Eigenvalue problems, solution

In the study of stationary states in quantum chemistry, one would normally introduce boundary conditions, as for instance, (a) = (b) = 0 and solve the resulting eigenvalue problem. Solutions occur only for certain values of E = n, so-called eigenvalues, and the corresponding solutions (x) are called eigenfunctions. 

In 2- or 3-dimensional turning chatter analysis, the characteristic equation of the system results in an eigenvalue problem solution of... 

Techniques for Nonlinear Eigenvalue Problems—Solutions of the Nonlinear Schrodinger-Poisson Eigenvalue Problem in 2 and 3 Dimensions. 

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

Another step that is common to most, if not all, approaches that compute orbitals of one fomi or anotiier is tlie solution of matrix eigenvalue problems of the fomi... 

The solution of any such eigenvalue problem requires a number of computer operations that scales as the dimension of the F matrix to the third power. Since the indices on the F matrix label AOs, this means... 

This is because no four-indexed two-electron integral like expressions enter into the integrals needed to compute the energy. All such integrals involve p(r) or the product p(/)p(r) because p is itself expanded in a basis (say of M functions), even the term p(r)p(r) scales no worse than tvF. The solution of the KS equations for the KS orbitals ([). involves solving a matrix eigenvalue problem this... 

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

The two sets of coeflicien ts, one for spin-up alpha electrons and the other for spin-down beta electrons, are solutions of iw O coupled matrix eigenvalue problems ... 

An eigenvalue problem is a homogeneous equation of the second land, and solutions exist only for certain A. 

At the energy minimum, each electron moves in an average field due to the Other electrons and the nuclei. Small variations in the form of the orbitals at this point do not change the energy or the electric field, and so we speak of a self-consistent field (SCF). Many authors use the acronyms HF and SCF interchangeably, and I will do so from time to time. These HF orbitals are found as solutions of the HF eigenvalue problem... 

Here, c is a column vector of LCAO coefficients and e is called the orbital energy. If we start with n basis functions, then there are exactly n different c s (and e s) and the m lowest-energy solutions of the eigenvalue problem correspond to the doubly occupied HF orbitals. The remaining n — m solutions are called the virtual orbitals. They are unoccupied. 

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. 

In the unrestricted treatment, the eigenvalue problem formulated by Pople and Nesbet (25) resembles closely that of closed-shell treatments.-On the other hand, the variation method in restricted open-shell treatments leads to two systems of SCF equations which have to be connected in one eigenvalue problem (26). This task is not a simple one the solution was done in different ways by Longuet-Higgins and Pople (27), Lefebvre (28), Roothaan (29), McWeeny (30), Huzinaga (31,32), Birss and Fraga (33), and Dewar with co-workers (34). 

Keller, H. B. Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, In Applications of Bifurcation Theory, Rablnowltz,P., Ed. Academic Press New York, 1977. 

If the matrix A is positive definite, i.e. it is symmetric and has positive eigenvalues, the solution of the linear equation system is equivalent to the minimization of the bilinear form given in Eq. (64). One of the best established methods for the solution of minimization problems is the method of steepest descent. The term steepest descent alludes to a picture where the cost function F is visualized as a land-... 

H. Rutishauser, Solution of eigenvalue problems with the LR-transformation. Appl. Math. Ser. Nat. Bur. Stand., 49 (1958) 47-81. 

The term that prevents an exact solution of the BO eigenvalue problem is the electron-electron repulsion that couples the motion of pairs of electrons. Without... 

The information obtainable upon solution of the eigenvalue problem includes the orbital energies eK and the corresponding wave function as a linear combination of the atomic basis set xi- The wave functions can then be subjected to a Mulliken population analysis<88) to provide the overlap populations Ptj ... 

Given ng and 1(R, p), Eqs. (7.32a, b) can be integrated successively from r = R to a large value of r. By definition Z(R, p) = -y where yis the recombination probability in presence of scavenger. Only for the correct value of ydo the solutions of (7.32a, b) smoothly vanish asymptotically as r—-o° otherwise, they diverge. Thus, the mathematics is reduced to a numerical eigenvalue problem of finding the correct value of I(R, p). 

Lanczos Method for the Numerical Solution of Large Sparse Generalized Symmetric Eigenvalue Problems. 

The solutions for the loading vectors pj and qj are found by solving two eigen vector/eigenvalue problems. Let Sx= cov(X), SY= cov(Y), and SXY= cov(X, Y) be the sample covariance matrices of X and Y, and the sample covariance matrix between X and Y (a matrix mx x mY containing the covariances between all x- and all y-variables), respectively. Also other covariance measures could be considered, see Section 2.3.2. Then the solutions are (Johnson and Wichem 2002)... 

While details of the solution of the quantum mechanical eigenvalue problem for specific molecules will not be explicitly considered in this book, we will introduce various conventions that are used in making quantum calculations of molecular energy levels. It is important to note that knowledge of energy levels will make it possible to calculate thermal properties of molecules using the methods of statistical mechanics (for examples, see Chapter4). Within approximation procedures to be discussed in later chapters, a similar statement applies to the rates of chemical reactions. 

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