The Discrete Variable Representation

In order to solve the nuclear Schrddinger equation, a suitable representation of the wavefunction and of the operators is needed. In this section, the variational 

The idea behind the DVR method [8-11] is to use a representation in terms of localized functions obtained by transformation from a global basis [12], Usually, bases constructed from orthogonal polynomials, noted F x), which are solution of one dimensional problems such as the particle in a box (Chebyshev polynomials) or the harmonic oscillator (Hermite polynomials), are used. These polynomial bases verify the general relationship 

Equation (4.9) demonstrates that 1/ is a unitary matrix. Therefore, by multiplying Eq. (4.10) by U from the left and by 1/+ from the right, one obtains that l/ diagonalizes the matrix of the coordinate operator X 

The function a(x) is localized around the DVR grid point Xa and is exactly zero at all the other grid points. This is seen by evaluating Eq. (4.13) at a DVR grid point X/  

As seen in this last equation, in the DVR, the wavefunction is represented by its values at the grid points jCq,. The matrix elements of the KEO in the DVR are exactly given by 

---

A recent numerical development is the introduction of the slow or smooth variable discretization (SVD) technique [101-103]. In the diabatic-by-sector method, the basis functions to expand the total wavefunction are fixed within each sector. In the SVD method, the hyperangular basis functions are constructed using the discrete variable representation (DVR) [104], The requirement is only that the total wavefunction be smooth in the adiabatic parameter p. By expanding the hyperradial wavefunctions using DVR basis functions, a new set of hyperangular basis functions are determined and they... 

H. Wei and T. Carrington Jr., /. Chem. Phys., 97, 3029 (1992). The Discrete Variable Representation for a Triatomic Molecule in Bond Length-Bond Angle Coordinates. 

The potential energy surface (PES) of ammonia has been studied repeatedly by many authors ([1-11], and references therein), and continues to be an object of active theoretical interest. Most authors start their analysis with an abinitio (or semi-empirical) calculation of the PES and then perform an additional refinement to achieve an agreement between the calculated and experimental vibrational frequencies. Lately, the discrete variable representation has received particular attention and is currently one of the preferred methods [3,7,8,10-12],... 

The RKR potential may be tested against the input G(v) and B(v) values by exact solution of the nuclear Schrodinger equation [see Wicke and Harris, 1976, review and compare various procedures, e.g., Numerov-Cooley numerical integration (Cooley, 1961), finite difference boundary value matrix diagonaliza-tion (Shore, 1973), and the discrete variable representation (DVR) (Harris, et al., 1965)]. G(v) + y00 typically deviates from EVjj=o by < 1 cm-1 except near dissociation. Bv may be computed from Xv,J=o(R) by... 

In the later case, this equation can be solved by using the discrete variable representation... 

G. C. Groenenboom and D. T. Colbert, Combining the discrete variable representation with the 5-matrix Kohn method for quantum reactive scattering, J. Chem. Phys. 99 9681 (1993). 

Matyus, E., Czako, G., Sutcliffe, B.T. Csaszar, A.G. Vibrational energy levels with arbitrary potentials using the Eckart-Watson Hamiltonians and the discrete variable representation, J. Chem. Phys. 2007, in press. 

No readily useful analytical solutions are available for the system (7) and we resort to an expansion of the amplitude functions in a basis set. The discrete variable representation is a convenient means and we chose to employ a localized basis associated with the Lobatto quadrature rule. It is convenient then to choose units and displacement such that the interval [r,p] equals the standard one, [-1,1]. A basis of n+1 Lagrange interpolation functions is defined from the Legendre polynomial Pniq) as follows... 

The integration rule overestimates this integral by more than 100% and thus it is necessary to be aware of the range of validity and to impose constraints in the variational expression where the discrete variable representation is employed. 

Another way to evaluate the expressions appearing in Eqs. (34 and 37) as well as related partial sums is closely related to the discrete variable representation of reaction probabilities as formulated by Seideman and Miller [148-150]. We have already seen that the sum... 

Both time-dependent and time-independent quantum theories can be employed to study three-atom, four-atom and polyatomic reactions with respect to nuclei motion. Clever features of the time-dependent wave packet theory are the use of absorbing potentials and the grid basis representations (the fast Fourier transformation and the discrete variable representation). 

The methodology of molecular quantum dynamics applied to non-adiabatic systems is presented from a time-dependent perspective in Chap. 4. The representation of the molecular Hamiltonian is first discussed, with a focus on the choice of the coordinates to parametrize the nuclear motion and on the discrete variable representation. The multi-configuration time-dependent Hartree (MCTDH) method for the solution of the time-dependent Schrddinger equation is then presented. The chapter ends with a presentation of the vibronic coupling model of Kdppel, Domcke and Cederbaum and the methodology used in the calculation of absorption spectra. 

The two most frequently used grid methods to solve the Schrodinger equation are the discrete variable representation [19-21] (DVR), and the fast Fourier transform method [6, 22] (FFT). [Please see Chapter 3 for a critical comparison between the two grid methods.] In this dissertation, we exclusively use DVR because it allows us to tailor the grid, in a simple fashion, to the shape of the physical and absorbing potentials. 

After discussing the MCTDH equations of motion, a practicable scheme has to be developed which facilitates the numerical evaluation of these equations. The single-particle functions have to be represented by a finite set of numbers. This is most conveniently achieved by employing a collocation scheme of the fast Fourier transform (FIT) or of the discrete variable representation (DVR) type. The single-particle functions of the /fth degree of freedom are then represented by their values on a set of grid points. 

---