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Molecular Modelingin Undergraduate
Chemistry Education
Warren J. HehreWavefunction, Inc.
18401 Von Karman Ave., Suite 370Irvine, CA 92612
Alan J. ShustermanDepartment of Chemistry
Reed College3203 S. E. Woodstock Blvd.
Portland, OR 97202
Introduction
Chemistry . . .by Experiment
Chemists make useful materials (pharmaceuticals,polymers, etc.). Typically, they do this by seeking toestablish connections with molecular structure.
Measurement
Preparation and Characterization
by Molecular Modeling
The goal is exactly the same as is the procedure. Thedifficult step (materials preparation) has been removed.
Simulation
Definition
"Laws of Nature"
Materials
Why is molecular modelingimportant to the
teaching of chemistry?
• Models are what we teach. Students need to learnto “think like a molecule”. To do this they need to“see” what a molecule sees and “feel” what amolecule feels. Models give us the best and mostdirect view of the molecular world.
• Modeling is the best tool for learning aboutchemical theory. VSEPR, Lewis structures, HückelMO are all crude attempts to convert good theoriesinto chemical predictions. Modern computationalmethods give a much more accurate assessment oftheoretical predictions.
• Models are easy to use, inexpensive, safe. Modelingis a student-friendly educational too. It is not justfor “experts”.
Should molecular modelingreplace experimental
chemistry?
Of course not!
The goals of chemistry are not changed by molecularmodeling. On a practical level we want to learn howto make things (synthesis) and how to figure out whatthings are made of (analysis). On an intellectual levelwe want to understand the “rules” that describechemical behavior.
Molecular modeling is, like NMR, a tool for achievingthese goals. Since two of the goals - synthesis andanalysis - are experimental, they cannot (and shouldnot) be done away with. However, modeling doeschange the way we do syntheses and analysis. And, itspeaks directly to the intellectual goals of chemistry.
A modern chemical education still requires practicaltraining in experimentation, but it requires training inmodeling too.
Molecular Models
Electron Densities
Electron densities show the locations of electrons.Large values of the density will first reveal atomicpositions (the X-ray diffraction experiment) and thenchemical bonds, while smaller values will indicateoverall molecular size.
increasing electron density
decreasing electron density
Electron Densities
Unlike conventional structure models, electrondensities assume no prior knowledge about bondingand, therefore, can be used to elucidate bonding. Forexample, the electron density for diborane shows noevidence for boron-boron bonding.
vs.
Electron Densities
Electron densities allow description of the bonding intransition states where there can be no (direct)information from experiment. For example, theelectron density for the transition state for pyrolysis ofethyl formate, leading to formic acid and ethene, showsthat the CO bond has nearly fully cleaved and that thehydrogen is midway between carbon and oxygen.
O
C C
H
OC
H
HH H
H
ethyl formate
formic acid
ethene
O
C C
OC
H
H
H HH H
Electrostatic Potentials
The electrostatic potential is the energy of interactionof a point positive charge (an electrophile) with thenuclei and electrons of a molecule. Negativeelectrostatic potentials indicate areas that are prone toelectrophilic attack. For example, a negativeelectrostatic potential of benzene (left) shows thatelectrophilic attack should occur onto the π system,above and below the plane of the ring, while thecorresponding electrostatic potential for pyridine (right)shows that an electrophile should attack the nitrogenin the σ plane, and not the π system of the ring.
The “electrophilic chemistry” of these two seeminglysimilar molecules is very different.
Electrostatic Potential Maps
A sufficiently small value of the electron densityprovides overall molecular size and shape (as givenby a conventional space-filling or CPK model). Theelectrostatic potential can then be mapped onto theelectron density by using color to represent the valueof the potential. The resulting model simultaneouslydisplays molecular size and shape and electrostaticpotential value. For example, the electrostatic potentialof benzene can be mapped onto the electron density.Colors toward “red” indicate negative values of theelectrostatic potential, while colors toward “blue”indicate positive values of the potential.
electron density
+
electrostatic potential
electrostaticpotential map
Electrostatic Potential Maps
An electrostatic potential map conveys informationabout the distribution of charge in a molecule. Forexample, the electrostatic potential map of thezwitterionic form of β-alanine shows the negativecarboxylate (red) and the positive ammonium (blue)termini separated by the neutral (green) carbon chain.This is consistent with the usual resonance structure.
H3N+CH2CH2CO2–
Electrostatic Potential Maps
An electrostatic potential map also gives informationabout delocalization of charge. For example, theelectrostatic potential map for perpendicular benzylcation (right) shows that the postive charge (blue) islocalized on the benzylic carbon, while the charge inthe planar cation (left) is fully delocalized. This agreeswith conventional resonance arguments.
+
+
+
+
+
Molecular Orbitals
Molecular orbitals, solutions of the approximatequantum mechanical equations of electron motion, aremade up of sums and differences of atomic solutions(atomic orbitals), just like molecules are made up ofcombinations of atoms.
Molecular orbitals for very simple molecules mayoften be interpreted in terms of familiar chemicalbonds, for example, in acetylene,
σ π π
or of nonbonded lone pairs, for example, in sulfurtetrafluoride.
Molecular Orbitals
Unoccupied molecular orbitals may also provideuseful information. For example, the lowest-unoccupied molecular orbital (LUMO) in planarbenzyl cation (left) indicates that the positive chargeis delocalized away from the benzylic carbon ontothe ortho and para ring positions. The LUMO inperpendicular benzyl cation (right) resides almostentirely on the benzylic carbon. This agrees withconventional resonance arguments.
H H+
+
+++
H H H H H H H H
Molecular Orbitals
Why should molecular orbital descriptions be usedinstead of conventional Lewis structures?
1. Molecular orbitals descriptions are often morecompact than Lewis structures.
2. Molecular orbital descriptions offer quantitativeinformation about molecular charge distributions.Lewis descriptions are strictly qualitative.
3. Molecular orbital descriptions are more generallyapplicable than Lewis descriptions.
Molecular Orbital Maps
Molecular orbitals may also be mapped onto electrondensity surfaces. For example, a map of the lowest-unoccupied molecular orbital (LUMO) ofcyclohexenone, where the “blue spots” indicatemaximum values of the LUMO, reveals likely sitesfor nucleophilic attack, and anticipates both the“carbonyl chemistry” and “Michael chemistry” knownfor enones.
O
H
Nu
ONuHO
Nu Nu
Michael addition carbonyl addition
Models that Move
Models need not be limited to static pictures. “Movies”can be used to depict vibrations in stable molecules,for example, in water.
H HO
H HO
H HO
Models that Move
Motion along the reaction coordinate provides detailsabout mechanism. For example, motion along thereaction coordinate for the pyrolysis of ethyl formateshows the simultaneous transfer of the hydrogen atomto the carbonyl oxygen along with carbon-oxygenbond cleavage.
O
C C
H
OC
H
HH H
H
ethyl formate
formic acid
ethene
O
C C
OC
H
H
H HH H
Molecular ModelingWorkbook
Molecular Modelingin the Curriculum
“Doing chemistry” with molecular modeling is amulti-step progress . . . not so different from doingexperimental chemistry.
Define Problem
Build Models
Do Calculations
Analyze Results
Given a “full” curriculum, the question that needs tobe answered is how much of this process to turn overto students.
The Workbook Approach
This leaves only the analysis of modeling results (andlearning the chemistry that follows from these results)to the student.
Define Problem
Build Models
Do Calculations
Analyze Results
The advantage of this approach is that it requires thefewest resources (hardware and software support,student training), while guaranteeing high qualitymodels and maximum student-model contact.
The Molecular ModelingWorkbook for
ORGANIC CHEMISTRY
A collection of over 200 problems arranged bychapters that parallel the contents of your organicchemistry textbook.
1 Lewis Structures and Resonance Theory2 Acids and Bases3 Reaction Pathways and Mechanisms4 Stereochemistry5 Alkanes and Cycloalkanes6 Nucleophilic Substitution and Elimination7 Alkenes and Alkynes8 Alcohols and Ethers9 Ketones and Aldehydes. Nucleophilic Addition10 Carboxylic Acid Derivatives. Nucleophilic Substitution11 Enolates as Nucleophiles12 Conjugated Polyenes and Aromaticity13 Electrophilic and Nucleophilic Aromatic Substitution14 Nitrogen-Containing Compounds15 Heterocycles16 Biological Chemistry17 Free Radicals and Carbenes18 Polymers19 Spectroscopy20 Mass Spectrometry21 Pericyclic Reactions
The Molecular ModelingWorkbook for
ORGANIC CHEMISTRY
Problems in each chapter address essential topics.
Chapter 11 Enolates as Nucleophiles1 Keto/Enol Tautomerism2 H/D Exchange Reactions3 What Makes a Good Enolate?4 Enolate Acidity, Stability and Geometry5 Kinetic Enolates6 Real Enolates7 Enolates, Enols and Enamines8 Enolates are Ambident Nucleophiles9 Silylation of Enolates10 Stereochemistry of Enolate Alkylation11 Enolate Dianions12 Aldol Condensation13 Dieckmann Condensation
The Molecular ModelingWorkbook for
ORGANIC CHEMISTRY
Each problem uses one or more models, and studentsare required to look at and query these models to solvethe problem. The models are contained on a CD-ROMthat comes with the workbook, and can be viewed ona Mac or PC.
All models provide many types of informationobtained from molecular orbital calculations,including structure, energy and atomic charges. Manymodels also provide molecular orbitals, electrondensity surfaces and electrostatic potential mapsamong other graphical displays.
Models include many types of molecular species,including conformers, reactive intermediates,transition states and weak complexes. A number ofmodels involve sequences that can be animated.
ConceptualBackground
Potential Energy Surfaces
A potential energy surface is a plot of energy vs.reaction coordinate. It connects reactants to productsvia a transition state.
energy
reaction coordinate
equilibrium structures
transition state structure
Energy minima correspond to equilibrium structures .
The energy maximum corresponds to a transitionstate structure.
transition state
reactants
products
Potential Energy Surfaces
energy
reaction coordinate
"thermodynamics"
"kinetics"
The relative energies of equilibrium structures givethe relative stabilities of the reactant and product (thethermodynamics of reaction).
The energy of the transition state relative to theequilibrium structures provides information about therelative difficulties going on between them (thekinetics of reaction).
transition state
reactants
products
Potential Energy Surfaces
A complete reaction pathway may comprise severalsteps and involve several different transition states andhigh-energy reactive intermediates.
energy
reaction coordinate
rate limiting step
Such a diagram describes the mechanism of areaction, the rate-limiting step for which proceedsvia the highest-energy transition state.
transition state
reactants
products
transition state
intermediate
Potential Energy Surfaces
Molecular modeling is primarily a tool for calculatingthe energy of a given molecular structure. Thus, thefirst step in designing a molecular modelinginvestigation is to define the problem as one involvinga structure-energy relationship.
There are two conceptually different ways of thinkingabout energy.
Molecular MechanicsA “Chemist’s” Model
Molecular mechanics describes the energy of amolecule in terms of a simple function which accountsfor distortion from “ideal” bond distances and angles,as well as and for nonbonded van der Waals andCoulombic interactions.
Coulombic
δ−
δ+
angledistortions
dihedraldistortions
van der Waals
distancedistortions
E = Σbonds
bond stretching +Σ angle bending +
+ Σ van der Waals + Σ Coulombic
pairs of atoms
pairs of atoms
angles
Σ dihedral twisting
dihedral angles
energy of nonbonded interactions
energy of distortion from idealized values
Quantum MechanicsA “Physicist’s” Model
Quantum mechanics describes the energy of amolecule in terms of interactions among nuclei andelectrons as given by the Schrödinger equation. Thesolutions (“wavefunctions”) for the hydrogen atomare the familiar s, p, d . . . atomic orbitals.
8 2m-h2
2 + V(x,y,z) ψ(x,y,z) = E ψ(x,y,z)
∆
total energypotential energy
wavefunction describing the system
kinetic energy
The square of the wavefunction gives the probabilityof finding an electron. This is the electron density, asobtained from an X-ray diffraction experiment.
Quantum Mechanics
While the Schrödinger equation is easy to write downfor many-electron atoms and molecules, it isimpossible to solve. Approximations are needed.
Schrödinger Equation HΨ = EΨ
Practical Molecular Orbital Methods
assume that nucleidon’t move
separate electron motions
get electron motion inmolecules by combiningelectron motion in atoms
“Born-OppenheimerApproximation”
“Hartree-FockApproximation”
“LCAO Approximation”
Molecular Modelingin Lecture
Molecular Modeling in Lecture
Molecular models can be introduced into almost anychemistry lecture. They not only liven the discussionwith “pretty pictures”, but more importantly encouragestudents to see and think for themselves. They freeteachers from the “limits of the chalkboard” and allowexamination and discussion of “real” molecules.
Visualizing Chemical Bonds
Are the different kinds of chemical bonds to whichchemists refer (nonpolar covalent, polar covalent andionic) fundamentally different? Look at electrondensities.
H-H H-F H-Li
The size of the electron density surface indicates thesize of the electron cloud. The H electron cloud islargest in HLi and smallest in HF. This is persuasiveevidence that the atoms in these molecules do not“share” electrons equally.
Visualizing Chemical Bonds
A clearer picture comes from electrostatic potentialmaps, where the color red demarks regions with excessnegative charge and the color blue demarks regionswith excess positive charge.
H-H H-F H-Li
Hydrogen fluoride and lithium hydride look verysimilar, except that the hydrogen in the former ispositively charged while the hydrogen in the latter isnegatively charged.
The SN2 Reaction
Molecular modeling provides insight into the familiarSN2 reaction.
N C N C CH3 + ICH3 I– –
An animation of the reaction clearly shows theinversion at carbon, but there are other importantquestions.
The SN2 Reaction
Why does cyanide attack from carbon and notnitrogen? Doesn’t this contradict the fact that nitrogenis more electronegative than carbon?
Look at the highest-occupied molecular orbital ofcyanide. This is where the “most available” electronsreside.
N≡C
It is more heavily concentrated on carbon, meaningthat cyanide is a carbon nucleophile.
The SN2 Reaction
Why does iodide leave following attack by cyanide?
Look at the lowest-unoccupied molecular orbital ofmethyl iodide. This is where the electrons will go.
It is antibonding between carbon and iodine meaningthat the CI bond will cleave during attack.
The SN2 Reaction
We tell students that bromide reacts faster with methylbromide than with tert-butyl bromide because of stericeffects, that is, increased crowding of the transitionstate.
This is not true. Space-filling models of the twotransition states show both to be uncrowded.
What is true is that the carbon-bromine distance inthe transition state in the tert-butyl system is largerthan that in the methyl system.
2.5Å 2.9Å
The SN2 Reaction
This leads to an increase in charge separation as clearlyshown by electrostatic potential maps for the twotransition states.
This is the cause of the decrease in reaction rate.
Flexible Molecules
Interconversion of anti and gauche forms of n-butaneis readily visualized “on a blackboard” usingconventional structure models.
Trans-1,2-dimethylcyclohexane undergoes the sametype of conformational change as n-butane. Thedifficulty for the lecturer is that this change is not easilyportrayed “on a blackboard”. This obstacle is readilyovercome with molecular models.
diaxial diequatorial
Flexible Molecules
energy
reaction coordinate
TS 2(eclipsed)
twist-boat(staggered)diaxial
(staggered)
diequatorial(staggered)
TS 1(eclipsed)
An animation shows that the conformational changein trans-dimethylcyclohexane is not so different fromthat in n-butane. There are two distinct steps in thering flip mechanism, but each involves rotation aboutcarbon-carbon bonds. The three minima all correspondto structures in which bonds are staggered. The twotransition states involve eclipsing interactions.
Intermolecular Interactions
Acetic acid is known to form a stable hydrogen-bonded dimer. What is it’s structure?
H
OC
O OC
OHCH3H3C
A
H
OC
O
OHC
OCH3
H3C
B
H
OC
OH3C
COH
OCH3
C
Instead of giving students the right structure, give themthe tools to find it for themselves.
Intermolecular Interactions
H
OC
O OC
OHCH3H3C
A
H
OC
O
OHC
OCH3
H3C
B
H
OC
OH3C
COH
OCH3
C
Energy Tool
Energies follow the order A<C<B (A is best). The alertstudent will be surprised. This is because six-membered rings (as in C) are common while eight-membered rings (as in A) are not.
Intermolecular Interactions
Electrostatic Potential Tool
Look at an electrostatic potential map for acetic acid.
Which atom is positively charged and most likely toact as a hydrogen-bond donor?
Which atom is most negatively charged and mostlikely to act as a hydrogen-bond acceptor?
The model gives the answers and allows assignmentof the proper dimer structure.
Intermolecular Interactions
Apply these same “tools” to a related question whereyou don’t know the answer (or where you “know” thewrong answer).
What is the crystal structure of benzene?
or
stacked
perpendicular
: :
:
:
Intermolecular Interactions
The energy tool shows that the “stacked” benzenedimer dissociates into the two benzenes, while theperpendicular dimer sticks together. It does not showwhy.
The electrostatic potential tool clearly shows thatstacking the rings results in unfavorable electrostaticinteractions, while a perpendicular arrangement ofbenzene rings results in favorable electrostaticinteractions between π and σ systems.
Molecular Modelingin the Laboratory
The Laboratory Approach
Teachers who want their students to “get their handsdirty” will find that students can learn much from“hands-on” modeling. Here, most of the steps involvedin actually “doing chemistry” are left up to the student.
Define Problem
Build Models
Do Calculations
Analyze Results
The “laboratory approach” offers students and teachersmuch greater flexibility than the “workbookapproach”. Because it puts students directly in contactwith actual molecular modeling software, it teachesthem that calculations, like experiments, are notinstantaneous, and that good “experimental design”is important.
The Laboratory Book ofComputational Organic
Chemistry
A collection of over 80 experiments covering topicsrelevant to both elementary and advanced organicchemistry courses. The experiments require accessto a modeling program, such as MacSPARTAN orPC SPARTAN.
Is Thiophene Aromatic?
Objective:
To quantify the “extra stability” of thiophene due toaromaticity.
Background:
Hydrogenation of benzene to 1,3-cyclohexadiene isslightly endothermic, whereas the correspondingreactions taking 1,3-cyclohexadiene to cyclohexeneand then to cyclohexane, are both significantlyexothermic.
+ H2 + H2
DH = 6 kcal/mol DH = -26 kcal/mol
+ H2
DH = -28 kcal/mol
The first hydrogen addition “trades” an H-H bondand a C-C π bond for two C-H bonds, but in so doingdestroys the aromaticity of benzene, whereas thesecond hydrogen addition “trades” the same bondsbut does not result in any loss of aromaticity.Therefore, the difference in the heats of hydrogenationof benzene and 1,3-cyclohexadiene (or cyclohexene)corresponds to the aromatic stabilization of benzene.This difference is approximately 33 kcal/mol.
elementary advanced
Is Thiophene Aromatic?
Procedure:
Calculate the energetics of hydrogen addition tothiophene and to the intermediate dihydrothiophene.
S S+ H2
S+ H2
The difference provides a measure of the aromaticityof thiophene.
Questions:
Is thiophene as aromatic as benzene? Half as aromatic?
Gas and Aqueous-PhaseBasicities of Methylamines
Objective:
To quantify changes in nitrogen base strength fromthe gas phase into water and to provide rationalizationfor these changes.
Background:
Experimentally, increasing methyl substitution greatlyincreases amine basicity in the gas phase, but leads todifferent results in water.
Energies of: BH+ + NH3 → B + NH+, kcal/mol
B ∆Hrxn(gas) ∆Hrxn(H2O)
CH3NH2 9 2
(CH3)2NH 16 2
(CH3)3N 19 1
elementary advanced
4
Gas and Aqueous-PhaseBasicities of Methylamines
Procedure:
Calculate gas-phase energetics of proton transferreactions for B=methylamine, dimethylamine andtrimethylamine. Calculate solvation energies for allneutral and ionic species, and estimate energies foraqueous phase proton transfer reactions.
Examine electrostatic potential maps. Specifically,examine changes in charge distributions andaccessibility to solvent in both neutral and ionic species.
Questions:
Where are solvent effects greater - on the neutralmolecules or on the ions? What is the cause for reversalin basicity?
Thermodynamic vs. KineticControl of Intramolecular
Radical Additionto Multiple Bonds
Objective:
To assign thermodynamic or kinetic origin to theproducts in an intramolecular radical addition reaction.
Background:
Cyclization of hex-5-enyl radical can either yieldcyclopentylmethyl radical or cyclohexyl radical.
or
While the latter might be expected (it should be lessstrained, and 2° radicals are generally more stable than1° radicals), the opposite is normally observed, e.g.,
+Br +AIBN
17% 81% 2%
Bu3SnH
elementary advanced
Thermodynamic vs. KineticControl of Intramolecular
Radical Additionto Multiple Bonds
Procedure:
Calculate the energies of both cyclopentylmethyl andcyclohexyl radicals. Assume that the reaction isthermodynamically controlled, and use the Boltzmannequation (with T = 298K) to calculate the productdistribution.
Locate transition states for cyclization reactionsleading to cyclopentylmethyl and cyclohexyl radicals.Assume that the reaction is kinetically controlled, anduse the transition state energies together with theEyring equation to calculate the product distribution.
Questions:
Is the cyclization reaction under thermodynamic orkinetic control or is it not possible to tell?
Molecular Recognition.Hydrogen-Bonded Base Pairs
Objective:
To model hydrogen-bonding between DNA base pairsin terms of charge-charge interactions. To identifynucleotide mimics.
Background:
The genetic code is “read” through the selectiveformation of hydrogen-bonded complexes, or Watson-Crick base pairs, of adenine (A) and thymine (T)(forms A-T base pair), and guanine (G) and cytosine(C) (forms G-C base pair). The importance of thishydrogen bond-based code lies in the fact that virtuallyall aspects of cell function are regulated by properbase pair formation, and many diseases can be tracedto “reading” errors, i.e., the formation of incorrect basepairs.
elementary advanced
Molecular Recognition.Hydrogen-Bonded Base Pairs
Procedure:
Calculate geometries, energies and electrostaticpotential maps for MeT, MeA, MeC and MeG.
HN NCH3
O
O
CH3
N
N
NH2
N
N
CH3
N NCH3
O
H2N
HN
N
O
N
N
CH3
H2N
1-methylthymine(MeT)
9-methyladenine(MeA)
1-methylcytosine(MeC)
9-methylguanine(MeG)
Identify electron-rich and electron-poor sites thatwould be suitable for hydrogen bonding (consider onlysites in the plane of the molecules). Draw all possiblebase pairs that involve two or three hydrogen bonds(do not limit yourself to Watson-Crick pairs).
Choose one system (your best guess for the mostfavorable base pair involving two hydrogen bonds ormost favorable base pair involving three hydrogenbonds). Obtain the geometry and energy of this basepair and calculate the total hydrogen bond energy.
Molecular Recognition.Hydrogen-Bonded Base Pairs
Procedure:
Calculate the geometries, energies and electrostaticpotential maps for AP and NA, heterocyclic moleculesthat mimic the hydrogen bonding properties of DNAnucleotides.
HN
O
H2N N
6-amino-2-pyridone(AP)
N-naphtharidinyl acetamide(NA)
N NH
O
Identify electron-rich and electron-poor sites for eachto see if either would be a suitable partner for one ofthe bases in your chosen base pair. Obtain the geometryand energy of the resulting hydrogen-bonded adductand calculate the total hydrogen bond energy.
Questions:
Does it mimic bind as tightly as the “natural” base?
Proper Role ofMolecular Modeling
Focus on Chemistry
Modeling is a tool for doing chemistry. Molecularmodeling is best treated in the same way as NMR - asa tool, not a goal. A good model has the same value asa good NMR spectrum. Molecular modeling allowsyou to “do” and teach chemistry better by providingbetter tools for investigating, interpreting, explaining,and discovering new phenomena.
Don’t be Afraid
Modeling is accessible. Anyone can build a usefulmodel.
Modeling is “hands on”. Molecular modeling, likeexperimental chemistry is a “laboratory” science, andmust be learned by “doing” and not just reading.
Modeling does not need to be intimidating. Theunderpinnings of molecular modeling (quantummechanics) are certainly intimidating to manychemists, but so too are the underpinnings of NMR.Using molecular modeling should be no moreintimidating than obtaining an NMR spectrum.
Modeling is not difficult to learn and do. Molecularmodeling is easy to do given currently availablesoftware (probably easier than taking an NMRspectrum). The difficulty lies in asking the rightquestions of the models and properly interpreting whatcomes out of them.
Stick with Standard Models
Standard models are a must. While there is only oneexperimental “result”, different models will yielddifferent results. It is necessary to define a smallnumber of “standard models” and to fully documentthe performance of each. These should range fromvery simple, which may not be very accurate but areapplicable to very large systems, to very sophisticated,which are accurate but may be applicable only to verysimple systems.
While the models are not perfect, they can be of greatvalue in learning and researching chemistry. They willcontinue to improve and in time it will be possible tomodel important chemical quantities as accurately asthey can be measured.
