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Computational Chemistry QMA_C06.qxd 8/12/08 9:11 Page 172 Computational 6 chemistry The central challenge In this chapter we extend the description of the electronic structure of molecules presented in Chapter 5 by introducing methods that harness the power of computers to calculate elec- 6.1 The Hartree–Fock formalism tronic wavefunctions and energies. These calculations are among the most useful tools 6.2 The Roothaan equations used by chemists for the prediction of molecular structure and reactivity. The computational methods we discuss handle the electron–electron repulsion term in the Schrödinger equa- 6.3 Basis sets tion in different ways. One such method, the Hartree–Fock method, treats electron–electron The first approach: interactions in an average and approximate way. This approach typically requires the numer- semiempirical methods ical evaluation of a large number of integrals. Semiempirical methods set these integrals to zero or to values determined experimentally. In contrast, ab initio methods attempt to 6.4 The Hückel method revisited evaluate the integrals numerically, leading to a more precise treatment of electron–electron 6.5 Differential overlap interactions. Configuration interaction and Møller–Plesset perturbation theory are used to account for electron correlation, the tendency of electrons to avoid one another. Another The second approach: ab initio computational approach, density functional theory, focuses on electron probability densit- methods ies rather than on wavefunctions. The chapter concludes by comparing results from differ- ent electronic structure methods with experimental data and by describing some of the 6.6 Configuration interaction wide range of chemical and physical properties of molecules that can be computed. 6.7 Many-body perturbation theory The field of computational chemistry, the use of computers to predict molecular The third approach: density structure and reactivity, has grown in the past few decades due to the tremendous functional theory advances in computer hardware and to the development of efficient software pack- ages. The latter are now applied routinely to compute molecular properties in a wide 6.8 The Kohn–Sham equations variety of chemical applications, including pharmaceuticals and drug design, atmo- 6.9 The exchange–correlation spheric and environmental chemistry, nanotechnology, and materials science. Many energy software packages have sophisticated graphical interfaces that permit the visualization of results. The maturation of the field of computational chemistry was recognized by Current achievements the awarding of the 1998 Nobel Prize in Chemistry to J.A. Pople and W. Kohn for their contributions to the development of computational techniques for the elucidation of 6.10 Comparison of calculations and experiments molecular structure and reactivity. 6.11 Applications to larger molecules The central challenge I6.1 Impact on nanoscience: The structures of nanoparticles The goal of electronic structure calculations in computational chemistry is the I6.2 Impact on medicine: Molecular solution of the electronic Schrödinger equation, @Ψ = EΨ, where E is the electronic recognition and drug design energy and Ψ is the many-electron wavefunction, a function of the coordinates of all the electrons and the nuclei. To make progress, we invoke at the outset the Born– Checklist of key ideas Oppenheimer approximation and the separation of electronic and nuclear motion Discussion questions (Chapter 5). The electronic hamiltonian @ is Exercises Problems QMA_C06.qxd 8/12/08 9:11 Page 173 6 COMPUTATIONAL CHEMISTRY 173 N N N N $2 e e nn Ze2 e e2 @ =−∑ ∇2 − ∑∑ I +1 ∑ (6.1a) r Electron 2 i ππεε2 12 244me i i I 0rIiij≠ 0r ij Electron 1 rB2 rA1 where rA2 rB1 • the first term is the kinetic energy of the Ne electrons; Nucleus ARAB Nucleus B • the second term is the potential energy of attraction Fig. 6.1 The notation used for the description of molecular between each electron and each of the Nn nuclei, with electron i hydrogen, introduced in the brief illustration preceding Section at a distance rIi from nucleus I of charge ZIe; 6.1 and used throughout the text. • the final term is the potential energy of repulsion between two electrons separated by rij. To keep the notation simple, we introduce the one-electron 1 operator The factor of –2 in the final sum ensures that each repulsion is counted only once. The combination e2/4πε occurs throughout 0 $2 =− ∇2 − 11 + computational chemistry, and we denote it j0. Then the hamilton- hiij0 (6.2) 2meAB rrii ian becomes NNe NeN nN e which should be recognized as the hamiltonian for electron i $2 Z 1 + =− ∇2 −I +1 in an H molecule-ion. Then @ ∑ i j0∑∑2 j0 ∑ (6.1b) 2 2me rIi≠ r ij i i I ij j @ =++hh 0 (6.3) We shall use the following labels: 12r 12 We see that the hamiltonian for H2 is essentially that of each Species Label Number used + electron in an H2-like molecule-ion but with the addition of = the electron–electron repulsion term. l Electrons i and j 1, 2,... Ne = Nuclei I A, B,... Nn Molecular orbitals, ψ m = a, b,... It is hopeless to expect to find analytical solutions with a = hamiltonian of the complexity of that shown in eqn 6.1, even Atomic orbitals used to construct the o 1, 2,... Nb molecular orbitals (the ‘basis’), χ for H2, and the whole thrust of computational chemistry is to formulate and implement numerical procedures that give ever more reliable results. Another general point is that the theme we develop in the sequence of illustrations in this chapter is aimed at showing explicitly how to use the equations that we have presented, and 6.1 The Hartree–Fock formalism thereby give them a sense of reality. To do so, we shall take the The electronic wavefunction of a many-electron molecule is a simplest possible many-electron molecule, dihydrogen (H2). Ψ function of the positions of all the electrons, (r1,r2,...). To Some of the techniques we introduce do not need to be applied formulate one very widely used approximation, we build on the to this simple molecule, but they serve to illustrate them in a material in Chapter 5, where we saw that in the MO description simple manner and introduce problems that successive sections of H2 we supposed that each electron occupies an orbital and show how to solve. One consequence of choosing to develop ψ ψ that the overall wavefunction can be written (r1) (r2).... a story in relation to H2, we have to confess, is that not all the Note that this orbital approximation is quite severe and loses illustrations are actually as brief as we would wish; but we many of the details of the dependence of the wavefunction on decided that it was more important to show the details of each the relative locations of the electrons. We do the same here, with little calculation than to adhere strictly to our normal use of the two small changes of notation. To simplify the appearance of the term ‘brief’. ψ ψ ψ ψ expressions we write (r1) (r2)...as (1) (2)....Next, we suppose that electron 1 occupies a molecular orbital ψ with l A BRIEF ILLUSTRATION a spin α, electron 2 occupies the same orbital with spin β, and so The notation we use for the description of H2 is shown in on, and hence write the many-electron wavefunction Ψ as the = = Ψ = ψ α ψ β Fig. 6.1. For this two-electron (Ne 2), two-nucleus (Nn 2) product a (1) a (2)....The combination of a molecular ψ α molecule the hamiltonian is orbital and a spin function, such as a (1), is the spinorbital ψ α introduced in Section 4.4; for example, the spinorbital a $2 1111 j @ =−() ∇2 +∇2 −+++j + 0 should be interpreted as the product of the spatial wavefunction 2m 1 2 0 rrrr r ψ α ψ α = ψ α eA1B1A2B212 a and the spin state , so a (1) a(1) (1), and likewise QMA_C06.qxd 8/12/08 9:11 Page 174 174 6 COMPUTATIONAL CHEMISTRY for the other spinorbitals. We shall consider only closed-shell molecules but the techniques we describe can be extended to open-shell molecules. A simple product wavefunction does not satisfy the Pauli principle and change sign under the interchange of any pair of electrons (Section 4.4). To ensure that the wavefunction does satisfy the principle, we modify it to a sum of all possible per- mutations, using plus and minus signs appropriately: Ψ = ψ α ψ β ...ψ β − ψ α ψ β ...ψ β + ... a (1) a (2) z (Ne) a (2) a (1) z (Ne) (6.4) There are N ! terms in this sum, and the entire sum can be Fig. 6.2 A schematic interpretation of the physical interpretation e of the Coulomb repulsion term, eqn 6.7a. An electron in orbital represented by the Slater determinant (Section 4.4): ψ ψ a experiences repulsion from an electron in orbital m where it |ψ |2 ψψαβψ β has probability density m . aa()11 ()$$ z () 1 ψψαβψ β 1 aa()22 () $$ z(()2 Ψ = %% %(6.5a) where f1 is called the Fock operator. This is the equation to solve N ! ψ e %% % to find a; there are analogous equations for all the other occu- ψψαβψ β pied orbitals. This Schrödinger-like equation has the form we aa()NNee ()$$ z () N e should expect (but its formal derivation is quite involved). Thus, The factor 1/Ne! ensures that the wavefunction is normalized f1 has the following structure: if the component molecular orbitals ψ are normalized. To save m f = core hamiltonian for electron 1 (h ) the tedium of writing out large determinants, the wavefunction 1 1 + average Coulomb repulsion from electrons 2, 3,..
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