Eigenfunction cosine

The problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

Recall that the bond vectors and not the loci of beads with respect to a common origin. Thus we must choose sines rather than cosines as eigenfunctions for chains with free ends. Also, since Eq. (10) is concerned only with the averages of the bond vectors, there are only N normal modes, the translational diffusion mode being automatically excluded. 

For a symmetric top, symmetry requires the dipole moment to lie along the symmetry axis, so that two of the three principal-axis components of d must vanish. In deriving the symmetric-top wave functions in Section 5.5, we assumed that the c axis was the symmetry axis hence to use the eigenfunctions (5.68) to find the selection rules, we must take da = db — 0, dcJ= 0. For a symmetric top, we thus must evaluate only the three integrals IXOc, lYoc anc Azof The three relevant direction cosines are given in (6.64) and Problem 5.15 they are independent of x- Since the integral... 

If, instead, the box is chosen symmetrically about the origin, then the eigenfunctions become the (even) cosine functions for n odd ... 

The components may be expressed in either a space-fixed axis system (p) ora molecule-fixed system (q). The early literature used cartesian coordinate systems, but for the past fifty years spherical tensors have become increasingly common. They have many advantages, chief of which is that they make maximum use of molecular symmetry. As we shall see, the rotational eigenfunctions are essentially spherical harmonics we will also find that transformations between space- and molecule-fixed axes systems, which arise when external fields are involved, are very much simpler using rotation matrices rather than direction cosines involving cartesian components. 

A contour plot is shown in Fig. 7.8. Note that this function is cylindrically-symmetrical about the z-axis with a node in the x, y-plane. The eigenfunctions 21 1 are complex and not as easy to represent graphically. Their angular dependence is that of the spherical harmonics 7i i, shown in Fig. 6.4. As deduced in Section 4.2, any linear combination of degenerate eigenfunctions is an equally-valid alternative eigenfunction. Making use of the Euler formulas for sine and cosine,... 

To compute the interacting RPA density-response function of equation (32), we follow the method described in Ref. [66]. We first assume that n(z) vanishes at a distance Zq from either jellium edge [67], and expand the wave functions (<) in a Fourier sine series. We then introduce a double-cosine Fourier representation for the density-response function, and find explicit expressions for the stopping power of equation (36) in terms of the Fourier coefficients of the density-response function [57]. We take the wave functions <)),(7) to be the eigenfunctions of a one-dimensional local-density approximation (LDA) Hamiltonian with use of the Perdew-Zunger parametrization [68] of the Quantum Monte Carlo xc energy of a uniform FEG [69]. 

Thus the matrix is block diagonal and there are two sets of eigenfunctions (Fig. 21.1), namely cosine type functions and sine type functions S ( ))), which are linear combinations of the base functions (Eq. (21.9)... 

An obvious solution to minimize the number of grid points NK is to introduce symmetry. Consider for example an inversion point such as the point x = 0 in the harmonic oscillator. The eigenfunctions can be classified as being either even or odd with respect to parity i j(<7) = i i(— < ) one restricts the calculation to one class of functions the computational effort can be reduced by a factor of 2 by using a fast cosine transform for even functions and a fast sine transform for odd functions (52). The same symmetry considerations should work for other types of grids. 

The matrix elements involving a cosine and a sine function vanish and so the Hamiltonian matrix has two blocks. We have verified that the ground state comes out from the cosine block for which the vibronic Hamiltonian of eq (7) gives rise to the following matrix representation (each submatrix has dimension N X N, where N is the number of free rotor eigenfunctions utilized ... 

The factor 2 is included so that the two temperature profiles, from Eqs. (2.106) and (2.107), will have the same overall horizontal temperature change for the same value of B. Equation (2.107) describes very well the temperature variation in hydrocarbon reservoirs it is also a solution to the energy equation where heat transfer is by pure conduction, i.e., the solution to = 0 (see Example 2.10). The advantage of using the cosine temperature distribution is that cos(n7i(r-t- Wj2)IW) is orthogonal to the eigenfunctions of the problem, cos(m7r(r+ VP/2)/W0. This reduces the number of infinite sum cosines to two terms, m = 0 and m n. 

For example, the block-reactor eigenfunctions must include the sine functions as well as the cosine, and in the spherical and cylindrical cases, the angular harmonics are required. 

Only one kind of function has a constant value for the wavelength a pure sinusoidal wave (which includes the functions sine, cosine, and e ). Therefore, the only states that are eigenfunctions of K are waves of the form sin(/cx + (f>), where the wavenumber k is inversely proportional to the wavelength. 

From this result an intrinsic eigenfunction matrix is kno to be equal to an eigenfunction matrix of M,. Finally, angular momentum M(a,P,y), havmg direction cosines eosa, eosp, and eosy, can be written as... 

The theoretical model for photodetachment is similar to that used to describe photodissociation outlined in the last section. As illustrated in Fig. 3.7, the initial wave packet on the neutral PES was chosen as the ground vibrational state of cis-HOCO, which has a lower energy than its tram counterpart. The anion vibrational eigenfunction was determined on a newly developed anion PES at the same CCSD(T)-F12/AVTZ level [130], as used to construct the neutral PES [100, 101]. The neutral wave packet was propagated to yield probabilities to both the HO-I-CO and H-I-CO2 asymptotes with a flux method [108] and the cosine Fourier transform of the Chebyshev autocorrelation function yielded the energy spectrum [44]. The discretization of the Hamiltonian and wavepacket, and the propagation were essentially the same as in our recent reaction dynamics study [107]. 

In such cases, the function is called an eigenfunction of the operation of second differentiation. One may take "of one s own" or "same" as a loose translation of the German word eigen. It is one s own function, or the same function, that is produced on the operation of second differentiation. The constant, C, that shows up multiplying the original function is the eigenvalue. In contrast, A(x) is not an eigenfunction of the operation d/dx since the first order differentiation of the cosine function produces a different function, the sine function. 

In order for the wavefunction, v, for this system to be an eigenfunction of the Hamiltonian, tj/ must be a function such that taking its second derivative yields the same function. Possible functions include sine, cosine, or the mathematically equivalent complex exponential (see the footnote on page 16). 

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