Linear combination of harmonics

The standard scheme has been the most popular one for the last 20 years or so. The harmonic scheme gets its name because if the bath is harmonic (more precisely, if F can be represented as a linear combination of harmonic coordinates), then this scheme is exact (16,19). By comparing these schemes to results from exactly solvable models we have concluded that the standard scheme is really not very accurate at high frequencies and that the harmonic and Egelstaff schemes are more promising (5,6). A fourth scheme satisfying detailed balance,... 

Historically, the most common technique used has been the linear variation method. In this procedure, the wave functions are expressed as linear combinations of harmonic-oscillator basis functions... 

The product of two spherical harmonics of the same argument can be written as a linear combination of harmonics according to the equation... 

In the classical limit (hcoj k T), the reaction coordinate X t) in each quantum state can be described as a Gaussian stochastic process [203]. It is Gaussian because of the assumed linear response. As follows from the discussion in Section II.A, if the collective solvent polarization follows the linear response, the ET system can be effectively represented by two sets of harmonic oscillators with the same frequencies but different equilibrium positions corresponding to the initial and final electronic states [26, 203]. The reaction coordinate, defined as the energy difference between the reactant and the product states, is a linear combination of the oscillator coordinates, that is, it is a linear combination of harmonic functions and is, therefore, Gaussian. The mean value is = — , for state 1 and = , for state 2, respectively. We can represent Xi(r) and X2 t) in terms of a single Gaussian stochastic process x(t) with zero mean as follows ... 

The harmonic-oscillator wave functions are given by a Hermite polynomial times an exponential factor (Problem 4.19b). By virtue of the expansion postulate, any well-behaved function/(x) can be expanded as a linear combination of harmonic-oscillator energy eigenfunctions ... 

In our analytic basis set approach for solving the APH surface function Hamiltonian q. (38) we use a linear combination of harmonic oscillators in each arrangement channel times Legendre polynomials. 

A string does not usually move as described by a single harmonic. A linear combination of harmonics can satisfy the wave equation ... 

The fact that a linear combination of solutions can be a solution to the wave equation is called the principle of superposition. When a string in a musical instrument is stmck or bowed, it moves according to some linear combination of harmonics. A violin sounds different from a piano because of the difference in the strengths of various harmonics. 

In the general case and are anharmonic vibrational wave functions, p is the dipole moment operator. The anharmonic wave functions can be represented as a linear combination of harmonic basic functions [155]... 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

The representation of tp(x, t) by the sine function is completely equivalent to the cosine-function representation the only difference is a shift by A/4 in the value of X when t = 0. Moreover, any linear combination of sine and cosine representations is also an equivalent description of the simple harmonic wave. The most general representation of the harmonic wave is the complex function... 

One of the procedures employed for the determination of the MEP is the CD method [19], This method introduces a harmonic restraint on the reaction coordinate, which is a linear combination of the distances between the atoms involved in the reaction to perform an optimization along a proposed reaction path. In this case the reaction coordinate is given by the expression ... 

In the partial wave theory free electrons are treated as waves. An electron with momentum k has a wavefunction y(k,r), which is expressed as a linear combination of partial waves, each of which is separable into an angular function Yi (0. ) (a spherical harmonic) and a radial function / L(k,r),... 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

As stated before, the coefficients /, g, and h are linear combinations of the components of the tensors yeee, y eem, and ymee. For a system with Ccc, symmetry (i.e., chiral, isotropic surface symmetry) and -polarized second-harmonic light detected in transmission, these coefficients are given by (For the complete set of equations, see previous sections.)... 

In Appendix A2, we have formally applied the perturbation method to find the energy levels of a d ion in an octahedral environment, considering the ligand ions as point charges. However, in order to understand the effect of the crystalline field over d ions, it is very illustrative to consider another set of basis functions, the d orbitals displayed in Figure 5.2. These orbitals are real functions that are derived from the following linear combinations of the spherical harmonics ... 

The functions ylmp are linear combinations of the complex spherical harmonic functions Ylm. Including normalization, the latter are defined as... 

For sites of cubic symmetry the point-group symmetry elements mix the spherical harmonic basis functions. As a result, linear combinations of spherical harmonic functions, referred to as Kubic harmonics (Von der Lage and Bethe 1947), must be used. 

As the traceless quadrupole moments are linear combinations of the spherical harmonic quadrupole moments, the corresponding expressions follow directly... 

The spherical harmonic functions constitute a complete set of functions in the spherical point group. A product of two spherical harmonics such as ytyj must therefore be a linear combination of spherical harmonic functions. An example of such an expression is... 

The delta function corresponds to Einstein s equation, which says that the kinetic energy of the emitted electron Ef equals the difference of the photon energy h(a and the energy level of the initial state of the sample, The final state is a plane wave with wave vector k, which represents the electrons emitted in the direction of k. Apparently, the dependence of the matrix element 1 j) on the direction of the exit electron, k, contains information about the angular distribution of the initial state on the sample. For semiconductors and d band metals, the surface states are linear combinations of atomic orbitals. By expressing the atomic orbital in terms of spherical harmonics (Appendix A),... 

The b2, b 1 and a2 blocks are formed in a similar manner. The eigenvalues of each of these blocks provide the squares of the harmonic vibrational frequencies, the eigenvectors provide the normal mode displacements as linear combinations of the symmetry adapted... 

The technical conditions on f are quite reasonable if a physical situation has a discontinuity, we might look for solutions with discontinuities in the function f and its derivatives. In this case, we might have to consider, e.g., piecewise-defined combinations of smooth solutions to the differential equation. These solutions might not be linear combinations of spherical harmonics. 

Any linear combination of the three functions (6.85) is an eigenfunction of the vibrational Hamiltonian (within the harmonic-oscillator approximation) and represents a possible state. There are three linearly independent linear combinations of special importance these are... 

The Onk and Q k are operators which are respectively linear combinations of spherical harmonics and expansions in terms of Cartesian coordinates, 1 = 2 for d-orbitals, 3 for /-orbitals. The parameters B k and A k are, of course, specialized forms of the general form given in equation (2), but including the evaluation of the relevant radial integrals. 

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