Nature of the Chemical Bond
 Creators
 Goddard, William A., III
Abstract
Chapter 1 Summary: It is assumed that all students reading this material have had some course (e.g., the traditional semester of a juniorlevel physical chemistry course) presenting the basic elements of quantum mechanics with some treatment of the hydrogen atom, the harmonic oscillator, and angular momentum. This course will concentrate on the explanation of the structure and reactivity of molecules using quantum mechanical ideas. The explanations will stress qualitative and semiquantitative considerations with the emphasis on developing principles (based on quantum mechanics) that can be used to make reliable predictions on new systems (rather than merely rationalize known results). Chapter 1 is a review of materals that all students should have had previously, but with an emphasis on those points that will be important later in the course. The basic principles of quantum mechanics are summarized in § 1.1. A key idea here is that in, the classical description of an atom, the electron would collapse into the nucleus. The critical difference with the quantum description is that the kinetic energy is proportional to the average value of the square of the gradient of the wavefunction, [equation]. Consequently, for an electron sitting on the nucleus, the kinetic energy is infinite (since Vp is infinite). This forces the electron to remain distributed over a finite region surrounding the nucleus and prevents the collapse of the electron into the nucleus. Thus the quantum deschption is essential for stability of atoms. We will find in later chapters that modifications in the khetic energy (due to superposition of orbitals) also plays the key role in the formation of chemical bonds. Throughout ths course we will be searching for qualitative ideas concerning the sizes and shapes of wavefunctions and for simple ways of predicting the energy ordering of the states of a system. A useful concept here is the nodal theorem described in §1.3. Basically, this theorem tells us that the ground state of a system is everywhere positive [no nodal planes (zeros) interior to the boundaries of the system]. Chapter 2 Summary: In this chapter we consider the two states of H2+ [equation] [equation] (the LCAO wavefunctions) arising from bringing a proton up to the ground state of hydrogen, and we consider the two states of H2 [equation] [equation] (the VB wavefunctions) arising from bringing together two hydrogen atoms each in the ground state. As expected from the nodal theorem, the g state (symmetric) is the ground state for both systems. Indeed, in each case we And that the g state leads to bonding, while the u state leads to a repulsive potential curve. The g state of H2+ leads to an increase of the electron density in the bond region; however (contrary to popular belief), this leads to an increase in the electrostatic interactions, thus opposing bond formation. A bond is formed because of a very large decrease in the kinetic energy due to the molecular orbital having a significantly decreased gradient in the bond region. The bonding of the g state of H2 arises from the same term (modified by an additional overlap factor due to the second electron). The potential curves for both states of both molecules are dominated by exchange terms of the form [equation] [equation] for H2+ and [equation] [equation] for H2, where S is the overlap of the atomic orbitals. The quantity [tau] is the quantitative manifestation of the decreased kinetic energy (and increased potential energy) arising from interference of the [Chi][sub]t and [Chi][sub]r orbitals. It has the form [equation] for large R. Thus, at large R the bonding of H2+ is proportional to S, while the bonding of H2 is proportional to S^2. Consequently, for large R the bond energy of H2+ exceeds that of H2. For small R, where S~1, the bond energy of H2 is approximately twice that of H2+. The u states are far more repulsive than the g states are attractive (due to the 1 ± S and 1 ± S^2 terms in the denominators of [symbol] and [symbol]). We also examine the molecular orbital (MO) wavefunction for H2 [equation] which provides a simple description of the ground and excited states for small R. For large distances, the ionic terms implicit in the MO wavefunction lead to an improper description. The energy [epsilon][sub]0, of any approximate wavefunction [phi][sub]0, is an upper bound on the exact energy of the ground state E[sub]0, [equation] leading to the variational condition: If an approximate wavefunction (and hence the energy) is a function of some parameter [lamda], then the optimum wavefunction satisfies the (necessary) condition [equation]. Expanding the unknown wavefunction [phi] in terms of a basis [equation] and applying the variational condition leads to a set of matrix equations, HC = ESC for obtaining the optimum coefficients (i.e., wavefunction). More exact wavefunctions of H2+ are considered in §3.2, but in §3.3 we find that the description of bonding in terms of exchange energies is retained. In §3.5 we present an overview of three useful methods for wavefunctions: (a) The Hartree Fock (HF) method is a generalization of the MO wavefunction in which the wavefunction (ground state of a twoelectron system) is taken as [equation] and the orbitals [phi] optimized by solving the differential equation [equation] or the matrix equation [equation]. These equations are nonlinear and must be solved iteratively. (b) The generalized valence bond (GVB) method is a generalization of the VB method with the wavefunction taken of the form [equation] and the orbitals [phi][sub]a and [phi][sub]b optimized. This leads to two matrix equations [equation] [equation] analogous to the HF equations (3.518) and to two differential equations analogous to (3.520). (c) The configuration interaction (CI) method with the wavefunction taken of the form [equation]. For the ground state this wavefunction can always be written in terms of natural orbitals [equation] as [equation]. In §3.5.5 we find that the ground state of any two electron system is nodeless and symmetric [equation]. In §3.6 and §3.7 we find that the HF wavefunction accounts for all but about 1.1 eV of the energy for He and H2 and that a CI wavefunction with five NO'S accounts for all but about 0.15 eV. The four correlating NO'S for this wavefunction all involve one nodal plane. All these methods involve expansions in terms of basis sets. For He it is possible to obtain highly accurate HF and GVB wavefunctions with only two (slike) basis functions [the double valence (DV) basis] and for H, similar quality wavefunctions can be obtained with six basis functions (two s and one p on each center), [the DVP or double valence plus polarization basis].
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