Secular perturbations

Now suppose that we begin with the full ( perturbed ) secular determinant of the form... 

The predominant term in the perturbing potential V is of the form er, equal to the electric dipole moment operator. This is the origin of the selection rule that if ( 0, er i) = 0, the perturbed secular equation will not mix the states xpo and t/ i) so that the transition tpo ip i will not occur. 

Let us first consider the rotational interaction between discrete Rydberg states. For example, the and 1E+ levels of an up-complex of H2 are mixed by the 1-uncoupling term in HROT. Since AE == 0, the selection rule AO = 1 implies that AA = 1. The 2 x 2 1II 1E+ perturbation secular determinant for the Jth rotational levels, expressed in the case (a) basis, is, for l = 1,... 

Note that in the above treatment it was necessary to evaluate only two integrals of the crystal field potential, one over the d orbital [Eq. (7a)] and the second over the orbital [Eq. (7b)]. This derives from the group theory of high symmetry crystal fields, where for an octahedral or tetrahedral complex each of the d orbital expressions is a good wave function for the molecular Hamiltonian and gives an energy appropriate for all members of the 2s and g sets, respectively. For a lower symmetry complex Eq. (4) must be expanded into a perturbation secular determination [Eq. (9)1 with 25 matrix elements, H , involving all pairwise combina-... 

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. 

These rules follow directly from the quantum-mechanical theory of perturbations and the resolution of the secular equations for the orbital interaction problem. The (small) interaction between orbitals of significantly different energ is the familiar second order type interaction, where the interaction energy is small relative to the difference between EA and EB. The (large) interaction between orbitals of same energy is the familiar first order type interaction between degenerate or nearly degenerate levels. 

In order to see how accurate this perturbation treatment actually is, we have substituted numerical values for the S s directly into the secular equation, and then solved it rigorously by numerical methods. The calculations are not given in detail, since they are quite straightforward and proceed along well-known lines. The results are shown in Table I. 

Substituting equations (44), (47) and (48) into the perturbed Schrodinger secular equation produces the n-th order equation ... 

The determination of the coefficients Cay is not necessary for finding the first-order perturbation corrections to the eigenvalues, but is required to obtain the correct zero-order eigenfunctions and their first-order corrections. The coefficients Cay for each value of a (a = 1,2,. .., g ) are obtained by substituting the value found for from the secular equation (9.65) into the set of simultaneous equations (9.64) and solving for the coefficients c 2, , in terms of c i. The normalization condition (9.57) is then used to determine Ca -This procedure uniquely determines the complete set of coefficients Cay (a, y = 1,2, gn) because we have assumed that all the roots are different. 

The secular determinant as presented above involves the first-order perturbations of the Hamiltonian and the energy. More generally, it is formulated in terms of the Hamiltonian and the total energies of the perturbed system. From Eqs. (12) and (16),... 

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. 

However, perturbation-theoretic expressions such as Eqs. (1.24) and (2.7) are problematic in the degenerate case when donor and acceptor orbitals have equal energies.7 In this case we can directly formulate the interaction of orbitals a and b (with equal energies e A = b = e. and / ab = <a. F b ) in terms of a limiting variational model with 2x2 secular determinant... 

Tjon has also written an integral equation for the diagonal elements of Mt, in the representation in which z + F(0) is diagonal. He therefore assumes that the non-secular perturbation V = 2 F(8) has matrix elements with randomly varying... 

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... 

Since the value of M does not affect E ° the unperturbed levels are (2y+l)-fold degenerate. Hence, before forging ahead, we must be sure that we have the correct zeroth-order wave functions for the perturbation (4.36). (See Section 1.10.) It was noted in Section 1.10 that when the secular determinant is diagonal, we have the right zeroth-order functions. We now show that the functions (4.38) give a diagonal secular determinant. An off-diagonal element has the form... 

Show that if the overlap between torsional-vibration wave functions corresponding to oscillation about different equilibrium configurations is neglected, the perturbation-theory secular equation (1.207) for internal rotation in ethane has the same form as the secular equation for the Hiickel MOs of the cyclopropenyl system, thereby justifying (5.96)-(5.98). Write down an expression (in terms of the Hamiltonian and the wave functions) for the energy splitting between sublevels of each torsional level. 

The m — 6 system will again be used as an example. The guest molecules cause the mixing of the lowest (r = 0) wave function with three other wave functions derived from p = 1, p = 2, and p — 3, as described in the secular equation (14). If cx,. . ., c5 are the coefficients of the basis functions in order of increasing energy in the perturbed lowest state, we have, by perturbation theory for small a,... 

The situation is different if there is degeneracy between a MO (Om) of R and one (T B) of S. In this case there is a first order perturbation SE given by the secular equation ... 

In order to find the energy x, we use first order perturbation theory. The degenerate orbitals involved are the doubly occupied AO s of Cl- and I-, and the empty NBMO of ArCHa. The necessary matrix elements are found as before in terms of the CC1 and Cl resonance integrals fta, ft. Solving the resulting three-row secular equation, we find for the perturbed energies—... 

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