The orbital approximation

The classical theory of a many-electron atom would be an even harder proposition. At least with the planets, their gravitational forces on each other are much smaller than the attraction to the Sun. Atomic forces ate electrostatic, not gravitational, and proportional therefore to the charges on particles. The repulsion between negatively charged electrons is comparable in its combined effect to their attraction to the nucleus. Even in a two-electron helium atom, the classical trajectories of electrons would be extraordinarily complicated, and would not even approximately resemble the elliptical motion of the single particle. 

Given these difficulties of classical physics, it is not surprising that the quantum theory of many-electron atoms has no exact solutions either. Yet in some ways the quantum theory puts us in a more favourable situation. We do not try to compute detailed trajectories, because we accept that they do not exist in the classical sense. What the wavefiinction should tell us is a probability distribution. The fact that we require less specific information makes it easier to develop an approximation which is accurate enough to be 

Consider a helium atom with two electrons. The wavefunction depends now on the positions of both electrons, i.e. y/ rj,r2), where rj = (X, yh Z ) and r2 = (x2, y2 z2) are the coordinates of each electron. The simplest approximation is to write 

Interpreted in words, this says that the probability of finding one electron at point r and the other at r2 is equal to the product of two individual probability distributions. It describes a situation where the probabilities of finding the electrons at given points are independent of one another. 

Better approximations can be made, and numerical calculations leave no doubt that Schrodinger s equation works very accurately for many-electron atoms, as it does for hydrogen. However, the orbital approximation is good enough for most purposes, and it leads to the very appealing picture of a many-electron atom in which each electron occupies an orbital which is similar to, although not identical with, the orbitals which form the exact solutions of the hydrogen atom. 

When we are calculating the total wavefunction for an atom or molecule, the orbital approximation is used (Eq. 14.13). F, the total wavefunction, is considered as a product of one electron wavefunctions known as orbitals— that is, yrjil), cO)r etc. We usually think 

CHAPTER 14 ADVANCED CONCEPTS IN ELECTRONIC STRUCTURE THEORY 

Since both k and all the y/ s can be related to probabilities (refer back to Section 14.1.1), the orbital approximation defines a total probability as a product of individual probabilities. This is only true in probability theory if the individual events (y/ s) are independent, such as each flipping of a coin to get heads or tails is an independent event. With respect to electrons, this means that the probability that electron 1 will exist at a certain position in space is completely independent of the positions of electrons 2, 3,4, etc. The electrons movements are therefore not correlated, a severe approximation (see the discussion of electron correlation given later). This is therefore called an independent electron theory. 

Since the probability of the electron being somewhere in the universe is 1, it must be true that the integral over all space of the square of a one-electron wavefunction is 1. Hence, one tenet of quantum mechanics is that all orbitals must be normalized, defined as satisfying Eq. 14.14. 

Another important attribute is that the orbitals are orthogonal, meaning that any pair of orbitals have zero overlap. Orthogonality is expresssed by Eq. 14.15. This greatly simplifies the calculational process, but with no cost in accuracy, as we will see later in this section. 

---

Having obtained a mediocre solution to the problem, we now seek to improve it. The next step is to take two Gaussian functions parameterized so that one fits the STO close to the nucleus and the other contributes to the part of the orbital approximation that was too thin in the STO-IG case, the part away from the nucleus. We now have a function... 

I conclude that in many ab initio calculations the orbital approximation represents the only practical approach, but its proponents might benefit by moderating their claims to success. As an example of a recent exaggerated claim we find. 

The problems which the orbital approximation raises in chemical education have been discussed elsewhere by the author (Scerri [1989], [1991]). Briefly, chemistry textbooks often fail to stress the approximate nature of atomic orbitals and imply that the solution to all difficult chemical problems ultimately lies in quantum mechanics. There has been an increassing tendency for chemical education to be biased towards theories, particularly quantum mechanics. Textbooks show a growing tendency to begin with the establishment of theoretical concepts such as atomic orbitals. Only recently has a reaction begun to take place, with a call for more qualitatively based courses and texts (Zuckermann [1986]). A careful consideration of the orbital model would therefore have consequences for chemical education and would clarify the status of various approximate theories purporting to be based on quantum mechanics. 

Scerri, E. R. [1989] Transition Metal Configurations and Limitations of the Orbital Approximation , Journal of Chemical Education, 66(6), p. 481. 

The most usual starting point for approximate solutions to the electronic Schrodinger equation is to make the orbital approximation. In Hartree-Fock (HF) theory the many-electron wavefunction is taken to be the antisymmetrized product of one-electron wavefunctions (spin-orbitals) ... 

Based on the orbital approximations, it is clear that/(r) is the DFT analog of the frontier orbital regioselectivity for nucleophilic (f (r)) and electrophilic (/ (r)) attack. It is then reasonable to define a reactivity indicator for radical attack by analogy to the corresponding orbital indicator,... 

Whereas the one-electron exponential form Eq. (5.5) is easily implemented for orbital-based wavefunctions, the explicit inclusion in the wavefunction of the interelectronic distance Eq. (5.6) goes beyond the orbital approximation (the determinant expansion) of standard quantum chemistry since ri2 does not factorize into one-electron functions. Still, the inclusion of a term in the wavefunction containing ri2 linearly has a dramatic impact on the ability of the wavefunction to model the electronic structure as two electrons approach each other closely. 

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

We will first give a discussion of some results of general spin-operator algebra not much is needed. This is followed by a derivation of the requirements spatial functions must satisfy. These are required even of the exact solution of the ESE. We then discuss how the orbital approximation influences the wave functions. A short qualitative discussion of the effects of dynamics upon the functions is also given. 

Therefore, the four linearly independent functions we obtain in the orbital approximation can be arranged into two pairs of linear combinations, each pair of which satisfies the transformation conditions to give an antis5mimetric doublet function. The most general total wave function then requires another linear combination of the pair of functions. In this case Eq. (4.18) can be written... 

Our approximations so far (the orbital approximation, LCAO MO approximation, 77-electron approximation) have led us to a tt-electronic wavefunction composed of LCAO MOs which, in turn, are composed of 77-electron atomic orbitals. We still, however, have to solve the Hartree-Fock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3)) ... 

The first approximation made in simplifying the task of solving for F is the orbital approximation. Each of the M electrons of the molecule is assumed to be described by a molecular orbital function ip the total wave function for the electronic state is the product of 0 s for the individual electrons (Equation A2.2).c... 

Step 4. Rank the orbitals approximately by their energy, and draw them as energy levels, one above the other, with the starting material on the left and the product on the right (Fig. 3.1). 

Equation 5.4 is often known as Koopmans Theorem. Equations 5.3 and 5.4 are approximate, as they not only depend on the orbital approximation, but also neglect the fact that exciting or removing one electron will cause some disturbance to the others. 

Owing to the indistinguishability of electrons, the wavefunction of a molecule s electron-cloud must be antisymmetric in the coordinates of the electrons. Hence, in the orbital-approximation, the wavefunction of a molecule (whose state corresponds to a set of complete electronic shells) can be expressed as a Slater-determinant, each column or row of which is written in terms of a single spin-orbital 8>. As pointed out, however, by Fock 9> and Dirac 10>, and later stressed by Lennard-Jones n> and Pople 12 the orbitals of a Slater-determinant are not uniquely determined, mathematically. 

Adding to or substracting from each other the rows or columns of a determinant does not alter the expanded determinant. Any orbital-set of a Slater-determinant can be replaced, therefore, by any linearly independent combination of the orbitals, without altering the determinant (except perhaps by a numerical factor). For example, the wavefunction y> for the four valence-shell electrons of a carbon atom in a 5S state may be written, in the orbital approximation, either in terms of the 2 s, 2 px,... 

In the light of the orbital approximation, a total wavefimction for the system of two separated hydrogen atoms can be written as... 

Until this point, the consideration of electron-electron repulsion terms has been neglected in the molecular Hamiltonian. Of course, an accurate molecular Hamiltonian must account for these forces, even though an explicit term of this type renders exact solution of the Schrddinger equation impossible. The way around this obstacle is the same Hartree-Fock technique that is used for the solution of the Schrddinger equation in many-electron atoms. A Hamiltonian is constructed in which an effective potential of the other electrons substitutes for a true electron-electron reg sion term. The new operator is called the Lock operator, F. The orbital approximation is still used so that F can be separated into i (the total number of electrons) one-electron operators, Fi (19). 

The periodic structure of the elements and, in fact, the stability of matter as we know it are consequences of the Pauli exclusion principle. In the words of A. C. Phillips Introduction to Quantum Mechanics, Wiley, 2003), A world without the Pauli exclusion principle would be very different. One thing is for certain it would be a world with no chemists. According to the orbital approximation, which was introduced in the last Chapter, an W-electron atom contains N occupied spinoibitals, which can be designated a, In accordance with the exclusion principle,... 

The first term in Eq. [3] is the kinetic energy operator, the second term is the electron-nucleus attraction energy operator and the third term stands for the electron-electron repulsion energy operator. In HF theory Eq. [2] is solved by means of the orbital approximation (1) the many-electron wavefunction P is represented with a determinant of one-electron wavefunctions r, (Slater determinant) satisfying the antisymmetric behavior of the many-electron wavefunc-... 

Under the first assumption, each electron moves as an independent particle and is described by a one-electron orbital similar to those of the hydrogen atom. The wave function for the atom then becomes a product of these one-electron orbitals, which we denote P (r,). For example, the wave function for lithium (Li) has the form i/ atom = Pa ri) Pp r2) Py r3). This product form is called the orbital approximation for atoms. The second and third assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and the unknown one-electron orbitals. These equations must be solved by iteration until a self-consistent solution is obtained. (In spirit, this approach is identical to the solution of complicated algebraic equations by the method of iteration described in Appendix C.) Like any other method for solving the Schrodinger equation, Hartree s method produces two principal results energy levels and orbitals. 

The LCAO method extends to molecules the description developed for many-electron atoms in Section 5.2. Just as the wave function for a many-electron atom is written as a product of single-particle AOs, here the electronic wave function for a molecule is written as a product of single-particle MOs. This form is called the orbital approximation for molecules. We construct MOs, and we place electrons in them according to the Pauli exclusion principle to assign molecular electron configurations. 

Computer calculations of molecular electronic structure use the orbital approximation in exactly the same way. Approximate MOs are initially generated by starting with trial functions selected by symmetry and chemical intuition. The electronic wave function for the molecule is written in terms of trial functions, and then optimized through self-consistent field (SCF) calculations to produce the best values of the adjustable parameters in the trial functions. With these best values, the trial functions then become the optimized MOs and are ready for use in subsequent applications. Throughout this chapter, we provide glimpses of how the SCF calculations are carried out and how the optimized results are interpreted and applied. 

The 3c and 2c bonding descriptions look different, but so long as both orbitals are doubly occupied in each case, they are in fact equivalent, hi the orbital approximation any set of occupied orbitals may be replaced by a linear combination of them without changing the overall many-electron wavefunction. The two 2c MOs of Fig, lb can be formed by making linear combinations of the 3c MOs in Fig, la. and conversely the 3c MOs could be reconstructed by combining the 2c MOs. The two pictures show different ways of dissecting the total electron distribution into contributions from individual pairs, but as electrons are completely indistinguishable such dissections are arbitrary and do not predict any observable differences. 

The orbital approximation itself suggests that the many-electron wave-function v / can be written as a product of one-electron functions, < (i), called orbitals. ... 

To make a general point now, I think it is still of great value to question the status of the orbital approximation even if one eventually returns to using it in a realistic manner in chemistry. This is because the eventual use of the orbital approximation is greatly improved by such questioning.9... 

---