Properties of the Secular Equations

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 

Partial Derivatives and Polarizability Coefficients Expansion of (8) yields a polynomial, the characteristic or secular polynomial, whose roots are determined by the values of the parameters , vw- The ground state energy (12) is likewise a function of the (a,j3) parameter values, as are all quantities such as AO coefficients in the MO s, charges q bond orders p t, etc. It is possible, therefore, to specify the h partial derivative with respect to any or at an arbitrary point defined by a set of values (a,j8) in the parameter space, and to make expansions such as 

Some reference must now be made to the properties of the basic set of equations for the special case of alternant hydrocarbons. The secular polynomial A e) acquires important analytical properties when all a s and all jS s take common values ao and o and when, in addition, the molecule is alternant in the sense of the Coulson-Rushbrooke (1940) theorem. The following results are well known  

Energy level diagrams have seldom been used to show how the disposition of levels changes with variations of the a and j8 parameters describing the system, and how such processes can be related to spectroscopic changes and postulated reaction mechanisms. Yet, they can often give a qualitative understanding of the effects of molecular modifications with a minimum of effort, and for this reason at least deserve mention. 

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 

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The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

This shows that in the limit of large systems, the thermodynamics of the system is determined by the largest root of the secular equation. In particular, -kgT In as seen in Eq. (7.1.30) is an extensive property, i.e., it is proportional to the number of units. 

We may obtain one root of this equation at once. Since the other elements of the row and the column which contains C — W are all zero, C — W is a factor of the determinant and may be equated to zero to obtain the root W = C. The other three roots may be obtained by solving the cubic equation which remains, but inspection of the secular equation suggests a simpler method. Determinants have the property of being unchanged in value when the members of any row are added to or subtracted from the corresponding members of any other row. The same is true of the columns. We therefore have... 

Symmetry has taken us to a point where still quintic, quartic, and quadratic secular equations must be solved. However, a closer look at this equations shows that they can easily be solved. Apparently, a further symmetry principle is present, which leads to simple analytical solutions of the secular equations. Triphenylmethyl is an alternant hydrocarbon. In an alternant, atoms can be given two different colors in such a way that all bonds are between atoms of different colors hence, no atoms of the same color are adjacent. A graph with this property is bipartite, and its eigenvalue spectrum obeys the celebrated Coulson-Rushbrooke theorem [16]. 

A Special Set of Coordinates. It has been previously mentioned that six of the ZN roots of the secular equation (11), Sec. 2-2, have the value zei o. This will now be proved. The basis of the proof is that there are six modes of motion of zero frequency, namely, the three translations and three rotations. The roots of the secular equation are properties of the molecule and not of the particular coordinate system used to set up the equation. Consequently, a special set of coordinates (Ri, fllz, , [Pg.17]

Properties. It is of considerable importance to examine the nature of die solutions obtained above. It is evident from Eq. (9), Sec. 2-2, that each atom is oscillating about its equilibrium position with a simple harmonic motion of amplitude Aik — Kkhk, frequency x /27t, and phase e. Ihirthermore, corresponding to a given solution X of the secular equation, i he frequency and phase of the motion of each coordinate is the same, but I lie amplitudes may be, and usually are, different for each coordinate. On account of the equality of phase and frequency, each atom reaches its position of maximum displacement at the same time, and each atom pa.sscs through its equilibrium position at the same time. A mode of ibration having all these characteristics is called a normal mode of vibra-iion, and its frequency is known as a normal, or fundamental, frequency of (he molecule. 

The properties of the minors of the secular determinant of an alternant hydrocarbon may again be used to show that the integrals for which the index is even in (44) and odd in (45) and (46) are zero. It follows that the finite change Aq is an odd function, of Sa, while AFg and Apgt are even. Any inequalities between values of any index for two different positions u), as defined in equations (31) to (34) which arise as first terms of the corresponding infinite series in (44) to (46), persist term-by-term in the expression for the exact finite changes (Baba, 1957). In consequence, the broad agreement with experiment found earlier in the description of ionic and radical reactions by the approximate method carries over to the exact form. 

First, it is possible to simplify the secular equation (2) by means of symmetry. It can be shown by group theory (140) that, in general, the integrals Hi and Si are nonzero only if the orbitals < , and j have the same transformation properties under all the symmetry elements of the molecule. As a simple example, the interaction between an s and a pn orbital which have different properties with respect to the nodal plane of the pn orbital is clearly zero. Interaction above the symmetry plane is cancelled exactly by interaction below the plane (Fig. 13). It is thus possible to split the secular determinant into a set of diagonal blocks with all integrals outside these blocks identically zero. Expansion of the determinant is then simply the product of those lower-order determinants, and so the magnitude of the... 

Similar methods have been used to integrate thermodynamic properties of harmonic lattice vibrations over the spectral density of lattice vibration frequencies.21,34 Very accurate error bounds are obtained for properties like the heat capacity,34 using just the moments of the lattice vibrational frequency spectrum.35 These moments are known35 in terms of the force constants and masses and lattice type, so that one need not actually solve the lattice equations of motion to obtain thermodynamic properties of the lattice. In this way, one can avoid the usual stochastic method36 in lattice dynamics, which solves a random sample of the (factored) secular determinants for the lattice vibration frequencies. Figure 3 gives a typical set of error bounds to the heat capacity of a lattice, derived from moments of the spectrum of lattice vibrations.34 Useful error bounds are obtained... 

This is simpler than 7.1-15, in that many of the terms in the lOth-order polynomial equation which will result on expanding the determinant will now be equal to zero. Nevertheless, the basic, awkward fact is that a lOth-order equation still has to be solved. This is not a task to be confronted with pleasurable anticipation without the use of a digital computer it would be a protracted, tedious job. Fortunately, in this case and all others in which the molecule possesses symmetry, the secular equation can be factored—that is, reduced to a collection of smaller equations—by using the symmetry properties in the right way. The method of symmetry factoring will now be explained and illustrated. 

We turn, finally, to the task of finding the actual expressions for the occupied MOs in order that we may compute such properties of the groundstate electron distribution as the n bond orders. For example, y/[Awhich has the energy (1 4- V2)/ , is neither y/A>, which has the energy 2/ , nor which has the energy 0. It is a linear combination of both, and the problem is to find the appropriate mixing coefficients. As explained earlier, we do this by returning to the simultaneous equations from which the secular equations arose. For the orbitals of A symmetry, we have... 

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... 

Together with the method outlined above, the Siegert quantization scheme discussed by Atabek and Lefebvre (36), and the secular equation method (19,39), it describes the predissociation purely as a scattering phenomenon which is completely characterized by the properties of the (continuum) wavefunction at the energy in question. 

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