Atoms secular equation

By the criterion of Exercise 2-9, is an eigenvalue of the matrix in a and p. There are two secular equations in two unknowns for ethylene. For a system with n conjugated sp carbon atoms, there will be n secular equations leading to n eigenvalues . The family of , values is sometimes called the spectrum of energies. Each secular equation yields a new eigenvalue and a new eigenvector (see Chapter 7). 

The variational problem is to minimize the energy of a single Slater determinant by choosing suitable values for the MO coefficients, under the constraint that the MOs remain orthonormal. With cj) being an MO written as a linear combination of the basis functions (atomic orbitals) /, this leads to a set of secular equations, F being the Fock matrix, S the overlap matrix and C containing the MO coefficients (Section 3.5). 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

Since P depends on the solution of the secular equation, which in turn depends on P, it is clear that we must solve iteratively for the molecular orbitals. In general, we will consider only the first few iterations and start the first iteration with = ZM, where is the effective charge of the nuclear core of the pth orbital (for more than one orbital per atom we have ZA = EM(y4) Zfi). The potential surface of the system is then approximated by... 

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

Since the nuclei are identical Haa = Hbb and since H is hermitian Hab = Hba. For normalized Is atomic wave functions the integrals Saa = Sbb = 1. The secular equation therefore reduces to... 

The diagrams are best understood in terms of the apparent repulsion between the energy levels of combining systems, which can easily be related to a perturbation treatment of the secular equations. For example, two carbon atom ir electron levels (1) and (2) with energies ao would interact to remove the degeneracy... 

The value of po for a particular state of an atom appears as a root of the secular equation, and is therefore determined by diagonalization of the matrix Ti/, /, equation (19), a matrix which is independent of po- Thus the values of the orbital exponents k are completely determined by solution of the secular equation. In the lowest approximation, we can represent the ground-state of... 

Problem 11-13. Write down and solve the Hiickel approximation to the secular equation for the vr system of ethylene. Noting that the energy of two electrons on non-interacting carbon atoms is 2o , show that the binding energy of the two vr electrons is —2/3. 

In the derivation of normal modes of vibration we started with a set of displacements of individual atoms. By determining the reducible representation Ltot and decomposing it, we calculated the number of normal modes of each symmetry species. We could determine what these modes are by solving a secular equation. We could alternatively have used projection operators to determine the symmetry-adapted combinations. 

All of the above conventions together permit the complete construction of the secular determinant. Using standard linear algebra methods, the MO energies and wave functions can be found from solution of the secular equation. Because the matrix elements do not depend on the final MOs in any way (unlike HF theory), the process is not iterative, so it is very fast, even for very large molecules (however, fire process does become iterative if VSIPs are adjusted as a function of partial atomic charge as described above, since the partial atomic charge depends on the occupied orbitals, as described in Chapter 9). 

In his valuable paper Molecular Energy Levels and Valence Bonds Slater developed a method of formulating approximate wave functions for molecules and constructing the corresponding secular equations.1 Let a,b, repreamt atomic orbitals, each occupied by one valence electron, and a and 0 represent the electron spin functions for spin orientation -f i and — J, respectively. Slater showed that the following function corresponds to a valence-bond structure with bonds a-----b, c---d, and so forth ... 

Write a computer program that finds the principal moments and principal axes of inertia for a molecule. Do not use matrix diagonalization instead, solve the secular equation by using the formula for the roots of a cubic equation. The input to the program is the set of atomic masses and coordinates in an arbitrary system with axes not necessarily at the center of mass. 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

Each such vibration (6.32) is called a normal mode of vibration. For each normal mode, the vibrational amplitude Aim of each atomic coordinate is constant, but the amplitudes for different coordinates are, in general, different. The nature of the normal modes depends on the molecular geometry, the nuclear masses, and the values of the force constants ujk. The eigenvalues m of U determine the vibrational frequencies the eigenvectors of U determine the relative amplitudes of the vibrations of the q, s in each normal mode, since Ajm / A im = Ijm/L- For H2° here are 9-6-3 normal modes, and the solution of (6.17) and (6.18) yields the vibrational modes shown in Fig. 6.1. For some molecules, two or more normal modes have the same vibrational frequency (corresponding to two or more equal roots of the secular equation) such modes are called degenerate. For example, a linear triatomic molecule has four normal modes, two of which have the same frequency. See Fig. 6.2. The general classical-mechanical solution (6.30) is an arbitrary superposition of the normal modes. 

---