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Computational Chemistry

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Molecular mechanics

Previous chapter - Computational Chemistry

Introduction

A good introduction is Wikipedia:molecular mechanics.

In molecular mechanics we treat a group of molecules as a classical collection of balls and springs rather than a quantum collection of electrons and nuclei. This means we can readily make physical models and have these physical models turned into computer programs.

There is a hierarchy of models, the minimal being atoms as hard spheres of radius equal to the covalent radius and using VSEPR (Valence Shell Electron Repulsion) for the lonepairs. Angles are approximately determined by best mutual avoidance in the hierarchy lone pairs > bond pairs. The electronegativities of atoms $\chi$ can be used to increase the flexibility of the model. Substituents with greater electronegativity decrease the size and repulsiveness of bond pairs. The model is incomplete because of neglect of bonding overlap and there is no description of any $\pi$ bonds the molecule may have. Also we are making invalid additivity assumptions illustrated as follows.

The longest bondlengths are adjacent to electron pairs as in the classic example of F-F. ( HF is 91.7 pico-metres, FF 141.7 (covalent radii H 37.0 pm, F 71.0pm).) HF is quite clearly shorter than average and FF longer. The Fluorine covalent radius F value determined as 1/2 F-F, will be longer than the value determined from fluoro-alkanes. Once you move to using a computer it is usual to go straight to a full molecular mechanics model with a force field such as MMx, Merck or AMBER.

The data which describes how the balls and springs behave is known as the force field. The earliest force fields were only for the atoms C, H, O and N and gradually added extra atoms, atom types and functional groups. Rappe and coworkers at CalTec, Los Angeles have produced a deductive paradigm for producing a whole periodic table force field. Everything necessary to code it is in the literature. Recent versions of CHARMM also have a whole periodic table force field. The Merck Forcefield which is currently thought to be the most accurate, has more metal elements than the older forcefields.

Force fields

A formal description of a typical force field follows.

harmonic force constant.

$V_{s}=\sum _{l}{\frac {1}{2}}k_{l}{(l-l_{0})}^{2}$

optional anharmonic force constant.

$-\sum _{l}{\frac {1}{3!}}k_{l}^{anh}{(l-l_{0})}^{3}$

angle bending constants.

$+\sum _{\theta }{\frac {1}{2}}k_{\theta }\left\{{(\theta -\theta _{0})}^{2}-k_{\theta }^{'}{(\theta -\theta _{0})}^{3}\right\}$

Fourier representation of torsional potential.

$+\sum _{n}{\frac {1}{2^{n}}}k_{\omega }(1+s\cos(n\omega ))+\sum _{\omega }{\frac {1}{2}}k_{\omega }(1+s\cos(\omega ))$

Out of plane bending for trigonal groups.

$+\sum _{\Gamma }{\frac {1}{2}}k_{\Gamma }{(\Gamma -\pi )}^{2}$

Van der Waals terms.

$\sum _{i<j}\left\{{\frac {A_{ij}}{r_{ij}^{12}}}-{\frac {B_{ij}}{r_{ij}^{6}}}\right\}$

electrostatic interactions.

$\sum _{i<j}{\frac {q_{i}q_{j}}{r_{ij}^{2}}}$

What do the force field components mean?

Let us first consider bond lengths. The most important number is $r_{0}$ the average bondlength. Then we have the harmonic force constant, which is the rigidity of the bond, obtainable from either IR spectroscopy or an energy derivative electronic structure calculation. The anharmonic constants are always negative corresponding to a steeper slope on the repulsive, compressive side and a shallowing out on moving towards dissociation. Bending constants are complicated and not so subject to simple rules of thumb. Four body dihedral functions are of course periodic and can be almost any shape with a number of minima related to valency and lonepair content of the four entities.

Pair potentials

The pair potential term is always clearly divided into a repulsive and an attractive part. In van der Waals complexes it is this repulsive function which ultimately prevents the molecules locking together usually giving the peculiarity of a positive van der Waals energy which seems counterintuitive. (Perhaps it should be relabelled, Lennard-Jones energy, or pair energy.)

For some purposes the ${\frac {1}{r^{12}}}$ term in the pair energy is too steep and a softer repulsive potential the Buckingham potential is used:

The Buckingham potential

$E_{vdw}=\sum {Ae^{-Br}-Cr^{-6}}$

Another modification to the pair potential is the addition of an inverse $r^{10}$ term, used in some programs to simulate the extra attractive energy of the hydrogen bond.

The Lennard-Jones potential

$E(r)=4\epsilon \left({{\left({\frac {\sigma }{r}}\right)}^{12}-{\left({\frac {\sigma }{r}}\right)}^{6}}\right)$

$\epsilon$ is the energy at the pair minimum. The location of the pair minimum is at $r=2^{1/6}\sigma$ i.e. $1.122462048\sigma$.

MOLWEB, the software written by MG to do interfacing and MM style property calculations, internally uses either the MM2 atom types or an atom type formed from the atomic number multiplied by 100, giving the possibility of 100 different atom types. This is not a ludicrous requirement if one thinks of transitional metal complexes where differentiation between half a dozen oxidation states, several coordination numbers and high and low spin environments might be required.

The ${\frac {A}{r^{12}}}-{\frac {B}{r^{6}}}$ is often expressed using the 2 parameters of an energy $\epsilon$ and a distance $\sigma$.

Table 1 - The MM2 atom types

U. Burkert and N. L. Allinger, Molecular Mechanics, ACS Monograph 177, (ACS,Washington,1982). A. K. Rappé and C. J. Casewit, Molecular Mechanics across Chemistry, (University Science Books, California, 1997).

1	Carbon sp3	16	S (+)onium
2	Carbon sp2	17	S (>S=O)
3	C (sp2)(>C=O)	18	S (>SO2)
4	Carbon sp	19	Silicon
5	Hydrogen	20	Lone Pair
6	Oxygen (-O)	21	H (O-H)alc
7	Oxygen (=O)	22	C cyc-prop
8	N (sp3)	23	H (H-N)Amin
9	N (sp2)	24	H (COOH)
10	N (sp)	25	P (Phosphine)
11	Fluorine	26	B Trigonal
12	Chlorine	27	B Tetrahed
13	Bromine	28	H-vinyl-alc
14	Iodine	29	P (Phosphate)
15	Sulphur

Notice the absence of metals from this table. Rappé and coworkers at CalTec, Los Angeles have produced a deductive paradigm for producing a whole periodic table force field (Rappé and Casewit, 1997).

If you do not have access to a code with this in it everything necessary to code it is in the literature. It is believed recent versions of CHARMM also have a whole periodic table force field. The new Merck Forcefield (Halgren,1996)), which is currently thought by some to be the most accurate, has more metal elements than the older forcefields.

Practical Problems

i) Conformations of cyclohexanes

Check that the available force fields predict the correct energy differences between chair and skew-boat forms of cyclohexane. Confirm that the boat is impossible to make as it is not a minimum on the potential surface. Trial boat forms of cyclohexane are rather difficult to make. One possible route is to start with benzene, put the carbons into the right place after removing the double bonds. After that it is necessary to add hydrogens to resaturate valency with H-ADD. Another way is to pull the chair form down from the menu and delete the 2 hydrogens at the foot of the chair. Move the carbon with 'MXY', and then replace the hydrogens.

Establish whether 1-chlorocyclohexane prefers the axial or equatorial positions. (Make the first conformer and optimise for energy. Then invert at the 1 centre to get the other conformer and check the energy). Try this for both MM2 and AMBER force fields.

2) Geometry of cis-Chlordane

Use your bibliographic skills to get a model of cis-Chlordane, (whatever that is). Minimize it with MM2. Calculate its energy when solvated in water and chloroform, using continuum models, (if available).

Bibliography

• U. Burkert and N. L. Allinger, Molecular Mechanics, ACS Monograph 177, (ACS,Washington,1982).
• A. Hinchliffe, Molecular Modelling for Beginners, (Wiley,2003) ISBN 13: 9780470843109.
• A. K. Rappé and C. J. Casewit, Molecular Mechanics across Chemistry, (University Science Books, California, 1997).

References

• U. Burkert and N. L. Allinger, Molecular Mechanics, ACS Monograph 177, (ACS,Washington,1982).
• T. A. Halgren, J. Comput. Chem., 17, 490 (1996).
• A. K. Rappé and C. J. Casewit, Molecular Mechanics across Chemistry, (University Science Books, California, 1997).

Next Chapter - Molecular dynamics

Molecular dynamics

Previous chapter - Molecular mechanics

Introduction to Molecular dynamics

You will remember from your spectroscopy that vibrations, and that includes internal rotations, are quantized. In MD we use a classical force field and ignore this quantization putting in classical energy according to Newton's Laws. This might at first sight appear to be totally unsound in view of all we have been taught about quantum theory but in fact what it means is we are replacing a summation with a Monte Carlo integration. Only in very special cases is there a dramatic difference. Situations with strong force constants and light particles i.e. protons are likely to show this. If, and this is a big if, the trajectories of the particles are chaotic all the phase space of the molecular vibrations and in particular internal rotational conformers, will be covered. Only if this has occurred do we have a temperature. (A temperature is a single number which allows one to reproduce the occupations of the states of the system using Boltzmann statistics. If there is no Boltzmann distribution, there is no temperature.)

One focus of research here has been the development of methods which allow a reliable temperature to be created for gas phase molecular systems. (Condensed phases with plentiful collisions are self randomizing and the Boltzmann distribution is rapidly established.) Simply apply Newtons laws of motion. In one dimensional form they are:

$f=ma$

$v=u+at$

$s=ut+{\frac {1}{2}}at^{2}$

s is a new position. Force is $-{\frac {dV_{s}}{dx}}$. $t+\delta t$ then run for n timesteps.

Verlet algorithm

$s(t+\delta t)=s(t)+v(t)\delta t+{\frac {1}{2}}a(t)\delta t^{2}+....$

$s(t-\delta t)=s(t)-v(t)\delta t+{\frac {1}{2}}a(t)\delta t^{2}-....$

add these together then :

$s(t+\delta t)=2s(t)-s(t-\delta t)+a(t)\delta t^{2}-....$

These are applied to the 3N-6 dimensional hypersurface of a single molecule, or to all free coordinates of an infinitely repeating system.

Single molecule dynamics

Applications in liquid crystals and biological chemistry often involve dynamics on a single molecule in an infinite vacuum. There are considerable technical problems here both in establishing a temperature and in ensuring the 3N-6 dimensional phase space is covered with populations according to the Boltzmann distribution. In practice this can only be assured in special circumstances and for proteins there is the continual phenomenon of sticking in a given neighbourhood of conformations. In this case the timescales of the physical processes of changing between energetically equivalent conformations may be longer than the simulation. Putting molecules in a box with inert gas bumping into it is one way of trying for conformational equilibrium.

Table - Timescales of molecular processes

Secs	name	Process
10-1		Surface reorganizations
10-3	milliseconds	Protein Folding
10-8	10 nanoseconds	NMR
10-9	to 10+3	Brownian dynamics
10-9	to 100	Lattice Monte-Carlo
10-12	picoseconds	Fast vibrations
10-15	femtoseconds	Electronic excitation
		(molecular dynamics)

A good simulation regime for a teaching exercise is 10 femtoseconds with C-H bond vibrations frozen out. Research requires considerably more care and much longer runs! It would not be a good idea to investigate protein folding using a conventional MD program and a 0.5 femtosecond time step.

The finite box size

A finite box is also a problem, but a minimum image convention allows for infinitely repeating boxes in the 3 Cartesian directions. Software for infinite boxes and the AMBER and GROMOS force field exists within CCP5 (The DLPOLY package). There are problems with electrostatic interactions. The Coulomb Interaction goes as inverse $r$ to the power one. This dies away very slowly over several images of the box before dying away to insignificance. This must be handled by a complicated procedure known as an Ewald Sum.

Liquid simulations often calculate G-of-r, not to be confused with the quantum theory matrix G-of-R. This is the radial probability density, which is experimentally accessible via neutron diffraction. It can therefore act as simulation quality check on whether the right liquid solvation shell structure is being generated.

A good description of modern simulation techniques is in Allen and Tildesley.

Problems

i) A Gas phase liquid crystal molecule

Make a liquid crystal prototype by adding a (CH2)n tail onto phenol's acidic proton. Run a dynamic simulation at various temperatures to see the side chain move about.

2) Histogramming

Make hexamine-diamine. Use ANALYZE to monitor the distance between the N-atoms. Use DYNAMIC and MonSt to create a histogram on this N to N distance. Repeat this with a flexible dihedral angle in a molecule of interest to you.

Bibliography

• M. P. Allen, and D. J. Tildesley, Computer Simulation of Liquids,(Oxford University Press, Oxford,1987).

Next Chapter - Molecular quantum mechanics

Molecular quantum mechanics

Previous chapter - Molecular dynamics

Introduction

Our applications of quantum theory here involve solving the wave equation for a given molecular geometry. This can be done at a variety of levels of approximation each with a variety of computing resource requirements.

Our applications of quantum theory here involve solving the wave equation for a given molecular geometry. This can be done at a variety of levels of approximation each with a variety of computing resource requirements. We are assuming here a vague familiarity with the Self Consistent Field wavefunction and its component molecular orbitals.

$<\Psi _{0}|{\hat {H}}|\Psi _{0}>$

$<\Psi _{0}|K.E.+V_{nuc}+\sum _{i<j}{\frac {1}{r_{ij}}}|\Psi _{0}>$

The $ij$ summation indices are over all electron pairs. It is the ${\frac {1}{r_{ij}}}$ which prevents easy solution of the equation, either by separation of variables for a single atom, or by simple matrix equations for a non spherical molecule.

The electron density $\rho (x,y,z)$ corresponds to the $N$-electron density $\rho (N)$. If we know $\rho (N-1)$ we can solve $<\Psi _{0}|{\hat {H}}|\Psi _{0}>$ So we guess $\rho (N)$ and solve $N$ independent Schrödinger equations. Unfortunately each solution then depends on $\rho (N)$ which we guessed. So we extrapolate a new $\rho ^{'}(N)$ and solve the temporary Schrödinger equation again. This continues until $\rho$ stops changing. If our initial guessed $\rho$ was appropriate we will have the SCF approximation to the ground state.

This can be done for numerical $\rho$ or we can use LCAO (Linear Combination of Atomic Orbitals) in an algebraic form and integrate into a linear algebraic matrix problem. This use of a basis set is our normal way of doing calculations.

Our wavefunction is a product of molecular orbitals, technically in the form of a Slater determinant in order to ensure the antisymmetry of the electronic wavefunction. This has some technical consequences which you need not be concerned with unless doing a theoretical project. Theoreticians should make Szabo and Ostlund their bedtime reading.

$\Phi _{k}(x_{1},x_{2},x_{3},.....x_{N})~=~{\frac {1}{\sqrt {N!}}}~determinant$

$\psi _{A}(x_{1})\psi _{B}(x_{1})..........\psi _{N}(x_{1})$

$\psi _{A}(x_{2})\psi _{B}(x_{2})..........\psi _{N}(x_{2})$

$......$

$......$

$\psi _{A}(x_{N})<th>\psi _{B}(x_{N})..........\psi _{N}(x_{N})$

When $\Psi$ is expanded in terms of the atomic orbitals $\chi$ the troublesome ${\frac {1}{r_{ij}}}$ term picks out producted pairs of atomic orbitals either side of the operator. This leads to a number of four-centre integralsof order $n^{4}$. These fill up the disc space and take a long time to compute.

Bibliography

• A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, (Macmillan, New York 1989).
• Computational Quantum Chemistry, Alan Hinchliffe,(Wiley, 1988).
• Tim Clark, A Handbook of Computational Chemistry, Wiley (1985).
• Cramer C.J., Essentials of Computational Chemistry,Second Edition,John Wiley, 2004.
• Jensen F. 1999, Introduction to Computational Chemistry,Wiley, Chichester.
• Web link on Hartree-Fock theory http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html

Next Chapter - Semiempirical quantum chemistry

Semiempirical quantum chemistry

Previous chapter - Molecular quantum mechanics

Semi-empirical Philosophies

There is a difference in philosophy between the Pople and Dewar schools. Dewar's parameterisation aims to absorb all the missing parts of the model where SCF is inadequate, (Correlation , relativity, algebraic approximation), into the semi-empirical parameters, whereas Pople's initial idea was to produce cheap practical equivalents of minimal Slater orbital basis SCF calculations which would have all the mathematical niceties and limits of Hartree-Fock theory. In recent times Dewar's ideas and programs have prevailed because computing power has increased to make the real SCF wavefunction routinely computable for all but the largest molecules so MOPAC style programs are used for the largest molecules where Hartree-Fock is still uncomputable.

Programs for linear scaling SCF, where small approximations are made to make the effort scale linearly with the size of the molecule rather than scaling to the power of 4 dictated by the integrals evaluation, are hoping to make uncomputable Hartree-Fock problems a thing of the past.

A Brief Technical Description of Semi-Empirical Methods

The ZDO approximation, (Zero Differential Overlap). The first approximation is core-valence separation. The core electron MOs are independent of molecular environment and can be parameterised out. This is a good approximation, the all valence approximation - effective nuclear charges (Za) adjust for the missing inner electrons.

Basis functions belong to individual atoms in semi-empirical methods. In all methods For NDDO, (the most complete approximation i.e. nearest the full Hartree-Fock solution) i.e. all 3 and 4 centre integrals disappear. In CNDO the least complete method the Kroneka deltas are so only integrals like survive. In INDO 1-centre exchange integrals are added to the CNDO set

Remember 2J - K in Hartree-Fock theory.

The Invariance Problem

To insure the values of calculated properties do not change with rotated orientations of the molecule:

         (sAsA|pxBpxB)  =  (sAsA|pyBpyB)

        and

         (sAsA|pxBpxB)  =  (sAsA|sBsB)

This is a considerable approximation.

Next Chapter - Geometry optimization

Geometry optimization

Previous chapter - Semiempirical quantum chemistry

Geometry Optimization

Important features to notice are that the potential is steeper on the inner side and shallower especially as it moves out towards dissociation. This concept is extended to polyatomic molecules where we have a potential energy surface in $n$ dimensions. ($n$ is 3N-6 where N is the number of atoms. The -6 comes from the translations and rotations of the molecule, the trivial vibrations referred to in the MOPAC printouts.) The bottom of the potential well is at the equilibrium bond length. In this region the potential function resembles a parabola $E={\frac {1}{2}}fr^{2}+c$

Optimization of the geometry of a diatomic is trivial if we can calculate the total energy we follow it downwards as a function of varying $r$ by Newton like procedures until we have points either side of the minimum. From this position we can keep bisecting the interval and using quadratic interpolation until the required accuracy is reached.

A vast improvement to this procedure has come from being able to differentiate the Hamiltonian with respect to the 3N nuclear position coordinates, ( $x,y,$ and $z$ on each atomic centre ). All the modern quantum chemistry programs now use this technique. Newton-Raphson like methods are used to descend down the surface and old gradients are remembered to refine the optimisation. Once in the quadratic region swift convergence to any accuracy is assured.

Cartesian Coordinates and Z-Matrices

The boring problem of molecular geometries.

We have talked so far about geometries assuming they were expressed as matrices of the $x,y,z$ coordinates of the individual atoms. Sometimes a more useful and efficient way is to express the same unique geometrical arrangement as the bond lengths and bond angles of the molecule. This has factored out the 6 degrees of freedom corresponding to rotation and translation of the molecule.

Practical Ways of Making Molecular Geometries

To write out the $z$-matrix for a medium sized molecule by inspection is a possible but arduous task. To make the cartesian coordinates without computerized assistance is impossible. In practice one uses a molecular modelling system like the MACROMODEL ( Columbia University, New York ), available on UNIX systems. The molecular modelling system uses one of several standard force-fields to refine the built-up geometry. (Notice the terminology force-fields for classical calculations like MM2, hamiltonians for quantum methods like AM1 (Austin Method-1).)

The force-field calculation has all the connectivity of the molecule established by the modelling system so generating the matrices involves walking the tree of connectivities in the most chemically sensible way picking off the leaves, (Hydrogen atoms), with their dihedral angles as $z$-matrix entries. In computing terminology trees grow downwards not up. The nodes of a tree are usually the atoms. Terminal nodes are called leaves.

n-pentane in Tree Form

                                        C
                                      . . . .
                                    .   .   .   .
                                  C     H    H   H
                                . . .
                              .   .   .
                             C    H    H
                           . . .
                         .   .   .
                        C     H    H
                      . . .
                    .   .   .
                   C    H    H
                 . . .
               .   .   .
              H    H    H

The theoretical study of the connectivity of chemistry is called Chemical Graph Theory and is important in areas such as computer assisted chemical synthesis and bibliographic searching.

Polar Coordinates

                          z
                          |
                          |
                          |               * (atom)i
                          | theta(i)    . .
                          |           .   .
                          |))       .     .
                          |  )    . r(i)  .
                          |   ) .         .
                          |   .           .
                          | .             .
                          0------------------------y
                         /   .            .
                        /     ( .         .
                       /(   ((     .      .
                      /  (((          .   .
                     /   phi(i)          ..
                    /                     .
                   /
                  /
                 /
                /
              x

Notice that there is a handedness to the axis system, following the right hand rule. The $z$-axis, (the principal axis), is the thumb, first finger gives $x$, second finger $y$. This convention is best adhered to as though left handed coordinate systems will work if consistent great trouble can be caused by stereochemistry, (proteins are made of only L-amino acids|).

Carbon Dioxide

                      :z
                      :
                      :
                      :
       00             CC           00
      0  0 ----------C  ----------0  0  ----------x
      0  0 ----------C  ----------0  0
       00 1           CC1          00 2
                    /
                   /
                  /
                 /
                y

By inspection the cartesian coordinates can be written down

	x	y	z
C	0	0	0
O1	r	0	0
O2	-r	0	0

The symmetry point group is $D_{\infty h}$ but in quantum chemistry calculations often the algorithms require the use of Abelian, (~non degenerate~), groups only so $D_{2}h$ is used for carbon dioxide. ($D_{2}h$ is the ab-initio quantum chemist's favourite group as it has 8 irreducible representations making the calculation much smaller than the more common $C_{s}$ or $C_{2}v$ groups, e.g. Quinone with only two unique heavy atoms. The principal axis of rotation is almost always defined as the $z$-axis.

Methyl Chloride

In methyl chloride the C-Cl bond is 1.8 angstroms the C-H bond 1.094 angstroms and the CL-C-H angle is 110.6667 degrees. in the diagram the Z-axis is along the C-CL bond.

                       CL
                        :
                        :
                        :
                        :
                        :
                        C --------------------y
                       /=.
                      / = .
                    //  =  ..            (x out of page)
                   //   =   . .
                 / /    =    . .
                / /    H(3)   . .
               /-/             .-.
              H(1)             H(2)

The obvious atom to make the origin is C. Cl will then be 1.8 up the $z$-axis. All the hydrogens will have the $z$-coordinate $1.094~*~\cos ~110.67$ i.e. $-~(1.094~*~\cos ~69.33)$.

H(3) has a $y$ coordinate zero and an $x$ coordinate negative, (that is in a right-handed cartesian framework, see the Polar Coordinates diagram).

It is best always to use the conventional right handed axes. The memory trick is to think of a sheet of graph paper, $x$ is across as usual and $y$ up the paper. Positive z then comes out of the paper. Interestingly failure to get this right does not affect any results of calculations other than those of optical activity etc. but a more serious problem is when coordinates are used as building blocks. For instance if you combined a set of amino-acid coordinates from your calculation with some from a crystal structure or the molecular model builder you could inadvertently end up mixing D and L forms in the same peptide.

So the above negative $x$ value will be $1.094~*~\sin ~69.33$. Now remembering this value as '$xc$', we can easily get the $x$ and $y$ coordinates of H2 and H3. These are related by rotations of 120 degrees and have identical coordinates except for a change of sign in the $y$-direction. Now the $y$-coordinate will be + or - $xc~*~\sin ~120$. The $x$-coordinate is cos~120.

Useful Table for Remembering Common Trig Values

0	0	90
30	${\frac {1}{2}}$	60
45	${\frac {1}{\sqrt {2}}}$	45
60	${\frac {\sqrt {3}}{2}}$	30
90	1	0

Sines read down the table and cosines up.

With the MACROMODEL modelling system one would bet the icon for CH4 and prod one of the hydrogens with the 'Cl' icon. The MM2 force-field could then be used to optimise the geometry.

Benzene

The calculation of the geometry for benzene is greatly simplified by placing the origin at the centre of symmetry. The carbon atoms are 1.39 angstroms away from the centre of symmetry and the hydrogens are 2.47 angstroms away. The triangle formed between two adjacent carbons and the centre of symmetry is an equilateral triangle. Cos and sin of 60 can be used to calculate the coordinates of H2 and C2. All the rest are related by sign changes.

        Y                                      H1
        :                                      *
        :                                      *
        :                         H6           *            H2
        :                          *          1*           *
        *                            *       *   *1.39   *
      * : *                            *   *       *   *
    *   :   *                            *6         2*
  *     :     *                          *           *
  *     :.....*.........X                *           *
  *           *                          *           *
  *           *                          *5         3*
    *       *                          *   *       *   *
      *   *                          *       * 4 *       *
        *                          *           *           *
                                  H5           *            H3
                                               *
                                               *
                                               H4

                 (Z-axis vertically out of page)
 Benzene
 Centre          0.000    0.000    0.000
 C               0.000    1.390    0.000
 H               0.000    2.470    0.000
 C               1.204    0.695    0.000
 H               2.139    1.235    0.000
 C               0.000   -1.390    0.000
 H               0.000   -2.470    0.000
 C              -1.204   -0.695    0.000
 H              -2.139   -1.235    0.000
 C               1.204   -0.695    0.000
 H               2.139   -1.235    0.000
 C              -1.204    0.695    0.000
 H              -2.139    1.235    0.000
 **

One of the problems with the above coordinates which I worked out with a calculator is the precision. The atoms on the axes have infinite precision, (~zero|~), whereas the atom positions worked out have only 4-figures, (atoms 2, 3, 5, and 6). This manifests itself as a slight slippage from the full $D_{6}h$ symmetry, which will cause degenerate orbitals and equal atomic properties to be not quite equal.

Such unexpected occurrences in computational results are very common though the effects of only finite precision on both input data and computation are not usually serious. In quantum chemistry we need larger than usual accuracy, obtained by double precision arithmetic, because we are concerned with differences in energy between entities which contain large amounts of core electronic energy which 'uses~up' the left hand significant figures of the total energy. (Compared with high energy phenomena most of chemistry is concerned with small differences in energy.) If you were working out the coordinates of a molecule containing a phenyl group you would probably use the molecular modelling system but there are cases where there is high symmetry where you would start from highly symmetrical coordinates as above.

Projections

Newman Projection                             Sawhorse Projection
*****************                             *******************
                                                                   .H6
                                                                  .
               *)                                                .
               *  )                                             * C2
               *    )                                         .. .
               * 27  )   @                         H1      . .   .
  @           .* .    )@                           .    .  .      .
    @      .   *    .@                             .   H4.        .
      @        *)                                  .   .           H5
        @.     *  )   .                            . .
               *120                             C1 *
        .      *   )   .                         .   .
             *   *)                            .       .
         . *       *  .              Z       .           .
         *           *              /       H2            H3
      *    .        .  *           /
    *         .  .       *        /
  *          @             *     /
                                :------X      Z       Y
            @                   :        (    ^      /         )
                                :        (    ^    /           )
           @                    :        (    ^ /              )
                                Y        (    *---------X      )
                                         (                     )

Newman projection is the most useful for quantitative working with dihedral angles. Programs exist to print out the dihedral angles from crystal structures or MACROMODEL coordinates.

Local Origins in Ring Systems

It is often desirable to describe a subsystem and shunt it into geometrical position. This rotation / translation process can be describes by a vector and three Euler Angles. A very suitable orientation for a model fragment is to make any aromatic part as planar as possible, (perhaps by least squares fitting). The origin of the cartesian system can be set to be the centre of the ring defined by the centre of mass assuming all the ring atoms to be carbon and no other atoms considered). The reason for this is obvious if one thinks of a C₆H₄I- group. The centre of mass would be in the C-I bond and the centre of the ring system is a much more practical origin.

There are many ways of defining the Euler Angles. QM uses the three angles $\alpha ,\beta$ and $\gamma$ which are rotations about $z$, $y^{'}$ then $z^{'}$ (i.e. $z$, new $y$, then the new $z$ axis.

$\cos(\alpha )$	$\sin(\alpha )$	0
$-\sin(\alpha )$	$\cos(\alpha )$	0
0	0	1

then

$\cos(\beta )$	0	$\sin(\beta )$
0	1	0
$-\sin(\beta )$	0	$\cos(\beta )$

then

$\cos(\gamma )$	$\sin(\gamma )$	0
$-\sin(\gamma )$	$\cos(\gamma )$	0
0	0	1

Here is a piece of FORTRAN code for the resulting transformation matrix from the product of these three rotations.

  tran (1,1) = (cos (alpha) * cos (beta) * cos (gamma)) _
  - sin (alpha) * sin (gamma)
  tran (2,1) = (cos (alpha) * cos (beta) * sin (gamma)) _
  + (sin (alpha) * cos (gamma))
  tran (3,1) = - cos (alpha) * sin (beta)
  tran (1,2) = - ( (sin (alpha) * cos (beta) * cos (gamma)) _
  +  ( cos (alpha) * sin (gamma))  )
  tran (2,2) = - ( sin (alpha) * cos (beta) * sin (gamma) ) _
  + ( cos (alpha) * cos (gamma) )
  tran (3,2) = sin (alpha) * sin (beta)
  tran (1,3) = sin (beta) * cos (gamma)
  tran (2,3) = sin (beta) * sin (gamma)
  tran (3,3) = cos (beta)

The optimization

Once you have your trial geometry, made by a graphical program or by trigonometry as describes previously one of the powerful quantum or molecular mechanics programs will solve the geometry by minimizing the (3N-6) free coordinates.

In many cases this does not give a unique geometry only one of many possible conformations.

Next Chapter - Applications of molecular quantum mechanics

Applications of molecular quantum mechanics

Previous chapter - Geometry optimization

Frontier Orbital Theory

(The Fukui Theory of Reactivity and Selection)

One of the most important pieces of information which can be obtained from the molecular eigenvectors is the Fukui Reactivity Indices. For normal conditions chemical reactivity there are two orbitals which are specially important, the occupied orbital which is highest in energy and the unoccupied orbital which is lowest in energy. These are known as the Frontier Orbitals as they are either side of the occupied / virtual divide. (The empty orbitals which are produced from diagonalising the Fock-Matrix are known as the virtual orbitals.)

The terminology your will come across is the HOMO, Highest Occupied Molecular Orbital, and the LUMO, Lowest Unoccupied Molecular Orbital. Initially nucleophiles attack molecules by placing their surplus electrons, (typically a lone- pair orbital), into the LUMO. The attacking lone pair orbital is then involved as an electron donor and the LUMO an electron acceptor. This is a description in the terminology of Perturbation Molecular Orbital theory. Of course there is a symmetry here in that what is nucleophilic attack by one molecule is electrophilic attach by the other.

Chemical sense is used to decide which label to place on the reaction but we now have a quantitive way to label the reaction in terms of the HOMO and LUMO energies of the reacting species.

In the Fukui theory reactivity indices are calculated from the square of the coefficients of the frontier orbitals. The nucleophilic reactivity indices come from the LUMO coefficients and the electrophilic from the HOMO. For free radical attack the indices are

$(c_{(HOMO)}^{2}+c_{(LUMO)}^{2})/2$

Hückel Calculation on Pyridine Output

  ***************************************************************
  * Pyridine                                                    *
  ***************************************************************

  N C2/C2 C3/C3 C4/C4 C5/C5 C6/C6 N/*
  E(N) = 1.5
  **


  ATOM LABELS
  ***********

      N           C2          C3          C4          C5

      C6


  ******************************
  *                            *
  *          6 ATOMS           *
  *                            *
  *          6 ELECTRONS       *
  *                            *
  ******************************





  SECULAR DETERMINANT
  *******************

   N    C2   C3   C4   C5   C6

  N        E    1    0    0    0    1

  C2       1    X    1    0    0    0

  C3       0    1    X    1    0    0

  C4       0    0    1    X    1    0

  C5       0    0    0    1    X    1

  C6       1    0    0    0    1    X



  ENERGIES IN TERMS OF BETA
  *************************

         1           2           3           4           5

      2.525982    1.430795    1.000000   -0.594211   -1.000000

         6

     -1.862566






  EIGENVECTOR MATRIX                                        PART    1


         1           2           3           4           5


  N          0.754598   -0.432035    0.000000   -0.443012    0.000000
  C2         0.387102    0.014950   -0.500000    0.463881   -0.500000
  C3         0.223215    0.453424   -0.500000    0.167369    0.500000
  C4         0.176735    0.633808    0.000000   -0.563333    0.000000
  C5         0.223215    0.453424    0.500000    0.167369   -0.500000
  C6         0.387102    0.014950    0.500000    0.463881    0.500000



  EIGENVECTOR MATRIX                                        PART    2


         6


  N         -0.218330
  C2         0.367074
  C3        -0.465370
  C4         0.499708
  C5        -0.465370
  C6         0.367074


  TOTAL ENERGY IN TERMS OF BETA =     9.913554
  ********************************************

  EXCITATION ENERGY       1.594 BETA
                  4.46 eV
                277.79 nM
                 430.7 kJ per Mole


  CHARGE DENSITIES WRT 1 ELECTRON PER ORBITAL
  *******************************************

      N           C2          C3          C4          C5

     -0.512145    0.199857   -0.010837    0.134105   -0.010837

      C6

      0.199857





  BOND ORDERS
  ***********
  N    --- C2           0.5713
  N    --- C6           0.5713
  C2   --- C3           0.6864
  C3   --- C4           0.6537
  C4   --- C5           0.6537
  C5   --- C6           0.6864


  CHARGE DENSITIES IN EXCITED STATE
  *********************************

  ( WTR 1 ELECTRON PER ORBITAL )


      N           C2          C3          C4          C5

     -0.708404    0.234672    0.211150   -0.183239    0.211150

      C6

      0.234672





  BOND ORDERS IN EXCITED STATE
  ****************************
  N    --- C2           0.3658
  N    --- C6           0.3658
  C2   --- C3           0.5140
  C3   --- C4           0.5594
  C4   --- C5           0.5594
  C5   --- C6           0.5140

The reactivity of pyridine is discussed in Streitwieser and Heathcock p1022 onwards. It can be seen from the charge densities that there is positive charge at the ortho (2,5) and para (4) positions, negative charge at the meta (3) position. Electrophiles overwhelmingly attack at the meta position. Both the charge density and the frontier orbitals reinforce here. (The HOMO orbital has equal 0.5 coefficients at ortho and meta positions, but ortho is deactivated by being positively charged.

Nucleophiles attack the LUMO orbital at C2 or C6 (ortho). The coefficients of LUMO at ortho and para are about equal but the ortho position has a greater positive charge which presumably makes it more attractive to electrophiles. (This discussion is about the beginning of the reaction. Sometimes factors which affect the transition state are more important and the predictions from charge densities etc. may not apply when the reaction has proceeded nearer the energy barrier.) In many aromatic systems this kind of frontier orbital analysis works well though. With respect to our discussions of ab-initio quantum chemistry often working in an empty universe with only two molecules involved, it is quite encouraging that Hückel Theory can give rationalizations of reactions which are going on in nitrating mixture, (conc. sulphuric and nitric acids), at 100C plus.

Charge Transfer Interactions

The so called charge transfer complexes are particularly amenable to frontier orbital analysis. In these complexes the acceptor's LUMO is energetically and geometrically accessible to the donor's HOMO and a small fraction of an electron (typically about 0.05) is passed across. The complex has more conjugation than the separated systems and this often moves the lowest excitation into the visible region so charge-transfer complexes are frequently strongly coloured. ( Streitwieser and Heathcock P.832.

Properties Which can be Calculated by Quantum Mechanics

Molecular Geometries and Force Constants.

We have talked about this previously. It is of overriding importance as molecules spend most of their time in the ground state equilibrium geometry, so molecular geometry optimization is usually the first process in any modelling calculation. The force constants lead to the normal coordinates which describe the molecular vibrations. The traversing of a reaction coordinate is sometimes described in rather sophisticated treatments as a molecular vibration of the super-molecule. (When calculating interactions between two molecules the geometry which represents the two molecules in proximity is called the supermolecule. The interaction energy is the energy of the supermolecule minus the sum of the energies of the two free species.)

Atomic Population Analysis and Dipole Moments.

Atomic population analysis allows the assignment of $\delta ^{+}$ and $\delta ^{-}$ charges to the component atoms. This is the charge separation caused by the different electronegativities of atoms in different environments. e.g. Azulene which has only carbon and hydrogen atoms in it but has a largeish dipole moment caused by quantum effects. This chemical phenomenon is even accessible to the simplest quantum method, H\"uckel Theory.

As well as the contribution to the dipole from atomic charges there is another contribution caused by distortion of charge on each atom. The semi-empirical programs separate the dipole into these components by labelling them 'POINT-CHARGE' and 'HYBRID' contributions.

Extract from Nitrobenzene MOPAC / AM1 Run

(Mopac see (Stewart, 1990).)

    NET ATOMIC CHARGES AND DIPOLE CONTRIBUTIONS

    ATOM NO.   TYPE          CHARGE        ATOM  ELECTRON DENSITY
    1          C          -0.0880          4.0880
    2          C          -0.1397          4.1397
    3          C          -0.0673          4.0673
    4          C          -0.1312          4.1312
    5          C          -0.0678          4.0678
    6          C          -0.1394          4.1394
    7          H           0.1443          0.8557
    8          H           0.1483          0.8517
    9          H           0.1711          0.8289
   10          H           0.1710          0.8290
   11          H           0.1484          0.8516
   12          N           0.5674          4.4326
   13          O          -0.3586          6.3586
   14          O          -0.3585          6.3585
    DIPOLE           X         Y         Z       TOTAL
    POINT-CHG.    -2.598    -4.652    -0.037     5.328
    HYBRID         0.044     0.078     0.001     0.090
    SUM           -2.554    -4.574    -0.036     5.239

Dipoles are often caused by electronegativity differences between the component atoms of the molecule. In some cases however dipoles are caused by entirely electronic quantum effects where the molecule is composed of nothing but carbon and hydrogen. A good example of this is azulene, a fused pair of a five and a seven membered ring. The H\"uckel 4~N~+~2 rule predicts the stability of C6H6 (benzene), C7H7(+) (Tropylium), and C5H5(-) (cyclopentadienyl). The chemistry of these compounds is in no conflict with the theory, C5H5(-) being a stable very common ligand in inorganic chemistry, (it is abbreviated Cp), Ferrocene Fe(C5H5)2 being the first such 'sandwich' compound to be made. Tropylium forms stable ionic salts, e.g. C7H7(+)Br(-).

  HUCKEL CHARGE DENSITIES FOR AZULENE
  ***********************************

  C1          C2          C3          C4          C5

  -0.027428   -0.027428    0.145054    0.013553    0.129999

  C6          C7          C8          C9          C10

  0.013553    0.145054   -0.172879   -0.046600   -0.172879


                     .
                    .                         .
   ....              3(+)                        ..
      .. 4... ... ...  ...                    ..
        .                 ...                10(-)
       .                     ... 2... ... ... ..
      .                          .              ..
     .                           .                .
  ........5(+)                         .                 ..9........
    ..                           .                 ..
     ..                          .               ..
      ..                         1              .
       ..                    ...  ... ... ... 8(-)
        .6                ...                 .
      ... ... ... ...7(+).                    ..
    ..                .                        ..
                    ..




Note the alternation of charges with bond connectivity, which
is typical of quantum effects.



  BOND ORDERS
  ***********
  C1   --- C2           0.4009
  C1   --- C7           0.5858
  C1   --- C8           0.5956
  C2   --- C3           0.5858
  C2   --- C10          0.5956
  C3   --- C4           0.6640
  C4   --- C5           0.6389
  C5   --- C6           0.6389
  C6   --- C7           0.6640
  C8   --- C9           0.6560
  C9   --- C10          0.6560

Electronic Spectra

Electronic spectra can be investigated by several methods the simplest of which of which have the disadvantage that the wavefunction being used is better for the ground state of the molecule than the excited state and this introduces some imbalance into the calculation. Many ab-initio programs now have MCSCF calculations to explore such things analytically. It is however half a PhD project to do it properly.

Ionisation Potentials

Ionisation potentials are often calculated by the approximate Koopmans' Theorem. This says that the ionisation potential of an electron in a given orbital is just minus the orbital energy. This is simple both to calculate and to interpret but incorrect in that it takes no account of relaxation processes which occur. When ionisation has happened the remaining N-1 electrons can relax to see more of the positive nuclei rather than just staying where they are. This relaxation is effectively instantaneous on the laboratory timescale and so the Koopmans energies are systematically too small. This is the inverse of the process which makes excitation energies in general too large when calculated in the virtual orbital (VO) approximation.

The experimental technique for looking at ionization potentials is photoelectron spectroscopy. This comes in several varieties with different acronyms such as ESCA. The core (1s) ionization potential, despite being tightly bound to the nucleus and taking no part in chemical bonding is still subject to a chemical shift. The simplest example of this is the azide anion, N₃. It is charged

   - + -
   N=N=N

leading to two photoelectron spectrum peaks at -399eV and -405eV in the ratio 2:1. The practical non-SI unit of spectroscopy at this energy range is the electron volt (eV), 1 eV = 96.4853 kJ per mole, and is about a quarter of the energy needed to break a chemical bond.

Electron Affinities and Excited States

The same problem of using MO coefficents optimised by the SCF procedure for the ground state configuration applies to the Electron Affinity and to using the virtual orbitals as prototype excited states. Nevertheless this is something which we often do as it is the simplest picture and can be worked on using our readily available ground state semi-empirical wavefunction.

The definitive solution to the above problems is through the use of the electron and polarization propagators.

Problems

naphalene and azulene

Use a molecular gui like Macromodel make trial geometries for naphalene and azulene. Optimise the geometry. Do a single point calculation at the Huckel level and at the AM1 semiempirical level. (A single point calculation is the quantum mechanics at a pre-specified geometry, in this case the MM2* geometry, often either i) the gas phase experimental geometry, ii) the crystal structure geometry from X-ray or neutron diffraction.

Look at the orbital energies and node patterns. Locate both crystal structures on the database and compare their characteristics with the MM2* geometry. Does azulene have C2v symmetry? It should have experimentally in the gas phase but some methodologies break the symmetry and produce only Cs symmetry with alternating single and double bonds. There is a paper on the solution of this problem at the Hartree-Fock level and with correlated methods. (Hartree-Fock gets it wrong.) MP2 and above is right. If you were sufficiently interested you could use a literature search program to find out if the correlated single-determinant method DFT, (Density Functional Theory), gets this structure right.

VSEPR and electronic properties mini project

This is a mini-project rather than a short problem and involves running the abinitio program of your choice.

The motivation of this computer experiment is to draw together various threads from chemistry about shape, symmetry, electronegativity and electric moments so you will have to consider things like why UF₆ is a gas and water and HF hydrogen bond but in the context of the rather dry looking numbers from computer programs which actually mean a lot and can match experiment.

The experiment involves calculation of the electronic structures of CH₄, NH₃, H₂O, HF, BeH₂, BF₃, H₃N-BF₃, SF₆, XeF₄, XeF₄O and IF₇.

In order to complete this experiment you will need access to a modern molecular quantum mechanics program. You will need to do a bit of literature work and use some common sense to get starting points and experimental numbers for comparison with theory. Starting molecular geometries can always be either calculated or estimated as the ability to guess what the value of a number might be approximately is a neccessary scientific skill. (Crude examples of this are estimating a bond length as the sum of the covalent radii of the atoms, regardless of the environment of the atoms in the current molecule or saying that the chemical shift of an atom is the usual average for that functional group, in the same way as we habitually use average bond energies.) Reference where you get your covalent radii from, there are several sources. You can start with the NH₃, BF₃ and H₃N-BF₃ and IF₇ (the difficult unusual molecule).

Unless you have a GUI you will need to learn a little bit about a file editor to prepare data files. vi is one of several, pico another. You typically prepare a file for running and submit it to the batch system. (Though many calculations will run in seconds, when preparing work for publication you must always use the best calculation you can afford, and these are typically several hours or even days long.) The quality of the calculation depends on the number of atom based basis orbitals you put in and the cost of the calculations typically increases by the number of basis orbitals to the power 3 and a bit. (You double the basis and the calculation takes 8 times longer.) In a Self Consistent Field (SCF) calculation the usual procedure is to store the many electron repulsion integrals created by the 1/r_(1,2) operator in the Hamiltonian on disc. This disc space then becomes a limiting resource which stops the improvement of the calculation. If you want to use lots of f-orbitals on I and d-orbitals on F in IF₇ it will fail by filling the disc. Even on NH₃ you might use the unfilled d-orbitals in the basis because they allow the nitrogen atom (Another way of looking at this is that the shape of the lone pair and to a lesser extent the tighter bonding orbitals is improved by the extra atomic orbitals.) to be polarized in its bonded environment. There is a mode of execution of most quantum chemistry programs called direct. In this case electron repulsion integrals are calculated on the fly and not stored. This causes the run to take much longer but not to require very much disc space. (This same technique is always used when calculating the 3N-6 position derivatives of electron repulsion integrals during a geometry optimisation as by using a matrix inversion technique these are only required once and not iterated with as in a SCF calculation). If you are really fanatical about doing the definitive calculation on a molecule you might have to use direct and a 32 hour run to use the largest basis to get the best possible calculation. This is not however necessary to get something good out of this experiment.

Estimate the bond lengths in these molecules by adding together the covalent radii. Find the experimental geometries of as many of them as you can. (One of the best compilations of geometries is Landolt-Börnstein. Use your abinitio program to optimize the geometries of the ones you cannot find an experimental geometry for. Here is the experimental geometry of XeF₄O below. The units are Angstrom, 10 to the -10 of a metre. You will notice the experimental precision of the Xe=O bond length, and the artificially high precision of the fluorine coordinates which have been subjected to a trigonometrical calculation using 64-bit floating point accuracy! (You will also notice the sloppy way the editor has not recorded the terminal page number of the pre-internet paper and so you might have to go to the library and look it up at some time.)

XeF4=O   J.F. Martins, E.B. Wilson,  J Mol Spectrosc 26, 410 (1968)
#   Xe=O   1.703   Xe-F  1.900  O=Xe-F 91.8degree
O               0.0000000000    0.0000000000    1.703
XE              0.0000000000    0.0000000000    0.0000000000
F               1.8990624647    0.0000000000   -0.0596804422
F              -1.8990624647    0.0000000000   -0.0596804422
F               0.0             1.8990624647   -0.0596804422
F               0.0            -1.8990624647   -0.0596804422
**

Preparing data for programs like GAUSSIAN, DALTON, GAMESS or CADPAC can be difficult If you have to use the raw data structure. Most programs now have a GUI by which the data preparation can be simplified, even automated.

Answer the following questions in a short report. Give some references and use a style which includes the paper titles and both starting and ending page numbers, (just to get used to the fiddly style used by some journals which waste your time if you loose the end page number!). Many Web and CD journals use this style as they are not wasting paper but it actually does make the reference list easier to navigate.

(Q1) Report the first non-zero moment of these molecules i.e. if the molecule has a dipole moment, the dipole, if it has a no dipole, the quadrupole. Two of these molecules are so symmetrical that they have no dipole or quadrupole. What is the name of the next moment, (rather obscure).

Note that for the dipole moment of XeF₄O it is not obvious which direction the dipole will point as both fluorine and oxygen are electronegative and the O-Xe-F angle is close to 90 degrees. However as there is no symmetry along the O=Xe bond so there must be a non-zero dipole.

(Q2) What effect does the lack of a dipole or quadrupole have on the physical properties of these molecules?

(Q3) If there is no dipole moment what forces cause the molecule to condense from gas to a liquid, (or in the case of CO₂ straight to a solid at 1 atmosphere). What is the relevance of this to the separation of the isotopes of Uranium? Which two factors fortunately make this extremely difficult, (hint. somebody's Law is important in one, how you make Uranium a gas in the other)?

The orbital energies are an approximation to the ionization potential. (Technically this is known as Koopmans' Theorem.) The orbital energies are usually in atomic units, but experimentalists use the electron volt (eV).

(Q4) See which electron comes out first in NH₃, HCl, IF₇ and XeF₄. Comment on what you expected and what actually happens.

(Q5) SF₆ is 23900 times more potent as a greenhouse gas than CO₂. Calculate its infrared spectrum and look at the orbital energies to see how the binding of electrons influences the molecule's reactivity. Examine the vibrational normal modes with respect to the group tables in a book such as Atkins Physical Chemistry or Cotton Group Theory. See how the degeneracies in the group table correspond to both the vibrational degeneracies and the degeneracies in the ionization potentials and therefore the photoelectron spectrum.

Confirm the VSEPR rules by starting BF₃ and AlF₃ from trigonal geometries like ammonia. Hopefully this optimization will go planar! Calculate the vibrational spectrum of these molecules in their correct geometry.

(Q6) Calculate the vibrational spectrum of flat ammonia and see what happens.

There are methods of estimating the moments of these molecules from electronegativity and shape alone. In general there are two geometries of the molecule we are interested in. The experimental equilibrium geometry and the optimised geometry of the biggest calculation we can afford.

(Q7) Calculate the proton and fluorine shieldings in the molecules which have F and H.

(Q8) Calculate the (H,H) spin-spin coupling constant in NH3 and the (F,F) constant in XeF₄, using the SCF wavefunction (as standard). If you have the computing capacity you could calculate the (H,H) spin-spin coupling constant with an expensive electron correlated calculation, which is actually necessary to get good answers for this exotic property.

Some Experimental Geometries (VSEPR)

NH3 <HNH> 104.9067 theta= 113.7213 R=0.1.024275 G.H.F.D.&A.J.S, J.C.P. 75,1253.
N    0.0            0.000000    0.1449084
H1    0.93773726    0.0           -0.26714530
H2             -0.4688686300    0.8121042892   -0.2671453000
H3             -0.4688686300   -0.8121042892   -0.2671453000
**

BF3   Kuchitsu K, Konaka S. JCP 45 4342 (1966)
B               0.0000000000    0.0000000000    0.0000000000
F               1.3120000000    0.0000000000    0.0000000000
F              -0.6560000000    1.1362253298    0.0000000000
F              -0.6560000000   -1.1362253298    0.0000000000
**


AlCl3 2.049 V.P. Spiridonov, A.G. Gershikov, E.Z. Zasorin, N.I. Popenko,
# A.A.Ivanov, L.I. Ermolayeva High. Temp. Sci. 14 (1981) 285.
Al 0 0 0
Cl  2.049
CL             -1.0245000000    1.7744860524    0.0000000000
CL             -1.0245000000   -1.7744860524    0.0000000000
**

Notice that when dealing with geometries of ammonia and phosphine the most useful angle is the lone pair to X-H angle, (theta), not the H-N-H angle.

Bibliography for VSEPR

(1) A Chemistry VSEPR website: http://www.shef.ac.uk/chemistry/vsepr/

(2) Different editions of J. E. Huheey, Inorganic Chemistry.

(3) C. J. Cramer, Essentials of computational chemistry: theories and models, (Chichester, John Wiley, 2002).

(4) F. Jensen, Introduction to Computational Chemistry, (Wiley, Chichester, 1999).

(5) The DALTON website: http://www.kjemi.uio.no/software/dalton/dalton.html

(6) Landolt-Börnstein, New series II, Volume 7, Structure Data of Free Polyatomic Molecules, ed. K. H. Hellwege and A. M. Hellwege (Springer, Berlin, 1976). Landolt-Börnstein, New series II, Volume 15 Supplement to Volume II/7, Structure Data of Free Polyatomic Molecules, ed. K. H. Hellwege and A. M. Hellwege (Springer, Berlin, 1987).

References

• J. J. P. Stewart, J. Comp. Aided Mol. Design, Vol. 4, 1 (1990).
• A. Streitwieser and C. H. Heathcock,Introduction to Organic Chemistry, Third Edition, p.236, (Macmillan,New York,1985).

Next Chapter - Applications of molecular modelling

Data bases and modelling

Previous chapter - Chemical informatics

This is a stub entry for databases

Examples of chemical databases are: CSD (the Cambridge Structural Database), ICSD (Inorganic Crystal Structure Data files), PDB (the Brookhaven Protein Data Bank), ELYS (Electrolyte Solutions Database), SPECINFO (IR and NMR Spectroscopy) and ACDRX (Fine Chemicals).

There are extensive organic synthesis data bases such as Comprehensive Heterocyclic Chemistry, Theilheimer Database, Chiras (Asymmetric Synthesis and SPG protecting group database.

Next Chapter - Macromolecular chemistry

Drug Design

Previous chapter - Macromolecular chemistry

Computer Aided Drug design

Quantum Pharmacology

This area covers the use of electronic structure theories to de novo design of drugs, extraction of structure activity relationships and development of pharmacophores to rationalise the drug's mechanism. Computations, proteomics and genomics have advanced considerably since Richard's book and maybe new definitions are needed.

W. G. Richards, Quantum Pharmacology, (Butterworths,London,1977).

QSAR

A lot of work goes into developing Quantitative Structure Activity Relationships. This can be done by regression analysis even where there is no adequate model of the active site of the process. Some examples are being extracted from literature which we have in the Chemistry Library, (much literatuire of relevance is in biological and pharmacological journals. Some QSAR work uses classic physical organic chemistry such as the Taft and Hammett equations, whereas in recent work all manner of unusual properties may be incorporated into a QSAR.

There is even interest in exotic artificial intelligence technologies such as neural nets. A new technique in this area is known as Support vector machines.

Hansch, C. and Leo, A. Exploring QSAR: ACS Professional Reference Book, (American Chemical Society, Washington, DC,1995).

Neural Nets to Design Drugs

This is a comparatively new area of chemistry / chemo-informatics but it already has a classic textbook:

J. Zupan and J. Gasteiger, Neural Networks in Chemistry and Drug Design: An Introduction, 2nd Edition, Wiley, 1999, ISBN: 3-527-29779-0.

Bibliography

• edited Andrew Vinter and Mark Gardner Molecular modelling and drug design, (Boca Raton:CRC Press,1994), ISBN 0849377722.

Next Chapter - Continuum solvation models

Continuum solvation models

Previous chapter - Drug Design

Introduction to Continuum Solvation Models

Most reactions we are concerned with actually go on in some kind of fluid medium rather than in the gas phase at zero degrees Kelvin. (There is the interesting but impractical philosophical point that the commonest reaction in the universe is H2 + proton >> H3+ at 3 degrees Kelvin, (the remnant of the big bang temperature), and very low pressure. Most of the universe is made of Hydrogen and most of its matter is in the vast spaces between the stars in galaxies.)

There are ways of dealing with the solvent medium problem, none of them fully satisfactory but some of them very good for dilute solutions. Read the electrochemistry and ionic atmosphere sections of Atkins' Physical Chemistry for a detailed discussion of some of the issues.

Continuum plus Quantum Mechanics

Our method of choice will abandon a description of the solvent as discrete molecules but will replace it by a continuous dielectric. This is like a jelly which can be electrostatically strained by non zero potentials which both screen and interact with the quantum part of the system.

The Dielectric Constant

The dielectric constant, a dimensionless number which is a ratio of electrical permittivities against the permittivity of free space determines the chemical characteristics of the solvent. Large numbers are polar solvents. Small numbers apolar. Large numbers mean that the Coulomb terms are attenuated and though the energy still goes off as 1/r the values are smaller than in free space.

The following table is a conflation of data from the following primary and secondary sources which need to be consulted over the precise values to use in research.

Citations for Dielectric Constant Sources

M. Witanowski, W. Sicinska, Z. Biedrzycka, Z. Grabowski, and G. A. Webb, J. Chem. Soc., Perkin Trans. 2, 619 (1996); A. d' Aprano, A. Capalbi, M. Iammarino, V. Mauro, A. Princi, B. Sesta, J. Solution Chem., Vol. 24, 227, (1995); P. W. Atkins, Physical Chemistry, Oxford University Press (4th Edition); and CRC Handbook of Chemistry and Physics, ed. D. R. Lide, 77th edition, (CRC Press, Boca Raton,1996).

The numbers do not always agree, even though they nominally have up three decimal points of accuracy. A single list of recommended numbers has somewhat arbitarially been chosen from the above sources.

Table of Relative permittivities, (dielectric constant), at 25 deg. C.

Free space	1.00	Ammonia (liquid)	16.9~
Methane	1.70	Acetone	19.75
Cyclohexane	1.87	Ethanol	24.20
Dioxane	2.19	Methanol	30.71
Carbon tetrachloride	2.21	Nitrobenzene	34.82
Benzene	2.25	CH3CN	36.05
Diethyl-ether	3.89	CH3NO2	36.48
CHCl3	4.806	DMSO	45.80
CH2Cl2	8.54	Water	78.54
Hydrogen sulphide (liquid)	9.26

The molecule lives in a shape where the dielectric is 1, i.e. free space, and outside the quantum zone a continuous dielectric with a permittivity chosen according to the system being modelled. The energies of interaction are most negative when the dielectric constant is high. This can be rationalized by image charges in the dielectric being large near the junction and are screened away rapidly but are near to their opposite values in polar groups in the molecule. Clearly energy per atom, when decomposed will follow the atoms with highest Mulliken charge population.

Such a calculation would allow the prediction of the $\Delta H$ of the acetic acids discussed earlier but of course would still give the wrong answer because of the entropic terms being entirely absent from the model.

The solvation model for Macromodel uses functional group derived charges but of course explicitly uses the molecular mechanics shape of the molecule.

Dielectric Screening

For some molecular mechanics application, particularly the calculation of long-range Coulomb forces inside macromolecules, a dielectric constant other than 1 may be used. (A constant of 4 is often used in protein modelling). A true quantum treatment would allow for the internal screening by the electrons but molecular mechanics can only deal with this by some kind of fix. Typically a dielectric constant equivalent to an alkane is used.

Next Chapter - At the edge of Biology, Genomics and Proteomics

At the edge of Biology, Genomics and Proteomics

Previous chapter - Continuum solvation models

The latest method of obtaining lead molecules for drug design involves using genome information to generate the virtual proteins and then generate inhibitors for the proteins by molecular mechanics / dynamics methods. The problem is there is often not enough knowledge of what the proteins coded for actually do. So far the computer programs can generate more lead molecules than can be sensibly tested by the traditional cell assay methods.

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