Secular solutions/terms

If we reexamine the governing equation, (4-230), for Re x = 0, we see that the mathematical difficulty for co = a>0 is that the forcing function is actually a solution of the homogeneous equation. The forcing function in this case is known as a secular term. Obviously, when a secular forcing term is present, a particular solution cannot exist in the form C sin 7 as is possible for co >0 - see, for example (4-232) - but must instead take the form Cl cos 7. Indeed, substituting this form into the governing equation, (4-230), with co = two, we find... 

This is an eigenvalue problem of the form of Eq. III.45 referring to the truncated basis only, and the influence of the remainder set is seen by the additional term in the energy matrix. The relation III.48 corresponds to a solution of the secular equation by means of a modified perturbation theory,19 and the problem is complicated by the fact that the extra term in Eq. III.48 contains the energy parameter E, which leads to an iteration procedure. So far no one has investigated the remainder problem in detail, but Eq. III.48 certainly provides a good starting point. 

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.

This determinant or its equivalent algebraic expansion is known as the secular equation. In equation (9.12) the parameters Ci correspond to the unknown quantities Xi in equation (9.13) and the terms (//, — WSu) correspond to the coefficients au- Thus, a non-trivial solution for the N parameters c, exists only if the determinant with elements Hu — Su) vanishes... 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

In this section we discuss, in general terms, certain properties of the secular equations (8) which provide insight into the various methods employed in defining reactivity indices. The energy levels arise from solution of the equation... 

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

Solution of the secular equation amounts to finding the roots of an iVth order equation in E. The N roots are the energies of the N molecular orbitals the forms of the orbitals in terms of the basis atomic orbitals 9are found by substituting each value of E, in turn, back into Equations A2.13 and solving for the c s using the additional condition that each MO tf)t is to be normalized,... 

Case 1 Orbitally degenerate ground term 2T2g needs a solution of a 6 x 6 secular equation... 

This arises from the 2T2gOh-term and needs a solution of 6 x 6 secular equation that is... 

Case 5 Elongated tetragonal bipyramid, small Aax X, orbitally degenerate ground term needs a solution of the 9x9 secular equation (which is factored) as above. The lowest nonmagnetic multiplets are separated by <534 = s(A) - s(A) owing to the configuration interaction with excited multiplets of the same symmetry. 

Orbitally degenerate groimd term 2 g interacts with excited 2T2g term, which needs a solution of 10 x 10 secular equation that is factored. The multiplets have energy (/ = A0 + X, (twice) s(F8) A0 - A/2 + (3/2)X2/A0, (four times) e(F8) -(3/2)2.2/A0, (four times, ground)... 

The expansion coefficients and energy eigenvalues are found from solutions of the secular equation Hnn — SnnE = 0. The nonzero matrix elements Hnn can be expressed in terms of the coefficients of the potential... 

The solution of the secular equation Fy —F5y = 0 requires the evaluation of the constituent matrix terms Fy. The Fy s are, however, themselves functions of the coefficients of the atomic orbitals amt through Pjel and therefore can only be evaluated by solving the secular equation. The Hartree-Fock procedure thus requires that a preliminary guess be made as to the values of the molecular population distribution terms Pici these values are then used to calculate the matrix elements Fy and thence solve the secular determinant. This, in turn, provides a better approximation to the wave function and an. .improved set of values of Pm. The above procedure is repeated with this first improved set and a second improved set evaluated. The process is repeated until no difference is found between successive improved wave functions. Finally, it may be shown that when such a calculation has been iterated to self-consistency the total electronic energy E of a closed shell molecule is given by... 

For the bonding in heteronuclear diatomics we start from a pseudodegenerate term of the two different atoms at Q = 0 and get the same secular equation (2) with the solutions (7) as in any other cases of the PJT effect. Again, one of the solutions leads to bonding with the same decrease in symmetry, as in the homonuclear case (Fig. 2b), and this picture can be easily extended to the bonding of several different atoms or groups of atoms. 

For a given set of atomic orbitals Xp, the one- and two-electron integrals (Equation 6.26 and Equation 6.28) can be calculated, and all terms of the Fock matrix in the AO basis, I jlv (Equation 6.25), would be known except for the density matrix, Ppo, which depends on the LCAO-MO coefficients as solutions of the secular equation (Equation 6.23) and thus on F. Accordingly, the density matrix depends on the evaluation of the Fock matrix, which depends on the elements of the density matrix. 

The trivial solution K = 0 represents a vibration without amplitude and is not interesting in terms of physics. Further solutions are obtained from the secular equation... 

This is recognized as a secular equation which gives the solution to the problem in internal coordinates. This is the original formulation proposed by Wilson, leading to the term GF matrix method which is often encountered in the literature. The relation between internal and normal coordinates is given by... 

As a result, the secular determinant for this function becomes a 10 by 10 matrix, the solution of which is not trivial however, the value of the lowest energy term, E, gives a more accurate value which is in accord with the variational principle. The use of still larger basis sets will push the value even lower. 

A procedure similar to that outlined in the elementary theory of flexion allows the determination of the normal modes. However, this method is not only tedious but also has the inconvenience that some terms in the secular equation depend explicitly on the material properties, that is, on the modulus. Instead of developing a solution of Eq. (17.132) in the classical way, it is more convenient to establish a method based on comparison of the apparent and real viscoelastic moduli (11,12). The basic idea is to compare Eq. (17.132) with the Laplace transform of Eq. (17.85), which is... 

In the previous chapters we sketched an elementary model of the chemical bond occurring between atoms in terms of a simple Hiickel theory mostly involving solution of 2 x 2 secular equations. The theory, first concerned with cr-bonding in H,. H2, He,. He2, was next extended to a- and 71-bonding in first-row homonuclear diatomics and to the study of multiple bonds, the fundamental quantity being a bond integral /3, whose form is... 

The normal-mode frequencies are obtained by solving an equation, the secular equation, that is the condition that must be satisfied if the molecule is to have harmonic modes of vibration. Since the constant terms in this equation are determined in part by the molecular structure, the frequency-structure correlation appears explicidy in the solution of the secular equation. These terms also depend on the potential-energy changes during the vibrations, and therefore the force field associated with such displacements from equilibrium must be known. 

The true solution (7) exhibits two time scales a fast time t 0(1) for the sinusoidal oscillations and a slow time t - l/ over which the amplitude decays. Equation (17) completely misrepresents the slow time scale behavior. In particular, because of the secular term tsint, (17) falsely suggests that the solution grows with time whereas we know from (7) that the amplitude A = (l - ) c decays exponentially. 

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