might be).

- Calc: Specialised databases deriving from archived calculations or algorithms (Corina)

- Calc: Sources such as electronic journals e.g. DOI: 10.1021/ja061400a or 10.1021/ic0519988

- Calc: Digital Repositories. These are only just now starting to be used.

- Coordinate Systems: For N atoms in a system to be modelled, at least 3N-6 coordinates are needed to specify the system geometrically. These coordinates can either be XYZ (resulting in 3N coordinates, of which 6 are normally redundant, corresponding to translations and rotations of the molecule), Internal or "Z" matrix coordinates;

H

O  0.96  1

O  1.4   2   111   1

H  0.96  3   111   2   90    1

A version of the latter called redundant internal coordinates (DOI: 10.1063/1.462844) which take full advantage of the properties of the energy Hessian and are particularly useful for bridged polycyclic compounds and cage structures, and tends to be the method of choice for most modern programs.

Some simple modelling methods (Hückel) need only the atom connectivity, and not the geometric information. Other modelling methods abandon the atom as the smallest unit whose coordinate needs to be known, and use larger scale approximations such as protein backbone positions, or even spherical or ellipsoidal approximations to whole molecules. For specialised cases (where group theoretical information is used/required to e.g.speed up calculations) symmetry adapted coordinates can be specified using exact symmetry restrictions (Gaussview is a program that can symmetrize a coordinate set). A Web site for handling coordinate symmetry even allows you to determine the symmetry group by providing XYZ coordinates.

- Coordinate File Types: Historically, various computer file formats were developed to described these coordinates, of which the best known are the "Molfile", the "PDB" and the "XYZ" formats. The MDL Molfile is really a database, not modelling format, and can lead to difficulties for small molecule modellers:

h2o2.mol

4  3  0  0  0                 1 V2000

0.1332    0.6883    2.1950 O   0  0  0  0  0

0.2562    0.6410    0.9013 O   0  0  0  0  0

0.8290    1.3074    2.5089 H   0  0  0  0  0

0.2935   -0.3133    0.6690 H   0  0  0  0  0

1  2  1  6  0  0

1  3  1  0  0  0

2  4  1  0  0  0

M  END

- Issues with the above format include the fixed precision (4 decimal places) of the coordinates

- Each atom is defined only by a line break (which can be lost on passage between different computers)

- The atom connectivity descriptors (yellow on black above) are fixed width, and break for more than 100 atoms!

The "XYZ" file is used almost entirely for animating molecular vibrations. The PDB format contains much more information about bio-molecules, but is poorly suited for small molecules (note the 3 decimal places for coordinates).

SEQRES   1 A  467  GLY ALA MET ALA SER SER VAL LEU VAL THR GLN GLU PRO

SEQRES   2 A  467  GLU ILE GLU LEU PRO ARG GLU PRO ARG PRO ASN GLU GLU

HET    COA    101      48

HETNAM     COA COENZYME A

HETNAM     MAH 3-HYDROXY-3-METHYL-GLUTARIC ACID

FORMUL   5  COA    4(C21 H36 N7 O16 P3 S1)

HELIX    1   1 PRO A  444  LEU A  449  1                                   6

HELIX    2   2 SER A  463  LYS A  474  1                                  12

SHEET    1   A 4 LYS A 549  ALA A 556  0

SHEET    2   A 4 VAL A 530  LEU A 546 -1  N  GLY A 539   O  MET A 555

CISPEP   1 GLY A  542    PRO A  543          0         0.61

CRYST1   75.297  130.182   92.547  90.00 106.48  90.00 P 1 21 1      8

ATOM      1  N   PRO A 439      -7.194 -13.702  30.538  1.00 76.06           N

ATOM      8  N   ARG A 440      -7.440 -15.246  28.234  1.00 76.37           N

A more modern (and open, ie non proprietary) example is the CML format, which is an extensible format which can carry as much (molecular modelling) information as is needed:

<cml:molecule xmlns:cml="http://www.xml-cml.org/schema/cml2/core">

<cml:metadataList title="generated automatically from Openbabel">

<cml:metadata name="dc:creator" content="OpenBabel version 1-100.1"/>

<cml:metadata name="dc:description" content="CCSD(T)//CCSD/6-31G(d) Gaussian 09 optimised geometries"/>

</cml:metadataList>

<cml:atomArray atomID="a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14" elementType="C C O O C C C C H H H H H H" formalCharge="0 0 0 0 0 0 0 0 0 0 0 0 0 0" x3="0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000100" y3="0.675900 -0.675900 -1.705300 1.705300 -1.701000 1.701000 -0.722800 0.722800 1.111600 -1.111600 -1.147000 1.147000 2.739700 -2.739700" z3="-1.572000 -1.572000 -0.678200 -0.678200 0.682200 0.682200 1.617300 1.617300 -2.568500 -2.568500 2.622800 2.622800 1.006400 1.006400"/>

<cml:bondArray atomRef1="a1 a1 a1 a2 a2 a3 a4 a5 a5 a6 a6 a7 a7 a8" atomRef2="a2 a4 a9 a3 a10 a5 a6 a7 a14 a8 a13 a8 a11 a12" order="2 1 1 1 1 1 1 2 1 2 1 1 1 1"/>

</cml:molecule>

The advantage of such modern (XML-based) formats is that e.g. molecular coordinates and properties can be embedded in a variety of delivery systems, including podcasts, Microsoft Office (CML4Word) and Journals.

3. Molecular Visualisation. Once 3D coordinates are available, they can be visualised, an important aid to interpretation of molecular modelling:

- Wireframe, Ball and Stick and Spacefill for small and medium sized molecules

- Ribbon for protein, nucleotide and carbohydrate structures to render the tertiary molecular structures, Polyhedral modes for eg ionic lattices.

- Isosurfaces, which are generated from the sizes of atoms, and onto which can be colour coded further properties such as MOs, charges etc.

- Animation to view molecular vibrations and the time dependent properties of molecules such as (intrinsic) reaction coordinates, protein folding dynamics, etc.

- Integration and Scripting. Programs such as Jmol allow seamless integration of models as part of lecture courses, electronic journals, podcasts, etc, and increasingly elaborate scripting of the models to illustrate scientific points.

4. Molecular Structure Analysis. Once a visual model is available, simple "heuristics" (rules) can be applied. These include:

- Detecting close atom contacts (defined as two atoms approaching significantly closer than the sum of their van der Waals distances (10.1021/jp8111556 and 10.1524/zkri.2009.1158), due to e.g. hydrogen bonding,

- Detecting bond lengths and the pattern of bond length alternation (e.g. aromaticity or anti-aromaticity, see 10.1021/ol703129z or anomeric effects)

- Detectinge.g. stereoelectronic effects such as atom antiperiplanarity or ring planarity

- Mapping to the simple ideas of e.g.arrow pushing which organic chemists tend to develop (and carry around in their head).

- IsoSTAR is a "data mining" method which can detect structural patterns in a large number of related structures.

- The Importance of being bonded (DOI: 10.1038/nchem.373): A sub-area of molecule modelling devoted to discussing what a chemical bond is, and how to characterise it,

5. Molecular Structure and Property Prediction

- Simple rule-based (heuristic) methods (e.g. CORINA) for approximations to molecular structures. Can be applied to most of the known 51,000,000 molecules, but is unlikely to lead to "new effects".

- Molecular Mechanics methods for larger molecules such as organics, carbohydrates, peptides, DNA oligomers, metal ion binding and some organometallics, zeolites, and complex ionic lattices.

- Semi-empirical Quantum mechanics methods for obtaining wavefunctions of molecules up to the size of medium sized enzymes.

- Ab initio and Density Functional Quantum Mechanics for accurate reproduction of geometries, and for e.g. weak molecular interactions such as hydrogen bonds, non-classical bonds (e.g. agostic interactions in organometallic species).

- Ab initio Coupled-cluster and double-hybrid correlated theories, particularly for "difficult" molecules such as e.g ozone, FOOF, aromatics, transition states, etc.

- Topological analysis of the wavefunction-derived electron density (AIM, ELF, etc) for defining bonding, and spectroscopic predictions.

6. Molecular Reactivity and Potential energy surfaces

- Hückel theories (heuristics) for aromaticity, frontier orbital analysis and simple relative energetics

- Extended Hückel theories for orbital correlation diagrams and metal systems.

- Semi-empirical, ab initio and density functional theories (DF) for

- reactions and bond formation/cleavage,

- potential energy surfaces and transition state modelling,

- electron density distributions and stereoelectronic properties,

- excited state properties and reactions,

- intermolecular properties,

- molecular surfaces and charge distributions (electrostatic potentials, etc).

- Multi-reference methods for exploring excited states, conical intersections etc.

The next two aspects of modelling are largely beyond the scope of the current lecture course:

- Molecular Solvation and Condensed phase properties Supermolecule and condensed phase models of specific and bulk solvation effects.

- Molecular Dynamics and Simulations. Theories of free energies and reaction kinetics.

Typical Molecular Modelling Software Tools

The "tools of the trade" have gradually evolved from physical models (Dreiding, CPK, etc) and calculators, including the use of programmable computers (starting around 1956 with the introduction of the first scientific programming language called Fortran), computers as visualisation aids (around 1970-), computers running commercially written analysis "packages" such as e.g. Sybyl (around 1984-) and most recently integration using Internet based tools and Workbenches (1994-) based on languages such as HTML, JavaScript, Java and C++. A Forum for discussing such tools, and other general queries is the Computational Chemistry List (CCL).

A typical selection of molecular modelling teaching tools available within the department is listed below.

1. Mercury: A (free) Crystallographic unit cell viewer and editor.

2. Jmol: A (free) Web-browser applet that can display molecules, and some of their properties such as surfaces, spectra, vibrations, etc.

3. Ghemical: an OpenSource molecular editing and molecular mechanics program. Superceded by Avogadro.

4. ChemDraw/ChemBio3D: Molecule editor and 3D geometry molecular mechanics/quantum mechanics optimisation and display tool (STEREO ENABLED)

5. Gaussview+Gaussian 09: Ab initio quantum mechanics editors and programs.

6. DS Viewer Pro (STEREO ENABLED)

7. VMD visualisation of molecular dynamics (STEREO ENABLED)

Return to overview| Forward to visualization|Forward to Mechanics| Forward to MO Reactant|Forward to MO TS| Forward to MO Advanced|

(c) H. S. Rzepa 1998-2012. No reproduction rights granted to this material without permission.

Difficulties with the Molfile

1. Z-cartesian coordinate may be zero for planar molecules (depending on how the molecule is oriented). If the Z coordinate is specified as 0.00000, some programs that process Molfiles may assume that this means this information is absent rather than zero, and act as if a 3D model does not actually exist!

2. Another undesirable feature of the Molfile is that each atom is defined by a line break. Since line breaks are not the same on different Computer operating systems, transfer between such system can often "break" the Molfile.

3. The Molfile defines exactly how atoms are connected to each other, and by which kind of bond. But this connection table may not be correct, or the bond types may not be adequate to map onto a molecular mechanics force field parameter. Thus it is quite often the case that a Molfile may not work correctly when subjected to mechanics modelling. Quantum mechanics modelling on the other hand will entirely ignore the connection table, and work out where the bonds are and of what type from first principles.

Techniques of Molecular Modelling: Transition States

Transition states fall into several broad categories:

1. Single bond rotations, preserving the 2-electron character of the bond (i.e. dihedral driving), and angle deformations which do not change electronic character at the atoms

2. Bond formation/cleavages which predominantly follow the two-electron "arrow pushing" formalisms (wavefunction dominated by a single configuration comprising doubly filled molecular orbitals) and which can be handled by Hartree-Fock models. If such arrow pushing results in charge separation (or its loss), then solvation models will have to be added.

3. Bond formation/cleavages and double bond rotations, where two-electron arrow pushing is not followed (homolytic bond cleavage, asynchronous bond cleavage/rotation forming biradicals) in which the wavefunction comprises two or more electronic configurations whose relative contributions change as the reaction proceeds. These are known as multi-configuration (MCSCF) reference states.

4. More complex points in the potential surface which include regions where different electronic (excited) states meet (conical intersections), and where the molecular dynamics (trajectories) of the atoms controls the reactions as well as simply the potential energy surfaces

Molecular mechanics can sometimes be used for 1 above, but molecular orbital methods have to be used for the rest of these cases.

Case Study 6: The transition state approach for aromatic electrophilic substitutions

The previous approach, which relied on the Hammond postulate to infer the properties of the transition state for the reaction is indirect. A better approach is to locate the actual transition state. The easiest way of doing this is to define one bond, in this case the distance between the N of a NO_2+ ion and the carbon it attaches to in an aromatic ring) as a reaction coordinate, and calculate the (Quantum mechanical) energy of the system as a function of this bond length. In the schematic energy profile so obtained, point 1 shows the nitronium ion forming a weak π-π complex with the aromatic in which the ion and the aromatic are face-to-face in their relative orientation. Transition state 2 is the highest in energy; point 3 represents the geometry of the "Wheland" intermediate. At this point, one has to select a different reaction coordinate, and one can now complete the the reaction via transition state 4, going to final product 5.

Case study 6 continued: The Transition states

From this reaction profile, one can select the approximate geometry of the transition 2 or 4 and use a precise algorithm such as Eigenvector Following to fully optimise its geometry. This is then subjected to a calculation of the 2nd derivatives of the energy with respect to the (3N) coordinates of the molecule. This matrix is diagonalised, and shown to have all positive roots (eigenvalues) except ONE. This is the reaction coordinate normal mode. The eigenvalue energy can be converted to a vibrational frequency, and the eigenvectors of this root of the matrix can be visualised (animated).

This procedure was followed for Pyridine rather than benzene (see see DOI: 10.1021/ja021152s for details of calculations for the prototypic reaction of a nitronium ion with benzene, where less is more, i.e. this simple prototype is in fact quite complex). The reaction coordinate normal mode obtained for the transition state clearly shows the required C-N bond formation in the 3-position.

Nitration of Pyridine

TS for 3-Nitration.

Vector correspond to C-N bond formation.

TS for (apparent) 4-Nitration. +4.6 kcal/mol higher.

Vectors correspond to a 3- to 5- migration.

The Wheland intermediate must now break down by proton removal to form the final product. This too should be studied by the transition state model, using the C-H and H-base distances as a reaction coordinate. Only then can one determine whether the C...N bond formation, the C...H bond cleavage, or a synchronous combination of both is actually the rate determining or highest energy step in the pathway.

If the electrophile is voracious, it tends to seek to acquire electrons as quickly as possible. This can favour Wheland intermediates (provided they are not destabilized, as above), as with E-X= CF_3CO_2F (E+=F+) for which an intrinsic reaction coordinate is shown below:

If the electrophile E+ is less voracious, and has a counterion X- which can act as the required base, it is perfectly possible for the Wheland intermediate to vanish entirely. Such merging of the two transition states and one intermediate into a single concerted pathway showing just a single transition state is shown below in the form of an intrinsic reaction coordinate (IRC), which charts the energy of the system as a function of the degree of reaction for E+=NO+ and X=CF_3CO_2-;

Aromatic electrophilic substitution as part of a catalytic process

An real example of such a reaction profile is found in the mechanism of metal-catalysed ring closure as shown below. Here would involve the attack by an electrophilic carbon on the carbon of phenyl group as illustrated with the red arrow below. In this mechanism, the electrophilic C-C bond formation (TS2) is actually concurrent with the proton removal by the metal ligand (TS4), i.e. TS2 and TS4 are one and the same, and the Wheland intermediate has no existence! By combining the two steps, the molecule does its best to avoid loss of too much aromaticity, which the existence of a discrete Wheland intermdiate would imply.

See DOI: http://dx.doi.org/10.1039/b913295c and this blog post for more details

Case Study 5: Revisited. The Diels-Alder transition state

This involved the catalysis of the Diels-Alder reaction inside the cavity of a crystal. An interesting question to ask is how the transition state is influenced by the surrounding cavity. We can use the above approach to locate this transition state both outside and inside the cavity.

Unlike the transition state for electrophilic aromatic nitration, the Diels-Alder has TWO bonds forming. The above approach could be extended by performing a two-dimensional reaction path search, varying the length of both bonds systematically to construct a 2D map. However, experience shows one can in fact adopt a short cut. It turns out that very many C-C bond forming reactions show a C-C bond length in the region 2.1-2.3Å. A good approximation is to calculate just the single point in the 2D map by restraining both forming C-C bonds to 2.2Å, whilst fully minimising all the remaining geometric variables (in practice this is done in the Gaussian program by specifying the keyword opt=modredundant and adding the two bond specifications at the end of the geometry input).

In the second step, the precise values of the two C-C bonds can now be themselves optimised by the Eigenvector Following method, involving calculating the force constant (Hessian) matrix (keyword opt(calcfc,ts)). This matrix is now quite likely to have a single negative root, as a consequence of having pre-optimised the geometry with the two C-C bonds constrained, and the EF method in effect implements the constraints that the energy of the total system will be maximised by following the vectors of this single (-ve) root, whilst simultaneously minimising the energy for all the other (+ve) 3N-7 roots of the Hessian. This argument of course presupposes that BOTH bonds have a length of 2.1-2.3Å at the transition state. Whilst simple Diels-Alders are so symmetrical, the topic has been controversial in the past, and there is good reason to suppose that larger cycloadditions such as e.g. π6s + π8s may be very unsymmetrical (DOI: 10.1021/ja00768a085). The result can be seen below for the and Diels-Alder transition states.

One interesting question to ask; how does the cavity affect the properties of the transition state. Thus outside the cavity, the transition state procedure (ωB97XD/6-31G(d) predicts that the two C-C bonds forming are 2.11 and 2.47Å, some small asymmetry tending towards a zwitterionic transition state with a C- stabilized by an adjacent ester group and a carbocation stabilised by allylic resonance. In the cavity the bond lengths are 2.13 and 2.50Å, indicating that the cavity does indeed have a small influence on the reaction enclosed within it (in effect acting as solvent). The point is further emphasized by the values of ΔH‡ and ΔG_298‡:

System

ΔH‡

ΔG‡

Isolated reaction

15.5

29.5

Cavity reaction

16.5

20.0

Using Ln k/T = 23.76 – ΔG/RT and t1/2 = Ln 2/k; 20.0 kcal/mol corresponds to a half life of ~40 hour at room temperature (298K), more or less what is observed. For more details, see here.

Back to scales|| Back to Visualisation| Back to Mechanics|Back to MO Reactants| Forward to MO Advanced|

(c) H. S. Rzepa 1998-2012. No reproduction rights granted to this material without permission.

Techniques of Molecular Modelling: Stereochemistry

Case Study 8: Enol Borinates in Stereospecific C-C bond formation. Studying Asymmetric Induction

A key stage in the reaction sequence shown as part of the problem is the following step involving C-C bond formation to an aldehyde, mediated by the chiral auxilliary X=di-isopinocampheyl (see DOI: 10.1016/S0040-4020(01)85588-5 and 10.1016/S0040-4020(01)86409-7)

Other than the chiral auxiliary, all the reagent components are achiral. However, two entirely new chiral centres are formed in the product. Modelling can help us understand two features of these new chiral centres and why they form so specifically.

1. their relative stereochemistry: R,R vs R,S (and their enantiomers S,S vs S,R in the absence of any other chiral centre)

2. their absolute stereochemistry: R,R vs R,S vs S,S vs S,R in the presence of another chiral centre X

The Relative Stereochemistry

A 3D model for the basic framework (i.e. replacing all substituents with H) of the transition state must be constructed. Since this involves bond formation and cleavage, a QM (Quantum Mechanics) based model must be used, in this case the AM1 semi-empirical method (Table). The transition state can be located using a variety of methods. Since the reaction involves two bonds making and one breaking, its difficult to construct a 1D or 2D grid. The easiest way is to guess the approximate lengths of the breaking/making bonds (about 1.9 - 2.1A), and optimise the geometry with these values constrained (using eg the opt(modredundant) keyword in Gaussian). The resulting geometry can then be fully optimised with no such constraints, but this time with the use of appropriate transition state location algorithm such as the eigenvector following method and specifying a transition state to be found (opt(calcfc,ts)).

These calculations ( ) reveal that both and of the transition state are possible. A full calculation of the 2nd derivative matrix, followed by appropriate mass weighting, will give the frequencies of the various normal modes of the molecule. One is shown as -ve (imaginary), being the transition state normal mode. A program such as JMol can be used to animate the form of this normal mode.

One next needs to understand how the various substitution sites interact sterically. To do this, methyl groups are inserted into the various positions (including one for the chiral auxilliary X) and the energies of some of the various possible isomers are calculated.

The is again the most stable. is formed with the thus avoiding steric congestion from the . This particular steric interaction in effect defines which π face of the carbonyl group is used to form the new C-C bond. Forming the C-C bond from the alternative face of the carbonyl would place the This interacts sterically with the chiral auxilliary and is 3 kcal/mol higher in energy, and hence not formed. Another way of avoiding steric congestion is for the to form. The methyl adopts an equatorial position, but it is eclipsed rather than staggered with the and hence the most stable boat isomer is 3 kcal/mol higher than the chair.

0.0

3.0

0.0

4.7

2.8

1.5

0.0

2.9

0.0

The Absolute Stereochemistry

The methyl group X must now be replaced with a chiral auxilliary. Many such auxiliaries can be used (most derived from natural products that are enantiomerically pure). To start with a simple model, will use the (un-natural) group -CHClI which contains three ligands of quite different size. Its chirality is defined in these models as S. When inserted into the model, two diastereoisomers with carbon chirality of or can be formed. The calculations reveal that one of these is about 1.5 kcal/mol more stable than the other, which is the correct order of magnitude to result in chiral induction. A more "natural" chiral group derived from camphor can now be inserted to form the final models and in this instance a rather larger energy difference is obtained. Using this last value in the equation ΔΔG = -RT Ln [R,S]/[S,R], we get [R,S]/[S,R] = ~130, an enantiomeric excess of better than 99%.

Case Study 9: The Asymmetric Strecker Synthesis

The Strecker synthesis represents a rapid way of constructing amino acids of specified configuration (see case study 3). The reaction comes in two parts. The first involves building a sulfoxide-based (S)-chiral auxilliary to an imine. The second part of the reaction involves nucleophilic attack on this imine with an aluminium cyanide reagent to form an (S,S) diastereoisome, via a postulated cyclic six-membered ring transition state, and subsquent hydrolysis of the cyano group to an enantiopure amino acid. Can this mechanism be verified using molecular modelling?

A: Is the currently acepted mechanisms for the Strecker reaction a good model?

System

Relative free energy

digital Repository

The carried out by Franklin Davis involves the following precursors, the N-(Phenylsulfonyl) (3,3-dichlorocamphoryl) oxaziridine (derived from a natural product in pure enantiomeric form, 10.1021/ja00030a045), from which an oxygen atom is delivered to produce enantiomerically pure sulfoxide. The transition states for the diastereomeric transition states are controlled by the 3,3-dichloro functionality, preventing rotation about the O...S bond. The a N=CH...O hydrogen bond in the (SS) isomer is at least in part responsible for diastereoselection (cf HUGHAS)

4.5

10042/to-144

0.0

10042/to-145

The Davis mechanism for the Strecker proper now involves the aluminium reagent forming an , which reacts with the sulfoxide to form which undergoes asymmetric transfer of cyanide via TS1, a six-membered ring transition state.

There is only one snag: no actual such transition state can be located at the B3LYP/cc-pVTZ level!

B: A modified mechanism. Will it be any better?

System

Relative free energy

digital Repository

The story is now taken over by Pedro Merino (10.1016/S0957-4166(02)00832-7), where we see the suggestion that the chair six-membered ring assumed by Davis et al (TS1) cannot exist unless an additional molecule of water/solvent is involved, as in . Had transfer of CN occured as Davis described, the Aluminum would have ended up with an empty coordination site, and this is now prevented by intermolecular SN2 like displacement by water/solvent.

The problem with this mechanism is that a bimolecular mechanism incurs an entropy penalty, which as we have seen before adds about 10-12 kcal/mol to the ΔG for the reaction! This contributes to the calculated barrier of 36 kcal/mol, which is too high to be a facile reaction at room temperature.

This mechanism too is broken. Back to first principles. Davis assumed that the active reagent is monomeric R_2Al-CN, which is formed by opening a presumed aluminium dimer. Checking the crystal structures of related compounds suggests instead a (e.g. 3) where both the N AND the C of the cyano group can coordinate to the Al; dissociation of this trimer could lead to either an Al-CN or an Al-NC species.

C: A new, improved mechanism that fits the facts!

Coordination to 1 can now give essentially isoenergetic or complex with the chiral imine-sulfoxide.

This leads to a new model, , which involves a 7-membered ring in which an iso-cyano group also plays the role of the solvent in TS1, and the resulting unimolecular nature saves the entropy term that was needed for TS1a.

23.9

10042/to-120

22.2

10042/to-122

TS2 has a more favourable Dunitz angle (104° vs 112° for TS1a). The product 4 can be fully decomposed by treatment with water, via transition state TS4. The barrier for this reaction, although requiring the now notorious 10 kcal/mol of entropic contribution, is nevertheless lower than TS2. Because the entropic step comes much later in the mechanism, it is no longer relevant!.

The form of TS2 is 1.7 kcal/mol higher (B3LYP/cc-pVTZ) than the form, due predominantly to the former being disfavoured due to the between a hydrogen and the phenyl group. In stark contrast, using TS1a as one's model, the (S,R) diastereoisomer is instead 0.8 kcal/mol lower than the (S,S).

D: Tidying up the loose ends

The calculations show that TS2 is favoured over by about 17 kcal/mol, as is the product 4 over product 5.

The effect of replacing an alkyl group on the Al catalyst with an ethoxy or methoxy group has the effect of increasing the barrier to the reaction, and hence slowing it down (making it more selective). With the model TS1a, such substitution is predicted to have the opposite effect, decreasing the barrier by 2.7 kcal/mol. With model TS2, the barrier is predicted to increase by 1.9 kcal/mol, thus accounting for the selectivity.

QED. THE END

back to scales|| Back to Visualisation|Back to Mechanics|Back to MO Reactants| Back to MO Transition states|

(c) H. S. Rzepa 1998-2012. No reproduction rights granted to this material without permission.

Molecular Modelling Problem.

Electrophilic substitution in furan can be modelled by protonation. At least three possible protonated isomers are possible; two are shown below (what is the third?)

1. Calculate the energies of the three protonated forms of furan at various levels of theory, including MM2 Mechanics (ChemBio3D), Semi-Empirical (PM6), RHF Ab initio and correlated B3LYP density functional. Which technique(s) agree with the known electrophilic substitution pattern of furans in the 2-position. How "expensive" in terms of computer time are the various techniques?

2. What happens with other heterocycles, i.e. pyrroles (N), thiophenes (S), oxazoles (NO)?

3. Does benzo substition on the heterocycle (i.e. benzofuran or indole) change the pattern?

4. Do you get the same prediction if you inspect merely the HOMO (highest occupied molecular orbital) of the reagent?

5. If you have performed a Gaussian calculation, you will have the .fchk file. From this you can generate a molecular electrostatic potential, using the DOS command cubegen 0 potential=scf indole.fchk test.cube. The cube file can then ve viewed using ChemBio3D or Gaussview. Does the MEP give the same result as the other techniques?

6. If you apply a transition state model by using a reaction coordinate and an electrophile such as nitronium cation, do the results change?

Instructions and Manuals for using Programs

The problems can be attempted using Gaussian 03 (Ab initio) and the Chemistry SCAN resource. Details on how to prepare a program input are found here.

(c) Copyright H. S. Rzepa, 1999-2009.

Table of Contents

Intro

Scales

Visualisation

Mechanics

MO Reactants

MO TS

Stereo

Problems

seven lectures

The helical structure of DNA

Environmentally Friendly Herbicides.

Chiral Resolution using the Pirkle Reagent

Nanotechnology and Molecular cavities: Molecular self-assembly of Nanobiotics, Dihydrogen bonding

3 revisited: Where IS the H-bond in the Pirkle Reagent?

The mechanism and stereochemistry of Carbocation rearrangements

Design of organic crystals: A molecular anvil

3 revisited: Where ARE the electrons in the Pirkle Reagent?

2 revisited: How do orbitals interact within a molecule?

4 revisited: The structure of DNA

5 revisited: Non-classical Carbocations

Regiospecificity: The mechanism of electrophilic aromatic substitutions

Nitration of Pyridine

Aryl C-C bond formation using a metal catalyst

6 revisited: The Diels Alder reaction

Stereospecificity: Asymmetric C-C bond formation via Enol Borinates.

Stereospecificity in the Strecker amino acid synthesis.

Stereospecificity in Catalytic polymerisation: The Ziegler-Natta mechanism and the role of agostic i

2011 Exam Paper

outline answer

Molecular Modelling

can be viewed here.

Barton

van der Waals

electrostatic

the joined up analysis

Erich Hückel

10.1021/jp900056w

this display

difficulties for small molecule modellers

http://dx.doi.org/10.1039/b913295c