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# Molecular mechanics

** Previous chapter - ** Computational Chemistry

### Introduction

editA 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 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 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

editA formal description of a typical force field follows.

harmonic force constant.

optional anharmonic force constant.

angle bending constants.

Fourier representation of torsional potential.

Out of plane bending for trigonal groups.

Van der Waals terms.

electrostatic interactions.

#### What do the force field components mean?

editLet us first consider bond lengths. The most important number is 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

editThe 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 term in the pair energy is too steep and a softer repulsive potential the Buckingham potential is used:

#### The Buckingham potential

edit

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

#### The Lennard-Jones potential

edit

is the energy at the pair minimum.
The location of the pair minimum is at *i.e.*
.

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 is
often expressed using the
2 parameters of an energy and a distance .

### Table 1 - The MM2 atom types

editU. 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 sp^{3} | 16 | S (+)onium |
---|---|---|---|

2 | Carbon sp^{2} | 17 | S (>S=O) |

3 | C (sp^{2})(>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 (sp^{3}) | 23 | H (H-N)Amin |

9 | N (sp^{2}) | 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

editi) 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

edit- 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

edit- 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**

edit
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:

s is a new position.
Force is . then run for *n* timesteps.

Verlet algorithm

add these together then :

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

editApplications 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

editSecs | 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 10^{0} | 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

editA 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 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

editi) 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

edit- 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

editOur 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.

The summation indices are over all electron pairs. It is the 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 corresponds to the -electron density
. If we know we can solve
So we *guess* and solve independent Schrödinger equations.
Unfortunately each solution then depends on which we guessed.
So we extrapolate a new and solve the temporary Schrödinger equation again. This continues until stops changing. If our initial guessed was
appropriate we will have the SCF approximation to the ground state.

This can be done for numerical 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.

When is expanded in terms of the atomic orbitals the troublesome
term picks out producted pairs of atomic orbitals
either side of the operator. This leads to a number of *four-centre integrals*of order . These fill up the disc space and take a long time to compute.

### Bibliography

edit- 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

editThere 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

editThe 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

editTo 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

editImportant 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 dimensions.
( 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

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 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, ( and 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

edit*The boring problem of molecular geometries.*

We have talked so far about geometries assuming they were expressed as matrices of the 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

editTo write out the -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 -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

edit
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

editz | | | * (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 -axis, (the principal axis), is the thumb, first finger gives ,
second finger . 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

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

#### Methyl Chloride

editIn 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 -axis.
All the hydrogens will have the -coordinate i.e.
.

H(3) has a coordinate zero and an 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, is across as usual and 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 value will be . Now remembering this value as ' ', we can easily get the and 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 -direction. Now the -coordinate will be + or - . The -coordinate is cos~120.

#### Useful Table for Remembering Common Trig Values

edit0 | 0 | 90 |
---|---|---|

30 | 60 | |

45 | 45 | |

60 | 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

editThe 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 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

editNewman 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

editIt 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 C_{6}H_{4}I- 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 and which are rotations about , then (i.e. , new , then the new axis.

0 | ||
---|---|---|

0 | ||

0 | 0 | 1 |

then

0 | ||
---|---|---|

0 | 1 | 0 |

0 |

then

0 | ||
---|---|---|

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

editOnce 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

edit(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

#### Hückel Calculation on Pyridine Output

edit*************************************************************** * 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

editThe 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

edit#### Molecular Geometries and Force Constants.

editWe 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.

editAtomic population analysis allows the assignment of and 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

edit(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

editElectronic 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

editIonisation 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, N_{3}.
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

editThe 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

edit#### naphalene and azulene

editUse 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

editThis 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 UF_{6} 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 CH_{4}, NH_{3}, H_{2}O, HF, BeH_{2},
BF_{3}, H_{3}N-BF_{3},
SF_{6}, XeF_{4}, XeF_{4}O and IF_{7}.

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 NH_{3},
BF_{3} and H_{3}N-BF_{3}
and IF_{7} (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 IF_{7} it will fail by filling the disc.
Even on NH_{3} 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 XeF_{4}O 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 XeF_{4}O 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 CO_{2} 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 NH_{3}, HCl, IF_{7} and XeF_{4}.
Comment on what you expected and what actually happens.

(Q5) SF_{6} is 23900 times more potent as a greenhouse gas than
CO_{2}.
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 BF_{3} and
AlF_{3} 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 XeF_{4},
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)

editNH3 <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

edit(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

edit- 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

editExamples 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**

edit
### Quantum Pharmacology

editThis 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

editA 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

editThis 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

edit*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**

edit
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

editOur 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

editThe 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

editM. 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 | CH_{3}CN | 36.05 | |

Diethyl-ether | 3.89 | CH_{3}NO_{2} | 36.48 | |

CHCl_{3} | 4.806 | DMSO | 45.80 | |

CH_{2}Cl_{2} | 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 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

editFor 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.

# Bibliography

## Computational Chemistry

edit- The multi-volume and definitely library living:
*The Encyclopedia of Computational Chemistry*, editor in chief: Paul von Ragué Schleyer, Wiley, Chichester, (1998). **Clark**, Tim,*A Handbook of Computational Chemistry*, Wiley (1985).**Cook**, D. B.,*A Handbook of Computational Quantum Chemistry*, Oxford University Press, (1998).**Cramer**C.J.,*Essentials of Computational Chemistry,Second Edition*,John Wiley, (2004).**Dronskowski**R.,*Computational Chemistry of Solid State Materials*, Wiley-VCH, (2005).**Hinchliffe**, Alan,*Computational quantum chemistry*, Wiley, Chichester, (1988). 541.383 (H).**Leach**, Andrew R.,*Molecular Modelling: Principles and Applications*, Addison Wesley Longman Ltd., (1996).**Jensen**F.,*Introduction to Computational Chemistry*, Wiley, Chichester, (1999).**Young**, David,*Computational Chemistry: A practical guide for applying techniques to real world problems*, Wiley-Interscience, (2001).**Ramachandran**, K.I.,**Deepa**, G. and**Krishnan Namboori**, P.K.,*Computational Chemistry and Molecular Modeling Principles and applications*Springer-Verlag GmbH (2008),[1]

## Quantum Chemistry

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**André**, Jean-Marie, J.**Delhalle**and J. L.**Bredas**,*Quantum Chemistry Aided Design of Organic Polymers*World Scientific (Singapore), (1991). - P. W.
**Atkins**,*Molecular Quantum Mechanics, 2nd edition*, (Oxford University Press,1983). - P. W.
**Atkins**,*Quanta, A Handbook of Concepts*, (Oxford University Press,1991). -
**Baggott**, Jim,*The meaning of quantum theory : a guide for students of chemistry and physics*, Oxford, (1994). -
**Calais**, Jean-Louis,*Quantum chemistry workbook : basic concepts and procedures in the theory of the electronic structure of matter*, (1994). -
**Chandra**, A.K.,*Introductory quantum chemistry. - 3. ed., 1. repr. - New Delhi,*Tata McGraw-Hill, (1989). -
**Chisholm**, Colin D.,*Group theoreticel techniques in quantum chemistry*C. D. H. Chisholm. - London, (1976). -
*Conceptual trends in quantum chemistry*, ed. by E. S.**Kryachko**... - Dordrecht, Kluwer, (1994). -
**Davidson**, Ernest R.,*Reduced density matrices in quantum chemistry*Ernest Roy Davidson. - London, (1976). -
**Denaro**, A. R.,*A foundation for quantum chemistry*, A. R. Denaro. - 1. publ. - London, Butterworths, (1975). -
**Dykstra**, Clifford E.,*Introduction to quantum chemistry*, Clifford E. Dykstra. - Englewood Cliffs, NJ , Prentice-Hall, (1994). -
**Epstein**, Saul T.,*The variation method in quantum chemistry*, by Saul T. Epstein. - New York, NY, Acad. Press, (1974). -
**Eyring**, Henry, John**Walter**and George E.**Kimball**,*Quantum Chemistry*, 17th. impression. - New York, NY, Wiley, (1965). -
**Fernández**, Francisco M.,*Algebraic methods in quantum chemistry and physics*, Francisco M. Fernández ; Eduardo A. Castro. - Boca Raton, (1996). -
**Flurry**, Robert L.,*Quantum chemistry : an introduction*, Robert L. Flurry. - Englewood Cliffs, NJ , Prentice Hall, (1983). -
**Goodrich**, Frank Chauncey,*A primer of quantum chemistry*, F. C. Goodrich. - Repr. - New York, NY, Wiley, (1982). -
**Haken**, Hermann,*Molecular physics and elements of quantum chemistry : introduction to experiments and theory*, Hermann Haken ; Hans (1995). -
**Hammond**, B. L., W. A.**Lester**and P. J.**Reynolds**,*Monte Carlo methods in Ab Initio Quantum chemistry*, B. Singapore , World Scientific (1994). -
**Hanna**, Melvin W.,*Quantum mechanics in chemistry*, Melvin W. Hanna., Benjamin/Cummings, (1969). - W.
**Hehre**... (et al.),*Ab initio molecular orbital theory*, (Wiley,1986), (Library Office 541.22.) -
**Hinchliffe, Alan**and Robert W.**Munn**,*Molecular electromagnetism*, (Chichester:Wiley,1985), ISBN 047110292x Invalid ISBN -
*Quantum chemistry, classic scientific papers*, transl. and ed. by Hinne**Hettema**. - Singapore, World Scientific, (1997). -
**Hinchliffe**, Alan,*Ab initio determination of molecular properties*, (Bristol:Hilger,1987), ISBN 0852745230 -
**Hinchliffe, Alan**,*Modelling molecular structures*, (Chichester:Wiley,1996). ISBN 0471959219 -
**Jørgensen**, Poul, and Jens**Oddershede**,*Problems in quantum chemistry*, Addison-Wesley, (1983). -
**Jørgensen**, Poul, and J.**Simons**,*Second Quantization-Based Methods in Quantum Chemistry*, (Academic,New York,1981). -
**Johnson**, Charles S.,*Problems and solutions in quantum chemistry and physics*, Charles S. Johnson ; Lee G. Pedersen. - New York , Dover (1986). **Kryachko**, Eugene S. and Eduardo V.**Ludeña***Energy density functional theory of many-electron systems*, (Dordrecht;London:Kluwer Academic,1990).-
**Levine**, Ira N.,*Quantum chemistry*, Ira N. Levine. - Internat. student Edition Prentice Hall, (1991). -
**Linderberg**, Jan,*Propagators in quantum chemistry*, Jan Linderberg ; Yngve Öhrn. - London, Acad. Press, (1973). -
*Local Density Approximations in quantum chemistry and solid state physics*, ed. by Jens Peter Dahl ... - New York, Plenum Pr. (1984). -
**Lowe**, John P.,*Quantum chemistry*, John P. Lowe. - Student ed., Acad. Pr., (1989). -
**McWeeny**, Roy,*Methods of Molecular Quantum Mechanics*, 2nd Edition, (Academic,1989). -
**McQuarrie**, Donald A.,*Quantum chemistry*, Donald A. McQuarrie. - Mill Valley, Calif. , Univ. Science Books -
**McQuarrie**, Donald A.,*Quantum chemistry.*Solutions manual to accompany "Quantum chemistry". - 1984. - 241 S. (1984). -
**Matthews**, Philip S. C.,*Quantum chemistry of atoms and molecules*Philip S. C. Matthews. - Cambridge, Cambridge Univ. Pr., (1986). -
**Minkin**, Vladimir I.,*Quantum chemistry of organic compounds : mechanisms of reactions*, V. I. Minkin ; B. Ya. Simkin ; R. M. Minyaev. - Berlin (1990). -
**Náray-Szabó**, Gábor,*Applied quantum chemistry.*- Dordrecht : Reidel, (1987). -
**Pauling**, Linus, and E. Bright**Wilson**,*Introduction to quantum mechanics : with applications to chemistry*, New York (1985). -
**Pauncz**, Ruben,*The symmetric group in quantum chemistry*, Ruben Pauncz. - Boca Raton, Fla., CRC Press, (1995). - F. L.
**Pilar**,*Elementary Quantum Chemistry*, 2nd edition, (McGraw-Hill,1990). -
**Prasad**, Ram K.,*Quantum chemistry*, R. K. Prasad. - 1. publ. - New York Wiley, (1992). -
**Quinn**, Charles M.,*An introduction to the quantum chemistry of solids*Charles M. Quinn. - Oxford , Clarendon Press, (1973). -
**Roos**, Bjorn (Ed.),*Lecture Notes in Quantum Chemistry, I*, (Springer-Verlag,Berlin,1992). B. O. Roos (Ed.),*Lecture Notes in Quantum Chemistry, II*, (Springer-Verlag,Berlin,1994). -
*Quantum chemistry approaches to chemisorption and heterogeneous catalysis, ed.*by F.**Ruette**. - Dordrecht , Kluwer, (1992). - J.M.
**Seminario**and P.**Politzer**,*Modern Density Functional Theory: A Tool For Chemistry*, (Elsevier,1995). J.M.**Seminario**,*Recent Developments and Applications of Modern Density Functional Theory*, (Elsevier,1996). -
**Schaefer**, Henry F. III,*Quantum chemistry :the development of ab initio methods in molecular electronic structure theory*, (Oxford:Clarendon Press,1984), ISBN 0198551835. -
**Simons**, Jack,*Quantum mechanics in chemistry*, Jack Simons ; Jeff Nichols. - New York, NY, Oxford Univ. Press, (1997). - J. C.
**Slater**,*Quantum Theory of Atomic Structure*, Vols. 1 and 2, (McGraw-Hill, New York, 1960). -
*Strategies and applications in quantum chemistry* -
**Surján**, Péter R.,*Second quantized approach to quantum chemistry : an elementary introduction.*- Berlin, Springer, (1989). - A.
**Szabo**and N. S.**Ostlund**,*Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory*, (Macmillan, New York 1989), ISBN 0486691861.