Equation, secular

What has been said previously about the Hartree-Fock method is only a sort of general theory. The time has now arrived to show how the method works in practice. We have to solve the Hartree-Fock-Roothaan equation (cf. Chapter 8, pp. 365 and 453). 

Electronic Motion in the Mean Field Periodic Systems 

The Fock matrix element is equal to (noting that (x p Fx ) = Fpq depends on the difference s between the vectors Rj and Ry)  

The same can be done with Spq and therefore the Hartree-Fock-Roothaan secular equation (see p. 453) has the form  

As a matter of fact, all depends on how distant the unit cells j and / are. We have used the fact that F exhibits the crystal symmetry and the sums over j all give the same result, independent of /. 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

Is a Cartesian force constant, g and f.. are the elements of the Wilson G and E matrices, respectively  

The first order approximation of ln(s/s )f through Eq.(26), j = 1, gives values of ln(s/s )f within 10% of the exact solution, Eq. (13), even for D/H Isotope effects at 300 K. Eq.(26) is an even better approximation to ln(s/s )f of heavier elements. Within this limitation, we can now derive the first order rules of Isotope chemistry by substitution of Equations (31) and (31a) into Eq. (26). They are  

2) Isotope effects between different compounds occur only 

4)Isotope effects are cumulative (first rule of the geometric mean) 

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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. 

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 ... 

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). 

The matrix elements ot — Ej and p are not variables in the minimization procedure they are constants of the secular equations with units of energy. Note that all elements in the matrix and vector are real numbers. The vector is the set of coefficients for one eigenfunction corresponding to one eigenvalue, Ej. From Eq. (7-24),... 

These are just the secular equations shown in equation set (7-2) with F in place of H and the stacked matrix Eq. (7-6) of eigenvectors in place of a single eigenvector. In matrix notation... 

With the approximation that 5 = 0, the secular equations of Equation (7.45) become... 

Equation (3-111) is the secular equation for this problem. [Pauling and Wilson discuss the meaning of the term secular in this context.] Expansion of the determinant gives a polynomial in X that is called the characteristic equation ... 

The roots of the characteristic equation are called the eigenvalues of the secular equation. 

For Scheme XVIII, find the secular equation, the characteristic equation, and the eigenvalues. 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

As there is one equation (4.5) for each i, the variational problem is transformed into solving a set of Cl secular equations. Introducing the notation Hjj = ( H ) the matrix equation becomes... 

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