Introduction to Quantum Mechanics 157

(b) Chemisorption of ethyne on silicon: Q. Liu and R. Hoffmann, J. Am. Chem. Soc., 1995, 117, 4082. (c) A carbon/sulfur polymer: H. Genin and R. Hoffmann. J. Am. Chem. Soc., 1995, 117, 12328.

[61](a) Q Liu, and R Hoffmann, J. Am,, Chem Soc., 1995, 117, 10108. (b) : P. Alemany and R. Hoffmann, J. Am, Chem Soc., 1993, 115, 8290; Mn clusters: D. M. Proserpio, R. Hoffmann, G. C. Dismukes, J. Am, Chem Soc., 1992, 114, 4374.

[62]J. H. Ammeter, H.-B. Bürgi, J C. Thibeault, R. Hoffmann, J. Am, Chem Soc., 1978, 100, 3686.

[63]Superior results from EHM compared to MINDO/3 and MNDO, for nonplanarity of certain C/C double bonds: J. Spanget-Larsen and R Gleiter, Tetrahedron, 1983, 39, 3345.

[64]EHM modified to give good geometries: S. L Dixon and P. C. Jurs, J. Comp. Chem., 1994, 15, 733.

4.6 EASIER QUESTIONS

1.What do you understand by the term quantum mechanics?

2.Outline the experimental results that led to quantum mechanics.

3.What approximations are used in the SHM?

4.How could the SHM Fock matrix for 1,3-butadiene be modified in an attempt to recognize the fact that the molecule has, formally anyway, two double bonds and one single bond?

5.What are the most important kinds of results that can be obtained from Hückel calculations?

6.Write down the simple Hückel Fock matrices (in each case using and 0, and 0, – 1 and 0) for: (1) the pentadienyl radical (2) the cyclopentadienyl radical

(3)trimethylenemethane, (4) trimethylenecyclopropane (5) 3-methylene- 1,4-pentadiene.

7.The SHM predicts the propenyl cation, radical and anion to have the same resonance energy (stabilization energy). Actually, we expect the resonance energy to decrease as we add electrons; why should this be the case?

8.What molecular feature cannot be obtained at all from the simple Hückel method? Why?

9.List the differences between the underlying theory of the SHM and the EHM.

10.A 400 × 400 matrix is easily diagonalized. How many carbons would an alkane have for its EHM Fock matrix to be 400 × 400 (or just under this size)? How many carbons would a (fully) conjugated polyene have if its SHM Fock matrix were 400 × 400?

4 . 7 HARDER QUESTIONS

1.Do you think it is reasonable to describe the Schrödinger equation as a postulate of quantum mechanics? What is a postulate?

158 Computational Chemistry

2.What is the probability of finding a particle at a point?

3.Suppose we tried to simplify the SHM even further, by ignoring all interactions i,

(ignoring adjacent interactions instead of setting them What effect would this have on energy levels? Can you see the answer without looking at a matrix or determinant?

4.How might the i, j-type interactions in the simple Hückel Fock matrix be made to assume values other than just – 1 and 0?

5.What is the result of using as a reference system for calculating the resonance energy of cyclobutadiene, not two ethene molecules, but 1,3-butadiene? What does this have to do with antiaromaticity? Is there any way to decide if one reference system is better than another?

6.What is the problem with unambiguously defining the charge on an atom in a molecule?

7.It has been reported that the extended Hückel method can be parameterized to give good geometries. Do you think this might be possible for the simple Hückel method? Why or why not?

8.Give the references to a journal paper that used the SHM, and one that used the EHM, within the last decade. Give an abstract of each paper.

9.The ionization energies usually used to parameterize the EHM are not ordinary atomic ionization energies, but rather valence-state AO ionization energies (VSAO ionization energies). What does the term “valence state” mean here? Should the VSAO ionization energies of the orbitals of an atom depend somewhat on the hybridization of the atom? In what way?

10.Which should require more empirical parameters: a molecular mechanics force field (chapter 3) or an EHM program? Explain.

Chapter 5

Ab initio calculations

“I could have done it in a much more complicated way”, said the

Red Queen, immensely proud.

Lewis Carroll, ca. 1870.

5.1 PERSPECTIVE

Chapter 4 showed how quantum mechanics was first applied to molecules of real chemical interest (pace chemical physics) by Erich Hückel, and how the extension of the simple Hückel method (SHM) by Hoffmann gave a technique of considerable usefulness and generality, the extended Hückel method (EHM). The SHM and EHM are both based on the Schrödinger equation, and this makes them quantum mechanical methods. Both depend on reference to experimental quantities (i.e. on parameterization against experiment) to give actual values of calculated parameters: the SHM gives energy levels in terms of a parameter which we could try to assign a value by comparison with experiment (actually the results of SHM calculations are usually left in terms of while the EHM needs experimental valence ionization potentials to calculate the Fock matrix elements. The need for parameterization against experiment makes the SHM and the EHM semiempirical (“semiexperimental”) theories. In this chapter, we deal with a quantum mechanical approach that does not rely on calibration against measured chemical parameters and is therefore called ab initio [1] meaning “from the first”, from first principles (it is true that ab initio calculations give results in terms of fundamental physical constants – Planck’s constant, the speed of light, the charge of the electron

– that must be measured to obtain their actual numerical values, but a chemical theory could hardly be expected to calculate the fundamental physical parameters of our universe).

Ab initio calculations 161

repulsion term(s). Thus, for the helium atom the Schrödinger equation (cf. section 4.3.3, Eqs (4.36) and (4.37)) is, in SI units

Here m is the mass (kg) of the electron, e is the charge (coulombs, positive) of the proton (= minus the charge on the electron), the variables and are the distances (meters) of electrons 1 and 2 from the nucleus, and from each other, Z = 2 is the number of protons in the nucleus, and is something called the permitivity of empty space; the factor is needed to make SI units consistent.

Hamiltonians can be written much more simply by using atomic units. Let’s take Planck’s constant, the electron mass, the proton charge, and the permitivity of space as the building blocks of a system of units in which and are numerically equal to 1 and the numerical values of physical constants are always dependent on our system of units). These

are the units of angular momentum, mass, charge, and permitivity in the system of atomic units. In this system, Eq. (5.4) becomes

Using atomic units simplifies writing quantum-mechanical expressions, and also means that the numerical (in these units) results of calculations are independent of the currently accepted values of physical constants in terms of kg, coulombs, meters, and seconds (of course, when we convert from atomic to SI units we must use accepted SI values of m, e, etc.). The atomic units of energy and length are particularly important to us. We can get the atomic unit of a quantity by combining and to give the expression with the required dimensions. The atomic units of length and energy, the bohr and the hartree, turn out to be:

The bohr is the radius of a hydrogen atom in the Bohr model (section 4.2.5), or the most probable distance of the electron from the nucleus in the fuzzier Schrödinger picture (section 4.2.5). The hartree is the energy needed to move a stationary electron one bohr distant from a proton away to infinity.The energy of a hydrogen atom, relative to infinite proton/electron separation as zero, is hartree: the potential energy is – 1 h

162 Computational Chemistry

consists of five terms, signifying (Fig. 5.1) from left to right: the kinetic energy of electron 1, the kinetic energy of electron 2, the potential energy of the attraction of the nucleus (charge Z = 2) for electron 1, the potential energy of the attraction of the nucleus for electron 2, and the potential energy of the repulsion between electrons 1 and 2. Actually this is not the exact Hamiltonian, for it neglects effects due to relativity and to magnetic interactions such as spin–orbit coupling [3]; these effects are rarely important in calculations involving lighter atoms, say those in the first two full rows of the periodic table (up to about chlorine). Relativistic quantum chemical calculations will be briefly discussed later. Thewavefunction is the “total”, overall wavefunction of the atom and can be approximated, as we will see later for molecular HF calculations, as a combination of wavefunctions for various energy levels. The problem with solving

hydrogen atom equation, could be solved exactly [4]. This problem arises in any system with three or more interacting moving objects, and in fact the many-body problem is an old one even in classical mechanics, going back to eighteenth century studies in celestial mechanics. The impossibility of an analytic solution to polyelectronic systems prompted Hartree’s approach to calculating wavefunctions and energy levels for atoms.

Hartree’s method was to write a plausible approximate polyelectronic wavefunction (a “guess”) for an atom as the product of one-electron wavefunctions:

This function is called a Hartree product. Here is a function of the coordinates of all the electrons in the atom, is a function of the coordinates of electron 1, is a function of the coordinates of electron 2, etc.; the one-electron functions etc. are called atomic orbitals (molecular orbitals if we were dealing with a molecule). The initial guess, is our zeroth approximation to the true total

Ab initio calculations 163

wavefunction zeroth because we have not yet started to refine it with the Hartree process; it is based on the zeroth approximations etc. To apply the Hartree process we first solve for electron one a one-electron Schrödinger equation in which the electron–electron repulsion comes from electron one and an average, smeared-out electrostatic field calculated from due to all the other electrons. The only moving particle in this equation is electron one. Solving this equation gives an improved version of We next solve for electron 2 a one-electron Schrödinger equation with electron two moving in an average field due to the electrons ofcontinuing to electron moving in a field due to . This completes the first cycle of calculations and gives

Repetition of the cycle gives

The process is continued for k cycles till we have a wavefunction and/or an energy calculated from that are essentially the same (according to some reasonable criterion) as the wavefunction and/or energy from the previous cycle. This happens when the functionsare changing so little from one cycle to the next that the smeared-out electrostatic field used for the electron–electron potential has (essentially) ceased to change. At this stage the field of cycle k is essentially the same as that of cycle k – 1, i.e. it is “consistent with” this previous field, and so the Hartree procedure is called the self-consistent-field-procedure, which is usually abbreviated as the SCF procedure.

There are two problems with the Hartree product of Eq. (5.7). Electrons have a property called spin, among the consequences of which is that not more than two electrons can occupy one atomic or molecular orbital (this is one statement of the Pauli (section 4.2.6) exclusion principle). In the Hartree approach we acknowledge this only in an ad hoc way, simply by not placing more than two electrons in any of the component orbitals that make up our (approximate) total wavefunction Another problem comes from the fact that electrons are indistinguishable. If we have a wavefunction of the coordinates of two or more indistinguishable particles, then switching the positions of two of the particles, i.e. exchanging their coordinates, must either leave the function unchanged or change its sign. This is because all physical manifestations of the wavefunction must be unchanged on switching indistinguishable particles, and these manifestations depend only on its square (more strictly on the square of its absolute value, i.e. on to allow for the fact that may be a complex, as distinct from a real, function). This should be clear from the equations below for a two-particle function: