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Quantum chemistry in Molecular Modeling



[Indonesia-Google-Translation]


5.1 Why use Quantum Chemical methods ?

Quantum-chemical methods are more general than empirical methods

5.2 The Schrödinger equation

The recipe for the calculation of the electronic wavefunction

5.3 Hartree-Fock 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 low-level methods for organic molecules

5.7 Semi-empirical quantum chemistry

Further approximations and introduction of empirical parameters

5.8 Quality of semi-empirical results

Low-level ab initio methods are usually better, but much more time-consuming

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 non-standard 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 semi-empirical 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 "non-standard" 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. Nàray-Szabò, 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 semi-empirical quantum-chemical 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. Nàray-Szabò, 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, McGraw-Hill, 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.

*          Born-Oppenheimer approximation; separation of the motion of nuclei and electrons.

*          orbital approximation; the electrons are confined to certain regions of space.

The Born-Oppenheimer approximation implies the separation of nuclear and electronic wavefunctions, the total wavefunction being a product of the two :
Born-Oppenheimer :

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 "del-squared"). The total energy in the Born-Oppenheimer 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 Hartree-Fock 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 : Hartree-Fock Self-Consistent 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 electron-nucleus 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 one-electron spin-orbitals.

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 self-consistent solution (a "Self-Consistent Field") is obtained, which can be expressed as a solution to the Hartree-Fock 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 "bra-ket" notation :

The Pauli principle states that the wavefunction must change sign when two independent electronic coordinates are interchanged :

For a two-electron system the spin-orbitals 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 Hartree-Fock calculation are readily expressed in one- and two-electron integrals. For the ground state it is :

Here we have used the following abbreviations :

The two-electron integral (ii|jj) which describes the repulsion between two electrons each localized in one orbital is called a Coulomb integral, (ij|ij) 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.

Next paragraph, 5.4 Limitations of the HF method; Electron correlation
Previous paragraph 5.2 The Schrödinger equation
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

5.4 Limitations of the HF method; Electron correlation.

Restricted Hartree-Fock SCF theory has some painful shortcomings. Consider for example the dissociation of the H2 molecule :

H+    +    H-   <------ H-H -------->  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 Hartree-Fock limit.
The difference between the exact energy (determined by the Hamiltonian) and the HF energy is known as the correlation energy: Ecorrelation = Eexact - EHF < 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 Hartree-Fock calculations (post-HF methods). We will very briefly discuss

*          Configuration Interaction (CI),

*          Møller-Plesset Perturbation Theory and

*          Multi-Configuration 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 SCF-calculation :

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 Hartree-Fock limit. Thus, the theoretical limit of the exact (time-independent, non-relativistic) 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 (CI-SD). The energy, calculated as the expectation value of the Hamiltonian for CISD is :

To perform the calculation one needs the two-electron integrals over Molecular Orbitals. The computation of these is very time-consuming, 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 time-consuming.
Moreover truncated CI is not size-consistent, 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 CI-singles (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 : Møller-Plesset 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 (6-31G* or better). Subsequent MP-levels MP3, MP4 (usually MP4 SDQ) are more complicated and much more time-consuming.
For example, for pentane (C5H12) with the 6-31G(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 HF-orbitals are optimized simultaneously with a "small" CI.
This can be used to study problems where the Hartree-Fock method is inappropriate (e.g. when there are low-lying 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 two-electron 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. Pays-Bas 1992, 111, 138-143

[10] Foresman, J.B.; Head-Gordon, 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 : Hartree-Fock Self-Consistent 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 6-31G* 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/6-31G* level.
The frequencies are used to calculate the zero-point 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 Hartree-Fock limit, that is to the complete Schrödinger equation for the motionless molecule.
Finally, the zero-point 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 6-31G* 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 :

           STO-3G  3-21G   6-31G*//STO-3G 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 zero-point 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.; Head-Gordon, 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.

Chapter 5. Quantum chemistry in Molecular Modeling

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. Hartree-Fock 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 present-day hardware and software, that is HF and MP2 methods with basis sets usually limited to the 6-31G(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/6-31G* or MP2/6-31G* are considered good and reliable methods for the determination of the geometries of organic molecules. In many cases the smaller basis sets 3-21G or even STO-3G 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 modest-basis 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 (6-31G* 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 Diels-Alder 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, McGraw-Hill, 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 Semi-empirical 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.


Chapter 5. Quantum chemistry in Molecular Modeling

5.7 Semi-empirical quantum chemistry

Ab initio quantum chemical methods are limited in their practical applicability because of their heavy demands of cpu-time and storage space on disk or in the computer memory.
At the Hartree-Fock level the problem is seen to be in the large number of two-electron 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 quantum-chemical 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 semi-empirical methods [6, 18].

The semi-empirical methods are based on the Hartree-Fock approach. A Fock-matrix is constructed and the Hartree-Fock 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 two-electron integrals vanish :

Another common feature of semi-empirical methods is that they only consider the valence electrons.
The core electrons are accounted for in a core-core repulsion function, together with the nuclear repulsion energy.
In the most popular semi-empirical 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 semi-empirical 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 semi-empirical methods differ in the details of the approximations (e.g. the core-core 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 semi-empirical 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 semi-empirical 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 two-electron integrals is replaced by a single diatomic parameter (the resonance integral), so that no search for a self-consistent field is necessary (nor possible). These methods have proven extremely valuable in qualitative and semi-quantitative MO theories of pi-electron systems and of organometallic systems [3].
For pi-electron systems ZDO treatments have been developed that take only pi-centers (p-atomic orbitals) into account, but do perform the SCF calculation. An example is the Pariser-Parr-Pople 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 pi-electron calculations are used to adjust the force constants and equilibrium values of bond lengths to the prevailing bond order. The pi-bond 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 pi-electron calculation is done, and the bond-orders 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 pi-centers are calculated.

When the geometry changes too much, the pi-electron treatment is repeated to adjust the force field to the new situation. For the pi-electron calculation the pi-system is treated as if it is planar. Otherwise the bond order for a twisted semi-single bond would become smaller as the bond is twisted more, and the "restoring force" towards planarity (conjugation) would vanish.

References:

[3] G. Nàray-Szabò, 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. Pays-Bas 1992, 111, 138-143

[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 semi-empirical 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.


Chapter 5. Quantum chemistry in Molecular Modeling

5.8 Quality of semi-empirical results

Semi-empirical 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 low-level ones, are more widely applicable.

For molecular structure and heats of formation of closed-shell 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 low-level ab initio calculations are usually better, but very much more time-consuming.

Trends in vibrational frequencies in related systems are usually reproduced by semi-empirical 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 Semi-empirical quantum chemistry
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

5.9 Solvation

The effect of a solvent can be incorporated in quantum-chemical 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 (Self-Consistent Reaction Field, SCRF)[7].
The model thus contains the quantum-mechanical 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 semi-empirical 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 semi-empirical results
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

5.10 Properties derived from the wavefunction

The electronic wavefunction which is computed in ab initio as well as semi-empirical 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 off-diagonal 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. Nàray-Szabò, 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, McGraw-Hill, 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. Pays-Bas 1992, 111, 138-143

[10] Foresman, J.B.; Head-Gordon, 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.; Head-Gordon, 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.


3C, Solvent effects

All semi-empirical 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 SN1 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.


3B, Frontier Molecular Orbital theory

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: Diels-Alder 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 Diels-Alder 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.


3D, Electrostatic interactions

(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.