Solving the Matrix Equation

Since we know the basis functions and the effective hamiltonian (in principle, at least), we are in a position to evaluate the elements in H and S. How do we then find C and E  

Let us first treat the sinplified situation where our basis set of functions is orthonormal, either by assunption or design. Then all the off-diagonal elements of S (which correspond to overlap between different basis functions) are zero, and all the diagonal elements are unity because of normality. In short, S = 1. Therefore, Eq. (9-17) becomes 

we want a set of coefficients that correspond to normalized MOs. We have seen earlier that, for an orthonormal basis set, this requires each MO to have coefficients satisfying the equation (assuming real coefficients) 

Furthermore, we know that any two different MOs must be orthogonal to each other. That is CiCj = 0,i j. All this may be summarized in the matrix equation 

the coefficient matrix is orthogonal. In the more general case in which coefficients may be complex, C is unitary, i.e., C C = 1. Our problem then is, given H, find a unitary matrix C such that HC = CE with E diagonal. 

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The absorbance A is proportional to 1 through Beer s law (see Computer Projeet 2-1). The analytical problem is to solve the matrix equation... 

With the advent of modern software tools, however, tools such as MATLAB and even the older language, APL, matrix operations can be coded directly from the matrix-math expressions, and then it becomes near-trivial to create and solve the matrix equations on-the-fly, so to speak, and calculate the coefficients for any derivative using any desired polynomial, and computed over any odd number of data points. 

As you learned in the previous sections, LU decomposition with built-in partial pivoting, followed by backsubstitution is a good method to solve the matrix equation Ax = b. You can use, however, considerable simpler technics if the matrix A has some special structure. In this section we assume that A is symmetric (i.e., AT = A), and positive definite (i.e., x Ax > 0 for all x 0 you will encounter the expression x Ax many times in this book, and hence we note that it is called quadratic form.) The problem considered here is special, but very important. In particular, estimating parameters in Chapter 3 you will have to invert matrices of the form A = X X many times, where X is an nxm matrix. The matrix X X is clearly symmetric, and it is positive definite if the columns of X are linearly independent. Indeed, x (x" X)x = (Xx) (Xx) > 0 for every x since it is a sum of squares. Thus (Xx) (Xx) = 0 implies Xx = 0 and also x = 0 if the columns of X are linearly independent. 

The method (ref. 2) is based an solving the matrix equation = Ax, where is not a fixed right-hand side, but a vector of variables 1 2 " > with completely "free" values. To solve the equation for x in terms of notice that an 0 due to positive definiteness of A, since an = (e ) Ae. We can therefore solve the first equation for x, and replace x by the resulting expression in the other equations ... 

Solving the matrix equation Ax = b by LU decomposition or by Gaussian elimination you perform a number of operations on the coefficient matrix (and also on the right-hand side vector in the latter case). The precisian in each step is constrained by the precision of your computer s floating-point word that can deal with numbers within certain range. Thus each operation will introduce some round-off error into your results, and you end up with same... 

Though (2.33) is the well form of the Newton-Raphson correction formula, it is more efficient to solve the matrix equation for d k by decomposition and backward substitution. 

Solve the matrix equation above for the four diffusion velocities V. (These diffusion velocities should be identical to the ones calculated in the previous exercise.) Verify that the sum of the calculated diffusive mass fluxes is zero. 

Density operator equations were converted into coupled integrodifferential equations suitable for numerical processing, and an extended Runge-Kutta algorithm has been implemented for solving the matrix equations in diadic form. A similar procedure can be followed for the original density matrix. 

Hess et al.119 utilized a Hamiltonian matrix approach to determine the spin-orbit coupling between a spin-free correlated wave function and the configuration state functions (CSFs) of the perturbing symmetries. Havriliak and Yarkony120 proposed to solve the matrix equation... 

This can be seen by trying to solve the matrix equation... 

The Fock equations can most easily be solved by solving the matrix equations... 

The partial derivatives of the flow rates which appear in these expressions may be evaluated by solving the matrix equations given below. The partial derivatives of F/s with respect to the 0/s may be computed by use of the matrix equation obtained by partial differentiation of the members of Eq. (4-10) with respect to any 6, say 0k... 

Smith and Missen [20] exemplified the solution to obtain the stoichiometric coefficients of this equation by the matrix method [21], A reacting system consisting of a set of chemical reaction equations is represented by a formula matrix A = (Oki), where the element Uki of this matrix is the subscript of the chemical element k in the compound i occurring in the reaction equation. Consider a matrix N = (v,j) of stoichiometric coefficients, where Vij is the coefficient of the species i in the chemical equation j. The matrix N is obtained by solving the matrix equation... 

To solve the matrix equation the method of the alternating regression is used. Initially the mass spectra are filled with random numbers. The peak shape curve (concentration curve) is calculated from the overlapped specttum. In the next step, the peak shape curve is treated as given and from this the mass spectra are calculated. In a not resolved region, the mass spectra should not change at all. In the iteration step from the newly obtained mass spectra, the peak shapes are calculated. 

Therefore, the basic algorithm solves the one-electron problem (4.269) iteratively guess the position for each electron, i.e., guess (C), then guess the average potential that an electron feels from the rest of electrons in the system, i.e., guess (F), solve the matrix equation, i.e., diagonalize to a new (C), form a new (F), repeat the procedure until the one-electronic wave function becomes consistent with the field produced by it and other electrons. 

At this point the problem of determining the Hartree-Fock molecular orbitals and orbital energies , involves solving the matrix equation FC = SCe. To proceed, however, we need an explicit expression for the Fock matrix. It is first of all necessary, however, to introduce the concept of a density matrix. 

The principles of Stokesian dynamics were developed by Durlofsky, Brady, and Bossis in the 1980s (Durlofsky et al. 1987 Brady and Bossis 1988). The computational costs increase with the third power of particle number (N ). In 2001, Sierou and Bossis introduced a modified numerical procedure for solving the matrix equations, called the accelerated Stokesian dynamics, which considerably reduced the computational effort (A logAO-... 

The expansion coefficients are calculated by solving the matrix equation of the CO LCAO method for crystals, see (4.67) ... 

FIGURE 8.3 Behavior of the integrand I z) in the case of cubic potential. Solid squares represent the numerical results obtained by solving the matrix equation, Equation (8.46). Solid line is the exact/(z). (Taken from Reference [44] with permission.)... 

Listing 12.26 shows a code segment for using the ADI iterative approach for solving the matrix equations associated with a two dimensional PDE. The differential equation in this example is similar to that in Listing 12.25 but is ... 

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