Solving the Secular Determinant

The parameters are defined in equations 45-48. The parameters A, B and T are negative quantities if the orientation of the basis orbitals is defined as shown in diagram 43, and the same is true for AA if we assume that the energy of the cr-orbital lies below that of the two jr-orbitals as indicated in presentation 44. The orbital energies j are obtained by solving the secular determinant given by equation 49, which yields the solutions given in equations 50. 

It is now possible to solve the secular determinant for the energy levels of these 7t orbitals with the assumptions already listed (Sec. II.A). For simplicity, overlap is neglected, and if Q denotes the Coulomb term of an isolated carbon 2pn atomic orbital, we put W—Q — E. Then the determinant Hjj-SUE =0 may be evaluated using the above symmetry orbitals. For example, the matrix element //13 is obtained as follows ... 

It is now preferable to solve the secular determinant directly with the aid of a computer but expansion is clearer for the reader. 

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... 

Upon obtaining the energies of the molecular orbitals by solving the secular determinant [eq. (3.1.17)], we are now ready to proceed to determine the coefficients ci and C2 of eq. (3.1.16). To do this we need to solve the secular equations... 

As mentioned previously in Chapter 3, when we treat the bonding of a molecule by applying molecular orbital theory, we need to solve the secular determinant... 

The energies of the ground (Egs) and excited (Ees) states that are obtained by solving the secular determinant by the variation method (and constraining the mixing coefficients to the normalized a2 + b2 = 1) can be expressed as [13] ... 

In an HMO calculation the reductions due to deviation from maximum (pd)v overlap may be introduced by the use of perturbation theory ll9) instead of solving the secular determinant directly. As usual, the Hiickel bond-intergral parameter (3 v is assumed to be proportional to the overlap integral of the corresponding bond 6>7). The reduction is therefore proportional in our case to the cosine of the angle of deviation y (10). 

In the evaluation of E (17) is preferred to either (15) or (16), since the e, are already determined in solving the secular determinant and only one-elecfron terms are involved in computing the H,-... 

Solve the secular determinant and generate the eigenvalues and coefficients of the LCAO-MOS over sto-3g representations of equation 6,3. 

The molecular orbital energies in this two orbital case, e, (/ = 1,2). are obtained by solving the secular determinant (equation 1.30) shown in equation 2.5 for this particular example... 

The energies of the Hiickel molecular orbitals ate obtained by solving the secular determinant ... 

For a symmetric rotor, in the present approximation, only the z component of in, the vibrational angular momentum, needs to be considered. The problem may be treated as a perturbation employing zero-order wave functions which are products of rigid rotor and harmonic oscillator functions. When the molecule is in a state such that vka + Vkb — 1, where Qka and Qw> are degenerate, it is necessary to solve the secular determinant... 

The approximation made in (32) breaks down if two or more of the frequencies, and therefore X s are nearly equal. Suppose Xg X . Then it is necessary to employ the off-diagonal perturbation terms coupling these frequencies and solve the secular determinant,... 

Sec. 9-1] methods of solving the secular determinant 9-1. Characteristic Values and Characteristic Vectors... 

Sec. 9-4] methods op solving the secular determinant easily shown that... 

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