Derivative vibrational wavefunctions

Figure 1 gives an example of derivative vibrational wavefunctions obtained from the DNC procedure. Using a quite realistic ab initio vibrational potential, Uo R), for the HF molecule, the basic NC approach jdelded the vibrational wavefunaions [Po(R)] for the ground and first two excited states, and these are plotted in the top parts of Figure 1. The first derivatives of the wavefunctions with respect to the strength of an axial electric field [Po (R)] are plotted for each state immediately below, and then the second derivatives of the wavefunaions [Po (R)] are below those. Notice that the first derivative of the ground state wavefunaion is similar (ignore phase) to the unperturbed wavefunaion for the first excited state. This is because the mixing of zeroeth-order vibrational states on application of an external field is primarily with adjacent level states, i.e., one level above or one level below.

One of the most familiar uses of dipole derivatives is the calculation of infrared intensities. To relate the intensity of a transition between states with vibrational wavefunctions i/r and jfyi it is necessary to evaluate the transition dipole moment... 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

The cover picture is derived artistically from the potential-energy profile for the dynamic equilibrium of water molecules in the hydration layer of a protein see A. Douhal s chapter in volume 1) and the three-dimensional vibrational wavefunctions for reactants, transition state, and products in a hydride-transfer reaction (see the chapter by S.J. Benkovic and S. Hammes-Schiffer in volume 4). 

Approximate deperturbed curves can be derived from unperturbed vibrational levels far from the energy of the curve crossing region. The overlap factor between vibrational wavefunctions is calculable numerically. (Note that a Franck-Condon factor is the absolute magnitude squared of the overlap factor.) From Eq. (3.3.5) and the experimental value of an initial trial... 

Let us ignore the R2 rotational part (R=J-L-S) of this operator, which leads to off-diagonal matrix elements that are proportional to J(J + 1) but still very small compared to the matrix elements of the remaining radial term (Leoni, 1972). The effect of the derivatives with respect to R on the electronic and vibrational wavefunctions, both of which depend on R, is given by... 

The vibrational wavefunction is often written as x = C/E, where is normalized with respect to dR (as opposed to R2dR as for x). Then the derivative of the vibrational function with respect to R results in two terms. One of these terms exactly cancels the term in (2/R) d< /dR) of Eq. (3.3.10), and the matrix element simplifies to... 

As emphasized, one of the advantages of this model is that it provides explicit wavefunctions which can be used in the computation of expectation values for various operators of interest. Due to limitations of space, we cannot reproduce here the complete set of vibrational wavefunctions obtained in the HCN calculation [76]. However, the typical outcome of the algebraic procedure can be outlined. We obtain a polyad of levels labeled by the numbers Vj and Ig of Eq. (4.56). Each polyad contains a number of local states, such as those listed in Eq. (4.57). The numerical diagonalization of the Hamiltonian matrix is performed separately for each polyad. Thus the eigenvectors derived represent the vibrational wavefunctions in the local basis. A possible outcome of the analysis of the HCN molecule could therefore be given by the following sequence of numbers ... 

To derive these and other conclusions, we need to specify the molecular states 7>. Since we are interested in vibrational spectra associated with coherent electronic absorption-emission processes, we represent the molecular states by products of electronic and vibrational wavefunctions... 

Pij(r) for a given pair of atoms could in principle be calculated from a knowledge of the vibrational wavefunctions of the molecule and the population of each excited vibrational state, i.e. the Boltzmann factors for the excited vibrational states. The vibrational wavefunctions should (again in principle) be derived from a potential energy expression which includes anharmonic terms. 

Figure 1 The forms of the zeroeth-order (A), first derivative (B), and second derivative (C) wavefunctions of HF for the ground ( = 0) vibrational state left series), the first excited ( = 1) vibrational state middle series), and the second excited (n = 2) vi brational state right series) where differentiation has been carried out with respect to the strength of a uniform electric field along the molecular axis.

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

It is reasonably well established that the non BO coupling term involving second derivatives of the electronic wavefunction contributes less to the coupling than does the term (-ih-3 rk/dRa) (-ib k 9Ra)/nia having first derivatives of the electronic and vibration-rotation functions. Hence, it is only the latter terms that will be discussed further in this paper. 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

As detailed in Section 2, we have derived and programmed the expression for line strengths of individual rotation-vibration transitions of XY3 molecules the line strengths depend on the vibronic transition moments entering into equation (70). With the theory of Section 2, we can simulate rotation-vibration absorption spectra of XY3 molecules. In computing the transition wavenumbers, line strengths, and intensities we use rovibronic wavefunctions generated as described in Ref. [1]. 

The SCF-MI algorithm, recently extended to compute analytic gradients and second derivatives [18,41], furnishes the Hartree Fock wavefunction for the interacting molecules and also provides automatic geometry optimisation and vibrational analysis in the harmonic approximation for the supersystems. The Ml strategy has been implemented into GAMESS-US package [42]. 

Hartree-Fock models are well defined and yield unique properties. They are both size consistent and variational. Not only may energies and wavefunctions be evaluated from purely analytical (as opposed to numerical) methods, but so too may first and second energy derivatives. This makes such important tasks as geometry optimization (which requires first derivatives) and determination of vibrational frequencies (which requires second derivatives) routine. Hartree-Fock models and are presently applicable to molecules comprising upwards of 50 to 100 atoms. 

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