Atomic orbitals , trial wavefunctions

Observables calculated from approximate wavefunctions as in Equation 1.13 are called expectation values, an expression used in probability theory. In practice, we will always have to be satisfied with approximate wavefunctions. How can we choose between different approximations And if our trial wavefunction has adjustable parameters (such as the coefficients of atomic orbitals in molecular orbitals see Section 4.1), how can we choose the adjustable parameters best values Here, Rayleigh s variation theorem is of great value. It tells us that the expectation value for the ground state energy E, (E ), calculated from an approximate wavefunction (P is always larger than the true energy E (Equation 1.14). Proof of the variation theorem is given in textbooks on quantum mechanics.18... 

Let us now indicate how local-scaling transformations can be used in order to carry out the constrained minimization of the kinetic energy functional [85-88]. The strategy that we have adopted is first to select a Slater determinant such as the one appearing in Eq. (73), as the trial wavefunction which generates the particular orbit d c Sn- For the case of atoms, the one-particle orbitals ( >g,i r) from which this Slater determinant is constructed are explicitly given by g,nim(r) = Rg,ni r)Yiimi(0,), where the subindex i has been replaced by the quantum numbers n, /, m. The radial functions are expanded as... 

The procedure is called the linear combination of atomic orbitals (LC AO) approximation and can be used for molecules of any size. H2 is a special case in that a wavefunction can be found that will solve the Schrodinger equation exactly, yet the MO approach will be used so that molecular orbitals can be derived. The simplest trial function for the H2+ system is written ... 

In this calculation, the trial wavefunction is a linear combination of the H Is and the Li 2s and 2p atomic orbitals. Theoretical background, a flowchart, and subroutines are again provided, and the students write the main program. After a successful run, the students compare the valence electron orbital energy with the first ionization energy of LiH. They also calculate the charge densities on Li and H and determine the dipole moment. This exercise is based on the more complete calculation of Karo and Olsen.i ... 

D11.1 Our comparison of the two theories will focus on the manner of construction of the trial wavefunctions for the hydrogen molecule in the simplest versions of both theories. In the valence bond method, the trial function is a linear combination of two simple product wavefunctions, in which one electron resides totally in an atomic orbital on atom A. and the other totally in an orbital on atom B. See eqns I l.l and 11.2, as well as Fig. 11.2. There is no contribution to the wavefunction from products in which both electrons reside on either atom A or B. So the valence bond approach undervalues, by totally neglecting, any ionic contribution to the trial function. It is a totally covalent function. The molecular oibital function for the hydrogen molecule is a product of two functions of the form of eqn 11.8, one for each electron, that is. 

Although it is possible to obtain exact solutions to the simplified Schrodinger equation given in equations (8.8) and (8.9), the resulting wavefunction is complicated, and it provides little insight into the wave-fimctions that might be used for other diatomic molecules. For this reason it wiU be more instructive to examine trial wavefunctions that have been constructed by a linear combination of hydrogen Is atomic orbitals ... 

The use of this type of trial wavefunction can be justified by the following argument. When the electron is very close to nucleus A, it experiences a coulombic attraction towards nucleus A which is far greater than that towards nucleus B. The wavefunction in this region is therefore expected to resemble a hydrogen Is orbital, centred on nucleus A. Similarly, when the electron is very close to nucleus B, the wavefunction is expected to resemble a Is orbital centred on nucleus B. Thus, by combining these two atomic wavefunctions it should be possible to produce trial wavefunctions which are fairly close to the true wavefunctions. 

Because there is more than one electron in the H2 molecule, we cannot solve Equations (10.12) and (10.13) exactly. As a first approximation, let us assume that the wavefunction for H2 is some linear combination of the I s AOs of the two isolated H atoms, as given by Equation (I0.I4), where y/ is the wavefunction of our molecular orbital and (p is the wavefunction for a Is AO. The constants c and C2 are simply weighting factors. These are the adjustable parameters of our trial wavefunction. Because the energies of the two I s AOs for each H atom are identical, c = c- in this example. 

In this trial function, f has the same value in each atomic orbital. This is not a necessary restriction. There is no physical reason for not choosing the more general trial function where orbitals with different are used. Symmetry requires that such a function be written 2 lulls 2) + ls"(l)ls (2)]2 F2[, (j) (2) - /3(l)cr(2)]. This type of function is called a split shell wavefunction. It gives a lower energy for He than does the function (7-26). However, for most quanrnm-chemical calculations split shells are not used, the gain in accuracy usually not being commensurate with the increased computational effort. 

The Rayleigh-Ritz variational theory is the basis for so-called variational methods in which an estimate of the energy of a system is calculated for an approximate trial wavefunction usually assembled from combinations of atomic orbitals. Expectation values of the energy may be calculated accurately for many trial wavefunctions and are upper bounds to the true energy. If the parameters of the trial wavefunctions are varied systematically, the lowest upper bound to the energy for a particular form of trial wavefunction may be determined (thus the term variational ). The trial functions must satisfy certain restrictions such... 

The electron affinity, which is very small for the Fe atom (0.15 eV), has so far not been reliably calculated. However, even the essentially zero affinity obtained is a tremendous improvement from the uncorrelated value of -2.36 eV. One of the reasons for the small remaining errors is that only simple trial functions were used. In particular, the determinants were constructed from Hartree-Fock orbitals. It is known that the Hartree-Fock wavefunction is usually more accurate for the neutral atom than for negative ion, and we conjecture that the unequal quality of the nodes could have created a bias on the order of the electron affinity, especially when the valence correlation energy is more than 20 eV. One can expect more accurate calculations with improved trial functions, algorithms, and pseudopotentials. 

The wavefunction for an SCF calculation is one or more antisymmetrized products of one-electron spin-orbitals. We have already seen (Chapter 5) that a convenient way to produce an antisymmetrized product is to use a Slater determinant. Therefore, we take the trial function to be made up of Slater determinants containing spin-orbitals (j>. If we are dealing with an atom, then the (f> s, are atomic spin-orbitals. For a molecule, they are molecular spin-orbitals. 

---