Secularization

As with any basis method the a(k, G) coefficients are detennined by solving a secular equation. 

The eigenvalues E of It can be detemiined from the Hamiltonian matrix by solving the secular equation... 

In solving the secular equation it is important to know which of the off-diagonal matrix elements " I wanish since this will enable us to simplify the equation. 

Figure Bl.7.18. (a) Schematic diagram of the trapping cell in an ion cyclotron resonance mass spectrometer excitation plates (E) detector plates (D) trapping plates (T). (b) The magnetron motion due to tire crossing of the magnetic and electric trapping fields is superimposed on the circular cyclotron motion aj taken up by the ions in the magnetic field. Excitation of the cyclotron frequency results in an image current being detected by the detector electrodes which can be Fourier transfonned into a secular frequency related to the m/z ratio of the trapped ion(s).

In this approach [51], the expectation value ( T // T ) / ( T ) is treated variationally and made stationary with respect to variations in the C and. coeflScients. The energy fiinctional is a quadratic function of the Cj coefficients, and so one can express the stationary conditions for these variables in the secular fonu... 

The value of detennines how much computer time and memory is needed to solve the -dimensional Sj HjjCj= E Cj secular problem in the Cl and MCSCF metiiods. Solution of tliese matrix eigenvalue equations requires computer time that scales as (if few eigenvalues are computed) to A, (if most eigenvalues are... 

In this approximation, the angle r does not appear explicitly in the vibronic secular equation. From Eq. (76) it follows that... 

For the variational calculations of the vibronic spectrum and the spin-orbit fine structure in the X H state of HCCS the basis sets involving the bending functions up to 0i = 02 = 11 with all possible and I2 values are used. This leads to the vibronic secular equations with dimensions 600 for each of the vibronic species considered. The bases of such dimensions ensure full... 

To determine the vibrational motions of the system, the eigenvalues and eigenvectors of a mass-weighted matrix of the second derivatives of potential function has to be calculated. Using the standard normal mode procedure, the secular equation... 

The expansion coefficients C . are determined by solving the Cl secular equation... 

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

Later in this book, we shall need to find the roots of the secular matrix... 

This equation is a quadratic and has two roots. For quantum mechanical reasons, we are interested only in the lower root. By inspection, x = 0 leads to a large number on the left of Eq. (1-10). Letting x = leads to a smaller number on the left of Eq. (1-10), but it is still greater than zero. Evidently, increasing a approaches a solution of Eq. (1-10), that is, a value of a for which both sides are equal. By systematically increasing a beyond 1, we will approach one of the roots of the secular matrix. Negative values of x cause the left side of Eq. (1-10) to increase without limit hence the root we are approaching must be the lower root. 

This is a pair of simultaneous equations in Ai and A2 called the secular equations... 

In what immediately follows, we will obtain eigenvalues i and 2 for //v / = Ei ) from the simultaneous equation set (6-38). Each eigenvalue gives a n-election energy for the model we used to generate the secular equation set. In the next chapter, we shall apply an additional equation of constr aint on the minimization parameters ai, 2 so as to obtain their unique solution set. 

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). 

If we divide each element of the secular matrix by p and perform the substitution... 

For the equation set to be linearly dependent, the secular determinant must be zero... 

The advantage of the method just described is that it can be generalized to molecules of any size. Setting up quite complicated secular mahices can be reduced to a simple recipe. A computer scheme can be used to diagonalize the resulting matrices by an iterative series of rotations. 

Write the secular matrix, compute the eigenvalues, and draw the energy level diagram for fulvcnc. 

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