Computational chemistry, as explored in this book, will be restricted to quantum mechanical descriptions of the molecules of interest. This should not be taken as a slight upon alternative approaches, principally molecular mechanics. Rather, the aim of this book is to demonstrate the power of high-level quantum computations in offering insight towards understanding the nature of organic molecules-their structures, properties, and reactions-and to show their successes and point out the potential pitfalls. Furthermore, this book will address applications of traditional ab initio and density functional theory methods to organic chemistry, with little mention of semi-empirical methods. Again, this is not to slight the very important contributions made from the application of Complete Neglect of Differential Overlap (CNDO) and its progeny. However, with the ever-improving speed of computers and algorithms, ever-larger molecules are amenable to ab initio treatment, making the semi-empirical and other approximate methods for treating the quantum mechanics of molecular systems simply less necessary. This book is therefore designed to encourage the broader use of the more exact treatments of the physics of organic molecules by demonstrating the range of molecules and reactions already successfully treated by quantum chemical computation. We will highlight some of the most important contributions that this discipline has made to the broader chemical community towards our understanding of organic chemistry.
We begin with a brief and mathematically light-handed treatment of the fundamentals of quantum mechanics necessary to describe organic molecules. This presentation is meant to acquaint those unfamiliar with the field of computational chemistry with a general understanding of the major methods, concepts, and acronyms. Sufficient depth will be provided so that one can understand why certain methods work well, but others may fail when applied to various chemical problems, allowing the casual reader to be able to understand most of any applied computational chemistry paper in the literature. Those seeking more depth and details, particularly more derivations and a fuller mathematical treatment, should consult any of three outstanding texts: Essentials of Computational Chemistry by Cramer, Introduction to Computational Chemistry by Jensen, and Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory by Szabo and Ostlund.
Quantum chemistry requires the solution of the time-independent Schrvdinger equation,
[??][PSI]([R.sub.1], [R.sub.2] ... [R.sub.N], [r.sub.1], [r.sub.2] ... [r.sub.n]) = E]PSI] ([R.sub.1], [R.sub.2] ... [R.sub.N], [r.sub.1], [r.sub.2] ... [r.sub.n]), (1.1)
where [??] is the Hamiltonian operator, [PSI]([R.sub.1], [R.sub.2] ... [R.sub.N], [r.sub.1], [r.sub.2] ... [r.sub.n]) is the wavefunction for all of the nuclei and electrons, and E is the energy associated with this wavefunction. The Hamiltonian contains all operators that describe the kinetic and potential energy of the molecule at hand. The wavefunction is a function of the nuclear positions R and the electron positions r. For molecular systems of interest to organic chemists, the Schrvdinger equation cannot be solved exactly and so a number of approximations are required to make the mathematics tractable.
1.1 APPROXIMATIONS TO THE SCHRVDINGER EQUATION: THE HARTREE-FOCK METHOD
1.1.1 Nonrelativistic Mechanics
Dirac achieved the combination of quantum mechanics and relativity. Relativistic corrections are necessary when particles approach the speed of light. Electrons near heavy nuclei will achieve such velocities, and for these atoms, relativistic quantum treatments are necessary for accurate description of the electron density. However, for typical organic molecules, which contain only first- and second-row elements, a relativistic treatment is unnecessary. Solving the Dirac relativistic equation is much more difficult than for nonrelativistic computations. A common approximation is to utilize an effective field for the nuclei associated with heavy atoms, which corrects for the relativistic effect. This approximation is beyond the scope of this book, especially as it is unnecessary for the vast majority of organic chemistry.
The complete nonrelativistic Hamiltonian for a molecule consisting of n electrons and N nuclei is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
where the lower case indexes the electrons and the upper case indexes the nuclei, h is Planck's constant, [m.sub.e] is the electron mass, [m.sub.I] is the mass of nucleus I, and r is a distance between the objects specified by the subscript. For simplicity, we define
[e'.sup.2] = [e.sup.2]/4[pi][[epsilon].sub.0]. (1.3)
1.1.2 The Born Oppenheimer Approximation
The total molecular wavefunction [PSI](R, r) depends on both the positions of all of the nuclei and the positions of all of the electrons. Because electrons are much lighter than nuclei, and therefore move much more rapidly, electrons can essentially instantaneously respond to any changes in the relative positions of the nuclei. This allows for the separation of the nuclear variables from the electron variables,
[PSI]([R.sub.1], [R.sub.2] ... [R.sub.N], [r.sub.1], [r.sub.2] ... [r.sub.n]) = [PHI]([R.sub.1], [R.sub.2] ... [R.sub.N])[psi]([r.sub.1], [r.sub.2] ... [r.sub.n]). (1.4)
This separation of the total wavefunction into an electronic wavefunction c(r) and a nuclear wavefunction [PHI](R) means that the positions of the nuclei can be fixed and then one only has to solve the Schrvdinger equation for the electronic part. This approximation was proposed by Born and Oppenheimer and is valid for the vast majority of organic molecules.
The potential energy surface (PES) is created by determining the electronic energy of a molecule while varying the positions of its nuclei. It is important to recognize that the concept of the PES relies upon the validity of the Born-Oppenheimer approximation, so that we can talk about transition states and local minima, which are critical points on the PES. Without it, we would have to resort to discussions of probability densities of the nuclear-electron wavefunction.
The Hamiltonian obtained after applying the Born-Oppenheimer approximation and neglecting relativity is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
where [V.sup.nuc] is the nuclear-nuclear repulsion energy. Equation (1.5) is expressed in atomic units, which is why it appears so uncluttered. It is this Hamiltonian that is utilized in computational organic chemistry. The next task is to solve the Schrvdinger equation (1.1) with the Hamiltonian expressed in Eq. (1.5).
1.1.3 The One-Electron Wavefunction and the Hartree-Fock Method
The wavefunction [psi](r) depends on the coordinates of all of the electrons in the molecule. Hartree proposed the idea, reminiscent of the separation of variables used by Born and Oppenheimer, that the electronic wavefunction can be separated into a product of functions that depend only on one electron,
[psi]([r.sub.1], [r.sub.2] ... [r.sub.n]) = [[phi].sub.1]([r.sub.1]) [[phi].sub.2]([r.sub.2]) ... [[phi].sub.n]([r.sub.n]). (1.6)
This wavefunction would solve the Schrvdinger equation exactly if it were not for the electron-electron repulsion term of the Hamiltonian in Eq. (1.5). Hartree next rewrote this term as an expression that describes the repulsion an electron feels from the average position of the other electrons. In other words, the exact electron-electron repulsion is replaced with an effective field [V.sup.eff.sub.i] produced by the average positions of the remaining electrons. With this assumption, the separable functions [[phi].sub.i] satisfy the Hartree equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)
(Note that Eq. (1.7) defines a set of equations, one for each electron.) Solving for the set of functions [[phi].sub.i] is nontrivial because [V.sup.eff.sub.i] itself depends on all of the functions [[phi].sub.i]. An iterative scheme is needed to solve the Hartree equatioons. First, a set of function ([[phi].sub.1], [[phi].sub.2] ... [[phi].sub.n]) is assumed. These are used to produce the set of effective potential operators [V.sup.eff.sub.i] and the Hartree equations are solved to produce a set of improved functions [[phi].sub.i]. These new functions produce an updated effective potential, which in turn yields a new set of functions [[phi].sub.i]. This process is continued until the functions [[phi].sub.i] no longer change, resulting in a self-consistent field (SCF).
Replacing the full electron-electron repulsion term in the Hamiltonian with [V.sup.eff] is a serious approximation. It neglects entirely the ability of the electrons to rapidly (essentially instantaneously) respond to the position of other electrons. In a later section we will address how to account for this instantaneous electron-electron repulsion.
Fock recognized that the separable wavefunction employed by Hartree (Eq. 1.6) does not satisfy the Pauli Exclusion Principle. Instead, Fock suggested using the Slater determinant
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
which is antisymmetric and satisfies the Pauli Principle. Again, an effective potential is employed, and an iterative scheme provides the solution to the Hartree-Fock (HF) equations.
1.1.4 Linear Combination of Atomic Orbitals (LCAO) Approximation
The solutions to the Hartree-Fock model, [[phi].sub.i], are known as the molecular orbitals (MOs). These orbitals generally span the entire molecule, just as the atomic orbitals (AOs) span the space about an atom. Because organic chemists consider the atomic properties of atoms (or collection of atoms as functional groups) to still persist to some extent when embedded within a molecule, it seems reasonable to construct the MOs as an expansion of the AOs,
[[phi].sub.i] = [k.summation over ([mu])]c.sub.i]mu][[chi].sub.[mu]], (1.9)
where the index [mu] spans all of the atomic orbitals [chi] of every atom in the molecule (a total of k atomic orbitals), and [c.sub.i]mu]] is the expansion coefficient of AO [[chi].sub.[mu]] in MO [[phi].sub.i]. Equation (1.9) thus defines the linear combination of atomic orbitals (LCAO) approximation.
1.1.5 Hartree-Fock-Roothaan Procedure
Taking the LCAO approximation for the MOs and combining it with the Hartree-Fock method led Roothaan to develop a procedure to obtain the SCF solutions. We will discuss here only the simplest case where all molecular orbitals are doubly occupied, with one electron that is spin up and one that is spin down, also known as a closed-shell wavefunction. The open-shell case is a simple extension of these ideas. The procedure rests upon transforming the set of equations listed in Eq. (1.7) into the matrix form
FC = SC]epsilon], (1.10)
where S is the overlap matrix, BLDBLD is the k x k matrix of the coefficients [c.sub.i]mu]], and [epsilon] is the k x k matrix of the orbital energies. Each column of BLDBLD is the expansion of [[phi].sub.i] in terms of the atomic orbitals [[chi].sub.[mu]]. The Fock matrix F is defined for the [mu]v element as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)
where [??] is the core Hamiltonian, corresponding to the kinetic energy of the electron and the potential energy due to the electron-nuclear attraction, and the last two terms describe the coulomb and exchange energies, respectively. It is also useful to define the density matrix (more properly, the first-order reduced density matrix),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)
The expression in Eq. (1.12) is for a closed-shell wavefunction, but it can be defined for a more general wavefunction by analogy.
The matrix approach is advantageous, because a simple algorithm can be established for solving Eq. (1.10). First, a matrix X is found that transforms the normalized atomic orbitals [[chi].sub.[mu]] into the orthonormal set [[chi].sub.[mu]]',
[[chi].sub.[mu]]' = [k.summation over ([mu])]X][chi].sub.[mu]], (1.13)
which is mathematically equivalent to
X†SX = 1, (1.14)
where X† is the adjoint of the matrix X. The coefficient matrix BLDBLD can be transformed into a new matrix C',
C' = [X.sup.-1BLD. (1.15)
Substituting C = XC' into Eq. (1.10) and multiplying by X† gives
X†FXC' = X†SXC'[epsilon] = C']epsilon] (1.16)
By defining the transformed Fock matrix
F' = X†FX, (1.17)
we obtain the simple Roothaan expression
F'C' = C']epsilon]. (1.18)
The Hartree-Fock-Roothaan algorithm is implemented by the following steps:
1. Specify the nuclear position, the type of nuclei, and the number of electrons.
2. Choose a basis set. The basis set is the mathematical description of the atomic orbitals. We will discuss this in more detail in a later section.
3. Calculate all of the integrals necessary to describe the core Hamiltonian, the coulomb and exchange terms, and the overlap matrix.
4. Diagonalize the overlap matrix S to obtain the transformation matrix X.
5. Make a guess at the coefficient matrix BLDBLD and obtain the density matrix D.
6. Calculate the Fock matrix and then the transformed Fock matrix F'.
7. Diagonalize F' to obtain C' and [epsilon].
8. Obtain the new coefficient matrix with the expression C = XC' and the corresponding new density matrix.
9. Decide if the procedure has converged. There are typically two criteria for convergence, one based on the energy and the other on the orbital coefficients. The energy convergence criterion is met when the difference in the energies of the last two iterations is less than some preset value. Convergence of the coefficients is obtained when the standard deviation of the density matrix elements in successive iterations is also below some preset value. If convergence has not been met, return to Step 6 and repeat until the convergence criteria are satisfied.
One last point concerns the nature of the molecular orbitals that are produced in this procedure. These orbitals are such that the energy matrix [epsilon] will be diagonal, with the diagonal elements being interpreted as the MO energy. These MOs are referred to as the canonical orbitals. One must be aware that all that makes them unique is that these orbitals will produce the diagonal matrix [epsilon]. Any new set of orbitals [[phi].sub.i]' produced from the canonical set by a unitary transformation
[[phi].sub.i]' = [summation over (j)][U.sub.ji][[phi].sub.j] (1.19)
will satisfy the Hartree-Fock (HF) equations and produce the exact same energy and electron distribution as that with the canonical set. No one set of orbitals is really any better or worse than another, as long as the set of MOs satisfies Eq. (1.19).
(Continues...)
Excerpted from Computational Organic Chemistry by Steven M. Bachrach Copyright © 2007 by John Wiley & Sons, Inc.. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.