Harmonic oscillator basis set

Following a standard procedure for determining stable eigenvalues (see, e.g , 24), we solved the vibrational secular equation at several values of the factor n which scales the frequencies In the harmonic oscillator basis set. Plots of eigenvalue versus n were then constructed and the stable eigenvalues were Identified by visual examination of these plots. The Identification was straightforward for the resonances corresponding to uco S. ... 

In all of the calculations reported here we have assumed identical oscillator frequencies for the perturbing mode in the ground and excited electronic states. Furthermore, in constructing the vibronic wave functions given by Eq. (65) we have employed a harmonic oscillator basis set which included all functions with v 10. 

For class-1 states, a simple harmonic representation of U leads to a complete set of eigenfunctions ( ) this harmonic oscillator basis set is used to diagonalize equation (6). In this case, it is sufficient to construct U( 4>k) using a standard approach involving mass fluctuation (or nuclear ) coordinates and the corresponding electronic state dependent Hessian. The higher terms in the Taylor expansion define anharmonic contributions to the transition moments. These diabatic states are confining and only one stationary point in -space would be found for each... 

Using equations (28) and (29) we obtain a set of equations for the new expansion coefficients B . The new expansion has better convergence properties than the power series expansion. The functions (x) are harmonic oscillator wave-functions. Thus the hermite expansion involves an expansion around the classical path in a harmonic oscillator basis set. The hermite-corrected GWP can be used in a basis set expansion so as to approach the exact quantum theory fi"om the classical path limit in a systematic fashion. Hence in this sense the method is complementary to the multitrajectory approach discussed below. The advantage of the hermit basis is however that it is an orthorgonal basis and therefore somewhat simpler to work with in practical computations. ... 

The above classification of asymmetric potential functions is convenient for comparison of different molecules or as a systematic basis for making an initial fit to experimental data. However, when the Schrodinger equation is being solved by the linear variation method with harmonic-oscillator basis functions, it may not provide the best choice of origin for the basis function. For example, a better choice in the case of an asymmetric double-minimum oscillator, where accurate solutions are required in both wells, would be somewhere between the two wells. Systematic variation of the parameters may still be made as outlined above, but the origin should be translated before the Hamiltonian matrix is set up. The equations given earlier... 

Diagonalize the full Hamiltonian in this space. The results can be presented in terms of the intervals G(v -F J) or can be analyzed into the spectroscopic constants. A modification of this method using a Morse function basis set has proved to be superior to the harmonic oscillator basis. ... 

To solve this equation, an appropriate basis set ( >.,( / ) is required for the nuclear functions. These could be a set of harmonic oscillator functions if the motion to be described takes place in a potential well. For general problems, a discrete variable representation (DVR) [100,101] is more suited. These functions have mathematical properties that allow both the kinetic and potential energy... 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, harmonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the trajectory, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

The low-lying excited states of the hydrogen molecule conhned in the harmonic potential were studied using the configuration interaction method and large basis sets. Axially symmetric harmonic oscillator potentials were used. The effect of the confinement on the geometry and spectroscopic constants was analyzed. Detailed analysis of the effect of confinement on the composition of the wavefunction was performed. 

Henry s group is also involved in theoretical studies to determine sources of local mode overtone intensity. These investigators have developed a very successful approach that uses their harmonically coupled anharmonic oscillator local mode model to obtain the vibrational wavefunctions, and ab initio calculations to obtain the dipole moment functions. The researchers have applied these calculations to relatively large molecules with different types of X-H oscillator. Recently they have compared intensities from their simple model to intensities from sophisticated variational calculations for the small molecules H20 and H2CO. For example, for H2CO they generated a dipole moment function in terms of all six vibrational degrees of freedom.244 This comparison has allowed them to determine the quality of basis set needed to calculate dipole moment... 

Quantum-chemical calculations for pyrylium including one, two, or three water molecules using DFT and 6-31 + G(d,p) basis set revealed that the aromaticity (estimated by harmonic oscillator stabilization energy, HOSE natural resonance theory, NRT harmonic oscillator model of aromaticity, HOMA and nucleus-independent chemical shifts, NICS) is not influenced by water molecules [82],... 

A theoretical evaluation of the aromaticity of the pyrones pyromeconic acid, maltol, and ethylmaltol along with their anions and cations was carried out at several levels (Hartree-Fock, SVWN, B3LYP, and B1LYP) using the 6-311++G(d,p) basis set <2005JP0250>. The relative aromaticity of these compounds was evaluated by harmonic oscillator model of aromaticity (HOMA), nucleus-independent chemical shifts (NICSs), and /6 indexes and decreases in the order cation > neutral molecule > anion. 

The resultant optimization equations cannot be solved analytically. Numerically, it is convenient to expand each field in an orthonormal basis set un(a>,x) (e.g., harmonic oscillator eigenfunctions) ... 

It follows from the preceding discussion that the equilibrium constant for complex formation evaluated using the rigid rotor-harmonic oscillator approximation, with molecular constants derived from ab initio SCF calculations with a medium basis set (of DZ quality), is not very accurate. Comparison of the AG° values calculated using extended and medium basis sets indicates that the major uncertainty in AG is derived from AH . TASP is not as dependent on the basis set used. Furthermore, it is evident that the entropy term plays an extremely important rote in complex formation neglecting it may result not only in quantitative, but even in qualitative failure. 

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