Quantum chemistry in Molecular Modeling
[IndonesiaGoogleTranslation]
5.1 Why use Quantum Chemical methods ?
Quantumchemical
methods are more general than empirical methods
5.2 The Schrödinger equation
The
recipe for the calculation of the electronic wavefunction
5.3 HartreeFock SCF theory
Part a: The independent particle approximation
Part b:Molecular Orbitals,
basis sets
5.4 Limitations of the HF method;
Electron
correlation
CI, MP2 and MCSCF
5.5 Energy calculations
Conversion of
absolute energies to heats of formation
Isodesmic reactions
5.6 Quality of ab initio results
Performance of lowlevel methods for
organic molecules
5.7 Semiempirical quantum
chemistry
Further approximations and
introduction of empirical parameters
5.8 Quality of semiempirical
results
Lowlevel ab initio methods are usually
better, but much more timeconsuming
5.9 Solvation
The medium can be represented as a
dielectric continuum
5.10 Atomic charges
Not a physical concept !
Several methods
exist for attributing charge to individual atoms
"For calculating molecular
properties, quantum chemistry seems to be the obvious
tool to use. Calculations that do not use the
Schrödinger equation are acceptable only to the extent
that they reproduce the results of high level quantum
mechanical calculations." (U. Burkert & N.L. Allinger,
"Molecular Mechanics", 1982)
5.1 Why use Quantum Chemical methods
?
Many aspects of molecular
structure and dynamics can be modeled using classical
methods in the form of molecular mechanics and dynamics.
The classical force field is based on empirical results,
averaged over a large number of molecules. Because of
this extensive averaging, the results can be good for
standard systems, but there are many important questions
in chemistry that can not at all be addressed by means
of this empirical approach. If one wants to know more
than just structure or other properties that are derived
only from the potential energy surface, in particular
properties that depend directly on the electron density
distribution, one has to resort to a more fundamental
and general approach : quantum chemistry.
The same holds for all nonstandard cases for which
molecular mechanics is simply not applicable.
Quantum chemistry is based on the
postulates of Quantum Mechanics. In this chapter we
shall recall some basic aspects of the theory of quantum
chemistry with an emphasis on their practical
implications for the molecular modeler, and we will try
to answer the questions of when to use a quantum
chemical method instead of molecular mechanics, which
quantum chemical method to choose, and what to
expect from the quality of the results.
In quantum chemistry, the system is
described by a wavefunction which can be
found by solving the Schrödinger equation. This equation
relates the stationary states of the system and their
energies to the Hamiltonian operator, which can be
viewed as the recipe for obtaining the energy associated
with a wavefunction describing the
positions of the nuclei and electrons in the system. In
practice the Schr?dinger equation cannot
be solved exactly and approximations have to be made, as
we shall see below. The approach is called "ab initio" when
it makes no use of empirical information, except for the
fundamental constants of nature such as the mass of the
electron, Planck's constant etc., that are required to arrive
at numerical predictions. Do not confuse "ab initio" with
"exact" ! In
spite of the necessary approximations, ab initio theory
has the conceptual advantage of generality, and the
practical advantage that (with experience) its successes
and failures are more or less predictable.
The major disadvantage of ab initio quantum chemistry
are the heavy demands on computer power.
Therefore, further approximations have been applied for
a long time which
go together with the introduction of empirical
parameters into the theoretical model. This has led to a
number of semiempirical quantum chemical methods, which
can be applied to larger systems, and give reasonable
electronic wavefunctions so that
electronic properties can be predicted. Compared with
ab initio
calculations their reliability is less and their
applicability is limited by the requirement for
parameters, just like in molecular mechanics.
In general, one should apply quantum
chemistry for "small" systems, which can be treated at a
very high level, when electronic properties are sought
(electric moments, polarizabilities, shielding
constants in NMR and ESR, etc.) and for "nonstandard"
structures, for which no valid molecular mechanics
parameters are available. Examples are conjugated pi
systems, organometallic compounds and
other systems with unusual bond or atom types, excited
states, reactive intermediates, and generally structures
with unusual electronic effects.
Quantum chemistry is the subject of
many excellent textbooks. The
one by Levine [1] has a good reputation, and devotes
much attention to practical computational methods. Several chapters in the series Reviews in Computational
Chemistry [2] put quantum chemical calculations in a
molecular modeling perspective. In the interesting book of G. NàraySzabò,
P.R. Surjàn and J.G. Angyàn [3]
theoretical chemistry is discussed in relation to
experimental chemistry. An
account of the development of "model chemistries" based
on ab
initio calculations is given by W.J. Hehre, L. Radom, P. von R.
Schleyer
and J.A. Pople [4]. A more advanced textbook is that of
Szabo and
Ostlund
[5]. Useful guides in the use of
ab initio
and semiempirical quantumchemical methods are the
manual of Spartan [6] and the book published by Gaussian
Inc. [7].
References:
1.
Levine Quantum Chemistry, 4th ed., 1991
2.
Reviews in
Computational Chemistry, Volumes 1 to
4, K.B. Lipkowitz and D.B. Boyd,
eds., VCH publishers.
3.
G. NàraySzabò, P.R. Surjàn and J.G.
Angyàn Applied Quantum Chemistry, Reidel, 1987
4.
W.J. Hehre, L. Radom, P. von R.
Schleyer
and J.A. Pople Ab
initio molecular orbital theory, Wiley, 1986
5.
Szabo and Ostlund Modern Quantum Chemistry,
McGrawHill, 1989.
6.
Spartan User's Guide, version 3.0,
Wavefunction, Inc., 1993.
7.
Foresman, J.B.; Frisch,
A. Exploring Chemistry with
Electronic Structure Methods: A Guide to Using
Gaussian, Gaussian Inc., 1993.
Next paragraph, 5.2 The Schrödinger
Equation Back to Chapter 3 of the
Computational Chemistry Course.
Chapter 5 MM Syllabus 1995 MODIFIED
November 8, 1995 Fred Brouwer, Lab. of
Organic Chemistry, University of Amsterdam.
Chapter 5. Quantum chemistry in Molecular
Modeling
5.2 The Schrödinger
equation
The energies and
wavefunctions of stationary
states of a system are given by the solutions of the
Schrödinger Equation : In this equation
is the Hamiltonian operator which in this
case gives the kinetic and potential energies of a
system of atomic nuclei and electrons. As we shall see
below it is analogous to the classical kinetic energy of
the particles and the Coulomb electrostatic interactions
between the nuclei and electrons. is a wavefunction, one of the
solutions of the eigenvalue equation. This
wavefunction depends on the
coordinates of the electrons and the nuclei. The
Hamiltonian is composed of three parts : the kinetic energy of
the nuclei, the kinetic energy of the electrons, and the
potential energy of nuclei and electrons.
Schrödinger equation : Hamiltonian :
Four approximations are commonly
(but not necessarily) made :
time independence;
we are looking at states that are stationary in time.
neglect of relativistic
effects; this is warranted unless the velocity of the
electrons approaches the speed of light, which is the
case only in heavy atoms with very high nuclear charge.
BornOppenheimer approximation;
separation of the motion of nuclei and electrons.
orbital approximation;
the electrons are confined to certain regions of space.
The BornOppenheimer approximation
implies the separation of nuclear and electronic wavefunctions,
the total wavefunction being a product
of the two
: BornOppenheimer :
The motivation behind this is that
the electrons are so much lighter than the nuclei that
their motion can easily follow the nuclear motion. In
practice, this approximation is usually valid. From this
point we will look at the electronic wavefunction which is obtained by solving the electronic Schrödinger equation
:
This equation still contains the
positions of the nuclei, however not as variables but as
parameters.
The electronic Hamiltonian contains
three terms
: kinetic energy, electrostatic interaction
between electrons and nuclei, and electrostatic
repulsion between electrons. In order to simplify
expressions and to make the theory independent of the
experimental values of physical constants, atomic units
are introduced : e = 1 charge of
electron m = 1 mass of the
electron = 1 Planck's constant divided by 2 pi Derived atomic units of length and
energy are : 1 bohr = 1 hartree = J = 627.51 kcal/mol
With these units the electronic
Hamiltonian is
:
The symbol is the Laplace operator (also called "delsquared"). The total
energy in the BornOppenheimer model is obtained by
adding the nuclear repulsion energy to the electronic
energy :
The total energy defines a potential
energy hypersurface E=f(Q) which can be
used to subsequently solve a Schrödinger equation for
the nuclear motion :
In the following section we shall
deal with the important problem of solving the
electronic Schrödinger equation.
Next paragraph, 5.3 HartreeFock SCF
theory Previous paragraph 5.1 Why use
Quantum Chemical methods ? Chapter 5 MM Syllabus 1995 MODIFIED
November 8, 1995 Fred Brouwer, Lab. of
Organic Chemistry, University of Amsterdam.
Chapter 5. Quantum chemistry in Molecular
Modeling
5.3 Solving the electronic
Schrödinger equation : HartreeFock SelfConsistent
Field theory.
As we have seen, the electronic
Hamiltonian contains two terms that act on one electron
at a time, the kinetic energy and the electronnucleus
attraction, and a term that describes the pairwise
repulsion of electrons. The latter depends on the
coordinates of two electrons at the same time, and has
turned out to be a practical computational bottleneck,
which can be passed only for very small systems :
To avoid this problem the
independent particle approximation is introduced : the interaction
of each electron with all the others is treated in an
average way. Suppose :
Then the Schrödinger equation which
initially depended on the coordinates x (representing spatial and spin
coordinates) of all electrons can be reduced to a set of
equations :
The wavefunctions are called oneelectron spinorbitals.
The obvious problem is that for each
electron the potential due to all other electrons has to
be known, but initially none of these is known. In
practice trial orbitals are used which are
iteratively modified until a selfconsistent solution (a
"SelfConsistent Field") is obtained, which can be
expressed as a solution to the HartreeFock equations :
It is important to realize that convergence of the SCF procedure is
by no means guaranteed. Many techniques have been
developed over the years to speed up convergence, and to
solve even difficult cases. In practice, difficulties
often occur with systems with an unusual structure,
where the electrons "do not know where to go". The eigenvalues are interpreted as
orbital energies. The orbital energies have an
attractively simple physical interpretation : they give
the amount of energy necessary to take the electron out
of the molecular orbital, which corresponds to the
negative of the experimentally observable ionization
potential (Koopmans' Theorem):
In addition to being a solution of
the electronic Schrödinger equation the wavefunction
must be normalized and satisfy the Pauli principle. The
normalization condition is connected with the
interpretation of the wavefunction as a
distribution function which when integrated over entire
space should give a value of one :
in "braket" notation :
The Pauli principle states that the
wavefunction must change sign
when two independent electronic coordinates are interchanged :
For a twoelectron system the
spinorbitals and (in which sigma is either alpha or beta spin
state) can be combined as follows :
According to the definition of a
determinant this antisymmetrized product is
equal to :
This type of wavefunction is known as a
Slater determinant, commonly abbreviated as :
An important property of the SCF
method is that its solutions satisfy the Variation
Principle, which states that the expectation value of
the energy evaluated with an inexact wavefunction is always higher
than the exact energy :
As a consequence the lowest energy
is associated with the best approximate wavefunction and
energy minimization is equivalent with wavefunction
optimization. The energies of
Slater determinants from a HartreeFock calculation are
readily expressed in one and twoelectron integrals.
For the ground state it is :
Here we have used the following
abbreviations
:
The twoelectron integral (iijj) which describes the
repulsion between two electrons each localized in one
orbital is called a Coulomb integral, (ijij) for which a classical
picture cannot be drawn so easily is called the Exchange
integral.
In many cases it is advantageous to
apply the restriction that electrons with opposite spin
pairwise
occupy the same spatial orbital. This leads to the
Restricted Hartree Fock method (RHF), as opposed
to the Unrestricted version (UHF).
An important advantage of the RHF method is that the
magnetic moments associated with the electron spin
cancel exactly for the pair of electrons in the same
spatial orbital, so that the SCF wavefunction is an eigenfunction of
the spin operators and . Note that the UHF
wavefunction is more flexible
than the RHF wavefunction, thus can
approximate the exact solution better and give a lower
energy. In practice RHF is mostly used for closed shell
systems, UHF for open shell species. RHF models for open
shell systems and more advanced models can used when
necessary. The total energy for
a closed shell ground state RHF model can be written
as :
The orbital energy in this case
is :
This paragraph is continued in part b with the
subjects: Molecular Orbitals : the LCAO/MO
method, Basis functions and Standard basis sets.
Chapter 5. Quantum chemistry in Molecular
Modeling
5.4 Limitations of the HF method;
Electron correlation.
Restricted HartreeFock SCF theory has
some painful shortcomings. Consider for example the
dissociation of the H2 molecule :
H^{+} +
H^{}
< HH > H.
+ H.
A "dissociation catastrophe" occurs
because the separated hydrogen atoms cannot be described
using doubly occupied orbitals, so that H2 tends to
dissociate in H+ and H, which can be described with a
doubly occupied orbital on H. This problem does not
occur in the UHF method, but this method has the
disadvantage that it does not give pure spin states.
An additional limitation of the HF
method in general is that due to the use of the
independent particle approximation the instantaneous
correlation of the motions of electrons is neglected,
even in the HartreeFock limit. The difference between the exact
energy (determined by the Hamiltonian) and the HF energy
is known as the correlation
energy: E_{correlation} = E_{exact}  E_{HF} < 0
Even though EHF is approximately 99%
of Esub>exact the difference
may be chemically important.
Several approaches are known that
try to calculate the correlation energy after HartreeFock
calculations (postHF methods). We will very briefly
discuss
Configuration Interaction (CI),
MøllerPlesset Perturbation
Theory and
MultiConfiguration SCF (MCSCF or
CASSCF).
HF theory gives a wavefunction which is
represented as a Slater determinant. In the conceptually
simple Configuration Interaction
(CI) method a linear combination of Slater
determinants is constructed, using the unoccupied
"virtual" orbitals from the SCFcalculation :
The total wavefunction is written as :
In principle, the exact correlation
energy can be obtained from a full CI calculation in
which all configurations are taken into consideration.
Unfortunately this is not
possible for all but the smallest systems. Moreover, the
problem is aggrevated when the size of
the basis set is increased, on the way towards the HartreeFock
limit. Thus, the theoretical limit of the exact
(timeindependent, nonrelativistic) Schrödinger
equation cannot be reached.
Even for small systems the number of
excited configurations is enormously large. A popular
way to truncate the CI expansion is to consider only
singly and doubly excited configurations (CISD). The
energy, calculated as the expectation value of the
Hamiltonian for CISD is :
To perform the calculation one needs
the twoelectron integrals over Molecular Orbitals. The
computation of these is very timeconsuming, even when
the integrals over AO's are available :
In general, CI is not the practical
method of choice for the calculation of correlation
energy because full CI is not possible, convergence of
the CI expansion is slow, and the integral
transformation timeconsuming. Moreover truncated CI is not
sizeconsistent, which means that the calculation of two
species at large separation does not give the same
energy as the sum of the calculations on separate
species. This is because a different selection of
excited configurations is made in the two calculations.
An advantage of the CI method is that it is variational, so
the calculated energy is always greater than the exact
energy.
Although CI is not recommendable as
a method for ground states CIsingles (CIS) has been
advocated as an approach to computation of excited state
potential energy surfaces [10].
A different approach to electron
correlation has become very popular in recent years : MøllerPlesset Perturbation Theory. The basic idea is that the
difference between the Fock operator and the exact
Hamiltonian can be considered as a perturbation :
Corrections can be made to any order
of the energy and the wavefunction :
The most popular method is the
lowest level of correction, MP2.
An enormous practical advantage is
that MP2 is fast (of the same order of magnitude as
SCF), while it is rather reliable in its behavior, and
size consistent. A disadvantage is that it is not variational, so
the estimate of the correlation energy can be too large.
In practice MP2 must be used with a reasonable basis set
(631G* or better). Subsequent MPlevels MP3, MP4
(usually MP4 SDQ) are more complicated and much more
timeconsuming. For example, for
pentane (C5H12) with the 631G(d) basis set (99 basis
functions) an MP2 energy calculation took about 4 times
the amount of time needed for SCF, while MP4 took almost
90 times that time [7].
Multiconfiguration SCF (MCSCF) or
Complete Active Space SCF (CASSCF) is a special method
in which HForbitals are optimized
simultaneously with a "small" CI. This can be used to study problems
where the HartreeFock method is
inappropriate (e.g. when there are lowlying excited
states), or to generate a good starting wavefunction for
a subsequent CI calculation.
The MCSCF method requires
considerable care in the selection of the basis set and
especially the active space, and should not be
considered for routine use. In
contrast to the HF, MPn and CI methods, MCSCF
does not provide a
"model chemistry" because each problem requires
different choices. MCSCF methods
are essential for the study of processes in which
transitions between potential energy surfaces occur,
such as in photochemical reactions [11, 12]. A
combination of MP2 with MCSCF has recently been explored
by Roos et
al. [13]. This seems to be a very promising method for
excited states.
Other methods to determine the
correlation energy are under development. At this point it is useful to note
another promising development, that of density functional theory. This
is a method in which the twoelectron integrals are not
computed in the conventional way. Application of this
approach to molecular systems is still in its infancy,
but rapid developments are to be expected in the next
few years, in particular driven by the desire to be able
to compute larger systems, e.g. metal complexes and
organometallic compounds.
References:
[7] Foresman, J.B.; Frisch, A.
Exploring Chemistry with Electronic
Structure Methods: A Guide to Using Gaussian, Gaussian
Inc., 1993.
[8] D. Feller and E.R Davidson,
in Reviews in
Computational Chemistry, K.B. Lipkowitz and D.B. Boyd,
eds., VCH, 1990, pp. 1  43.
[9] Brouwer, A.M.; Bezemer, L.;
Jacobs, H.J.C.,
Recl. Trav. Chim. PaysBas
1992, 111, 138143
[10] Foresman, J.B.; HeadGordon,
M.; Pople,
J.A.; Frisch, M.J.,
J. Phys. Chem., 1992, 96, 135  149
[11] Palmer, I.J.; Ragazos, I.N.;
Bernardi,
F.; Olivucci, M.; Robb, M.A.,
J. Am. Chem. Soc. 1993,115, 673 
682
[12] Olivucci, M.; Ragazos, I.N.;
Bernardi,
F.; Robb, M.A.,
J. Am. Chem. Soc. 1993, 115, 3710 
3721
[13] Roos, B.O.; Andersson, K.;
Fülscher,
M.P.,
Chem. Phys. Lett., 1992, 192, 5  13
Next paragraph, 5.5 Energy
calculations Previous paragraph 5.3 Solving the
electronic Schrödinger equation : HartreeFock SelfConsistent
Field theory. Chapter
5 MM Syllabus 1995 MODIFIED November 8, 1995 Fred Brouwer, Lab. of Organic
Chemistry, University of Amsterdam.
Chapter 5. Quantum chemistry in Molecular
Modeling
5.5 Energy calculations
Ab initio
calculations give the absolute energy of the system of
fixed nuclei and moving electrons. These are large
numbers, for example for cyclohexane the HF energy
with the 631G* basis set is 234.2080071 a.u., which is
equal to 146967.86 kcal/mol. Thus, the chemically significant
energy quantities of a few kcal/mol are very much
smaller than the computed quantity, and high accuracy is
required.
The absolute energy is not a
directly useful quantity. It can however be used to
calculate the Heat of Formation with a reasonable
accuracy. According to G1 and G2 theories [14, 15] the
molecular structure and vibrational frequencies are
first determined at the HF/631G* level. The frequencies are used to
calculate the zeropoint energy. Then, the geometry is
further optimized at the MP2 level. Subsequently, basis
set effects and correlation energies are calculated at
various levels of theory, to allow an extrapolation
(using small empirical contributions !) to the
limits of full CI and the HartreeFock limit, that is
to the complete Schrödinger equation for the motionless
molecule. Finally, the
zeropoint vibrational energy is added.
This procedure can account for heats of formation with
an accuracy of < 2 kcal/mol, which rivals the quality
of experimental data.
Other authors [16] calculate the
heat of formation based on the 631G* calculation and
bond increments, similar to the way MM2 deals with this.
This is a much less elaborate
procedure than the G1 and G2 theories, but it is
essentially empirical. The empirical corrections needed
in G1 and G2 are of a very "mild" kind, they are not
related to the structure of the species, but only depend
on the number of electrons.
The isodesmic reaction approach allows a
fairly accurate calculation of the heat of reactions,
even at the HF level. Isodesmic reactions are
defined as transformations in which the numbers of bonds
of each formal type are conserved, and only the
relationships among the bonds are altered [4, 6]. For
example
:
CH4 + CH3CH2OH > CH3CH3 + CH3OH (1)
CF4 + 3 CH4 > 4 CH3F (2)
Energy changes (kcal/mol) for these
two reactions are
:
STO3G 321G 631G*//STO3G experimental
deltaE (1) 2.6 4.8 4.1 5.0 (5.7)
deltaE (2) 53.5 62.4 49.6 49.3 (52.8)
(The experimental numbers in
parentheses are without correction for zeropoint energy
changes).
References:
[4] W.J. Hehre, L. Radom, P. von R.
Schleyer
and J.A. Pople
Ab initio
molecular orbital theory, Wiley, 1986
[6] Spartan User's Guide, version
3.0,
Wavefunction, Inc., 1993.
[14] Pople, J.A.; HeadGordon, M.;
Fox, D.J.; Raghavachari, K.; Curtiss,
L.A.,
J. Chem. Phys., 1989, 90, 5622 
5629.
[15] Curtiss, L.A.; Raghavachari,
K.; Trucks, G.W.; Pople, J.A.,
J. Chem. Phys., 1991, 94, 7221.
Next paragraph, 5.6 Quality of ab initio
results Previous paragraph 5.4 Limitations
of the HF method; Electron
correlation.
Chapter 5 MM Syllabus 1995 MODIFIED
November 8, 1995 Fred Brouwer, Lab. of
Organic Chemistry, University of Amsterdam.
5.6 Quality of ab initio results
One of the most useful features of
ab initio
MO theory is that it allows the definition of "model
chemistries". A
theoretical model chemistry entails a method
(e.g. HartreeFock or MP2 etc.) and
a basis set. The philosophy of
a model
chemistry is that it should be uniformly applicable and
tested on as many systems as possible to learn about its
performance. This turns out to be useful, because the
reliability and accuracy of model chemistries can be
systematically assessed in this way.
In the rest of this section we will
focus on the performance of model chemistries that can
be practically applied for organic molecules with
presentday hardware and software, that is HF and MP2
methods with basis sets usually limited to the 631G(d) level. For
more detailed comparisons see references 4, 6, 8 and 17.
Geometry
As far as equilibrium geometry is
concerned, HF and MP2 ab initio models even with
modest basis sets lead to excellent results. At present,
HF/631G* or MP2/631G* are considered good and reliable
methods for the determination of the geometries of
organic molecules. In many cases the smaller basis sets
321G or even STO3G can give useful results. The bond
length calculated at the HF level is usually
overestimated by ca. 0.01  0.02 Å as a result of the
neglect of electron correlation. For examples see
references 4, 6, 8 and 17.
For transition metal compounds and
organometallics the results
are less satisfactory. Because of the size of such
systems, adequately large basis sets cannot be applied
with the present generation of computers and programs.
Moreover, electron correlation can be important. It is
conceivable that a model chemistry based on Density
Functional Theory will become available which covers
this area of chemistry.
Vibrational frequencies
Due to the availability of analytic
second derivatives of HF and MP2 wavefunctions, the
calculation of vibrational frequencies and
normal modes of organic molecules has become almost a
routine matter [4, 6, 7]. It turns out that the
results even with modestbasis HF models are quite good.
The frequencies are consistently
overestimated, which is due to the neglect of
correlation energy and of anharmonicity. Uniform
scaling of the computed frequencies by a factor of 0.89
? 0.01 gives a good agreement for
most cases. For MP2 the scaling factor should be closer
to 1.0. Of course, with
dedicated empirical force fields a better fit of the
spectra can be achieved, however at the expense of
generality and of great effort.
Energies
The accurate computation of absolute
or relative energies remains a major challenge. Even
conformational energy differences and barriers are not
reliably computed with HF or MP2 models using small
basis sets (631G* or smaller). Of course the demands on accuracy
are very high in this case. On the other hand, a
comparison between related systems (e.g. predicting a
substituent effect) can often
be made quite well. Energies of
reactions can be predicted relatively accurately,
especially for isodesmic processes. When the number of
formal bonds changes, electron correlation methods are
essential. As mentioned in section 5.5,
successful methods have been developed to estimate heats
of formation on the basis of ab initio results.
Chemical reactions
The quality of the prediction of
structures of transition states can hardly be verified
by comparison with experiments, so the only way is to
look at convergence of the computed values with
increasing sophistication of the method employed. Energies of transition states can be
related to experimental activation energies. In practice theoretical methods can
at best to predict relative activation energies, in
other words the selectivities of reactions.
Often this is chemically more significant than the
precise number
: usually it doesn't really matter whether a
product ratio is 50 : 1 or 100 : 1. To be able to
predict whether products will be formed in a ratio of
3:1 or 1:3 one
needs relative TS energies with an accuracy of
roughly 1 to 2 kcal/mol, and this is feasible for some
reactions, such as the DielsAlder reaction [6].
References:
[4] W.J. Hehre, L. Radom, P. von R.
Schleyer
and J.A. Pople
Ab initio molecular orbital theory,
Wiley, 1986
[5] Szabo and Ostlund
Modern Quantum Chemistry,
McGrawHill, 1989.
[6] Spartan User's Guide, version
3.0,
Wavefunction, Inc., 1993.
[7] Foresman, J.B.; Frisch, A.
Exploring Chemistry with Electronic
Structure Methods: A Guide to Using Gaussian, Gaussian
Inc., 1993.
[8] D. Feller and E.R Davidson,
in Reviews in Computational Chemistry,
K.B. Lipkowitz and D.B. Boyd,
eds., VCH, 1990, pp. 1  43.
[17] Boyd, D.B.,
in Reviews in Computational Chemistry,
K.B. Lipkowitz and D.B. Boyd,
eds., VCH, Vol. 1, 1990, 321  354.
Next
paragraph, 5.7 Semiempirical quantum
chemistry Previous
paragraph 5.5 Energy calculations Chapter 5 MM Syllabus 1995 MODIFIED
November 8, 1995 Fred Brouwer, Lab. of
Organic Chemistry, University of Amsterdam.
5.7 Semiempirical quantum chemistry
Ab initio quantum chemical methods are
limited in their practical applicability because of
their heavy demands of cputime and storage space on
disk or in the computer memory. At the HartreeFock level the
problem is seen to be in the large number of
twoelectron integrals that need to be evaluated.
Without special tricks this is proportional to the
fourth power of the number of basis functions. In
practice this can be reduced to something close to the
third power for larger molecules, e.g. because use is
made of the fact that integrals between orbitals
centered on distant atoms need not be calculated because
they will be zero anyway.
Still, the size of systems that can
be treated is limited, and this holds much more strongly
for correlated treatments. MP2
for example formally scales with the fifth power of the
number of basis functions. Therefore there is a place
for more approximate methods that retain characteristics
of the quantumchemical approach, in particular the
calculation of a wavefunction from which
electronic properties can be derived. In this section we
will present a brief overview of commonly used
semiempirical methods [6, 18].
The semiempirical methods are based
on the HartreeFock approach. A
Fockmatrix is constructed
and the HartreeFock equations are
iteratively solved. The approximations are in the
construction of the Fock matrix, in other words
in the energy expressions. Recall how the Fock matrix
elements are expressed as integrals over atomic basis
functions :
in which P is the density matrix :
To simplify matters drastically, the
Zero Differential Overlap (ZDO) approximation assumes :
which implies that
This can be justified when the
atomic basis orbitals are orthogonalized
(Löwdin
orthogonalization). As a result of the ZDO approximation
many twoelectron integrals vanish :
Another common feature of
semiempirical methods is that they only consider the
valence electrons. The core
electrons are accounted for in a corecore repulsion
function, together with the nuclear repulsion energy. In the most popular semiempirical
methods used today (MNDO, AM1 and PM3) the ZDO
approximation is only applied to basis functions on
different atoms. This is called the NDDO approximation
(Neglect of Diatomic Differential Overlap). The
resulting Fock matrix elements are
given in ref. 6 and discussed in detail in ref. 18. The next step is to replace many of
the remaining integrals by parameters, which can either
have fixed values, or depend on the distance between the
atoms on which the basis functions are located. At this
stage empirical parameters can be introduced, which can
be derived from measured properties of atoms or diatomic
molcules.
In the modern semiempirical methods the parameters are
however mostly devoid of this physical significance:
they are just optimized to give the best fit of the
computed molecular properties to experimental data. For
more technical details see references 6 and 18. Different semiempirical methods
differ in the details of the approximations (e.g. the
corecore repulsion functions) and in particular in the
values of the parameters. Note that in contrast to
molecular mechanics, only parameters for single atoms
and for atom pairs are needed. The number of published
parameters increases steadily. The semiempirical methods can be
optimized for different purposes. The MNDO, AM1 and PM3
methods were designed to reproduce heats of formation
and structures of a large number of organic molecules.
Other semiempirical methods are specifically optimized
for spectroscopy, e.g. INDO/S or CNDO/S, which involve
CI calculations and are quite good at prediction of
electronic transitions in the UV/VIS spectral
region.
Some even more approximate methods
are still quite useful. In the Hückel and Extended Hückel methods
the whole sum over twoelectron integrals is replaced by
a single diatomic parameter (the resonance integral), so
that no search for a selfconsistent field is necessary
(nor possible). These methods have proven extremely
valuable in qualitative and semiquantitative MO
theories of pielectron systems and of organometallic
systems [3]. For pielectron
systems ZDO treatments have been developed that take
only picenters (patomic orbitals) into account, but
do perform the SCF calculation. An example is the PariserParrPople method,
which involves a CI calculation as well. This method is
very successfully used to predict the optical absorption
spectra of conjugated organic molecules [19]. In the MM2 and MM3 programs
pielectron calculations are used to adjust the force
constants and equilibrium values of bond lengths to the
prevailing bond order. The pibond order between two
atoms is simply the sum over MOs of the product of the
coefficients of the basis functions on the atoms in the
MO, multiplied by the occupation number of the MO :
For a given geometry the pielectron
calculation is done, and the bondorders computed. Then
the force field is adjusted : the force
constants for stretching and torsion are scaled and the
equilibrium bond length for the bonds between the
picenters are calculated.
When the geometry changes too much,
the pielectron treatment is repeated to adjust the
force field to the new situation. For the pielectron
calculation the pisystem is treated as if it is planar.
Otherwise the bond order for a twisted semisingle bond
would become smaller as the bond is twisted more, and
the "restoring force" towards planarity (conjugation)
would vanish.
References:
[3] G. NàraySzabò, P.R. Surjàn and J.G.
Angyàn
Applied Quantum Chemistry, Reidel, 1987
[6] Spartan User's Guide, version
3.0,
Wavefunction, Inc., 1993.
[9] Brouwer, A.M.; Bezemer, L.;
Jacobs, H.J.C.,
Recl. Trav. Chim. PaysBas 1992, 111,
138143
[18] Stewart, J.J.P.,
in Reviews in Computational Chemistry,
K.B. Lipkowitz and D.B. Boyd,
eds., VCH, Vol. 1, 1990, 45  82.
Next
paragraph, 5.8 Quality of semiempirical
results Previous
paragraph 5.6 Quality of ab initio results
Chapter 5 MM Syllabus
1995 MODIFIED November 8, 1995 Fred Brouwer, Lab. of Organic
Chemistry, University of Amsterdam.
5.8 Quality of semiempirical
results
Semiempirical methods are
parameterized on the basis of selected properties of a
selected set of molecules. A reasonable performance can
be expected for related compounds, but for problems that
are not covered by the "training set" the reliability is
limited. In this respect ab initio methods, even
lowlevel ones, are more widely applicable.
For molecular structure and heats of
formation of closedshell molecules MNDO, AM1 and PM3
are quite good. Practical experience has shown that for
some particular problems one of the three performs
markedly better than the others, but in general the most
recent methods AM1 and PM3 are preferred. PM3 is
parameterized for a greater number of elements, but
sometimes the parameters are based upon a very small set
of data. The mean absolute errors of the bond lengths
between heavy atoms are reported to be 0.036 Å for PM3
and slightly greater for AM1 and MNDO. The error in bond
angles is 3  4 degrees [6]. Even lowlevel ab initio
calculations are usually better, but very much more
timeconsuming.
Trends in vibrational frequencies in
related systems are usually reproduced by semiempirical
calculations, but the errors are not as nicely
systematic (and therefore correctable) as in ab initio
calculations. When energy is concerned the results are
not completely satisfactory, in spite of the fact that
the methods are parameterized to reproduce heats of
formation.
Reference: [6]
Spartan User's Guide, version 3.0, Wavefunction, Inc., 1993.
Next
paragraph, 5.9 Solvation Previous
paragraph, 5.7 Semiempirical quantum chemistry
Chapter 5 MM Syllabus
1995 MODIFIED November 8, 1995 Fred Brouwer, Lab. of Organic
Chemistry, University of Amsterdam.
5.9 Solvation
The effect of a solvent can be
incorporated in quantumchemical calculations most
easily by considering it as a continuous dielectric
medium, characterized by a dielectric constant. The
electric field caused by the molecule induces a
polarization of the medium, which in turn acts on the
electrons in the molecule (SelfConsistent Reaction
Field, SCRF)[7]. The model thus
contains the quantummechanical description of the
molecule and a classical medium. The problem is to
choose where to locate the boundary between quantum
system and medium. In the Gaussian programs a simple
approximation is used in which the volume of the solute
is used to compute the radius of a cavity which forms
the hypothetical surface of the molecule. Spartan offers
solvation
models for the semiempirical Hamiltonians. This method, in which the molecular
surface is constructed from atomic (Born) radii,
requires parameters for each atom [6, 20]. In most
cases, solvation hardly affects the
structure of a molecule (relative to the gas phase), but
in cases of polar molecules, zwitterions or ions, the
relative energies can be changed dramatically [6].
References:
[6] Spartan User's Guide, version
3.0,
Wavefunction, Inc., 1993.
[7] Foresman, J.B.; Frisch, A.
Exploring Chemistry with Electronic
Structure Methods: A Guide to Using Gaussian, Gaussian
Inc., 1993.
[20] Cramer, C.J.; Truhlar, D.G.,
Science, 1992, 256, 213  217.
Next
paragraph, 5.10 Properties derived from the wavefunction Previous
paragraph, 5.8 Quality of semiempirical results
Chapter 5 MM Syllabus
1995 MODIFIED November 8, 1995 Fred Brouwer, Lab. of Organic
Chemistry, University of Amsterdam.
5.10 Properties derived from the
wavefunction
The electronic wavefunction which is
computed in ab initio as well as
semiempirical quantum chemical methods can be used to
derive observable quantities of a molecule, but it can
also be analyzed and used to rationalize certain
chemical phenomena.
electrical properties
The electric dipole moment
? of a molecule can be calculated
directly from the positions of the nuclei and the
electronic wavefunction [6]:
The dipole moment can be viewed as
the first term of an expansion of the electric field due
to the molecule, the next higher term being the quadrupole
moment. It is also possible to obtain the dipole moment
and polarizabilities directly as
derivatives of the energy with respect to a uniform
electric field [21]. The electrostatic potential of the
molecule represents the interaction between the charge
distribution of the molecule and a unit point charge
located at some position p :
Calculation of the molecular
electrostatic potential at the surface of the molecule
(described by the total electron density) can indicate
how the molecule will interact with polar molecules or
charged species. Visualization of this can be nicely
accomplished using color coding [6].
Atomic charges
Although concepts like atomic point
charges or bond dipoles are widely used in molecular
mechanics, there is no unique definition of atomic
charge in a molecule. All ways to attribute a part of
the electron density to individual atoms are to a
certain extent arbitrary. As a first analysis, or as a
way to compare related systems, Mulliken Population Analysis
can be applied. The electron density distribution (the
probability of finding an electron in a volume element
dr) is :
Integrated over entire space this
gives the total number of electrons (S?v is the overlap):
This can be separated into diagonal
and offdiagonal terms, where the former represent the
net population of the basis orbitals and the latter are
make up the overlap population.
In the Mulliken scheme the overlap
population is simply shared between the contributing
atoms, which leads to the following charge for each
basis orbital
:
Summing of the charges in the orbitals
associated with each atom gives the atomic charge. An important disadvantage of the
Mulliken
population analysis is that extended basis sets can lead
to unphysical results, e.g. charges of more than 2e,
which result from the fact that the basis orbitals
centered at one atom actually describe electron density
close to another nucleus. Population Analysis based on
Natural Atomic Orbitals does not have
this problem. An approach which
may be physically more relevant is to fit charges at the
atomic positions to the molecular
electrostatic potential measured at a grid of
points. This still leaves some arbitrariness in the
choice of the grid, and the procedure is computationally
much more demanding than the other types of population
analysis.
References:
[1] Levine
Quantum Chemistry, 4th ed., 1991
[2] Reviews in Computational
Chemistry,
Volumes 1 to 4, K.B. Lipkowitz and
D.B. Boyd, eds., VCH publishers.
[3] G. NàraySzabò, P.R. Surjàn and J.G.
Angyàn
Applied Quantum Chemistry, Reidel, 1987
[4] W.J. Hehre, L. Radom, P. von R.
Schleyer
and J.A. Pople
Ab initio molecular orbital theory,
Wiley, 1986
[5] Szabo and Ostlund
Modern Quantum Chemistry,
McGrawHill, 1989.
[6] Spartan User's Guide, version
3.0,
Wavefunction, Inc., 1993.
[7] Foresman, J.B.; Frisch, A.
Exploring Chemistry with Electronic
Structure Methods: A Guide to Using Gaussian, Gaussian
Inc., 1993.
[8] D. Feller and E.R Davidson,
in Reviews in Computational Chemistry,
K.B. Lipkowitz and D.B. Boyd,
eds., VCH, 1990, pp. 1  43.
[9] Brouwer, A.M.; Bezemer, L.;
Jacobs, H.J.C.,
Recl. Trav. Chim. PaysBas 1992, 111,
138143
[10] Foresman, J.B.; HeadGordon,
M.; Pople,
J.A.; Frisch, M.J.,
J. Phys. Chem., 1992, 96, 135  149
[11] Palmer, I.J.; Ragazos, I.N.;
Bernardi,
F.; Olivucci, M.; Robb, M.A.,
J. Am. Chem. Soc. 1993,115, 673 
682
[12] Olivucci, M.; Ragazos, I.N.;
Bernardi,
F.; Robb, M.A.,
J. Am. Chem. Soc. 1993, 115, 3710 
3721
[13] Roos, B.O.; Andersson, K.;
Fülscher,
M.P.,
Chem. Phys. Lett., 1992, 192, 5  13
[14] Pople, J.A.; HeadGordon, M.;
Fox, D.J.; Raghavachari, K.; Curtiss,
L.A.,
J. Chem. Phys., 1989, 90, 5622 
5629.
[15] Curtiss, L.A.; Raghavachari,
K.; Trucks, G.W.; Pople, J.A.,
J. Chem. Phys., 1991, 94, 7221.
[16a] Allinger, N.L.; Schmitz,
L.R.; Motoc, I.; Bender,
C.; Labanowski,
J.,
J. Phys. Org. Chem., 1990, 3, 732 
736.
[16b] L.R. Schmitz et
al.
J. Phys. Org. Chem., 1993, 6, 551;
Heteroatom Chem. 1992, 3, 69;
J. Comput. Chem., 1992, 13, 838;
J. Phys. Org. Chem., 1992, 5, 225;
J. Am. Chem. Soc., 1992, 114, 2880.
[17] Boyd, D.B.,
in Reviews in Computational Chemistry,
K.B. Lipkowitz and D.B. Boyd,
eds., VCH, Vol. 1, 1990, 321  354.
[18] Stewart, J.J.P.,
in Reviews in Computational Chemistry,
K.B. Lipkowitz and D.B. Boyd,
eds., VCH, Vol. 1, 1990, 45  82.
[19] Suzuki, H.
Electronic Absorption Spectra and
Geometry of Organic Molecules, Academic Press,
1967.
[20] Cramer, C.J.; Truhlar, D.G.,
Science, 1992, 256, 213  217.
[21] Dykstra, C.E.; Augspurger,
J.D.; Kirtman, B.; Malik, D.J.,
in Reviews in Computational Chemistry,
K.B. Lipkowitz and D.B. Boyd,
eds., VCH, Vol. 1, 1990, 83  118.
First page,
table of contents Previous
paragraph 5.9 Solvation Back to Chapter
3 of the Computational Chemistry Course.
Chapter 5 MM Syllabus 1995 MODIFIED
November 8, 1995 Fred Brouwer, Lab. of
Organic Chemistry, University of Amsterdam.
All semiempirical programs (MOPAC,
AMPAC, VAMP) have incorporated
solvent models. (Short note
in Dutch.) In
MOPAC the COSMO model has been implemented, which is
invoked by the keyword EPS=n.nn. The activation energy of almost any
reaction will be influenced by the polarity of the
solvent. However, in a comparison of two reactions, the
effects may be similar, and cancel if the difference is
calculated.
In paragraph 2A
we mentioned an S_{N}1 reaction, the
dissociation of methyl bromide, which doesn't display a
maximum in energy in the range shown. Now repeat this
calculation after adding the keyword EPS=78.4 (submit to
power queue) and try to
locate a maximum. Also study the
effect of substitution at the central carbon: ethyl
bromide, isopropyl bromide, tert.butyl bromide.
For an extensive presentation of all
existing models (very nicely illustrated), see Mike Colvin's
page on this subject.
This is part C of chapter 3: More
theory Previous part, B: FMO
theory Next part,
D:
Electrostatic interactions Back to Chapter 3
Contents page.
Apart from electrostatic
interactions, the overlap between orbitals may favour the
reaction between an electron donor and an electron
acceptor. A high (in energy) lying occupied orbital in
the donor may overlap with a low lying empty one in the
acceptor, leading to a net stabilization.
The strength of the interaction is
determined by:
the energy difference between the two
orbitals
involved (the smaller the better).
the amount of overlap between the orbitals (the
larger the better).
These effects can account for the
subtle differences that are found in facial selectivity,
caused by
steric effects which hinder overlap, or by
asymmetry of a pi orbital, which favours overlap
on that side where the orbital is larger.
Example: DielsAlder reaction of
fluorocyclopentadiene, which
takes place on the side where the F resides. Sterically the
other side would be preferred, so an electronic effect
must play a role. We can investigate the orbital
concerned, the HOMO of fluorocyclopentadiene, and
other cyclopentadienes.
See also the VRML representation of
frontier orbitals (HOMO, LUMO) in a
series of DielsAlder reactions, at Imperial
College, and a
related paper.
This is paragraph B of chapter 3:
More theory Previous paragraph A: Quantum
chemistry Next
paragraph C: Solvent
effects Back to Chapter 3
Contents page.
(preliminary text)
Reactions very often involve an
electron donating reagent that attacks a positive
center. Moreover, electrostatic interactions may
determine the relative orientation of two reacting
species, and therefore the selectivity. The distribution of charge can be
expressed as an electrostatic potential map, showing the
interaction a probe charge would experience from the
nuclei and the electron cloud. The calculation of atomic charges
from the electron distribution is subject to discussion;
i.e. the quality of the algorithms which assign the
'molecular' electrons to a particular atom (population
analysis).
This is part D of chapter 3: More
theory Back to Chapter
3 Contents page. Forward to Chapter
4, Programs used.
