The Eigenvalue Problem

This algorithm entirely avoids multiplications with zeros and repeated evaluation of identical a, and b constants. The most time consuming part is the search through the paths reaching vt and vr in order to evaluate the respective sets It and /r - Timings for evaluating the product 

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The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

The eigenvalue problem defined by equations (12.56) and (12.37) has been studied by Lee and Luss l79j and, more recently, in considerable detail by Villadsen and Michelsen When - I it is easy to show... 

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. 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

In solving the eigenvalue problem for the energy operator //op, we have previously always introduced a complete basic set Wlt in which the eigenfunction has been expanded ... 

The eigenvalue problem with respect to the energy i/op is now conveniently solved by means of the orthogonalized A-adapted set 0 i 0 2, 0 3,. . . Using the "turn-over rule" and Eq. III.92, we... 

The single sum in Eq. III.99 is here replaced by the sum of m such single sums, depending on the orthogonalization. The eigenvalue problem (Eq. III.21) is in this way reduced to the form... 

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). 

In the approach of Dewar and co-workers (34), termed the half-electron method , a physical model is considered in which an unpaired electron is replaced by two hypothetical half-electrons of opposite spin. For radicals containing one unpaired electron, the eigenvalue problem of this method is, in our opinion, identical with the method of Longuet-Higgins and Pople (27) ... 

In the unrestricted Hartree-Fock method, a single-determinant wave function is used with different molecular orbitals for a and jS spins, and the eigenvalue problem is solved with separate F and F matrices. With the zero differential overlap approximation, the F matrix elements (25) become... 

In order to obtain nonzero spin densities even on hydrogen atoms in tt radicals, one has to take the one-center exchange repulsion integrals into account in the eigenvalue problem. In other words, a less rough approximation than the complete neglect of differential overlap (CNDO) is required. This implies that in the CNDO/2 approach also, o and n radicals have to be treated separately (98). 

The eigenvalue problem for the Laplace operator in the rectangle Go subject to the first kind boundary conditions... 

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator i. y = - -... 

The eigenfunction of the vibrational ground state is calculated on the ab initio 2D So potential energy surface by solving the eigenvalue problem. 

From here, the goal consists to find the eigenvalues and the eigenvectors of the perturbed system, which we denote as the sets (E,) and i> respectively. That is, the target is focused into solving the eigenvalue problem ... 

The characteristic-value problem - more often referred to as the eigenvalue problem - is of extreme importance in many areas of physics. Not only is it the very basis of quantum mechanics, but it is employed in many other applications. Given a Hermitian operator a, if their exists a function (or functtofts) g such that... 

The eigenvalue problem was introduced in Section 7.3, where its importance in quantum mechanics was stressed. It arises also in many classical applications involving coupled oscillators. The matrix treatment of the vibrations of polyatomic molecules provides the quantitative basis for the interpretation of their infrared and Raman spectra. This problem will be addressed tridre specifically in Chapter 9. 

The eigenvalue problem can be described in matrix language as follows. Given a matrix ff, determine the scalar quantities X and the nonzero vectors U which satisfy simultaneously file equation... 

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 ... 

Looking at (110), it appears that the spectral density /Sf(oo, aG = 0) ex is composed of two sub-bands, as is also shown by the four sample spectra of Fig. 10 which were computed for various parameters A and A. The frequency and intensity of these two sub-bands do not depend on the temperature, and are similar to those which may be obtained within the simpler undamped treatment. Solving the eigenvalue problem (111), we obtain... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

The eigenvalue problem for H as it stands in Eq. (3.13) can rarely be solved. However, one can do a series of approximations that brings the Hamiltonian H to a more manageable form (1) One can neglect the dependence of p and Pyg on normal coordinates. This brings H to the form... 

With these approximations, the rotational and vibrational motion is completely separated and the eigenvalue problem... 

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